Fast Open Circuit Voltage Estimation of Lithium-Ion Batteries Using a Relaxation Model and Genetic Algorithm

Battery Open Circuit Voltage (OCV) is of fundamental characteristic for enabling battery modeling and states estimation. However, the traditional OCV measurement method takes a very long time to make the battery reaches its equilibrium, which is rather inconvenient and cannot be performed online for battery energy storage application. Motived by this, this paper proposes an effective method for fast OCV estimation in the relaxation process. In this work, a novel relaxation model is designed for capturing the voltage response of a battery during relaxation time and the Genetic Algorithm (GA) is further applied for optimizing the model parameters and acquiring accurate OCV estimation results. Experimental results confirm the validity of the proposed method under different State of Charges (SOCs), current rates, ambient temperatures, and aging conditions. The results suggest that the proposed method can accurately and quickly estimate battery OCV, which only takes 10 minutes of measurement data (more than 2 hours for the traditional method) and the maximum estimation error is limited to merely 1.8 mV.

descriptions, we realize that the aforementioned OCV tests 78 are still quite time-consuming and inconvenient for battery 79 applications, especially, for the cases when OCV is expected 80 to be obtained within a short time. 81 It is noted that EVs are often stopped during a traffic jam 82 or traffic light, and the battery current is close to zero when 83 the EV stops. There leaves a chance that the OCV could be 84 estimated during a short interruption period, which facilitates 85 onboard battery OCV acquisition. A straightforward way for 86 battery OCV estimation is to utilize the characteristic of the 87 voltage responses during relaxation time. Meng et al. [28] 88 propose a novel multiple correction approach for battery 89 OCV estimation, which has been proven to be feasible on a 90 LiFePO 4 battery with different SOCs. However, the method 91 suffers from a trouble tuning procedure of the parameters, 92 which is not convenient for practical usage. Pei et al. [35] also 93 develop a voltage relaxation model to estimate the terminal 94 voltage of a battery. However, the relaxation voltage has a 95 very strong nonlinear characteristic, which complicates the 96 curving fitting process. 97 As an alternative choice, many researchers focus on esti-98 mating battery OCV using the Equivalent Circuit Models 99 (ECMs). The reason is that the ECMs have a simple structure, 100 while they can capture the main voltage dynamics of a battery. 101 Duong et al. [36] propose a Multiple Adaptive Forgetting 102 Factors based RLS (MAFF-RLS) method for identifying the 103 parameters of an ECM, which obtains the OCV from a 40 Ah 104 LiFePO 4 battery. Yang [37] first estimates the OCV and the 105 RC circuits of an ECM simultaneously using an evolutionary 106 algorithm. Zhou et al. [38] proposed a weighted voltage relax-107 ation model consisting of two parallel resistor-capacitor (RC) 108 components for fast OCV estimation. By taking a short rest 109 period (less than 30 minutes), the maximum OCV estimation 110 error is limited to 4 mV through all the tests. It can be seen 111 that the estimation accuracy is low due to the limited ECM 112 modeling ability.

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In this work, a novel voltage relaxation model is proposed 114 for describing the dynamic response of a battery during the 115 rest process. In comparison with the traditional Thevenin 116 model, the proposed relaxation model is more accurate for 117 simulating battery terminal voltage in relaxation time. For 118 obtaining the best results, the Genetic Algorithm (GA) is 119 further carried out for optimizing the model parameters, 120 which shows an excellent performance in dealing with the 121 nonlinear effects. The validity of the proposed method is 122 verified experimentally in terms of accuracy and robustness 123 with two batteries, which also considers both the temperature 124 variations and the battery aging effect. The main contribu-125 tions of this work are listed as follows:

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(1). The Li-ion battery OCV can be accurately estimated 127 within 10 minutes by using the proposed voltage relaxation 128 model, whose parameters are adjusted in a GA framework.

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(2). The validation of the proposed method is proved not 130 only on different SOCs but also with the variation of temper-131 atures and battery aging status.

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The remainder of this paper is organized as fol-133 lows. Section II introduces the proposed relaxation model. 134 Section III elaborates the procedures of parameter optimiza-135 tion with the GA. Experimental results are carried out in 136 Section IV. The main conclusions are given in Section V.

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In this section, the experimental setup for measuring the 139 OCVs of the batteries is introduced first. The voltage relax-140 ation behavior of a battery is investigated afterward. A novel 141 relaxation model is further carried out to describe the 142 dynamic characteristics of the battery in relaxation time.

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A. EXPERIMENTAL SETUP

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The experimental tests are carried out on two LiFePO 4 batter-145 ies with a 3.2 V nominal voltage to validate the performance 146 of the proposed fast battery OCV estimation method. The 147 upper and lower cut-off voltages of the batteries are 3.6 V 148 and 2 V, respectively. The specifications of the batteries are 149 listed in Tab. 1. As shown in Fig. 1, the battery test platform 150 includes a thermal chamber to control the ambient temper-151 ature, a Chroma 17011 test station to charge and discharge 152 the battery, a host computer to generate the control signal 153 and store the measurement data. In this study, the sampling 154 frequency is set to 1 Hz.

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In this section, we have tested Cell A to measure the OCV 156 with different SOCs and current rates. The voltage and current 157     where the ambient temperature is set to 25 • C. It can be seen 159 that the battery is discharged with a 0.5 C rate with a 10%   consists of instantaneous voltage variation, which is caused 167 by the Ohmic resistance, and the dynamic variation, which is 168 caused by the kinetic effect and ion transfer, etc. For obtaining 169 the OCV of a battery, it has to take a long time (several 170 hours or even days) for reaching the equilibrium state due 171 to the slow process of the internal chemical and physical 172 reaction. Consequently, the cut-off voltage of a battery cannot 173 immediately meet the OCV without a long relaxation time.  In this subsection, a relaxation model is proposed to simulate 182 the terminal voltage variation of a battery during the relax-183 ation process.

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The Thevenin model is the most used battery model as 185 it has a simple structure and provides acceptable modeling 186 accuracy under various operating conditions. As shown in 187 Fig. 6, R 0 is the Ohmic resistance, which consists of the 188 electrolyte resistance and electrode material resistance, etc. 189 R p and C p are the electrochemical polarization capacitance 190 VOLUME 10, 2022   Here we define τ t as the time constant at the time of t, 201 which is determined by, where U oc is battery OCV, U t and U t−1 are the terminal  The proposed relaxation battery model contains an RC net-216 work. Both the polarization capacitance C p,t and the polariza-217 tion resistance R p,t are designed as time-varying parameters. 218 The governing equation of the proposed relaxation battery 219 model is expressed as, where U t and U t−1 are the simulated voltages from the relax-222 ation battery model at the time of t and t-1. τ t is subjected to a 223 linear function, where a and b are the polynomial coefficients 224 of the linear function.

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It can be seen that a, b, and U oc are the parameters to 226 be identified. The method for identifying the parameters is 227 discussed in the following section.

III. THE PROPOSED OCV ESTIMATION METHOD
The model parameters can be obtained by fitting the termi-231 nal voltage measurements with the output voltages from the 232 relaxation battery model. Here we define a parameter vector, 233 which is expressed as θ = a b U oc T . A least-square based 234 estimator is designed for estimating the model parameters, 235 which is expressed as, whereθ is the estimated parameter vector, U t is the measured 238 battery terminal voltage,Û t is the model simulated voltage, 239 The fitness function is further presented to compare the 242 model outputÛ t and the measured U t , which is expressed 243 as, whereŪ t is the average value of U t over the relaxation time 246 period.

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To obtain a reasonable parameter identification result, the 248 GA is presented for optimizing the model parameters, which 249 is further discussed in the following subsection. , [39], [40].

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The evolutionary process of a population is introduced in  The flowchart of the GA for optimizing the model parame-268 ters is shown in Fig. 10. The calculation is executed according 269 to the fitness function. If the fitness value is not satisfied, 270 GA takes the selection, crossover, and mutation for updating 271 the parameters. The repetition terminates when the fitness 272 value is larger than the boundary value and the output values 273 are regarded as the final identified model parameters.

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Considering a balance between practicability and modeling 276 accuracy, we only take 600 s relaxation voltage measurements 277 to establish the relaxation model and optimize the model 278 parameters in this work. The experimental results concerning 279 battery OCV estimation and the fitted terminal voltages are 280 shown in Fig. 11. It can be seen that the simulated voltage 281 plots almost identical curves in comparison with the voltage 282 measurements, which confirms the modeling accuracy of the 283 proposed relaxation model. Meanwhile, the estimated battery 284 OCV is very close to the reference, which proves the accuracy 285 of the proposed method for OCV estimation.

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To further investigate the proposed fast OCV estimation 287 method under different operating conditions, additional tests 288 concerning different SOCs and current rates are carried out in 289 this work. The experimental results with 0.5 C,1 C, and 2 C 290 current rates are shown in Figs. 12, 14, and 16. It can be seen 291 VOLUME 10, 2022      Batteries' aging effects are commonly described as capac-306 ity losses. As shown in Table 1, Cell B has the same specifica-307 tions as Cell A, while the capacity of Cell B is lower than the 308 initial ones. Meanwhile, the thermal effects are investigated 309 by testing Cell A under the ambient temperature of 40 • C. For 310 controlling variables, the current rate is selected as 0.5 C in 311 this subsection.     Table. 2. It can be seen that the maximum errors in [28], [35], proposed in this work only takes 10 minutes of the relaxation 326 time, and the maximum estimation error is limited to 1.8 mV. 327 The above results confirm the superiority and practicability 328 of the proposed method.

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This study proposes an effective method for estimating bat-331 tery OCV within a short relaxation time period. A novel 332 relaxation model is designed for characterizing the voltage 333 response of a battery during the relaxation process. The pro-334 posed relaxation model can correctly simulate the terminal 335 voltage in relaxation time, which significantly outperforms 336 the traditional Thevenin model in terms of accuracy. The GA 337 can effectively deal with the nonlinear effect, which is applied 338 for optimizing the model parameters and obtaining the best 339 OCV estimation results.

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Experimental tests have verified the effectiveness of the 341 proposed method under different SOCs, current rates, aging 342 status, and ambient temperatures. The proposed method 343 shows excellent performance for estimating battery OCV, 344 which takes only 10 minutes of measurement data, and the 345 maximum estimation error is limited to 1.8 mV.