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A Parameter Identification Method for a Battery
Equivalent Circuit Model

2011-01-1367
Published
04/12/2011

Shugang Jiang
A&D Technology Inc.
Copyright © 2011 SAE International
doi: 10.4271/2011-01-1367

ABSTRACTGood battery modeling is critical for energy management ofelectric vehicles and hybrid electric vehicles. Because of its
simplicity and satisfactory performance, equivalent circuit
models are widely used in this area. A frequently adopted
equivalent circuit model is one that consists of an open-circuit
voltage and a resistor in series with two sets of parallel resistorcapacitor combinations. This model performs well indescribing battery transient behavior due to the dynamics ofsuch physical phenomena as mass transport effects and double
layer effects.
Generic methods for obtaining the parameters of this model
involve analyzing the battery voltage behavior under stepchanges of load current. The fact that the model has two time
constants places a challenge on parameter identification. Somemost often used method makes use of the property that each of
the two time constants plays a dominant role at different stagesof the battery voltage response, and calculates the model
parameters accordingly. For such method, the results are
greatly influenced by the partition of faster and slower
dynamics of the battery, and selection of the data points used
for the calculation. Moreover, because majority of the testingdata is not used towards calculating the parameters, theobtained model might not reflect the overall battery
characteristics well and consequently might not give highfidelity predictions. For other methods that use nonlinear curve
fitting or genetic algorithm for parameter searching, thesuccessful implementation greatly depends on the proper
setting of initial values and searching space.
A novel method of parameter identification for the equivalentcircuit model is presented in this paper. It makes use of aregression equation which is linear in variables that can bemeasured or calculated from the test. With this approach, alltesting data during the relaxation period of a constant current
pulse discharge or charge test is used towards obtaining themodel parameters and the calculation is in the sense of leastsquared error. Application of the method to real battery testingdata is presented. The example indicates that the method gives
very good results with modeling error of less than 0.5%.

INTRODUCTION
With ever growing concerns on environmental issues and thedepletion of fossil fuel supply, the automotive industry has put
plenty of effort into the development of Electric Vehicles (EV)and Hybrid Electric Vehicles (HEV) [1]. For both forms of
implementation the battery plays a critical role in affecting theoverall vehicle performance [2]. A well designed energy
management system, including battery management isfundamental for obtaining the highest possible emission
reduction, significant fuel economy improvement, and better
vehicle performance. The battery management functions
monitor the battery voltage, current and temperature etc.,
estimate the status of the battery, and perform appropriate
actions including cell balancing and thermal management etc.,
to guarantee that the battery is operating in the proper range for
optimum vehicle performance.
Being able to accurately calculate the battery state of charge(SOC) over wide range of operating conditions is theprerequisite for almost all battery management functions [3,4].
To determine battery SOC, the relationship among SOC,
battery open circuit voltage (OCV), and battery input to output
dynamic characteristics, i.e., battery voltage v.s. current
behavior, needs to be established. There are various approaches
for setting up this mathematical representation, two major
types of which are electrochemical modeling [5, 6, 7] andequivalent circuit modeling [8, 9, 10, 11, 12, 13, 14, 15, 16].
Electrochemical models are based on the battery physicalconstruction and chemistry. While these models can be
extremely accurate in describing the battery behavior, usually
they are computationally time-consuming and not suitable forreal-time control oriented applications. Different from theGratis copy for Shugang Jiang
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electrochemical approach, the equivalent circuit model
describes the effects of the chemical processes of major interest
using electrical circuit elements, such as voltage source,
resistors and capacitors etc. to approximate the battery
dynamics.
Since battery dynamic effects can range as wide as from MHz
to mHz, i.e., the time domain response can span frommicroseconds up to hours, days and even longer [17], one
needs to pay attention to the targeted time range of the battery
behavior when using the equivalent circuit model approach.
For example, while the electromagnetic effects are very fast,
such long-term effects as the cycling effect, reversible effectand aging effect have time constants from hours to years. The
equivalent circuit model discussed in this paper is often used todescribe the influence of such electrochemical phenomena as
double layer effects and mass transport effects, which havetime ranges from milliseconds to hours and have big influences
on battery performance in EV and HEV applications.
Something deserves noting is although a physical explanation
can be designated between the electric circuit components ofthe model and the actual battery dynamics effects, an equivalentcircuit model is constructed mainly to match experimental
data. Such a model, while simple and intuitive, provides
sufficiently accurate information that serves the need of realtime battery management when correctly designed.
The widely used equivalent circuit model [11, 12, 13, 14, 15,16] studied in this paper is shown in figure 1. It consists ofbattery OCV Voc, ohmic resistance of the connectors, electrodes
and electrolyte R0, and two sets of parallel resistor-capacitor
combination R1, C1 and R2, C2 representing the mass transporteffects and the double layer effects respectively [13, 15].
Commonly the time constants of the two dynamics differ by atleast an order of magnitude.

parameters (R1, C1 and R2, C2) are more difficult to identify
because two time constants are involved in the model.
Some paper takes advantage of the fact that the two time
constants differ greatly in magnitude, thus partitions the test
data into the short time constant dominant segment and thelong time constant dominant segment, and uses the battery
voltage response in each segment to estimate the model
parameters [13]. However, partition of the segments, andselection of the data points used for the calculation could bevery arbitrary and thus the parameter values determined mightvary greatly, and good performance of the model is notguaranteed. Moreover, since majority of the testing data except
the selected data points is not used in the calculation, theoverall battery characteristics might not be well reflect andconsequently the model might not give high-fidelity
predictions. Some other papers make use of the nonlinear
curve fitting techniques [14] or genetic algorithm for searchingof the parameters inside the defined space [16]. While thesemethods might lead to model parameters of good quality, theyneed users to have good experience in setting up the initial
parameter values and the searching space.
This paper presents a novel parameter identification method
for this battery equivalent circuit model. Different from theabove mentioned approaches, this method makes use of aregression equation which is linear in variables that can bemeasured or calculated from the test, and takes into account all
testing data during the relaxation period of constant current
pulse tests towards obtaining the model parameters. Application
to real testing data indicates that the method gives good results
and the modeling error is less than 0.5%.
The paper is organized as follows. First, a detailed descriptionof the parameter identification method is given. Then the test
setup and procedure for obtaining the data needed for model
parameter identification is described. Next, application of thedesigned method on the obtained testing data is presented, andthe result is analyzed. The last section summarizes the paper.

PARAMETER IDENTIFICATION

Figure 1. Battery Equivalent Circuit Model.
To establish battery models of good quality, model parametersneed be accurately calculated from test data. To identify theparameters, usually specific tests, such as constant current
pulse discharge and charge tests at various SOC and current
rates are performed. The ohmic resistance R0 can be calculated
easily based on the immediate battery voltage change before
and after the current step [13, 14, 15]. However, the other

Parameter identification of the battery equivalent circuit model
includes determination of the battery OCV, the ohmic
resistance, and the parallel resistor-capacitor parameters at
various SOC. The tests performed are usually constant current
pulse discharge or charge tests. Since the model parameters arefunctions of battery SOC, for parameter identification at each
SOC, the discharge or charge pulse is set short so that the SOC,
and all the model parameters can be considered constant during
the testing.
While the method for determining the battery OCV and theohmic resistance R0 is the same as in several papers mentioned
before [13, 14, 15], it is included here for completeness. As to
identification of the remaining parameters (R1, C1 and R2, C2)
that involve the two time constants, the approach presented

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here is based on establishing and making use of a regression
equation which is linear in variables that can be measured or
calculated from the test results.

(8)

Referring to Figure 1 where i is the current (positive fordischarge and negative for charge), and Vt the battery voltage,
it is evident:

(1)

(2)

(3)
Figure 2. Constant Current Pulse Discharge Test.

(4)
The two time constants of the system are:

(5)

(6)
Take a discharge test for example. As shown in Figure 2,
assume the battery has been at rest for sufficiently long
before a step change of magnitude I occurs for the current i
at t = T0 = 0. The corresponding battery voltage has an
immediate drop at time T0 due to the ohmic resistance. Then it
enters the smoothly dropping portion which is governed by
the time constants of the two resistor-capacitor combinations.
When the current pulse ends at t = T1, the battery voltage has
an immediate jump, again due to the ohmic resistance, andthen enters the relaxation stage where the voltage levels off
exponentially to the OCV. Assume the test data is recorded
until t = T2.

B. Identification of τ1 and τ2
To determine the remaining parameters, the time constants ofthe two resistor-capacitor combinations are calculated first.
Since the pulse duration is quite short, the test data during thepulse might not contain sufficient information for correctly
extracting these two time constants. On the contrary, therelaxation period may last minutes or even hours, thus thebattery voltage behavior during the relaxation gives the bestindication of the dominant time constants of the battery. Basedon this observation, the test data from time
to time T2 isused for the identification of τ1 and τ2. During this period thecircuit is in zero input response. Denote the initial states as:

(9)

(10)
Then from (3) and (4) we have:(11)

A. Identification of Voc and R0
For determination of Voc, since the battery has been at rest forsufficiently long and the test starts from zero state, we have:

(12)
(7)
As to R0, since the immediate battery voltage change when thecurrent step occurs is due to the ohmic resistance, it can becalculated as:

Since i = 0, (1) and (2) gives:(13)

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DefineNote in this equation U, X, and Y are all available frommeasurement and calculation. Expressing it in matrix format
gives:(14)

This gives:

(15)

(22)

(16)

This is used for identification of the time constants τ1 and τ2, as
well as the initial states V10 and V20. Take all the sampling
points tk, k = 1,2,…, n during the period fromto T2 andobtain:While from (13) it is also obtained:

Since both terms on the right side of equation (16) are available,
U (t − T1) becomes a known variable.
To establish a regression equation that can be used forparameter identification, the following two variables aredefined and calculated from the test data:
Then from (22) we have
(17)

(18)
By using (15) and (17), it is obtained:(23)
Denote(19)

Subsequently by using (18) and (19), we have:

(24)
(20)
By using any two of the three equations (15), (19) and (20), we
can express
andin other terms. When these
expressions are substituted into the third equation, a regression
equation is obtained as follows, which has no the exponential
terms in it:

(21)

(25)

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(33)

(26)
(34)

The equation (23) is(27)
and the least squared error solution is(28)
After the values of P are obtained, τ1, τ2 and V10, V20 can becalculated from (26).

C. Identification of R1, C1 and R2, C2

After the time constants τ1, τ2 and the initial states V10, V20 are
identified, R1, C1 and R2, C2 can be calculated as follows.
Consider the battery voltage behavior during the pulse. From(3) and (4) it is obtained that for t ∈ [T0, T1]

Since V10 and V20 are already obtained, R1 and R2 can becalculated from (33) and (34), and subsequently C1 and C2 canbe calculated from (5) and (6).
Note τ1 and τ2 are obtained based on all the testing data during
the relaxation period, while the subsequent identification of
R1, R2 is mainly based on matching the initial states for thezero input response, thus the performance of the identified
parameters in modeling the behavior during the pulse might
not be as good as in modeling the behavior of the relaxation
period.
Also deserving of note is that the model parameters forcharging and discharging will differ at each SOC. However,the designed parameter identification method works for bothcases.

APPLICATION EXAMPLE
(29)

(30)

Thus at time T1

(31)

(32)
which are exactly the initial states for the relaxation period.
That is:To study the performance of the designed parameter
identification method, it is applied to the testing results of a 1.2
V, 1.22 Ah NiMH battery cell. The test setup is as shown in
Figure 3. The battery cell under test is connected to a
programmable cycler. A data acquisition and control systemcommunicates with the cycler for command setting as well asbattery current and voltage sampling. The system also samples
the temperature of the battery. Sampling rate is set to 100 Hz
for the tests.
Tests are performed for both discharge and charge, and various
SOC. Before each test, the battery has been at rest for at least
1 hour. At the start of each test the battery voltage is recorded,
which is considered as the OCV at this SOC. Then a constantdischarge or charge current of about 1C is applied. The current
pulse lasts about 20 seconds and then the battery returns to idle
period again. The tests are performed in the SOC range
between 90% and 10%, in 10% intervals.
Take as an example the results of a test at 50% SOC using adischarge current of 1.15 A, and pulse duration of 21.4 seconds.
The OCV is 1.2771 V. When the discharge pulse starts, aGratis copy for Shugang Jiang
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battery voltage drop of 0.0410 V is observed. This gives thevalue of R0 of 0.0356 ohm. The test data is sampled for over2500 seconds, including the pre-pulse, during pulse, andrelaxation stages. From all the data during the relaxation
period, it is identified that the two time constants are τ1 =1109.7 sec, τ2 = 45.1 sec, and the initial states are obtained as
V10 = 0.0066 V, V20 = 0.0075 V. R1, R2 and C1, C2 aresubsequently identified using the designed method and theresults are R1 = 0.2988 ohm, R2 = 0.0173 ohm, C1= 3713.6 F,
C2= 2607.5 F.

Comparison of the model output and the measured results isshown in Figure 4 for the whole test period. Figure 5 shows theresults for 200 seconds to give more details of the performance.
It is evident the model matches the testing data very well andthe maximum error between the measured and calculated
voltage is only 0.0023 V, or 0.18% of the OCV at this SOC.
The model parameters are functions of SOC. Parameter
identification results for various SOC levels are summarized
as in Table 1. In addition to all the parameter values, themaximum modeling error, both in volt and in percentage of thecorresponding OCV, is included in the table as well. It can beseen that for the SOC range from 20% to 90%, all the modelingerrors are less than 0.5%. While for the 10% SOC, the error is
as big as 2.8%. This may indicate that at very low level of
SOC, the structure of the equivalent circuit model might not beable to reflect the battery behavior well. Relationship of themodel parameters with respect to SOC is shown in Figure 6.
Note that because some parameters deviate greatly inmagnitude, for display purpose a factor of 10 is used for thosewith smaller values.

Figure 3. Battery Test Setup.

Figure 4. Measured and Calculated Battery Voltage.

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Figure 5. Details of Measured and Calculated Battery Voltage (for 200 Seconds).

Table 1. Identified Battery Model Parameters.

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A novel method which makes use of a regression equation thatis linear in variables that can be measured or calculated frombattery test is designed for the parameter identification of awidely used battery equivalent circuit model. With thisapproach, all testing data during the relaxation period of aconstant current pulse discharge or charge test is used towards
obtaining the model parameters and the calculation is in thesense of least squared error. Application of the method to real
battery testing data shows that the method gives very goodmodeling performance and the error is less than 0.5%.

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Figure 6. Model Parameters as Functions of SOC.

SUMMARY/CONCLUSIONS

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It is important to accurately identify the parameters of a battery
model in order to fulfill good battery management functions.

11. Gao, Lijun, Liu, Shengyi and Dougal, Roger A., Dynamic
Lithium-Ion Battery Model for System Simulation, IEEE

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Transactions on Components and Packaging Technologies,
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CONTACT INFORMATION
Dr. Shugang Jiang
A&D Technology Inc.
4622 Runway Blvd.
Ann Arbor, MI 48108 USA
sjiang@aanddtech.com
SAE

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