State of Charge and Parameter Estimation of Electric Vehicle Batteries

Transcription

1 State of Charge and Parameter Estimation of Electric Vehicle Batteries By Richard Alfonso Bustos Bueno A Thesis Presented to The University of Guelph In partial fulfillment of requirements for the degree of Master of Applied Science in Mechanical Engineering Guelph, Ontario, Canada Richard Alfonso Bustos Bueno, March, 2018

2 ABSTRACT STATE OF CHARGE AND PARAMETER ESTIMATION OF ELECTRIC VEHICLE BATTERIES Richard Bustos University of Guelph, 2018 Advisor: S. Andrew Gadsden, Ph.D., P.Eng., P.M.P. Lithium-ion batteries have gained enormous popularity as energy storage elements for the vast majority of rechargeable electric systems, including electric vehicles. To make Lithium-Ion batteries safer and more reliable, a large number of battery models have been developed to estimate their current state of charge (SOC). The SOC determines the driving distance of electric vehicles. The primary objective of this thesis is the analysis of popular battery models found in literature, and to provide a foundation for developing intelligent control and estimation strategies. Four equivalent circuit models (ECMs) are analyzed with a thermal model: The Rint, Thevenin, PNGV, and DP models. Furthermore, three electrochemical models are presented: full order, single particle, and 3-parameter single particle models. Estimation techniques such as the Kalman, extended Kalman, and

3 unscented Kalman filters are applied to the ECMs to increase estimation accuracy. Additionally, interactive multiple models are analyzed for fault detection and diagnosis.

4 iv DEDICATION TO ANDREA

5 v ACKNOWLEDGEMENTS First and foremost, I would like to thank my supportive advisory committee and give a special thanks to Dr. Gadsden who has been very understanding and have encouraged me at every step of the way in the development of this thesis. Also, I would like to thank Dr. Gadsden for the funding and time invested in this research. I would also like to thank Hung Yi (Steven) Chiang for lending me his brain from time to time to discuss some topics of this thesis and for all of his support. Thank you to my family for their ongoing love and support. I am blessed to have you for family. To my parents who have nurtured and guide me through this roller coaster called life, thank you. I would never be who I am today if it had not been for you. To my brother, thank you for all of the resources send my way. To the rest of my family, I thank you for all of your thoughts, blessings, and support. Thanks to my wife who has supported me endlessly with her love and to whom I dedicate this thesis. Finally, I would like to thank my daughter who continually reminds me to enjoy the little things in life.

6 vi ABSTRACT...ii DEDICATION...iv ACKNOWLEDGEMENTS... v LIST OF TABLES...xi LIST OF FIGURES... xiii LIST OF ABREVIATIONS... xxi LIST OF NOMENCLATURES... xxiii 1. INTRODUCTION Introduction Purpose of Study Contributions Organization of Thesis LITERATURE REVIEW Battery technology and Electric vehicle Battery technology Battery Terminology Internal Resistance Cell, module and packs Charge rate Terminal Voltage Open Circuit Voltage Capacity... 12

7 vii State of Charge (SOC) Depth of discharge (DOD) Cut off Voltage Life Cycle State of Health (SOH) Operating Principle of Li-Ion Batteries The Battery Management System (BMS) Battery Modeling Behavioral Model Equivalent Circuit Model (ECM) Electrochemical Models Battery Thermal Management System (BTMS) Battery aging in for Li-Ion SOC estimation techniques Discharge Test Ampere-Hour Counting Open Circuit Voltage Impedance Spectroscopy State Estimation Techniques Fault detection methods EQUIVALENT CIRCUIT MODELS Introduction Rint Model Thevenin Model PNGV... 33

8 viii 3.5 Dual Polarity Thermal Model Electrochemical Model Introduction Full Order Electrochemical Model (Doyle-Fuller-Newman (DFN) model) Transport in Solid Phase Transport in electrolyte Electrical Potential Butler- Volmer Kinetics Single Particle Model (SPM) Governing Equations Three-Parameter Single Particle Model (3-Parameter SPM) Estimation Methods Estimation Theory Kalman Filter (KF) Extended Kalman Filter (EKF) Unscented Kalman Filter (UKF) Interactive Multiple Model Strategy Computer Experiments Model Calibrations... 71

9 ix ECM Parameter Optimization Rint Model Thevenin Model PNGV Model DP Model Thermal Model Final Remarks regarding calibration Electrochemical Model Calibration SPM model Calibration Parameter SPM Calibration Final Remarks regarding calibration Urban Dynamometer Driving Schedule (UDDS) Setup Results Rint Model Thevenin Model PNGV Model DP Model DFN model SPM Parameter SPM Dynamic Stress Test (DST) Setup Results Rint Model Thevenin Model PNGV Model

10 x DP Model DFN model SPM Parameter SPM BMS and Fault Detection Setup Results Rint Model Aged battery Increased core thermal resistance DP Model Aged battery Increased core thermal resistance CONCLUSIONS Summary of Research Future Work and Recommendations APPENDIX B: MATLAB Code: Thevenin Model APPENDIX C: MATLAB Code: PNGV Model APPENDIX D: MATLAB Code: DP Model APPENDIX E: MATLAB Code: SPM APPENDIX F: MATLAB Code: 3-Parameter SPM APPENDIX G: MATLAB Code: Functions

11 xi LIST OF TABLES Table 1: Thermal Model Parameters Table 2: Optimized Battery Resistance at different SOC levels Table 3: Rint model RMSE Output Voltage estimate Table 4: Thevenin Model Parameter Values Table 5: Thevenin model RMSE Output Voltage estimate Table 6: Optimized Parameter Values for PNGV model Table 7: PNGV model RMSE Output Voltage estimate Table 8: Optimized Parameter Values for the DP model Table 9: DP model RMSE Output Voltage estimate Table 10: Thermal model parameter values [82] Table 11: Electrochemical Models Parameter Values Table 12: RMSE Results for 3-Parameter SPM Vs. DFN model Table 13: Initial Conditions for model and KF algorithms Table 14: RMSE for KF state estimates using UDDS current Table 15: RMSE values of Thevenin model state estimates using EKF and UKF methods Table 16: RMSE values of PNGV model state estimates using EKF and UKF methods Table 17: RMSE values of DP model state estimates using EKF and UKF methods Table 18: RMSE for EKF-UKF estimates using Rint model with DST current Table 19: RMSE for EKF-UKF estimates using Thevenin model with DST current Table 20: RMSE of EKF-UKF estimates for PNGV model

12 xii Table 21: RMSE EKF-UKF estimates for DP model using DST current Table 22: Test Cases to be perform using the IMM strategy with the Rint model and DP model Table 23: RMSE values of KF and KF-IMM under aged battery test case Table 24: RMSE values for KF and KF-IMM under increased thermal resistance test 159 Table 25: RMSE EKF, UKF, EKF-IMM, UKF - IMM for aged battery case Table 26: RMSE EKF, UKF, EKF IMM, UKF IMM Temperature Fault

13 xiii LIST OF FIGURES Figure 1: Ragone plot of various energy storage technologies [16] Figure 2: Electrochemical functionality of a Li-Ion battery [24] Figure 3: Common used cathode and anode electrodes in Li-Ion technology [15] Figure 4: Safety Operating window for lithium ion Battery [26] Figure 5: Rint Model Circuit Configuration [70] Figure 6: Hysteresis model of the battery [73] Figure 7 Schematic diagram of Thevenin Model [70] Figure 8: PNGV model: Circuit configuration [76] Figure 9: DP model: Circuit Schematic [70] Figure 10: Battery Thermal Model [82] Figure 11: 1D-spatial Model of a Li-ion Battery [84] Figure 12: Electrode as a sphere approximation with radius Rk [24] Figure 13: Governing equation of DFN model at each region [89] Figure 14: SPM model of a Li-ion Cell [87] Figure 15: Three Parameter Model as an ECM [91] Figure 16: State Estimation General Block Diagram [92] Figure 17: Predictor-Corrector Method for KF [96] Figure 18: Estimation Algorithm for the IMM [106] Figure 19: Experimental Input Current Profile [110] Figure 20: Experimental Output Voltage Profile for Optimization [110] Figure 21: Rint Model in Simulink Used for Parameter Optimization Figure 22: Rint model OCV curve for all SOC levels... 74

14 xiv Figure 23: Rint Simulated Output Voltage Figure 24: Thevenin Simulink Model used for optimization of the model's parameters Figure 25: Cth Parameter Curve at various levels of SOC Figure 26: Rth Parameter Values at various levels of SOC Figure 27: Experimental Output Voltage Vs. Thevenin Model Voltage Output Figure 28: PNGV Simulink Model for Parameter Optimization Figure 29: Optimized 1/Uocv Curve at all levels of SOC Figure 30: Optimized Cpn Curve Figure 31: Experimental Output Voltage Vs. PNGV Model Output Voltage Figure 32: Simulink DP model for Parameter Optimization Figure 33: Cpc polynomial curve at all levels of SOC Figure 34: Cpa polynomial curve at all levels of SOC Figure 35: Experimental Output Voltage Vs. DP model Output Voltage Figure 36: SOC estimate using coulomb counting method Figure 37: Input current for the electrochemical models [111] Figure 38: Output Voltage of the DFN model Figure 39: SOC of the Cathode based on DFN model Figure 40: SOC of the anode based on DFN model Figure 41: SOC anode DFN Vs. SPM Simulation Figure 42: SOC cathode DFN Vs. SPM Simulation Figure 43: Output Voltage of the SPM Figure 44: Cathode SOC using 3 Parameter SPM... 91

15 xv Figure 45: Anode SOC using 3 Parameter SPM Figure 46: Output Voltage using the 3 Parameter SPM Figure 47: SOC estimate using coulomb counting method Figure 48: UDDS Speed Profile in Miles per hours Figure 49: UDDS current demanded from battery for EV vehicles using a LFP Li-Ion battery Figure 50: Noise Example for SOC Figure 51: SOC state profile using UDDS current Figure 52: Core Temperature state profile using UDDS current Figure 53: Surface temperature state profile using UDDS current Figure 54: Output Voltage profile using UDDS current Figure 55: Rint model state error covariance Figure 56: Rint model Kalman Gain for all states Figure 57: Estimate of the voltage across the capacitor Cth using EKF and UKF methods Figure 58: SOC estimate using the Thevenin model with EKF and UKF methods Figure 59: Tcore and Tsurface state profiles using Thevenin model with EKF and UKF methods Figure 60: Output Voltage using the Thevenin model with EKF and UKF methods Figure 61: Thevenin Model State Error Covariance using (EKF) Figure 62: Thevenin State Error Covariance (UKF) Figure 63: Thevenin Model Kalman Gain (EKF) Figure 64: Thevenin model Kalman Gain (UKF)

16 xvi Figure 65: SOC estimate using the PNGV model with EKF and UKF methods Figure 66: VUocv estimate using the PNGV model with EKF and UKF methods Figure 67: Vpn estimate using the PNGV model with EKF and UKF methods Figure 68: Vout estimate using the PNGV model with EKF and UKF methods Figure 69: PNGV model core temperature Figure 70: PNGV model surface temperature Figure 71: PNGV model state error covariance using EKF Figure 72: PNGV model state error covariance using UKF Figure 73: PNGV model Kalman gain using EKF Figure 74: PNGV Kalman gain using UKF Figure 75: Vpc state estimate with EKF and UKF Figure 76: Vpa state estimate with EKF and UKF Figure 77: Vout DP model using EKF and UKF methods Figure 78: DP model core temperature Figure 79: DP model surface temperature Figure 80: DP model state error covariance using EKF Figure 81: DP state error covariance using UKF Figure 82: DP model Kalman Gain (EKF) Figure 83: DP model Kalman Gain UKF Figure 84: SOC anode using UDDS current Figure 85: SOC cathode using UDDS current Figure 86: Output voltage using UDDS current profile Figure 87: SPM Vs DFN model output voltage using UDDS curent

17 xvii Figure 88: SPM Vs DFN model SOC cathode using UDDS profile Figure 89: SPM Vs. DFN model SOC anode using UDDS profile Figure 90: DFN model Vs. 3-Parameter SPM output voltage (UDDS) Figure 91: DFN model Vs. 3-Parameter SPM SOC anode (UDDS) Figure 92: DFN model Vs. 3-Parameter SPM SOC cathode (UDDS) Figure 93: DFN model Vs. 3-Parameter SPM average SOC anode (UDDS) Figure 94: DFN model Vs. 3-Parameter SPM average SOC cathode (UDDS) Figure 95: DST Current Profile Figure 96: SOC estimate Rint model using DST current demand Figure 97: Core Temperature Profile using DST current Figure 98: Surface temperature estimate using DST current Figure 99: Vout estimate using DST current Figure 100: Rint Model State Error Covariance with DST current profile Figure 101: Rint Kalman Gain Profile using DST current profile Figure 102: SOC estimate using DST current profile Figure 103: Vth estimate using DST current Figure 104: Battery s core temperature using DST current (Thevenin model) Figure 105: Battery's surface temperature using DST current (Thevenin model) Figure 106: Vout estimate using DST current Figure 107: Thevenin State Error Covariance using DST current profile (EKF) Figure 108: Thevenin Kalman Gain using DST current Profile (EKF) Figure 109: Thevenin State Error Covariance using DST current profile (UKF) Figure 110: Thevenin Kalman Gain using DST current profile (UKF)

18 xviii Figure 111: SOC estimate using DST current Figure 112: VUocv estimate using DST current Figure 113: Vpn estimate using DST current Figure 114: Vout estimate using DST current Figure 115: Battery's core temperature using DST current (PNGV) Figure 116: Battery's surface temperature using DST current (PNGV) Figure 117: PNGV State Error Covariance EKF using DST current profile Figure 118: PNGV Kalman Gain EKF using DST current profile Figure 119: PNGV State Error Covariance UKF using DST current profile Figure 120: PNGV Kalman Gain UKF using DST current profile Figure 121: SOC estimate using DST current Figure 122: Vpa estimate using DST current Figure 123: Vpc estimate using DST current Figure 124: Battery's core temperature using DST current profile (DP model) Figure 125: Battery's surface temperature using DST current profile (DP model) Figure 126: Vout estimate using DST current Figure 127: DP State Error Covariance EKF using DST current profile Figure 128: DP Kalman Gain EKF using DST current profile Figure 129: DP State Error Covariance UKF using DST current profile Figure 130: DP Kalman gain UKF using DST current profile Figure 131: SOC anode using DST current Figure 132: SOC cathode using DST current Figure 133: Output voltage using DST current profile

19 xix Figure 134: SPM Vs DFN model output voltage using DST curent Figure 135: SPM Vs DFN model SOC cathode using DST current Figure 136: SPM Vs. DFN model SOC anode using DST current Figure 137: DFN model Vs. 3-Parameter SPM output voltage (DST) Figure 138: DFN model Vs. 3-Parameter SPM SOC anode (DST) Figure 139: DFN model Vs. 3-Parameter SPM SOC cathode (DST) Figure 140: DFN model Vs. 3-Parameter SPM average SOC anode (DST) Figure 141: DFN model Vs. 3-Parameter SPM average SOC cathode (DST) Figure 142: Estimated terminal voltage using KF and IMM KF under aged battery test Figure 143: Estimated SOC of the battery using KF and KF-IMM under aged battery test Figure 144: Mode probability of the KF-IMM under aged battery test Figure 145: Rint State Error Covariance KF-IMM Fault Aged Figure 146: Rint Kalman Gain KF-IMM Fault Aged Figure 147: Estimated core temperature under increased thermal resistance test Figure 148: Estimated surface temperature under increased thermal resistance Figure 149: Mode probability of the KF-IMM under increased thermal resistance test 158 Figure 150: Rint State Error Covariance KF-IMM Fault Temperature Figure 151: Rint Kalman Gain KF-IMM Fault Temperature Figure 152: Output Voltage of all estimation methods for aged battery test case Figure 153: Estimated SOC for aged battery test case Figure 154: Mode probability of the EKF-IMM for aged battery test case

20 xx Figure 155: Mode probability of the UKF-IMM for aged battery test case Figure 156: DP State Error Covariance EKF-IMM Fault Aged Figure 157: DP Kalman Gain EKF-IMM Fault Aged Figure 158: DP State Error Covariance UKF-IMM Fault Aged Figure 159: DP Kalman Gain UKF-IMM Fault Aged Figure 160: Tcore with Fault (Increased in thermal resistance) Figure 161: Surface temperature with fault Figure 162: Mode probability EKF Figure 163: Mode probability (UKF) Figure 164: DP State Error Covariance EKF-IMM Fault injected Figure 165: DP Kalman Gain EKF-IMM Fault injected Figure 166: DP State Error Covariance UKF-IMM Fault injected Figure 167: DP Kalman Gain UKF-IMM Fault Injected

21 xxi LIST OF ABREVIATIONS Abbreviation Li-ion EV ICE HEV LCO LMO NMC LFP NCA LTO ECM SPKF CKF SVSF AEKF DOD BMS BTMS SOA SOC SOH OCV SOH SOF RUL KF EKF UKF IMM MMAE RC PNGV DP UDDS DST RMSE Definition Lithium-ion Electric vehicle Internal combustion engine Hybrid electric vehicles Lithium cobalt oxide Lithium-manganese oxide Nickel manganese cobalt oxide Lithium iron phosphate Lithium nickel cobalt aluminum oxide Lithium titanate Equivalent circuit model Sigma point-kalman filter Cubature Kalman filter Smooth variable filter Adaptive extended Kalman filter Depth of discharge Battery management system Battery thermal management system Safe operation area State of charge State of health Open circuit voltage State of health State of function Remaining useful life Kalman filter Extended Kalman filter Unscented Kalman filter Interactive multiple model Multiple model adaptive estimation Resistor-capacitor Partnership for a new generation of vehicles Dual polarity model Urban dynamometer driving schedule Dynamic stress test Root mean square error

22 xxii CC CV USCAR DFN SPM 1D PDAE PDE DAE Constant current Constant voltage US council for automotive research Doyle-Fuller-Newman Single particle model One dimensional Partial differential algebraic equations Partial differential equations Differential algebraic equations

23 xxiii LIST OF NOMENCLATURES Symbol Definition c e Electrolyte concentration [molm 3 ] Maximum solid phase concentration for each electrode (k = p, n) [mol m 3 ] c s,k Concentration of solid phase lithium. (k = p, n) [mol m 3 ] c s,k Volume-average concentration of solid phase lithium. (k = p, n) [mol m 3 ] q s,k Volume-averaged concentration of flux. (k = p, n) [A m 2 ] D s,k Solid phase diffusion coefficient of Li-ions for each electrode (k = p, n) [m 2 s 1 ] D e,k Electrolyte phase lithium ion diffusion coefficient [m 2 s 1 ] F Faraday s constant [C mol 1 ] J k current density at the surface of the spherical particle for each particle in electrode (k = p, n) [A m 2 ] c k,max k k Rate of constant for each electrode (k = p, n) [m 2.5 mol 0.5 s 1 ] R gas Gas constant [J mol 1 K 1 ] R k,p Radius of particle for each electrode (k = p, n) [m] S k Electroactive surface area for each electrode (k = p, n) [m 2 ] T Temperature [K] U k Open circuit potential for each electrode (k = p, n) [V] V(t) Cell Voltage [V] Cathodic transfer coefficient α a,k α a,k η k Anodic transfer coefficient Over-potentials for the lithium ion intercalation reaction for each electrode (k = p, n) [V]. Unless specified. Φ k Potential reaction for each electrode (k = p, n) [V] Active material volume fraction in the electrode (k = p, n) Electrolyte phase volume fraction. (k = p, n) L Thickness of electrode plate [m] a Specific interfacial area [m 1 ] ε s,k ε e,k

24 xxiv i 0 Exchange current density [Am 2 ] R f Solid-Electrolyte interface layer film resistance [Ωm 1 ] 0 t a Transfer number for the anion A Electrode plate Area [m 2 ] R collector Resistance of the current collector plate [Ω] A Linear system matrix A thermal Linear system matrix for the thermal model A ECM Linear system matrix for the equivalent circuit model B Input gain matrix C Linear measurement matrix f Nonlinear system function F Linearized system matrix g Nonlinear measurement matrix H Linearized measurement matrix i Index value I Identity matrix j Index value k Step value (recursion algorithms) K Kalman gain (filters) m Number of measurements n Number of states P State error covariance matrix Q System noise covariance matrix R Measurement noise covariance matrix S Innovation covariance matrix T Sample rate u Input of the system (vector/scalar) v Measurement noise (vector/scalar) w System noise (vector/scalar) W Weight function (filters) x States or parameters (scalar or vector) X Propagated states (Sigma point filter) y Artificial measurement (vector/scalar) z Measurement (vector/scalar) k + 1 k Subscript, a priori (before the fact) value k + 1 k + 1 Subscript, a posteriori (after the fact) value p i,j Mode transition matrix (designer value) μ j Mode probabilities μ i j Mixing probabilities j Likelihood function a T or a Transpose of a ^ Estimate of state or parameter Applied/Supplied current I s

25 xxv C n R battery R th C th U th I th R pa R pc C pa C pc U pa U pc S r R pn C pn T core T surface Nominal capacity Battery resistance Thevenin resistance Thevenin capacitance Voltage across Thevenin elements Current through Thevenin elements Dual polarity polarization resistance Dual polarity concentration resistance Dual polarity polarization capacitance Dual polarity concentration capacitance Voltage across the polarization RC branch (dual polarity model) Voltage across the concentration RC branch (dual polarity model) Heat generation rate PNGV model resistance PNGV model capacitance Thermal model core temperature Thermal model surface temperature

26 1 CHAPTER 1 INTRODUCTION

27 2 1. INTRODUCTION 1.1 Introduction In today s society, high importance is being placed on the stress levels that technology puts on the environment. This factor has pushed automobile technology to an eco-friendlier solution which is electric vehicles (EV) [1]. When compared to petroleum-based vehicles using the internal combustion engine (ICE), EVs have a significantly less footprint depending on their fuel source [2]. In an EV, batteries store the electrical energy in an electrochemical reaction for later use. There are several types of batteries in the industry. The most popular are lead-acid, nickel, alkaline and lithium-ion [1]. Lithium has become very popular because it is light metal, has the greatest electrochemical potential and provides the largest specific energy per weight [1]. Current lithium-ion battery technology allows EV to cover about km per battery charge. Unfortunately, batteries' full load capacity degrades over time as they are subject to charging cycles, resulting in a lower driving range throughout its lifetime [3] In addition, the SOC and voltage measurement from the cell s terminal is crucial information to determine the available energy in the battery, which can be used to determine the available driving range of the EV. Unfortunately, it is not possible to have direct measurements of the SOC [2]. The main difficulties are listed below: The battery pack of EVs has hundreds of cells connected in series, the different accumulated potential of each cell is different to each other making it hard to have unified compensation or elimination methods.

28 3 Voltage measurement requires high precision as other parameters are estimated based on the voltage measurement. Required voltage precision is around 1 mv to have low carried errors [3]. Furthermore, the components used to build the Lithium-Ion battery significantly reduces the safe operating area (SOA) of the battery. The SOA of Li-Ion cells is bound by current, temperature and voltage [3]. Some outcomes if these boundaries are exceeded are [3]: Cells might burst into flames if overcharged or work outside a safe temperature range. Cells might be permanently damaged if allowed to be over-discharged. Cells lifetime will be affected if they are discharged at too high current or charged too fast. Cells might be damaged if operated at high pulse currents for too long. A battery management system (BMS) is responsible for calculating several parameters such as the battery s SOC, state of health (SOH), temperature, state of function (SOF), remaining useful life (RUL) and communicating with other vehicle components and subsystems [1]. The SOC of a battery can help estimate current driving range and prevent the battery pack from over charge and over discharge [3]. These parameters are not easy to calculate for, as it would demand extensive computing times, expensive instrumentation or have the vehicle stopped preventing its application in real time [1] [4]. A solution is to generate estimates of the parameters,

29 4 which requires a quality model, and an accurate estimation strategy for the various scenarios: temperature, power demands, and state of function [3] [5] Purpose of Study There is a high desired to use batteries in the automobile technology to significantly reduce emissions generated by the ICE driving vehicles. Research to optimize the performance of Li-Ion within the battery s SOA has been extensive [3]. The desired to accurately model the Li-Ion battery for use in power management system has open tremendous line of research were various models has been developed and studied. Some battery models include: ECM, Electrochemical models, behavioral models, hydraulic models, among others [6] [7] [8] [9]. Often these models are called high fidelity due to the high accuracy required to capture critical nonlinear capacity effects of the battery for implementation with the power management system [10]. The main purpose of this research work is the study of Li-Ion mathematical models found in the literature and the comparison and analysis of the performance of the models when implemented in EV technology. Furthermore, this work will serve as a basis for the development of novel strategies for monitoring the condition and health of EV Batteries. Condition monitoring involves estimating the system s states over an operational period, where abnormal values or significant changes would indicate a fault [11]. The final goal is the development of intelligent condition monitoring strategies with the ability to adapt over time and to be able to predict and anticipate future environments. A system with such condition monitoring traits will have considerable advantages over standard condition

30 5 monitoring systems including improved robustness to uncertainties and disturbances, increased accuracy regarding estimated versus actual states and reduced failures due to system faults. These strategies will later be applied to the BMS which monitors the critical parameters previously mentioned Contributions In this thesis, various Li-Ion models and SOC estimation strategies are compared and analyzed. The following is a list of research objectives. Objectives pursued during the development of this research: Provide with a literature review of EV battery technology. Development and analysis of Li-Ion battery models. Analysis of the performance of the models using industry tests Application of estimation techniques to improve accuracy of the battery s parameter estimates. Implementation of the IMM for fault detection Organization of Thesis This thesis consists of 6 chapters: Chapter 1: The purpose of study and contribution are presented. Chapter 2: Reviews current battery technologies and applications to electric vehicles. Moreover, this chapter discusses the use of a BMS to ensure the safety of use of Li-Ion batteries and it presents different methods used for fault detection in electrical systems.

31 6 Chapter 3: In this chapter, the thermal model and various ECMs are introduced. Chapter 4: In this chapter, the electrochemical models are presented. Chapter 5: Estimation algorithms are discussed for estimation of the battery s SOC. In addition, the IMM algorithm is introduce for fault detection. Chapter 6: The experimental setup for each model and their corresponding results are presented for each experimental case. Chapter 7: The accuracy of the SOC estimation techniques based on the ECMs is discussed. Furthermore, a conclusion is drawn based on the performance of each model implementing the estimation techniques described in Chapter 5.

32 7 CHAPTER 2 Literature Review

33 8 2.0 LITERATURE REVIEW The increasing number of sales of electric vehicles and high demand of consumer electronics in conjunction with today s society concern on the environment has push research into the development of a reliable ecofriendly solution for portable energy storage devices. Great focus has been placed into battery technology with Li-Ion batteries becoming one of the most popular technologies due to their specific energy per weight [12]. This chapter will provide with an overview of current Li-Ion battery technology in the automobile industry Battery technology and Electric vehicle The automobile has been one of the most useful technologies developed by humankind. It provides with a practical way to cover great distances daily. EV and hybrid electric vehicles (HEV) has become very popular due to their significant reduction on toxic emission compared to ICE driving vehicles. Thus, great focus has been placed in the development of battery technology as it is the main source of power in EVs Battery technology Batteries store the electrical energy in an electrochemical reaction for later use. There are several types of batteries in the industry. The most popular are lead-acid, nickel cadmium, and lithium-ion. Lead-acid batteries are one of the most popular and oldest technologies in batteries. The first lead-acid assembly is attributed to Gaston Plante over 150 years ago. In today s time, Lead-acid batteries have high reliability and low cost due to their mature technology

34 9 [13]. Among the disadvantages of this technology is the low specific energy and the short service life [14]. Nickel cadmium battery on the other hand have high specific energy with no degradation for deep charge/discharge cycles; however, cadmium is very toxic and it has high recycling cost associated to it which is why its application has been limited. There exist other non-toxic compositions of nickel batteries, however, they suffer of memory, high self-discharge, high cost and shortened life due to the development of dendrites [14]. Li-Ion is among the newest technologies still under research and development. Li-Ion show promising application to EV due to their high specific energy, and high voltage operation; however, cost still remain high and their life is significantly compromised by deep discharge cycles and are significantly affected to temperature [14] [15]. A comparison across battery technologies is possible by comparing their specific power and energy density on the Ragone plot [16]. In a Ragone plot, the y-axis denotes the amount of energy available, while the x-axis denotes how fast that energy may be accessible. A point in this plot, represents the amount of time for which the energy (per mass) on the y-axis may be accessed at given power on the x-axis. Furthermore, that time (hours) is derived by the ratio between the energy and the power density of the element [17]. Figure 1 depicts various battery technologies as well as the ICE, gas turbine and capacitor technologies. It can be seen that the ICE remains as a desirable technology to be used; however, due to their hazardous emissions Li-Ion batteries are the next choice

35 10 of preference. Furthermore, it can be seen that Li-Ion have high specific energy but low power density, which is why batteries are combined with high power devices, resulting in hybrid systems that achieve the desirable power characteristics [16]. Figure 1: Ragone plot of various energy storage technologies [16] Li-Ion batteries are rechargeable battery types meaning that it allows for reversible chemical reactions. Moreover, Li-Ion batteries possess low self-discharging rates. They are becoming popular for EV applications and have been slowly replacing lead-acid batteries technologies where the heavy lead plates and acid electrolyte is changed to a Li-Ion carbon anode and li-ion phosphate cathode. The main reason behind the change in technology is the capability of Li-Ion batteries to provide the same voltage as lead-acid, resulting in no modifications to the system they are incorporated to [14]. Currently in the US, li-ion batteries are used for vehicles such as the hybrid Chevrolet Volt, Honda fir EV Ford focus electric, Mitsubishi i-miev, Nissan Leaf, and the Tesla Roadster.

36 11 Unfortunately, batteries' full load capacity degrades over time as they are subject to charging cycles, resulting in a lower driving range throughout its lifetime. The following section covers various important battery terminology: Battery Terminology Internal Resistance The internal resistance of a battery limits the amount of power that is delivered from the battery. Often internal resistance is designed to be small to allow for more power to be delivered; however, the internal resistance increases over time due to sulfation and grid corrosion of the battery [12] [18]. Furthermore, the amount of energy available in the battery or state of charge (SOC), affects the apparent internal resistance of the battery. Li-ion has higher resistance at a full charge stage and at the end of discharge but little resistance in between [19] Cell, module and packs Often HEV contain high power battery packs. The smallest battery unit inside a Li-Ion battery pack is the cell. These cells are arranged in series and parallel to achieve a required voltage and capacity. Often the first arrangement of cells is called a module which are arranged as well in series and parallel to create the battery pack. It is important to mention that each cell has its own SOC, capacity and internal resistance [20] Charge rate

37 12 Charge rates are used to describe and compare different batteries discharge currents. The C-rate is a normalized value against the battery capacity, which differs across batteries Terminal Voltage Terminal voltage refers to the voltage across the battery terminal when a load is applied to the battery. This voltage often varies with respect to SOC and discharge/charge currents Open Circuit Voltage This voltage refers to the voltage across the battery s terminals with no load applied. There is a direct relationship between this voltage and SOC Capacity The coulometric capacity, is defined as the total Amp-hours available when the battery is discharged at a given C-rate from 100 SOC to cut-off voltage. Capacity decreases with increasing C-rate [21] State of Charge (SOC) When referring to HEV, or EV, the SOC shows the available battery capacity as a percentage of maximum capacity. In others, it measures the amount of electric energy in the battery. SOC has a value from 0 to 1 representing a fully discharged battery to a fully charged battery, respectively. This parameter can be compared to the fuel tank gauge of an ICE vehicle [22] Depth of discharge (DOD)

38 13 The DOD is a measurement of the percentage of battery capacity that has been used. In others, is represented as the opposite of SOC and may be calculate as DOD = 1 SOC Cut off Voltage Cut off voltage refers to the minimum allowable voltage at which the battery is referred as empty Life Cycle This number refers to the amount of times the battery may be charge and discharged before its retired State of Health (SOH) The SOH measures the irreversible degradation that occurs in the battery performance due to cycling and aging. SOH allows for an easy comparison with a healthy battery. The best way to evaluate the health of a battery is by testing its capacity [12]. Some factors that affect the battery's capacity are the temperature at different SOC, the level of SOC it is charged every time, cycling, and depth of discharge (DOD) [12].Furthermore, The SOH of a battery serves as an indicator of the battery life conditions between its beginning of life to retirement. Often SOH is a unit less value similar to the SOC that indicates the available remaining use. Often, batteries are retired when their overall capacity or maximum power drops to 80, which is referred as capacity fading and power fading [22] Operating Principle of Li-Ion Batteries

39 14 The lithium Ion cell is an electrochemical cell composed of a positive electrode, separator, negative electrode and electrolyte solution. The negative and positive electrode are referred as anode and cathode respectively. Furthermore, the negative electrode plate in Li-Ion batteries is often connected to a copper current collectors while the positive electrode is connected to an aluminum current collector plate [23]. Figure 2 depicts the components and functionality of the Li-Ion battery. During charge, lithium ions travel from the cathode to the anode using the electrolyte as medium. Once at the electrode the electrons make use of the collector plates to move through the external circuit (load). The main purpose of the Li-Ion cell is to store electro-chemical energy to be used later [24]. Figure 2: Electrochemical functionality of a Li-Ion battery [24] There is a selection of materials that may be used for the cathode electrode plate. The most popular are described on Figure 3 with their corresponding specific energy.

40 15 Figure 3: Common used cathode and anode electrodes in Li-Ion technology [15]. As it can be seen on Figure 3, often the anode contains graphite (Li x C 6 ) that serves as an intercalation material (reversible inclusion or insertion of molecules in the material); whereas the cathode changes to obtain a higher specific energy [25]. The most commonly used cathodes are: Lithium Cobalt Oxide (LCO), Lithium Manganese Oxide (LMO or Li-manganese), Lithium Nickel Manganese Cobalt Oxide (NMC), Lithium Iron Phosphate (LFP), Lithium Nickel Cobalt Aluminum Oxide (NCA), and Lithium Titanate (LTO) [25]. 2.2 The Battery Management System (BMS) A battery management system (BMS) is responsible for calculating several parameters such as the battery s SOC, SOH, temperature, state of function (SOF), remaining useful life (RUL) and communicating with other vehicle components and subsystems [12]. The SOC of a battery can help estimate current driving range and

41 16 prevent the battery pack from over charge and over discharge. SOH can be used to track capacity and resistance of the battery. RUL is crucial as it determines when a battery should be replaced as well as the potential resell value of the battery [12] [26]. For example, if a battery's temperature is exceeding its desired operating temperature, the BMS will signal the cooling system to start. It is evident the importance of an accurate value of these parameters for the sake of efficiency of the entire system. These parameters are not easy to calculate for, as it would demand extensive computing times, expensive instrumentation or have the vehicle stopped preventing its application in real time [27]. A solution is to generate estimates of the parameters, which requires a quality model, and good estimation strategy for the various scenarios: temperature, power demands, and state of function [26]. Some suggested models that have been found in the literature are equivalent circuit based models, electrochemical-based models, hydrodynamics model, black-box models and tabulated data models [28]. Some estimation techniques used are the simple correction, weighted fusion algorithm, Kalman filtering (KF), Extended Kalman filtering (EKF), unscented Kalman filter (UKF), Gaussian filters, sliding mode observer, and Neuro-fuzzy techniques [29] [30] [31] [32]. Furthermore, the components used to build the Lithium-Ion battery significantly reduces the safe operating area (SOA) of the battery. The SOA of Li-Ion cells is bound

42 17 by current, temperature and voltage [26]. Some outcomes if these boundaries are exceeded are: Cells might burst into flames if overcharged or work outside a safe temperature range Cells might be permanently damaged if allowed to be over-discharged Cells lifetime will be affected if they are discharged at too high current or charged too fast. Cells might be damaged if operated at high pulse currents for too long Lastly, the BMS is also responsible for cell balancing which address the issue that cells do not have the same levels of SOC, capacity, and internal resistance; significantly reducing the overall capacity of the lithium-ion battery and affecting the ability of the battery pack to be charge and discharge [33] Battery Modeling Battery modeling is crucial for the development of a robust BMS. Often the decision of selecting a model depends on how much complexity, and accuracy is desired. Complex models will also have a higher parameterization complexity as more parameters are needed to capture higher order of nonlinearities. This section explores different battery models found in the literature and were categorized as Behavioral, Equivalent Circuit and Electrochemical Behavioral Model

43 18 Behavioral models make use of empirical data and often neglect the underlying physical or electrochemical behavior of the battery. Their dependency on data and look up tables has labelled them as tabular models. One of the most established behavioral models is Peukert s law [34]: I s PC t = Constant Where, I s is the discharge current, t is the maximum discharge time and PC is known as the Peukert s Coefficient which varies from 1 to 2 [34]. The battery capacity is then calculated using C n1 = C n ( I PC 1 n ) I n1 Where C n1 represents the capacity remaining at the given discharge current I n1. Another well-established Behavioral model is the Shepherd model [35]. It uses the following equation to predict the terminal voltage during charging/discharging conditions: E(t) = E 0 + R α i(t) + K 1 q s (t) Where, E 0 refers to the initial cell voltage, R α is the cell s internal resistance, q s (t) is the instantaneous stored charge and K 1 is a constant. Since its introduction, further modification has been done to this equation which introduced more K 1,2 constants to capture higher nonlinearities resulting in the following models: Unnewehr model, Nernst model and Plett who combined various of the

44 19 previously mentioned models to derive a self-correcting model which accounted for hysteresis effects of the battery [35] Equivalent Circuit Model (ECM) ECM are models based on electrical components (e.g. ideal voltage sources, resistors, capacitors etc.) to simulate the behaviour of a battery. Most ECM models are semi-empirical models where the process of calculating the final output is based on a previously derived parameter curves based on the battery s SOC. The design of ECM is fairly easily and may vary greatly depending on the desired output accuracy [1]. Moreover, computational power and simulation time required for these models is low compared to other models. This trait makes them more suitable for fast calculations, with acceptable accuracy. Thus, making them perfect to be used and implemented in a battery management system (BMS) [36]. Below is a summary of common traits among ECMs: The main values used to simulate the state of charge is open circuit voltage (OCV) and the battery voltage. The states of battery and initial values are identified by characterization method. The electrical, and/or chemical behaviour of the battery represents through the use of electrical components [37] Electrochemical Models

45 20 The Electrochemical models use partial differential equations to describe lithium diffusion inside the electrolyte and electrode [38]. These models are complex and require extensive computational power; however, they are preferred for analysis as it is based on the physical phenomenon that occurs in the battery. To reduce the computational demand of this models, reduced models are often developed. Another drawback of implementing these models it's their number of parameters [38]. In the development of an electrochemical model, the movement of Ions in and out of an interstitial site in a lattice is modeled. In simple words, the ions may be store in a lattice. Batteries are often modeled as two lattice system were ions can travel from one site to the other. In electrochemical models the battery is discretize into 4 main parts: porous negative electrode, porous positive electrode, separator and electrolyte. Figure 2, shows the different parts of the battery. To reduce complexity, the diffusion dynamics in one dimension are considered for the model and the following parameters are evaluated throughout the process: State Variables: i s (x, t) The current in the solid electrode. i e (x, t) The current in the electrolyte. Φ s (x, t) The electric potential in the solid electrode. Φ e (x, t) The electric potential in the electrolyte. The molar flux j n (x, t) of lithium at the surface.

46 21 The concentration of lithium in electrolyte, c e (x, t), and the electrode c s (x, r, t). These parameters are evaluated because lithium may exist in two phases: Solid Lithium that is store in the electrode (Negative or Positive) and dissolved lithium that exists at the electrolyte. Common Assumptions made for the model are: Spherical solid particles exist along the X-axis. The movement of ions is modeled by the insertion of lithium ions in and out of. these spherical solid particles. Solid particles are immersed in the electrolyte. The terminal voltage may be calculated based on the electric potential of the solid Lithium: V(t) = Φ s (0 +, t) Φ s (0, t) Furthermore, in a state value model, the input current I s (s) is related to the model s outputs. Afterwards, these outputs are combined in sub-models to determine the terminal voltage. The most commonly used electrochemical model is the single particle model were the electric potential generated by the particle is tracked from the positive solid electrode to the negative solid electrode. The following equation summarizes the relationship between the parameters and the input current [39].

47 22 Φ s (x, t) x = i e(x, t) I(t) σ Φ e (x, t) x = i e(x, t) k + 2RT d ln fc F (1 e c 0 a ) (1 + d lnc e (x, t)) ln c e(x, t) x C e (x, t) = t x (D C e (x, t) e ) + 1 (t 0 a i e (x, t)) x Fε e x C s (x, r, t) t = 1 r 2 r (D sr 2 C s(x, r, t) ) r i e (x, t) = afj x n (x, t) j n (x, t) = i 0(x, t) [e (α af RT η s (x,t)) e ( α cf RT η s (x,t)) ] F In the literature, these equations have been solved using Laplace transformations [40], impedance model formulation [41], orthogonal decomposition [42], Padé approximation [43]. Furthermore, an electrochemical-thermal-coupled model have been developed in [44] Battery Thermal Management System (BTMS) Thermal management systems in EV is critical to the performance of the vehicle as high and low temperatures may reduce the battery's operation, efficiency, charge acceptance, apparent energy, power, safety, life expectancy and life cycle cost [45] [46]. Based on the Arrhenius equation, at high temperatures, the battery's aging increases

48 23 exponentially due to rapid chemical reactions [45]. Moreover, the battery's apparent capacity is significantly affected by temperature. Furthermore, Figure 4 shows some of the things that may happened to the battery if the battery is operated outside of its safe temperature range. Figure 4: Safety Operating window for lithium ion Battery [26] In industry, BTMS may be categorized based on how they interact with the system. Among these categories are: cooling BTMS, cooling/heating BTMS, Air-based or liquid based BTMS, direct versus indirect BTMS, passive and active BTMS [47]. In literature, BTMS have been designed based on control oriented modelling implementing, a linear optimal controller, feedback optimal control, PID and Rule based controllers [48] [49] [50] [51] Battery aging in for Li-Ion

49 24 Battery retirement is based according to the system s application. Scanning devices may work with batteries of 60 capacity. Most batteries are replaced at 80 capacity to prevent system failure [52]. Due to the importance of preventing failure, medical and military device s battery is replaced too soon. Manufacturers often use a cycling or stamp date system to determine when a battery should be retired; however, using this system can prove to be inefficient as the battery's health is not tested resulting in an early replacement of the battery [12]. Some factors that contribute to battery aging are overcharging and excessive discharging of the battery. These cycles create irreversible reaction forces in the ions after the solid particles in the cell are saturated [45]. Overcharging causes, a battery to significantly increase its temperature and pressure that could lead to fire, short circuits, and cell damage. In vehicles, battery failure is highly related to the driving habits of the owner. A continuous assessment of the battery is crucial to prevent system failure and allow maximum use of the battery SOC estimation techniques Extensive research has been done on SOC estimation techniques as it is critical for determining the performance of the battery as well as determining the remaining energy in the battery. This section discusses common SOC techniques found in the literature.

50 Discharge Test The discharge test is the most intuitive and reliable test to determine the SOC of the battery. In this test, the battery is fully discharged under controlled conditions (e.g. temperature). The battery s discharge time is used to calculate the SOC. The downside of this test is the time required to perform the test and the decreased of life of the battery due to the large DOD, thus this test cannot be implemented on-board a system and is left for laboratory experiments Ampere-Hour Counting This technique is the most commonly used for determining SOC. This method is based on the fact that the battery s SOC is related to the input current. Thus, if the Initial SOC is known one can add/subtract based on the current demand/supplied to the battery to determine the remaining SOC level. The following formula depicts this method [26]: SOC = SOC 0 1 t I C s dτ n t 0 Where, soc 0 is the initial SOC, C n is the nominal capacity, and I s is the discharge current. Even though it s a clever method there are some issues associated with it: High dependence on the measured current may result in high accumulated error if there are errors associated with the current measurement due to the integral action. This method may have to be re-calibrated as the battery ages. Not all current that goes in/out is stored/released due to losses.

51 26 All these issues may be solved by implementing an accurate current sensor (expensive) and setting a predefined re-calibration point with a correcting factor (losses) for charge and discharge cycle [26]. This yields a new equation: SOC = SOC 0 η i I C s dτ n t t 0 Where η i represents the correcting factor and i refers to the charge or discharge cycle. It has been well documented that if these setbacks are account for a high accuracy is attainable [53] Open Circuit Voltage This method is based on the relationship between terminal voltage, open circuit voltage and SOC. If the battery is left to rest the terminal voltage decays to the open circuit voltage (OCV). At the OCV the SOC may be derived by using look-up tables as there is a interrelation between OCV and SOC. Often this SOC is used as a recalibration point for the Ampere counting method [54]. The main drawback of this method is the required time for the terminal voltage to reach the OCV. Thus, its impractical for the use in EV technology unless the vehicle is left to rest overnight Impedance Spectroscopy This method is based on creating a set of data by testing the cell impedance over a wide range of Alternating current frequencies under different SOC levels. Then the value of SOC is inferred on-board by the use of specialized equipment and using the set

52 27 of data previously created [55] [56]. This method is not desirable due to the cost associated with the instrumentation required State Estimation Techniques In this technique, an estimation method is applied in conjunction with the Ampere- Hour Counting method to accurately estimate the SOC level of the battery. Some of the techniques include Kalman filter (KF), Extended Kalman filter (EKF), Unscented Kalman filter (UKF), Cubature Kalman filter (CKF), Smooth Variable Structure filter (SVSF), etc [57] [58] [59] [60] [35]. 2.3 Fault detection methods Another consideration when designing a BMS is fault detection and isolation. The idea is that most of the estimated parameters accuracy are dependent on the employed sensors. The purpose of this function is to attempt to identify a faulty sensor and isolate it from the estimation process [61]. In the literature, an adaptive extended Kalman filter (AEKF) was applied in conjunction with an equivalent circuit model. In this scheme, the AEKF estimate outputs from the voltage of the cells and current were compared to the battery s sensor measurements to generate residuals which were fed into a fault decision block [62]. Furthermore, a fault detection method implementing the Luenberger observer was proposed in conjunction with a two-state thermal model by [63]. Moreover, A multiple model adaptive estimation (MMAE) technique has been implemented to detect and identify several fault types while estimating for SOC [64].

53 28 CHAPTER 3 Equivalent Circuit Models

54 EQUIVALENT CIRCUIT MODELS 3.1 Introduction A Li-Ion battery pack is composed of smaller batteries called cells as they are arranged in series and parallel to achieve the required voltage and capacity. Each cell has its own SOC, capacity and internal resistance making it intuitively to represent a Li- Ion battery pack as an electric circuit. This circuit representation of the battery is known as the equivalent circuit model. This model is popular due to its simplicity and fast computing time making it a good candidate for control-oriented applications [65] [66]. Thus, making them perfect to be used and implemented in a BMS [36]. ECM are models based on electrical components such as ideal voltage sources, resistors, and capacitors to simulate the behaviour of the battery [67]. The vast majority of ECM models are semi-empirical models where the process of calculating your final outputs is based on two distinctive activities. First, the OCV with respect to SOC must be known beforehand by performing laboratory experiments where the battery is put through discharge and resting cycles [68]. Second, the ECM parameters such as resistors and capacitors must be calibrated through I/O data using parameter identification techniques such as least square methods [69]. A disadvantage of this type of the ECM is that these models do not rely on underlying physics of the battery, thus making it impossible to calculate for the power fading, capacity fading and aging effect of the battery. In this thesis, four different ECM models were identified and selected for analysis which will be detailed in the following subsections.

55 Rint Model The Rint model is the simplest ECM. It is used to approximate the output voltage of the battery based on the 2 most critical parameters: Open Circuit Voltage (OCV) based on the SOC of the battery, and equivalent Internal Resistance of the battery [70]. In simple terms, the battery is modelled as an ideal voltage source (OCV) and the resistor represent the energy loss when releasing that energy. Figure 5, depicts the Rint Model circuit. Figure 5: Rint Model Circuit Configuration [70] The output voltage of the ECM, V out, or also called terminal voltage, V t, is calculated as follows: V t = V OCV (SOC) I s R Battery A disadvantage of this model is the parameter dependence on SOC levels, and temperature. Parameter curves have to be derived offline experimentally using constant current (CC) and constant voltage (CV) charge and discharge cycles while subject to a constant temperature [71]. The experimental setup is often composed of a battery test system, a thermal chamber for environment control, and a host computer. Similar experiments are performed at different temperatures and lookup tables are created for

56 31 the models. Afterwards, the data is processed using software such as MATLAB for parameter identification. Other parameter identification techniques include genetic algorithms, and least square method [72]. Lastly, the battery output voltage suffers from hysteresis making the R Battery vary when charging and discharging [73]. Figure 6, illustrates a model of the hysteresis effect in Li-Ion batteries. Figure 6: Hysteresis model of the battery [73]. 3.3 Thevenin Model The Thevenin model is often called the 1 st order RC model, because it consists of a parallel resistor-capacitor (RC) circuit in series with a resistance, R Battery. This model is similar to the Rint model with the addition of a RC branch. Figure 7, depicts the Thevenin ECM.

57 32 Figure 7 Schematic diagram of Thevenin Model [70] The RC branch is composed of R th and C th elements which represent the polarization resistance and the transient response during charging and discharging of the battery. The polarization resistance is used so that the battery does not provide energy right away to the system. In simple words, the RC circuit represents the time the battery takes to release the total amount of power [74]. Furthermore, U th represents the voltage across the RC branch, I th represents the current passing through the C th and U L represents the terminal voltage, V t, of the ECM. Lastly, the Thevenin model may be represented by the following ODE and corresponding terminal voltage equation [75]: U th = U th R th C th + I th C th U L = U oc U th I s R battery Which may be represented into discrete state space form: [U th,k+1 ] = [1 T R th C th ] [U th,k ] + [ T C th ] I s,k

58 33 y k = OCV(SOC) RI s,k U th,k 3.4 PNGV The US government and the US council for automotive research (USCAR) established the Partnership for a New Generation of Vehicles (PNGV) model in 1993 based mainly on the polarization characteristics of the battery [76]. Moreover, this model has been adopted in the PNGV battery test Manual, which details test to be perform to batteries. The program addresses improvements in national competitiveness in manufacturing and in the implementation of energy saving innovations in passenger vehicles [77]. The PNGV model is illustrated in Figure 8. The model consists of 5 parameters: E, C 0, R 0, R 1 and C 1 which correspond to physical traits of the battery. E represents the OCV of the battery. C 1 represent the capacitance of the parallel plates and the diffusion effects of the battery. R1 represents the non-linear resistance of the battery due to the contact resistance between the plates and the electrolyte. Lastly, C 0 depicts the OCV variation generated by the accumulation of the load current and scales the battery storage capacity [76].

59 34 Figure 8: PNGV model: Circuit configuration [76]. The PNGV model may be represented by the following state space form: 1 [ U ] = [ 0 1 ] [ U 0 C ] + 0 U U 1 C 1 R [ C 1 ] I input With terminal voltage define as: U t = U 0 + U 1 + R 0 I + E With discrete forms defined as: [ U 1 0 0,k+1 ] = [ U 1,k T ] [ U 0,k ] + U R 1 C 1,k 1 [ T C 0 I T s,k C 1 ] U t,k = U 0,k U 1,k R 0 I s,k + E(SOC) 3.5 Dual Polarity The Dual Polarity (DP) model is an enhancement to the Thevenin model. The Thevenin model cannot accurately capture both of the polarization characteristics of the

60 35 battery by using one RC branch. The DP model attempts to capture the concentration polarization and the electrochemical polarization separately by using 2 different RC branches, hence its name dual polarity [70]. Figure 9, shows the DP model circuit schematic. Figure 9: DP model: Circuit Schematic [70] The DP model may be studied by breaking it into three pieces: the OCV, resistances R 0, R pa, R pc which describe the battery s internal resistance, electrochemical polarization, and polarization resistance, respectively; lastly, the capacitances C pa, and C pc implemented to characterize the transient response during the transfer of power to/from the battery and the electrochemical and concentration polarization separately [70]. Finally, the circuit behaviour may be described by the following set of equations: U pa = U pa R pa C pa + I L C pa U pc = U pc R pc C pc + I L C pc U L = U ocv U pa U pc I L R 0

61 36 Which may be represented by the following discrete state space form: [ U pa,k+1 U pc,k+1 ] = [ 1 T R pa C pa T R pc C pc] [ U pa,k U pc,k ] + [ T C pa T C pc] I s U L,k = U ocv,k (SOC) U pa,k U pc,k I L,k R Thermal Model Li-ion batteries have good performance at room temperature. If the temperature of the battery is too high or too low its performance is affected significantly. Thermal management systems ensure that the battery remains within the SOA preventing it from bursting into flames or significantly reducing its life expectancy [78]. Temperature models have been developed for thermal control purposes. Among the models found in the literature are: lumped capacitance thermal models, numerical and analytical models, and equivalent circuit thermal models [79] [80] [81]. In this thesis, the model proposed by [82] was implemented for thermal simulation of the battery. The thermal model used for determining the core and surface temperature of the cell is shown in Figure 10. This model is based on a radially distributed thermal model of a cylindrical battery cell. Using classical heat transfer theory, the heat transfer can be described by a pair of first order partial differential equations: T c = 1 C c (S r + T s T c R c )

62 37 T s = 1 C s ( T f T s R u T s T c R c ) where T s and T c are the surface and core temperature, respectively. T f is the coolant temperature of the system (Input). Furthermore, S r = (OCV V T )I input, where, OCV is the open circuit voltage, V T is the terminal voltage and I s is the supplied current to the model. Figure 10: Battery Thermal Model [82] The core s resistance models the amount of heat flux leaving or entering the core and the capacitor models the core s ability to store heat. The surface of the battery was modeled similarly, yielding four parameters depicted in Table 1: R c, C c, R u, C u. Table 1: Thermal Model Parameters Thermal Model Core C core R core Surface C surface R surface States T core T surface Inputs I s T f

63 38 [82]: The following is the state space representation of the two differential equations 1 1 R A = c C c C c R c 1 1 ( ) ( R c C s C s R c R s ) B = ( (OCV V t ) C c C s R s ) u = [I s T f]

64 39 CHAPTER 4 Electrochemical Model

65 Electrochemical Model 4.1 Introduction As previously mentioned, ECMs do not provide sufficient information regarding the battery s state due to their lack of physics in the model, resulting in a model that cannot yield information about the battery s power fading, capacity fading and aging effect of the battery. To overcome this, electrochemical models were developed. These models are able to provide more information regarding the battery s state when compared to ECMs [83]. Electrochemical models are based on the battery s dynamics when charging and discharging. To develop these models, the battery is often discretized along spatial planes. For a one dimensional (1D)-spatial model, the battery is broken down into three regions, namely, negative electrode, separator, and positive electrode. Furthermore, these three regions are submerged in a liquid solution known as electrolyte. The negative electrode (anode) is often some Li-ion chemistry whereas the positive electrode (cathode) remains as graphite. Figure 11, illustrates the 1-D-spatial model of a battery [24].

66 41 Figure 11: 1D-spatial Model of a Li-ion Battery [84] At full charge, the negative electrodes are saturated with lithium ions in solid form, that is, the concentration inside each electrode is at its maximum. On the other hand, the positive electrodes are fully depleted of lithium ions. During a discharge cycle, the lithium ion inside each negative electrode travels to the surface of the electrode and diffuses (changes to liquid form) to travel across the separator region using the electrolyte. The diffusion is possible because the electrodes are porous. Once the lithium ion reaches a positive electrode, it diffuses from the surface into the electrode and changes to solid form, thus increasing the concentration of the positive electrode. The opposite happens during a charging process. The process by which the ion enters the electrode is known as intercalation. It is evident that a relationship between concentration of lithium and SOC can be derived. For each side (positive/negative), it is said to have a 100 SOC when its saturated and 0 SOC when its fully depleted [24].

67 42 It must be distinguished that the concentration of lithium ions at the surface of the electrode is often related to the immediately available power of the battery and the amount of lithium concentration inside the electrode is known as the bulk concentration of the battery, thus total available power of the battery [24]. Furthermore, at low current rates, the surface concentration of lithium is approximately the same as the bulk concentration. However, at high current rates, the diffusion phenomena cannot keep up with the consumption of surface lithium ion concentrations, resulting in a lower concentration of lithium ion at the surface than inside the electrode. Lastly, this difference in reactions causes the battery to behave as a dynamic system where gradients are generated at all regions and within the electrode [24]. Based on the previous analysis of the battery s operation, the battery can be modelled as a dynamic system with input current and output terminal voltage, where the terminal voltage may be measured and the internal dynamics of the battery are mainly governed by diffusion phenomena [85]. Moreover, state of interest inside the battery are the lithium concentrations at the different regions and the SOC of the battery may be determined by calculating the bulk concentration of lithium ions. Lastly, the bulk concentration may be correlated to the terminal voltage of the battery. A measurement of SOC based on solid lithium concentrations may be defined as [86]: SOC(t) = c s,average(t) c s,max (1) where, c s,average (t) is the average concentration of lithium ions in solid form (from the core of the electrode to the surface) and c s,max is the maximum is the saturation amount of lithium ions.

68 43 The following section will describe the full dynamics of the battery with respect to the gradients of concentration across all the regions aforementioned. 4.2 Full Order Electrochemical Model (Doyle-Fuller-Newman (DFN) model) Electrochemical Li-ion battery models are often derived from porous electrode and concentration solution theories [87]. These theories were developed by Newman, Tiedemann, and Doyle in which the charge/discharge process of a cell structure was mathematically described using a 1-D spatial plane [87]. The 1-D model describes the cell s dynamics along the x-axis and ignores the other two axes. This approximation is accurate for cell structures with a large cross sectional area and low current demand/supplied due to difference in sizes between the x-axis and the other two axes [24]. The following sections will provide with a brief summary of the governing equations of lithium concentrations across the discretized three regions Transport in Solid Phase Across the four regions, lithium may exist in two disjoint states, that is, the solid phase inside the electrode material and the liquid phase in the electrolyte solution [88]. For deriving the governing equations, the solid and electrolyte phase are superimposed. Figure 12 illustrates the electrode with its corresponding boundary.

69 44 Figure 12: Electrode as a sphere approximation with radius R k [24] Fick s law of diffusion may be used to describe the amount of concentration of lithium inside the electrode material with the assumption that the electrode has a spherical shape [24] [88]: c s,k (x, r, t) = D s,k c s,k(x, r, t) t r 2 r (r2 ) r with boundary and initial conditions (2) c s,k ( D s,k r ) (3) = 0 r=0 c s,k ( D s,k r ) (4) = J k (x, t) r = R s,k c s,k (x, r, 0) = c s,k,0 (5) where, k refers to either positive or negative electrodes Transport in electrolyte

70 45 For the concentration of lithium in the electrolyte, a governing equation was derived based on the changes in the gradient diffusive flow of the ions. The following formula describes the dynamics [24] [88]: c e,k (x, t) ε k = t x (D c e,k (x, t) eff,k ) + a x k (1 t + )J k (x, t) where k may represent the 3 regions: positive side, separator, or negative side. (6) The boundary conditions of (6) also account for zero ion-flux at the current collectors located at the edge of each side of the battery. The boundaries are given by: c e,p ( D eff,p x ) c e,n = ( D eff,n x ) = 0 x=0 x=l Furthermore, four additional boundary equations are required to describe the (7) dynamics at each interface (electrode-separator, etc.). These boundaries are based on continuity of the flux and concentration of the electrolyte [24] [88] [89]. c e,p ( D eff,p x ) c e,s = ( D eff,s x=l x ) = 0 (8) x=l+ p p c e,s ( D eff,s x ) = ( D c e,n x=(l p +L s ) eff,n x ) = 0 (9) x=(l p +L s ) + With initial conditions: (c e,p ) x=lp = (c e,p ) x=lp + (10) (c e,p ) x=(lp +L s ) = (c e,p) x=(lp +L s ) + (11) c e,k (x, 0) = c e,k,0 (12) Where D eff,k is the effective diffusion coefficient and may be calculated using the brugg Bruggman relation D eff,k = Dε k k that accounts for the path of the ions follows through the porous media. Furthermore, a k refers to the specific electrode surface area and be defined as [24] [88] [89]:

71 46 a k = 3 (1 ε R k ε f,k ) s,k Where ε refers to the volume fraction of the electrode. Moreover, equation (13) (13) accounts for the porosity of the electrode Electrical Potential The following equations are based on the relationship between the electrode potential and the current inside the electrode. Ohm s law. The conservation of charge in the solid phase of the electrode is described using 2 Φ s,k (x, t) σ eff,k x 2 = a k FJ k (x, t) With boundary conditions at the current collectors being proportional to the (14) current density [24] [88] [89]: Φ s,p ( σ eff,p x ) x=0 Φ s,n = ( σ eff,n x ) = I x=l Φ s,p ( σ eff,p x ) Φ s,n = ( σ eff,n x ) = 0 x=l x=(l p p +l s ) The effective electronic conductivity may be defined as: (15) (16) σ eff,k = σ k (1 ε k ε f,k ) (17) Combining Kirchhoff s law with Ohm s law in the electrolyte phase yields: σ eff,k Φ s,k (x, t) x k eff,k Φ e,k (x, t) x + 2k eff,k(x, t)rt (1 t F + ) lnc e,k = I x (18) Where k eff,k and t + are functions of electrolyte concentration but often t + is assumed to be constant and k eff is approximated using the Bruggman relation with a polynomial.

72 47 The boundary conditions for (18) based on continuity for Φ e,k may be defined as: ( k Φ e,p eff,p x ) Φ e,n = ( k eff,n x ) = 0 x=0 x=l (19) ( k Φ e,p eff,p x ) Φ e,s = ( k eff,s x=l x ) = 0 x=l+ p p (20) Φ e,s ( k eff,s x ) = ( k Φ e,n x=(l p +L s ) eff,n x ) = 0 (21) x=(l p +L s ) + Taking into account that no solid phase potential Φ e,k (x, t) and molar flux J k (x, t) exist in the separator region, results in the removal of these terms from equations (6), and (17) for the separator regions Butler- Volmer Kinetics The molar flux J k (x, t) depends on the concentration c s,k of lithium ions in the electrode k, the concentration of lithium ions in the electrolyte, and the intercalation over-potential μ s,k (x, t) using the Butler-Volmer equation [24] [88] [89]. The equation may be represented as: μ s,k (x, t) = Φ s,k (x, t) Φ e,k (x, t) U k (θ k (x, t)) (22) Where U k represents the open circuit potential, specific to the battery chemistry. Furthermore, the open circuit potential is often derived in laboratories. Moreover, θ k (x, t) is the ratio of concentration at the corresponding electrode. Similar to equation (1) it may found by using the following equation:

73 48 θ k (x, t) = c s,k,surf(x, t) c s,k,max (23) Lastly, the Butler-Volmer equation describing the relationship between the current density, concentrations, and over-potential is given by Newman and Thomas- Aleya [24] [88] [89]: J k (x, t) = K k (c s,k,max c s,k,surf ) 0.5 (c s,k,surf ) 0.5 c 0.5 e,k (x, t) [e ( 0.5F R gas T μ s,k (x,t)) (24) 0.5F ( R e gas T μ s,k (x,t)) ] In summary, the electrochemical model is defined by equations (2), (6), (14), (18), (24) and the corresponding open circuit potential equations of the battery. Taking into account positive and negative sides, the final model is defined by fourteen nonlinear partial differential algebraic equations (PDAE) with 14 unknowns [89], that is, c s,p, c s,n, c e,p, c e,s, c e,n, Φ s,p, Φ s,n, Φ e,p, Φ e,s, Φ e,n, U p, U n, J p, J n. Based on the amount of PDAEs that are required to solve for the full electrochemical model, it is evident that this model requires significant amount of computational power. Furthermore, its computational time is highly dependent on both time sample and space discretization. Lastly, Figure 13 depicts the governing equations of this models at each region.

74 49 Figure 13: Governing equation of DFN model at each region [89] 4.3 Single Particle Model (SPM) One approach to reduce the complexity and computational power required to solve the full electrochemical model is to coarse the discretized 1D-spatial domain as much as possible, resulting in a model that only has one node for each region, that is, one node for the positive electrode, one node for the separator and one node for the negative electrode [90]. This reduced model is known as the single particle model (SPM). Figure 14 depicts the SPM model.

75 50 made [90]: Figure 14: SPM model of a Li-ion Cell [87] To further reduce the dynamics of the model, the following assumptions are Quantities at the nodes are averages over the entire domain. The spatial and time characteristics at the separator region are approximately 0, that is, c e x 0, and c e t 0. This assumption is valid if the applied current, I, is small or the ionic conductivity, k k, is very large. Furthermore, this results on a constant concentration of lithium ions in the electrolyte. All unknown states are assumed to be scalar and uniform. Thermal effects are ignored These assumptions lead to a significantly reduced model that can be divided into the following calculations: Spherical particle diffusion calculation, SOC calculation, and voltage calculation.

76 Governing Equations (PDE) [90]. The spherical particle diffusion is governed by two partial differential equations c k s (x, t) = 1 t r 2 r (D s k r 2 c s k (x, t) ) r (25) With boundary conditions: ( c s k r ) = 0 r = 0 (26) ( c s k r ) = J k,t n (t) = I(t) (27) r = R D k p s Fa k L k And Initial Condition: c s k (r, 0) = c s 0,k (r) (28) Where k refers to the positive or negative electrodes of the SPM. The output voltage, V(t), is given by: V(t) = Φ s0 + Φ s0 (29) where Φ s0 + and Φ s0 are the electric potential in the solid positive and negative electrode, respectively. Furthermore, these electric potentials are given by: Φ k s0 (t) = 2RT I(t) F sinh 2a ( k L k r eff c 0 e c k ss (t) k (c ss,max + U k (c k ss (t)) + R f k I(t) c k a ss (t)) k L k ) (30)

77 52 Where k represents either positive or negative electrode, and c k ss (t) is the surface concentration of lithium ion, which may be approximated by evaluating the solid concentration at the edge of the electrode, e.g. c s k (R p k, t). 4.4 Three-Parameter Single Particle Model (3-Parameter SPM) Even though the model described on section 4.3 greatly reduces the model described on section 4.2, it is undesirable to solve for a PDE due to its high computational time. The three parameter SPM reduces the PDE of the SPM to ordinary differential equations (DAE) [91]. The reduction starts by assuming that the lithium concentration of the single particle may be represented with the following polynomial [91]: c s (r, t) = a(t) + b(t) ( r 2 ) + d(t) ( r 4 (31) ) R p R p Equation (31) takes into account the boundary condition of equations (26). Moreover, by using equation (31), equation (27) becomes [91]: 2D s R p b(t) + 4D s R p d(t) = j n (t) (32) The volume average concentration and volume averaged concentration flux may be express as [91]:

78 53 c s (t) = R p r = 0 3 r2 R p 2 c s(r, t)d ( r R p ) (33) q s(t) = R p r=0 3 r2 R p 2 d dr c s(r, t)d ( r R p ) (34) Evaluating equation (31) at the surface (r = R p ) yields, c s (r, t) = a(t) + b(t) + d(t) (35) The coefficients on equation (35) may be solve by using equations (33 35), resulting in [91]: a(t) = 39 4 c ss(t) 35 4 c s(t) + 3q s(t)r p (36) b(t) = 35c ss (t) + 35c s(t) + 10q s(t)r p (37) c(t) = c ss(t) c s(t) 7q s(t)r p (38) Substituting equations (36 38) into equation (31), results in: c s (r, t) = 39 4 c ss(t) 35 4 c s(t) 3q s(t)r p + ( 35c ss (t) + 35c s(t) + 10q s(t)r p ) ( r 2 ) R p (39) + ( c ss(t) c s(t) 7q s(t)r p ) ( r 4 ) R p Volume averaging equation (25), results in [91]: R p r=0 3 r2 R2 [ c s(r, t) p t 1 r 2 r (D c s(r, t) s )] d ( r ) = 0 r R p Substituting equation (39) with (40), results on the molar flux equation [91]: (40)

79 54 d dt c s(t) = 3 J n(t) (41) R p Differentiating equation (25) and volume averaging we obtain: c 3 r2 [ s (r, t) 1 R2 t r 2 r (D c s(r, t) R p s )] r d ( r ) r=0 p r R p Substituting equation (39) into equation (42), results in: = 0 (42) d dt q s(t) + 30 D s R p 2 q (t) J n (t) R p 2 = 0 (43) Lastly, the surface concentration of lithium in the particle c ss (t) is obtained by substituting equations (37 38) into (32) [91]: c ss (t) = c s(t) + 8R p 35 q s(t) R p D s J n (t) (44) The model s state space is shown below: A = D s,n R p,n D s,p [ R 2 p,p ] 3 R p D p s Fa p L p 45 2R 2 p D p s Fa p L p B = 3 R p D p s Fa p L p 45 [ 2R 2 p D p s Fa p L p ]

80 55 In summary, the three-parameter SPM is given by equations (41), (43), and (44). Initial conditions for (41), and (43) during a single discharge may be approximated to c s (r, 0) = c 0,k s (r) and q s(0) = 0. Figure 15 depicts the implementation of the three parameter SPM. Figure 15: Three Parameter Model as an ECM [91]

81 56 CHAPTER 5 Estimation Methods

82 Estimation Methods In this chapter, all the estimation strategies used for estimating the SOC of the battery are discussed. The following filters were implemented in this thesis: Kalman Filter (KF), Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF). 5.1 Estimation Theory Estimation theory is a branch of statistics that focuses on generating estimated values of parameters using measured empirical random data [92]. Estimation theory has been applied to various fields such as science, and engineering. Furthermore, it is most likely that some form of estimation algorithm is implemented in most of the devices that are used in our daily life. In industry, there is high demand on techniques that allow to extract quality information from measurement signals. Quality information allows for cost reduction and increase of the overall robustness and reliability of the system [93]. This type of problem is called the parameter estimation problem. The state estimation problem consists of three parts: the physical system, the measurement system, and the state estimator. Figure 16 shows the general block diagrams of the state estimation process, where both the physical system and the measurement system will add noise to the signal Z which will be used by the state estimator to derive an estimate of the true states of the physical system.

83 58 Figure 16: State Estimation General Block Diagram [92] In this thesis, the physical system to be analyzed is the battery and the states will vary depending on the selected ECM but may include SOC, and values of the polarization effects (capacitors) as it can be seen in Chapter 3. The KF is the selected state estimation tool to be implemented in conjunction with the battery models. These estimation tools will be discussed in the following sections. 5.2 Kalman Filter (KF) The KF was introduced by Rudolf Emil Kalman back in 1960 [94]. Kalman proposed an optimal recursive estimator solution to the discrete data linear filtering problem. Even though Kalman s algorithm is based on advanced stochastic theory, the end result can be summarized into two sets of equations: time update equations (Predictor) and the measurement update equations (Corrector). Moreover, the KF equations can be easily implemented in digital systems [95]. Figure 17 depicts the Predictor-Corrector algorithm of the KF. The states of the system are estimated using the system model and input, or in other words the estimate is projected forward in time of the current state estimate to obtain the a priori, meaning

84 59 prior to state estimates for the next time step. A correction term is then added based on the innovation term, thus forming the updated or a posterior, meaning after the fact, state estimates. The corrector step is responsible for the feedback, in other words, for adding the new measurement into the a priori estimate to generate an improved version of it [94] [95] [96]. Figure 17: Predictor-Corrector Method for KF [96]. KF of a dynamic model and measurement model can be described by the following two equations [97]: x k+1 = A k x k + B k u k + w k z k+1 = C k+1 x k+1 + v k+1 Where the variables A, B, C correspond to the system matrix, input matrix and the output matrix, respectively. Furthermore, x is the system state vector, z is the measurement output, w is the system noise, v is the measurement noise and k represents the sampling time. The KF was developed with the assumption that w and v

85 60 are independent of each other, and have zero mean Gaussian distribution. The following equations depict these two properties of the noise [95]. p(w)~n(0, Q) p(v)~n(0, R) Where Q and R are the corresponding system and measurement noise covariance, respectively [95]. The following equation sets describe the process of the time and measurement update within the KF algorithm: Time Update (Predictor) { x k+1 = A k x k + B k u k P k+1 k = A k P k k A T k + Q k K k = P k+1 k C T k [C k P k+1 k C T k + R k ] 1 Measurement Update (Corrector) { x k+1 k+1 = x k+1 k + K k [z k C k x k+1 k ] P k+1 k+1 = [1 K k C k ]P k+1 k In the time update stage, P k+1 k represents the prediction of the state error covariance matrix. High values of P indicate high levels of uncertainty whereas low values of P indicate low levels of uncertainty or in other words high level of confidence about the state estimate [98]. In the measurement update stage, K k defines the Kalman gain, which is used to update the state estimate and [C k P k+1 k C T k + R k ] 1 is called the Innovation covariance sometimes referred as S k+1. Lastly, the a posteriori state error covariance is updated. This process is repeated iteratively throughout the linear state estimation problem.

86 61 Finally, the KF yields the optimal solution to the linear estimation problem by minimizing the mean-square error of the state estimate; however, some assumptions have to be hold for it to yield the optimal solution. These assumptions are listed below [96]: The system, gain, and measurement matrices are known. The system and measurement noise are Gaussian and white. The system and measurement covariance are exactly known. Even though it is rarely for all of these assumptions to be satisfied, the KF has proven to yield good results [95]; however, the KF may become unstable if the assumptions are violated significantly [99]. 5.3 Extended Kalman Filter (EKF) Extensive research has been done to the KF and it has proven to be a reliable mathematical tool that enhances the computational efficiency of the systems. Furthermore, its cost for implementation is significantly cheaper than enhancing the system with more accurate sensors. In reality, most of the systems are highly nonlinear systems. Therefore, it was natural for the KF to be extended to tackle nonlinear systems. The nonlinear system may be described by the following two equations: x k+1 = f(x k, u k, w k ) (5.3.1) z k+1 = g(x k+1, u k, v k+1 ) (5.3.2)

87 62 Where equation (5.3.1) represent the state equation at the previous time step, k to the future time step k + 1, and equation (5.3.2) represents the output equation which relates the state to the output, both w k and v k represent the system and measurement noise and are assumed to be white noise with zero mean and known covariance Q and R respectively. The EKF linearizes the estimation around the current estimate using partial derivatives (1 st Order Taylor series expansion) of the system and measurement functions to compute estimates. By linearizing, the EKF may introduce instability as it may overlook system nonlinearities that were not capture with the model [100] [101] [102]. The following equations depict the approximation: f(x k, u k, w k ) f(x k, u k, 0) + f(x k, u k, 0) x k x k =x k (x k x k) (5.3.3) g(x k+1, u k+1, v k+1 ) g(x k+1, u k+1, 0) + g(x k+1, u k+1, 0) x k+1 x k =x k+1 (x k+1 x k+1 ) (5.3.4) Substituting equations (5.3.3) and (5.3.4) into equations (5.3.1), (5.3.2), yields the matrices A k 1 and C k A k = f(x k, u k, 0) x k x k =x k+1 k C k = g(x k+1, u k+1, 0) x k+1 x k =x k+1 k The new time update and new measurement update stage equations are [96]:

88 63 Time Update (Predictor) { x k+1 k = f(x k k, u k, w k ) P k+1 k = A kp k k A kt + Q k K k+1 = P k+1 k C k+1 T T [C kp k+1 k C k+1 + R k+1 ] 1 Measurement Update (Corrector) { x k+1 k+1 = x k+1 k + K k+1 [z k+1 h(x k+1, u k+1, v k )] P k+1 k+1 = [1 K k+1 C k+1 ]P k+1 k The EKF has proven to yield good results when the consist consists of low order nonlinearities [103]. For systems with high order nonlinearities other forms of the KF has proven to outperform the EKF. 5.4 Unscented Kalman Filter (UKF) The UKF is a form of sigma point Kalman filter (SPKF) which is based on a statistical linear regression strategy which linearizes the nonlinear model statistically [102]. In simple steps, SPKF techniques generate a certain number of points referred as sigma points from the projected probability distribution of the states. These points are then projected using the nonlinear system model, to obtain the a posteriori estimate for probability distribution. The main difference between the SPKF and the EKF is that the SPKF does not calculate the Jacobian matrices of the system in an attempt to linearize the system often yielding better results than the EKF if the system consists of high order nonlinearities. The main reason behind the higher accuracy is because the SPKF attempts to captures more nonlinearities than the EKF. In the literature, one of the most popular type of SPKF is the UKF. Furthermore, it has been shown that the UKF performs significantly better than the EKF [102] [104]. The following is a summary of the UKF algorithm [105]:

89 64 Intialization: Prediction: X 0,k k = x k k (5.4.1) W 0 = k (n + k) x k k + ( (n + k)p k k ) W i = 1 [2(n + k)] X i+n,k k = x k k ( (n + k)p k k ) W i = 1 [2(n + k)] i i (5.4.2) (5.4.3) (5.4.4) (5.4.5) (5.4.6) Update: X i,k+1 k = f(x i, u k, w k ) (5.4.7) 2n x k+1 k = W i,k k, u k i=0 2n P k+1 k = W i (X i,k+1 k x k+1 k )(X i,k+1 k x k+1 k ) T i=0 (5.4.8) (5.4.9) Z i,k+1 k = h(x i,k+1 k, u k ) (5.4.10) 2n (5.4.11) Z k+1 k = W i Z i,k+1 k i=0 2n P zz.k+1 k = W i (Z i,k+1 k z k+1 k )(Z i,k+1 k z k+1 k ) T i=0 2n P xz.k+1 k = W i (X i,k+1 k x k+1 k )(Z i,k+1 k z k+1 k ) T i=0 (5.4.12) (5.4.13) 1 K k+1 = P xz.k+1 k P zz.k k (5.4.14) x k+1 k+1 = x k+1 k + K k+1 (z k+1 z k+1 k ) (5.4.15) T P k+1 k+1 = P k+1 k K k+1 P zz,k+1 k K k+1 (5.4.16)

90 65 The set of initialization equations first approximate the n-dimensional random variable x k with mean x k k and covariance P k k by using 2n + 1 sigma points. Then the first n number of sigma points are calculated using equations (5.4.3) and (5.4.4) while the remaining set of points are calculated using (5.4.5) and (5.4.6). The parameter k is referred as a design value, often hold a value less than 1. The factor ( (n + k)p k k ) i represents the i th row or column of the matrix square root of (n + k)p k k, and W i is the weight that is associated with the i th sample point [105]. The prediction stage set of equations first propagate the sigma points through the nonlinear system model using equations (5.4.7) and (5.4.8) with their corresponding weights to calculate the predicted state estimate. Afterwards, the state error covariance of the prediction is calculated using (5.4.9). Lastly, the next set of sigma points are propagated through the nonlinear measurement model (5.4.10) and (5.4.11). The update stage starts by calculating the innovation using (5.4.12). Similarly, the cross covariance, meaning between the state and measurement, is calculated using (5.4.13), followed by calculating the Kalman gain K k+1 using (5.4.14). Lastly, the a posteriori state estimate and their corresponding state error covariance is updated based of (5.4.15) and (5.4.16). Even though all these calculations may seem extensive, the UKF has about the same computational demand as the EKF because it does not linearize the nonlinear functions, which is computationally expensive [102]. 5.5 Interactive Multiple Model Strategy

91 66 In reality, many systems change dynamics due to different operating regimes. For example, if the level of liquid inside a tank is to be estimated, a model representing a static level in the tank will differ from a model representing the dynamics of a tank with liquid being poured into it. For cases such as this, it would be beneficial to have an adaptive estimation strategy where the algorithm adapts itself to certain types of uncertainties or models to minimize the state estimation error. In this thesis, the Interactive Multiple Model (IMM) will be discussed. The IMM is a type of Multiple Model (MM) algorithm which has the characteristics described previously [106]. In the IMM algorithm the estimate is calculated for each possible current model, with a mixed initial condition [106]. The input to the filter matched to M j is obtained from an interaction of the r filters, which consists of the missing of the estimates x i,k k and weightings μ i j,k k, referred as mixing probabilities [106]. Figure 18, depicts the flow of the IMM algorithm with corresponding mixing stage and mode probability stage.

92 67 Figure 18: Estimation Algorithm for the IMM [106] A summary of the IMM algorithm broken down into 5 steps is shown below [96]: Calculation of the mixing probablities: μ i j,k k = 1 c j p ij μ i,k (5.5.1) mixing stage: r c j = p ij μ i,k i=1 (5.5.2) r x 0j,k k = x i,k k μ i j,k k i=1 r P 0j,k k = μ i j,k k {P i,k k + (x i,k k x 0j,k k )(x i,k k x 0j,k k ) T } i=1 (5.5.3) (5.5.4)

93 68 mode matched filtering: Λ j,k+1 = N(z k+1 ; z j,k+1 k, S j,k+1 ) (5.5.5) 1 1 Λ j,k+1 = exp ( 2 e T j,z,k+1e j,z,k+1 k ) S 2πS j,k+1 j,k+1 Abs (5.5.6) mode probability update: μ j,k = 1 c Λ j,k+1 p ij μ i,k r i=1 r r c = Λ j,k+1 p ij μ i,k j=1 i=1 (5.5.7) (5.5.8) state estimate and covariance combination: r x k+1 k+1 = μ j,k+1 x j,k+1 k+1 j=1 r P k+1 k+1 = μ j,k+1 {P j,k+1 k+1 j=1 + (x j,k+1 k+1 x k+1 k+1)(x j,k+1 k+1 x k+1 k+1) T } (5.5.9) (5.5.10) The first step calculates the mixing probabilities, in other words the probability of the system in the current model, i, and the probability of switching to the other model, j, at the next step. These are calculated by using equations (5.5.1) and (5.5.2). The calculated mixing probabilities, μ i j,k k, in conjunction with the previous matched states x i,k k and covariance s P i,k k are used in the mixing stage for the filter

94 69 matched to M j. The mixed initial conditions are calculated using equations (5.5.3) and (5.5.4). The following step is mode-matched filtering, which uses the output of the mixing probabilities stage as inputs to the filters in this stage. In addition, each filter requires the measurement z k+1 and input to the system u k. Using all this information, the likelihood functions are calculated using equations (5.5.5) and (5.5.6). Next, the mode probability is updated using the likelihood functions calculated on the previous stage. These equations are (5.5.7) and (5.5.8). Lastly, like all KF algorithms the overall state estimate and covariance are calculated using equations (5.5.9) and (5.5.10). These last two equations represent the output of the recursive algorithm ((5.5.1) - (5.5.8)). The IMM has been successfully applied for Nonlinear target tracking where the system was modelled using two models one for coordinated turns and one for linear motion [107]. The IMM has also been applied to fault detection problems and diagnosis [108]. In this thesis, the IMM will be used for fault detection.

95 70 CHAPTER 6 Computer Experiments

96 Computer Experiments In this chapter, all the computer experiments are presented including parameter optimization results for the ECMs: Rint, Thevenin, PNGV, Dual polarity. Moreover, the models were simulated using the Urban Dynamometer Driving Schedule (UDDS) and the Dynamic Stress Test (DST) current profiles, which are industry standard tests for EV Li-ion batteries. Lastly, the fault detection algorithm is discussed in the final section of this chapter. 6.1 Model Calibrations In this section, all the battery models were calibrated. The ECMs parameter were calibrated based on I/O experimental data found in the literature. The electrochemical models were calibrated based on a DFN model code with experimental data found in the literature ECM Parameter Optimization There is extensive research regarding the battery s behaviour and response to changes of SOC levels [109]. This behavior is reflected on the ECMs by representing the parameter values with a curve dependent on SOC. The curve or values at different levels of SOC has to be derive for each parameter in the ECM. In this thesis, the parameter values were optimized by modifying the MATLAB Li-ion battery optimization model file [110]. The MATLAB file uses input data to optimize the model s parameter values accordingly to fit the output data. Figure 19 depicts the Input current profile used and

97 72 Figure 20 illustrates the output voltage profile of the battery used for parameter optimization of the models. Figure 19: Experimental Input Current Profile [110] Rint Model Figure 20: Experimental Output Voltage Profile for Optimization [110]

98 73 As per section 3.2 of this thesis, the Rint model has 1 parameters to be optimized, R Battery. Figure 21 illustrates the Simulink model used for optimization of the Rint model parameters. Figure 21: Rint Model in Simulink Used for Parameter Optimization Table 2 summarizes the values of the internal resistance obtained after running the simulation. Table 2: Optimized Battery Resistance at different SOC levels SOC OCV(V) R Battery (Ω)

99 74 As it is shown on Table 2, the R battery may be approximated by setting the resistance to Furthermore, the corresponding OCV curve was derived using the MATLAB curve fit toolbox, which may be access using the command cftool. Figure 22, illustrates the OCV curve used for the simulation of the Rint model. Figure 22: Rint model OCV curve for all SOC levels Figure 23 shows the terminal voltage obtained by using the obtained parameter value. It is clear from Figure 23 that the Rint model is unable to capture all the nonlinearities of the battery; however, it still yields good approximations. The main problem with this model is that there is no relaxation of the battery being modelled which reflects as straight output voltages or instantaneous jumps in voltage.

100 75 Figure 23: Rint Simulated Output Voltage Table 3, shows the Root Mean Square Error (RMSE) of the output voltage between the Rint model and the experimental data. Table 3: Rint model RMSE Output Voltage estimate Root Mean Square Error V out Thevenin Model For the Thevenin model the following parameters were optimized: R th, and C th. Figure 24, depicts the Simulink model used for parameter optimization of the Thevenin model.

101 76 Figure 24: Thevenin Simulink Model used for optimization of the model's parameters The optimized parameter values found for the Thevenin model using the Simulink model are shown on Table 4. Table 4: Thevenin Model Parameter Values SOC OCV(V) R Battery (Ω) R th (Ω) C th (F) , , , , , , , , , , , As shown on Table 4, the parameter values for R th, and C th vary slightly for which no single value may be assign to represent the parameter for all levels of SOC. The parameter values were represented by a curve using the MATLAB curve fit toolbox with command call cftool. A 9 th degree polynomial was set to represent each

102 77 parameter. Figure 25 and Figure 26 depict the parameter curves derived for C th and R th respectively. Figure 25: C th Parameter Curve at various levels of SOC Figure 26: R th Parameter Values at various levels of SOC

103 78 Next, the model was simulated using the equations described previously in section 3.3 of this thesis. Figure 27 illustrates the output voltage of the Thevenin model compared to the experimental voltage used for calibration. It can be seen that the Thevenin model do not match the bottom features of the experimental voltage profile. This inaccuracy may be explained by the time response that was added to the model (Capacitor) which does not allow for instantaneous drops of voltage. Table 5 shows the corresponding RMSE value of the Thevenin model. Figure 27: Experimental Output Voltage Vs. Thevenin Model Voltage Output Table 5: Thevenin model RMSE Output Voltage estimate Root Mean Square Error V out PNGV Model

104 79 Similar to the previous models, the PNGV model was optimized using the Simulink model shown in Figure 28. The parameters optimized were: R battery, C pn, R pn, and 1 U which values are illustrated on Table 6. The PNGV capacitance values for ocv C pn and 1 U vary significantly for all levels of SOC, making the system nonlinear. ocv Figure 28: PNGV Simulink Model for Parameter Optimization Table 6: Optimized Parameter Values for PNGV model SOC OCV(V) R Battery (Ω) C pn (F) R pn (Ω) 1 (F) , , , ,761, , ,386, , ,846, , ,842, , ,122, , ,095, , ,152, , ,681, , ,791, , ,430, U ocv

105 80 The parameter curves for C pn and 1/U ocv were derive using the MATLAB curve fit toolbox with a 9 th degree polynomial. Figure 29 and Figure 30 show the curve of the two parameters. Figure 29: Optimized 1/U ocv Curve at all levels of SOC Figure 30: Optimized C pn Curve

106 81 The output Voltage of the PNGV model compared to the experimental voltage is shown on Figure 31. Furthermore, the RMSE of the output voltages is shown in Table DP Model Figure 31: Experimental Output Voltage Vs. PNGV Model Output Voltage Table 7: PNGV model RMSE Output Voltage estimate Root Mean Square Error V out The DP model parameters to optimize were: C pa, R pa, C pc, R pc and R battery. The Simulink model depicted on Figure 32 was used for optimization of the parameter values. The corresponding values of the parameters found after optimization are displayed on Table 8. As it can be seen on Table 8, the capacitance values of C pa and C pc vary significantly for all SOC levels.

107 82 Figure 32: Simulink DP model for Parameter Optimization Table 8: Optimized Parameter Values for the DP model SOC OCV(V) R Battery (Ω) R pa (Ω) C pa (F) R pc (Ω) C pc (F) , , , , , , , , , , , , , , , , , , , , , ,051, As shown on Table 8, the parameters with significant changes are C pc and C pa for which two polynomial curves were derived using the MATLAB curve fit toolbox. The curves are depicted on Figure 33 and Figure 34.

108 83 Figure 33: C pc polynomial curve at all levels of SOC Figure 34: C pa polynomial curve at all levels of SOC Furthermore, the output voltage of the DP model Vs. the experimental voltage is depicted on Figure 35. Moreover, the corresponding RMSE value is shown on Table 9.

109 Thermal Model Figure 35: Experimental Output Voltage Vs. DP model Output Voltage Table 9: DP model RMSE Output Voltage estimate Root Mean Square Error V out For the thermal model the parameters of interest are depicted on Table 10. This thermal model was used across all simulated ECMs. Furthermore, a value of 25 C was used for T f to simulate the battery inside a laboratory with air conditioner on. The parameter values used for simulation are shown on Table 10 and were adapted from [82]: Table 10: Thermal model parameter values [82] Name Symbol Unit Value Conduction R core C/W 1.26 Resistance Heat Capacity at C core J/K 270 Core Convection R surface C/W 1.5 Resistance Heat capacity at skin C surface C/W 19

110 Final Remarks regarding calibration The selected current profile represents a full discharge of the battery (e.g. SOC from 100 to 0). The estimated SOC discharge is based on coulomb counting and is the same across all models. Figure 47 depicts the estimated SOC of the battery with initial value of 100 based of the current profile for calibration described in section 5.1. SOC k+1 = SOC intial/k T sample C Battery capacitance Figure 36: SOC estimate using coulomb counting method Lastly, it is noticeable that the only parameters that are sensitive to different SOC levels are the capacitors in each ECM. This result was expected as the capacitors were introduced in the model to capture the nonlinearities of the battery Electrochemical Model Calibration As covered in chapter 4 of this thesis, electrochemical models required significant amount of parameter values which are dependent on the battery s chemistry

111 86 and physical characteristics [91]. To calibrate the electrochemical models of this thesis, the DFN model code was adapted from [111]. The code models the full order model by discretizing the solid diffusion (electrode) PDEs along the radius of the electrode, r, using 3 rd order Padé approximations [112] [84]. Furthermore, the remaining PDEs are solved along the x axes using central difference theorem [113], resulting in a conservation of lithium mass. The end result is a system of DAEs. Figure 37 shows the profile of the input current used for the DFN model code and Figure 38 shows the corresponding output voltage of the model. Figure 37: Input current for the electrochemical models [111]

112 87 Figure 38: Output Voltage of the DFN model To obtain this result the following assumptions were made: Initial voltage of the battery was 3.9 Volts, meaning an initial concentration of on the negative side and on the positive side of the battery. Furthermore, the output SOC of the battery is shown on Figure 39 and Figure 40. Recall from chapter 4 that the overall SOC of the battery is based on the SOC of the anode, that is, Figure 40. Figure 39: SOC of the Cathode based on DFN model

113 88 Figure 40: SOC of the anode based on DFN model Lastly, the parameter values used for the DFN model are shown on Table 11. These parameter values were used for all the electrochemical models. Table 11: Electrochemical Models Parameter Values Symbols Value Units L [n,s,p] , , 70 [m] 10 6 R p,[n,p] , [m] ε s,[n,p] 0.55, a s,[n,p] 3ε s,[n,p] [ m2 R p,[n,p] m 3] D s,[n,p] , [ m2 s ] F Faraday [ Coulomb ] mol Area current collector 1 [m 2 ] R gas 8.31 J [ mol K ] c s,[n,p],max 30555, [ mol m 3 ] Volt cutoff[max,min] 4.5, 2.5 [V] SPM model Calibration For the SPM model, the MATLAB function pdepe was used to solve for the solid diffusion PDE of the SPM model. Furthermore, the electrode radius was

114 89 discretized into ten pieces. The simulation results are depicted on Figure 41, Figure 42, and Figure 43. Figure 41 and Figure 42 show the corresponding anode and cathode SOC results, respectively. It can be seen that both models follow the same trend; however, the SPM anode SOC shows a significantly lower amount of SOC but settles to the same SOC as the DFN model once the battery is relaxed, that is, no current is demanded. On the other hand, the cathode SOC does not settle to the same SOC value as the DFN model at it seems to always lag when compared to the DFN model s cathode SOC. Lastly, the output voltage of the SPM, shown on Figure 43, follows the trend of the DFN model output voltage but its amplitude spikes are significantly lower. Figure 41: SOC anode DFN Vs. SPM Simulation

115 90 Figure 42: SOC cathode DFN Vs. SPM Simulation Figure 43: Output Voltage of the SPM Parameter SPM Calibration The 3-Parameter SPM was simulated on MATLAB. The state space described on chapter 4 was discretized for simulation purposes. Moreover, Figure 44, Figure 45, and Figure 46 illustrate the simulation results. For all simulation results, it can be a there is little difference between the output of the DFN model and the 3-Parameter model.

116 91 Figure 44: Cathode SOC using 3 Parameter SPM Figure 45: Anode SOC using 3 Parameter SPM Figure 46: Output Voltage using the 3 Parameter SPM Lastly, the RMSE results of the 3-Parameter SPM are shown on Table 12. It can be seen that there is little RSME with the highest being for the anode SOC.

117 92 Table 12: RMSE Results for 3-Parameter SPM Vs. DFN model RMSE Output Voltage SOC Anode SOC Cathode Final Remarks regarding calibration The selected current profile represents a full discharge of the battery (e.g. SOC from 100 to 0). The estimated SOC discharge is based on coulomb counting and is the same across all models. Figure 47 depicts the estimated SOC of the battery with initial value of 100 based of the current profile for calibration described in section 5.1. SOC k+1 = SOC intial/k T sample C Battery capacitance Figure 47: SOC estimate using coulomb counting method Lastly, it is noticeable that the only parameters that are sensitive to different SOC levels are the capacitors in each ECM. This result was expected as the capacitors were introduced in the model to capture the nonlinearities of the battery.

118 Urban Dynamometer Driving Schedule (UDDS) The UDDS is a driving profile used by the USA Environmental Protection Agency (EPA) to determine vehicle s emission and fuel economy testing. Figure 48 illustrates the driving profile in mph. The UDDS test was selected to be used with the ECM and electrochemical models. Figure 48: UDDS Speed Profile in Miles per hours Setup The current demand from the battery based on the UDDS test for a LFP Li-Ion battery was extracted from [114] and is shown on Figure 49. This current profile was used as the input for the ECMs and the electrochemical models.

119 94 Figure 49: UDDS current demanded from battery for EV vehicles using a LFP Li-Ion battery Furthermore, to show the full potential of the estimation techniques implemented with the models, noise was introduced to the models using the MATLAB command mvnrnd. This function generates a random vector based of inputs mean and covariance. As mentioned earlier in this thesis, noise will be assumed white with zero mean (μ) and covariance Q, and R for the corresponding system and measurement noise covariance. The values of Q and R used for simulation are shown below: Q = diagonal(10 7 ) R = diagonal(10 3 ) An example of the noise generated by the mvnrnd function using Q and R is shown on Figure 50.

120 95 Figure 50: Noise Example for SOC The initial conditions for all models and KF algorithms are shown on Table 13. Table 13: Initial Conditions for model and KF algorithms x intial x 0 0 P 0 0 C battery V out Initial Conditions State Specific 0 to model SOC 100 T core 20 C T surface 20 C Same as x initial 10 Q (F) 4.12 (V) Results In this subsection, the results from all the ECM and electrochemical models using the UDDS current profile are discussed Rint Model

121 96 For the Rint model, due to its linearity and assumptions while calibrating the model, the KF was applicable for this model. The states for this model are SOC, T core and T surface. Figure 51, Figure 52, Figure 53, and Figure 54 show the output of the model using a KF method. It can be seen from these figures that the KF yields a good estimate of the states when subject to noise. Moreover, the KF is able to follow the trend of the output voltage with low error as it is shown on Table 14. Lastly, the temperature shown in Figure 52, and Figure 53 illustrates that a forced cool air is able to cool down the battery as the core temperature is slowly dropping whereas the temperature of the outside of the battery is drastically dropping as it is fully expose to the cooling air. The reason behind this drop is because the thermal model was developed based on internal heat generation (e.g. IR 2 ). For this simulation, the current profile does not remain high for a significant amount of time to generate enough heat. Figure 51: SOC state profile using UDDS current

122 97 Figure 52: Core Temperature state profile using UDDS current Figure 53: Surface temperature state profile using UDDS current

123 98 Figure 54: Output Voltage profile using UDDS current Table 14: RMSE for KF state estimates using UDDS current RMSE KF state estimates and output voltage V out SOC T core T surface Lastly, the state error covariance (P) and the Kalman gain (K) for all states is shown on Figure 55 and Figure 56. It can be seen from these figures that P and K converge fairly fast and remain low throughout the simulation.

124 99 Figure 55: Rint model state error covariance Figure 56: Rint model Kalman Gain for all states Thevenin Model The EKF and UKF estimation strategies were applied to the Thevenin model due to its inherited nonlinearities from the C th parameter. Figure 57, Figure 58, and Figure 59 show the state estimates of the Thevenin model using the EKF and UKF methods.

125 100 Table 15 depicts the high accuracy of these methods for estimating the states of the system with about same performance. Figure 57: Estimate of the voltage across the capacitor C th using EKF and UKF methods Figure 58: SOC estimate using the Thevenin model with EKF and UKF methods

126 101 Figure 59: T core and T surface state profiles using Thevenin model with EKF and UKF methods A comparison of the simulated terminal voltages is illustrated on Figure 60. Based on Table 15, the models show high accuracy when estimating the terminal voltage while subject to noise.

127 102 Figure 60: Output Voltage using the Thevenin model with EKF and UKF methods The state error covariance for the EKF and UKF are shown on Figure 61 and Figure 62. The values for the state error covariance for both estimation strategies converge and show similar profiles. Moreover, the K implemented by the EKF and UKF is shown on Figure 63 and Figure 64, respectively. The K values similar profiles as the state error covariance except they are lower in magnitude. Figure 61: Thevenin Model State Error Covariance using (EKF)

128 103 Figure 62: Thevenin State Error Covariance (UKF) Figure 63: Thevenin Model Kalman Gain (EKF)

129 104 Figure 64: Thevenin model Kalman Gain (UKF) Table 15: RMSE values of Thevenin model state estimates using EKF and UKF methods RMSE EKF and UKF state estimates and Output Voltage EKF UKF V out 1.05e e-06 SOC 1.93e e-05 T core 3.03e e-05 T surface 1.60e e PNGV Model Similar to the Thevenin model, EKF and UKF were used to estimate for the states of the system. The PNGV model is nonlinear because of its parameters 1/U ocv and C pn which represent the two capacitors in the circuit. Figure 65, Figure 66, and Figure 67 show the state profiles of the PNGV model when subject to the UDDS current. The parameter 1/U ocv of the model when using EKF seems to be corrupted by noise causing a deviation from the true value at the terminal voltage which can be

130 105 detailed on Figure 68. This error is evident by looking at the RMSE values shown on Table 16. Figure 65: SOC estimate using the PNGV model with EKF and UKF methods Figure 66: V Uocv estimate using the PNGV model with EKF and UKF methods

131 106 Figure 67: V pn estimate using the PNGV model with EKF and UKF methods Figure 68: V out estimate using the PNGV model with EKF and UKF methods

132 107 Figure 69: PNGV model core temperature Figure 70: PNGV model surface temperature Table 16: RMSE values of PNGV model state estimates using EKF and UKF methods RMSE EKF and UKF state estimates and Output Voltage EKF UKF V out V Uocv 9.99e e-06

133 108 V pn SOC T core T surface Lastly, the state error covariance and Kalman gain are shown on Figure 71 and Figure 72, respectively. It can be seen from Figure 71 that the state error covariance for the first state using EKF method corrupted by noise is higher than the other states. Moreover, all state error covariance for EKF and UKF converge to a low value. Figure 71: PNGV model state error covariance using EKF

134 109 Figure 72: PNGV model state error covariance using UKF The K used for both EKF and UKF methods are shown on Figure 73 and Figure 74. Both K remained low throughout the simulation with the exception of the V Uocv when using the EKF method, which means it suggest to do a full correction based on the observed measurements. Figure 73: PNGV model Kalman gain using EKF

135 110 Figure 74: PNGV Kalman gain using UKF DP Model Similar to the previous models the DP model is nonlinear due to parameters C pa and C pc. Figure 75 and Figure 76 show the corresponding two voltages V pa and V pc. It can be seen from these figures that the EKF sometimes fails to yield a good estimate but quickly recovers to follow the true value. Figure 77 illustrates the output voltage of the model using the EKF and UKF estimation strategies. Moreover, based on Table 17, both estimation strategies yield good results.

136 111 Figure 75: V pc state estimate with EKF and UKF Figure 76: V pa state estimate with EKF and UKF

137 112 Figure 77: V out DP model using EKF and UKF methods Figure 78: DP model core temperature

138 113 Figure 79: DP model surface temperature Table 17: RMSE values of DP model state estimates using EKF and UKF methods RMSE EKF and UKF state estimates and Output Voltage EKF UKF V out V pa V pc SOC T core T surface The DP model state error covariance using EKF and UKF methods is shown on Figure 80 and Figure 81. Moreover, Figure 82 and Figure 83 show the Kalman gain profiles used by the model. Each state error covariance and Kalman gain is converging. Furthermore, the spike on the first state can be correlated to the spike in the output discussed previously.

139 114 Figure 80: DP model state error covariance using EKF Figure 81: DP state error covariance using UKF

140 115 Figure 82: DP model Kalman Gain (EKF) Figure 83: DP model Kalman Gain UKF DFN model The UDDS current profile was used for the DFN model. When compared to the ECM output figures, a similar trend can be seen based on Figure 84 and Figure 85. Furthermore, Figure 86 shows the corresponding output voltage profile which is similar to the output voltage seen by the ECMs. The differences seen between the ECM and

141 116 the DFN model output is because the two classes of models are based of different batteries; however, a similar trend can be observed. Recall that for electrochemical models, the SOC of the battery is based of the SOC of the anode, that is, the battery is discharging. Figure 84: SOC anode using UDDS current Figure 85: SOC cathode using UDDS current

142 117 Figure 86: Output voltage using UDDS current profile SPM In this section, the UDDS profile was inputted to the SPM. The results of the simulation are depicted on Figure 87, Figure 88, and Figure 89, corresponding to the terminal voltage, cathode SOC and anode SOC, respectively. It can be seen that the SPM output provides with a good estimate of the terminal voltage, considering that the SPM only calculates an average concentration of lithium to determine the SOC of the battery.

143 118 Figure 87: SPM Vs DFN model output voltage using UDDS curent Figure 88: SPM Vs DFN model SOC cathode using UDDS profile

144 119 Figure 89: SPM Vs. DFN model SOC anode using UDDS profile Parameter SPM Lastly, the UDDS current was applied to the 3-Parameter SPM. The 3-Parameter SPM yielded better results than the SPM model. The model s outputs are illustrated on Figure 90, Figure 91, Figure 92, Figure 93 and Figure 94. Figure 90: DFN model Vs. 3-Parameter SPM output voltage (UDDS)

145 120 Figure 91: DFN model Vs. 3-Parameter SPM SOC anode (UDDS) Figure 92: DFN model Vs. 3-Parameter SPM SOC cathode (UDDS)

146 121 Figure 93: DFN model Vs. 3-Parameter SPM average SOC anode (UDDS) Figure 94: DFN model Vs. 3-Parameter SPM average SOC cathode (UDDS) 6.3 Dynamic Stress Test (DST) The DST is a variable power discharge cycle commonly used to evaluate the service life of batteries. For this test, the battery is charged fully and the temperature stabilized, then the test is applied without letting the battery rest. This test is often used to establish the capacity of the battery and value of maximum power step [115] Setup

147 122 The DST current profile is depicted on Figure 95. This profile was adjusted to represent a full discharge of the battery capacity used in this thesis Results Figure 95: DST Current Profile Rint Model The state estimates of the Rint model are shown on Figure 96, Figure 97, and Figure 98. The KF shows good performance throughout the simulation for all states.

148 123 Figure 96: SOC estimate Rint model using DST current demand Figure 97: Core Temperature Profile using DST current

149 124 Figure 98: Surface temperature estimate using DST current The terminal voltage of the simulation using the Rint model is illustrated on Figure 99. The KF yields a good estimate of the terminal voltage despite the noise added to the system. Table 18 shows the RMSE values for the estimate of the terminal voltage and the states showing that the KF has great accuracy in presence of noise.

150 125 Figure 99: V out estimate using DST current Table 18: RMSE for EKF-UKF estimates using Rint model with DST current RMSE EKF and UKF state estimates and Output Voltage KF V out SOC T core T surface The state error covariance and Kalman gain are shown on Figure 100 and Figure 101, respectively. Both the state error covariance and Kalman gain converge relatively fast and remain low throughout the simulation.

151 126 Figure 100: Rint Model State Error Covariance with DST current profile Figure 101: Rint Kalman Gain Profile using DST current profile Thevenin Model

152 127 The Thevenin model state estimates are shown on Figure 102 and Figure 103. Furthermore, Figure 104 and Figure 105 show the corresponding cell s core and surface temperature. Lastly, Figure 106 shows the output voltage of the model. Figure 102: SOC estimate using DST current profile

153 128 Figure 103: V th estimate using DST current Figure 104: Battery s core temperature using DST current (Thevenin model)

154 129 Figure 105: Battery's surface temperature using DST current (Thevenin model) Figure 106: V out estimate using DST current

155 130 Next, the RMSE values are illustrated on Table 19. It is clear both the EKF and UKF perform well but there is a noticeable difference with respect to the estimated V th. Table 19: RMSE for EKF-UKF estimates using Thevenin model with DST current RMSE EKF and UKF state estimates and Output Voltage EKF UKF V out 5.94e e-05 V th 4.50e e-05 SOC 3.95e e-05 T core 3.65e e-05 T surface 2.09e e-05 Figure 107 and Figure 108 depict the state error covariance and Kalman gain when implementing the EKF method. It can be seen low error with respect to the covariance and small changes on the Kalman gain throughout the simulation.

156 131 Figure 107: Thevenin State Error Covariance using DST current profile (EKF) Figure 108: Thevenin Kalman Gain using DST current Profile (EKF)

157 132 Figure 109 and Figure 110 illustrate the state error covariance and the Kalman gain when implementing the UKF estimation strategy. It can be seen that the UKF plots are more stable that of the EKF plots. Figure 109: Thevenin State Error Covariance using DST current profile (UKF)

158 133 Figure 110: Thevenin Kalman Gain using DST current profile (UKF) PNGV Model The PNGV state estimates are depicted on Figure 111, Figure 112, and Figure 113. The EKF and UKF have great performance throughout the simulation.

159 134 Figure 111: SOC estimate using DST current Figure 112: V Uocv estimate using DST current

160 135 Figure 113: V pn estimate using DST current Figure 114: V out estimate using DST current

161 136 Figure 115: Battery's core temperature using DST current (PNGV) Figure 116: Battery's surface temperature using DST current (PNGV)

162 137 The RMSE values for the EKF, and UKF are illustrated on Table 20. The table shows that the UKF performed better than the EKF. Table 20: RMSE of EKF-UKF estimates for PNGV model RMSE EKF and UKF state estimates and Output Voltage EKF UKF V out V Uocv 1.00e e-06 V pn SOC T core T surface Figure 117, and Figure 118 depict the State error covariance and the Kalman gain experienced throughout the simulation. It can be seen that the state error covariance converges quickly with low error and the Kalman gain remains low throughout the simulation. V Uocv state has great accumulated error which reflects on the state error covariance and the Kalman is set to 1 to try to reduce the error.

163 138 Figure 117: PNGV State Error Covariance EKF using DST current profile Figure 118: PNGV Kalman Gain EKF using DST current profile

164 139 Figure 119, and Figure 120 show the state error covariance and the Kalman gain experience throughout the simulation. It is noticeable that the UKF shows better performance in terms of state error covariance and Kalman gain. Figure 119: PNGV State Error Covariance UKF using DST current profile Figure 120: PNGV Kalman Gain UKF using DST current profile

165 DP Model The DP model states are illustrated on Figure 121, Figure 122, and Figure 123. It can be seen for the state estimates of V pa and V PC that the EKF sometimes generate spikes on the profile but then quickly recovers to the correct state estimate. The EKF spikes can also be seen on the terminal voltage profile. On the other hand, the UKF performs well throughout the simulation. Figure 121: SOC estimate using DST current

166 141 Figure 122: V pa estimate using DST current Figure 123: V pc estimate using DST current

167 142 Figure 124: Battery's core temperature using DST current profile (DP model) Figure 125: Battery's surface temperature using DST current profile (DP model)

168 143 Figure 126: V out estimate using DST current Despite the noticeable spikes on the state estimate profiles, the EKF RMSE remains low and similar to the UKF RMSE values. Table 21: RMSE EKF-UKF estimates for DP model using DST current RMSE EKF and UKF state estimates and Output Voltage EKF UKF V out V pa V pc SOC T core T surface

169 144 Figure 127, and Figure 128 illustrate the state error covariance and Kalman gain experienced by the simulation when implementing the EKF. It can be seen that the state error covariance experiences some spikes which may be related to the state errors shown on previous figures. Furthermore, the error quickly returns to normal values. With respect to the Kalman gain, a similar behavior is noticeable. Figure 127: DP State Error Covariance EKF using DST current profile

170 145 Figure 128: DP Kalman Gain EKF using DST current profile Figure 129 and Figure 130 shows the profiles of the state error covariance and the Kalman gain experienced throughout the simulation by the UKF method. The UKF state error covariance and Kalman gain are more stable than the EKF, that is, no spikes are noticeable.

171 146 Figure 129: DP State Error Covariance UKF using DST current profile Figure 130: DP Kalman gain UKF using DST current profile

172 DFN model The DST current profile was used with the DFN model. Figure 84, Figure 85 and Figure 86 depict the corresponding anode SOC, cathode SOC and output voltage, respectively. Figure 131: SOC anode using DST current Figure 132: SOC cathode using DST current

173 148 Figure 133: Output voltage using DST current profile SPM In this section, the DST profile was used with the SPM. Figure 87, Figure 88, and Figure 89 illustrate the results obtained. Even though the SPM only calculates for an average of the lithium concentration to determine the battery s SOC, a good performance can be seen from the model s output voltage. Figure 134: SPM Vs DFN model output voltage using DST curent

174 149 Figure 135: SPM Vs DFN model SOC cathode using DST current Figure 136: SPM Vs. DFN model SOC anode using DST current Parameter SPM In this section, the DST current was used with the 3-Parameter SPM. The 3- Parameter SPM calculates for the average SOC (similar to the SPM) and the SOC of the anode and cathode. Figure 137, Figure 138, Figure 139, Figure 140, and Figure 141 show the corresponding model outputs: terminal voltage, anode SOC, cathode SOC, average anode SOC, and average cathode SOC. Similar results as the SPM can be

175 150 observed from the average SOC plots. A high accuracy can be seen from the 3- Parameter SPM model by observing Figure 137. Figure 137: DFN model Vs. 3-Parameter SPM output voltage (DST) Figure 138: DFN model Vs. 3-Parameter SPM SOC anode (DST)

176 151 Figure 139: DFN model Vs. 3-Parameter SPM SOC cathode (DST) Figure 140: DFN model Vs. 3-Parameter SPM average SOC anode (DST)

177 152 Figure 141: DFN model Vs. 3-Parameter SPM average SOC cathode (DST) 6.4 BMS and Fault Detection The UDDS current profile was used to simulate the faults of the system. The current profile was adjusted to prevent over-discharging of the battery in the case were the battery was aged, which in reality, the BMS unit would intervene and block the usage of the battery to prevent dangerous levels of SOC. Table 22, shows the fault test cases to be used for the IMM strategies. Table 22: Test Cases to be perform using the IMM strategy with the Rint model and DP model Faults Injected Aged 80 of total capacity R core Increased by factor of 10 For the aged battery test case the total capacity of the battery was reduced to 80. This reduction in battery capacity represents an industry standard of replacing battery at about 20 reduction of their health [52]. The R core was increased until a significant increase in temperature was achieved Setup

178 153 Similar to the experimental setup and throughout this thesis, noise will be assumed white with zero mean (μ) and covariance Q, and R. The values of Q and R used for the simulation are shown below: Q = diagonal(10 9 ) R = diagonal(10 4 ) These changes were made to further depict the power of the IMM. Otherwise, the test cases would have had to be increased further (e.g. Aged to 70 and R_core increased by factor of 20) to clearly show the offset between the EKF, UKF and the IMM Results Rint Model The Rint model was selected to show the application of the IMM strategy with the KF. The two cases to be studied are: aged battery and increased in core thermal resistance based of Table Aged battery For the Rint model, the voltage started to significantly drop after half of the simulation. The reason behind this is that the same amount of current demanded in the first half now would discharge the battery more. Aged batteries may be thought of a healthy battery represented as a reservoir of water with some sand building inside. Figure 142 and Figure 143 show the output of the model under aged battery testing. It can be seen that the KF fails to accurately estimate the terminal voltage and the SOC of the battery; whereas, the IMM performs well throughout the simulation.

179 154 Furthermore, Table 23 shows the RMSE values of the KF-IMM and it is evident that the KF-IMM has significant less error when estimate for the SOC of the battery. Figure 142: Estimated terminal voltage using KF and IMM KF under aged battery test Figure 143: Estimated SOC of the battery using KF and KF-IMM under aged battery test The mode probability of the KF-IMM is shown on Figure 144. It can be seen that the KF-IMM switches between models after half-way of the simulation to better estimate for the SOC.

180 155 Figure 144: Mode probability of the KF-IMM under aged battery test Table 23: RMSE values of KF and KF-IMM under aged battery test case RMSE KF for Aged Case KF KF-IMM V out SOC Figure 145, and Figure 146 show the state error covariance and the Kalman gain of the KF-IMM during simulation. The state error covariance shows stability and convergence despite of the constant changes in its profile. The Kalman gain shows a similar profile when compared to the state error covariance.

181 156 Figure 145: Rint State Error Covariance KF-IMM Fault Aged Figure 146: Rint Kalman Gain KF-IMM Fault Aged Increased core thermal resistance For this test case the core thermal resistance of the model was increased by a base of Table 22.This fault would indicate that the battery is heating up, which should force the surface temperature of the model to increase as well.

182 157 The estimates for T core and T surface of the battery under increased thermal resistance are shown on Figure 147 and Figure 148. These are the only states to be significantly affected by the increased in R core. It can be seen in these figures that the battery is no longer cooling down. Furthermore, the surface temperature of the battery is increasing significantly; whereas, the core remains about the same temperature after the fault is injected into the system. Moreover, it can be seen that the KF estimation methods fail to accurately track the true temperature values; on the other hand, the KF- IMM still performs well after the fault. Figure 147: Estimated core temperature under increased thermal resistance test

183 158 Figure 148: Estimated surface temperature under increased thermal resistance The mode probability of the KF-IMM is shown on Figure 149. It is evident that the KF-IMM detects the fault immediately and switches to the model with full dynamics. Figure 149: Mode probability of the KF-IMM under increased thermal resistance test Lastly, the RMSE error is shown on Table 24. It can be seen that the KF-IMM shows great accuracy compared to the KF method.

184 159 Table 24: RMSE values for KF and KF-IMM under increased thermal resistance test KF and KF-IMM RMSE values under Increase thermal resistance test KF KF-IMM T core T surface Figure 150, and Figure 151 illustrate the state error covariance and Kalman gain experience by the KF-IMM during the simulation. It can be seen that the state error covariance experiences spikes however it converges. The Kalman gain shows a similar profile to the state error covariance. Figure 150: Rint State Error Covariance KF-IMM Fault Temperature

185 160 Figure 151: Rint Kalman Gain KF-IMM Fault Temperature DP Model The DP model was selected to illustrate the application of the IMM in conjunction with the EKF and UKF estimation strategies. The two cases to be studied are: aged battery and increased in core thermal resistance based of Table Aged battery Figure 152 and Figure 153 show the terminal voltage and SOC of the battery throughout the simulation. It was expected to see a significant drop on SOC after halfway the simulation (fault injected at half time of the simulation) due to the reduction of battery capacity. The terminal voltage curve also shows the same drop in voltage about half way of the simulation due to its high dependency on SOC levels. It can be seen from both pictures that the EKF and UKF fail to accurately estimate the correct level of SOC, thus failing to yield the correct result of output voltage of the battery. The EKF and UKF are still able to follow the trend of the true value due to their access to the measurement values. Furthermore,

186 161 Table 25 shows the RMSE values for the applied estimation strategies and it is clear that the IMM strategies outperform the EKF and UKF by looking at the SOC RMSE value. Figure 152: Output Voltage of all estimation methods for aged battery test case Figure 153: Estimated SOC for aged battery test case

187 162 Lastly, the mode probability for the EKF-IMM and UKF-IMM are shown on Figure 154 and Figure 155. It is evident that both the EKF-IMM and the UKF-IMM are able to detect the injected fault immediately. Figure 154: Mode probability of the EKF-IMM for aged battery test case Figure 155: Mode probability of the UKF-IMM for aged battery test case Table 25: RMSE EKF, UKF, EKF-IMM, UKF - IMM for aged battery case RMSE EKF, UKF, EKF-IMM and UKF-IMM for Aged Case

188 163 EKF UKF EKF-IMM UKF-IMM V out SOC Figure 156, and Figure 157 depict the state error covariance and the Kalman gain experienced by the EKF-IMM. The state error covariance seems to converge but it seems to vary throughout the simulation. In terms of the Kalman gain, the graph shows a similar profile as the state error covariance. Figure 156: DP State Error Covariance EKF-IMM Fault Aged Figure 157: DP Kalman Gain EKF-IMM Fault Aged

189 164 Figure 158, and Figure 159 show the profiles of the state error covariance and Kalman gain experience by the UKF-IMM during the simulation. The state error covariance looks more stable than the EKF-IMM. The Kalman gain seems to remain flat for the UKF-IMM. Figure 158: DP State Error Covariance UKF-IMM Fault Aged Figure 159: DP Kalman Gain UKF-IMM Fault Aged Increased core thermal resistance For this test case the core thermal resistance of the model was increased by a factor of 10 (e.g. R core = 10 R core ). This fault would indicate that the core of the battery

190 165 is heating up, which should force the surface temperature of the model to increase as well. Figure 160 and Figure 161 depict the temperature profiles of the battery with the injected fault. It can be seen that the normal estimation strategies fail to adapt to new changes of the model s dynamics. The curve profile of the EKF and UKF attempt to follow the true value by making used of the measurement information; however, they yield wrong results because they lack the full dynamics of the system. The IMM methods on the other hand still perform with high accuracy by making use of the second model with full dynamics. The RMSE values for all the estimation methods used are shown on Table 26 and it is clear that the IMM strategies outperform the normal estimation strategies. Lastly, Figure 162 and Figure 163 show the mode probability of switching between modes (Normal conditions Faulty). It is clear that the IMM has detected the change in model after half of the simulation.

191 166 Figure 160: T core with Fault (Increased in thermal resistance) Figure 161: Surface temperature with fault

192 167 Figure 162: Mode probability EKF Figure 163: Mode probability (UKF) Table 26: RMSE EKF, UKF, EKF IMM, UKF IMM Temperature Fault RMSE EKF, UKF, EKF-IMM and UKF-IMM for increased thermal core resistance case EKF UKF EKF-IMM UKF-IMM T core T surface

193 168 Figure 164 and Figure 165 show the state error covariance and Kalman gain experience by the EKF-IMM and UKF-IMM throughout the simulation. It is noticeable there is some spikes which may reflect the evaluation between both models; however, the error converges and stabilizes after the fault is detected. Figure 164: DP State Error Covariance EKF-IMM Fault injected Figure 165: DP Kalman Gain EKF-IMM Fault injected