State of Charge Estimation for Lithium-Ion Power Battery Based on H-Inﬁnity Filter Algorithm

applied sciences

Article State of Charge Estimation for Lithium-Ion Power Battery Based on H-Inﬁnity Filter Algorithm

Lan Li 1, Minghui Hu 1,*, Yidan Xu 1, Chunyun Fu 1,* , Guoqing Jin 2 and Zonghua Li 2

1 State Key Laboratory of Mechanical Transmissions, School of Automotive Engineering, Chongqing University, Chongqing 400044, China; [email protected] (L.L.); [email protected] (Y.X.) 2 Chongqing Changan Automobile Co., Ltd., Chongqing 400023, China; [email protected] (G.J.); [email protected] (Z.L.) * Correspondence: [email protected] (M.H.); [email protected] (C.F.)

Received: 20 July 2020; Accepted: 10 September 2020; Published: 13 September 2020

Abstract: To accurately estimate the state of charge (SOC) of lithium-ion power batteries in the event of errors in the battery model or unknown external noise, an SOC estimation method based on the H-infinity filter (HIF) algorithm is proposed in this paper. Firstly, a fractional-order battery model based on a dual polarization equivalent circuit model is established. Then, the parameters of the fractional-order battery model are identified by the hybrid particle swarm optimization (HPSO) algorithm, based on a genetic crossover factor. Finally, the accuracy of the SOC estimation results of the lithium-ion batteries, using the HIF algorithm and extended Kalman filter (EKF) algorithm, are verified and compared under three conditions: uncertain measurement accuracy, uncertain SOC initial value, and uncertain application conditions. The simulation results show that the SOC estimation method based on HIF can ensure that the SOC estimation error value fluctuates within 0.02 in any case, and is slightly affected by environmental and other factors. It provides a way to ± improve the accuracy of SOC estimation in a battery management system.

Keywords: lithium-ion power batteries; fractional-order model; state of charge estimate; H-inﬁnity ﬁlter; hybrid particle swarm optimization algorithm

1. Introduction The state of charge (SOC) estimation is the most basic function of battery status monitoring. SOC represents the remaining power of the battery, and accurate SOC estimations are of great significance in improving the battery efficiency and safety performance [1,2]. Currently, domestic and foreign scholars have been researching different SOC estimation methods for lithium-ion batteries, which can be divided into four major categories [3]: estimation methods based on characteristic parameters of lithium-ion batteries, estimation methods based on ampere-hour integration, data-driven estimation methods, and model-based estimation methods. The derivation process of SOC estimation for lithium-ion batteries involves many characteristic parameters, such as open circuit voltage (OCV), residual capacity, and electrochemical impedance. There is a certain mapping relationship between these characteristic parameters and SOC. The latter can be estimated by establishing this mapping relationship. The method of obtaining the battery residual capacity by a discharge test is considered to be the most direct method to determine the SOC value of lithium-ion batteries [4]. In addition to using the current residual capacity as a parameter, M. Einhorn et al. [5] used a linear interpolation method to estimate the battery’s SOC based on the functional relation curve between the SOC and OCV. However, this type of method can only be applied under specific environmental constraints, such as laboratories, since it is impossible to carry out long-term constant current discharges or static states during the motion of electric vehicles.

Appl. Sci. 2020, 10, 6371; doi:10.3390/app10186371 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 6371 2 of 18

Therefore, characterization parameters such as residual capacity and OCV cannot be determined [6,7]. A. J. Salkind et al. [8] optimized the parameters of the established fuzzy model through experimental data. This fuzzy model can fit the relationship between the battery charge state and electrochemical impedance spectroscopy (EIS) values, and the estimated error of the SOC can be controlled within 5%. However, online measurement of EIS is not easy, and varies with battery type and experimental conditions. The exact relationship between EIS and SOC and the repeatability of the online measurement of EIS needs to be studied further. The ampere-hour integration [9], also known as the coulomb counting method, incurs a low calculation cost, as well as enables fast calculation. It is the estimation method of the SOC currently adopted by most electric vehicles. However, the estimation method based on ampere-hour integration requires high accuracy requirements from the initial parameters of the battery model. It is difficult to meet the sampling accuracy and frequency requirements of the sensor during the actual application of the battery. Data-driven estimation methods can be divided into three categories according to different algorithms: support vector machine (SVM) [10], neural network (NN) [11], and the fuzzy control method [12]. These methods are similar to the black box model in the battery modeling process. In order to achieve high estimation accuracy and strong adaptability, a large amount of data is usually required for training. Moreover, the selection of training data is very strict, while requiring high precision in data fitting, as well as meeting the applicability of non-linear models. The data that does not meet the requirements will make it difficult to train the model, which will affect the accuracy and, consequently, the SOC estimation to a great extent. The model-based SOC estimation method is self-adjusted by the difference between the battery’s measured voltage and the model’s output voltage. This is done to overcome the error caused due to uncertainty, and achieve the purpose of improving the estimation accuracy of the SOC. It is a robust closed-loop estimation algorithm. Common methods include the impedance method [13], particle filter (PF) [14], Kalman filter (KF) [15], and improved algorithm [16–18]. Among them, the PF algorithm can provide better results in a non-Gaussian white noise system, but its operation intensity is higher than that of the KF algorithm. The KF algorithm is widely used, because it can find the optimal solution for a linear Gaussian system. A simple linear system can use the ordinary KF algorithm [19], whereas for complex nonlinear models, the extended Kalman filter (EKF) algorithm has been studied and applied extensively [20–22]. Based on the equivalent circuit model, Plett [23,24] used Kalman filter series algorithms (KF, EKF, unscented Kalman filter (UKF)) to achieve the identification of model parameters and SOC estimation at the same time. Sun et al. [25] improved the traditional EKF algorithm, and realized that when the input and output of the system changed, the noise covariance matrix in the algorithm responded accordingly to self-correct the noise. Thus, the influence of noise on the accuracy of the traditional EKF algorithm is overcome; however, the EKF algorithm requires that the input noise of the model be white noise with known statistical characteristics. It is greatly affected by the measurement accuracy of the sensor during vehicle operation, and has not been applied in real vehicles. To solve the above problems, the H-infinity filter (HIF) algorithm based on the minimum-maximum criterion is adopted in this study instead of the traditional EKF algorithm. This algorithm takes into account the time-varying element of battery parameters, and does not require the details of process noise and measurement noise. Therefore, the HIF can expand the scope of application of the SOC estimation algorithm and improve the robustness of the algorithm under real vehicle operating conditions. Chen et al. [26] used the HIF algorithm to estimate the SOC of lithium-ion batteries, and the method was tested by real-time experimental data of batteries. However, it ignores the measurement noises and random disturbances. Shu et al. [27] proposed an adaptive H-infinity filter (AHIF) algorithm to predict the SOC of lithium-ion batteries. The proposed algorithm is comprehensively validated within a full operational temperature range of battery, and with different aging status. Even so, it ignores the effects of application conditions on SOC estimation. In [28–30], the HIF is also applied to the Appl. Sci. 2020, 10, 6371 3 of 18

Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 18 model parameters identification and state estimation. The results show that it has better performance 96 estimation. The results show that it has better performance than UKF and EKF. However, none of than UKF and EKF. However, none of the above algorithms comprehensively analyze the influence 97 the above algorithms comprehensively analyze the influence of various uncertain factors (i.e., noises, of various uncertain factors (i.e., noises, SOC initial value, and application conditions) on the SOC 98 SOC initial value, and application conditions) on the SOC estimation results of lithium-ion batteries. estimation results of lithium-ion batteries. 99 To explore the influence of different uncertainties (i.e., noises, SOC initial value, and application To explore the influence of different uncertainties (i.e., noises, SOC initial value, and application 100 conditions) on the estimation results when HIF is used for SOC estimation, the following chapters of conditions) on the estimation results when HIF is used for SOC estimation, the following chapters 101 this paper are arranged as follows. In the second part, a fractional-order battery model of the of this paper are arranged as follows. In the second part, a fractional-order battery model of the 102 lithium-ion battery is established. The hybrid particle swarm optimization (HPSO) algorithm based lithium-ion battery is established. The hybrid particle swarm optimization (HPSO) algorithm based on 103 on a genetic crossover is used to identify the parameters of the built model to obtain the initial values a genetic crossover is used to identify the parameters of the built model to obtain the initial values of 104 of the model parameters. The accuracy of the model parameter identification under different the model parameters. The accuracy of the model parameter identification under different working 105 working conditions is also verified. In the third part, the HIF based on the minimum-maximum conditions is also verified. In the third part, the HIF based on the minimum-maximum criterion is 106 criterion is proposed to estimate the SOC of the lithium-ion battery. The robustness of the algorithm proposed to estimate the SOC of the lithium-ion battery. The robustness of the algorithm is verified 107 is verified under three conditions: uncertain measurement accuracy, uncertain initial value of SOC, under three conditions: uncertain measurement accuracy, uncertain initial value of SOC, and uncertain 108 and uncertain application conditions. Hereafter, we compared the SOC estimation results obtained application conditions. Hereafter, we compared the SOC estimation results obtained using the EKF 109 using the EKF algorithm. The fourth part presents the conclusion of the article. algorithm. The fourth part presents the conclusion of the article.

110 2.2. EstablishmentEstablishment ofof Fractional-OrderFractional-Order ModelModel andand ParameterParameter IdentificationIdentification 111 TheThe establishmentestablishment of of an an eff ectiveeffective battery battery model mode is thel is premise the premise and foundation and foundation for studying for studying battery 112 SOC.battery Lower SOC. model Lower accuracy model willaccuracy directly will reduce directly theprecision reduce the ofthe precision SOC estimation of the SOC algorithm, estimation and 113 evenalgorithm, directly and lead even to the directly divergence lead ofto thethe estimation divergence algorithm of the estimation in serious cases.algorithm From in the serious perspective cases. 114 ofFrom EIS, the a circuit perspective composed of ofEIS, fractional a circuit components composed canof betterfractional fitthe components impedance can characteristics better fit the of 115 theimpedance battery, socharacteristics that they can of be the applied battery, in batteryso that principlethey can analysis,be applied model in battery establishment, principle and analysis, state 116 estimationmodel establishment, [31]. and state estimation [31].

117 2.1.2.1. EstablishmentEstablishment ofof Fractional-OrderFractional-Order ModelModel 118 AccordingAccording toto thethe detaileddetailed analysisanalysis ofof EISEIS inin [32[32],], thisthis studystudy establishedestablished thethe fractional-orderfractional-order 119 equivalentequivalent circuit circuit model model of aof lithium-ion a lithium-ion battery, battery, as shown as shown in Figure in 1Figure. R0 represents 1. R0 represents the Ohmic the internal Ohmic 120 resistanceinternal resistance of the battery, of theR1 battery,and R2 are R1 theand pure R2 are resistive the pure impedance, resistiveCPE impedance,stands for CPE the constantstands for phase the 121 element,constant andphaseW element,represents and the W Warburg represents element. the Warburg The terminal element. voltage The terminal and OCV voltage of the and battery OCV are of 122 representedthe battery are by Urepresentedd and Uocv, by respectively, Ud and Uocv and, respectively,I is the current and valueI is the of current the circuit. value of the circuit.

R1 R2

CPE1 CPE2

+ Ud - Uocv

123 Figure 1. Schematic diagram of fractional-order equivalent circuit model. 124 Figure 1. Schematic diagram of fractional-order equivalent circuit model.

125 In this circuit, the impedance of the fractional-order element can be described by the following 126 equation: Appl. Sci. 2020, 10, 6371 4 of 18

In this circuit, the impedance of the fractional-order element can be described by the following equation:  1  ZCPE = α  1 C1s  = 1  ZCPE2 β (1)  C2s  1 ZW = Wsγ where, C1, C2, and W indicate the parameters of fractional-order elements; α and β represent the fractional order of CPE1 and CPE2 components. γ is the fractional order of the Warburg components. The fractional diﬀerential equation of the parallel circuit of CPE and resistance can be expressed as:

1 1 α ( ) = ( ) + ( ) D UCPE1 t UCPE1 t I t (2) −R1C1 C1

1 1 β ( ) ( ) ( ) D UCPE2 t = UCPE2 t + I t (3) −R2C2 C2 The fractional diﬀerential equation of the Warburg element is shown in Equation (4):

1 DγU (t) = I(t) (4) W −W According to the definition, the system state of the SOC as a fractional-order model is presented in Equation (5) as follows: η D1SOC(t) = I(t) (5) −Cn where Cn is the battery rated capacity, η is the coulombic efficiency, and η is desirable 1 for lithium-ion batteries. According to Kirchhoff’s law of current and voltage, we obtain

U = Uocv I(t)R U U UW (6) d − 0 − CPE1 − CPE2 − where Uocv is the OCV of the battery, which is a monotonic function of the SOC. In summary, the fractional-order equivalent circuit model of a lithium-ion battery can be established by Equations (1)–(6). A complex lithium-ion battery system can be described by the simple structure and limited parameters of the model. The parameters to be identiﬁed in the fractional model are shown in Equation (7), below: θ = [R0 R1 C1 R2 C2 W α β γ] (7)

2.2. Lithium-Ion Battery Test Experiment Battery test experiments are the premise of battery model establishment and state estimation, and are an indispensable step in battery research. In this paper, the battery test system is shown in Figure2. The test object is a universal A123, ternary lithium-ion soft packet battery cell. The main parameters are listed in Table1. Appl. Sci. 2020, 10, 6371 5 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 18

Computer Battery Computer Battery

Ethernet Ethernet

Test Test BTS-5V100A cable Incubator BTS-5V100A cable Incubator 148 148 149 Figure 2.2. Software and hardware systemsystem forfor batterybattery testtest experiment.experiment. 149 Figure 2. Software and hardware system for battery test experiment. Table 1. Main performance parameters of universal A123 battery. 150 Table 1. Main performance parameters of universal A123 battery. 150 Table 1. Main performance parameters of universal A123 battery. Nominal Nominal Charging Cut-Off Discharge Cut-Off Charging Cut-Off Nominal Nominal Charging Cut-Off Discharge Cut-Off Charging Cut-Off NominalCapacity (Ah) NominalVoltage (V) ChargingVoltage Cut-Off (V) DischargeVoltage Cut-Off (V) ChargingCurrent (A) Cut-Off Capacity（Ah） Voltage（V） Voltage（V） Voltage（V） Current（A） Capacity（28Ah） Voltage 3.7（V） Voltage 4.2（V） Voltage 2.5（V） Current 1.25 （A） 28 3.7 4.2 2.5 1.25 28 3.7 4.2 2.5 1.25 151 InIn orderorder toto identifyidentify thethe modelmodel parameters,parameters, aa seriesseries ofof experimentsexperiments onon lithium-ionlithium-ion batteriesbatteries werewere 151 In order to identify the model parameters, a series of experiments on lithium-ion batteries were 152 conductedconducted withwith reference toto relevant national standardsstandards and the USABC Electric Vehicle Battery Test 152 conducted with reference to relevant national standards and the USABC Electric Vehicle Battery Test 153 ProceduresProcedures Manual.Manual. TheThe batterybattery testtest flowflow chartchart ofof thethe relevantrelevant experimentsexperiments isis shownshown inin FigureFigure3 .3. 153 Procedures Manual. The battery test flow chart of the relevant experiments is shown in Figure 3.

Start Start Peak power ① Constant current and constant voltage Peak power ① Constant current and constant voltage test charging to charge cut-off voltage. Static capacity test charging to charge cut-off voltage. ② Constant current discharging to discharge Static testcapacity ② Constant cut-off voltage. current discharging to discharge test Refer to the USABC cut-off voltage. DST Refertest to themanual USABC ① Constant current discharging to discharge DST test manual ① Constant cut-off voltage. current Batterydischarging shelved to discharge for 1 h. Double pluse cut-off voltage. Battery shelved for 1 h. Doubletest pluse ② Charging with duration of 10% of battery test ②capacity. Charging Battery with durationshelved forof 10%2 min. of battery FUDS capacity. Battery shelved for 2 min. FUDS ③ Repeat step ② until the voltage reaches ③ Repeat the charge step cut-off② until voltage. the voltage Battery reaches shelved HPPC the for charge12 h. cut-off voltage. Battery shelved HPPC for 12 h. ④ Discharging with duration of 10% of End ④battery Discharging capacity. with Battery duration shelved of 10% for 2of min. End battery capacity. Battery shelved for 2 min. ⑤ Repeat step ④ until the voltage reaches ⑤ Repeat the discharge step ④ cut-off until the voltage. voltage reaches 154 the discharge cut-off voltage. 154 155 FigureFigure 3.3. Battery test ﬂowflow chart.chart. 155 Figure 3. Battery test flow chart. 156 According to the experimental data obtained by the double-pulse test, the relationship curve 156 According to the experimental data obtained by the double-pulse test, the relationship curve 157 between OCV and SOC can be obtained by [33] using the empirical Equation (8), as shown in Figure 157 between OCV and SOC can be obtained by [33] using the empirical Equation (8), as shown in Figure 158 4. 158 4. Appl. Sci. 2020, 10, 6371 6 of 18

Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 18 According to the experimental data obtained by the double-pulse test, the relationship curve between OCV and SOC can be obtained by [33] using the empirical Equation (8), as shown in Figure4. =+ +1 + + − USOCCCSOCCOCV ( )01 21 CSOCC 3 ln( ) 4 ln(1 SOC ) (8) U (SOC) = C + C SOC + C SOC+ C ln(SOC) + C ln(1 SOC) (8) OCV 0 1 2 SOC 3 4 − 159

4.2

4.1 Experimental data 4 Fitted curve 3.9

3.8 OCV (V) OCV 3.7

3.6

3.5

3.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 160 SOC 161 Figure 4. OpenOpen circuit voltage (OCV)–state of charge (SOC) ﬁttingfitting curve.

162 2.3. Model Parameter Iden Identificationtification Based on HPSO 163 The parameter identification identification link is very important before the model is successfully applied. applied. 164 The parameter identificationidentification results directly aaffectffect thethe accuracyaccuracy ofof thethe model.model. Because of the obvious 165 time-varying non-linear and complex structure of the lithium-ion battery system, there are many 166 parameters involvedinvolved in in operation, operation, and and most most parameter parameter values values cannot becannot directly be measured.directly measured. Therefore, 167 parameterTherefore, identificationparameter identification is a difficult problemis a diffic inult the problem battery modeling in the battery process. modeling For high-dimensional process. For 168 complexhigh-dimensional optimization complex problems, optimization this paper problems, proposes this an paper HPSO proposes algorithm an based HPSO on algorithm a genetic based cross 169 factor.on a genetic The flowchart cross factor. of the The algorithm flowchart is of shown the algorithm in Figure is5. shown in Figure 5. Referring to the genetic algorithm (GA), the genetic crossover factor is introduced into the classic particle swarm optimization (PSO). After sorting the fitness of the particles from high to low, it is used to select the particles with the fitness of the top half in each generation. As part of the next generation of updated particle swarms, the particles in the second half of fitness are paired as crossover factors. The position of the exchange between the particles is called the intersection, which is randomly set. At the intersection point, the particles pair and exchange with each other to produce two completely new offspring. By comparing the fitness of the offspring particles with that of their parent particles and sorting them, the top half of the particles with high fitness values are selected to enter the next generation of updated particle swarm so as to realize the update of the particle swarm. The introduction of genetic crossover factors enriches the diversity of population particles, solves the problem that the classical PSO algorithm is prone to fall into the local optimality, improves the global searching capability of the parameter identification algorithm, and further improves the fitting degree of the battery model. Appl. Sci. 2020, 10, 6371 7 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 18

Calculate particle fitness

Sort the fitness of the particles YN

Half-fit particles The latter half of the less fit particles

Randomly set intersection points, cross pairing Calculate the New progeny New progeny fitness of the particles particles parent particles

Compare particle swarm particle with the

Go directly to the updated parent

Select the first half of particles with good fitness Regroup for updates

The updated new 170 particle swarm 171 Figure 5. Flow chart of hybrid particle swarm filtering ﬁltering algorithm.

172 2.4. Fractional-OrderReferring to the Model genetic Accuracy algorithm Analysis (GA), the genetic crossover factor is introduced into the 173 classicIn particle this selection, swarm the optimization dynamic stress (PSO). test After (DST) sorting condition the fitness test data of the of theparticles universal from A123 high batteryto low, 174 it is used to select the particles with the fitness of the top half in each generation. As part of the next at 25 ◦C ambient temperature is used as training data to identify the parameters of the established 175 fractional-ordergeneration of updated model. Throughparticle swarms, different identificationthe particles methodsin the second and di halffferent of simulationfitness are conditions,paired as 176 thecrossover accuracy factors. and robustnessThe position of of the the fractional-order exchange betw modeleen the are particles analyzed. is called the intersection, which 177 is randomlyThe identification set. At the resultsintersection obtained point, using the part the classicalicles pair PSO and algorithmexchange arewith shown each other in Table to produce2. 178 two completely new offspring. By comparing the fitness of the offspring particles with that of their 179 parent particles Tableand sorting 2. Parameter them, recognition the top half results of the of particles ordinary particlewith high swarm fitnes algorithm.s values are selected to 180 enter the next generation of updated particle swarm so as to realize the update of the particle swarm. 181 The introductionR0 Rof1 genetic crossoverC1 factorsR2 enrichesC2 the diversityWA of population particles,β solves γ the 182 problem0.003 that the 0.0001 classical PSO 960 algorithm 7.12 is prone 1112 to fall into 500 the local 0.61 optimality, 0.13 improves 0.64 the 183 global searching capability of the parameter identification algorithm, and further improves the 184 fittingThe degree identification of the battery results model. obtained using the HPSO algorithm are shown in Table3.

185 2.4. Fractional-OrderTable Mo 3.delParameter Accuracy recognition Analysis results of hybrid particle swarm algorithm. 186 In this selection, the dynamic stress test (DST) condition test data of the universal A123 battery R0 R1 C1 R2 C2 W α β γ 187 at 25 °C ambient temperature is used as training data to identify the parameters of the established 0.001 0.05 5431 5.31 4758 2122 0.59 0.22 0.12 188 fractional-order model. Through different identification methods and different simulation 189 conditions, the accuracy and robustness of the fractional-order model are analyzed. 190 The identification results obtained using the classical PSO algorithm are shown in Table 2. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 18

191 Table 2. Parameter recognition results of ordinary particle swarm algorithm.

R0 R1 C1 R2 C2 W Α β γ 0.003 0.0001 960 7.12 1112 500 0.61 0.13 0.64

192 The identification results obtained using the HPSO algorithm are shown in Table 3.

193 Table 3. Parameter recognition results of hybrid particle swarm algorithm.

Appl. Sci. 2020, 10, 6371 8 of 18 R0 R1 C1 R2 C2 W α β γ 0.001 0.05 5431 5.31 4758 2122 0.59 0.22 0.12

194 AccordingAccording toto the the identification identification results results of the of abovethe above two algorithms, two algorithms, the model the accuracy model isaccuracy simulated is 195 simulatedand verified and under verified the three under working the three conditions working of conditions the dynamic of the stress dynamic test (DST), stress federal test (DST), urban drivingfederal 196 urbanschedule driving (FUDS), schedule and hybrid (FUDS), pulse and power hybrid characteristic pulse power (HPPC). characteristic The simulation (HPPC). results The are simulation shown in 197 resultsFigures 6are–8 .shown It can be in seen Figures from 6–8. Figures It can6–8 bethat, seen under from di Figuresfferent operating 6–8 that, conditions,under different the voltage operating error 198 conditions,value output the by voltage the fractional-order error value output model by fluctuates the fractional-order slightly around model zero. fluctuates When the slightly input around current 199 zero.has a When large mutation,the input current the voltage has a error large value mutation, is large, the butvoltage can alsoerror be va controlledlue is large, at but approximately can also be 200 controlled0.05 V. Therefore, at approximately the improved 0.05 V. fractional-order Therefore, the modelimproved is more fractional-order accurate and model suitable is more for diaccuratefferent 201 andworking suitable conditions. for different working conditions.

4.5 Experimental data 4 PSO HPSO

3.5 3.76 Voltage (v)

3.72 3 0 500 1000 1500 2000 2500 3000 3.68

1360 1370 1380 1390 1400 Time (s) 0.1 PSO 0.05 HPSO 0

Error (v) Error -0.05

-0.1 0 500 1000 1500 2000 2500 3000 202 Time (s) 203 Appl. Sci. 2020FigureFigure, 10, x 6.FOR 6. ModelModel PEER voltage voltageREVIEW and and simulation simulation error error under under dynamic dynamic stress stress test (DST) conditions. 9 of 18

204 4.2 Experimental data 4 PSO 3.8 HPSO

Voltage (v) Voltage 3.6 3.76

3.4 3.72 03.68 500 1000 1500 2000 2500 3000 3.64 Time (s) 1365 1370 1375 1380 0.1 PSO 0.05 HPSO 0 Error (v) -0.05

-0.1 0 500 1000 1500 2000 2500 3000 205 Time (s) 206 Figure 7.7. ModelModel voltage voltage and and simulation simulation error error under under federal federal urban drivingurban scheduledriving schedule (FUDS) condition. (FUDS) 207 condition.

4.2 Experimental data 4 PSO 3.8 HPSO

3.6 3.64

Voltage (v) Voltage 3.4 3.60 3.2 0 5000 10000 15000 20000 25000 3.56 20500 20600 20700 20800 20900 21000 Time (s) 0.05 PSO 0.025 HPSO

Error (v) Error -0.025

-0.05 0 5000 10000 15000 20000 25000 208 Time (s) 209 Figure 8. Model voltage and simulation error under hybrid pulse power characteristic (HPPC) 210 conditions.

211 Table 4 shows the root mean square error (RMSE) value of the model output voltage 212 corresponding to the classical PSO algorithm and the HPSO algorithm under different working 213 conditions. It can be seen from Table 4 that the parameter identification method based on the HPSO 214 algorithm further improved the accuracy of the fractional model.

215 Table 4. Root mean square error under different working conditions.

RMSE HPPC DST FUDS PSO 0.0061 0.0050 0.0057 HPSO 0.0036 0.0022 0.0035

216 3. SOC Estimation Based on the HIF Algorithm 217 The Kalman series of filtering algorithms have been supported, widely researched and applied 218 owing to their high accuracy, low computational complexity, and good robustness. However, they 219 have two limitations that are difficult to overcome: (1) It is necessary to ensure that the input noise is 220 white noise, and the statistical characteristics of the noise are approximately known. In practice, the Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 18

4.2 Experimental data 4 PSO 3.8 HPSO

Voltage (v) Voltage 3.6 3.76

3.4 3.72 03.68 500 1000 1500 2000 2500 3000 3.64 Time (s) 1365 1370 1375 1380 0.1 PSO 0.05 HPSO 0 Error (v) -0.05

-0.1 0 500 1000 1500 2000 2500 3000 205 Time (s)

206 Appl.Figure Sci. 2020 ,7.10 ,Model 6371 voltage and simulation error under federal urban driving schedule (FUDS)9 of 18 207 condition.

4.2 Experimental data 4 PSO 3.8 HPSO

3.6 3.64

Voltage (v) Voltage 3.4 3.60 3.2 0 5000 10000 15000 20000 25000 3.56 20500 20600 20700 20800 20900 21000 Time (s) 0.05 PSO 0.025 HPSO

Error (v) Error -0.025

-0.05 0 5000 10000 15000 20000 25000 208 Time (s) 209 FigureFigure 8. 8. ModelModel voltage voltage and andsimulation simulation error errorunder underhybrid hybridpulse power pulse characteristic power characteristic (HPPC) 210 conditions.(HPPC) conditions.

211 TableTable 4 4shows shows the the root root mean mean square square error (RMSE) error value(RMSE) of thevalue model of outputthe model voltage output corresponding voltage 212 correspondingto the classical to PSO the algorithm classical andPSO the algorithm HPSO algorithm and the HPSO under algorithm diﬀerent working under different conditions. working It can 213 conditions.be seen from It Tablecan be4 seenthat thefrom parameter Table 4 that identiﬁcation the parameter method identification based on themethod HPSO based algorithm on the further HPSO 214 algorithmimproved further the accuracy improved of the the fractional accuracy model. of the fractional model.

215 TableTable 4. 4. RootRoot mean mean square square error error under under different diﬀerent working conditions.

RMSERMSE HPPC HPPC DST DST FUDS FUDS PSOPSO 0.0061 0.0061 0.0050 0.0050 0.0057 0.0057 HPSOHPSO 0.0036 0.0036 0.0022 0.0022 0.0035 0.0035

216 3.3. SOC SOC Estimation Estimation Based Based on on the the HIF HIF Algorithm Algorithm 217 TheThe Kalman Kalman series series of of filtering filtering algorithms algorithms have have been been supported, supported, widely widely researched researched and and applied applied 218 owingowing to theirtheir highhigh accuracy, accuracy, low low computational computational complexity, complexity, and and good good robustness. robustness. However, However, they they have 219 havetwo limitationstwo limitations that arethat di arefficult difficult to overcome: to overcome: (1) It is(1) necessary It is necessary to ensure to ensure that the that input the input noise noise is white is 220 whitenoise, noise, and the and statistical the statistical characteristics characteristics of the of noise the noise are approximately are approximately known. known. In practice, In practice, the noise the is often not a white noise that satisfies the Gaussian distribution, and the statistical characteristics are difficult to obtain. (2) The accuracy of the estimation is very dependent on the accuracy of the established model. When the accuracy of the model is insufficient or the accuracy of the model gradually decreases with the dynamic changes of the working process, the filtering error becomes increasingly larger. To overcome the Kalman series filtering algorithm’s requirements for white noise and its dependence on accurate modeling, an HIF with poor model accuracy, uncertain SOC initial value, and stronger robustness of various colored noise interference is introduced.

3.1. State Space Equation of the Lithium-Ion Battery The establishment of system state space equations is the basis for controlling the parameters of dynamic systems in modern control theory. Establish a mathematical model that reﬂects the internal structure of the system and characterizes the system through the state equation and the output equation. Owing to the time-varying non-linearity of the battery system, its state space equation can be expressed as: xk+1 = Akxk + Bkuk + ωk (9) Appl. Sci. 2020, 10, 6371 10 of 18

yk = Ckxk + Dkuk + υk (10) This is the first step in the derivation of the state space equation of a lithium-ion battery to determine the relevant variable parameters of the battery system. According to the established battery = [ ] model, the state variable of the system is set as xk VCPE1,k VCPE2,k VW,k SOCk . The input of the system is u = Ik, which is the current state of the battery. The output is y = Ud,k, which is the terminal voltage of the battery. After determining the parameters, according to the fractional-order model established in Section 2.1, the state space equations of Equations (8) and (9) are derived, where Ak, Bk, Ck, and Dk are defined as follows: α  α 1   T  α T 0 0 0 C  R1C1   1   −   β   β 1   T   0 β T R C 0 0   C  A =  − 2 2  , B =  2γ  k   k  T   0 0 γ 0   W  (11)    −   0 0 0 1   ηT  Cn d f (SOC ) − C = [ 1 1 1 k ] , D = R k − − − dSOC k − 0 The implementation of SOC estimation is to design the corresponding filter algorithm based on the above-mentioned state space equations to eliminate the effects of noise ωk and υk as much as possible. With improvements in the accuracy of the system state estimation, the SOC estimation achieves the desired effect.

3.2. SOC Estimation Based on HIF Algorithm

To judge the accuracy of the estimated object xk, the HIF algorithm deﬁnes a cost function J:

NP1 − x xˆ 2 k k Sk k=0 k − k J = (12) NP1 x xˆ 2 + − ( ω 2 + υ 2 ) 0 0 P 1 k Q 1 k R 1 k − k 0− k=0 k k k− k k k−

The matrices P0, Qk, Rk, and Sk in the equation are all symmetric positive definite matrices, which are set by the designer for specific problems. Designers use these matrices to reflect the degree to which different types of noise affect the system. In general, we expect to perform similar processing on the weights of different types of noises to avoid the impact on the accuracy of estimation when a certain type of noise is ignored. It can be seen from Equation (11) that the cost function of HIF is a relative proportional value, and its numerator and denominator are related to the estimated value error and the overall noise of the system. The significance of establishing the cost function is to reflect the proportional relation between the estimation error and the noise interference. In order to meet the design requirements, the cost function has an upper bound, which satisfies the following condition:

J < 1/θ

To solve the above optimization process, the algorithm designer must ﬁnd a suitable xk method to minimize the denominator of the cost function J. However, the reality is contrary to the designer’s goal; it hopes to produce a speciﬁc special initialization state x0, noise ωk, and υk, so as to maximize the cost function J. Therefore, this problem becomes a maximum-minimum problem, that is, when ωk, υk, and x0 make the cost function J maximum, the appropriate xk should be selected to minimize the cost function Appl. Sci. 2020, 10, 6371 11 of 18

J. By re-integrating Equations (11) and (12) and combining it with the output Equation (9) of the state space, Equation (13) can be obtained:

NX1 1 2 − 2 1 2 2 x0 xˆ0 + x xˆ ( ω + y C x D u ) < 0 (13) P 1 k k Sk k Q 1 k k k k k R 1 − θk − k 0− k − k − θ k k k− k − − k k− k=0

By solving the maximum-minimum problem of the cost function, a recursive relation satisfying Equation (12) can be obtained, as shown below:

 1  K = A P (I θS P + CTR 1C P )− CTR 1  k k k k k k k− k k k k−  − xˆ + = A xˆ + B u + K (y C xˆ D u ) (14)  k 1 k k k k k k − k k − k k  T 1 1 T  P + = A P (I θS + C R C P )− A + Q k 1 k k − k k k− k k k k Similar to the KF algorithm, in order to facilitate calculation and application of the SOC estimation method based on the HIF, the recursive relation (Equation (14)) is divided into two stages: prediction and update. The speciﬁc calculation process is shown in Table5.

Table 5. Battery SOC estimation algorithm based on H ﬁlter algorithm. ∞ The Speciﬁc Calculation Process Is as Follows: ( xk = f (xk 1, uk 1) + ωk 1 Building a nonlinear system: − − − yk = g(xk, uk) + υk h i Initialization: xˆ = E(x ), P+ = E (x xˆ )(x xˆ )T 0 0 0 0 − 0 0 − 0 When k 1, 2, ... , , calculation ∈ { ∞ } First step: prediction stage + System status estimation: xˆ− = f (xˆ , uk 1) k k 1 − Error covariance prediction: P = A− P+ AT + Q k− k 1 k 1 k 1 k 1 Second step: update stage − − − − = ( ) Innovation matrix: ek yk g xˆk−, uk − 1 Gain matrix: K = A P (I θS P + CTR 1C P )− CTR 1 k k k− − k k− k k− k k− k k− xˆ+ = xˆ K e System state correction: k k− 1 k k + − 1 Error covariance correction: P = A P (I θS + CTR 1C P )− AT + Q k k k− − k k k− k k− k k

3.3. SOC Estimation Accuracy Verification After describing in detail the principle of the HIF and the specific calculation process, in order to verify the robustness of the algorithm under the three conditions of uncertain measurement accuracy, initial SOC value uncertainty, and uncertain application conditions, this section uses a file in MATLAB software to write the HIF and EKF running procedures, and carries out simulation operations in Simulink. Based on the obtained experimental data, accuracy verification and comparative analysis are conducted on the estimation results of the SOC of the lithium-ion battery using HIF and EKF. In the current literature, the reference values used for battery SOC estimation algorithm verification are all obtained by the ampere-hour integration method [34]. In order to eliminate the measurement error of the current sensor as far as possible, a high-precision battery testing equipment with a stability of 0.1% of FS is adopted, and the battery is fully placed before the test. Therefore, this study uses the approximate SOC calculation value obtained by the ampere-hour integration method as the real SOC value.

3.3.1. Uncertainty of Measurement Accuracy Owing to the limitation of sensor acquisition accuracy, there are some acquisition errors in the current and voltage data acquired during vehicle driving. At present, the acquisition error of the mainstream current sensor is within 1%, and the voltage acquisition accuracy is within 5 mV. In order ± Appl. Sci. 2020, 10, 6371 12 of 18

to fully ensure the reliability of the verification, this study evaluates the estimation effect of the HIF in the event of inaccurate measurement by adding colored noise interference to the current and voltage data. Figure9 shows that when colored noise is added to the current and voltage data, the SOC estimation results obtained by HIF and EKF have different degrees of deviation. Since the robustness of HIF algorithm is better than EKF, the SOC estimation results based on HIF are slightly better than that based on EKF. The estimation error of the former is within 0.02, while that of the latter is within Appl. Sci. 2020, 10, x FOR PEER REVIEW ± 13 of 18 0.04. ± 1 0.9 Measured data 0.8 EKF 0.7 H∞ 0.8 0.6 SOC 0.5 0.75 0.4 0.3 0.7 0 1500 2000 2500 3000 0.65 Time (s) 1000 1100 1200 1300 1400 1500 0.04 EKF 0.02 H∞ 0 Error -0.02

-0.04 0 500 1000 1500 2000 2500 3000 299 Time (s) 300 FigureFigure 9. 9.ComparisonComparison of ofSOC SOC estimation estimation results results and and erro errorsrs based based on on DST DST conditions conditions (colored (colored noise noise 301 addedadded to tocurrent current and and voltage). voltage).

302 3.3.2.3.3.2. Uncertainty Uncertainty of ofSOC SOC Initial Initial Value Value 303 InIn actual actual use, use, it it is didifficultﬃcult toto obtain obtain an an accurate accurate initial initial SOC SOC value. value. Therefore, Therefore, the algorithm the algorithm should 304 shouldhave thehave ability the ability to track to thetrack system the system state when state thewhen initial the SOCinitial value SOC is value inaccurate, is inaccurate, and correct and thecorrect initial 305 theSOC initial value. SOC To value. evaluate To evaluate the convergence the convergence ability of ability the HIF of underthe HIF uncertain under uncertain initial values, initial thevalues, initial 306 thevalue initial of value the SOC of the is set SOC to 0.7 is set (the to exact 0.7 (the initial exact value initial of the value SOC of is the 1), SOC and theis 1), results and the of its results convergence of its 307 convergencecurve and errorcurve are and shown error inare Figure shown 10 in. Figure 10. Figure 10 shows that in the case of the inaccurate initial SOC value, the SOC values estimated by the two algorithms1 can quickly converge to the accurate value after algorithm calculation. However, for the convergence0.9 rate, HIF needs approximately 1.5 s to make the SOCMeasured converge data to the initial value, which is 50%0.8 higher than the 3 s of the EKF algorithm. Moreover, the estimatedEKF error of HIF ﬂuctuates 0.7 H∞ within 0.02 after1 convergence, which proves that the algorithm is not sensitive to the setting of the ± 0.6 initial valueSOC 0.5 of SOC,0.9 can calculate the true value recursively, and has relatively better robustness. 0.4 0.3 0.8 00.7 1500 2000 2500 3000 0 10 20 Time (s)

0.3

0.2 EKF H∞ 0.1 Error 0

-0.1 0 500 1000 1500 2000 2500 3000 308 Time (s) 309 Figure 10. Comparison of SOC estimation results and errors under inaccurate SOC initial value 310 based on DST conditions.

311 Figure 10 shows that in the case of the inaccurate initial SOC value, the SOC values estimated 312 by the two algorithms can quickly converge to the accurate value after algorithm calculation. 313 However, for the convergence rate, HIF needs approximately 1.5 s to make the SOC converge to the 314 initial value, which is 50% higher than the 3 s of the EKF algorithm. Moreover, the estimated error of 315 HIF fluctuates within ±0.02 after convergence, which proves that the algorithm is not sensitive to the Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 18

1 0.9 Measured data 0.8 EKF 0.7 H∞ 0.8 0.6 SOC 0.5 0.75 0.4 0.3 0.7 0 1500 2000 2500 3000 0.65 Time (s) 1000 1100 1200 1300 1400 1500 0.04 EKF 0.02 H∞ 0 Error -0.02

-0.04 0 500 1000 1500 2000 2500 3000 299 Time (s) 300 Figure 9. Comparison of SOC estimation results and errors based on DST conditions (colored noise 301 added to current and voltage).

302 3.3.2. Uncertainty of SOC Initial Value 303 In actual use, it is difficult to obtain an accurate initial SOC value. Therefore, the algorithm 304 should have the ability to track the system state when the initial SOC value is inaccurate, and correct 305 the initial SOC value. To evaluate the convergence ability of the HIF under uncertain initial values, 306 Appl.the initial Sci. 2020 value, 10, 6371 of the SOC is set to 0.7 (the exact initial value of the SOC is 1), and the results13 of of its 18 307 convergence curve and error are shown in Figure 10.

1 0.9 Measured data 0.8 EKF 0.7 H∞ 0.6 1

SOC 0.5 0.9 0.4 0.3 0.8 00.7 1500 2000 2500 3000 0 10 20 Time (s)

0.3

0.2 EKF H∞ 0.1 Error 0

Appl. Sci. 2020, 10-0.1, x FOR PEER REVIEW 14 of 18 0 500 1000 1500 2000 2500 3000 316308 setting of the initial value of SOC, can calculate the trueTime value(s) recursively, and has relatively better 317309 robustness.Figure 10.10. ComparisonComparison of of SOC SOC estimation estimation results results and and errors errors under under inaccurate inaccurate SOC initialSOC initial value basedvalue 310 onbased DST on conditions. DST conditions. 318 3.3.3. Uncertainty of Application Conditions 3.3.3. Uncertainty of Application Conditions 319311 TheFigure applicability 10 shows ofthat the in HIFthe casealgorithm of the inaccurateunder different initial operatingSOC value, conditions the SOC valueswas evaluated estimated 320312 usingby the DST,The two applicability FUDS, algorithms and HPPC of thecan HIFconditions.quickl algorithmy converge It can under be to seen di thefferent fromaccurate operating Figures value 11 conditions andafter 12 algorithm that was under evaluated calculation. DST usingand 321313 FUDSDST,However, conditions, FUDS, for and the the HPPC convergence current conditions. and rate, voltage ItHIF can fluctuationsneeds be seen approx from areimately Figures relatively 1.5 11 s andsevere. to make 12 thatThe the undernoise SOC interference DSTconverge and FUDSto ofthe 322314 theconditions,initial battery value, model the which current in this is 50% case and higher is voltage obvious, than fluctuations the and 3 its noof the longer are EKF relatively exhibits algorithm. severe. the Moreover,characteristics The noise the interference estimatedof zero mean. error of In the of 323315 addition,batteryHIF fluctuates modelthe model within in thisparameters ±0.02 case after is change obvious, convergence, to anda large it which noextent longer proves with exhibits the that current the the algorithm characteristicsmutation. is Figuresnot sensitive of zero13 and mean.to 14 the 324 showIn addition, that the the EKF model algorithm parameters is changeused to to esti a largemate extent the battery with the SOC current under mutation. the DST Figures and 13 FUDS and 14 325 conditions.show that The the EKFsimulated algorithm output is used voltage to estimate of the batte the batteryry model SOC deviates under greatly the DST from and the FUDS real conditions. voltage 326 value,The simulated so that when output the voltage current of the pulse battery is large, model the deviates SOC greatlyestimation from accuracy the real voltage is not value, high, soand that 327 fluctuateswhen the within current ±0.04. pulse is large, the SOC estimation accuracy is not high, and fluctuates within 0.04. ± 60 4.2 Voltage Current

0 3.9 ) V ( e g

-60 3.6 Current (A) Current Volta

-120 3.3 0 500 1000 1500 2000 2500 3000 328 Time (s) 329 FigureFigure 11. 11. CurrentCurrent and and voltage voltage diagram diagram under under DST DST conditions. conditions.

150 5 Voltage Current 75 4.5

0 4 Current (A) Current -75 3.5 (V) Voltage

-150 3 0 500 1000 1500 2000 2500 3000 330 Time (s) 331 Figure 12. Current and voltage diagram under FUDS condition.

1 0.9 Measured data 0.8 EKF 0.7 H∞

0.6 0.8

SOC 0.5 0.4 0.3 0 500 1000 1500 2000 2500 3000 0.7 1000 1100 1200 1300 1400 1500 Time (s) 0.04 EKF 0.02 H∞ 0 Error -0.02

-0.04 0 500 1000 1500 2000 2500 3000 332 Time (s) Appl.Appl. Sci. Sci. 2020 2020, 10, 10, x, FORx FOR PEER PEER REVIEW REVIEW 1414 of of18 18

316316 settingsetting of of the the initial initial value value of of SOC, SOC, can can calculate calculate th the etrue true value value recursively, recursively, and and has has relatively relatively better better 317317 robustness.robustness.

318318 3.3.3.3.3.3. Uncertainty Uncertainty of of Application Application Conditions Conditions 319319 TheThe applicability applicability of of the the HIF HIF algorithm algorithm under under different different operating operating conditions conditions was was evaluated evaluated 320320 usingusing DST, DST, FUDS, FUDS, and and HPPC HPPC conditions. conditions. It Itcan can be be seen seen from from Figures Figures 11 11 and and 12 12 that that under under DST DST and and 321321 FUDSFUDS conditions, conditions, the the current current and and voltage voltage fluctuations fluctuations are are relatively relatively severe. severe. The The noise noise interference interference of of 322322 thethe battery battery model model in in this this case case is isobvious, obvious, and and it itno no longer longer exhibits exhibits the the characteristics characteristics of of zero zero mean. mean. In In 323323 addition,addition, the the model model parameters parameters change change to to a largea large ex extenttent with with the the current current mutation. mutation. Figures Figures 13 13 and and 14 14 324324 showshow that that the the EKF EKF algorithm algorithm is isused used to to esti estimatemate the the battery battery SOC SOC under under the the DST DST and and FUDS FUDS 325325 conditions.conditions. The The simulated simulated output output voltage voltage of of the the batte batteryry model model deviates deviates greatly greatly from from the the real real voltage voltage 326326 value,value, so so that that when when the the current current pulse pulse is islarge, large, the the SOC SOC estimation estimation accuracy accuracy is isnot not high, high, and and 327327 fluctuatesfluctuates within within ±0.04. ±0.04.

6060 4.24.2 VoltageVoltage CurrentCurrent )

0 3.9 )

0 3.9 V V ( ( e e g g -60 3.6 Current (A) Current -60 3.6 Volta Current (A) Current Volta

-120-120 3.33.3 0 0 500500 1000 1000 1500 1500 2000 2000 2500 2500 3000 3000 Time (s) 328328 Time (s) Appl. Sci. 2020, 10, 6371 14 of 18 329329 FigureFigure 11. 11. Current Current and and voltage voltage diagram diagram under under DST DST conditions. conditions.

150150 5 5 VoltageVoltage CurrentCurrent 7575 4.54.5

0 0 4 4 Current (A) Current Voltage (V) Voltage Current (A) Current -75-75 3.53.5 (V) Voltage

3 -150-150 3 0 0 500500 1000 1000 1500 1500 2000 2000 2500 2500 3000 3000 Time (s) 330330 Time (s) 331331 FigureFigureFigure 12. 12. 12. Current CurrentCurrent and and and voltage voltage voltage diagram diagram diagram under under under FUDS FUDS FUDS condition. condition. condition.

1 1 0.90.9 MeasuredMeasured data data 0.80.8 EKFEKF H∞ 0.70.7 H∞ 0.6 0.6 0.80.8 SOC SOC 0.50.5 0.40.4 0.30.3 0 0 500500 1000 1000 15001500 2000 2000 2500 2500 3000 3000 0.70.7 10001000 1100 1100 1200 1200 1300 1300 1400 1400 1500 1500 TimeTime (s) (s) 0.040.04 EKF 0.02 EKF 0.02 H∞H∞ 0 0 Error Error -0.02-0.02

-0.04-0.04 0 500 1000 1500 2000 2500 3000 Appl. Sci. 2020, 10, x FOR0 PEER REVIEW 500 1000 1500 2000 2500 3000 15 of 18 Time (s) 332332 Time (s) 333 Figure 13. Comparison of SOC estimation resultsresults and errors underunder DST conditions.

1 0.9 Measured data 0.8 EKF 0.7 H∞ 0.8 0.6

SOC 0.5 0.4 0.3 500 1000 00.7 1500 2000 2500 3000 1100 1200 1300 1400 1500 Time (s) 0.04 EKF 0.02 H∞ 0 Error -0.02

-0.04 0 500 1000 1500 2000 2500 3000 334 Time (s) 335 Figure 14. Comparison of SOC estimation resultsresults and errors under FUDS conditions.conditions.

336 ItIt cancan bebe seenseen fromfrom FiguresFigures 1515 andand 1616 thatthat thethe HPPCHPPC operatingoperating conditionsconditions areare constantconstant currentcurrent 337 pulsepulse discharges,discharges, and thethe currentcurrent ﬂuctuatesfluctuates periodically.periodically. The HPPC condition is relatively stable, 338 compared withwith thethe aboveabove twotwo conditions,conditions, soso thethe SOCSOC estimationestimation accuracyaccuracy isis relativelyrelatively high.high. The SOC 339 estimation errorerror based based on on the the EKF EKF can can be controlledbe controlled within within0.02, ±0.02, and theand SOC the estimationSOC estimation error basederror ± 340 based on the HIF can be controlled within ±0.01, which further verifies the applicability of the HIF 341 algorithm under different working conditions.

20 4.5 Voltage 10 Current 4

0 3.5

-10 3 Voltage (V) Voltage Current (A) -20 2.5

-30 2 0 5000 10000 15000 20000 25000 30000 342 Time (s) 343 Figure 15. Current–voltage diagram under HPPC conditions. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 18

333 Figure 13. Comparison of SOC estimation results and errors under DST conditions.

1 0.9 Measured data 0.8 EKF 0.7 H∞ 0.8 0.6

SOC 0.5 0.4 0.3 500 1000 00.7 1500 2000 2500 3000 1100 1200 1300 1400 1500 Time (s) 0.04 EKF 0.02 H∞ 0 Error -0.02

-0.04 0 500 1000 1500 2000 2500 3000 334 Time (s) 335 Figure 14. Comparison of SOC estimation results and errors under FUDS conditions.

336 It can be seen from Figures 15 and 16 that the HPPC operating conditions are constant current 337 pulseAppl. discharges, Sci. 2020, 10, 6371 and the current fluctuates periodically. The HPPC condition is relatively stable,15 of 18 338 compared with the above two conditions, so the SOC estimation accuracy is relatively high. The SOC 339 estimation error based on the EKF can be controlled within ±0.02, and the SOC estimation error on the HIF can be controlled within 0.01, which further veriﬁes the applicability of the HIF algorithm 340 based on the HIF can be controlled within± ±0.01, which further verifies the applicability of the HIF 341 algorithmunder di underﬀerent different working working conditions. conditions.

20 4.5 Voltage 10 Current 4

0 3.5

-10 3 Voltage (V) Voltage Current (A) -20 2.5

-30 2 0 5000 10000 15000 20000 25000 30000 342 Time (s) 343 Appl. Sci. 2020, 10, x FOR PEERFigureFigure REVIEW 15. 15. Current–voltage Current–voltage diagram diagram under under HPPC HPPC conditions. conditions. 16 of 18

1 0.8 Measured data EKF 0.6 H∞ 0.4 SOC 0.2

0 5000 10000 15000 20000 25000 30000 1550 1600 1650 1700 1750 1800 Time (s) 0.04 EKF 0.02 H∞ 0 Error -0.02

-0.04 0 5000 10000 15000 20000 25000 30000 344 Time (s) 345 FigureFigure 16. 16. ComparisonComparison of of SOC SOC estimation estimation result resultss and and errors errors under under HPPC HPPC conditions. conditions.

346 4.4. Conclusions Conclusions 347 AimingAiming at thethe SOCSOC estimation estimation problem problem of electricof electric vehicle vehicle power power lithium-ion lithium-ion batteries, batteries, this paper this 348 paperproposes proposes an SOC an estimationSOC estimation method method based based on the on HIF the algorithm.HIF algorithm. Firstly, Firstly, based based on theon the EIS EIS of the of 349 thelithium-ion lithium-ion battery, battery, a fractional-order a fractional-order model model based ba onsed the on dual the polarizationdual polarization equivalent equivalent circuit circuit model 350 modelwas established. was established. Without Without reducing redu thecing accuracy the accuracy of the model, of the it model, not only it not simplifies only simplifies the model the structure, model 351 structure,but also reduces but also the reduces amount the of calculation.amount of calculat Then, theion. parameters Then, the of parameters the fractional-order of the fractional-order battery model 352 batteryare identified model byare theidentified HPSO algorithmby the HPSO based algorithm on the geneticbased on crossover the gene factor.tic crossover It solves factor. the It problem solves 353 thethat problem the classic that PSO the classic algorithm PSO easily algorithm falls easily into the fall locals into best, the local improves best, improves the global the search global ability search of 354 abilitythe parameter of the parameter identification identification algorithm, andalgorithm, further and improves further the improves fitting degree the fitting of the batterydegree model.of the 355 batteryFinally, model. the accuracy Finally, of the the accuracy SOC estimation of the SOC results estimation of lithium-ion results of batteries lithium-ion using batteries HIF and using EKF HIF are 356 andverified EKF and are compared verified underand compared three conditions: under uncertainthree conditions: measurement uncertain accuracy, measurement uncertain SOCaccuracy, initial 357 uncertainvalue, and SOC uncertain initial value, application and uncertain conditions. application The simulation conditions. results The show simulation that the results SOC estimationshow that method based on HIF can ensure that the SOC estimation error value fluctuates within 0.02 in any 358 the SOC estimation method based on HIF can ensure that the SOC estimation error value± fluctuates 359 withincase, and ±0.02 is slightly in any a ffcase,ected and by environmentalis slightly affected factors by and environmental other factors. factors It provides and aother way tofactors. improve It 360 providesthe accuracy a way of SOCto improve estimation the accuracy in a battery of SO managementC estimation system. in a battery management system.

361 Author Contributions: Conceptualization: M.H.; methodology: M.H.; validation: Y.X.; formal analysis: L.L. and 362 Y.X.; investigation: L.L., G.J., and Z.L.; writing—original draft: L.L.; writing—review and editing: M.H. and 363 C.F.; supervision: M.H. and C.F.; project administration: G.J. and Z.L.; funding acquisition: M.H. and C.F. All 364 authors have read and agreed to the published version of the manuscript. 365 Funding: This work was supported by the National Key R&D Program of China [No. 2018YFB0106102], and the 366 Major Program of Chongqing Municipality [No. cstc2018jszx-cyztzxX0007]. 367 Conflicts of Interest: The authors declare no conflicts of interest. The funders had no role in the design of the 368 study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to 369 publish the results. 370 Abbreviations SOC state of charge HIF H-infinity filter HPSO hybrid particle swarm optimization EKF extended Kalman filter OCV open circuit voltage EIS electrochemical impedance spectroscopy SVM support vector machine NN neural network PF particle filter Appl. Sci. 2020, 10, 6371 16 of 18

Author Contributions: Conceptualization: M.H.; methodology: M.H.; validation: Y.X.; formal analysis: L.L. and Y.X.; investigation: L.L., G.J., and Z.L.; writing—original draft: L.L.; writing—review and editing: M.H. and C.F.; supervision: M.H. and C.F.; project administration: G.J. and Z.L.; funding acquisition: M.H. and C.F. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Key R&D Program of China [No. 2018YFB0106102], and the Major Program of Chongqing Municipality [No. cstc2018jszx-cyztzxX0007]. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Abbreviations

SOC state of charge HIF H-infinity filter HPSO hybrid particle swarm optimization EKF extended Kalman filter OCV open circuit voltage EIS electrochemical impedance spectroscopy SVM support vector machine NN neural network PF particle filter KF Kalman filter AUKF adaptive unscented Kalman filter AHIF adaptive H-infinity filter CPE constant phase element GA genetic algorithm DST dynamic stress test FUDS federal urban driving schedule HPPC hybrid pulse power characteristic RMSE root mean square error

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