1

Battery Identification Based on Real-World DataMiao Zhang, Student Member, IEEE, Zhixin Miao, Senior Member, IEEE, Lingling Fan, Senior Member, IEEE

Abstract—In this paper, system identification is carried out fora 20 kWh battery using real-world measurements data. State-of-charge (SOC) and open-circuit voltage (VOC ) relationshipwill be obtained using least square estimation (LSE) non-linearregression. In addition, how to estimate SOC using currentmeasurements and how to estimate the equivalent circuit’s RCparameters are carried out using autoregressive exogenous (ARX)models. The respective ARX models are first derived. Estimationof the ARX coefficients is then carried out. Finally, parameterrecovery is conducted to find out parameters with physicalmeanings, e.g., RC values. With the identified VOC and SOCrelationship and RC parameters, we built a simulation modelin MATLAB/Simpowersystems. With the measured current datafrom the real-world as the input, the simulation model gives theterminal DC voltage as the output. This output is compared withthe real-world DC voltage measurements data and the matchingdegree is satisfactory.

Index Terms—state-of-charge (SOC), least-square-estimation(LSE), autoregressive exogenous (ARX) model, battery equivalentmodel, system identification, data analysis.

I. INTRODUCTION

IN recent years, batteries are used more and more. The targetbattery pack in this paper is located in St. Petersburg of

Florida with the characteristics listed in TABLE I. It serves asan energy storage device. It is charged by Photovoltaic (PV)in the morning and discharged in the afternoon to mitigateelectric power consuming. Fig. 1 shows the raw data obtainedfrom meters, including measured terminal DC voltage (VL),measured terminal DC current (IL) and state-of-charge (SOC)in time-series. The time span is 22 days and sample timeis 1 minute. The data extraction for analysis relies on dataprogramming in Python.

TABLE IBATTERY MAIN CHARACTERISTICS

Rated Capacity 20 kWhRated Power 5 kWCell Rated Capacity 400 Ah

This paper has three objectives. The first one is to obtainopen-circuit voltage (VOC) to SOC relationship from voltagemeasurements and SOC data. The second one is to estimateSOC from current measurements. The third one is to estimatethe equivalent circuit’s RC parameters from the measuredcurrent and SOC.

There are at least two major systematic methods for batterysystem estimation: Kalman filter-based estimation and least-square estimation (LSE). For battery system identification,Kalman filter-based estimation, including Extended Kalmanfilter (EKF) [1], [2], Unscented Kalman filter (UKF) [3]–[5],

M. Zhang, Z. Miao and L. Fan are with Dept. of Electrical Engineering,University of South Florida, Tampa FL 33620. Email: zmiao@usf.edu

have been widely used in state-of-charge (SOC) estimationand parameters identification. Kalman filter based method isa way to estimate the time-varying dynamic system withGaussian noise. Kalman filter can be implemented online.On the other hand, LSE is chosen as a fast and efficientpolynomial estimation method to identify battery systems in[6]–[8]. By approximating derivatives discrete data, discrete-time ARX models can be found [9], [10]. With an ARX model,a linear LSE problem can be formulated and the parameters ofthe ARX model can be estimated. In [9], [10], autoregressiveexogenous (ARX) model are applied to generator system pa-rameters identification. There are very few papers to estimatebattery systems parameters using ARX model. Our paper fillsthe gap.

This paper is organized as follows. In Section II, VOCand SOC relationship will be obtained using LSE non-linearregression. How to estimate SOC using current measurementsand how to estimate the equivalent circuit’s RC parametersare carried out using ARX model in Section III and IV,respectively. In Section V, with the identified VOC to SOCrelationship and RC values, we build a simulation model inMATLAB/Simpowersystems for validation. The conclusion isgiven in Section VI.

II. ESTIMATION OF OPEN CIRCUIT VOLTAGE AND SOCRELATIONSHIP

The relationship curve of open-circuit-voltage and SOC isusually used as a criterion to describe the battery health status.In [3], [8], [11]–[13], the experiment currents are kept constantso to get VOC under different SOCs easily. However, thebattery terminal current (IL) and voltage (VL) in our data varywith weather condition and power demand in real-time. Thebattery terminal voltage (VL) equals the open-circuit-voltage(VOC) when no current flows, as shown in Fig. 1 and pointedby the red arrows.

Normally, the open-circuit-voltage (VOC) of a battery isinfluenced by SOC, working temperature T and the numberof cycles C, as shown in (1).

VOC = f(SOC, T, C) (1)

In our case, the VOC and SOC are acquired at each earlymorning when the battery environment temperature varies littleaccording to the historic record. Also, considering 15 cycles in22 days, the influence of cycle C can be neglected. Then (1)is simplified as (2). The triangle markers are used to representthe extracted 13 pairs of VOC and SOC data in Fig. 2.

VOC = f(SOC) (2)

We can assume a target relationship function as a combina-tion of exponential and polynomials. It is expressed in (3).

2

Time (days)

May 03 May 05 May 07 May 09 May 11 May 13 May 15 May 17 May 19 May 21 May 23 May 25

VL (

V)

48

50

52

54

(c)

May 03 May 05 May 07 May 09 May 11 May 13 May 15 May 17 May 19 May 21 May 23 May 25

IL (

A)

-40

-20

0

20

40(b)

May 03 May 05 May 07 May 09 May 11 May 13 May 15 May 17 May 19 May 21 May 23 May 25

SO

C (

%)

0

50

100(a)

SOC calibration

Fig. 1. (a) State-of-charge (SOC). (b) Battery terminal current measurements. (c) Battery terminal voltage measurements.

VOC =a · eb·SOC + p1 · SOC7 + p2 · SOC6 + p3 · SOC5+

p4 · SOC4 + p5 · SOC3 + p6 · SOC2 + p7 · SOC+

p8

(3)

We aim to find the coefficients: a, b, p1, ...p8 that can bestfit the curve to the given data points. The objective functionis

mina,b,p1,··· ,p8

n∑i=1

(VOC(a, b, p1, · · · , p8)− VOCm)2, (4)

where VOC is the estimation from SOC, VOCm is the voltagemeasurements.

By applying curve fitting toolbox in MATLAB, we get thecoefficients of (3) and a single-variable function is used torepresent the curve, as shown in (5).

VOC =− 4 · e−0.3·SOC + 9.431× 10−12 · SOC7

− 2.981× 10−9 · SOC6 + 3.541× 10−7 · SOC5

− 1.899× 10−5 · SOC4 + 3.965× 10−4 · SOC3

+ 9.775× 10−4 · SOC2 − 0.08582 · SOC + 52.32(5)

This function is plotted in Fig. 2 and it is shown that thecurve fits the data points very well.

III. SOC ESTIMATION

In this Section, the general discrete-time ARX model struc-ture will be first explained. SOC estimation and the currentmeasurements (IL) relationship will be converted to discretetime based on Coulomb counting method.

SOC (%)

0 20 40 60 80 100

Vo

c (

V)

48

49

50

51

52

53

54

55

Measurement

Estimation

Fig. 2. SOC vs. VOC measured points and estimated nonlinear relationship.

A. ARX Model Structure

A general polynomial ARX model structure can be ex-pressed as equation (6):

A(z)y(k) = B(z)u(k − nk) + e(k), (6)

where• u(k) is the system inputs.• y(k) is the system outputs.• nk is the system delay.• e(k) is the White-noise system disturbance.• A(z) and B(z) are polynomial with ka and kb orders

respect to the backward shift operator z−1 and definedby the following equations:

A(z) = 1 + a1z−1 + · · ·+ akaz

−ka (7)

B(z) = b0 + b1z−1 + · · ·+ bkb−1z

−(kb−1) (8)

3

Fig. 3 depicts the signal flow of an ARX model. Giventime series no-delay (nk = 0) measurements of the input andoutput, say from k step to N step, an overestimated problemcan be formulated as (9) based on (6): 1

A(z)B (z)u + y

e

Fig. 3. ARX model signal flowchart.

y(k)

y(k + 1)...

y(N)

=

y(k − 1) y(k − 2) ... y(k − ka) u(k) u(k − 1) ... u(k − (kb − 1))y(k − 2) y(k − 3) ... y(k − ka − 1) u(k − 1) u(k − 2) ... u(k − kb)

......

. . ....

......

. . ....

y(N − 1) y(N − 2) ... y(N − ka) u(N) u(N − 1) ... u(N − (kb − 1))

−a1

...−akab0b1...

bkb−1

+ e(k)

(9)B. ARX Model-based SOC Estimation

Theoretically, the state of charge is a relative quantity thatdescribes the ratio of the remaining capacity to the nominalcapacity of a battery. The Coulomb counting is a methoddeveloped from this concept, which estimates SOC by themeasurements of total current accumulation. It can be givenby

SOC = SOC0 +

∫ηILdt

CN, (10)

where SOC0 is the initial value of the SOC, η is theCoulombic efficiency, and CN is the nominal capacity.

In our case, the battery had been fully discharged andexperienced a long time of self-discharge. Shown in Fig. 1,IL was kept 0 while VL kept decreasing before May 5th. Thisis due to the effect of battery self-discharging. The SOC0

was calibrated to zero on May 5th as shown in Fig. 1 (a) andpointed by the black arrow.

Convert (10) to discrete-time as:

SOC(k) = SOC(k − 1) +η∆t

CNIL(k − 1) (11)

Assume η∆tCN

as a constant number b1. From (11), thetransfer function can be expressed by:

y

u=

b1z−1

1− z−1(12)

Applying this to discrete-time ARX model, we can get thetransfer function with order of [1 1 1]. This yields to

A(z)SOC(k) = B(z)IL(k) + e(k), (13)

where

A(z) = 1− z−1

B(z) = 0.004615z−1

With above discrete ARX transfer function, SOC estimationcan be expressed as:

SOC(k) = SOC(k− 1) + 7.69× 10−5∆t · IL(k− 1) + e(k)(14)

where

SOC(0) = 0

∆t = 60seconds

Given IL as input, we can compare ARX model based SOCsimulation output with SOC raw data in Fig. 4. The SimulatedARX model output matches the raw data very well with afitting degree of 98.59% and a mean-square error 0.06522.

Time (sec) ×105

0 2 4 6 8 10 12 14 16

SO

C (

%)

0

20

40

60

80

100

Simulation

Raw

Fig. 4. Comparison of SOC ARX model simulation output and SOC rawdata.

IV. EQUIVALENT CIRCUIT PARAMETERS IDENTIFICATION

In this section, the battery equivalent circuit is first pro-posed. The dynamic equations of two RC branches are con-verted to discrete-time in z-domain. Finally, estimation ofARX Model coefficients are carried out and RC parametersrecovery is conducted with physical meaning.

A. Equivalent Circuit Modeling

Among battery equivalent circuits, the Thevenin-basedmodel is widely used in [3], [8], [11], [12], [14]–[17] sinceit can not only bridge SOC to open-circuit voltage, but alsosimulate the transient response of load changing. It consistsof two parts. One part is open-circuit voltage(VOC), whichin function of state-of-charge (SOC) as shown in Eq. (5).VOC is presented by a voltage-controlled voltage source inFig. 5. Another part is the RC network, including one ohmic

4

+− VOC(SOC)

R0

R1

− +V1

R2

− +V2

IL+

C1 C2

+

−

VL

Fig. 5. Schematic of the battery equivalent circuit.

resistance R0 and two paralleled RC branches (R1, C1 andR2, C2), are responsible for short-time and long-time constantsof the step response. The proposed equivalent model is a tradeoff between accuracy and complexity.

B. Discretization of Dynamic Equations

As assumed in Fig. 5, IL is the terminal current with apositive value in the charging process and negative value inthe discharging process. Two RC parallel branches in proposedmodel can be expressed as following differential equations:

C1dV1(t)

dt=−V1(t)

R1+ IL (15a)

C2dV2(t)

dt=−V2(t)

R2+ IL (15b)

The state space model is:

[V1

V2

]=

[ −1R1C1

0

0 −1R2C2

]︸ ︷︷ ︸

A

×[V1

V2

]+

[ 1C11C2

]︸ ︷︷ ︸

B

×IL (16a)

VL − VOC =[1 1

]︸ ︷︷ ︸C

×[V1

V2

]+ R0︸︷︷︸

D

×IL (16b)

From (16a) and (16b) we have the expression as:

x = Ax+Bu (17a)

y = Cx+Du (17b)

where x =

[V1

V2

], u = IL is the input, and y = VL − VOC is

the output. VOC is assumed as known and can be found using(5). A discrete-time form of (17a) is arranged as (18), wherek = 1, 2, 3 . . .

x(k + 1) = x(k) + (Ax(k) +Bu(k)) · h, (18)

where h is time interval. Substitute x(k + 1) by z · x(k):

z · x(k) = x(k) + (AH + I) · x(k) +Bh · u(k) (19)

−→ x(k) = [zI − (Ah+ I)]−1Bh · u(k) (20)

For the output, substitute (20) into (17b):

y(k)

u(k)= C · [zI − (Ah+ I)]−1Bh+D (21)

The corresponding transfer function G(z) of (21) is:

G(z−1) =b0 + b1z

−1 + b2z−2

1 + a1z−1 + a2z−2, (22)

where a1, a2, b0, b1 and b2 are the coefficients relate to RCparameters (23a)∼(23e). In our measurements data, the timeinterval h is 60 seconds.

a1 =h

R1C1+

h

R2C2− 2 (23a)

a2 = − h

R1C1− h

R2C2+

h2

R1R2C1C2+ 1 (23b)

b0 = R0 (23c)

b1 =h

C1+

h

C2+

hR0

R1C1+

hR0

R2C2− 2R0 (23d)

b2 = R0 +h2

R1C1C2+

h2

R2C1C2+

h2R0

R1R2C1C2

− h

C1− h

C2− hR0

R1C1− hR0

R2C2

(23e)

C. ARX Model-based RC Estimation

In this case, the setting of ARX model is assumed asfollows:

A(z)y(k) = B(z)u(k) + e(k), (24)

where

A(z) = 1 + a1z−1 + a2z

−2

B(z) = b0 + b1z−1 + b2z

−2

• Inputs: u(k) = IL• Outputs: y(k) = VL − Voc• Sample time: Ts = 60 seconds• Total time: S = 19 days• Order of the polynomial A(z): ka = 2• Order of the polynomial B(z) + 1: kb = 3• Input-output delay: nk = 0

Using Matlab identification toolbox we can solve ARXModel with 68.82% simulation focus and 0.0099 mean-square-error (MSE). The ARX model simulated output and VL−VOCevaluated output are compared and shown in Fig. 6. And thetransfer function we can get is:

G(z−1) =0.009229− 0.01419z−10.004967z−2

1− 1.817z−1 + 0.8168z−2(25)

Applying the 3rd, 10th and 16th days data into (VL − Voc)ARX model to validate the reliability of coefficients we gotin (25). All these three groups coefficients are compared andlisted in TABLE II, shown that the variation of coefficientsis acceptable so (25) is reliable to identify equivalent circuitparameters.

Substitute the coefficients in (25) into the system of equa-tions (23a)-(23e). We can get the solution of circuit RC param-eters are R0 = 0.009188Ω, R1 = 0.0068Ω, R2 = 0.0140Ω,C1 = 7.926×106F , C2 = 2.38×104F . In TABLE III, the RC

5

×105

0 2 4 6 8 10 12 14

IL (

A)

-40

-20

0

20

40(a)

Time (seconds) ×105

0 2 4 6 8 10 12 14

VL

-Vo

c (

V)

-1

-0.5

0

0.5

1

1.5(b)

Estimation

ARX Simulation

Fig. 6. (a) ARX model input IL. (b) Comparison of VL − VOC estimation and ARX model simulation

×105

0 2 4 6 8 10 12 14

SO

C (

%)

0

50

100(a)

×105

0 2 4 6 8 10 12 14

IL (

A)

-40

-20

0

20

40(b)

Time (seconds) ×105

0 2 4 6 8 10 12 14

VL

(V

)

50

51

52

53

54

55(c)

Measurement

Simulation

Fig. 7. (a) State-of-charge (SOC). (b) IL input for current source. (c) Comparison of voltage measurements data and voltage simulation.

TABLE IIARX MODEL COEFFICIENTS FROM DIFFERENT TIME PERIODS

a, b 19 days 3rd day 10th day 16th day Max Deviationa1 -1.817 -1.879 -1.876 -1.887 3.8%a2 0.8186 0.8789 0.8759 0.8872 8.3%b0 0.009229 0.009308 0.008698 0.00833 9.7%b1 -0.01419 -0.01584 -0.01464 -0.01401 11.6%b2 0.0049676 0.006543 0.005945 0.005675 31.7%

parameters which estimated from different periods are listed.Theoretically, the RC parameters are multi-variable functionsof current, SOC, temperature and cycle number. It will lead the

TABLE IIIRC PARAMETERS ESTIMATED FROM DIFFERENT TIME PERIODS

R,C 19 days 3rd day 10th day 16th dayR1(Ω) 0.0529 0.1320 0.0518 0.0495R2(Ω) 0.0137 0.0127 0.0131 0.0162R0(Ω) 0.009229 0.009308 0.008698 0.00833C1(F ) 1.04 × 106 5.54 × 105 1.4454 × 106 6.74 × 105

C2(F ) 2.38 × 104 3.89 × 104 3.67 × 104 3.33 × 104

RC parameters variation without considering above all factors.

6

V. VALIDATION TESTBED

A simulation model, which shown in Fig. 8, was builtto validate the proposed battery model performance. In thismodel, we set SOC and current (IL) as two inputs. VOC is infunction of SOC in (5) as a voltage-controlled voltage source.IL is the original measurements current acting as a currentsource on the terminal side. R,C values estimated from 19 daysdata is applied to R0, R1, R2, C1 and C2. The terminal voltagein the Simulink model is obtained and compare it with theoriginal battery measured voltage. All the inputs and outputsare presented in Fig. 7. By comparing the results, the meansquared error (MSE) is 0.01V . The simulation measured DCvoltage fit to raw measured DC voltage very well.

Fig. 8. Simulation testbed for validation.

VI. CONCLUSION

In this brief, system identification progress has been carriedout for a 20 kW.h battery pack using real-world measurementsdata. VOC to SOC relationship has been obtained by usingleast square estimation (LSE) non-linear regression. In addi-tion, how to estimate SOC using current measurements andhow to estimate the equivalent circuit’s RC parameters werecarried out using ARX model. Finally, with the identified VOCto SOC relationship and RC parameters, we built a simulationmodel in MATLAB/Simpowersystems. With the measuredcurrent data from the real-world as the input, the simulationmodel gives the terminal DC voltage as the output. This outputis compared with the real-world DC voltage measurementsdata and the matching degree is satisfactory.

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