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

: 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 a ﬀ ected by environmental and other factors. It provides a way to improve the accuracy of SOC estimation in a battery management system.


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.

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In this circuit, the impedance of the fractional-order element can be described by the following 125 equation: In this circuit, the impedance of the fractional-order element can be described by the following equation: Ws γ (1) where, C 1 , C 2 , and W indicate the parameters of fractional-order elements; α and β represent the fractional order of CPE 1 and CPE 2 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: The fractional differential equation of the Warburg element is shown in Equation (4): According to the definition, the system state of the SOC as a fractional-order model is presented in Equation (5) as follows: where C n 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 where U ocv 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:

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 Figure 2. The test object is a universal A123, ternary lithium-ion soft packet battery cell. The main parameters are listed in Table 1.  (8), as shown in Figure  Figure 2. Software and hardware system for battery test experiment. In order to identify the model parameters, a series of experiments on lithium-ion batteries were conducted with reference to relevant national standards and the USABC Electric Vehicle Battery Test Procedures Manual. The battery test flow chart of the relevant experiments is shown in Figure 3.  Figure 3. Battery test flow chart. 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 Figure 4.

Model Parameter Identification Based on HPSO
The parameter identification link is very important before the model is successfully applied. The parameter identification results directly affect the accuracy of the model. Because of the obvious time-varying non-linear and complex structure of the lithium-ion battery system, there are many parameters involved in operation, and most parameter values cannot be directly measured. Therefore, parameter identification is a difficult problem in the battery modeling process. For high-dimensional complex optimization problems, this paper proposes an HPSO algorithm based on a genetic cross factor. The flowchart of the algorithm is 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. enter the next generation of updated particle swarm so as to realize the update of the particle swarm.

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The identification results obtained using the classical PSO algorithm are shown in Table 2.
190 Figure 5. Flow chart of hybrid particle swarm filtering algorithm.

Fractional-Order Model Accuracy Analysis
In this selection, the dynamic stress test (DST) condition test data of the universal A123 battery at 25 • C ambient temperature is used as training data to identify the parameters of the established fractional-order model. Through different identification methods and different simulation conditions, the accuracy and robustness of the fractional-order model are analyzed.
The identification results obtained using the classical PSO algorithm are shown in Table 2. Table 2. Parameter recognition results of ordinary particle swarm algorithm. The identification results obtained using the HPSO algorithm are shown in Table 3. Table 3. Parameter recognition results of hybrid particle swarm algorithm. According to the identification results of the above two algorithms, the model accuracy is simulated and verified under the three working conditions of the dynamic stress test (DST), federal urban driving schedule (FUDS), and hybrid pulse power characteristic (HPPC). The simulation results are shown in Figures 6-8. It can be seen from Figures 6-8 that, under different operating conditions, the voltage error value output by the fractional-order model fluctuates slightly around zero. When the input current has a large mutation, the voltage error value is large, but can also be controlled at approximately 0.05 V. Therefore, the improved fractional-order model is more accurate and suitable for different working conditions. The identification results obtained using the HPSO algorithm are shown in Table 3. 192 Table 3. Parameter recognition results of hybrid particle swarm algorithm.   conditions. It can be seen from Table 4 that the parameter identification method based on the HPSO 213 algorithm further improved the accuracy of the fractional model.   Table 4 shows the root mean square error (RMSE) value of the model output voltage corresponding to the classical PSO algorithm and the HPSO algorithm under different working conditions. It can be seen from Table 4 that the parameter identification method based on the HPSO algorithm further improved the accuracy of the fractional model.

SOC Estimation Based on the HIF Algorithm
The Kalman series of filtering algorithms have been supported, widely researched and applied owing to their high accuracy, low computational complexity, and good robustness. However, they have two limitations that are difficult to overcome: (1) It is necessary to ensure that the input noise is white noise, and the statistical characteristics of the noise are approximately known. In practice, the noise 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.

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: x k+1 = A k x k + B k u k + ω k (9) 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 x k = [V CPE 1 ,k V CPE 2 ,k V W,k SOC k ]. The input of the system is u = I k , which is the current state of the battery. The output is y = U d,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 A k , B k , C k , and D k are defined as follows: 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 ω k and υ k as much as possible. With improvements in the accuracy of the system state estimation, the SOC estimation achieves the desired effect.

SOC Estimation Based on HIF Algorithm
To judge the accuracy of the estimated object x k , the HIF algorithm defines a cost function J: The matrices P 0 , Q k , R k , and S k 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: To solve the above optimization process, the algorithm designer must find a suitable x k 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 x 0 , noise ω k , and υ k , so as to maximize the cost function J.
Therefore, this problem becomes a maximum-minimum problem, that is, when ω k , υ k , and x 0 make the cost function J maximum, the appropriate x k should be selected to minimize the cost function 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: By solving the maximum-minimum problem of the cost function, a recursive relation satisfying Equation (12) can be obtained, as shown below: 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 Table 5. Table 5. Battery SOC estimation algorithm based on H∞ filter algorithm.

The Specific Calculation Process Is as Follows:
Building a nonlinear system: System state correction:x + k =x − k−1 K k e k Error covariance correction:

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.

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 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. Figure 9 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 ±0.04.

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In actual use, it is difficult to obtain an accurate initial SOC value. Therefore, the algorithm 303 should have the ability to track the system state when the initial SOC value is inaccurate, and correct 304 the initial SOC value. To evaluate the convergence ability of the HIF under uncertain initial values, 305 the initial value of the SOC is set to 0.7 (the exact initial value of the SOC is 1), and the results of its 306 convergence curve and error are shown in Figure 10.   Figure 10 shows that in the case of the inaccurate initial SOC value, the SOC values estimated

Uncertainty of SOC Initial Value
In actual use, it is difficult to obtain an accurate initial SOC value. Therefore, the algorithm should have the ability to track the system state when the initial SOC value is inaccurate, and correct the initial SOC value. To evaluate the convergence ability of the HIF under uncertain initial values, the initial value of the SOC is set to 0.7 (the exact initial value of the SOC is 1), and the results of its convergence curve and error are shown in Figure 10. Figure 10 shows that in the case of the inaccurate initial SOC value, the SOC values estimated by the two algorithms can quickly converge to the accurate value after algorithm calculation. However, for the convergence rate, HIF needs approximately 1.5 s to make the SOC converge to the initial value, which is 50% higher than the 3 s of the EKF algorithm. Moreover, the estimated error of HIF fluctuates within ±0.02 after convergence, which proves that the algorithm is not sensitive to the setting of the initial value of SOC, can calculate the true value recursively, and has relatively better robustness.
should have the ability to track the system state when the initial SOC value is inaccurate, and correct 304 the initial SOC value. To evaluate the convergence ability of the HIF under uncertain initial values, 305 the initial value of the SOC is set to 0.7 (the exact initial value of the SOC is 1), and the results of its 306 convergence curve and error are shown in Figure 10.

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However, for the convergence rate, HIF needs approximately 1.5 s to make the SOC converge to the 313 initial value, which is 50% higher than the 3 s of the EKF algorithm. Moreover, the estimated error of 314 HIF fluctuates within ±0.02 after convergence, which proves that the algorithm is not sensitive to the 315 Figure 10. Comparison of SOC estimation results and errors under inaccurate SOC initial value based on DST conditions.

Uncertainty of Application Conditions
The applicability of the HIF algorithm under different operating conditions was evaluated using DST, FUDS, and HPPC conditions. It can be seen from Figures 11 and 12 that under DST and FUDS conditions, the current and voltage fluctuations are relatively severe. The noise interference of the battery model in this case is obvious, and it no longer exhibits the characteristics of zero mean. In addition, the model parameters change to a large extent with the current mutation. Figures 13 and 14 show that the EKF algorithm is used to estimate the battery SOC under the DST and FUDS conditions. The simulated output voltage of the battery model deviates greatly from the real voltage value, so that when the current pulse is large, the SOC estimation accuracy is not high, and fluctuates within ±0.04.

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It can be seen from Figures 15 and 16    It can be seen from Figures 15 and 16 that the HPPC operating conditions are constant current pulse discharges, and the current fluctuates periodically. The HPPC condition is relatively stable, compared with the above two conditions, so the SOC estimation accuracy is relatively high. The SOC estimation error based on the EKF can be controlled within ±0.02, and the SOC estimation error based on the HIF can be controlled within ±0.01, which further verifies the applicability of the HIF algorithm under different working conditions.

Conclusions
Aiming at the SOC estimation problem of electric vehicle power lithium-ion batteries, this paper proposes an SOC estimation method based on the HIF algorithm. Firstly, based on the EIS of the lithium-ion battery, a fractional-order model based on the dual polarization equivalent circuit model was established. Without reducing the accuracy of the model, it not only simplifies the model structure, but also reduces the amount of calculation. Then, the parameters of the fractional-order battery model are identified by the HPSO algorithm based on the genetic crossover factor. It solves the problem that the classic PSO algorithm easily falls into the local best, improves the global search ability of the parameter identification algorithm, and further improves the fitting degree of the battery model. Finally, the accuracy of the SOC estimation results of lithium-ion batteries using HIF and EKF 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 factors and other factors. It provides a way to improve the accuracy of SOC estimation in a battery management system.