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Relaxationmodeloftheopen-circuitvoltageforstate-of-chargeestimationinlithium-ionbatteries

ARTICLEinIETELECTRICALSYSTEMSINTRANSPORTATION·DECEMBER2013

DOI:10.1049/iet-est.2013.0020

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LeiPei

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Published in IET Electrical Systems in TransportationReceived on 8th April 2013Revised on 3rd July 2013Accepted on 26th July 2013doi: 10.1049/iet-est.2013.0020

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ISSN 2042-9738

Relaxation model of the open-circuit voltage forstate-of-charge estimation in lithium-ionbatteriesLei Pei, Rengui Lu, Chunbo Zhu

School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin 150001,

People’s Republic of China

E-mail: lei.pei@hotmail.com

Abstract: The open-circuit voltage (OCV) of batteries is a crucial characteristic parameter that reflects many aspects of a battery’sperformance, such as capacity, state-of-charge (SOC) and state-of-health. OCV is most widely used to determine the SOC whenthe battery works in a charge-depleting state. However, the application of the OCV to SOC estimation can be difficult because ofthe need for a long rest time for full relaxation. In this study, based on the analysis on the curve shape of battery voltage relaxation,a new adaptive model for simulating the voltage relaxation process is developed to predict the final static OCV in a few minutesinstead of via the traditional long-term rest method. Avoiding this disadvantage, the SOC can be deduced from the predicted OCVvia the corresponding relationship obtained in a short amount of time. A working condition experiment is performed to validatethe new methods and the results are very accurate.

1 Introduction

Rechargeable batteries play a significant role in poweringmany portable applications that have high electric powerrequirements. For battery electric vehicles (BEV), hybridelectric vehicles (HEV) and plug-in hybrid electric vehicles(PHEV), using the lithium-ion (Li-ion) as a storage systemis currently the best choice based on comprehensiveconsideration of its energy and power density and cycle-life[1–5]. Therefore an accurate determination of thestate-of-charge (SOC), which indicates the availablecapacity of cells, is requested to enable us to manage thebattery in its optimal state.Many studies have been done in recent years to improve

SOC estimation [6, 7]. Battery SOC can be calculated bycoulomb counting method, which is simple and direct [8].However, accumulated errors of current sensors andcolumbic efficiency will lead to the error of SOCestimation; and the estimation error cannot be self-correctedin dynamic processes because of the open-loop nature ofthe coulomb counting method. On the other hand, thismethod depends on the initial value for SOC. Othermodel-based method with filter algorithms, such as extendKalman filter (EKF), is introduced in this field to improvethe accuracy of SOC estimations by removing measurementnoise [9]. The EKF method is based on an equivalentcircuit model and realises SOC estimation by means ofestablishing state-space representation. However, thismethod strongly depends on the precision of the batterymodel and the system noise. Inappropriate information

matrices of the system noise may lead to obvious errors anddivergence.Another method focuses on the open-circuit voltage (OCV)

in equilibrium, which is found to be a good indicator of SOCfor Li-ion and lead-acid batteries. It can be used to determinethe initial SOC value and correct the estimation results of thedynamic methods (e.g. the coulomb counting technique). Inthe conventional OCV method, the battery voltage onlyequals the static OCV when the battery is underopen-circuit conditions and the voltage has been relaxed toits equilibrium. However, a long rest time is always neededto reach this static status [6, 7, 10]. Hence, even in thecharge-depleting state of the BEVs and PHEVs, the longrelaxation time severely limits the application of thetraditional OCV method for estimating SOC via a storedOCV–SOC relation.To overcome the shortcoming of the conventional OCV

method, some meaningful studies have been done toshorten the time of obtaining the OCV in equilibrium [10],but more accurate relaxation model and its applicableconditions still need to be studied. In this paper, a newrapid OCV method is proposed. Two features of the newalgorithm are presented below. First, a new model for theover-potential relaxation process is presented to ensure thatthe cells’ OCV can be predicted quickly and accurately inthe first several minutes under open-circuit conditions.Second, according to the given OCV–SOC relation, theusable ranges of OCV are marked off to limit the SOCestimation offset caused by the potential maximumprediction error of OCV.

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2 Experiment design

In this research, the cells with LiFePO4 cathodes andLiMn2O4 cathodes were chosen as the research objects [11].The tests were carried out on single cells and divided intotwo parts. The first part acquires the cell’s OCV behaviourof relaxation. The goal of the second part is to validate theeffect of the new algorithm. The first part was onlyperformed on the LiFePO4 cell, and the second part wasperformed on both of the LiFePO4 and LiMn2O4 cells.

2.1 Relaxation behaviour

Four relaxation behaviour tests were performed underdifferent SOC points, charge–discharge processes. Each testcan be divided into two steps: the preparation step and thecharacteristic step. During the first step, the LiFePO4 cellswere fully charged (SOC = 1) with constant current-constantvoltage (CCCV) (constant current 0.5 C charge until thevoltage reaches 3.65 V; then, constant voltage 3.65 Vcharge until the current is below 0.05 C) and discharged tothe presupposed SOCs (shown in Table 1) with 1 Cconstant current after a 30 min rest. During the second step,the tests were taken between two 24 h rests with differentconstant currents, charge–discharge processes. The detailsof the four tests are given in Table 1.

2.2 Validation test

The test was administered to validate the efficacy of the newOCV algorithm. First, the cell was fully charged by CCCVand then rested for 30 min. Second, the federal urbandriving schedule (FUDS) test was administered for 1 h,followed by a rest of 24 h. Finally, the cell was dischargedto the cutoff voltage (2.5 V for LiFePO4 and 3 V forLiMn2O4) by a 1 C constant current to determine theremaining capacity in the cell.

2.3 Measurement equipment

All tests were performed with a channel of the Arbininstruments’ BT2000 test bench (18 V, ±10 A for themedium current range and ±100 A for the high-currentrange), which has a voltage measurement accuracy of±0.01% and a current measurement accuracy of ±0.02% (onthe full-scale value of both ranges). Moreover, all the testswere performed at the room temperature (25 ± 5°C).

3 Results and discussion

3.1 Relaxation model

At the beginning of the battery switching from working toopen-circuit state, the cell’s actual voltage does not reach the

Table 1 Details of characteristic experiment steps

Test Presupposed SOCin step 1

Charge–discharge process in step 2

(a) 0.9 discharge to SOC = 0.8 with 1 C(b) 0.5 discharge to SOC = 0.4 with 1 C(c) 0.6 charge to SOC= 0.8 with 0.5 C(d) 0.9 discharge to SOC = 0.79 with 1 C,

and simulate regeneration chargeto SOC = 0.8 with 2 C

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equilibrium state at once because its internal electrochemicaland mechanical processes do not disappear immediatelywith the current cutoff. The voltage without full relaxationcannot represent the cell’s OCV that coincides with thebattery electromotive force (EMF). The difference betweenthe non-equilibrium voltage and OCV is brought by theremnant over-potential after current interruption. Althoughthe overpotential will degrade along with the relaxationprocess, this process still takes a long time (e.g. 20 h).Known from all four tests referred to in Section 2.1, as

shown in Fig. 1, the four curves correspond to the results ofthe four tests. From the contrast between curve (a) and (b),corresponding to the relaxation processes of the batteriesthat had undergone the same charge–discharge process butreached the different SOCs, a subtle but distinct differencein the relaxation behaviour can be found. In the same way,comparing another groups of the curves – (a), (c) and (d)had different charge–discharge processes – it is safe to saythat, when the battery is in non-equilibrium, it is almostimpossible to shorten the time to reach OCV by removing afixed deviation from the actual voltage at a fixed time or tocalculate it from a fix-parameter relaxation model [12–14].To solve the problem described above, based on the

analysis on the curve shape of the battery voltagerelaxation, a new adaptive OCV relaxation model wasproposed that can accurately predict the OCV in the firstfew minutes (e.g. 10 min) without previously storedparameters. According to (1), in the new model, URLX isthe result of fitting the cell’s relaxation voltage measured(URLX) at time t, OCV is the prediction value of OCV andA, B, C and D are the parameters used to fit the overpotential.

URLX = OCV+ Ct + D

t2 + At + B(1)

When the cell is in the open-circuit state, denoteU1

RLX, . . . , UNRLX as the measured URLX values that are

obtained at the moments of t1 to tN. The OCV and othermodel parameters, including A, B, C and D, are found bybest fitting the new relaxation model to all the samplegained points (ti, Ui

RLX) in the least-squares sense. Then, theparameters obtained and the current time tN are broughtback into the model to calculate the UN

RLX. Only if the errorbetween the measurement UN

RLX and the UNRLX is less than

a sensor error value (e.g. 1 mV) could the results predictedbe adopted. Furthermore, another two conditions also needto be met at the same time: first, the coefficient ofdetermination for fitting is higher than 95% according to

Fig. 1 Contrast of the relaxation processes under differentconditions

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Fig. 3 Schematic illustration of the SOC estimation method

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(2); second, the errors between the measured and predictedcells’ voltages are normally distributed with the mean = 0,according to (3)

r2 = 1−∑

U iRLX − Ui

RLX

( )2∑

U iRLX − URLX

( )2 ≥ 0.95 (2)

URLX − URLX

( ) � N 0, s( ) (3)

Using this new model to fit the four tests above, the results aregiven in Fig. 2a–d. In this figure, the true OCV is obtained bystanding for 24 h, and the mark ‘○’ (‘ × ’) is supposed to bethe adopted (un-adopted) predictive results of OCV. Fromthese analyses of predictive results, the model was provedto fit all relaxation OCV curves well under any conditions,which is difficult for a common resistance–capacitance(RC) model [13–15] and other models [10]. In addition, theerror between OCV and OCV can be less than 3 mV in thefirst 15 min. Herein, 3 mV is less than the measurementaccuracy of the common sensor for a single cell (0.1% FSRfor voltage ≤5 V).

3.2 New OCV method

A new process of using OCV to estimate SOC is shown inFig. 3. First, once the circuit is disconnected, therelationship between OCV and SOC is obtained with the

Fig. 2 Relaxation processes after

a 1 C (constant current) discharge from SOC = 0.9 to 0.8b 1 C discharge from SOC = 0.5 to 0.4c 0.5 C charge from SOC = 0.6 to 0.8d 1 C discharge from SOC = 0.9 to 0.79 and then 2 C charge to SOC = 0.8

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consideration of practical working conditions (e.g.temperature), and the OCV is predicted by the newrelaxation model. Second, it is necessary to determinewhether the prediction value of OCV is in the availablevoltage range of the current SOC–OCV-related curve.Herein, take the LiFePO4 cell for example. Considering theOCV flats of LiFePO4 shown in Fig. 4a, in the availablerange, the error of SOC estimation that caused by themaximum error of the OCV prediction can be accepted.The results of the validation test are shown in Fig. 5.Fig. 5a gives the progress of voltage values during theentire validation test, plotted against time. Observed fromFig. 5b that corresponds to the marked rectangle region inFig. 5a, the adopted OCV can be given in 10 min by the

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Fig. 4 OCV against SOC at room temperature

a For LiFePO4 batteryb For LiMn2O4 battery

Fig. 5 Validation test for LiFePO4 battery

a FUDS test for 1 hb Predictive OCV during the relaxation processc Available OCV ranged SOC estimation error comparison between different methodsHerein, the true SOC value is 60.9% that is measured by the experiment of constant current discharge

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relaxation model, and its error is less than 3 mV. Additionally,it can be seen that the curve of URLX fits the relaxation processof the measured voltage very well in Fig. 5b. The availableOCV ranges are shown in Fig. 5c. The red (blue) lineindicates the possible SOC estimation error caused by thepositive (negative) 5 mV OCV error at each OCV pointunder the current SOC–OCV relation. Only when the

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possible SOC error is less than 5% (the green areas) will anSOC value be estimated by the OCV. At last, the predictiveerrors of SOC from the traditional OCV method,second-order RC model [15], and the new method are givenin Fig. 5d. From this figure, we can safely conclude that thenew method has higher estimation accuracy and a shorterrest time.

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Fig. 6 Validation test for LiMn2O4 battery

a FUDS test for 1 hb Predictive OCV during the relaxation processc Available OCV ranged SOC estimation error comparison between different methodsHerein, the true SOC value is 69.2% that is measured by the experiment of constant current discharge

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To verify the method’s universality, the new method isapplied to the validation test for the LiMn2O4 cell and theestimation results are shown in Fig. 6. This figure highlightsthat the new method also works well in the aspects of therelaxation process fitting, the OCV and SOC estimation.Compared with the LiFePO4 cell, the LiMn2O4 cell has asteeper slope of the OCV–SOC curve, which can be foundin Fig. 4b. The steeper slope brings about the larger OCVavailable range and more accurate SOC estimation results asshown in Figs. 6c and d, respectively.

4 Conclusions

The OCV behaviours of Li-ion batteries were studied in thispaper via a series of experiments. A new adaptive OCVrelaxation model was presented to shorten the traditionalrest time of several hours for obtaining the OCV in theequilibrium state. The new model can predict theequilibrium OCV quickly (≤15 min) and accurately(≤3 mV) via an on-line fit to the relaxation curve.Moreover, the predicted OCV is determined to be acceptedor not depending on whether it belongs to the reasonableregion. Finally, considering all the factors, the specificprocess of how to use the identified OCV to estimate theSOC was given, and the validation experiments showsatisfactory results for the new OCV method. Its applicationensures that the practicality of the OCV method in on-lineSOC estimation is significantly improved.

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The thorough studies on the physical law of the OCVrelaxation behaviour and the impacts of temperature andaging in the OCV relaxation process will be performed inthe future.

5 Acknowledgments

This research was supported by the National HighTechnology Research and Development Program of China(grant no. 2012AA111003) in part and the National EnergyTechnology Research and Application of EngineeringDemonstrative Project of China (grant no. NY20110703-1)in part. The author would also like to thank the reviewersfor their corrections and helpful suggestions.

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