A Method for State of Charge and State of Health Estimation of LithiumBatteries Based on an Adaptive Weighting Unscented Kalman Filter

Abstract

This paper considers the estimation of SOC and SOH for lithium batteries using multi-innovation Levenberg–Marquardt and adaptive weighting unscented Kalman filter algorithms. For parameter identification, the second-order derivative of the objective function to optimize the traditional gradient descent algorithm is used. For SOC estimation, an adaptive weighting unscented Kalman filter algorithm is proposed to deal with the nonlinear update problem of the mean and covariance, which can substantially improve the estimation accuracy of the internal state of the lithium battery. Compared with fixed weights in the traditional unscented Kalman filtering algorithm, this algorithm adaptively adjusts the weights according to the state and measured values to improve the state estimation update accuracy. Finally, according to simulations, the errors of this algorithm are all lower than 1.63 %, which confirms the effectiveness of this algorithm.

1. Introduction

Lithium batteries have significant advantages, such as good safety performance, excellent discharge characteristics, and high energy density [1,2]. These qualities make them crucial to the future direction of energy production. To ensure high reliability and operational safety, the battery management system (BMS) must track and control the electrical system in real-time [3,4]. The state of charge (SOC) and state of health (SOH) are crucial factors in the BMS that influence the battery’s energy management distribution. SOC refers to the current remaining charge of the battery, and its effective estimation is essential to the operating process of a lithium battery. SOH means the health of the lithium battery, usually expressed as resistance or capacity. In summary, the accuracy of SOC and SOH estimation helps to better manage and maintain a lithium battery, ensuring the safety and reliability of the battery [5,6].

Lithium battery models can be used to describe the electrical behavior and characteristics of the battery, which can enhance understanding of how lithium batteries work, facilitating prediction of the SOC and lifetime of lithium batteries. Currently, there are four prevalent types of battery models: electrochemical models [7,8,9], neural network models [10], alternating current (AC) impedance models [11,12], and the equivalent circuit model [13]. Compared with the AC impedance and electrochemical models, the equivalent circuit model can simplify the complex internal structure and improve computational efficiency. It can directly reflect the dynamic processes and state changes inside the battery. Compared to the neural network model, the equivalent circuit model has enhanced precision in predicting the SOC, the SOH, and the remaining useful life (RUL) based on known internal battery parameters [14,15], and black-box effects, and over-fitting can be avoided [16].

The open circuit voltage method is one of the most prevalent methods for estimating the SOC, but the results estimated by this method can be confounded by environmental factors and require a longer test time. Therefore, the method is generally not considered for practical applications. Zhang et al. proposed a particle filter in the battery SOC and the open circuit voltage (OCV) estimation, which can avoid the influence of noise generated during battery terminal voltage measurement and current drift [17]. The ampere-time integration method requires the use of high-precision test equipment to obtain accurate data. Moreover, the initial values are not easy to determine and this model is not suitable for rapid testing of the lithium battery state in a short period of time. Liu et al. linked the EKF and Ah-integral algorithms to apply the fuzzy-extended Kalman filter-ampere-hour algorithm. The serious estimation error in the strong nonlinear interval of the EKF algorithm and the accumulated error generated by the sampling accuracy of the ampere-time integration algorithm may both be successfully resolved by the fused algorithm [18]. Support vector machines are based on vectors to construct a decision boundary, so data interference has a more significant impact on them than other methods; therefore, it is important to ensure thorough data cleaning. Kim et al. utilized basic building blocks, such as the residual and leverage statistics, to represent the Cook distance, using simple descriptive statistics and approximation measures. They proposed application of the deletion method and the infinitesimal perturbation method to address outlier effects in vector machines [19]. The neural network has many parameters to identify, which can lead to a more complex network structure that is not easily optimized, moreover, the model itself is less interpretable. Talha et al. proposed a neural network (NN) simplified state estimation algorithm that can directly estimate the SOC and SOH online in the absence of the internal parameters of the battery. Finally, the reliability of this algorithm was demonstrated through experiments [20].

The model-based method offers higher flexibility compared to conventional and data-driven methods. Combining models and filters to estimate the SOC is a prevalent modeling method. Based on the dual-polarization model, Tan et al. proposed the EKF algorithm to calculate the SOC and SOP of lithium batteries. Simulations showed that the EKF algorithm can achieve high predictive values under any operating conditions, reflecting the effectiveness of this algorithm [21]. However, this method is not suitable for solving problems with special noise and unknown uncertainty. In order to solve the problem of low accuracy of the EKF algorithm, Li et al. proposed the forgetting factor stochastic gradient-multi-innovation extended Kalman filter (FFSG-MIEKF) algorithm for parameter identification and SOC estimation. Simulations were used to demonstrate the effectiveness of this algorithm [22]. However, the MIEKF algorithm introduces covariance matrix repair and an update operation, which increases the calculation required and the complexity of the algorithm. El Din et al., combining an unscented transform and Kalman filter using covariance estimation for the sampling point state noise and observation noise, proposed an unscented Kalman filter (UKF) to improve the state estimation accuracy. Finally, experiments were undertaken to confirm the effectiveness of this algorithm [23]. As the UKF algorithm needs to use sigma points, the amount of computation and the accuracy of calculation are affected during state estimation. In order to reduce the effect of interference on the SOC accuracy, in this paper, the mean and observation weights are adaptively adjusted according to the state and observation residual vectors so that this algorithm has better weight stability in state estimation.

The parameter estimation of battery models is very important for battery performance analysis and the parameters of battery models can be identified through some identification methods [24,25,26,27,28,29] from observation data such as the gradient-based algorithms [30,31,32,33,34] and the least squares-based identification methods [35,36,37,38]. In practical applications, the SOH plays a very important role in maintaining the performance and extending the life of a battery. Relatively speaking, the SOH and the internal resistance have a more direct link, so the internal resistance is chosen as the state variable for estimating the SOH [39,40,41]. In current research, most studies have considered the SOC and SOH separately, with the interaction between them rarely considered in the process of estimating the state parameters of the battery. Consequently, we propose an adaptive weighting unscented Kalman filter algorithm (AWUKF) algorithm. This algorithm is based on the SOC for SOH estimation, and can greatly reduce the problem of decreased accuracy of state estimation caused by environmental changes. The main contributions of the paper are as follows:

• The AWUKF algorithm reselects the particles using the unscented transformation, and the adaptive weights are brought into the UKF framework to reduce the sensitivity to noise.
• This algorithm is utilized to attenuate the deviation of the SOC values under measurement and state noise disturbances; moreover, the SOC is used as a basis for estimating the SOH values.
• Simulations are carried out under different experimental conditions to assess the practicality and effectiveness of this algorithm.

The paper is organized as follows: Section 2 describes the circuit model of the lithium battery. The construction of the multi-innovation Levenberg–Marquardt (MILM) algorithm to determine the internal parameters of the battery is described. Section 3 considers the AWUKF algorithm, and uses this algorithm for joint estimation of the SOC and SOH. Section 4 subjects the traditional EKF, the UKF, and the AWUKF algorithms to experimental simulation under different experimental conditions. Finally, Section 5 summarizes the main conclusions and presents some perspectives.

2. Battery Model

2.1. Integral Second-Order RC Model

$V_{ocv}$ is the open-circuit voltage, $V_0$ is the terminal voltage, and I is the load current. The first RC network describes the polarization characteristics, $R_{p1}$ denotes the polarization resistance, and $C_{p1}$ represents the polarization capacitance. The second RC circuit represents the dynamic behavior of the lithium battery related to the concentration polarization, $R_{p2}$ represents the concentration polarization resistance, and $C_{p2}$ represents the capacitance [42,43,44]. The second-order RC equivalent circuit model is shown in Figure 1.

The electrical characteristics of the model can be expressed as

$$\begin{array}{ccc}\hfill V_0& =& V_{ocv}-V_R-V_{p1}-V_{p2},\hfill \\ \hfill V_{p1}& =& I{R}_{p1}-C_{p1}\frac{\mathrm{d}{V}_{p1}}{\mathrm{d}t},\hfill \\ \hfill V_{p2}& =& I{R}_{p2}-C_{p2}\frac{\mathrm{d}{V}_{p2}}{\mathrm{d}t},\hfill \end{array}$$

where $V_{p1}$ and $V_{p2}$ are the terminal voltages of the $R_{p1}{C}_{p1}$ and $R_{p2}{C}_{p2}$ circuits, respectively.

By Laplace transformation, we get the electrical characteristic equation:

$$\begin{array}{c}\hfill V_0\left(s\right)-V_{ocv}\left(s\right)=I\left(s\right)\left[R_0+\frac{R_{p1}}{1+C_{p1}{R}_{p1}s}+\frac{R_{p2}}{1+C_{p2}{R}_{p2}s}\right].\end{array}$$

(1)

It follows the discretization expression:

$$\begin{array}{c}\hfill \frac{V_0\left(s\right)-V_{ocv}\left(s\right)}{-I\left(s\right)}=\frac{R_0(1+{\tau }_{1}s)(1+{\tau }_{2}s)+R_{p1}(1+{\tau }_{2}s)+R_{p2}(1+{\tau }_{1}s)}{(1+{\tau }_{1}s)(1+{\tau }_{2}s)}.\end{array}$$

(2)

Applying the bilinear transformation to (2), we get the differential equation for the system:

$$\begin{array}{c}\hfill V_0\left(t\right)={\theta }_1{V}_0(t-1)+{\theta }_2{V}_0(t-2)+{\theta }_{3}I\left(t\right)+{\theta }_{4}I(t-1)+{\theta }_{5}I(t-2)+{\theta }_6,\end{array}$$

(3)

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\theta }_1=-({\tau }_1+{\tau }_2),\hfill \\ {\theta }_2=-{\tau }_1{\tau }_2,\hfill \\ {\theta }_3=-(R_0+R_{p1}+R_{p2})\hfill \\ {\theta }_4=-(R_0{\tau }_1+R_0{\tau }_2+R_{p1}{\tau }_2+R_{p2}{\tau }_1),\hfill \\ {\theta }_5=-R_0{\tau }_1{\tau }_2,\hfill \\ {\theta }_6=(1+{\tau }_1+{\tau }_2+{\tau }_1{\tau }_2)V_{ocv},\hfill \end{array}\right.\end{array}$$

(4)

where ${\tau }_1$ and ${\tau }_2$ represents the time constants, which can be obtained by $R_{p1}{C}_{p1}$ and $R_{p2}{C}_{p2}$, respectively. ${\theta }_1$ to ${\theta }_6$ are equivalent expressions of the correlation coefficients. The internal resistance $R_0$, the polarization resistors $R_{p1}$ and $R_{p2}$, and the polarization capacitors $C_{p1}$ and $C_{p2}$ are obtained based on the measured data. Then, the optimization algorithm is used to identify ${\theta }_i$$(i=1,2,3,4,5,6)$. The parameter optimization algorithm is described in detail in the next section. The proposed parameter estimation algorithms in this paper are based on the identification model in (3). Many identification methods are derived based on the identification models of the systems [45,46,47,48,49] and these methods can be used to estimate the parameters of other linear systems and nonlinear systems [50,51,52,53,54] and can be applied to other fields [55,56,57,58,59] such as chemical process control systems.

2.2. Parameter Identification Based on MILM Method

The traditional LM algorithm combines gradient descent insensitivity to initial values and Gaussian–Newton global optimality characteristics. Based on this property, the solution efficiency of the LM algorithm in between the two methods can be determined [60]. Therefore, based on multi-innovation theory, by extending a single innovation into a multi-innovation vector containing both past moment and current moment data, an MILM algorithm is proposed to improve the accuracy of battery parameter identification. The specific process used is as follows:

First, the recognition model is as follows:

$$\begin{array}{c}\hfill V_0\left(t\right)={\phi }^{\mathrm{T}}\left(t\right)\theta +\nu \left(t\right),\end{array}$$

(5)

where $V_0$ are the observed output data, $\phi \left(t\right)={[V_0(t-1),V_0(t-2),I\left(t\right),I(t-1),I(t-2),1]}^{\mathrm{T}}$ is composed of the terminal voltage and current input data, $\theta ={[{\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4,{\theta }_5,{\theta }_6]}^{\mathrm{T}}$ is the parameter vector to be identified, and $\nu \left(t\right)$ is zero mean random noise.

According to the LM algorithm, the quadratic criterion function for the nonlinear system is given by

$$\begin{array}{c}\hfill J\left(\theta \right)=\frac{1}{2}{\parallel V_0\left(t\right)-{\phi }^{\mathrm{T}}\left(t\right)\theta \parallel }^2.\end{array}$$

(6)

According to (5) and (6), the first-order partial derivative and the second-order partial derivative of $J\left(\theta \right)$ with respect to $\theta$ are defined as

$$\begin{array}{ccc}\hfill \nabla J\left(\theta \right)& =& \frac{\partial J\left(\theta \right)}{\partial \theta }=-{\phi }^{\mathrm{T}}\left(t\right)\nu \left(t\right),\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill H\left(\theta \right)& =& \frac{{\partial }^{2}J\left(\theta \right)}{\partial \theta \partial {\theta }^{\mathrm{T}}}=\phi \left(t\right){\phi }^{\mathrm{T}}\left(t\right),\hfill \end{array}$$

(8)

According to (7) and (8), the introduction of the damping factor $\mu$ decreases the sensitivity to the initial values problem and enables the Gaussian–Newton characteristics to converge quickly around the optimum. Thus, we can obtain the LM algorithm:

$$\widehat{\theta }\left(t\right)=\widehat{\theta }(t-1)-{[H\left(\widehat{\theta }\left(t\right)\right)+\mu I]}^{-1}\nabla J\left(\widehat{\theta }\left(t\right)\right).$$

Using the multi-innovation identification theory [61,62,63,64,65,66,67,68], the single innovation is expanded into multi-innovation:

$$\begin{array}{ccc}\hfill \nu \left(t\right)& =& V_0\left(t\right)-{\phi }^{\mathrm{T}}\left(t\right)\theta ,\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \nu (p,t)& =& \left[\begin{array}{c}\nu \left(t\right)\\ \nu (t-1)\\ \dots \\ \nu (t-p+1)\end{array}\right]=\left[\begin{array}{c}{V}_0\left(t\right)-{\phi }^{\mathrm{T}}\left(t\right)\theta \\ V_0(t-1)-{\phi }^{\mathrm{T}}(t-1)\theta \\ \dots \\ V_0(t-1)-{\phi }^{\mathrm{T}}(t-1)\theta \end{array}\right],\hfill \end{array}$$

(10)

where p is the innovation length.

The input data matrix $\Phi (p,t)$ and output terminal voltage vector $V_0(p,t)$ are defined as

$$\begin{array}{ccc}\hfill \Phi (p,t)& =& {[\phi \left(t\right),\phi (t-1),\dots ,\phi (t-p+1)]}^{\mathrm{T}},\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill V_0(p,t)& =& {[V_0\left(t\right),V_0(t-1),\dots ,V_0(t-p+1)]}^{\mathrm{T}}.\hfill \end{array}$$

(12)

Combining (9)–(12), the multi-innovation vector can be expressed as

$$\nu (p,t)=V_0(p,t)-{\Phi }^{\mathrm{T}}(p,t)\widehat{\theta }(t-1),$$

(13)

where the state $\nu (p,t)$ is optimized by expanding $V_0\left(t\right)$ to $V_0(p,t)$ and $\phi \left(t\right)$ to $\Phi (p,t)$. Therefore, the MILM algorithm can capture more information, and identifying the parameters is more accurate than for the LM algorithm.

In summary, the MILM algorithm is as follows:

$$\begin{array}{ccc}\hfill \widehat{\theta }\left(t\right)& =& \widehat{\theta }(t-1)-{[H\left(\widehat{\theta }\left(t\right)\right)+\mu I]}^{-1}\nabla J\left(\widehat{\theta }\left(t\right)\right),\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill I& =& \mathrm{eye}\left(size\left(Q\right)\right),Q=H\left(\widehat{\theta }\left(t\right)\right),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \nabla J\left(\widehat{\theta }\left(t\right)\right)& =& \frac{\partial J\left(\theta \right(t\left)\right)}{\partial \theta }=-{\Phi }^{\mathrm{T}}(p,t)\nu (p,t),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill H\left(\widehat{\theta }\left(t\right)\right)& =& \frac{{\partial }^{2}J\left(\theta \left(t\right)\right)}{{\partial }^2\theta }=\Phi (p,t){\Phi }^{\mathrm{T}}(p,t).\hfill \end{array}$$

(17)

3. The SOC and SOH Estimation Method

3.1. SOC Estimation Based on the AWUKF Algorithm

This paper proposes an adaptive weight method to adjust the system noise and measurement noise. Compared with the fixed weight in traditional UKF, the method adaptively adjusts the state estimation weight based on the state and measurement residual vector, which can improve the SOC estimate accuracy and convergence.

From the battery model, the state relationship equation is defined as

$$\left\{\begin{array}{c}\dot{x}=Ax+BI,\hfill \\ y=Cx+DI,\hfill \end{array}\right.$$

where

$$\begin{array}{ccc}\hfill A& =& \left[\begin{array}{ccc}1& 0& 0\\ 0& {\mathrm{e}}^{-\alpha }& 0\\ 0& 0& {\mathrm{e}}^{-\alpha }\end{array}\right],\hfill \\ \hfill B& =& \left[\begin{array}{c}-\frac{{\int }_{t0}^t\eta \mathrm{d}t}{Q_N}\\ R_{p1}-R_{p1}{\mathrm{e}}^{-{\alpha }_1}\\ R_{p2}-R_{p2}{\mathrm{e}}^{-{\alpha }_2}\end{array}\right],{\alpha }_1=\frac{\Delta t}{R_{p1}{C}_{p1}},{\alpha }_2=\frac{\Delta t}{R_{p2}{C}_{p2}},\hfill \\ \hfill C& =& [\frac{d_{ocv}}{d_{soc}}-1-1],D=-R_0,\hfill \\ \hfill x& =& \left[\begin{array}{c}SOC\\ V_{p1}\\ V_{p2}\end{array}\right],y=V_0,x\in {\mathbb{R}}^3.\hfill \end{array}$$

The state equation and measurement equation are given by

$$\begin{array}{ccc}\hfill x_{k+1}& =& f(x_k,u_k)+{\omega }_k,\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill y_{k+1}& =& h(x_{k+1},u_{k+1})+{\upsilon }_k,\hfill \end{array}$$

(19)

where $f(x_k,u_k)$ is the nonlinear state equation, $h(x_k,u_k)$ is the nonlinear measurement equation, $u_k$ is the current, $x_k$ is the state variable, subscript k denotes the kth discrete sampling, ${\omega }_k$ is the state noise, and ${\upsilon }_k$ is the measurement noise. According to (18) and (19), the calculation process of the AWUKF algorithm is as follows.

The first values of the state and the error covariance matrix are given by

$$\begin{array}{ccc}\hfill {\widehat{x}}_0& =& \mathrm{E}\left[\left(x_0\right)\right],\hfill \\ \hfill P_0& =& \mathrm{E}\left[(x_0-{\widehat{x}}_0){(x_0-{\widehat{x}}_0)}^{\mathrm{T}}\right].\hfill \end{array}$$

Select $2n+1$ sample points:

$$\left\{\begin{array}{c}{x}_{k-1}^0={\widehat{x}}_{k-1},\hfill \\ x_{k-1}^{\left(i\right)}={\widehat{x}}_{k-1}+\left(\sqrt{(n+\lambda )P_{k-1}}\right),i=1,2,\dots ,L,\hfill \\ x_{k-1}^{\left(i\right)}={\widehat{x}}_{k-1}-\left(\sqrt{(n+\lambda )P_{k-1}}\right),i=L,L+2,\dots ,2L,\hfill \end{array}\right.$$

(20)

where i is the number of sample points.

Calculation of the weights coefficient:

$$\left\{\begin{array}{c}\lambda ={\alpha }^2(n+\kappa )-n,\hfill \\ {\omega }_c^{\left(0\right)}=\frac{\lambda }{n+\lambda },{\omega }_c^{\left(i\right)}=\frac{\lambda }{2(n+\lambda )},i=1,2,\dots ,2L,\hfill \\ {\omega }_m^{\left(0\right)}=\frac{1}{n+\lambda }+1-{\alpha }^2+\beta ,{\omega }_m^{\left(i\right)}=\frac{1}{2(n+\lambda )},i=1,L+1,\dots ,2L,\hfill \end{array}\right.$$

(21)

where $\lambda$ is a scaling coefficient, $\kappa$ is an optional parameter, ${\omega }_m$ is the weighted value of the particle mean value, ${\omega }_c$ is the weighted value of the particle variance, and subscript i represents the ith discrete sampling. In the traditional algorithm, the weighting coefficient is a fixed value, but in practical applications, both the measurement noise and the state noise are random, being easily affected by temperature, voltage, and other factors. Therefore, an adaptive weights coefficient is proposed to adjust the state noise and the measurement noise.

Using (18), (20) and (21), the a priori state values and state covariance matrix for the $2n+1$ sample points are denoted by

$$\begin{array}{ccc}\hfill x_{k|k-1}^{\left(i\right)}& =& f(x_{k-1}^{\left(i\right)},u_{k-1}),i=0,1,2,\dots ,2L,\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\widehat{x}}_{k|k-1}& =& \sum _{i=1}^M{\omega }_m^{\left(i\right)}{x}_{k|k-1}^{\left(i\right)},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill P_{k|k-1}& =& \sum _{i=1}^M{\omega }_c^{\left(i\right)}({\widehat{x}}_{k|k-1}-x_{k|k-1}^{\left(i\right)}){({\widehat{x}}_{k|k-1}-x_{k|k-1}^{\left(i\right)})}^{\mathrm{T}}+Q_{k-1},\hfill \end{array}$$

(24)

where the system noise covariance matrix uses $Q_{k-1}$ at moment $k-1$.

According to the one-step predicted values in (23) and (24), the new sampling points are produced by the unscented transform:

$$x_{k|k-1}^{\left(i\right)}=[{\widehat{x}}_{k|k-1},{\widehat{x}}_{k|k-1}+\sqrt{(n+\lambda )P_{k|k-1}},{\widehat{x}}_{k|k-1}-\sqrt{(n+\lambda )P_{k|k-1}}].$$

(25)

Combining (19) and (25), the mean values and covariance matrix of the predicted observation are obtained by the weighted summation:

$$\begin{array}{ccc}\hfill y_{k|k-1}^{\left(i\right)}& =& h(x_{k|k-1}^{\left(i\right)},u_k),i=0,1,2,\dots ,2L,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\widehat{y}}_{k|k-1}& =& \sum _{i=1}^M{\omega }_m^{\left(i\right)}{y}_{k|k-1}^{\left(i\right)},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill P_{({\widehat{y}}_k,{\widehat{y}}_k)}& =& \sum _{i=1}^M{{\omega }_c}^{\left(i\right)}({\widehat{y}}_{k|k-1}-y_{k|k-1}^{\left(i\right)}){({\widehat{y}}_{k|k-1}-y_{k|k-1}^{\left(i\right)})}^{\mathrm{T}}+R_k,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill P_{({\widehat{x}}_k,{\widehat{y}}_k)}& =& \sum _{i=1}^M{{\omega }_c}^{\left(i\right)}({\widehat{x}}_{k|k-1}-x_{k|k-1}^{\left(i\right)}){({\widehat{y}}_{k|k-1}-y_{k|k-1}^{\left(i\right)})}^{\mathrm{T}},\hfill \end{array}$$

(29)

where $P_{(y_k,y_k)}$ is the predicted measurement covariance matrix, $P_{(x_k,y_k)}$ is the state and measurement cross-covariance matrix, and $R_k$ is the observation noise covariance matrix at moment k.

According to (28) and (29), update the Kalman gain by

$$K_k=P_{({\widehat{x}}_k,{\widehat{y}}_k)}{P}_{({\widehat{y}}_k,{\widehat{y}}_k)}^{-1}.$$

(30)

Use (23), (24), (27) and (30) to update the system state vector and covariance matrix:

$$\begin{array}{ccc}\hfill {\widehat{x}}_k& =& {\widehat{x}}_{k|k-1}+K_k(y_k-{\widehat{y}}_{k|k-1}),\hfill \\ \hfill P_{k|k}& =& P_{k|k-1}-K_k{P}_{({\widehat{y}}_k,{\widehat{y}}_k)}{K}_k^{\mathrm{T}}.\hfill \end{array}$$

Adaptive weights are given by

$$\left\{\begin{array}{c}\Delta x_j=x_k-{\widehat{x}}_{k|k-1},\hfill \\ \Delta y_j=y_k-{\widehat{y}}_{k|k-1},\hfill \\ \parallel \Delta x_j\parallel =\sqrt{\Delta x_j^{\mathrm{T}}\Delta x_j},\hfill \\ \parallel \Delta y_j\parallel =\sqrt{\Delta y_j^{\mathrm{T}}\Delta y_j},\hfill \\ {\lambda }_j=\parallel \Delta x_j\parallel \parallel \Delta y_j\parallel ,\hfill \\ {\omega }_j=\frac{{\lambda }_j}{{\sum }_{k=0}^{2L}\parallel \Delta x_j\parallel \parallel \Delta y_j\parallel },\hfill \\ {\omega }_j={\omega }_m^{\left(i\right)}={\omega }_c^{\left(i\right)},j=1,2,3,\dots ,2L.\hfill \end{array}\right.$$

3.2. SOH Estimation

The SOH of a lithium battery degrades over time until it can no longer meet the power and energy requirements. So, we need to predict the SOH of lithium batteries to ensure their safety and reliability. With increase in the number of charge and discharge cycles, a lithium battery will inevitably degrade. Therefore, the SOH of a battery can be indirectly computed using the ohmic internal resistance and the maximum available capacity.

Based on the SOH estimation of the ohmic internal resistance, the state of charge and $V_{ocv}$ of the battery are taken as known values, and the state space equation of the system is established as

$$R_0(k+1)=R_0\left(k\right)+r\left(k\right),k=1,2,3,\dots ,2L,$$

(31)

where r is the process noise of the system, the mean is 0, and the covariance is ${\sigma }_r$.

The observation equation is as follows:

$$V_0\left(k\right)=V_{ocv}\left(k\right)-I\left(k\right)R_0\left(k\right)-V_{p1}-V_{p2}+\epsilon \left(k\right),$$

(32)

where $\epsilon$ is the observed noise of the system, the mean value is 0, and the covariance is ${\sigma }_{\epsilon }$.

According to (31) and (32), define the state relationship:

$$\begin{array}{ccc}\hfill x_1(k+1|t)& =& x_1\left(k\right)+r\left(k\right),\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill y_1\left(k\right)& =& E{x}_1\left(k\right)+F+\epsilon \left(k\right),\hfill \end{array}$$

(34)

where $E=-I\left(k\right)$ and $F=V_{ocv}\left(t\right)-V_{p1}-V_{p2}$. The SOC and SOH estimation algorithm proposed in this paper can combine some estimation algorithms [69,70,71,72] for studying new methods for linear and nonlinear stochastic systems with colored noises [73,74,75,76,77] and can be applied to other literatures such as industrial control systems [78,79,80,81,82], the information processing and communication systems and so on.

4. Simulation Experiment

4.1. Experimental Condition

When utilizing the MILM-AWUKF algorithm for lithium battery parameter identification and state estimation, it is necessary to construct a test platform for experimentation. This platform consists of a programmable temperature chamber, a battery operated device, and a computing machine. The subject is a lithium battery. The battery specifications are shown in

Table 1. Battery information.

Information	Type	Data
Rate	Voltage	3.2 V
Operation	Voltage	2.7 V–4.2 V
Capacitance		3600 mAh
Rate	Charge current	0.72 A
Max	Discharge current	3.6 A

.

This paper uses the standard constant-current-constant-voltage charging and discharging mode, presenting a nonlinear relationship curve between SOC and OCV. Using sixth-order fit, the SOC-OCV fitting curve is presented in Figure 2 and the fitting equation in (35).

$$\begin{array}{c}\hfill \begin{array}{cc}{V}_{ocv}\left(SOC\right)\hfill & =6.4291\ast {SOC}^6-14.0566\ast {SOC}^5+9.0118\ast {SOC}^4-0.5504\ast {SOC}^3\\ & -0.9967\ast SO{C}^2+0.8016\ast SO{C}^1+3.5414.\hfill \end{array}\end{array}$$

(35)

4.2. Battery Parameters Identification Results

According to the second-order RC equivalent circuit model, the MILM algorithm is used to identify the resistance and capacitance parameters of the equivalent circuit model. The parameter identification results are shown in the figure. Figure 3 shows the MILM parameter identification under the federal urban driving schedule (FUDS). The results show that the parameter identification values converge quickly to stable values at a count of 112 and fluctuate within a small error range. Figure 4 shows the predicted and true values of the terminal voltage. It is clear that the predicted values of the terminal voltage closely follow the true values, with root mean square error (RMSE) values of about 0.0606% and mean absolute error (MAE) values of about 0.0345%. Figure 5 shows that the terminal voltage error range is consistently within 0.5%, indicating that this algorithm can characterize the dynamic characteristics and internal parameters of the battery model very well.

Figure 6 shows the MILM parameter identification results under dynamic stress testing (DST). Unlike the FUDS experiment, the equivalent model parameter identification results are not smooth and exhibit relatively large fluctuation except for $R_0$. The equation of each parameter is derived from (4), which is associated with the characteristics of the input voltage and current under the DST experiment. We infer from these results that the voltage and current do not vary at most moments and the voltage and current values are too large at a certain moment.

However, from subsequent comparison of the predicted and true values of the terminal voltage, we conclude that the errors obtained when using this algorithm are still very small under complex conditions, verifying the reliability and safety of this algorithm.

Figure 7 and Figure 8 present the predicted values and errors of the terminal voltages during DST. The predicted values of the terminal voltages converge to the true values at 187 sampling periods, with an error of about 0.015%, which represents a small range of change. In summary, even though the fluctuation in the identification parameters is large under current shock, it can be seen from the RMSE value of 0.012% and the MAE value of 0.002758% that there is minimal deviation between the predicted and true values of the voltages, which enables us to rely on these predictions with more confidence for the SOC and SOH estimation decisions. This accurate estimation can improve the system’s understanding of the battery state, thus enhancing the performance and efficiency of the battery management system, reflecting the algorithm’s excellent performance in modeling and predicting complex systems.

4.3. SOC Prediction Results

First, the initial values of SOC are set to 0.93. From Figure 9 and Figure 10, with regard to the convergence speed and accuracy, it can be concluded that the EKF has the worst estimation performance, the AWUKF algorithm is better, and the UKF algorithm performance is between these two algorithms. In terms of SOC prediction, the SOC estimated by the AWUKF algorithm overlaps with the true values after 217 cycles in the battery state retention time interval, and the SOC estimated by the UKF algorithm overlaps with the true value for the first time after 662 cycles. In terms of error, After 106 cycles, the error of the AWUKF algorithm remains at about 0.1% and the error of the UKF algorithm remains within 3.5%. The SOC estimation accuracy under the DST experimental conditions shows that the proposed algorithm can not only improve the dynamic performance, but can also increase the estimation convergence speed.

The two indicators RMSE and MAE are used for evaluation. As shown in

Table 2. SOC estimation results under DST conditions.

	RMSE	MAE
AWUKF	0.0133	0.0011
UKF	0.0299	0.0211
EKF	0.0442	0.0350

, the values of RMSE and MAE for the AWUKF algorithm are about 1%. Therefore, the estimation error can be greatly reduced by using this algorithm for state estimation.

Firstly, the initial values of SOC are set to 0.95; then, the EKF, UKF, and AWUKF algorithms are used to estimate the SOC under FUDS experimental conditions. Figure 11 shows the SOC prediction results using the different algorithms. It can be seen that the curves of the AWUKF and UKF algorithms are closest to the true values. However, with delay in time, compared to the UKF algorithm, the AWUKF algorithm can track the actual values well. Figure 12 shows the error curves for the different algorithms. It is clear that the UKF algorithm has an increasing error, while the AWUKF algorithm error becomes smaller and smaller, with a maximum error of about 2.5% after 5000 cycles. This confirms the accuracy and convergence of this algorithm.

As can be seen from

Table 3. SOC estimation results under FUDS conditions.

	RMSE	MAE
AWUKF	0.0285	0.0188
UKF	0.0584	0.0324
EKF	0.0907	0.0774

, using RMSE and MAE to evaluate SOC estimation errors, the AWUKF algorithm has the smallest error among the three algorithms. This indicates that this model has good predictive ability for SOC estimation of lithium batteries.

4.4. SOH Prediction Results

Combining (33) and (34), using the EKF, UKF, and AWUKF algorithms to calculate the internal resistance. From Figure 13 and Figure 14, it can be seen that the EKF algorithm provides the worst estimation curve. The estimation curves of the UKF and AWUKF algorithms are very close. Because the resistor itself is small, as can be seen by examination of the local zoomed-in image, the AWUKF algorithm is closer to the true value. It is clear from the error curves that the AWUKF algorithm quickly converges to a range close to the true values, and the error decreases.

Figure 15 and Figure 16 show the values of the internal resistance and error curves estimated by each algorithm under FUDS experimental conditions. It can be seen from the figures that the highest error of the AWUKF algorithm is 0.0041, that of the UKF algorithm is 2.15 times higher than the AWUKF algorithm, and that of the EKF algorithm is 4.68 times higher.

5. Conclusions

In this paper, an equivalent circuit model is adopted to simplify the chemical processes inside lithium batteries. The parameters of lithium batteries are obtained using the MILM algorithm, which provides superior parameter estimation results and lays a foundation for SOC estimation. The AWUKF algorithm is applied to address the sensitivity to noise issue by adaptively adjusting the sigma point weights based on state and measurement residual vectors, which improves the convergence speed of SOC estimation. The simulation indicates that the SOC estimation error of the AWUKF algorithm is below 5.79% for the FUDS experiment and 0.0376 % for the DST experiment. At the same time, the SOH of the battery is estimated using the AWUKF. From the results, we can conclude that the AWUKF algorithm shows faster convergence. In other words, this algorithm has strong applicability under any experimental conditions, and can achieve the detection and estimation of electric vehicles battery properties in real-time.