Electrothermal Battery Pack Modeling and Simulation

Transcription

1 Electrothermal Battery Pack Modeling and Simulation A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Benjamin J. Yurkovich, B.S. Department of Mechanical Engineering * * * * * The Ohio State University 2010 Master s Examination Committee: Prof. Yann Guezennec, Adviser Prof. Giorgio Rizzoni

2 c Copyright by Benjamin J. Yurkovich 2010

3 ABSTRACT Much attention as been given to the study of Li-Ion batteries for their use in automotive applications such as Hybrid Electric Vehicles (HEV), Plug In Hybrid Electric Vehicles (PHEV), and pure Battery Electric Vehicles (BEV). The battery packs that are used in these applications are capable of delivering tens to hundreds of kilowatts for extended periods of time which requires packs made out of many batteries put in series to increase voltage, and also in parallel, to increase pack capacity. Hence, automotive battery packs are inherently large, distributed systems which must function as a system, subject to all the cells behaving as identically as possible. Due to the large number of cells (possibly hundreds) and the physical extent (large pack or multiple modules), cells-to-cells differences exist due to manufacturing differences, thermal gradients, possibly different aging histories, leading to the need for Battery Management Systems (BMS) to continuously balance the State of Charge (SoC) of all the cells. This thesis addresses the dynamic simulations of these large distributed battery systems to quantify the dynamic trajectories of each of the cells and the impact of pack topology, as well as statistical differences between cells, on the non-ideal behavior of the pack. In this thesis, we present a battery pack model and simulation methodology in order to study some of the operation characteristics of different battery pack configurations, under different temperatures, different initial SoCs, and different current ii

4 profiles. It is often assumed that the dynamic behavior of a battery pack can be approximated by a scaled up model of a single cell. Unfortunately, this assumption fails due to small manufacturing differences on each cell, the state of health (SoH) of each cell and non-uniform thermal conditions inside a battery pack. We begin by deriving a computationally efficient analytical single cell battery dynamic model with scheduled parameters (scheduled on SoC and temperature) that are used in a larger pack. Building on the single cell battery model, a general pack model is derived. Next, we introduce a computationally efficient methodology to formulate and solve the complex dynamic system representing a full battery pack or module, while allowing each cell to have parameter variations (small statistical manufacturing variability, imposed thermal gradients and/or large faults, i.e., damaged cells). The battery pack model and simulation methodology developed here may be extended to support any topology and pack sizing. Using the derived pack model and simulator and parameters identified experimentally for a single cell, a large number of simulations are carried out in parallel on a computing cluster to investigate the statistical behavior of the cell unbalance in the presence of appropriately distributed statistical variations in parameters (Monte- Carlo-like approach). In this thesis, both series of parallel strings and strings of cells in parallel are examined. For both topologies, simulations corresponding to charge sustaining HEV profiles and also charge depleting PHEV profiles are considered as well as three representative temperatures. From this vast array of simulations, appropriate statistical metrics of the divergence of the individual cell dynamic trajectories are then extracted and correlated to the statistics of the variability in input parameters. This information is invaluable for designing appropriate control algorithms in iii

5 battery management systems (BMS), as well as to perform battery-pack diagnostics. We define metrics in order to quantify the SoC divergence between cells within the pack and current splits between the cells and strings in parallel. In addition, an example pack is presented with a simulated damaged cell in order to show the effect that one damaged cell has on an entire pack. iv

6 This work is dedicated to my parents v

7 ACKNOWLEDGMENTS I would first and foremost like to thank my adviser, Dr. Yann Guezennec, for the opportunities that he gave me over my many years working at the Center for Automotive Research. From the first simple MATLAB exercises he assigned to me when I was high school working as an intern all the way to assigning me the task of completing this thesis, he has given me the opportunity and guidance to develop and hone my technical skills. I would like to thank my father. Without his guidance, support, and wisdom, I would never have been able to complete this work. Not only did he help me with this specific work, he also provided advice in school, research, business, and life; he was always there to support me and guide me. It is truly a blessing to have such a supportive father. I would like to thank my friends (and colleagues) who have been a major reason for my success. Their support and help have meant a great deal to me. I would particularly like to thank Yiran Hu, who not only provided me with expert advice and guidance in the fields of mathematics, physics, and engineering throughout my years at CAR, but whose friendship means a great deal to me. Thank you. I would like to thank the numerous students, staff, and faculty at the Center for Automotive Research. I wish that I could thank each one personally in these vi

8 acknowledgments, but there are too many to count who have helped me throughout my years at CAR. I would also like to thank my family, who have always supported me and throughout the entire process of this work. Lastly, I would also like to acknowledge the Ohio Supercomputer Center, which helped me complete this work. vii

9 TABLE OF CONTENTS Page Abstract Dedication Acknowledgments List of Tables ii v vi xi List of Figures xiii Chapters: 1. Introduction to Battery Systems, Battery Packs, and Applications Introduction Battery Systems Battery Chemistries Battery Packs in Automotive Applications PHEV and HEV Differences HEV Battery Packs PHEV Battery Packs Literature Review Single Cell Battery Models Battery Pack Models Summary and Thesis Outline Battery System Experimentation Introduction Challenges in Battery Experimentation viii

10 2.3 Station Hardware Architecture Power Supply Electronic Load Environment Control Station Software Architecture Initialization Layer: Control Process Initialization Layer: DAQ Process System Runtime Layer: Control Process System Runtime Layer: DAQ Process Application Layer: Control Process Summary Electrothermal Modeling and Identification of Li-Ion Battery Chemistries Introduction Choosing the Battery Model Obtaining a Closed-form Solution Identification of Model Coefficients Optimization Summary Battery Pack Modeling and Simulation Introduction Battery Pack Model Derivation Explanation of Notation Model Derivation Pack Simulator Simulating a String of Parallel Cells Configuration Battery Pack Simulation Examples Current Profile Example Results Verifying the Model and Simulation Summary Methods and Applications Monte Carlo Simulation and Statistical Analysis Simulation Profiles Battery Pack Configurations Battery Pack Variability Modeling Cell Variability ix

11 5.1.5 Temperature and SoC Choices Summary of Simulations Parallelization of Simulations Analysis Results Example Diagnostic Battery Pack Application Perturbed Battery Pack Simulation Results Summary Conclusions and Recommendation Contributions Future Work Appendices: A. Standard Deviation Input to Output Relationship Tables Bibliography x

12 LIST OF TABLES Table Page 5.1 Simulation time requirements Percent variation for capacity parameter for {PS,HEV,C, 25,50,3} example pack from Monte Carlo simulations RMS for current splits for {PS,HEV,C, 25,50,3} example pack from Monte Carlo simulations A comparison of the % SoC Deviation per hour statistics for {PS,HEV,C, 25,50,3} example pack with 10,000 simulations per set and 1000 simulations per simulation set A comparison of the % SoC Deviation per hour statistics for {SP,PHV,C, 25,70,1} example pack with 10,000 simulations per set and 1000 simulations per simulation set Standard deviation of input to standard deviation of output ratios where ψ < Standard deviation of input to standard deviation of output ratios ψ > Standard deviation of input to standard deviation of output ratios where 6 < ψ < Standard deviation of input to standard deviation of output ratios where 4 < ψ < Percent variations on each cell in the battery pack RMS for current splits xi

13 5.12 SoC Ratios of each cell in the battery pack A.1 Standard deviation of input to standard deviation of output relationships of PS packs A.2 Standard deviation of input to standard deviation of output relationships of SP packs A.3 Standard deviation of input to standard deviation of output ratios of PS packs A.4 Standard deviation of input to standard deviation of output ratios of SP packs xii

14 LIST OF FIGURES Figure Page 1.1 Panasonic NiMH modules from a Prius battery pack Converted Prius containing a 5kWh Li-Ion Battery Pack SoC usage range of a typical PHEV [17] Capacity drop for different Li-Ion battery measurements over a temperature range [17] Software System Layers Hardware and software system architecture Equivalent Circuit Model for a Single Battery Cell Battery response to a series of asymmetric steps at -15 o C Battery response to a series of asymmetric steps at 5 o C The three different configurations of automotive battery packs Graphical Representation of Notation Current Profile Current Split Close Up of Current Split Example of 3P1S Battery Pack SoC xiii

15 4.7 Pack Voltage of an Example 3P1S Battery Pack Current profile for HEV C-Rate profile for HEV Current profile histogram for HEV C-Rate histogram for HEV SoC change from 50% for HEV Profile Current profile for PHEV C-Rate profile for PHEV Current profile histogram for PHEV C-Rate histogram for PHEV SoC change from 80% for PHEV profile Parallel string configuration (3P8S) String of parallel cell configuration (8S3P) Parameter distribution for σ=1% standard deviation Parameter distribution for σ=2% standard deviation Parameter distribution for σ=3% standard deviation Cluster architecture SoC Ratio for a 3P8S pack with linear fit example Current split for {PS,HEV,C, 25,50,3} example pack from Monte Carlo simulations xiv

16 5.19 SoC for each cell in {PS,HEV,C, 25,50,3} example pack from Monte Carlo simulations Percent SoC deviation trends for {PS,HEV,C, 25,50,3} example pack from Monte Carlo simulations Probability Density Functions of % SoC deviations per hour where ζ = 1 ˆσ for {SP, HEV, C, 25,50, 1}, {SP,HEV, C, 25, 50, 2}, {SP, 2π HEV, C,25, 50, 3} with varied σ Probability Density Functions of % SoC deviations per hour where ζ = 1 ˆσ for {PS, HEV, C, 25, 50, 1}, {PS,HEV,C,25,50,2}, {PS, 2π HEV, C, 25, 50, 3}, {PS, PHV, C, 25, 50, 1}, {PS, PHV, C, 25, 50,2}, {PS, PHV, C, 25, 50,3} with varied σ Probability Density Functions of % SoC deviations per hour where ζ = 1 ˆσ for {PS,HEV,R,45,50,1}, {PS,HEV,R,45,50,2}, {PS,HEV,R,45,50,3}, 2π {PS, HEV, C, 45, 50, 1}, {PS, HEV, C, 45, 50, 2}, {PS, HEV, C, 45, 50, 3} with varied σ and profile Standard deviation of input to standard deviation of output ratios where ψ = ˆσ for all combinations of simulations σ 5.25 RMS of current split for {PS, *, C, 25, 50, *} with varied profile and σ where blue is the first string in pack, green is the second string, and red is the third string RMS of current split for {SP, *, R, 45, 70, *} with varied σ and parameter variation where blue is the first string in pack, green is the second string, and red is the third string RMS of current split for {PS, HEV, *, 25, 50, *} with varied σ and parameter variation where blue is the first string in pack, green is the second string, and red is the third string RMS of current split for {SP, HEV, *, 25, 50, *} with varied σ and parameter variation where blue is the first string in pack, green is the second string, and red is the third string xv

17 5.29 RMS of current split for {PS, PHV, C, *, *, 2} with varied temperature and SoC where blue is the first string in pack, green is the second string, and red is the third string Current split plot of severely perturbed pack Current split close up plot of severely perturbed pack SoC plot of severely perturbed pack SoC Ratio plot of severely perturbed pack xvi

18 CHAPTER 1 INTRODUCTION TO BATTERY SYSTEMS, BATTERY PACKS, AND APPLICATIONS 1.1 Introduction Recently, Li-ion and NiMH batteries have received much attention for enabling Hybrid Electric Vehicles (HEV), Plug-in Hybrid Electric Vehicles (PHEV) and battery dependent Electric Vehicles (EV) due to their high power and energy densities. In these automotive applications, batteries are capable of delivering tens or hundreds of kilowatts, hence necessitating large battery packs composed of multiple series strings of cells (on the order of 100), and often involve multiple strings in parallel to increase the pack capacity as well as to decrease the aging severity of the current seen by each cell. All the cells are subject to a very dynamic current demand and a wide thermal environment (typically -20C o to +60C o ). The issue is to develop an appropriate battery pack simulation which accurately captures these dynamics. In essence, therefore, a battery pack is but one vital component in a large, complex system. But even when considered in and of itself, the pack is a complex, large-scale system consisting of a myriad of individual dynamical systems, at the electrochemical level (within an individual cell), at the cell level, at the string level, and at the overall 1

19 level when integrated into a large pack. Nonetheless, it is often assumed that the dynamic pack behavior can be approximated by a scaled-up model of a single cell, hence implying that all cells are the same. This assumption fails on multiple grounds due to (small) manufacturing differences, unavoidable thermal gradients and hence different degradation due to aging in each cell (see [16]). This results in progressive cell unbalance and can possibly lead to catastrophic pack failure if not properly mitigated. In this thesis, we introduce a computationally efficient methodology to formulate, model, and analyze the complex dynamic system representing a full battery pack, while allowing each cell in the pack to have parameter variations. This dynamic battery pack model representing a dynamic system with coefficients that are scheduled on both state of charge (SoC) and temperature is then subject to typical current profiles as seen in vehicles. Using the results from these simulations, we can develop comparison metrics to accurately gauge and compare SoC deviation of individual cells in different battery packs that vary in temperature, state of charge, variation, current profile, and pack configuration. By compiling large amounts of simulation data of large packs comprised of single cells with parameter perturbations (using Monte Carlo methods), we will gather statistical information on the behavior of battery packs, which could potentially allow for the extension of the pack concept to include battery management system (BMS) design and diagnostic algorithm development. Many methods for battery management have been provided in the literature (see [4] for more discussion on BMS), as well as SoC and state of health (SoH) estimation (see [13, 30, 32]) that are also important 2

20 topics for which a low order battery pack model could be used. Hence, all practical battery packs incorporate a Battery Management System (BMS) to constantly (mostly when idle) re-balance the cells. By developing a lumped parameter distributed dynamic battery pack model which captures the individual evolution of each cell, it is possible to look at the effects of individual battery cell SoC drift within a large pack due to cell-to-cell non-uniformities. It is hoped that this simulation tool can lead to design optimization of the cell configuration within the pack, better BMS algorithm designs, and model-based estimation and fault diagnostics algorithms to mitigate aging effects in packs due to cell unbalance. 1.2 Battery Systems Using a single cell in a large application such as a vehicle is impossible because most likely it is either technologically infeasible (most chemistries yield cell voltages between 1.2V and 4V) or not economically viable (large for most high capacity cells). For this reason, we first take a look at the different battery chemistries as well as a brief history of electrification of transportation as well as battery packs that are currently available commercially Battery Chemistries With the rising interest in clean, renewable energy solutions, battery technology has become a popular area of research in the scientific and engineering communities. Battery technology has applications in both stationary and mobile power systems. In a stationary environment, battery applications have few constraints with regard to weight and size. In the mobile application, however, batteries have many highly 3

21 restrictive constraints and requirements such as size, weight, cost, reliability, and performance. Especially in HEV, PHEV, EV applications, batteries must be sized appropriately to optimize cost, weight, performance, and reliability. Unlike stationary power applications, where historically the main battery chemistry used is lead-acid (Pb-Acid) batteries, automotive traction applications require smaller, lighter battery chemistries possessing higher energy density such as Nickel Metal Hydride (NiMH) or Lithium Ion (Li-Ion). Because of this, much focus of battery research is in these chemistries, and are typically found in current production HEVs (NiMH) or upcoming vehicles (Li-Ion). 1.3 Battery Packs in Automotive Applications The electrification of modern vehicles is not a new concept. In the 1990s GM released the EV1 (based on GM s concept EV, Impact, unveiled in 1990), a fully electric vehicle. Due to economic issues and the fact that it was never offered as a broadly marketed product, however, the EV1 never did gained wide-spread success. The EV-1 contained a large battery pack (about 18kWh) of Pb-Acid batteries, adding significant weight to the vehicle. In 1999, GM released a second generation EV1 that contained Panasonic NiMH batteries (about 26kWh). Because of the apparent lack-luster success of GM s EV1, other automotive manufacturers were hesitant to bring fully electric vehicles to the market. Instead, auto makers such as the Honda released HEVs which incorporated an internal combustion engine (ICE) with an electric motor and energy storage. Today many different auto makers have multiple HEVs in their portfolio. The most popular and well-known HEV, the Toyota Prius, contains a 200V (Gen III, 2004) 4

22 NiMH Battery Pack which contains cells manufactured by Panasonic (see Figure 1.1). Each cell is 6.5Ah and is packaged into a module containing six cells. Twenty-eight of these modules make up the entire 1.3kWh battery pack. The entire battery pack is configured as one string of cells in series (as all commercial HEV packs are configured currently). Figure 1.1: Panasonic NiMH modules from a Prius battery pack Although currently there are no large scale production EVs or PHEVs, there are conversion kits that allow for an auxiliary battery pack (typically a Li-Ion battery pack) to be added to a stock HEV (i.e. a Prius, see Figure 1.2) converting the HEV to a PHEV. By late 2010, auto makers plan to put PHEVs into production, such as the Chevy Volt, Mitsubishi imiev, and the Nissan Leaf PHEV and HEV Differences The requirements for HEV and PHEV battery packs are significantly different. This is mainly due to the fact that a typical HEV battery pack sees a different type 5

23 Figure 1.2: Converted Prius containing a 5kWh Li-Ion Battery Pack of load demand than a typical PHEV battery pack [3]. One fact still remains, however: new battery technologies (i.e. NiMH and Li-Ion) found in HEVs and PHEVs are very expensive, often taking up more than half of the entire bill of materials of the vehicle (especially in PHEV applications, which require higher capacity battery packs) HEV Battery Packs A typical HEV battery pack, depending on hybrid configuration, has significantly higher loads due to the fact that the battery pack is normally used for vehicle acceleration assist (via an electric motor) and/or to offset the amount of torque production required from the ICE. This allows for better fuel efficiency and a downsized ICE. Because the HEV battery pack is not necessarily required to store a great amount of energy, the battery packs normally have high peak power capabilities and lower 6

24 capacity characteristics. The result of this is that the battery pack operates in a relatively narrow SoC (State of Charge) range [3]. This operation can be accomplished because the ICE (as well as some regenerative breaking using the electric motor as a generator) replenishes the energy used in the battery pack during discharge. Most HEV battery packs in production to date are composed of NiMH batteries. This is due to the fact that NiMH battery technology has matured faster than the Li-Ion battery technology, and the fact that NiMH batteries have high energy density (compared to traditional Pb-Acid batteries) and show minimal capacity loss at low operating temperatures [5]. These battery packs are almost exclusively single string packs (that is, all the battery cells are configured in series) PHEV Battery Packs Unlike HEV battery packs, PHEV Battery packs typically require more capacity because the battery packs in PHEVs are designed to be the main source of energy for vehicle mobility. In most cases, the ICE acts as a safety catch or range-extender (in the case of the battery pack becoming depleted) and an auxiliary power unit for the vehicle. In addition, PHEV and EV packs require that multiple strings of cells in a series configuration are connected in parallel in order to increase the capacity. Because the ICE cannot support the complete replenishing of the energy of the battery pack, the pack must be recharged from the electrical grid (see [8]). Therefore, we see that typical PHEV operation uses a wide range of SoC (95% to 25% SoC, see Figure 1.3). The use of the entire battery SoC range poses some problems in terms of battery pack life estimation (SoH estimation, see [28]), fault diagnostics, and battery management (SoC estimation) [29, 21, 22]. 7

25 Figure 1.3: SoC usage range of a typical PHEV [17] Due to recent advances in Li-Ion battery technology, their success in consumer electronics, and that they have an extremely high energy density, Li-ion batteries have become the accepted battery technology for PHEVs and EVs. In addition to safety concerns, the downside of all batteries is the severe capacity drop that is seen when operating at low temperatures (below 0 o C as seen in Figure 1.4; in later chapters we will see more information regarding multi-temperature identification of NiMH and Li-Ion cells will be presented). Much research has been done in the area of identification and modeling of Li-Ion cells for automotive applications at The Ohio State University s Center for Automotive Research and is used in this work. 8

26 Figure 1.4: Capacity drop for different Li-Ion battery measurements over a temperature range [17] 9

27 1.4 Literature Review Single Cell Battery Models A larger amount of published work exists regarding single cell battery models. The work in this thesis, in fact, is built upon a large amount of recently published findings regarding a specific method of single cell battery model identification. The modeling methodology that is used in this work (see [11]), is merely one of many different modeling techniques. There have been several types of models used to capture input to output behaviors of batteries. One class of models is electrochemical models. Electrochemical models are built upon the physical makeup of the battery and use diffusion principles and other elements from physics and chemistry [15]. The fundamental models characterize the fundamental mechanism of battery power generation. Electrochemical models often contain systems of coupled partial differential equations that can be difficult to simulate/solve and are typically not suitable for control design or models that require evaluation of multiple battery models (see [15] for examples). Because of the proprietary nature of most readily available batteries, the parameters needed to fit the model are often difficult, if not impossible, to obtain. Thus, often times a simpler approach is taken to modeling a single cell battery [25, 9]. A simpler alternative approach to the electrochemical first principle models is the equivalent circuit based models. In such models, the electrochemical elements in the battery are replaced by an electrically equivalent process. For example, charge transfer across a boundary can be represented by parallel resistor and capacitor and ion diffusion can be represented by waves propagating on a transmission line. Therefore, the construction of the equivalent circuit can be obtained via electrochemical impedance 10

28 spectroscopy (EIS) [2] because different elements in the battery show different behaviors depending on the frequency range. Equivalent circuit models obtained this way show good input to output response when magnitude and frequency range of the input excitation signal are limited [19]. Though simpler than electrochemical models, equivalent circuit models can still be difficult to work with due to the inclusion of distributed elements such as the transmission line or pure frequency domain quantities like the Warburg impedance [10]. By lumping elements and assuming homogeneity in certain dimensions, an equivalent circuit model can be simplified by including only resistors, capacitors and a voltage source [23]. The advantage to this type of model is that the dynamic model equations are all ordinary differential equations. When operating conditions such as temperature and SoC are restricted and the input excitation signal is limited in bandwidth and magnitude, the inaccuracy of the model can be very manageable when looking to be used in larger applications with multiple battery models. By scheduling the model parameters with respect to the operating conditions, the validity of the model can be expanded to include all necessary conditions [11] Battery Pack Models Although a large amount of published work exists regarding single cell battery modeling, there is a noticeable lack of publications regarding battery pack modeling, particularly battery pack modeling that includes multiple battery cell models. The literature that exists mainly addresses battery packs that are configured in a single string. In fact, often times, a battery pack is modeled with a single cell battery model, and its parameters are scaled in order to meet the requirements for an entire pack 11

29 [14]. In [24], a battery pack is modeled, but no dynamics are considered for each individual battery cell in the series string. Modeling a battery pack in a single string is a relatively simple endeavor, particularly if a single cell battery model exists. If an equivalent circuit model single cell battery model is used, it is clear that the order of the model can in essence be infinitely extended to represent an infinitely long string of batteries in series. The equivalent circuit models need only to be summed in order to obtain the voltage of the overall pack [31]. A challenge, however, is presented when trying to model a battery pack with multiple parallel strings of battery cells. As noted in previous work [31], the main challenge when modeling battery strings in parallel occurs when trying to calculate the current split that occurs between strings when a battery pack is under operation (or, if unmanaged, sitting idle). It follows naturally that because all the string cells are slightly different than a nominal functioning battery pack, the current through each string will be slightly different in order to satisfy natural laws of current flow. It is most likely for this reason that modeling battery packs containing parallel strings poses a unique challenge and has not been widely researched. There does exist work that notes the use of parallel strings that make up an entire battery pack, but there is either no discussion on modeling [26] or current splits between parallel cells [27]. One reference, in particular, presents an interesting viewpoint on battery packs with parallel strings [18]. In this work, a case is presented that multiple batteries in strings are unnecessary arguing that the only reason to put multiple cells in strings is to increase battery pack voltage. Instead, it is argued that DC/DC converters can be used to regulate the voltage and only parallel cells are necessary to increase the capacity of the pack. Because DC/DC converters are 12

30 used, it is assumed that the voltage will remain the same for each of the strings and the current will be controlled, thus making the current split equal among all strings. Although there is merit to such an idea, the cost of implementation would be potentially insurmountable, due to the many DC/DC converters required in the application. This is a very likely reason as to why such implementations do not exist on a wide scale basis in commercial applications. It should be noted that there does exist a large amount of literature on thermal management of battery packs (see [12] for example), but such applications are not directly addressed in this thesis. The main reason for this is because thermal management is highly dependent on the topology and geometric configuration of a battery pack. The model and methodology presented in this thesis are general enough to be applied to any geometric topology, thus not needing discussions regarding specifics of battery pack topology will therefore not be given. However, it can be safely stated that thermal management hardware is always implemented at the module or pack level and hence cell-to-cell thermal gradients will always occur. 1.5 Summary and Thesis Outline In this chapter, a brief history of the electrification of transportation was presented. In addition, an introduction to battery chemistries and the different types of automotive battery packs existing today were discussed. In addition, a brief literature review introducing the model and methodologies given in this thesis was presented. The work presented in this thesis is extending upon the isothermal, constant coefficient battery pack model that was presented in [31]. We will extend the battery pack model to include coefficients which are scheduled on SoC and temperature. The 13

31 remainder of this thesis is organized as follows. In the next chapter, we discuss battery experimentation and the challenges that are present when obtaining experimental data. In Chapter 3, we derive a single cell analytic battery model and describe the identification techniques used to obtain the model coefficients Following this, Chapter 4 gives a battery pack model and simulation methodology. In chapter 5, we present methods and applications using the battery pack model and simulator. Finally, in Chapter 6, the thesis is summarized and the contributions are identified. 14

32 CHAPTER 2 BATTERY SYSTEM EXPERIMENTATION 2.1 Introduction In this chapter, we describe the experimental battery systems that will be referenced when developing a battery model parameter identification methodology in Chapter 3. This chapter is included in order to give the reader some insight into the process of developing such a battery testing system. 2.2 Challenges in Battery Experimentation Testing batteries can be a very tedious and long process requiring many days and hours to complete simple tests. This is due mainly to the fact that batteries are electrochemical systems that vary greatly from battery sample to battery sample. Due to this variation, it is often difficult to predict the behavior of the battery during a test, resulting in many failed tests and ruined batteries. Single battery cells, particularly Li-Ion and NiMH chemistries that are most commonly tested by the equipment used for this work, have a theoretical cell voltage that is relatively low ( 3.3V for LiFePO 4 and 1.2V for NiMH), resulting in voltage 15

33 measurement challenges. These challenges are particularly noticeable when attempting to measure the voltage of a cell. Due to this phenomenon of a low nominal cell voltage, a measurement error of just a few millivolts could result in a battery cell SoC calculation to be off by almost 10% SoC! To make matters more difficult, the battery test stations used in this work contain a switching power supply to support the requirement of charging the battery cell during testing. The switching frequency in the power supply generates noise in the electrical circuit, and because the supply must be electrically connected to the battery directly, the voltage measurement on a battery is inherently noisy. Thus, it is easy to see that signal noise plays the part of a very challenging adversary in the task of obtaining accurate and consistent battery testing data. To combat this noise, many analog low pass filters are used during the collection process, as well as digital filters in the post processing of the data. 2.3 Station Hardware Architecture Each battery testing station is equipped with an electronic programmable load, a programmable power supply, computer, and data acquisition Power Supply A 3.3kW AC-DC switching power supply is used to charge the battery cells. The power supply contains a controller that supports serial RS232 communication that allows for the programming of current and voltage. 16

34 2.3.2 Electronic Load A 1.2kW programmable electronic load is used to discharge the battery cells. The load contains a controller that supports serial RS232 communication that allows for the programming of current and voltage Environment Control A peltier junction with an accompanying controller is used to programmatically control the temperature of a battery cell. For low temperatures (below 0C o C), an environmental chamber is used. 2.4 Station Software Architecture The software architecture for each battery testing station is built on the MATLAB interpreter engine, which allows for easy access to program variables during runtime, as well as dynamic function creation (if needed). The software system is split up into three main layers: System Initialization, System Runtime, and Application (see Figure 2.1). Figure 2.1: Software System Layers 17

35 The Initialization Layer is responsible for all bootstrapping and resource allocation that is needed for the system to begin running. This includes initializing system timers, communication mediums, and system states. The System Runtime Layer is responsible for running the actual system functions. The Application Layer contains user-defined applications that are responsible for the specifics of the actual battery testing. Most of the MATLAB functions used in the system will not be discussed in the documentation. A non-conventional MATLAB application programming scheme is used to run the entire system architecture. Although somewhat slower than the conventional application programming scheme, the scheme employed is easily modified and debugged, allowing for rapid Application Layer Development and easy global variable implementation. Since many of the objects used throughout the architecture (namely in the Application Layer) are used in many different places, a global variable implementation must be employed. To do this, MATLAB scripts are utilized instead of functions that make global variable support cumbersome due to function scoping. In addition to the abstract architecture scheme, a two instance (process) MAT- LAB scheme is used (see Figure 2). Because MATLAB versions 2006b and 2007a are single-threaded, it is desirable to have two instances running: one instance of MAT- LAB to carry out control of the system, and one instance of MATLAB to conduct Data Acquisition (DAQ). By completely decoupling both operations, and also duplicating DAQ hardware, a level of redundancy and reliability is added to the system. Thus, even though it is possible to have communication between the two MATLAB processes (either using shared memory or socket communication), the two MATLAB 18

36 processes operate independently from one another. In addition, using two processes of MATLAB allows for the OS to exploit the multi-core processor found on each machine. Figure 2.2: Hardware and software system architecture Initialization Layer: Control Process Initialization of the MATLAB Runtime Engine is easily done by placing a script in MATLAB s root startup directory. Each station s startup script is unique to the station due to the fact that UDP (protocol run on IEEE 802.3) is used as a communication protocol requiring a unique IP address. All System Runtime scripts and Application Layer scripts are shared across the network. Instrument Communication Initialization Instrument Communication is done via RS232. Both pieces of testing equipment (the Load and the Supply) are connected to the same RS232 bus. This allows for 19

37 one communication method and one Serial Resource to be used. Due to the OS, the COM port for each station is for the instrument Serial Port. The COM ports are specified in the initialization file for each station. Control Reporting Communication The station reporting is mainly used to report on Application Layer Operation. Thus, actual reporting is completely Application Layer Specific, and it is the responsibility of the application program implementer to define the semantics of the reporting. Part of the reporting architecture allows the messages sent by the testing nodes to be accepted by the master and then relayed to a cell phone utilizing SMS technology. It should be noted that the SMS software located on the master is not written in MATLAB, but in C#, since certain networking libraries were required that MATLAB did not support. The physical transport layer that is used for Station Reporting is Ethernet and the transport layer is UDP. The UDP provides an excellent medium to use for communication, since it does not require any additions to the wire harness. In addition, as long as the messaging occurs under the same firewall, 100% message reliability can be assumed. Although no protocol is currently defined, the port scheme is implemented in such a way that two-way communication can occur between the master and each station. Analog and Digital I/O Initialization The Data Acquisition toolbox is used for initializing the Analog Input and Digital I/O objects. It is easy to see that initialization of the object is mostly handled in the 20

38 National Instrument s M-Series API. The MATLAB analog input Digital I/O objects are started upon initialization in order to use the peekdata function during system runtime, which allows for manually written data acquisition rather than relying on the engine located on the physical data acquisition hardware due to the fact that the control process does not require rigorous data acquisition. The Analog Input signals are meant for the system layer to use as safety checks and for the application layer to utilize for feedback to be used for application control. The Digital I/O is used to signal the DAQ process to start and stop. MATLAB Process Reporting In order to have a record of all input and output that occurs in the MATLAB command window, the diary command is used. All the command windows input and output are stored in the file. This is used in case of a crash to diagnose problems, and also for Application Layer traceability. The diary is turned on upon initialization Initialization Layer: DAQ Process The DAQ process objects and variables are initialized upon system startup. Analog Input and Digital I/O Initialization The DAQ process uses the same MATLAB Data Acquisition Toolbox as the control process and is initialized in the same way. Three different Analog Input signals are initialized: current, voltage, and temperature of the battery. 21

39 MATLAB Process Reporting Just as the diary command was utilized in the control process, the same occurs for the DAQ process, allowing for constant reporting and logging of the data acquisition events System Runtime Layer: Control Process The System Runtime Layer is responsible for system safety, processing communication requests, and general purpose I/O. There is a suite of methods that can be used in user-defined programs in the Application Layer (in the control process), as well as scripts that handle callbacks and other runtime tasks. Control Timer Functions and Responsibilities MATLAB timers are utilized to control the runtime of the system. The timer period is set to 1 second. This allows all possible tasks written into the callback function of the timer to be executed within the time before the next timer callback execution. The responsibilities of the Timer are to provide a layer of safety checks and to provide signal processing and feedback mechanisms. Control Monitoring and Safety Safety checks are built directly into the runtime of the system to ensure battery integrity and overall lab safety. The cell voltage is monitored and checked against a maximum and a minimum voltage threshold. In the event that either threshold is breached, the system terminates all running application processes and sends an error report to the master node. 22

40 2.4.4 System Runtime Layer: DAQ Process The System Runtime Layer is the fundamental building block of the system. DAQ Timer Functions and Responsibilities Depending upon the rate at which the signals are sampled, the DAQ uses a timer to clock the data collection rate. The DAQ process requests the samples from the hardware acquisition engine and stores them in memory where they are written to disk incrementally. DAQ File I/O In order to ensure data integrity and to minimize data loss in the event of software failure, the DAQ process leaves open a file stream to a specified data file and, during runtime, incrementally writes collected data to disk. The files written can be accessed by the master via the shared node network. In the event of a system failure, the file that is in the process of being written to is closed automatically in order to avoid file corruption. The data is written in equal sized blocks, thus minimizing the chances of leaving a file unbalanced. File imbalance occurs when only a portion of a signal channels are written to a line in the file, thus leaving an unbalanced sample line Application Layer: Control Process As shown above, the software architecture is generalized enough to allow for a user defined application layer. In addition, MATLAB provides a very good software platform to enable user defined applications. The application layer program structure 23

41 is based on a Finite State Machine implementation, utilizing the MATLAB timer objects managed in the System Runtime Layer. Application Layer Tools There are a number of usable functions for an application layer client. These functions include the ability to start and stop data collection, instrument command, and retrieve signals for feedback and monitoring purposes. An Application Layer Programming Template is provided in the form of a Finite State Machine to the client programmer. The client, using the template, has the tools to implement any type of battery testing profile, including the ability to implement control algorithms using signal feedback. Selected User Defined Applications The software system is designed to allow client users to easily build their own battery tests. An entire toolbox of instrument control and data acquistion is available to them. It is relatively easy to understand the simple state machine that exists that allows for the control of the overall battery testing system. 2.5 Summary In this chapter, we have presented the battery experimentation hardware and software, and have described some of the challenges in battery testing. The battery testing systems described in this chapter are the systems used for the battery model coefficient identification that will be developed and used in the following chapters. Development for this work, as well as for a large body of testing work for other 24

42 researchers at the Center for Automotive Research, represents a contribution in and of itself, and is thus archived in this thesis. 25

43 CHAPTER 3 ELECTROTHERMAL MODELING AND IDENTIFICATION OF LI-ION BATTERY CHEMISTRIES 3.1 Introduction This chapter presents a brief yet important summary of the basic battery model structure and the methodology of identifying key parameters of the model. Brevity is exercised because this work has appeared in detail in [11], yet we feel it is important for the studies that follow in the remainder of the thesis. In order to derive a battery pack model, it is necessary to first start with the fundamental building block of the model: an individual battery cell. As discussed previously, it is common in the literature to see battery packs modeled in a black box fashion, avoiding explicit dynamics attributable to individual cells. However, in order to model the battery pack in enough detail to capture the internal dynamics and therefore to provide a foundation for advanced characterization and simulation, individual cells must be modeled. The rest of this chapter, therefore, focuses on the fundamental building block of the pack model. 26

44 3.2 Choosing the Battery Model We have chosen to use a Randle equivalent circuit model to model a single cell. The reason for this is two-fold. First, alternative approaches to modeling a battery, such as electrochemical modeling (see [20, 15] for examples and further explanation), often involve complex partial differential equations to describe the highly dynamic response of the battery as well as ion diffusion and other electrochemical properties. While desirable and descriptive from a mathematical and physical viewpoint, such representations are cumbersome for simulation and control as well as for incorporation into larger systems. Second, the automotive industry tends to prefer a model-based approach for control algorithm implementation in the BMS, which requires simplified, low-ordered models. Thus, based on extensive studies [24, 11] we have chosen a 2 nd order Randle model for the single battery cell. The model structure we have employed is a version of the equivalent circuit battery model, which consists of an internal resistance, parallel RC circuits, and an ideal voltage. Figure 3.1 shows the basic equivalent circuit model employed. The voltage V (t) across the battery is given by V (t) = E 0 (t) R 0 I(t) v c1 (t) v c2 (t), (3.1) where E 0 (t) is the ideal voltage source (which essentially models the OCV and depends on time through the SoC dependence), I(t) is the current input (as seen by the battery), and v ci (t) (for i = 1, 2) is the voltage across the capacitor/resistor combinations. For our purposes, v ci (t) satisfies the first order differential equation dv ci dt = 1 RC v ci(t) + 1 I(t). (3.2) C 27

45 Figure 3.1: Equivalent Circuit Model for a Single Battery Cell It should also be noted that this equation can be also be expressed as a first order dynamic system in state variable form dv ci dt = A i v ci + A i B i I(t), (3.3) where A i = 1/(R i C i ) and B i = R i are the inverse of time constant and input coefficients, respectively. R i and C i (consequently A i and B i ) are assumed to be dependent in some fashion on operating conditions. The precise nature of this dependence, and its justification, are discussed later when we discuss the parameter identification. 28

46 A little more explanation is required to throughly explain the OCV element. In this model, the OCV is a function of the SoC. We have found through experimental results that the OCV can have a very small variation with respect to the temperature. For our purposes, however, we assume this difference is insignificant and very small, so we have chosen to not parameterize the OCV as a function of the temperature. In addition, the SoC element in the model has a specific form for a fixed temperature. We have chosen a particular form to model the OCV due to the fact that the battery terminal voltage drops significantly as the SoC approaches 0% and increases greatly as SoC approaches 100% [16]. In the middle portion of the SoC (around 30% to 85%), the relationship between SoC and terminal voltage is approximately linear. Due to this behavior, the OCV is modeled by a double exponential function given as V oc (z) = V 0 + α(1 exp( βz)) + γz + ζ(1 exp( ɛ )), (3.4) 1 z where z is the SoC in the defined range z [0, 1] and α, β,..., ɛ are tunable parameters which can be found through optimization techniques [11]. The SoC and capacity, also elements of the model, are addressed next. First, define the capacity C n to be the amount of amp-hours that can be taken from the battery at a discharge rate of 1C at room temperature after a complete charge (that is, the battery is at 100% SoC as defined by the manufacturer of the battery). Using this definition, the SoC can then be defined using simple Coulomb counting (current integration) as ż = C n I(t). (3.5) It should also be noted that the same equation is used to compute the SoC for experimental datasets and simulation profiles. Over a long period of time, this method 29

47 of SoC calculation can become inaccurate (like all integration based techniques), and therefore should be avoided for SoC estimation on board a vehicle. For our purposes, however, the datasets for experimentation as well as simulation are kept short and processed carefully Obtaining a Closed-form Solution Due to the fact that our final goal is to develop a battery model that can be used to create a multi-cell battery pack and accompanying simulator, we must be aware of the fact that there will exist multiple battery models that must be simulated in parallel. Thus, it is necessary to derive a closed form solution of the battery model in order to avoid having to use numerical ODE solvers during simulation. In addition, the method of simulation lends itself well to a closed form solution of a battery model, as we will see later. In order to obtain a closed-form solution to this differential equation for a battery simulation, and ultimately to specify V (t), we assume that E 0 (t) and I(t) are constant across a time step; this is reasonable because the current profile to be used in simulation will be sampled at very small intervals (for example, 0.01 or 0.1 seconds). Thus we will consider the current input to be a step function (defined as having a value of I(t) across the time interval of interest). In deriving the closed-form solution, we simplify our notation for the voltage across each capacitor by suppressing the second subscript for the v ci in Equation 3.1. In doing so, the voltage across the capacitor is the solution to v c (t) = βv c (t) + γi(t) (3.6) where β = 1 RC and γ = 1 C. 30

48 Rearrange and multiply both sides of this expression by e βt to get e βt ( v c (t) + βv c (t)) = e βt γi(t). (3.7) Recognizing the fact that we can rewrite the right-hand side of Equation 3.8 as e βt γi(t) = d dt (eβt v c (t)), (3.8) d dt (eβt v c (t)) = βe βt v c (t) + e βt v c (t). (3.9) Substituting Equation 3.9 in Equation 3.8, and integrating from t 0 to t we obtain, e βt v c (t) e βt 0 v(t 0 ) = t t 0 e βτ γi(τ)dτ. (3.10) Since I(t) is specified to be constant across an interval, we can rearrange the expression to give v c (t) = e β(t t 0) v c (t 0 ) + e βt t Evaluating the integral term on the right-hand side, we find t 0 e βτ γi(τ)dτ. (3.11) e βt t e βτ γi(τ)dτ = γi(t) t 0 β = γi(t) β (1 e β(t t 0) ) (3.12) Thus, the expression for a constant forcing current across a specified time interval is given as v c (t) = e β(t t 0) v c (t 0 ) + γi(t) β (1 e β(t t 0) ), (3.13) for t > t 0. In fact, if the interval is constant (that is, if T = t t 0 ), then v c (t) = e βt v c (t 0 ) + γi(t) β (1 e βt ). (3.14) Clearly, we can use the v c expression derived above in both dynamic expressions in Equation (3.1) for v c1 and v c2, characterized by identical dynamics. 31

49 Ultimately the expression in Equation 3.14 is inserted into Equation 3.1, for each resistor-capacitor pair, to form the single cell battery model for use in the pack model (the subject of Chapter 4). For usefulness in simulation, the next step is identification of the relevant parameters; this process is briefly summarized in the next section. 3.3 Identification of Model Coefficients In this section we sketch the procedure for identifying model coefficients, according to the model structure described in the previous section. Because the system identification methodology was developed and described in [11], and because it is not a major topic of this thesis (indeed, it is only a tool), we omit most of the detail and direct the reader to the indicated references. As seen in the open literature, using the model described earlier is a relatively common thing to do. The difference in our methodology, however, is how we choose our coefficients. Rarely do we see model coefficients that are based on actual identification methods involving intense experimental tests due to the fact that battery experimentation is very intensive and challenging (as described in the previous chapter). Instead, coefficients are often based on manufacturers supplied data and educated guesses. These methods are clearly not suitable for our purposes, and thus we propose an in depth methodology for model identification using Linear Parameter Varying Systems and Genetic Algorithms. It should be noted that the methodology was developed in collaboration with Yiran Hu at the Center for Automotive Research and a more intense look at the methodology can be found in [11]. A majority of the parameters of the model formulated previously depend heavily on the operating conditions of the battery. The operating condition of the battery 32

50 is determined primarily by the SoC, temperature, and direction of the current input. Because of this fact, we have devised schemes for scheduling model coefficients based on critical parameters. To illustrate this idea, we show in Figure 3.2 the voltage response of a lithium phosphate battery excited with a set of asymmetric discharging and charging steps at -15 o C. The nature of the tests can be found in [11]. As can be seen, the voltage of the battery varies greatly between SoC end points, which suggests that the coefficients describing the dynamics of the battery should be scheduled on SoC. In addition, Figure 3.3 presents the same set of current steps for the battery at a higher temperature of 5 o C, where it can be seen that the voltage response at 5 o C differs from that of -15 o C. This indicates that the coefficients should also be scheduled on temperature. Lastly, to show that the coefficients should also be scheduled on current direction, we can see that at the end of the dataset shown in Figure 3.2 there exists an asymmetry between the charge and discharge cycles. 4.5 battery voltage over step profile at 15 o C terminal voltage [V] significant voltage drop at low SoC time [s] x 10 4 Figure 3.2: Battery response to a series of asymmetric steps at -15 o C 33

51 4 battery voltage over step profile at 5 o C terminal voltage [V] time [s] x 10 4 Figure 3.3: Battery response to a series of asymmetric steps at 5 o C Due to the fact that the dependence of the coefficients on the parameters is, for the most part, unknown and nonlinear, the structure of the function used for scheduling should be robust enough to support an indeterminate shape, but also should be reasonable to evaluate. Therefore, it is reasonable to define two sets of coefficients for current direction scheduling (one for charging and the other for discharging). For the case when the current is equal to 0, we use the charging coefficients after a charging event, and discharging coefficients after a discharging event. For the scheduling of SoC and temperature, we will use linear spline functions, which are linear piecewise functions defined over a hypercube [7, 6]. We define the linear spline function domain to be arbitrarily fine, allowing us to approximate any continuous function on that domain. Since we cannot arbitrarily choose linear functions on each partition chosen 34

52 due to continuity constraints, we will use a representation called a P-spline. For more information on P-splines, see [11]. It should be noted that the coefficients scheduled on both SoC and temperature use a 2D linear spline parameterization, whereas those that only depend on temperature use a 1D linear spline representation. Because the model equations are linear, that is, the algebraic output equation contains the only nonlinearity, the overall system with these parametrized coefficients is a quasi-linear parameter varying system [10]. The model shown in (3.3) clearly has a large number of unknown coefficients that must be identified. Due to the fact that the model is a continuous time differential equation in state variable form with nonlinear elements in the output, optimization methods such as least squares [1] used for linear parameter varying discrete systems are not immediately applicable. Because of this, we have chosen to identify the coefficients using an optimization procedure where the objective is to minimize the error between the measured battery terminal voltage and the model terminal voltage. The cost function to be minimized is the sum of squared errors at the sample points. The work in [10] describes good results for this methodology when compared with various other norms Optimization As mentioned previously, the coefficient identification procedure uses a genetic algorithm as an optimization method, as used in [10]. Specifically, the identification process is broken into two segments: 1. identify a constant parameter model for the temperature of interest; and 35

53 2. employ the obtained constant parameter model from the first step as the initial point to find the SoC-dependent model. The reason this process is used is to handle the large number of unknowns that make up the fully parameterized model. Extending this process to support identification for the multi-temperature case, the following steps are used from [11]: 1. Choose a desired model order. Then design and collect modeling datasets at different temperatures, including one at room temperature. 2. Apply single temperature identification for the room temperature dataset. This not only provides a model for the specific condition represented by that dataset, but also provides an open circuit voltage function to be used for other temperatures. 3. Using the OCV function identified for the room temperature dataset, identify the dynamic model equation coefficients for the other temperatures. Assuming the OCV is not dependent on temperature, the OCV curve is found once. 4. Interpolate the single temperature models using linear spline functions. In other words, if a coefficient is dependent on both SoC and temperature, then perform a secondary least squares fitting of a 2D linear spline function using the identified single temperature coefficients as the data. If a coefficient is not dependent on SoC but is different for each temperature, then a 1D linear spline is used. 5. Perform a global optimization over all the datasets using the interpolated coefficients as the starting point. While some single temperature coefficients are very smooth with respect to temperature, others are not. Thus, in a sense the 36

54 interpolation process degrades optimality of the model coefficients. Therefore, it is important to re-optimize the coefficients jointly to regain global optimality. The most convenient result of this identification procedure is that before each optimization step, a good starting point always exists. For a global model with hundreds of unknown coefficients, lack of good starting point will often keep the optimization from finding a reasonable solution. For a deeper look at the identification results, error metrics, and input selection, see [11]. 3.4 Summary In this chapter, we have introduced a viable battery model to be used later in a larger battery pack model. The model structure chosen was a Randle equivalent circuit model for its simplicity and relatively good accuracy. In addition to the model, a methodology was discussed that will allow for the identification of the model coefficients for accurate representation of a battery cell. Using these developments, we can now move to deriving the battery pack model and simulation methodology. 37

55 CHAPTER 4 BATTERY PACK MODELING AND SIMULATION 4.1 Introduction In the previous chapter, we showed the battery model derivation that is the fundamental building block of the battery pack model. Using the single cell battery model, we will build a larger model that will allow us to model an entire battery pack with multiple configurations. The configurations include single series strings, strings of parallel cells (abbreviated SP), or parallel strings (abbreviated PS). Figure 4.1 shows the three different configurations that are possible. In addition to the battery pack model derivation, we will explain the simulator and show single pack simulation results. We will also present metrics to analyze the battery pack simulations which will be used later in a Monte Carlo application. To showcase the robust nature of our identification techniques, as well as the flexibility of the battery pack model, we will present one set of parameters based on a smaller Li-Ion Iron Phosphate battery. We will use a different cell when presenting our results in later chapters. 38

56 Figure 4.1: The three different configurations of automotive battery packs 4.2 Battery Pack Model Derivation In most (if not all) current HEV architectures, the battery pack is made up of a single string of cells. For example, the Toyota Prius contains 32, NiMH 6-cell modules in a single string which makes up its entire battery pack. Modeling and simulating a string of cells is a relatively easy task, since modeling a string of cells is essentially just increasing the model order number of a single cell to match the number of cells that exist in the battery string. Therefore, we see that no further derivation is required, since we already derived in the previous chapter a model for a single cell. As discussed in the introduction, PHEV battery architectures vary slightly from HEV battery architectures. PEV and PHEV battery packs are most commonly configured as either strings of parallel cells or parallel strings in order to increase overall pack capacity since most individual cells (most commonly Li-Ion) do not provide sufficient energy capacity to support the necessary range requirements for the vehicle. It 39

57 is for this reason that we focus on the model derivation for a battery pack in a parallel configuration and because it is more economically feasible (and thermally desirable) to use the multiple cell in parallel rather than a single, large capacity cell Explanation of Notation We will begin by describing the notation that is used for the battery pack. With the intent of making notation and battery pack representation clear, we will present the battery pack in a matrix-like topology where columns of batteries are connected in series and rows of batteries are not electrically connected other than by the overall parallel string connection. In so doing, we introduce subscripting notation for the ideal voltages for each battery as E i,j where i is the notation for a battery in the i th row, and j is the notation for a battery in the j th column of the battery pack. We can do this because in this work, we do not address the issue of pack topology, since we only assume an imposed temperature profile, as opposed to also including a thermal model to calculate heat generation of the battery. A diagram showing a graphical description of this notation can be seen in Figure 4.2. The current input (as seen by each battery) is denoted as α i, for the current through the i th string (or battery column). For our purposes, V i (t) represents the string voltages for each string in a pack, and V ck[i,j] (t) represents the voltage across each k th capacitor in the model. Likewise, R 0[i,j] represents the internal resistance of the (ij) th battery, R k[i,j] represents the k th resistor of the (ij) th battery in the pack model, and C k[i,j] represents the k th capacitor of the (ij) th battery cell model in a pack. 40

58 Figure 4.2: Graphical Representation of Notation Model Derivation One of the main challenges in designing a battery pack model with parallel strings to be used in simulation is defining the current splits during simulation runtime. Obviously, if all the cells in a pack were identical, then the current split between the strings would be easy to calculate; the current through each string would simply be obtained by dividing the overall input current by the number of batteries (or independent strings) that are connected in parallel. We know, however, that in a real 41

59 pack the current will split between the strings in an uneven fashion due to the fact that the laws of physics must be upheld. The current split calculation is complicated further by the fact every cell is a dynamic system. Thus, we must devise a model representation that allows for current splits between the strings to be calculated during the runtime of the simulation. Deriving the Model for a 2P1S Battery Pack For illustrative purposes, we begin the description of the modeling of the battery pack with a model of a 2P1S pack (that is, a pack with two batteries in parallel). Recalling the closed form solution of a single cell battery model derived in section 3.2.1, we can say that the voltages corresponding to two batteries, call them V 1 (t) and V 2 (t), each for a 2nd order equivalent circuit battery model, are given as V 1 (t) = E 1,1 V c1[1,1] e β 1[1,1]t V c2[1,1] e β 2[1,1]t +α 1 [ R 0[1,1] + R 1[1,1] (1 e β 1[1,1]t ) +R 2[1,1] (1 e β 2[1,1]t )] (4.1) and V 2 (t) = E 1,2 V c1[1,2] e β 1[1,2]t V c2[1,2] e β 2[1,2]t +α 2 [ R 0[1,2] + R 1[1,2] (1 e β 1[1,2]t ) +R 2[1,2] (1 e β 2[1,2]t )], (4.2) where β k[i,j] = 1 R 0[i,j] C k[i,j] and V ck[i,j] is the voltage of the capacitor containing the current (amp) history of the battery. Since the battery strings are connected in parallel, we know that V 1 (t) = V 2 (t). (4.3) 42

60 Finally, to include the current input (forcing function as seen by the parallel combination of batteries), we note that a simple application of Kirchhoff s current law renders I(t) = α 1 (t) + α 2 (t), (4.4) which is constant across a simulation time step. Through the definition of each battery model in parallel it is now easy to see that the entire system can be represented as a solution of linear equations. To simplify notation, we will define Φ j = R 0[1,j] + R 1[1,j] (1 e β 1[1,j]t ) + R 2[1,j] (1 e β 2[1,j]t ) (4.5) and Γ j = E 1,j + V c1[1,j] (1 e β 1[1,j]t ) + V c2[1,j] (1 e β 2[1,j]t ). (4.6) From Equations (4.4), (4.5), and (4.6), we can construct a system of linear equations to represent the current split between two batteries in a parallel pack connection: [ α1 α 2 ] = [ Φ1 Φ Extending to a Generalized Form ] 1 [ Γ2 Γ 1 I ]. (4.7) From this example of a 2P1S pack, it should be clear how we can extend this methodology to a generalized form. Clearly, by adding more strings in parallel to the battery pack, we will increase the size of the Φ matrix (while maintaining it as a square matrix). For an arbitrary number of parallel strings, the Φ matrix will always take the structure of a bidiagonal matrix with a row of 1 s representing the current. This is due to the fact that we set each battery string voltage equal to the next string 43

61 in the pack. Thus, we see that the general form for the current splits in a battery pack will be α 1 α 2 α 3.. α n = Φ 1 Φ Φ 2 Φ Φ 3 Φ Φ n 1 Φ n Γ 2 Γ 1 Γ 3 Γ 2.. Γ n Γ n 1 I, (4.8) where n is the number of parallel strings in the battery pack, α i is the current through each string at a simulation time step, and I is the total current seen by the entire pack at a simulation time step. To this point, we have only shown a pack with one battery in a series string. However, it is an obvious extension to add batteries to any of the strings in the pack model. Because the single cell battery model is linear, we simply sum up all battery voltages evaluated in a single string given by each individual single cell battery model. Doing this gives Φ i = and Γ i = m ( R 0[i,j] + R 1[i,j] (1 e β1[i,j]t ) + R 2[i,j] (1 e β2[i,j]t )) (4.9) j=1 m (E i,j + V c1[i,j] (1 e β 1[i,j]t ) + V c2[i,j] (1 e β 2[i,j]t )), (4.10) j=1 where i is the i th parallel string and m is the number of batteries in the i th string. With this battery pack model representation, we can move to the simulation of the battery pack. 44

62 4.3 Pack Simulator One of the main challenges that presents itself when simulating the battery pack is when to calculate the current split. Since the SoC is calculated according to the input current, we find that during the simulation we can either calculate the SoC first, or calculate the current split first. In either case, we find that one of the two calculations lags by a simulation time step. We justify this, however, by making two assumptions: 1. The current input to the system is constant over a simulation time interval. 2. The SoC (and therefore the E 0 ) of each battery does not change significantly over a simulation time interval. We must make the first assumption in order to use our battery model defined in the previous sections. The second assumption can be made because we assume that the time interval is small enough so that the there are no large SoC or E 0 changes from one simulation instance to the next (sampling on the order of 10Hz). Both of these assumptions are reasonable for this application, as we will see later. The simulation of the battery pack algorithm is carried out as follows: 1. Define battery characteristics for each individual cell. 2. Set up initial conditions for each battery (i.e. set SoC/OCV equal to represent a pack at equilibrium and capacitor voltages (V ck[i,j] ) set to 0V to simulate a completely relaxed battery.) 3. Evaluate first simulation time step using first value of input current profile with initial battery parameters and SoC. 45

63 4. Begin simulating current profile. 5. Calculate battery parameters and OCV 1 as a function of instantaneous battery SoC, temperature, current input, and individual battery characteristics. 6. Construct Φ and Γ matrices using perturbed or base battery parameters. 7. Use the Φ and Γ matrices to solve for the current split of the next simulation time step. 8. Evaluate the SoC, V c1[i,j], and V c2[i,j] for each battery in the pack. As we can see, the last step in the algorithm involves evaluating the SoC, which is to be used in the next simulation step. Because we assume the SoC does not change drastically over a time interval (and that the current is constant across a time step) we can be assured that the calculation of the SoC one step behind the rest of the simulation will not effect the overall outcome. Due to the analytical nature of the battery model and the fact that the solution of the current splits is based on a solution to a set of linear equations, we can easily implement the entire simulator in MATLAB. The implementation in MATLAB allows for more flexibility in terms of data storage and model tracking, as well as controlling the exact nature of simulation calculations that often pose problems in graphical simulator packages such as Simulink Simulating a String of Parallel Cells Configuration Earlier in the chapter we claimed that simulating a battery pack in a string of parallel cells configuration (SP) was achievable with using our methodology. Clearly, 1 The OCV can be calculated based on the previous step s SoC because it is assumed that the battery SoC does not change drastically over a simulation time step 46

64 simulating a parallel strings (PS) configuration is straight forward using the simulation procedures defined previously. A string of parallel cells configuration, however, is not immediately obvious. Recalling the Figure 4.1, we see that in an SP configuration, there are interconnections between each of the cells as opposed to the PS configuration which has only two connections between each of the strings. Consider a 3P4S pack for example. In the PS configuration case, there are three strings of four battery cells, with only two connections per string (on the top of the string and the bottom of the string as the way we have defined our general pack topology). The current splits at the top of each string, and each battery in the string sees the same current as the groups of batteries in the string. In the SP case, however, each battery is connected to the batteries above and below itself as well as the battery (or batteries) that are located in its row. These interconnections cause each battery s state to be affected not only by the other three batteries in the string (as in the PS case), but also by the three batteries in its row. This does not allow us to use the same simulation methodology described above to be used because it was derived assuming a string of batteries with no interconnections. Thus, we must simulate an SP pack differently. Still using the 3P4S battery pack in an SP configuration as an example, imagine if we separated each of the rows into four distinct packs, thereby leaving four individual 3P1S packs. We know that we can now simulate each of the 3P1S packs using the PS simulation methodology. If we assume the input current is identical at each of the 3P1S pack connections, we can simply simulate each pack separately and then combine their results at the end of the simulation to make up the entire pack. Therefore, in 47

65 order to simulate a battery pack in an SP configuration with m strings in parallel and n batteries in a string, one must 1. separate the pack into n individual battery packs of m 1 (or mp1s) size, 2. simulate each m 1 pack as if it was in a PS configuration, and 3. combine the simulated m 1 packs into the original pack size of m n (or mpns). Thus, we see that both PS and SP packs can be simulated using our simulation methodology described previously. 4.4 Battery Pack Simulation Examples To show that the model and simulator function as described, a specific example is presented. The example that is shown is of a 3P1S pack using model coefficients derived from an A123 Li-Ion Iron Phosphate cell with a nominal capacity of 2.3Ah, and nominal voltage of 3.2V. Although the pack model can clearly handle a large number of batteries in a string, it is more instructive to show only one in a string (to simplify tedious notation). However, from the derivation in the previous sections, it should be clear that adding an arbitrary number of batteries to a string is a simple process. The model coefficients were identified using the methodology described in the previous chapter as well as in [11]. We present this specific chemistry in order to show the robustness of the identification and pack model because a different Li-Ion chemistry will be employed in the following chapter. 48

66 10 5 Current [A] Time [sec] Figure 4.3: Current Profile Current Profile The input current profile that will be used for this example is taken from an actual Toyota Prius driving cycle. The driving cycle chosen from the data collected mimics what is a relatively typical profile for a battery pack in an HEV application. We downsampled the profile to fit our sampling time (10Hz) and applied an FIR filter to clean the noise from the signal. In addition, the profile is scaled in order to be more realistic for the size of currents that a battery pack the size of the simulation pack would endure. Figure 4.3 shows the standard input current profile Example Results In this example, the individual battery parameters have been generated to mimic a natural variation in manufacturing, thus forcing the current split between the three 49

67 parallel strings to be unequal as seen in Figure 4.4 (the overall input current can be seen in Figure 4.3). A formal definition of how we have defined battery parameter variability is discussed in Chapter 5. For this example, we just apply a different random perturbation of the R 0 parameter in each of the three batteries. As we can see, there is an unequal distribution of current caused by pack variability. Figure 4.5 shows a closer look at the current splits shown in Figure 4.4. We can also see in Figure 4.5 that there exists only one base back current (denoted by the dotted line) in the plot. This is due to the fact that the base pack has no random variation applied to the model coefficients, thus causing the current split to be equal for each string. It should be noted that the current splits are distributed around the base current split B v 1 B 2 v B 3 v 0.5 B v 1 B 2 v B 3 v B base 0 B base C Rate (C) 0 5 C Rate (C) Time [sec] Time [sec] Figure 4.4: Current Split Figure 4.5: Close Up of Current Split A more descriptive plot is the SoC plot (Figure 4.6), which is the simulation element with which we will be mostly concerned with in the next chapter. We can 50

68 SOC B 1 v B v 2 B v 3 B base SOC (%) Time [sec] Figure 4.6: Example of 3P1S Battery Pack SoC see that there is a significant SoC divergence from the beginning of the profile even though each battery began in an assumed resting state with equilibrated SoC. In the next chapter, we will define metrics to better analyze the SoC divergence and current splits in order to give a more concrete way of analyzing battery pack behavior Verifying the Model and Simulation A sufficient way of verifying that the model and simulation are working correctly is by analyzing the voltage plot of the entire pack. Figure 4.7 shows the three string voltages of the example 3P1S battery pack. We notice that all three string voltages 51

69 are equal, thus confirming that our model and simulation give expected results. In addition, we can verify that the three base pack currents (α 1, α 2, α 3 ) add up to the input current I. In our example case, then, as long as I(t) = α 1 (t) + α 2 (t) + α 3 (t) (4.11) and V 1 (t) = V 2 (t) = V 3 (t), (4.12) over the entire simulation time, where V i are the voltages of the strings in the pack, the simulation has been completed successfully. In general for a battery pack with n strings in parallel where n 2, if we can show that I(t) = n α k (t) (4.13) k=1 and n V 1 (t) V k (t) = 0, (4.14) k=2 across the entire length of the simulation profile, then the simulation can be assumed to have been completed successfully. It should be noted that although we have not provided experimental validation of the overall pack, such exhaustive experimental validation has been conducted for the individual battery cell model and coefficient identification (see [11] for validation). It is in this light that we feel particularly confident that our overall pack model is accurate. In addition, it should also be realized that in order to experimentally validate such a pack, extensive work must be done to identify each individual battery model coefficients in the pack. Even then we are not assured that we will be able to validate exactly the battery pack model. 52

70 Voltage (V) Time [sec] Figure 4.7: Pack Voltage of an Example 3P1S Battery Pack 4.5 Summary In this chapter, we used the battery model derived in the previous chapter to derive a battery pack model of an arbitrary size in either a single string configuration, a parallel string configuration, or a string of parallel cells configuration. Using the model, we designed a simulator that allows us to simulate a battery pack in order to analyze the battery pack behaviors, such as current splits between the strings and SoC deviation. We also showed an example simulation of a 3P1S battery pack. In the next chapter, we will show methods and applications using the derived model and simulator as well as develop some tools in order to better quantify battery pack behavior. 53

71 CHAPTER 5 METHODS AND APPLICATIONS As in all research, it is helpful to show the applicability of the results. In this chapter, we will show a few different methods and applications using the modeling methodologies and simulation techniques developed in the previous chapters. First, we will show statistical Monte Carlo simulations to analyze the SoC deviation of a battery module from start to finish. We will then show an example of a battery module fault, and how the model could potentially be used for diagnostics. 5.1 Monte Carlo Simulation and Statistical Analysis Using the model and simulation methodology shown previously, we will demonstrate an application and also provide an analysis of the results. The application that will be demonstrated will be a Monte Carlo simulation, simulating over a million different battery modules (that can be connected together to make up a complete battery pack) that vary in random parameter perturbation, temperature, SoC, configuration, and load profile. By doing this, the effects of these various battery pack elements can be analyzed to show how SoC divergence occurs over the different pack combinations. We will begin by discussing how the different variations were chosen. 54

72 5.1.1 Simulation Profiles In order to capture the different types of modes of operations that electric vehicles experience in normal driving conditions, it was decided that both an HEV profile and PHEV profile would be used for simulations. HEV Profile Design The HEV current profile was designed from a experimental data taken from a Toyota Prius. From the hours of data collected on the Prius, 457 seconds were chosen. The data subset was then filtered using an FIR filter for smoothing and was scaled to reflect a 60 Ah battery pack (3P8S or 8S3P pack with 20Ah cells). The resulting profile is shown in Figure 5.1. It should be noted that the HEV profile represents the charge sustaining mode of operation. As it can be seen in Figure 5.2, the C-rates of the current are relatively high (around 7C), common in HEV operation. However, in Figure 5.3, we see that the high currents are infrequent, thus helping maintain the charge sustaining nature of the HEV profile as shown in Figure

73 Current[A] Time [sec] Figure 5.1: Current profile for HEV 56

74 C Rate Time [sec] Figure 5.2: C-Rate profile for HEV 57

75 Samples Figure 5.3: Current profile histogram for HEV Samples Figure 5.4: C-Rate histogram for HEV 58

76 Change in SoC from 50% Figure 5.5: SoC change from 50% for HEV Profile 59

77 PHEV Profile Design PHEV experimental data is not readily available because there are no production PHEVs existing today in the marketplace. The data that was used to design our PHEV profile was taken from a PHEV simulator designed at the Center for Automotive Research. By taking a 457 second sample from the overall profile and scaling to fit our 60Ah pack, we designed our charge depleting profile as shown in Figure Current[A] Time [sec] Figure 5.6: Current profile for PHEV Different from the HEV profile, the PHEV profile shows lower C-rates (a maximum of around 2C) as shown in Figure 5.7. This is representative of normal PHEV operation. In the histogram shown in Figure 5.9, we can see that the majority of 60

78 C Rate Time [sec] Figure 5.7: C-Rate profile for PHEV currents do not exceed a rate of C/2. The projected SoC profile (calculated based on our 60Ah battery pack) in Figure 5.10 shows a charge depleting nature, very representative of PHEV operation. Figure 5.9 shows a histogram of the C-rates for the PHEV profile. As we can see, the majority of currents seen by the PHEV pack are very low which is characteristic of PHEV packs [17] Battery Pack Configurations Because the main focus of this work is based on the simulation and modeling of parallel strings of batteries, we will restrict our module configurations to the two different types that are feasible in an automotive battery pack: Parallel String Configuration (see Figure 5.11) and String of Parallel Cells Configuration (see Figure 5.12). 61

79 Samples Figure 5.8: Current profile histogram for PHEV These figures are also shown in Chapter 4. In addition, we will size each module to contain 3 strings in parallel, and 8 cells in series. The motivation for this is based on the fact each cell is 20Ah, giving a total module capacity of 60Ah, and an overall module voltage of 28.8V (assuming a nominal cell voltage of 3.6V). These cell statistics are taken from an EIG Lithium Ion battery, whose parameters are identified using the methodology described previously. The 20Ah cell is chosen as prototypical of power cells used to assemble PHEV modules, such as the ones made by A123 Systems Battery Pack Variability Every battery pack is unique, which is a product of manufacturing variability, battery state of health (SoH), and instantaneous conditions, such as current and 62

80 Samples Figure 5.9: C-Rate histogram for PHEV temperature. Therefore, it is of interest to analyze the effects that the variability of the cells have on the SoC divergence in a battery pack. SoC divergence can have a significant effect on battery life and overall safety if improperly managed. Thus, by analyzing multiple different variability conditions within a battery pack, insight can be gained into how the SoC diverges so that such knowledge can be incorporated into BMS designs Modeling Cell Variability One of the more attractive elements of the model design is the fact that the model is based on a cell level description, rather than an overall pack, mimicking a battery 63

81 Change in SoC from 80% Figure 5.10: SoC change from 80% for PHEV profile pack in reality. Therefore, it is easy to see that applying variability on a cell level is relatively straight forward and realistic. We model cell variability based mostly on differences in manufacturing. Traditionally, manufacturing variability can be assumed to be distributed normally over a large number of samples. It is for this reason that it was decided to use a normal distribution to generate such variability among battery cells. It is reasonable to assume that since manufacturing variability normally distributed over a large sample, the standard deviation will not be very large, perhaps only on the order of one or two percent. Therefore, by generating normal values with three different standard deviations - 1%, 2%, and 3%- to represent variability, an accurate analysis can be performed. It should be noted that we will refer to the 64

82 standard deviations of variability in percent. This is merely done for convenience as it will be seen in notation definitions later. Choosing the Model Parameters to Vary Applying the variability within the model is not as obvious as it initially seems. The model is an equivalent circuit model which is an abstract way of representing a battery. There are parameters in the equivalent circuit model that represent physical elements (such as E 0 representing the OCV), however it is not obvious which of the parameters should be varied. It was observed through experimental data that the physical elements showing the most variability are the internal resistance (able to be calculated indirectly by voltage and current experimental data, and directly calculated through Electrochemical Impedance Spectroscopy (EIS) tests) and capacity (able to be experimentally calculated). Therefore, the parameter R 0 was chosen, since it most closely models the internal resistance of the battery, and the capacity parameter was chosen, which appears during simulation and is used to calculate each individual battery SoC. Applying Cell Variability in the Model With the parameters to be varied and accompanying standard deviations (represented by σ) chosen, we can generate normally distributed random data sets to represent variation of the parameters. The three different standard deviations (1%, 2%, and 3%) are shown in Figures 5.13, 5.14, and Each data set contains 240,000 normally distributed random numbers (10,000 3P8S battery modules, or 240,000 different cells). 65

83 Figure 5.11: Parallel string configuration (3P8S) 66

84 Figure 5.12: String of parallel cell configuration (8S3P) 67