Download Electrical Battery Model for Use in Dynamic Electric Vehicle

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
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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
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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
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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.
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¦
¦
¦
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
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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
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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]
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