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Optimal Sizing of Modular Multilevel Converters Thèse Amin Zabihinejad Doctorat en génie électrique Philosophiae Doctor (Ph. D.) Québec, Canada © Amin Zabihinejad , 2017

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Page 1: Optimal Sizing of Modular Multilevel Converters

Optimal Sizing of Modular Multilevel Converters

Thèse

Amin Zabihinejad

Doctorat en génie électrique

Philosophiae Doctor (Ph. D.)

Québec, Canada

© Amin Zabihinejad , 2017

Page 2: Optimal Sizing of Modular Multilevel Converters

Optimal Sizing of Modular Multilevel Converters

Thèse

Amin Zabihinejad

Sous la direction de:

Philippe Viarouge, directeur de recherche

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iii. Résumé

L’électronique de puissance a pénétré depuis quelques décennies les applications à forte

puissance dans de nombreux domaines de l’industrie électrique. Au-delà de l’apparition des

technologies d’interrupteur à forte puissance commutable en moyenne tension, ces

applications imposaient également des avancées dans le domaine des topologies de

convertisseurs statiques : les principaux défis à affronter concernaient l’atteinte de niveaux

de tension compatibles avec le domaine de puissance des applications, l’augmentation de la

fréquence de commutation apparente en sortie afin d’augmenter la bande passante de la

commande, de réduire la taille des éléments de filtrage et de limiter les harmoniques de

courant injectés dans le réseau d’alimentation. Les topologies de convertisseurs modulaires

multiniveaux (MMC) sont issues de cette problématique de recherche : elles permettent grâce

à l’association de cellules de commutation d’atteindre des niveaux de tension exploitables en

grande puissance avec les technologies d’interrupteurs existantes, de limiter les fréquences

et les pertes de commutation des interrupteurs élémentaires tout en maîtrisant la distorsion

harmonique totale (THD). La modularité, la redondance, les degrés de liberté et les

fonctionnalités des MMC leur permettent aussi d’augmenter la tolérance aux défauts. Ils

pénètrent à présent une large gamme d'applications comme le transport à courant continu en

haute tension (HVDC), les systèmes d'énergie renouvelable, les entraînements à vitesse

variables de grande puissance, la traction ferroviaire et maritime ainsi que des applications

spécifiques très contraignantes en matière de performance dynamique comme les systèmes

d’alimentation des électro-aimants dans les accélérateurs de particules.

Les topologies MMC sont composées de cellules de commutation élémentaires utilisant des

interrupteurs électroniques tels que le Thyristor à Commande Intégrée (IGCT) standard ou

les dernières génération d’IGBT. Les convertisseurs MMC ont fait l’objet de nombreux

travaux de recherche et de développement en ce qui concerne les topologies, la modélisation

et le calcul du fonctionnement en régime permanent et transitoire, le calcul des pertes, le

contenu harmonique des grandeurs électriques et les systèmes de commande et de régulation.

Par contre le dimensionnement de ces structures est rarement abordé dans les travaux publiés.

Comme la grande majorité des topologies de convertisseurs statiques, les convertisseurs

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MMC sont composés non seulement d’interrupteurs mais aussi d’organes de stockage

d’énergie de type composants diélectriques (condensateurs) et magnétiques (inductances,

coupleurs) qui sont essentiels pour assurer la conversion des grandeurs électriques en entrée

et en sortie. Ces composants ont une forte influence sur la taille, le volume et le rendement

des convertisseurs et le dimensionnement optimal de ces derniers résulte souvent de

compromis entre la taille des composants passifs, la fréquence et la puissance commutable

par les interrupteurs élémentaires.

Le travail de recherche présenté dans ce mémoire concerne le développement d’une

méthodologie de dimensionnement optimal et global des MMC intégrant les composants

actifs et passifs, respectant les contraintes des spécifications de l’application et maximisant

certains objectifs de performance. Cette méthodologie est utilisée pour analyser divers

compromis entre le rendement global du convertisseur et sa masse, voire son volume. Ces

divers scénarios peuvent être également traduits en termes de coût si l’utilisateur dispose du

prix des composants disponibles. Diverses solutions concurrentes mettant en œuvre un

nombre de cellules spécifique adaptées à des interrupteurs de caractéristiques différentes en

termes de calibre de tension, de courant et de pertes associés peuvent ainsi être comparées

sur la base de spécifications d’entrée-sortie identiques. La méthodologie est appliquée au

dimensionnement d’un convertisseur MMC utilisé comme étage d’entrée (« Active Front-

end » : AFE) d’une alimentation d’électro-aimant pulsée de grande puissance.

Dans une première partie, une méthode de calcul rapide, précise et générique du régime

permanent du convertisseur MMC est développée. Elle présente la particularité de prendre

en compte la fréquence de commutation contrairement aux approches conventionnelles

utilisant la modélisation en valeurs moyennes. Cet outil se révèle très utile dans l’évaluation

du contenu harmonique qui est contraint par les spécifications, il constitue le cœur de

l’environnement de conception du convertisseur.

Contrairement aux convertisseurs conventionnels, il existe des courants de circulation dans

les convertisseurs MMC qui les rendent complexe à analyser. Les inductances de limitation

incorporées dans les bras de la topologie sont généralement volumineux et pénalisants en

termes de volume et de masse. Il est courant d’utiliser des inductances couplées afin de

réduire l'ondulation , la THD et la masse. Dans le travail présenté, un circuit équivalent des

inductances couplée tenant compte de l'effet de saturation est développé et intégré à

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l’environnement. L’utilisation d’inductances couplée augmente la complexité de l'analyse du

fonctionnement et la précision de leur méthode de dimensionnement est critique pour

l’optimisation globale du convertisseur. Un modèle analytique de dimensionnement de ces

composants a été développé et intégré dans l’environnement ainsi qu’un modèle de

complexité supérieure qui utilise le calcul des champs par éléments finis.

La méthodologie de conception optimale et globale proposée utilise une procédure

d’optimisation non linéaire avec contraintes qui pilote l’outil de calcul de régime permanent,

le modèles de dimensionnements à plusieurs niveaux de complexité des composants passifs

ainsi que d’autres modules permettant de quantifier les régimes de défaut. Pour pallier à la

précision réduite des modèles analytiques, une approche d'optimisation hybride est

également implantée dans l’environnement. Dans la boucle d'optimisation hybride, le modèle

de dimensionnement des inductances peut être corrigé par le modèle de complexité

supérieure qui utilise le calcul des champs. On obtient ainsi un meilleure compromis entre la

précision de la solution optimale et le temps de convergence de la méthode itérative

d’optimisation globale.

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iv. Abstract

In the last decades, power electronics has penetrated high power applications in many areas

of the electrical industry. After the emergence of high-voltage semiconductor switch

technologies these applications also required advances in the field of static converter

topologies: The main challenges were to achieve voltage levels compatible with the

application power domain, to increase the apparent switching frequency at the output, to

increase the control bandwidth, to reduce the size of the elements of filtering and of limiting

the current harmonics injected into the supply network. The topologies of multi-level

modular converters (MMC) are based on this research problem: they enable the use of

switching cells to achieve high power levels that can be used with existing switch

technologies, frequencies and switching losses of the elementary switches while controlling

the total harmonic distortion (THD). Modularity, redundancy, degrees of freedom and MMC

functionality also allow them to increase fault tolerance. They now penetrated a wide range

of applications, such as high-voltage DC (HVDC), renewable energy systems, high-speed

variable speed drives, rail and marine traction, and very specific applications in terms of

dynamic performance such as electromagnet power systems in particle accelerators.

MMC topologies are composed of elementary switching cells using electronic switches such

as the standard Integrated Control Thyristor (IGCT) or the latest generation of IGBTs. MMC

converters have been the subject of extensive research and development work on topologies,

modeling, and calculation of steady-state and transient operation, loss calculation, the

harmonic content of electrical quantities and systems control and regulation functions. On

the other hand, the dimensioning methodology of these structures is rarely addressed in the

published works.

Like most static converter topologies, MMC converters are composed not only of switches

but also passive components of energy storage devices (capacitors) and magnetic (inductors,

couplers) that are essential to ensure the conversion of the input and output electrical

quantities. These components have a strong influence on the size, the volume and the

efficiency of the converters and the optimal dimensioning of the latter often result from a

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compromise between the size of the passive components, the frequency and the power

switchable by the elementary switches.

The research presented in this thesis concerns the development of an optimal and

comprehensive design methodology for MMCs integrating active and passive components,

respecting the constraints of the application specifications and maximizing certain

performance objectives. This methodology is used to analyze the various trade-off between

the overall efficiency of the converter and its mass, or even its volume. These various

scenarios can also be translated into cost if the user has the price of the available components.

Various competing solutions using a specific number of cells adapted to switches with

different characteristics in terms of voltage, current, and associated losses can thus be

compared on the basis of identical input-output specifications. The methodology is applied

to the dimensioning of an MMC converter used as an active front-end (AFE) input of a high-

power pulsed solenoid power supply.

In the first part, a fast, precise and generic method for calculating the steady-state model of

MMC converter is developed. It has the particularity of taking into account the switching

frequency as opposed to conventional approaches using modeling in mean values. This tool

is very useful in evaluating the harmonic content that is constrained by the specifications, it

is the heart of the design environment of the converter.

Unlike conventional converters, there are circulation currents in MMC converter structure

that make it complex to analyze. The inductors which are used in the arms of the topology

are generally bulky and expensive in terms of volume and mass. It is common to use coupled

inductors to reduce ripple, THD, and mass. In the presented work, an equivalent circuit of

coupled inductances considering the saturation effect is developed and integrated. The use of

coupled inductors increases the complexity of the analysis and the precision of its sizing

method is critical for the overall optimization of the converter. An analytical model for the

dimensioning of these components has been developed and integrated as well as a higher

complexity model which uses the finite element method calculation.

The proposed optimal and global design methodology uses a nonlinear optimization

procedure with constraints that drive the steady-state computing tool, multi-level design

models of passive component complexity, and other modules to quantify the fault state. To

compensate the low precision of the analytical models, a hybrid optimization approach is

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also implemented. In the hybrid optimization loop, the inductance-sizing model can be

corrected by the higher complexity model that uses finite element computation. A better

compromise is thus obtained between the precision of the optimal results and convergence

time of the iterative global optimization method.

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List of Contents

iii. Résumé ...................................................................................................... iii

iv. Abstract .................................................................................................... vi

List of Contents ............................................................................................................... ix

List of Tables xv

List of Figures ............................................................................................................... xvi

List of Symbols .............................................................................................................. xxi

CHAPTER I .................................................................................................................... 1

1 Introduction to Multilevel converters ........................................................... 1

1.1. Introduction ..................................................................................................... 1

1.2. Relevant State of the Art and Problem Description........................................... 4

1.3. History and MMC Definition ........................................................................... 7

1.4. Description of Multilevel structures ................................................................. 9

1.4.1. Neutral Point Clamped (NPC) ..................................................................... 9

1.4.2. Flying capacitor......................................................................................... 10

1.4.3. Cascaded Multilevel Converters ................................................................ 12

1.5. Applications and Industrial Relevance ............................................................17

1.5.1. Multilevel converters and renewable energy .............................................. 19

1.5.2. Multilevel converters and HVDC and FACT systems ................................ 21

1.5.3. Multilevel converters and Marine propulsion ............................................. 24

1.5.4. Traction motor drive .................................................................................. 24

1.5.5. Losses analysis of multilevel converters .................................................... 25

1.6. Main Objective: Converter sizing methodology applied to MMC structures of

static converters ............................................................................................................26

1.7. Secondary Objectives......................................................................................27

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1.7.1. Specify the global optimization approach for Optimization of MMC static

converters ................................................................................................................. 27

1.7.2. Applying the converter sizing method to an industrial application ............. 30

1.7.3. Finding the optimal solution for the MMC AFE converter ......................... 32

1.7.4. Using the Converter sizing method for other Applications ......................... 33

1.8. Conclusion ......................................................................................................34

CHAPTER II...................................................................................................................35

2 Design of MMC converter based on the load specification ........................................35

2.1. Introduction ....................................................................................................35

2.2. Calculation of MMC converter variables.........................................................36

2.2.1. Converter and sub-module topology .......................................................... 36

2.2.2. Converter inputs and outputs variables ...................................................... 38

2.2.3. Semiconductor sizing in steady state ......................................................... 39

2.2.4. Passive components sizing in steady state .................................................. 41

2.3. Investigation of adjustable parameters of multilevel converter ........................43

2.3.1. Converter topology .................................................................................... 43

2.3.2. Number of sub-modules per arm ............................................................... 43

2.3.3. Passive component values ......................................................................... 44

2.3.4. IGBT selection .......................................................................................... 45

2.4. Investigation of converter performance and limitations ...................................45

2.4.1. Converter Losses and efficiency ................................................................ 45

2.4.2. Power quality and harmonic Investigation ................................................. 46

2.4.3. Converter volume and mass....................................................................... 47

2.5. Comprehensive analysis of MMC converter ....................................................47

2.5.1. Circuit Analysis......................................................................................... 48

2.5.2. Electromagnetic Analysis .......................................................................... 49

2.5.3. Thermal Analysis ...................................................................................... 50

2.6. MMC dimensioning Analysis .........................................................................51

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2.6.1. Capacitor dimensioning analysis ............................................................... 51

2.6.2. Inductance dimensioning analysis ............................................................. 52

2.7. MMC Multiphase Analysis .............................................................................53

2.7.1. Global analysis using analytical model ...................................................... 53

2.7.2. Global analysis using modified analytical model ....................................... 55

2.8. Optimal sizing of MMC converter ..................................................................56

2.9. Conclusion ......................................................................................................60

CHAPTER III .................................................................................................................61

3 Circuit Model of Modular Multilevel AFE..............................................61

3.1. Introduction ....................................................................................................61

3.2. Steady-State Average Model of Modular Multilevel Active-Front-End Converter

62

3.2.1. Sub-module circuit analysis ....................................................................... 62

3.2.2. Single phase average model parameters ..................................................... 63

3.2.3. Average switching function ....................................................................... 64

3.2.4. Circulating current and capacitor voltage ripple estimation ........................ 66

3.2.5. Advantages and disadvantages of steady-state average model of MMC Active-

Front-End ................................................................................................................. 68

3.3. Time-domain steady-state model of Modular Multilevel Active-Front-End .....69

3.3.1. Sub-module switching function ................................................................. 69

3.3.2. Time-domain state equations ..................................................................... 69

3.3.3. Proposed time-domain model .................................................................... 70

3.4. Steady-State Model Verification using Simulink .............................................72

3.5. Conclusion ......................................................................................................74

CHAPTER IV .................................................................................................................76

4 Electromagnetic and Dimensioning Analysis of Passive Components ...76

4.1. Introduction ....................................................................................................76

4.2. Capacitor Dimensioning Analysis ...................................................................76

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4.2.1. Capacitor Mass Function ........................................................................... 77

4.2.2. Transient Equivalent Model of Capacitor .................................................. 78

4.3. Dimensioning Analysis of Inductor .................................................................80

4.3.1. Core Topologies ........................................................................................ 80

4.3.2. Magnetic Equivalent Circuit (type 1) ......................................................... 81

4.3.3. Magnetic Equivalent Circuit (type 2) ......................................................... 82

4.3.4. Inductance and Resistance Estimation ....................................................... 84

4.3.5. Volume and Mass Function ....................................................................... 86

4.4. Inductor Thermal Analysis ..............................................................................86

4.4.1. Inductor Losses ......................................................................................... 86

4.4.2. Thermal Model of Inductor ....................................................................... 92

4.5. Investigation the effect of core saturation ........................................................94

4.5.1. Finding the Mathematical Core Magnetizing Function .............................. 95

4.5.2. Inductance circuit Equation Considering Core Saturation .......................... 95

4.6. Finite Element Analysis of Coupled Inductors ................................................96

4.6.1. Magneto-static Analysis using Finite Element Method .............................. 96

4.6.2. Correction of Analytical Model using Finite Element Method ................... 98

4.7. Conclusion .................................................................................................... 100

CHAPTER V ................................................................................................................. 102

5 Converter Analysis in the Fault Condition ........................................... 102

5.1. Introduction .................................................................................................. 102

5.2. Investigation of Standard Defects in MMC Converter ................................... 102

5.2.1. DC Link Fault ......................................................................................... 102

5.2.2. Sub-module Fault .................................................................................... 103

5.2.3. Inductance Fault ...................................................................................... 104

5.3. Close Loop Control of MMC converter using Simulink ................................ 104

5.4. Investigation of Converter Performance in Defect Condition ........................ 105

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5.5. Combination of Time-Domain Steady-State Model and Faults in the unit package

..................................................................................................................... 109

5.5.1. Sub-module faults investigation .............................................................. 109

5.5.2. Proposed global optimization considering fault analysis .......................... 111

5.6. Conclusion .................................................................................................... 111

CHAPTER VI ............................................................................................................... 113

6 Optimal Design of Modular Multilevel Converter ................................ 113

6.1. Introduction .................................................................................................. 113

6.2. Optimization algorithm using numerical solver ............................................. 114

6.3. Load Specification of the MMC Active Front End converter application ....... 116

6.4. Constraints Calculation ................................................................................. 117

6.4.1. Sub-module capacitor voltage ripple........................................................ 117

6.4.2. THD ........................................................................................................ 117

6.4.3. Semiconductor Losses ............................................................................. 117

6.4.4. Inductor Losses ....................................................................................... 118

6.5. Goal function ................................................................................................ 118

6.6. MMC Optimization using analytical circuit model ........................................ 119

6.6.1. Optimization algorithm ........................................................................... 119

6.7. Optimal design of modular multilevel converter using dimensioning model .. 121

6.7.1. Global Mass Minimization Algorithm ..................................................... 121

6.8. Hybrid Optimization Model using 2-D FEM ................................................. 123

6.8.1. Hybrid Global Optimization Algorithm ................................................... 124

6.8.2. Hybrid Global Optimization Algorithm considering fault margin ............ 125

6.9. Conclusion .................................................................................................... 126

CHAPTER VII .............................................................................................................. 128

7 Investigation of Optimization Results ................................................... 128

7.1. Introduction .................................................................................................. 128

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7.2. High power IGBT specifications ................................................................... 129

7.3. Optimization results using proposed time-domain circuit model.................... 130

7.4. Mass Minimization of Modular Multilevel Converter ................................... 134

7.4.1. Selection of inductor core topology ......................................................... 136

7.4.2. Optimization using 3.3KV/1500A IGBT ................................................. 137

7.4.3. Optimization using 6.5KV/750A IGBT ................................................... 140

7.5. Optimization Results using Hybrid Analytical Model ................................... 143

7.5.1. Optimization results using 3.3KV/1500A IGBT ...................................... 144

7.5.2. Optimization results using 6.5 kV/750 A IGBT ....................................... 148

7.6. Parameter sensitivity analysis ....................................................................... 150

7.6.1. Sensitivity analysis of maximum Temperature Rise ................................. 151

7.6.2. Investigation the effect of maximum Flux Density on Converter Mass .... 153

7.6.3. Investigation the effect of maximum THD on Converter Mass ................ 154

7.6.4. Investigation of the effect of Capacitor Voltage Ripple on Converter Mass ...

................................................................................................................ 156

7.6.5. Sensitivity analysis converter mass against Fault margin ......................... 157

7.7. Conclusion .................................................................................................... 159

CHAPTER VIII ............................................................................................................ 160

8 Conclusion and Future Researches ....................................................... 160

8.1. Conclusion ............................................................................................... 160

8.2. Future Researches .................................................................................... 163

REFERENCES ............................................................................................................... 165

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List of Tables

Table 1.1: Some of the today’s MMC projects ..................................................................18

Table 1.2: ABB SVC Light for electrical transmission grids ..............................................23

Table 3.1 The MMC converter parameters and operating point .........................................72

Table 4.1: Calculated coefficients using fitting algorithm ..................................................78

Table 4.2: Rac/Rdc of copper conductor with 13.21mm diameter in 50Hz, 100Hz and 200Hz

using three estimation methods ...................................................................91

Table 4.3: Calculation of conductor copper losses using the estimated AC resistances and

error calculation compared to utilization of DC resistance ..........................92

Table 5.1: Simulation parameters of MMC converter in simpower .................................. 106

Table 5.2: Normal operation of a half-bridge sub-module................................................ 109

Table 5.3 Investigation of Open-Circuit Fault in T1 ........................................................ 110

Table 5.4 Investigation of Open-Circuit Fault in T2 ........................................................ 110

Table 5.5 Investigation of Short-Circuit Faults in T1 or T2 .............................................. 111

Table 6.1: Load specification of MMC Active Front End converter application............... 116

Table 6.2 The main optimization constraints ................................................................... 120

Table 6.3: List of optimization variables ......................................................................... 122

Table 6.4: List of main constraints in optimization algorithm .......................................... 123

Table 7.1 Technical specifications of high power IGBTs ................................................. 130

Table 7.2: Comparison of total mass of two different inductor core topologies ................ 137

Table 7.3: Converter parameter and thermal coefficient .................................................. 151

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List of Figures

Figure 1.1: Multilevel neutral point clamped converter....................................................... 9

Figure 1.2: Multilevel flying capacitor topology ................................................................11

Figure 1.3: Multilevel cascaded converter with elementary converters ..............................12

Figure 1.4: Typical configuration of sub-module (Commutation Cell) ...............................13

Figure 1.5: Single-leg multilevel converter ........................................................................14

Figure 1.6: Multi-leg multilevel converter .........................................................................15

Figure 1.7: Single-leg two levels converter with interleaving inductors .............................16

Figure 1.8: MMC converter with interleaving inductors ....................................................17

Figure 1.9: Single-line schematic diagram of the proposed multilevel converter-based wind

energy conversion system [28] ...................................................................20

Figure 1.10: Multilevel inverter topology proposed for solar energy systems [32] .............21

Figure 1.11: The modular cascaded multilevel converter used by Siemens Company for

HVDC application [33] ..............................................................................22

Figure 1.12: Three-phase five-level structure of a diode-clamped multilevel converter [36]

...................................................................................................................23

Figure 1.13: Multilevel converter in electric marine propulsion system [39] .....................24

Figure 1.14: 6-level diode-clamped back to back converter for traction motor drive [15] ...25

Figure 1.15: Optimization and verification loop ................................................................28

Figure 1.16: Optimization and verification loop ................................................................30

Figure 1.17: Multi-megawatt power supply of the PS Booster ...........................................31

Figure 1.18: Electromagnet Current pulsed Cycle delivered by the Multilevel H-bridge

converter ....................................................................................................31

Figure 1.19: Corresponding capacitor voltage cycle in the DC bus ....................................31

Figure 1.20: Example of High Power Converter in CERN complex...................................33

Figure 2.1 Modular multilevel active front-end converter ..................................................37

Figure 2.2 a)half-bridge sub-module configuration b)full-bridge sub-module configuration

...................................................................................................................37

Figure 2.3 Inputs and output variables of active front-end converter ..................................38

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Figure 2.4 Thermal dissipation circuit of IGBT .................................................................41

Figure 2.5 MMC circuit model and input/output variables .................................................48

Figure 2.6 Analytical electromagnetic model of arm inductance and its input/output .........49

Figure 2.7 Proposed model for electromagnetic analysis includes analytical model and finite

element analysis .........................................................................................50

Figure 2.8 The analytical thermal model of MMC converter .............................................51

Figure 2.9 The magnetic core topology .............................................................................52

Figure 2.10 Global analysis plan of MMC converter using analytical models ....................54

Figure 2.11 Global analysis plan of MMC converter using finite element correction loop .55

Figure 2.12 First proposed optimization plan of MMC converter.......................................57

Figure 2.13 Second proposed optimization plan using analytical inductor model ...............58

Figure 2.14 Third optimization plan includes the correction loop using FEM ....................59

Figure 3.1: MMC-based topology of the Multi-megawatt power supply of Fig.1.17 ..........62

Figure 3.2: half-bridge sub-module topology .....................................................................63

Figure 3.3: Single phase average equivalent circuit ...........................................................64

Figure 3.4: initializing diagram using average steady-state model .....................................71

Figure 3.5: Diagram of Time-domain analytical model .....................................................72

Figure 3.6: The sub-module capacitor current using analytical model and Simulink ..........73

Figure 3.7: The sub-module capacitor voltage using analytical model and Simulink ..........73

Figure 3.8: The lower and upper arm current using analytical model and Simulink ...........74

Figure 3.9: The input line current using analytical model and Simulink .............................74

Figure 4.1 Cylindrical capacitor of VISHAY Company designed for power electronic

applications ................................................................................................77

Figure 4.2: Capacitor weight versus the capacitance and maximum voltage value .............78

Figure 4.3 The simple model of high power capacitor .......................................................79

Figure 4.4 The modified high power capacitor model ........................................................80

Figure 4.5: The proposed core topologies for independent and dependent mutual inductances

...................................................................................................................81

Figure 4.6 The equivalent magnetic circuit of inductance (type 1) .....................................81

Figure 4.7: The inductor core topology and sizing parameters ...........................................83

Figure 4.8 Magnetic flux lines in the core and leakage flux lines .......................................85

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Figure 4.9: AC resistance of round copper conductor versus conductor diameter using exact

equation .....................................................................................................89

Figure 4.10: AC resistance of round copper conductor versus conductor diameter using

simplified equation .....................................................................................89

Figure 4.11 AC resistance of round copper conductor versus conductor diameter using IEC

standard equation .......................................................................................90

Figure 4.12: Finite element analysis of skin effect .............................................................91

Figure 4.13: Inductor equivalent thermal circuit ................................................................92

Figure 4.14: The B-H curve of iron sheet core ...................................................................94

Figure 4.15: Finite element analysis of coupled arm inductance ........................................97

Figure 4.16: The flowchart of the proposed correction approach ..................................... 100

Figure 5.1: The DC link fault and currents path in the converter ...................................... 103

Figure 5.2: Various kind of sub-module faults ................................................................. 104

Figure 5.3 Close loop control diagram of MMC converter............................................... 105

Figure 5.4: DC Link voltage variation via various converter faults .................................. 106

Figure 5.5: Line current variation via various converter faults ......................................... 107

Figure 5.6: Upper inductor current variation via various converter fault .......................... 108

Figure 5.7: Magnetic flux density of inductor core via various converter fault ................. 108

Figure 6.1: Implementation of Global Optimization Algorithm with Microsoft Excel ...... 115

Figure 6.2: Optimization flowchart of MMC converter.................................................... 120

Figure 6.3: The proposed global optimization algorithm using analytical model .............. 121

Figure 6.4: The proposed hybrid optimization algorithm ................................................. 124

Figure 6.5 Global optimization algorithm considering fault margin ................................. 125

Figure 6.6 Hybrid optimization algorithm considering the fault margin ........................... 126

Figure 7.1: Electric energy stored in the capacitors versus the number of sub-module per arm

................................................................................................................. 131

Figure 7.2: Magnetic energy stored in the inductors versus the number of sub-module per

arm ........................................................................................................... 131

Figure 7.3: Total energy stored in the converter versus the number of sub-module per arm

................................................................................................................. 132

Figure 7.4: Total converter efficiency versus the number of sub-module per arm ............ 132

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Figure 7.5: Optimal switching frequency versus the number of sub-modules per arm ...... 133

Figure 7.6: Sub-module capacitor ripple versus capacitor energy and sub-module number

................................................................................................................. 133

Figure 7.7: Contour of capacitor ripple versus capacitor energy and sub-module number 134

Figure 7.8: THD and total efficiency versus coupling factor ............................................ 134

Figure 7.9: MMC topology using 3.3 kV/1500 A IGBT .................................................. 135

Figure 7.10: MMC topology using 6.5 kV/750 A IGBT .................................................. 136

Figure 7.11: Optimal arm inductance value versus number of sub-modules per arm ........ 138

Figure 7.12: Optimal sub-module capacitor value versus number of sub-modules per arm

................................................................................................................. 138

Figure 7.13: Optimal total inductor mass versus number of sub-modules per arm ............ 139

Figure 7.14: Optimal total capacitor mass versus number of sub-modules per arm .......... 139

Figure 7.15: Optimal converter mass versus number of sub-modules per arm .................. 140

Figure 7.16 The optimal arm inductance versus the number of sub-modules per arm ....... 141

Figure 7.17 Optimal value of the sub-module capacitor versus the number of sub-modules

per arm ..................................................................................................... 141

Figure 7.18 Total inductor mass versus the number of sub-modules per arm ................... 142

Figure 7.19 Total capacitor mass versus the number of sub-modules per arm .................. 142

Figure 7.20 Total converter mass versus the number of sub-modules per arm .................. 143

Figure 7.21: The optimal arm inductance value versus the sub-modules per arm ............. 144

Figure 7.22: The optimal sub-module capacitor value versus the sub-modules per arm ... 145

Figure 7.23: The optimal arm inductance mass versus the sub-modules per arm .............. 146

Figure 7.24: The optimal capacitor mass versus the sub-modules per arm ....................... 146

Figure 7.25: The optimal converter mass versus the sub-modules per arm ....................... 147

Figure 7.26: Total converter efficiency versus the sub-modules per arm .......................... 147

Figure 7.27 Total inductor mass versus the number of sub-modules per arm ................... 148

Figure 7.28 The total sub-module capacitor mass versus the number of sub-module per arm

................................................................................................................. 149

Figure 7.29 Total converter mass versus the number of sub-modules per arm .................. 150

Figure 7.30 Total converter efficiency versus the number of sub-modules per arm .......... 150

Figure 7.31: Optimal inductor mass versus maximum temperature rise ........................... 152

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Figure 7.32: Total converter mass versus the maximum temperature rise ........................ 152

Figure 7.33: The contour of optimal inductor mass versus the sub-modules number and the

maximum flux density .............................................................................. 153

Figure 7.34 The discontinuity value versus number of sub-modules per arm ................... 154

Figure 7.35 The THD value of input current versus the arm inductance value ................. 155

Figure 7.36 The total inductor mass versus the arm inductance value .............................. 156

Figure 7.37: The total converter mass versus the capacitor ripple .................................... 157

Figure 7.38 Total converter mass sensitivity against fault margin for 3.3KV IGBT ......... 158

Figure 7.39 Total converter mass sensitivity against fault margin for 6.5KV IGBT ......... 158

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List of Symbols

𝐴𝑐 Core section of magnetic circuit [𝑚2]

𝐴𝑐𝑢 Wire section area [𝑚2]

𝐴𝑒 The effective value of core section of magnetic circuit [𝑚2]

𝐴𝑤𝑇 Total copper section of each winding [𝑚2]

𝑎 Core window width [𝑚]

𝑏 Core window height [𝑚]

𝐵 Magnetic flux density [𝑇]

𝐵ℎ1𝑚𝑎𝑥 Maximum flux density of first harmonic [𝑇]

𝐵ℎ2𝑚𝑎𝑥 Maximum flux density of second harmonic [𝑇]

𝐵𝑚𝑎𝑥 Maximum magnetic flux density [𝑇]

𝐵𝑙𝑖𝑛𝑒𝑎𝑟 Linear flux density of the core [𝑇]

𝐵𝑛𝑜𝑙𝑖𝑛𝑒𝑎𝑟 Flux density in the saturation region [𝑇]

𝐵𝑠𝑎𝑡 Saturation flux density [𝑇]

𝑐 Inductor core depth [𝑚]

𝐶 Capacitor value [𝐹]

𝐶𝑎𝑟𝑚 Arm capacitor value [𝐹]

𝐶𝑠𝑚 Sub-module capacitor value [𝐹]

𝐶1 The capacitance between the anode and cathode of the

capacitor

[𝐹]

𝐶2 Correction capacitor [𝐹]

𝑑 Core Width [𝑚]

𝑑𝑐 Conductor diameter [𝑚]

𝐸𝑐𝑚𝑎𝑥 Maximum energy stored in capacitors [𝐽]

𝐸𝑐𝑎𝑝 Electrical energy of capacitors [𝐽]

𝐸𝑐𝑜𝑛𝑣 Total energy stored in converter [𝐽]

𝐸𝑖𝑛𝑑 Magnetic energy stored in the inductors [𝐽]

𝐸𝑜𝑛𝑖 Turn-on energy losses of IGBT [𝐽]

𝐸𝑜𝑛𝑑 Turn-off energy losses of diode [𝐽]

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𝐸𝑜𝑓𝑓𝑖 Turn-off energy losses of IGBT [𝐽]

𝐸𝑜𝑓𝑓𝑑 Turn-off energy losses of diode [𝐽]

𝐸𝑢 Unit capacitor voltage [𝑉]

𝐸𝑃 Energy Power ration

𝑓𝑠 AC source frequency [𝐻𝑧]

𝑓𝑠𝑤 Switching frequency [𝐻𝑧]

𝐹𝑇 Fault time constant [𝑠𝑒𝑐]

ℎ Even-order harmonics

ℎ𝑓𝑒 Heat transfer coefficient of iron core

ℎ𝑐𝑢 Heat transfer coefficient of copper

ℎ𝑓𝑒−𝑐𝑢 Heat transfer coefficient between core and copper

𝐻 Magnetic field value [𝐴/𝑚]

𝐻𝑙𝑖𝑛𝑒𝑎𝑟 Linear magnetic field [𝐴/𝑚]

𝐻𝑠𝑎𝑡 Saturation magnetic field [𝐴/𝑚]

𝐼𝑎 AC phase current of phase a (peak) [𝐴]

𝐼1 The current of first winding [𝐴]

𝐼2 The current of second winding [𝐴]

𝑖𝑎𝑏𝑐 Instantaneous three phase AC line currents [𝐴]

𝐼𝑎𝑢 Upper arm current of phase a (peak) [𝐴]

𝐼𝑎𝑙 Lower arm current of phase a (peak) [𝐴]

𝑖𝑐𝑒 Collector-emitter current of IGBT [𝐴]

𝐼𝑐𝑢 Capacitor current of upper sub-modules (peak) [𝐴]

𝐼𝑐𝑢𝑖 Capacitor current of ith upper sub-modules (peak) [𝐴]

𝐼𝑐𝑙 Capacitor current of lower sub-module (peak) [𝐴]

𝐼𝑐𝑙𝑖 Capacitor current of ith lower sub-module (peak) [𝐴]

𝐼𝑐𝑖𝑟𝑐 Circulation current (peak) [𝐴]

𝐼𝑐𝑖𝑟𝑐ℎ2 Second harmonic of circulation current (peak) [𝐴]

𝐼𝑑𝑐 DC link current (peak) [𝐴]

𝑖𝑑𝑐 Instantaneous DC link current [𝐴]

𝐼𝑐𝑢ℎ1 Main component of capacitor current (peak) [𝐴]

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𝐼𝑐𝑙ℎ1 Main component of capacitor current (peak) [𝐴]

𝐼𝑐𝑖𝑟𝑐𝑑𝑐 DC component of circulation current (peak) [𝐴]

𝐼𝑓 Fault current (peak) [𝐴]

𝑖𝐹𝐷 Diode forward current [𝐴]

𝐼𝐿 Inductor current (peak) [𝐴]

𝐼𝑛𝑓 No fault current [𝐴]

𝑖𝑜𝑢𝑡 Instantaneous load current [𝐴]

𝐼𝑎𝑟𝑚𝑟𝑚𝑠 Effective value of arm current [𝐴]

𝐼𝑟𝑚𝑠 Effective value of first winding current [𝐴]

𝐼𝑟𝑚𝑠ℎ1 Effective value of first harmonic of winding current [𝐴]

𝐼𝑟𝑚𝑠ℎ2 Effective value of second harmonic of winding current [𝐴]

𝐼𝑟𝑚𝑠ℎ4 Effective value of fourth harmonic of winding current [𝐴]

𝐼𝑠1 Main switch current [𝐴]

𝐼𝑠2 Bypass switch current [𝐴]

𝐼𝑠𝑚 Submodule current [𝐴]

𝐼𝑢 Upper arm current [𝐴]

𝐽 Current density of inductor [𝐴/𝑚2]

𝐾𝑚𝑢 Estimated Coupling factor

𝑘𝑤 Filling factor of inductor winding

𝐾ℎ Hysteresis losses coefficient

𝐾𝑒 Eddy current losses coefficient

𝐾𝑐 Correction coefficient of analytical model

𝐾𝑠 AC resistance coefficient

𝐾𝑠𝑎𝑡 Core saturation coefficient

𝑙𝑐 The magnetic length of the core considering air gap [𝑚]

𝑙𝑚 The magnetic length of the core [𝑚]

𝑙𝑒 Effective value of magnetic circuit length [𝑚]

𝑙𝑔 Air gap length [𝑚]

𝑙𝑔1 Left and right leg air gap length [𝑚]

𝑙𝑔2 Center-leg air gap [𝑚]

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𝑙𝑖𝑠𝑜 Isolation thickness [𝑚]

𝐿 Inductance value [𝐻]

𝐿11 Self-inductance value [𝐻]

𝐿12 Mutual inductance value [𝐻]

𝐿21 Mutual inductance value [𝐻]

𝐿𝑎𝑛 Inductance value from analytical model [𝐻]

𝐿𝑓𝑒𝑚 Inductance value from FEM calculation [𝐻]

𝐿𝑐 Inductance of capacitor [𝐻]

𝑀 Mutual inductance [𝐻]

𝑀𝑖𝑛𝑑 Inductance total mass [𝐾𝑔]

𝑀𝑐𝑎𝑝 Capacitor mass [𝐾𝑔]

𝑀𝑐𝑎𝑝−𝑇 Capacitor total mass [𝐾𝑔]

𝑀𝑤𝑖𝑛𝑑𝑖𝑛𝑔 Copper mass [𝐾𝑔]

𝑀𝑐𝑜𝑟𝑒 Core mass [𝐾𝑔]

𝑀𝐼𝐺𝐵𝑇 IGBT mass [𝐾𝑔]

𝑀𝐼𝐺𝐵𝑇−𝑇 Total IGBT mass [𝐾𝑔]

MLT Mean length per turn [𝑚]

𝑁 Number of parallel arms

𝑛 Inductor turn number

𝑛1 First winding turn number

𝑚 Number of sub-modules per arm

𝑃𝑎𝑐 Active power of AC side [𝑊]

𝑃𝑑𝑐 Power of DC side [𝑊]

𝑃𝑐𝑢 Copper losses [𝑊]

𝑃𝑐𝑢𝑇 Total inductor copper losses [𝑊]

𝑃𝑒 Eddy current losses [𝑊]

𝑃ℎ Hysteresis losses [𝑊]

𝑃𝐼𝐺𝐵𝑇−𝑇 Total IGBT Losses [𝑊]

𝑃𝑐 Total conduction losses of semiconductor switch [𝑊]

𝑃𝑐𝑖𝑇 Total conduction losses of IGBT [𝑊]

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𝑃𝑐𝑖 Conduction losses of IGBT [𝑊]

𝑃𝑐𝑑𝑇 Total conduction losses of diode [𝑊]

𝑃𝑐𝑑 Conduction losses of diode [𝑊]

𝑃𝑠𝑤 Total switching losses of semiconductor switch [𝑊]

𝑃𝑠𝑤𝑖 Switching losses of IGBT [𝑊]

𝑃𝑠𝑤𝑑 Switching losses of diode [𝑊]

𝑞 Conductor constant

𝑅1 Left side core reluctance of the core [𝐻−1]

𝑅2 Right side core reluctance of the core [𝐻−1]

𝑅3 Center core reluctance of the core [𝐻−1]

𝑅4 Right side core reluctance of the core [𝐻−1]

𝑅5 Left side core reluctance of the core [𝐻−1]

𝑅𝑎 Capacitor equivalent resistance [𝑜ℎ𝑚]

𝑅𝑎𝑐 AC resistance of the winding [𝑜ℎ𝑚]

𝑅𝑎𝑐1 AC resistance of the conductor in the main frequency [𝑜ℎ𝑚]

𝑅𝑎𝑐2 AC resistance of the conductor in the second harmonic [𝑜ℎ𝑚]

𝑅𝑎𝑐4 AC resistance of the conductor in the fourth harmonic [𝑜ℎ𝑚]

𝑅𝑎𝑔 Air gap reluctance [𝐻−1]

𝑅𝑙𝑔1 Center leg air gap reluctance [𝐻−1]

𝑅𝑙𝑔2 Side leg air gap reluctance [𝐻−1]

𝑅𝑏 Modeling resistance [𝑜ℎ𝑚]

𝑅𝑐 Dielectric leakage resistance [𝑜ℎ𝑚]

𝑅𝑐𝑜𝑟𝑒 Magnetic core reluctance [𝐻−1]

𝑅𝐿1 Inductor winding resistance [𝑜ℎ𝑚]

𝑅𝑇 Total magnetic reluctance [𝐻−1]

𝑅2𝑎 Compensation resistance [𝑜ℎ𝑚]

𝑅2𝑐 Compensation resistance [𝑜ℎ𝑚]

𝑅𝑐𝑢−𝑎𝑖𝑟 Thermal resistivity of copper [𝑚2℃/𝑊]

𝑅𝑑𝑐 DC resistance of the winding [𝑜ℎ𝑚]

𝑅𝑓𝑒−𝑎𝑖𝑟 Thermal resistivity of core [𝑚2℃/𝑊]

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𝑅𝑓𝑒−𝑐𝑢 Thermal resistivity of core and copper [𝑚2℃/𝑊]

𝑅𝑗𝑐 Junction-case thermal resistance [𝑚2℃/𝑊]

𝑅𝑐𝑠 Case-Sink thermal resistance [𝑚2℃/𝑊]

𝑅𝑠𝑎 Sink-air thermal resistance [𝑚2℃/𝑊]

𝑟 Conductor radius [𝑚]

𝑆𝑎𝑙 Switching function of lower arm

𝑆𝑎𝑢 Switching function of upper arm

𝑆𝑐𝑢 Copper external area [𝑚2]

𝑆𝑓𝑒 Core external area [𝑚2]

𝑆𝑓𝑒−𝑐𝑢 The area between the core and copper [𝑚2]

𝑆𝑚 Modulation index

𝑆𝑛 Nominal power [𝑉𝐴]

𝑆𝑢𝑖 Switching function of ith submodule of upper arm

𝑆𝑙𝑖 Switching function of ith submodule of lower arm

𝑇𝑎 Ambient temperature [℃]

𝑇𝑐𝑢 Copper temperature [℃]

𝑇𝑐𝑢𝑚𝑎𝑥 Maximum copper temperature [℃]

𝑇𝑓𝑒 Core temperature [℃]

𝑇𝐻𝐷 Total harmonic distortion [%]

𝑣𝑎𝑏𝑐 Instantaneous three phase AC voltage [𝑉]

𝑉𝑐 Submodule capacitor voltage [𝑉]

𝑣𝑐𝑒 Collector-emitter voltage of IGBT [𝑉]

𝑉𝑐𝑢 Upper side submodule capacitor voltage [𝑉]

𝑉𝑐𝑢𝑖 Upper side ith submodule capacitor voltage [𝑉]

𝑉𝑐𝑙 Lower side submodule capacitor voltage [𝑉]

𝑉𝑐𝑙𝑖 Lower side ith submodule capacitor voltage [𝑉]

𝑉𝑐𝑜𝑟𝑒 Total core volume [𝑚3]

𝑣𝐹 Forward conduction voltage of diode [𝑉]

𝑣𝑑 D-axis AC voltage [𝑉]

𝑉𝑑𝑐 DC link voltage (mean value) [𝑉]

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𝑣𝑑𝑐 Instantaneous DC link voltage [𝑉]

𝑉𝑙 Lower arm voltage [𝑉]

𝑉𝐿 Inductance voltage [𝑉]

𝑉𝐿−𝐿,𝑟𝑚𝑠 Line to line rms voltage [𝑉]

𝑉𝑢𝑇 Total upper submodule voltage [𝑉]

𝑉𝐿𝑢 Upper arm inductance voltage [𝑉]

𝑉𝐿𝑙 Lower arm inductance voltage [𝑉]

𝑉𝑙𝑖 Voltage of ith submodule in lower arm [𝑉]

𝑉𝑙𝑠 Lower side submodule voltage [𝑉]

𝑉𝐼𝐺𝐵𝑇 IGBT voltage value [𝑉]

𝑉𝑜 Terminal voltage [𝑉]

𝑉𝑜𝑢𝑡 Out put converter voltage [𝑉]

𝑣𝑞 Q-axis voltage of AC side [𝑉]

𝑉𝑠𝑚 Submodule terminal voltage [𝑉]

𝑉𝑢 Upper arm voltage [𝑉]

𝑉𝑢𝑇 Total upper submodule voltage [𝑉]

𝑉𝑢𝑖 Voltage of ith submodule in upper arm [𝑉]

𝑉𝑢𝑠 Upper side submodule voltage [𝑉]

𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 Total copper volume [𝑚3]

𝑥𝑠 AC resistance IEEE constant

𝑦𝑠 AC resistance IEEE constant

𝜔1 Circular frequency of AC source [𝑟𝑎𝑑/𝑠𝑒𝑐]

𝜃1 Phase of the main component of arm current [𝑟𝑎𝑑]

𝜃2 Phase of second harmonic of arm current [𝑟𝑎𝑑]

Δ𝑉𝑐𝑢ℎ1 Main component of upper sub-module capacitor voltage [𝑉]

Δ𝑉𝑐𝑙ℎ1 Main component of lower sub-module capacitor voltage [𝑉]

Δ𝑉𝑐𝑢ℎ2 Second harmonic of upper sub-module capacitor voltage [𝑉]

Δ𝑉𝑐𝑙ℎ2 Second harmonic of lower sub-module capacitor voltage [𝑉]

Δ𝑉𝑐𝑢ℎ3 Third harmonic of upper sub-module capacitor voltage [𝑉]

Δ𝑉𝑐𝑙ℎ3 Third harmonic of lower sub-module capacitor voltage [𝑉]

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Δ𝑉𝑎𝑟𝑚 Arm voltage ripple [𝑉]

Δ𝑉𝑎𝑟𝑚𝑑𝑐 DC component of arm voltage [𝑉]

Δ𝑉𝑎𝑟𝑚ℎ2 Second harmonic of arm voltage [𝑉]

Δ𝑉𝑎𝑟𝑚ℎ4 Fourth harmonic of arm voltage [𝑉]

Δ𝑡 Fault period [𝑠𝑒𝑐]

Δ𝑇 Inductor temprature rise [℃]

𝜇0 Vacuum permeability [𝐻/𝑚]

𝜇𝑟 Core permeability

𝜂 Converter efficiency [%]

𝜙 Core flux [𝑊𝑏]

𝜙1 Core flux of first winding [𝑊𝑏]

𝜙11 First winding flux of first winding current [𝑊𝑏]

𝜙12 First winding flux of second winding current [𝑊𝑏]

𝜙21 Second winding flux of first winding current [𝑊𝑏]

𝜌𝑐𝑢 Electrical resistivity [𝛺.𝑚]

𝜌0 Copper electrical resistivity constant at 25℃ [𝛺.𝑚]

𝐷𝑐𝑜𝑟𝑒 Iron volumetric mass [𝑘𝑔/𝑚3]

𝐷𝑤𝑖𝑛𝑑𝑖𝑛𝑔 Copper volumetric mass [𝑘𝑔/𝑚3]

𝛼 Hysteresis losses field constant

휀25 Copper resistivity coefficient at 25℃

𝛽 Eddy current losses frequency constant

𝛾 Eddy current losses field constant

𝜎 Copper conductivity [𝛺.𝑚]−1

𝜆 Thermal conduction factor of the isolation

𝛿 Air gap correction coefficient

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1

CHAPTER I

1 Introduction to Multilevel converters

1.1. Introduction

Recently, multilevel converters have emerged as an interesting solution in the power

industry. The general structure of the multilevel converter is to synthesize a sinusoidal

voltage from several levels of voltages, typically obtained from capacitor voltage sources.

For several years, multilevel voltage source converters allow working at the high voltage

level and producing a quasi-sinusoidal voltage waveform. Classical multilevel topologies

such as NPC and Flying Capacitor VSIs were introduced twenty years ago, and are widely

used in Medium Power applications such as traction drives nowadays. In the scope of High

Voltage AC/DC converters, the Modular Multilevel Converter (MMC), proposed ten years

ago, by Professor R. Marquardt from the University of Munich (Germany), appeared

particularly interesting for HVDC transmissions. On the base of the MMC principle, this

thesis considers different topologies of elementary cells, which make the High Voltage

AC/DC converter more flexible to achieve different voltage and current levels.

Trends in power semiconductor technology indicate a trade off in the selection of power

devices in terms of switching frequency and voltage sustaining capability. New multi-level

high-power converter topologies have been proposed using a hybrid approach involving

integrated gate commutated thyristors (IGCTs) or gate turned off thyristors (GTOs) and

insulated gate bipolar transistors (IGBTs) operation in synergism. The new multilevel power

conversion concept combines the flexibility of the frequency converter with the robustness

of the industrial active neutral point clamped converter (ANPC) to generate multilevel

voltages.

In recent years, the industry has begun to demand higher power equipment, which now

reaches the megawatt level. Controlled ac drives in the megawatt range are usually connected

Page 30: Optimal Sizing of Modular Multilevel Converters

2

to the medium-voltage network. Today, it is hard to connect a single power semiconductor

switch directly to medium voltage grids (2.3, 3.3, 4.16, or 6.9 kV). For these reasons, a new

family of multilevel converters has emerged as the solution for working at higher voltage

levels [1].

Multilevel converters are one of the best solutions in order to use in the high voltage

applications. Increasing the demands for high power converters in order to connect to the

grid for renewable energy systems, HVDC and other applications made multi-level

converters topology more suitable than two-level PWM rectifier [2].

Using multilevel converters is increasing more and more, especially in high power industries

of electric power conversion. Multilevel converters are customized for a wide range of

applications, such as extruders, compressors, conveyors, crushers, pumps, grinding mills,

fans, rolling mills, blast furnace blowers, gas turbine starters, mixers, mine hoists, reactive

power compensation, marine propulsion, high-voltage direct-current (HVDC) transmission,

hydro pumped storage, wind energy conversion, and railway traction [2].

Multilevel converters provide us a lot of challenges and offer a wide range of possibilities.

Researchers are trying to further improve them in the fields of efficiency, reliability, power

density, simplicity, and cost of multilevel converters and its dimension.

Recently, the interleaved converters with coupled-inductors have been widely used in

medium- to low-power applications, mainly to increase the output current, while the current

ripple in power devices reduces. Using the interleave technique; the size of converter’s

passive components (inductances and capacitances) and the harmonic content of output

voltage are considerably decreased with respect to the classical approach.

There are many publications in the field of MMC converters. In the past decade, due to

increasing the demands of MMC converters, researchers investigated the multilevel

converter from a different point of view. These converters have been analyzed in order to use

in the various applications such as HVDC, electric vehicle, renewable energy system and etc.

also, new modulation approaches and modern control strategies have been proposed to

increase the power quality and decrease the harmonics.

One of the interesting subjects is to estimate the losses of multilevel converters. Due to its

complex topology, researchers proposed various losses models based on its topology and

switch type. According to use the multilevel converter in high power application, its final

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3

dimension is very big and bulky. Also, utilization of different components such as the

semiconductor switch, coupled inductor, heat sinks, capacitors, bus bars and other

accessories makes it very difficult to implement. The analysis which is used to find the best

component value of the multilevel converter in order to achieve the minimum dimension is

called “converter sizing analysis.” Therefore, the dimensioning analysis will be an important

part of the design procedure.

In literature, there are few publications about the global dimensioning methodology of MMC

converters, especially in high power application. It seems there are two points, which affect

the lack of researches. First, academic researchers do not have access to the information of

high power applications. Secondly, dimensioning design data is a part of confidential

engineering documents of the companies and therefore, highly unlikely to be published

publicly.

The main objective of this dissertation is to propose and implement of the converter sizing

of the static converters applied to MMC structures. Providing a platform, which analyzes and

executes the global optimal dimensioning algorithm, utilizing the specifications of a MMC

industrial converter application, verification of algorithm parameters, finding the optimal

solution for the industrial application and extending the algorithm for other applications are

the secondary objectives.

This dissertation introduces the converter sizing methodology in order to optimize the MMC

converters. Chapter 1 presents a definition for MMC converter and explains the history of

modular multilevel converters. Then it introduces different topologies of MMC converter

and investigates their advantages and disadvantages. In addition, the important industrial

application of MMC converters with an industrial example is presented.

In chapter 2, the main Ph.D. objectives are explained. The converter sizing model is defined

to implement to the high power MMC converters. Also, the validation tool in order to verify

our methodology is introduced.

In chapter 3, the average steady-state model and its advantages and weakness are presented.

Then, to enhance the model performance, the time-domain steady-state model is introduced

and investigated. Finally, in order to validate the proposed model, the outputs are compared

with the outputs of a Simulink model.

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4

In chapter 4, the methodology of MMC converter optimization is investigated and clarified.

The plan of the optimization loop is proposed and detail sections of the global optimization

loop are introduced and explained. Also, the software which must be used to integrate the

optimization loop is identified and the validation method with the specifications of the MMC

converter industrial application is investigated.

The dimensioning analysis of converter components is presented in chapter 5. Using an

analytical approach, the mathematical mass function of capacitor and inductor are extracted

and added to the optimization algorithm. Then, an optimization is done to minimize the total

converter mass regarding the mass functions.

In order to increase the analytical model accuracy especially in the case of the magnetic

model, a novel hybrid optimization model is presented in chapter 6. It consists a combination

of analytical model and finite element approach which intensely increases the model

accuracy while the optimization time does not increase so much. The mass minimization is

repeated using new optimization algorithm and the result is discussed.

In chapter 7, the optimal converter is investigated and analyzed in the defect condition in

order to minimize the converter damage in fault condition. This study is done using two

different methods. The first one is to send the optimal parameters to Simulink and make a

co-simulation in the defect condition using Matlab/Simulink. The second one is to add the

defect analysis to the time domain model to calculate the extra constraints while the

optimization is running.

Finally, in chapter 8, the future works in the field of the high power modular multilevel

converter is investigated. The optimization results and the proposed approach is investigated

and summarized.

1.2. Relevant State of the Art and Problem Description

In response to the demands of high-power systems and to supply the requirements of the

industrial processes powered by large electric ac drive systems, two different solutions are

proposed by power electronics researchers:

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5

• Conventional two-level voltage/current source topologies comprising high-

voltage/current-rated power switches based on developing and immature high-

voltage semiconductor technology (currently 8 kV and 6 kA);

• Multilevel power converters covering a power range from several MW to tens of

MW based on matured semiconductor technology of medium-voltage/current rated

power switches (currently 1.2 kV up to 6.6 kV) [3].

Although the first approach leads to the simplicity of power and control circuit, two-level

power converters suffer from major disadvantages of augmented price of newer high-power

semiconductors and power quality concerns, specifically as going higher in the power ranges.

In turn, multilevel power converters bring many technical advantages such as extended power

range due to the capability of the multilevel topologies to handle the voltage and power in

the range of several kV and MW utilizing reliable medium voltage insulated gate bipolar

transistors (IGBTs), improved harmonic content of the switched output voltage, and hence

increased power quality, increased reliance on power converter operation owing to possible

fault-tolerant feature, lowered electromagnetic interference and upgraded electromagnetic

compatibility, lowered switching losses, enhanced efficiency, and reduced amount of output

filter, etc. [3-7].

Nowadays, three commercial topologies of multilevel voltage-source inverters were

introduced as classical topologies: neutral point clamped (NPC) [8], cascaded H-bridge

(CHB) [9], and flying capacitors (FCS) [10]. Among these inverter topologies, cascaded

multilevel inverter reaches the higher output voltage and power levels (13.8 kV, 30 MVA)

and the higher reliability due to its modular structure.

In the field of high power application, multilevel converters with a high number of levels

seem to be the most suitable types, because of the need for series connection of

semiconductors in combination with low voltage distortion on the line side. There are many

other important aspects have to be taken into account for these applications. MMC converter

is the technique of using standard commutation cells in series and parallel compositions in

order to achieve higher voltage and current with conventional semiconductor switches.

The main technical and economic aspects of the development of multilevel converters are

[11]:

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6

1. Modular realization

• Scalable to different power voltage levels

• Independent of the fast development of semiconductor switches

• Developing power devices

• Expandable to any number of voltage steps

• Dynamic division of voltage to the power devices

2. Multilevel waveform

• low total harmonic distortion

• use of approved devices

• redundant operation

3. Investment and life cycle cost

• standard components

• modular construction

4. High availability

5. Failure management

Since 50 years ago that multilevel conversion was introduced, several multilevel converters

are used to increase power and reduce THD. Because of the limited current capability of the

cables and semiconductor devices, high power systems need a type of converters which able

to operate with more than a thousand hundred volts.

The multilevel converters provide us the possibility of working at a high-voltage and high

current level, with a better efficiency and power quality. In high power systems, the current

and voltage ratings can easily go beyond the range the existing semiconductor switches.

Multicell interleaved converters which are connected in parallel or series is an interesting

solution for high power application. However, extra measures should be taken for equal

sharing of the current or voltage among the parallel or series devices. In multilevel structures,

due to the interleaved modulation technique, it is possible to achieve a series of advantages

[12], such as:

• Quasi-sinusoidal AC voltage waveform

• Low harmonic impact

• Reduced costs for the filtering elements

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7

• Possible direct connection to the MV grid

• Reduction of semiconductor losses due to a very low single-switching frequency per

device

Multilevel converters include an array of power semiconductors and capacitor voltage

sources, the output of which generate voltages with stepped waveforms. The commutation

of the switches permits the addition of the capacitor voltages, which reach a high voltage at

the output, while the power semiconductors need to be able to withstand only partial of the

total voltage.

For completeness and a better understanding of the advances in multilevel technology, it is

necessary to review the classic multilevel converter topologies. However, in order to focus

the content of this thesis on the most recent advances and ongoing research lines, well-

established topologies will only be briefly introduced and referred to existing literature.

1.3. History and MMC Definition

History of multilevel inverters began in 1975 with Baker and Bannister. This first patent

described a converter topology capable of producing a multilevel voltage by connecting

single phase inverter in series.

The multilevel converter has been developed to compensate the shortcomings in solid-state

switching device ratings and technology so that they can be applied to high-voltage electrical

systems. The special topology of multilevel voltage converters allows them to obtain high

voltages with low harmonics without the use of transformers [11].

Since last two decades, the demand for medium and high voltage power converters has grown

to provide medium and high voltage output with the low harmonic rate. The application of

these converters is HVDC links, static VAR compensators, traction motor variable speed

drives and active filtering. Multilevel power converters have been introduced and presented

as a solution in high voltage and medium voltage applications. MMC provides a cost-

effective solution in the medium and high voltage energy management market. A multilevel

converter has several advantages over a conventional two-level converter that uses high

switching frequency Pulse Width Modulation (PWM). The attractive advantages of a

multilevel converter can be briefly summarized as follows:

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8

Harmonic distortion: Multilevel converter is based on energy conversion using small

voltage steps, their output waveforms are close to the sinusoidal wave. Hence, it contains less

harmonic distortion.

Electromagnetic compatibility: Multilevel converters not only can generate the output

voltage with very low distortion but also can reduce the dv/dt stresses; therefore,

Electromagnetic Compatibility (EMC) problems can be reduced.

Common-Mode (CM) voltage: Multilevel converters produce smaller CM voltage;

therefore, the stress in the bearings of a motor connected to a multilevel motor drive can be

reduced. Furthermore, CM voltage can be eliminated by using advanced modulation

strategies [13].

Input current: The input current of multilevel converters has very low distortion.

Switching frequency: Multilevel converters operates at both fundamental switching

frequency as well as high switching frequency PWM [1, 11].

Multilevel converters have some disadvantages. One of the most important disadvantages is

the greater number of power semiconductor switches needed. Although lower voltage rated

switches can be utilized in a multilevel converter, each switch requires a related gate drive

circuit. This may cause the overall system to be more expensive and complex.

The number of multilevel converter topologies has been introduced during the last two

decades. The researches have concentrated on novel converter topologies and unique

modulation schemes. Moreover, three different major multilevel converter structures have

been reported in the literature: cascaded H-bridges converter with separate dc sources, diode

clamped (neutral clamped), and flying capacitors (capacitor clamped). Furthermore,

abundant modulation techniques and control schemes have been developed for multilevel

converters such as sinusoidal pulse width modulation (SPWM), selective harmonic

elimination (SHE-PWM), space vector modulation (SVM), and others. In addition, many

multilevel converter applications focus on industrial medium-voltage motor drives [1, 14,

15], utility interface for renewable energy systems [16], flexible AC transmission system

(FACTS) [17], and traction drive systems [18].

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9

1.4. Description of Multilevel structures

In the following, classic topologies will be referred to those that have extensively been

analyzed and documented and have been commercialized and used in practical applications

for more than a decade. The more important configurations of multilevel converters are:

1. Neutral Point Clamped (NPC)

2. Flying capacitor

3. Cascaded Multilevel Converters

1.4.1. Neutral Point Clamped (NPC)

The most commonly used multilevel topology is the diode clamped inverter, in which the

diode is used as the clamping device to connect the dc bus voltage in order to take steps in

the output voltage. In the natural point clamped topology, the diodes are the key difference

between the two-level inverter and the three-level inverter. In Figure 1.1, a single-phase

three-level and four levels version is shown, but it is possible to increase the number of level

and legs (phase). In the three levels topology, using two diodes, it is possible to convert the

voltage to half the level of the dc-bus voltage. In general, the voltage across each capacitor

for an N level diode-clamped inverter at steady state is1

dcV

N .

Figure 1.1: Multilevel neutral point clamped converter

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10

In general, for an N level diode-clamped inverter, for each leg 2(N-1) switching devices, (N-

1)(N-2) clamping diodes and (N-1) dc link capacitors are required. Increasing the number of

levels leads to increase the number of diodes and the number of switching devices and makes

the system impracticable to implement. In the PWM converters, the main constraint will be

the reverse recovery of the clamping diode.

The component which characterizes this topology is the diode necessary to clamp the

switching voltage to the half level of the DC bus, which is split into three levels by two series

of connected bulk capacitors. In this topology, the middle point is also called the neutral

point. By increasing the number of levels, the voltage which the diodes have to sustain rises.

For a specific diode rating voltage, more devices are necessary to withstand the whole

voltage. Therefore, if the number of voltage levels that the system can impose is N, 2(N-1)

diodes are necessary. For high-DC voltages, the system becomes less convenient due to the

huge number of diodes.

1.4.2. Flying capacitor

The Flying Capacitor is another multilevel topology, which is suitable for high-power

applications. This topology is composed of the series connection of capacitor clamped

switching cells. Figure 1.2 shows the topology of the multilevel flying capacitor.

The flying capacitor topology has some advantages when compared to the diode-clamped

inverter. It does not need to clamping diodes. In addition, the flying capacitor inverter has

switching redundancy within the phase, which is used to balance the flying capacitors so that

only one dc source is needed.

The flying capacitor and diode clamped inverter have the same problem in implementation.

A large number of bulk capacitors must be used and install. The voltage rating of each

capacitor must be the same as the main power switch rating, an N level converter will require

a total of (𝑁 − 1)(𝑁 − 2)/2 clamping capacitors per phase in addition to (𝑁 − 1) main dc

bus capacitors.

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11

Figure 1.2: Multilevel flying capacitor topology

This topology also has other disadvantages which have limited its utilization. First of all, it

needs the converter initialization. Before the converter can be modulated by any modulation

scheme, the capacitors need to be charged with the required voltage level as the initial

voltage. This complicates the modulation process and becomes a hindrance to the operation

of the converter. The capacitor voltages must also be regulated under normal operation in a

similar way to the capacitors of a diode clamped converter. Another problem of this topology

is the rating of the capacitors. The capacitors have large fractions of the dc bus voltage across

them.

The two topologies analyzed present a better reduction in the harmonics. Despite the

improvements which they are able to reach, these kinds of multilevel converters present a

series of limitations. For this reason, they did not succeed in these HV-application demands.

Also, there are some problems which limit the utilization of this topology in industrial

applications.

1. Unwanted EMI disturbances generated by a very high slope (di/dt) of the arm

currents

2. The DC bulk capacitor stores a huge quantity of energy which leads to damages

under faulty conditions.

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12

3. The stored energy of the concentrated DC capacitor at the DC-Bus results in

extremely high surge currents and subsequent damage if short circuits at the DC-

Bus cannot be excluded.

4. Harmonics on the AC current must always be suppressed.

1.4.3. Cascaded Multilevel Converters

These structures are characterized by a series connection of elementary converters that are

normally identical. Each cell corresponds to a voltage level according to the particular

modulation technique. It is possible to achieve the desired voltage waveform according to

the imposed reference. Figure 1.3 shows the cascaded structure with elementary converter

modules.

Figure 1.3: Multilevel cascaded converter with elementary converters

The cascaded structures ensure the modularity of the system by ensuring series industrial

production. Due to modularity, they do not present upper DC voltage limits. In fact, it is

possible to add more series cells to sustain the desired voltage. The converter is a composition

of series-connected elementary cells. This converter offers the possibility to regulate the

active and reactive power independently. Each phase is composed of two groups of

elementary cells (1…N and N+1…2N), called branches. Each branch conducts the half-phase

current. The advantages of using Cascaded Multilevel Converters are:

• Each arm conducts half current and in continuous conduction mode.

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13

• Arm inductances contribute to limit faulty conditions.

• The bulk capacitor is not necessary because there are two terminal cells.

• Each capacitor cell voltage can be controlled very slowly with respect to the current

regulator.

• The DC link voltage can be controlled by the converter.

The typical structure of an MMC and the configuration of a Sub-Module (SM) are shown in

figure 1.4. This sub-module is known as general commutation cell. Each sub-module is a

simple chopper cell which is composed of two IGBT switches (T1 and T2), two anti-parallel

diodes (D1 and D2) and a capacitor C. The commutation cell limits the maximum switch

voltage. So, it is possible to achieve high voltage by a series connection of commutation cells

(sub-modules).

Icu T1

T2

Iu

C +Vsm

-

+Vc

-

D1

D2

Figure 1.4: Typical configuration of sub-module (Commutation Cell)

Page 42: Optimal Sizing of Modular Multilevel Converters

14

SM1

SM2

SM1

SM2

L1

L1

Vac

E

E

-Vdc

+Vdc

Figure 1.5: Single-leg multilevel converter

The configuration with T1 and T2 both ON should not be considered because it creates a

short circuit across the capacitor. Also, the configuration with T1 and T2 both OFF is not

useful as it produces different output voltages depending on the current direction.

In an MMC the number of steps of the output voltage is related to the number of series

connected sub-modules. In order to show how the voltage levels are generated, in the

following, reference is made to the simple three level MMC configuration shown in figure

1.5.

In this case, in order to get the positive output, +E, the two upper SMs 1 and 2 are bypassed.

Accordingly, for the negative output, - E, the two lower SMs 3 and 4 are bypassed. The zero

state can be obtained through two possible switch configurations. The first one is when the

two SMs in the middle of a leg (2 and 3) are bypassed, and the second one is when the end

SMs of a leg (1 and 4) are bypassed. It has to be noted that the current flow through the SMs

that are not bypassed determining the charging or discharging of the capacitors depending on

the current direction. Therefore, in order to keep the capacitor voltages balanced, both zero

Page 43: Optimal Sizing of Modular Multilevel Converters

15

states must be used alternatively. The principle of operation can be extended to any multi-

level configuration.

In this type of inverter, the only states that have no redundant configurations are two states

that generate the maximum positive and negative voltages, +E and –E. For generating the

other levels, in general, there are several possible switching configurations that can be

selected in order to keep the capacitor voltages balanced. In MMC of Figure 1.6, the

switching sequence is controlled so that at each instant only N SMs are in the on-state. As an

example, if at a given instant in the upper arm SMs from 2 to N are in the on-state, in the

lower arm only one SM will be in on-state. It is clear that there are several possible switching

configurations. Equal voltage sharing among the capacitor of each arm can be achieved by

control of bypassed SMs during each sampling.

In the applications that we need to pass more than 10 kA, the nominal IGBT current is not

sufficient to pass the high value of current. The parallel arm topology was emerged to solve

this problem. Figure 1.6 shows the multilevel converter with parallel arms. The switching

function of the parallel arms will be the same. The arm inductances limit the circulation

current between the parallel arms.

+Vdc

L1

L2

E

E

VacL1

L2

-Vdc

...

Figure 1.6: Multi-leg multilevel converter

Another constraint of converters in high power application is their capability to suffer the

high value of current. The parallelization of the commutation cells by coupled inductors is

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16

an effective approach in order to obtain a high current output with low current switches [19].

The interleaving technique which is the dual of the series connection of commutation cells

to achieve high voltage consists of connecting converters in parallel, with synchronized and

complementary operation, connected to the same load and with the same power source.

Figure 1.7 shows a single-leg two-level converter with interleaving inductors. Interleaved

converters can be classified in two ways: without magnetic coupling and with magnetic

coupling [20].

-V

+V

SM1

SM1'

-V

+V

SM2

SM2'

-V

+V

SMn

SMn'

Vout

Figure 1.7: Single-leg two levels converter with interleaving inductors

The converters are using an output filtering structure with magnetic coupling. These filtering

structures are usually employed to minimize the mass of the converters in a significant way.

These filters are normally sized to work with a constant number of phases. To optimize the

magnetic components, it is necessary to take into account some physical constraint, as the

iron sections of the magnetic circuits and saturation problems. These problems are

challenging, especially when the number of parallel cells is one of the variables [21].

The multicell interleaving converter with small inductance has proved to be desirable for

voltage regulator modules (VRMs) with low voltages, high currents, and fast transients.

Integrated magnetic components are used to reduce the size of the converter and improve the

efficiency. Researchers mentioned that with the proper design, coupling inductors can

improve both the steady-state and dynamic performances of VRMs with easier

manufacturing [22].

Interleaved power converters are used in many different conversion systems involving

various topologies (series or parallel) and related to different fields or loads. Researchers deal

Page 45: Optimal Sizing of Modular Multilevel Converters

17

with interleaved parallel commutation cells using coupling transformers with a possibly high

number of cells.

In high power applications, a composition of series and parallel connection of commutation

cells which provides the possibility of working in the high voltage and high current. This

configuration is known as a multilevel multicell converter (MMC). Figure 1.8 shows an

MMC converter which is consisted of a composition of the series and parallel commutation

cells.

-V

+V

SM1

SMn

SM1'

SMn'

SM1

SMn

Vout

-V

+V

SM1'

SMn'

-V

+V

SM1

SMn

SM1'

SMn'

Figure 1.8: MMC converter with interleaving inductors

1.5. Applications and Industrial Relevance

Researchers presented various topologies with different characteristics. Multilevel converters

make possible to connect to the medium and high voltage grids. In addition, they can connect

to the separated DC sources such as photovoltaic cells and battery package. Multilevel

converters increase the number of switching vectors, therefore they create the possibility to

use modern switching approach and design the progressive controller. There are many

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18

research papers, which classified the modern control of multilevel converters. Recently,

multilevel converters are introducing as a good choice in the case of renewable energy

systems. Because of its scalable technology, it is possible to work in very high power

application. Todays, multilevel converters are the best choice in HVDC and FACT projects.

Another application which is very interesting in literature is electric vehicles. The modular

structure of multilevel converters is ideal to connect to the battery and photovoltaic sources.

MMC has received great attention because of its modular structure, which gives many

benefits as a multilevel converter. Lately, MMC technology has been used in large HVDC

transmissions, and hence well suited for high voltage structures. It is therefore natural to

include MMC in studies regarding high voltage installations. Mentionable projects are the

new cable connection between Norway and Denmark (Skagerrak 4 [18]), Trans Bay Cable

in San Fransisco [23], German offshore wind projects (HelWin, BorWin, DolWin), and the

cable connection between Sweden and Lithuania [24]. Table 1.1 demonstrates some of the

latest projects that include MMC technology. Both Siemens and ABB have developed HVDC

concepts which use some sort of modular multilevel converter topologies. The two concepts

are called HVDC Plus (Siemens) and HVDC Light (ABB).

Table 1.1: Some of the today’s MMC projects

Site Contractor Power (MW)

Trans-Bay San Francisco (2010) Siemens

400

Skagerrak 4 (2014) ABB

700

DolWin1 Germany (2013) ABB

800

DolWin2 Germany (2015) ABB

900

BorWin2 Germany (2013) Siemens

800

NordBalt Swe-Lit (2013) ABB

700

Losses analysis has been a popular subject in the multilevel converter literature. Researchers

presented some solutions to reduce the semiconductor losses. One of the most important

solutions is the multicell converters, which employs the coupled inductors. Some researchers

tried to optimize the converter parameters like ripple, efficiency and etc.

Many researchers tried to introduced and develop the multilevel topologies such as diode-

clamped inverter (neutral-point-clamped), capacitor-clamped (flying capacitor), and

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19

cascaded multicell with separate dc sources. In the recent years, new multilevel topologies

like asymmetric hybrid cells and soft-switched multilevel converters have been presented [1,

25].

Also, multilevel converters have been compared with two-level converters in simulations to

investigate the advantages of using multilevel converters [25]. The symmetrical and

asymmetrical multilevel inverter has been classified by researchers. Both types are very

effective and efficient for improving the quality of the inverter output voltage [26].

On the other hand, researchers presented the weakness of these topologies. One of the major

limitations of multilevel converters is the voltage unbalancing between different levels. The

voltage balancing techniques between different levels normally involve voltage clamping or

capacitor charge control. There are several ways to provide the voltage balance in multilevel

converters [13, 25].

Due to the special topology of multilevel converters, various switching strategies and control

principles have been proposed and developed in the literature and advantages and

disadvantages were classified [25, 26].

The important section of research in the field of the multilevel converter is modular

converters which are suitable for very high voltage applications, especially network interties

in power generation and transmission [11].

The Modular Multilevel Converter (MMC) was represented as a scalable technology making

high voltage and power capability possible. The mathematical model of MMC was presented

with the aim to develop the model converter control system [27].

1.5.1. Multilevel converters and renewable energy

The general function of the multilevel inverter is to synthesize the desired AC voltage from

several levels of DC voltages. For this reason, multilevel converters are ideal for connecting

either in series or in parallel an AC grid with renewable energy sources such as photovoltaics

or fuel cells or with energy storage devices.

Researchers introduced multilevel converters to control the frequency and voltage output of

renewable energy because of its fast response and autonomous control [16]. In addition, they

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20

represented the multilevel converters as a significant tool to control the real and reactive

power flow from a utility connected to the renewable energy source.

The rotor of a doubly-fed induction generator (DFIG) driven by a wind turbine needs rotor

excitation. In a variable-speed wind energy conversion system (WECS), the mechanical

frequency of the generator varies, and to keep the stator voltage and frequency constant, the

rotor voltage and its frequency have to be varied. Thus, the system requires a power

conversion unit to supply the rotor with a variable frequency voltage that keeps the stator

frequency constant irrespective of the wind speed. Also, in the case of permanent magnet

generators, a high power multilevel converter is used to convert the output AC voltage of the

generator to a fixed frequency voltage to supply the grid [28]. Figure 1.9 shows the single-

line diagram of multilevel converter proposed for wind energy conversion system [29].

Figure 1.9: Single-line schematic diagram of the proposed multilevel converter-based wind energy

conversion system [28]

Researchers are interested in the structure of Cascaded H-bridge DC-AC in order to use with

photovoltaic cells. In this structure, each module needs to connect to a DC source. Thus,

modular multilevel converters can be an ideal topology to connect to the PV panels [30]. The

multilevel converter is realized using a multicell topology where the total AC output of the

system is formed by series connection of several full-bridge converter stages. The dc links of

full bridges are supplied by individual DC-DC isolation stages which are arranged in parallel

concerning the DC input of the total system [31]. Figure 1.10 shows the multilevel topology

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21

using full bridge commutation cells which are utilized to make a connection between solar

panels and single phase grid [32].

Figure 1.10: Multilevel inverter topology proposed for solar energy systems [32]

1.5.2. Multilevel converters and HVDC and FACT systems

Over the twenty-first century, HVDC transmissions will be a key point in green electric

energy development [33]. Modular multilevel converter (MMC) was proposed by many

researchers for high voltage AC/DC power conversion applications, such as HVDC and

FACTS [34]. Multilevel voltage source converters allow working at a high voltage level and

draw a quasi-sinusoidal voltage waveform. Classical multilevel topologies such as NPC and

Flying Capacitor VSIs were introduced twenty years ago, and are nowadays widely used in

Medium Power applications such as traction drives. In the scope of High Voltage AC/DC

converters, the Modular Multilevel Converter (MMC) appeared particularly interesting for

HVDC transmissions. On the base of the MMC principle, modular topologies of elementary

cells make the High Voltage AC/DC converter more flexible and easily suitable for different

voltage and current levels [33, 34]. The first modular multilevel converter for HVDC

application went operational in 2010. Figure 1.11 shows the cascaded multilevel topology

used for HVDC application by Siemens Company.

Page 50: Optimal Sizing of Modular Multilevel Converters

22

Figure 1.11: The modular cascaded multilevel converter used by Siemens Company for HVDC

application [33]

The advantages of multilevel topologies were proven in practice: efficiency comparable to

thyristor-based HVDC plants, very low space consumption due to the absence of filters and,

most important for the customer, a very good reliability. The same success could be achieved

for Static VAR Compensators (SVC) based on the modular multilevel concept. Both products

will certainly play a vital role in meeting the future challenges imposed upon the transmission

grids, like the integration of large amounts of renewable energy sources or increasing the

power transmission capability of the existing grids [35]. In the case of FACT applications,

various types of converter topologies are proposed. The most popular topology in this

application is neutral point clamped converters. Figure 1.12 shows a five level NPC

converter. In [36], a five-level FCMC is selected and designed for a 6.6 kV STATCOM.

Page 51: Optimal Sizing of Modular Multilevel Converters

23

Figure 1.12: Three-phase five-level structure of a diode-clamped multilevel converter [36]

Recently, ABB Company introduced two generations of SVC light for electrical transmission

grids [37]. In the first generation, ABB utilized 3-level NPC topology, while in the new

generation multi-level chain link (full bridge commutation cells) topology was used.

Table 1.2: ABB SVC Light for electrical transmission grids

First Generation Next Generation

Introduced 1997 2014

VSC tech. 3-level NPC Multi-level chain link

Converter voltage ≤ 35 kV ≤ 69 kV

Power range per block +/- 90 MVAr +/- 360 MVAr

Losses Medium Low Losses Medium Low Losses Medium Low

Active filtering Yes, up to 9th harm. Yes, up to 9th harm.

Need for filters Yes, high-pass No (depending on design)

DC capacitor Common Distributed

IGBT ABB StakPak, 2.5 kV, 1600 A ABB StakPak, 4.5 kV, 1800 A

Page 52: Optimal Sizing of Modular Multilevel Converters

24

1.5.3. Multilevel converters and Marine propulsion

Marine propulsion systems are undergoing rapid development with a significant thrust

towards the use of electric propulsion and the integration of auxiliary and propulsion power

systems. Multilevel converters are very popular for high power AC drives for large

electric/hybrid vessel. The marine propulsion drive systems usually are in the range of

medium power. The NPC converter is proposed in order to improve the system reliability

and increase its expansion flexibility [38]. The multilevel inverter can generate a high-quality

output voltage with less switching frequency and losses since only the small power cells of

the inverter operate at high switching rate. In the case of marine propulsion, increasing the

quality of power generated by multilevel inverter causes to less torque variation and smooth

propulsion. Furthermore, in this application, converter efficiency, reliability and power

quality are important. Figure 1.13 shows the NPC configuration in the electric propulsion

system [39].

Figure 1.13: Multilevel converter in electric marine propulsion system [39]

1.5.4. Traction motor drive

In the rapid development process of a high-speed electrified railway, power quality problems

in the traction power grid have become increasingly deteriorative. Increasing the usage of

electric railways in transportation systems augments the demands of high power converters.

Therefore, the power quality prolusion related to the high-power converters has increased

more and more [4]. Power traction and load characteristics of electrified railway give rise to

lots of power quality problems, such as negative-sequence current (NSC), harmonic and

Page 53: Optimal Sizing of Modular Multilevel Converters

25

voltage fluctuation. These will not only deteriorate power quality in traction power grid but

also threaten the safe and economical operation of power system [40-42]. The multilevel

converters were proposed as an appropriate solution to enhance the power quality of electric

railways. The main advantage of this kind of topology is that it can generate almost perfect

current or voltage waveforms because it is modulated by amplitude instead of pulse-width.

That means that the pulsating torque generated by harmonics can be eliminated, and power

losses into the machine due to harmonic currents can also be eliminated [43]. Another

advantage of this kind of drive is that the switching frequency and power rating of the

semiconductors are reduced considerably [15]. Figure 1.14 shows a 6-level diode-clamped

back to back converter that was proposed for traction drive application.

Figure 1.14: 6-level diode-clamped back to back converter for traction motor drive [15]

1.5.5. Losses analysis of multilevel converters

Some applications require light, efficient and reliable converters. Multilevel converters have

some potential advantages due to their lower output harmonic distortion and also the lower

device voltage rating requirements. Multilevel converters were compared to conventional

converters in terms of power losses and harmonic distortion of the output waveforms. Also,

the effects of modulation strategy which plays a significant role in the switching loss

distribution were investigated by some researchers [44].

Some researchers compared different structures of the cascaded multilevel converter with

IGBT technologies with the intention of minimizing power loss. The total power losses in

Page 54: Optimal Sizing of Modular Multilevel Converters

26

the IGBTs and diodes of each cell in the chain were estimated [45]. The hard-switching

transients of the power semiconductors at high commutation voltage cause high switching

losses and a poor harmonic spectrum which produces additional losses in the machine [46].

1.6. Main Objective: Converter sizing methodology applied to

MMC structures of static converters

The main objective of this research is to define and develop a converter sizing analysis

algorithm in order to optimize the dimension of power stack of MMC structures of static

converters. Converter sizing or converter dimensioning is an approach, which is used to find

the optimal dimension of a power electronic converter with respect to the inputs and outputs

specifications to achieve the best possible performance. Generally, the converter-sizing

procedure consists of several steps.

• Finding suitable multilevel topology

• Number of converter levels

• Switch sizing based on the specification and circuit model

• Passive component sizing includes capacitor, inductors

The first step is to propose an appropriate converter topology. With the certain source and

load specifications, it is possible to utilize different topologies. After choosing topology,

converter sizing algorithm tries to find the size of semiconductor switches, dimension of

electromagnetic components such as coupled inductors and specification of passive

components such as the capacitors, cables, fuses and etc. Furthermore, in the case of modular

MMC converters, converter sizing algorithm calculates the optimal number of parallel and

series commutation cells in order to achieve the maximum efficiency. Due to a limited

number of power IGBTs and converter topologies, the power semiconductor switches and

the converter topologies will be considered as discrete variables in the optimization

procedure. Therefore, the converter-sizing algorithm must be done for each power switch

and converter topology in order to find the best performance. The most important constraints

are:

• Capacitor voltage ripple

Page 55: Optimal Sizing of Modular Multilevel Converters

27

• Maximum switch losses

• Circulation current

• THD in AC side

• Efficiency

In this project, it is considered that the converter topology is determined. The converter sizing

procedure is used to determine the switch size, the dimension of coupled inductors, the

specification of passive components and number of series and parallel commutation cells.

Because of the limited number of IGBTs in the field of high power application, the

optimization algorithm will repeat for each IGBT to find the best performance. In these

project, we will focus on the MMC converter industrial application and its load specifications

and the sizing algorithm will calculate the optimal dimensions of power components.

Converter sizing analysis is a comprehensive analysis approach which consists of different

sub-methods from related knowledge. It is composed of circuit analysis, electromagnetic and

thermal analysis, mechanical analysis and manufacturing rules. Converter sizing analysis of

a multilevel multicell converter is a subject, which has rarely been studied, especially in high

power applications. Generally, the dimensioning analysis data are the confidential documents

of the companies. This may be the reason why this area has not been studied well in the

literature. Furthermore, it is impossible to verify the dimensioning analysis of the high-power

applications by university laboratories.

1.7. Secondary Objectives

1.7.1. Specify the global optimization approach for Optimization of MMC

static converters

In this thesis, we will propose an optimization model for MMC converters in order to

minimize its volume and mass in high power applications. This model consists of circuit

analysis, magnetic and thermal analysis and optimization program which work together to

find the best dimension.

Calculation of converter dimension is dependent on the total efficiency, total dimension and

manufacturing cost. The switch losses, inductor losses, inductor dimension, total efficiency,

Page 56: Optimal Sizing of Modular Multilevel Converters

28

and dimension are not independent variables. Therefore, an optimization loop must be

employed in order to find the number of levels, arm and inductor value, which obtain minimal

losses and dimension. Additionally, in each optimization loop, we have an internal

verification loop in order to achieve the precise result. The converter sizing of multilevel

converters is composed of several analysis methods, which need to work together.

• Electrical analysis

• Magnetic analysis

• Thermal analysis

• Dimensioning analysis

• Cost analysis

The converter-sizing algorithm of the multilevel converter has two main optimization cores.

The first core optimizes the semiconductor switches rating and the number of cells. It works

based on the circuit analytical model. There is an internal verification loop, which transfers

the circuit data to a circuit analysis software in order to verify the core results. The second

core finds the optimal dimension of the passive components (coupled inductors and

capacitors) which is offered by the first core. The analysis in this core is more complicates

than the first one. It receives the circuit data of inductors and capacitors and tries to optimize

the inductor losses, capacitor ripple and the dimension.

Figure 1.15: Optimization and verification loop

Dim

ensio

n V

ariab

les

Circuit Analysis

Electromagnetic Analysis

Thermal Analysis

Dimensioning Analysis

Op

timal

Dim

ensio

ns

Solver

Constraints Goal Function

Page 57: Optimal Sizing of Modular Multilevel Converters

29

Figure 1.15 shows the main parts of the optimization loop which is used in the converter-

dimensioning algorithm. The inductor and capacitor value and the dimensions are the outputs

of the second core. The passive components variables must return to the first core in order to

complete the global optimization loop. The design of the inductors is done in the second core.

Thus, the output of this core must be verified by an advanced software. In order to verify the

inductor analysis output, two internal verification loops have been used. The electromagnetic

verification loop is composed of a finite element electromagnetic analysis which receives the

data from the core and corrects the design parameters to achieve the response. With the same

approach, a thermal verification loop is used to verify the thermal design.

Optimal dimensioning is an art more than to be a science. It makes a connection between

classical analysis approach and industrial manufacturing methods. In this dissertation, a

converter sizing methodology will be proposed in order to find the optimal dimension of

passive components that lead to minimize the material consumption and cost.

Dimensioning analysis needs to collect some technical and experimental data of high power

semiconductors, heat sink systems, isolation materials and inductor core. This information

must be formulized in order to use in the optimization algorithm.

The optimization core generally uses the analytical analysis of the system. It decreases the

solution time, which is very important in the complex issues. Analytical approaches have

some errors in their results in comparison to the actual data. In order to eliminate the error,

an internal verification loop will be connected to each optimization loop. Each verification

loop consists of a software, which corrects the error coefficient until the result of analytical

approach is fitted to the simulation results. The analytical analysis consists of circuit analysis,

electromagnetic analysis, and thermal analysis. Thus, we need to connect to three software

in order to verify the circuit data, electromagnetic design, and thermal analysis. Figure 1.16

shows the proposed converter-sizing algorithm with verification loop applied to MMC static

converters.

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30

fswNL

abcdlgJ

Converter Optimization

Core

InductorOptimization

Core

L,M

Pcu,Pcore

2D-3D FEM Magnetic

Verification

Initial Value

2D-3D FEM Thermal

Verification

Circuit Verification

Dimensioning

data

Figure 1.16: Optimization and verification loop

1.7.2. Applying the converter sizing method to an industrial application

Using multi-level converters will lead to the lower voltage stress across semiconductor

devices, lower switching frequency and less harmonic distortion in AC side. Additionally,

regarding design the appropriate controller, multi-level converters provide more adjustable

states.

The simplified power diagram of the specific MMC industrial application is shown in Figure

1.17. This multi-megawatt power supply is used in the PS Booster, a circular particle

accelerator of the CERN accelerator complex (European Organisation for Nuclear Research

in Geneva Switzerland) [75]. An Active AC/DC Front End (AFE) converter is supplying

electrical power to a DC/DC H-bridge Multilevel converter that is feeding the accelerator

electromagnets ring.

The capacitor bank in the DC bus is oversized in order to be used as a storage capacitor to

exchange the energy with the electromagnets during their current pulsed operation. The rated

electromagnet current pulsed cycle and the corresponding capacitor voltage waveform are

presented in Fig 1.17 & 1.18 respectively [75]. The multilevel H-bridge converter provides

the current of the load which is adjusted with a current controlled controller. The current of

the load changes from 300 A to 6000 A in less than 0.3 second. The extreme change of load

Page 59: Optimal Sizing of Modular Multilevel Converters

31

current affects the DC link voltage and drops it. In this condition, the conventional controller

does not have satisfactory performance. Also, the controller must set the reactive power to

zero to decrease the mis-effect of the converter on the grid, especially in the low switching

frequencies.

Figure 1.17: Multi-megawatt power supply of the PS Booster

Figure 1.18: Electromagnet Current pulsed Cycle delivered by the Multilevel H-bridge converter

Figure 1.19: Corresponding capacitor voltage cycle in the DC bus

Filtercircuit

Multilevel Rectifier

Transformer

MultilevelH-Bridge

Converter

Load

(18M

W)

Gri

d

18KV, 50 Hz 18KV:2KV

AFEActive

Front End

MultilevelActive

rectifier

DC BusStorage

Capacitor

Page 60: Optimal Sizing of Modular Multilevel Converters

32

The initial topological structure of the Active Front End converter (AFE) presented in [75]

was a three phase Neutral Point Clamped (NPC) converter. In this document, the replacement

of this NPC based AFE by a MMC is investigated as an application example of the proposed

design methodology.

Due to the variation of output current, it is necessary to do a precise investigation in order to

find the optimal size of the active and passive component. In addition, the DC link voltage

oscillation is another problem, which affects the performance of the power module and

controller. This will be possible using a converter sizing analysis, which considers all rules,

constraints, and limitations.

In high power applications, the role of switch losses will be very important. It plays a

significant role in order to increase the power of the converters. Decreasing the switching

frequency is one of the prevalent tasks to decrease the switch losses. This limitation forces

us to choose a low switching frequency for this application. The value of switching frequency

depends on the maximum losses of IGBT, which is the summation of conduction and

switching losses.

Due to high power rating, it is not possible to implement the converters with the classical

approach. There is not any semiconductor switch, which endures in this voltage and current

rating. Therefore, we need to use series and parallel configurations. Voltage limitation of the

semiconductor switches will be resolved using multilevel topologies whereas its current

limitation will remain. In the literature, the multi-leg (multi-phase) interleaved converter was

proposed for this problem. Therefore, in this application, we need to use a composition of

multilevel multicell topology to achieve our desired outputs. The power quality, final

dimension and implementation cost are the main constraints in this design approach. The

number of switches per arm must be determined. It is dependent on the DC bus voltage and

switches characteristics.

1.7.3. Finding the optimal solution for the MMC AFE converter

Result verification is an important part of high power application research. Due to the

difficulty of proving the validity of the result, researchers that work in the academic

laboratories are not interested in working in the field of dimensioning analysis of high power

Page 61: Optimal Sizing of Modular Multilevel Converters

33

converters. It is impossible to manufacture the high-power converters in the university

laboratory. Therefore, result validation is usually an important obstacle against the

development of academic research in the field of high power converters.

Figure 1.20: Example of High Power Converter in CERN complex

Fortunately, it was possible to have access to the specifications and technical information of

the industrial application presented in [75]. Figure 1.20 shows one of high power converter

cubicles on the CERN complex.

After result verification of our converter-sizing algorithm with load specification of the

MMC AFE converter application of Fig. , we need to propose an optimal solution for the

next generation of this application. The optimal dimensioning method ensures us to achieve

the best performance with minimum material and manufacturing cost. In this optimization,

we consider that the converter topology was already chosen and the optimization algorithm

tries to find the switch specification, the number of levels, the number of parallel cells, cables,

capacitor specifications, and inductor dimensions.

1.7.4. Using the Converter sizing method for other Applications

The next goal of this research is to extend our methodology in order to design and upgrade

other high power converters applications. It will be possible to extend our algorithm to these

converters with small variations.

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34

1.8. Conclusion

To outline this section, there are many types of research about multilevel multicell converters.

Increasing the demands of high power converters causes to increase the number of researches

in this field. Developing the multilevel multicell topology, control system, losses reduction,

harmonic distortion and enhancement of transient behavior is the most important field of

research.

In high power applications, one of the important issues is to minimize the final volume and

the mass of the converter in order to decrease the manufacturing cost. Also, the consumer

expects to achieve the best possible desired outputs in return.

In this dissertation, a systematic approach will be proposed in order to achieve the converter

desired outputs with minimum mass, volume, and price. This type of analysis is called the

converter sizing analysis. The converter sizing analysis is an optimization between switch

and cable specifications, number of levels, number of parallel cells, capacitors size, and

inductor dimension with respect to different constraints in order to minimize the final mass

or volume of MMC converter. The details and methodology of this analyzing method will be

clarified in the next chapters.

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35

CHAPTER II

2 Design of MMC converter based on the load

specification

2.1. Introduction

This chapter explains the design procedure of an active-front-end MMC converter. The

design procedure is an algorithm, which leads to determining the components value and size

regarding the nominal operating point. Unlike the conventional converters, modular

multilevel converters have many parameters to determine. The selection of converter

topology, sub-module capacitor, electromagnetic components and number of sub-modules

are known as the most important issues in the design procedure

The selection of converter topology depends on the converter application and the technical

and manufacturing constraints. The nominal voltage and current, power quality and total

efficiency are some of the parameters, which are important to choose the topology. In

addition, the sub-module topology is chosen dependent on the application.

The passive components such as capacitor and inductor are the main parts of converter

topology. There is an internal interaction between the components value and circuit variables

that made it very complex to investigate. The circuit analysis approach, which estimates the

circuit variables in transient and steady state condition.

The arm inductance is one of the important components of MMC converter. The arm

inductance should be investigated not only in term of circuit evaluation but also in term of

electromagnetic analysis. Hence, the MMC converter needs an extra analysis in comparison

with conventional converters.

Page 64: Optimal Sizing of Modular Multilevel Converters

36

The maximum voltage of IGBT is the most important factor that affects the number of sub-

modules. In high voltage application, increasing the number of a series sub-module is a

solution to endure the voltage. In addition, increasing the number of sub-modules leads to

higher power quality of converter. On the other side, the increasing of sub-module augments

the converter complexity and price.

In this chapter, the design procedure of MMC converter is explained. The converter and sub-

module topology and the input/output variables of the converter are investigated. A sizing

procedure is introduced to design the passive components and IGBTs in steady state and fault

condition. The adjustable parameters are introduced that should be design based on the

nominal. In order to calculate the converter efficiency, THD, and volume, a simple approach

has been presented. Finally, the advanced analysis tools are presented in order to investigate

the converter in term of circuit, electromagnetic and thermal analysis. In addition, utilizing

the analysis tools, a global optimization algorithm has been introduced to find the best

converter performance using the converter model and analysis tools.

2.2. Calculation of MMC converter variables

2.2.1. Converter and sub-module topology

MMC converters have a modular structure and each converter arm is composed of several

sub-modules in series. This modularity provides the possibility to employ the MMC

converter in a wide range of voltage from medium voltage to very high voltage applications.

Also, each converter arm includes an inductor that reduces the harmonics and the circulation

current between the arms. Figure 2.1 represents the topology of a modular multilevel active

front-end converter with parallel arms. The parallel arms topology is used to provide the

possibility to pass high current values, which are higher than IGBT nominal currents. In the

parallel topology, the voltages are the same while the currents are divided by the number of

parallel arms. Hence, in this chapter, to simplify the equations, the single leg topology is

investigated. There are various sub-module topologies that are used in the MMC converters.

The half-bridge and full bridge are the most important sub-module that is employed in high

power application. Figure 2.2 shows the circuit configuration of half-bridge and full bridge

sub-modules. The full-bridge sub-module includes four switches and one bypass capacitor

while half-bridge sub-module includes two switches and on bypass capacitor. The full bridge

Page 65: Optimal Sizing of Modular Multilevel Converters

37

configuration generates more voltage steps in combination with other sub-modules. Hence,

it provides better power quality and lower harmonics. In addition, the input current is divided

between two arms and it is possible to employ the switch with lower power. On the other

side, full bridge configuration needs more semiconductor switches and more drivers that

made it bulky and expensive.

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+

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Page 66: Optimal Sizing of Modular Multilevel Converters

38

2.2.2. Converter inputs and outputs variables

If an active front-end converter is considered as a black box, in the steady-state condition,

the active power relation in the AC side and DC side are written as below:

𝑃𝑎𝑐 = 𝑣𝑎𝑖𝑎 + 𝑣𝑏𝑖𝑏 + 𝑣𝑐𝑖𝑐 (2.1)

𝑃𝑑𝑐 = 𝑖𝑑𝑐𝑣𝑑𝑐 (2.2)

It was assumed that the harmonics are negligible. Figure 2.3 represents the input and output

parameters of a three-phase active front-end converter regardless of the converter topology.

c

ia

ib

ic

idc

vdc

va

C

iout

R L

vb R L

vc R L

a

b

c

n

Figure 2.3 Inputs and output variables of active front-end converter

The circuit equation of input and output circuit are written as below:

𝑣𝑎𝑏𝑐 = (𝑅 + 𝐿𝑝)𝑖𝑎𝑏𝑐𝑠 + 𝑣𝑎𝑏𝑐𝑛 (2.3)

𝐶𝑝𝑣𝑑𝑐 = 𝑖𝑑𝑐 − 𝑖𝑜𝑢𝑡 (2.4)

Where 𝑝 =𝑑

𝑑𝑡 and 𝑣𝑎𝑏𝑐𝑛 is the phase voltages between the converter terminals (abc) and

neutral point (n) , 𝑖𝑎𝑏𝑐𝑠 are three phase line currents, 𝑅, 𝐿 are line resistance and inductance

respectively..

Regarding the switching index, the relation between the input and output variables for the

fundamental frequency is defined in the DQ axis as below:

𝑣𝑞 = 𝑅𝑖𝑞𝑠 +𝜔𝑒𝐿𝑖𝑑𝑠 +𝑚𝑞

2𝑣𝑑𝑐

(2.5)

𝑣𝑑 = 𝑅𝑖𝑑𝑠 −𝜔𝑒𝐿𝑖𝑞𝑠 +𝑚𝑑

2𝑣𝑑𝑐

(2.6)

3

4(𝑚𝑞𝑖𝑞𝑠 +𝑚𝑑𝑖𝑑𝑠) = 𝑖𝑑𝑐

(2.7)

Page 67: Optimal Sizing of Modular Multilevel Converters

39

Where 𝑚𝑞 and 𝑚𝑑 are the average switching index on q and d axis, respectively and

𝑖𝑑𝑠𝑎𝑛𝑑 𝑖𝑞𝑠 are d and q axis currents. Also, the active and reactive power on AC side.

𝑃𝑖𝑛 =3

2(𝑉𝑞𝑠𝑖𝑞𝑠 + 𝑉𝑑𝑠𝑖𝑑𝑠)

(2.8)

𝑄𝑖𝑛 =3

2(𝑉𝑞𝑠𝑖𝑑𝑠 − 𝑉𝑑𝑠𝑖𝑞𝑠)

(2.9)

2.2.3. Semiconductor sizing in steady state

The semiconductor switch is chosen regarding its nominal voltage and current and its losses

function. In the half-bridge configuration, the capacitor voltage is the maximum voltage that

should be endured by IGBTs. The capacitor voltage is a DC part with a ripple. The IGBT

voltage must be greater than 𝑉𝑐𝑚𝑎𝑥.

𝑉𝑐𝑚𝑎𝑥 =

𝑉𝑑𝑐𝑚+𝑅𝑖𝑝𝑝𝑙𝑒

2

(2.9)

where 𝑅𝑖𝑝𝑝𝑙𝑒 is the difference between maximum and minimum voltage of the capacitor.

The current of IGBTs in half bridge configuration depends on the switching function and

load specification. The arm current, 𝐼𝑎𝑟𝑚 is composed of phase current and circulating

current.

𝐼𝑎𝑟𝑚 = 𝐼𝑐𝑖𝑟𝑐 ±𝐼𝑎2

(2.10)

Where 𝐼𝑎is the instantaneous value of line current and 𝐼𝑐𝑖𝑟𝑐 is the the instantaneous value of

circulation current in arm. The arm current is divided between two IGBT regarding to the

modulation index. The current of sub-module switches 𝐼𝑠1, 𝐼𝑠2 are computed as below

𝐼𝑠1 = Sm. 𝐼𝑎𝑟𝑚 (2.11)

𝐼𝑠2 = (1 − 𝑆𝑚)𝐼𝑎𝑟𝑚 (2.12)

Where 𝑆𝑚 is the modulation Index of the sub-module. In the next sections, the precise

calculation of the capacitor voltage and IGBT current is presented and clarified.

The IGBT losses are the most important parameter in order to select the appropriate switch.

High power IGBT are manufactured by companies based on specific applications to provide

a balance between conduction losses and switching losses. The switching and conduction

losses of IGBTs depends on the device structure. Fundamental and early device structures

Page 68: Optimal Sizing of Modular Multilevel Converters

40

include symmetric blocking IGBTs and asymmetric blocking IGBTs. Symmetric structures,

also called “reverse blocking,” have inherent forward and reverse blocking capabilities,

which make them well suited for AC applications such as matrix (AC-to-AC) converters or

three-level inverters. Asymmetric structures maintain only forward blocking capability and

offer a lower on-state voltage drop than symmetric IGBTs. This makes them ideal for DC

applications like a variable speed motor control, where an anti-parallel diode is used across

the device allowing operation in only the first quadrant of i-v characteristics.

Faster IGBTs that achieve a higher PWM frequency will reduce ripple current and the

required filters can be made smaller because the output waveform is closer to the desired

waveform. Comparing the two device families to determine suitability for different

applications is based on several key parameters: conduction losses (𝑉𝐶𝐸𝑆𝐴𝑇 (for IGBT), 𝑉𝐹

(for diode)) and switching losses (ETS (for IGBT), 𝑄𝑅𝑅(for diode). In general, the RC-DF

devices [47] have at least 50 percent lower switching loss (ETS (mJ)) values when compared

to the lower frequency devices. The RC-D devices [47] stand out in terms of lower

𝑉𝐶𝐸𝑆𝐴𝑇 values and 𝑉𝐹 values which are the thermal or conduction losses, at the expense of

the higher switching losses [48].

Each IGBT switch includes an IGBT and a freewheel diode. The conduction losses of the

switch are the summation of IGBT and diode conduction losses. The conduction losses

depend on the conduction resistance, saturation voltage and the current of IGBT and

freewheel diode.

𝑃𝑐𝑖 = 𝑣𝑐𝑒𝑖𝑐𝑒 + 𝑟𝑐𝑖𝑐𝑒2 (2.13)

𝑃𝑐𝑑 = 𝑣𝐹𝑖𝐹𝐷 + 𝑟𝑑𝑖𝐹𝐷2 (2.14)

𝑃𝑐 = 𝑃𝑐𝑖 + 𝑃𝑐𝑑 (2.15)

where 𝑖𝑐𝑒and 𝑖𝐹𝐷 are the effective current of IGBT and reverse diode, respectively. 𝑟𝑐 and 𝑟𝑑

are resistance of IGBT and diode in conduction cycle. The switching losses depends on the

switching frequency 𝑓𝑠𝑤 and commutation energy of IGBT.

𝑃𝑠𝑤𝑖 = (𝐸𝑜𝑛𝑖 + 𝐸𝑜𝑓𝑓𝑖)𝑓𝑠𝑤 (2.16)

𝑃𝑠𝑤𝑑 = (𝐸𝑜𝑛𝑑 + 𝐸𝑜𝑓𝑓𝑑)𝑓𝑠𝑤 ≈ 𝐸𝑜𝑛𝑑𝑓𝑠𝑤 (2.17)

Where 𝐸𝑜𝑛 and 𝐸𝑜𝑓𝑓 are the energy losses in one switching period. Figure 2.4 shows the

thermal dissipation circuit of IGBT. The maximum temperature of junction is determined by

manufacturer. The junction temperature is calculated as below:

Page 69: Optimal Sizing of Modular Multilevel Converters

41

(𝑇𝑗 − 𝑇𝑎) =𝑃𝑐 + 𝑃𝑠𝑤

𝑅𝑗𝑐 + 𝑅𝑐𝑠 + 𝑅𝑠𝑎

(2.18)

Where 𝑇𝑗 is the junction temperature, 𝑇𝑎 is the ambient temperature, 𝑅𝑗𝑐 is the thermal

resistance of junction-case, 𝑅𝑐𝑠 is the thermal resistance of case-sink and 𝑅𝑠𝑎 is the thermal

resistance of sink-air.

Rjc Rcs Rsa

PcPsw Ta

Tj Tc Tr

Figure 2.4 Thermal dissipation circuit of IGBT

2.2.4. Passive components sizing in steady state

A. Selection of sub-module capacitor

In MMC converters, the electric energy is stored in sub-module capacitors. The maximum

energy stored in capacitors ECmax is determined by the rated converter power Sn and the

energy-power ratio(𝐸𝑃) [49].

𝐸𝑃 =𝐸𝑐𝑚𝑎𝑥𝑆𝑛

(2.18)

The energy-power ratio varies from EP = 10 J/kVA to 50 J/kVA and depends on the converter

application. Lower values mean a reduction of the converter cost while it leads to higher

voltage ripples in the DC-link circuit.

The design procedure of an active front-end converter starts by selection of two main

converter parameters. It is the rated converter power 𝑆𝑛 and RMS value of the line-to-line

voltage 𝑉𝐿−𝐿,𝑟𝑚𝑠 at the ac side of the converter or the voltage 𝑉𝑑𝑐at the dc side of the

converter. Assuming that in the MMC there are no redundant sub-modules, the relation

between AC side and dc side voltages is given as

𝑉𝐿−𝐿,𝑟𝑚𝑠 =√3𝑆𝑚2

𝑉𝑑𝑐

√2

(2.19)

where the modulation index 𝑆𝑚 can be changed from 0 up to 2/√3.

Page 70: Optimal Sizing of Modular Multilevel Converters

42

The maximum energy stored in sub-module capacitors of the three-phase MMC consisting

of 6m sub-modules is given by:

𝐸𝑐𝑚𝑎𝑥 = 6𝑚𝐶𝑠𝑚2(𝑉𝑑𝑐𝑚)2

= 3𝐶𝑎𝑟𝑚𝑉𝑑𝑐2

(2.20)

Where 𝐶𝑎𝑟𝑚 = 𝐶𝑠𝑚/𝑚 . Hence the arm capacitance 𝐶𝑎𝑟𝑚 can be calculated using the energy-

power ratio EP.

𝐶𝑎𝑟𝑚 =𝐸𝑐𝑚𝑎𝑥3𝑉𝑑𝑐

2 = 𝐸𝑃𝑆𝑛3𝑉𝑑𝑐

2 (2.21)

The value of 𝐶𝑎𝑟𝑚 could not increase so much due to the ratio of 𝑆𝑛/𝑉𝑑𝑐2 , which affects the

dimension of current, limits the rated current of the converter, load and other components.

The maximum energy-stored in capacitors (𝐸𝑐𝑚𝑎𝑥) and the energy-power ratio (𝐸𝑃) are the

parameters that are important to determine the sub-module capacitor.

B. Arm inductance selection

In the structure of MMC converter, the role of the arm inductors 𝐿𝑎𝑟𝑚 in the MMC is to

suppress any high-frequency components of the arm currents caused by differences in upper

and lower arm voltages. The voltage difference between upper and lower arms exists due to

the different switching function of upper and lower switches.

The arm inductances value Larm have been chosen regarding a number of parameters. The

exact value of the arm inductance depends on the sub-module capacitor voltage 𝑉𝑑𝑐/𝑚, the

modulation technique, the switching frequency and an additional controller optionally used

for suppressing the circulating current.

It is assumed that the open loop sinusoidal PWM is used as the modulation technique and the

circulating current is not suppressed by any other control methods. It means that the

circulating current has to be suppressed only by the proper selection of the arm inductance

Larm. The arm inductance must be calculated in term of avoiding resonances that occur in the

circulating current for the given arm capacitance 𝐶𝑎𝑟𝑚.

𝐿𝑎𝑟𝑚 =1

𝐶𝑎𝑟𝑚𝜔22(ℎ2 − 1) + 𝑆𝑚

2 ℎ2

8ℎ2(ℎ2 − 1)

(2.22)

Where ℎ is the existing even-order harmonics (ℎ = 2, 4,…).

The proper selection procedure should be restricted to the particular values of harmonic order

and modulation indices, which are possible for the converter application. In a situation when

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43

the modulation index 𝑆𝑚 is limited during the converter operation to values much closer to

the maximum value 𝑆𝑚 = 1. The best value concerning the RMS value of the circulating

current can be obtained by performing a simulation of the MMC averaged model.

2.3. Investigation of adjustable parameters of multilevel

converter

The design of multilevel converters depends on lots of parameters that should be considered

to achieve a high-performance converter.

2.3.1. Converter topology

Multilevel topologies provide different properties in term of efficiency, harmonic and losses,

complexity, volume, and mass. Neutral point clamped converter, flying capacitor and

modular multilevel converter are the most important topologies of multilevel converters. The

selection of topology depends on the load specification and its application. In the medium

power applications, the neutral point clamped converter is used because of its performance

and reliability. By increasing the converter voltage where the number of levels must be

augmented to endure high voltage, the modular converter is the best choice for high voltage

applications. Because of modular structure, there is no limit to achieving high voltage values.

Also, MMC converters provide the possibility of harmonic cancellation using their internal

passive filter.

2.3.2. Number of sub-modules per arm

The number of sub-module per arm is the most important parameter which separates

conventional and multilevel converters. By increasing the converter voltage level, there is no

semiconductor that suffers the converter voltage. The converter voltage is divided between

the sub-modules. Hence, the maximum voltage of IGBT must be greater than the DC link

voltage divided by the number of sub-module per arm 𝑚.

𝑉𝐼𝐺𝐵𝑇 >𝑉𝑑𝑐𝑚

(2.23)

Increasing the number of sub-modules leads to reduce the sub-module voltage and switch

voltage. The switching and conduction losses of IGBT depend on the voltage [50, 51] and it

Page 72: Optimal Sizing of Modular Multilevel Converters

44

could be reduced by increasing the number of series sub-modules per arm. In result,

according to the IGBT voltage limit, increasing the number of sub-modules can affect and

reduce the switching losses.

The number of sub-modules per arm can effectively decrease the converter harmonic and

enhance the power quality of the system. In high power application, regarding the high

voltage and current values, the switching losses of semiconductor switches is high. Hence,

to avoid high switching losses and respect the thermal limit of semiconductors, a low

switching frequency is utilized. Semiconductor losses are composed of conduction and

switching losses. Increasing the switching frequency leads to increase the switching losses.

In high power application, due to the high value of voltage and current, the low switching

frequency is employed to avoid high switching losses. On the other hand, utilizing low

switching frequency leads to diminishing the converter power quality [52, 53]. Multilevel

converters provide the possibility to work with the low switching frequency without

decreasing the power quality. The effect low switching frequency could be compensated by

increasing the number of commutation cells [52, 53]. The frequency that emerges in the

converter output is the switching frequency multiplied by the number of series sub-modules

per arm. Hence, in the case of high voltage converters where a high number of sub-modules

is used, the low switching frequency is employed to avoid high switching losses while high

power quality is achieved.

2.3.3. Passive component values

MMC converters have passive components (sub-module capacitors and arm inductors) in

their topology. The value of circulation current depends on the capacitor and inductor value.

Therefore, the passive component value affects the converter variable and in result converter

performance. The main role of sub-module capacitor is to balance the sub-module voltage

and divide the DC link voltage between sub-modules. Changing the current direction of the

capacitor leads to vary the capacitor voltage and generate the voltage ripple, which could

make unbalance voltage condition. Increasing the sub-module capacitor value directly

reduces the voltage ripple.

In MMC topology, arm inductance is used to control and diminish the arm circulation current

due to the imbalance voltage between upper and lower arms. Also, it works as a passive filter

Page 73: Optimal Sizing of Modular Multilevel Converters

45

at the converter input that eliminates or reduces the current harmonics. The arm inductance

directly affects the THD value and IGBT losses by minimizing the circulation currently.

Also, in the fault condition, the arm inductance limits the fault current.

Unlike the conventional converters, passive components play an important role in the

structure of MMC converters. They play not only the role of the passive filter but also the

fundamental role in converter circuit performance. Selection of proper value for the passive

components depends on the converter specifications, constraints, and its application. The

passive component values will be computed using a comprehensive circuit analysis model.

2.3.4. IGBT selection

The selection of IGBT specification is another parameter that could be determined.

Unfortunately, in term of high power application, the number of IGBTs that are introduced

by the manufacturer is not so much. Hence. The IGBT selection is a discrete parameter and

there are not lots choices for it. The design procedure of MMC converter could be repeated

for the available IGBTs in order to find the best specification.

2.4. Investigation of converter performance and limitations

The term of converter performance is generally defined regarding its application. The

converter performance could be defined as total efficiency, converter power quality, total

losses, converter volume or mass and etc.

2.4.1. Converter Losses and efficiency

MMC converter losses include IGBT losses and inductor losses. The IGBT losses are

composed of switching and conduction losses. Each sub-module includes two IGBT: main

IGBT and bypass IGBT. The conduction losses of each IGBT includes the collector losses

and diode losses. Also, the IGBT switching losses include the IGBT and diode switching

losses. The total IGBT losses are calculated as below:

{𝑃𝑐𝑖𝑇 = 6𝑚(𝑃𝑐𝑖1 + 𝑃𝑐𝑖2)𝑃𝑐𝑑𝑇 = 6𝑚(𝑃𝑐𝑑1 + 𝑃𝑐𝑑2)

(2.24)

{𝑃𝑠𝑤𝑖 = 12𝑚𝑓𝑠𝑤(𝐸𝑜𝑛𝑖1 + 𝐸𝑜𝑓𝑓𝑖1 + 𝐸𝑜𝑛𝑖2 + 𝐸𝑜𝑓𝑓𝑖2)

𝑃𝑠𝑤𝑑 = 12𝑚𝑓𝑠𝑤(𝐸𝑜𝑛𝑑1 + 𝐸𝑜𝑛𝑑2)

(2.25)

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46

𝑃𝐼𝐺𝐵𝑇−𝑇 = 𝑃𝑐𝑖𝑇 + 𝑃𝑐𝑑𝑇 + 𝑃𝑠𝑤𝑖 + 𝑃𝑠𝑤𝑑 (2.26)

The total IGBT losses depends on the switching frequency (𝑓𝑠𝑤) and number of sub-modules

per arm (𝑚).

The inductor losses are composed of copper losses of the winding and core losses. The copper

losses depend on the inductor current and current frequency. In the first step, the effect of

frequency is neglected. The total copper losses are calculated as below:

𝑃𝑐𝑢𝑇 = 6 𝑅𝐿𝐼𝑎𝑟𝑚𝑟𝑚𝑠2 (2.27)

The core losses include hysteresis and eddy current losses. The core losses depend on the

frequency and magnetic flux density of the core. The total hysteresis and eddy current losses

are calculated as below:

𝑃ℎ = 6𝐾ℎ𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝐵ℎ1.𝑚𝑎𝑥α + 2𝑓𝑠𝐵ℎ2.𝑚𝑎𝑥

α ) (2.28)

𝑃𝑒 = 6𝐾𝑒𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝛽𝐵ℎ1.𝑚𝑎𝑥γ

+ (2𝑓𝑠)β𝐵ℎ2.𝑚𝑎𝑥

γ) (2.29)

where 𝛼 is the hysteresis losses constant and 𝛽 𝑎𝑛𝑑 𝛾 are eddy current losses constants,

which are dependent on material magnetic properties. The converter efficiency is known as

most important criterion that shows the converter performance. The total converter efficiency

is:

η =𝑃𝑑𝑐

𝑃𝑑𝑐 + 𝑃𝐼𝐺𝐵𝑇−𝑇 + 𝑃𝑐𝑢𝑇 + 𝑃ℎ + 𝑃𝑒× 100

(2.30)

2.4.2. Power quality and harmonic Investigation

One of the most important advantages of MMC converter is to eliminate the current

harmonics that are injected into the grid by the power electronic converter. MMC converters

utilize low switching frequency to minimize the switching losses. The multilevel structures

transport and shift the voltage to high-frequency spectrum.

There are two different harmonic frequencies in MMC structures. The double-frequency

component of single-phase ac power is emerged in the phase and compose the low-order

harmonics of MMC converter. On the other hand, the current harmonic generation due to the

switching frequency that was shifted to high spectrum frequency is the high-frequency part

of MMC harmonics. Low frequencies are more expensive in terms of filters because the

components of filters are large and costly. The arm inductances should be designed in order

Page 75: Optimal Sizing of Modular Multilevel Converters

47

to limit the low-order harmonics which can affect the converter losses. In term of high

frequency, researchers proposed several methods to control and diminish the high-frequency

harmonics [54].

2.4.3. Converter volume and mass

MMC converter is composed of multiple sub-modules that are series connected to sustain the

high voltage values. By increasing the converter voltage, the total converter volume and size

will be an important issue due to the high number of components. Hence, the converter mass

estimation is a criterion in converter design and optimization.

The converter mass depends on the number of sub-modules. Each sub-module consists of

two IGBTs and one capacitor. The total mass of IGBTs (𝑀𝐼𝐺𝐵𝑇) and sub-module capacitors

(𝑀𝑐𝑎𝑝) is calculated as below:

𝑀𝐼𝐺𝐵𝑇−𝑇 = 12𝑚𝑀𝐼𝐺𝐵𝑇 (2.31)

𝑀𝑐𝑎𝑝−𝑇 = 6𝑚𝑀𝑐𝑎𝑝 (2.32)

The value of IGBT and capacitor mass are written in the datasheet. Unlike the IGBT and

capacitor, the number of arm inductances is not dependent on the number of sub-module per

arm. The inductor mass is composed of core mass and winding mass. The total inductance

mass (𝑀𝑖𝑛𝑑) is written as below:

𝑀𝑖𝑛𝑑 = 6(𝑉𝑐𝑜𝑟𝑒𝐷𝑐𝑜𝑟𝑒 + 𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔𝐷winding) (2.33)

2.5. Comprehensive analysis of MMC converter

Due to the complexity of MMC structures, a comprehensive analysis includes the circuit,

electromagnetic and thermal model must be done to achieve the reliable results. The circuit

model determines the electrical quantity of MMC circuit regarding the load specification,

IGBT model, and switching function. The electromagnetic model estimates the magnetic

variables of inductors regarding the core parameters and nominal values. The thermal model

investigates the heat distribution which is produced via converter losses in the inductor and

semiconductors.

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48

2.5.1. Circuit Analysis

The converter circuit model provides the quantity of all electrical variables versus the

operating load values, component value, and switching function. In the case of MMC

converter, there is a circular interaction between the passive component values and the

electrical quantity of the converter that made it complex to investigate. Figure 2.5 shows the

input/output variables of MMC circuit model of an active front-end converter. Various circuit

models were proposed by the researchers to explain the circuit behavior of MMC converter

and estimate the important electrical variables that are important to determine the converter

performance.

In the literature, various circuit models have been proposed for MMC converter. The MMC

circuit model based on the average switching function are the most popular models [55-58].

The average model neglects the effect of switching frequency. Hence, it is not accurate

enough for harmonic study [59-61]. In this research, a time-domain MMC circuit model is

presented and developed to estimate the ripple and THD value.

MMC Circuit

Model and

Fault Analysis

Vabc

Idc

S(t)

fsw m

Csm

Larm

ωs

Iabc

Is1,Is2

Icirc

VCsm

Voltage ripple

THD

IL

Ifault

Switch losses

Spe

cifi

ctio

ns

Co

nve

rte

r to

po

logy

N.

.

.

Pas

sive

co

mp

on

en

t v

alu

e

M

IGB

T

Spe

cifi

cati

on

s Vmax

Imax

Psw

Inputs

Stead

y-state Variab

les

Stead

y-state Pe

rform

ance

Inductor losses

Efficiency

Energy stored

Outputs

Figure 2.5 MMC circuit model and input/output variables

Page 77: Optimal Sizing of Modular Multilevel Converters

49

2.5.2. Electromagnetic Analysis

Unlike the conventional converters, the electromagnetic components are the part of MMC

converter topology. The electromagnetic components a circuit model which is used in circuit

analysis and a magnetic model that calculates the magnetic parameters of the electromagnetic

core.

The electromagnetic model of inductor investigates the magnetic behavior of inductor core

and the winding. The electromagnetic model estimates the inductance, resistance, magnetic

core flux, losses, volume and mass regarding the core size and winding parameters. The

magnetic analysis is a complex analysis which is generally done using numerical solution

approaches. The simplest way to analyze a magnetic component is to use the analytical

model. The analytical model explains the magnetic behavior of an electromagnetic system

using simple mathematical equations. The analytical model accuracy is not high but, it is an

appropriate tool in the initial design procedure. Figure 2.6 shows the input/output of the

analytical model of arm inductance which calculates the inductance value regarding the core

and winding parameters.

Dimension model

of arm

inductances

IL1

µr

Larm

Jabcdg

M

Bmax

RL

Pcu

Pcore

Vind

Mind

Spe

cifi

ctio

ns

Co

re t

op

olo

gy

IL2

Win

din

g

n1

n2

Inputs

Electro

magn

etic V

ariable

s

Pe

rform

ance

Outputs

η%

Figure 2.6 Analytical electromagnetic model of arm inductance and its input/output

The major weakness of the analytical model is the low model accuracy especially in the case

of electromagnetic analysis. Various numerical solution approach is used to analyze the

electromagnetic structures. Finite element method (FEM) is the most popular approach which

is used to solve the electromagnetic issues.

Page 78: Optimal Sizing of Modular Multilevel Converters

50

Finite element method provides high accuracy results in comparison with the analytical

model. Finite element method is a time-consuming analysis method which is not suitable for

the initial step of design. In this research, an innovative model is presented that is a

combination of analytical model and finite element method to provide the accuracy and

analysis speed at the same time. Figure 2.7 shows the proposed model topology which utilizes

the analytical model and finite element analysis at the same time.

FEM

softwareLfem=Larm

Correction

coefficient

no

Finish

Dimension model

of arm

inductances

IL1

µr

Larm

Jabcdg

M

Bmax

RL

Pcu

Pcore

Vind

Mind

Spe

cifi

ctio

ns

Co

re t

op

olo

gy

IL2

Win

din

g

n1

n2

InputsE

lectrom

agne

tic Variab

les

Pe

rform

ance

Outputs

η%

yes

Figure 2.7 Proposed model for electromagnetic analysis includes analytical model and finite

element analysis

2.5.3. Thermal Analysis

The thermal losses which are generated in the components increase the components

temperature and might leads to destroying them. The IGBTs and arm inductances are major

heat sources in the converter topology. In the case of IGBT, the thermal limitation and its

dissipation function are described in the technical datasheet. The cooling system is designed

based on the technical data and the IGBT losses.

Page 79: Optimal Sizing of Modular Multilevel Converters

51

In the case of an inductor, the thermal losses are generated in the winding and the magnetic

core. Hence, the thermal analysis should be done to estimate the maximum temperature rise.

Figure 2.8 shows the proposed thermal model based on the analytical approach. The model

output is the temperature rise while the inputs are the calculated losses and cooling system

specifications.

Thermal Model

Pcu

Pcore

Psw

IGBT

cooling

Inductor

cooling

Co

nve

rte

r lo

sses

Co

olin

g

InputsTe

mp

erature rise

ΔTIGBT

ΔTcore

ΔTcu

Outputs

Figure 2.8 The analytical thermal model of MMC converter

2.6. MMC dimensioning Analysis

The last analysis which should be done to determine the MMC converter performance is

dimensioning or sizing analysis. The circuit value of passive components does not represent

the component dimension. The passive component dimension depends on lots of technical

and manufacturing and thermal parameters that must be considered to achieve the real size.

Hence, a physical dimensioning analysis is employed by designers to estimate the size of

components. The total converter size which is composed of capacitor, inductor, and

semiconductor switches and cooling system can be considered as converter performance.

2.6.1. Capacitor dimensioning analysis

The capacitor is an industrial component which is manufactured by the industrial companies

base on the standard tables. Actually, regarding the product of each manufacturer, there is a

limited number of capacitors, especially in high voltage applications. Hence, the capacitor

value must be sized regarding the available products.

Power electronic capacitors are designed regarding their applications [62]. Power electronic

capacitors are generally manufactured based on the Polypropylene/Polyester film and

Page 80: Optimal Sizing of Modular Multilevel Converters

52

Metallized polypropylene film technologies in order to minimize the internal inductance and

eliminate the high-frequency noises. The capacitor container is a robust rectangular or

cylinder case.

The volume or mass of capacitor depends on the capacitor type (technology), capacitance

value and capacitor voltage. Investigation of industrial products shows that in low voltage,

the capacitor volume strongly depends on the capacitance. In high voltage, increasing the

capacitor voltage intensely affect the capacitor mass.

2.6.2. Inductance dimensioning analysis

Unlike the capacitors, inductors that are used in power electronic are not pre-designed

products. Power electronic designers calculate the technical parameters of the inductor to

make an order to the manufacturing companies.

The magnetic core and winding are the most important parts of the inductor. The magnetic

cores are designed with various shapes and topologies. The most popular and simplest

topology which is used to design inductor is shown in figure 2.9. The core volume is

calculated as below:

𝑉𝑐𝑜𝑟𝑒 = 𝑐[(𝑎 + 2𝑑)(𝑏 + 2𝑑) − 𝑎𝑏] (2.34)

𝑀𝑐𝑜𝑟𝑒 = 𝑉𝑐𝑜𝑟𝑒𝐷𝑐𝑜𝑟𝑒 (2.35)

d

a

b

d

d

g

c

Figure 2.9 The magnetic core topology

Page 81: Optimal Sizing of Modular Multilevel Converters

53

The core parameters should be designed with respect to the magnetic parameters such as

maximum flux density and core losses and the thermal constraints.

The inductor winding is another part of an inductor that affects the inductor mass. The

average length of one turn is calculated as below:

MLT = 2𝑐 + 4𝑎 + 2𝑑 (2.36)

The total copper volume and mass are estimated as below:

𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑛.𝐼𝐿𝐽. MLT

(2.37)

𝑀𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔𝐷𝑤𝑖𝑛𝑑𝑖𝑛𝑔 (2.38)

where 𝐷𝑤𝑖𝑛𝑑𝑖𝑛𝑔 is the density of the winding’s material.

2.7. MMC Multiphase Analysis

As it was mentioned, MMC converter is composed of various components that need several

analysis models to design. The design of electrical quantity must be done at the same time

that passive components are designed. This kind of analysis which investigates all technical

parameters in a unique design procedure is known as global analysis approach. In the global

analysis, all models should be executed in the same algorithm. The models interact together

to design all MMC variables considering all technical or manufacturing constraints.

2.7.1. Global analysis using analytical model

Analysis of an MMC converter is composed of circuit, electromagnetic and thermal analysis.

Each model includes several inputs/outputs. Some inputs of the models come from the

outputs of other models. For example, the circuit model provides the inductance current value

for the electromagnetic model, while electromagnetic model determines the self and mutual

inductances which are utilized in circuit model calculation.

The circuit and electromagnetic models estimate the switch losses and inductor losses

respectively. The calculated losses are the inputs of the thermal model which estimates the

maximum temperature rise in the semiconductors and inductors.

Page 82: Optimal Sizing of Modular Multilevel Converters

54

MMC Circuit

Model and

Fault analysis

Vabc

Idc

S(t)

fsw m

Csm

Larm

ωs

Iabc

Is1,Is2

Icirc

VCsm

Voltage ripple

THD

IL

Ifault

Switch losses

Spe

cifi

ctio

ns

Co

nve

rte

r to

po

logy

N

.

.

.

Pas

sive

co

mp

on

en

t v

alu

e

M

IGB

T

Spe

cifi

cati

on

s Vmax

Imax

Psw

Inputs

Stead

y-state Variab

les

Stead

y-state Pe

rform

ance

Inductor losses

Efficiency

Energy stored

Outputs

Dimension model

of arm

inductances

IL1

µr

Larm

Jabcdg

M

Bmax

RL

Pcu

Pcore

Vind

Mind

Spe

cifi

ctio

ns

Co

re t

op

olo

gy

IL2

Win

din

g

n1

n2

Electro

magn

etic V

ariable

s

Pe

rform

ance

η%

Thermal Model

Pcu

Pcore

Psw

IGBT

cooling

Inductor

cooling

Co

nve

rte

r lo

sses

Co

olin

g

Tem

peratu

re rise

ΔTIGBT

ΔTcore

ΔTcu

Figure 2.10 Global analysis plan of MMC converter using analytical models

Figure 2.10 shows the plan of global analysis approach which represents the models,

inputs/outputs variables and interconnections. The circuit model gets the nominal operating

point, passive components values and switching function to calculate all electrical quantity

of the converter. The electromagnetic model gets the magnetic core size and parameters to

Page 83: Optimal Sizing of Modular Multilevel Converters

55

calculate the self and mutual inductance that is used by circuit model. Finally, the thermal

model estimates the temperature rise based on the calculated losses and cooling system

parameters.

2.7.2. Global analysis using modified analytical model

In order to compensate the low accuracy of an analytical model of the inductor, a correction

loop has been added to the plan to correct the model parameters regarding the finite element

results.

MMC Circuit

Model and

Fault analysis

Vabc

Idc

S(t)

fsw m

Csm

Larm

ωs

Iabc

Is1,Is2

Icirc

VCsm

Voltage ripple

THD

IL

Ifault

Switch losses

Spe

cifi

ctio

ns

Co

nve

rte

r to

po

logy

N

.

.

.

Pas

sive

co

mp

on

en

t v

alu

e

M

IGB

T

Spe

cifi

cati

on

s Vmax

Imax

Psw

Inputs

Stead

y-state Variab

les

Stead

y-state Pe

rform

ance

Inductor losses

Efficiency

Energy stored

Outputs

Dimension model

of arm

inductances

IL1

µr

Larm

Jabcdg

M

Bmax

RL

Pcu

Pcore

Vind

Mind

Spe

cifi

ctio

ns

Co

re t

op

olo

gy

IL2

Win

din

g

n1

n2

Electro

magn

etic V

ariable

s

Pe

rform

ance

η%

Thermal Model

Pcu

Pcore

Psw

IGBT

cooling

Inductor

cooling

Co

nve

rte

r lo

sses

Co

olin

g

Tem

peratu

re rise

ΔTIGBT

ΔTcore

ΔTcu

FEM

softwareLfem=Larm

Correction

coefficient

no

Finish

yes

Figure 2.11 Global analysis plan of MMC converter using finite element correction loop

Page 84: Optimal Sizing of Modular Multilevel Converters

56

Figure 2.11 shows the proposed plan of global analysis of MMC converter that employs the

finite element correction loop. This plan provides more accurate results compared to the

previous plan.

2.8. Optimal sizing of MMC converter

By increasing the demands of high power converters in medium and high power applications,

MMC converters were changing to complex, bulky and expensive structures. At this time,

mass minimization was emerged to reduce and optimize the volume of passive components

such as capacitors and inductors. In the case of MMC converter, the mass minimization

algorithm is dependent on the circuit operation, electromagnetic and thermal functionalities.

The passive components make the converter very bulky especially by increasing the number

of sub-modules in high voltage application. The most important goal of converter

optimization is to minimize the converter volume by optimal selection of passive components

regarding the technical and manufacturing constraints.

In this dissertation, three optimization plans are presented and investigated. The first plan is

to minimize the passive component value regardless of its mechanical volume and size. In

this plan, the inductance value, sub-module capacitor value, number of sub-modules per arm

and switching frequency are optimization variables that should be calculated by the solver.

The capacitor voltage ripple, total harmonic distortion, and maximum temperature rise are

the most important constraints. In this plan, the mechanical volume of passive components

is neglected. Hence, it is important to define a criterion which has relation to the mechanical

volume. The energy stored in the components is the best criterion to estimate the mechanical

volume. The electric energy stored in the capacitors and magnetic energy stored in the

inductors are chosen as goal function which must be minimized to achieve the optimal design.

Figure 2.12 shows the first plan of MMC optimization using energy stored in the converter.

Page 85: Optimal Sizing of Modular Multilevel Converters

57

MMC Circuit

Model and

Fault analysis

Vabc

Idc

S(t)

fsw m

Csm

Larm

ωs

Iabc

Is1,Is2

Icirc

VCsm

Voltage ripple

THD

IL

Ifault

Switch lossesSp

eci

fict

ion

s

Co

nve

rte

r to

po

logy

N

.

.

.P

assi

ve

com

po

ne

nt

val

ue

M

IGB

T

Spe

cifi

cati

on

s Vmax

Imax

Psw

InputsStea

dy-state V

ariable

sStea

dy-state P

erfo

rman

ce

Inductor losses

Efficiency

Energy stored

Outputs

Non-linear

Solver

Total mass function

Capacitor energy

Csm

Const

rain

ts

Init

ial V

alues

Op

tim

izat

ion

par

am

eter

s

Inductor energy

Larm,M

Figure 2.12 First proposed optimization plan of MMC converter

The design of magnetic components is complex and depends on lots of parameters. It is not

possible to investigate and design an MMC converter without considering the inductor

topology and core and winding parameters. In the second plan, the analytical model of arm

inductances and capacitor volume estimation block are added to the optimization plan. The

core size and winding parameters are added to the optimization variables vector. Also, some

new constraints concerning the magnetic flux in the core are added to the constraints vector.

The most important change of the second plan in comparison with the first one is the goal

function. Utilizing the analytical model of inductor and capacitor mass function, the total

converter mass is chosen as main goal function. The optimization solver tries to minimize

the total converter mass by adjusting the optimization variables. Figure 2.13 shows the

second plan of optimization that employs the analytical model of inductors.

Page 86: Optimal Sizing of Modular Multilevel Converters

58

abcdg

Non-linear

Solver

Init

ial

Val

ues

Op

tim

izat

ion

par

am

eters

Total mass function

Capacitor mass

Csm

Co

nst

rain

ts

MMC Circuit

Model and

Fault analysis

Vabc

Idc

S(t)

fsw m

Csm

Larm

ωs

Iabc

Is1,Is2

Icirc

VCsm

Voltage ripple

THD

IL

Ifault

Switch losses

Spe

cifi

ctio

ns

Co

nve

rte

r to

po

logy

N

.

.

.

Pas

sive

co

mp

on

en

t v

alu

eM

IGB

T

Spe

cifi

cati

on

s Vmax

Imax

Psw

Inputs

Stead

y-state Variab

les

Stead

y-state Pe

rform

ance

Inductor losses

Efficiency

Energy stored

Outputs

Dimension model

of arm

inductances

IL1

µr

Larm

Jabcdg

M

Bmax

RL

Pcu

Pcore

Vind

Mind

Spe

cifi

ctio

ns

Co

re t

op

olo

gy

IL2

Win

din

g

n1

n2

Electro

magn

etic V

ariable

s

Pe

rform

ance

η%

Thermal Model

Pcu

Pcore

Psw

IGBT

cooling

Inductor

cooling

Co

nve

rte

r lo

sses

Co

olin

g

Tem

peratu

re rise

ΔTIGBT

ΔTcore

ΔTcu

Figure 2.13 Second proposed optimization plan using analytical inductor model

Page 87: Optimal Sizing of Modular Multilevel Converters

59

MMC Circuit

Model and

Fault analysis

Vabc

Idc

S(t)

fsw m

Csm

Larm

ωs

Iabc

Is1,Is2

Icirc

VCsm

Voltage ripple

THD

IL

Ifault

Switch losses

Spe

cifi

ctio

ns

Co

nve

rte

r to

po

logy

N.

.

.

Pas

sive

co

mp

on

en

t v

alu

e

M

IGB

T

Spe

cifi

cati

on

s Vmax

Imax

Psw

Inputs

Stead

y-state Variab

les

Stead

y-state Pe

rform

ance

Inductor losses

Efficiency

Energy stored

Outputs

Dimension model

of arm

inductances

IL1

µr

Larm

Jabcdg

M

Bmax

RL

Pcu

Pcore

Vind

Mind

Spe

cifi

ctio

ns

Co

re t

op

olo

gy

IL2

Win

din

g

n1

n2

Electro

magn

etic V

ariable

s

Pe

rform

ance

η%

Thermal Model

Pcu

Pcore

Psw

IGBT

cooling

Inductor

cooling

Co

nve

rte

r lo

sses

Co

olin

g

Tem

peratu

re rise

ΔTIGBT

ΔTcore

ΔTcu

FEM

softwareLfem=Larm

Correction

coefficient

no

Finish

yes

Init

ial

Val

ues

Op

tim

izat

ion

par

am

eters

Non-linear

Solver

Total mass function

Capacitor mass

Csm

Perf

orm

ance f

un

ctio

n,

dim

en

sio

n a

nd

co

nst

rain

ts

Figure 2.14 Third optimization plan includes the correction loop using FEM

The accuracy of inductor analytical model is not sufficient enough to be sure about the

optimization result especially in the case of inductor parameters. Hence, the analytical model

accuracy must be evaluated using a precise approach such as finite element method. In this

plan, a correction loop is designed to modify the analytical model parameters in each solution

step using finite element results. This approach increases the model accuracy while the

optimization time does not increase so much. Figure 2.14 shows the third plan of optimization

Page 88: Optimal Sizing of Modular Multilevel Converters

60

which includes the correction loop using finite element method. The analytical model

parameters are modified in each cycle regarding the FEM results.

2.9. Conclusion

In this chapter, the initial design of an MMC converter using the nominal load specifications

was presented. The IGBT selection, sub-module capacitor value and arm inductance sizing

were the most important point in MMC design procedure. Regarding the variety of

components in MMC topology, several analyses should be done to verify the converter

performance in terms of circuit functionality, magnetic performance, and thermal dissipation.

In order to achieve the reliable results, all analyses should be integrated into a unique model

as global analysis model which includes all technical variables and analysis tools.

Due to the high number of variables that should be determined and the number of technical

constraints, the optimization idea has been introduced and investigated. Three optimization

plans with a different degree of complexity were presented and studied. The inputs/outputs,

constraints, interconnections and goal function were introduced. Also, the method of

integration with a non-linear solver was represented.

In the next chapters, the analysis tools that were introduced in this chapter will be investigated

in detail. Also, the optimization plans will be employed to minimize the MMC converter

volume and mass with respect to the technical and manufacturing constraints. The

optimization result will be compared and discussed to find the accuracy of different models.

Page 89: Optimal Sizing of Modular Multilevel Converters

61

CHAPTER III

3 Circuit Model of Modular Multilevel AFE

3.1. Introduction

The average steady-state model of the converter is an analytical model, which widely used

in power electronic applications. Regardless to the conventional converters, there is a

circulating current in the MMC converter arms. The value of circulating current is dependent

on the operating point and component values. It makes a circular interaction between the

circuit variables such as arm current, capacitor voltage ripple, and component values. This

circular interaction made the MMC converter complex to analyze. Hence, the average steady-

state model of Modular Multilevel is a significant tool in order to estimate the parameter

values in a steady-state condition. The average steady-state model of Modular Multilevel

AFE is explained in this chapter. The circuit model investigates the multicell (series sub-

modules) and multi-leg (parallel arms). In order to simplify the analysis, the single leg

topology is investigated. In the multi-leg topology, the voltages are the same while the

currents are divided by the number of parallel arms.

Page 90: Optimal Sizing of Modular Multilevel Converters

62

Figure 3.1: MMC-based topology of the Multi-megawatt power supply of Fig.1.17

Figure 3.1 shows the MMC-based alternative topology of the multi-megawatt power supply

presented in Figure 1.17 and [75]. The design of the MMC Active Front End converter on

the left of Figure 3.1 will be investigated using the methodology proposed in this document

only.

3.2. Steady-State Average Model of Modular Multilevel Active-

Front-End Converter

The most popular circuit model of the converter which is employed by power electronic

designer is the average model. The switching function of IGBTs is a discrete function. The

average model uses the average switching function in the circuit model calculation. The

average model provides the converter electrical quantities in steady-state. It represents a

suitable approximation of converter variables in a steady-state condition. The accuracy of the

average model increases by increasing the switching frequency.

3.2.1. Sub-module circuit analysis

Figure 3.2 shows the half-bridge sub-module topology and the important sub-module

variables. Each converter arm is composed of the number of the series sub-module. Each

sub-module consists of two IGBTs and one capacitor as shown in figure 3.2. The sub-module

Transformer

Multilevel Multicell H-Bridge Converter

Load

(18M

W)

Gri

d

18KV, 50 Hz 18KV:2KV

SM1

SMN

SM1

SMN

..

SM1

SMN

SM1

SMN

..

SM1

SMN

SM1

SMN

..

Phase A

Phase B

Phase C

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

SM1

SMN

.

.

.

.

.

.

Multilevel Multicell Three-Phase Rectifier

SM1

SMN

SM1

SMN

..

SM1

SMN

SM1

SMN

..

SM1

SMN

SM1

SMN

..

Phase A

Phase B

Phase C

SM1

SMN

SM1

SMN

..

SM1

SMN

SM1

SMN

..

SM1

SMN

SM1

SMN

..

Phase A Phase B Phase C

Page 91: Optimal Sizing of Modular Multilevel Converters

63

current is divided between two IGBTs dependent on the switching function. According to the

KCL, the IGBTs currents are calculated as below:

𝐼𝑠𝑚(𝑡) = 𝐼𝑠1(𝑡) + 𝐼𝑠2(𝑡) (3.1)

𝐼𝑠1(𝑡) = −𝐼𝑐𝑢(𝑡) (3.2)

Icu S1

S2

Ism

C +Vsm

-

+Vc

-

D1

D2

Is2

Is1

Figure 3.2: half-bridge sub-module topology

3.2.2. Single phase average model parameters

Figure 3.3 shows the single-phase average model of modular multilevel AFE and the

important variables which should be calculated to solve the average model. Each arm is

modeled by the total sub-module voltage and arm inductance which could be coupled or not.

According to the KCL:

𝐼𝑎𝑢 = 𝐼𝑎 + 𝐼𝑎𝑙 (3.1)

{𝐼𝑐𝑖𝑟𝑐 = 𝐼𝑎𝑢 −

𝐼𝑎2

𝐼𝑐𝑖𝑟𝑐 = 𝐼𝑎𝑙 +𝐼𝑎2

(3.2)

Page 92: Optimal Sizing of Modular Multilevel Converters

64

Figure 3.3: Single phase average equivalent circuit

In order to calculate the precise values, it is important to estimate the circulating current. The

existing literature has proven that the most significant component of the circulating current

is the second-order harmonic [23].

𝐼𝑐𝑖𝑟𝑐 =𝐼𝑑𝑐3+ 𝐼𝑐𝑖𝑟𝑐

ℎ2 cos (2𝜔1𝑡 + 𝜃2) (3.3)

Where, 𝐼𝑑𝑐 is the current of DC side, 𝐼𝑐𝑖𝑟𝑐ℎ2 is the peak value of second order circulation

current. Thus, the upper arm current for a three-phase converter is defined as below:

{𝐼𝑎𝑢(𝑡) =

𝐼𝑑𝑐3+ 𝐼𝑐𝑖𝑟𝑐

ℎ2 𝑐𝑜𝑠(2𝜔1 + 𝜃2) +𝐼𝑎2𝑐𝑜𝑠(𝜔1𝑡 + 𝜃1)

𝐼𝑎𝑙(𝑡) =𝐼𝑑𝑐3+ 𝐼𝑐𝑖𝑟𝑐

ℎ2 𝑐𝑜𝑠 (2𝜔1 + 𝜃2) −𝐼𝑎2𝑐𝑜𝑠(𝜔1𝑡 + 𝜃1)

(3.4)

The upper and lower arm currents consist of DC source current divided by three, phase

current divided by two and the circulating current.

3.2.3. Average switching function

The internal sub-module variables such as current and voltage are dependent on the switching

function. In order to reduce the harmonics, the sinusoidal PWM switching is used. Generally,

Vus

.

.

R

L

R

L

M

Vls

Vdc/2

Vdc/2

Ia

Idc

Iau

Ial

Icirc

Page 93: Optimal Sizing of Modular Multilevel Converters

65

the switching function is not a pure sinusoidal waveform. If the number of sub-modules is

high enough or the switching frequency is high enough, the harmonic components in the

switching function can be ignored. Hence, the average switching waveform will be a

sinusoidal waveform. The upper and lower arm sub-module switching function will be

written as below:

{𝑆𝑎𝑢 =

1

2(1 − 𝑆𝑚𝐶𝑜𝑠 𝜔1𝑡)

𝑆𝑎𝑙 =1

2(1 + 𝑆𝑚𝐶𝑜𝑠 𝜔1𝑡)

(3.5)

where 𝑆𝑚 is the modulation index.

Considering to the half-bridge configuration, the current passes through the capacitor when

the lower switch is ON and the upper is off. The capacitor current waveform is calculated as

below:

{𝐼𝑐𝑢(𝑡) = 𝑆𝑎𝑢(𝑡)𝐼𝑎𝑢(𝑡)

𝐼𝑐𝑙(𝑡) = 𝑆𝑎𝑙(𝑡)𝐼𝑎𝑙(𝑡)

(3.6)

The sub-module capacitor current waveform consists of DC component, the main frequency

component, and second-order and third-order harmonics:

𝐼𝑐𝑢𝑑𝑐(𝑡) = 𝐼𝑐𝑙

𝑑𝑐(𝑡) =1

6𝐼𝑐𝑖𝑟𝑐𝑑𝑐 −

1

8𝑆𝑚𝐼𝑎𝑐𝑜𝑠𝜃1

(3.7)

𝐼𝑐𝑢ℎ1(𝑡) = −𝐼𝑐𝑙

ℎ1(𝑡)

= −1

6𝑆𝑚𝐼𝑐𝑖𝑟𝑐

𝑑𝑐 cos(𝜔1𝑡) +𝐼𝑎4cos(𝜔1𝑡 + 𝜃1)

−𝐼𝑐𝑖𝑟𝑐2𝑆𝑚𝑐𝑜𝑠(𝜔1𝑡 + 𝜃2)

(3.8)

𝐼𝑐𝑢ℎ2(𝑡) = 𝐼𝑐𝑙

ℎ2(𝑡) =𝐼𝑐𝑖𝑟𝑐2cos(2𝜔1𝑡 + 𝜃2) −

𝐼𝑎8𝑆𝑚𝑐𝑜𝑠(2𝜔1𝑡 + 𝜃1)

(3.9)

𝐼𝑐𝑢ℎ3(𝑡) = −𝐼𝑐𝑙

ℎ3(𝑡) = −𝐼𝑐𝑖𝑟𝑐4𝑆𝑚𝑐𝑜𝑠(3𝜔1𝑡 + 𝜃2)

(3.10)

The DC part of capacitor current must be zero in a steady-state condition. So,

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66

𝜃1 = 𝑐𝑜𝑠−1(8𝐼𝑑𝑐6𝑆𝑚𝐼𝑎

) (3.11)

3.2.4. Circulating current and capacitor voltage ripple estimation

To calculate the capacitor voltage, it is sufficient to multiply the capacitor current by the

capacitor impedance. By eliminating the DC section of the result, the capacitor voltage ripple

is determined. The capacitor voltage ripple consists of main frequency, second-order and

third-order harmonics. By adding three terms of ripple, the total capacitor voltage ripple is

calculated for each sub-module. The importance of capacitor voltage ripple comes from the

limitation of semiconductor rating. Finding the maximum capacitor voltage is a criterion to

select the semiconductor switch. On the other hand, to find the precise losses function, the

capacitor voltage ripple plays an important role [21], [22].

∆𝑉𝑐𝑢ℎ1(𝑡) = −∆𝑉𝑐𝑙

ℎ1(𝑡)

=−𝑆𝑚𝐼𝑐𝑖𝑟𝑐

𝑑𝑐

6𝜔1𝐶𝑠𝑚sin(𝜔1𝑡) +

𝐼𝑎4𝜔1𝐶𝑠𝑚

sin(𝜔1𝑡 + 𝜃1)

−𝑆𝑚𝐼𝑐𝑖𝑟𝑐2𝜔1𝐶𝑠𝑚

sin (𝜔1𝑡 + 𝜃2)

(3.12)

∆𝑉𝑐𝑢ℎ2(𝑡) = ∆𝑉𝑐𝑙

ℎ2(𝑡)

=𝐼𝑐𝑖𝑟𝑐

4𝜔1𝐶𝑠𝑚sin(2𝜔1𝑡 + 𝜃2) −

𝐼𝑎𝑆𝑚16𝜔1𝐶𝑠𝑚

sin(2𝜔1𝑡 + 𝜃1)

(3.13)

∆𝑉𝑐𝑢ℎ3(𝑡) = −∆𝑉𝑐𝑙

ℎ3(𝑡) =−𝐼𝑐𝑖𝑟𝑐𝑆𝑚12𝜔1𝐶𝑠𝑚

sin (3𝜔1𝑡 + 𝜃2) (3.14)

Where ∆𝑉𝑐𝑢ℎ1, ∆𝑉𝑐𝑢

ℎ2, ∆𝑉𝑐𝑢ℎ3 are first, second and third voltage harmonics of capacitor. It should

be noted that the DC part of sub-module capacitors in steady state condition is 𝑉𝑑𝑐

𝑚, where 𝑚

is the number of series sub-modules per arm.

The phase voltage ripple is the summation of all sub-module capacitors voltage ripple

multiplied by the switching function. The phase voltage ripple is composed of the pair

harmonics and the odd harmonics were eliminated. The output voltage ripple consists of DC

component, second-order and fourth order harmonics [24].

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67

∆𝑉𝑎𝑟𝑚 = [𝑚

2−𝑚

2𝑆𝑚𝑐𝑜𝑠(𝜔1𝑡)] (∆𝑉𝑐𝑢

ℎ1 + ∆𝑉𝑐𝑢ℎ2 + ∆𝑉𝑐𝑢

ℎ3)

+ [𝑚

2+𝑚

2𝑆𝑚𝑐𝑜𝑠(𝜔1𝑡)] (∆𝑉𝑐𝑙

ℎ1 + ∆𝑉𝑐𝑙ℎ2 + ∆𝑉𝑐𝑙

ℎ3)

(3.15)

The voltage ripple function is a valuable analytical function which is important to estimate

the grid-side THD value. This function is used to estimate the THD constraint in the

optimization procedure or it could be an appropriate criterion to design the external filter.

∆𝑉𝑎𝑟𝑚𝑑𝑐 = −

𝑚𝑆𝑀𝐼𝑎8𝜔1𝐶𝑠𝑚

𝑠𝑖𝑛𝜃1 +𝑚𝑆𝑚

2 𝐼𝑐𝑖𝑟𝑐4𝜔1𝐶𝑠𝑚

𝑠𝑖𝑛𝜃2 (3.16)

∆𝑉𝑎𝑟𝑚ℎ2 =

𝑚𝑆𝑚2 𝐼𝑐𝑖𝑟𝑐

𝑑𝑐

12𝜔1𝐶𝑠𝑚sin(2𝜔1𝑡) −

3𝑚𝑆𝑚𝐼𝑎16𝜔1𝐶𝑠𝑚

sin(2𝜔1𝑡 + 𝜃1)

+𝑚𝐼𝑐𝑖𝑟𝑐(1 +

76 𝑆𝑚

2 )

4𝜔1𝐶𝑠𝑚sin (2𝜔1𝑡 + 𝜃2)

(3.17)

∆𝑉𝑎𝑟𝑚ℎ4 =

𝑚𝑆𝑚2 𝐼𝑐𝑖𝑟𝑐

24𝜔1𝐶𝑠𝑚sin (4𝜔1𝑡 + 𝜃2)

(3.18)

It is noted that the circulating current consists of second-order harmonic. Therefore, to

estimate the circulating current (𝐼𝑐𝑖𝑟𝑐), the second-order harmonic of the voltage is utilized:

𝐼𝑐𝑖𝑟𝑐 cos(2𝜔1𝑡 + 𝜃2) =∆𝑉𝑎𝑟𝑚

ℎ2

−𝑗2𝜔12(𝐿 −𝑀)=

∆𝑉𝑎𝑟𝑚ℎ2

−𝑗4𝜔1(𝐿 − 𝑀)

(3.20)

According to Eq.(3.17), the circulating current 𝐼𝑐𝑖𝑟𝑐 is calculated as below:

𝐼𝑐𝑖𝑟𝑐 cos(2𝜔1𝑡 + 𝜃2) (1 −𝑚(1 +

76𝑆𝑚

2 )

16𝜔12(𝐿 + 𝑀)𝐶𝑠𝑚

)

=𝑚𝑆𝑚

2 𝐼𝑐𝑖𝑟𝑐𝑑𝑐

96𝜔12(𝐿 + 𝑀)𝐶𝑠𝑚

cos(2𝜔1𝑡)

−3𝑚𝑆𝑚𝐼𝑎

64𝜔12(𝐿 + 𝑀)𝐶𝑠𝑚

cos(2𝜔1𝑡 + 𝜃1)

(3.21)

By solving the above equation, the magnitude and phase of the circulating current are

obtained as below:

𝐼𝑐𝑖𝑟𝑐 =√(𝐵𝑐𝑜𝑠𝜃1 + 𝐶)2 + (𝐵𝑠𝑖𝑛𝜃1)2

1 − 𝐴

(3.22)

< 𝜃2 = atan (𝐵𝑠𝑖𝑛𝜃1

𝐵𝑐𝑜𝑠𝜃1 + 𝐶)

(3.23)

Where

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68

{

𝐴 =

𝑚(1 +76 𝑆𝑚

2 )

16𝜔12(𝐿 −𝑀)𝐶𝑠𝑚

𝐵 =−3𝑚𝑆𝑚𝐼𝑎

64𝜔12(𝐿 −𝑀)𝐶𝑠𝑚

𝐶 =𝑚𝑆𝑚

2 𝐼𝑐𝑖𝑟𝑐𝑑𝑐

96𝜔12(𝐿 −𝑀)𝐶𝑠𝑚

(3.24)

According to Eq.(3.22), there is a point of discontinuity. On the other word, the circulating

current goes to infinitive value if 𝐴 = 1. Hence, a new constraint is added to avoid the

instability. The capacitor and arm inductance values should be adjusted as below to avoid the

instability.

𝐶𝑠𝑚(𝐿 −𝑀) ≠𝑚(1 +

76𝑆𝑚

2 )

8𝜔12

(3.25)

3.2.5. Advantages and disadvantages of steady-state average model of

MMC Active-Front-End

Because of the circular interaction, MMC converter analysis is complicated behavior.

Generally, the numerical solvers are used to simulate and analyze the MMC converters. This

is a complex and time-consuming analysis approach which is not suitable for component

selection and optimization. The average model provides the significant information of the

converter which could be very helpful for power electronic designers. Also, it could be an

appropriate model to use optimization loop. On the other hand, the average steady-state

model neglects the switching frequency effect which is important especially in high power

application which utilizes low switching frequency. The most important weakness of the

average model is in the harmonic calculation.

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69

3.3. Time-domain steady-state model of Modular Multilevel

Active-Front-End

The average model neglects the effect of switching function and switching frequency. High

power converters use low switching frequency. Hence, neglecting the switching frequency

leads to diminishing the model accuracy. The time-domain steady-state model is presented

in order to resolve the average model weakness and increase the accuracy. It estimates the

waveform of the important parameters considering the real switching function. Using time-

domain model, it is possible to calculate the more accurate harmonic and ripple values.

3.3.1. Sub-module switching function

The time -domain steady-state model employs real sub-module switching function which is

a discrete function. The switching function could be written as below:

1 2

1 2

1 , , ,( )

0 , , 0, 0

sm cu u cu

sm cu

S ON S OFF V V I IS t

S OFF S ON V I

(3.26)

The sinusoidal PWM (SPWM) is utilized to reduce the harmonic. By comparing a sinusoidal

waveform with a high-frequency sawtooth waveform, the switching function is generated.

3.3.2. Time-domain state equations

In order to analyze the converter in time-domain, the state equations must be extracted. Using

the state equations, it is possible to calculate all converter circuit parameters in the time

domain. The first step is to calculate the sub-module capacitor current from the arm current.

{𝐼𝑐𝑢𝑖 = 𝑆𝑢𝑖(𝑡)𝐼𝑎𝑢(𝑡)

𝐼𝑐𝑙𝑖 = 𝑆𝑙𝑖(𝑡)𝐼𝑎𝑙(𝑡)

(3.27)

In each sub-module, the capacitor current and capacitor voltage could be calculated using

switching function (𝑆𝑢 , 𝑆𝑙).

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70

{

𝑉𝑐𝑢𝑖 =1

𝐶𝑠𝑚∫𝑆𝑢𝑖(𝑡)𝐼𝑎𝑢(𝑡)

𝑉𝑐𝑙𝑖 =1

𝐶𝑠𝑚∫𝑆𝑙𝑖(𝑡)𝐼𝑎𝑙(𝑡)

(3.28)

Also, using the capacitor sub-modules and the switching function, it is possible to determine

the sub-module terminal voltage. Also, the total sub-module voltage is the summation of all

sub-modules voltage in each arm.

{𝑉𝑢𝑖(𝑡) = (1 − 𝑆𝑢𝑖(𝑡))𝑉𝑐𝑢𝑖(𝑡)

𝑉𝑙𝑖(𝑡) = (1 − 𝑆𝑙𝑖(𝑡))𝑉𝑐𝑙𝑖(𝑡)

(3.29)

{

𝑉𝑢𝑇(𝑡) = ∑𝑉𝑢𝑖(𝑡)

𝑁

𝑖=1

𝑉𝑙𝑇(𝑡) = ∑𝑉𝑙𝑖(𝑡)

𝑁

𝑖=1

(3.30)

According to KVL, the arm inductance voltage is calculated versus total sub-module voltage

and phase voltage of the source. The upper and lower arm inductance are calculated versus

the inductance voltages (𝑉𝐿) and inductance (𝐿) and coupling factor (𝑀). The calculated

currents are utilized in the next step of the numerical solution approach.

{𝑉𝐿𝑢(𝑡) =

𝑉𝑑𝑐2− 𝑉𝑎(𝑡) − 𝑉𝑢𝑇(𝑡) + 𝑉𝑛

𝑉𝐿𝑙(𝑡) =𝑉𝑑𝑐2− 𝑉𝑙𝑇(𝑡) + 𝑉𝑎(𝑡) − 𝑉𝑛

(3.31)

{

𝐼𝑢(𝑡) =

𝐿2

𝐿2 +𝑀2∫𝑉𝐿𝑢(𝑡)𝑑𝑡 +

𝑀2

𝐿2 +𝑀2∫𝑉𝐿𝑙(𝑡)𝑑𝑡

𝐼𝑙(𝑡) =𝐿2

𝐿2 +𝑀2∫𝑉𝐿𝑙(𝑡)𝑑𝑡 +

𝑀2

𝐿2 +𝑀2∫𝑉𝐿𝑢(𝑡)𝑑𝑡

(3.32)

By elimination of 𝐼𝑐𝑖𝑟𝑐, the input current of phase A is calculated,

𝐼𝑎(𝑡) = 𝐼𝑢(𝑡) − 𝐼𝑙(𝑡) (3.33)

3.3.3. Proposed time-domain model

Utilizing the state equations, a time domain model will be developed in order to use as an

analytical model. In order to simulate the converter in a steady-state condition, the state

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71

variables should be initialized. There is a possibility to initialize the state variables using

average steady-state model. Figure 3.4 shows the diagram using an average steady-state

model which is used to initialize the state variables.

Average Switching Functions

Average Switching Functions

IaIa

IcircIcirc

+

±

Iau,Ial

11+

- ×

× Submodule terminal voltages

Submodule terminal voltages

Capacitor Currents

Capacitor Voltages

Capacitor Voltages

Total upper sub. VoltagesTotal lower sub. Voltages

Total upper sub. VoltagesTotal lower sub. Voltages

Upper inductor voltageLower inductor voltage

Upper inductor voltageLower inductor voltage

Figure 3.4: initializing diagram using average steady-state model

The initial value of sub-module capacitors and arm inductances are estimated using average

steady-state model. This is a useful tool to eliminate the transient part of the simulation.

Figure 3.5 shows the complete analytical model which consists of the average model to

initialize and the state space model to simulate. The most important advantages of the

proposed model are:

1- Fast initialization

2- Using real SPWM switching function

3- Accurate steady-state simulation using state variables

4- Harmonic investigation

5- Investigation of voltage and current ripple

6- Accurate semiconductor losses calculation

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72

Figure 3.5: Diagram of Time-domain analytical model

3.4. Steady-State Model Verification using Simulink

An important section of introducing a model is to validate and verify the model accuracy.

The proposed model is implemented using visual basic programming code in Excel. The

Microsoft Excel provides an appropriate graphic space which simplifies the modeling and

parameter adjusting. The Excel cells support mathematical functions. Also, it connects to the

visual basic code sheets. In order to validate the outputs of the proposed model, the MMC

converter is simulated using Simulink/Matlab.

Table 3.1 The MMC converter parameters and operating point

Parameter Value Parameter Value

Nominal Power 2.5 MW Number of module per arm 6

AC Line Voltage 2000 V Number of parallel arm 1

DC Link Voltage 5000 V Sub-module capacitor 11.1 mF

Power Factor 1 Arm inductance 4.22 mH

Nominal Frequency 50 Hz Switching frequency 755 Hz

Load SpecificationPout, Vac,Iac,cosɸ,Vdc,...

InitilizationInductance, Capacitor, Modulation Index,...

Estimation of required parametersCirculation current, line current, dc current

Calculation of submodule Capacitor current and Voltage

𝑉𝑐𝑢𝑡𝑜𝑡𝑎𝑙 =∑𝑉𝑐𝑢 𝑖

𝑚

𝑖=1

𝑉𝑐𝑙𝑡𝑜𝑡𝑎𝑙 =∑𝑉𝑐𝑙 𝑖

𝑚

𝑖=1

Calculation of arm currents

𝐼𝑢𝐼𝑙

= 𝐿 −1 𝑉𝑢 𝑡𝑜𝑡𝑎𝑙𝑉𝑙 𝑡𝑜𝑡𝑎𝑙

Calculation of line Current, THD and capacitor voltage

ripple

𝐼𝑙𝑖𝑛𝑒, THD, ∆𝑉𝑐

Calculation of Losses, Efficiency and Energy

Stored in the converter components

𝑆𝑎𝑢𝑆𝑎𝑙𝑓𝑠𝑤

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73

Table 3.1 shows the MMC converter parameters and the operating point which is used in the

simulation. The simulation has been done using Simulink/Matlab and Excel/VB

programming. Then the output waveforms are compared. The sub-module capacitor currents

simulated by Excel vb code and Simulink model have been compared in figure 3.6. The open

loop controller using SPWM switching function has used.

a) Visual basic code

b) Simulink/Matlab

Figure 3.6: The sub-module capacitor current using analytical model and Simulink

Figure 3.7 shows the sub-module capacitor voltage waveforms which have obtained from the

analytical model and Simulink model. The comparison proves the analytical model accuracy.

Figures 3.8 and 3.9 show the upper/lower arm currents and the input line current respectively.

Also, the THD calculation proves the analytical model accuracy again. Using Simulink, the

THD value of the line current is about 0.7% while the analytical model estimates 0.64%.

a) Visual Basic code

b) Simulink/Matlab

Figure 3.7: The sub-module capacitor voltage using analytical model and Simulink

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74

a) Visual basic code

b) Simulink/Matlab

Figure 3.8: The lower and upper arm current using analytical model and Simulink

a) Visual basic code

b) Simulink/Matlab

Figure 3.9: The input line current using analytical model and Simulink

3.5. Conclusion

In this chapter, a circuit model of MMC converter was investigated and developed. The

average model was presented as most popular converter model. In high power application,

the low switching frequency is used to avoid high switch losses. The accuracy of average

switching function is suitable for high switching frequencies. Therefore, the accuracy of the

average model is not sufficient enough in MMC converters, especially in low switching

frequency. In the high power applications, the low switching frequency is utilized in order to

minimize the switching losses. To enhance the model accuracy, a time-domain circuit model

was introduced which employs the real switching function. The proposed model provides

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75

high accuracy results especially in term of harmonic analysis. The accuracy of the proposed

model was evaluated using Simulink/MATLAB.

In the next chapter, the electromagnetic analytical model of arm inductances will be

presented. The dimensioning analysis of passive components will be investigated and

formulated.

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76

CHAPTER IV

4 Electromagnetic and Dimensioning Analysis of

Passive Components

4.1. Introduction

In chapter 4, the electromagnetic arm inductance and dimensioning analysis of the passive

components are presented. In this chapter, the mass function of the main components is

extracted in order to achieve the total converter mass function. The mass function of IGBTs

and their cooling system could be extracted from the datasheet. The mass function of the

capacitor is related to the capacitance, capacitor nominal voltage, and the capacitor type.

Regarding the products of a manufacturer, an analytical equation is fitted to estimate the

capacitor mass value. In term of arm inductance, the dimensioning model will be a part of

the electromagnetic analysis model.

In this chapter, the analytical electromagnetic model of arm inductances is reviewed and

investigated based on the electrical variables of the MMC converter. The electromagnetic

model should calculate the magnetic quantities such as inductance, magnetic flux, core and

winding losses regarding the core size and winding parameters.

4.2. Capacitor Dimensioning Analysis

In this section, the mass function of power capacitor is extracted. Also, the losses function of

the capacitor is calculated.

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77

4.2.1. Capacitor Mass Function

Capacitors are industrial components which are designed and manufactured in the standard

packages and sizes. Power electronic designer generally utilizes the standard capacitors in

their applications. Therefore, the mass function must be extracted based on the product

datasheets. In this research, the cylindrical capacitors of VISHAY Company are chosen.

Figure 4.1 Cylindrical capacitor of VISHAY Company designed for power electronic applications

Figure 4.1 shows the cylindrical capacitor of VISHAY Company designed for power

electronic applications. The nominal voltage of capacitors is between 880 V and 2200 V.

Also, the capacitances are from 30 μF to 1000 μF.

The most important variables which affect the capacitor mass are the capacitance and

maximum voltage. Figure 4.2 shows the contour of capacitor mass versus the capacitance

and maximum voltage according to the product's datasheet. A second-order linear function

has been fitted to the points which are obtained from company datasheet. Table 4.1 shows

the function coefficients which are calculated using curve fitting algorithm.

𝑊𝑒𝑖𝑔ℎ𝑡 = 𝑝00 + 𝑝10𝑉𝑑𝑐 + 𝑝01𝐶 + 𝑝20𝑉𝑑𝑐2 + 𝑝02𝐶

2 + 𝑝11𝑉𝑑𝑐𝐶 (4.1)

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78

Figure 4.2: Capacitor weight versus the capacitance and maximum voltage value

Table 4.1: Calculated coefficients using fitting algorithm

Function coefficient Value

𝑝00 1.005

𝑝10 -0.00119

𝑝01 -3550

𝑝20 3.87E-07

𝑝02 43200

𝑝11 5.27

4.2.2. Transient Equivalent Model of Capacitor

Various electrical models have been represented to simulate the electrical and thermal

behavior of the capacitor. In this research, two different models are used with different

complexity.

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79

Model 1: the simplest model that was presented for an electrical model of the capacitor is

shown in figure 4.3. The capacitance 𝐶1 represents total capacitance between the anode and

the cathode terminals. Resistance 𝑅𝑎 includes different terms: terminal resistance, tabs

resistance, foils resistance, resistance of the impregnated electrolyte paper, dielectric

resistance, and tunnel-electrolyte resistance. 𝑅𝑐 represents the leakage current that depends

on the quality of the dielectric material. Inductance 𝐿𝑐 is especially dominated by the loop

area from the terminals and tabs outside of the active winding. Resistance 𝑅𝑏 is essential to

have in a physical representation of the model [63, 64]. The Laplace formulation

corresponding to this model is:

𝑍1(𝑠)

=(𝑅𝑎 + 𝑅𝑏)𝑅𝑐𝐿c𝐶1𝑠

2 + (𝑅𝑎𝑅𝑏𝑅𝑐𝐶1 + 𝐿c𝑅𝑐 + 𝐿c𝑅𝑏 + 𝐿c𝑅𝑎)𝑠 + (𝑅𝑎 + 𝑅𝑐)𝑅𝑐𝑅𝑐𝐿c𝐶1𝑠2 + (𝑅𝑏𝑅𝑐𝐶1 + 𝐿c)𝑠 + 𝑅𝑏

(4.2)

Rc

C1 Lc

Rb

Ra

Figure 4.3 The simple model of high power capacitor

Model 2: the simple model is not accurate especially by changing the capacitor temperature.

In order to enhance the model accuracy throughout the temperature range, a modified

capacitor model has been proposed [63, 64]. Figure 4.4 represents the modified capacitor

model that provides more accurate results in a wide range of temperature.

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80

Rc

C1 Lc

Rb

RaC2

R2a

R2c

Figure 4.4 The modified high power capacitor model

The element values are identified at the maximum temperature of the component for which

the impedance is weakest (85℃). The elements (𝐶2, 𝑅2𝑎 , 𝑅2𝑐), are added in series to this

model in order to take into account the influence of the reduction of temperature and the

shape of the curve Z versus frequency.

The Laplace function corresponding to the modified capacitor model is:

𝑍2(𝑠) = 𝑍1(𝑠) + 𝑍02(𝑠) (4.3)

Where

𝑍02(𝑠) =𝐶2𝑅2𝑎 + 𝑅2𝑐

𝐶2(𝑅2𝑎 + 𝑅2𝑐)𝑠 + 1

(4.4)

4.3. Dimensioning Analysis of Inductor

The dimension analysis is done to provide the mass function of arm inductance. This analysis

is based on the analytical model of the inductor.

4.3.1. Core Topologies

Various core topologies which are suitable for MMC converter application are investigated.

Each topology provides some advantages and some disadvantages. In the case of coupled

inductor, two basic core topologies are utilized. Figure 4.5 shows two core topologies that

bring suitable technical and manufacturing properties.

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81

a) Independent mutual inductance (type 1) b) Dependent mutual inductance (type 2)

Figure 4.5: The proposed core topologies for independent and dependent mutual inductances

The first topology provides the possibility to determine the coupling factor 𝑀 independently,

while the final mass will be higher because of the center yoke. In the second topology, the

coupling factor is not an independent variable. The coupling factor will be so close to one

regarding to the leakage flux. Instead, it has lower mass in compare with the first one. In this

application, the second topology is chosen to minimize the total converter mass.

4.3.2. Magnetic Equivalent Circuit (type 1)

The magnetic equivalent circuit provides important magnetic variables of inductance core.

The first step is to calculate the core reluctances. Figure 4.6 shows the equivalent magnetic

circuit of inductance with the core type 1.

N1I1 N2I2

Rag1 Rag1Rag2

R1 R2

R3

R5R4

Φ1 Φ2

Figure 4.6 The equivalent magnetic circuit of inductance (type 1)

d

b

a

d/2

lg1lg2lg1

a

b

lg

d

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82

If 𝑅1 = 𝑅2and 𝑅4 = 𝑅5 then, the total reluctance value of each winding based on the

geometry in figure 4.5.a is calculated as below:

𝑅𝑇 = (𝑅1 + 𝑅𝑎𝑔1 + 𝑅4) + ((𝑅3 + 𝑅𝑎𝑔2)‖(𝑅2 + 𝑅5 + 𝑅𝑎𝑔1)) (4.5)

where

𝑅1 = 𝑅2 =2(𝑎 +

32𝑑 + 𝑏)

𝜇0𝜇𝑟 𝑐 𝑑

(4.6)

𝑅3 =𝑏 − 𝑙𝑔2𝜇0𝜇𝑟 𝑐 𝑑

(4.7)

𝑅4 = 𝑅5 =2(𝑎 + 𝑑)

𝜇0𝜇𝑟 𝑐 𝑑

(4.8)

𝑅𝑎𝑔1 =2𝑙𝑔1𝜇0 𝑐 𝑑

(4.9)

𝑅𝑎𝑔2 =2𝑙𝑔2𝜇0𝑐 𝑑

(4.10)

To calculate the magnetic flux that passes through the windings,

𝛷1 =𝑁1𝐼1𝑅𝑇

(4.11)

Φ2 =𝑁2𝐼2𝑅𝑇

(4.12)

Φ12 = Φ21 =Φ1(𝑅3+𝑅𝑙𝑔2)

𝑅2 + 𝑅5 + 𝑅𝑙𝑔1

(4.13)

Also, the magnetic flux density is calculated as below:

𝐵 =𝜙1 + 𝜙12

𝐴𝑐=(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑁1𝐼1 + (𝑅3+𝑅𝑙𝑔2)𝑁1𝐼1

𝑐. 𝑑(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑅𝑇/2

(4.14)

4.3.3. Magnetic Equivalent Circuit (type 2)

The magnetic circuit analysis is done to calculate the important parameters such as flux, flux

density, reluctance and etc. Figure 4.7 shows the core topology and parameters of the core.

The core reluctance is calculated as below:

𝑅𝑐 =𝑙𝑐

𝜇0𝜇𝑟𝐴𝑐

(4.15)

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83

where 𝑙𝑐 is the magnetic core length, 𝐴𝑐 is the cross-sectional area of the core, 𝜇0 is the

vacuum permeability and 𝜇𝑟 is the relative magnetic permeability of the material. The

magnetic core length and cross-sectional area are computed as below:

𝑙𝑐 = 2𝑎 + 2𝑏 + 4𝑑 (4.16)

𝐴𝑐 = 𝑐𝑑 (4.17)

where 𝑎 is the width of the core window, 𝑏 is the height of the core window, 𝑐 is the core

depth and 𝑑 is the core width. The total reluctance is the summation of core reluctance and

airgap reluctance. The airgap reluctance and total reluctance are calculated as below:

𝑅𝑎𝑔 =2𝑙𝑔𝜇0𝐴𝑐

(4.18)

𝑅𝑇 = 𝑅𝑐 + 𝑅𝑎𝑔 =𝑙𝑐 + 2𝜇𝑟𝑙𝑔𝜇0𝜇𝑟𝐴𝑐

(4.19)

where 𝑙𝑔 is the airgap length.

ab

lg

d

c

lc

Figure 4.7: The inductor core topology and sizing parameters

The magnetic flux which passes through the core and the magnetic flux density is calculated

as below:

𝜙 =𝑛1𝐼1 + 𝑛2𝐼2

𝑅𝑇

(4.20)

𝐵 =𝜙

𝐴𝑐=𝜇0𝜇𝑟(𝑛1𝐼1 + 𝑛2𝐼2)

(𝑙𝑐 + 2𝜇𝑟𝑙𝑔)

(4.21)

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84

4.3.4. Inductance and Resistance Estimation

According to the magnetic flux value that was calculated in the previous section, the

inductance value and the mutual inductance are:

𝐿11 =𝜙1𝐼1=𝑛12

𝑅𝑇

(4.22)

𝐿12 = 𝐿21 =𝜙11𝐼2

(4.23)

The winding resistance value is calculated as below:

𝑅𝑑𝑐 =𝜌𝑐𝑢𝐽

𝐼𝑟𝑚𝑠1(2𝑎 + 2𝑑 + 2𝑐)𝑛1

(4.24)

where 𝜌𝑐𝑢 is the copper resistivity, 𝐽 is the current density.

In the analytical model of the inductor, the most complex parameter to compute is the mutual

inductance. In the case of center column topology, the magnetic flux which generated by the

winding is divided between two magnetic paths. It helps to estimate the coupling factor. In

the centerless core topology, the magnetic flux that passes through the first and second

windings are the same. Hence, the self-inductance and the mutual inductance are equal. On

the other words, the coupling factor is equal to one. In reality, there is some flux leakage in

the inductor structure and the coupling factor will be always less than one. The flux leakage

which is dependent on the material, shape, and size of the electromagnetic core. To estimate

the coupling factor in for this application, a simulation has done using electromagnetic

analysis software. Figure 4.8 shows the magnetic flux lines and leakage flux lines. In this

simulation, 91% percent of the flux generated by first winding passes through the second

winding. Hence, in this model, the mutual inductance is a function of self-inductance (𝐿12 =

𝐾𝑚𝑢𝐿11), in this research 𝐾𝑚𝑢 = 0.9 that could be estimated using electromagnetic analysis

software.

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85

Figure 4.8 Magnetic flux lines in the core and leakage flux lines

The most important constraints of the inductor are the maximum flux density and the core

window area. Each magnetic material has a maximum flux density to avoid the saturation.

The core flux density must be lower than these values in the case of utilizing center column

and centerless core topology, respectively.

{

𝐵𝑚𝑎𝑥 ≥

(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑛1𝐼1 + (𝑅3+𝑅𝑙𝑔2)𝑛1𝐼1𝑐. 𝑑(𝑅2 + 𝑅5 + 𝑅𝑙𝑔1)𝑅𝑇

𝐵𝑚𝑎𝑥 ≥𝑛(𝐼1 + 𝐼2)𝜇0𝜇𝑟𝑙𝑐 + 2𝜇𝑟𝑙𝑔

(4.25)

The core window area and total winding cross-section are calculated as below:

𝐶𝑜𝑟𝑒 𝑤𝑖𝑛𝑑𝑜𝑤 𝑎𝑟𝑒𝑎 = 𝑎𝑏 (4.26)

𝐴𝑤𝑇 = 𝑛𝐼𝑟𝑚𝑠𝐽

(4.27)

Also, the core window area should be greater than the total winding cross-section. In the first

type topology, each winding is placed in a window, while in the second topology, the core

window must consist of two windings.

{

𝑘𝑤 ∗ 𝑎 ∗ 𝑏 >𝑛 ∗ 𝐼𝑟𝑚𝑠

𝐽

𝑘𝑤 ∗ 𝑎 ∗ 𝑏 >2 ∗ 𝑛 ∗ 𝐼𝑟𝑚𝑠

𝐽

(4.28)

where 𝑘𝑤 is the filling factor of the winding.

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86

4.3.5. Volume and Mass Function

Regarding the analytical model, a mass function is proposed to estimate the total inductor

mass dependent on the core and winding size. The inductor core volume is calculated for two

different core topologies, respectively as below:

{𝑉𝑐𝑜𝑟𝑒 = 𝑐(2𝑎 + 3𝑑)(2𝑑 + 𝑏) − 2𝑎𝑏 − (𝑙𝑔2 − 𝑙𝑔1)𝑐𝑑

𝑉𝑐𝑜𝑟𝑒 = 𝑐 ((𝑎 + 2𝑑)(𝑏 − 𝑙𝑔 + 2𝑑) − 𝑎𝑏)

(4.29)

The copper volume is dependent on the winding turn number and wire cross-section.

Regarding the inductor topology, the copper volume for each winding is:

{

𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑛(4𝑎 + 2𝑐 + 2𝑑)𝐴𝑟𝑚𝑠𝐽

𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 = 𝑛(4𝑎 + 2𝑐 + 𝑑)𝐴𝑟𝑚𝑠𝐽

(4.30)

Also, considering the volumetric mass density, the total inductor mass is obtained:

𝑀𝑖𝑛𝑑 = 𝐷𝑐𝑜𝑟𝑒𝑉𝑐𝑜𝑟𝑒 + Dwinding𝑉𝑤𝑖𝑛𝑑𝑖𝑛𝑔 (4.31)

4.4. Inductor Thermal Analysis

Thermal analysis of inductor is important to calculate the final inductor size. The inductor

loss consists of core losses and winding losses as heat source increase the inductor

temperature. In addition to the magnetic analysis, a thermal analysis should be done to

determine the thermal dissipation and maximum temperature rise in the inductor material.

4.4.1. Inductor Losses

The inductor losses are divided to core losses and copper losses. The core loss consists of

hysteresis and eddy current losses. Based on the literature, the hysteresis and eddy current

losses are calculated as below:

𝑃ℎ = 𝐾ℎ𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝐵ℎ1.𝑚𝑎𝑥α + 2𝑓𝑠𝐵ℎ2.𝑚𝑎𝑥

α ) (4.32)

𝑃𝑒 = 𝐾𝑒𝑉𝑐𝑜𝑟𝑒(𝑓𝑠𝛽𝐵ℎ1.𝑚𝑎𝑥γ

+ (2𝑓𝑠)β𝐵ℎ2.𝑚𝑎𝑥

γ) (4.33)

where 𝐾ℎ is the hysteresis constant, 𝐾𝑒 is the eddy current constant.

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87

The electric resistivity of copper augments by increasing its temperature. Hence, the copper

losses will increase. The copper resistivity versus copper temperature is written as below:

𝜌𝑐𝑢 = 𝜌0(1 + 휀25(𝑇𝑐𝑢 − 25)) (4.34)

where 𝜌0 is the copper resistivity in 25℃. On the other hand, the copper losses and in results

the copper temperature depends on the electric resistivity. It makes a circular interaction

which makes it complex to compute. Regarding to our goal which is to achieve the minimum

inductor mass, the copper temperature that is used to compute the copper resistivity is

considered as maximum copper temperature in order to compute the worst-case copper

losses. Therefore, Eq.(4.24) is rewritten as below:

𝜌𝑐𝑢 = 𝜌0(1 + 휀25(𝑇𝑐𝑢𝑚𝑎𝑥 − 25)) (4.35)

where 𝑇𝑐𝑢𝑚𝑎𝑥 is the maximum copper temperature which can be determined regarding to

some technical limitations such as isolation class. The copper losses of the winding depend

on the winding resistance and effective current of the winding. According to the skin effect

of the conductors, the copper resistance depends on the current frequency. The copper

resistance will augment by increasing the frequency of the current. The copper losses can be

calculated regarding to the Superposition Principle in the linear circuit. The inductor current

is consisted of DC current, main component (50 Hz), second-order component (100 Hz) and

fourth-order component (200 Hz). Hence, the copper loss for each winding is calculated as

below:

𝑃𝑐𝑢 = 𝑅𝑑𝑐𝐼𝑑𝑐2 + 𝑅𝑎𝑐1𝐼𝑟𝑚𝑠ℎ1

2 + 𝑅𝑎𝑐2𝐼𝑟𝑚𝑠ℎ22 + 𝑅𝑎𝑐4𝐼𝑟𝑚𝑠ℎ4

2 (4.36)

The winding DC resistance is calculated as below:

𝑅𝑑𝑐 =𝑛 𝜌0(1 + 휀25(𝑇𝑐𝑢𝑚𝑎𝑥 − 25)).MLT

𝐴𝑐𝑢

(4.37)

where 𝑀𝐿𝑇 is length of winding and 𝐴𝑐𝑢 is the conductor cross section.

In order to calculate the AC resistance, several equations have been presented in the literature

[65-68]. The most popular equation which calculates the AC resistance of round wires based

on the zero-Kelvin function is [67]:

𝑅𝑎𝑐 =√2

𝜋𝑑𝑐𝜎𝛿

𝑏𝑒𝑟(𝑞)𝑏𝑒𝑖′(𝑞) − 𝑏𝑒𝑖(𝑞)𝑏𝑒𝑟′(𝑞)

[𝑏𝑒𝑟′(𝑞)]2 + [𝑏𝑒𝑖′(𝑞)]2

(4.38)

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88

where 𝑑𝑐is the conductor diameter, 𝑞 =𝑑𝑐

√2𝜎 , 𝜎 is the copper conductivity, ber and bei being,

respectively, the real and imaginary parts of the zero-order Kelvin functions of first kind, and

ber’ and bei’ their derivatives. The real and imaginary parts of the ith-order Kelvin functions

of first kind and their derivatives are as follow [69],

𝑏𝑒𝑟(𝑖, 𝑞) = (𝑞

2)𝑖

∑cos [(

3𝑖4 +

𝐾2)𝜋

]

𝐾! (𝑖 + 𝐾)!𝐾≥0

(𝑞2

4)

𝐾

(4.39)

𝑏𝑒𝑖(𝑖, 𝑞) = (𝑞

2)𝑖

∑sin [(

3𝑖4 +

𝐾2)𝜋

]

𝐾! (𝑖 + 𝐾)!𝐾≥0

(𝑞2

4)

𝐾

(4.40)

𝑏𝑒𝑟′(𝑖, 𝑞) =𝑏𝑒𝑟(𝑖 + 1, 𝑞) + 𝑏𝑒𝑟(𝑖 + 1, 𝑞)

√2

+ (𝑖

𝑞) . 𝑏𝑒𝑟(𝑖, 𝑞)

(4.41)

𝑏𝑒𝑖′(𝑖, 𝑞) =𝑏𝑒𝑖(𝑖 + 1, 𝑞) − 𝑏𝑒𝑖(𝑖 + 1, 𝑞)

√2+ (

𝑖

𝑞) . 𝑏𝑒𝑖(𝑖, 𝑞)

(4.42)

Regarding the complexity of this equation, it is not suitable for the optimization loop. Hence,

in the literature, some approximate formulas derived from Eq.4.38 such as below [68]:

𝑅𝑎𝑐 =1

𝜋𝜎𝛿 (1 − 𝑒−𝑟𝛿) [2𝑟 − 𝜎 (1 − 𝑒−

r𝜎)]

(4.43)

where r is the conductor radius.

Also, the international standard (IEC 60287-1-1) [66] provides an effective and accurate

approach to estimate the AC resistance of a solid round wire in the air. Using IEC standard,

the AC resistance is calculated as below:

𝑅𝑎𝑐 = 𝑅𝑑𝑐(1 + 𝑦𝑠) [Ω/m] (4.44)

The skin effect factor 𝑦𝑠 is calculated as,

𝑦𝑠 =𝑥𝑠4

192 + 0.8𝑥𝑠4

(4.45)

where,

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89

𝑥𝑠4 = (

8𝜋𝑓𝐾𝑠𝑅𝑑𝑐107

)2

(4.46)

and 𝐾𝑠 = 1 in the case of a solid round conductor. However, according to [66], to obtain

accurate results is only applicable when xs ≤ 2.8.

Figure 4.9: AC resistance of round copper conductor versus conductor diameter using exact

equation

Figure 4.10: AC resistance of round copper conductor versus conductor diameter using simplified

equation

0.9

1.1

1.3

1.5

1.7

1.9

2.1

9 11 13 15 17 19 21 23 25 27 29 31 33 35

Rac/R

dc

Copper diameter (mm)

Copper resistivity versus current frequency (Kelvin function)

f=50Hz

f=100

f=200Hz

0.9

1.1

1.3

1.5

1.7

1.9

2.1

9 11 13 15 17 19 21 23 25 27 29 31 33 35

Rac/R

dc

Copper diameter (mm)

Copper resistivity versus current frequency (Simplified)

f=50Hz

f=100

f=200Hz

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90

Figures 4.9, 4.10 and 4.11 represent the variation of AC resistance of copper conductor versus

the conductor diameter calculated using Kelvin Function, simplified equation, and IEC

standard equation, respectively.

Figure 4.11 AC resistance of round copper conductor versus conductor diameter using IEC

standard equation

In order to verify the analytical equation that was proposed to estimate the AC resistance, a

finite element analysis has been done. Figure 4.12 shows the finite element simulation of skin

effect. Figure 4.12.and 4.12.b show the magnetic flux density and current density in wire

cross-section respectively. The wire diameter is 13.1mm, the current is 528A RMS and the

frequency is 50Hz

0.9

1.1

1.3

1.5

1.7

1.9

2.1

9 11 13 15 17 19 21 23 25 27 29 31 33 35

Rac/R

dc

Copper diameter (mm)

Copper resistivity versus current frequency (IEC standard)

f=50Hz

f=100

f=200Hz

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91

a) Magnetic flux density in wire section

b) current density in the wire section

Figure 4.12: Finite element analysis of skin effect

Table 4.2 shows the values of 𝑅𝑎𝑐/𝑅𝑑𝑐 of a round copper conductor with the diameter of

13.21 mm which were calculated using three methods for 50Hz, 100Hz and 200Hz.

Table 4.2: 𝑅𝑎𝑐/𝑅𝑑𝑐 of copper conductor with 13.21mm diameter in 50Hz, 100Hz and 200Hz using

three estimation methods

Frequency (Hz) Kelvin Function Simplified IEC Standard FEM

0 1.00 1.00 1.00 1.00

50 1.002 1.062 1.002 1.002

100 1.009 1.113 1.009 1.0091

200 1.037 1.202 1.037 1.0371

Also, regarding the current value of each frequency, the total copper losses are computed

using Eq.4.36. Table 4.3 shows the copper losses in each frequency and the error compared

to utilizing the same resistance in AC current. The IEC standard equation is not very complex

to implement, Hence, it will be appropriate to employ in the analytical model. It provides

suitable accuracy compared to the simplified equation.

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92

Table 4.3: Calculation of conductor copper losses using the estimated AC resistances and error

calculation compared to utilization of DC resistance

Frequency

(Hz)

Current (A)

Kelvin Function

Copper

losses(W)

Simplified

Copper

losses(W)

IEC Standard

Copper

losses(W)

FEM

Copper losses

(W)

0Hz, 166A 198.27 198.27 198.27 198.27

50Hz, 528A 2001.56 2121.86 2001.56 2001.55

100Hz, 476A 1632.97 1801.445 1632.97 1632.97

200Hz, 211A 331.22 384.22 331.56 331.25

Error 8.4% 15% 8.42% 8.4%

4.4.2. Thermal Model of Inductor

The thermal model simulates the thermal dissipation and the temperature rise in a different

node of the system. The temperature rise is a function of time and it depends on the thermal

capacity of the materials and the heat source value. It was assumed that the thermal

distribution in the materials is homogenous and the internal thermal resistivity is neglected.

Pfe Pcu

Ta

Rfe-air

Rfe-cu Rcu-airTfe

Tcu

Figure 4.13: Inductor equivalent thermal circuit

Figure 4.13 shows the thermal equivalent circuit of the inductor. The core loss is divided to

the vertical and horizontal parts regarding the volume.

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93

The temperature values of the core and copper are calculated as below:

𝑇𝑓𝑒 − 𝑇𝑎 =𝑅𝑓𝑒−𝑎𝑖𝑟(𝑅𝑓𝑒−𝑐𝑢 + 𝑅𝑐𝑢−𝑎𝑖𝑟)𝑃𝑓𝑒 + 𝑅𝑐𝑢−𝑎𝑖𝑟𝑃𝑐𝑢

𝑅𝑓𝑒−𝑎𝑖𝑟 + 𝑅𝑓𝑒−𝑐𝑢 + 𝑅𝑐𝑢−𝑎𝑖𝑟

(4.47)

𝑇𝑐𝑢 − 𝑇𝑎 =𝑅𝑓𝑒−𝑎𝑖𝑟𝑅𝑐𝑢−𝑎𝑖𝑟𝑃𝑓𝑒 + (𝑅𝑓𝑒−𝑎𝑖𝑟 + 𝑅𝑓𝑒−𝑐𝑢)𝑅𝑐𝑢−𝑎𝑖𝑟𝑃𝑐𝑢

𝑅𝑓𝑒−𝑎𝑖𝑟 + 𝑅𝑓𝑒−𝑐𝑢 + 𝑅𝑐𝑢−𝑎𝑖𝑟

(4.48)

where 𝑅𝑓𝑒−𝑎𝑖𝑟 , 𝑅𝑓𝑒−𝑐𝑢 , 𝑅𝑐𝑢−𝑎𝑖𝑟 are the thermal resistivity between core-air, core-winding and

winding-air respectively.

The thermal resistance between the materials are calculated as below:

𝑅𝑓𝑒−𝑎𝑖𝑟 =1

ℎ𝑓𝑒𝑆𝑓𝑒

(4.49)

𝑅𝑐𝑢−𝑎𝑖𝑟 =1

ℎ𝑐𝑢𝑆𝑐𝑢

(4.50)

where 𝑆𝑓𝑒 , 𝑆𝑐𝑢 are the contact surface between two materials; core-air, and copper-air

respectively. There is an isolation space between the winding and the core. Hence, the

isolation is consisted of isolation materials and immobile air. Hence, the thermal conduction

between the winding and the core is weak. The thermal resistivity between the winding and

the core is calculated as below:

𝑅𝑓𝑒−𝑐𝑢 =1

𝜆

𝑙𝑖𝑠𝑜𝑆𝑐𝑢

(4.51)

where 𝜆 is the thermal conduction factor of isolation and 𝑙𝑖𝑠𝑜 is the isolation thickness.

𝑆𝑓𝑒 = 4𝑑(2𝑑 + 𝑎 + 𝑐) + 2𝑐(2𝑑 + 𝑎) (4.52)

𝑆𝑐𝑢 =4𝑑𝐴𝑐𝑢𝑏

𝑐 + 4𝑑𝑏 (4.53)

Also, ℎ𝑓𝑒 , ℎ𝑐𝑢 are the dissipation factors; core-air and copper-air. The value of dissipation

factors is dependent on the cooling system specifications. In the case of natural air cooling

system, the value of dissipation factors 15 W/m^2/°C. Utilizing the ventilation air cooling

system, the dissipation factor can be augmented four times and it can be increased to eight

times using water cooled system.

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94

4.5. Investigation the effect of core saturation

Power electronic designers often consider that inductance core has linear characteristic and

neglect the nonlinearity and saturation region. Hence, dependent on the maximum flux

density in the core, a nonlinearity will emerge in inductance function.

Figure 4.14: The B-H curve of iron sheet core

Figure 4.14 shows a specific B-H curve of the iron sheet which is utilized in inductor

production. The B-H curve is usually linearized to use in the mathematical calculation. Power

electronic engineers prefer to keep the flux density in the linear part while increasing the flux

density can be lead to reduce the inductor size. On the other hand, in the transient state or

fault condition, a converter might enter to the nonlinear region. Therefore, performance

analysis of the converter with a nonlinear inductor will be important. In order to add the effect

of nonlinearity to the model, the relation between voltage and current must be calculated.

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95

4.5.1. Finding the Mathematical Core Magnetizing Function

If the inductance core is linear or the magnetic flux density is kept in the linear range, 𝑉𝐿 =

𝑛𝑑𝜙

𝑑𝑡 where 𝜙 =

𝑛𝐼𝐿

𝑅𝑇, 𝑅𝑇 =

𝑙𝑒

𝜇0𝜇𝑟𝐴𝑐. Therefore, in the case of linear inductance, the inductance

voltage will be 𝑉𝐿 = 𝐿𝑑𝐼𝐿

𝑑𝑡. In the case of non-linear core inductance, to find the relation

between the voltage and current of the inductor, we must pass several steps. In the first step,

the magnetic field (𝐻), must be computed.

𝐻 =𝑛𝐼𝐿𝑙𝑒

(4.54)

𝐻𝑙𝑖𝑛𝑒𝑎𝑟 =𝐵𝑙𝑖𝑛𝑒𝑎𝑟𝜇0𝜇1

(4.55)

𝐻𝑠𝑎𝑡 =𝐵𝑠𝑎𝑡𝜇0𝜇𝑟

(4.56)

where 𝑙𝑒 is the effective magnetic length, 𝑛 number of turn, 𝐵𝑙𝑖𝑛𝑒𝑎𝑟 maximum linear magnetic

flux density and 𝐵𝑠𝑎𝑡 is the saturation flux density. To calculate the magnetic flux density, a

hyperbolic function provides the best fitting factor to show nonlinear effect of the core.

𝐾𝑠𝑎𝑡 = (tanh−1(

𝐵𝑠𝑎𝑡𝐵𝑙𝑖𝑛𝑒𝑎𝑟

))/𝐻𝑙𝑖𝑛𝑒𝑎𝑟 (4.57)

𝐵 = 𝐵𝑠𝑎𝑡tanh (𝐾𝑠𝑎𝑡𝐻) (4.58)

4.5.2. Inductance circuit Equation Considering Core Saturation

Using the nonlinear flux function, the inductor voltage function will be as below:

𝑉𝐿 = 𝑛𝑑𝜙

𝑑𝑡

=𝑛2𝑘𝑠𝑎𝑡𝐴𝑒𝐵𝑠𝑎𝑡

𝑙𝑒sech2 (

𝐾𝑠𝑎𝑡𝑛𝐼𝐿𝑙𝑒

)𝑑𝐼𝐿𝑑𝑡

(4.59)

we can easily extend this approach for coupled inductors. In the case of coupled inductors,

inductor voltage will be calculated as below:

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96

𝑉𝐿 =𝑛1𝑑𝜙𝑇𝑑𝑡

= 𝑛1𝑑

𝑑𝑡(𝐵𝑠𝑎𝑡𝐴𝑒 tanh (

𝐾𝑠𝑎𝑡𝑛1𝐼1𝑙𝑒1

+⋯+𝐾𝑠𝑎𝑡𝐾𝑛𝑛1𝐼1

𝑙𝑒𝑛))

= (𝑛12𝐾𝑠𝑎𝑡𝐴𝑒𝑙𝑒1

𝑑𝐼1𝑑𝑡

+⋯+𝐾𝑁𝑛1

2𝐾𝑠𝑎𝑡𝐴𝑒𝑙𝑒𝑛

𝑑𝐼𝑛𝑑𝑡) sech2(

𝐾𝑠𝑎𝑡𝑛1𝐼1𝑙𝑒1

+⋯+𝐾𝑛𝐾𝑠𝑎𝑡𝑛1𝐼𝑛

𝑙𝑒𝑛)

(4.60)

In order to add the saturation effect, it is sufficient to modify the flux density relation as

below:

𝐵𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 = 𝐵𝑠𝑎𝑡 tanh−1(

𝐾𝑠𝑎𝑡𝑛(𝐼1 + 𝐼2)

𝑙𝑚 + 2𝜇𝑟𝑙𝑔)

(4.61)

4.6. Finite Element Analysis of Coupled Inductors

The finite element method (FEM) is a numerical technique for solving problems which are

described by partial differential equations or can be formulated as functional minimization.

A domain of interest is represented as an assembly of finite elements. Approximating

functions in finite elements are determined in terms of nodal values of a physical field which

is sought. A continuous physical problem is transformed into a discretized finite element

problem with unknown nodal values. For a linear problem, a system of linear algebraic

equations should be solved. Values inside finite elements can be recovered using nodal values

[70].

The finite element approach could be employed in 2-D or 3D space. 2-D finite element has

fewer nodes to solve in compare with the 3-D finite element approach. Therefore, it is faster

and needs less computer memory to solve. In the case of symmetrical volume on the z axis,

2-D finite element approach could be an effective solution to accelerate the simulation time

and decrease the required computer memory.

4.6.1. Magneto-static Analysis using Finite Element Method

In this section, the finite element analysis using a software is explained. The finite element

software calculates the precise magnetic field density, self-inductance, and the magnetic

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97

coupling factor. Also, the copper and magnetic losses are calculated by the software. There

is a number of electromagnetic analysis software which use the finite element method to

solve the partial differential equations. The main sections of a finite element software are:

1- Geometry

2- Generate mesh

3- Solve equations

4- Show result

The finite element software is complex and expensive. In this project, a free version of FEM

software is used to analyze the coupled inductor. This version is an open source version and

free for the academic researches. The most advantages of this software are the possibility to

execute the netlist code. The netlist code consists of geometry parameters, material

parameters, and mesh constraints are written as an LUA file. In this project, the notepad++

is used to write the LUA file. The LUA file is executed by FEM software. This is a useful

possibility to make a connection between the EXCEL and FEM software. Figure 4.15 shows

the magnetic flux lines and magnetic flux density of the inductor.

a) Magnetic flux lines

b) Magnetic flux density

Figure 4.15: Finite element analysis of coupled arm inductance

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98

4.6.2. Correction of Analytical Model using Finite Element Method

The analytical model of coupled inductor provides an appropriate platform to integrate the

optimization loop. The accuracy of the analytical model will reduce by increasing the length

of the inductor air gap. Hence, in most of the operating points, the analytical model accuracy

is not sufficient enough for the optimization algorithm. The precise value should be

calculated using a finite element method. On the other hand, utilizing the finite element

software in the optimization loop made it very slow and some time difficult to converge. To

find the global optimal point, solver chooses the initial values from a different region to avoid

falling in a local point. If the finite element model is employed directly in the optimization

loop, it takes so much time to find the global points. In this research, an innovative solution

has been proposed to resolve the time and accuracy issues at the same time.

The FEM analysis results are employed in order to verify and correct the analytical model

parameters. Utilization the FEM software in the optimization loop, intensely increase the

optimization time and makes it complex to converge. On the other hand, the analytical model

does not provide accurate results at all operating points. In this research, a hybrid

optimization algorithm has been proposed and developed to modify the analytical model with

FEM software in order to decrease the optimization time and increase the model preciseness.

2-D finite element method is employed to analyze the inductance and modify the analytical

model. A Hybrid optimization model consisted of the analytical model and finite element

approach is employed to increase the model accuracy without increasing the optimization

time.

The air gap length is known as the most important variable which affects the accuracy of the

analytical model. Therefore, the inductance relation is defined for the analytical model and

FEM analysis as below:

𝐿𝑎𝑛 =𝑛2𝜇0𝜇1𝐴𝑐𝑙𝑚 + 2𝜇𝑟𝑙𝑔

(4.62)

𝐿𝑓𝑒𝑚 =𝑛2𝜇0𝜇1𝐴𝑐

𝑙𝑚 + 2𝐾𝑐𝜇𝑟𝑙𝑔

(4.63)

If we consider that 𝛿 =𝜇𝑟𝑙𝑔

𝑙𝑚, then the correction factor 𝐾𝑐 and model error 𝐸 will be:

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99

𝐾𝑐 =𝐿𝑎𝑛(1 + 𝛿) − 𝐿𝑓𝑒𝑚

𝛿𝐿𝑓𝑒𝑚

(4.64)

𝐸 =𝐿𝑓𝑒𝑚 − 𝐿𝑎𝐿𝑓𝑒𝑚

× 100 (4.65)

Figure 4.16 shows the flowchart of the proposed correction loop, which was employed to

combine the analytical model and finite element method. It consists of two separated loops;

the optimization loop and correction loop. The analytical model is utilized to solve and find

the optimal values while the correction loop is used to modify the analytical model and

eliminate the error. In this method, the optimization loop will be repeated for more than 1000

times while the correction loop is called for 3 or 4 times. Using multi-start algorithm, we can

find the global optimal point and avoid falling in local optimal points. This is exactly the

point, which makes difficult to use the finite element model in optimization. The solver

chooses different initial points to find the global optimal point. If we use the finite element

in optimization without suitable initial values, it may go to the region without an optimal

point that leads to missing the optimization time. In this approach, the initial evaluation has

done using analytical model, then the variables are sent to finite element software to validate.

This innovative method effectively decreases the optimization time and increases the

converging probability.

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100

FEM

software

Dimension model of

arm inductances

IL1

µr

Larm

Jabcdg

M

Bmax

RL

Pcu

Pcore

Vind

Mind

Spe

cific

tio

ns

Co

re t

op

olo

gy

IL2

Win

din

gn1

n2

Electromagne

tic Variab

les

Perfo

rmance

Outputs

η%

yes

Error calculation

Calculate correction factor

Figure 4.16: The flowchart of the proposed correction approach

4.7. Conclusion

In this chapter, the dimension analysis of MMC passive components is presented. The mass

equation of sub-module capacitor depends on the capacitance value and nominal capacitor

voltage. The capacitor mass equation is determined regarding product datasheet of the

manufacturer. The dimension analysis of the arm inductances is dependent on

electromagnetic analysis.

The analytical electromagnetic model of the arm inductances is investigated and developed

regarding MMC specifications. Utilizing the analytical model, the inductance value, mutual

inductance, magnetic flux, core and winding losses are calculated. Also, the electromagnetic

model of coupled inductor and its circuit model were investigated.

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101

Finally, the finite element analysis of the arm inductance was presented as an accurate

approach to analyzing the magnetic components. The dimensioning variables are sent to

finite element software to generate the model geometry. The inductance model is supplied

using a current source to compute the electrical and magnetic parameters. Utilizing the finite

element approach is complex and time-consuming. In this chapter, a hybrid model was

presented to correct the analytical model parameters and enhance its accuracy. In the next

chapter, the converter analysis in the fault condition is presented.

Page 130: Optimal Sizing of Modular Multilevel Converters

102

CHAPTER V

5 Converter Analysis in the Fault Condition

5.1. Introduction

The steady-state model guarantees the appropriate converter functionality in the nominal

operating point. Regarding the nonlinearity of the magnetic core, the functionality of MMC

converter should be studied in the fault condition. In the final design, a margin should be

considered to minimize the components damage in defect condition. In this chapter, the

standard MMC converter faults are introduced and investigated. Also, two different methods

are proposed to analyze the defect condition. The first one is to use a Simulink model in order

to simulate the defect condition and measure the circuit parameters. The second one is to add

the analytical model of the fault condition to the proposed time-domain model. In order to

investigate the fault condition, the standard faults are designed and analyzed using state

equations. The results could be added to the constraints vector.

5.2. Investigation of Standard Defects in MMC Converter

In this section, the standard defect type which is happened in MMC converter is investigated.

Based on the literature, three kinds of converter fault have been introduced and investigated.

The faults might happen in the sub-modules, arm inductance or at the converter output. The

most important faults are DC link fault, sub-module short circuit or open circuit and arm

inductance fault [71-73].

5.2.1. DC Link Fault

The first kind of faults that were studied in the literature is the DC bus fault [72]. The DC

bus dependent on its application might be a short or a long DC line. The short circuit might

happen on the different points of the DC line. Therefore, the fault’s impedance is different.

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103

The most important effect of DC link fault is to affect the DC line voltage and increase the

arm currents. Figure 5.1 shows the current paths in an MMC converter when a DC link fault

happens. DC link fault strongly affects the arm currents. Hence, it might lead to destroy the

semiconductors and saturate the arm inductances.

Figure 5.1: The DC link fault and currents path in the converter

5.2.2. Sub-module Fault

There are two different types of the switching device fault in a sub-module: open-circuit fault

which appears due to lifting of the bonding wires in a switch module caused by over-

temperature or aging and usually does not cause additional serious damage to the system if

the protection system functions well; short-circuit faults are caused by wrong gating signals,

overvoltage, or high temperature and could cause additional damage to other components in

the circuit, so the short circuit fault should be treated fast and carefully. Generally, the

hardware overcurrent protection which stops the operation of the system is the most common

solution for short-circuit faults [17]. However, for MMC, the overcurrent appears in the

faulty module rather than flowing through the whole arm or phase which means it is the faulty

module that needs to be bypassed by the action of overcurrent protection devices and the

system operation can continue without stopping. The failure configurations of the open-

circuit and short-circuit fault in a sub-module concerning on the failure of the switching

devices T1, T2 are shown in Figure 5.2.

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104

D1

C

D2

Vc

IcT1

T2

Ism

Vsm

a) Short-circuit of

bypass IGBT

D1

C

D2

T1

T2

Vsm

IsmVc

Ic

b) Short-circuit of the

main IGBT

D1

C

D2

T1

T2

Vsm

IsmVc

Ic

c) Open circuit of

sub-module IGBTs

Figure 5.2: Various kind of sub-module faults

5.2.3. Inductance Fault

There are two types of inductor fault that are investigated in literature; short-circuit and open-

circuit. Instantaneous changes in inductor parameters are unphysical. Therefore, when the

Inductor enters the faulted state, short-circuit and open-circuit voltages transition to their

faulted values over a period of time, according to the following formula:

𝐼𝐿 = 𝐼𝑓 − (𝐼𝑓 − 𝐼𝑛𝑓)sech (Δ𝑡

𝐹𝑇)

(5.1)

For short-circuit types of faults, the conductance of the short-circuit path also changes

according to a 𝒔𝒆𝒄𝒉(𝚫𝒕/𝑭𝑻) function from a small value (representing an open-circuit path)

to a large value. where Δ𝑡 is the time since the onset of the fault condition and 𝐹𝑇 is the time

constant associated with the fault transition.

5.3. Close Loop Control of MMC converter using Simulink

In power converters, there is a current control loop which limits the maximum current in the

fault condition. The inductance of non-ideal core inductors is reduced by increasing the

current. In the fault condition where the current is massively increased, there are serious

concerns about the inductors functionality. The inductance reduction intensely affects the

converters circuit performance and may lead to component failure. In this situation, the

recommissioning is usually so expensive and time-consuming. Hence, it is important to

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105

verify the component safety in the fault condition. There are various types of faults that

should be investigated in MMC converter.

The performance of optimal inductor must be evaluated in the fault condition in order to

make sure about the converter and inductance stability in a fault condition. A close loop

controller has employed to control the DC link voltage while the dc current is changing.

Figure 5.3 shows the close loop control which is utilized to control the MMC converter.

MMC Converter

+

Vdc

-

Grid

Vdc*

Vdc

PID ×

Kv

Vabc,Iabc

Vabc

abc/dq0

abc/dq0

Iabc

Park Trans.

Park Trans.

Id

Iq

Vd

Vq

PID

PID

dq0/abc

1

0.5

fsw/m

(m-1)fsw/m

.

.

.

Submodule pulses

+

+

++

-

-

-

+

Figure 5.3 Close loop control diagram of MMC converter

5.4. Investigation of Converter Performance in Defect Condition

Using Simulink, the performance of MMC converter and its components is evaluated in

defect condition. Figure 5.4 and 5.5 show the DC link voltage and AC line current of MMC

converter in a complete converter cycle. The faults happen at t=0.5. The normal condition

was compared to the faults condition.

Figures 4.9, 4.10 and 4.11 represent the variation of AC resistance of copper conductor versus

the conductor diameter calculated using Kelvin Function, simplified equation, and IEC

standard equation, respectively.

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106

Table 5.1: Simulation parameters of MMC converter in simpower

Parameters Value

Number of sub-modules per arm (m) 3

Sub-module Capacitor value 3.5 𝜇𝐹

Inductance value 2.83 𝑚𝐻

Coupling factor 0.9

DC voltage 5000 V

AC voltage L-L 2000 V

Figure 5.4: DC Link voltage variation via various converter faults

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107

Figure 5.5: Line current variation via various converter faults

In the normal condition, inductor current does not pass 2500A in the full load. The sub-

module short circuit or DC link fault cause to jump the current to 5000 A. Therefore, the

inductor core might enter to the saturation region. It leads to increase the circulation current

and damage the IGBTs and inductor. In order to verify the core saturation, the maximum flux

density in the core should be evaluated. Figure 5.7 shows the magnetic flux density of the

core in the normal and fault condition. It should be noted that in the coupled inductor

topology, the magnetic flux density is affected by upper and lower arm currents at the same

time. In the normal condition, the maximum flux density is about 0.9T, while in the fault

condition it goes to 1T which is not a critical condition for the electromagnetic core.

Therefore, the optimized inductor parameters could pass the fault condition.

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108

Figure 5.6: Upper inductor current variation via various converter fault

Figure 5.7: Magnetic flux density of inductor core via various converter fault

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109

5.5. Combination of Time-Domain Steady-State Model and

Faults in the unit package

In order to investigate the MMC converter under a fault condition, it is necessary to review

the sub-module operation in the fault condition. First of all, the normal operation of half-

bridge sub-module must be investigated. Table 5.1 shows the normal operation of half-bridge

sub-module in four sub-module statuses.

5.5.1. Sub-module faults investigation

Normal operation: In normal operation, as listed in Table 5.2, when the arm current 𝑖𝑎𝑟𝑚 is

positive, if T1 is turned on, T2 is turned off in the sub-module module which means 𝑆 = 1,

the current flows through D1 and C, the capacitor is charged; otherwise, T1 is turned off, T2 is

turned on, the current will go through T2 and the capacitor voltage maintains stable; when

𝑖𝑎𝑟𝑚 is negative, T1 is turned off, T2 is turned on, the current flows through D2; oppositely,

𝑆 = 0, the current will go through T1 and C, the capacitor is discharged.

Table 5.2: Normal operation of a half-bridge sub-module

St

No.

Current Status Gate Current goes

through

capacitor Capacitor

voltage

1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Charged Increased

2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 T2 Bypassed Stable

3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 T1 and C Discharged Decreased

4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 Bypassed Stable

Open-circuit fault in T1: as shown in Table 5.3, the sub-module operates as normal when the

arm current 𝑖𝑎𝑟𝑚 > 0, the arm current still goes through D1 and C to charge the capacitor

when the gating signal 𝑆 = 1 and the arm current flows through T2 to bypass the capacitor

when the gating signal 𝑆 = 0; when 𝑖𝑎𝑟𝑚 < 0, the module is in normal operation when 𝑆 =

0, the arm current flows through D2 and the capacitor voltage is stable; however, when the

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110

gating signal 𝑆 = 1, the arm current will be forced to go through D2 instead of T1 and C in

the normal condition;

Table 5.3 Investigation of Open-Circuit Fault in T1

St

No.

Current Status Gate Current goes

through

capacitor Capacitor

voltage

1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Charged Increased

2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 T2 Bypassed Stable

3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 D2 Bypassed Stable

4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 Bypassed Stable

Open-circuit fault in T2: the open-circuit fault is shown in Table 5.4, the sub-module

operates as normal when the arm current 𝑖𝑎𝑟𝑚 > 0 and 𝑆 = 1, if T1 is turned off, T2 is turned

on, the arm current is forced to go through D1 and C to charge the capacitor instead of T2 to

bypass the capacitor; when the arm current 𝑖𝑎𝑟𝑚 < 0, the module is in normal operation;

Table 5.4 Investigation of Open-Circuit Fault in T2

St

No.

Current Status Gate Current goes

through

capacitor Capacitor

voltage

1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Charged Increased

2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 D1 and C Charged Increased

3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 T1 and C Discharged Decreased

4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 Bypassed Stable

Short-circuit fault in T1 or T2 : as shown in Table 5.5, when the short-circuit fault happens

in T1 (T2), the sub-module operates as normal if the corresponding IGBT T1 (T2) is turned on

and the complementary IGBT T2 (T1) is turned off; when the complementary IGBT T2 (T1) is

turned on, the capacitor discharged through the capacitor discharging loop which is formed

by the short-circuited T1 (T2), the complementary IGBT T2 (T1) and the capacitor C. Due to

the small-time constant of the capacitor discharging loop, the capacitor discharged very

quickly which leads to the rapid declines of capacitor voltage and the large short-circuit

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111

current in the faulty module. Generally, the faulty module is bypassed and the arm current

goes through the switch used to do overcurrent protection. However, with MMC topology,

the arm current will go through D1 to charge C when the arm current is positive and go

through D2 to discharge C with negative arm current. The capacitor voltage of the faulty

module changes from zero to a small value compared to the normal capacitor voltages.

Table 5.5 Investigation of Short-Circuit Faults in T1 or T2

St

No.

Current Status Gate Current goes

through

capacitor Capacitor

voltage

1st 𝐼𝑠𝑚 > 0 T1 on, T2 off S=1 D1 and C Bypassed Increased

2nd 𝐼𝑠𝑚 > 0 T1 off, T2 on S=0 D1 and C Bypassed Increased

3rd 𝐼𝑠𝑚 < 0 T1 on, T2 off S=1 D2 and C Bypassed Decreased

4th 𝐼𝑠𝑚 < 0 T1 off, T2 off S=0 D2 and C Bypassed Decreased

It can be seen that various module performances are caused by different faults which mean

that the different faults can be identified by analyzing the performance of the system.

Moreover, the system performance under fault conditions can be limited by changing the

gating signals which makes the below fault identification possible.

5.5.2. Proposed global optimization considering fault analysis

Another approach is to make a combination of transient model and fault analysis. In this

method, while the transient model is calculating the value of the normal operation, the fault

model calculates the important parameters in the fault condition. The value of the converter

parameters in the fault condition is considered as constraints of the optimization algorithm.

5.6. Conclusion

Power electronic designers design the converters base on the nominal operating point and

usually neglect the fault condition. In high power application, the fault occurrence could lead

to costly damages. Hence, in high power application, the fault condition must be noted in

design procedure in order to minimize the damages and reduce the cost of fault.

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112

In this chapter, various kinds of the faults in MMC structure was studied and their effects on

the sub-module voltage and arm current were investigated. To simulate the fault condition, a

close loop control system was proposed and explained. The Saturable model of coupled

inductances which was presented in the previous chapter was employed in the converter

model. The fault condition was simulated using Simulink/MATLAB to investigate the

important converter variables in the fault condition. One of the most important variables in

the fault condition is the magnetic flux density of the arm inductance core. The magnetic flux

density is the criterion to determine the saturation condition of inductance core.

In the next chapter, various optimization scenarios with different complexity will be proposed

and developed in order to minimize the final volume and mass of MMC converter. The results

of different optimization algorithm are compared and discussed to achieve the best

optimization approach.

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113

CHAPTER VI

6 Optimal Design of Modular Multilevel Converter

6.1. Introduction

In this chapter, different optimization scenarios of MMC converter are proposed and

developed. The major optimization goal is to minimize the converter size with respect to the

technical and manufacturing constraints. Each optimization algorithm consists of a

mathematical model, optimization variables, constraints, goal function and the solver. In term

of MMC converter, the converter model is divided to circuit model, electromagnetic model,

thermal model, and dimensioning model.

In this research, various optimization algorithms with different degree of complexity have

been proposed. The first optimization algorithm employs the proposed time-domain circuit

model which was explained in chapter 3 as converter model and neglects the structure of the

arm inductance and capacitor size. It investigates the inductor and capacitor as circuit

components. This algorithm is fast and provides a suitable guideline to get closer to the

optimal point. The weakness of this approach is to neglect the electromagnetic characteristics

of the inductances and its dimension.

The arm inductance is the bulkiest part of MMC converter which strangely affects the total

converter mass. In chapter 4, the electromagnetic and thermal model of the model of arm

inductances was presented. Utilizing the proposed circuit model, electromagnetic model, and

thermal model, a new optimization algorithm has proposed and developed. Unlike the

previous algorithm, the core and winding parameters of arm inductance are considered as

part of optimization variables. Utilizing the analytical dimensioning model of inductor and

capacitor, the total converter mass is calculated as goal function.

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114

The electromagnetic model of inductors is not accurate enough especially in terms of using

airgap in the core topology. In chapter 4, finite element method was introduced as an effective

tool to analyze the electromagnetic components. A hybrid correction loop was proposed to

modify the analytical model parameters and enhance the model accuracy. The third

optimization algorithm employs the proposed hybrid correction loop to enhance the accuracy

of optimization results. Finally, results of various optimization algorithm are discussed and

investigated.

6.2. Optimization algorithm using numerical solver

Numerical solvers are the mathematical tools which are used in lots of engineering domain

such as mathematics, engineering, science, business, and economics. One of the interesting

application of the numerical solvers is to use in the optimization problems. In the

optimization issues, an engineering issue is formulated using mathematical approaches called

analytical model to represent the situation. There are lots of software which were designed

to implement the optimization issues. Microsoft excel provides an appropriate environment

and functions to implement the optimization problems.

Microsoft excel 2013 supports three solvers, Linear, nonlinear and evolutionary (Genetic

algorithm based). Using Microsoft excel leads to visualize the optimization loop and will be

a very suitable platform to implement the problem. Also, we have a possibility to make a

connection between Microsoft Excel and MATLAB software in order to communicate and

use their software facilities. Using the link between Microsoft Excel and MATLAB, we are

able to use the optimization toolbox of the MATLAB in the Excel environment. Therefore,

we will have a chance to utilize the modern optimization function and all Matlab functions.

It will be an attractive option in future researches. Figure 6.1 shows the image of excel sheet

which is consist of different parts of the optimization problem.

Optimization toolbox of MATLAB is another solution to implement an optimization

problem. MATLAB software is a professional and powerful mathematical package which

consists of recent numerical functions. Some researchers, use Microsoft excel to implement

the analytical model, afterward the model is sent to the MATLAB optimization toolbox in

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115

order to solve and optimize the problem. An optimization loop generally consists of the

following components:

1- Optimization variables: Optimization variables are independent variables which are

chosen to optimize. The decisions of the problem are the variables which must be

changed by the numerical solver to find the optimal point.

2- The objective of the problem is expressed as a mathematical expression based on

decision variables. The objective may be maximizing or minimizing by changing the

decision variables.

3- Constraints: The limitations or requirements of the problem are expressed as

inequalities or equations in decision variables. The constraints might be a simple value

or a mathematical function which depends on lots of parameters.

Figure 6.1 shows the proposed global optimization algorithm for modular multilevel

converters. The Excel cells are connected together via mathematical equations.

Figure 6.1: Implementation of Global Optimization Algorithm with Microsoft Excel

Microsoft excel represented three solver functions in order to solve the different types of

models; linear solver, nonlinear solver and evolutionary which employs genetic algorithm.

The linear solver is used to solve the models which consist of linear objective function and

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116

linear constraints in decision variables, it is called linear programming model. The nonlinear

solver is used to solve the nonlinear programming model consists of a nonlinear objective

function and nonlinear constraints. The advantage of Microsoft Excel is the possibility of

visualization of the model, constraints and goal function.

6.3. Load Specification of the MMC Active Front End converter

application

The next step is to determine the load specification based on the application. In order to

investigate the application using the proposed optimization algorithm, it is necessary to use

the load specifications of the converter which must be studied. In the application, there is no

nominal point and the output power changes from zero to 30MW in each cycle. Hence, the

average power should be considered as a nominal point. The average converter power is

2.5MW. Table 6.1 shows the converter load specifications.

Table 6.1: Load specification of MMC Active Front End converter application

Converter Parameter Value

Nominal Power 2.5 MW

AC Line Voltage 2000 V

DC Link Voltage 5000 V

Power Factor 0.8

Nominal Frequency 50 Hz

THD < 2%

Efficiency > 95%

ΔT < 50 ℃

Capacitor voltage ripple < 20%

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117

6.4. Constraints Calculation

One of the important parts of optimization loop is to choose and calculate the appropriate

constraints. The optimization finds the optimal values concerning the technical and

manufacturing constraints. The main constraints related to the application and method which

is used to calculate is presented as below:

6.4.1. Sub-module capacitor voltage ripple

The DC link voltage is divided between the series sub-modules. The DC part of sub-module

capacitor voltage is 𝑉𝑑𝑐

𝑚 in steady-state condition. The voltage ripple increases the maximum

capacitor voltage and converter harmonic and reduces the converter stability. Therefore, it

should be limited in the specific band. The sub-module capacitor voltage must not be greater

than the IGBT voltage. The capacitor voltage ripple is considered as a percentage of dc part.

The IGBT voltage is chosen based on the maximum capacitor voltage, generally with 50%

safety margin.

6.4.2. THD

Total harmonic distortion (THD) is an important parameter which shows the converter power

quality. Reducing the THD depends on lots of parameters such as switching frequency, a

switching method, the number of sub-module per arm, arm inductance, input filter and

capacitor value. High power converters generally use low switching frequency to minimize

the IGBT losses. Hence, reducing the THD will lead to increase the passive component

values. On the other word, there is a trade-off between the THD and MMC component value

especially arm inductance.

6.4.3. Semiconductor Losses

According to the IGBT datasheet, each IGBT can dissipate a maximum thermal loss which

depends on the cooling system. Therefore, the IGBT loss will be an important constraint

which guarantees the semiconductor safety. On the other hand, IGBT loss intensely affects

the converter efficiency. In some applications, the IGBT loss must be limited to achieve

higher converter efficiency. To calculate the IGBT loss, the technical data of power IGBT is

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118

necessary. The number of power switches which are appropriate for our application is

limited. Therefore, the optimization has been done with two IGBTs that their specifications

are closer to our application. The technical data of ABB IGBT (5SNA 1500E330305) was

utilized to calculate the precise IGBT losses value. The nominal value of voltage and current

of this switch is 3.3 kV and 1500 A respectively.

6.4.4. Inductor Losses

The inductor loss is should be limited regarding the thermal dissipation function. There is a

maximum allowable temperature rise which must be considered to avoid the inductor fault.

On the other hand, it affects the total converter efficiency and reduces the converter

performance.

6.5. Goal function

In this thesis, the main goal function is to minimize the total converter volume and mass.

Mass optimization needs the component mathematical mass function to put in the

optimization loop. It needs a dimensioning analysis in order to obtain all mass functions. The

dimensioning analysis will be done in the next chapters. Therefore, the mass function should

be replaced by a circuit criterion which could explain the mass function.

The energy stored in the capacitor and inductor are suitable functions in order to estimate the

capacitor and inductor mass. In each converter arm, there are 𝑚 number of series sub-

modules, while there is on inductor per arm. The total electric and magnetic energy stored in

capacitors and arm inductance are calculated as below:

𝐸𝑐𝑎𝑝 = 3𝑚𝐶𝑠𝑚𝑉𝑐2 (6.1)

𝐸𝑖𝑛𝑑 =3

2𝐿(𝐼𝑢

2 + 𝐼𝑙2) + 3𝑀𝐼𝑢𝐼𝑙

(6.2)

Finally, the total converter energy stored could be calculated as goal function.

𝐸𝑐𝑜𝑛𝑣 = 𝐸𝑐𝑎𝑝 + 𝐸𝑖𝑛𝑑 (6.3)

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119

6.6. MMC Optimization using analytical circuit model

The first optimization algorithm utilizes the proposed time-domain circuit model to

determine the optimal value of capacitor and inductor value of MMC converter. In addition

to the passive components value, the switching frequency and the number of sub-modules

per arm are considered as optimization variables. In this algorithm, the magnetic parameters

of inductors are neglected. Hence, an energy criterion is chosen to estimate the size of

inductors and capacitors. Also, the total harmonic distortion, total converter efficiency, and

sub-module capacitor ripple are the most important constraints of this optimization

algorithm.

6.6.1. Optimization algorithm

Figure 6.2 shows the optimization algorithm based on the analytical circuit model which is

proposed for MMC converter. The most important role of this algorithm is to find the optimal

circuit value of the sub-module capacitor and arm inductance. The optimization variables are

sub-module capacitor 𝐶𝑠𝑚, arm inductance 𝐿, 𝑀, switching frequency 𝑓𝑠𝑤 and modulation

index. The circuit analytical model of MMC regarding to the load specifications and some

constant values organize the main core of the algorithm. With the same reason, the main

constraints are the circuit constraints. The constraints calculation is done to estimate the

important limitation such as ripple, THD, losses and DC bus voltage and finally the goal

function is calculated. The circuit model does not obtain the size or dimension of any

components. Therefore, it is necessary to define some criteria which are proportional to the

components dimension. The capacitors and inductors store the energy in their structure. This

energy is proportionally depends on it size. Hence, the best parameter is to use the maximum

energy stored in the capacitor and inductor to estimate the size and dimension. The final part

of optimization algorithm is the solver. The solver is a calculation engine which solves the

nonlinear equation using advanced numerical approach. The nonlinear solver searches to find

the optimal values to minimize the total energy stored in the converter. Table 6.2 shows the

optimization constraints.

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120

MMC Circuit

Model and

Fault analysis

Vabc

Idc

S(t)

fsw m

Csm

Larm

ωs

Iabc

Is1,Is2

Icirc

VCsm

Voltage ripple

THD

IL

Ifault

Switch losses

Spe

cifi

ctio

ns

Co

nve

rte

r to

po

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N

.

.

.

Pas

sive

co

mp

on

en

t v

alu

e

MIG

BT

Sp

eci

fica

tio

ns Vmax

Imax

Psw

Inputs

Stead

y-state Variab

les

Stead

y-state Pe

rform

ance

Inductor losses

Efficiency

Energy stored

Outputs

Non-linear

Solver

Total mass function

Capacitor energy

Csm

Co

nst

rain

ts

Vo

ltag

e ri

pp

le,

TH

D,e

ffic

ien

cy,

swit

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oss

es

Init

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Val

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Op

tim

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ion

par

amet

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(L,M

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)

Inductor energy

Larm,M

Figure 6.2: Optimization flowchart of MMC converter

Table 6.2 The main optimization constraints

Constraints Value

DC bus voltage 5000 V

Efficiency >95%

THD <2%

Capacitor voltage ripple 0.2 ∗

𝑉𝑑𝑐𝑚

Number of series sub-modules 3< 𝑚 < 8

Number of parallel branches 1 < 𝑁 < 3

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121

6.7. Optimal design of modular multilevel converter using

dimensioning model

The mass function of capacitor and inductors are added to the optimization algorithm and the

optimization is done to minimize the total converter mass. The optimization results should

be discussed and investigated.

6.7.1. Global Mass Minimization Algorithm

By increasing the demands of medium and high power converters, MMC converters were

changing to complex, bulky and expensive structures. In this time, mass minimization was

emerged to reduce and optimize the volume of the main components such as capacitors and

inductors. In the case of MMC converter, the mass minimization algorithm is dependent on

the circuit operation, electromagnetic and thermal functionalities.

Figure 6.3: The proposed global optimization algorithm using analytical model

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122

Accordingly, the global optimization algorithm has been proposed to consider all technical

issues and constraints. Figure 6.3 shows the proposed mass minimization algorithm includes

circuit, electromagnetic and thermal analytical model. The mass minimization algorithm

consists of four main sections; mathematical model, goal function, constraints and numerical

solver.

Given that in MMC topology, the circuit initial values are dependent on the component

values, an extra part as initializing section has been added to the algorithm. The initializing

section employs the steady-state model to find the circuit initial values. It should be

considered that the converter does not stay in a constant operating point. Hence, after

initialization, the transient model will be started to calculate the state variables. In the same

time, the electromagnetic model of coupled inductors is employed in order to estimate the

inductor parameters and core size. Also, thermal model investigates the thermal distribution

in the various converter components and thermal exchange with the cooling system.

Table 6.3: List of optimization variables

Variables definition

𝐶𝑠𝑚 Sub-module capacitor

𝑓𝑠𝑤 Switching Frequency

A Core window width

B Core window height

C Core depth

D Core Width

G Air gap

J Current density

n Turn Number

The constraints are the important part of optimization loop to determine the technical and

manufacturing limits of MMC converter. The constraints values are determined by

mathematical equations and compared to model outputs. If the outputs do not satisfy the

constraints, the solver chooses other values for optimization variables. If the outputs are

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123

placed in the allowed region, the goal function is calculated using optimization variable. This

subroutine will repeat until the optimization loop finds the global optimal point of goal

function.

Table 6.4: List of main constraints in optimization algorithm

Constraints Value

Total efficiency >95%

Capacitor voltage ripple < 0.2𝑉𝑑𝑐/𝑚

THD < 2%

𝐵𝑚𝑎𝑥 0.9T

Core window area > 𝐴𝑐𝑢−𝑡𝑜𝑡𝑎𝑙

Δ𝑇 < 60

Utilizing the circuit and electromagnetic model, the switch and inductor losses are calculated

in the transient state. Therefore, the total converter efficiency was calculated as a constraint

in the optimization algorithm. The optimization must determine the sub-module capacitor,

inductor value, and size. Table 6.3 shows the optimization variables which have to optimize

by the solver. Table 6.4 shows the constraints values in this optimization.

6.8. Hybrid Optimization Model using 2-D FEM

Utilizing the air gap leads to reduce the analytical model accuracy. Finite element analysis is

used to increase the analysis accuracy. Utilization of finite element model in optimization

procedure increases the time of convergence. It will be a trade-off between the optimization

time and result accuracy. The hybrid optimization model is proposed and developed to

increase the result accuracy without increasing the optimization time.

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124

6.8.1. Hybrid Global Optimization Algorithm

A hybrid optimization algorithm is presented using a combination of the analytical model

and finite element method. The most important advantage of this method is to achieve high

accuracy and low convergence time in comparison with other optimization algorithms.

By a combination of global optimization algorithm and the proposed correction loop, a hybrid

optimization algorithm is presented which provides the advantages of conventional global

optimization algorithm while the results accuracy has been enhanced.

Figure 6.4: The proposed hybrid optimization algorithm

Figure 6.4 shows the proposed hybrid optimization algorithm which provides the optimal

converter size. After an optimization cycle, outputs are sent to the software to analyze using

finite element method. The software results are compared to the analytical results and the

model will modify to minimize the error. This subroutine is repeated until achieving the same

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125

results by analytical model and software. The optimization algorithm estimates the optimal

value of these variables to achieve the minimum size.

6.8.2. Hybrid Global Optimization Algorithm considering fault margin

Figure 6.5 shows the global optimization of MMC converter considering fault margin. The

outputs of the fault calculation block are sent to the constraint block. These new constraints

increase the converter margin against the fault condition.

Steady State Model

Nominal Specifications

InitializingSet Initial Values

MMC circuit Analytical Model

Optimization Variables

Electromagnetic Analytical Model of coupled Inductors

Thermal ModelOutputs satisfy

the constraints?

Circuit Constraints

Electromagnetic Constraints

Thermal Constraints

Goal function(Converter or inductor Mass Minimization)

Nonlinear Solver

Optimal Values

Initializing

Analytical Model Constraints

Numerical Solver

No

yesGoal Function

IGBT open circuit

IGBT short circuit

Inductor short circuit

Inductor open circuit

Fau

lt c

alc

ula

tio

n

Fault condition model

Figure 6.5 Global optimization algorithm considering fault margin

Figure 6.6 shows the proposed hybrid optimization algorithm considering the fault margin.

This algorithm is more complex and slower in comparison to the previous algorithm. The

main optimization loop and auxiliary correction loop affect the outputs of the fault model at

the same time. Utilizing this mode, the output of optimization is calculated regarding actual

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126

core specifications such as nonlinear permeability and saturation effect. The fault margins

will minimize the converter damage in the failure time.

Steady State Model

Nominal Specifications

InitializingSet Initial Values

MMC circuit Analytical Model

Optimization Variables

Electromagnetic Analytical Model of coupled Inductors

Thermal Model

Outputs satisfy the

constraints?

Circuit Constraints

Electromagnetic Constraints

Thermal Constraints

Goal function(Converter or inductor Mass Minimization)

Nonlinear Solver

Optimal Values

Initializing

Analytical Model Constraints

Numerical Solver

No

yes

Goal Function

Sending inductor size to FEM software

FEM Analysis

La=Lfem

Generate model correction factor

No

Verified Optimal Values

Yes

IGBT open circuit

IGBT short circuit

Inductor short circuit

Inductor open circuit

Fau

lt c

alc

ula

tio

n

Fault condition model

Figure 6.6 Hybrid optimization algorithm considering the fault margin

6.9. Conclusion

In this chapter, the proposed optimization plans for MMC converter were introduced and

developed. In the first section, the general parts of an optimization loop are introduced. Each

optimization loop is composed of optimization variables, mathematical model, constraints,

goal function and solver. Depending on the application, the structure of optimization

algorithm should be changed.

In this project, the optimization algorithm has chosen with different complexity levels. The

first algorithm employed the converter circuit model to find the optimal circuit value of

passive components. In this algorithm, the mechanical dimensioning of components was

neglected and the design process should be done regarding the circuit values. The goal

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127

function is the total energy stored in the converter which is the summation of electrical energy

stored in the capacitors and the magnetic energy stored in the inductors. The most important

advantages of this approach are the simplicity and fast converging that could be used to find

the best optimal bond of variables. The optimization variables are the sub-module capacitor,

arm inductance, switching frequency and the modulation index.

The second algorithm employed the circuit model, electromagnetic model and thermal model

in the unique shell in combination with the dimensioning model in order to minimize the total

converter volume and mass. Unlike the first algorithm, this algorithm not only calculates the

circuit values but also the dimension parameters of the passive components. Because the high

number of optimization variables and constraints, the complexity and convergence time of

optimization algorithm is higher in comparison with the first algorithm.

The analytical model of arm inductance does not provide high precision results. The accuracy

of the analytical model varies by changing the air gap value. In order to correct the model

parameters and enhance its accuracy, a novel correction loop was presented and utilized in

the optimization loop. In this approach, the analytical model parameters are modified

regarding the finite element analysis results. This approach enhances the model accuracy,

while the optimization time does not increase so much.

Finally, the proposed hybrid optimization algorithm was combined with the fault condition

model of MMC converter. The fault analysis adds a margin to the variables in order to suffer

the fault condition in a small-time period. This margin leads to minimize the converter

damage if a fault happens.

In the next chapter, the optimization results using the proposed optimization algorithms are

presented and discussed. Two power IGBT is used in the optimization in order to investigate

the switch influence.

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128

CHAPTER VII

7 Investigation of Optimization Results

7.1. Introduction

In the previous chapter, several optimization plans with the aim of minimization volume of

MMC converter were explained in details and developed regarding the MMC Active Front

End converter application. The optimization plans were presented with a different level of

complexity. The algorithms with lower complexity have a higher probability to converge.

The result of each algorithm are used by another as a guide to initialize the optimization

variables. This approach increases the probability of finding a solution and converging of

optimization loop.

The first optimization plan works based on the converter circuit model. The proposed circuit

model calculates the time-domain waveform of each electrical variables in a sinusoidal cycle.

In this plan, the mechanical dimensioning of components was neglected. The optimization

algorithm just tries to find the optimal circuit value of passive components. The goal function

is the total energy stored in the converter that is the summation of electric energy stored in

the capacitors and the magnetic energy stored in the inductors. The major constraints are the

capacitor voltage ripple, AC current THD, and the total converter efficiency.

The first optimization plan neglects the component dimensions and the magnetic parameters.

In the second plan, the analytical electromagnetic model of arm inductances and the

dimensioning model of passive components were added to the optimization algorithm. The

electromagnetic model estimates the magnetic variables regarding the core size and winding

parameters. In the optimization variables, the inductance value was replaced by the core and

winding parameters of the inductor. Also, the new constraints concerning the magnetic

analysis and inductor manufacturing were added to the constraint vector. Unlike the first plan

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129

that a criterion was assumed as goal function, in this optimization plan, the dimensioning

model estimates the total converter volume and size using the dimensioning model.

The analytical electromagnetic model of inductor does not provide precise results compare

to the real data. Hence, to enhance the model accuracy, a composition of the analytical model

and finite element method was employed. In this approach, an extra loop was added to the

optimization algorithm. In each iteration, the inductor parameters are sent to the finite

element software and the results are used to modify the analytical model parameters to

enhance the model accuracy. This approach increases the model accuracy while the

optimization time does not increase very much.

As it was mentioned in chapter 2, because of the limit number of IGBT switches in high

power application, the type of IGBT could not be considered as an optimization variable. In

this research, regarding the application, two high power IGBTs with different specifications

were selected to use. The optimization is repeated for each IGBT and the results were

recorded for each of them in order to find the best switch.

In this chapter, the optimization results are investigated and discussed to the complexity of

algorithms. The results of simpler algorithms can be used as the initial values of more

complex algorithms. This solution increases the convergence probability of optimization.

Finally, the sensitivity of converter mass versus several parameters are studied and discussed.

7.2. High power IGBT specifications

As it was mentioned in chapter 2, in high power application the number of IGBTs is not

unlimited. Regarding to the nominal values and load specifications, there are a limited

number of IGBTs to choose. Hence, the IGBT is a discrete variable, which could not be

considered as optimization variables. The best solution is to do optimization using a specific

IGBT and then repeat it with the other ones.

In this dissertation, two high power IGBTs with different specifications from ABB Company

were employed to design the MMC converter [50, 74]. Table 7.1 shows some specifications

of selected IGBTs.

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130

Table 7.1 Technical specifications of high power IGBTs

Parameters 5SNA1500E330305

5SNA 0750G650300

Collector-emitter voltage 3300 V 6500 V

DC collector current 1500 A 750 A

Peak collector current 3000 A 1500 A

Total power dissipation 14700 W 9500 W

Junction operating temperature -50 ℃ to 150 ℃ -50 ℃ to 125 ℃

Short circuit current 6400 A 3400 A

IGBT thermal resistance junction to case 0.0085 K/W 0.011 K/W

Diode thermal resistance junction to case 0.017 K/W 0.021 K/W

IGBT thermal resistance case to heatsink 0.009 K/W 0.009 K/W

Diode thermal resistance case to heatsink 0.018 K/W 0.018 K/W

Dimensions 190×140×38 mm 190×140×48

Turn-on switching energy (𝐸𝑜𝑛) 2150 mJ @125 ℃ 6400 mJ @125 ℃

Turn-off switching energy (𝐸𝑜𝑓𝑓) 2800 mJ @125 ℃ 5300 mJ @125 ℃

IGBT forward resistance 1.5 mΩ 3 mΩ

Diode forward resistance 0.75 mΩ 2.2 mΩ

On state collector/emitter voltage 0.95 V 1.55 V

On state diode voltage 1.125 V 1.9 V

7.3. Optimization results using proposed time-domain circuit

model

In this section, the optimization result using circuit model is investigated. The optimization

has been done for two high power IGBTs that were introduced in the previous section. The

optimization procedure was repeated for a different number of sub-modules and parallel

arms. The number of sub-modules per arm was changed between 3 and 8 and number of

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131

parallel arms was changed between 1 and 3. The optimized capacitor energy stored is shown

in figure 7.1. The capacitor energy stored is the criterion to estimate the total capacitor size

which should be installed. The total capacitor size is almost constant and is independent of

the number of sub-modules and parallel arms. It seems that the capacitor size is dependent

on the voltage ripple, which will be explained in the next section.

Figure 7.1: Electric energy stored in the capacitors versus the number of sub-module per arm

Figure 7.2: Magnetic energy stored in the inductors versus the number of sub-module per arm

Figure 7.2 shows the magnetic energy stored in the inductors which represent the total size

of the inductors. The magnetic energy stored is decreased by increasing the number of sub-

modules. The reduction rate of magnetic energy increases in higher sub-module numbers.

Also, figure 7.3 shows the total energy stored in the converter which is the summation of

electric and magnetic energies.

3 4 5 6 7 8

N=1 103585 100184 105540 102949 106808 99888

N=2 98158 104942 105160 105185 107066 105528

N=3 100318 108120 106945 102918 105662 102333

80000

90000

100000

110000

120000

Ener

gy S

tore

d in

Cap

acit

ors

(J)

Number of Series Modules

3 4 5 6 7 8

N=1 3719 1351 1321 1152 797 799

N=2 3348 1482 1215 902 843 676

N=3 3520 1765 1243 1107 811 830

0

2000

4000

Ener

gy s

tore

d in

Co

up

led

ind

uct

ors

(J

)

Number of Series Modules

Page 160: Optimal Sizing of Modular Multilevel Converters

132

Figure 7.3: Total energy stored in the converter versus the number of sub-module per arm

Figure 7.4: Total converter efficiency versus the number of sub-module per arm

Figure 7.4 shows the total converter efficiency versus the number of series sub-modules. The

total converter efficiency is reduced by increasing the number of sub-modules and parallel

arms. By increasing the number of modules, optimization procedure decreases the switching

frequency to reduce the losses and keep the efficiency higher than 95%. Therefore, by

increasing the number of sub-modules per arm, it is possible to reduce the total converter size

respecting to the efficiency and THD constraints. Figure 7.5 shows the optimal switching

frequency. Increasing the switching frequency leads to reduce the total harmonic distortion

and enhance the converter power quality. It leads to smaller passive filter size and hence the

smaller converter volume. On the other hand, increasing the switching frequency augments

the IGBT switching losses and reduces the converter efficiency. In the constraint vector, the

3 4 5 6 7 8

N=1 107304 101535 106861 104101 107605 108007

N=2 101507 106425 106375 106088 107910 106205

N=3 103839 109885 108189 104025 106474 103164

80000

90000

100000

110000

120000

Tota

l En

ergy

Sto

red

in C

on

vert

er

(J)

Number of Series Modules

3 4 5 6 7 8

N=1 96.99 96.72 96.51 95.93 95.44 95.36

N=2 96.67 96.33 95.4 95.2 95.24 95.25

N=3 96.2 95.45 95.16 95.07 95.06 95.29

94

94.5

95

95.5

96

96.5

97

97.5

Tota

l Eff

icie

ncy

(%)

Number of Series Modules

Page 161: Optimal Sizing of Modular Multilevel Converters

133

total converter efficiency is kept higher than 95%. Therefore, in order to compensate the

switching losses, the switching frequency must be reduced.

Figure 7.5: Optimal switching frequency versus the number of sub-modules per arm

Figure 7.6 presents a better recognition of the effect of capacitor voltage variation on the

capacitor voltage ripple. The contour of capacitor voltage ripple versus capacitor energy

stored and a number of sub-module is shown in figure 7.7 while the arm inductance is 2.3

mH and the switching frequency is 1000 Hz. It shows that the rate of ripple reduction is

decreased by increasing the capacitor energy stored. It means that to achieve lower ripple,

more capacitor volume is needed. Also, it is possible to reduce the ripple by increasing the

number of sub-modules.

Figure 7.6: Sub-module capacitor ripple versus capacitor energy and sub-module number

3 4 5 6 7 8

N=1 1000 994 947 947 947 828

N=3 1000 948 814 696 598 496

N=2 996 949 947 884 650 647

0

200

400

600

800

1000

1200

Swit

chin

g Fr

equ

ency

(H

z)

Number of Series Modules

Page 162: Optimal Sizing of Modular Multilevel Converters

134

Figure 7.8 shows the effect of magnetic coupling between the upper and lower arm

inductances on the line current THD and total efficiency. It shows that the negative coupling

factor enhances the converter efficiency while reduces the power quality. On the other side,

the positive coupling factor leads to lower THD and efficiency.

Figure 7.7: Contour of capacitor ripple versus capacitor energy and sub-module number

a) THD versus coupling

b) Efficiency versus coupling

Figure 7.8: THD and total efficiency versus coupling factor

7.4. Mass Minimization of Modular Multilevel Converter

To minimize the MMC converter mass, the dimensioning model of the passive component

must be added to the optimization algorithm. In addition, in the case of arm inductance, the

electromagnetic model of arm inductance should be considered. The electromagnetic model

Page 163: Optimal Sizing of Modular Multilevel Converters

135

provides the magnetic parameters versus the core and winding size and parameters. Utilizing

these two models, it makes possible to estimate the component mass and change the goal

function to total converter mass.

This optimization has done utilizing two different high power IGBTs from ABB

semiconductor products. The IGBT specifications and technical data were used in the

optimization algorithm to estimate the losses. Two IGBTs with different nominal voltage and

current are employed in this research. The first one has 3.3 kV and 1500 A. regarding the

nominal value of the converter, the IGBT current is suitable but its voltage is less than the

maximum sub-module voltage. Regarding the DC link voltage and the capacitor voltage

value, at least three series sub-modules must be used to endure the converter and sub-module

voltages. The second IGBT has 6.5 kV and 750 A. The nominal current of this IGBT is not

sufficient enough to endure the converter arm current. Hence, at least two parallel branches

must be employed. In term of IGBT voltage, two series sub-modules are sufficient to work

safely. Figures 7.9 and 7.10 show the proposed MMC topologies in the case of using different

IGBTs.

SM1

SMN

SM1

SMN

.

.Ia

R

L

R

L

M

Iau

Vu1

VuN

Vl1

VlN

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VcuN

Vcl1

VclN

SM1

SMN

SM1

SMN

.

.Ib

R

L

R

L

M

Ibu

Vu1

VuN

Vl1

VlN

Ial

SM1

SMN

SM1

SMN

.

.Ic

R

L

R

L

M

Icu

Vu1

VuN

Vl1

VlN

Idc

Vdc/2

Vdc/2

Figure 7.9: MMC topology using 3.3 kV/1500 A IGBT

Page 164: Optimal Sizing of Modular Multilevel Converters

136

SM1

SMN

SM1

SMN

.

.Ia

R

L

R

L

M

Iau

Vu1

VuN

Vl1

VlN

Vcu1

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Vcl1

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SM1

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SM1

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.

.Ib

R

L

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L

M

Ibu

Vu1

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VlN

Ial

SM1

SMN

SM1

SMN

.

.Ic

R

L

R

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M

Icu

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VuN

Vl1

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SM1

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SM1

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.

.Ia

R

L

R

L

Iau

Vu1

VuN

Vl1

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Vcu1

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Ial

SM1

SMN

SM1

SMN

.

.Ib

R

L

R

L

Ibu

Vu1

VuN

Vl1

VlN

SM1

SMN

SM1

SMN

.

.Ib

R

L

R

L

Ibu

Vu1

VuN

Vl1

VlN

Figure 7.10: MMC topology using 6.5 kV/750 A IGBT

7.4.1. Selection of inductor core topology

In chapter 4, two inductor core topologies were introduced and their dimensioning models

were extracted. In the first topology, there is the possibility to adjust the coupling factor,

while in the second one, the coupling is not an independent variable and it depends on the

core parameters. Using the second topology, the coupling factor will be close to the unit

coupling factor.

The circuit analysis of MMC converter shows that the high coupling factor enhances the

converter performance in terms of ripple, THD, and efficiency. Hence, in this application,

the coupling factor must be kept at the maximum to optimize the performance. In order to

maximize the coupling factor in the center leg topology, the middle air gap must be increased.

The best coupling will be obtained by elimination of the center leg. Therefore, we will

achieve the centerless topology.

The main optimization goal in this research is to minimize the total converter mass. Hence,

the inductor mass as the important part of the converter must be minimized. Utilizing the

dimensioning model proposed in chapter 4, the inductor mass using two proposed topologies

was calculated in the same circuit specifications. Table 7.2 shows the total inductor mass of

center leg and centerless topologies. The results show that with the same specifications, the

Page 165: Optimal Sizing of Modular Multilevel Converters

137

centerless topology provides lower total mass. Hence, the optimization will continue using

centerless topology.

In the case of multi-leg topology, the separated inductor core for each leg is used. For

example in the case of two parallel legs, two inductor cores per phase were utilized.

Table 7.2: Comparison of total mass of two different inductor core topologies

Inductance

(mH)

Coupling

factor

Mass -type 1

(Kg)

Mass-type2 (Kg)

1 mH 0.92 354 420

2mH 0.91 630 725

3mH 0.91 1025 1180

7.4.2. Optimization using 3.3KV/1500A IGBT

The optimization algorithm solves the mathematical equations of the analytical model to find

the best value of the optimization variables considering technical and manufacturing

constraints. The number of sub-modules per arm is not considered as an optimization

variable. Hence, the optimization has been repeated for the different sub-modules number.

Figure 7.11 and 7.12 show the optimal arm inductance and sub-module capacitor values,

respectively. The arm inductance value decreases by increasing the number of sub-modules.

By increasing the number of the sub-module, the current harmonic reduces, therefore a

smaller arm inductance is required. On the other hand, the sub-module capacitor value

augments by increasing the number of sub-modules. Connecting the capacitor in series leads

to the smaller equivalent capacitor. Therefore, to achieve the same voltage ripple the

capacitance should be augmented. It is clear that utilizing the coupled inductor strongly

affects the arm inductance and capacitor value and decreases required passive components.

Page 166: Optimal Sizing of Modular Multilevel Converters

138

Figure 7.11: Optimal arm inductance value versus number of sub-modules per arm

Figure 7.12: Optimal sub-module capacitor value versus number of sub-modules per arm

Figure 7.13 shows the arm inductance mass that is computed according to the analytical

model. The inductor mass decreases by increasing the sub-module number, especially in the

lower sub-module numbers. Also, in the case of using coupled inductor, the inductor mass

strongly decreases in comparison to the uncoupled inductor.

Figure 7.14 shows the total sub-module capacitor mass that is calculated regarding the

capacitor mass function that was obtained from the fitting algorithm. The capacitor mass

depends on the capacitance and the capacitor voltage. In this application, according to the

load specifications, the capacitor mass increases on the small sub-module numbers. In this

state, the capacitance augments and therefore the capacitor mass increases. Then, in the

Page 167: Optimal Sizing of Modular Multilevel Converters

139

higher sub-module numbers, the capacitor mass decreases. By increasing the sub-module

number, the capacitor voltage reduced. It leads to reduce the capacitor mass according to the

capacitor mass function. The interesting point in this figure is the effect of coupled inductor

on capacitor mass. The total capacitor mass in the state of using coupled inductor is lower in

comparison to the state of using uncoupled inductors. On the other word, utilizing the coupled

inductors, it is possible to reduce the inductance and capacitor mass at the same time.

Figure 7.13: Optimal total inductor mass versus number of sub-modules per arm

Figure 7.14: Optimal total capacitor mass versus number of sub-modules per arm

Page 168: Optimal Sizing of Modular Multilevel Converters

140

Figure 7.15 shows the total converter mass which consists of inductance, capacitor and

semiconductor mass. The total converter mass decreases by increasing the sub-modules

number. Also, it is clear that utilizing the coupled inductor strongly reduces the total

converter mass.

Figure 7.15: Optimal converter mass versus number of sub-modules per arm

7.4.3. Optimization using 6.5KV/750A IGBT

The next optimization has done using 6.5KV/750A IGBT. Regarding the IGBT

specifications, the MMC converter must have at least two parallel branches in each arm.

Hence, the arm current is divided by two. The IGBTs current and inductor current are almost

divided by two. It affects the capacitor and inductor values. Also, the losses parameters of

this IGBT is different which affects the switch losses and converter efficiency.

Regarding IGBT voltage, the optimization is started with two sub-modules per arm and two

parallel branches in each arm. The main constraints are the capacitor voltage ripple less than

20%, the THD current less than 2% and Δ𝑇 less than 60℃. Figure 7.16 shows the optimal

arm inductance value versus the number of series sub-modules per arm in the case of using

coupled and non-coupled inductors. The arm inductance value strongly reduces by increasing

the number of sub-modules. Also, the reduction rate of inductance versus the sub-module

number decreases in high sub-module numbers. In comparison with the results of pervious

IGBT, the arm inductance value is higher. All parts of arm current are divided by two except

Page 169: Optimal Sizing of Modular Multilevel Converters

141

the second order part of circulation current. It leads to bigger arm inductance value to

eliminate the harmonics and provide the same converter power quality.

Figure 7.16 The optimal arm inductance versus the number of sub-modules per arm

Figure 7.17 shows the optimal sub-module capacitor values versus the number of sub-

modules per arm. The capacitor value augments by increasing the sub-module numbers. The

increase rate of the capacitor is almost linear while utilizing of coupled inductance decreases

the capacitor value and enhances the converter performance. Due to decreasing the sub-

module current, the capacitor ripple will decrease. Hence, the sub-module capacitor will be

lower compared to the previous IGBT.

Figure 7.17 Optimal value of the sub-module capacitor versus the number of sub-modules per arm

Page 170: Optimal Sizing of Modular Multilevel Converters

142

The investigation of circuit value of the components is not sufficient to explain the

component mass. Figure 7.18 shows the total inductor mass versus the number of sub-

modules per arm in the case of coupled and non-coupled inductor. Increasing the number of

sub-modules per arm leads to decrease the inductor mass. Also, utilization of coupled

inductor provides lower inductor mass.

Figure 7.18 Total inductor mass versus the number of sub-modules per arm

Figure 7.19 Total capacitor mass versus the number of sub-modules per arm

Figure 7.19 shows the total sub-module capacitor mass versus the number of sub-modules

per arm. The capacitor mass depends on the capacitance and capacitor voltage. As it was

1159

612

435359

314202 213

191 171

1470

962

650

511 467405 426 408 404

0

200

400

600

800

1000

1200

1400

1600

2 3 4 5 6 7 8 9 10

Ma

ss (

Kg

)

Number of submodules per arm

Total Inductor mass

coupled

uncoupled

379401

505439

384

298

201150

140

401

480

591 591545

508

380

280235

0

100

200

300

400

500

600

700

2 3 4 5 6 7 8 9 10

Ma

ss (

Kg

)

Number of submodules per arm

Total Capacitor mass

coupled

uncoupled

Page 171: Optimal Sizing of Modular Multilevel Converters

143

shown in figure 7.19, the variation of capacitor mass is not linear. The capacitance value and

capacitor voltage do not increase in the same direction. Hence, first, the capacitor mass

increases due to increase the capacitance and then it reduces due to voltage reduction.

The total converter mass includes the inductor mass, the sub-module capacitor mass, and

IGBTs. Figure 7.20 shows the total converter mass versus the number of sub-modules per

arm. The curve of total converter mass is a descending curve versus the number of sub-

modules per arm. The total converter mass is higher compared to the previous optimization.

Figure 7.20 Total converter mass versus the number of sub-modules per arm

7.5. Optimization Results using Hybrid Analytical Model

In chapter 6, the proposed hybrid analytical model of arm inductance and the proposed

optimization algorithm using hybrid model was introduced and investigated. The hybrid

model is an innovative approach to enhance the model accuracy while the optimization time

does not increase very much. In this section, the optimization results using hybrid

optimization model are explained and discussed. Also, the effect of some important

constraints such as temperature rise, flux density, THD and voltage ripple on converter mass

are investigated.

1592

10651003

943 872703

607 602 601

1929

1528

1301 1247 1178 1116970 949 929

0

500

1000

1500

2000

2500

2 3 4 5 6 7 8 9 10

Ma

ss (

Kg

)

Number of submodules per arm

Total Converter mass

coupled

uncoupled

Page 172: Optimal Sizing of Modular Multilevel Converters

144

7.5.1. Optimization results using 3.3 kV/1500 A IGBT

The optimization results using the proposed hybrid algorithm are shown in Figures 7.21 to

7.26. The nominal values and constraints are the same as previous analysis. Figure 7.21 and

7.22 show the optimal inductance and capacitor values in states of utilizing coupled and

uncoupled inductors versus a number of series sub-modules per arm respectively. In both of

them, the use of coupled inductors reduces the required inductance and capacitor values

which may lead to lower converter cost. By increasing the number of sub-modules, the

inductance value reduces while capacitor value is increasing. Also, it should be considered

that the reduction rate of inductance is more in lower sub-module numbers while the rate of

increase of capacitor value will be higher by rising the number of series sub-modules. It

means the midpoints could be the desired points.

Figure 7.21: The optimal arm inductance value versus the sub-modules per arm

4.8

3.062.62

2.191.83

1.35 1.2 1.05

2.56

0.806 0.801 0.659 0.496 0.49 0.46 0.45

0

1

2

3

4

5

6

3 4 5 6 7 8 9 10

Ind

uct

ance

valu

e (m

H)

Number of sub-modules per arm

Arm inductance value

uncoupled

coupled

Page 173: Optimal Sizing of Modular Multilevel Converters

145

Figure 7.22: The optimal sub-module capacitor value versus the sub-modules per arm

The electromagnetic model of inductor provides a mathematical relation between the

inductance value and its physical size. The hybrid optimization algorithm uses an analytical

model to calculate and finite element analysis to verify and modify the model and finally, it

estimates the physical size of the optimal inductor. In the case of a capacitor, the final mass

depends on the type of capacitor and its technical data which provides by manufacturing

company. Utilizing the capacitor value and maximum voltage, it is possible to find a

mathematical function which estimates the capacitor mass dependent on the capacitance and

voltage.

Figure 7.23 and 7.24 show the estimated inductor and capacitor mass versus the number of

sub-modules per arm which were calculated using hybrid optimization algorithm. The curve

of inductor mass is descending the same as the inductor value in figure 7.21. Also, the

inductor mass reduces in presence of coupled inductors. Figure 7.23 shows the capacitor

mass which is estimated based on capacitor datasheet. The curve of capacitor mass is

different from the curve of capacitor value in figure 7.22. First, the capacitor voltage is high

and capacitance is low. As a result, capacitor mass is low. By increasing the number of the

sub-module, the voltage reduces while capacitance increases. The variation of capacitor mass

depends on the mass function that is provided by the manufacturer. In this case, it leads to

increase the capacitor mass. In continue, the capacitor value increases while the capacitor

4.546.07

7.75

9.53

11.5

14.1

16.718.2

3.134.2

5.656.65

7.828.89

10.111.25

0

4

8

12

16

20

3 4 5 6 7 8 9 10

cap

acit

or

valu

e (

mF)

Number of sub-modules per arm

Sub-module capacitor value

uncoupled

coupled

Page 174: Optimal Sizing of Modular Multilevel Converters

146

voltage effectively reduces by increasing the number of sub-modules. Therefore, the effect

of voltage overcomes the capacitor value and the capacitor mass decreases.

Figure 7.23: The optimal arm inductance mass versus the sub-modules per arm

Figure 7.24: The optimal capacitor mass versus the sub-modules per arm

1090

699603

520453

396 385 374

726

227 192 159 144 133 124 118

0

200

400

600

800

1000

1200

3 4 5 6 7 8 9 10

Mas

s (K

g)

Number of sub-modules per arm

Total inductor mass

uncoupled

coupled

518

579605 590

539

480

410 388354

393427

384

318

213172

145

0

100

200

300

400

500

600

700

3 4 5 6 7 8 9 10

Mas

s (K

g)

Number of sub-modules per arm

Total capacitor mass

uncoupled

coupled

Page 175: Optimal Sizing of Modular Multilevel Converters

147

Figure 7.25: The optimal converter mass versus the sub-modules per arm

Figure 7.26: Total converter efficiency versus the sub-modules per arm

According to the figure 7.23 and 7.24, by increasing the number of the sub-module, it is

possible to reduce the total converter mass. Figure 7.25 shows the total converter mass which

includes capacitor, inductor and IGBT mass. It is clear that the converter mass is reduced by

increasing the number of sub-module per arm. Also, utilizing coupled inductor intensely

affects and reduces the converter mass. It should be noted that the total converter efficiency

1652

1336 12811197

1094992 926 907

1124

678 692 630 564462 427 408

0

400

800

1200

1600

2000

3 4 5 6 7 8 9 10

Mas

s (K

g)

Number of sub-modules per arm

Total converter mass

uncoupled

coupled

97.68

97.05

96.4

95.73

95.04 95.11 95 95.01

97.897.44

96.74

96.19

95.59

95.01 95 95

93.5

94

94.5

95

95.5

96

96.5

97

97.5

98

98.5

3 4 5 6 7 8 9 10

Effi

cien

cy (%

)

Number of sub-modules per arm

Total converter efficiency

uncoupled

coupled

Page 176: Optimal Sizing of Modular Multilevel Converters

148

is reduced too. Figure 7.26 shows the total converter efficiency in case of using coupled and

uncoupled inductors. Total converter efficiency is dependent on IGBT losses and inductor

losses. The switch losses increase and as a result, the efficiency is reduced by increasing the

number of sub-modules and IGBTs. However, the total efficiency could be mentioned as a

constraint to find the minimum converter mass. In this research, the total efficiency should

be kept greater than 95%.

7.5.2. Optimization results using 6.5 kV/750 A IGBT

This section represents the optimization result using hybrid optimization algorithm and

second type IGBT (6.5 kV/750 A). Utilizing the 6.5 kV IGBT, it is possible to decrease the

number of sub-modules to two sub-modules. Due to the nominal current of the converter, at

least two parallel branches must be employed in each arm.

Figure 7.27 Total inductor mass versus the number of sub-modules per arm

Figure 7.27 shows the total inductor mass versus the number of sub-modules per arm in term

of using coupled and non-coupled inductors. The value of total inductor mass is lower

compared to the optimization using the analytical model. The correction loop strongly

enhances the accuracy of the analytical model. Also, it should be considered that the coupled

inductor intensely reduces the total inductor mass.

1095

565

335295

242177 185 165 162

1320

862

598

445372

315 335 303 285

0

200

400

600

800

1000

1200

1400

2 3 4 5 6 7 8 9 10

Mas

s (K

g)

Number of sub-modules per arm

Total Inductor mass

coupled

uncoupled

Page 177: Optimal Sizing of Modular Multilevel Converters

149

Figure 7.28 shows the total sub-module capacitor mass versus the number of sub-modules

per arm that was computed using hybrid optimization algorithm. Unlike the inductor mass,

the capacitor mass does not change very much compared to the optimization results using

analytical model.

Figure 7.28 The total sub-module capacitor mass versus the number of sub-module per arm

Figure 7.29 shows the total converter mass that is the summation of inductor mass, capacitor

mass, and IGBT mass. The total converter mass is significantly higher in a low number of

series sub-modules than the same condition using another IGBT. By increasing the number

of sub-modules, the first IGBT provides lower converter mass. Also, it is clear that utilizing

of coupled inductors effectively reduce the converter size and enhance the converter

performance.

Figure 7.30 shows the total converter efficiency versus the number of sub-modules per arm

in terms of using coupled and non-coupled inductors. By increasing the number of sub-

modules, the switch losses will increase and the total converter efficiency is reduced. Hence,

the optimization algorithm decreases the switching frequency to keep the efficiency more

than 95%.

350

398

500

421382

304

195158 142

392

485

598 589543 528

372

286242

0

100

200

300

400

500

600

700

2 3 4 5 6 7 8 9 10

Mas

s (K

g)

Number of sub-modules per arm

Total Capacitor mass

coupled

uncoupled

Page 178: Optimal Sizing of Modular Multilevel Converters

150

Figure 7.29 Total converter mass versus the number of sub-modules per arm

Figure 7.30 Total converter efficiency versus the number of sub-modules per arm

7.6. Parameter sensitivity analysis

Sensitivity analysis is a type of study that investigates and computes the goal function

sensitivity versus the different design parameters. This study provides very interesting results

15031050

951861

798 684612 584 594

1770

14341312

11791089 1046

939850 817

0

200

400

600

800

1000

1200

1400

1600

1800

2000

2 3 4 5 6 7 8 9 10

Mas

s (K

g)

Number of sub-modules per arm

Total Converter mass

coupled

uncoupled

97.88

97.12

96.22

95.3995 95 95 95 95

97.84 97.24

96.51

95.74

95 95 95 95 95

93.5

94

94.5

95

95.5

96

96.5

97

97.5

98

98.5

2 3 4 5 6 7 8 9 10

Effi

cien

cy (%

)

Number of sub-modules per arm

Total converter efficiency

uncoupled

coupled

Page 179: Optimal Sizing of Modular Multilevel Converters

151

that could use in the design process. In the case of MMC converter, especially in high power

applications, designers are confronted by many parameters and variables which make

complex to select the appropriate parameters to design. The sensitivity study presents the

sensitivity of goal function for one parameter while others are constant.

In this section, the sensitivity of total converter mass function against some important

parameters such as maximum temperature rise, maximum core flux density, maximum THD,

maximum capacitor voltage ripple, fault margin and inductor core material are investigated

and discussed.

7.6.1. Sensitivity analysis of maximum Temperature Rise

In the single objective optimization algorithm, there is a goal function and number of

constraints. In this research, the main goal function is to minimize the total converter mass.

Other functions such as temperature rise, capacitor voltage ripple, and THD are considered

as constraints. Therefore, the objective sensitivity versus constraints variation should be

investigated. Table 7.3 shows the converter parameters that are used in optimization.

Table 7.3: Converter parameter and thermal coefficient

Parameters Value Unit

Number of sub-modules 3-8

DC Link Voltage 5000 V

Capacitor voltage ripple 0.2𝑉𝑑𝑐/𝑚 V

Copper thermal exchange 80 𝑊/𝑚2𝐾

Iron thermal exchange 80 𝑊/𝑚2𝐾

Ambient temperature 40 ℃

Modulation Index 65 %

One of the most important constraints which intensely affects the converter mass is the

maximum temperature rise. The optimization has been done by changing the maximum

temperature rise as a constraint in the different sub-module number per arm. Figure 7.31

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152

shows the optimal inductor value versus the maximum temperature rise. Increasing the

maximum temperature rise intensely reduces the inductor mass.

Figure 7.31: Optimal inductor mass versus maximum temperature rise

Also, the total converter mass versus the maximum temperature rise is shown in figure 7.32.

The optimal mass converter is reduced by increasing the maximum temperature rise. The

high value of the capacitor mass in comparison with the inductor mass leads to move the

results versus the sub-module numbers.

Figure 7.32: Total converter mass versus the maximum temperature rise

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153

7.6.2. Investigation the effect of maximum Flux Density on Converter

Mass

One of the most important constraints of the arm inductance is the maximum core flux

density. The maximum flux density is dependent on the material and its B-H characteristic.

Regarding the saturation effect, power electronic designers try to keep the operating point in

the linear region. The silicon electrical steel which is used to design the electrical machine

and transformer is in the linear region until 0.9T and in the saturation region until 1.8T.

Increasing the maximum flux density of the core might lead to reducing the inductor mass

while it might lead to increase the core losses. Figure 7.33 shows the contour of optimal

inductor mass versus the sub-module number and the maximum flux density of the core. It

is clear that the inductor mass is reducing while the maximum flux density is increased.

Therefore, it proves the effect of maximum flux density in reducing the arm inductance mass.

Figure 7.33: The contour of optimal inductor mass versus the sub-modules number and the

maximum flux density

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154

7.6.3. Investigation the effect of maximum THD on Converter Mass

Finding the relation between the component mass and the converter THD is dependent on

lots of parameters such as passive component values, circulating current, switching

frequency, the number of sub-module per arm, switching index, switch characteristics and

etc. Hence, extracting the relation between THD and converter mass will be more complex

than other constraints. Regarding the circulating current formula (Eq. 3.25) which was

investigated in chapter 3, shows that there is a point of discontinuity that could lead to

augment the circulating current. The circulating current is composed of the second harmonic

which increases the THD value. The point of discontinuity is dependent on the switching

index, capacitor value, self-inductance and mutual inductance and the number of sub-module

per arm. Figure 7.34 shows the point of discontinuity for a different number of sub-modules.

The value of arm inductance and sub-module capacitor must be chosen to avoid this point.

Figure 7.34 The discontinuity value versus number of sub-modules per arm

Figure 7.35 shows the THD value of three phase current versus the arm inductance value and

number of sub-module per arm. It is clear that the THD value reduces by increasing the arm

inductance. Also, the THD value is smaller in the higher sub-modules per arm. But it does

not sufficient to be sure about the performance of the converter. The inductor mass should

be investigated to avoid the point of discontinuity.

0.00E+00

4.00E-06

8.00E-06

1.20E-05

1.60E-05

2.00E-05

3 4 5 6 7 8 9 10

(L-M

)Csm

The number of sub-modules per arm

The point of discontinuty (A=1)

Page 183: Optimal Sizing of Modular Multilevel Converters

155

Figure 7.35 The THD value of input current versus the arm inductance value

Figure 7.36 shows the total inductor mass versus the arm inductance and sub-module number

per arm. Theoretically, the inductor mass should be reduced by decreasing the arm inductance

value if they work at the same operating point. But, in the case of MMC converter, the

circulating current value could affect the inductor mass. If 𝐴 ≫ 1 𝑜𝑟 𝐴 ≪ 1 then the inductor

mass is dependent on the inductance value. The results show that the inductor mass reduces

by decreasing the arm inductance except one for point. At 𝑚 = 3 when the arm inductance

becomes smaller than 1.5 𝑚𝐻 the inductor mass augments. This is the effect of discontinuity

which strongly increases the circulating current.

0

2

4

6

8

10

12

1 1.5 2 2.5 3 3.5 4 4.5

THD

(%

)

Arm inductance value (mH)

THD of input current versus inductance value

m=3 m=4

m=5 m=6

m=7 m=8

Page 184: Optimal Sizing of Modular Multilevel Converters

156

Figure 7.36 The total inductor mass versus the arm inductance value

7.6.4. Investigation of the effect of Capacitor Voltage Ripple on Converter

Mass

The constraint that strongly affects the capacitor mass is the voltage ripple of the sub-module

capacitors. To enhance the converter power quality and the capacitor safety, the capacitor

ripple should be minimized. The voltage ripple reduction augments the capacitor mass

regarding the capacitor mass function. Figure 7.37 shows the optimal total capacitor mass

versus the voltage ripple percentage of sub-module capacitors. The capacitor mass

augmentation is not linear. The capacitor mass and the final converter price intensely

increases to achieve the low voltage ripple rates. Hence, the constraints must be selected

carefully to achieve a suitable balance between the converter quality and the final price.

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5

ind

uct

ance

Mas

s (k

g)

Arm inductance (mH)

The Inductor mass versus arm inductance value

m=3 m=4 m=5

m=6 m=7 m=8

Page 185: Optimal Sizing of Modular Multilevel Converters

157

Figure 7.37: The total converter mass versus the capacitor ripple

7.6.5. Sensitivity analysis converter mass against Fault margin

In chapter 5, the fault margin of MMC converter and the proposed optimization algorithm

that considers the fault margin was investigated and explained. The fault margin guarantees

the functionality of the components in a transient fault condition. The fault margin is defined

based on a percentage of nominal converter values. Various types of fault affect different

variables such as capacitor voltage, inductor current and IGBT’s voltage/current. The fault

model computes all variables in a fault condition and it computes the converter mass

regarding the fault values and fault margin.

Utilizing the proposed optimization model with fault margin block, the total converter mass

is computed versus the number of sub-modules per arm and different fault margin value. The

optimization has done using two different IGBTs. Figure 7.38 shows the total converter mass

using 3.3 kV IGBT versus the number of sub-modules per arm and different fault margins.

Figure 7.39 shows the same results for 6.5 kV IGBT. The results show that the converter

mass sensitivity is higher in a lower number of sub-modules. Increasing the number of sub-

modules reduces the converter mass sensitivity to the fault condition. Hence, to increase the

converter stability and reliability, the number of sub-modules should be increased.

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158

Figure 7.38 Total converter mass sensitivity against fault margin for 3.3KV IGBT

Figure 7.39 Total converter mass sensitivity against fault margin for 6.5KV IGBT

Page 187: Optimal Sizing of Modular Multilevel Converters

159

7.7. Conclusion

In the previous chapter, several global optimization algorithms were proposed and explained in order

to minimize the MMC converter volume and mass. The optimization algorithms were presented with

a different level of complexity. The first optimization model employs the time-domain circuit model

to compute the circuit value of passive components and minimize the total energy stored in the

converter. In the second optimization algorithm, the electromagnetic model of arm inductance and

dimensioning model of passive components were added to the optimization algorithm. The final

optimization algorithm has an internal correction loop using finite element approach in order to

enhance the electromagnetic model accuracy. Also, the fault margin calculation block was added to

the optimization algorithm to compute the components parameters based on the fault condition.

In this chapter, the optimization results of MMC converter using variously proposed optimization

algorithm were presented, investigated and discussed. The optimization has done using two

high power IGBT with different nominal values. The optimization results show that by

increasing the number of sub-modules per arm, the total inductor mass is reduced. The total

capacitor mass depends on the capacitor mass function that is provided by the manufacturer.

The total converter mass which is consisted of inductor mass, capacitor mass and IGBTs is

reduced by increasing the sub-module number per arm.

Also, the optimization has done in terms of utilizing coupled and non-coupled inductor. The

results prove that utilization of coupled inductors strongly increases the MMC performance

and reduce the inductor and converter mass. Another interesting point is the effect of coupled

inductors on the sub-module capacitor value. The lower sub-module capacitor is needed

when the coupled inductors are employed. Finally, a parameter sensitivity analysis has done

to determine the effects of several parameters on the total converter mass.

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160

CHAPTER VIII

8 Conclusion and Future Researches

8.1. Conclusion

In the field of electrical energy conversion, Modular Multilevel Converter (MMC) have

emerged in the recent years as an attractive solution for high/medium power and voltage

applications. These topologies of power electronics converters can now be used up to

900MW and higher in AC/DC & DC/AC applications like High-Voltage Direct Current

(HVDC) light transmissions systems, Flexible Alternating Current Transmissions Systems

(FACTS), hydro pumped storage, wind energy conversion, marine propulsion and railway

traction drives.

The MMC topologies are composed of elementary commutation cells using standard

Integrated Gate-Commutated Thyristor (IGCT) or insulated gate bipolar transistors (IGBT)

associated with capacitors. They are easily scalable to be adapted to different voltage and

current levels. Their modularity can be used to improve the redundancy and the fault-tolerant

operation, and to implement the multilevel control concept to provide a specified AC

waveform quality with a reduced size of filtering magnetic components.

The main objective of this research project is to propose a generic and versatile sizing

methodology of MMC converters based on a global optimization approach constrained by

the specifications of each application. With such an approach, it is possible to a fixed set of

input-output specifications of the application to compare the performances of different MMC

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161

topologies in terms of converter volume and weight including passive (magnetic &

capacitive) components, cell cooling and insulation system, the number of cells and switches,

global efficiency and power quality. The methodology will be validated on different

applications including a high pulsed power supply for particle accelerator electromagnets

with an active front-end.

In this dissertation, a systematic optimization approach has been proposed and developed in

order to minimize high power modular multilevel converters considering the technical,

thermal and manufacturing constraints. Capacitors volume, arm inductances size and a

number of sub-modules per arm are the most important parameters which affect the final

converter mass. In the case of variable load converters, the steady-state model is not

appropriate to analyze the converter. Hence, the transient converter model should be

employed to analyze MMC circuit. The steady-state model is utilized in the initializing step.

The analytical model of the converter and coupled inductors are utilized as a part of

optimization algorithm to find the capacitor value, arm inductance value, and size and a

number of sub-modules per arm. The electromagnetic core has saturation characteristic in

reality. The electromagnetic analytical model accuracy is not sufficient enough in

comparison with the real data to present the saturation effect. On the other side, utilization of

complex model such as finite element model in the optimization loop is time-consuming. In

this research, a novel hybrid optimization algorithm by a combination of analytical model

and finite element model has been proposed to investigate the saturation effect of

electromagnetic core and enhance the model preciseness. The proposed hybrid optimization

algorithm was employed to optimize the high power Active-Front-End converter for the

application. The result proves that couple inductors reduce the converter mass and improve

the total performance. Also, according to the possibility of core saturation, the performance

of MMC converter has been evaluated in a fault condition.

Chapter 1 presented a brief history of the modular multilevel converter. Some of the

important researches on the field of MMC converter has presented and studied. In the

literature, MMC converters were investigated from different points of view. Literature

investigation proved that the global optimization of MMC converter is a subject that was

rarely investigated in the previous works. Then the most important types and topologies were

introduced and their advantages and limitations were discussed. The structure of neutral point

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162

converter, flying capacitor, and modular converter were investigated. Also, MMC converters

were sorted base on its applications. Finally, the goals of this dissertation were explained and

the methodology and approach which should be employed were introduced.

In chapter 2, the systematic design procedure of MMC converter was presented and reviewed.

First of all, the important parameters and variables of MMC converter were determined. The

adjustable MMC parameters and variables were discussed and explained. Various analysis

tools that are needed to investigate the MMC structure are introduced. The conventional

approach which is used to design and size the MMC components were investigated. Finally,

several integrated analysis models with different level of complexity which consisted of

circuit model, electromagnetic model, thermal model, and dimensioning model were

proposed and presented in order to global analysis of MMC converter.

Chapter 3 presented a modified circuit model of MMC converter which is suitable to estimate

time-dependent variables such as ripple and THD. The conventional average model of MMC

converter neglects the effect of switching frequency and uses the average switching function

to calculate the steady-state values. A time-domain circuit model was proposed and

developed in order to calculate the precise value of time-dependent parameters such as

capacitor voltage ripple, circulating current and total harmonic distortion. The accuracy of

the proposed model was verified using Simulink/Matlab.

The circuit model analyzes the arm inductances as circuit component and neglects the

magnetic and dimensioning variables. Chapter 4 presented the analytical electromagnetic

model of arm inductances. The electromagnetic model estimates the magnetic variables such

as magnetic flux, inductance value, reluctance and magnetic losses regarding the core and

winding parameters. In addition, the magnetic and circuit model of saturable inductance in

the case of coupled and non-coupled inductors. Utilizing the analytical models, the

dimensioning model of inductor and capacitors were extracted. Also, the thermal model of

arm inductances and semiconductor switches were presented in chapter 4. Finally, the finite

element analysis method was presented in order to achieve more precise results in arm

inductance analysis.

In chapter 5, the MMC converter in the fault condition was investigated. In high power, it is

very important to control and protect the system against the faults in order to minimize the

damage and cost. Hence, the converter should be designed with a fault margin to increase the

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163

component capability to endure in the fault condition for a short time. In chapter 5, various

standard MMC converter faults were presented and investigated. Also, the behavior of

different converter variables in the fault condition was studied. Finally, a close loop control

system was proposed and simulated to control the converter in the fault condition.

In chapter 6, the most important parts of an optimization algorithm were presented and

investigated. Various strategies of MMC converter optimization were proposed and

developed. The proposed optimization algorithms were sorted regarding its complexity level.

The first optimization algorithm employs the time-domain circuit model to minimize the total

energy stored in the converter. It neglects the dimension of passive components. The second

optimization model utilizes the circuit model in combination with the electromagnetic model

of arm inductances and dimensioning model of passive components. This optimization

algorithm minimizes the total converter mass and searches to find the optimal size of

capacitor and inductor. The last optimization model utilizes an internal correction loop using

finite element method in order to correct the analytical model parameters based on the finite

element analysis results. It is an innovative approach which leads to enhance the model

accuracy while the optimization time does not increase very much.

In chapter 7, the optimization results of the various proposed optimization algorithms were

presented, investigated and discussed. The total inductor mass, capacitor mass, and converter

mass were calculated versus the number of sub-modules per arm in terms of using coupled

and non-coupled inductors. The THD of AC current, capacitor voltage ripple, and total

converter efficiency were considered as the constraints. Finally, the sensitivity of total

converter mass versus the variation of several variables was investigated.

8.2. Future Researches

The climate change and global warming increase the tendency to employ the renewable

energies such as the wind and solar generation. In the last decade, demands too high power

electronic converters have intensely augmented in order to use in renewable generation,

electric railways, HVDC and other power applications. Emerging the modular multilevel

converters increased the tendency to employ high power converters which provide the high

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164

global efficiency and power quality. MMC converter provides high power quality using low

switching frequency that leads to low switching losses and high converter efficiency.

Regarding the high number of components, MMC converters have two weakness in

comparison to the conventional converters; high volume of passive components and high

possibility to enter to fault condition due to the number of components. By increasing the

converter voltage, the number of sub-modules an as result, the number of components will

intensely increase. It leads to increase the size of the converter and made it so expensive. On

the other hand, increasing the number of components augments the chance of fault and

malfunction in the converter. It reduces the reliability of the converter.

As a future activity, it is recommended to investigate the fault tolerant optimization that

considers a margin for each component to resist under the fault condition. Regarding the

modularity of MMC converter, it will lead to minimizing the converter maintenance and

repairing cost. Also, according to the new information, the appropriate protection setting

could be chosen for each section.

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REFERENCES

[1] J. Rodriguez, J.-S. Lai, and F. Z. Peng, "Multilevel inverters: a survey of topologies,

controls, and applications," IEEE Transactions on industrial electronics, vol. 49, no.

4, pp. 724-738, 2002.

[2] S. Kouro et al., "Recent advances and industrial applications of multilevel

converters," IEEE Transactions on industrial electronics, vol. 57, no. 8, pp. 2553-

2580, 2010.

[3] V. Dargahi, A. K. Sadigh, M. Abarzadeh, S. Eskandari, and K. A. Corzine, "A new

family of modular multilevel converter based on modified flying-capacitor multicell

converters," IEEE Transactions on Power Electronics, vol. 30, no. 1, pp. 138-147,

2015.

[4] M. Z. Youssef, K. Woronowicz, K. Aditya, N. A. Azeez, and S. S. Williamson,

"Design and development of an efficient multilevel DC/AC traction inverter for

railway transportation electrification," IEEE Transactions on Power Electronics, vol.

31, no. 4, pp. 3036-3042, 2016.

[5] D. Montesinos-Miracle, M. Massot-Campos, J. Bergas-Jane, S. Galceran-Arellano,

and A. Rufer, "Design and control of a modular multilevel DC/DC converter for

regenerative applications," IEEE transactions on power electronics, vol. 28, no. 8,

pp. 3970-3979, 2013.

[6] M. Saeedifard and R. Iravani, "Dynamic performance of a modular multilevel back-

to-back HVDC system," IEEE Transactions on power delivery, vol. 25, no. 4, pp.

2903-2912, 2010.

[7] M. Hagiwara, K. Nishimura, and H. Akagi, "A medium-voltage motor drive with a

modular multilevel PWM inverter," IEEE Transactions on Power Electronics, vol.

25, no. 7, pp. 1786-1799, 2010.

[8] A. Nabae, I. Takahashi, and H. Akagi, "A new neutral-point-clamped PWM inverter,"

IEEE Transactions on industry applications, no. 5, pp. 518-523, 1981.

[9] P. W. Hammond, "A new approach to enhance power quality for medium voltage

drives," in Petroleum and Chemical Industry Conference, 1995. Record of

Conference Papers., Industry Applications Society 42nd Annual, 1995, pp. 231-235.

[10] T. A. Meynard, H. Foch, P. Thomas, J. Courault, R. Jakob, and M. Nahrstaedt,

"Multicell converters: basic concepts and industry applications," IEEE transactions

on industrial electronics, vol. 49, no. 5, pp. 955-964, 2002.

[11] A. Lesnicar and R. Marquardt, "An innovative modular multilevel converter topology

suitable for a wide power range," in Power Tech Conference Proceedings, 2003 IEEE

Bologna, 2003, vol. 3, p. 6 pp. Vol. 3.

[12] M. Fioretto, G. Rubino, L. Rubino, N. Serbia, P. Ladoux, and P. Marino, "The

efficiency in interleaved structures based on VSI topologies," in AFRICON, 2013,

2013, pp. 1-5.

Page 194: Optimal Sizing of Modular Multilevel Converters

166

[13] J.-S. Lai and F. Z. Peng, "Multilevel converters-a new breed of power converters,"

IEEE Transactions on industry applications, vol. 32, no. 3, pp. 509-517, 1996.

[14] M. F. Aiello, P. W. Hammond, and M. Rastogi, "Modular multi-level adjustable

supply with parallel connected active inputs," ed: Google Patents, 2001.

[15] L. M. Tolbert, F. Z. Peng, and T. G. Habetler, "Multilevel converters for large electric

drives," IEEE Transactions on Industry Applications, vol. 35, no. 1, pp. 36-44, 1999.

[16] L. M. Tolbert and F. Z. Peng, "Multilevel converters as a utility interface for

renewable energy systems," in Power Engineering Society Summer Meeting, 2000.

IEEE, 2000, vol. 2, pp. 1271-1274.

[17] L. M. Tolbert, F. Z. Peng, and T. G. Habetler, "A multilevel converter-based universal

power conditioner," IEEE Transactions on Industry Applications, vol. 36, no. 2, pp.

596-603, 2000.

[18] L. M. Tolbert, F. Z. Peng, and T. G. Habetler, "Multilevel inverters for electric vehicle

applications," in Power Electronics in Transportation, 1998, 1998, pp. 79-84.

[19] A. Laka, J. A. Barrena, J. Chivite-Zabalza, and M. RODRIGUEZ-VIDAL,

"Parallelization of two three-phase converters by using coupled inductors built on a

single magnetic core," Przegląd Elektrotechniczny, vol. 89, no. 3a, pp. 194--198,

2013.

[20] R. Hausmann and I. Barbi, "Three-phase multilevel bidirectional DC-AC converter

using three-phase coupled inductors," in 2009 IEEE Energy Conversion Congress

and Exposition, 2009, pp. 2160-2167.

[21] K. Guépratte, P.-O. Jeannin, D. Frey, and H. Stephan, "High efficiency interleaved

power electronics converter for wide operating power range," in Applied Power

Electronics Conference and Exposition, 2009. APEC 2009. Twenty-Fourth Annual

IEEE, 2009, pp. 413-419.

[22] P.-L. Wong, P. Xu, P. Yang, and F. C. Lee, "Performance improvements of

interleaving VRMs with coupling inductors," IEEE Transactions on Power

Electronics, vol. 16, no. 4, pp. 499-507, 2001.

[23] S. Energy. (2011). Living Energy - Siemens Debuts HVDC Plus with San

Fransisco’s Trans Bay Cable.

[24] ABB. (2011). ABB HVDC Reference projects.

[25] A. Nordvall, "Multilevel inverter topology survey," 2011.

[26] R. A. Ahmed, S. Mekhilef, and H. W. Ping, "New multilevel inverter topology with

a minimum number of switches," in TENCON 2010-2010 IEEE Region 10

Conference, 2010, pp. 1862-1867.

[27] E. N. Abildgaard and M. Molinas, "Modelling and control of the modular multilevel

converter (MMC)," Energy Procedia, vol. 20, pp. 227-236, 2012.

[28] E. Ahmed and S. Yuvarajan, "Hybrid renewable energy system using DFIG and

multilevel inverter," in Green Technologies Conference, 2012 IEEE, 2012, pp. 1-6.

[29] S. Debnath and M. Saeedifard, "A new hybrid modular multilevel converter for grid

connection of large wind turbines," IEEE Transactions on Sustainable Energy, vol.

4, no. 4, pp. 1051-1064, 2013.

[30] K. R. Chakravarthi and S. G. Basha, "Design & Simulation of 11-level Cascaded H-

bridge Grid-tied Inverter for the application of Solar Panels," IJSEAT, vol. 2, no. 1,

pp. 015-021, 2014.

Page 195: Optimal Sizing of Modular Multilevel Converters

167

[31] H. Ertl, J. W. Kolar, and F. C. Zach, "A novel multicell DC-AC converter for

applications in renewable energy systems," IEEE Transactions on industrial

Electronics, vol. 49, no. 5, pp. 1048-1057, 2002.

[32] Y. Cao and L. M. Tolbert, "11-level cascaded H-bridge grid-tied inverter interface

with solar panels," in Applied Power Electronics Conference and Exposition (APEC),

2010 Twenty-Fifth Annual IEEE, 2010, pp. 968-972.

[33] N. Serbia, "Modular Multilevel Converters for HVDC power stations," Institut

National Polytechnique de Toulouse-INPT, 2014.

[34] J. Liang, A. Nami, F. Dijkhuizen, P. Tenca, and J. Sastry, "Current source modular

multilevel converter for HVDC and FACTS," in Power Electronics and Applications

(EPE), 2013 15th European Conference on, 2013, pp. 1-10.

[35] H.-J. Knaak, "Modular multilevel converters and HVDC/FACTS: A success story,"

in Power Electronics and Applications (EPE 2011), Proceedings of the 2011-14th

European Conference on, 2011, pp. 1-6.

[36] K. Fujii, U. Schwarzer, and R. W. De Doncker, "Comparison of hard-switched multi-

level inverter topologies for STATCOM by loss-implemented simulation and cost

estimation," in 2005 IEEE 36th Power Electronics Specialists Conference, 2005, pp.

340-346: IEEE.

[37] A. Jan R Svensson, Corporate Research. (2014). ABB’s HVDC and SVC Light

technology and applications.

[38] J. Li, A. Huang, S. Bhattacharya, and S. Lukic, "ETO light multilevel converters for

large electric vehicle and hybrid electric vehicle drives," in 2009 IEEE Vehicle Power

and Propulsion Conference, 2009, pp. 1455-1460.

[39] C. Hodge, "Modern applications of power electronics to marine propulsion systems,"

in Power Semiconductor Devices and ICs, 2002. Proceedings of the 14th

International Symposium on, 2002, pp. 9-16.

[40] S. M. M. Gazafrudi, A. T. Langerudy, E. F. Fuchs, and K. Al-Haddad, "Power quality

issues in railway electrification: A comprehensive perspective," ieee transactions on

industrial electronics, vol. 62, no. 5, pp. 3081-3090, 2015.

[41] G. W. Chang, H.-W. Lin, and S.-K. Chen, "Modeling characteristics of harmonic

currents generated by high-speed railway traction drive converters," IEEE

Transactions on Power Delivery, vol. 19, no. 2, pp. 766-773, 2004.

[42] F. Ma, Z. He, Q. Xu, A. Luo, L. Zhou, and M. Li, "A Multilevel Power Conditioner

and Its Model Predictive Control for Railway Traction System."

[43] A. Dell'Aquila, M. Liserre, V. Monopoli, and C. Cecati, "Design of a back-to-back

multilevel induction motor drive for traction systems," in Power Electronics

Specialist Conference, 2003. PESC'03. 2003 IEEE 34th Annual, 2003, vol. 4, pp.

1764-1769.

[44] K. Papastergiou, P. Wheeler, and J. Clare, "Comparison of losses in multilevel

converters for aerospace applications," in 2008 IEEE Power Electronics Specialists

Conference, 2008, pp. 4307-4312.

[45] M. Rehman-Shaikh, P. Mitcheson, and T. Green, "Power Loss Minimization in

Cascaded Multi-Level Converters for Distribution Networks," in Industrial

Electronics Society, 2007. IECON 2007. 33rd Annual Conference of the IEEE, 2007,

pp. 1774-1779.

Page 196: Optimal Sizing of Modular Multilevel Converters

168

[46] S. S. Fazel, "Investigation and comparison of multi-level converters for medium

voltage applications," Technische Universität Berlin, 2007.

[47] Infineon, "RC-Drives, RC-Drives Fast an d RC-Drives Automotive," Application

Note, vol. Revision 1.1, 2015.

[48] B. J. Baliga, Fundamentals of power semiconductor devices. Springer Science &

Business Media, 2010.

[49] M. Zygmanowski, B. Grzesik, and R. Nalepa, "Capacitance and inductance selection

of the modular multilevel converter," in Power Electronics and Applications (EPE),

2013 15th European Conference on, 2013, pp. 1-10.

[50]

https://library.e.abb.com/public/b47d6dffcf024d33a38f4f2bb2ed1896/5SNA

%200750G650300_5SYA%201600-04%2003-2016.pdf, "ABB HiPak IGBT

Module 5SNA 0750G650300," ABB Library, vol. No. 5SYA, 2014.

[51] ABB, "ABB HVDC Reference projects," 2011.

[52] S. Safari, A. Castellazzi, and P. Wheeler, "The impact of switching frequency on

input filter design for high power density matrix converter," in 2014 IEEE Energy

Conversion Congress and Exposition (ECCE), 2014, pp. 579-585.

[53] D. Stepins, "Examination of influence of periodic switching frequency modulation in

dc/dc converters on power quality on a load," in 2008 11th International Biennial

Baltic Electronics Conference, 2008, pp. 285-288.

[54] A. Shojaei and G. Joos, "An improved modulation scheme for harmonic distortion

reduction in modular multilevel converter," in Power and Energy Society General

Meeting, 2012 IEEE, 2012, pp. 1-7.

[55] A. Beddard, C. E. Sheridan, M. Barnes, and T. C. Green, "Improved Accuracy

Average Value Models of Modular Multilevel Converters," IEEE Transactions on

Power Delivery, vol. 31, no. 5, pp. 2260-2269, 2016.

[56] G. J. Kish and P. W. Lehn, "Modeling Techniques for Dynamic and Steady-State

Analysis of Modular Multilevel DC/DC Converters," IEEE Transactions on Power

Delivery, vol. 31, no. 6, pp. 2502-2510, 2016.

[57] J. Xu, A. M. Gole, and C. Zhao, "The Use of Averaged-Value Model of Modular

Multilevel Converter in DC Grid," IEEE Transactions on Power Delivery, vol. 30,

no. 2, pp. 519-528, 2015.

[58] H. Saad et al., "Dynamic Averaged and Simplified Models for MMC-Based HVDC

Transmission Systems," IEEE Transactions on Power Delivery, vol. 28, no. 3, pp.

1723-1730, 2013.

[59] A. Dekka, B. Wu, V. Yaramasu, and N. R. Zargari, "Model Predictive Control With

Common-Mode Voltage Injection for Modular Multilevel Converter," IEEE

Transactions on Power Electronics, vol. 32, no. 3, pp. 1767-1778, 2017.

[60] B. Chen, Y. Chen, C. Tian, J. Yuan, and X. Yao, "Analysis and Suppression of

Circulating Harmonic Currents in a Modular Multilevel Converter Considering the

Impact of Dead Time," IEEE Transactions on Power Electronics, vol. 30, no. 7, pp.

3542-3552, 2015.

[61] D. Jovcic and A. A. Jamshidifar, "Phasor Model of Modular Multilevel Converter

With Circulating Current Suppression Control," IEEE Transactions on Power

Delivery, vol. 30, no. 4, pp. 1889-1897, 2015.

Page 197: Optimal Sizing of Modular Multilevel Converters

169

[62] V. Company, "CAPACITORS - POWER - HEAVY CURRENT (ESTA) - POWER

ELECTRONIC," http://www.vishay.com/capacitors/power-heavy-current/power-

electronic/.

[63] J. Prymak et al., "Capacitor EDA Models with Compensations for Frequency,

Temperature, and DC Bias," in CARTS-CONFERENCE–Electronic Components

Association, 2010.

[64] F. Perisse, P. Venet, G. Rojat, and J.-M. Rétif, "Simple model of an electrolytic

capacitor taking into account the temperature and aging time," Electrical

Engineering, vol. 88, no. 2, pp. 89-95, 2006.

[65] J.-R. Riba, "Analysis of formulas to calculate the AC resistance of different

conductors’ configurations," Electric Power Systems Research, vol. 127, pp. 93-100,

2015.

[66] I. Standard, "Electric Cables–Calculation of Current Rating–Part 1: Current rating

equations (100% load factor) and calculation of losses, Section 1: General,"

Publication IEC-60287-1-11994.

[67] A. Reatti and F. Grasso, "Solid and Litz-wire winding non-linear resistance

comparison," in Circuits and Systems, 2000. Proceedings of the 43rd IEEE Midwest

Symposium on, 2000, vol. 1, pp. 466-469.

[68] M. Al-Asadi, A. Duffy, A. Willis, K. Hodge, and T. Benson, "A simple formula for

calculating the frequency-dependent resistance of a round wire," Microwave and

optical technology letters, vol. 19, no. 2, pp. 84-87, 1998.

[69] F. W. Olver, NIST Handbook of Mathematical Functions Hardback and CD-ROM.

Cambridge University Press, 2010.

[70] G. Nikishkov, "Introduction to the finite element method," University of Aizu, 2004.

[71] X. Li, Q. Song, W. Liu, H. Rao, S. Xu, and L. Li, "Protection of nonpermanent faults

on DC overhead lines in MMC-based HVDC systems," IEEE Transactions on Power

Delivery, vol. 28, no. 1, pp. 483-490, 2013.

[72] L. Bin, H. Jiawei, T. Jie, F. Yadong, and D. Yunlong, "DC fault analysis for modular

multilevel converter-based system," Journal of Modern Power Systems and Clean

Energy, pp. 1-8, 2016.

[73] H. Liu, K. Ma, P. C. Loh, and F. Blaabjerg, "Online fault identification based on an

adaptive observer for modular multilevel converters applied to wind power

generation systems," Energies, vol. 8, no. 7, pp. 7140-7160, 2015.

[74]

https://library.e.abb.com/public/d740233e818310bc83257ca9002c95b1/5SN

A%201500E330305%205SYA%201407-07%2002-2014.pdf, "5SNA 1500E330305

HiPak IGBT Module," ABB library, vol. Doc. No. 5SYA 1407, 2014.

[75] Genton C.,Rocca S., Boattini F." Control development for an 18 MW pulsed power

converter using a real-time simulation platform", 17th European Conference on Power

Electronics and Applications, Geneva, Switzerland, 2015, DOI 10.1109/EPE.2015.7309258