virtual crank angle based cylinder pressure sensor

Virtual Crank Angle based

Cylinder Pressure Sensor

CHRISTOPHER RINGSTRÖM

Master of Science Thesis Stockholm, Sweden 2017

Virtual Crank Angle based

Cylinder Pressure Sensor

Christopher Ringström

Master of Science Thesis MMK 2017

KTH Industrial Engineering and Management

Machine Design, Division of Internal Combustion Engines

SE-100 44 STOCKHOLM

I

Examensarbete MMK 2017: 150 MFM 170

Virtuell Vevvinkel baserad Cylindertryck Sensor


Godkänt

Examinator

Andreas Cronhjort

Handledare

Ola Stenlåås

Uppdragsgivare

Scania CV AB

Kontaktperson

Ola Stenlåås

Sammanfattning Förbränningsåterkoppling är ett aktuellt forskningsområde inom utvecklingsarbetet för att minska utsläpp och öka verkningsgraden hos förbränningsmotorer. Cylindertryck är en viktig parameter att mäta . Ett alternativ är att använda en tryckgivare men det skulle vara mer kostnadseffektivt att kunna uppskatta trycket baserat på vevvinkeln som redan idag mäts i motorer. Därav utvecklades en virtuell sensor för uppskattning av cylindertrycket genom detta examensarbete. Studien har berört hur tryck spår, bitvis noggranna för kompressionen och expansionen, från en stel vevaxelmodell kan kompletteras med modeller för värmeavgivningen från förbränningen för att erhålla ett fullt tryck spår. För att kunna bygga och utveckla modellerna utvecklades en metod för att bestämma det indikerade arbetet baserat på den uppmätta varvtalssignalen som beror mycket på hur förbränningen skett och är därmed en viktig parameter vid modellerande av värmeavgivningen.

Det indikerade arbetet kunde uppskattas genom att jämföra den kinetiska effekten med den effekt som kolvarna totalt bidrog med. Det upptäcktes att offseten mellan kurvorna motsvarade effekten av förlusterna och lasten som därmed kunde bestämmas vid de punkter där momentet från cylindrarna var i jämvikt. Den kinetiska effekten beräknades från varvtalssignalen medan effekten från kolvarna uppskattades genom att använda isentropiska tryckkurvor för kompressionen och expansionen, innan och efter förbränningen respektive. Relativt noggranna resultat erhölls för arbetspunkterna med ett varvtal på 800 rpm medan större avvikelser inträdde vid högre varvtal. Anledningen till detta var att torsionssvängningar influerade varvtalssignalen mer vid högre varvtal. Resultaten kunde förbättras genom lokal medelvärdesbildning av den kinetiska effekten vid de decelerationer som sker efter förbränningen i respektive cylinder.

II

Torsionens inverkan på vevaxelns dynamik uppskattades genom att använda estimeringar av förvridningen av vevaxeln från en dynamisk vevaxelmodell. Uppskattningen tycktes vara tillräckligt noggrann inom vissa intervall men det var inte möjligt att avlägsna torsionssvängningarna i kinetiska effektspåret för hela cykeln. Uppskattningen av indikerat arbete kunde därför inte förbättras genom denna torsionsuppskattning. Torsionen var vidare studerad i form av frekvens och amplitud av svängningarna inom varvtalssignalen. Inga tydliga samband kunde säkerställas mellan svängningarna och arbetspunkternas varvtal och last. Detta tyder på att torsionen är för komplex att förutse. Vidare, då invertering av den dynamiska vevaxelmodellen tidigare visat sig ge en oriktig lösning kunde modellen istället itereras för att bestämma tryck spåret likt tidigare gjort för den stela vevaxelmodellen. Torsionssvängningarna influerade dock fortfarande det resulterande tryck spåret. Det finns stor potential att förbättra den virtuella sensorn om torsionen kan uppskattas noggrant för alla arbetspunkter.

Värmeavgivningen från förbränning var först modellerad som isochorisk och isobarisk i två respektive modeller. Dessa modeller gav information om gränsvärdena för tändningen genom att finna den tändning för modellerna som resulterade i samma arbete som det tidigare estimerade indikerade arbetet. Därefter anpassades en Wiebe funktion så att den resulterande tryckderivatan minsta-kvadrat anpassades till tryckderivatan från vevaxelmodellen under den sena förbränningen där vevaxelmodellen var mest noggrann. Wiebe funktion gav en bra anpassning till den senare diffusiva förbränningen men var inte tillräcklig för att beskriva den förblandade förbränningen. Slutligen anpassades två Wiebe funktioner där den diffusiva förbränningen anpassades likt för singel Wiebe-funktions anpassningen medan den förblandade förbränningen anpassades så att det resulterande arbetet stämde med det uppskattade indikerade arbetet. För att få bättre resultat bestämdes den förblandade förbränningens start och duration från uppskattningen av den kinetiska effekten innan anpassningen.

Den virtuella trycksensorn och de flesta av dess delmodeller var mest noggranna för arbetspunkterna vid låga varvtal. Slutsatsen var att det var främst på grund av torsionssvängningarnas påverkan på insignalerna till delmodellerna som noggrannheten föll för de högra varvtalen. Genom denna studie visades det att deltrycksspåret från vevaxelmodellen kunde kompletteras med en modell för värmeavgivningen för att slutligen få en bättre uppskattning av hela tryck spåret där singulariteten vid TDC kunde undvikas.

Nyckelord: Virtuell sensor, Cylindertryck, Värmeavgivning, Wiebe anpassning, Indikerat arbete,

Diesel motor

III

Master of Science Thesis MMK 2017: 150 MFM 170

Virtual Crank Angle based Cylinder Pressure Sensor


Approved

Examiner

Andreas Cronhjort

Supervisor

Ola Stenlåås

Commissioner

Scania CV AB

Contact person

Ola Stenlåås

Abstract Closed-loop combustion control is an on-going field of research for improving reducing engine emissions and increasing efficiency. Cylinder pressure is a key parameter to monitor for combustion feedback. Measuring pressure with a transducer is an option, although being able to estimate the pressure based on the crank angle measurement instead would be beneficial in terms of costs. A virtual crank angle based pressure sensor was therefore developed within this thesis. It was studied how the in-cylinder pressure trace for a full closed cycle could be modelled from a pressure trace from a rigid crankshaft model, the angular velocity measurement and heat release modelling. The pressure trace from the crankshaft model was subjected to a singularity at TDC and torsional oscillations, it was therefore of interest to study whether the singularity could be avoided by modelling the heat release. Further on, the indicated work and total heat released during combustion were estimated from the angular velocity measurements as they are important parameters for determining the heat release trace.

It was found that the indicated work could be approximated by comparing the kinetic power trace, obtained from the measured angular velocity, with the piston power trace, estimated using isentropic pressure curves for the compression and expansion within the cylinder. Accurate results were obtained for operating points at 800 rpm while large deviations were seen for higher speeds as a consequence of larger torsional effect on the angular velocity trace; on the form of perturbed oscillations. The results could be improved from local averaging of the kinetic power trace at the occasions of deceleration, although it could be concluded that only the low speed operating were still accurate enough.

IV

The kinetic power trace was attempted to be corrected for torsional power using angular displacement estimations of the crankshaft nodes from a dynamic crankshaft model. Even though the model seemed to capture the torsional behaviour at parts of the cycle, the oscillations could not be completely removed and it was determined that the final work estimate could not be improved from the torsional power estimate. The torsion was further studied regarding frequency and amplitude of the oscillations within the angular velocity and acceleration trace. No clear relations between the torsional behaviour and operating speed and load could be concluded. Further, since inversion of the dynamic crankshaft model for pressure estimation resulted in an improper solution since before, the model was iterated instead. The pressure trace could thereby be derived accounting for torsion, however the trace still contained oscillations which highlights the challenge of estimating the torsion accurately. The torsion is a complex phenomenon to describe and further development of a model for estimating the torsion with high accuracy for all operating points would improve the virtual pressure sensor significantly.

The heat release was, as a first step, modelled as isochoric and isobaric. These models gave information of the limits of SOC by comparing the indicated work from the resulting pressure trace with the work estimate from the angular velocity measurement. Further, one Wiebe function was parametrised such that the resulting pressure derivative during late combustion was adapted to the trace from the crankshaft model in a least-square sense. This allowed for better adaption as the partial pressure trace was subjected to torsional oscillations. The fitted Wiebe function described the diffusive combustion well but missed out the shape of the premixed combustion. Lastly, a double Wiebe function parametrisation was done where the diffusive combustion function was fitted to the late combustion data and the premixed combustion function was adapted such that the resulting indicated work matched the estimated work. To receive more accurate results, the premixed SOC and duration had to be approximated beforehand from the kinetic power trace.

The virtual pressure sensor and most of the sub models were most accurate for low speed operating points. It was concluded that the reason is most probably the torsional effect on the input data to all sub models. It was shown that the crankshaft model can be complemented with heat release estimations which improved the final pressure trace and removed the singularity present around TDC.

Keywords: Virtual sensor, In-cylinder pressure, Heat release, Wiebe parametrisation, Indicated

work

V

FOREWORD

I am greatly thankful for all the help that I have been offered throughout this thesis. I would like to thank all the involved people for their friendly support and inspirational discussions and suggestions. It has helped a lot during the endeavor to reach my goals and objectives.

I am very grateful to Scania for giving me the opportunity to do this thesis work which has given me much new knowledge and experience as well as many new colleagues. I would first like to thank Ola Stenlåås for his well-structured and educational supervision of this thesis and for always finding time for discussions. I would also like to thank all the coworkers at NESC and adjacent groups for their supportive suggestions during particularly challenging occasions. I would especially like to thank Stephan Zentner, Henrik Berggren and Carlos Jorques Moreno for their help and expertise of combustion and heat release estimations. Further on, I am also thankful to Andreas Cronhjort for his quick responses when anything was unclear regarding the project phases of the thesis.

Further I would like to thank my girlfriend Charlotte for her never ending support, faith in my work and all the motivation she brings to me. I am also thankful to my family, Mats, Åsa and Lovisa for always helping me no matter what. Lastly I would like to thank my grandfather Clas for inspiring me and developing my interest for technology and mechanics and especially engines.


Södertälje, May 2017

VI

Table of Contents

Sammanfattning .............................................................................................................................................................. I

Abstract ........................................................................................................................................................................ III

Foreword ....................................................................................................................................................................... V

Introduction .................................................................................................................................................................... 1

Problem formulation ................................................................................................................................................... 2

Research questions ..................................................................................................................................................... 2

Methodology .............................................................................................................................................................. 2

Delimitations ........................................................................................................................................................... 2

Related work ............................................................................................................................................................... 2

Heat release modelling ............................................................................................................................................ 3

Pressure modelling based on heat release ............................................................................................................... 4

Crankshaft dynamics modelling .............................................................................................................................. 4

Neural network-based models ................................................................................................................................. 5

Frame of reference ......................................................................................................................................................... 5

Heat release ................................................................................................................................................................ 5

The isochoric, the isobaric and the mixed mode cycles .......................................................................................... 5

The Wiebe function ................................................................................................................................................. 6

Two Wiebe function fits ......................................................................................................................................... 6

Ignition delay .............................................................................................................................................................. 6

In-cylinder thermodynamics ....................................................................................................................................... 6

Specific heat ratio ................................................................................................................................................... 6

Compression stroke pressure contribution .................................................................................................................. 6

Crankshaft dynamics .................................................................................................................................................. 7

Translational motion of oscillating masses ............................................................................................................. 7

Torque from rotating and oscillating masses .......................................................................................................... 7

Kinetic and torsional energy within a crankshaft in motion.................................................................................... 7

Implementation .............................................................................................................................................................. 8

Experimental set-up ....................................................................................................................................................... 8

Model development ........................................................................................................................................................ 8

Full cycle and zones without combustion ................................................................................................................... 9

Isentropic pressure traces ........................................................................................................................................ 9

Exhaust pressure ..................................................................................................................................................... 9

Gross indicated torque and work........................................................................................................................... 10

Pressure estimation from dynamic crankshaft model............................................................................................ 11

Torsional vibrations .............................................................................................................................................. 11

Modelling of the heat release trace from combustion ............................................................................................... 11

Isochoric and isobaric heat release ........................................................................................................................ 11

VII

Premixed SOC and duration ................................................................................................................................. 11

Normal distributed HRR for rough pressure trace estimation ............................................................................... 12

Volume correction ................................................................................................................................................ 12

Late combustion HRR Wiebe function fit ............................................................................................................ 12

Parametrised heat release with two Wiebe functions ............................................................................................ 13

Results and discussion ................................................................................................................................................. 14

Filtering of the logged data ....................................................................................................................................... 14

Full cycle and zones without combustion ................................................................................................................. 15

Specific heat ratio ................................................................................................................................................. 16

Exhaust pressure ................................................................................................................................................... 16

Indicated work and torque..................................................................................................................................... 17

Local kinetic power averaging for improved indicated work estimation .............................................................. 18

Including energy stored due to torsion .................................................................................................................. 20

Determination of pressure trace iteratively using a dynamic crankshaft model .................................................... 21

Heat release trace adaption ....................................................................................................................................... 21

SOC interval from isochoric and isobaric heat release ......................................................................................... 21

Total heat release estimate through assumed normal distributed heat release rate ............................................... 22

Volume correction for deformations ..................................................................................................................... 23

One Wiebe function fit to late combustion pressure ............................................................................................. 23

Two Wiebe function fit ......................................................................................................................................... 26

Tolerance analysis .................................................................................................................................................... 28

Conclusions .................................................................................................................................................................. 31

Recommendations for future research .......................................................................................................................... 33

References .................................................................................................................................................................... 34

Notations ...................................................................................................................................................................... 36

Definitions/Abbreviations ............................................................................................................................................ 36

Appendix ...................................................................................................................................................................... 37

2017-06-02

2017-06-02

Development of Virtual Crank Angle based In-cylinder Pressure Sensor

Christopher Ringström Division of Internal Combustion Engines

KTH-Royal Institute of Technology, Stockholm, Sweden

Supervisor: Ola Stenlåås Engine Control Software

Scania CV AB, Södertälje, Sweden

Introduction

Emission regulations within the automotive industry are used as a measure to reduce the internal combustion engines’ effect on global warming and public health. With the introduction of the Euro VI commercial vehicle regulations in 2014, the limits became increasingly stringent together with an introduction of a number of additional regulations [1]. As a consequence of the regulations but also the customer demand for more fuel efficient vehicles, engines are becoming progressively complex, especially in terms of control algorithms, in order to meet the requirements [2].

Combustion diagnostics and control are on-going research and development fields within the endeavour of meeting the demands. A trend towards more extensive combustion diagnostics and closed loop combustion control (CLCC) with combustion parameters as feedback signal has given room for more optimised combustion. Monitoring in-cylinder pressure during engine operation is a valuable part of this as it gives much information of the combustion and is directly related to the produced torque on the crankshaft. Some of the advantages include misfiring detection, engine health monitoring, improved fuel economy, lower emissions and acoustic improvements. Therefore, combustion diagnostics and especially pressure monitoring plays an important role in the development of less emitting engines with no loss in performance [3].

Mounting a pressure transducer in the cylinder to directly measure the pressure is an alternative for monitoring pressure. It is however problematic as the operational environment in the cylinder is harsh and assuredly limits the transducers lifetime [4]. The pressure signal may also be noisy needing signal processing which can give a less reliable measurement. Systems for validating the health of the sensor would also be needed to assure acceptably accurate measurements.

Consequently, approaches to measure the in-cylinder pressure indirectly based on other measurements have been made. The objective is to develop a more reliable and cost efficient virtual pressure sensor with the possibility of implementation in production vehicles. Gustafsson investigated in a study [5] the possibility of estimating the pressure based on crank angle degree (CAD) measurements (angular position and its first and second time derivative) by building a physical model of the crankshaft to estimate the torque generated by the cylinders and thereby the pressure within the cylinders. It was shown that the pressure could accurately be modelled for parts of the compression stroke and most of the expansion stroke. There was a problem of torsional perturbations, especially at higher speeds. The model also had a singularity at TDC caused by the inverted sliding mechanism relating force on the piston to the generated gas torque [5].

In another study by Johansson [6], a model was built to estimate the heat release based on measured cylinder pressure and crank angle. The model was based on the first thermodynamic law relating heat release to the cylinder volume and pressure. A parametrisation model based on three Wiebe functions was also built [6]. The heat release estimation based on pressure showed good results. By inverting the heat release model, the Wiebe models accuracy could be validated. It was found that CA10, CA50 and pmax could be estimated with good accuracy compared to experimental data although CA90 showed larger deviations. The model demanded good reconstruction of the compression pressure and cylinder volume for each CAD.

This study aims to evaluate how these two approaches can be combined to develop a virtual CAD based pressure sensor. The sensor should be capable of estimating the in-cylinder pressure during a full engine cycle and the study includes the evaluation of how accurate it would be.

2017-06-02

Problem formulation

Research has shown that a crankshaft model can predict the in-cylinder pressure well during part of the compression stroke and most of the expansion stroke, however not around TDC with good accuracy. While it has also been shown that CAD resolved heat release prediction can also be used to approximate the pressure for the whole combustion cycle. The main aim for this thesis is to evaluate whether a mechanical model of the crankshaft for pressure estimation based on time resolved crank angle information can be combined with a thermodynamic pressure model based on heat release prediction for improved accuracy. The project addresses thereby how the CAD based virtual pressure sensor can be improved for pressure estimation for the whole engine cycle and how the accuracy is affected. The study also treats how the model can be improved based on the heat release prediction and its input parameters, for example injection timing and duration, common rail pressure and fuel type and quality. The long term goal is to implement a model of a CAD resolved in-cylinder pressure sensor in an ECU and aspects of limited computational power shall be kept in mind. As a side objective, evaluation of the model accuracy to the engine operating conditions and applicability of the model to all cylinders is of interest.

Research questions

How can an in-cylinder pressure estimator based on the crank angle signal be built valid for the whole cylinder cycle including piston position around TDC and how accurate would such a model be?

Sub research questions:

1. How can models already built for heat release and heat release parametrisation and crankshaft dynamics be further developed to estimate the in-cylinder pressure for the whole cylinder cycle?

2. How accurate can such a model be and what factors, including approach of heat release modelling, would affect the accuracy and to what extent?

Methodology

The chosen scientific method for this thesis is quantitative quasi-experiments. A quasi-experiment aims to study if and how a specific treatment of a process affects its outcome [7]. In this study, the crankshaft model already built by Gustafsson [5] will be the base model to be combined with pressure estimations from heat release models. The treatment to study is how the heat release is modelled and combined with the crankshaft model and how this affects the accuracy of the complete model; which is the output of the quasi-experiment. Therefore, as a first step basic heat release models based on the Otto and Diesel cycle will be combined with the crankshaft model. Secondly, the heat release model and

parametrisation by Johansson [6] will be tested. Lastly, after an evaluation of whether an even more improved heat release model can be built and implemented will be done. It will then be possible to show whether the crankshaft model can be improved by combining it with a heat release model and whether improved heat release models can further improve the final models accuracy.

A validation of each model will be done using available experimental data. The data consists of in-cylinder pressure and CAD measurements from a HD engine operated at different operating conditions. Data from simulations of the models will be validated using statistical methods determining how accurate the models can predict reality. A mutual comparison of the models will then indicate whether a change in accuracy can be detected from the treatment (change in complexity of the model).

The treatment consists of changes of controllable factors of the model, for example theoretical approach of heat release estimation. There is, however, a risk that in reality, certain uncontrollable factors might have an effect that the models do not cover and which is not observed in the validation process. Therefore a further objective of the study is to conduct a tolerance analysis to analyse the effect on the result from errors in the input values and controllable factors and to what extent; including TDC offset, cylinder volume variations during operation and limitations from the heat release model and the pressure estimations before and after TDC. This study will also be made using quantitative experiments.

Delimitations

The study is limited by the following:

• The model will only estimate in-cylinder pressure and heat release. Combustion control will not be included in the model.

• The model shall be built based on parameters of the reference Scania engine. The model’s applicability to other engine models will not be studied in depth.

• The model will be built in Matlab/Simulink. Model implementation in an engine ECU is not included.

• The experimental data for model validation originates from test cell running of the reference Scania engine.

• Only non-dimensional and quasi-dimensional thermodynamic engine cycles will be considered.

Related work

There are many existing models developed for pressure estimation in diesel engines. Many share the focus of modelling the combustion and heat release as accurate as possible as it directly affects the pressure trace. Therefore the first part is focused on the different heat release models proposed by researches through the years. Secondly, models of pressure estimation based on modelled heat release are

2017-06-02

presented and lastly other types of pressure estimation models are dealt with. Regarding complexity of modelling, phenomenological models have been prominent as they are thermodynamically zero or quasi dimensional meaning that less computational power is generally needed compared to a multi-dimensional CFD simulation but they can still give satisfying predictions [8]. When the purpose of the model is ECU implementation, there is always a trade-off between computational power and accuracy.

Heat release modelling

Modelling the heat release with one or more Wiebe functions is a popular method used within the engine development field [2, 9]. The Wiebe function predicts the burn rate and can be applied to any engine using any fuel. It is an empirical approach and the models coefficients are not constant for different operating points [10].

There are several papers that have used Wiebe functions to model the burn rate and so the heat release. In a study by Maroteaux et al. [11] such a model was developed and it was found that for an engine operating with a single injection, a double Wiebe approach gave accurate results compared to test data. In another study by the same authors, the same approach was applied for multi-injection cycles. It was found that the model quickly became complex and the results showed that building a model based on double Wiebe function is difficult to make accurate for the engines complete speed range. Their models did however show good correlation between estimated pressure and measured pressure, although the model over-estimated the heat release which gave a slight over-estimation of pressure during the expansion stroke. For all compared pressure traces, the difference during post combustion was less than 5% for all tested points except one which had an error level of 20% [12].

Further, in the same study, it was found that the ignition delay is challenging to estimate and needs calibration based on both rail pressure, EGR rate, engine speed and start of injection [12]. In another research paper that also evaluated the use of the Wiebe function, it was found that the estimated combustion phasing and maximum in-cylinder pressure was generally accurate while the start of combustion showed more uncertainty [9].

Chamela and Orthaber developed another model predicting ROHR based on purely mixing controlled combustion (known as MCC) [13]. The model was built due to the identified limited capability of the Wiebe approach to consider the effects of high pressure fuel injection on the early combustion. The rate of heat release is based on estimation of available fuel for combustion from injected mass and estimation of the kinetic turbulent energy from the intake swirl, squish flow and fuel jet. The model is based on the assumption that the local density of turbulent kinetic energy directly affects the local mixing rate which primarily influences the rate of fuel oxidation.

From model verification with experimental data, the study shows that the model predicts the heat release rate well. However the model showed deviations for higher loads. The ROHR was over predicted between 13 CA and end of injection and then under predicted after end of injection, although giving a valid cumulative heat release. The conclusion was that the combustion retarding effect that comes from the fuel interacting with the cylinder walls is not included in the model which might be the cause of the deviations [13].

A single-zone phenomenological model evaluating IMEP, peak pressure, NO-emissions and noise was built by Barba et. al in 2000 [14]. In their paper, they mention Chamela’s mixing-controlled combustion model and highlights its limited capacity of simulating the premixed combustion. Therefore the model includes three key processes, namely ignition delay, premixed combustion and mixing-controlled combustion and a focus is on low computational time and uncomplicated calibration of coefficients. The ignition delay is predicted based on a calculation of fuel evaporation time and chemical ignition delay. The physical evaporation delay is based on a presented equation for fuel droplet size estimation. By including mean injection velocity, Reynolds- and Weber-number the results can be further improved. Arrhenius type correlation is then used for the chemical delay estimation. To account for the difference in ignition delay for the pre- and main-combustion events, two sets of the parameters used for the delay estimations were established. Through comparison with a two-zone model it is claimed that the results are satisfactory [14]. The fuel mass from the ignition delay model is then used to predict the heat release from pre-mixed combustion with modelling of flame propagation speed and radius of combustion zone. While for the mixing-controlled combustion, a frequency model is used with the mixing frequency calculated based on estimated turbulence and characteristic mixing length. Both max pressure and IMEP could be predicted accurately but it was found that the model was very sensitive to input parameters like rail pressure and injection timing [14]. The model is however regarded as extensive and can be difficult to implement in a computer programme [10].

In a study 2006 by Xavier et. al [10], a new model for diesel combustion was developed as a consequence of the improvements in injection technology and common rail systems for improved combustion. In their model they separated the fuel in three states, injected but not “prepared” fuel mass, “prepared” fuel mass and burnt fuel mass. A set of equations were used for calculating these masses, the “prepared” fuel mass could be calculated by a characteristic preparation time, and the combustion rate could be calculated by the prepared fuel mass and an empirical combustion component. It was found that the characteristic preparation time varies during the combustion process. Using an inversed method to compute the time from in-cylinder pressure and with comparison to reference data from a CFD simulation a sub-model for the characteristic time could be made. Through

2017-06-02

comparison with CFD calculations and experimental test bench results, it was found that the model seems accurate [10].

Another phenomenological model developed by Arsie and co-workers in 2004 [8], includes ignition delay and combustion for a sequence of pilot, pre and main injections to simulate the pressure cycle. Three sub-models for injection delay, the mixing process and the combustion has been modelled for each injection type by determining empirical coefficients based on engine parameters [8].

Pressure modelling based on heat release

A common way of relating pressure to heat release is to set up the first law of thermodynamics for a closed system as proposed by Heywood [15], which has been used in several studies [2, 16, 17]. The approach includes assumptions that the medium in the cylinder is homogenously mixed, neglecting crevices and leakages (blow-by) [17]. Estimation of the specific heat ratio of the cylinder medium and the heat transfer to the cylinder walls are two challenges. The heat transfer is conventionally estimated through the Woschni correlation for convective heat transfer [17].

Through studies it has been shown that the estimation of the specific heat ratio has a large effect on the errors induced by the model [6, 18]. It is also difficult to model as it is affected by both temperature and charge composition. Although it has also been shown that the temperature effect is much larger. In one of the studies, it was found that estimating heat release from pressure though the thermodynamic relation gives an apparent heat release absolute error of 4 % from variation of the ratio by 1 % [18]. Johansson [6] showed that calculations of the specific heat ratio based on at least temperature is vital for avoiding large errors which confirms that it has a large impact.

Through a sensitivity analysis of the thermodynamic model by Johansson [6], it was found that correct angular phasing and specific heat ratio estimation is of major importance for reducing errors. In another study conducted by Rosvall [19], it was found that the CAD can differ a lot for the different cylinders during engine operation due to flexing and torsion of the crankshaft. The study showed that through torsion modelling, the CAD could be estimated for each cylinder giving more accurate results.

In another sensitivity study by Lapuerta et. al [20], one stressed conclusion was that errors in cylinder volume estimation can have a large effect, meaning calculations of combustion chamber deformations can improve the model much. Aronsson et. al concluded in two studies [21, 22] that deformations of the cylinder volume can have a large impact on heat release calculations if not counted for, although it was found that it had a low impact on load calculations. By including a spring constant in the cylinder volume calculations, it was found that the distortion from pressure and inertial effects could be accounted for. West showed in a study

[23] that the combustion chamber volume typically differ from the ideal volume by a few percent. It was shown that this difference could be corrected for through calculations of the different causes to the deviation, for example chamber deformation and strain of connecting rod.

Also of interest is the compression contribution of the in-cylinder pressure. It is often estimated as a polytrophic process [24].

Crankshaft dynamics modelling

Another way of modelling the in-cylinder pressure is to approximate the gas torque from a cylinder by modelling the crankshafts dynamics. This was done by Gustafsson [5] like described earlier and the method has also been attempted and evaluated in a study by Liu and co-workers [25]. However, the problem of a singularity at TDC is encountered in both studies. An interpolation method was however used to estimate the pressure profile around TDC [25]. The method used a single Wiebe function which was developed to estimate the mass fraction of burnt fuel. It improved the model significantly but showed deviations as the multiple injection strategy could not be captured by the single Wiebe function. Through comparison with experimental data gathered for different operating conditions, it was found that the error of peak pressure position estimation was less than 5 CAD and the maximum error in peak pressure estimation was 11,2 %.

Another problem with the pressure trace obtained by Gustafsson was that torsion had been neglected as a rigid crankshaft model was used, resulting in torsional oscillations within the pressure trace. Therefore, a dynamic crankshaft model was introduced by Gustafsson [5] and Rosvall [19] which was proven to capture the torsion of the crankshaft to a certain extent as the angular acceleration of the flywheel could be calculated more accurately than for the stiff crankshaft model. To improve the partial pressure trace retrieved from the stiff crankshaft model, it was attempted to invert the dynamic crankshaft model instead to be able to estimate the gas torque from each cylinder with the acceleration as input. As the dynamic crankshaft model has been proven to capture the torsional effect, rationally, the pressure trace from an inverted model should contain less perturbed oscillations from the torsion.

Gustafsson attempted the method suggested by Thor [9], where the state-space was transformed into 8 transfer functions representing the response of each input on the flywheel acceleration. With one of the transfer functions inverted, the MISO system is turned into a SISO system instead if the pressure in the cylinders which are not in combustion can be approximated beforehand. Even though the same approach was used, Gustafsson’s crankshaft MISO system output angular velocity instead of torque on the flywheel like in Thor’s model, which means that the transfer functions would have quite a different structure. The inverted

2017-06-02

transfer function was found to be improper, and the method could therefore not be used.

Neural network-based models

In a study by Wang et al. [4] the complexity of a physical model for in-cylinder pressure was avoided by instead using a neural network-based approach. The so-called VCPS (virtual cylinder pressure sensor) consists of individual and independent estimators of variables related to the pressure. In the study two estimators for maximum rising rate position for pressure and maximum pressure value were developed. The models were built based on the chosen major input variables which were injection timing and injection fuel mass for maximum rising rate position and additionally intake manifold pressure for the maximum pressure value. Training data from extensive gathering of experimental data was then used to create the final model. It was shown that the model had the same order of accuracy as a physical pressure sensor for steady state conditions.

However, the disadvantages with such a model is its limited input variables, the lack of capability to estimate a continuous pressure trace, its increasing complexity if accurate predictions during transient conditions is demanded and the large experimental data set needed for the model development [4]. This type of approach was also been used in other studies [26, 27]. A further similar approach was done by Johnsson as he used a complex radial basis function with the ability to generalise the knowledge of the training data for pressure estimations for different operating conditions [28].

Frame of reference

The following theoretical reference frame presents the equations and theory used within the research work described in the previous section.

Heat release

As described, a number of different ways of estimating and modelling heat release in a simplified manner has been introduced during the years [29]. The first approaches dates back far in time when the ideal Otto cycle and the ideal diesel cycle were invented. These are very simplified cycles and with time more realistic estimations have been developed.

The isochoric, the isobaric and the mixed mode cycles

The Otto cycle consists of an isentropic compression stroke from BDC to TDC, an isochoric (constant volume) heat addition at TDC, then an isentropic expansion stroke to BDC and lastly an isochoric heat loss. This means that the heat release from the combustion happens instantaneously. The total heat released can be calculated from Equation 1.

�� = �� = �� (1)

where �� is the mass of the injected fuel, � is the combustion efficiency and �� is the lower heating value. Further, � is the in-cylinder mass, � is the specific heat at constant volume and Δ� is the change in temperature from the combustion [30].

Moving on to the Diesel cycle, it is similar to the Otto cycle in terms of isentropic compression and expansion and isochoric heat loss, however the heat addition occurs during the first part of the expansion with constant pressure; referred to as isobaric heat addition. Equation 1 is valid for the Diesel cycle if the specific heat for constant pressure (cp) is used instead of the specific heat for constant volume (cv). If the initial temperature and pressure, the in-cylinder mass (after fuel injection for a diesel engine), air/fuel ratio, specific heats, combustion efficiency and compression ratio is known, the pressure and temperature trace can be calculated for the whole Otto and Diesel cycle. Noteworthy is that a diesel and a SI engine has normally a very high combustion efficiency not uncommonly close to 100%. By modelling the combustion as partly isochoric and then isobaric, a more realistic heat release trace can be obtained [30].

The thermodynamic efficiency of two cycles can be calculated with Equation 2 [30].

� = 1 − �� ⇒

�� , �� = 1 − �� !�" − �#$�� !�% − �&$ = 1 − !�" − �#$!�% − �&$

�,�'�(�) = 1 − �� !�" − �#$��*!�% − �&$ = 1 − !�" − �#$+!�% − �&$ (2)

where T is the temperature at the locations specified by the indices: 1 refers to intake, 2 to SOC, 3 to EOC and 4 to EVO. Further, γ is the specific heat ratio and m is the cylinder mass.

For isochoric heat release the amount of heat released during combustion can be calculated from the pressure difference between SOC and EOC according to Equation 3.

��, = -./!0123$ − .4/!0123$5 ∙ 7!0123$ ∙ � 89 :�'�; < (3)

where Pb is the post-combustion pressure trace, Pub is the pre-combustion pressure trace, cv is the specific heat for constant volume, R is the gas constant, Mair is the molar mass of air, V is the in-cylinder volume trace and θSOC is the crank angle at start of combustion.

For the case of isobaric the total heat released can be calculated using Equation 4.

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��, = -7!0=23$ − 7!0123$5 ∙ .4/!0123$ ∙ �*89 :�'�; < (4)

where cp is the specific heat at constant pressure and θEOC is the crank angle at EOC.

The Wiebe function

The single Wiebe function is written according to Equation 5.

>/ = 1 − ?@A�BCACDEFG∆CDEFG IFJKL (5)

where xb is the burn fraction, θ is the instantaneous crank angle, θcomb is the crank angle at start of combustion and ∆θcomb is the combustion duration. a and m are adjustable parameters for adapting the model. The heat release can then be calculated by multiplying the burnt fraction with the mass of the injected fuel and the fuels lower heating value. The function needs calibration of the factors a, m, θcomb and ∆θcomb [12].

Two Wiebe function fits

As the combustion in a diesel engine can be divided into different phases a single Wiebe function may not be accurate enough to describe the full combustion and a common practice is therefore to derive two functions, one for the premixed combustion and one for the diffusive combustion. If multiple injections are used, a double Wiebe function is needed for each injection and calibrated respectively [12].

When parametrising the heat release with two Wiebe functions it is common to use a weighting factor describing the amount of fuel burnt during the premixed phase. The burn rate is then commonly described using Equation 6.

>/ = M>/* + !1 − M$>/� (6)

where M is the weight factor which is the fraction between fuel amount burnt during diffusive combustion and the total amount of injected fuel, >/* is the mass fraction of burnt fuel through premixed combustion and >/� is the mass fraction of burnt fuel through diffusive combustion [11].

Ignition delay

The chemical time delay can be estimated by an Arrhenius correlation dependent on the cylinder pressure, equivalence ratio and temperature according to Equation 7.

O�� = �# ∙ P�Q ∙ @ RR��L�S ∙ ?�T �⁄ (7)

where Φ is the equivalence ratio, p is pressure and T is temperature. The parameters c0, c3 and TA needs calibration with measured data [14].

In-cylinder thermodynamics

The thermodynamic equation proposed by Heywood [15] and used in many studies of pressure or heat release modelling can be seen in Equation 8.

V�� = ++ − 1 ∙ R ∙ V7 + 1+ − 1 ∙ 7 ∙ VR + V�W (8)

where Qhr is heat release, Qw is heat transfer through cylinder walls, γ is mean specific heat ratio for the cylinder medium, p is pressure and V is volume. A common way of estimating the wall heat release is to use the Woschni correlation for convective heat transfer [31]. Details on the parameters that can be used in Woschni’s equation can be seen in [6]. To estimate the heat loss using Woschni’s equation, both the in-cylinder pressure and temperature needs to be known.

Specific heat ratio

The specific heat ratio is dependent on the composition of the medium and its temperature. In engine applications the equivalence ratio and temperature can be used to calculate the ratio although it has been shown that temperature has a relatively larger effect and, commonly, for general purposes the specific heat ratio is only calculated as a function of temperature [32]. In a study by Brunt et. al [31], Equation 9 was used to calculate the specific heat in a diesel engine application.

+ = 1,35 − 6,0 ∙ 10A\ ∙ � + 1,0 ∙ 10A] ∙ �& (9)

where γ is the specific heat ratio and T is temperature in K.

The specific heat ratio is sometimes assumed to be constant for simplicity and is commonly set to 1,33 for pure air. Another way of calculating the specific heat is to use the NASA polynomial, for more details see [6].

Compression stroke pressure contribution

The in-cylinder pressure during the compression stroke can be modelled as a polytrophic process according to Equation 10 [24].

.̂ !0$ = .'_��`� B7'_��`�7!0$ Ia = .'_��`� B�'_��`��!0$ I bbcK (10)

where Pintake, Tintake and Vintake is the pressure, temperature and volume when the intake valve is closed, γ is the polytrophic index and θ is the crank shaft position. Commonly, the compression and expansion process in an engine is modelled as isentropic processes where Equation 10 applies with the polytrophic index equal to the specific heat ratio of the in-cylinder mixture [30].

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Crankshaft dynamics

Using Newton’s second law of motion, the crankshaft can be described by Equation 11.

def + geh + ie = jklm + jnopqrpst + juslv + jwlmm (11)

where J, D and K are the inertia, damping and stiffness for the crankshaft for each cylinder, θ and its derivatives are the angular position, velocity and acceleration of the crankshaft for each position of the cylinders and lastly the torques T for each cylinder. The multi-body system is usually expressed as a state-space matrix model [24]. For further details on how each component can be derived see Gustafsson’s work [5]. The crankshaft model used to obtain the partial pressure traces used within this study consists of a MISO system originating from the torque equation, for further details on the systems appearance see Gustafsson’s [5] and Rosvall’s [19] studies.

Translational motion of oscillating masses

The translational motion of the pistons can be described by the angular velocity of the crankshaft according to Equation 12.

VxVy = VxV0 z = {z

|}x~� 0 + {� x~� 0 ��x 0

�1 − {&�& x~�& 0�� (12)

where s is the position of the piston, z is the angular velocity of the crankshaft, r is the crank radius, l is the connecting rod length and 0 is the angular position of the crank.

Torque from rotating and oscillating masses

The inertial torque from the crank mechanism is challenging to model. The piston and piston pin can be assumed to oscillate in a translational motion relatively the crankshaft, while the connecting rod is more complicated to model as it both rotates and oscillates relatively the centre line of the crankshaft. A common way of modelling the inertial torque from the crank mechanism is to model the connecting rod as two masses, one at the piston pin and one at the crank pin experiencing pure translational motion and rotational motion relative to the crankshaft respectively. The relation of the two masses is such that the total is the same as the connecting rod mass and the location of the centre of mass is kept the same. Including the translational motion of the piston gives the expression for the torque from the crank mechanism, shown in Equation 13 [33].

�̂ �(( = −!��!0$ + ��{&$0f − 12 V��!0$V0 0h & = = − @�� BVxV0I& + ��{&L 0f − @�� VxV0 V&xV0&L 0h &

(13)

where mA is the mass of the piston, piston pin and the part of the connecting rod mass oscillating and mB is the part of the mass of the connecting rod rotating. The inertia of the crankshaft would further be included in the factor multiplied with the crank angle acceleration to receive the total inertial torque from the parts in motion.

Kinetic and torsional energy within a crankshaft in motion

A crankshafts kinetic energy is described by its angular velocity and the inertia of the different parts in rotational motion in accordance with Equation 14.

�`'_��'�,� � = 12 ��( ∙ z& + 6 ∙ 12 ��^,� �z& (14)

where ��( is the inertia of the crankshaft, ��^,� � is the inertia of the rotating part of the crank mechanism and z is the angular velocity of the crankshaft. The oscillating parts of the crank mechanism in each cylinder can be described by its translational motion and its mass in accordance with Equation 15.

�`'_��'�, (� = � 12 � (�,' VxVy& =�

'�#� 12 � (�,' BVxV0 zI'

&�'�#

(15)

where � (�,' is the mass of the oscillating parts of the crank mechanism and Vx/Vy is the translational velocity which can be calculated with Equation 12.

The torsional energy can be estimated as the spring energy if the crankshaft is modelled as a mass-spring-damper system where the spring constants of the nodes are known using Equation 16.

�� (' _ = � 12]

'�#�'�#,'∆0'& (16)

where ki+1,i is the torsional stiffness between the cranks and the crankshaft and ∆0' is the angular displacement between the cranks.

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Implementation

In this study, experimental data was used as reference data and different models were developed and combined in different ways to reach the project aims. How the experimental data was gathered and the procedure of model building throughout the study is explained in this chapter. To facilitate perceiving the developed models, the sub models for calculations for the full closed cycle and within the zones without combustion will be explained in a first section of the model development chapter and the different heat release modelling approaches will then follow in a second section.

Experimental set-up

The experimental data used within this thesis was gathered during the experimental tests conducted during the research projects by Gustafsson [5], Johansson [6], Rosvall [19] and Rugland [34]. The tested engine was a Scania 13L Euro 6 engine with the specifications presented in Table 1.

Table 1. Specifications of the engine used in the conducted tests of which the data used within this project was gathered.

Engine Scania 13L Euro 6 engine

Emission standard EURO VI

Number of cylinders 6

Firing sequence 1-5-3-6-2-4

Displacement volume 12,74 dm3

Stroke 160 mm

Bore 130 mm

Connecting rod length 255 mm

The monitored parameters used for the simulations of this work was:

• In-cylinder pressure in cylinder 1 and 6. • Crank angle • Intake and exhaust manifold pressure • Intake and exhaust manifold temperature

The in-cylinder pressure was measured at a resolution of 0.1 CAD with a water-cooled Kistler 7061B (cylinder 1) and an uncooled AVL GU24D (cylinder 6), both are of the piezoelectric type. The crank angle was measured at the flywheel with an optical crank angle encoder with a resolution of 0.5 CAD; from interpolation a resolution of 0.1 CAD was obtained.

During the tests, the engine was run at 36 steady-state operating points. The operating points consisted of the combination of the speeds of 2000, 1900, 1600, 1200, 1000 and 800 rpm and the loads of 100, 75, 50, 25 and 0 % of the maximum load at the particular speed and it was also motored for each speed. The operating points were covered by starting

at the highest speed and load. The testing sequence was to reduce the speed in steps in descending order for each load that was in turn reduced in descending order. For validating the different sub models and the final virtual sensor developed within this thesis, the data from all operating points at the speeds of 2000, 1600, 1200 and 800 rpm was used as reference data. Each testing point was considered to be in steady-state when the exhaust temperature had stabilised. The testing was carried out for two fuel types, namely Euro VI diesel reference fuel (with 7% RME) and B100 biodiesel (100% RME).

Model development

The model development process consisted of building of sub models and continuous verification and validation of the sub models results and accuracy. At first, isochoric and isobaric heat release models were built and integrated with the crankshaft pressure model. Following, one Wiebe function was fitted to estimated late combustion heat release. Then a two Wiebe function fit was done by requiring the final indicated work to match the indicated work estimated from the angular velocity measurements. Along the Wiebe function fitting work, a model for estimating the indicated work was built. Further, the torsional behaviour of the crankshaft was studied in an attempt to improve the virtual sensor further. The different models gave different useful results and are therefore part of the final model for the virtual pressure sensor. The working principle of the virtual sensor can be seen in Figure 1.

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Figure 1. Diagram of the algorithm of calculations to reach the final pressure trace from the measured crankshaft, position, velocity and acceleration.

As it can be seen in the figure, the assumption of isochoric and isobaric heat release as well as the late combustion heat release rate are used to improve the estimation of the heat release. Different approaches to retrieve a full heat release curve estimate were attempted, which was then used to calculate the final pressure trace for the full closed cycle. The independent models were developed individually and their working principle will be explained in this chapter.

Full cycle and zones without combustion

All the models consisting of either averaged calculations of the full closed cycle or models valid before and after combustion are explained in this section

Isentropic pressure traces

Due to oscillations perturbing the pressure signal from the crankshaft model originating from torsion of the crankshaft [5] and due to the model singularity obstructing the pressure estimation partly before the combustion, isentropic pressure curves were calculated pre and post combustion. From assuming the cylinder to be an adiabatic closed system and with known pressure at parts of the compression and expansion within the engine cycle, a pressure curve for the unburnt and burnt mixture could be extrapolated for the whole interval from IVC to EVO using Equation 10, where the specific heat ratio was used as the polytrophic index. The specific heat ratio was at first estimated by rearranging Equation 10 to get Equation 17 and using the partial pressure traces.

+ = �� B.!0#$.!0&$I�� B7!0&$7!0#$I (17)

where θ1 is the crank angle position prior to the crank angle position θ2.

The specific heat ratio calculation demands high accuracy of the pressure trace. Since the partial pressure trace was perturbed with torsional oscillations the specific heat ratio was in a second approach calculated using Equation 9 which gave more accurate values of the ratio. It was however found that the specific heat ratio estimation did not perform well enough for all operating points when the isentropic pressure curves were compared to the measured pressure. Therefore, the specific heat ratio model that was finally used was the NASA model. Further, with known temperature at the intake manifold and the exhaust manifold, temperature curves could iteratively be calculated from IVC to EVO for an unburnt fuel mixture and vice versa for a burnt fuel mixture which allowed for estimation of the temperature dependent specific heat ratio.

However, it was found that the standard exhaust pressure measurements generally were lower than the actual in-cylinder pressure at the defined EVO, which will be further elaborated upon in the results. This meant that the isentropic expansion curve had to be estimated in another way as a first step and later when the indicated work estimate had been calculated, an improved isentropic expansion curve could be derived with a corrected exhaust pressure value reflecting the cylinder pressure at EVO instead.

Exhaust pressure

To extrapolate the isentropic pressure traces, boundary conditions are needed. During regular operation of the engine, the only available pressure measurements will be the boost and exhaust pressure. For this reason, the isentropic pressure curves were calculated from IVC and EVO with these pressures as boundaries. For improved accuracy of the

Crankshaft model

0, 0h , 0f

Gross indicated work and load estimation

Late combustion

Isochoric and isobaric HR

Wall heat losses

Cumulative heat release

Heat release estimation through Wiebe function parametrisation

Energy equation

Closed cycle pressure trace

Volume

Volume correction

Rough full pressure curve

estimation

Pressure at EVO

Isentropic pressure curves first derivation

Isentropic pressure curves second derivation

Premixed comb. SOC and duration

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pressure traces, it is important that the boundary pressures match the actual in-cylinder pressure at IVC and EVO. The boost pressure will not be equal to the actual cylinder pressure since pressure losses are present as the intake gas passes through the valve and in the contrary, the temperature of the cylinder walls, assuming warm engine operation, will be much higher resulting in an increased pressure as the boost gas warms up compared to the pressure in the intake manifold where the boost pressure sensor is located. The same applies for the exhaust pressure, which will deviate from the in-cylinder pressure at EVO. There will be oscillations in the exhaust pressure as the exhaust valves continuously open and close and as the exhaust pressure sensor monitors an averaged pressure value, the pressure at EVO will deviate slightly.

As the test results showed that the exhaust pressure sensor reading deviated much more from the in-cylinder pressure measurements compared to the boost pressure sensor, the boundary pressure at EVO was corrected for with a lookup table containing the pressure deviation at different operating points. As it also was found that the pressure deviations speed and load dependence was similar to the torque curve of the engine, it was scaled to the pressure offset. This can allow for more accurate isentropic expansion reconstruction without the need for much calibration.

Gross indicated torque and work

Once a full pressure trace is retrieved, the corresponding work transferred from the gas to the piston can be calculated from integration of the pressure over the volume. The gross indicated work is a useful parameter as it can directly show whether the pressure trace for the compression and expansion stroke of the cylinder is correct since the corresponding piston work from the pressure must match the gross indicated work. Therefore, if the gross indicated work can be calculated from another source of information other than the pressure, it can be used to improve the estimation of the in-cylinder pressure.

The piston work can easily be calculated from the gas torque which in turn can be estimated from the angular acceleration of the crankshaft, as shown by Gustafsson [5]. To facilitate the estimation of the gas torque and avoid using the measured pressure, the torques from the cylinders in the gas exchange phase were assumed to cancel and the contributing torque from the cylinders in compression pre-combustion and expansion post-combustion was calculated from the derived isentropic pressure curves. The inertial torques and the oscillating mass torques could be calculated according to Equation 13. Further, by assuming a constant friction, load and auxiliary systems torque acting on the crankshaft, motivated by the steady state operating point, the inertial torque could be assumed to directly correlate with the cylinder gas torques; only offset by the other torques. Since the load is needed to move from the inertial torques to the gas torque, another approach was needed.

Instead the kinetic energy of the moving parts of the engine was studied and compared to the total piston work calculated with the assumption that the pressure trace in all cylinders was the same as the trace measured in cylinder 6. By approximating the kinetic power curve and the piston power curve and finding the offset between the curves, the power loss to load and friction could be retrieved. Following, the net indicated work from one cylinder could be calculated as the product of the power loss and 120 [CAD]. The estimation method was developed along the process of studying the energy and power estimations and the method will therefore be further explained in the results section. A schematic representation of the method for calculating the indicated work can be seen in Figure 2.

Figure 2. Schematic representation of the order of the different calculations to retrieve an estimation of the indicated work.

The total piston power could be approximated before and after the combustion (outside the blind zone) by using the isentropic curves to estimate the pressure inside the cylinders in compression and expansion stroke. By calculating the derivative of the total piston work approximation, a piston power approximation was retrieved. Following, it was found that the crank angle locations where the power approximation became zero could iteratively be found using the secant method. Further on, the kinetic energy of the engines moving parts could be calculated from the measured angular velocity of the flywheel; using Equation 14 and 15. The kinetic power could then be estimated either numerically by calculating the derivative of the kinetic energy using the symmetric difference quotient, or it could be calculated by multiplying the angular velocity with the inertial torques derived from Equation 13.

0, 0h , 0f Kinetic energy and

power of engine parts in motion

Isentropic pressure curves

Piston work and power

Crank angle with zero piston power

Power to load, friction and auxiliaries

Net indicated work and torque

Isentropic expansion curve

improvement

Exhaust pressure correction

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The torque contribution of the rotating parts and the oscillating parts were considered.

The kinetic power could then be extracted at the locations with zero piston power, which is equivalent to the power loss due to load, friction and auxiliary systems. Finally the indicated work could be estimated from the retrieved power loss. The location of the determined points is visualised in Figure 15 for clarification.

Correcting the kinetic power for torsion

As the angular velocity measurement will be affected by the energy stored due to torsion, the kinetic energy and the equivalent power will also be affected by the torsional on the form of perturbed oscillations. By estimating the energy stored due to torsion during a full cycle, it can be added to the kinetic power trace to remove the oscillations. This is necessary as if the offset of the kinetic power trace to the piston power trace is calculated at a moment where the rate of torsional energy change is large, an error is introduced in the estimation of the indicated work and torque. It is not desired to include the torsional power as the torsional energy is not lost, only momentarily stored as the crankshaft has been twisted and in turn is released as the shaft unwinds.

With the dynamic crankshaft model consisting of a mass-body-spring model with 9 nodes, the torsional energy and power during an engine cycle could be simulated. The retrieved torsional power could then be added to the kinetic power to remove the torsional oscillations. The more accurately the torsional power can be estimated, the more accurate the indicated work and torque can be approximated with the method used.

Pressure estimation from dynamic crankshaft model

In an attempt to account for torsion during the derivation of the pressure trace from the mechanical crankshaft model, the dynamic crankshaft model was studied. Instead of inverting the MISO system, the cylinder pressure traces were iteratively adapted using an optimising algorithm minimising the least-square error of the angular acceleration. The error was defined as the difference between the measured angular acceleration and the output acceleration from the MISO system.

The MISO iteration approach was performed for different cases. Firstly only the pressure trace of cylinder 6 was optimised to adapt the acceleration within the cylinder’s closed cycle. Then each cylinder’s pressure trace was optimised in the firing order to obtain an adapted acceleration trace for the full engine cycle. This was firstly done without accounting for angular offsets of the cylinders due to torsion and then repeated accounting for the offsets. Finally the method was repeated twice with corrections of the load torque giving a constant angular velocity, between each full optimisation of the pressure traces. The reason behind the last case is that the change of the pressure traces will alter the

angular velocity trace which should stay on a constant level as the operating points are stationary. Not perfect estimations of the auxiliary loads and friction calls for correction of the load torque for a resulting constant angular velocity.

Torsional vibrations

The oscillations within the measured angular velocity were studied regarding oscillating frequency and descending amplitude. The purpose was to analyse whether the torsion of the crankshaft could be predicted based on angular velocity. Such information could possibly be used to correct for torsion in the partial pressure trace from the crankshaft model and also within the kinetic energy power calculations.

Modelling of the heat release trace from combustion

The different approaches of modelling the heat release are explained in this section.

Isochoric and isobaric heat release

As a first step of heat release modelling, the combustion was assumed to be an isochoric and isobaric process in two respective models. The two isentropic pressure curves were used pre and post combustion and for the case of isochoric heat release, the pressure difference between the two pressure curves at SOC could be calculated together with the trapped cylinder mass allowing for calculation of the amount of heat released according to Equation 3. In a similar manner, the total heat released from isobaric combustion can be determined using Equation 4. By using the lower heating value of the fuel and combustion efficiency, the stoichiometric ratio could be calculated and following the mass of injected fuel and gas composition of the burnt mixture could be found. The shape of the heat release trace directly influences the in-cylinder pressure and thereby the piston work. Therefore, the SOC could be approximated for the isochoric and isobaric heat release models such that the piston work corresponded to the indicated work which was calculated prior from the angular velocity.

Premixed SOC and duration

The start of combustion was roughly estimated from the derivative of the kinetic power. Before the start of combustion, the shape of the derivative of the kinetic power follows the shape of the derivative of the motored piston power. Just before TDC, both derivatives increases as the lever length is significantly reduced resulting in an increasing piston power. After TDC this increase in the motored piston power starts decreasing. The kinetic power, however, keeps on increasing similarly to the actual piston power as an effect of the combustion adding energy. The SOC was estimated by finding the location where the difference between the second derivative of the kinetic power and the second derivative of

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motored piston power becomes below a threshold meaning that the motored piston power increase has decreased more than the indicated power derivative and combustion must have started. The different power curves with their derivatives are illustrated in Figure 3 for clarification.

Figure 3. Kinetic power, motored piston power and actual piston power with combustion (top), derivative of the respective trace (middle) and second derivative and each trace (bottom) at 800 rpm and 50 % load in cylinder 6 clearly presenting how the kinetic power trace follows the actual cylinder power after combustion.

To roughly approximate EVO, the location where the second derivative of the kinetic power became smaller than a threshold was determined. This location is roughly where the derivative of the kinetic power starts to decrease significantly as a result from the ended premixed combustion which is no longer supplying energy at a high rate. Another estimation approach of SOC was also done where the ignition delay is firstly estimated according to Equation 7 and then added to the start of injection location to get an approximation of SOC. This method adds another input parameter (SOI) which may be inaccurate and also adds the assumption that Equation 7 is valid for the particular combustion type which is an important aspect to take into account.

Normal distributed HRR for rough pressure trace estimation

By assuming the heat release rate to be normal distributed, suggested to be a more realistic HRR trace than the isobaric and isochoric heat releases by Johansson [30], a rough pressure trace for the closed cycle could be obtained. To define the HRR distribution, values for the heat release duration, start of combustion and total accumulated heat release are needed. By estimating the SOC according to the previous section and by assuming the combustion duration to be 30 CAD, the total heat release could be iteratively

calculated such that the equivalent gross indicated work to the resulting pressure trace matched the pre-estimated indicated work. The used numerical approach was the secant method. To calculate the equivalent indicated work from the heat release rate trace, the energy equation, Equation 8, was used to first estimate the pressure trace. The equation was reordered such that the pressure derivative could be calculated with the forward Euler method. The indicated work could in turn be estimated by integrating the pressure over the volume. To use the energy equation, the wall heat losses had to be approximated which was done using Woschni’s convective heat transfer equation. As the pressure was estimated in steps, the temperature could be estimated using the ideal gas law which together with the pressure could be used in the Woschni equation. From this model a continuous pressure trace, a total heat release and a wall heat loss estimation were obtained.

Alternative total heat release estimation

Another approach of estimating the total heat release was to estimate wall heat losses, exhaust heat losses and the indicated work for one engine cycle. The sum of these energies would then approximately be the total heat release from combustion. A wall heat loss estimation was obtained from the rough pressure trace model and the indicated work was estimated from the angular velocity. By estimating the trapped cylinder mass, the exhaust heat losses could be approximated as an isochoric heat rejection from the condition at EVO to IVC using the difference between the temperatures of the intake and the exhaust gases.

Volume correction

Due to the high pressures within the cylinder, large stresses and resulting deformations are present which means that a solely geometrical estimation of the cylinder volume will be erroneous. The volume estimation could however be corrected by using the work by West [23]. The volume model calculates deformations of the engine components affecting the cylinder volume and estimates the actual volume. The pressure within the cylinder and temperatures of the components are needed for this estimation. With the pressure trace roughly estimated, the volume could thereby be corrected for. The assumed temperatures where used according to recommended approximate temperatures during regular warm engine operation [23].

Late combustion HRR Wiebe function fit

Using the thermodynamic equation for a closed system, Equation 8, the pressure trace during the expansion stroke could be used to get an estimation of the heat release rate for that particular crank angle interval. The pressure trace from the rigid crankshaft model was most accurate during late combustion and expansion. It was at first attempted to fit this heat release information with one Wiebe function, Equation 5, to describe the burn rate. This was done by minimising the summed squared error of the late combustion heat release and

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the functions heat release trace by adapting the Wiebe parameters. This method is ideal if the late combustion pressure trace is highly accurate.

A second approach was also attempted to parametrise a Wiebe function to the late combustion pressure data. The function was instead fitted such that the corresponding pressure derivative’s least-square error to the derivative of the late combustion pressure was minimised; see Figure 28 for an illustration of the resulting pressure derivative fit. To obtain the pressure derivative from one Wiebe function, the heat release was calculated from the burn rate described by the Wiebe function. The burn rate was in turn used in the energy equation with the volume trace to obtain the pressure trace. Iteratively the in-cylinder temperature was calculated using the ideal gas law which then allowed for the Woschni wall heat loss estimation. Then the derivative of the pressure trace was numerically calculated.

Parametrised heat release with two Wiebe functions

In next step of the progression of the quasi-experiment, a better heat release parametrisation of the diesel combustion was attempted through fitting of two Wiebe functions. The purpose was to parametrise a second function to describe the premixed combustion. The working principle of the second function adaption can be seen in Figure 4.

Since the pressure curve obtained from the crankshaft state-space system did not give reliable values close to TDC, no pressure information during the premixed combustion could be obtained from the torque model. The same approach used for the one Wiebe function fit could therefore not be attempted for the second Wiebe function fit. Instead, the second Wiebe function was parametrised such that the resulting indicated work matched the estimated work. This was done using a least-square approach. The function parametrisation was

iterated until minimal values were obtained for both the difference between the resulting indicated work from the parametrisation and the approximated indicated work from the CAD data and the fit to the late combustion trace which was obtained from the first Wiebe function fit.

The two Wiebe function structure used can be seen in Equation 6 where the weighting parameter defines the relation between the amount of injected fuel combusted during the premixed and the diffusive combustion. The weighting function changes the other parameters of the diffusive combustion function which is why the parameters of both Wiebe functions had to be adapted and the parameters from the one Wiebe fit not could be used directly. Therefore the adaption was performed such that the resulting heat release trace from the two Wiebe functions with the weighting parameter matched the one Wiebe function fit within the late combustion interval.

The start of combustion and duration of the premixed combustion were estimated beforehand like explained earlier and the shape parameter, the weight parameter and the parameters for the diffusive combustion function were adapted in the optimising algorithm.

Indicated work

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Kinetic energy work

Premixed SOC and duration

Isentropic pressure curves

Isochoric and isobaric HR

One Wiebe

function fit

Late combustion HR

Crankshaft model

Second Wiebe function fit

Figure 4. Working principle of the second Wiebe function fit where the adaption minimises the error of the resulting indicated work from the estimated indicated work and the late combustion heat release trace from the first Wiebe function fit.

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Results and discussion

The results obtained from the simulations of the built models and the subsequent discussions are presented in the following section. The results will be separated in four sections where the first section explains filtering of the test cell data. Then, similarly to the model development chapter, the second section will include the models for the full cycle and the zones without combustion and the third section includes the heat release models. Lastly, the tolerance analysis is presented and discussed.

Filtering of the logged data

The pressure and the crank angle traces contained a considerable amount of noise and had to be filtered before they could be used for calculations and simulations. A spectrogram of the power of the frequency content of the angular velocity trace for the operating point of 800 rpm and 50 % load can be seen in Figure 5.

Figure 5. Single-sided amplitude spectrum of crankshaft angular velocity from a fast fourier transform of the signal.

As it can be seen, both signals contain high frequency noise, especially the CAD signal. From further analysis of the frequency content of the crankshafts angular velocity trace for different operating points, it was found that it depends heavily on the engine speed. In the zoomed-in image of the spectrogram of the CAD signals frequency content, Figure 6, it can be seen that frequencies of 1, 3, 6, 9 and 12 periods per crank angle revolution stands out significantly. The reason is that since the engine has six cylinders, mutually phase shifted 120º, combustion occurs three times per revolution which gives rise to three velocity peaks per revolution. The frequency content above 12 periods per revolution seems more randomly distributed and it was therefore decided to filter it out.

Figure 6. Zoomed in frequency spectrum of the angular velocity trace showing clearly the frequencies with largest amplitude; namely 1, 3, 6, 9 and 12 periods per revolution.

To filter out the frequency content above 12 periods per revolution, a 10th order low-pass Butterworth filter was used with a cut-off frequency of 15 periods per revolution. It was determined that the ending points towards the boundaries for one cycle became inaccurate if a mean trace from the CAD data of 50 cycles was calculated before filtering. Instead, the full signal from the cycles was filtered at first and subsequently the mean trace could be obtained with accurate boundaries. In Figure 7, plots of the frequency content and the signal traces of the unfiltered and filtered angular velocity signal can be seen for comparison. As it can be seen, the unfiltered signal contains a lot of high frequency noise. This is also clear in Figure 8 presenting the velocity signal, where large fluctuations can be seen along the curves.

Figure 7. Comparison of the frequency content of the unfiltered and the filtered angular velocity trace where a 10th order low-pass Butterworth filter has been used with a cut-off frequency of 15 periods per revolution showing that the unfiltered signal contains a lot of high frequency noise that is effectively removed by the filter.

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Figure 8. Comparison of the unfiltered and filtered velocity trace where a 10th order low-pass Butterworth filter with a cut-off frequency of 15 periods per revolution has been used.

The filter effectively removes the high frequency noise from the signal which can be seen both in the velocity trace, Figure 8, and the spectrogram, Figure 7. The frequencies below 15 periods per revolution are affected somewhat but not significantly, which can be seen in the zoomed-in spectrum in Figure 7. This filtering allows for more accurate estimations of the angular acceleration and is therefore necessary.

The nature of the frequency content of the measured pressure signal is very different in comparison to the velocity signal. It was found that similarly to the velocity signal, the frequency content of the pressure signal is also dependent on the engine speed. After comparing the frequency content of pressure traces from different operating points it was found that the amplitude for the frequencies up to a certain upper limiting frequency had a similar shape independent of operating point. The general case was that the lower frequencies had the largest amplitudes. The frequencies in between the low frequencies and the upper limiting frequency had fairly large amplitude, although roughly for the same frequency there was a drop in amplitude which then increases again. This can be seen for the operating point of 800 rpm and 50 % load in Figure 9, where the filtered signal is also included. The first assumption was that the frequencies above the upper limiting frequency mainly consisted of noise while the frequencies below contained the actual pressure information.

Figure 9. Frequency content of the pressure signal from the operating point at 800 rpm and 50% load obtained from a fast fourier

transform. Clearly showing the location of the cut-off frequency just below 80 periods per revolution where the amplitude of the frequencies in the filtered signal drops and stays on low level.

From further analysis it was found that the closing of the intake valve causes high frequency ripples in the pressure signal. The oscillations can be seen in Figure 10. By estimating the period of the oscillations which is also indicated in the figure, it was determined that the oscillating frequency was around 80 periods per revolution. These oscillations would not be needed for the purpose of this study; therefore, it was decided to low-pass filter the signal with a cut-off frequency just below the pressure ripple frequency and where the amplitude of the frequencies falls. In Figure 9, it can be seen that the amplitude of the filtered signal’s frequency content drops around 80 periods per revolution and all higher frequencies have been filtered out.

Figure 10. Pressure ripple during intake valve closing clearly present in the measured pressure in cylinder 1 which has been filtered out by a 10th order Butterworth filter with a cut-off frequency of 77 periods per revolution; the measurements originate from the operating point of 800 rpm and 50 % load. The period of the oscillations is illustrated.

It was found that a cut-off frequency of 77 periods per revolution gave good results for the operating point of 800 rpm and 50 % load. The most optimal cut-off frequency for the other operating points varied a little but were around the same frequency. Since the filter also seemed to filter out the noise sufficiently well for the other operating points, it was decided that the filter performed sufficiently well. If it is important that the pressure signal also includes high frequency pressure information, the cut-off frequency can be selected with more caution and would need to be adaptable to more parameters than the engine speed to be adaptable for any operating point. For the purpose of this study, the designed filter performed sufficiently well.

Full cycle and zones without combustion

Results from the models valid for an averaged cylinder cycle or before and after combustion are included in this section.

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Isentropic pressure traces

The isentropic pressure curves were necessary to derive as the partial pressure curves from the rigid crankshaft model contains oscillations originating from the torsional vibrations of the crankshaft which the model is incapable of describing. Different attempts of estimation of the specific heat ratio, cylinder volume and exhaust pressure affected the precision of the isentropic pressure curves and these will be discussed.

Specific heat ratio

The specific heat ratio was estimated in four ways. The first approach was to use the partial pressure curve from the crankshaft model during the compression phase to estimate the ratio. The resulting heat ratio was far too inaccurate as the torsional oscillations within the pressure trace has a large impact on the specific heat ratio estimation when using Equation 17. Instead three other approximations were attempted, a constant gamma, a temperature dependent gamma model adapted for diesel combustion and a model based on NASA polynomials [6]. The resulting isentropic compression traces from the three modelling approaches can be seen in Figure 11.

Figure 11. Visual representation of the accuracy of the pre-combustion isentropic compression curve for the three different specific heat ratio estimations (NASA model, temperature dependent model and constant value) for the operating points specified in the legend for cylinder 6. The NASA model gave the most accurate results.

The NASA model gave the most prominent results, which can be seen in the figure where the pressure curves originating from the NASA model follows the measured pressure curves better than the curves from the other approaches does. The reason is most probably that the NASA model considers both the effects of temperature and the composition of the in-cylinder mixture which both affect the specific heat ratio.

Therefore, the NASA model was chosen for the virtual sensors specific heat ratio calculations. It can, however, be noticed in the figure that the NASA model still deviates from the measured pressure, especially for the operating point at 1200 rpm and 100 % load. This is because of the deviation of the boundary condition. The measured boost temperature and pressure which has been used for the derivation of the isentropic compression curves will deviate from the actual cylinder temperature and pressure at IVC. This deviation will remain for the whole trace as it can be seen. The same discussion applies for the fully burnt fuel mixture’s isentropic pressure trace. The NASA model also includes simplifications regarding the composition of the gas. The cylinder mass is assumed to only consist of air and burnt gases. The reaction is assumed to only be between hydrocarbons and oxygen meaning that the concentrations of CO, NO and NO2 in the exhaust are assumed negliable. Further, the residual gas fraction was assumed to be zero for the compression curve. These assumptions deviate from the reality and also contribute to the error.

For the low speed operating points, the partial pressure trace from the crankshaft model for both the compression and expansion is fairly accurate at certain intervals. The offset of the isentropic curves could therefore be corrected for at these operating points for improved accuracy.

Exhaust pressure

From correcting the exhaust pressure measurement such that it was compliant with the actual in-cylinder pressure at EVO, the isentropic expansion trace could be significantly improved. The largest pressure offset was seen at 100 % load and 1200 RPM in speed which gave a deviation of around 10 bar, which in turn gave a pervading offset of the isentropic expansion trace. The exhaust pressure correction was therefore vital, especially for the higher load operating points.

From scaling the engines torque curve to match the pressure deviations, the exhaust pressure could still be corrected for but with a slight reduction in accuracy as the torque curve does not coincide perfectly at all operating points; even though it has be scaled. Using the torque curve instead of storing the information of the pressure offset at different operating points can save calibration effort and data storage space. The pressure offset and the scaled torque curve can be seen in Figure 12.

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Figure 12. Difference between in-cylinder pressure at EVO and the exhaust pressure measured with the standard exhaust pressure sensor for the different operating points and scaled engine torque curve.

The torque curve is expected to match the pressure offset due to the nature of the engine torque and its exhaust pressure. The torque produced by the engine is directly related to the cylinder pressure. The pressure in turn, can be seen as dependent on two main parts, the motored pressure caused by the change of the cylinder volume during operation and the boost pressure, and the heat release caused by fuel combustion. In other words, the boost pressure and the heat release directly influences the engine-out torque. Further on, the boost pressure depends on the turbo compression of the charge and is controlled based on speed. The heat release in turn depends on the injection of the fuel which, simplified, is controlled based on the torque demand and the exhaust lambda value which in turn is dependent on the boost pressure. The cylinder pressure of course also depends on the cylinder pressure at IVC and the heat release. Consequently, the engine torque and the exhaust pressure both depend on the boost pressure and the heat release which is controlled based on the operating point. Therefore, they are expected to behave similarly and it is therefore suitable to scale the torque curve to receive a sufficient map for exhaust pressure correction.

Indicated work and torque

It was found that the derivative of the kinetic energy had the same shape as the derivative of the total piston work, the curves were only offset; as it can be seen in Figure 14. This offset was the derivative of the energy lost to the load and friction which was assumed constant. Therefore, by finding this offset, the piston work could be calculated as the energy lost to load and friction for 120°.

The kinetic energy and power approximations included the oscillating engine parts as it was found that their contribution affected the kinetic power curve noticeably much. This can be seen in Figure 13 where the piston power is compared to the kinetic power estimated for only the rotating parts and estimated for both the rotating and the oscillating parts; for the

operating point at 800 rpm and 50 % load. As it can be seen, the shape of the kinetic power curve matches the shape of the piston power curve better when the contribution of the oscillating masses is considered. To verify whether the oscillating masses had a significant effect, the RMS of the difference between the derivative of the piston power trace and the derivative each kinetic power trace were calculated respectively and compared. For the particular operating point at 800 rpm and 50 % load, a reduction of around 8 % of the RMS of the difference could be seen from including the oscillating masses. This means that the shape of the kinetic power trace is more similar to the piston power trace as the difference between the derivatives of the kinetic and piston power traces is reduced when the oscillating masses are considered. The conclusion is that the oscillating masses have a notable effect that has to be included in the kinetic energy and power calculations for better accuracy.

Figure 13. Piston power, kinetic power of rotating engine parts and kinetic power of both rotating and oscillating parts plotted during one engine cycle at 800 rpm and 50 % load.

The power transferred to the crankshaft from the pistons calculated from the isentropic pressure curves and the derivative of the kinetic energy of the engines moving parts can be seen in Figure 14.

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Figure 14. Kinetic energy compared to piston work (top) and derivative of kinetic energy compared to power from pistons and approximated piston power using the isentropic pressure curves.

In the figure it can be seen that the shape of the derivative of the kinetic energy is similar to the total power from the pistons but with the offset. This offset is caused by load and friction which approximately is constant resulting in the approximately constant power lost to friction and the load on the engine. This explains why the total piston energy curve has the same shape, in terms of location of peaks and valleys, as the kinetic energy but is increasing constantly unlike the kinetic energy that stays on a constant level. It can also be seen that the approximation of the power from the pistons using the isentropic pressure curves shows good correlation at the intervals without combustion in any cylinder. The accuracy of the isentropic pressure curves at the moments without combustion is important as it directly influences the work estimate. As the isentropic pressure curves only represent the in-cylinder pressure curve for unburnt and fully burnt injected fuel, the equivalent piston work during combustion is not reliable, this is however not a problem as there is no combustion at the locations where the cylinder torque is cancelling.

After the kinetic and piston power curves had been calculated, the locations with zero piston power were numerically determined. Analytical expressions for determining the locations were derived and can be seen in the appendix. These quickly became complex and non-linear due to the trigonometric relations of the crank mechanism and the expression could not be solved for the angle. These points can be seen in Figure 15.

Figure 15. The kinetic power trace and the piston power trace calculated for the operating point at 800 rpm and 50 % load with indicated points that where numerically determined to retrieve the power lost to the load, friction and auxiliary systems. Number one shows the first points that were determined where the total piston power is zero and number two shows the points where the power loss is finally retrieved.

The kinetic power trace has perturbed oscillations during the deceleration after and during late combustion. From inspecting Figure 14, this can be seen in the intervals -80 to -20 CAD and 50 to 100 CAD. Further, in Figure 15 it can be seen that the locations where the piston power cancels are within these intervals heavily influenced by the oscillations. These oscillations can give deviations amongst the indicated work estimations for the six locations where the torque from the cylinders cancel; which can be seen in Figure 16.

Local kinetic power averaging for improved indicated work estimation

As the crankshaft torsion causes oscillations in the kinetic power trace, a linear regression on the form of a least square fit was done for the kinetic power trace during the moments of deceleration of the crankshaft. The torsional oscillations effect on the indicated work approximation could then be avoided to a certain extent and the estimation could be improved. As it can be seen in Figure 16, the partial traces of deceleration in the kinetic power trace obtained for the operating point at 800 rpm and 50 % load, have been linearly fitted. The points where the piston torques cancel were then numerically determined on the linearly fitted lines to attain the power loss during the deceleration after combustion in each cylinder. The average value of the six power loss estimates was calculated as the final power loss estimate. This method gave improved estimations of the summed power of the losses and the load as the torsional oscillations could be avoided to a certain extent.

Crank angle [CAD]

Crank angle [CAD]

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Figure 16. Local linear regression of kinetic power to avoid the torsional effect on the indicated work estimation for a full cycle at 800 rpm and 50 % load.

This can be more clearly seen in the zoomed plot of the deceleration during late expansion of cylinder 3 and 6 which can be seen in Figure 17.

Figure 17. Local linear regression of kinetic power trace zoomed in at deceleration at late expansion of cylinders 3 and 6.

It was found that the indicated work estimate was improved by the local averaging, especially for the operating points at lower speeds. As the final work estimate was the mean value of the estimates from each deceleration after combustion in each cylinder, see the indicated markers in Figure 16, the mean error and standard deviation of the six estimates for each operating point was calculated. It was found that the deviation between the six estimates was reduced after the linear regression. By introducing local averaging, the mean value of the standard deviation of the work estimates for each operating point above 0 % load could be approximately halved. The same was found for only the operating points at 800 rpm although the deviation was lower both before and after averaging compared to the deviation for all operating points combined. This is an improvement as the different estimates for each operating point deviated significantly before the averaging as a result of the large difference in torsional effect of each cylinder. A further discussion on the nature of the torsional effect will follow. The piston work from each cylinder is however expected to mutually deviate somewhat due to differences in injected fuel mass, cylinder trapped mass,

cylinder volume etc. Such differences is of course of interest to estimate. However, the large difference in the work estimate is caused by torsion, which is clearly seen in Figure 16 as the difference in oscillating amplitude during the deceleration. That difference causes an error to the work estimate which is not desirable to include which is why the reduced deviation between the estimates for each operating point is beneficial.

The accuracy of the indicated work estimate is presented as the relative error to the actual work calculated from integrating the measure pressure over the calculated volume which is presented in Figure 18. It can be seen that the error for the higher speeds is very large and the indicated work estimate is therefore not reliable. There are two main causes of the unreliable estimate at higher speeds. Partially it is because of the large torsional effect on the kinetic power trace; which is highly subjected to oscillations as a consequence. Further, the isentropic expansion trace will be offset as the partial pressure trace from the dynamic crankshaft model is not accurate at higher speeds; which is also an effect of the torque model’s inability to describe the torsion accurately enough. The unreliable work estimate will in turn reduce the accuracy of the isentropic expansion trace that is improved by the exhaust pressure correction which is based on the work estimate; the sequence of the sub models can be seen in Figure 1. The isentropic expansion trace can therefore not be improved from the indicated work estimate meaning an iterative approach to improve the estimation is also not possible. For the lower speeds, especially at 800 rpm, the estimate is much more accurate with a mean absolute error of only around 15 J.

Figure 18. Relative error of the indicated work estimates where local averaging has been used and a mean value for all cylinders has been calculated and the measured pressure from cylinder 6 has been used to calculate the reference indicated work.

For comparison, the indicated work estimate error was calculated relative the equivalent work to the measured pressure in cylinder 1. The indicated work error for cylinder 1 can be seen in Figure 19.

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Figure 19. Relative error of the indicated work estimates where local averaging has been used and a mean value for all cylinders has been calculated and the reference indicated work has been calculated from the pressure from cylinder 1.

From comparing the work estimate errors in Figure 18 and Figure 19 it can be noted that the work is overestimated for the lower speeds and generally a little less underestimated for the higher speeds for cylinder 1 compared to cylinder 6. The reason is that the pressure within cylinder 1 was larger for the operating points meaning more work was generated from cylinder 1. Since the work estimate is averaged for all cylinders, the value of the error is increased for cylinder 1. The estimation error has thus increased for the operating points at 800 rpm but decreased for the other operating points. This highlights that assuming the same indicated work from all cylinders introduces an error as the pressure traces for the cylinders will not be identical in reality.

Including energy stored due to torsion

As the energy stored due to torsion affects the shape of the kinetic energy curve, the indicated work estimation could be improved if the kinetic power curve could be corrected with the torsional power. This is especially important for operating points with larger engine speeds where the torsional oscillations affect the kinetic power curve a lot. As the MISO system describing the crankshaft dynamics continuously calculates the relative displacement of the nodes, the energy stored due to torsion could be estimated using Equation 16. In Figure 20, the displacement of each node relative to the flywheel is presented. Further, the equivalent torsional power together with the kinetic energy calculated from the measure angular velocity of the crankshaft can be seen in Figure 21.

Figure 20. Angular displacement between each node and the flywheel during one cycle at 800 rpm and 50 % load calculated from the output of the crankshaft state-space system.

Figure 21. Estimated torsional power, kinetic power from the crankshaft model and the measured angular velocity and the piston power calculated from measured in-cylinder pressure for engine operation at 800 rpm and 50 % load.

As it can be seen in Figure 21, the torsional power calculated from the displacements output from the MISO system seems to be offset compared to the torsional oscillations within the kinetic energy calculated from the angular velocity. This is believed to be an effect of inaccuracies in the model properties describing the flexing behaviour of the system. In the chapter discussing the dynamic crankshaft model and its ability to describe the system a further discussion on the models torsional properties will follow. It can also be seen that the torsional power is larger after combustion in the cylinders further away from the flywheel (1, 2 and 3) compared to the ones closer. The reason to this is that as the cylinders further away from the flywheel suddenly produces a lot of torque as the combustion occurs, a larger part of the crankshaft will be distorted. Hence more energy should naturally be bound in the crankshaft due to the torsion. As the estimated torsional power seemed to be offset, the trace was delayed before adding it to the kinetic power approximation. The resulting trace can be seen in Figure 22.

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Figure 22. Offset torsional power and resulting power trace from addition of the kinetic power from measured angular velocity and the torsional power trace for engine operation at 800 rpm and 50 % load.

The power trace resulting from adding torsional energy to the kinetic energy trace seems to have been improved at parts. However, the torsional estimation does not seem to capture the actual torsion perfectly as the trace is degraded during the expansion stroke of for example cylinder 1. The indicated work is estimated at the locations where cylinder 3 and 6 expands. These partial traces seem to have been improved slightly which can give a better indicated work approximation.

Determination of pressure trace iteratively using a dynamic crankshaft model

In an attempt to avoid the torsional oscillations of the partial pressure trace from the rigid crankshaft model, the pressure trace was iteratively adapted as an input to the dynamic crankshaft model. The model includes torsion along the crankshaft and the goal was to verify if the oscillations could be removed from the pressure trace if the model could be used for the pressure estimation.

Iterations of the crankshaft model to adapt a full pressure trace was found to be computationally heavy and time consuming. Nonetheless, it was of interest to verify if more accurate pressure trace could be obtained from the crankshaft model. For such a case, optimising the iterative approach or further studying the possibilities of inverting the MISO system would be more motivated.

The starting trace for the optimisation was the motored isentropic pressure trace which was then iteratively changed within a limited interval until the best acceleration trace match was obtained. The boundaries and resolution of the adapted trace had a significant effect on the end results. The pressure trace interval limits was the motored pressure as a lower limit and the fully burnt isentropic pressure trace added with 5 bar as the upper limit of the expansion and the motored pressure added with 10 bars during the compression; which gave the best results. The pressure trace for the case where only the pressure trace of cylinder 6 was adapted such that the model output acceleration matched the measured acceleration can be seen in Figure 23.

Figure 23. Resulting pressure trace of cylinder 6 from iterative calculation until the same acceleration as measured was reached. The measured pressure and the trace from the stiff crankshaft model can be seen for comparison.

As it can be seen, the trace is partly improved for the compression and the singularity is avoided. However, the torsional oscillations remain in the trace, especially clear during early compression and late expansion. Many trials were conducted where the pressure trace of all cylinders were adapted in the firing order and other trails where the adaption was repeated multiple times and others where the cylinders were phased continuously with the calculated twist of the crankshaft. None of the trial gave better results as the oscillations still remained in the pressure trace.

This could either be a consequence of an inability of the dynamic crankshaft model to describe the torsional behaviour of the crankshaft or another phenomenon is causing the oscillations at these locations.

Heat release trace adaption

The results from the heat release models are included in this section.

SOC interval from isochoric and isobaric heat release

By analysing the effect of location of SOC on the piston work for the isobaric and isochoric heat release, the limits to SOC for the actual heat release could be defined. It was found that, for the operating point at 50 % load and speed of 800 rpm, the resulting pressure from isobaric heat release could not deliver enough piston work for any location of SOC. On the other hand, the isochoric heat release could deliver more piston work than the actual work for the operating point within a certain interval. Piston work as function of SOC for the two models can be seen in Figure 24.

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Figure 24. Piston work as function of SOC for isochoric and isobaric heat release respectively for the operating point of 800 rpm and 50 % load. The actual gross indicated work calculated from the pressure measured in cylinder 6 shows that only the isochoric heat release can give enough work for this operating point.

As it can be seen in the Figure 25, the SOC could be chosen such that the pressure trace from the isochoric heat release resulted in the same piston work as estimated beforehand. The maximum work for the pressure trace from the isobaric heat release was obtained for SOC at TDC which was the closest value to the actual work. This was expected as the Otto cycle has a larger theoretical efficiency than the diesel cycle. This can be seen in Equation 2, cp will always be larger than cv which means that the temperature after combustion will be much lower after isobaric heat release rather than isochoric. The resulting pressure curves for the two respective types of heat release and SOC can be seen in Figure 25.

Figure 25. Pressure trace from isochoric heat release resulting in the same piston work calculated from the measured pressure trace (left)

and the trace from isobaric heat release giving the maximal piston work which is still below the actual work.

The real combustion will neither be isochoric nor isobaric, rather will it have a certain trace somewhere in-between the two. By calculating the SOC that results in the same amount of piston work as the actual indicated work for the two types of heat release, an interval of possible SOC locations for the actual heat release trace is retrieved. As the isobaric heat release would not give enough work for any SOC, only the isochoric heat release limits the SOC.

Anyhow, as shown in these results, the estimated pressure trace shows some clear improvements compared to the pressure trace from the crankshaft model. Even if the heat release is not very accurate, the singularity is avoided while the more accurate partial pressure data before and after combustion can be maintained giving a full pressure trace of the closed cycle. The next step was to further analyse if more realistic heat release traces could be derived for improved results.

Total heat release estimate through assumed normal distributed heat release rate

To be able to estimate the heat release trace to improve the pressure trace, the total accumulated heat release had to be estimated. It was found that the approach of using the derived isentropic pressure curves together with a rough approximation of the heat release to estimate the accumulated heat release iteratively requiring the resulting indicated work to match the work estimated since before gave best results. This rough heat release approximation was also necessary to be able to estimate the wall heat losses and was also useful for calculating the cylinder volume with correction for deformations. The relative error of the total heat release estimation for a number of operating points can be seen in Figure 26. The error was defined as the difference between the total heat release estimation and the actual total heat release calculated using the energy equation and the measured pressure trace.

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Figure 26. Relative error of the total accumulated heat release estimation obtained through assuming a normal distributed heat release and requiring the resulting pressure trace to correspond to the same indicated work as estimated beforehand.

As it can be seen in Figure 26, the error distribution and dependence on the speed correlates directly with the error of the indicated work estimation (Figure 18). This is natural as the total heat release is estimated through the requirement of the same resulting indicated work. This means that any errors in the indicated work estimate will also directly influence the total heat release estimate with an error. As the indicated work estimate was highly erroneous for higher speeds and more accurate for lower speeds, so is the total heat release estimate.

Volume correction for deformations

As the cylinder volume directly influences the actual in-cylinder pressure, the estimation of the cylinder volume was corrected for deformations as it from the beginning was estimated purely geometrically. This corrected volume could give a more accurate pressure curve and was therefore attempted. Through the work by West [23], the rough full pressure trace could be used with assumed values of the temperatures of the cylinder block, cylinder liner, crankshaft, piston and cylinder head for thermal expansion estimation. It was found that the volume including deformations deviates quite a lot relative to the geometrical volume estimation. To validate how well the rough pressure estimation could be used for the volume correction, the resulting volume trace difference to the solely geometrically calculated volume was compared to when the measured pressure was used instead, which can be seen in Figure 27.

Figure 27. Relative difference between the volume trace corrected for deformations and the geometrically calculated volume trace with the rough pressure trace estimation and with the measured pressure for comparison for cylinder 6 at 800 rpm and 50 % load.

The corrected volume trace shows a smaller volume during compression than calculated geometrically. This is an effect of the thermal expansions of the engine components which was considered since the engine was operated in a warm state during the experiments. As the pressure increases during combustion, the estimated volume increase relative the geometrical volume. During warm engine operation, the volume displacement from pressure and inertial forces is counteracted by the thermal expansions [23]. If the volume is corrected for only deformations from pressure and inertial forces or calculated for cold engine operation, the volume estimation would be larger than the geometrically calculated volume. The volume trace corrected for deformations deviates from the geometrically calculated volume the most around TDC; in relative terms. As it can be seen in Figure 27, the volume trace calculated using the estimated pressure trace follows the trace calculated from the measured pressure fairly well but deviates slightly around the combustion zone. The deviation between the two traces is still far smaller than the deviation to the geometrically calculated volume which is an improvement. The effect of the volume estimation is further studied in the tolerance analysis.

One Wiebe function fit to late combustion pressure

The next step of trying to improve the pressure trace was to describe the heat release with a Wiebe function as the literature had pointed out the function to describe the burn rate of diesel combustion well. Due to the torsional oscillations within the pressure signal, the calculated heat release from the pressure trace from the crankshaft model became irrational as sudden pressure drops and peaks during the expansion corresponds to heavy heat release fluctuations. Fitting a Wiebe function to the irrational heat release information did not give accurate results.

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Instead, one Wiebe function was fitted such that the equivalent late combustion pressure derivative was adapted to the pressure derivative trace from the crankshaft model. Through this method, the shape of the diesel combustion heat release is maintained which does not allow for the pressure derivative fluctuations, but it is still adapted to the partial pressure derivative trace such that the late combustion information is kept. The resulting pressure derivative fit from the operating point at 800 rpm and 50 % load can be seen in Figure 28. This allowed for avoiding the pressure oscillations caused by the torsion while adapting the Wiebe function.

Figure 28. Adapted pressure derivative from parametrising one Wiebe function to fit the pressure derivative data from the crankshaft model in a least-square sense.

The resulting heat release rate trace from the one Wiebe function fit together with the cumulative heat release and the resulting pressure trace can be seen in Figure 29 for the operating point at 800 rpm and 50 % load. The estimated in-cylinder temperature, wall heat losses and the PV-diagram for the same cylinder and operating point can also be seen in Figure 29.

As it can be seen, the late combustion heat release rate trace resulting from the one Wiebe function fit correlates well with the measured trace during the late combustion. This was expected as one Wiebe function is adequate to describe either the premixed combustion or the diffusive combustion. As the function is fitted to HRR data of only the late combustion, it captures the heat release during the diffusive combustion.

However the heat release during the premixed combustion is missed out. This can be seen in the pressure trace in the bottom left plot in Figure 29, where the measured pressure is subjected to a fast pressure increase shortly after TDC which is not the case for the single Wiebe function. It is therefore necessary to include as second Wiebe function to describe the early combustion. For the second Wiebe function fit more information of the early combustion event is needed. The SOC interval limited by the isochoric and isobaric heat release was used during the Wiebe fit but it was not enough information to describe the early combustion. It can also be noted that the measured heat release trace has a small increment before TDC which is the typical effect of the pilot combustion. Neither the one nor the two Wiebe function fit can capture this combustion event neither and a third Wiebe function is needed

Figure 29. Resulting HRR, cumulative HR, pressure traces, in-cylinder temperature, wall heat losses and PV-diagram from one Wiebe function fit such that the resulting pressure derivative trace matched the partial pressure trace from the crankshaft model and from two Wiebe function fits such that the resulting indicated work matched the estimated for cylinder 6 at the operating point of 800 rpm and 50 % load.

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if it is to be included. Information about the pilot combustion is also needed for the third fit in that case.

Further on, comparing the resulting pressure trace to the measured trace, it can be seen that the estimation from the Wiebe function fit to the estimated late combustion heat release rate is much more accurate than from assuming isochoric or isobaric heat release. The pressure, like the heat release trace, during the late combustion correlates, of course, better than the pressure during the pilot and the premixed combustion. The same applies for the temperature estimation that correlates well with the temperature calculated from the measured pressure before and after the early combustion. The wall heat losses estimation is mildly affected by the lack of an accurate description of the early combustion.

The one Wiebe function fit was done for all the operating points run with diesel fuel to determine the performance of the approach. To see how speed and load affects the accuracy, the combustion parameters maximum pressure, crank angle at maximum pressure, CA10, CA50 and CA90 was extracted. The parameters errors relative to the actual values calculated from the measured pressure were then plotted for the operating points.

The errors of the maximum pressure estimates can be seen in Figure 30. It was found that for higher speeds, the maximum pressure estimation accuracy seem to be reduced. This is most probably because of the reduced accuracy of the late combustion pressure trace obtained from the crankshaft model, which is caused by the inability of the rigid crankshaft model to describe the torsional effects. Another cause is probably the reduced accuracy of the estimate of the indicated work at higher speeds which affects the total heat release estimation and thereby the accumulated heat release trace and the resulting pressure trace.

Figure 30. Relative error of the maximum pressure estimate from the one Wiebe function fit (red) and from the two Wiebe function fits (blue) for a number of operating points and for cylinder 6.

The most accurate results were obtained for the operating points at 800 rpm. The accuracy of the maximum pressure crank angle location estimation can further be seen in Figure 31. It can be seen in the figure that the most accurate crank angle estimates are retrieved at the low speed operating points; which is the multiples of 4. It can be noted that operating point 3, 5, 6 and 14 are all close to TDC. This is seemingly because of the inaccurate total heat release estimate. From examining Figure 26 it can be noted that the total heat release estimates for these specific operating points are very imprecise.

Figure 31. Crank angle location of maximum pressure estimation compared to the actual location extracted from the measured pressure trace of cylinder 6 at the operating points 1 to 16. Where each 4 operating points in order are descending in speed and each group of 4 points are descending in load.

Further, with the heat release trace modelled the CA10, CA50 and CA90 could be estimated. The estimates were compared for the different operating points to the reference values retrieved from calculating the heat release from the measured

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pressure with the energy equation, which can be seen in Figure 32.

Figure 32. CA10, 50 and 90 estimates for the operating points 1 to 16 together with the actual values calculated from the heat release trace derived from the measured pressure trace of cylinder 6.

As it can be seen in the figure, the CAx estimates accuracy is also reduced for higher speeds. For the operating points at 2000 rpm (operating point 1, 5, 9 and 13), all CAx estimates are unreliable. All CAx estimates are most accurate for the operating points at 800 rpm, CA10 and 90 seem to be generally a little early while CA50 seem to give best results but generally a little late estimates. It can also be noted that the CA90 estimate gives too early predictions for higher speeds while CA10 and CA50 seem to give too late predictions. As the late heat release parametrisation is directly dependent on the accuracy of the expansion pressure estimate, the reduced accuracy of the pressure trace for higher speeds explains the reduced accuracy of the CAx estimates.

This was not expected as the Wiebe function was adapted to the partial pressure trace from the late combustion. It was anticipated that the late combustion heat release would be captured well and would therefore result in an accurate CA90 estimate. A plausible explanation to why this is the case is that the CA90 value is heavily dependent on the value CA50. Even if the shape of the heat release trace is captured well during the late combustion, if the estimate of amount of combusted fuel before the late combustion is inaccurate, so will the CA90 estimate be. This could explain why a correct estimate of CA90 was made for operating point 12, where also the CA50 estimate was accurate. If CA50 is accurate and a good fit of the heat release trace for late combustion is made, the CA90 accuracy is improved.

Two Wiebe function fit

The two Wiebe function parametrisation was challenging as a result of limited data about the premixed combustion. The premixed Wiebe function fit needed a number of constraints in

order to retrieve adequate results. It was found that the SOC needed to be constrained because heat release before TDC will give a negative contribution to the indicated work which is why an adaption without constraints allows for too early SOC which can still give the same indicated work. Additionally, only the indicated work error is not enough information to adapt both the shape of and the duration of the premixed combustion. Therefore, best results were obtained when the premixed combustion SOC and duration were estimated beforehand. The most accurate heat release fits were obtained for the operating points at 800 rpm. The resulting heat release rate, cumulative heat release and the pressure trace for the operating point at 800 rpm and 50 % load can be seen in Figure 29 together with the one Wiebe fit for comparison.

As it can be seen in Figure 29, the introduction of a second Wiebe function improved the estimation of the premixed combustion for the particular operating point. Including a description of the premixed combustion smooths out the instantaneous heat release at SOC for the single Wiebe fit. However, only the indicated work was not enough information to adapt the function.

The introduction of a second Wiebe function brings an additional three parameters for the function and a weighting factor giving a total of 7 parameters to fit. It was therefore necessary to estimate SOC and duration of the premixed combustion prior to the adaption. The estimation of SOC for the kinetic power trace gave fairly good approximations for some operating points but was found to not be fully accurate for all operating points. The start of combustion was therefore estimated using the model for ignition delay and the ECU SOI parameter.

To visually demonstrate the difference between operating points, the resulting heat release and pressure for a load sweep at 800 rpm and a speed sweep for the load of 50 % is presented in Figure 33.

It was found that the estimation of the indicated work, the total heat released and SOC and duration of the premixed combustion affects the accuracy of the pressure estimation significantly. Therefore, the accuracy is reduced a lot for the operating points with inaccurate indicated work estimations; the operating points with higher speed. This is clear for the operating point at 2000 rpm and 50 % load (light green in the right plots in Figure 33) where the total heat release estimate is far too low, which complies with the results presented in Figure 26. This in combination with a less inaccurate partial pressure derivative trace from the crankshaft model results in an instantaneous heat release very early to reach the total heat release early which is not the case at all. For the lower speeds, each sub model gives more accurate estimates and the final traces are therefore also more accurate. In the case of the operating point at 800 rpm and 50 % load, it can be seen in the figure (blue lines in right plots), that the total heat release is far more accurate, together with a more accurate partial pressure derivative trace, the shape of the late combustion is

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maintained as well. With the more accurate indicated work estimate, also the premixed combustion is fairly accurately described as well, although it starts a little late but then increases quickly, seen in the heat release rate in the top right plot.

The output combustion parameters were also studied and compared to the reference data. The relative maximum pressure error was not clearly reduced for the double Wiebe parametrisation compared to the single Wiebe fit which can be seen in Figure 30 presenting the relative error of the maximum pressure estimation.

The resulting estimates of the location of the maximum pressure can be seen in Figure 34. A likewise plot of the locations of CA10, CA50 and CA90 can further be seen in Figure 35.

Figure 34. Crank angle location of maximum pressure estimation from a number of operating points from the two Wiebe function fit for cylinder 6.

Figure 33. Resulting heat release and pressure traces from the one and two Wiebe function fits compared to the measured traces for a load sweep for steady operating points at 800 rpm in the plots to the left and equivalent plots from a speed sweep for steady operating points at 50 % load. The traces originating from the one Wiebe function fits are dashed.

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Figure 35. Estimates of CA10, 50 and 90 from the two Wiebe function fits to late combustion pressure data and indicated work estimate compared to the actual values calculated from the measured in-cylinder pressure.

The second Wiebe function fit does not improve the estimation of the combustion parameters at higher speeds. This is because the accuracy of the indicated work estimation has a direct effect on the combustion parameter estimation accuracy. Introducing more accurate heat release assumptions does not compensate for this error. Since the indicated work estimations for higher speeds is inaccurate, so is the heat release and pressure reconstruction. The CA10, CA50 and CA90 follow the same trend as for the one Wiebe function parametrisation case. It can be noted that the estimates of CA10 is generally more accurate for the two Wiebe function fit compared to the one Wiebe function fit. The mean absolute error of the CA10 estimate for all operating points was reduced by introducing a second Wiebe function fit. This seems reasonable since the introduced second Wiebe function

is meant to describe the early premixed combustion which would directly influence the CA10 values.

Nevertheless, the accuracy for the low speed operating points, at 800 rpm has been improved. By analysing the results it was concluded that for the lower speed the virtual pressure sensor can fairly accurately measure the maximum pressure.

By introducing the estimated indicated work in the adaption of the Wiebe functions, the final pressure trace for the cylinder cycle can be improved. This shows that by extracting as much information about the combustion as possible from the partial pressure trace, the measured angular velocity and by making assumption of the heat release the singularity within the crankshaft model output pressure can not only be avoided but also calculated fairly accurately. The singularity was avoided already during the first heat release estimates, which the isochoric and isobaric heat release assumptions. This gave improvements around TDC compared to the partial crankshaft model pressure trace, although significant deviations were still present due to the absence of proper validity of the heat release models.

Improvements were observed as heat release assumptions more valid to diesel combustion were taken. Introducing a Wiebe function fit gave more accurate pressure traces. At least for the low speed operating points where the partial pressure traces were not heavily subjected to torsional oscillations. Utilising the partial pressure trace from the mechanical crankshaft model to optimise a heat release trace within a thermodynamic model shows how these two models can be combined. It also shows how the data from different domains can be utilised and combined to arrive at a final result with less information loss. This was further highlighted as the two Wiebe functions fits were done where the estimated indicated work was included which originated from the angular velocity measurement.

The conclusion was that the more valid the heat release assumption was, the more accurate the final pressure trace would be for the operating points where the partial pressure trace and the angular velocity was not highly perturbed with oscillations; which was the lower speed operating points.

Tolerance analysis

To analyse how sensitive the final virtual pressure model is to inaccuracies within input parameters and choice of models for certain calculations within the virtual sensor model, a tolerance analysis was done. The virtual sensor used the double Wiebe parametrisation approach. The input parameters of the model that were expected to have a significant effect on the model error were defined and varied between the two realistic deviations (upper and lower limit) that can be expected for the parameters. The input sensor measurements of boost pressure, boost temperature, exhaust temperature and crank angle position can all contain measurement errors and are important parameters used for the calculations of the

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virtual pressure sensor and were therefore included in the analysis. The virtual sensor error for the tolerance analysis was defined as the difference between the output combustion parameter estimates when the altered input values or models were used and the output estimates when the nominal input values were used. Both the maximum error and the RMS errors were calculated. The analysis does not show the actual virtual sensor error compared to test data but the error introduced to the model by altering the input values or the sub models.

As the induction sensor, measuring the crank angle position can be mounted slightly offset, the crank angle position can contain a constant delay. The tolerance for the mounting angle of the sensor is ±0,5 ° and with a maximum delay of 0,1 ° of the induction sensor the maximum tolerance for crank angle delay is ±0,6 ° [19] which can be the delay of the CAD signal and was therefore tested in the sensitivity analysis. Further, a typical boost temperature sensor has an accuracy of ±2 K and boost and exhaust pressure sensor have an accuracy of ±0,05 bar [6]. Therefore, these assumed tolerances were used for the pressure and temperature sensitivity analysis. The resulting RMS errors can be seen in Figure 36. The resulting maximum errors were similar and therefore not included in the figure for clarity.

Figure 36. Root mean square error of estimates of combustion parameters from nominal values to examine the sensitivity of the virtual pressure sensor model for all operating points with load above 0% for cylinder 6.

As it can be seen in the results from the tolerance analysis, the choice of specific heat ratio (γ) affects all the results significantly. The temperature dependent specific heat ratio model does not differ notably much from the constant specific heat. It seems like an error is introduced in the two cases with other specific heat ratio models not considering the gas composition, unlike the NASA model which includes an estimation of the mass fraction burnt.

The choice of specific heat ratio modelling gave largest deviations, although the crank angle delay also had a large influence on the accuracy of the virtual sensor. It can be

observed that the crank angle delay influences the CAx estimates in the same order as the specific heat ratio models. The maximum pressure and total heat release was not affected as much.

The temperature and pressure sensor reading in the intake and exhaust manifold also has an effect on the virtual sensors accuracy. The pressure sensors have a larger effect on the maximum pressure estimate than the temperature sensor; the same applies for the total heat release estimate. The boost temperature has a likewise effect on the crank angle estimate of max pressure location and CAx as the pressure sensor readings.

Regarding the crank angle combustion information, CA90 is affected the most through the introduced errors. Johansson et. al also came to the conclusion that the CA90 estimate was most sensitive to errors within the model developed in the study [6], which agrees with the results. The reason why the input measurement errors have a large effect on all estimates is because the measurements all affect the isentropic pressure curves that are in turn used throughout the virtual sensor’s different calculations. All the inputs and models tested in the tolerance analysis are key parameters within the virtual sensor and their accuracy and validity directly affects the accuracy of the virtual sensor.

An important discovery is that large errors of the estimates arise as a consequence from not correcting the geometrical volume for deviations from deformations. This indicates that the volume estimate has a large impact on the end result. An accurate volume trace estimation is therefore necessary for accurate results which implies that considering the deformations of the engine components for volume estimation is vital to acquire the most accurate results possible.

As it was found that inaccurate results were obtained at higher speeds, the equivalent tolerance analysis results for only the operating points at 800 rpm are presented in Figure 37. The virtual sensor proved to be most valid for the low speed operating points and it is therefore of interest to test the sensors sensitivity for those specific points only.

RM

SE

PmaxCA PmaxCA10CA50CA90

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Figure 37. Root mean square error of estimates of combustion parameters from nominal values to examine the sensitivity of the virtual pressure sensor model for to operating points at 800 rpm and load above 0 % for cylinder 6.

From comparing the low speed tolerance analysis results with the results for all speeds, it can be determined that the virtual sensor is generally less sensitive to input errors at 800 rpm than at higher speeds. All estimate errors from all tested input measurement errors and models except the crank angle delay are significantly lower for the low speed points. The crank angle delay introduces errors of the same order for the low speed points. The reason why the specific heat ratio models do not give as large errors for the low speed operating points is probably because the models are more valid to the temperature range and gas composition present in the cylinder for the lower speeds.

The results suggests that the accuracy of the virtual sensors input measurements and sub models is not as critical if the sensor is only to be used for low speed operating points. If the full range is to be covered, however, the accuracy of all input parameters is of vital importance for the accuracy of the virtual sensor. Further on, as the CA delay has a large effect on all operating points it could be advantageous to reduce the tolerance for the mounting angle of the induction sensor. A possibility is also to correct for the offset if it can be measured or determined beforehand.

The sensitivity of the virtual sensor’s accuracy of total heat release estimation for lower loads is another area of interest as it could potentially be useful for misfire detection. Therefore, the equivalent sensor’s total heat release estimation errors from the introduced input errors for the operating points at 25 % load and 0 % load are presented in Figure 38.

Figure 38. The virtual sensor's sensitivity of total accumulated heat release estimation to input measurement and submodel errors and inaccuracy for the operating points at 25 % and 0 % load, calculated for cylinder 6, and the difference between these errors.

As seen in the table, the sensitivity of the virtual sensor for operating points with 25 % load and 0 % load is of the same order, although the resulting errors for the points at 25% load is generally a little larger. The sum of the maximum errors is significantly large, however the root mean square value is not as high compared to the actual heat release values. Like discussed earlier, the torsional oscillations give large error for the higher speeds, so if that problem can be further reduced, this approach of heat release estimation could potentially be used for misfire detection.

Figure 39. Summed error distribution for the different speeds of the operating points at 25 % load and 0 % load clearly presenting the sensitivity dependence on angular speed.

To further analyse the model’s sensitivity dependence on operating speed, the summed error for the operating points at 25 % and 0 % load for the different speeds are presented in Figure 39. It can clearly be seen that the model is less sensitive for the lower speed which agrees with the earlier findings. This means that attempts of detecting misfire by estimating the total heat release using the presented method would be most reliable for lower operating speeds.

RM

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PmaxCA PmaxCA10CA50CA90

RM

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Qtot 25

Qtot 0

Diff

200016001200800

Sum

med

err

or [

J]

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25 % load0 % load

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Conclusions

Through this study, it has been shown that a partial pressure trace, obtained from a torque model of the crankshaft, can be complemented by modelling the heat release from the combustion such that a full pressure trace for the closed-cycle is retrieved without a singularity around TDC. The choice of heat release modelling and the combustion information that can be established affects how accurate the final pressure trace becomes.

It has been shown that the indicated work can be estimated from the angular velocity measurement by estimating the kinetic power and determining it at locations where the total piston power becomes zero as the torques from the cylinders cancel. The indicated work is very beneficial to know as it contains averaged information of the pressure traces from the cylinders and therefore averaged information about the combustion. There was a deviation between the six indicated work estimates from each cycle as a consequence from the torsional effect on the crankshaft resulting in perturbed oscillations in the angular velocity trace. Through local averaging of the kinetic power trace during the crankshaft decelerations the standard deviation for all operating points above 0 % load was halved. It was also attempted to estimate the torsional power trace using a dynamic crankshaft model to be able to remove the oscillations from the kinetic power trace. The final power trace could be improved at parts but the oscillations could not be removed fully. The conclusion is that for this specific purpose, the dynamic crankshaft model cannot perfectly describe the torsional behaviour of the crankshaft. Anyhow, the indicated work estimate was most accurate for operating points at lower speeds. The error becomes very large for higher speeds and is therefore regarded as unreliable for those operating points.

With an estimate of the indicated work, the total accumulated heat release could be approximated. The most accurate results were retrieved by assuming the heat release rate to be normal distributed and by estimating the SOC and assuming a combustion duration to be able to iteratively determine the total heat release necessary to achieve the same indicated work as estimated beforehand. The resulting total heat release estimate accuracy correlated directly with the accuracy of the indicated work estimate indicating that if the indicated work can be estimated accurately, so can the total heat release.

Different heat release modelling approaches were attempted to study how accurate the virtual sensor could estimate the pressure. By assuming the heat release to be isobaric and isochoric a full closed-cycle pressure trace could be obtained after calculating isentropic pressure traces before combustion and for the gas mixture after full combustion. It was found that correcting the exhaust pressure measured with the exhaust pressure sensor to get a more accurate value of the in-cylinder pressure at EVO was necessary for an accurate isentropic expansion trace. Further, the specific heat ratio model based on NASA polynomials gave the best results. The resulting

pressure traces were not very accurate within the combustion zone, although as they represent the limiting cases regarding heat release, the latest possible SOC could be determined in the case of the isochoric heat release. This could be done by comparing the resulting indicated work from the pressure trace with isochoric heat release to the pre-estimated work.

To get improved pressure traces, fitting Wiebe functions could be done. The partial pressure traces from the crankshaft model can be used to adapt one Wiebe function describing the late combustion heat release. Further adapting two Wiebe functions to the late combustion heat release and such that the resulting indicated work matches the estimated work, even more realistic heat release traces can be obtained. Adapting two Wiebe functions demands some information about the premixed combustion though. The kinetic power can be used to roughly estimate the premixed combustion duration which together with a SOC estimate makes it possible to fit both Wiebe functions.

It was found that fitting of two Wiebe functions to describe the heat release gives the best results regarding deriving a full pressure trace. This demands good accuracy of the estimates of the indicated work, the total heat release, the partial pressure trace from the crankshaft and the SOC and duration of the premixed combustion. The mechanical crankshaft model includes information about the late combustion heat release that can be utilised in the thermodynamic model for heat release modelling either as a one or two Wiebe function fits. Best results were obtained for the lower engine speeds as the torsional effects on the angular velocity measurement and the partial pressure trace from the stiff crankshaft model is not as prominent and does not result in as large errors.

The best results for the final virtual pressure sensor were obtained at the operating points at 800 rpm. The errors quickly becomes larger for higher speeds as a consequence of the reduced accuracy of all sub models used prior to the final combustion parameter estimations. It was found that oscillations perturbing both the velocity trace and the partial pressure trace caused errors in the virtual pressure estimates. These oscillations originate from the torsion of the crankshaft and have a larger impact for operating points with higher speed. As the velocity trace is used for the indicated work estimate and the partial pressure trace is necessary for the late combustion HR estimation, the accuracy of the virtual sensor for higher velocities is low. Different attempts to remove the torsional oscillations were conducted. Using the dynamic crankshaft model, the torsional power was estimated. The power trace did not match the perturbed oscillations in the kinetic power trace through the full cycle and could therefore not improve the indicated work estimate. Analyses of the angular acceleration trace were also conducted in an attempt to study whether the torsional oscillations can be predicted beforehand from the crank angle measurements. No certain trends could be determined for the different operating points.

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Through the tolerance analysis it was found that the virtual sensor is most sensitive to the choice of specific heat ratio modelling. The upper tolerance of the induction sensor mounting angle, causing a crank angle delay also has a significant effect on the virtual sensors output parameters. Not correcting the cylinder volume for deformations was also found to have a notable effect which highlights the importance of estimating the volume trace well for improved accuracy of the sensor. The virtual sensor was significantly less sensitive for the lower speed operating points.

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Recommendations for future research

This study has included calculations within many domains that are related to the power flow within an engine. The virtual pressure sensor includes estimates of both the heat release from combustion to the final work and power output from the engine with estimates of heat losses and cylinder temperatures along the way. There is always possible to improve simulations in many ways by improving each of the estimates and improve assumptions such that they are more realistic. However, the accuracy of the final model can of course be improved by developing the partial estimations that have shown most errors and those areas are therefore recommended to study further for the next step in the development of the virtual pressure sensor and are listed below.

• Further analysis and modelling of the torsion could allow for correction of the torsional power in the indicated work estimation and the crankshaft model could either be improved or complemented to remove the torsional oscillations that perturb the output partial pressure trace. It has through the study been shown that the torsional effect is difficult to calculate and predict based on the angular velocity trace and the dynamic crankshaft model could not capture the torsion perfectly and its accuracy was reduced for higher velocities. If either an improved dynamic crankshaft model could be developed or a separate torsion model could be built, the virtual pressure sensor could as a result also be improved.

• Further development of the predictions of the premixed start of combustion and duration would allow for better possibilities of good parametrisations of two Wiebe functions to describe the heat release traces for different operating points. Further analysis of the angular velocity trace could be beneficial as ideally the premixed combustion parameters should be based on measurements and not phenomenological models as the combustion feedback should be as close to the actual combustion information as possible for closed-loop combustion control.

• If the sub models developed within this thesis can be improved from a more accurate torsion model and better premixed combustion SOC and duration estimates can be made, the next step of the virtual pressure sensor development is to analyse its performance for varying speed and load during operation. If adequate results can be obtained, it is then of interest to further study how the sensor needs to be developed to be valid for the full truck drivetrain. This would introduce even more torsional effects on the crankshaft dynamics that would need to be studied and included in the sub models.

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Notations Definitions/Abbreviations

cp Specific heat at constant pressure

cv Specific heat at constant volume

D Damping

J Inertia

K Stiffness

m Mass

mf Injected fuel mass

p Pressure

Qhr Heat released from combustion

QLHV The lower heating value of the fuel

Qw Heat transfer to cylinder walls

RMS Root mean square

RMSE Root mean square error

T Temperature

Tgas Gas Torque (other index indicates source of torque)

V Volume

xb Burnt mass fraction

γ Specific heat ratio

ηc Combustion efficiency

eh Angular velocity

ef Angular acceleration

θ Instantaneous crank angle

θcomb Crank angle at start of combustion

κ Polytrophic index

τch Chemical time delay

Φ Equivalence ratio

BDC Bottom Dead Centre

CA Crank Angle

CAD Crank Angle Degree

CAx Crank Angle at x % fuel burnt

CLCC Closed Loop Combustion Control

ECU Electronic Control Unit

EOC End Of Combustion

EVO Exhaust Valve Opening

HD Heavy-Duty

IMEP Indicated Mean Effective Pressure

IVC Intake Valve Closing

MCC Mixing-Controlled Combustion

ROHR Rate Of Heat Release

SOI Start Of Injection

TDC Top Dead Centre

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Appendix

Analytical expressions for cylinder torque cancelling

By setting the expressions for the torque from one cylinder in compression and one in expansion (offset 120 CAD between the cylinders) with isentropic relations, an analytical expression is obtained which is ideally solved for the angle. Either, the tangential forces on the cranks of the crankshaft is calculated and then the crank radius is used to calculate the torque or the piston force and velocity is calculated to derive an expression of the torque. The expression when using the tangential force can be seen in Equation 18.

|�}.-0� ^*5 1

@7!0 − 120$7-0� ^*5 La − .��^�� B�2I& � ∙ { sin-0 − 120 + M!0 − 120$5cos-M!0 − 120$5 =

|�}.-0��*5 1

@ 7!0$7-0��*5La − .��^�� B�2I& � ∙ { sin!0 + M!0$$cos!M!0$$ (18)

where the common variables can be seen in the notations list and B is the cylinder bore, r is the crank radius, β is the angle between the connecting rod and the piston centre line expressed in Equation 19 and the expression for the volume V can be seen in Equation 20

M = arcsin(19 sin(0)) (19)

where R is the ratio between the connecting rod length and the crank radius.

7(0) = 7� + 7�2 89 + 1 − �9& − sin&(0) − cos(0)< (20)

where Vc is the combustion bowl volume and Vd is the displacement volume. When instead using the piston force and piston velocity, the expression in Equation 21 is obtained.

.� ^*7� ^*a

|�}7� +7�2

{ @1 + 9 − cos 80 − 2�3 < − �9& − sin& 80 − 2�3 <L{��Aa{|}sin B0 − 2�3 I + {�

sin 80 − 2�3 < cos80 − 2�3 <�1 − {&�& sin& 80 − 2�3 < �

� = .��*7��*a @7�

+ 7�2 {-1 + 9 − cos0 − √9& − sin& 05{ LAa {|}sin(0) + {� sin 0 cos0

�1 − {&�& sin& 0��

(21)

where l is the connecting rod length between the connections to the piston and the crank, Pcomp and Vcomp is the boundary pressure and volume to the isentropic compression pressure and Pexp and Vexp is the boundary pressure and volume to the isentropic expansion pressure trace.

virtual crank angle based cylinder pressure sensor

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