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Electrical Battery Model for Use in Dynamic Electric Vehicle Simulations Ryan C. Kroeze, Philip T. Krein University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering 1406 W. Green St. Urbana, IL 61801 Tel.: 217-333-6592, Fax: 217-333-1162 Email: [email protected], [email protected] developed through the National Renewable Energy Laboratory (NREL) [1] is a simulator based on static maps that reflect steady-state behavior of vehicle subsystems. A dynamic simulator [2]-[3] offers a more detailed look, based on dynamic equations of each subcomponent (the engine, battery, inverter, motor, and transmission) on microsecond time scales. The Simulink component block of the battery model in [2-3] was designed for vehicles with lead-acid battery packs. The lead-acid model is insufficient for accurate simulations of vehicle drivetrains with different battery-pack chemistries. Here, a three time-constant dynamic electric battery model for lead-acid, nickel metal hydride (NiMH), and lithium ion (Li-ion) batteries is proposed and developed for use within a dynamic HEV simulator. The electrical model must both accurately represent the terminal voltage, state-of-charge (SOC), and power losses of each battery type, without excessive simulation times. The paper includes experimental validation of the models and simulator at several levels. Abstract - Simulation of electric vehicles, hybrid electric vehicles, and plug-in hybrid electric vehicles over driving schedules within a full dynamic hybrid and electric vehicle simulator requires battery models capable of predicting state-of-charge, I-V characteristics, and dynamic behavior of different battery types. A battery model capable of reproducing lithium-ion, nickel-metal hydride, and leadacid I-V characteristics (with minimal model alterations) is proposed. A battery-testing apparatus was designed to measure the proposed parameters of the battery model for all three battery types and simulate driving schedules with a programmed source and load configuration. A multiple time-constant battery model was used for modeling lithiumion batteries; verification of time constants in the seconds to minutes and hour ranges has been shown in numerous research articles and a time constant in the millisecond range is verified here with experiments. Lack of significant time constants in the millisecond range is validated through direct testing. A modeled capacity-rate effect within the state-of-charge determination portion of the proposed model is verified experimentally to ensure accurate prediction of battery state of charge after lengthy driving schedules. The battery model was programmed into a Matlab/Simulink environment and used as a power source for plug-in hybrid electric vehicle simulations. Results from simulations of lithium-ion battery packs show that the proposed battery model behaves well with the other subcomponents of the vehicle simulator; accuracy of the model and prediction of battery internal losses depends on the extent of tests performed on the battery used for the simulation. Extraction of model parameters and their dependence on temperature and cycle number is ongoing, as well as validation of the Simulink model with hardwarein-the-loop “driving schedule” cycling of real batteries. II. In the search for a battery model to meet the above specifications, electrochemical, mathematical, and electrical models of Li-ion, NiMH, and lead-acid battery cells have all been reviewed. Electrochemical models [4]-[7] typically are computationally time-consuming due to a system of coupled time-varying partial differential equations. Such models are best suited for optimization of the physical design aspects of electrodes and electrolyte [4]. Although battery current and voltage are related to microscopic information about the battery cell (e.g., concentration distribution), parameters for electrochemical models are not provided by manufacturers and require extensive investigation [4], [8]. Mathematical models [9], [10] use stochastic approaches or empirical equations which can predict runtime, efficiency, and capacity. However, these models are inaccurate (5-20% error) and have no direct relation between model parameters and the I-V characteristics of batteries. As a result, they have limited value in circuit simulation software [8]. Electrical models [8], [11]-[17] are the most intuitive for use in circuit simulations. They often emphasize Keywords: plug-in hybrid electric vehicle, lithium-ion battery, vehicle simulator, battery model I. INTRODUCTION Electrochemical batteries are used in conjunction with an auxiliary power unit (APU) to power hybrid electric vehicles (HEV). Plug-in hybrid electric vehicles (PHEV) and electric vehicles (EV) can be treated as a subclasses of HEVs in which the batteries function in different ways. In order to better understand losses, thermal characteristics, and durability of various drivetrains, a dynamic simulator is a key tool. The ADVISOR program 978-1-4244-1668-4/08/$25.00 ©2008 IEEE BACKGROUND 1336 Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply. Thevenin equivalents and impedances, or provide runtime models. Combinations are often used. Thevenin models [11]-[13] assume the open-circuit voltage is constant and use a series resistor and an RC parallel network to track battery response to transient loads [8]. An increase in the number of parallel RC networks can increase the accuracy of the predicted battery response. However, between the SOC and time-constant dependence on cycle number and temperature, prediction errors for estimating run time and SOC tend to be high. Impedance-based models, like Thevenin models, are accurate only for a fixed SOC and temperature setting; hence their accuracy when predicting dc response and battery runtime is limited [14]. Impedance spectroscopy can be used to fit a complicated equivalent network to measured impedance spectra in order to validate the time constants found in the Thevenin models. Runtime-based electrical models use continuous or discrete-time implementations to simulate battery runtime and dc voltage response in SPICEcompatible simulations for constant current discharges. Inaccuracy increases as the load currents vary [15]. Combinations of these circuit models (in particular the Thevenin and run-time models) can take advantage of the positive attributes of each [8], [16], [17] such that accurate SOC prediction, transient response, run-time, and temperature effects can be obtained. III. t SOC[i(t ), T (t ), ncycle , t ] SOCinitial ³ f1[i (t )] (1) 0 u f 2 [T (t )] u f 3[ncycle ] u i(t )dt Vterminal Voc ( SOC , T ) ibat (t ) u Rint ( SOC , T ) (2a) ibat (t ) u Rtransient (W s ,W m .W h ) Rtransient ( Rs ( SOC )e Rm (SOC )e Rh (SOC )e 1 t Rs ( SOC )Cs ( SOC ) (2b) 1 t Rm ( SOC )Cm ( SOC ) 1 t Rh ( SOC )Ch ( SOC ) ) Eq. (1) for SOC can be modeled as the left circuit in Fig. 1 when temperature and cycle number are given (e.g. short vehicle drive cycles at a low speed and grade). Here (1) is normalized to battery capacity, so VSOC lies between 0 and 100% and represents the SOC of the battery [8]. I battery is the battery load, modeled as a MODEL The model in [8] is capable of predicting run time and IV performance for portable electronics, but is not accurate for the transient response to short-duration loads (less than 1 s). As a result, it does not predict accurately the SOC throughout drive cycles for HEV simulations. Fast time constants of Li-ion batteries have been shown in [13], [16], [17] and are necessary to determine the losses within a battery pack during vehicle drive cycles. Accurate determination of the discharge capacity, which is a function of the discharge rate i (t ) , temperature f 2 [T (t )] , and cycle number f 3 [ ncycle ] [16], must also current source, while Rself disch arg e and Rcap _ fade are paralleled to Ccapacity and represent normalized selfdischarge and battery capacity correction factor [16]. Vter min al relates to the parameters shown in the right circuit of Figure 1, a multiple time-constant approach ( W sec , W min , and W hour ) for modeling the transient behavior of the terminal voltage where each parameter is a function of the SOC [8]. This battery model has been implemented in the Simulink environment. include a rate factor, f1[i (t )] . The rate factor accounts for a decrease in capacity due to unwanted side reactions [8], [18] as the current increases. The proposed model for predicting SOC, terminal voltage, and power losses of Li-ion, NiMH, and lead-acid batteries is shown in Figure 1. It is governed by equations representing SOC and Vter min al : IV. DATA Measurements of the circuit model parameters of Fig. 1 for different batteries are found using a battery testing apparatus which consists of data loggers, an electronic load, power supply, and other equipment. Each component is controlled via Labview 7.1, which is used to program test sequences and analyze the recorded Figure 1 Proposed electrical battery model for use in HEV simulation at a constant temperature 1337 Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply. measurements. The open circuit voltage ( VOC ) and 0.1 terminal voltage versus SOC at room temperature was tested for five Panasonic CGR18650A 3.7 volt, 2200mAh cylindrical Li-ion batteries using a constant current discharge profile at several different charge rates. Each of the parameters in the model of Fig. 1 is a nonlinear function of the various effects. For purposes of the model, each is represented as a polynomial function of SOC up to sixth order, with coefficients given by: Parameter a 0 + a1* SOC a 2 * SOC 2 ... Rseries Rseconds Resistance (ohm s) 0.08 Rminutes Rhours 0.06 0.04 0.02 0 0.1 (3) The values in Table 1, when inserted into (3), formulate the open-circuit voltage as the SOC changes. The resulting expressions can be used within modeling programs for efficient computation of the open-circuit voltage. Estimates of W min (seconds/minutes range), W hour , 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 SOC Figure 2 Resistance values versus SOC for all discharging current resistive circuit parameters 1 values for Rseries , Rt _ s , Rt _ m , and Rt _ h during discharge currents are shown in Figure 2. The functions of the resistive components versus SOC are interpolated values between the measured data points; model simulations may use either the interpolated values between the measured points (i.e. a look-up table) or the best-fit polynomial functions from the data in Table 1. To check for time constants smaller than W sec , and the corresponding resistive and capacitive circuit parameters are shown in [8] and [13]. Both the W sec and W min time constants have been measured here using charging and discharging currents at intervals of 5% SOC. The series resistance and resistive and capacitive components used to model W sec and W min are shown as a function of SOC for discharging and charging currents when the values in Table 3 are inserted into equation 3. (D) and (C) in Table 3 denote time constants measured during discharging and charging current profiles, respectively. Determination of the longest time constant, W hour , and its resistive and capacitive which would be expected to have effects on dynamics of vehicle power electronics, a buck converter switching at 10 kHz was placed between a single Panasonic CGR18650A cell and a load. Figure 3 depicts the measured battery voltage, current, converter output, and switching function for a load of 1.5 :. The waveform shows a voltage drop corresponding to the switching frequency, caused by the Rseries component of the components is a time-consuming process, hence the values found for discharging currents will be used to model W hour for charging currents as well. The measured battery model. The voltage drops do not show evidence of exponential behavior on this time scale. Hence time scales faster than W sec do not impact the model. TABLE 1 CIRCUIT PARAMETER FUNCTION VALUES FOR THE OPEN-CIRCUIT VOLTAGE AND TIME CONSTANTS Parameter a0 a1 a2 a3 a4 a5 a6 Voc 3 13.433 -90.038 284.33 -453.64 355.88 -108.97 Rseries (D) 0.0482 0.144 -0.4577 0.4965 -0.1297 -0.049 0 Rt _ s (D) 0.1457 -1.0586 4.867 -10.131 9.5077 -3.3024 0 Ct _ s (D) 0.7472 9.0675 -30.712 32.551 -7.358 -3.4621 0 Rt _ m (D) 1.7423 -33.837 228.96 -712.75 1118.1 -859.2 -.12 Ct _ m (D) 239.49 -14755 264236 -825025 947250 -371150 0 Rseries (C) 0.051 0.2078 -1.1148 2.1656 -1.7766 0.5251 0 Rt _ s (C) 0.1134 0.9721 5.0929 -11.278 10.914 -3.845 0 Ct _ s (C) 0.9936 14.521 -91.976 199 -179.67 -57.923 0 Rt _ m (C) 1.6561 -35.02 246.62 -78559 1250.7 -970.92 292.72 Ct _ m (C) 216.52 28308 -166348 431168 -491497 199631 0 Rt _ h 0.1484 -1.1978 3.5946 -4.3618 1.8335 0 0 Ct _ h 493923 3E+07 -1E+08 2E+08 -9E+07 0 0 1338 Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply. Open Circuit Voltage (volts) 4.2 62.5 degrees Celcius 27 degrees Celcius 3 degrees Celcius 4 3.8 3.6 3.4 3.2 3 2.8 1 Figure 3 Battery voltage (500 mV/ div), load current (500 mA/ div), converter dc output (5 V/div), and fswitch (10 kHz), respectively. Figure 5 Determination of the rate factor, f1[i (t )] , is shown internal voltage drops due to the internal resistance and discharge rate, 2Cref Voc Rint u (i2 C iC / 25 ) . Variables a and b correspond to the SOC at which the voltage reaches the minimum (specified by manufacturer) for 2Cref and 2C discharge curves, the corresponding discharge current [16] and must be taken into account to accurately predict SOC during drive cycles with large load transients. Tables 2 and 3 include the measured values of the rate factors for discharging and charging currents, respectively, at rates of C/25, C/5, C/2, and C. Trials of the Li-ion battery open circuit voltage at a temperature of 3°C, 27°C, and 62.5°C are shown in Figure 5. The temperature coefficient ( f 2 [T (t )] ) can C Voltage [volts] 4 3.8 3.6 0 Figure 4 0.2 0.4 0.6 SOC Determination of rate factor, ( 0.8 f1[i (t )] ), iload 0.0808 0.4389 1.0886 2.1603 f1[iload ] 1 0.9946 0.96495 0.9232 iload -0.0838 -0.4386 -1.0988 -2.202 f1[iload ] 1 0.99324 0.9803 0.9665 RESULTS simulated within the model and taken from [8]. Results of the four simulations are shown in Fig. 7. Fig. 7a confirms that the model vehicle (see [3] for model details) was able to follow the FUDS schedule during the simulations. The decrease in SOC (from an initial SOC of 90%) is shown in Fig. 7b for each vehicle. The results are optimistic because [8] does not include a rate factor. Fig. 7c shows the resistive losses within each 3.4 b Open circuit voltage versus SOC at constant temperatures of 3°C, 27°C and 62.5°C. V. Battery Voltage Discharge Curves 3 0 0.2 Tests have been performed in Simulink to ensure accurate model performance during driving schedules. Fig. 6 depicts the subcomponents of the vehicle simulator, including new battery storage model, as well as the flow of variables between each subcomponent. Simulations for four different HEV configurations, corresponding to plug-in hybrids with Li-ion pack sizes of 25, 50, 75, and 100 kg were performed on the Federal Urban Driving Schedule (FUDS). TCL PL-383562 850mAh polymer Li-ion batteries were used in a series and parallel configuration to model the battery pack. Equations for all battery parameters (except W sec ) were 4.4 C/25 Discharge Curve with resistive losses 2C 0.4 TABLE 3 RATE FACTORS FOR VARIOUS CHARGE CURRENTS be found the same way the rate factor ( f1[i (t )] ) was 3.2 SOC TABLE 2 RATE FACTORS FOR VARIOUS DISCHARGE CURRENTS respectively. The ratio of b to a is equal to f1[i (t )] for C/25 C/5 C/2 0.6 found. At a higher temperature, there is very little alteration of the open-circuit voltage as compared to the room temperature case; a drop in open-circuit voltage can be seen for the 3°C case, and accurate results at low temperatures can be achieved by including a voltage drop equation within the model. As the temperature changes, a voltage versus SOC equation will be used to determine the voltage drop at a certain temperature and SOC. in Figure 3 for the 2C discharge rate. A reference curve for the 2C discharge rate is shown as the dotted line in Figure 4, and represents VOC with included battery 4.2 0.8 a 1 from measurements of voltage vs. SOC at different discharge rates. 1339 Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply. ¦ ¦ ¦ Figure 6 High-level model of EV and PHEV simulators in the MATLAB environment battery pack, and Fig. 7d shows losses of all four vehicles compared with the road power. The proposed battery model successfully represents Li-ion batteries within a PHEV throughout driving schedules. Data extracted from the results provide insight on the effects of various driving behaviors and the short-term transients on losses within a battery pack. Simulations on computers with Intel’s Core 2 Duo processor take approximately 2 min per 1 s of driving schedule; this is ten times longer than driving simulation times with the previous Thevenin-based lead-acid battery model [2]. This increase is caused by algebraic loops introduced into the Simulink model from differential equations describing the three time constants. a. S peed profile b. S OC 30 1 V EH 1 25 FUDS 1510 SOC[%] speed [m/s] 0.9 20 15 0.8 10 V EH 1 0.7 V EH 2 V EH 3 5 V EH 4 0 0 500 1000 0.6 1500 0 200 400 time [s] c . B attery loss es 1500 x 10 V EH 1 1000 1200 1400 d. V EH 1 road power and los s es 2 V EH 2 V EH 3 Road Pow er Battery Los s es 1.5 Power [kW] V EH 4 Power [W] 4 600 800 Time [s] 1000 500 1 0.5 0 0 20 40 60 80 Time [s ] 100 0 120 0 20 40 60 80 Time [s] 100 Figure 7 (a-d) Simulation of VEH 1, 2, 3, and 4 on the FUDS with battery pack mass as the variable 1340 Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply. 120 4.25 T e rm in a l V o lta g e (V o lts ) Results of verification of the Simulink model against the actual response of the tested Li-ion batteries are shown in Figures 8-11. Figure 8 depicts the simulated battery terminal voltage and measured voltage fluctuations versus time when both the model and batteries are put through a simulated city/highway driving schedule. After the 38 min. driving schedule, the error between the actual and simulated terminal voltage is 0.199%. This correlates to a SOC deviation of the simulated model to the actual of 1.00%. Figure 9 is the first 500 s of the simulated driving schedule where the battery model terminal voltage tracks the measured LI-ion terminal voltage. Discrepancies between the terminal voltage of the simulated and measured batteries near the upper peaks of the voltage during charging periods (simulated regenerative breaking) are inconclusive. Although it would appear that the resistive constants during cycle charging may be inaccurate, the measured data points occur at approximately every second, while the simulated driving schedule has the current charging peaks at below one second time scales. Hence, the measuring apparatus is too slow to pick up all details of fluctuations of the real battery during the driving schedule. The lower terminal voltage peaks match up well since the driving schedule maintains the discharging current peaks at a rate above one second. Figures 10 and 11 display the simulated road power and measured Li-ion output power during a trial using the 22 min.Federal Urban Driving Schedule. The Simulink Li-ion battery model, along with the measured model circuit parameters shown previously was placed into the University of Illinois Electric Vehicle Simulator to run the driving schedule [3]. Figure 10 depicts the road power divided by the number of batteries in the entire pack (in this case 6831 batteries) for both the simulated and measured power into and out of the battery. Tracking of the measured battery output power against the Simulink model road power per battery is accurate. Figure 11 is a close-up of the output power from 4000 to 5000 s of the FUDS and just as with Figure 8 peak output power has a limited time fidelity. 4.15 4.1 4.05 4 0 100 200 300 400 500 Time (s) Figure 9 First 500 seconds of simulated and measured battery terminal voltage versus time for city/highway driving schedule The simulated peak road power per battery occurs during intervals less than a second, hence the measurement apparatus will not pick up those peaks during the measured Li-battery trial. VI. CONCLUSION The proposed battery model will accurately represent lithium-ion battery behavior within a dynamic HEV simulator. Experimental parameters found from bench tests characterize Li-ion cells well at a constant temperature and cycle life, and the effects on those parameters caused by fluctuating temperature can easily be modeled in a dynamic simulator through open circuit voltage characterization at a few temperatures. Simulink models of battery pack/cell response to simulating driving schedules have been verified using a hardwarein-the-loop battery cycle testing apparatus; the simulated SOC, terminal voltage and output power response to the FUDS and another formulated city/highway driving schedule tracks the measured responses. Although both the University of Illinois HEV simulator and the 14 4.25 12 Simulated Voltage Measured Voltage 4.2 Simulated Road Power Measured Road Power 10 4.15 Road Power (Watts) Terminal Voltage (Volts) Simulated Voltage Measured Voltage 4.2 4.1 4.05 4 8 6 4 2 0 3.95 -2 3.9 3.85 0 -4 500 1000 1500 Time (s) 2000 -6 0 2500 Figure 8 Simulated and measured battery terminal voltage versus time for city/highway driving schedule 200 400 600 800 Time (seconds) 1000 1200 Figure 10 Simulated and measured road power per battery versus time for FUDS 1341 Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply. 1400 [5] 5 4 Simulated Road Power Measured Road Power Road Power (Watts) 3 [6] [7] 2 1 [8] 0 -1 [9] -2 -3 800 850 900 Time (seconds) 950 1000 [10] [11] Figure 11 200 seconds of simulated and measured road power versus time for the FUDS [12] proposed battery model are capable of providing simulations which include self-discharge and batterycycling degradation, the simulation times necessary to get results over these effects on various battery types are currently too long to validate the measured affects. The accurate response of the battery model now allows for extensive tests over the efficiency and losses within the battery pack during various driving schedules and alteration in vehicle parameters. [13] [14] ACKNOWLEDGEMENT [15] This work was supported in part by the Power Affiliates Program at the University of Illinois. [16] REFERENCES [1] K. B. Wipke, M. R. Cuddy, S. D. 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Doerffel, “Rapid test and non-linear model characterization of solid-state lithium-ion batteries,” J. Power Sources, vol. 130, pp. 266-274, 2004. R. Rao, S. Vrudhula, and D. N. Rakhmatov, “Battery modeling for 1342 Authorized licensed use limited to: GOVERNMENT COLLEGE OF TECHNOLOGY. Downloaded on December 31, 2009 at 04:42 from IEEE Xplore. Restrictions apply.