Estimating the state of charge of lithium-ion batteries using different noise inputs

ABSTRACT


INTRODUCTION
With the growth of public demand and government support, technological development strongly encouraged the use of electric vehicles.Lithium-ion batteries (LIBs) are drawing more research interest due to their environmental friendliness, higher energy density, higher power output, and longer lifespan [1], [2].Vehicles that are powered by new energy sources are commonly equipped with LIBs.Therefore, obtaining an accurate battery state of charge (SOC) estimate becomes a challenging task for safe battery operation [3], [4].The estimation of the SOC inside the battery management system (BMS) has the potential to enhance both the reliability and efficiency of the system.However, estimating SOC is significantly influenced by complicated factors related to self-discharge, discharge current, and battery aging, which leads to an imprecise estimation of SOC [5].
Currently, several approaches have emerged to estimate battery SOC.In general, the ampere-hour know as AH method is commonly employed because of its ease of implementation [6].However, in practical application, this method is susceptible to errors caused by noise and random interference which tend to accumulate.As a result, various model-based algorithms have been proposed to correct those random errors.
The model-based methods provide a consistent performing method, like an equivalent circuit model, such as employing an equivalent circuit model in combination with state estimation computations.Among these methods, the Kalman filter is the commonly employed model.Nevertheless, the linear Kalman filter ISSN: 2088-8694  Estimating the state of charge of lithium-ion batteries using different noise inputs (Anas El Maliki)

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(LKF) [7], [8] an only be applied to simple linear systems.To address nonlinear system applications, recent research has led to the development of extensions to the Kalman filter.In particular, the extended Kalman filter (EKF) [9], unscented Kalman filter (UKF) [10], and cubature Kalman filter (CKF) [11].Utilizing the EKF can reduce the time to convergence in SOC estimation, yet it amplifies the computing burden on the battery management system (BMS) [12].Fang et al. [13] suggested an iterative EKF method to estimate the SOC, which has shown improvements.However, the algorithm's robustness is limited when identifying and updating parameters such as battery capacity or internal resistance.Xiong et al. [14] employed an EKF approach to assess the SOC in a vanadium redox flow battery.The method uses measurement of applied currents and terminal voltages to predict the SOC, achieving a maximum error of 5.5% in SOC estimation.Sun et al. [15] and Tian et al. [16] proposed a novel approach to improve the accuracy of SOC estimation by integrating the first-order resistor-capacitor (RC) equivalent model with the EKF.They reported that this approach reduced the root mean square error (RMSE) of SOC estimation by 43.34 while only slightly increasing computational time by 4.59%.
The assessment of the EKF performance depends roughly on the key parameters Q and R, indeed Wang et al. [17] and Zhao et al. [18] show that picking these values provide a big challenge as the determination of the noise remains random and difficult.However, these values impact significantly on the estimation error and the convergence rate of the EKF process.The major contribution of this paper is to evaluate how the Q and R matrices influence the EKF estimation.
In this work, we proposed a method for estimating SOC using an extended Kalman filter in combination with the Thevenin battery model.Additionally, we present and discuss the implementation of the EKF using the MATLAB software.Moreover, we utilize the new European driving cycle (NEDC) [19] as a load profile for online SOC estimation.The organization of this paper is as follows: i) Section 2 provides details of the mathematical modeling of the Lithium-ion battery and its parameter identification method used; ii) Followed by a demonstration of the state-of-charge estimation method employed; iii) Section 3 presents the proposed method deployed on a MATLAB Simulink program, along with a discussion of the simulation data; iv) In section 4, we discuss the results obtained from the previous section; and v) Section 5 presents the conclusion.

MATHEMATICAL MODEL 2.1. Lithium-ion battery model
The most common battery models are electrochemical models (EM) and equivalent circuit models (ECM).An electrochemical model can be used to characterize external characteristics as well as to simulate changes in distribution and internal characteristics.Any physical meaning that can be attributed to a process can be represented by these changes.Nonetheless, since the electrochemical parameters and partial differential equations require considerable amounts of computation, electrochemical models are not commonly used in practice to assess the reliability of estimates of SOC.Equivalent circuit models, on the other hand, are more widely used for this purpose.
An equivalent circuit model represents a battery's external properties using hardware circuit elements like capacitors, resistors, and current loads.Due to their ease of use, these models are widely used to simulate battery performance.One of the most popular equivalent circuit models used is Thevenin model.Thevenin model shown in Figure 1 consists of voltage source a parallel RC circuit and a rint model.The main concept behind using Thevenin model is to characterize the battery's behavior by representing it with an equivalent resistance in series and a voltage source.The Thevenin model allows for better characterization of the dynamic properties of the battery compared to simpler models like rint model.

Parameter identification
In this subsection, the hybrid pulse power characterization (HPPC) test and the recursive least square with forgetting factor (FFRLS) algorithms are conceived for the estimation of online battery parameters.The HPPC test is commonly used to determine the OCV-SOC relation.However, the test also provides a way to identify the parameter values from the ECM model which is derived from the offline parameter identification method [20], [21].For performing the HPPC test, the outlined steps in Table 1, must be followed.

Table 1. HPPC test steps
Step The test steps Step 1 Keep the battery cell in a temperature control at 25 °C for four hours Step 2 Load the cell with a constant current 1 C up to 4.2 V, then switch to a constant voltage 4.2 V until the current ≤ 0.05 C Step 3 Give the cell a one hour rest Step 4 Unload the cell with a current 1 C until 90% SOC.Wait until 1 hour, then discharge the cell at a current of 3 C for 10 s, put the cell to rest for 30s, then load it with a current 2.25 C for 10 s Step 5 Now perform the same steps 4) for various SOC (80%, 70%..., 10%) As a result, we adopt a sextic polynomial as a fit to the relation.Using  0 ~6 as a constants, as (2).
On the other hand, the FFRLS is employed to determine the battery parameters.Generally, recursive least square (RLS) algorithms are derived from the least square (LS) algorithm, and the basic principle is given by (3).Based on the Thevenin model described in Figure 1, the transfer function of the battery impedance is given by the following electrical equation with respect to the Laplace domain as (3).
A bilinear transformation is used to map as (3) to the Z plane.The transformation is represented as (4).
The model parameters can be obtained as (5).
Estimating the state of charge of lithium-ion batteries using different noise inputs (Anas El Maliki) 11

State of charge estimation methods by EKF
As a standard method of estimating the battery's system of cells, we commonly use the EKF method.Due to its model of a nonlinear system, its approximation can accurately estimate system states across a wide range of operations [22].Using the EKF, you get first-order polynomial accuracy where both quadratic and higher order terms are discarded.In addition to improving the algorithm's ability to handle nonlinear systems, EKF also adds flexibility to the algorithm, further improvements are needed to handle complex state monitoring problems in practical applications such as Li-ion batteries.
In the scenario of estimating SOC with an extended Kalman filter, the linear equation for the statespace model is requested at every time point near the latest estimation of SOC.To implement the EKF equations, the battery model must first be obtained.The input signal in this case is the charge/discharge current while the output is the voltage of the battery.The discrete state-space battery model equation for a nonlinear system can be as (7).
The state along with the output, is predicted via nonlinear battery models.At time step , the nonlinear battery model is linearized through the predicted state  ̂ − to obtain the matrices  ̂,  ̂, and  ̂.These matrices are used when calculating and updating the covariance matrix of the state estimation errors and Kalman gain.This process leads to the main purpose of predicting  ̂ − and   − , − =  −1   +  (10) and the update process is as (11), (12), and (13).
̂ + =  ̂ − +   (  −  ̂ − − ) Where   represents the Kalman gain vector, R is the covariance matrix of the zero-mean Gaussian measurement noise   , Q is the covariance matrix of the zero-mean Gaussian process noise   , and   is the covariance matrix of the state estimation error.

METHOD
The proposed model consists of three parts, as illustrated in Figure 2. The first part is the data input, where the input data is initialized.The FFRLS algorithm and the HPPC test are considered to determine the initial values for variables  0 to  6 and ( 0 ,   ,   ) in the system.These values play a crucial role in predicting the load state.Moving on to the second part, known as the Thevenin model, it involves two substeps.In step A, the model estimates certain parameters.Then, in step B, the SOC-OCV calculation is performed, which estimates the terminal voltage defined by (1).The third part is the SOC estimation, where the EKF algorithm is utilized.This estimation process comprises two steps.First, the model calculates the predicted state and current state.Next, the process incorporates initial noises Q and R as inputs.Finally, the state filter determines the resulting estimated SOC value.Notably, the proposed MATLAB model's current profile is derived from the new European driving cycle (NEDC) [23].These three interconnected parts work together to predict and estimate various parameters and the state of charge in the system under study.The NEDC current profile illustrated in Figure 3 has been used to construct the simulation of the battery using the proposed model.Based on the key parameters on both Tables 2 and 3, Figure 4 presents Thevenin model estimation of terminal voltage.Additionally, Figure 5 demonstrates the empirical model for estimating the state of charge.The HPPC test is conducted to gain the connection between OCV and SOC.The results obtained are fitted via a sixth-order polynomial with  0~6 shown at Table 2 [24].The system parameter identification is employed to obtain the parameter of the model on the basis of the SOC-OCV curve.Here the result of the parameters based on the FFRLS function is given in Table 3 [25].
Although the EKF estimation is adopted to estimate the SOC, their initial values of the Kalman parameters are determined as (14).
In accordance with (14), those values can be considered inputs to the EKF algorithm, as can be seen in Figure 6, which shows a comparison between the AH empirical method and the EKF SOC estimation of the NEDC test profile load for an electric vehicle, as well as Figure 7, which shows the respective errors of the methods.the proposed model, SOC results are influenced by three key parameters: i) Qa, ii) Qb, and iii) R. Additionally, two other parameters X and Y have a direct effect on these three key parameters.This relationship can be expressed as   =  − , and similarly for Qb and R, and vice versa.

Case 1: variation of Qa and Qb input noise parameters
In this case, the SOC was estimated using a parametric study.The R parameter is kept constant with a value of  = 2 −1 .To vary the input noise parameters, we use   =  − and   =  − , where X and Y are subparameters that can be adjusted independently.The simulation of the EKF estimation using these parameters was performed under the NEDC load profile.The research aimed to evaluate the performance of the conventional EKF algorithm in estimating SOC for a specific system.The obtained results were presented in Figures 8 and 9, using the input values specified in Table 4.The analysis included comparisons between the estimated SOC values, as well as the corresponding errors introduced by the EKF algorithm.
To investigate the impact of the Y factor we conducted a comprehensive analysis by varying its values from 1 to 6.The results of this study were quite promising, as we observed in both Figures 8 and 9 a remarkable reduction in the maximum SOC estimation error.Initially recorded at the error decreased significantly to 0.8971% as we made adjustments to the Y factor.This improvement in accuracy was further validated by a corresponding reduction in the RMSE, which dropped from 0.0075 to 0.0068.These findings demonstrate a substantial enhancement in the SOC estimation precision, underscoring the significance of the Y factor in refining the algorithm's performance.

Case 2: variation of Qa input noise parameters
In this study, we estimation the SOC using a parametric study.The objective was to systematically investigate the system's behavior over a range of input parameters.For this example, we kept the parameters   = 5 −3 and  = 2 −1 constant, while we varied the parameters X and Y of   according to   =  − .Based on the input values specified in Table 5, we simulated the EKF estimation under the NEDS load profile.Figures 10 and 11 present the results obtained.
In this study, Figure 10 shows the measured and estimated SOC based on the input values provided in Table 5.Additionally, Figure 11 show the corresponding SOC errors.By comparing these results with other algorithms, it is evident that this algorithm produces a large error when   = 2 −1 .However, by varying the Y factor between 1 and 4, we found that the maximum SOC estimation error was reduced from 1.8826% to 0.8856%, and the RMSE was reduced from 0.0140 to 0.0068.

Case 3: variation of Qb input noise parameters
Through a parametric study, an estimation of SOC was performed.In the following example, the   = 2 −8 and  = 2 −1 parameter are kept constant, while X and Y sub parameters of   =  − is varied.Using the parameters specified in Table 6, we simulated EKF estimation under the NEDS load profile.The results obtained are presented in Figures 12 and 13.According to Table 6, Figures 12 and 13 shows the measured and estimated SOC, along with the errors resulting from the conventional EKF algorithm.With   = 5 −1 , this algorithm induces a very small error in comparison with other EKF algorithms.As a result of the Y factor being varied from 6 to 1, the maximum SOC estimation error has decreased from 0.8970% to 0.4554%, while the RMSE has decreased from 0.0068 to 0.0030.

Case 4: variation of R input noise parameters
A parametric study was conducted to estimate the SOC using the EKF method.In the following example, we kept the parameters   = 2 −8 and   = 5 −1 constant while varying the sub-parameters X and Y of  =  − .These parameters were used as inputs in Table 7 to simulate the EKF estimation under a specific load profile known as the NEDS load profile.The results are shown in Figures 14 and 15.
The provided data in Table 7 serves as input values for the EKF algorithm, which is then illustrated in Figures 14 and 15.These figures represent both the measured and estimated SOC values, as well as the conventional EKF errors used for system evaluation.In comparison to other algorithms, this particular EKF algorithm shows a higher error of 0.5652% for  = 5 −1 .However, the error reduces significantly when the X factor is varied between 1 and 7. Specifically, the maximum error associated with SOC estimation decreases from 0.5051% to 0.3082%, and the RMSE reduces from 0.0034 to 0.0020.This improvement indicates the effectiveness of the EKF algorithm as the X factor is adjusted.

CONCLUSION
To enhance the SOC estimation method's accuracy, we utilized the Thevenin model with an extended Kalman filter.Additionally, we introduced the new European driving cycle (NEDC) for testing purposes.The simulated data results demonstrate that our proposed model can predict SOC with a root mean square error (RMSE) of approximately 0.68%.During the implementation of the EKF algorithm, During the implementation of the EKF algorithm, we observed that the initial noise values of both the process covariance matrix Q and the measurement noise covariance matrix R significantly affect the state estimation process.To analyze this effect, we varied the initial noise matrices (Q and R) in MATLAB/Simulink.The comparative results indicate that the SOC estimation accuracy was notably improved, reducing the maximum SOC estimation error from 1.8826% to 0.3082%, and the RMSE from 0.0140 (1.4%) to 0.0020 (0.2%).

Figure 2 .
Figure 2. Flowchart of the proposed method

Figure 8 .Figure 9 .
Figure 8. SOC estimation results under varied subparameters X and Y of Qa & Qb

Figure 10 .
Figure 10.SOC estimation results under varied subparameters X and Y of Qa Figure 11.SOC estimation error results under varied sub-parameters X and Y of Qa

Table 2 .
OCV-SOC fitting results at 25 °C 3. New European driving cycle (NEDC) load profile Int J Pow Elec & Dri Syst ISSN: 2088-8694  Estimating the state of charge of lithium-ion batteries using different noise inputs (Anas El Maliki) 13

Table 3 .
Model parameters at 25 °C

Table 4 .
RMSE of SOC estimation under various Qa and Qb sub-parameters

Table 5 .
RMSE of SOC estimation under different Qa sub-parameters

Table 7 .
RMSE of SOC estimation under different R sub-parameters