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A Model Parameter Identification Method forBattery Applications

2013-01-1529
Published
04/08/2013

Xiao Hu, Lewis Collins and Scott Stanton
ANSYS Inc.

Shugang Jiang
A&D Technology Inc
Copyright © 2013 ANSYS Inc.
doi:10.4271/2013-01-1529

ABSTRACTDue to growing interest in hybrid and electric vehicles, thebattery, being one of the critical components, is receiving a
lot of attention from designers and researchers. Two batterymodeling approaches, though seemingly different, share thesame mathematical challenge of robust non-linear curvefitting. The two methods are battery equivalent circuit modeland battery system level thermal modeling using the linear
time-invariant (LTI) method. Both modeling approaches
involve curve-fitting testing data or data from advanced
models to identify four parameters in a circuit model
consisting of two pairs of RC elements. Such curve-fitting ismathematically a non-linear least-squares (LS) problem.
Standard methods like the Levenberg-Marquardt (LM)method can be used for non-linear curve-fitting, but the LM
method is known to be sensitive to initial conditions. Due tothe unique features of the two pairs of RC values in themodel, the curve-fitting problem can be reformulated into alinear LS problem. Solution from the linear LS problem canthen be used as an initial condition for the LM method for
greater accuracy. Since the initial conditions from the linear
LS problem are already close to the minimum, the sensitivity
issue associated with the LM method is mitigated.

I. INTRODUCTION
Due to environmental concerns and the depletion of fossil
fuels, the automotive industry has invested heavily in electricvehicles (EV) and hybrid electric vehicles (HEV). For bothtypes of vehicles, the battery plays a critical role in affecting
the overall vehicle performance. Numerical simulation
becomes an indispensable tool for battery designers andresearchers. Since the battery is a multi-physics application,

its simulation or modeling involves many different
disciplines, including but not limited to electrochemical
modeling, thermal modeling, electrical circuit modeling,
structural modeling, etc. Two of the modeling efforts, though
seemly different, share the same mathematical problem of
non-linear curve-fitting. One is the battery equivalent circuit
modeling and the other is battery system-level thermal
modeling.
The battery equivalent circuit model has gained popularity
among system-level design engineers due to its ease of use
and its capability of representing state-of-charge, I-V
characteristics, and dynamic behavior of a battery system[1,2]. A commonly used equivalent circuit model consists ofan open-circuit voltage source, a resistor in series, and twopairs of parallel resistor-capacitor (RC) elements as shown in
Fig. 1. The series resistance can be determined quite easily.
The main challenge is to identify the parameters of the twoparallel RC's through curve-fitting test data. The simplicity ofthe battery equivalent circuit model and yet its satisfactory
results rely on good curve-fitting to test data to identify
values of the two RC's.
System-level thermal modeling using the linear timeinvariant (LTI) approach, originated from electronics coolingapplications [3,4], has recently been adopted successfully tobattery cooling [5, 6, 7]. This method treats the thermal
problem as an LTI system. In building a simple LTI system,
two pairs of RC network similar to those from the battery
equivalent circuit model are used [5, 6, 7]. An example ofsuch an LTI model with two pairs of RC network is shown in
Fig. 2. One needs to identify the values of the RC's throughcurve-fitting the impulse or step response of the RC network
to that of the original thermal system.

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II. MATHEMATICAL STATEMENT
OF THE PROBLEM
For models using Fig. 1 and Fig. 2, the RC values are
determined by curve-fitting to test data or results from othersimulation models. In Ref. [10], Jiang shows that theanalytical solution of the RC circuit starting right after thepulse discharge is the following:(1)

Fig. 1. A battery equivalent circuit model.

Therefore, one needs to determine the values for parameters
V10, τ1, V20, τ2 from a given set of test or simulation results.
Consider a case having m sets of data (t1, U1),…,(tm, Um)with m>4. This process then becomes a non-linear leastsquares (LS) problem. Formally, we are seeking a local
minimizer for

(2)
where

and
. ƒi are functions defined asfollows:

Fig. 2. An LTI model for battery thermal system.
For the two battery applications mentioned above, eventhough the problems addressed and physics involved are quite
different, the essential mathematics involved in both cases is
non-linear curve-fitting. For general non-linear curve-fitting
problems, standard methods like the Levenberg-Marquardt(LM) [8,9] can be used. However, the LM method is known
to be sensitive to initial conditions. Based on the uniquefeatures of the two pairs of RC employed in the model, Jiang[10] introduced a novel method of performing the curvefitting by turning the non-linear curve-fitting problem into alinear one followed by solving a quadratic equation. While
Jiang's method works well for a lot of cases, a few issues areobserved in some applications. The current paper proposes animproved approach to resolve the observed issues.
The paper is organized as follows. Section II formally defines
the mathematical problem. Section III introduces Jiang's
method. Section IV demonstrates the issues associated with
Jiang's method and the modifications used to improve themethod. A complete solution using modified Jiang's method
as an initial condition for the LM method is then proposed.
This hybrid approach shows improvement over Jiang's
approach and LM method used individually. Finally, Section
V is the conclusion.

(3)
where
is defined in Equ (1) with being [V10, τ1,
V20, τ2]. For a general non-linear curve-fitting problem,
namely a general function instead of the specific form of Equ(3), the standard LM method [8-9] can be used. However, theperformance of the LM method is sensitive to initialconditions, which are necessary since the LM method
involves iteration. (The LM method is described briefly inAppendix A for reference.)
Another general method is to perform the curve-fitting in thefrequency domain using vector-fitting (VF) [11,12]. Thisapproach involves identifying the transfer function of the
proposed model with the sampled Fourier transform of theimpulse response of the modeled system. While this is a very
powerful and general technique, it is more involved to
implement.
Because of the specific form of function U shown in Equ (1)and salient features of test results, certain techniques havebeen developed for this non-linear curve-fitting problem forbattery equivalent-circuit models. A commonly used method
makes use of the property that each of the two time constants
plays a dominant role at different stages of the battery voltage
response, and calculates the model parameters accordingly[13]. For such a method, the process is quite manual and theresults are greatly influenced by the partition of faster andTHIS DOCUMENT IS PROTECTED BY U.S. AND INTERNATIONAL COPYRIGHT.
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slower dynamics of the battery and the selection of the data
points used for the calculation. This method does not apply tobattery thermal LTI modeling since the required data property
is not possessed by battery thermal systems.
In this paper, a simple and novel method of parameter
identification is presented. It makes use of a regression
equation which is linear in variables that can be measured or
calculated from the test or other advanced models. Such amethod was originally proposed by Jiang [10]. While Jiang's
method works well for a lot of cases, there are issues
observed in some cases. Modifications are made to Jiang's
method in this paper to overcome the observed issues. Afterthe solution from the modified Jiang's method is obtained, it
is used as the initial condition for the LM method. The LM
method is used because results from the modified Jiang's
method do not satisfy the LS solution of the original problem,
namely Equ (2). With the combination of the modified Jiang's
method and the LM method, the final solution satisfies the LS
problem and the approach becomes less sensitive to the initialcondition since the initial condition is already close to the
optimum solution.

III. JIANG'S METHOD FOR
PARAMETER IDENTIFICATIONThe starting point of Jiang's method is two functions, X andY, defined by the following two equations:

(4)

(5)
Substitution of U from Equ (1) into Equ (4) and (5) gives thefollowing expressions for X and Y, respectively,(6)(7)
Solving for e−t/τ1 and e−t/τ2 using Equ (1) and (6) andsubstituting the results into Equ (7) gives the final expression
for Y,(8)

When the m sets of experimental data points of (ti, Ui) areused to evaluate t, U, X, and Y in Equ (8), one obtains mequations with four unknowns. These m equations can be
written in a matrix form shown below,(9)
The solution to the above equation is obtained by solving alinear LS problem. After Equ (9) is solved, V10, τ1, V20, τ2
can be obtained by solving a quadratic equation for τ1 and τ2
and a linear set of two equations for V10 and V20. It isinteresting to note that a non-linear LS problem has beenchanged to a linear LS problem using Jiang's approach.

IV. MODIFICATION OF JIANG'SMETHOD
There are a few problems observed using Jiang's method insome cases. First of all, though not often, the matrix in Equ(9) can be rank deficient for certain cases. This issue can besolved quite easily by using singular value decomposition(SVD) based LS solver rather than QR-based LS solver. So,
all linear LS problems solved in this paper use the SVDbased LS solver. When the matrix is not rank deficient, theSVD-based LS solver gives the same results as the QR-based
solver. When the matrix is rank deficient, the SVD-based
solver is stable and gives a least-squares and least-norm
solution.
Secondly, the method only ensures a good fit for the function
Y, not the original function U that is ultimately needed. In
some cases, a good fit for the function Y does not translate toa good fit for the function U. Large errors of more than 100%(in the sense of 2-norm) have been observed for the function
U even when the error for Y is quite small. Fig. 3 and 4 showsuch an example. It is clear from Fig. 3 that the fit for Y isquite good. However, the fit for U shown in Fig. 4 is rather
poor near t = 0 as indicated in Fig. 4b). This result can be
explained if one notices that the area difference between thetwo curves in Fig. 4 is quite small even though the values of
the two curves near t = 0 are quite different. But Jiang's
method essentially compares the area rather than the values
for the function U. This is clear from how X and Y are definedin Equ (4) and (5).

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By using Equ (10) rather than Equ (9), we are forcing thesum of V10 and V20 to be the correct value of U0. This helps
avoid the issue observed near t = 0 associated with Jiang's
original approach. The corresponding matrix form for Equ(10) is the following:(11)
With the above modification, the fitting results for thefunctions Y and U shown in Fig. 3 and 4 are repeated in Fig. 5
and 6, respectively. Visually, there is no difference for thefitting of the function Y using the original method and themodified one, but the improvement for the more importantfunction U is quite obvious near t = 0 as can be seen bycomparing Fig. 6 with Fig. 4.

Fig. 3. A result for function Y using Jiang's method

Another observation with Jiang's method is accuracy. Resultsfrom the method do not satisfy Equ (2), so it is not a LS
solution to the problem. In order to obtain the more accurate
LS solution, the LM method is used with the initialconditions from the above modified Jiang's method.
It is worthwhile to mention that Jiang's approach only works
well with two RC pairs. When more than two pairs are used,the counterparts of Equ (10) and (11) become toocomplicated to be useful.

a). Zoomed-out viewb). Zoomed-in view near t = 0
Fig. 4. A result for function U using Jiang's method.
This issue can be resolved quite nicely with one observation.
From Equ (1), it is clear that V10+V20 = U0, where U0
represents the function U at t = 0. After this relation issubstituted into Equ (8), one obtains the following new
expression for Y,(10)

Fig. 5. A result for function Y using modified Jiang's
method.

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a good fit as shown in Fig. 9. Such a method is robust anddoes not suffer the sensitivity issue associated with timedomain fitting using the LM method. But the vector-fitting
method is more involved. Frequency-domain fitting is beyond
the scope of this paper and will not be discussed further here.

a). Zoomed-out viewFig. 7. A function U due to cross heating that results incomplex V10, τ1, V20, τ2 using Jiang's method or the
improved version.

b). Zoomed-in view near t = 0
Fig. 6. A result for function U using modified Jiang's
method.
As a final observation, both the original and modified Jiang's
method could generate complex (as opposed to real) solutions
for V10, τ1, V20, τ2 after the linear LS problem is solved fromEqu (9) or (11). It was observed that this happens more often
when the step response data does not reach steady-state. But
it can occur even when the step response data approachsteady-state, and this is especially true when the curve to be
fitted looks like the one shown in Fig. 7. While such a curve
may not show up in the battery circuit model, it shows up in
the LTI thermal model when cross-heating is involved. The
author has not found a method to resolve this issue within theframework of Jiang's method. For such a curve, one can use
the LM method directly and cope with the sensitivity issue.
Results from LM method using two rather randomly selected
initial conditions are shown in Fig. 8 to demonstrate thesensitivity issue. The first initial condition gives an error of
2.5% while the second gives an error of 7.6% (based on the
2-norm in both cases). Note that even the worse result ofusing initial condition 2 might be acceptable since crossheating makes a small contribution to the final solution andthe curve-fitting accuracy is thus less important for thesecurves from cross-heating. If higher accuracy is desired,though, the more powerful vector-fitting method [11] in thefrequency domain can be used. For the curve shown in theFig. 7, fitting results from the vector-fitting method generate

Fig. 8. Fitting results using LM method for two different
initial conditions.

Fig. 9. Fitting results using vector-fitting method.

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The results shown so far are all for battery thermal LTI
modeling. As mentioned before, this approach works just as
well for battery equivalent circuit modeling. Fig. 10 shows
curve-fitting results for the function U calculated from abattery pulse discharge using the Newman electrochemistrymodel [14, 15, 16]. Fig. 11 shows curve-fitting results for thefunction U provided by A&D Technology from test results.
Note that the values for U to be fitted can be generated either
from more advanced models or from test data.

more accurate LM method. The combined approach makes
the method less sensitive to initial conditions compared withusing the LM method alone and more accurate than without
using the LM method.
The approach however is not without issues. The main
problem observed is that the parameters could be complex forbattery cell cross-heating curves. Fortunately, accurate fitting
for these curves is not as critical since cross-heating usually
does not contribute significantly to the final temperature
solution. If high accuracy is desired, a good fit can beachieved by using frequency domain fitting methods, thoughsuch methods are more involved to implement.

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Fig. 10. A result for function U using the modified
Jiang's method. Original U values are obtained from theNewman electrochemistry model.

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Fig. 11. A result for function U using the modified
Jiang's method. Original U values are provided by A&D
Technology.

V. CONCLUSION
Two battery applications share the same mathematicalchallenges of non-linear curve-fitting for parameter
identification. Non-linear curve-fitting can be solved using
the standard LM method, but the LM method is sensitive toinitial conditions. A novel method which makes use of aregression equation that is linear in the variables that can bemeasured or calculated from testing or from other advanced
models is proposed for the parameter identification. Resultsfrom this method are then used as the initial condition for the

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Estimation on Nonlinear Parameters,” SIAM J. Appl. Math.
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State Space Thermal Model for HEV/EV Battery Using

THIS DOCUMENT IS PROTECTED BY U.S. AND INTERNATIONAL COPYRIGHT.
It may not be reproduced, stored in a retrieval system, distributed or transmitted, in whole or in part, in any form or by any means.
Downloaded from SAE International by Xiao Hu, Monday, March 18, 2013 07:58:54 PM

Vector Fitting,” IEEE Transportation Electrification
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CONTACT INFORMATION
Dr. Xiao Hu
ANSYS Inc.
xiao.hu@ansys.com

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APPENDIX
Appendix A: Levenberg-Marquardt Method for Battery Curve-Fittingwith m > n. We are seeking a local minimizer for

Given a vector function(A1)
For the battery application, the vector function is, (m > 4), whose components are defined as follows:(A2), and (ti, yi) are testing data or data from other models at m different points,

where

A linear approximation forin the neighborhood offor a small

is the following:

(A3)
where

is the Jacobian defined by:(A4)
Inserting Equ (A3) into Equ (A1) gives the following:(A5)
For a given , Equ (A5) is a function of . This function is defined as, shown below,(A6)
The gradient of L is the following:(A7)
Setting
following:equals to zero provides a mininizer. This is called the Gauss-Newton method, and the resulting equation is the(A8)
The above equation is equivalent to the following LS problem. Numerically, it is better to solve the following LS problem rather than
(A8).

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In the LM method, it is suggested to use a damped Gauss-Newton method. The step is defined by the following modification to Equ
(A8)

(A10)
When µ is very small, then, and the LM method goes back to the Gauss-Newton method. When µ is large,(A11)
i.e. a short step in the steepest descent direction. So, the LM method can be thought of as a combination of the steepest descent and theGauss-Newton method. When the current solution is far from the minimum, the algorithm behaves like the steepest descent method,which is slow in convergence but guaranteed to converge. When the current solution is close to the minimum, it becomes the GaussNewton method.
The value for μ has to be modified based on the solution from the current iteration. The choice of initial μ value is not critical as thesolution is not sensitive to the initial value. During the iteration the size of μ can be updated by the gain ratio defined below,(A12), is a good approximation to, and the damping may be reduced. A small value of ρ,A large value of ρ indicates that
including negative values, indicates that we should increase the damping factor and thereby increase the penalty on large steps. It wasdemonstrated in Nielsen [17] that the following strategy in general works well,

A detailed description of the LM method can be found in [9-10, 17]. The algorithm can be summarized in the following pseudo-code.

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