pmu-based state estimation for hybrid ac and dc...

PMU-based State Estimationfor Hybrid AC and DC Grids

WEI LI

Doctoral ThesisStockholm, Sweden 2018

TRITA-EECS-AVL-2018:23ISBN 978-91-7729-723-9

KTH Royal Institute of TechnologySchool of Electrical Engineering and Computer Science

Department of Electric Power and Energy SystemsSE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i elkrafttekniktorsdag den 12 april 2018, klockan 10:00, i sal F3, Kungliga Tekniska högskolan,Lindstedtsvägen 26, Stockholm.

© Wei Li, April 2018. All rights reserved.

Tryck: Universitetsservice US AB

Abstract

Power system state estimation plays key role in the energy management systems(EMS) of providing the best estimates of the electrical variables in the grid that arefurther used in functions such as contingency analysis, automatic generation control,dispatch, and others. The invention of phasor measurement units (PMUs) takes thepower system operation and control into a new era, where PMUs’ high reportingrate and synchronization characteristics allow the development of new wide-areamonitoring, protection, and control (WAMPAC) application to enhance the grid’sresiliency. In addition, the large number of PMU installation allows the PMU-onlystate estimation, which is ready to leap forward today’s approach which is based onconventional measurements.

At the same time, high voltage direct current (HVDC) techniques enable totransmit electric power over long distance and between different power systems,which have become a popular choice for connecting variable renewable energy sourcesin distant locations. HVDCs together with another type of power electronic-baseddevices, flexible AC transmission system (FACTS), have proven to successfullyenhance controllability and increase power transfer capability on a long-term cost-effective basis. With the extensive integration of FACTS and HVDC transmissiontechniques, the present AC networks will merge, resulting in large-scale hybrid ACand DC networks. Consequently, power system state estimators need to considerDC grids/components into their network models and upgrade their estimationalgorithms.

This thesis aims to develop a paradigm of using PMU data to solve stateestimations for hybrid AC/DC grids. It contains two aspects: (i) formulating thestate estimation problem and selecting a suitable state estimation algorithm; (ii)developing corresponding models, particularly for HVDCs and FACTS.

This work starts by developing a linear power system model and applying thelinear weighted least squares (WLS) algorithm for estimation solution. Linear networkmodels for the AC transmission network and classic HVDC links are developed. Thislinear scheme simplifies the nonlinearities of the typical power flow network modelused in the conventional state estimations and has an explicit closed-form solution.However, as the states are voltage and current phasors in rectangular coordinates,phasor angle is not an explicit state in the modeling and estimation process. Thisalso limits the linear estimators’ ability to deal with the corrupt angle measurementsresulting from timing errors or GPS spoofing. Additionally, it is cumbersome toselect state variables for an inherently nonlinear network model, e.g., classic HVDClink, when trying to fulfill its linear formulation requirement.

In contrast, it is more natural to use PMU measurements in polar coordinatesbecause they can provide an explicit state measurement set to be directly used inthe modeling and estimation process without form changes, and more importantly,it allows detection and correction for angle bias which emerges due to imperfectsynchronization or incorrect time-tagging by PMUs. To this end, the state estimationproblem needs to be formulated as a nonlinear one and the nonlinear WLS is applied

iv

for solution. We propose a novel measurement model for PMU-based state estimationwhich separates the errors due to modeling uncertainty and measurement noise sothat different weights can be assigned to them separately. In addition, nonlinearnetwork models for AC transmission network, classic HVDC link, voltage sourceconverter (VSC)-HVDC, and FACTS are developed and validated via simulation.

The aforementioned linear/nonlinear modeling and estimation schemes belongto static state estimator category. They perform adequately when the system isunder steady-state or quasi-steady state, but less satisfactorily when the system isunder large dynamic changes and the power electronic devices react to these changes.Testing results indicate that additional modeling details need to be included toobtain higher accuracy during system dynamics involving fast responses from powerelectronics. Therefore, we propose a pseudo-dynamic modeling approach that canimprove estimation accuracy during transients without significantly increasing theestimation’s computational burden. To illustrate this approach, the pseudo-dynamicnetwork models for the static synchronous compensator (STATCOM), as an exampleof a FACTS device, and the VSC-HVDC link are developed and tested.

Throughout this thesis, WLS is the main state estimation algorithm. It requiresa proper weight quantification which has not been subject to a sufficient attentionin literature. In the last part of thesis, we propose two approaches to quantify PMUmeasurement weights: off-line simulation and hardware-in-the-loop (HIL) simulation.The findings we conclude from these two approaches will provide better guidancefor selecting proper weights for power system state estimation.

Sammanfattning

Uppskattning av kraftsystemets tillstånd spelar en nyckelroll i energihanter-ingssystemen (EMS) för att göra den bästa uppskattningen av elektriska variabler inätet som används i analysfunktioner som beredskapsanalys, automatisk genera-tionskontroll med flera. Uppfinnandet av fasmätningsenheter (PMU) tar drift ochkontroll av kraftsystem till en ny era, där PMU:ers höga rapporteringstakt ochsynkroniseringsegenskaper möjliggör utvecklandet av ny övervakning, skydd ochkontroll över stora områden (WAMPAC) för att förhöja nätets elasticitet. Dessu-tom möjliggör ett stort antal PMU installationer tillståndsuppskattningar enbartbaserat på dessa PMU:er, vilket kan förbättra dagens metod som använder sig avkonventionella mätningar.

Samtidigt gör högspänd likström (HVDC) det möjligt att överföra elkraft överlånga avstånd och mellan olika kraftsystem, vilket har blivit ett populärt alternativför att ansluta förnyelsebara energikällor på avlägsna platser. HVDC-ledningartillsammans med en annan typ av kraftelektronik-baserade enheter, flexibla ACöverföringssystem (FACTS), har visats att framgångsrikt förhöja styrbarheten samtöka kraftöverföringskapaciteten på ett långsiktigt kostnadseffektiv sätt. Med enomfattande integration av FACTS och HVDC överföringstekniker, kommer denuvarande AC näten att sammanfogas, vilket resulterar i storskaliga hybrida AC ochDC nät. Följaktligen behöver kraftsystems estimatorer beakta DC nät/komponenteri sina nätverksmodeller samt uppgradera sina uppskattningsalgoritmer.

Den här avhandlingen syftar till att utveckla ett paradigm av att använda datafrån PMU för att göra tillståndsuppskattningar för AC/DC nät. Den innehåller tvåaspekter: (i) formulering av tillståndsuppskattningsproblemet och val av passandeuppskattnings-algoritm; (ii) utveckla motsvarande modeller, speciellt för HVDC ochFACTS.

Det här verket börjar med att utveckla en linjär kraftsystemsmodell och tillämparden linjära viktade minsta-kvadrat-metoden (WLS) för uppskattningslösning. Linjäranätverksmodeller för AC nätet och klassiska HVDC länkar är utvecklade. Denhär linjära proceduren förenklar olinjäriteten i den typiska kraftnätverksmodellensom används i konventionella tillståndsuppskattningar och har en explicit lösningpå sluten form. Dock, eftersom tillstånden är spännings- och strömfasvektorer irektangulära koordinater, är fasvinkeln inte ett explicit tillstånd i modellerings-och uppskattningsprocessen. Det begränsar även de linjära estimatorernas förmågaatt hantera de korrupta vinkelmätningarna som är en produkt av tidsfel ellerklockförskjutning i GPS:en. Dessutom är det besvärligt att välja tillståndsvariablerför en immanent olinjär nätverksmodell, t.ex. klassisk HVDC-länk, och försökauppfylla linjära formuleringskrav.

Det är däremot mer naturligt att använda PMU mätningar i polära koordinatereftersom de kan tillhandahålla en explicit tillståndsmätning som kan användasdirekt i modellerings- och uppskattningsprocessen utan att ändra form, och ännuviktigare, så tillåter den detektering och korrektion för systematiska avvikelseri vinkeln som uppstår på grund av ofullkomlig synkronisering eller felaktig tid-

vi

taggning av PMU:er. För detta ändamål behöver tillståndsuppskattningsproblemetformuleras som ett olinjärt problem och olinjär WLS används för lösning. Vi föreslåren ny mätningsmodell för PMU-baserad tillståndsuppskattning som separerar felensom uppstår på grund av modellosäkerhet och på grund av mätbrus så att de kantilldelas olika vikter separat. Dessutom är olinjära nätverksmodeller för AC nät,klassisk HVDC-länk, så kallad Voltage source converter (VSC)-HVDC, och FACTSutvecklade och validerade via simuleringar.

De tidigare nämnda linjära och olinjära modellerings- och uppskattningssche-mana tillhör kategorin för statiska tillståndsestimatorer. De fungerar adekvat närsystemet är i stabilt, eller kvasi-stabilt, tillstånd, men mindre acceptabelt när sys-temet är under stora dynamiska förändringar då kraftelektronikenheter regerarpå dessa förändringar. Testresultat indikerar att ytterligare modelleringsdetaljermåste inkluderas för att erhålla högre precision under systemtransient dynamiksom involverar snabba responser från kraftelektronik. Därför föreslår vi en pseudo-dynamisk modelleringsmetod som kan förbättra uppskattningsprecisionen undertransienter utan att signifikant öka uppskattningens beräkningsbörda. För att il-lustrera detta tillvägagångssätt är de pseudo-dynamiska nätverksmodellerna förden statiska synkrona kompensatorn (STATCOM), som ett exempel på en FACTSenhet, och VSC-HVDC-länken utvecklade och testade.

Genom hela denna avhandling är WLS den huvudsakliga tillståndsuppskat-tningsalgoritmen. Den kräver en lämplig viktkvantifiering, vilket inte har fått till-räcklig uppmärksamhet i litteraturen. I den sista delen av avhandlingen föreslårvi två metoder för att kvantifiera vikter för PMU-mätningar: off-line simuleringoch Hardware-in-the-loop (HIL) simulering. De resultat vi erhåller från dessa tvåtillvägagångssätt kommer att ge bättre vägledning för hur man ska välja lämpligavikter för uppskattning av kraftsystemstillstånd.

To Meng, Emma, and my parents.

Acknowledgements

I would like to firstly express my gratitude to my main advisor Prof. Luigi Vanfrettifor your persisting support, guidance and encouragement in my work. You introducedme to the world of research with the master thesis project in 2011. I am extremelygrateful that you give me the perfect blend of guidance and independence duringthis journey.

I would like to thank Prof. Joe H. Chow for hosting me at Rensselaer PolytechnicInstitute during Jan.-May, 2015. I had a great and fruitful time there and got toknow lots of awesome people. Your sincerity and passion for research will continueinfluencing me later on.

The five years of PhD study at KTH have been the most memorable time in mylife so far. Electric power and energy systems department has been such a joyfulplace to live and work. I thank all the colleagues for fun time during lunches andfikas. I thank all my officemates, Yuwa, Francisco, Maxime, Almas, Farhan, Jan, Elisand Evelin for creating a pleasant working atmosphere. I thank Yalin, Meng S. andZhao for native speaking chats. Special thanks go to Almas and Viktor for helpingme set up the hardware test in SmarTS Lab. Thanks also go to the administratorsfor always being helpful.

Finally, I would like to thank my parents for always believing in and supportingme, and my husband Meng for your genuine and everlasting love, as well as mydaughter Emma for your limitless trust and beautiful smiles.

Wei LiStockholm, April 2018

ix

Contents

Acknowledgements ix

Contents xi

1 Introduction 11.1 Motivating examples . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Power system state estimation . . . . . . . . . . . . . . . 11.1.2 Phasor measurement units . . . . . . . . . . . . . . . . . . 31.1.3 High voltage direct current . . . . . . . . . . . . . . . . . 4

1.2 Contributions and outline . . . . . . . . . . . . . . . . . . . . . . 61.3 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Power system state estimation overview 112.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Power system structure . . . . . . . . . . . . . . . . . . . 112.1.2 System operating conditions . . . . . . . . . . . . . . . . . 122.1.3 Power system state estimation . . . . . . . . . . . . . . . 12

2.2 PMUs and phasor state estimators . . . . . . . . . . . . . . . . . 132.2.1 SCADA-based and PMU-based SEs . . . . . . . . . . . . 13

2.3 State estimations for HVDC and FACTS . . . . . . . . . . . . . . 142.4 Static, forecasting-aided, and dynamic SEs . . . . . . . . . . . . . 16

2.4.1 Forecasting-aided state estimation . . . . . . . . . . . . . 162.4.2 Dynamic state estimation . . . . . . . . . . . . . . . . . . 17

2.5 Transmission system, and distribution system SEs . . . . . . . . 182.6 Centralized, distributed, and multi-area SEs . . . . . . . . . . . . 20

3 Conventional state estimations and test systems 213.1 Conventional state estimations . . . . . . . . . . . . . . . . . . . 21

3.1.1 Measurement’s distribution: Gaussian . . . . . . . . . . . 213.1.2 Objective function: weighted least squares . . . . . . . . . 223.1.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . 23

3.2 Test systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

xi

xii Contents

3.2.1 9-bus system . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 KTH-Nordic 32 system . . . . . . . . . . . . . . . . . . . 253.2.3 6-bus hybrid AC/DC system . . . . . . . . . . . . . . . . 253.2.4 VSC-based HVDC transmission link . . . . . . . . . . . . 253.2.5 Synthetic measurement generation . . . . . . . . . . . . . 27

4 Linear network models 294.1 AC transmission network . . . . . . . . . . . . . . . . . . . . . . 30

4.1.1 Network model . . . . . . . . . . . . . . . . . . . . . . . . 304.1.2 Measurement model . . . . . . . . . . . . . . . . . . . . . 31

4.2 Classic HVDC link . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.1 Network model . . . . . . . . . . . . . . . . . . . . . . . . 314.2.2 Measurement model for hybrid AC/DC systems . . . . . . 32

4.3 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3.1 LSE for the 9-bus system . . . . . . . . . . . . . . . . . . 334.3.2 LSE for the 9-bus hybrid AC/DC system with a classic

HVDC link . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3.3 LSE for the KTH-Nordic 32 system . . . . . . . . . . . . 364.3.4 LSE for the KTH-Nordic 32 hybrid AC/DC system with

a classic HVDC link . . . . . . . . . . . . . . . . . . . . . 364.3.5 Effect of less DC measurements . . . . . . . . . . . . . . . 374.3.6 Effect of measurement noise . . . . . . . . . . . . . . . . . 39

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Nonlinear network models 415.1 AC transmission network . . . . . . . . . . . . . . . . . . . . . . 425.2 Classic HVDC link . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2.1 Network model . . . . . . . . . . . . . . . . . . . . . . . . 435.2.2 Measurement model for hybrid AC/DC systems . . . . . . 475.2.3 Considerations for practical application . . . . . . . . . . 505.2.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3 VSC-HVDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3.1 VSC substation model . . . . . . . . . . . . . . . . . . . . 675.3.2 VSC control modes . . . . . . . . . . . . . . . . . . . . . . 695.3.3 Point-to-point VSC-HVDC link model . . . . . . . . . . . 705.3.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4 FACTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4.1 Shunt devices . . . . . . . . . . . . . . . . . . . . . . . . . 725.4.2 Series devices . . . . . . . . . . . . . . . . . . . . . . . . . 755.4.3 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Pseudo-dynamic network modeling and examples 816.1 Pseudo-dynamic concept . . . . . . . . . . . . . . . . . . . . . . . 82

Contents xiii

6.2 STATCOM model . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3 VSC-HVDC model . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3.1 VSC substation model . . . . . . . . . . . . . . . . . . . . 846.3.2 Point-to-point VSC-HVDC link model . . . . . . . . . . . 87

6.4 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.4.1 STATCOM models’ comparison and validation using real

PMU data . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.4.2 STATCOM model in two test systems . . . . . . . . . . . 906.4.3 VSC-HVDC model . . . . . . . . . . . . . . . . . . . . . . 92

6.5 Computation performance . . . . . . . . . . . . . . . . . . . . . . 956.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7 Quantifying PMU measurement weights 977.1 Quantification approaches . . . . . . . . . . . . . . . . . . . . . . 99

7.1.1 Off-line simulation . . . . . . . . . . . . . . . . . . . . . . 997.1.2 Hardware-in-the-loop simulation . . . . . . . . . . . . . . 100

7.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2.1 Off-line simulation . . . . . . . . . . . . . . . . . . . . . . 1017.2.2 Hardware-in-the-loop simulation . . . . . . . . . . . . . . 106

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

8 Conclusion 111

Bibliography 113

Chapter 1

Introduction

Electricity is so ubiquitous in our daily life that we take it for granted like air webreathe. Electric power system is the network behind to supply, transfer, store andconsume electric power. Failure or outage of electric power system can result intremendous risk to the environmental and public safety. An accurate estimation ofsuch system’s operating states is of critical importance for further tasks like frequencycontrol and load balancing. The development and installation of phasor measurementunits (PMUs) help to provide high-resolution, time-synchronized measurement acrossa large geographic area. Moreover, the unprecedented development in high voltagedirect current (HVDC) transmission technology exhibits a great need for high-performance real-time state estimation techniques.

1.1 Motivating examples

This chapter presents the motivation of this thesis work via giving three examples todemonstrate the importance of power system state estimation, the advantage of usingPMUs for state estimation, and the necessity of including power electronic-baseddevices (e.g., HVDCs, FACTS) into the state estimation, respectively. This reflectsthe three main elements in the thesis title “PMU-based State Estimation forHybrid AC and DC Grids".

1.1.1 Power system state estimation

Power grids become more complex with increasing electricity consumption andrenewables exploitation. For instance, from 2000 to 2016, more than 11,655 megawattsof new generating capacity have been added to the New York State’s electric system,and more than 2,765 MW of transmission capability have been added as well, whichis shown in Fig. 1.1. In 2016, the total generating capacity reaches 38,576 MWand the total circuit miles of transmission come to 11,124 miles (≈ 17,902 km). Inaddition, New York’s largest demand response program, Special Case Resources, isprojected to be capable of offering up to 1,248 MW [1].

1

2 Introduction

Figure 1.1: New Transmission in New York State: 2000-2016 (source [1])

Despite power grids generally are designed to meet demand extremes and handlethe worst-case situation, to ensure power system reliability is always a challenge.Transmission system operators (TSOs) or independent system operators (ISOs) haveoverall responsibility to maintain the short-term balance between production andconsumption of electricity. Therefore, they control and monitor grids around theclock in their control rooms.

A picture of the control center at the New York ISO is shown in Fig. 1.2. Therethe computer systems receive 50,000 data points about every six seconds, andoperators monitor regional activity on a 2,300-square-foot video wall. Mandatoryreliability standards have been put in place for the thousands of entities involved inthe operation of the country’s electric systems [2].

Across the world, tasks in power system control rooms are similar: monitoringelectricity zooming through the national or regional grid and electric power exchangewith neighboring grids. This task is conducted with the help of power system stateestimation (PSSE), which provides the optimal estimation of the system’s statusbased on the collected measurements and the presumed network model. Sequentially,exploiting the estimation results, operators enable to constantly coordinate thesupply and demand of electricity to ensure enough electricity is available to keepthe lights on without overloading transmission lines. If the system is out of balanceor the flow of electricity is interrupted, it can damage equipment or cause poweroutages.

1.1. Motivating examples 3

Figure 1.2: Control center at the New York Independent System Operator: (i) featuresa 2,300-square-foot video wall—the largest utility installation in North America; (ii)displays more than 3,000 live status points—presenting line flows, limits, transformerloading, voltages, and generator output; (iii) customizes the regional electric systeminformation, weather and lightning-strike data, load forecast, etc. to address systemdynamics [3].

In 2003, the worst blackout of the United States started with a sagging powerline in Ohio that shorted out after touching a tree branch. A series of human errorsand a computer problem plunged about 50 million people into darkness from NewYork City to Toronto and cost the United States economy about $6 billion [2].Operators carry out extensive trainings in simulation labs to devise strategies toensure system reliability when subjected to such faults. Fast and accurate stateestimation techniques would enable operators to detect such faults faster and thusreact in a more timely fashion.

1.1.2 Phasor measurement unitsThe performance of power system state estimation relies on the properties of thecollected measurements, such as measurement quantities, sampling rate, accuracyand variance, synchronization, etc. At present, phasor measurement units (PMUs)are the most accurate and advanced time-synchronized technology available. Theyprovide voltage and current phasor and frequency information, synchronized withhigh precision to a common time reference provided by the global positioning system(GPS). Implementing PMUs can enhance the accuracy and computational efficiencyof power system state estimation. Figure 1.3 gives an intuitive comparison betweenthe data from a PMU and the data from the traditional supervisory control anddata acquisition (SCADA) system.

For the past decade, PMUs installation and deployment have dramatically in-creased. By 2014, the American Recovery and Reinvestment Act (ARRA) investment

4 Introduction

Figure 1.3: Data comparison between PMU and SCADA [4].

resulted in total number of installed PMUs to more than 1,100, offering nearly 100percent observability of the transmission system. Based on a report of North Ameri-can SynchroPhasor Initiative (NASPI), up till 2017 summer, over 2,500 networkedPMUs have been installed and connected in America synchrophasor networks, andthis number continues to grow [5]. In China, by the end of 2013, the number ofPMUs installed at substations and power plants by the State Grid Corporationof China (SGCC) was 2,027 [6]. The Powergrid of India is installing about 1,700PMUs covering all 400 kV and above voltage level substations and major generatingstations [7].

With the rapid deployment of PMUs, using only PMU data for power systemstate estimation is feasible and promising. However, more research need to be carriedout in order to utilize PMU data in a more efficient and reliable fashion.

1.1.3 High voltage direct current

Presently, alternating current (AC) power technology is dominant in electricitygeneration, transmission and distribution. For high voltage transmission systems,compared to HVAC, HVDC techniques enable (i) low loss interconnection over largedistances, (ii) connection between asynchronous grids, (iii) connection to remoteenergy resources and loads, and (iv) flexibility to accommodate variable renewableenergy. In addition, over a specific distance, called as break-even distance, HVDCline becomes cheaper than HVAC. The break-even distance for overhead lines isaround 600 km and for submarine lines it is around 50 km.

A report from Bloomberg new energy finance (BNEF) [9] shows the globalHVDC capacity has experienced an exponential growth since 2010, and by 2025the capacity will double the number in 2015. Based on the “Compendium of allHVDC projects 2009” [10] prepared by the SCB4 (HVDC and Power Electronics)committee of CIGRE, the number of HVDC link projects in Africa, Australia and

1.1. Motivating examples 5(3

00 k

V)

R tock

Ringhals

K pe -ham

Göteborg

almöKarlshamn

Norrköping

Örebro

Oskarshamn

HasleStavanger

Bergen RjukanOslo

Stockholm

Nea

Trondheim

Tunnsjødal

Umeå

Sundsvall

Røssåga

Ofoten

Narvik

SVERIGE

NORGE

FINLAND

Loviisa

OlkiluotoViborg

Kristiansand

Rauma

Forsmark

0 100 200 km

Luleå

Vasa

Tammerfors

Kemi

Uleåborg

KielLübeck

Slupsk

Eemshaven

N

Klaipeda

STAMNÄTET FÖR EL 2017

Riga

Vilnius

ESTLAND

Tallinn

LETTLAND

400 kV ledning

275 kV ledning

220 kV ledning

HVDC (likström)

Samkörningsförbindelse för lägre spänning än 220 kVPlanerad/under byggnad

Vattenkraftstation

Värmekraftstation

Planerad/under byggnad

Transf./kopplingsstation

Vindkraftpark

Helsingfors

kraftledningar, 160 transformator- och kopplingsstationerDet svenska stamnätet för el består av 15 000 km

och 16 utlandsförbindelser.

Flensburg

DANMARK

Åbo

Rovaniemi

LITAUEN

os

nn

M

ö

Güstrow

Figure 1.4: Map of the Swedish national grid (source [8])

6 Introduction

Oceania, Asia, Europe, North America, and South America was 2, 4, 15, 14, 10, and2 up till 2009; and the number of back-to-back HVDC projects reached 30 in total.This compendium has not been updated since then, while Wikipedia provides a listof HVDC projects in December 2017, which can act as an informal reference for theupdated information. A rough count on the HVDC link projects in Africa, Australiaand Oceania, Asia, Europe, North America, and South America is 3, 5, 55, 64, 21,and 3; and the number of back-to-back HVDC projects reaches 53 in total [11].

Inside Europe, to allow long-distance transmission for renewables and cross-border exchange of power, HVDC becomes the key to an integrated European powernetwork. For instance, the HVDC links (indicated by purple lines) of NorthernEurope are shown in Fig. 1.4. HVDC can provide fast, precise and flexible controlof transmission flow, which greatly improves European grid’s reliability, capacityand efficiency.

With the extensive integration of HVDC transmission techniques, the presentAC dominant networks will merge to large-scale hybrid AC and DC networks. Tofacilitate meeting this challenge, power system state estimators need to adapt DCgrids into the network models and upgrade the estimation algorithms. More researchin this field is warranted.

1.2 Contributions and outline

In this section, the main contributions and outline of this thesis are summarizedincluding the publications that each chapter is based upon.

Chapter 2 — Power system state estimation overviewThis chapter introduces the background and the history of power system stateestimation. How to include PMU measurements and consider power electronic-baseddevices for state estimation is also discussed. In addition, this chapter provides anovel perspective to overview power system state estimation through demonstratingdifferent categorizations. Depending on the timing and evolution of the estimates,state estimation schemes can be classified into static, forecasting-aided, and dynamicstate estimations. The target system of the state estimation function decides whetherit is a transmission system or a distribution system state estimation. Whetherdistributing the estimation computation among different geographical areas classifiesstate estimations into centralized, distributed, and multi-area state estimations.

Chapter 3 — Conventional state estimations and test systemsThis chapter focuses on the formulation and derivation of state estimation methods,particularly the formulation and solution of the conventional state estimations. Inaddition, the test systems that are developed and implemented for case studiesare introduced. All of the aforementioned contents provide a background for thefollowing chapters.

1.2. Contributions and outline 7

Chapter 4 — Linear network models

Using synchronized phasor measurements makes it possible to formulate the stateestimation problem as a linear one. This chapter presents the linear network andmeasurement models for both the AC transmission network and the classic HVDClink. The above results have been published in:

• W. Li and L. Vanfretti. Inclusion of Classic HVDC links in a PMU-BasedState Estimator. IEEE PES General Meeting, Washington DC, USA, 2014.

Chapter 5 — Nonlinear network models

This chapter introduces a novel measurement model for PMU-based state estimation,which separates the network model equations from measurements so that differentweights can be assigned to them separately. Corresponding nonlinear network modelsfor the AC transmission network, classic HVDC link, VSC-HVDC, and FACTSdevices are presented. The above results have been published in:

• W. Li and L. Vanfretti. A PMU-based state estimator considering classicHVDC links under different control modes. Sustainable Energy, Grids andNetworks, vol. 2, pp. 69-82, 2015.

• W. Li and L. Vanfretti. A PMU-based state estimator for networks containingVSC-HVDC links. IEEE PES General Meeting, Denver, USA, 2015.

• W. Li and L. Vanfretti. A PMU-based state estimator for networks containingFACTS devices. IEEE PowerTech, Eindhoven, Netherlands, 2015.

Note that the models and case studies for some FACTS devices presented in thisthesis have been modified from the corresponding paper.

Chapter 6 —Pseudo-dynamic network modeling and examples

This chapter presents the pseudo-dynamic modeling approach that leverages theexisting body of the static network model and includes the difference equations thatdescribe system dynamical properties, so that it can improve the state estimationaccuracy during transients without significantly increasing its computational burden.Moreover, pseudo-dynamic models for STATCOM and VSC-HVDC are demonstratedas examples. The comparison and validation for the STATCOM model using realPMU data were conducted at Rensselaer Polytechnic Institute under the supervisionof Prof. Joe. H. Chow. The above results have been published in:

• W. Li, L. Vanfretti and J. H. Chow. Pseudo-Dynamic Network Modeling forPMU Based State Estimation of Hybrid AC/DC Grids. IEEE Access, vol. 6,pp. 4006-4016, 2018.

8 Introduction

Chapter 7 — Quantifying PMU measurement weights

In power system state estimation, weighting is a practice of accounting for the confi-dence in the model and in a measurement. In order to quantify PMU measurementweights, two methods are proposed: off-line simulation and hardware-in-the-loop(HIL) simulation. The latter was conducted in SmarTS Lab at KTH. The aboveresults have been submitted to:

• W. Li and L. Vanfretti. Quantifying PMU Measurements Weights for PhasorState Estimation. IEEE Access, submitted, Feb. 2018.

Chapter 8 — Conclusions and future work

The thesis is concluded with a summary.

Other publications

The following publications are not covered in this thesis, but contain material thatmotivates the work presented here:

• L. Vanfretti, W. Li, A. Egea-Alvarez and O. Gomis-Bellmunt, Generic VSC-Based DC Grid EMT Modeling, Simulation, and Validation on a ScaledHardware Platform, IEEE PES General Meeting, Denver, USA, 2015.

• R. Rogersten, L. Vanfretti and W. Li, Towards Consistent Model Exchangeand Simulation of VSC-HVdc Controls for EMT Studies through C-CodeIntegration, IEEE PES General Meeting, Denver, USA, 2015.

• L. Vanfretti, N. A. Khan, W. Li, M. R. Hasan and A. Haider, Generic VSCand Low Level Switching Control Models for Offline Simulation of VSC-HVDCSystems, Electric Power Quality and Supply Reliability Conference, Rakvere,Estonia, 2014.

• M. R. Hasan, L. Vanfretti, W. Li and N. A. Khan, Generic High Level VSC-HVDC Grid Controls and Test Systems for Offline and Real Time Simulation,Electric Power Quality and Supply Reliability Conference, Rakvere, Estonia,2014.

• N. A. Khan, L. Vanfretti, W. Li, and A. Haider, Hybrid Nearest Level andopen loop control of Modular Multilevel Converters with N+1 and 2N+1 levels,EPE’14-ECCE Europe, Lappeenranta, Finland, 2014.

• R. Rogersten, L. Vanfretti, W. Li, L. Zhang, and P. Mitra, A QuantitativeMethod for the Assessment of VSC-HVdc Controller Simulations in EMTTools, IEEE PES ISGT Europe, Istanbul, Turkey, 2014.

1.3. Acronyms 9

• L. Vanfretti, W. Li, T. Bogodorova and P. Panciatici, Unambiguous PowerSystem Dynamic Modeling and Simulation using Modelica Tools, IEEE PESGeneral Meeting, Vancouver, Canada, 2013.

1.3 Acronyms

AC Alternative currentAVM Average value modelCSC Current source converterDC Direct currentDMU DC measurement unitDR Demand regulationDSSE Distribution system state estimationDySE Dynamic state estimationEKF Extended Kalman filterFACTS Flexible AC transmission systemFASE Forecasting-aided state estimationGPS Global positioning systemGW GigawattHVDC High voltage direct currentHVAC High voltage alternative currentIGBT Insulated-gate bipolar transistorIKF Iterative Kalman filterISO Independent system operatorLCC Line commutated converterLSE Linear state estimationMINLP Mixed-integer nonlinear problemMLE Maximum likelihood estimateMW MegawattNSE Nonlinear state estimationPCC Point of common couplingPDF Probability density functionPMU Phasor measurement unitPSSE Power system state estimationPWM Pulse width modulationRMS Root mean square

10 Introduction

RTU Remote terminal unitSCADA Supervisory control and data acquisitionSE State estimationSNR Signal-to-noise ratioSSSC Static synchronous series compensatorSTATCOM Static synchronous compensatorSVC Static var compensatorTCSC Thyristor controlled series compensatorTSO Transmission system operatorTSSE Transmission system state estimationUKF Unscented Kalman filterVSC Voltage source converterWAMPAC Wide-area monitoring, protection, and controlWLAV Weighted least absolute valueWLS Weighted least squares

Chapter 2

Power system state estimation overview

In this chapter, we firstly describe the background of power system state estimation,particularly its role in power system operation. With the installation of PMUs andunprecedented development of power electronic-based devices, it is inevitable to takethem into account for power system state estimation. To meet the diverse needsfor state estimation, measurements over consecutive instants or a system modelthat considers time evolution can be utilized in addition to the static measurementmodel, leading to the so-called forecasting-aided state estimation and dynamicstate estimation, respectively. Furthermore, state estimation is not exclusive fortransmission systems any more, but become applicable to distribution systems as well.The architecture of state estimations, i.e., models, formulations, and computation isno longer confined to a centralized methodology; a distributed or hierarchy schememay be a better choice in some cases. For each of the aforementioned aspects, somerelated research in literature is discussed. More background on the formulation andderivation of different methods will be presented in next chapter.

2.1 Background

Power system is a gigantic system covering a large geographical area and influencinga tremendous number of population. A stable and resilient power supply is the topfundamental facility for a society.

2.1.1 Power system structure

A typical power system consists of generation, transmission, and distribution systems.Electricity can be generated in different types of power plants depending on theused energy sources, e.g. coal, gas, hydro, wind, and solar, etc. This generatedelectricity is transported to distribution systems through the transmission system,which connects the producers with the consumers. To minimize power loss duringtransmission, transmission systems have high voltages. The receiving end of thetransmission system is equipped with substations where electricity is transformed

11

12 Power system state estimation overview

into lower voltages for distribution systems. Then the distribution system feeds theelectricity to the commercial and residential consumers.

Maintaining stability and functionality of such a complex system is the toppriority for most of TSOs or ISOs. Unlike water, electricity can not be stored in abucket for most cases, thus most electricity is used the instant it is created. Thedelicate art of balancing the grid is to ensure enough electricity is available to keepthe lights on without overloading transmission lines.

2.1.2 System operating conditions

According to [12], a system can be characterized as one of the three conditions—normal, emergency, and restorative. A system is said to be in the normal operatingcondition if both the load and operating constraints are satisfied. In the emergencycondition the operating constraints are not satisfied. In the restorative condition,the operating constraints are satisfied but not the load constraints. Even under thenormal operating condition, the system may not be secured in a sense that it canhit the operating constraints and degrade to emergency state when a contingencyfrom the predefined critical contingencies list occurs.

In order to obtain a comprehensive knowledge of the system operating condition,it is essential to conduct a power system security analysis, which includes monitoringof the system conditions, identification of the operating state, and determination ofthe necessary preventive actions in case the system state is found to be insecure. Asthe set of voltage phasors enables to fully specify the system, it is referred to as thestatic state of the system.

The first step is monitoring. Supervisory control systems were initially applied tomonitor and control the status of circuit breakers at the substations. And generatoroutputs and the system frequency were also monitored for purposes of automaticgeneration control and economic dispatch. Later the augment of real-time system-wide data acquisition capabilities led to the establishment of the first SCADAsystem. Nowadays, with the proliferation of PMU installations, time-synchronizedmeasurements with higher reporting rate and accuracy are available for most ofsubstations and even for distribution grids. Are these raw measurements enough forthe second step—identifying the system operating state?

The information provided by the monitoring step may not always be reliable dueto errors in measurements, telemetry failures, communication noise, etc. Furthermore,it may not be economically feasible to communicate all possible measurements evenif they are available from the transducers at the substations. In those cases, wouldit be helpful to apply the pre-known network information such as the admittancematrix?

2.1.3 Power system state estimation

Power system static state estimation was initiated by Fred Schweppe in 1960s withthe purpose of converting available information (direct meter readings plus other

2.2. PMUs and phasor state estimators 13

information) into an estimate of the present state of the power system [13, 14, 15].State estimation essentially is a data processing function, or generally speaking afilter. It seeks for the optimal solution based on the redundant measurements andthe assumed system model. Introducing the state estimation function broadenedthe capabilities of the SCADA system, leading to the establishment of the EnergyManagement Systems (EMS) whose functions such as contingency analysis, auto-matic generation control, load forecasting and optimal power flow, etc. heavily relyon the state estimation solution.

State estimators typically include the following functions: topology processor,observability analysis, state estimation solution, bad data processing, parameterand structural error processing [16]. Each function is an independent field thatreceives considerable attention from researchers. Moreover, other related researchsubjects, such as PMUs deployment, calibrating instrument, estimating networkmodel’s parameters, are also active. This thesis mainly contributes to the functionof state estimation solution, specifically in the phasor state estimation formulationand network modeling.

2.2 PMUs and phasor state estimators

The term phasor was firstly presented by Charles Proteus Steinmetz in 1893 whenhe presented a paper on simplified mathematical description of the waveformsof AC electricity [17]. In late 1970s, the symmetrical component distance relay(SCDR) was introduced, which proposed the recursive algorithm of symmetricalcomponent discrete Fourier transform (SCDFT) for measuring the positive sequencevoltages and currents accurately and with the response time of one cycle of thefundamental frequency. To spread this phasor calculator across power system forwide-area monitoring, a synchronized clock pulse was used for sampling at differentsites. Precise synchronization of sampling clocks became possible with the advent ofGPS satellite system [18]. All these techniques eventually led to the development ofPMUs in early-1980s by Dr. Arun G. Phadke and Dr. James S. Thorp at VirginiaTech.

PMUs can provide voltage and current phasors and frequency information withtypical reporting rates as high as 50/60/100/120 frames/second, and synchro-nized by the GPS, which makes the wide-area monitoring, protection, and control(WAMPAC) possible. Utilizing PMUs can significantly improve the performance ofWAMPAC, including real-time visualization of power systems, advanced early warn-ing system, post-contingency analysis, state estimation, real-time angular, voltage,and frequency control, inter-area oscillation damping, and so on.

2.2.1 SCADA-based and PMU-based SEsTraditionally, real-time measurements are collected through the remote terminalunits (RTUs) installed in the SCADA system. These measurements are usuallysampled and collected with a rate of several seconds and they are not synchronized.


In contrast, PMU measurements are GPS synchronized with a rate as high as50/60/100/120 frames/seconds.

The measured quantities from these two measurement source are different aswell. The typical measurement quantities from the SCADA system include activeand reactive power flows on transmission lines, active and reactive power injectionsat buses, and bus voltage magnitudes. For some cases, there are measurements forline current magnitude as well. While the PMU measurement quantities are mainlybus voltage phasors, line current phasors, and frequency.

Differences in measurement quantity lead to the differences in the formulation ofnetwork models. Generally, the network model for the conventional state estimationsis constructed based on the power flow model, which is of high nonlinearity. Incontrast, the network models for PMU-based state estimations connect the mea-surements for voltage and current phasors (in polar coordinates) with states, i.e.,voltage magnitude and angle. This relationship possesses much less nonlinearity.

Both of the aforementioned network models are nonlinear. But when using onlyPMU measurements, it is also possible to formulate a linear network model anda linear state estimator [19, 20] by applying the voltage and current phasors inrectangular coordinates, instead of polar coordinates. This design improves thecomputational efficiency, however, it loses the availability of directly exploitingindependent angle information, resulting in difficulties in angle error detection, aswell as obstacles for phase angle monitoring.

Phase angle information allows early identification of potential problems bothlocally and regionally [21]. For instance, monitoring angle separation or rate-of-change of angle separation between two buses or two parts of a grid can help todetermine the stress on the system. Moreover, angle information can be used formore accurate detection of nominal transfer capability (NTC) based on thermal,voltage, or stability limitations. Another critical application of phase angle is duringrestoration. The phase angle value across an opened tie line or an opened circuitbreaker would guide an operator in circuit breaker closing. All the aforementionedphase angle monitoring (PAM) functions require accurate phase angle estimationwhich can be achieved by phasor state estimators where phase angle is formulatedas an independent state [22, 23, 24].

Both linear and nonlinear network models are utilized in this thesis. Theiradvantages and disadvantages will be demonstrated in the following chapters.

2.3 State estimations for HVDC and FACTS

AC has been the preferred global platform for electricity transmission for the past100 years. But HVAC transmission has some limitations: (i) constraints on thetransmission capacity and distance, (ii) the impossibility of directly connecting twoAC power networks of different frequencies. With the rapid growth of exploitingvariable renewable energy sources and the flourish in access to electricity, newtechnologies for transmitting power over long distance and between different powersystems are expected to grow far beyond their current levels of deployment. The

2.3. State estimations for HVDC and FACTS 15

HVDC techniques fulfill this need.Electronic converters for HVDC are divided into two main categories. Line-

commutated converter (LCC) is made with electronic switches that can only beturned on. It used mercury-arc valves until the 1970s, or thyristors from the 1970s tothe present day. Voltage source converter (VSC) is made with switching devices thatcan be turned both on and off. The world’s first commercial HVDC transmissionusing VSC converters is the“Gotland HVDC Light” which commissions since 1999.VSCs use fully-controlled power devices, such as insulated-gate bipolar transistor(IGBT).

LCC-based HVDC, so called the classic HVDC technology is mature and hasbeen used worldwide. It is still irreplaceable for HVDC applications of high powerand voltage ratings, which recently can be up to 10 GW , with voltages up to1, 100 kV . VSC technology can be useful for interconnecting weak AC systems, forconnecting large-scale wind power to the grid, or for HVDC interconnections thatare likely to be expanded as multi-terminal direct current (MTDC) systems in future.Compared to the classic HVDC, VSC-HVDC technology offers a key advantage ofindependent control of active and reactive powers, together with additional benefitsin control flexibility and reliability [25]. Up to date, VSC-HVDC is widely andeffectively applied to interconnected remote generation plants, isolated remote loads,metropolitan areas, and offshore bays, etc [26]. The market for VSC-HVDC isgrowing fast in Europe, driven partly by the surge of the investment in offshorewind power.

Another type of power electronic-based devices that have been proven to enhancecontrollability and transfer capability is flexible AC transmission system (FACTS)[27, 28]. The concept of FACTS was introduced in 90s and soon was recognized asan indispensable asset to improve transmission quality and efficiency of existing ACgrids.

FACTS and HVDC can provide both steady state and dynamic control forpower systems [29]. For steady state control, FACTS and HVDC can providevoltage regulation, power flow management and control, congestion management,eliminating bottlenecks, and enhancement of transfer capability, etc. For dynamiccontrol, FACTS and HVDC can provide fast voltage support, rapid power flowcontrol and dynamic congestion management, fast controlled voltage and powercompensation, power oscillation damping, voltage stability control, and fault ride-through, etc. With the extensive integration of FACTS and HVDC transmissiontechniques, the present AC dominant networks will merge to large-scale hybrid ACand DC networks.

On the other hand, to efficiently control hybrid AC and DC networks overlarge geographical areas synchronized measurements are required. An technologyinnovation project initiated by Bonneville Power Administration (BPA), i.e.,TIP318: WAMS-Enhanced HVDC Control for Flexible and Stable Grid Operations [30],aims to increase transfer capability of Pacific AC and DC Intertie (PADI) usingsynchronized wide area measurements. This project points a direction of utilizingsimultaneous state information from both AC and DC parts of the network for


control purposes. To obtain the essential awareness of the states, hybrid AC andDC state estimators need to be developed by including FACTS and HVDCs intothe network models and upgrading the estimation algorithms.

HVDC modeling for power system state estimation dates back to 1980s when[31] first includes a classic HVDC link model into an AC system state estimation.[32] presents a relatively simplified AC/DC converter model and extends it into aMTDC model for state estimations. Both of them are developed for conventionalstate estimations, thus the usage of SCADA data makes them complex and withhigh nonlinearity. [33, 34] introduce a basic VSC model and a generic VSC-basedMTDC model for state estimation, respectively. [35] combines SCADA data withPMU data for hybrid state estimations for VSC-HVDC links.

Previous work on modeling FACTS for state estimation has been conductedin the past decade. [36, 37, 38, 39] present FACTS models for the conventionalstatic state estimations using SCADA data. [40, 41, 42] propose to use PMU datato conduct state estimations for the networks containing FACTS devices.

This thesis makes a great effort to develop network models for classic HVDC,VSC-HVDC, and FACTS that are suitable for PMU-based state estimation, in orderto achieve the utmost goal of wide-area hybrid AC and DC state estimation. Moreover,as these technologies play an important role in improving system controllability andflexibility, their real-time performance during transients also needs to be monitoredfor on-line operations. To this end, this thesis presents the pseudo-dynamic modelingapproach that can accurately represent devices with dynamical properties in bothsteady state and transient conditions.

2.4 Static, forecasting-aided, and dynamic SEs

Conventional static state estimations are based on the assumption that the system isoperating under normal conditions, known as quasi-static regime, where the systemvaries smoothly and slowly. Thus, static state estimations construct measurementmodels based on a single scan of measurements. On the other hand, forecasting-aidedstate estimations (FASEs) and dynamic state estimations (DySEs) gain redundantinformation in addition to the measurement model, via applying measurementsover consecutive instants and using a system model that considers time evolution,respectively.

2.4.1 Forecasting-aided state estimationForecasting-aided state estimation extracts valuable information from a consecutivesuccession of static states evolving in time. For this reason, it was called trackingstatic state estimation in early literature. The redundant information can help toenhance network observability analysis, overcome data missing, detect and correctbad data, and process network configuration and parameter errors.

The state forecasting process is formulated as a transition model, where theforecasted state is represented by the sum of the estimated state at previous time-step,

2.4. Static, forecasting-aided, and dynamic SEs 17

the trend behavior of the state trajectory, as well as the modeling uncertainty. Themeasurement model remains the same with the static state estimations. Its objectivefunction endures the weighted least squares (WLS) format while incorporates theforecasted state as an additional set of virtual measurements, weighted according tothe forecasting error covariance matrix.

Moreover, the forecasted state can be substituted into the measurement modelto compute the forecasted measurement. The difference between the forecastedmeasurement vector and the received measurement vector is defined as the innovationvector [43], which can be used for innovation analysis. High innovation acts as anindicator for possible anomalies, such as bad data, network configuration error,sudden change, network parameter error, etc.

The key technique of FASEs is to find the parameters that fit the transitionmodel to the historical data. The FASEs concept was firstly proposed by [44] in1971. However the static time evolution model, i.e., transition model, was an oversimplified one: the most recent estimated state is used as one-step ahead forecasting.Reference [45] developed a more appropriate transition model, in which the Holt’s(two parameters) linear exponential smoothing method is converted into a statespace representation. Similar exponential smoothing techniques are also implementedin [46]. Computational intelligence tools such as patten analysis [47], artificial neuralnetworks (ANNs) [48], and fuzzy control methods [49] also gain increasing popularity.A more comprehensive related work can be found in [50].

2.4.2 Dynamic state estimation

System dynamic model describes a system’s behavior during transient periods,and it can be represented by differential equations for continuous systems. Asthe measurements are sampled discretely, this continuous dynamic model can bediscretized to be the network model. This network model uses the system’s dynamicbehavior to predict the next time-step state, in contrast, the network model of FASEuses historical data to predict the next time-step state.

The term “dynamic estimator” can be traced back to the early 1970s [51].However, the time update model was oversimplified with a strong assumption thatthe system was in a quasi steady state, so as to avoid any serious attempt tomodel the time behavior of the system states. Compared to static state estimations,dynamic state estimations did not receive much attention until the last decade, whichwas mainly due to four reasons [52]. First, static state estimations were sufficientfor most cases in terms of filtering measurements. Second, the objective of dynamicstate estimations were not always clear. For instance, in early literature the statesfor dynamic state estimations were voltage magnitude and angle, the same with theconventional static state estimations. However, for many cases voltage magnitudeand angle are not dynamic states, and voltage phasors at different buses are noteven independent from each other. Third, the dynamic modeling was not alwaysavailable and practical. Lastly, the computational burden was significantly large.

From the last decade, the term “dynamic state estimation” refers to the estimation


for dynamic states and parameters (e.g., generator rotor angles and generator speed)in power system using an appropriate time update model. The Kalman filter family isthe most widespread technique for formulating and solving dynamic state estimations.It provides an efficient computational (recursive) means to estimate the state of aprocess, in a way that minimizes the mean of the squared error [53]. Its objectiveis to estimate a process by using a form of feedback control: the filter estimatesthe process state at some time and then obtains feedback in the form of (noisy)measurements. As such, the equations for the Kalman filter fall into two groups:time update equations and measurement update equations. For power system stateestimation applications, the time update equations contain critical low frequencyelectromechanical dynamics. They are responsible for projecting forward the currentstate and error covariance estimates to obtain a prediction for the next time step.The measurement update equations are formulated in the form of the measurementmodel that is used for the conventional static state estimations to correct theprediction.

For a nonlinear dynamic system, extended Kalman filter (EKF) [54] is oftenused for solving dynamic state estimation through linearization. To mitigate fist-order approximation errors of the EKF, iterative EKF (IEKF) [55] linearizes thesystem nonlinear equations iteratively to compensate for the higher-order terms.A derivative-free alternative to EKF is the unscented Kalman filter (UKF) [56].Its idea is to produce several sampling points (Sigma points) around the currentstate estimate based on its covariance. Then, it propagates these points through thenonlinear map to get more accurate estimation of the mean and covariance of themapping results.

The dynamic state estimations enable a dynamic view of power system in thecontrol room. The estimated dynamic states by DySE can help to improve thereal-time control scheme and provide an initialization for a look-ahead dynamicsimulation [54]. In addition, the prediction capacities of DySE automatically ensuresystem observability even in some stringent condition.

2.5 Transmission system, and distribution system SEs

Most state estimation algorithms are designed for improving the awareness oftransmission systems as they establish the backbones of power systems. However, thissituation has been gradually changed as more timely and accurate state estimationsfor distribution systems are needed to facilitate the demand response (DR) and thetwo-way power flow.

Since the last decade, the integration of intermittent renewable energy sources,the market deregulation and demand response, as well as the penetration of electricvehicles have increased the overall energy effectiveness, however, at the same time,they stress the grid, complicate the system operation, and introduce more possibilitiesfor frauds. This condition needs to be improved via distribution automation. Stateestimation for distribution systems enables to provide an essential awareness of

2.5. Transmission system, and distribution system SEs 19

distribution system operation so as to support the distribution automation.Compared to transmission systems, distribution systems lack proper infrastruc-

ture for state estimations. The number of installed telemetered devices is limited.Load data obtained from historical local profile and existing automated meterreading (AMR) devices have limited accuracy. Moreover, three phase imbalanceand low X/R ratios complicate the measurement functions, making the decoupledestimator not suitable.

Pioneer works on distribution system state estimation (DSSE) were conductedsince 1990s [57]. However, research and application in this field were not been trulybrought into fruition, probably due to the lack of proper infrastructure. Three typesof data sources for the distribution system state estimation are listed below [58].

• Equipment connectivity status from automated mapping and facility manage-ment (AM/FM) systems, geographical information system (GIS) and outagemanagement systems (OMS). An interface function has to be implemented toconvert these connection “maps” and attribution data from AM/FM/GIS tothe operational database structure via common information model (CIM) [59].

• Real-time voltage, current and power flow measurements from distributionautomations, SCADA, intelligence electronic devices (IEDs), and PMUs. Abig issue for these real-time measurement is the time skew problem. Unsyn-chronized measurements have to be incorporated to a same time reference.

• Customer interval demands and distributed energy resources (DERs) outputdata from customer information system (CIS) and meter data management sys-tem (MDMS). Based on the customer billing data and typical load profiles, cus-tomer load curve with stochastic contents are utilized as pseudo-measurementto improve the observability of the estimation. Advanced techniques, such asGaussian mixture model (GMM) [60] and artificial neural network (ANN) areproposed for modeling the load probability. Moreover, the smart meter datacan also be taken as pseudo-measurements due to its less frequent update.

Due to the distinct features of distribution systems, its state estimation schemesto some extent are different from transmission system state estimations (TSSEs)in terms of network model and objective function. For instance, in TSSEs, powersystem state estimation generally assumes the system operates in steady state underbalanced conditions. This assumption allows to use the positive sequence equivalentcircuit for modeling the entire power system. However, in practice, power flowimbalances often appear in distribution systems. To this end, three-phase stateestimators [57, 61] are more suitable to handle unbalanced power flow issues. Inaddition, due to the high fluctuations of the two-way power flows in distributionsystems, a Bayesian network model [62] is proposed for DSSE.

The objective function for DSSEs can be formulated as a basic weighted leastsquares (WLS) problem. Feeder branch currents in rectangular coordinates arechosen as the state variables as it is computationally efficient for radial networks. It


can also be formulated as a constrained optimization problem [63], or a nonlinearoptimization problem, such as a hybrid particle swarm optimization problem [64].Robust state estimations are also applied for distribution systems. To obtain astatistically robust load estimation, machine learning algorithms can also be used[65]. An M-estimator combining WLS and weighted least absolute value (WLAV) isproposed in [66] to suppress the effect of bad data. DSSEs can also be formulatedusing forecasting-aided or dynamic state estimation techniques, such as iteratedKalman filter (IKF) [67], extended Kalman filter (EKF), or unscented Kalman filter(UKF) [56]. Distributed or multi-area state estimations are also attractive candidatesfor DSSE solutions.

2.6 Centralized, distributed, and multi-area SEs

Power system state estimation is traditionally formulated and computed in a cen-tralized fashion at regional control centers. While now the deregulation of electricitymarket requires to monitor the power system over a very large geographical area. Inthis context, decentralized state estimation schemes can enhance the computationalperformance and the reliability of the estimation algorithms. But at the same timethey require more efficient and reliable communication techniques and need to solvetime skewness issue.

Both distributed and multi-area state estimations focus on interconnected sys-tems. The difference between them mainly lies on the structure of the state vector[68]: in the distributed state estimations several nodes or areas estimate a commonstate/parameter vector through local collaborations; while in the multi-area stateestimations the measurements of each area only relate to a small part of the wholestate/parameter vector. Multi-area state estimations can be formulated as either ahierarchical process [69] or a fully distributed manner [70].

Chapter 3

Conventional state estimations and testsystems

The previous chapter gives a general overview on state estimation problems. Thischapter will focus on the formulation and derivation of state estimation methods,particularly the formulation and solution of the conventional state estimations. Inaddition, the test systems that are developed and implemented for case studiesare introduced. All of the aforementioned contents provide a background for thefollowing chapters.

3.1 Conventional state estimations

Power system state estimation aims to find the best match between the real-time measurements and the system states, i.e. voltage phasors at buses. Thereby,most state estimators firstly formulate the mathematical model that describes therelationship between the system states and the measurements as:

z = h(x) + e, (3.1)

where z ∈ Rm is the measurement vector, x ∈ Rn is the unknown true state vector;m and n are the numbers of measurements and states (m ≥ n), respectively. h :Rn → Rm is a function relating the measured quantities to the state variables, whichis called the network model. e ∈ Rm is the unknown measurement error vector. As(3.1) contains measurement vector, it is usually called the measurement/observationmodel.

3.1.1 Measurement’s distribution: GaussianTo solve state estimation problems requires selecting an x that makes the observedz most likely to be observed, in other words, to find x that maximizes the likelihoodof the observed measurements z, i.e., maximum likelihood estimate (MLE). Asdifferent measurement sources can possess different likelihood functions, which are

21

22 Conventional state estimations and test systems

often defined by probability density function (PDF), we need to take measurement’sprobability distributions into account when selecting suitable state estimationalgorithms.

Most literature presumes the measurement noise has a Gaussian distribution, i.e.,e = [e1, e2, · · · , em] and ek ∼ N (0, σ2

i ), where N (0, σ2i ) is a Gaussian distribution

with mean 0 and covariance σ2i , ∀i = 1, 2, · · · ,m. For the ease of notation, let R

denote the diagonal covariance matrix of the measurement noises, which are assumedto be independent random processes. Thus the measurement z also has a Gaussiandistribution with the mean of h(x) and the same covariance matrix R. Then theprobability (also called likelihood) of observing z given state x can be computed as:

m∏i=1N (zi|h(x)i, σ2

i ) =m∏i=1

1√2πσ2

i

e− [zi−h(x)i]2

2σ2i , (3.2)

where h(x)i is the ith element of h(x), σ2i is the variance of the corresponding

ith measurement zi. In order to perform the optimization of the above likelihoodfunction, (3.2) usually is rewritten into logarithm as

L = ln

m∏i=1

1√2πσ2

i

e− [zi−h(x)i]2

2σ2i

= −1

2

m∑i=1

[zi − h(x)i]2

σ2i

− m

2 ln(2π)−m∑i=1

ln(σi),

(3.3)where ln(·) is the natural logarithm. Thus the maximization of the log-likelihoodfunction is transformed into the minimization of

∑mi=1

[zi−h(x)i]2σ2i

, which is exactlythe formulation of weighted least squares (WLS) problem [16]. This also implicatesthat WLS solution is equivalent to the MLE for minimizing an L2-norm cost function.

3.1.2 Objective function: weighted least squaresBasic WLS as derived in (3.3) serves as the most common objective function forsolving static state estimation problems. It is formulated as:

minimizex

J(x) = 12

m∑i=1

[zi − h(x)i]2

σ2i

= 12[z− h(x)]TR−1[z− h(x)]. (3.4)

In addition to the field measurements, there are two other kinds of measure-ments, known as the pseudo-measurement and the virtual measurement. Pseudo-measurements are manufactured data, such as generator output or substation loaddemand, that are based on the historical data or the dispatcher’s objective guesses.Virtual measurements are the information that does not require metering, e.g.,zero injection at a switching station. Three kinds of measurements own differentvariances, particularly the variance of the zero injection is zero given the correcttopology information, thereby the covariance matrix could become ill-conditioned.Specially when numerically solving WLS, the ill-condition problem becomes morestringent.

3.1. Conventional state estimations 23

In order to overcome the ill-conditioning, an alternative formulation was proposedin [71] where the virtual measurements are represented by equality constrains.Another method to improve numerical stability is Hachtel’s augmented matrixmethod [72], where the residuals are defined as independent variables.

WLS is the most common method used for power system state estimation owingto its computational efficiency and stability. However, it is sensitive to outliers,and a single outlier can distort the estimation results. To overcome this drawback,robust estimators are proposed, one of which is the weighted least absolute value(WLAV). Their objective functions are mathematically formulated and programmedas a mixed-integer non-linear problem (MINLP) [73].

3.1.3 Numerical solutionsThe numerical solution presented herein is based on the basic WLS formulationin (3.4). Its optimal solution can be found when the partial derivative of the objectivefunction equals to zero:

∂J(x)∂x = g(x) = −H(x)TR−1[z− h(x)] = 0, (3.5)

whereH(x) = ∂h(x)

∂xis the Jacobian matrix of the network model h(x). In order to find the root of thenonlinear function g(x) = 0, iterative numerical methods are used. For instance,Newton Method, is a powerful technique for solving equations numerically. For eachiteration, the linear approximation equation is formulated as:

∂g(xk)∂xk

(xk+1 − xk) = g(xk+1)− g(xk), (3.6)

where k denotes the number of iteration and

∂g(xk)∂xk

= −[∂H(xk)∂xk

]TR−1[z− h(xk)] + [H(xk)]TR−1[H(xk)].

Generally, the basic WLS ignores the second-derivative terms, namely the partialderivative ∂H(xk)

∂xk . Thus the updating rule in (3.6) is rewritten as:

G(xk)∆xk = [H(xk)]TR−1[z− h(xk)], (3.7)

where G(xk) = H(xk)TR−1H(xk) denotes the gain matrix and (3.7) is calledthe normal equation. Ideally, the normal equation can be numerically solved viacalculating the inverse of the gain matrix G(xk). However, as aforementioned, dueto the ill-condition of the weight matrix, the gain matrix can also become singular.Therefore, more robust linear algebra algorithms are implemented.


As the gain matrix G(x) is usually a symmetric, positive definite matrix, itcan be factorized using the Cholesky decomposition [74]. Cholesky decompositionis a special case for LU decomposition. QR decomposition, which is also knownas the orthogonal transformation method [75, 76], is numerically more robustthan LU decomposition. Additionally, the hybrid method [77] combines both QRdecomposition (orthogonal transformation) and Cholesky decomposition.

3.2 Test systems

There are four test systems—9-bus system, KTH Nordic 32 system, 6-bus sys-tem, and VSC-based HVDC transmission link—used in this thesis to validate theproposed network models and state estimation algorithms. They are modified orre-implemented to fit various simulation needs for different case studies.

3.2.1 9-bus systemWSCC 3-generator 9-bus test system is already available in Power System AnalysisToolbox (PSAT) [78]. Its one-line diagram is shown in Fig. 3.1. This 9-bus testsystem is modified and applied for three case studies.

• Linear network model of classic HVDC in Chapter 4, where a classic HVDClink replaces the original AC line between bus 7 and bus 8.

• Nonlinear network models of FACTS in Chapter 5, where different FACTSdevices are placed in different positions according to their specific functions.Generally, shunt devices are installed at bus 8, and series devices are installedon the line between bus 5 and bus 4.

• Pseudo-dynamic model of STATCOM in Chapter 6, where a STATCOM isinstalled at bus 8.

Bus5

Bus4

Bus8

Bus6

Bus9Bus7 Bus3Bus2

Bus1

Figure 3.1: 9-bus test system

3.2. Test systems 25

3.2.2 KTH-Nordic 32 system

The KTH-Nordic 32 system is a conceptualization of the Swedish power system andits neighbors. Its predecessor is the CIGRE “Nordic 32A” test network developedby K. Walve [79] and a system data set was proposed by T. Van Cutsem [80]. In[81] some adjustments to the system model and its parameters were made, sincethen the model is referred to as the KTH-Nordic32 system. For more details, thereader is referred to [81]. This system has already been implemented in PSAT by Y.Chompoobutrgool and its one-line diagram is shown in Fig. 3.2. This KTH-Nordic32 test system is modified and applied for three case studies.

• Linear network model of classic HVDC in Chapter 4, where a classic HVDClink replaces one of the original AC lines between bus 38 and bus 40.

• Nonlinear network models of classic HVDC in Chapter 5, where a classicHVDC link replaces the original AC line between bus 36 and bus 41.

• Pseudo-dynamic model of STATCOM in Chapter 6, where a STATCOM isinstalled at bus 43.

3.2.3 6-bus hybrid AC/DC system

The 6-bus hybrid AC/DC test system is designed with parallelized AC line andDC link so as to improve its robustness under stringent perturbations. Its one-linediagram is shown in Fig. 3.3. This test system is applied for two case studies.

• Nonlinear network model of classic HVDC in Chapter 5, where the classicHVDC link is placed between bus 3 and bus 4. This model is implemented inPSAT where the classic HVDC link model is available.

• Nonlinear network model of VSC-HVDC in Chapter 5, where the VSC-HVDClink is placed between bus 3 and bus 4. This model is implemented in mat-lab/Simulink where the VSC model is available.

3.2.4 VSC-based HVDC transmission link

There are many VSC-HVDC simulation models proposed in literature. However,in order to make the test system accessible by other researchers, the VSC-BasedHVDC Transmission Link model provided by matlab R2013b/Simulink is used togenerate the synthetic measurements for the case study in Chapter 6. A detaileddescription of the model and control strategy can be found in [82].


21 23

22 24

25 26

33 32

37

38 39

36

4140

43 42 45

46

4829 30

49

50

27 31

44

47

28

34

35

51

52

NORTH

EQUIV.

SOUTH

CENTRAL

1G

2G

3G

4G 5G

6G

7G

8G

9G

10G

11G

12G

13G14G

15G

16G

17G

18G

19G

20G

400 kV220 kV130 kV15 kV

SL

Figure 3.2: KTH-Nordic 32 test system [81]

3.2. Test systems 27

Busl Bus3 Bus4 Bus2

us us6

Figure 3.3: 6-bus test system

Figure 3.4: VSC-Based HVDC Transmission Link model [82]

3.2.5 Synthetic measurement generationIn all the case studies, synthetic measurements used for off-line state estimationsare obtained by running the time-domain simulations of the test systems in PSATor matlab/Simulink, depending on where the test system is implemented. Thesimulation results are then re-sampled with the rate of 20ms to imitate the PMUdata. Unless otherwise stated, for off-line state estimations, no measurement noise isadded because we focus on investigating the influence of network model on estimationaccuracy. Moreover, all the weights for network equations and measurements areassumed to be 1, and full measurement observability is satisfied so as to avoid theinfluences of weighting and measurement redundancy on estimation results.

Chapter 4

Linear network models

Using synchronized phasor measurements allows to formulate the state estimationproblem as a linear one. This chapter presents the linear network and measurementmodels for both AC transmission network and classic HVDC link.

As introduced in Chapter 3, measurement model for the conventional stateestimations is generally formulated as

z = h(x) + e. (4.1)Conventional state estimations are performed using measurements such as bus

voltage magnitude, active power flow and injection, reactive power flow and injection,current flow, etc. And the state vector includes bus voltage magnitude and angle.Therefore, the network model h(x) is a nonlinear function based on the power flowmodel. In contrast, as PMUs can measure voltage and current phasors directly, itis possible to formulate a linear network model using Kirchhoff’s circuit law. Butthe state variables, usually bus voltage phasors, need to be adjusted in rectangularcoordinates [19].

The linear measurement model for PMU-based state estimations is given byz = Ax + e, (4.2)

where A ∈ Rm×n is a constant matrix constructed based on Kirchhoff’s circuit law.With the proposed linear measurement model, the basic WLS objective function

(3.4) becomesminimize

xJ(x) = 1

2(z−Ax)TR−1(z−Ax). (4.3)

Differently from iteratively solving nonlinear WLS problems, linear WLS problemshave explicit closed-form solutions. The Moore-Penrose inverse, Cholesky decompo-sition, QR decomposition, singular value decomposition (SVD) are applicable tosolve linear WLS [83, 84], depending on how well-conditioned the matrices A andR are.

This chapter will focus on constructing matrix A from physical models ofAC transmission network and classic HVDC link [85]. Sequentially their linearmeasurement models for static PMU-based state estimation are formulated.

29

30 Linear network models

4.1 AC transmission network

4.1.1 Network model

An AC transmission network is a collection of AC lines/branches that can berepresented by π-models. Similarly to [22, 23], an AC transmission line comprises aline with series admittance and shunt admittance, as well as a transformer. All thesecomponents are enough to formulate the AC network model, which describes therelation between complex voltages at buses and complex currents flowing throughthe lines adjacent to these buses. This model is similar to those in [16, 78], buttaking the shunt admittances and transformers into consideration.

+

-

+

-

Rectifier Inverter

fVtVxV

fI tIxI

rV iV

Figure 4.1: An AC transmission line model [16, 78]

As shown in Fig. 4.1, the subscript f denotes the bus where current flows from(i.e., sending end) and t is the bus where current flows to (i.e., receiving end). Theith line is represented by a series admittance yi and shunt admittance yi0 in perunit. An ideal transformer is represented by the off-nominal tap ratio a : 1. In thecase of phase shifting transformers, a is a complex number. Consider a fictitiousbus x between the ideal transformer and the line series admittance, thus, for theassumed current directions, we have

Vx = 1aVf , Ix = a∗If .

The current Ix and It are given by

Ix = −yiVt + (yi + 12yi0)Vx, It = (yi + 1

2yi0)Vt − yiVx.

Substituting for Ix and Vx, we have

a∗If = −yiVt + 1a

(yi + 12yi0)Vf

It = (yi + 12yi0)Vt −

1ayiVf .

4.2. Classic HVDC link 31

Writing the above equations in a matrix form

[If

It

]=

yi+ 12yi0a2 − yi

a∗

−yia yi + 12yi0

︸︷︷︸

YAC

[Vf

Vt

]. (4.4)

More generally,I = YACV. (4.5)

Matrix YAC is essentially an admittance matrix that connects the bus voltagephasors with the measured line current phasors. Note that YAC is only part of thecoefficient matrix A in the measurement model as there are also measurements ofvoltage phasors.

4.1.2 Measurement model

For the AC-only state estimations, the state vector contains complex voltage phasorsat all buses while the measurement vector contains both current and voltage phasors.Thus the measurement model of an AC network is given by[

zVzI

]=[

Y1

YAC

]︸︷︷︸

A

V +[

eVeI

], (4.6)

where

Y1 =

1

. . .1

.Note that the voltage measurements are usually not available for all the buses, thusmatrix Y1 usually has less rows than columns.

4.2 Classic HVDC link

4.2.1 Network model

A simplified classic HVDC link model is shown in Fig. 4.2. It integrates DC voltagesand currents with AC voltages at the terminal buses, which can be measured byPMUs. This model avoids using active and reactive power variables to constructrelations.


+

-

+

-

Rectifier Inverter

fVtVxV

fI tIxI

rV iV

Figure 4.2: A simplified classic HVDC link model [78]

The subscript r refers to the rectifier terminal of the classic HVDC link and irefers to the inverter terminal. The network equations are given by [78]:

Vrdc = 0.995 ∗ 3√

2π ar|Vr| cosα− 3

πXrIdc,

Vidc = 0.995 ∗ 3√

2π ai|Vi| cos δ − 3

πXiIdc,

Idc = 1Rdc

(Vrdc − Vidc),(4.7)

where Vrdc, Vidc, and Idc refer to the rectifier side DC voltages, inverter side DCvoltages, and DC currents, respectively; |Vr| and |Vi| are the AC voltage magnitudesat the rectifier terminal bus and at the inverter terminal bus, respectively; ar andai are the tap ratios; α and δ are the firing angle and extinction angle; Xr and Xi

are the transformer reactances; Rdc is the resistance of the DC connection.For the hybrid AC/DC state estimation, in addition to the AC bus voltage

phasors, the rectifier side DC voltage, inverter side DC voltage, DC current andproducts |Vr| cosα and |Vi| cos δ are also state variables. Choosing |Vr| cosα and|Vi| cos δ as state variables is based on the context of developing a linear model.Therefore, the linear model of (4.7) is formulated as follows

000

=

0.995 ∗ 3

√2

π ar −1 − 3πXr

0.995 ∗ 3√

2π ai −1 − 3

πXi

1Rdc

− 1Rdc

−1

︸︷︷︸

YDC

|Vr| cosα|Vi| cos δVrdc

Vidc

Idc

.

(4.8)Since Vr and Vi are state variables of the AC network, they can be obtained afterevery estimation computation. Then, cosα and cos δ can be derived.

4.2.2 Measurement model for hybrid AC/DC systems

The measurement model of a hybrid AC/DC system combines the AC networkmodel and the classic HVDC link model as

4.3. Case study 33

zVz|Vr| cosαz|Vi| cos δ

zV rdczV idczIdczI000

︸︷︷︸Measurements

=

Y1

11

11

1YAC

YDC

︸︷︷︸

A

V|Vr|cosα|Vi|cos δ

Vrdc

Vidc

Idc

︸︷︷︸

States

+

eVe|Vr| cosαe|Vi| cos δ

eV rdceV idceIdceI

edc1edc2edc3

︸︷︷︸

Residuals

(4.9)

4.3 Case study

The 9-bus system, the KTH-Nordic 32 system and their modified hybrid AC/DCsystems are utilized for this case study. The 9-bus hybrid AC/DC system replacesthe original AC line between bus 7 and bus 8 in the 9-bus system with a classicHVDC link. The KTH-Nordic 32 hybrid AC/DC system replaces the original ACline between bus 38 and 40 in the KTH-Nordic 32 system with a classic HVDClink. Other parameters of the hybrid AC/DC systems remain the same as in theiroriginal AC systems. Unless otherwise stated, it is assumed that PMUs are installedat all the buses and the DC measurements (zV rdc, zV idc, zIdc, cosα and cos δ) aresampled synchronously and timestamped by GPS at the same rate as the PMUdata.

Estimation results are shown in such sequence: 9-bus system, 9-bus hybridAC/DC system, KTH-Nordic 32 system, and KTH-Nordic 32 hybrid AC/DC system.For each case, we show the results from two perspectives: all quantities at onetime instant (snapshot), and one quantity in a time series. Sequentially, the effectof having less DC measurements and the effect of measurement noises on stateestimation performance are investigated.

4.3.1 LSE for the 9-bus systemThe state vector for the 9-bus AC system is organized as follows

xAC = [V1 V2 V3 V4 V5 V6 V7 V8 V9]T .

The time domain simulation is performed in PSAT to generate the synthetic mea-surements for 10 s, and a 0.1 p.u. (10%) load increase is applied at bus 8 at t = 2 s .


Figures 4.3 and 4.4 show the estimation results for a single snapshot and for a timeseries, respectively. In the simulation environments, the true values of the statesare known. Consequently, the estimation error (e.g. |V |est − |V |true) and estimationresidual (e.g. |V |est − |V |meas) are equivalent. For each subfigure, the true value,the measurement and the estimated result are shown on the top and the estimationerror/residual is placed on the bottom. The estimation errors/residuals for the 9-bussystem are lower than 10−13 p.u. (or deg).

1 2 3 4 5 6 7 8 90.95

1

1.05

Bus No.

|V| (

p.u

.)

Vmag−true

Vmag−measVmag−est

1 2 3 4 5 6 7 8 90

2

4

6x 10 −16

Bus No.

Error (

p.u

.) Vmag−residual−error

(a) |V| at all buses

1 2 3 4 5 6 7 8 9−10

0

10

Bus No.

θ(d

eg)

Vang−true

Vang−measVang−est

1 2 3 4 5 6 7 8 90

0.5

1x 10 −14

Bus No.

Vang−residual−error

Erro

r (de

g)

(b) θ at all buses (bus 7 is set as the reference)Figure 4.3: LSE for the 9-bus system in a single snapshot

0 100 200 300 400 5001.01

1.015

1.02

time step

|V| (

p.u

.)

Vmag−trueVmag−measVmag−est

0 100 200 300 400 5000

1

2x 10 −15

time step

Error (

p.u

.) Vmag−residual−error

(a) |V| at bus 8

0 100 200 300 400 500−3.5

−3

−2.5

time step

θ(d

eg)

Vang−trueVang−measVang−est

0 100 200 300 400 500−5

0

5x 10 −13

time step

Vang−residual−errorError (deg)

Erro

r (de

g)

(b) θ at bus 8 (bus 7 is set as the reference)Figure 4.4: LSE for the 9-bus system in multiple snapshots

4.3.2 LSE for the 9-bus hybrid AC/DC system with a classicHVDC link

The state vector for the 9-bus hybrid AC/DC system is organized as follows

xAC/DC =[xAC |V7| cosα |V8| cos δ V7dc V8dc Idc

]T.

The same test scenario in Section 4.3.1 is implemented herein. Figures 4.5 and 4.6show the estimation results.

4.3. Case study 35

1 2 3 4 5 6 7 8 90.85

0.9

0.95

1

1.05

Bus No.

|V| (

p.u.

)


1 2 3 4 5 6 7 8 90

2

4

6x 10

−16

Bus No.

Error (

p.u.

)

Vmag−residual−error


1 1.5 2 2.5 30.95

1

1.05

Vrdc Vidc Idc

(p.u

.)

Hvdc−trueHvdc−measHvdc−est

1 1.5 2 2.5 30

0.5

1x 10

−13

Vrdc Vidc Idc

Erro

r (p.

u.)

Hvdc−residual−error

(b) DC state variables (before the perturbation)Figure 4.5: LSE for the 9-bus hybrid AC/DC system with a classic HVDC link in a singlesnapshot

0 100 200 300 400 5000.882

0.884

0.886

0.888

time step

|V| (

p.u.

)


0 100 200 300 400 5000

1

2

3

4x 10

−16

time step

Error (

p.u.

)


(a) |V| at bus 8

0 100 200 300 400 5000.98

0.982

0.984

0.986

0.988

time step

Vid

c (

p.u.

)


0 100 200 300 400 50010

−20

10−10

100

time step

Err

or (p

.u.)


(b) DC voltage on the inverter side (close tobus 8)

0 100 200 300 400 5000.9999

1

1

1.0001

1.0001

time step

I dc (

p.u.

)


0 100 200 300 400 50010

−15

10−10

10−5

time step

Erro

r (p.

u.)


(c) DC current

0 100 200 300 400 5000.911

0.912

0.913

0.914

0.915

time step

cos

α

cosα−turecosα−meascosα−est

0 100 200 300 400 50010

−20

10−10

100

time step

Error

cosα−residual−error

(d) cosαFigure 4.6: LSE for the 9-bus hybrid AC/DC system with a classic HVDC link in multiplesnapshots

In Fig. 4.5b, the DC states are computed before applying the perturbation, andthus the estimated states have small residuals in the order of 10−14 p.u. However,


as shown in Figs. 4.6b, 4.6c, and 4.6d, when the system is subject to a perturbation,the estimation residuals for the DC states increase owing to the limitation of thestatic network model. The static state estimator does not include a dynamic DClink model, thus its static network equations for the classic HVDC link may nothold during system dynamics, especially when its control scheme is active. This canalso be confirmed by noting that the estimation residuals decrease as the system’soscillations decay. Regardless, the static estimation residuals for the DC states arestill within an acceptable range. For AC states, the hybrid AC/DC state estimatorperforms as accurate as the AC-only state estimator.

4.3.3 LSE for the KTH-Nordic 32 systemThe state vector for the KTH-Nordic 32 system is organized as follows

xAC = [V1 V2 . . . V51 V52]T .

The time domain simulation is performed in PSAT to generate the synthetic mea-surements for 10 s, and a 0.5 p.u. (10%) load increase is applied at bus 40 at t = 2s. Figures 4.7 and 4.8 show the estimation results, which remain the same accuracyas that in the 9-bus system.

0 10 20 30 40 500.8

0.9

1

1.1

1.2

Bus No.

|V| (

p.u.

)


0 10 20 30 40 500

0.5

1

1.5

2x 10

−15

Bus No.

Error (

p.u.

)



0 10 20 30 40 50−100

−50

0

50

100

Bus No.

θ(d

eg)


0 10 20 30 40 50−4

−2

0

2

4x 10

−14

Bus No.


Erro

r (de

g)

(b) θ at all buses (bus 38 is set as the reference)

Figure 4.7: LSE for the KTH-Nordic 32 system in a single snapshot

4.3.4 LSE for the KTH-Nordic 32 hybrid AC/DC system with aclassic HVDC link

The state vector for the KTH-Nordic 32 hybrid AC/DC system is organized asfollows

xAC/DC = [xAC |V38| cosα |V40| cos δ V38dc V40dc Idc]T .

The same test scenario in Section 4.3.3 is applied here. Figures 4.9 and 4.10 show theestimation results. Observe that in Fig. 4.10d the true values of cosα are constant

4.3. Case study 37

0 100 200 300 400 5000.885

0.89

0.895

0.9

0.905

time step

|V| (

p.u.

)


0 100 200 300 400 5000

0.5

1

1.5

2x 10

−15

time step

Error (

p.u.

)


(a) |V| at bus 40

0 100 200 300 400 500−20

−19.5

−19

time step

θ(d

eg)


0 100 200 300 400 500−1

−0.5

0

0.5

1x 10

−13

time step


Erro

r (de

g)

(b) θ at bus 40 (bus 38 is set as the reference)Figure 4.8: LSE for the KTH-Nordic 32 system in multiple snapshots

because the classic HVDC link’s control scheme is active, however its estimatedvalues have a variation over time. This is because (i) this linear static model lacksrepresentation of the classic HVDC link’s dynamics as explained in Section 4.3.2,especially for its control scheme; (ii) cosα is not an independent state in this linearstate estimator and it is derived from the estimated states |Vr| cosα and Vr, thustheir estimation accuracies will directly influence cosα. The same condition alsoapplies to cos δ.

0 10 20 30 40 500.8

0.9

1

1.1

1.2

Bus No.

|V| (

p.u.

)


0 10 20 30 40 500

0.5

1

1.5

2x 10

−15

Bus No.

Erro

r (p.

u.)



1 1.5 2 2.5 3

2.45

2.5

2.55

2.6

2.65

Vrdc Vide Idc

(p.u

.)


1 1.5 2 2.5 30

1

2

3

4x 10

−12

Vrdc Vide Idc

Erro

r (p.

u.)


(b) DC states (before the perturbation)Figure 4.9: LSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDC linkin a single snapshot

4.3.5 Effect of less DC measurements

From (4.9) we can observe that as long as two out of five DC measurements (zV rdc,zV idc, zIdc, zcosα and zcos δ) and the AC terminal voltage magnitudes |Vr| and |Vi|are provided, the other three remaining DC states can be computed. This brings a


0 100 200 300 400 5000.84

0.845

0.85

time step

|V| (

p.u.

)


0 100 200 300 400 5000

1

2

3x 10

−15

time step

Erro

r (p.

u.)


(a) |V| at bus 40

0 100 200 300 400 5002.36

2.38

2.4

2.42

time step

Vid

c (

p.u.

)


0 100 200 300 400 5000

1

2

3x 10

−3

time step

Erro

r (p.

u.)



0 100 200 300 400 5002.3

2.35

2.4

2.45

time step

I dc (

p.u.

)


0 100 200 300 400 5000

1

2

3x 10

−6

time step

Err

or (p

.u.)


(c) DC current

0 100 200 300 400 5000.996

0.9965

0.997

0.9975

time step

cos

α

cosα−turecosα−meascosα−est

0 100 200 300 400 5000

0.5

1x 10

−3

time step

Error


(d) cosαFigure 4.10: LSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDClink in multiple snapshots

great advantage that only few DC measurements are needed during normal operatingconditions as AC measurements can be obtained using PMUs.

When a DC measurement is lost, the corresponding rows in the measurementvector, in the residual vector, and in the A matrix need to be removed. As longas the number of rows is equal or larger than the number of columns in A, all thestates can be solved out through the WLS. The more rows that A has, the higherredundancy that the estimation owns.

The same test scenario in Section 4.3.2 is applied herein. Only the DC measure-ments of cosα and cos δ are provided. As shown in Fig. 4.11, the residuals of theDC voltages and DC current increase while the residuals of cosα decrease. Evenwith a limited number of DC measurements it is still possible to obtain acceptableestimation results for the whole AC/DC grid.

4.3. Case study 39

1 1.5 2 2.5 30.95

1

1.05

1.1

Vrdc Vide Idc

p.u.

Hvdc−trueHvdc−est

1 1.5 2 2.5 30

0.5

1

1.5x 10

−9

Vrdc Vide Idc

Erro

r (p.

u.)


(a) DC state variables in a single snapshot(before the perturbation)

0 100 200 300 400 5000.975

0.98

0.985

0.99

time step

Vid

c (

p.u.

)


0 100 200 300 400 50010

−10

10−5

100

time step

Error (

p.u.

)



0 100 200 300 400 5000.99

1

1.01

1.02

1.03

time step

I dc (

p.u.

)


0 100 200 300 400 50010

−10

10−5

100

time step

Erro

r (p.

u.) Hvdc−residual−error

(c) DC current

0 100 200 300 400 5000.911

0.912

0.913

0.914

0.915

time step

cos

α

cosα−turecosα−est

0 100 200 300 400 5001.64

1.645

1.65

1.655

1.66x 10

−11

time step

Error


(d) cosαFigure 4.11: Effect of less DC measurements: LSE for the 9-bus hybrid AC/DC systemwith a classic HVDC link (no measurements of Vrdc, Vidc, and Idc are provided)

4.3.6 Effect of measurement noiseGaussian white noise is added to the synthetic measurements used for the stateestimation. The signal-to-noise ratio per sample is 75 dB and the same test scenarioin Section 4.3.3 is applied here. Figure 4.12 shows the estimation residuals increaseto the order of 10−4 compared to 10−15 achieved when without noise. Nevertheless,this estimation accuracy is acceptable for the signal-to-noise ratio applied.


0 10 20 30 40 500.8

0.9

1

1.1

1.2

Bus No.

|V| (

p.u.

)


0 10 20 30 40 500

1

x 10−4

Bus No.

Error (

p.u.

)



0 10 20 30 40 50−100

−50

0

50

100

Bus No.

θ(d

eg)


0 10 20 30 40 50−0.02

−0.01

0

0.01

0.02

Bus No.

Erro

r (de

g)


(b) θ at all busesFigure 4.12: Effect of measurement noise: LSE for the KTH-Nordic 32 system in a singlesnapshot

4.4 Summary

A linear PMU-only state estimator for hybrid AC/DC systems has been introducedin this chapter. Using Kirchhoff’s circuit laws, the proposed linear AC network modelsimplifies the nonlinearities of the typical power flow network model used in theconventional state estimations. A simplified classic HVDC model is reconstructedinto a linear formulation. All the AC and DC states are considered simultaneouslyfor solving the linear least squares problem. After presenting the network and mea-surement models of the hybrid AC/DC state estimator, case studies are performed.It shows when the system is experiencing transient dynamics, the estimation resultsfor the DC states are not as good as that during normal operating conditions. Thisis mainly attributed to the linear and static characteristics of the proposed classicHVDC link model. Finally, the effect of having less DC measurements and the effectof measurement noise are discussed.

Chapter 5

Nonlinear network models

In the last chapter, we have observed the limitation of using linear state estimators,which lies on the restraints of restructuring a nonlinear network model into linearequations. Hence, this chapter will develop nonlinear network equations for PMU-based state estimations.

For the conventional state estimations, the network model h(x) represents thephysical relationships between the measurements from SCADA system and thedesired unknown states under system steady state. Equation (3.1) assumes that h(x)is well-developed and it perfectly reflects the system’s behavior. However, this idealcase is rarely true in real life as the system always operates around the steady-state,which is defined as the normal operating condition, resulting in a small deviationbetween the real value of the states and the value from the model. This deviationis the modeling uncertainty and it reflects the model’s limitations to capture thereality.

Due to the lack of representation of the modeling uncertainty in (3.1), a novelmeasurement model for PMU-based state estimations is proposed [86, 87, 88].It accounts for both modeling uncertainty (due to modeling imperfections) andmeasurement noise (due to instrumentation and environmental noise, communicationnoise, etc). This measurement model is formulated as:

e =[

uv

]=[

f(x)x− z

], (5.1)

where u ∈ Rn and v ∈ Rm are the modeling uncertainty and the measurement noise,respectively. They are independently distributed from each other and E(uvT ) = 0.Lastly, f(x) ∈ Rn is the restructured network model.

Accordingly, the objective function of the basic WLS (3.4) is adjusted to

minimizex

12

( n∑i=1

1Qii

u2i +

m∑j=1

1Rjj

v2j

), (5.2)

where both u and v are assumed to have a Gaussian distribution, i.e., u ∼ N (0,Q)

41

42 Nonlinear network models

and v ∼ N (0,R). Qii is the ith diagonal element of the modeling uncertainty’s covari-ance matrix and Q = E(uuT ); Rjj is the jth diagonal element of the measurement’scovariance matrix and R = E(vvT ).

In order to solve the above WLS problem, the normal equation (3.7) used foriteratively updating the state variables is adjusted to

G(xk)∆xk = [H(xk)]TWe(xk), (5.3)

where W is the weighing matrix with Qiis and Rjjs on its diagonal, and the gainmatrix G(xk) = [H(xk)]TWH(xk).

As described by (5.1), the network model equations are separated from themeasurements. The upper half of the error vector, i.e., u, represents the modelingerrors of the network equations, thus weights based on the confidence in the model’saccuracy are individually assigned to them. For the lower half of the error vector,i.e., v, the measurement errors are for the PMU measured quantities: bus voltagemagnitude |V| and angle θ, line current magnitude |I| and angle δ, and even otheruser-defined quantities, depending on what the PMUs are measuring.

The advantage of using (5.1) over (3.1) lies in the flexibility of granting differentweights to different network model equations because they have inherently differentaccuracies due to disparate uncertainties of the model’s parameters.

This chapter will focus on constructing nonlinear network models f(x) of ACtransmission network, classic HVDC link [86], VSC-HVDC [87], and FACTS devices[88]. Furthermore, different control modes for some components are attempted to beincluded into the network model for supplying redundant information. Sequentiallyindividual case study is conducted.

5.1 AC transmission network

The physical model of an AC transmission line is the same as that for the linear stateestimation. However, the magnitudes and angles of bus voltages and line currentsbecome independent state variables for the nonlinear PMU-based state estimation.Thus the state variable vector x for AC system is

xac = [|V| |I| θ δ]T

In addition, each network equation in (4.4) has to be expanded into two equations.Hence the network model f(x) for an AC transmission line is formulated as

uac = fac(x) =

|Vi||yfi| cos(θi + φfi)− |Vj ||yfj | cos(θj + φfj)− |If | cos δf|Vi||yfi| sin(θi + φfi)− |Vj ||yfj | sin(θj + φfj)− |If | sin δf|Vi||yti| cos(θi + φti)− |Vj ||ytj | cos(θj + φtj)− |It| cos δt|Vi||yti| sin(θi + φti)− |Vj ||ytj | sin(θj + φtj)− |It| sin δt

,(5.4)


whereyfi =

yi + 12yi0

a2 ; yfj = yia∗

; yti = yia

; yfj = yi + 12yi0.

And θ, δ and φ are the angles of voltage phasor, current phasor and admittance,respectively. The lower part of the measurement model, i.e., v = x− z, is

vac =[|V| − z|V| |I| − z|I| θ − zθ δ − zδ

]T. (5.5)

The Jacobian matrix for the measurement model is

Hac(x) =

∂uac∂xac

∂vac∂xac

=

∂fac(x)∂|V|

∂fac(x)∂|I|

∂fac(x)∂θ

∂fac(x)∂δ

Y1

(5.6)

where

∂fac(x)∂|V|

=

|yfi| cos(θi + φfi) or −|yfj | cos(θj + φfj)|yfi| sin(θi + φfi) or −|yfj | sin(θj + φfj)|yti| cos(θi + φti) or −|ytj | cos(θj + φtj)|yti| sin(θi + φti) or −|ytj | sin(θj + φtj)

;

∂fac(x)∂|I|

= [− cos δ − sin δ − cos δ − sin δ]T ;

∂fac(x)∂θ

=

−|Vi||yfi| sin(θi + φfi) or |Vj ||yfj | sin(θj + φfj)|Vi||yfi| cos(θi + φfi) or −|Vj ||yfj | cos(θj + φfj)−|Vi||yti| sin(θi + φti) or |Vj ||ytj | sin(θj + φtj)|Vi||yti| cos(θi + φti) or −|Vj ||ytj | cos(θj + φtj)

;

∂fac(x)∂δ

= [|I| sin δ − |I| cos δ |I| sin δ − |I| cos δ]T .

Substituting (5.4), (5.5), (5.6) and the predefined weighting matrix W into (5.3),the successive update of x can be calculated.

5.2 Classic HVDC link

5.2.1 Network modelIt is more realistic to build a nonlinear network model for a classic HVDC link dueto its inherent nonlinearity. The proposed nonlinear model also includes the adjunctAC currents as state variables and takes the control modes as a supplement.

A simplified classic HVDC link model is shown in Fig. 5.1, where Ir and Iidenote the current phasors flowing from the AC side of the rectifier to the DC linkand from the DC link to the AC side of the inverter, respectively. Based on the


Figure 5.1: A simplified classic HVDC link model [78]

available PMU measurements and DC measurements, the network equations can beformulated as [31, 32, 78, 89]:

f(x) =

|Ir| −K ∗ 3√

2π ∗ arIdc

Vrdc −K ∗ 3√

2π ∗ ar|Vr| cos(θr − δr)

Vrdc −K ∗ 3√

2π ∗ ar|Vr| cosα+ 3

πXrIdc

|Ii| −K ∗ 3√

2π ∗ aiIdc

Vidc −K ∗ 3√

2π ∗ ai|Vi| cos(θi − δi)

Vidc −K ∗ 3√

2π ∗ ai|Vi| cos δ + 3

πXiIdc

Idc − 1Rdc

(Vrdc − Vidc)

, (5.7)

where K = 0.995 is a coefficient for a twelve-pulse AC/DC converter; ar and aiare the tap ratios on the rectifier side and the inverter side, respectively; α and δ1

(= π − γ2) are the firing angle (also called ignition delay angle) and the extinctiondelay angle, respectively; Xr and Xi are the transformers’ reactances on the rectifierside and on the inverter side, respectively. |Vr| and θr are the magnitude and angleof Vr; |Ir| and δr are the magnitude and angle of Ir, which is the current phasorflowing from the AC side of rectifier to the DC link; the same quantities withsubscript i apply to the inverter. For the above seven equations, there are seven DCquantities of concern, which are Vrdc, Vidc, Idc, cosα, cos δ, ar, and ai. Both ACand DC variables in (5.7) will become vectors when multiple classic HVDC linksare installed in the system.

5.2.1.1 Classic HVDC control modes

Equation (5.7) characterizes classic HVDC links under steady state without includingthe control modes. As the referenced values can be maintained by controllers duringsteady state, it is beneficial to include HVDC control modes into the network modelto provide more redundant information.

Generally, a classic HVDC link can control two variables at both the rectifier andthe inverter: the transformer tap ratio ar and the firing angle α on the rectifier side;the transformer tap ratio ai and the extinction angle γ on the inverter side. The firing

1δ here is a DC grid variable, which is different from the angle of an AC line current phasor.2γ is the extinction (advance) angle


angle α keeps a normal operation range within 15 to 20 with a minimum limit ofabout 5. The extinction angle γ maintains a minimum limit of 15 for 50 Hz and 18for 60 Hz. Controlling the firing/extinction angles is called grid/gate control, and isfar more rapid (1 ∼ 10 ms) than tap ratio control (5 ∼ 6 s per step) [78]. Therefore,firing/extinction angle control is used initially for rapid actions, probably followedby tap ratios’ changes to restore the converter quantities, firing and extinctionangles, to their normal ranges. Since the slow changes of tap ratios could be easilyestimated [23], they are not included into the control mode equations. Herein weassume that ar and ai are maintained at constant values and the firing/extinctionangles are used for control purposes.

The two most common control modes are briefly discussed below in order todevelop the control mode equations, more details can be found in [78, 89].

Rectifier current control mode (RCCM)

• When α > αmin, the rectifier maintains constant DC current by changing α.It is the normal constant current (CC) control mode (represented by AB inFig. 5.2 ).

• When α = αmin, the rectifier maintains the constant ignition angle (CIA)control mode (represented by FA).

• The inverter always maintains a constant extinction angle (CEA) γ = γref

control mode (represented by CD).

In this control mode, the intersection point E represents the normal operatingcondition.

Inverter current control mode (ICCM)

When the rectifier operates at a reduced voltage represented by F’A’B, CDrepresenting the inverter’s CEA operation would not intersect it. Therefore,

• The inverter maintains a constant current (represented by GH).

• The rectifier maintains a constant firing angle α = αmin (represented by F’A’).

The intersection point E’ represents the operating condition at a reduced rectifiervoltage.

Im in Fig. 5.2 denotes the current margin, which represents the difference betweenthe rectifier current reference and inverter current reference. It is usually set at 10%to 15% of the rated current to ensure that the two constant current characteristicsdo not cross each other due to errors in measurements or other causes [89].


Rectifier (CC)Inverter (CC)

Rectifier (CIA)

Inverter (CEA)

F

C

F’

Idc

Vdc

BH

D

GA

A’E’

E

Reducded vol.

Normal vol.

Im

Figure 5.2: Converter control steady-state characteristics [78, 89]

For static state estimation purposes, the control mode equations only considerthe equilibrium of each control mode, which are given by:

0 = C1 ∗ (Idc − Iref(re)

dc )0 = C2 ∗ (γ − γref )0 = C3 ∗ (Idc − Iref(in)

dc )0 = C4 ∗ (α− αmin),

(5.8)

where Iref(re)dc and Iref(in)

dc are the DC current references for the RCCM and ICCM,respectively; C = [C1 C2 C3 C4] is the control mode index. Ci = 1 indicates thecorresponding equation is activated, otherwise Ci = 0 indicates the correspondingequation will be removed out of the control mode equation set during the stateestimation. Hence, C = [1 1 0 0] and C = [0 0 1 1] refer to the RCCM and ICCM,respectively. Replacing γ and α with the state variables cos δ and cosα, respectively,(5.8) is rewritten as:

0 = C1 ∗ (Idc − Iref(re)dc )

0 = C2 ∗ (cos δ − cos δref )0 = C3 ∗ (Idc − Iref(in)

dc )0 = C4 ∗ (cosα− cosαmin),

(5.9)


5.2.1.2 Interface model between DC Links and AC grids

Since ar and ai are treated as constant parameters, (5.7) only involves five DCstates (Vrdc, Vidc, Idc, cosα, and cos δ), and AC voltage phasors Vr, Vi and currentphasors Ir, Ii. However, the current phasors Ir, Ii are not state variables of theAC network model as Vr and Vi. Hence, Kirchhoff’s current law is applied to theterminal buses that are adjacent to the rectifier or the inverter in order to formulatethe current phasors Ir, Ii with AC states as

Ir = −∑mj=1 Irj , Ii =

∑nj=1 Iij , (5.10)

where j denotes an AC bus to which bus r (or i) is connected through the AC linerj (or ij); m (or n) denotes the number of buses that are connected to bus r (or i).Consequently, the DC link and the AC grid can be interfaced through the voltagephasors Vr, Vi and the current phasors Ir, Ii.

This DC link and interface models are able to adapt to various topologies ofhybrid AC/DC grids. It can represent an embedded DC link in an existing AC gridor as an interconnection between two asynchronous AC grids. Furthermore, it isflexible to be extended to a multi-terminal DC (MTDC) grids [32].

5.2.2 Measurement model for hybrid AC/DC systemsFor the hybrid AC/DC state estimation, the AC network model, the classic HVDClink model, as well as their interface model need to be combined together. The statevariable vector x for a hybrid AC/DC system is

x = [|V| |I| θ δ |Ir| |Ii| δr δi Vrdc Vidc Idc cosα cos δ]T . (5.11)

Here voltage and current phasors are applied in polar coordinates, i.e. magnitudeand angle. Using phasors in polar coordinates takes two significant advantages:(i) PMU measurements can be directly used without coordinate change; (ii) moreimportantly, it allows angle bias detection and correction, which will be addressedin Section 5.2.3.3.

For a hybrid AC/DC grid, the network model f(x) is formulated as:

f(x) =

fac(x)fad(x)fdc(x)

, (5.12)

where fac(x) is(5.4)), and fad(x) and fdc(x) are formulated as

fad(x) =

|Ir| cos δr +

∑mj=1 | ˜Irj | cos δrj

|Ir| sin δr +∑mj=1 | ˜Irj | sin δrj

|Ii| cos δi −∑nj=1 |Iij | cos δij

|Ii| sin δi −∑nj=1 |Iij | sin δij

, (5.13)


fdc(x) =

K ∗ 3√

2π ∗ arIdc − |Ir|

K ∗ 3√

2π ∗ ar|Vr| cos(θr − δr)−Vrdc

K ∗ 3√

2π ∗ ar|Vr| cosα− 3

πXrIdc −Vrdc

K ∗ 3√

2π ∗ aiIdc − |Ii|

K ∗ 3√

2π ∗ ai|Vi| cos(θi − δi)−Vidc

K ∗ 3√

2π ∗ ai|Vi| cos δ − 3

πXiIdc −Vidc

Vrdc −Vidc −RdcIdcC1 ∗ (Idc − Iref(re)

dc ) + C3 ∗ (Idc − Iref(in)dc )

C2 ∗ (cos δ − cos δref) + C4 ∗ (cosα− cosαmin)

. (5.14)

The second part of the measurement model can be expressed as:

v =

vacvadvdc

=

xac − zacxad − zadxdc − zdc

, (5.15)

where vac is (5.5), vad and vdc are shown below

vad = xad − zad =[|Ir| − z|Ir| |Ii| − z|Ii| δr − zδr δi − zδi

]T,

vdc = xdc − zdc =

Vrdc − zVrdcVidc − zVidcIdc − zIdc

cosα− zcosα

cos δ − zcos δ

.


The Jacobian matrix for a hybrid AC/DC state estimation is :

H(x) =

∂fac(x)∂|V|

∂fac(x)∂ |I|

∂fac(x)∂θ

∂fac(x)∂δ 0 0

cos δr −|Ir| sin δrsin δr |Ir| cos δr

0 ∂fad(x)∂|I| 0 ∂fad(x)

∂δ cos δi −|Ii| sin δi 0sin δi |Ii| cos δi

−IK1 ∗ ar|Vr| sin(θr − δr)

∂fdc(x)∂|V| 0 ∂fdc(x)

∂θ 0 −I ∂fdc(x)∂xdc

K1 ∗ ai|Vi| sin(θi − δi)

I 0 00 I 00 0 I

,

(5.16)where

∂fad(x)∂ |I|

=

cos δrj or 0sin δrj or 0cos δij or 0sin δij or 0

, ∂fad(x)∂δ

=

−|Irj | sin δrj or 0|Irj | cos δrj or 0|Iij | sin δij or 0−|Iij | cos δij or 0

,

∂fdc(x)∂|V|

=

0K1 ∗ ar cos(θr − δr) or 0

K1 ∗ ar cosα or 00

K1 ∗ ai cos(θi − δi) or 0K1 ∗ ai cos δ or 0

000

,∂fdc(x)∂θ

=

0−K1 ∗ ar sin(θr − δr) or 0

00

−K1 ∗ ai sin(θi − δi) or 00000

,


∂fdc(x)∂xdc

=

0 0 K1ar 0 0−1 0 0 0 0−1 0 −K2Xr K1 ∗ ar|Vr| 00 0 K1ai 0 00 −1 0 0 00 −1 −K2Xi 0 K1 ∗ ai|Vi|1 −1 −Rdc 0 00 0 C1 + C3 0 00 0 0 C4 C2

,

K1 = 0.995 ∗ 3√

2π

, K2 = 3π.

The nonlinearities in the Jacobian matrix (5.16) are fewer than those in the Jacobianmatrix of a conventional state estimation, which will reduce the computationload. The high degree of sparsity also helps to decrease the computational effortsubstantially.

5.2.3 Considerations for practical application

In the perfect condition for the PMU-based state estimation, PMUs are installed atall buses so that all the bus voltage and line current phasors are measured. Thusthe system is not only fully observable but also of high redundancy. In addition, themeasurement noise is too small to be considered.

However, in real life this condition can rarely be true, and all the aforementionedissues have to be carefully considered since they may affect the estimation’s perfor-mance. Therefore, prior to implementing the proposed nonlinear PMU-based stateestimator, all these issues need to be carefully assessed. This section discusses howto analyze system observability with PMUs and the effect of redundancy, what isthe allowable measurement noise level of PMUs and DC grid measurements, andhow to deal with the phase mismatch owing to imperfect PMUs synchronization.

5.2.3.1 Observability analysis and measurement redundancy

Generally, observability analysis aims to determine whether there are observableislands within the network, and to isolate observable islands. System observability canbe analyzed in two main approaches: topological and/or numerical methods. Basictopological methods for conventional state estimations can be found in [16, 90, 91].A non-iterative numerical method is proposed in [92]. In addition, observability oftenacts as the criterion for PMU placement in power systems, which aims to maximizesystem observability with a minimum number of PMUs. If full observability for anentire network is required, the algorithms in [93, 94, 95, 96, 97] may be used foradding new PMUs.


However, observability analysis herein is performed for each individual island,aiming to define whether this island is observable or not as a portion of the powernetwork, and then developing independent state estimator models for each islandto be solved. Strictly defining observable islands for an entire system is out of thissection’s scope.

Since PMUs’ phasor measurements replace power flow and power injectionmeasurements from SCADA, new topological rules for AC systems are defined asfollows [94, 98]:

• R1: A bus with a PMU installed and any line extending from the bus areobserved.

• R2: Any bus that is incident to an observed line connected to an observed busis observed.

• R3: Any line joining two observed buses is observed.

• R4: If all the lines incident to an observed bus are observed, save one, then allof the lines incident to that bus are observed.

• R5: Any bus incident only to observed lines is observed.

To define the observability rules for classic HVDC links, it is necessary to accountfor all the states involved in each link. There are five DC states to be estimated,together with four AC states (|Vr|, |Vi|, θr, θi), and four AC/DC interface states(|Ir|, |Ii|, δr, δi). Observe that for every set of DC link equations, i.e. (5.14), thereare 13 states involved. Therefore, the hybrid AC/DC observability algorithm isextended by considering the DC network model:

• R6: The DC link is observable if all DC states are measured by meteringdevices.

• R7: If the DC states are not fully measured, at least 4 of 13 states involved ineach link model have to be known in order to make all the DC states observable.Note that the AC/DC interface state variables among the four measurementsdo not have to be measured directly. They can also be accounted as knownstates by being calculated from AC measurements.

However, not all combinations of four measurements will suffice. First, these fourmeasurements should not contain Idc and cos δ for RCCM, or Idc and cosα for ICCMsince the control mode equations have already provided references for the above statevariables. Second, the four measurements should not all come from only the rectifierside or only the inverter side. In addition, any of the four measured state variablesshould not be possible to be calculated by the other three measurements. Providinga classic HVDC link under RCCM, (5.14) can be reduced to a five-equation set as


follows: f1(|Vr|, θr, δr,Vrdc)f2(|Vr|, cosα,Vrdc)f3(|Vi|, θi, δi, Vidc)f4(|Vi|, Vidc)f5(Vrdc,Vidc).

. (5.17)

Based on (5.17), the combinations can be divided in two types: two measurementsfrom the rectifier side and two from the inverter side, three measurements from therectifier side and one from the inverter side. For the first type, all the combinationscan be inferred by the following steps:

• S1: The first measurement selected from inverter side can be |Vi| or Vidc.Knowing either of them can calculate the other one by using f4. Moreover,Vrdc will be known sequentially by f5.

• S2: Knowing either θi or δi can calculate the other. So far, all the states fromthe inverter side have been known.

• S3: Since Vrdc has been calculated, there are two equation f1 and f2 for fourunknown states, which are |Vr|, θr, δr, and cosα. Therefore, any two of thefour states can be selected, except for choosing|Vr| and cosα simultaneously.

• S4: In total, the number of combinations is 2× 2× (C(4, 2)− 1) = 20.

For the second type, the steps are as follows:

• S1: Select any three measurements among five states on the rectifier side exceptfor selecting |Vr|, cosα,Vrdc together. Hence, all the states on the rectifierside can be calculated associated with f1 and f2.

• S2: Vidc will be sequentially known by f5 and then |Vi| is known by f4.

• S3: Either θi or δi can calculate the other. So far, all the states from theinverter side have been known.

• S4: In total, the number of combinations is (C(5, 3)− 1)× 2 = 18.

All the proper combinations of four measurements to make a classic HVDC linkobservable are shown in Table 5.1. The upper part of Table 5.1 presents the firsttype of combinations, whereas, the lower part is for the second type of combinations.As PMUs measure synthetic magnitude and angle in phasor coordinates, in realityextra measurements out of each combination might be obtained incidentally. Forinstance, when using the combination of θr, δr, |Vi|, and θi; |Vr|, and |Ir| would beknown automatically when θr, δr are measured or calculated.

As indicated by the red marks in Table 5.1, in the case of no measurementredundancy, the DC states can still be computed even without any DC measurement.


Table 5.1: Proper combinations of four measurements to make a classic HVDC linkobservable

|Vr| θr δr Vrdc cosα |Vi| θi δi Vidc

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗


This brings a great advantage since AC measurements are readily provided byPMUs. When measurements are redundant, this can be taken as an advantage forcross-validation [99], as to be discussed in Section 5.2.3.3. In addition, the tap ratiosof the rectifier and inverter can also be included as DC states when needed as thereare four more equations than the number of DC states.

A complementary approach is to examine the numerical properties of the Jacobianmatrix of the hybrid AC/DC model, which is referred to as the numerical method.

rank((H)T(H)) = Nx, (5.18)

where H(x) is the Jacobian matrix; and Nx represents the number of state variables.Generally, the more rows that the Jacobian matrix has w.r.t. the number of columns,the higher redundancy offered by the PMU-based state estimation. As mentionedin [16, 23, 90], measurement redundancy is crucial for bad data detection andidentification.

When a measurement is lost, its corresponding rows in the measurement modelequation, in the Jacobian matrix, and in the weighting matrix will be removed. Aslong as (5.18) holds, all the states can be estimated by using the nonlinear WLSalgorithm.

A case study for a low measurement redundancy scenario is presented in Sec-tion 5.2.4.3 to investigate its effect on the PMU-based state estimation.

5.2.3.2 Measurement noise and the choice of weightings

In reality, it is inevitable to have noise in measurements due to (i) the instrumenttransformers, (ii) the cables connecting the instrument transformers to the sensors orA/D converters, and (iii) the sensors or A/D converter [90]. The standard uncertainty(σ) for each measurement is proportional to the specified maximum uncertainty(σmax) of the PMU with a coefficient of 1√

3 [100]. Nevertheless σmax varies fordifferent PMU vendors, and Table 5.2 gives an example [101].

It is assumed that the DC measurements have a σmax of 0.01% of reading or0.001% range. Then the σmax values are transfered into signal-to-noise ratios (SNRs)using (5.19) in order to add Gaussian noise to the true values.

SNR = 10 ∗ log10

( √3

σmax

)2

dB. (5.19)

As discussed in Chapter 3, when the measurement has a Gaussian distribution,the WLS solution is equivalent to the MLE if the inverse of each measurement

Table 5.2: σmax for different variables from one PMU vendor

|V | 0.02% of reading or 0.002% range|I| 0.03% of reading or 0.003% range

θ and δ 0.01 or 10% of range minimum


Table 5.3: SNRs and weights for different measurements

Meas. SNR Weighting|V | 78.75 dB 7.5 ∗ 107

|I| 75.23 dB 3.3 ∗ 107

θ and δ 79.93 dB 9.8 ∗ 107

DC states 84.77 dB 3 ∗ 108

covariance is used as its weight. Therefore, the SNRs and the weights of differentmeasurements are calculated and shown in Table 5.3.

The weightings for the network model equations depend on the modeling accuracyfor each component. A case study on measurement noise and weighting selection ispresented in Section 5.2.4.4, where the weights for the network model equations areall equal to the highest measurement weight.

5.2.3.3 Angle bias detection and correction by PMU-based stateestimation

Angle biases (or shifts) emerge due to imperfect synchronization or incorrect time-tagging by PMUs [22, 102, 103]. These phase angle errors have been observed fromrecorded data in several utilities [22, 23]. Reference [104] presents two time skewcases that result in angle biases. In one case, the GPS signal cable was looselyconnected so the signal was intermittent. Thus, the PMU time was not accurate,resulting in spikes on top of the correct angle values. The other case of angle biasoccurred due to drifting of the internal clock.

Different from measurement errors, angle bias does not have a normal distributionand its deviation varies within 1 ∼ 2 degrees, even 20 degrees in some extreme cases.In addition, angle bias could last for a few snapshots, which may contaminate othermeasurements.

Since magnitude and angle are independent states herein, angle biases can bedetected by using redundant measurements [22]. The vector of angle bias variablesΩ is included in the state vector as

x = [|V| |I| θ δ |Ir| |Ii| δr δi Vrdc Vidc Idc cosα cos δ Ω]T

whereΩ = [Ωθ Ωδ Ωr Ωi]

The voltage and current angles in the measurement error vector become

θ − zθ + Ωθ and δ − zδ + Ωδ

Similarly, the angles of the AC/DC interface currents in the measurement errorvector become

δr − zδr + Ωr and δi − zδi + Ωi


In order to correct the angle bias, it is required that

rank((H)T(H)) = Nx +NΩ, (5.20)

where NΩ represents the number of angle bias auxiliary state variables.This approach has been defined more formally in the control literature, and it

has been termed as cross-validation. Mathematical proofs for cross-validation rulescan be referred to [99].

Angle bias variable Ω greatly facilitates angle bias correction. Compared to com-mon bad data detection and correction methods, such as using normalized residuals[90], it does not need to perform additional calculations or to define a thresholdto determine bad data. In addition, it avoids the risk that the largest normalizedresidual method may fail in the detection of gross errors for the measurements thathave a large undetectable component [105]. In fact, this angle bias correction givesa huge flexibility for correcting angle bias no matter how large the bias is. Therequirement, however, is to have enough redundancy to accommodate the angle biasvariables Ω. Hence, if the power system has high measurement redundancy, thisapproach would be suitable.

Reference [102] applies the same algorithm for magnitude error detection andcorrection. However, the authors believe it is not necessary to use this method tocorrect magnitude errors, because the incorrect time tagging issues normally do notgive rise to deviations in the magnitude measurements. It is not worth incorporatingmeasurements redundancy to detect and correct minor magnitude errors.

A case study for the angle bias scenario is presented in Section 5.2.4.5.

5.2.4 Case study

The 6-bus system and the KTH-Nordic 32 system are modified for this case study.In the 6-bus system a classic HVDC link is established between bus 3 and bus 4.In the KTH-Nordic 32 system a classic HVDC link replaces the original AC linebetween bus 36 and bus 41.

The first two subsections present estimation results when all the weightings forboth network equations and measurements are assumed to be 1, and full measurementobservability is assumed. The following three subsections present simulation resultsfor a scenario with low measurement redundancy, a scenario with measurementnoises and corresponding weights, and a scenario where angle bias correction isperformed. Finally, simulation time and computation performance are discussed.

5.2.4.1 NSE for the 6-bus hybrid AC/DC system

A circuit breaker located on line 4 between bus 4 and 6 was opened at t = 5 s andafter three cycles it was re-closed at t = 5.06 s. The DC link was under the normaloperation condition with Irefdc = 0.506 p.u. and cos δref = 0.951 p.u. for RCCM.


1 2 3 4 5 60.95

1

1.05

Bus No.

|V|(

p.u

.)

1 2 3 4 5 60

1

2x 10

−16

Bus No.

Erro

r (p

.u.)

Vmag−true Vmag−meas. Vmag−est.

Vmag−estimation−residual


1 2 3 4 5 60

5

10

15

Bus No.

θ(d

eg)

Vang−true Vang−meas. Vang−est.

1 2 3 4 5 6

−5

0

5x 10

−15

Bus No.

Erro

r (d

eg)

Vang−estimation−residual

(b) θ at all buses (bus 2 is the reference)

1 1.5 2 2.5 3 3.5 4−0.5

0

0.5

1

|Ir| |Ii| δr δ i

p.u

.

Iri−true Iri−meas. Iri−est.

1 1.5 2 2.5 3 3.5 40

2

4

6x 10

−16

|Ir| |Ii| δrδi

Erro

r (p

.u.)

Iri−estimation−residual

(c) AC/DC interface states

1 1.5 2 2.5 3 3.5 4 4.5 5

0.6

0.8

1

Vrdc Vide Idc cosa cosg

p.u

.

Hvdc−true Hvdc−meas. Hvdc−est.

1 1.5 2 2.5 3 3.5 4 4.5 5

5

6

7

8

x 10−16


Erro

r (p

.u.)

Hvdc−estimation−residual

(d) DC statesFigure 5.3: NSE for the 6-bus hybrid AC/DC system with a classic HVDC link for asingle snapshot

And the state vector for the 6-bus hybrid AC/DC system is defined as follows

x = [|V1|, |V2| . . . |V6|, θ1, θ2, . . . θ6, |I1|, |I2| . . . |I12|, δ1, δ2, . . . δ12,

|Ir|, |Ii|, δr, δi, Vrdc, Vidc, Idc, cosα, cos δ]T .

Figure 5.3 shows the state estimation results for one single snapshot, and Fig. 5.4 isfor multiple snapshots.

As the network model is relatively similar to the model in PSAT, the residualsfor one single snapshot are extremely small, lower than 10−14p.u. However, whenthe system was subject to a perturbation, the estimation residuals increased asshown in Fig. 5.4. This is due to two reasons:


0 2 4 6 8 10 12 14 16 180.9

0.95

1

time (s)

|V|(

p.u

.).


0 2 4 6 8 10 12 14 16 1810

−20

10−10

100

time (s)

Erro

r (p

.u.)


(a) |V | at bus 4

0 2 4 6 8 10 12 14 16 182

4

6

time (s)

θ(d

eg)


0 2 4 6 8 10 12 14 16 1810

−20

10−10

100

time (s)

Erro

r (de

g) Vang−residual−error

(b) θ at bus 4 (bus 2 is the reference)

0 2 4 6 8 10 12 14 16 181.1

1.15

1.2

time (s)

Vid

c (p

.u.)

Vidc

−true Vidc

−meas. Vidc

−est.

0 2 4 6 8 10 12 14 16 1810

−20

10−10

100

time (s)

Erro

r (p

.u.)

Vidc

−residual−error

(c) DC Voltage at bus 4

2 4 6 8 10 12 14 16 180.94

0.96

0.98

1

time (s)

cosα

(p

.u.)

cosα−true cosα−meas. cosα−est.

0 2 4 6 8 10 12 14 16 1810

−20

10−10

100

time (s)

Error (

p.u

.)


(d) cosαFigure 5.4: NSE for the 6-bus hybrid AC/DC system with a classic HVDC link formultiple snapshots

• Lack of model and/or topology update during the perturbation. A circuit breakerwas opened and re-closed after three cycles. During this period, the networkmodel was not updated accordingly, resulting in a large estimation residual.

• The limitation of static network models. During the transient dynamic periodafter the perturbation, the controllers and/or the components that havedynamical properties respond to the changes. For the AC network model,the Kirchhoff’s circuit law is still valid given correct admittance matrix.(AC lines with FACTS devices can be a different case). However, for thestatic classic HVDC link model, as some of its states are directly controlledor indirectly affected by the controllers, it may not hold during transientdynamics. This indeed explains in Fig. 5.4 that the performance of the stateestimation gradually came back to a normal level after the drop, but still had


larger residuals compared to that in steady state. Nevertheless, the estimationresiduals are within an acceptable error range.

5.2.4.2 NSE for the KTH-Nordic 32 hybrid AC/DC system

Herein a 400 kV classic HVDC link is established between bus 36 and bus 41. A 0.4p.u. (10%) load increase was applied at bus 41 at t = 5 s. And the state vector forthe KTH-Nordic 32 hybrid AC/DC system is organized as follows

x = [|V1|, |V2| . . . |V53|, θ1, θ2, . . . θ53, |I1|, |I2| . . . |I160|, δ1, δ2, . . . δ160,

|Ir|, |Ii|, δr, δi, Vrdc, Vidc, Idc, cosα, cos δ]T .

10 20 30 40 500.8

0.9

1

1.1

Bus No.

|V|(

p.u

.)


10 20 30 40 50

5

10

15x 10

−15

Bus No.

|Err

or (p

.u.)


(a) |V | at all buses

10 20 30 40 50−100

−50

0

Bus No.


10 20 30 40 50−4−2

02

x 10−13

Bus No.

Erro

r (d

eg)


θ(d

eg)

(b) θ (bus 9 is the reference)

0

1

2

−1 |Ir| |Ii| δr δi

p.u

.

Iri−true Iri−meas. Iri−est.

0.5

1

1.5

2

x 10−13

|Ir| |Ii| δr δi

Erro

r (p

.u.)

Iri−estimation−residual

(c) AC/DC interface states

0.96

0.98

1


p.u

.


5

10

15

x 10−14


Erro

r (p

.u.)


(d) DC states

Figure 5.5: NSE for the KTH-Nordic 32 hybrid AC/DC with a classic HVDC link for asingle snapshot


0 2 4 6 8 10 12 14 160.848

0.85

0.852

0.854

time (s)

|V|(

p.u

.)


0 2 4 6 8 10 12 14 1610

−20

10−10

100

time (s)

Erro

r (p

.u.)


(a) |V | at bus 41

0 2 4 6 10 12 14 16

−66.5

−67

−67.5

−66

8 time (s)

Vang−true Vang−m Vang−est

0 2 4 6 10 12 14 1610

−20

10−10

100

8

time (s)

Erro

r (d

eg)


θ(d

eg)

(b) θ at bus 41 (bus 9 is the reference)

0 2 4 6 8 10 12 14 160.94

0.942

0.944

0.946

time (s)

Vid

c (p

.u.)

Vidc−true Vidc−meas. Vidc−est.

0 2 4 6 8 10 12 14 1610

−20

10−10

100

time (s)

Erro

r (p

.u.)

Vidc−residual−error

(c) DC Voltage at bus 41

2 4 6 8 10 12 14 16

0.9645

0.965

0.9655

0.966

time (s)

cosα

(p

.u.)

cosα−true cosα−meas. cosα−est.

0 2 4 6 8 10 12 14 1610

−20

100

time (s)

Error (

p.u

.)


(d) cosαFigure 5.6: NSE for the KTH-Nordic 32 hybrid AC/DC system with a classic HVDC linkfor multiple snapshots

As shown in Fig. 5.5, the estimation residuals are comparable to results in Section5.2.4.1, which indicates that the accuracy of the proposed state estimator is notaffected by the size of the test system. As this case study has no topology changeduring the perturbation, the estimation residual in Fig. 5.6 does not experience aleap as in Fig. 5.4. The increase of the estimation residual after the instance whenthe perturbation occurred is due to lack of representation of the system’s dynamicbehavior. Figure 5.6 shows that the estimation residuals are within an acceptableerror range even when the system is subject to a dynamic change.

5.2.4.3 NSE for the scenario without DC measurements

The same test scenario as in Section 5.2.4.1 was applied here. To decrease themeasurement redundancy, only the voltage phasor measurements at bus 3 and 4,


1 2 3 4 5 60.95

1

1.05

Bus No.

|V|(

p.u

.)


1 2 3 4 5 60

1

2x 10

−16

Bus No.

Erro

r (p

.u.)


(a) |V | at all buses for one snapshot

0.6

0.8

1


p.u

.

Hvdc−true Hvdc−meas.

1.5

2

2.5

3x 10

−15


Erro

r (p

.u.)


(b) DC states for one snapshot

0 2 4 6 8 10 12 14 16

0.96

0.98

1

1.02

time (s)

|V|(

p.u

.).


0 2 4 6 8 10 12 14 1610

−20

10−10

100

time (s)

Erro

r (p

.u.)


(c) |V | at bus 5

0 2 4 6 8 10 12 14 161.1

1.15

1.2

time (s)

Vid

c (p

.u.)

Vidc

−true Vidc

−meas. Vidc

−est.

0 2 4 6 8 10 12 14 1610

−20

10−10

100

time (s)

Erro

r (p

.u.)

Vidc

−residual−error

(d) DC Voltage at bus 4

Figure 5.7: Effect of reducing AC/DC measurements redundancy: NSE for the 6-bushybrid AC/DC system with a classic HVDC link

and current phasor measurements on lines 3, 4, 7, and 8, which are incident to eitherbus 3 or 4, were provided. There was no DC state measurement provided and theestimation results are shown in Fig. 5.7.

Comparing the results in Fig. 5.7 with that in Section 5.2.4.1, reducing measure-ments redundancy does not significantly influence the estimation performance aslong as the measurements can satisfy the observability requirements discussed inSection 5.2.3.1. This test scenario illustrates that when PMUs are available only atcritical boundary buses between the AC system and the DC link, it is possible toestimate DC states without having any DC measurements.


1 2 3 4 5 60.95

1

1.05

Bus No.

|V|(

p.u

.)


1 2 3 4 5 6

3

4

5

6

7x 10

−6

Bus No.

Erro

r (p

.u.)


(a) |V | at all buses

0.6

0.8

1


p.u

.


1

2

3

x 10−5


Erro

r (p

.u.)


(b) DC statesFigure 5.8: Effect of measurement noises: NSE for the 6-bus hybrid AC/DC system witha classic HVDC link for a single snapshot

5.2.4.4 Results for the scenario with measurement noises andcorresponding weights

The effect of measurement noises and weights selection were studied using the sametest scenario as in Section 5.2.4.1 by adding Gaussian white noise and using theweights in Table 5.3. Figure 5.8 shows the estimation results, in which the residualsare larger in comparison with the previous case in Section 5.2.4.1. Nevertheless, thestate estimates are acceptable for the applied signal-to-noise ratios.

5.2.4.5 Results for the scenario with angle bias correction

An example of angle bias correction was made for the 6-bus hybrid AC/DC system,where a 7.5 angle jump at bus 1 and 30 angle jump on line 1, which is incident tothe bus 1, were applied at t = 10 s and removed at t = 11 s as shown in Fig. 5.9.This test scenario shows that the proposed state estimator also has the ability ofcorrecting angle biases for hybrid AC/DC grids.

5.2.4.6 Comparison between the linear and nonlinear SEs

Chapter 4 presents a linear state estimation, while this chapter proposes a nonlinearone. A linear, or nonlinear WLS is determined by the measurement error vectore(x). In most cases, this vector presents the same linearity (or nonlinearity) as thenetwork model. The rationale to use the nonlinear state estimation is explainedbelow.

For the AC part, both algorithms use the same linear network model. However,since the state variables used in this chapter are in polar coordinates, each linearequation needs to be rewritten into two equations associated with trigonometric


1 2 3 4 5 60

10

20

Bus No.

θ(d

eg)

1 2 3 4 5 6

−5

0

5

x 10−15

Bus No.

Error (

deg

)



(a) Voltage angles for one snapshot (bus 2 isthe reference

2 4 6 8 10 12

−100

0

100

Line No.

δ (d

eg)

Iang−true Iang−meas. Iang−est.

2 4 6 8 10 120

2

4

6

x 10−14

Line No.

Error (

deg

)

Iang−estimation−residual

(b) Current angles for one snapshot

0 2 4 6 8 10 12 14 16 1810

20

30

40

time (s)

θ (d

eg)


0 2 4 6 8 10 12 14 16 1810

−20

10−10

100

time (s)

Error (

deg

)


(c) Voltage angle at bus 1 (bus 2 is the refer-ence)

0 2 4 6 8 10 12 14 16 18

−40

−20

0

20

time (s)

δ (d

eg)

Iang−true Iang−meas. Iang−est.

0 2 4 6 8 10 12 14 16 1810

−20

10−10

100

time (s)

Erro

r (d

eg)

Iang−residual−error

(d) Current angle on line 1 (between bus 1 andbus 3)

Figure 5.9: Angle bias correction: NSE for the 6-bus hybrid AC/DC system with a classicHVDC link


0 2 4 6 8 100.98

1

1.02

1.04

1.06

Bus No.

|V| (

p.u

.)

Vmag−true Vm−mag Vmag−est

0 2 4 6 8 100

1

2

3x 10

−3

Bus No.

Erro

r (p

.u.)


(a) Voltage magnitudes at all buses

0 2 4 6 8 10−10

−5

0

5

10

Bus No.

θ(d

eg)

Vang−true Vm−ang Vang−est

0 2 4 6 8 10−2

0

2

4

Bus No.

Erro

r (d

eg)


(b) Voltage angles at all busesFigure 5.10: LSE for the 9-bus test system when a 7.5 angle bias was introduced tobus 8 for a single snapshot

0 2 4 6 8 100.98

1

1.02

1.04

1.06

Bus No.

|V|(

p.u

.)

Vmag−true Vmag−m Vmag−est

0 2 4 6 8 100

1

2

3x 10

−16

Bus No.

Erro

r (p

.u.)


(a) Voltage magnitudes at all buses

0 2 4 6 8 10−10

−5

0

5

10

Bus No.

θ(d

eg)

Vang−true Vang−m Vang−est

0 2 4 6 8 10

−0.5

0

0.5

x 10−14

Bus No.

Erro

r (d

eg)


(b) Voltage angles at all busesFigure 5.11: NSE for the 9-bus test system when a 7.5 angle bias was introduced tobus 8 for a single snapshot

functions of the phasor angles. These trigonometric functions introduce nonlinearities.Although this brings additional computational burden, using phasors in polarcoordinates gives the significant advantage of allowing angle bias detection andcorrection.

To support this point, a comparison was performed on the same 9-bus ACtest system as in Chapter 4 by using the linear and nonlinear state estimators,respectively. A 7.5 angle bias was introduced at bus 8 and 30 angle biases at the


0

0.5

1

1.5

Vrdc Vide Idc

p.u

.

Hvdc−true Hvdcm Hvdc−est

0

1

2

3x 10

−10

Vrdc Vide Idc

Erro

r (p

.u.)


(a) DC states in linear SE for one snapshot

0.6

0.8

1


p.u

.


5

6

7

8x 10

−16


Erro

r (p

.u.)


(b) DC states in nonlinear SE for one snapshotFigure 5.12: Comparison between the linear (left) and nonlinear (right) classic HVDClink models for LSE and NSE, respectively

lines that are connected with bus 8 for both cases.Figure 5.10 shows the linear state estimation results for the voltage magnitudes

and angles when angle biases were introduced. The estimation accuracy for bothmagnitudes and angles reduces significantly compared to the results presented inChapter 4 when no angle bias was introduced: magnitude residuals from 10−16

to 10−3; angle residuals from 10−15 to 1 deg. This indicates that the angle biaseswere not successfully detected; moreover, they contaminated other angles and evenmagnitudes being estimated. Therefore, it is verified that the linear state estimatoris unable to detect and correct angle biases. Reference [106] also shows a simpleexample that draws the same conclusion.

On the contrary, the nonlinear state estimator presented in this chapter suc-cessfully detected and corrected the angle bias. As shown in Fig. 5.11, estimationaccuracy is similar to the case when no angle bias was introduced.

The limitations of linear state estimation was reported in [106], even in the casewhere there is full observability and redundant measurements (PMUs installed atall buses and measuring all the currents at each bus), the linear state estimator failsto provide the correct measurement residuals in the presence of an angle bias.

The DC network model can be simplified to a linear one as presented in Chapter4 by reducing the number of equations and using complex variables. However, thiswill lead to the following consequence: the state variables that must be used, forinstance |Vr| cosα, are not consistent with other variables, and this may lead tomatrix conditioning issues during the linear least squares solution.

A comparison between the linear classic HVDC link model as in Chapter 4 andthe nonlinear one was carried out and the results are shown in Fig 5.12. Compared


Table 5.4: Statistics of the computation time

Computation time No. of meas. sets (1000 in total ) %t ≥ 0.02s 4 0.4%

0.01 ≤ t < 0.02s 5 0.5%0.006 ≤ t < 0.01s 113 11.3%0.003 ≤ t < 0.006s 641 64.1%

t ≤ 0.003s 237 23.7%

to the nonlinear model’s estimation residual of 10−16, the linear one only achieves10−10.

5.2.4.7 Simulation time and computation performance

Referring to the estimation results in Section 5.2.4.1 and 5.2.4.2, it is observedthat most of the estimation errors are below 10−12 with relatively few iterations.Residuals that are higher than 10−12 can result from two reasons: (i) the limitationof the static network models, which can hardly represent the system’s behaviorduring transient dynamics; (ii) lack of model and/or topology update during theperturbations. For instance, in the test scenario of Section 5.2.4.1, the number ofiterations for each estimation snapshot remained around 2 when the system wasin steady state. During the first three cycles after the perturbation occurred, theiteration number was 37 at once, and then decreased to 4 after another three cycles.

The computation time statistics for the test scenario in Section 5.2.4.1 are shownin Table 5.4. Two hundred and fifty out of 1000 measurement sets were obtainedbefore the perturbation; the others were after the perturbation. The measurementsets for which the estimation computation time was below 0.003 s account for 23.7%of the whole measurement sets; more than 99.6% of the estimation computationsare faster than the measurement rate, which is 0.02 s. Computation time exceedingthe measurements interval mainly occurred during the perturbation period. Thetime-performance of the algorithm in a standard PC (2.80 GHz Intel Core processorrunning matlab R2012b) is already acceptable for real-time applications withminimum delay, and could be improved if the code is optimized and re-implementedin a low-level programming language such as C++. In addition, it demonstrates thefeasibility of implementing the estimation algorithm into a real-time application.

5.3 VSC-HVDC

VSC-based HVDC systems have been successfully deployed in various projectsglobally. Compared to the classic HVDC technology, VSC-HVDC technology offersa key advantage of independent control of active and reactive powers, together withadditional benefits in control flexibility and reliability [25].

5.3. VSC-HVDC 67

5.3.1 VSC substation model

To match the PMU measurements in AC systems, DC measurement units (DMUs)could be easily implemented by extending the functionalities of the acquisitionsystem used in converter stations to gather the DC measurements. Based on theknowledge that switching frequency of transistors in VSCs is high and so is thecontrol response, DMUs should be capable to provide high speed and GPS time-synchronized measurements, at least not slower than PMUs.

A basic diagram of a VSC substation is shown in Fig. 5.13. It comprises anAC/DC converter, DC capacitor(s), a phase reactor, a transformer and filter(s).The converter requires self-commutating switches, such as insulated-gate bipolartransistors (IGBT), which have a turn-on and turn-off capability. The DC capacitoron the DC side maintains the DC voltage across the VSC. However, in reality, DCvoltage may contain ripples due to the harmonic currents in the DC circuit generatedduring VSC operation. The interface transformer and phase reactors are installed toreduce the fault current and the harmonic current content in the AC current, andenable the control of active and reactive power flow. The AC filter aims to filteringthe high-order harmonics introduced by the commutation valve switching process[25].

In Fig. 5.13, V , I are the voltage and current phasors at Bus i, which is also thepoint of common coupling (PCC); Vf is the voltage phasor at the AC filter and Ifis the current phasor flowing into it; Vv and Iv are the voltage and current phasorsat the AC side of the converter; Zt and Zp are the impedances of the transformerand phase reactor, respectively.

dcV

dcI

vV

vI

pZ

Phase

reactorAC

filter

fV

fI

tZ

:1a

VI

Bus i

PCC

dciV

dciI

Phase

reactorAC

filter

fI

Bus i

PCCPhase

reactor AC

filterBus j

PCC

dcjI

dcjV

…...

…...

Figure 5.13: Basic diagram of a VSC substation

As indicated in Fig. 5.13, Bus i connects to the VSC substation through an ACline, which can be represented by the AC branch model to build up the relationamong V , I, Vf , If and Vv, Iv. Hence, the static network model is formulated as:

f(x) =

Vv − Vf − IvZpVf − 1

a V − aIZtIv − If − aI

. (5.21)


Furthermore, in order to find the relation between converter’s voltage and currentphasors with the DC voltage and current, the converter model needs to be included.For state estimation purposes, an average value model (AVM) is sufficient for aVSC, which avoids distinguishing different switching levels and modulation types,and instead it focuses on the voltage and current components of the fundamentalfrequency.

An AVM can be represented by a combination of controllable three-phase ACvoltage sources connected to the AC circuit and a controllable current sourceconnected to the DC circuit. These three phase voltages are controlled by theirrespective modulation indexes, namely: ma, mb and mc. Therefore, the relationshipbetween the AC and DC circuits can be formulated as:

vav = maVdc

vbv = mbVdc

vcv = mcVdc.

(5.22)

Besides, regardless of the power flow direction, the power consumption on one sidemust be always equal to the power injection from the other side. Thus,

Idc = vav iav + vbvi

bv + vcvi

cv

Vdc= mai

av +mbi

bv +mci

cv. (5.23)

All the AC variables in (5.22) and (5.23) use lowercase letters, which indicates theyare instant variables; Vdc and Idc use uppercase letters because when using the AVM,DC variables are relatively constant without considering the harmonics. Then theconverter model is formulated as

f(x) :[MvVdc − |Vv|(Kd2a)2VdcIdc − |Vv||Iv| cos(θv − δv)

]. (5.24)

where Mv is the modulation index, which is defined here as the ratio of the root-mean-square (RMS) value of the modulating wave, i.e., positive-sequence componentof the AC voltage at the converter, to the peak value of the carrier wave, i.e., thepole-to-pole DC voltage; Kd2a is the coefficient that transfers DC quantities to theAC base when using per unit values; θv and δv are the angles of the converter’svoltage and current, respectively. The first equation uses the modulation index tobuild the relation between the converter’s AC side voltage and DC voltage; whilethe second equation models the active power equality at the converter’s AC sideand DC side when omitting its internal losses.

In summary, combining (5.21) and (5.24) yields a VSC substation model forstatic state estimations. The state vector of an AC system with VSC substations isgiven by

x =[|V|, |I|, θ, δ, |Vf |, θf , |If |, δf , |Vv|, θv,Vdc, Idc

]T. (5.25)

5.3. VSC-HVDC 69

5.3.2 VSC control modesGenerally, VSCs contain a two-level control strategy. The high-level control providesaccurate voltage references for AVM (or for each sub-module of a detailed model),so as to maintain the control variables within an acceptable range. Among thehigh-level control scheme, vector-current control has been successfully applied onseveral VSC-HVDC link installations.

Inside vector-current control, it has a two-loop control scheme, referred to asthe outer control loop and the inner control loop. The outer control loop transfersthe VSC control references, i.e. P ref , Qref , V refdc or V refac , into converter currentreferences in dq coordinate as iref

dq = [irefd irefq ]T . The inner control loop transfersthe converter current references iref

dq = [irefd irefq ]T into converter bridge voltage asvref

dq = [vrefd vrefq ]T . Then, they are transformed to three-phase quantities, whichare the converter voltage references in three phases.

Finally, a low-level control scheme is utilized, such as pulse width modulation(PWM) algorithm, to regulate the switching signal generation via the voltagereferences generated by the high-level control, and further to provide switchingpulses for valves. This is out of the scope of this application because the converter ofan AVM is represented by a voltage source, where no switching device is considered.

Depending on the operation mode, reference irefd can be determined from eitherthe active power reference or the DC voltage reference. Similarly, reference irefqcan be determined from either the reactive power reference or the AC voltagereference. For each VSC substation, only one irefd and only one irefq are allowed tobe implemented. Thus, a choice of d control reference (choosing between Pref andV refdc ), and q control reference (choosing between Qref and V refac ) has to be made.Therefore, control modes for a VSC substation are formulated as follows:

0 = CP ∗ (|Vi| ∗ |Ii| ∗ cos(θi − δi)− P ref ) + CV d ∗ (Vdci − V refdci )0 = CQ ∗ (|Vi| ∗ |Ii| ∗ sin(θi − δi)−Qref ) + CV a ∗ (|Vi| − V refi ),

(5.26)

where C = [CP CV d CQ CV a] is the control model index. Ci = 1 indicates the cor-responding control mode is activated, otherwise Ci = 0 indicates the correspondingcontrol mode is disabled.

Note that to include the control modes into the state estimator, only theequilibrium condition of a system is taken into account, which means the controlledvariables are presumed to be equal to the references. There are two reasons to makethis assumption. First, the network model for the static state estimations doesnot include differential equations to represent system dynamics. Nevertheless, fastrate of the PMU-based state estimation will compensate for it as the Kirchhoff’slaws of the network model should hold at each measurement snapshot. Second,vector-current control is rather fast and robust, particularly, it is facilitated withmany saturation limiters to protect hardware from being damaged by peak currentsafter perturbations occur. Hence, the discrepancy of controlled variables is withina small range even during a perturbation, which would not greatly detract the


accuracy of the proposed control modes equations. Therefore, including controlmodes into the state estimator improves its robustness and redundancy.

In summary, by combining (5.21), (5.24) and (5.26), a VSC substation modelwith different control modes can be formulated as:

f(x) =

Vv − Vf − IvZpVf − 1

a V − aIZtIv − If − aIMvVdc − |Vv|(Kd2a)2VdcIdc − |Vv||Iv| cos(θv − δv)CP ∗ (|Vf | ∗ |If | ∗ cos(θf − δf )− P ref ) + CV d ∗ (Vdc − V refdc )CQ ∗ (|Vf | ∗ |If | ∗ sin(θf − δf )−Qref ) + CV a ∗ (|Vf | − V reff )

. (5.27)

5.3.3 Point-to-point VSC-HVDC link model

A single VSC may operate as a FACTS device, such as the static synchronouscompensator (STATCOM) and the static synchronous series compensator (SSSC),which will be introduced in next section. A more common application of VSCs is thepoint-to-point VSC-HVDC link, as shown in Fig. 5.14. Two VSC substations areconnected through a long distance DC cable. Since VSC is a bidirectional converter,it enables to change the power flow direction. One VSC acts the rectifier and theother one will act as an inverter, depending on the power flow control referencesand settings.

dcV

dcI

vV

vI

pZ

Phase

reactorAC

filter

fV

fI

tZ

:1a

VI

Bus i

PCC

dciV

dciI

Phase

reactorAC

filterBus i

PCCPhase

reactor AC

filterBus j

PCC

dcjI

dcjV

…...

…...

Figure 5.14: A point-to-point VSC-HVDC link model

For the point-to-point VSC-HVDC link, each converter can be modeled as a VSCsubstation as in (5.27). Additionally, the DC link can be modeled with a resistor.Thus,

f(x) =[Vdci − Vdcj + Idci ∗RdcIdci + Idcj

]. (5.28)

The control strategy for a point-to-point VSC-HVDC link needs to take bothconverter substations into account. Since only one DC voltage reference can be usedin the link, either rectifier or inverter would be assigned to control the DC voltage,and the other one has to choose the active power control for the d axis control. Withrespect to the q axis control, both substations can be operated under AC voltage

5.3. VSC-HVDC 71

control or reactive power control. Moreover, from each converter’s point of view,(5.26) still holds for d and q axis controls.

Therefore, by combining (5.27) and (5.28) the point-to-point VSC-HVDC linkmodel can be formulated.

5.3.4 Case studyThe 6-bus test system is developed in matlab/Simulink with a point-to-pointVSC-HVDC link placed between bus 3 and bus 4. Average value models are usedfor the VSCs. A circuit breaker located on line 4, between bus 4 and 6, was openedat t = 2 s and after three cycles it was re-closed at t = 2.06 s. Figure 5.15 showspartial estimation results for multiple snapshots.

1 1.5 2 2.5 3 3.5 40.8

0.9

1

Time (s)

|V|(

p.u

.) Vmag−true

Vmag−meas.

Vmag−est.

1 1.5 2 2.5 3 3.5 410

−5

100

Time (s)

Erro

r (p

.u.) Vmag−estimation−residual

(a) |V | at bus 4

1 1.5 2 2.5 3 3.5 4

0.95

1

1.05

Time (s)

Vrd

c(p

.u.)

Hvdc−true

Hvdcm

Hvdc−est

1 1.5 2 2.5 3 3.5 4

10−5

100

Time (s)

Erro

r (p

.u.)

Vrdc−residual−error

(b) DC voltage on the rectifier side

1 1.5 2 2.5 3 3.5 40.6

0.8

1

Time (s)

Idc (

p.u

.)

Hvdc−true

Hvdcm

Hvdc−est

1 1.5 2 2.5 3 3.5 410

−4

10−2

Time (s)

Erro

r (p

.u.)

Idc−residual−error

(c) DC currentFigure 5.15: NSE for the 6-bus hybrid AC/DC system with a point-to-point VSC-HVDClink for multiple snapshots

As shown in Fig. 5.15, the residuals between estimation results and “true values”(obtained from matlab/Simulink simulation and then processed to mimic phasordata) are maintained around 10−3, which is reasonable in the presence of ambigu-


ous modelings between matlab/Simulink model and state estimator model. Thissimulation study mimics a realistic condition that may be found in field installations.

On the other hand, when the system was subject to a perturbation, the estimationresiduals increased to 10−1. This is mainly due to the lack of representation of thetransient dynamics in the estimation model. Thus even when a system is oscillatingafter a perturbation occurs, the estimation model remains the same and is not ableto take the system’s dynamical behavior into account properly. How to consider thesystem dynamics when operating a static state estimation becomes an intriguingproblem and will be introduced in next chapter.

5.4 FACTS

FACTS has become an indispensable asset to improve transmission quality andefficiency of existing AC grids. Therefore, FACTS devices have to be properlymodeled and studied for power system operation and control.

Depending on the types of power electronic components being used, FACTSdevices can be classified as thyristor-based and converter-based devices. Thyristordevices have no gate turn-off capability while converter devices, i.e. voltage sourceconverters (VSCs) and current source converters (CSCs), have gate turn-off capability.This characteristic results in differences in control schemes. Thyristor-based FACTSinclude devices such as static var compensator (SVC) and thyristor controlledseries compensator (TCSC); converter-based FACTS include devices such as staticsynchronous compensator (STATCOM) and static synchronous series compensator(SSSC).

Both a thyristor-based and a converter-based devices with a similar controlfunction can have a similar response within their linear operating range (understeady state conditions). Therefore, it is reasonable to categorize FACTS devicesaccording to their functions. To this end, FACTS can be categorized as shunt devices(e.g., SVC and STATCOM), series devices (e.g., TCSC and SSSC), and combinedseries-series or series-shunt devices (e.g., UPFC).

This section will briefly present the network models of SVC, STATCOM, TCSCand SSSC for PMU-based state estimations. Other FACTS models can be easilyobtained by modifying or extending these models with minor changes.

5.4.1 Shunt devices

A shunt FACTS device serves as a static compensator such as SVC and STATCOM.The compensator varies its reactive output power to control the voltage at giventerminals of the transmission network so as to maintain the desired power flowunder possible system disturbances. This voltage control is performed on the busindependently from the individual lines connected to it [27]. In addition, as shuntdevices are not embedded in transmission lines, they do not need to sustain con-tingencies and dynamic overloads compared to series devices. On the other hand,

5.4. FACTS 73

shunt devices do not have the advantage of controlling power flow on lines and theyare bigger than series devices for a required MVA size.

In practice, SVC and STATCOM are not used as a perfect voltage regulator(VT = Vref ), but rather a droop controller (Ish = 1

Slope (VT − Vref )), where theterminal voltage can vary in proportion with the compensating current. This isrepresented in Fig. 5.16 by the linear slope between the terminal voltage and thecompensating current before the current hits its capacitive or inductive limit. Forterminal voltages that are out of the linear control range, the compensating currentof STATCOM stays at the maximum capacitive or inductive value; in contrast, SVCchanges in the manner of a fixed capacitor or inductor. Our static estimation modelsonly take into consideration the linear control range.

refV

VsX

stIstI

max

capImax

indI

Capacitive Inductive

refVV

stIstI

max

capImax

indI


1V

refVV

stIstI

max

capImax

indI


2V

refVV

stIstI

max

capImax

indI


1V

2V

refV

TV

shImax

capImax

indI


shI

SVCSTATCOM

STATCOM SVC

Figure 5.16: V-I characteristic of the SVC and the STATCOM [27]

5.4.1.1 Static Var Compensator

SVC is a controlled variable reactive impedance type var generator, which employsthyristor-controlled (TCR) reactors with fixed and/or thyristor-switched capacitors(FC and/or TSC). Figure 5.17a and 5.17b shows basic diagrams of FC-TCR andTSC-TCR, respectively. The firing angle (α) controls the turn-on period of thethyristor so as to vary the equivalent reactance of the SVC. To simplify the networkmodel and reduce the number of state variables, the controlled variable is assumedto be an equivalent susceptance, bSV C [78], as shown in Fig. 5.17c.

Its network model can be formulated as:

f(x) =[|Vi| − V refi −Kbsvc

], (5.29)

where V refi is the constant voltage which the SVC aims to maintain for bus i andVi is the bus voltage phasor; bsvc is the equivalent susceptance value. K is the slopeof the V-I linear characteristic, typically 1 − 5%. Therefore, the states of an AC


V

V

(a) Basic FC-TCR type staticvar generator

V

V

(b) Basic TSC-TCR type staticvar generator

Bus i

SVCb

(c) equivalent susceptancemodel

Figure 5.17: SVC model schemes [27, 78]

system with SVCs can be denoted as

x =[|V|, |I|, θ, δ,bsvc

]T, (5.30)

where |V|, |I|, θ and δ are AC system states.

5.4.1.2 Static Synchronous Compensator

STATCOM is a synchronous voltage source type static var generator which employsa switching power converter. It can be VSC-based, as shown in Fig. 5.18a, or CSC-based. Its model can be simplified as a current injection source for state estimationpurposes, as shown in Fig. 5.18b. The STATCOM current Ist is always in quadraturewith the bus voltage Vi, hence only reactive power is drawn or injected into theconnected bus.

Its network model can be formulated as:

f(x) =[|Vi| − V refi −K|Ist|

], (5.31)

where Ist is the equivalent current phasor generated by STATCOM; other notationsare the same with those used in SVC model. Therefore, the states of an AC system

Bus i

(a) VSC-based STATCOM

Bus i

stI

(b) equivalent current source model

Figure 5.18: STATCOM model schemes [27, 78]

5.4. FACTS 75

with STATCOMs can be denoted as

x =[|V|, |I|, θ, δ, |Ist|

]T, (5.32)

where |V|, |I|, θ and δ are AC system states.Both SVC and STATCOM operate to maintain the bus voltage by manipulating

the reactive power, but in different ways: SVC is modeled as an equivalent susceptancebsvc while STATCOM as a current source Ist.

5.4.2 Series devicesThe shunt compensation is effective in maintaining the desired voltage profile at buses.However, it is ineffective in controlling the actual transmitted power at a definedtransmission voltage. In contrast, series capacitive compensation is able to cancela portion of the reactive line impedance and thereby increases the transmittablepower. On the other hand, series FACTS devices have smaller size for the sameMVA rating of a shunt device, but they have to be designed to be able to sustaindifferent contingencies and overloads. In general, series FACTS can be classified intothyristor-control impedance type (e.g., TCSC) and converter-based voltage sourcetype compensators (e.g., SSSC).

5.4.2.1 Thyristor Controlled Series Compensator

A TCSC consists of the series compensating capacitor shunted by a thyristor-controlled reactor, as shown in Fig. 5.19a. It allows varying the series reactance of atransmission line to regulate the active power flow through the line. Although firingangle (α) is the control variable, an equivalent susceptance btc is used in order tosimplify the network model and reduce the number of states variables, as shown inFig. 5.19b. Thus, the network model of TCSC can be formulated as:

f(x) =[P refij − |Vi||Vj |(yij + btc) sin(θi − θj)

], (5.33)

where P refij is the constant active power flow reference on which the TCSC aims tomaintain; yij is the line admittance without considering the TCSC and btc is the

α

α

Bus i Bus j

ijy

(a) firing angle model

Bus i jTCSCb

ijyBus

(b) equivalent susceptance model

Figure 5.19: TCSC model schemes [27, 78]


ijLine

ijZ

Bus i Bus j

ijI SSSCV

(a) VSC-based SSSC

ijLine

ijy

Bus i Bus j

ijI SSSCV

(b) equivalent synchronous voltage sourcemodel

Figure 5.20: SSSC model schemes [27, 78, 108]

equivalent susceptance of the TCSC; θi and θj are the angles of voltage phasors Viand Vj , respectively.

Similarly to (5.29) and (5.31), (5.33) is added to the network model. In addition,the equations for the AC line where the TCSC is installed also need to be updatedaccordingly as the line impedance changes.

It is notable that a TCSC can be operated under different control modes.Maintaining a constant active power is a commonly used one. Apart from that,maintaining specific impedance level, or current magnitude through the line is alsofeasible [107]. This network model is developed based on the constant active powercontrol mode. For other control modes, it needs minor modification.

5.4.2.2 Static Synchronous Series Compensator

An SSSC is a converter-based FACTS device as shown in Fig. 5.20a. Its model canbe simplified as a series voltage source [27, 78, 108] for state estimation purposes, asshown in Fig. 5.20b. An SSSC is able to provide a constant compensating voltageindependently from variable line currents. This voltage Vss is always in quadraturewith the line current Iij , hence only its magnitude is controllable.

In some cases, the injected voltage is assigned larger than the voltage differencebetween the sending- and receiving-end, that is, if |Vss| > |Vi − Vj |, the power flowwill reverse. This bi-directional compensation capability distinguishes SSSCs fromother series devices. However, in many practical applications, only capacitive seriesline compensation is required.

An SSSC can be controlled in three different ways:

• in voltage compensation mode [27, 78], the SSSC can maintain the ratedcapacitive or inductive compensating voltage regardless of the changes in theline current. This voltage Vss is always in quadrature with the line current Iij ,hence only its magnitude is controllable.

• in impedance compensation mode [27, 78], the SSSC is fixed to the maximumrated capacitive or inductive compensating reactance.

5.4. FACTS 77

• some literature (e.g. [108]) also proposes to use SSSC for controlling the activepower flow of the line to a desired value.

In all the control methods, the active power injected into the transmission line bythe SSSC is zero unless the series VSC has an energy storage system connected tothe DC capacitor or it is connected to the DC bus of another VSC.

Under constant voltage control mode, an SSSC can be formulated as:

f(x) =

(±Vss + Vi − Vj) ∗ yij − Iij|Vss| − V refss

θss − δij − π/2

, (5.34)

where Vss is the equivalent synchronous voltage of an SSSC and V refss is the constantvoltage value which the SSSC aims to maintain; θss and δij are the angles of SSSCequivalent voltage Vss and line current Iij , respectively. Note that the ± symbolindicates the type of compensating voltage: a positive symbol indicates a capacitivecompensation and a negative symbol indicates an inductive compensation. For othercontrol modes, this model needs to be adapted.

Equation (5.34) replaces the corresponding (row) equations that are associatedwith the SSSCs in the AC network model, hence, the number of network equationsremains the same. As phasors in the proposed state estimator are represented bymagnitudes and angles, (5.34) is thus rewritten as follows:

f(x) =[|Vss|yij cos(δij + π/2 + γij) + |Vi|yij cos(θi + γij)− |Vj |yij cos(θj + γij)− |Iij | cos(δij)|Vss|yij sin(δij + π/2 + γij) + |Vi|yij sin(θi + γij)− |Vj |yij sin(θj + γij)− |Iij | sin(δij)

],

(5.35)where θ and δ are the angles of the voltage phasor and the line current phasor,respectively; γ is the angle of the line admittance.

5.4.3 Case studyThe 9-bus test system is used for this case. Shunt devices are installed at bus 8, andseries devices are installed on the line between bus 5 and bus 4.

5.4.3.1 Test for the SVC model

An SVC was installed at bus 8, where a 16.67% load increase (both active powerand reactive power) was applied at t = 2 s. Figure 5.21 shows the estimation resultsfor multiple snapshots.

As shown in Fig. 5.21, the estimation residuals for both the voltage magnitudeat bus 8 and the equivalent susceptance reach below 10−13 during steady state.When the perturbation occurs, the estimation residuals of the voltage magnitudemanage to maintain on the same level as before, which is due to the availability ofits measurement. However, the estimation residuals of the equivalent susceptance


1 1.5 2 2.5 3 3.5 4

1.02

1.025

1.03

time

|V|(p

.u.)

.

Vmag−trueVmag−mVmag−est

1 1.5 2 2.5 3 3.5 40

0.5

1x 10

−14

time

erro

r(p.

u.)


(a) |V | at bus 8

1 1.5 2 2.5 3 3.5 40.1

0.2

0.3

0.4

time

bsvc

(p.u

.)

bsvc−truebsvc−mbsvc−est

1 1.5 2 2.5 3 3.5 410

−20

10−10

100

time

erro

r(p.

u.)

bsvc−residual−error

(b) bsvc

Figure 5.21: NSE for the 9-bus system with an SVC for multiple snapshots

1 1.5 2 2.5 3 3.5 4

1.02

1.025

1.03

time

|V|(p

.u.)

.


1 1.5 2 2.5 3 3.5 40

2

4

6x 10

−15

time

Err

or(p

.u.)


(a) |V | at bus 8

1 1.5 2 2.5 3 3.5 40

0.2

0.4

time

|Ist|(

p.u.

)

|Ist|−ture|Ist|−est

1 1.5 2 2.5 3 3.5 410

−20

10−10

100

time

Err

or(p

.u.)

|Ist|−residual−error

(b) Ist

Figure 5.22: NSE for the 9-bus system with a STATCOM for multiple snapshots

are subject to a big leap and then come down to about 10−5. This big leap of theestimation residual implies the limited performance of the static state estimationwhen the system is under large dynamic changes and power electronic devices reactto these changes.

5.4.3.2 Test for the STATCOM model

The same scenario used for the SVC test was also applied to the test system witha STATCOM installed at bus 8. As shown in Fig. 5.22, the estimation performsrelatively close with that in the SVC test.

5.4. FACTS 79

1 1.5 2 2.5 3 3.5 40.85

0.9

0.95

1

1.05

time

|V|(

p.u

.).


1 1.5 2 2.5 3 3.5 410

−20

10−10

100

time

Erro

r (p

.u.)


(a) |V | at bus 5

1 1.5 2 2.5 3 3.5 420

40

60

time

|ytc

sc|(

p.u

.)

ytcsc−trueytcsc−mytcsc−est

1 1.5 2 2.5 3 3.5 410

−20

10−10

100

time

|ytc

sc|(

p.u

.)

ytcsc−residual−error

(b) ytc

Figure 5.23: NSE for the 9-bus system with a TCSC for multiple snapshots

1 1.5 2 2.5 3 3.5 40.85

0.9

0.95

1

1.05

time

|V|(

p.u

.).


1 1.5 2 2.5 3 3.5 410

−20

10−10

100

time

Erro

r (p

.u.)


(a) |V | at bus 5

1 1.5 2 2.5 3 3.5 40.4

0.5

0.6

time

|I|(p

.u.)

Imag−trueImag−mImag−est

1 1.5 2 2.5 3 3.5 410

−20

10−10

100

time

|I|(p

.u.)

Imag−residual−error

(b) |I| on the line between bus 5 and bus 4Figure 5.24: NSE for the 9-bus system with an SSSC for multiple snapshots

5.4.3.3 Test for the TCSC model

A TCSC was installed on the line between bus 5 and bus 4, and a fault was appliedat bus 3 from t = 2 s to t = 2.04 s. Figure 5.23 shows the estimation results formultiple snapshots, where ytc denotes the line admittance considering the TCSC.The residuals increase immediately at the instance when the fault was applied, fromaround 10−13 to 10−3.

5.4.3.4 Test for the SSSC model

The same scenario for TCSC was applied on the test system with an SSSC installedon the line between bus 5 and bus 4. Figure 5.24 shows the estimation results formultiple snapshots. The residuals increase immediately at the instance when thefault was applied, from around 10−13 to 10−3.


5.5 Summary

This chapter presents a novel way of formulating the measurement model of a PMU-based state estimator, where the network model is separated from measurementsin order to protect the network model from missing measurements and assigndifferent weights to them separately. Sequentially, network models of classic HVDClink, VSC-HVDC link, and FACTS are introduced. The proposed network modelsimplifies the nonlinearities of the conventional network model. Different controlmodes are also included to enrich network information. All the AC and DC statesare simultaneously considered to solve nonlinear WLS problem. After presentingthe network models for the state estimator, case studies for classic HVDC link,VSC-HVDC link, and FACTS are conducted individually to validate the proposednetwork models.

At the same time, it is noted that the estimation accuracy needs to be improvedwhen the system is under large dynamic changes. Testing results indicate thatadditional modeling details may need to be included to obtain higher accuracy whenthe system is under large dynamic changes and power electronic devices react tothese changes. This will be addressed in the following chapter.

Chapter 6

Pseudo-dynamic network modeling andexamples

As renewable energy integration brings intermittent fluctuations into power systems,state estimation applications face a greater need to capture system dynamics in afaster and more flexible way, which has been difficult for the static state estimations.On the other hand, most of the dynamic state estimations and forecasting-aidedstate estimations are computationally demanding, which introduces delays in theestimation calculation cycle [109]. When the calculation cycle is longer than thePMU data rate, the estimated results become less valuable for on-line applications,especially those with real-time requirements.

Therefore, we propose a pseudo-dynamic modeling approach [110] that canimprove the estimation accuracy during transients without significantly increasingthe estimation’s computational burden. This pseudo-dynamic modeling approachoffers the following advantages:

• High accuracy: it establishes difference equations from the dynamic model ofcertain components from both AC and DC grids, and then combines themwith the static network equations. This process is named as pseudo-dynamicnetwork modeling.

• Compatible modeling and implementation: the pseudo-dynamic network modelmaintains the basic structure of the static model, which greatly reduces theworkload of re-composing network models when updating existing algorithms.

• Fast computation: the algorithm used to solve static state estimations (i.e.,WLS)is directly applied to the pseudo-dynamic state estimation, which ensures thatthe computational speed will be minimally affected.

• Explicit representation of control modes: for devices with time-varying controlreferences, the pseudo-dynamic state estimation is capable of taking theircontrol modes into account, which would be challenging for static-only stateestimations.

81

82 Pseudo-dynamic network modeling and examples

At the same time, as the power electronics-based devices (e.g., FACTS andVSC-based HVDC) play an important role in improving the system control abilityand flexibility with the increased integration of renewable energy, their real-timeperformance during transients needs to be monitored in order to make full use ofthese devices in on-line operations. To this end, suitable pseudo-dynamic models thatcan represent power electronics-based devices for both steady state and transientconditions are developed. Two power electronics-based devices are used to illustratethe proposed approach—STATCOM, as an example of a FACTS device, and VSC-HVDC link.

6.1 Pseudo-dynamic concept

Network models for static state estimations describe the system topology andproperties under normal operating conditions, which can be considered to be in thequasi-stationary regime. However, when the system is under transient conditions, thestatic network model for certain components may no longer hold. If the models forsuch components are not replaced by the models that can represent their dynamicbehaviors, the estimation error would be large. In other words, when the systementers a transient condition due to a perturbation, if the system cannot restorea steady state before the next PMU snapshot comes, the static network modelwill conflict with the PMU measurement, leading to an inaccurate state estimationsolution.

Therefore, a new type of network model, called herein the pseudo-dynamicnetwork model, is proposed. It leverages the existing body of the network modeland includes the difference equations that describe the system dynamical properties.The following section explains how to formulate the pseudo-dynamic network model.

Typically, it is assumed that power system as a continuous dynamical systemcan be described by employing ordinary differential equations (ODEs). Even fora higher-order system, its higher-order ODEs can be converted into a larger setof first-order equations by introducing extra variables. However, for PMU-basedstate estimations, PMU data is streamed discretely over fixed and synchronous timeintervals. Therefore, a power system can be treated as a discrete dynamical system,where difference equations are used to update the state variables in discrete timesteps of the same size as the PMU data.

This discretization is similar to numerically solving differential equations, i.e.numerical integration using Euler’s method, where the states are updated whenknowing the starting point and the slope at it, and the error can be made smallif the step size is small enough and the interval of computation is finite. However,Euler’s method is insufficiently robust, and thus the Euler’s full-step modification,which belongs to a second-order Runge-Kutta method, is used next.

The difference equation used herein is formulated as

xk ≈ xk−1 + Ts2 (xk−1 + ˙xk), (6.1)

6.2. STATCOM model 83

where Ts is the sample time (step size). x = g(x) can be either a linear or anonlinear function of x, which is essentially the differential equation of the continuousdynamical system. The value of xk−1 is calculated by substituting xk−1 into g(x),which is denoted by g(x)k−1.

Equation (6.1) implies that the current value is determined by adding the previousvalue to the average increment during the time interval using the average of theslopes. In order to comply (6.1) with the generalized form of the network modelequation f(x), it is rewritten as

f(xk) : xk − xk−1 −Ts2[g(x)k−1 + g(x)k

]. (6.2)

The foregoing is the procedure of pseudo-dynamic network modeling and (6.2)is a pseudo-dynamic network model, which can be used to describe all differentcomponents with dynamical behavior. While the majority of the literature considersthe use of Kalman Filters or other types of observers for a similar purpose, thiswork chooses a simpler approach, which only requires few PMU measurements closeto a certain component and without the need of formulating a complex dynamicmodel. Two examples are presented in the following sections.

6.2 STATCOM model

This section introduces the pseudo-dynamic network model of a STATCOM. It isused as an example of showing how to develop a pseudo-dynamic network model fora controller.

1

K

T s

refV

V stI

Figure 6.1: Simplified STATCOM control block diagram

A simplified STATCOM’s control process can be represented by the block diagramshown in Fig. 6.1. The output is the current magnitude |Ist|, which varies the currentflow at the connected bus in order to change the reactive power flow and controlthe bus voltage. Ist is perpendicular to the bus voltage phasor, hence its angle canbe computed from the voltage angle. This controller model is given by

|Ist| =K

T(V ref − |V |)− 1

T|Ist|. (6.3)

Using the pseudo-dynamic network model described by (6.2), (6.3) can be rewrittenas:

f(xk) : (1+ Ts2T )|Ist,k|+

TsK

2T |Vk|−TsK

TV ref−(1− Ts2T )|Ist,k−1|+

TsK

2T |Vk−1|. (6.4)


Equation (6.4) embodies the dynamic relation between |V | and |Ist|, and replacesthe static relation in (5.31). This pseudo-dynamic network model of STATCOMwill be validated using the real PMU data in Sec. 6.4.1, and simulation studies inSec. 6.4.2.

The states of the new model are the same as those of the static model (5.32).The corresponding Jacobian matrix elements must be updated using

H(xk) : ∂h(xk)∂|Ist,k|

= 1 + Ts2T ,

∂h(xk)∂|Vk|

= TsK

2T . (6.5)

6.3 VSC-HVDC model

6.3.1 VSC substation model

In (5.24), the modulation index Mv defines the relation between Vdc and |Vv|.However, in reality |Vv| varies with the system dynamics; while Mv is a fixed valuethat depends on the control mode. Therefore, in order to improve the accuracy ofthe model especially during transients, |Vv| in (5.24) is replaced by |vref |, which isthe voltage reference of the converter’s bridges. This voltage reference is generatedby the converter’s control system.

The substation’s pseudo-dynamic model intends to represent the control processof the converter. As already explained in Section 5.3.2, the most common high-levelcontrol strategy for VSCs is vector-current control, which has a two-level controlscheme, so-called the outer active-reactive power and voltage loop (here abbreviatedto outer loop), and the inner current loop (here abbreviated to inner loop). Asshown in Fig. 6.2, the outer loop transfers the VSC control references, i.e., P ref ,Qref , V refdc and V refac , into the converter’s current references, iref

dq = (irefd irefq )T . Inthis loop, three-phase fundamental currents and voltages are transformed into dqcomponents in a synchronously rotating reference frame through Clark’s and Park’stransformations. Hence, all quantities become DC signals [111].

Depending on the converter’s operation mode, reference irefd can be determinedby either active power P ref or DC voltage V refdc . Similarly, reference irefq can bedetermined by either reactive power Qref or AC voltage V refac at the PCC. Foreach VSC substation, only one irefd and only one irefq can be utilized. An ordinaryintegral controller can be used for outer loop, which can be formulated using the

6.3. VSC-HVDC model 85

pseudo-dynamic equations (6.2) as:

f(xk) :irefd,k − irefd,k−1 −KP

Ts2

(P ref − Pk|Vf,k|

+ P ref − Pk−1

|Vf,k−1|

),

or irefd,k − irefd,k−1 −KV dc

Ts2

(2V refdc − Vdc,k − Vdc,k−1

);

irefq,k − irefq,k−1 −KQ

Ts2

(Qref − Qk|Vf,k|

+ Qref − Qk−1

|Vf,k−1|

),

or irefq,k − irefq,k−1 −KV ac

Ts2

(2V refac − Vac,k − Vac,k−1

);

(6.6)

where Pk = |Vk||Ik| cos(θk − δk), Qk = |Vk||Ik| sin(θk − δk).Next, the outputs of the outer loop become the inner loop’s input. The inner

loop transfers the current references irefdq = (irefd irefq )T into the voltage references

of the converter’s bridges vrefdq = (vrefd vrefq )T , which then are transformed into the

three-phase voltage references.

refPref

dcV

ref

dI

ref

qI

refQref

acV

ref

dV

ref

qV

ref

aV

ref

bV

ref

cV

Outer loop control Inner loop control Transformation

Figure 6.2: The vector-current control process

Based on the converter’s structure shown in Fig. 5.13, the fundamental voltage ofthe converter’s bridges equals to the voltage at PCC minus the voltage drop on thetransformer and the phase reactor. By neglecting the resistances of the transformerand the phase reactor, an ordinary PI controller can be used for the inner loop,which is formulated as:

vref = V − (Kp +Ki1s

)(iref − Iv)− j(Xt +Xp)iref , (6.7)

where vref = vrefd + jvrefq and iref = irefd + jirefq . Equation (6.7) can be rewrittenas:

vrefk = Vk −Kp

(irefk − Iv,k

)−Ki

1s

(irefk − Iv,k

)− j(Xt +Xp)irefk . (6.8)


In order to formulate (6.8) in the form of the pseudo-dynamic network model,assume that yk = Ki

1s

(irefk − Iv,k), then

yk = yk−1 +KiTs2

(irefk − Iv,k + irefk−1 − Iv,k−1

),

yk−1 = −vrefk−1 + Vk−1 −Kp(irefk−1 − Iv,k−1)− j(Xt +Xp)irefk−1.

Substituting yk and yk−1 into (6.8), then

f(xk) =vrefk − vrefk−1 − Vk + Vk−1 +Kp

(irefk − Iv,k − irefk−1 + Iv,k−1

)+Ki

Ts2

(irefk − Iv,k + irefk−1 − Iv,k−1

)+ j(Xt +Xp)

(irefk − irefk−1

).

(6.9)As all the states at step (k − 1) are known, they can be considered as a constantcomponent that is denoted by C. Hence, (6.9) can be simplified as

f(xk) = vrefk − Vk +(Kp +Ki

Ts2

) (irefk − Iv,k

)+ j(Xt +Xp)irefk + C. (6.10)

The above equation can be split into d and q components:

f(xk) =

vrefd,k − |Vk| cos(θk) +(Kp +Ki

Ts2) (irefd,k − |Iv,k| cos(δv,k)

)− (Xt +Xp)irefq,k + Cd

vrefq,k − |Vk| sin(θk) +(Kp +Ki

Ts2) (irefq,k − |Iv,k| sin(δv,k)

)+ (Xt +Xp)irefd,k + Cq

.(6.11)

Thus, the voltage reference of the converter’s bridges vref can be expressed by (6.6)and (6.11). Consequently (5.24) becomes

f(x) =[Kd2aMvVdc −

√(vrefd )2 + (vrefq )2

(Kd2a)2VdcIdc − |Vv||Iv| cos(θv − δv)

]. (6.12)

Therefore, (5.21), (6.12), (6.6), and (6.11) compose the pseudo-dynamic networkmodel for VSC substation. As a result, four more states are added for the pseudo-dynamic model in addition to the states of the static model 5.25:

x = [|V|, |I|, θ, δ, |Vf |, θf , |If |, δf , |Vv|, θv,Vdc, Idc, irefd , iref

q ,vrefd ,vref

q ]T . (6.13)

The Jacobian matrix needs to be updated w.r.t. (6.6), (6.11), and (6.12). Takingthe first equation of (6.6) as an example, its corresponding Jacobian matrix elementw.r.t. voltage magnitude is given by

H(xk) : ∂f(xk)∂|Vk|

= KPTs2|Ik|cos(θk − δk)

|Vf,k|.

6.4. Case study 87

1dcV

1dcI

dcR dcL

dcR dcL2dc

I2dcV

ccI

ccI

1dcC

1dcC2dcC

2dcC

Figure 6.3: DC circuit of a point-to-point VSC-HVDC link model

6.3.2 Point-to-point VSC-HVDC link model

Using the VSC substation model constructed above, other VSC models can bedeveloped, among which point-to-point VSC-HVDC link is of great interest. Tomodel it, one VSC substation acts as the rectifier and the other one as the inverter,depending on which side controls the active power flow. At each substation a largeDC capacitor is installed, and a DC cable connects two substations [112]. TheDC circuit of a point-to-point VSC-HVDC link is shown in Fig. 6.3 and the basicequations are:

f(x) =

Cdc1

dVdc1dt− (Idc1 − Icc)

Cdc2dVdc2dt− (Idc2 + Icc)

LdcdIccdt− (Vdc1 − Vdc2 −RdcIcc)

. (6.14)

The corresponding pseudo-dynamic equations are:

f(xk) =

Vdc1,k − Ts2Cdc1

Idc1,k + Ts2Cdc1

Icc,k − Vdc1,k−1 − Ts2Cdc1

Idc1,k−1 + Ts2Cdc1

Icc,k−1

Vdc2,k − Ts2Cdc2

Idc2,k − Ts2Cdc2

Icc,k − Vdc2,k−1 − Ts2Cdc2

Idc2,k−1 − Ts2Cdc2

Icc,k−1

(1 + TsRdc2Ldc )Icc,k − Ts

2Ldc Vdc1,k + Ts2Ldc Vdc2,k − (1− TsRdc

2Ldc )Icc,k−1 − Ts2LdcVdc1,k−1 + Ts

2LdcVdc2,k−1

.(6.15)

6.4 Case study

This section focuses on studying the performances of the PMU-based state estimationwhen using the static network model and the proposed pseudo-dynamic networkmodel. Subsection 6.4.1 uses real PMU data to compare and validate these twomodels in the case of a STATCOM. In Subsection 6.4.2, two test systems with aSTATCOM installed are built up and simulated in PSAT to generate the syntheticmeasurements for the accuracy comparison. At last, the VSC-Based HVDC Linkmodel provided by matlab/Simulink is used to generate the synthetic measurementsfor two test scenarios in Subsection 6.4.3.


6.4.1 STATCOM models’ comparison and validation using realPMU data

The real PMU data used in this subsection was recorded during a generator tripevent. The generator loss resulted in reducing the active power flow on the maintransfer paths in a neighboring system, and caused an increase in bus voltages.A STATCOM installed on the transfer paths reacted to the voltage change. TheSTATCOM bus voltage phasor, as well as its output current phasor, was measuredby a PMU with a reporting rate of 30 samples/second. Figure 6.4 shows the voltagemagnitude, current magnitude and angle differences.

0 50 100 150 200 250 3001

1.02

1.04

time [s]

Vm [p

u]

STATCOM voltage magnitude

0 50 100 150 200 250 300−0.5

0

0.5

time [s]

Im [p

u]

STATCOM current magnitude

0 50 100 150 200 250 300−100

0

100

time [s]

θ [d

eg

]

STATCOM VI angle differences

Figure 6.4: The STATCOM’s PMU measurements: voltage magnitude (top); currentmagnitude (middle); angle difference between voltage and current (bottom)

The PMU data is used to compute the STATCOM’s V-I curve as shown inFig. 6.5. Blue dots represent pre-fault operation points, which depict a linear V-Icharacteristic. When the fault occurred, several green dots scatter away from theV-I characteristic. Gradually the system reached another stable operation pointwhere the STATCOM current is about 0.25 p.u. Note that the STATCOM’s output

6.4. Case study 89

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

1.02

1.025

1.03

1.035

1.04

Im [pu]

Vm [p

u]

Pre−faulttransient oscillationreference changesnew operation pointvar reserveV−I characteristics

Figure 6.5: The STATCOM’s V-I curve computed using the PMU data

current switched from capacitive to inductive after the disturbance occurred, aimingto decrease the bus voltage. When the system was approaching the next steadystate, the system operator changed the voltage reference of the STATCOM in severalsteps, which can be seen from the saw-tooth variation in red (the voltage referencecorresponds to the voltage when the STATCOM current output equals to zero).The system reached a new operation point after the reference changed, which isrepresented by the light blue dots. As the system turned to a new steady statecondition, the system operator switched the STATCOM to the var reserve controlmode such that other slower voltage controls in the system can take over the reactivepower support.

Figure 6.5 shows the variation of operating conditions during the whole eventand clearly reveals the linear V-I droop relation of a STATCOM in steady-stateoperation. Using the pre-fault PMU data (0 to 69.33s), this linear V-I characteristicwas determined by applying the linear regression function polyfit in matlab, whichis shown by the black line in Fig. 6.5. The two coefficients of the linear predictor,slope and the intercept, represent the equivalent impedance and voltage referenceof the STATCOM, respectively. For the pre-fault steady state, Xs = 0.0285 andV ref = 1.0214.

Note that during the transients, the operating points scatter and do not exactlyfollow the V-I characteristic. For a certain |V |, the measured |Ist| and the readvalue from the V-I characteristic can deviate up to 0.04 p.u. This implies that thestatic STATCOM model, which is based on the V-I characteristic, would result indeviations during transients.

On the other hand, the pseudo-dynamic model can be a good choice to reflect theSTATCOM’s control process and its system dynamics. Using the transfer functionestimator tfest in matlab, the transfer function is estimated as 32.1981

0.0329s+1 , with


two control parameters K = 32.1981 and T = 0.0329, which are those in (6.3)-(6.5).The gain K theoretically is the reciprocal of the V-I characteristics’ slope,

which has been calculated for the pre-fault steady state, i.e. Xs = 0.0285. Itsinverse (32.1543) is quite close to the calculated K (32.1981), which verifies that ofthe transfer function. In addition, the time constant T is in accordance with thestatement of STATCOM in [27], which says “typically about 10-50 ms dependingon the var generator transport lag”.

In order to intuitively compare the STATCOM’s static model with its pseudo-dynamic model, PMU data of the voltage magnitude |V | is used as the arbitrary in-puts of the static model (see (5.31)) and the pseudo-dynamic model (see (6.3)-(6.5)),respectively. The static model applies the STATCOM’s linear V-I characteristic andthe pseudo-dynamic model applies its first-order control model whose parameterswere obtained above through model identification. These models’ outputs are thencompared to the |Ist| PMU measurement in Fig. 6.6.

69.5 70 70.5 71 71.5 72 72.5 73 73.5 74−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time [s]

|Ist

| [p

u]

PMU measurementpseudo−dynamic model outputstatic model output

Figure 6.6: PMU data, static model output and pseudo-dynamic model output for thecurrent at the STATCOM

Figure 6.6 compares results: the pseudo-dynamic model’ output |Ist| coincideswith the PMU data; while the static model’s output gradually deviates from thethe PMU data. The reason that the deviation is not prominent is because the timeconstant of the STATCOM is quite close to the PMU sampling rate, thereby itsdynamical trajecory can be nearly tracked by the PMU measurement. For a largerand more complex system, this deviation could be amplified further. An analysis oftheir residuals w.r.t. the PMU data is shown in Table 6.1.

6.4.2 STATCOM model in two test systemsNext, in order to further study the proposed STATCOM model and PMU-basedstate estimation algorithm, the 9-bus test system and the KTH-Nordic 32 testsystem are used. In the modified 9-bus test system, a STATCOM is installed at bus

6.4. Case study 91

Table 6.1: STATCOM models accuracy comparison

SE methods µ* δ** Max. residual

Pseudo-dynamic 0.0012 0.0056 0.0152

Static 0.0198 0.0085 0.0406*µ denotes the average value of the residuals w.r.t. the PMU data

**δ denotes the standard deviation of the residuals w.r.t. the PMU data

1.5 2 2.50.1

0.2

0.3

0.4

time (s)

|Ist|(

p.u.

)

|Ist|−m|Ist|−est

1.5 2 2.510

−20

10−10

100

time (s)

Err

or(p

.u.)


(a) |Ist| Static

1.5 2 2.50.1

0.2

time (s)

|Ist|(

p.u.

)


1.5 2 2.510

−20

10−15

10−10

time (s)

Err

or(p

.u.)


(b) |Ist| Pseudo-dynamicFigure 6.7: Static and Pseudo-dynamic SEs for the modified 9-bus system

8; in the modified KTH-Nordic 32 test system, the same STATCOM is installed atbus 43. The two key parameters are K = 25, T = 0.04.

6.4.2.1 Pseudo-dynamic SE for the modified 9-bus system

A 16.67% load increase (both active power and reactive power) at bus 8 was appliedat t = 2s. As shown in Fig. 6.7, for both cases the current magnitude residuals beforethe perturbation occurred are below 10−13p.u.; however, after the instance whenthe perturbation occurred, the static estimation residuals increase up to 0.1783p.u. then drop to 10−3p.u. while the pseudo-dynamic estimation successfully stayson the same accuracy level during transient dynamics. More estimation accuracyperformances are shown in Table 6.2.

6.4.2.2 Pseudo-dynamic SE for the modified KTH-Nordic 32 system

A 33% load increase (both active power and reactive power) at Bus 43 was appliedat t = 2s. As shown in Fig. 6.8, the static estimation and pseudo-dynamic estimationhold similar estimation accuracy of the current magnitude before the perturbation.However, after the perturbation occurs, the current magnitude estimated by the


1.5 2 2.5−0.2

0

0.2

0.4

0.6

Time (s)

|Ist|(

p.u.

)


1.5 2 2.50

0.2

0.4

Time (s)

Err

or(p

.u.)


(a) |Ist| Static

1.5 2 2.5−0.2

0

0.2

0.4

0.6

Time (s)

|Ist|(

p.u.

)


1.5 2 2.510

−20

10−15

10−10

Time (s)

Err

or(p

.u.)


(b) |Ist| Pseudo-dynamicFigure 6.8: Static and Pseudo-dynamic SEs for the modified KTH-Nordic 32 system

static estimation shows a residual up to 0.3666 p.u. In contrast, the pseudo-dynamicstate estimation gives a maximum residual of 5.0626×10−14p.u. Table 6.2 summariesthe results of these tests.

6.4.3 VSC-HVDC modelThe VSC-Based HVDC Transmission Link model provided by matlab R2013b/SimPowerSystems is used herein to generate the synthetic measurements used tovalidate the proposed VSC-HVDC model and the PMU-based state estimationalgorithm. Its rectifier uses active power and reactive power control, and its inverteruses DC voltage and reactive power control.

In the case study, all the control parameters preset in the SimPowerSystemsmodel are kept the same in the pseudo-dynamic state estimation model.

6.4.3.1 First test scenario

The inverter’s DC voltage reference dropped from 1 p.u. to 0.95 p.u. at t = 2.1s. Asshown in Fig. 6.9, the pseudo-dynamic state estimation performs more accuratelyby a factor of 1/1000 than the static state estimation not only during transients,but also during steady state. For instance, the voltage magnitudes on the rectifierside estimated by the static state estimation shows a maximum residual up to0.0279 p.u. while the pseudo-dynamic state estimation gives a maximum residual of7.6517× 10−8p.u.

6.4.3.2 Second test scenario

A three-phase line breaker on the inverter side was opened from t = 2.1s for 0.12s. Forthis larger perturbation, as shown in Fig. 6.10, the pseudo-dynamic state estimationperforms more accurately by a factor of 1/1000 than the static state estimation.

6.4. Case study 93Ta

ble6.2:

SEaccuracy

perfo

rmance

ontest

system

s

Bef

ore

pert

urba

tion

(µ,σ

)A

fter

pert

urba

tion

(µ,σ

)M

ax.

resi

dual

stat

icps

eudo

-dyn

amic

stat

icps

eudo

-dyn

amic

stat

icps

eudo

-dyn

amic

STA

TC

OM

test

1Res.|V

8|

(1.5

7e-1

6,1.

02e-

16)

(1.5

7e-1

6,1.

02e-

16)

(3.8

5e-1

6,1.

27e-

15)

(7.2

5e-1

6,1.

76e-

15)

7.11

e-15

8.22

e-15

Res.|Ist|

(-3.

84e-

15,

9.58

e-15

)(-

3.72

e-15

,2.

01e-

15)

(8.0

3e-0

4,0.

0127

)(-

3.57

e-15

,2.

16e-

14)

0.17

831.

05e-

13

STA

TC

OM

test

2Res.|V

43|

(-8.

71e-

17,

7.13

e-17

)(-

8.71

e-17

,7.

13e-

17)

(-2.

94e-

17,

2.12

e-16

)(-

2.39

e-17

,1.

95e-

16)

2.44

e-15

2.22

e-15

Res.|Ist|

(-2.

06e-

16,

6.37

e-15

)(2

.11e

-15,

1.83

e-15

)(0

.004

8,0.

0332

)(1

.57e

-15,

7.16

e-15

)0.

3666

5.06

e-14

VSC

-HV

DC

test

1

Res.|V

v|

(0.0

023,

0.00

24)

(5.4

5e-1

0,4.

74e-

09)

(0.0

187,

0.00

39)

(-5.

39e-

08,

1.78

e-08

)0.

0279

7.65

e-08

Res.|Iv|

(-0.

0091

,0.

0035

)(-

1.59

e-08

,4.

37e-

07)

(-0.

0135

,0.

0042

)(5

.60e

-08,

2.36

e-06

)0.

0214

1.04

e-05

Res.Vdc

(-0.

0012

,0.

0010

)(-

6.60

e-10

,1.

09e-

07)

(-0.

0094

,0.

0021

)(1

.05e

-08,

1.25

e-07

)0.

0141

4.44

e-07

Res.Idc

(0.0

108,

0.00

41)

(-3.

46e-

11,

2.00

e-09

)(0

.014

8,0.

0045

)(2

.00e

-08,

7.62

e-09

)0.

0237

2.81

e-08

VSC

-HV

DC

test

2

Res

.|Vv|

(0.0

023,

0.00

24)

(5.4

5e-1

0,4.

74e-

09)

(-0.

0131

,0.

0423

)(-

1.62

e-08

,3.

77e-

08)

0.02

071.

87e-

07R

es.|Iv|

(-0.

0091

,0.

0035

)(-

1.59

e-08

,4.

37e-

07)

(-0.

0059

,0.

0174

)(1

.18e

-06,

1.84

e-05

)0.

0824

8.15

e-05

Res

.Vdc

(-0.

0012

,0.

0010

)(-

6.60

e-10

,1.

09e-

07)

(0.0

026,

0.01

19)

(9.5

7e-0

7,9.

75e-

07)

0.04

003.

44e-

06R

es.Idc

(0.0

108,

0.00

41)

(-3.

46e-

11,

2.00

e-09

)(0

.006

8,0.

0307

)(1

.01e

-08,

5.23

e-08

)0.

1418

2.95

e-07


1 1.5 2 2.5 31

1.05

1.1

Time (s)

|Vv|(p

u)

Vvmag−trueVvmag−est.

1 1.5 2 2.5 30

0.01

0.02

0.03

Time (s)

Err

or(p

u)

Vvmag−estimation−residual

(a) |Vv| Static

1 1.5 2 2.5 31

1.05

1.1

1.15

Time (s)

|Vv|(p

u)

Vvmag−trueVvmag−est.

1 1.5 2 2.5 3−10

−5

0

5x 10−8

Time (s)

Err

or(p

u)

Vvmag−estimation−residual

(b) |Vv| Pseudo-dynamic

1 1.5 2 2.5 32.3

2.4

2.5

2.6

Time (s)

Vdc

(p.u

.)

Vdc−trueVdc−est.

1 1.5 2 2.5 30

0.005

0.01

0.015

Time (s)

Err

or(p

.u.)

Vdc−estimation−residual

(c) Vdc Static

1 1.5 2 2.5 32.3

2.4

2.5

2.6

Time (s)V

dc(p

.u.)

Vdc−trueVdc−est.

1 1.5 2 2.5 3−5

0

5x 10−7

Time (s)

Vdc

(p.u

.)

Vdc−estimation−residual

(d) Vdc Pseudo-dynamicFigure 6.9: Static and Pseudo-dynamic SEs for the VSC-HVDC link model: first testscenario

For instance, the DC current estimated by the static state estimation shows amaximum residual up to 0.1418 p.u. while the pseudo-dynamic state estimationgives a maximum residual of 2.9504× 10−7p.u. Table 6.2 summaries the results ofthese tests.

As it might be noted that in Figs. 6.7 and 6.8 both static and pseudo dynamicmodels maintain almost the same level of accuracy during normal operation (orderof 1.0× 10−14) while in Figs. 6.9 and 6.10 improvement during normal operationis seen for the pseudo dynamic model. The reason is the pseudo-dynamic modelis capable to include internal states that are calculated in real-time, such as thevoltage reference. In contrast, static models do not possess such flexibility.

6.5. Computation performance 95

1 1.5 2 2.5 30

0.5

1

1.5

Time (s)

|I v|(p.u

.)

Ivmag−trueIvmag−est.

1 1.5 2 2.5 30

0.05

0.1

Time (s)

Err

or(p

.u.)

Ivmag−estimation−residual

(a) |Iv| Static

1 1.5 2 2.5 30

0.5

1

1.5

Time (s)

|I v|(p.u

.)

Ivmag−trueIvmag−est.

1 1.5 2 2.5 3−1

0

1

2x 10−4

Time (s)

|I v|(p.u

.)

Ivmag−estimation−residual

(b) |Iv| Pseudo-dynamic

1 1.5 2 2.5 3−1

0

1

2

Time (s)

I dc(p

.u.)

Idc−trueIdc−est.

1 1.5 2 2.5 30

0.1

0.2

Time (s)

Err

or(p

.u.)

Idc−estimation−residual

(c) Idc Static

1 1.5 2 2.5 30

0.5

1

1.5

Time (s)

I dc(p

.u.)

Idc−trueIdc−est.

1 1.5 2 2.5 3−2

0

2

4x 10−7

Time (s)

I dc(p

.u.)

Idc−estimation−residual

(d) Idc Pseudo-dynamic

Figure 6.10: Static and Pseudo-dynamic SEs for the VSC-HVDC link model: secondtest scenario

6.5 Computation performance

One advantage of the proposed pseudo-dynamic PMU-based state estimation isthat it does not significantly increase the computation complexity and burden whencompared to the static state estimation. By comparing the number of iterationsand computation time of the static and the pseudo-dynamic PMU-based stateestimations, as shown in Table 6.3, it can be seen that the pseudo-dynamic stateestimation performs similarly to the static state estimation in terms of computationalspeed. These computations were carried out on the VSC-HVDC test scenario I usingan ordinary PC with an Intel(R) Core(TM) i7-2640M CPU @2.80GHz and a 8.00GB RAM, and using matlab R2013b.


Table 6.3: Computation performance comparison

PMU-basedSE method

Aver. comp. timeper snapshot

Aver. no. of iter. persnapshot

Largest no.of iteration

P-dynamic 4.754 ms 5.465 10

Static 3.115 ms 5.525 11

6.6 Summary

A PMU-based state estimator using a pseudo-dynamic network model is presentedherein. This method significantly improves the state estimation accuracy duringtransients as compared to the static PMU-based state estimation. In contrast tomost dynamic state estimation algorithms, it implements an iterative algorithmto update the estimated states instead of solving DAEs, which significantly savescomputational resources. Additionally, the pseudo-dynamic network model canbe easily constructed from the original static model and represent devices withtime-varying control references.

In addition, the pseudo-dynamic network models for STATCOMs and VSC-HVDCs are developed. These two models also provide valuable and practical insighton how to develop pseudo-dynamic network models for components and controllersin power systems.

The case studies provide sufficient evidence that the pseudo-dynamic stateestimation algorithm is capable of performing much more accurate estimation duringtransient dynamics without significantly increasing computational resources ascompared to the static state estimations. The STATCOM using real PMU datashows that the proposed modeling approach and the PMU-based state estimationalgorithm is applicable when real PMU data from actual power systems is available.

Chapter 7

Quantifying PMU measurement weights

The performance of power system state estimation (PSSE) relies on the propertiesof the model and those of the collected measurements, such as sampling rate,measurement accuracy and variance, etc. Weighting is a practice of accounting forthe confidence in the model and in a measurement. Particularly, weighted leastsquares (WLS) chooses the weights as inversely proportional to the measurementerror variances. This selection is straightforward only when the measurement noise isproperly quantified in a systematic fashion, which has not been subject to sufficientattention in literature.

As mentioned in Chapter 3, WLS regression resembles the maximum likelihoodestimators (MLE) [16, 113] based on the assumption of Gaussian measurement error.The weights are chosen as inversely proportional to the variances of measurementerrors. Most of related work assumes that the variances of PMU measurementsare fully-known or are chosen empirically, without addressing how the variancesare quantified in practice. For instance in [114], the standard deviation of eachmeasurement is defined as a linear function of the measurement value and the fullscale value of instrumentation. Reference [19] extended it to a PMU-based stateestimation algorithm. In another example of [24], the weights are both set to unityfor the measurements of bus voltage magnitude and angle, is set to the smaller valuebetween unity and the reciprocal of magnitude for the measurement of line currentmagnitude, and is set to 0.2 for the measurement of line current angle. Empiricalweighting selections may perform well for specific applications, but can be difficultto apply to general cases. Thus, we need a systematic mechanism to improve itsefficiency and applicability when quantifying the measurement variance, which hasrarely been addressed in literature.

Over the last two decades reweighted or adjusted techniques [115, 116, 117, 118]are introduced and integrated into WLS regressions for PSSE, with the aim to dealwith non-Gaussian distributions and dependent measurements [119], or to obtainan efficient estimation scheme with certain robustness to outliers [120]. The workin [115] proposes an iterative re-weighted least square method where given rotationsare implemented to solve a non-quadratic optimization problem, while the work

97

98 Quantifying PMU measurement weights

signal

generation

1,0.95,1 puA teaching tool

for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

signal

generation

RT-Lab

3

signal

streams

Opal-RT real-time

simulator

3

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3

/ 0

Read data

locally

PMU connection

tester

3

50*32 Hz

50Hz

Gaussian

noise

3rd

harmonics

Raw dataCurve

fitting

Detrended

data

pdf pdf pdf

signal

with

harmonicsA teaching tool

for phasor

measurement

estimation

Reference PMU

Sequence

Analyzer

Calculate the mean

and standard

deviation

Calculate the mean

and standard

deviation

Simulink/Matlab

3

50*32 Hz

50Hz

Gaussian

noise

Perfect

signal 3

Unbalanced

signal 3

Raw dataCurve

fitting

Detrended

data

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3

0

signal

with


for phasor

measurement

estimation

Reference PMU

Sequence

Analyzer

Calculate the mean

and standard

deviation

Calculate the mean

and standard

deviation

Simulink/Matlab

3

50*32 Hz

50Hz

Gaussian

noise

Perfect

signal 3

Unbalanced

signal 3

0

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3 Collect and

store data for

analysis

SEL-PDC-5073

3

Instrument

transformers

CT/PT

PMU PDC

Communication

channel &

update

PMU-based

state

estimation

3-phase grid

signal

Figure 7.1: Synchrophasor measurement chain: blue boxes are devices that may affectthe measurement uncertainty; the relation between the input and output of the red boxunder different noise levels is the main interest herein.

in [116] uses a similar scheme to integrate weighted least absolute value (WLAV) inPSSE. An algorithm is proposed in [117] that adjusts the covariance matrix to takeinto account of the network parameter uncertainty. Furthermore, a non-diagonalcovariance matrix is used in [118] to overcome measurement dependencies appearingin WLS regressions. On the other hand, an adaptive scheme is proposed in [121] toupdate the variances of the measurement noises, based on the calculated residualsover several past measurement snapshots. This result is further extended in [122] toimprove its computational efficiency and resilience against measurement redundancy.Although using the aforementioned recursive procedures to update measurementcovariance is suitable for on-line implementation, most of them still requires aninitialization procedure. Thus, a systematic mechanism to quantify the measurementnoise variances will benefit not only the classic WLS algorithm but also the adaptivealgorithms mentioned above.

Recently published paper [123] fills in this blank by performing theoreticalanalysis of uncertainty associated with the chain of involved devices. It concludeswith an algebraic expressions of measurement uncertainty contributed by threemajor sources of errors, i.e., instrument transformers, measurement devices, anddeadband. However, this analysis is conducted for the conventional measurementchain containing remote terminal units (RTUs) and associated devices, which havemajor differences from those used for the synchrophasor measurement chain.

This chapter focuses on the role that a PMU plays in the synchrophasor mea-surement chain. We quantify PMU measurement uncertainty resulting from phasorcalculation algorithms and the device’s internal filtering. Particularly, we examinethe PMU outputs’ variances and expected values given different input variances.This procedure can be represented by the red box in the synchrophasor measurementchain shown in Fig. 7.1.

In this chapter we will explore two approaches [124] to quantify PMU measure-ments: off-line simulation and hardware-in-the-loop (HIL) simulation. The off-linesimulation characterizes the statistical relation between the inputs and outputs ofPMUs for various input signals and the HIL simulation evaluates the impact ofincluding actual hardware of PMUs.

7.1. Quantification approaches 99

signal

generation


for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

signal

generation

RT-Lab

3

signal

streams

Opal-RT real-time

simulator

3

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3

/ 0

Read data

locally

PMU connection

tester

3

50*32 Hz

50Hz

Gaussian

noise

3rd

harmonics

Raw dataCurve

fitting

Detrended

data

pdf pdf pdf

signal

with


for phasor

measurement

estimation

Reference PMU

Sequence

Analyzer

Calculate the mean

and standard

deviation

Calculate the mean

and standard

deviation

Simulink/Matlab

3

50*32 Hz

50Hz

Gaussian

noise

Perfect

signal 3

Unbalanced

signal 3

Raw dataCurve

fitting

Detrended

data

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3

0

signal

with


for phasor

measurement

estimation

Reference PMU

Sequence

Analyzer

Calculate the mean

and standard

deviation

Calculate the mean

and standard

deviation

Simulink/Matlab

3

50*32 Hz

50Hz

Gaussian

noise

Perfect

signal 3

Unbalanced

signal 3

0

Figure 7.2: Set-up for the off-line simulations

7.1 Quantification approaches

As indicated by (5.2) in Chapter 5, quantifying weights in WLS is transformed toquantifying the variances of the modeling uncertainty and measurement noise. Inthis thesis, we focus on quantifying measurement noise. Evaluating the networkmodel uncertainties remains part of the future work.

Before developing methods to quantify these variances, the following questionsarise: How is the variance of a PMU output affected given different inputs noises?How sensitive are the output magnitude and angle w.r.t varying input variances? Andhow these would affect the choice of weightings in the WLS algorithm? Furthermore,how these answers would change if the system signal is of low quality, such as mixedwith harmonics or unbalanced in three phases? These questions are firstly studiedthrough off-line simulation.

Off-line simulation provides a basic relationship between the input’s varianceand the output’s expected value and variance. Furthermore, to evaluate the influ-ences of actual PMU hardware, an investigation is conducted through the secondquantification approach, using HIL simulation, where a physical PMU is included inthe simulation loop.

7.1.1 Off-line simulation

The off-line simulation focuses on evaluating the performance of the PMU’s outputwhen changing the variance of the input signal. It is conducted in matlab/Simulinkand the set-up is shown in Fig. 7.2.

The three red blocks represent three test scenarios: (i) perfect three-phase signalwith the amplitude of 1; (ii) third-order harmonic with 0.5 gain on each phase signal;and (iii) unbalanced three-phase signal with the amplitude of 1, 0.95, 1, respectively.Three test scenarios are also described in Table 7.1. The signal frequency for all thethree scenarios is 50 Hz and sampled at 1.6 kHz (i.e. 32 samples per cycle).

After the three-phase signal is generated, Gaussian noise is added to eachphase with the same variance but different initial seeds. For each scenario differentinput noise variances are assigned, i.e., [0, 0.001, 0.025, 0.05, 0.075, 0.1, 0.2, 0.5]. The


Table 7.1: Three test scenarios

Scenario I Perfect three-phase signalScenario II Third-order harmonic with 0.5 gain is added to each phaseScenario III Unbalanced three-phase signal having the amplitude of

1, 0.95 and 1

sampling frequency of the Gaussian noises is 1.6 kHz .Then the three-phase signal is sent to the PMU model. Two PMU models are

used in this step: (i) the sequence analyzer block [125] from matlab/Simulink,whose function is to compute the positive-, negative-, and zero-sequence componentsof a three-phase signal; (ii) a teaching tool for phasor measurement estimationproposed by [126]. The first model is taken as a reference PMU herein. Please notethat the PMU models in the off-line simulation only represent the phasor calculatorsegment rather than a whole physical PMU model.

The last step is to store two PMUs’ outputs and calculate their expected valuesand variances for both magnitude and angle. After organizing all these results, it isexplained how the PMU outputs change given different input variances for all thethree scenarios.

7.1.2 Hardware-in-the-loop simulationCompared to the off-line simulation, HIL simulation introduces a physical PMUinto the simulation loop with the purpose of imitating a field test. As the inputs ofphysical PMUs need to be real-time analog signals, a real-time simulator, Opal-RT’seMegaSim, in the SmarTS Lab [127] at KTH is used to generate the input signalsfor the PMU.

The set-up for the HIL simulation is shown in Fig. 7.3. It starts with developing athree-phase signal generation model in RT-LAB, a software that acts as the interfacebetween the Opal-RT real-time simulator and the users. After compiling and loadingthe model into the real-time simulator, it runs with a real-time clock.

The output of the simulator is a three-phase analog signal and it is sent to SEL-421 Protection Relay and PMU. Note that PMUs installed in real power systemsmeasure high voltage and current phasors through instrument transformers at lowvoltage and current levels (±300V , 0-5A), however the real-time simulator can notgenerate such values. Therefore the simulator’s outputs are sent to the low-levelinterface connections inside the PMU, which has a limit of 2.33 V rms [128].

Then the phasors calculated by the PMU is sent to the phasor data concentrator(PDC), SEL-PDC-5073, whose function is to receive and time-synchronize phasordata from multiple PMUs to produce a real-time, time-aligned output data stream.Because these experiments were carried out in a controlled environment, where lossof data and latency could be controlled, it is reasonable to neglect the possibleinfluences of both the PDC and the communication channels. Therefore, for practical

7.2. Results 101

signal

generation


for phasor

measurement

estimation [1]

Reference PMU

Sequence

Analyzer

Calculate the

standard deviations

for magnitude and

angle

Calculate the

standard deviations

for magnitude and

angle

Simulink/Matlab

signal

generation

RT-Lab

3

signal

streams

Opal-RT real-time

simulator

3

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3

/ 0

Read data

locally

PMU connection

tester

3

50*32 Hz

50Hz

Gaussian

noise

3rd

harmonics

Raw dataCurve

fitting

Detrended

data

pdf pdf pdf

signal

with


for phasor

measurement

estimation

Reference PMU

Sequence

Analyzer

Calculate the mean

and standard

deviation

Calculate the mean

and standard

deviation

Simulink/Matlab

3

50*32 Hz

50Hz

Gaussian

noise

Perfect

signal 3

Unbalanced

signal 3

Raw dataCurve

fitting

Detrended

data

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3

Collect data

SEL-PDC-5073

3Read data

locally

PMU connection

tester

3

0

signal

with


for phasor

measurement

estimation

Reference PMU

Sequence

Analyzer

Calculate the mean

and standard

deviation

Calculate the mean

and standard

deviation

Simulink/Matlab

3

50*32 Hz

50Hz

Gaussian

noise

Perfect

signal 3

Unbalanced

signal 3

0

signal

generation

RT-Lab

signal

streams

Opal-RT real-time

simulator

3 3 · Relay

· A/D

· Phasor

estiamtor

SEL-421

Protection Relays

and PMU 3 3 Collect and

store data for

analysis

SEL-PDC-5073

3

Figure 7.3: Set-up for the HIL simulation experiments

purposes, it is assumed that the PDC and the communication channels have no effecton the phasor computation, neither on the measurement uncertainty in our studycase. The influence of hardware on the measurement uncertainty is only associatedwith the PMU.

7.2 Results

7.2.1 Off-line simulationOff-line simulation is conducted for three scenarios as introduced in Subsection 7.1.1,and its results are shown in Figs. 7.4, 7.5 and 7.6.

Figure 7.4 shows the variances and the expected values of the PMU outputswhen varying the input’s variances for Scenario I. Figure 7.4a depicts the variancesof the output magnitude and angle. For the range of input σ2 ∈ [0 0.5], the figureindicates that both variances for magnitude and angle change linearly along with theinput variance. In order to verify this observation, we firstly apply the “fit" function[129] and the “poly1" [130] model to fit the curves in Fig. 7.4a. The fitting resultsare straighting lines passing through the origin, whose expressions are presentedin Table 7.2. Then these results are evaluated by the normalized root mean squareerror (NRMSE), which is defined as

NRMSE = RMSE|ymax − ymin|

=

√1n

∑ni=1(yi − yi)2

|ymax − ymin|, (7.1)

where yi is the value of ith data; yi is the corresponding value for yi on the fittedcurve; ymax and ymin are the maximum and minimum values among all the data. Bytaking account of the range of the data, NRMSE can help to compare the RMSEsamong different quantities. In this sense, NRMSE becomes a global evaluation indexused for all the three scenarios.

The NRMSE for both variances of magnitude and angle in Fig. 7.4a are approxi-mately in the order of 0.1% to 0.2%, which are sufficiently small. Hence, the fitted


0 0.1 0.2 0.3 0.4 0.50

0.002

0.004

0.006

0.008

0.01

0.012

σ2 of the input noise

σ2 of the output

Magnitude (volt)

refPMUPMU

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35


σ2 of the output

Angle (deg.)

refPMUPMU

(a)

0 0.1 0.2 0.3 0.4 0.51

1.001

1.002

1.003

1.004

1.005

1.006

1.007


µ of the output

Magnitude (volt)

refPMUPMU

0 0.1 0.2 0.3 0.4 0.5−0.25

−0.2

−0.15

−0.1

−0.05

0


µ of the output

Angle (deg.)

refPMUPMU

(b)Figure 7.4: PMU outputs with increasing input noise for Scenario I: Perfect three-phasesignal. (7.4a) (two left figures): covariances of the output magnitude (top) and angle(bottom). (7.4b) (two right figures): expected values of the output magnitude (top)and angle (bottom).

7.2. Results 103

Table 7.2: Off-line simulation results

Scenario I Fitted Curve NRMSE Typeσ2|V |ref

y = 0.0222x+ 1.281× 10−5 0.00115 linearσ2|V | y = 0.02257x+ 1.252× 10−5 0.00110 linearσ2θ ref y = 69.19x− 0.08718 0.000983 linearσ2θ y = 46.6x− 0.0516 0.00224 linearµ|V |ref y = 0.01116x+ 1 0.00792 linearµ|V | y = 0.01176x+ 1 0.00753 linearµθref y = −0.3249x0.5471 − 0.00372 0.00145 quasi square rootµθ y = −0.3157x0.5362 − 0.002976 0.00120 quasi square root

Scenario IIσ2|V |ref

y = 0.02156x+ 6.127× 10−6 0.000585 linearσ2|V | y = 0.0218x+ 6.128× 10−6 0.000574 linearσ2θ ref y = 66.14x− 0.03225 0.00103 linearσ2θ y = 41.49x− 0.01667 0.000877 linearµ|V |ref y = 0.0138x+ 1 0.0309 linearµ|V | y = 0.01446x+ 1 0.0298 linearµθref y = −0.5195x0.5072 − 0.00129 0.000351 quasi square rootµθ y = −0.4908x0.5008 − 0.000448 0.000201 quasi square root

Scenario IIIσ2|V |ref

y = 0.02156x+ 6.298× 10−6 0.000601 linearσ2|V | y = 0.0217x+ 6.319× 10−6 0.000593 linearσ2θ ref y = 68.43x− 0.03446 0.00106 linearσ2θ y = 42.92x− 0.01792 0.000913 linearµ|V |ref y = 0.01397x+ 0.9837 0.0306 linearµ|V | y = 0.01464x+ 0.9837 0.0295 linearµθref y = −0.5285x0.5074 − 0.00134 0.000361 quasi square rootµθ y = −0.4991x0.501 − 0.000492 0.000209 quasi square root


curves are valid. Furthermore, it implies the weighting matrix generated using a setof variances under the same noise level will be valid for most noise levels.

On the other hand, Fig. 7.4a shows that the angle variance changes dramaticallycompared to the magnitude variance, which can also be observed from the slopespresented in Table 7.2. This indicates that the angle derivation inside the phasorestimators are more sensitive to the inputs noises. Moreover, under the same inputnoise, different instrument models (refPMU and PMU phasor estimators) havedifferent impacts on angle variance. As a result, different weights are needed whendifferent instrument models are employed.

When the variance of the input noise is increased from 0 to 0.5, the expectedvalues of both the magnitude and angle in the output deviate from their initialvalues, which although is much less than the derivation in the variances. In Fig. 7.4b,it is observed the expected values of the output magnitudes are linear, however, whenusing the “poly1" model to fit these curves, their NRMSEs are in the order of 0.7%,which is larger than the values for variances. This is due to the slight nonlinearityin the beginning of the curves when the input variances are small. In contrast, theexpected value of the output angle is clearly nonlinear, with a similar appearanceof a negative square root function. Hence, “power2" [130] model is applied to fitthe curves and the calculated NRMSEs are in the order of 0.1% to 0.2%, which aresame with the values for variances.

Figures 7.5 and 7.6 show the variances and the expected values of PMU outputswhen varying the input’s variances for Scenario II and III, respectively. It can beseen from Fig. 7.5 that introducing harmonics only slightly affects the variances ofthe magnitude and the angle. However, it increases the absolute expected values ofboth magnitude and angle. Similarly, as shown in Figure 7.6, introducing unbalancedthree-phase signal affects mostly the absolute expected values of both magnitudeand angle. Therefore, a plausible conclusion is harmonics or unbalanced three-phaseconditions will mainly affect the expected values of output magnitude and angle.

In summary, the main observations from comparing Figs. 7.4-7.6 and Table 7.2are

• σ2|V |/σ

2θ is constant ∀σ2

input ∈ [0 0.5]. Therefore, the weighting matrix generatedusing a set of variances under the same noise level will be valid for most noiselevels.

• σ2|V | σ2

θ , ∀σ2input ∈ [0 0.5]. This implies that angle calculation inside phasor

estimators are more sensitive to the inputs noises.

• σ2θ(refPMU) > σ2

θ(PMU). It indicates that under the same input noise differentinstrument models (refPMU and PMU phasor estimators) have differentimpacts on angle variance.

• Harmonics or unbalanced three-phase conditions mainly affect the expectedvalues of output magnitude and angle.

7.2. Results 105

0 0.1 0.2 0.3 0.4 0.50

0.002

0.004

0.006

0.008

0.01

0.012


σ2 of the output

Magnitude (volt)

refPMUrefPMU with harmonicsPMUPMU with harmonics

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35


σ2 of the output

Angle (deg.)


(a)

0 0.1 0.2 0.3 0.4 0.51

1.002

1.004

1.006

1.008

1.01


µ of the output

Magnitude (volt)


0 0.1 0.2 0.3 0.4 0.5−0.4

−0.3

−0.2

−0.1

0


µ of the output

Angle (deg.)


(b)Figure 7.5: PMU outputs with increasing input noise for Scenario II: A third harmonicswith 0.5 gain is added to each phase. (7.5a) (two left figures): covariances of the outputmagnitude (top) and angle (bottom). (7.5b) (two right figures): expected values of theoutput magnitude (top) and angle (bottom).


0 0.1 0.2 0.3 0.4 0.50

0.002

0.004

0.006

0.008

0.01

0.012


σ2 of the output

Magnitude (volt)

refPMUrefPMU unbalanced 3ΦPMUPMU unbalanced 3Φ

0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35


σ2 of the output

Angle (deg.)

refPMUrefPMU unbalanced 3ΦPMUPMU unbalanced 3Φ

(a)

0 0.1 0.2 0.3 0.4 0.50.98

0.985

0.99

0.995

1

1.005

1.01

1.015


µ of the output

Magnitude (volt)

refPMUrefPMU unbalanced 3 ΦPMUPMU unbalanced 3 Φ

0 0.1 0.2 0.3 0.4 0.5−0.4

−0.3

−0.2

−0.1

0


µ of the output

Angle (deg.)

refPMUrefPMU unbalanced 3 ΦPMUPMU unbalanced 3 Φ

(b)Figure 7.6: PMU outputs with increasing input noise for Scenario III: Unbalanced three-phase signal with the magnitude of 1, 0.95 and 1. (7.6a) (two left figures): covariancesof the output magnitude (top) and angle (bottom). (7.6b) (two right figures): expectedvalues of the output magnitude (top) and angle (bottom).

7.2.2 Hardware-in-the-loop simulation

The results of HIL simulation are shown in Fig. 7.7 and Table 7.3. Note that twoy-axes are applied for Fig. 7.7a, where it can be observed the significant decreasein the output’s variances due to the PMU’s filtering stage. The PMU includesdifferent low-pass filters: one installed before the A/D conversion, and two after thecalculation of real and complex components, respectively [131].

The PMU filters also affect the expected values of the magnitude and angle in theoutputs. In order to evaluate these influences, the IEEE Standard for SynchrophasorMeasurements for Power Systems is utilized here. Specifically, the requirement for themaximum total vector error (TVE) of the steady-state synchrophasor measurementis 1%. This 1% criterion can be visualized as a small circle drawn on the end of the

7.2. Results 107

0 0.1 0.2 0.3 0.4 0.52 of the input noise

0

0.005

0.01

0.015

2 of the output

0

0.5

1

1.5

2

2 of the output10-4Magnitude (volt)

offlineHIL


0

10

20

30

40

2 of the output

0

0.2

0.4

0.6

0.8

2 of the outputAngle (deg.)

offlineHIL

(a)


0.97

0.98

0.99

1

1.01

of the output

Magnitude (volt)

offlineHIL


-0.3

-0.2

-0.1

0

0.1

of the output

Angle (deg.)

offlineHIL

(b)Figure 7.7: PMU outputs with increasing input noise for HIL simulation. (7.7a) (left-column figures): covariances of the output magnitude (top) and angle (bottom) whenvarying the variance of input noise. (7.7b) (right-column figures): expected values of theoutput magnitude (top) and angle (bottom) when varying the variance of input noise.

phasor. The maximum magnitude error is 1% when the error in phase is zero, andthe maximum error in angle is just under 0.573 when the error in magnitude iszero [131]. The objective is to determine the noise level under which the output willviolate the 1% criterion.

• Regarding the output magnitude, in order to meet the 1% criterion, thecondition |µ− 1|+σ ≤ 0.01 has to be satisfied. Since the expected value of themagnitude is smaller than 1 (due to the PMU’s filtering), the input signals willnot violate the 1% criterion when the output magnitude satisfies µ− σ ≥ 0.99.As shown in Table 7.3, this criterion is met when the variance of the inputnoise is between 0.1 and 0.2.

• Similarly, regarding the output angle, in order to meet the 1% criterion,|µ−0|+σ ≤ 0.537. Because the expected value is larger than 0, the input signals


Table7.3:H

ILsim

ulationresults

σ2input

00.01

0.0250.05

0.0750.1

0.20.5

σ2|V|

(8.574

77×

10−

6) 20.00191081

20.00305857

20.0043325

20.00525709

20.00599411

20.00818291

20.012705

2

σ2θ

0.0004527

20.10542

20.170788

20.24678

20.294144

20.343734

20.465666

20.773598

2

µ|V|

0.9983540.998581

0.9986140.998696

0.9986620.998645

0.9970860.979782

µθ

0.02826450.0290496

0.02837970.0244459

0.03144550.0270704

0.04640340.0395692

7.3. Summary 109

will not violate the 1% criterion when the output angle satisfies µ+ σ ≤ 0.537.As shown in Table 7.3, this criterion is met when σ2

input ≈ 0.2.

• In summary, based on the PMU used in this HIL simulation, injecting signalswith σ2

input ≥ 0.2 may violate the 1% criterion. Nevertheless, 0.1 is already arelatively large number for variances. As it is shown in Fig. 7.7b, the expectedvalues for both magnitude and angle remain considerably unchanged whenσ2input ≤ 0.1.

The off-line simulation studies in Section 7.2.1 indicate that the measurement’s vari-ance quantified around the operating point holds for most noise levels. Furthermore,the HIL simulation shows that the input noise needs to be bounded to σ2

input ≤ 0.1.

7.3 Summary

Two approaches to quantify the PMU measurement weights are carried out inthis chapter: off-line simulation and HIL simulation. First, in off-line simulationwe demonstrate how the expected value and variance of the PMU measurementchange given inputs with different variances. This is carried out using PMU modelsdescribed in [125] and [126]. Different scenarios are analyzed, including the perfectthree-phase signals, signals with harmonics, and unbalanced three-phase signals.A linear relationship between the input’s and output’s variances implies that thePMU measurement variance quantified at any normal operating condition is capableto generate a weighting matrix that is valid for most noise levels. In addition,angle derivation inside the phasor estimators is more sensitive to the input’s noisescompared to magnitude. These results are ideal as different scenarios are applied tothe PMU models that do not include the impact of hardware. Therefore, we examinethe influence of actual hardware on the previous results that are based purely onsimulation, via hardware-in-the-loop (HIL) simulation.

Chapter 8

Conclusion

At the beginning of this thesis, several practical examples were used to motivatethe need of fast and accurate power system state estimation techniques to facilitatemeeting the challenges of increasing penetration of renewable energy sources and howto exploit them through HVDC and power electronic-based devices. In particular,the feasibility of PMU-only state estimation is highlighted as a viable path forwardas the PMU installation and deployment is dramatically increasing. In addition, theemphasis is laid on the use of HVDC and other power electronic systems, whichrequire a state estimation solution for hybrid AC/DC grids.

These motivation examples were followed by an overview on power system stateestimation techniques in Chapter 2. The key roles that a state estimation plays in apower system and the development of state estimation schemes when using PMUsand considering power electronic devices were presented. To meet the diverse needsfor state estimation, measurements over consecutive instants or a system modelthat considers time evolution can be utilized in addition to the static measurementmodel, leading to the so-called forecasting-aided state estimation and dynamicstate estimation, respectively. Furthermore, state estimation is not exclusive fortransmission systems any more, but becomes applicable to distribution systems aswell. The architecture of state estimations, i.e., models, formulation and computation,is no longer confined to a centralized paradigm; distributed or hierarchical schemesmay be a better choice in some cases. For each of the aforementioned aspects, somerelated research in literature was discussed.

Chapter 3 focused on the formulation and derivation of state estimation methods,particularly the formulation and solution of the conventional state estimationapproach. In addition, the test systems that were developed and implemented forcase studies were introduced. These contents provided a background for the followingchapters.

Chapter 4 and 5 developed a paradigm of using PMU data to solve static stateestimation for hybrid AC/DC grids, but using different problem formulations andnetwork models. Chapter 4 attempted a linear scheme where the measurement modelwas linear and the linear WLS algorithm was applied for solution. In addition, linear

111

112 Conclusion

network models for AC transmission network and classic HVDC link were developed.Although linear WLS can be solved directly (i.e., without iterations), it concealsthe state variables to be in rectangular coordinates, which loses the advantage ofangle bias detection and correction. Therefore, in Chapter 5, state variables in polarcoordinates were used, leading to a nonlinear problem formulation. The nonlinearWLS algorithm was applied for its solution. At the same time, we proposed a novelmeasurement model that separates the errors due to modeling uncertainty andmeasurement noise so that different weights can be assigned to them separately.Nonlinear network models for AC transmission network, classic HVDC link, voltagesource converter (VSC)-HVDC, and FACTS were developed and tested.

The case studies in Chapter 4 and 5 showed that both linear and nonlinearsolution schemes performed adequately when the system was under steady-state orquasi-steady state, but less satisfactorily when the system was under large dynamicchanges and power electronic devices react to these changes. This implies thatadditional modeling details need to be included to obtain higher accuracy duringsystem transient dynamics. On the other hand, most dynamic state estimators andforecasting-aided state estimators are computationally demanding.

In Chapter 6, we proposed a pseudo-dynamic modeling approach that canimprove the estimation accuracy during transients without significantly increasingthe estimation’s computational burden. It maintains the formulation and algorithmused for the nonlinear state estimation in Chapter 5 and only adds differenceequations from dynamic models of certain components to the original static networkequations. This approach greatly reduces the workload of re-composing networkmodels when updating existing algorithms. To illustrate this approach, pseudo-dynamic network models for static synchronous compensator (STATCOM), as anexample of a FACTS device, and VSC-HVDC link were developed and validatedvia simulation.

Last but not least, in Chapter 7, we proposed two approaches to quantify PMUmeasurement weights: off-line simulation and hardware-in-the-loop (HIL) simulation.The findings provide better guidance for selecting proper weights for power systemstate estimation.

Bibliography

[1] Power trends 2016—the changing energy landscape. Technical report, NewYork Independent System Operator, 2016.

[2] Emily S. Rueb. How New York City gets its electricity.https://www.nytimes.com/interactive/2017/02/10/nyregion/how-new-york-city-gets-its-electricity-power-grid.html, 2017.

[3] W. Yeomans. Watching the grid control room situational awareness at theNYISO. In Advanced Energy Conference, April 2014.

[4] U.S. Energy Information Administration. New technology can improveelectric power system efficiency and reliability. https://www.eia.gov/todayinenergy/detail.php?id=5630, 2012.

[5] A. Silverstein. Sychrophasors & the grid. In NARUC Summer Meeting, July2017.

[6] C. Lu, B. Shi, X. Wu, and H. Sun. Advancing China’s smart grid: Phasormeasurement units in a wide-area management system. IEEE Power andEnergy Magazine, 13(5):60–71, Sept 2015.

[7] India country report—research, development, demonstration and deploymentof smart grids in India. Technical report, Government of India, Ministry ofScience and Technology, Department of Science and Technology, 2017.

[8] Svenska Kraftnät. Map of the national grid. https://www.svk.se/drift-av-stamnatet/stamnatskarta/, 2017.

[9] Global HVDC interconnector database. Technical report, Bloomberg NewEnergy Finance, 2016.

[10] CIGRE SCB4 (HVDC and Power Electronics) committee. Compendiumof all HVDC projects 2009. http://b4.cigre.org/Publications/Other-Documents/Compendium-of-all-HVDC-projects, 2009.

[11] Wikipedia. List of HVDC projects, December 2017.

113

114 Bibliography

[12] T. E. Dy Liacco. Real-time computer control of power systems. Proceedingsof the IEEE, 62(7):884–891, July 1974.

[13] F. C. Schweppe and J. Wildes. Power system static-state estimation, PartI: Exact model. IEEE Transactions on Power Apparatus and Systems, PAS-89(1):120–125, Jan 1970.

[14] F. C. Schweppe and D. B. Rom. Power system static-state estimation, PartII: Approximate model. IEEE Transactions on Power Apparatus and Systems,PAS-89(1):125–130, Jan 1970.

[15] F. C. Schweppe. Power system static-state estimation, Part III: Implementa-tion. IEEE Transactions on Power Apparatus and Systems, PAS-89(1):130–135,Jan 1970.

[16] A. Abur and A. G. Expósito. Power System State Estimation: Theory andImplementation. Marcel Dekker, 2004.

[17] C. P. Steinmetz. Complex quantities and their use in electrical engineering.In Proceedings of the American Institute of Electrical Engineers, 1893.

[18] A. G. Phadke. Synchronized phasor measurements—a historical overview.In IEEE/PES Transmission and Distribution Conference and Exhibition,volume 1, pages 476–479, Oct 2002.

[19] A. G. Phadke, J. S. Thorp, and K. J. Karimi. State estimlatjon with phasormeasurements. IEEE Transactions on Power Systems, 1(1):233–238, Feb 1986.

[20] A. G. Phadke and J. S. Thorp. Synchronized Phasor Measurements and TheirApplications. Springer, 2008.

[21] V. Terzija, G. Valverde, D. Cai, P. Regulski, V. Madani, J. Fitch, S. Skok,M. M. Begovic, and A. Phadke. Wide-area monitoring, protection and controlof future electric power networks. Proceedings of the IEEE, 99(1):80–93, Jan2011.

[22] L. Vanfretti, J. H. Chow, S. Sarawgi, and B. Fardanesh. A phasor-data-basedstate estimator incorporating phase bias correction. IEEE Transactions onPower Systems, 26(1):111–119, Feb 2011.

[23] S. G. Ghiocel, J. H. Chow, G. Stefopoulos, B. Fardanesh, D. Maragal, B. Blan-chard, M. Razanousky, and D. B. Bertagnolli. Phasor-measurement-basedstate estimation for synchrophasor data quality improvement and power trans-fer interface monitoring. IEEE Transactions on Power Systems, 29(2):881–888,March 2014.

Bibliography 115

[24] E. R. Fernandes, S. G. Ghiocel, J. H. Chow, D. E. Ilse, D. D. Tran, Q. Zhang,D. B. Bertagnolli, X. Luo, G. Stefopoulos, B. Fardanesh, and R. Robertson.Application of a phasor-only state estimator to a large power system usingreal PMU data. IEEE Transactions on Power Systems, 32(1):411–420, Jan2017.

[25] J. Arrillaga, Y. H. Liu, and N. R. Watson. Flexible Power Transmission: TheHVDC Options. John Wiley & Sons, Inc., 2007.

[26] V. G. Agelidis N. Flourentzou and G. D. Demetriades. VSC-based HVDCpower transmission systems: An overview. IEEE Transactions on PowerElectronics, 24(3):592–602, March 2009.

[27] N. G. Hingorani and L. Gyugyi. Understanding FACTS—Concepts andTechnology of Flexible AC Transmission Systems. IEEE Press, 2000.

[28] A-A. Edris and et al. Proposed terms and definitions for flexible AC transmis-sion system (FACTS). IEEE Transactions on Power Delivery, 12(4):1848–1853,Oct 1997.

[29] X. P. Zhang, L. Yao, B. Chong, C. Sasse, and K. R. Godfrey. FACTS andHVDC technologies for the development of future power systems. In 2005International Conference on Future Power Systems, Nov 2005.

[30] Bonneville Power Administration. TIP 381: WAMS-enhanced HVDC controlfor flexible and stable grid operations. https://www.bpa.gov/Doing\%20Business/TechnologyInnovation/TIPProjectBriefs/2017TX-TIP\%20381.pdf, Oct. 2016.

[31] H. R. Sirisena and E. P. M. Brown. Inclusion of HVDC links in AC power-system state estimation. IEE Proceedings C - Generation, Transmission andDistribution, 128(3):147–154, May 1981.

[32] Q. Ding, B. Zhang, and T. S. Chung. State estimation for power systemsembedded with FACTS devices and MTDC systems by a sequential solutionapproach. Electric Power Systems Research, 55(3):147 – 156, 2000.

[33] A. de la Villa Jaen, E. Acha, and A. G. Expósito. Voltage source convertermodeling for power system state estimation: STATCOM and VSC-HVDC.IEEE Transactions on Power Systems, 23(4):1552–1559, Nov 2008.

[34] J. Cao, W. Du, and H. F. Wang. The incorporation of generalized VSC MTDCmodel in AC/DC power system state estimation. In International Conferenceon Sustainable Power Generation and Supply (SUPERGEN 2012), pages 1–6,Sept 2012.

116 Bibliography

[35] E.A. Zamora-Cárdenas, C.R. Fuerte-Esquivel, A. Pizano-Martínez, and H.J.Estrada-García. Hybrid state estimator considering SCADA and synchronizedphasor measurements in VSC-HVDC transmission links. Electric PowerSystems Research, 133(Supplement C):42 – 50, 2016.

[36] B. Xu and A. Abur. State estimation of systems with embedded FACTSdevices. In 2003 IEEE Bologna Power Tech Conference Proceedings,, volume 1,June 2003.

[37] B. Xu and A. Abur. State estimation of systems with UPFCs using the interiorpoint method. IEEE Transactions on Power Systems, 19(3):1635–1641, Aug2004.

[38] C. Rakpenthai, S. Premrudeepreechacharn, and S. Uatrongjit. Power systemwith multi-type FACTS devices states estimation based on predictor–correctorinterior point algorithm. International Journal of Electrical Power & EnergySystems, 31(4):160 – 166, 2009.

[39] A. Zamora-Cárdenas and Claudio R. Fuerte-Esquivel. State estimation ofpower systems containing FACTS controllers. Electric Power Systems Research,81(4):995 – 1002, 2011.

[40] P. I. Bartolomey, S. A. Eroshenko, E. M. Lebedev, and A. A. Suvorov. Newinformation technologies for state estimation of power systems with FACTS.In 2012 3rd IEEE PES Innovative Smart Grid Technologies Europe (ISGTEurope), pages 1–8, Oct 2012.

[41] E.A. Zamora-Cárdenas, B.A. Alcaide-Moreno, and C.R. Fuerte-Esquivel. Stateestimation of flexible AC transmission systems considering synchronized phasormeasurements. Electric Power Systems Research, 106(Supplement C):120 –133, 2014.

[42] V. I. Presada, C. V. Cristea, M. Eremia, and L. Toma. State estimation inpower systems with FACTS devices and PMU measurements. In 2014 49thInternational Universities Power Engineering Conference (UPEC), pages 1–5,Sept 2014.

[43] J. Hasegawa K. Nishiya, H. Takagi and T. Koike. Dynamic state estimationfor electric power systems—introduction of a trend factor and detection ofinnovation processes. Electrical Engineering in Japan, 96(5):79–87, May 1976.

[44] F. C. Schweppe and R. D. Masiello. A tracking static state estimator. IEEETransactions on Power Apparatus and Systems, PAS-90(3):1025–1033, May1971.

[45] A. M. Leite da Silva, M. B. Do Coutto Filho, and J. F. de Queiroz. Stateforecasting in electric power systems. IEE Proceedings C - Generation, Trans-mission and Distribution, 130(5):237–244, September 1983.

Bibliography 117

[46] D. M. Falcao, P. A. Cooke, and A. Brameller. Power system tracking stateestimation and bad data processing. IEEE Transactions on Power Apparatusand Systems, PAS-101(2):325–333, Feb 1982.

[47] A. P. Alves da Silva, V. H. Quintana, and G. K. H. Pang. Solving dataacquisition and processing problems in power systems using a pattern analysisapproach. IEE Proceedings C - Generation, Transmission and Distribution,138(4):365–376, July 1991.

[48] J. C. S. Souza, A. M. Leite da Silva, and A. P. Alves da Silva. Data debuggingfor real-time power system monitoring based on pattern analysis. IEEETransactions on Power Systems, 11(3):1592–1599, Aug 1996.

[49] K.-R. Shih and S.-J. Huang. Application of a robust algorithm for dynamicstate estimation of a power system. IEEE Transactions on Power Systems,17(1):141–147, Feb 2002.

[50] M. B. Do Coutto Filho and J. C. Stacchini de Souza. Forecasting-aidedstate estimation—Part I: Panorama. IEEE Transactions on Power Systems,24(4):1667–1677, Nov 2009.

[51] A. S. Debs and R. E. Larson. A dynamic estimator for tracking the stateof a power system. IEEE Transactions on Power Apparatus and Systems,PAS-89(7):1670–1678, Sept 1970.

[52] P. Rousseaux, T. Van Cutsem, and T. E. Dy Liacco. Whither dynamic stateestimation? International Journal of Electrical Power & Energy Systems,12(2):104 – 116, 1990.

[53] G. Welch and G. Bishop. An introduction to the Kalman filter. Technicalreport, University of North Carolina at Chapel Hill, 2006.

[54] Z. Huang, K. Schneider, and J. Nieplocha. Feasibility studies of applyingKalman filter techniques to power system dynamic state estimation. In 2007International Power Engineering Conference (IPEC 2007), pages 376–382,Dec 2007.

[55] L. Fan and Y. Wehbe. Extended Kalman filtering based real-time dynamicstate and parameter estimation using PMU data. Electric Power SystemsResearch, 103:168 – 177, 2013.

[56] G. Valverde and V. Terzija. Unscented Kalman filter for power system dynamicstate estimation. IET Generation, Transmission Distribution, 5(1):29–37, Jan2011.

[57] M. E. Baran and A. W. Kelley. State estimation for real-time monitoring ofdistribution systems. IEEE Transactions on Power Systems, 9(3):1601–1609,Aug 1994.

118 Bibliography

[58] A. Primadianto and C. N. Lu. A review on distribution system state estimation.IEEE Transactions on Power Systems, 32(5):3875–3883, Sept 2017.

[59] G. R. Gray, J. Simmins, G. Rajappan, G. Ravikumar, and S. A. Khaparde.Making distribution automation work: Smart data is imperative for growth.IEEE Power and Energy Magazine, 14(1):58–67, Jan 2016.

[60] R. Singh, B. C. Pal, and R. A. Jabr. Distribution system state estimationthrough Gaussian mixture model of the load as pseudo-measurement. IETGeneration, Transmission Distribution, 4(1):50–59, January 2010.

[61] C. W. Hansen and A. S. Debs. Power system state estimation using three-phase models. IEEE Transactions on Power Systems, 10(2):818–824, May1995.

[62] Y. Hu, A. Kuh, T. Yang, and A. Kavcic. A belief propagation based power dis-tribution system state estimator. IEEE Computational Intelligence Magazine,6(3):36–46, Aug 2011.

[63] A. Souza, E. M. Lourenço, and A. S. Costa. Real-time monitoring of distributedgeneration through state estimation and geometrically-based tests. In 2010IREP Symposium Bulk Power System Dynamics and Control - VIII (IREP),pages 1–8, Aug 2010.

[64] S. Naka, T. Genji, T. Yura, and Y. Fukuyama. A hybrid particle swarmoptimization for distribution state estimation. IEEE Transactions on PowerSystems, 18(1):60–68, Feb 2003.

[65] J. Wu, Y. He, and N. Jenkins. A robust state estimator for medium voltagedistribution networks. IEEE Transactions on Power Systems, 28(2):1008–1016,May 2013.

[66] O. Chilard and S. Grenard. Detection of measurements errors with a distribu-tion network state estimation function. In 22nd International Conference andExhibition on Electricity Distribution (CIRED 2013), pages 1–4, June 2013.

[67] S. Sarri, M. Paolone, R. Cherkaoui, A. Borghetti, F. Napolitano, and C. A.Nucci. State estimation of active distribution networks: Comparison betweenWLS and iterated Kalman-filter algorithm integrating PMUs. In 2012 3rdIEEE PES Innovative Smart Grid Technologies Europe (ISGT Europe), pages1–8, Oct 2012.

[68] Y. F. Huang, S. Werner, J. Huang, N. Kashyap, and V. Gupta. State estimationin electric power grids: Meeting new challenges presented by the requirementsof the future grid. IEEE Signal Processing Magazine, 29(5):33–43, Sept 2012.

Bibliography 119

[69] A. Gómez-Expósito, A. de la Villa Jaén, C. Gómez-Quiles, P. Rousseaux, andT. Van Cutsem. A taxonomy of multi-area state estimation methods. ElectricPower Systems Research, 81(4):1060 – 1069, 2011.

[70] G. N. Korres. A distributed multiarea state estimation. IEEE Transactionson Power Systems, 26(1):73–84, Feb 2011.

[71] F. C. Aschmoneit, N. M. Peterson, and E. C. Adrian. State estimation withequality constraints. In Tenth PICA Conference, pages 427–430, May 1977.

[72] A. Gjelsvik, S. Aam, and L. Holten. Hachtel’s augmented matrix method -a rapid method improving numerical stability in power system static stateestimation. IEEE Transactions on Power Apparatus and Systems, PAS-104(11):2987–2993, Nov 1985.

[73] E. Caro and A. J. Conejo. State estimation via mathematical programming: acomparison of different estimation algorithms. IET Generation, TransmissionDistribution, 6(6):545–553, June 2012.

[74] An overview of sparse matrix techniques for on-line network applications.IFAC Proceedings Volumes, 20(6):19 – 25, 1987.

[75] A. Simoes-Costa and V. H. Quintana. A robust numerical technique for powersystem state estimation. IEEE Transactions on Power Apparatus and Systems,PAS-100(2):691–698, Feb 1981.

[76] A. Simoes-Costa and V. H. Quintana. An orthogonal row processing algorithmfor power system sequential state estimation. IEEE Transactions on PowerApparatus and Systems, PAS-100(8):3791–3800, Aug 1981.

[77] J. W. Gu, K. A. Clements, G. R. Krumpholz, and P. W. Davis. The solutionof ill-conditioned power system state estimation problems via the method ofPeters and Wilkinson. IEEE Transactions on Power Apparatus and Systems,PAS-102(10):3473–3480, Oct 1983.

[78] F. Milano. PSAT–Power System Anaylysis Toolbox Manual. 2010.

[79] Long term dynamics. Phase II. Final report—WG 38.02.08. Technical report,Cigre, 1995.

[80] T. Van Cutsem and L. Papangelis. Description, modeling and simulation resultsof a test system for voltage stability analysis. Technical report, University ofLiege, Nov. 2013.

[81] Y. Chompoobutrgool, W. Li, and L. Vanfretti. Development and implemen-tation of a nordic grid model for power system small-signal and transientstability studies in a free and open source software. In 2012 IEEE Power andEnergy Society General Meeting, pages 1–8, July 2012.

120 Bibliography

[82] MathWorks, Inc. VSC-based HVDC link. http://se.mathworks.com/help/physmod/sps/examples/vsc-based-hvdc-transmission-system-detailed-model.html?searchHighlight=vsc-based\%20hvdc\%20link\&s_tid=doc_srchtitle,2015.

[83] J. W. Demmel. Applied numerical linear algebra. Society for Industrial andApplied Mathematics, 1997.

[84] Å. Björck. Numrical methods for least squares problems. Society for Industrialand Applied Mathematics, 1996.

[85] W. Li and L. Vanfretti. Inclusion of classic HVDC links in a PMU-based stateestimator. In 2014 IEEE PES General Meeting | Conference Exposition, pages1–5, July 2014.

[86] Wei Li and Luigi Vanfretti. A PMU-based state estimator considering classicHVDC links under different control modes. Sustainable Energy, Grids andNetworks, 2:69–82, 2015.

[87] W. Li and L. Vanfretti. A PMU-based state estimator for networks containingVSC-HVDC links. In 2015 IEEE Power Energy Society General Meeting,pages 1–5, July 2015.

[88] W. Li and L. Vanfretti. A PMU-based state estimator for networks containingFACTS devices. In 2015 IEEE Eindhoven PowerTech, pages 1–6, June 2015.

[89] P. Kundur. Power System Stability and Control. McGraw-Hill Education,1994.

[90] A. Monticelli. State Estimation in Electric Power Systems—A GeneralizedApproach. Kluwer Academic Publishers, 1999.

[91] A. Monticelli and F. F. Wu. Network observability: Identification of observableislands and measurement placement. IEEE Transactions on Power Apparatusand Systems, PAS-104(5):1035–1041, May 1985.

[92] B. Gou and A. Abur. A direct numerical method for observability analysis.IEEE Transactions on Power Systems, 15(2):625–630, May 2000.

[93] J. Chen and A. Abur. Placement of PMUs to enable bad data detection instate estimation. IEEE Transactions on Power Systems, 21(4):1608–1615, Nov2006.

[94] T. L. Baldwin, L. Mili, M. B. Boisen, and R. Adapa. Power system observabilitywith minimal phasor measurement placement. IEEE Transactions on PowerSystems, 8(2):707–715, May 1993.

Bibliography 121

[95] B. Xu and A. Abur. Observability analysis and measurement placement forsystems with PMUs. In IEEE PES Power Systems Conference and Exposition,2004., volume 2, pages 943–946, Oct 2004.

[96] R. Emami and A. Abur. Reliable placement of synchronized phasor measure-ments on network branches. In 2009 IEEE/PES Power Systems Conferenceand Exposition, pages 1–6, March 2009.

[97] G. R. Krumpholz, K. A. Clements, and P. W. Davis. Power system observ-ability: A practical algorithm using network topology. IEEE Transactions onPower Apparatus and Systems, PAS-99(4):1534–1542, July 1980.

[98] D. J. Brueni and L. S. Heath. The PMU placement problem. SIAM Journalon Discrete Mathematics, 19(3):744–761, 2005.

[99] D. Gyllstrom, E. Rosensweig, and J. Kurose. On the impact of PMU placementon observability and cross-validation. In 2012 Third International Conferenceon Future Systems: Where Energy, Computing and Communication Meet(e-Energy), pages 1–10, May 2012.

[100] S. Chakrabarti and E. Kyriakides. PMU measurement uncertainty consid-erations in WLS state estimation. IEEE Transactions on Power Systems,24(2):1062–1071, May 2009.

[101] Arbiter Systems, Inc. Paso Robles, CA. Model 1133A power sentinel GPS-synchronized power quality revenue standard. http://www.arbiter.com/files/product-attachments/1133a_manual.pdf, 2014.

[102] D. Shi, D. J. Tylavsky, and N. Logic. An adaptive method for detection andcorrection of errors in PMU measurements. IEEE Transactions on SmartGrid, 3(4):1575–1583, Dec 2012.

[103] P. Yang, Z. Tan, A. Wiesel, and A. Nehorai. Power system state estimationusing PMUs with imperfect synchronization. IEEE Transactions on PowerSystems, 28(4):4162–4172, Nov 2013.

[104] Q. F. Zhang and V. M. Venkatasubramanian. Synchrophasor time skew: For-mulation, detection and correction. In 2014 North American Power Symposium(NAPS), pages 1–6, Sept 2014.

[105] N. G. Bretas, S. A. Piereti, A. S. Bretas, and A. C. P. Martins. A geometricalview for multiple gross errors detection, identification and correction in powersystem state estimation. IEEE Transactions on Power Systems, 28(3):2128–2135, Aug 2013.

[106] L. Vanfretti and J. H. Chow. Synchrophasor data applications for wide-areasystems. In Power Systems Computation Conference (PSCC), Aug. 2011.

122 Bibliography

[107] Thyristor controlled series compensation. Technical report, Cigre workinggroup 14.18, 1997.

[108] Static synchronous series compensator (SSSC). Technical report, Cigre workinggroup B4.40, 2009.

[109] N. Zhou and Z. Huang, et al. Capturing dynamics in the powergrid: Formulation of dynamic state estimation through data assimila-tion. http://www.pnnl.gov/main/publications/external/technical_reports/PNNL-23213.pdf, 2014.

[110] W. Li, L. Vanfretti, and J. H. Chow. Pseudo-dynamic network modeling forPMU based state estimation of hybrid AC/DC grids. IEEE Access, 6:4006–4016, 2018.

[111] L. Harnefors and H. P. Nee. Model-based current control of AC machinesusing the internal model control method. IEEE Transactions on IndustryApplications, 34(1):133–141, Jan 1998.

[112] S. Cole, J. Beerten, and R. Belmans. Generalized dynamic VSC MTDC modelfor power system stability studies. IEEE Transactions on Power Systems,25(3):1655–1662, Aug 2010.

[113] F. C. Schweppe and E. J. Handschin. Static state estimation in electric powersystems. Proceedings of the IEEE, 62(7):972–982, July 1974.

[114] J. J. Allemong, L. Radu, and A. M. Sasson. A fast and reliable state estima-tion algorithm for AEP’s new control center. IEEE Transactions on PowerApparatus and Systems, PAS-101(4):933–944, April 1982.

[115] R. C. Pires, A. Simoes Costa, and L. Mili. Iteratively reweighted least-squaresstate estimation through givens rotations. IEEE Transactions on PowerSystems, 14(4):1499–1507, Nov 1999.

[116] R. A. Jabar and B. C. Pal. Iteratively reweighted least-squares implementa-tion of the WLAV state-estimation method. IEE Proceedings - Generation,Transmission and Distribution, 151(1):103–108, Jan 2004.

[117] G. D’Antona. Power system static-state estimation with uncertain networkparameters as input data. IEEE Transactions on Instrumentation and Mea-surement, 65(11):2485–2494, Nov 2016.

[118] E. Caro, R. Mínguez, and A. J. Conejo. Robust WLS estimator using reweight-ing techniques for electric energy systems. Electric Power Systems Research,104(Complete):9–17, 2013.

[119] P. J. Green. Iteratively reweighted least squares for maximum likelihoodestimation and some robust and resistant alternatives. Journal of the RoyalStatistical Society. Series B (Methodological), 46(2):149–192, 1984.

Bibliography 123

[120] A. Luceño. Multiple outliers detection through reweighted least deviances.Computational Statistics & Data Analysis, 26(3):313 – 326, 1998.

[121] G. Liu, E. Yu, and Y. H. Song. Novel algorithms to estimate and adaptivelyupdate measurement error variance using power system state estimation results.Electric Power Systems Research, 47(1):57 – 64, 1998.

[122] S. Zhong and A. Abur. Auto tuning of measurement weights in WLS stateestimation. IEEE Transactions on Power Systems, 19(4):2006–2013, Nov2004.

[123] A. de la Villa Jaen, J. Beloso Martinez, A. Gomez-Expósito, and F. GonzálezVázquez. Tuning of measurement weights in state estimation: Theoreticalanalysis and case study. IEEE Transactions on Power Systems, PP(99):1–1,2017.

[124] W. Li and L. Vanfretti. Quantifying PMU measurements weights for phasorstate estimation. submitted to IEEE Access, 2018.

[125] MathWorks, Inc. Sequence analyzer. https://se.mathworks.com/help/physmod/sps/powersys/ref/sequenceanalyzer.html.

[126] D. Dotta, J. H. Chow, and D. B. Bertagnolli. A teaching tool for phasormeasurement estimation. IEEE Transactions on Power Systems, 29(4):1981–1988, July 2014.

[127] L. Vanfretti, M. Chenine, M. S. Almas, R. Leelaruji, L. Ängquist, and L. Nord-ström. Smarts lab—a laboratory for developing applications for WAMPACsystems. In 2012 IEEE Power and Energy Society General Meeting, pages1–8, July 2012.

[128] SEL-421 Relay—Protection and Automation System—Instruction Manual &User’s Guide.

[129] MathWorks, Inc. fit. https://se.mathworks.com/help/curvefit/fit.html.

[130] MathWorks, Inc. List of library models for curve and sur-face fitting. https://se.mathworks.com/help/curvefit/list-of-library-models-for-curve-and-surface-fitting.html\#btbcvnl. R2017b.

[131] IEEE Standard for Synchrophasor Measurements for Power Systems.