On-Line Parameter Identification and SOC Estimation for Lithium-Ion Batteries Based on Improved Sage–Husa Adaptive EKF

Abstract

For the Battery Management System (BMS) to manage and control the battery, State of Charge (SOC) is an important battery performance indicator. In order to identify the parameters of the LiFePO₄ battery, this paper employs the forgetting factor recursive least squares (FFRLS) method, which considers the computational volume and model correctness, to determine the parameters of the LiFePO₄ battery. On this basis, the two resistor-capacitor equivalent circuit model is selected for estimating the SOC of the Li-ion battery by combining the extended Kalman filter (EKF) with the Sage–Husa adaptive algorithm. The positivity is improved by modifying the system noise estimation matrix. The paper concludes with a MATLAB 2016B simulation, which serves to validate the SOC estimation algorithm. The results demonstrate that, in comparison to the conventional EKF, the enhanced EKF exhibits superior estimation precision and resilience to interference, along with enhanced convergence during the estimation process.

1. Introduction

In recent decades, environmental issues and the energy crisis have become increasingly prominent, prompting a gradual realization of the necessity for developing new energy sources for human life. This has driven research and development of new energy sources on a global scale. The growth of electric vehicles (EVs) can be attributed to the influence of these two factors. As illustrated in Figure 1 [1], by 2023, there will be around 14 million new EV registrations worldwide, bringing the total number of EVs on the road to 40 million. The EV industry is growing exponentially. Battery electric vehicles will account for 70% of the total EV fleet. These patterns show that even though EV markets are maturing, growth is still strong. Lithium-ion battery storage technology is critical to driving EVs.

LIBs possess several advantageous characteristics, including higher energy density, extended cycle life, superior environmental compatibility, and recyclability. Consequently, they are widely regarded as the optimal energy storage solution for electric vehicles (EVs). The estimation of the state of charge (SOC) in battery management systems (BMS) is considered to be one of the most vital and important factors in this field of study and has been extensively researched in recent years [2]. However, because batteries and electrochemical reactions are nonlinear and time-varying, it is not possible to measure SOC directly. In addition, the variety of factors that affect battery performance makes SOC estimation challenging [3]. The accurate estimation of the SOC provides valuable information regarding the remaining battery power and also assesses the reliability of the battery.

The commonly used SOC estimation methods can be mainly categorized into direct, data-driven method, observer-based, and model-based [4]. Direct measurement methods include the Coulomb counting method (CCM) [5], the open-circuit voltage-based SOC estimation method [6], and the Electrochemical impedance spectroscopy (EIS) method [7]. However, these methods are rarely applied on a large scale due to the long testing time and high testing accuracy requirements. Data-based SOC estimation models mainly include neural network methods, support vector machine (SVM) methods, etc. [8,9]. This approach requires the use of a large dataset to obtain the relationship between SOC and measured variables such as battery terminal voltage, operating current, external temperature, etc. to achieve an estimate of the battery SOC. However, the training data affects the accuracy of this method’s estimates. Observer-based techniques minimize the difference between the observed and actual states to estimate the battery state using a closed-loop feedback system. Observer-based techniques such as the Luhnberg observer, sliding mode observer, nonlinear observer, etc. are commonly used. However, improper controller settings for this method can lead to an inaccurate SOC estimation, and selecting the appropriate gain matrix is complicated. Because of its ease of use and computational efficiency, the equivalent circuit model (ECM) is often used in the model-based technique to generate state–space equations [10].Variations in operating temperature and battery age can affect the ECM characteristics, which include the OCV curve, capacity, internal resistance, polarization resistance, and polarization capacitance. The ECM is capable of more accurately simulating the dynamic and static characteristics of a battery under actual operating conditions with relatively simple components, including resistors, capacitors, and inductors. The nRC model consists of n parallel RC branches and a series of internal resistors (R) used to simulate transient reactions with different time constants related to diffusion, voltage source $Uoc$, and LIB charge transfer [11]. Observer-based methods include sliding mode observers, Luenberger observers, proportional-integral (PI) controllers, and the well-known KF (Kalman Filter) family of algorithms [12]. A practical and theoretical tool for optimizing the estimation of a system’s state variables is the KF algorithm. Through iterative computations, it recursively estimates and updates the state of the system using the state–space equations and input–output data. Several extensions have been developed for nonlinear systems, although the basic EKF is only applicable to linear systems. These include the extended Kalman filter (EKF) [13], the adaptive extended Kalman filter (AEKF) [14], the unscented Kalman filter (UKF) [15], and other algorithm variants [16]. As a consequence of the advancement of BMS hardware technology, the EKF algorithm is becoming increasingly prevalent in the estimation of SOC in actual vehicles. This is because it is capable of rapidly converging on the true value of SOC when an erroneous initial value is provided. This is because the Li-ion battery is a classic example of a nonlinear system, and the EKF method is often used to solve nonlinear system problems. The EKF method is ideal for predicting the SOC of EVs batteries, which have significant current variations, as it provides a fair compromise between accuracy and complexity. However, all algorithms have some limitations. Error analysis of EKF-based SOC estimation can help understand its limitations and further improve the robustness of the algorithm [17].

While an increasing number of studies have used the EKF method of SOC estimation to address the nonlinearities in battery modeling [18]. However, ambient noise affects the accuracy of SOC estimation in real-world applications, making it difficult to distinguish between system and observation noise, leading to inconsistent estimation results. Numerous adaptive filtering techniques that can estimate noise have been extensively studied in battery SOC estimation to overcome this limitation [19]. The Sage–Husa adaptive algorithm is an adaptive filtering method that reduces the variance of the filter. It estimates and corrects the measurement and process noise of the system in real time, improving the accuracy of the filter within certain limits [20]. However, it has the potential to lead to computational overload and filter scattering in the case of continuous noise growth. In the case of sudden current fluctuations, the Sage–Husa adaptive algorithm has some limitations and can misrepresent the positive features of the covariance matrix, leading to significant deviations between the estimated state and associated covariance matrix and the true values. One of the reasons for this phenomenon is that the observation noise R and the system noise Q lose their positive features during the iterative computation process. Consequently, a better approach to correcting the covariance matrix of system noise and observation noise is proposed for the current experiments. This work uses the multiplication and inverse operations of the matrix for the noise covariance matrix and optimizes the subtraction operation in the R and Q correction estimation. This reduces the complexity of the covariance matrix to a certain extent, reduces the computational load of the algorithm, accordingly, increases the computational speed and improves the accuracy of the SOC estimation.

2. Battery Model and Parameter Identification

2.1. Battery Mode

An Equivalent Circuit Model (ECM) is a circuit model consisting of an ideal resistor or capacitor. By modeling a battery as a resistor–capacitor (RC) circuit with different time constants, the ECM can be used to describe the voltage changes of a battery during charging and discharging. In an ECM, the voltage response of a battery on different time scales can be accurately modeled by connecting RC elements with different time constants in series. Each RC element represents the resistance and capacitance characteristics within the battery. ECM offers a straightforward calculation process and parameters with well-defined physical significance. This makes it a more suitable choice for integration into BMS than other models [21]. In order to maintain a balance between the complexity and accuracy of the battery model, a 2RC model is used in this paper, which consists of 2RC loops describing the charge transfer and polarization effects of the battery. The model is shown in Figure 2. This model has higher accuracy in online SOC estimation compared to other models, and the parameter identification is also more convenient and accurate than other models.

Figure 2 shows that $I$ is the battery loop current, $U\mathrm{o}\mathrm{c}$ is the battery open-circuit voltage and $U_t$ is the battery terminal voltage. $R_0$ is the ohmic internal resistance of the battery, $R_1$, $R_1$ and $C_1$, $C_2$ are the two polarization resistances and polarization capacitances of the battery. The functional equation of the model can be obtained according to Kirchhoff’s law as follows:

$$\left\{\begin{array}{l}{U}_{\mathrm{o}\mathrm{c}}=I\left(t\right)R_0+U_1+U_2+U_{\mathrm{t}}\\ I\left(t\right)={\displaystyle \frac{U_1}{R_1}}+C_1{\displaystyle \frac{\mathrm{d}{U}_1}{\mathrm{d}t}}\\ I\left(t\right)={\displaystyle \frac{U_2}{R_2}}+C_2{\displaystyle \frac{\mathrm{d}{U}_2}{\mathrm{d}t}}\end{array}\right.$$

(1)

2.2. On-Line Parameter Identification of Battery Model

A LiFePO₄ battery with a nominal capacity of 200 Ah was used in this study, and its specifications are listed in

Table 1. Battery parameters and test equipment.

Item	Values
Capacity	200 Ah
Charge/discharge cut-off voltage	3.65 V/2.5 V
Range of temperature	−20 °C–45 °C
Cycle life	>3000
Battery test system	Chroma 17010

. In general, the parameters of the ECM can be obtained by fitting the data of the polarization change process at different SOC. In this paper, data for parameter identification are obtained using a hybrid pulse power characterization (HPPC) test, and parameter identification is performed using the recursive least squares (RLS) method. The HPPC test protocol is shown in

Table 2. HPPC test procedure.

Step Number	Status
1	Charge the battery to 100% SOC by the CC-CV method
2	Resting for 4 h
3	Discharge at 1.5 C for 30 s
4	Resting for 60 s
5	Charge at 1.2 C for 30 s
6	Resting for 15 min
7	Discharge 10% SOC at 1/3 C
8	Resting for 4 h
9	From step 3 to step 8, cycle until the test is complete

and Figure 3 shows the voltage and current profiles obtained throughout the test.

The RLS method is a commonly used static system parameter estimation technique that performs parameter estimation by reducing the sum of squared errors between the estimates and the observed data. However, as the amount of data increases during the recursive process, the problem of data saturation may occur, which affects the algorithm’s performance in tracking rapidly changing parameters. To solve the data saturation problem in RLS, the forgetting factor recursive least square (FFRLS) online identification technique can be used. The effect of old data is reduced by introducing a forgetting factor to better accommodate rapidly changing parameters. This online identification technique based on real-time data is able to continuously update the ground model parameters and remain sensitive to the dynamic changes in the system, thus improving the accuracy and robustness of the identification results [22].

The following are the primary steps in parameter identification using FFRLS:

(1)

Equation (1) is Laplace transformed to obtain the frequency domain expression.

$$G\left(s\right)={\displaystyle \frac{U_{\mathrm{oc}}\left(s\right)-U_{\mathrm{t}}\left(s\right)}{I\left(s\right)}}=R_0+{\displaystyle \frac{R_1}{1+R_1{C}_{1}s}}+{\displaystyle \frac{R_2}{1+R_2{C}_{2}s}}$$

(2)

$$\left\{\begin{array}{l}U\left(s\right)=U_{\mathrm{oc}}\left(s\right)-U_{\mathrm{t}}\left(s\right)\\ {\tau }_1=R_1{C}_1\\ {\tau }_2=R_2{C}_2\end{array}\right.$$

(3)

(2)

Combining Equation (2) with Equation (3), the transfer function is given by

$$G(s)={\displaystyle \frac{U\left(s\right)}{I\left(s\right)}}={\displaystyle \frac{R_0{\tau }_1{\tau }_2{s}^2+\left(R_0{\tau }_1+R_0{\tau }_2+R_1{\tau }_2+R_2{\tau }_1\right)s+R_0+R_1+R_2}{{\tau }_1{\tau }_2{s}^2+\left({\tau }_1+{\tau }_2\right)s+1}}$$

(4)

Let:

$$\left\{\begin{array}{l}a=R_0\\ b={\tau }_1{\tau }_2\\ c={\tau }_1+{\tau }_2\\ d=R_0+R_1+R_2\\ e=R_0{\tau }_1+R_0{\tau }_2+R_1{\tau }_2+R_2{\tau }_1\end{array}\right.$$

(5)

(3)

To ensure consistency of the system before and after discretization, Equation (4) is discretized using the Z-transformation $s={\displaystyle \frac{2}{T}}\times {\displaystyle \frac{1-Z^{-1}}{1+Z^{-1}}}$ to obtain Equation (6). Where T is the sampling interval.

$$G\left(Z^{-1}\right)={\displaystyle \frac{\left(d{T}^2-2Te+4ab\right)Z^{-2}+\left(2d{T}^2-8ab\right)Z^{-1}+d{T}^2+2Te+4ab}{\left(T^2-2Tc+4b\right)Z^{-2}+\left(2T^2-8b\right)Z^{-1}+T^2+2Tc+4b}}$$

(6)

Let:

$$\left\{\begin{array}{l}{k}_1=-{\displaystyle \frac{2T^2-8b}{T^2+2Tc+4b}}\\ k_2=-{\displaystyle \frac{T^2-2Tc+4b}{T^2+2Tc+4b}}\\ k_3={\displaystyle \frac{d{T}^2-2Te+4ab}{T^2+2Tc+4b}}\\ k_4={\displaystyle \frac{2d{T}^2-8ab}{T^2+2Tc+4b}}\\ k_5={\displaystyle \frac{d{T}^2-2Te+4ab}{T^2+2Tc+4b}}\end{array}\right.$$

(7)

(4)

The system’s differential equation is obtainable.

$$U\left(t\right)=k_1\times U(t-1)+k_2\times U(t-2)+k_3\times I\left(t\right)+k_4\times I(t-1)+k_5\times I(t-2)$$

(8)

Let:

$$\left\{\begin{array}{l}\phi \left(t\right)={\left[U(t-1),U(t-2),I\left(t\right),I(t-1),I(t-2)\right]}^{\mathrm{T}}\\ \theta =\left[k_1,k_2,k_3,k_4,k_5\right]\end{array}\right.$$

(9)

(5)

Suppose that a, b, c, d, f are represented by (10)

$$\left\{\begin{array}{l}a={\displaystyle \frac{k_4-k_3-k_5}{1+k_1-k_2}}\\ b={\displaystyle \frac{T^2\left(1+k_1-k_2\right)}{4\left(1-k_1-k_2\right)}}\\ c={\displaystyle \frac{T\left(1+k_2\right)}{1-k_1-k_2}}\\ d=-{\displaystyle \frac{k_3+k_4+k_5}{1-k_1-k_2}}\\ e={\displaystyle \frac{T\left(k_3-k_3\right)}{1-k_1-k_2}}\end{array}\right.$$

(10)

(6)

The resistance and capacitance parameters $R_0$, $R_1$, $R_2$, $C_1$, and $C_2$ can be obtained by (11).

$$\left\{\begin{array}{l}{R}_0=a\\ R_1={\displaystyle \frac{{\tau }_1(d-a)+ac-e}{{\tau }_1-{\tau }_2}}\\ R_2=d-a-R_1\\ C_1={\displaystyle \frac{{\tau }_1}{R_1}}\\ C_2={\displaystyle \frac{{\tau }_2}{R_2}}\\ {\tau }_1,{\tau }_2={\displaystyle \frac{c\pm \sqrt{c^2-4b}}{2}}\end{array}\right.$$

(11)

Equation (9) leads to $U\left(t\right)={\phi }^{\mathrm{T}}\left(t\right)\theta$, the iterative process of the FFRLS algorithm is as follows.

$$\begin{array}{l}\left\{\begin{array}{l}K\left(k+1\right)={\displaystyle \frac{P\left(k\right)\phi \left(k+1\right)}{\left[\lambda +{\phi }^{\mathrm{T}}\left(k+1\right)P\left(k\right)\phi \left(k+1\right)\right]}}\\ e\left(k+1\right)=y\left(k+1\right)-{\phi }^{\mathrm{T}}\left(k+1\right)\theta \left(k\right)\\ \theta \left(k+1\right)=\theta \left(k\right)+K\left(k+1\right)e\left(k+1\right)\\ P\left(k+1\right)={\displaystyle \frac{1}{\lambda }}[I-K\left(k+1\right){\phi }^{\mathrm{T}}\left(k+1\right)P\left(k\right)]\end{array}\right.\end{array}$$

(12)

where $K$ is the gain of the system; $e$ is the prediction error of the system; $y$ is the observed value of the system; $\phi$ is the observation vector; $P$ is the error covariance matrix; $\theta$ is the matrix of parameters to be identified; $k+1$ denotes the value of the system at $k+1$ lambda is the forgetting factor. λ typically ranges from 0.95 to 1.00. The parameter identification findings stabilize with increasing forgetting factor, but the rate of convergence slows down [22].

We use the HPPC test technique to record the open circuit voltage in real time at different SOCs to determine the relationship between $U_{\mathrm{oc}}$ and SOC. SOC-OCV are two important metrics used to define and model batteries, and the relationship between SOC-OCV is typically nonlinear and can be represented by an OCV curve. Our aim is to find an OCV function that accurately estimates the battery state of charge for each SOC state. Figure 4 shows a plot of the relationship between $U_{\mathrm{oc}}$ and SOC resulting from a polynomial fit to the experimental data using MATLAB. When the polynomial coefficient is 9, the fitting result meets the accuracy criterion. Therefore, Equation (13) establishes the relationship between OCV and SOC.

$$U_{oc}=p_{1}SO{C}^9+p_{2}SO{C}^8+p_{3}SO{C}^7+p_{4}SO{C}^6+p_{5}SO{C}^5+p_{6}SO{C}^4+p_{7}SO{C}^3+p_{8}SO{C}^2+p_{9}SOC+p_{10}$$

(13)

where $p$ is the fitting coefficient.

As shown in Figure 5A, the curve of the experimental terminal voltage versus the simulated voltage, and Figure 5B shows the error curve of the voltage. It is shown that the maximum error is less than 1.5%, which more accurately reflects the actual condition of the battery.

3. EKF with the Improved Sage–Husa Adaptive Method

3.1. EKF Algorithm

Taylor series expansions are a prevalent methodology for approximating the SOC of lithium-ion batteries, largely due to their exceptional tracking capabilities [23]. They are employed in the state–space model of nonlinear systems for the EKF algorithm. Building a mathematical model of the battery is usually required to apply the EKF approach. In general, a system of differential equations that describe the properties of the battery under various operating situations is used to build the model. Moreover, the EKF approach is intended to estimate a nonlinear dynamic system’s state. It is an advancement of the traditional KF algorithm. The specific pushdown steps of the EKF are illustrated in Table 3. It updates the state estimate by employing a combination of estimated and measured values. In each iteration, the EKF utilizes the estimate of the current state to approximate a linearized version of the dynamics of the nonlinear system [24].

Based on the current integral method of SOC estimation, the prediction model is Equation (14)

$$SOC=SO{C}_0-\int {\displaystyle \frac{\eta Idt}{Q}}$$

(14)

where $SOC$ is current SOC, $SO{C}_0$ is initial SOC, $\eta$ is charge/discharge efficiency, $Q$ is battery capacity.

According to the ECM, the sampling time T is set to 1 s, let $\tau =RC$ and Equations (1) and (14) are discretized:

$$\left\{\begin{array}{l}SO{C}_k=SO{C}_{k-1}-\eta T{I}_{k-1}/Q\\ U_{1,k}=U_{1,k-1}\exp (-T/{\tau }_1)+R_1{I}_{k-1}(1-\exp (-T/{\tau }_1\left)\right)\\ U_{2,k}=U_{2,k-1}\exp (-T/{\tau }_2)+R_2{I}_{k-1}(1-\exp (-T/{\tau }_2\left)\right)\end{array}\right.$$

(15)

The state and observation equations of the battery model could be constructed by using the system’s SOC, $U_1$ and $U_2$ as the state variables, as indicated by Equations (16) and (17):

$$x_k=\left[\begin{array}{c}SO{C}_k\\ U_{1,k}\\ U_{2,k}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{e}\mathrm{x}\mathrm{p}(-T/{\tau }_1)& 0\\ 0& 0& \mathrm{e}\mathrm{x}\mathrm{p}(-T/{\tau }_2)\end{array}\right]x_{k-1}+\left[\begin{array}{c}-\eta T/Q\\ R_1(1-\exp (-T/{\tau }_1\left)\right)\\ R_2(1-\exp (-T/{\tau }_2\left)\right)\end{array}\right]I_{k-1}+w_{k-1}$$

(16)

$$U_t=Uoc\left(SO{C}_k\right)-U_{1,k}-U_{2,k}-R_0{I}_k+v_k$$

(17)

where ${\nu }_k$ is the observation noise brought on by the measurement error and ${\omega }_{k-1}$ is the system noise brought on by the modeling error. The following is a definition of the system parameters:

$$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{e}\mathrm{x}\mathrm{p}(-T/{\tau }_1)& 0\\ 0& 0& \mathrm{e}\mathrm{x}\mathrm{p}(-T/{\tau }_2)\end{array}\right]$$

(18)

$$B=\left[\begin{array}{c}-\eta T/Q\\ R_1(1-\exp (-T/{\tau }_1\left)\right)\\ R_2(1-\exp (-T/{\tau }_2\left)\right)\end{array}\right]$$

(19)

$$C=\left[\begin{array}{ccc}{\displaystyle \frac{\partial U_{oc,k}}{\partial SO{C}_k}}& -1& -1\end{array}\right]$$

(20)

$$D=\left[\begin{array}{c}-R_0\end{array}\right]$$

(21)

The observation equation can be obtained

$$x_k=A{x}_{k-1}+B{u}_{k-1}+w_{k-1}$$

(22)

The state equation can be obtained

$$y_k=C{x}_k+D{u}_k+v_k$$

(23)

Table 3. The EKF algorithm steps [23].

(1) Setting Equations

$$\left\{\begin{array}{l}{x}_k=f(x_{k-1},u_{k-1})+w_k\\ y_k=h(x_k,u_k)+v_k\end{array}\right.$$ (24) $f(x_{k-1})$ is the nonlinear state equation, $h(x_k,u_k)$ is the nonlinear output equation, the process noise is represented by ${\omega }_k$; the measurement noise is represented by ${\upsilon }_k$; and the two are independent of one another. The first-order Taylor expansion of Equation (25) should be performed: $$\left\{\begin{array}{l}f(x_{k-1},u_{k-1})\approx f({\widehat{x}}_{\mathrm{k}-1},u_{k-1})+\frac{\partial f\left(x_{k-1},u_{k-1}\right)}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}(x_{k-1}-{\widehat{x}}_{\mathrm{k}-1})\\ h(x_{k-1},u_{k-1})\approx h({\widehat{x}}_{\mathrm{k}-1},u_{k-1})+\frac{\partial h(x_{k-1},u_{k-1})}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}(x_{k-1}-{\widehat{x}}_{\mathrm{k}-1})\end{array}\right.$$ (25) We construct the state transfer matrix $A_k=\frac{\partial f(x_{k-1},u_{k-1})}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}$, the observation matrix $B_k=\frac{\partial h(x_{k-1},u_{k-1})}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}$, and the standard state space equation formulation may be obtained as follows: $$\left\{\begin{array}{l}{x}_k\approx A{x}_{k-1}+[f({\widehat{x}}_{\mathrm{k}-1},u_{k-1})-A_k{\widehat{x}}_{\mathrm{k}-1}]+w_k\\ y_k\approx C{x}_{k-1}+[h({\widehat{x}}_{\mathrm{k}-1},u_{k-1})-C_k{\widehat{x}}_{\mathrm{k}-1}]+v_k\end{array}\right.$$ (26) (2) Initialization

$${\widehat{x}}_0=E\left[x_0\right],P_0=E\left[(x_0-{\widehat{x}}_0)(x_0-{\widehat{x}}_0{)}^T\right]$$ (27) BEGIN PREDICT (3) Prior estimation update (3.1) Prior state estimation update

$${\widehat{x}}_{k-1}=f({\widehat{x}}_{k-1},u_{k-1})$$ (28) (3.2) Prior covariance estimation update $$P_{\mathrm{k}-1}=A{P}_{k-1}{A}^T+Q_{K-1}$$ (29) (4) Measurement update (4.1) Kalman gain

$$K_k=P_{k-1}{\widehat{C}}_k^T({\widehat{C}}_k{P}_{k-1}{\widehat{C}}_k^T+R_k{)}^{-1}$$ (30) (4.2) Posterior estimation update $${\widehat{x}}_k={\widehat{x}}_{\mathrm{k}-1}+K_k(y_k-h({\widehat{x}}_{k-1},u_k))$$ (31) (4.3) Covariance error update $$P_k=(I-L_k{C}_k)P_{k-1}$$ (32) K = k − 1. END PREDICT The ideal estimate is denoted by the superscript ^ in the equation above

Table 3. The EKF algorithm steps [23].

(1) Setting Equations

$$\left\{\begin{array}{l}{x}_k=f(x_{k-1},u_{k-1})+w_k\\ y_k=h(x_k,u_k)+v_k\end{array}\right.$$ (24) $f(x_{k-1})$ is the nonlinear state equation, $h(x_k,u_k)$ is the nonlinear output equation, the process noise is represented by ${\omega }_k$; the measurement noise is represented by ${\upsilon }_k$; and the two are independent of one another. The first-order Taylor expansion of Equation (25) should be performed: $$\left\{\begin{array}{l}f(x_{k-1},u_{k-1})\approx f({\widehat{x}}_{\mathrm{k}-1},u_{k-1})+\frac{\partial f\left(x_{k-1},u_{k-1}\right)}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}(x_{k-1}-{\widehat{x}}_{\mathrm{k}-1})\\ h(x_{k-1},u_{k-1})\approx h({\widehat{x}}_{\mathrm{k}-1},u_{k-1})+\frac{\partial h(x_{k-1},u_{k-1})}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}(x_{k-1}-{\widehat{x}}_{\mathrm{k}-1})\end{array}\right.$$ (25) We construct the state transfer matrix $A_k=\frac{\partial f(x_{k-1},u_{k-1})}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}$, the observation matrix $B_k=\frac{\partial h(x_{k-1},u_{k-1})}{\partial x_{k-1}}{|}_{x_{k-1}={\widehat{x}}_{\mathrm{k}-1}}$, and the standard state space equation formulation may be obtained as follows: $$\left\{\begin{array}{l}{x}_k\approx A{x}_{k-1}+[f({\widehat{x}}_{\mathrm{k}-1},u_{k-1})-A_k{\widehat{x}}_{\mathrm{k}-1}]+w_k\\ y_k\approx C{x}_{k-1}+[h({\widehat{x}}_{\mathrm{k}-1},u_{k-1})-C_k{\widehat{x}}_{\mathrm{k}-1}]+v_k\end{array}\right.$$ (26) (2) Initialization

$${\widehat{x}}_0=E\left[x_0\right],P_0=E\left[(x_0-{\widehat{x}}_0)(x_0-{\widehat{x}}_0{)}^T\right]$$ (27) BEGIN PREDICT (3) Prior estimation update (3.1) Prior state estimation update

$${\widehat{x}}_{k-1}=f({\widehat{x}}_{k-1},u_{k-1})$$ (28) (3.2) Prior covariance estimation update $$P_{\mathrm{k}-1}=A{P}_{k-1}{A}^T+Q_{K-1}$$ (29) (4) Measurement update (4.1) Kalman gain

$$K_k=P_{k-1}{\widehat{C}}_k^T({\widehat{C}}_k{P}_{k-1}{\widehat{C}}_k^T+R_k{)}^{-1}$$ (30) (4.2) Posterior estimation update $${\widehat{x}}_k={\widehat{x}}_{\mathrm{k}-1}+K_k(y_k-h({\widehat{x}}_{k-1},u_k))$$ (31) (4.3) Covariance error update $$P_k=(I-L_k{C}_k)P_{k-1}$$ (32) K = k − 1. END PREDICT The ideal estimate is denoted by the superscript ^ in the equation above

3.2. Sage–Husa EKF Algorithm

Typically, process noise and measurement noise are corrected throughout the SOC estimate process with EKF. However, in the actual process, the noise is frequently unknown, which might lead to mistakes in the SOC estimation. The Sage–Husa adaptive method may adaptively estimate the system noise. By the coupling of the Sage–Husa adaptive algorithm with the EKF, the system noise is adaptively assessed while conducting the SOC estimation, and the estimation accuracy of the SOC is improved.

Table 4. The Sage–Husa algorithm steps.

(1) Gian Factor $$d_k=(1-\mathrm{b})/(1-{\mathrm{b}}^{k+1})$$ (33) (2) System noise and system noise covariance matrix $$w_k=(1-d_k)w_{k-1}+d_k({\widehat{x}}_{k-1/k-1}-A_{k-1}{\widehat{x}}_{k-1/k-1})$$ (34) $$Q_k=(1-d_k)Q_{k-1}+d_k(K_{k-1}{e}_k{e}_k^T{K}_{k-1}^T+P_{x,k-1/k-1}-A_{k-1}{P}_{x,k-1/k-1}{A}_{k-1}^T)$$ (35) (3) Observed noise and observed noise covariance matrix $$v_k=(1-d_k)v_{k-1}+d_k(y_{k-1}-{\widehat{y}}_{k-1})$$ (36) $$R_k=(1-d_k)R_{k-1}+d_k(e_k{e}_k^T-C_k{P}_{x,k/k-1}{C}_k^T)$$ (37)

displays the algorithm’s stages.

In Table 4, $x_{k/k-1}$ indicates the predicted value of the state variable at moment $k$. $x_k$ indicates the optimal estimate of the state variable at moment $k$. b is a forgetting factor in the range 0.95 to 0.99, and when b = 1 there is no forgetting function.

3.3. Improved Sage–Husa EKF Algorithm

The Sage–Husa adaptive algorithm may cause the filtering results to diverge due to a sudden increase in the error by a factor of several, one of the reasons for this is because $R_k$ and $R_k$ lose their positivity during the iterative calculation of formulas. Equations (35) and (37) are simplified as shown in Equation (38):

$$\left\{\begin{array}{c}{Q}_k=(1-d_k)Q_{k-1}+d_k\left(K_{k-1}{e}_k{e}_k^T{K}_{k-1}^T\right)\\ R_k=(1-d_k)R_{k-1}+d_k\left(e_k{e}_k^T\right)\end{array}\right.$$

(38)

The flow of the FFRLS real-time parameter identification through the 2RC ECM and the combined SOC online estimation based on the improved Sage–Husa EKF algorithm is shown in Figure 6.

4. Simulation Verification and Analysis

A 200 Ah rated capacity LiFePO₄ battery is employed in the HPPC test condition to confirm the efficacy of the revised Sage–Husa EKF algorithm in calculating the SOC. Figure 7 and Figure 8 display the SOC estimate results and error curves of the EKF algorithm as well as the modified Sage–Husa EKF algorithm, of which this study is a MATLAB simulation.

The evaluation metric chosen is the Root Mean Square Error (RMSE), which is defined as follows:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{1}{n}}\sqrt{\sum _{k=1}^n{\left(S{\mathrm{o}\mathrm{c}}_k-\widehat{S}{\mathrm{o}\mathrm{c}}_k\right)}^2}$$

(39)

where the actual and estimated values of SOC are denoted by $S{\mathrm{o}\mathrm{c}}_k$ and $\widehat{S}{\mathrm{o}\mathrm{c}}_k$, respectively.

As shown in Figure 7, Real-SOC denotes the actual SOC value measured experimentally, EKF represents the SOC estimate acquired by the EKF algorithm based on the 2-RC equivalent circuit model, SAEKF denotes the SOC estimate obtained by the Sage–Husa EKF method, and ISAEKF denotes the SOC estimate obtained by the improved Sage–Husa EKF method. The figure illustrates that the EKF algorithm exhibits a deviation, and the inaccuracy is more noticeable, particularly when the current is constantly discharged to low SOC, the accuracy of each estimating technique stays good at the start of the discharge. Furthermore, the estimation accuracy of SAEKF and ISAEKF is comparatively high. ISAEKF is more accurate than EKF and SAEKF, particularly when it comes to estimating SOC in the late stage of discharge. The SOC estimate results of the improved Sage–Husa EKF method exhibit superior accuracy and stability than the EKF and the SAEKF algorithm. According to Figure 8, The discrepancy between the values derived using the various techniques and the real SOC values is used to explain the error curves. EKF denotes the difference between the actual SOC and the SOC estimate using the EKF algorithm; SAEKF denotes the difference between the SOC estimate from the Sage–Husa EKF algorithm and the actual SOC; ISAEKF denotes the difference between the SOC estimate from the improved Sage–Husa EKF algorithm and the actual SOC. The graph shows how the error of the EKF algorithm changes significantly as the SOC decreases, with the error peaks occurring mostly in the pulse-discharge interval. The estimation accuracy of the EKF algorithm decreases and the difficulty of calculating the SOC increases when the charge and discharge currents fluctuate significantly. Although the Sage–Husa EKF method, which corrects for noise covariance, has a more stable estimation error, the estimation error is larger compared to the improved Sage–Husa EKF method. In addition, the improved Sage–Husa EKF method is more computationally efficient and takes less time.

As shown in

Table 5. Error analysis of different SOC estimation methods.

Item	EKF	SAEKF	ISAEKF
maximum estimation error	3.51%	2.48%	0.72%
average estimation error	1.52%	0.83%	0.27%
RMSE	1.79%	0.97%	0.3%

, the maximum estimation error of the improved Sage–Husa EKF algorithm is 0.72%, the average estimation error is 0.27%, and the RMSE is 0.3%. This consistent stability emphasizes the reliability of the estimation process as the SOC estimation error remains relatively stable despite various external factors and challenging factors such as battery aging. However, the EKF algorithm has a maximum estimation error of 3.51%, average estimation error of 1.52%, and RMSE of 1.79%. The Sage–Husa EKF algorithm has a maximum estimation error of 2.48%, average estimation error of 0.83% and RMSE of 0.97%. In contrast, the improved Sage–Husa EKF algorithm can track the dynamic characteristics of the cell better than the EKF algorithm, enabling a more accurate calibration and thus improving the estimation accuracy. The results show that the improved Sage–Husa EKF algorithm based on the 2RC ECM can provide a relatively accurate and stable SOC estimation and prove its effectiveness.

5. Conclusions

In this paper, we investigate the estimation of a lithium-ion battery SOC based on the improved EKF algorithm with adaptive estimation of system noise using the improved Sage–Husa algorithm. MATLAB simulation software was used to set the parameters of the battery model based on the equivalent circuit model of a lithium-ion battery. This allows the model to accurately represent the charging and discharging characteristics of the battery, and the discrepancy between the simulation and experimental data is less than 1.5%. Furthermore, the technique avoids potential divergences in the filtering process and guarantees a positive characterization of the calculated process matrix. The accuracy of the algorithm has been verified in the HPPC environment by comparing the simulation results of the EKF and the improved Sage–Husa algorithms. The findings demonstrate that the improved Sage–Husa EKF method outperforms the EKF algorithm in terms of modeling of the dynamic features of battery charging and discharging, and it has a maximum estimate error of 0.72%. This algorithm also exhibits higher accuracy compared to the experimental results. The study’s enhanced Sage–Husa EKF algorithm estimates SOC with great efficacy and precision, laying the theoretical foundation for condition monitoring and offering a secure assurance for the engineering use of lithium-ion batteries.