Joint Estimation of SOC and SOH Based on Kalman Filter Under Multi-Time Scale

Abstract

Optimizing the accurate estimation algorithms for the State of Charge (SOC) and State of Health (SOH) of power batteries is crucial for improving the performance of electric vehicles. This paper takes lithium-ion batteries as the research object. The Singular Value Decomposition-Unscented Kalman Filter (SVDUKF) at a micro-time scale is used to estimate the battery’s State of Charge, and the traditional Extended Kalman Filter (EKF) at a macro-time scale is used to estimate impedance parameters and capacity. The two filters operate alternately, with the output of one serving as the input for the other, thereby establishing a joint estimation method for SOC and SOH based on the SVDUKF-EKF under a multi-time scale. The joint estimation method is verified under the Dynamic Stress Test (DST) condition and Federal Urban Driving Schedule (FUDS) condition. The results show that the SOH estimation error is within 2% under the DST condition and within 1% under the FUDS condition. The method exhibits high estimation accuracy and stability under both conditions.

1. Introduction

With the rapid development of electric vehicles, power batteries, as their core power source, have become increasingly critical [1]. The Battery Management System (BMS), which integrates battery state monitoring, protection, and control, is responsible for ensuring the safe, stable, and efficient operation of battery packs and constitutes a crucial link in achieving efficient battery management.

State estimation of batteries is one of the core functions of BMS. Accurate state estimation enables BMS to manage batteries in a more refined manner. It facilitates early warning for batteries, alleviates drivers’ range anxiety, reduces the operational risks of electric vehicles, extends battery service life, and also provides a foundation for the energy optimization of the entire vehicle. The estimation of battery State of Charge (SOC) and State of Health (SOH) is the foundation of battery state estimation [2]. SOC estimation directly affects State of Energy (SOE) estimation, and accurate SOC and SOH estimation is crucial for the prediction and control of State of Power (SOP). If the SOC is underestimated, the battery may enter the discharge phase in advance, leading to a decrease in SOP; conversely, the battery may fail to provide the required power output when needed. The accuracy of SOC estimation is heavily influenced by battery SOH. As batteries degrade, SOC estimation algorithms may produce large estimation errors. The inaccurate SOC estimations may mislead the battery SOH calibration. Therefore, simultaneous estimation of SOC and SOH is quite beneficial [3,4].

There are various forms of definitions for State of Charge (SOC), and the most widely used one is the ratio of the battery’s remaining capacity at the current moment to its total capacity [5], as shown in Equation (1).

$$\mathrm{SOC}={\displaystyle \frac{Q_{\mathrm{c}}}{Q_{\max }}}$$

(1)

In the equation, $Q_{\mathrm{c}}$ is the remaining capacity of the battery at the current moment, and $Q_{\max }$ is the maximum available capacity of the battery when it is fully charged at the current moment.

SOC is related to factors of a battery such as its temperature, charge–discharge rate, number of cycles, and degree of degradation [6]. Currently, the commonly used SOC estimation methods are the direct method, model-based method, and data-driven method.

The direct method estimates the battery SOC by directly measuring parameters such as voltage, current, and temperature. It mainly includes the Open Circuit Voltage (OCV) Method [7,8], Impedance Spectroscopy [9], and Ah Integration Method [10]. This method is relatively simple and easy to implement.

Model-based methods mainly utilize the physical or chemical models of batteries [11]. By measuring parameters such as the battery’s voltage, current, and temperature, filtering algorithms are used to estimate the battery’s SOC in real-time. Common models include Electrochemical Models [12], Equivalent Circuit Models [13], and Black-box Models [14]. The residual between the measured terminal voltage and the terminal voltage of the established model is used as a feedback signal to realize closed-loop correction of SOC estimation. Commonly used estimation algorithms mainly include Kalman Filter-Based Algorithms [15,16], Particle Filter-Based Algorithms [17], Least Squares-Based Filters [18], H∞ Filters [19], State Observers [20], Luenberger Observers [21], etc.

Data-driven methods regard the interior of the battery as a black box, without focusing on its internal physical and chemical changes, and do not need to establish a specific battery system model. They only need to collect relevant data, such as current, voltage, and temperature during battery operation, and obtain the estimated value of battery SOC through training [22]. This method requires a large amount of experimental data, and the data quality directly affects the accuracy of the final SOC estimation. This method mainly includes neural network methods [23,24], support vector machines [25], machine learning [26], etc.

The SOH of a battery refers to the changes in performance parameters, such as capacity, internal resistance, and lifespan, during its service life. It reflects the degree to which the battery’s key parameters deviate from the set parameters and serves as a crucial indicator for evaluating battery performance and reliability. There are mainly two definitions of SOH. One is SOH characterized by capacity degradation, as shown in Equation (2); the other is SOH characterized by internal resistance increase, as shown in Equation (3). This paper adopts the SOH characterized by capacity degradation. Generally, a battery is considered to have reached the end of its life when its capacity degrades to 80% of the initial capacity [27].

$$\mathrm{SOH}={\displaystyle \frac{C_{\max }}{C_n}}$$

(2)

In the equation, $C_{\max }$ is the maximum available capacity of the current battery, and $C_{\mathrm{n}}$ is the nominal capacity of the battery.

$$\mathrm{SOH}={\displaystyle \frac{R_{\mathrm{now}}-R_{\mathrm{new}}}{R_{\mathrm{old}}-R_{\mathrm{new}}}}$$

(3)

In the equation, $R_{\mathrm{now}}$ is the ohmic internal resistance of the battery at the current moment, $R_{\mathrm{old}}$ is the ohmic internal resistance of the battery at the end of its life, and $R_{\mathrm{new}}$ is the initial ohmic internal resistance of the battery.

The estimation methods of SOH mainly include experimental analysis methods, model-based methods [28], and data-driven methods [29,30].

The experimental method directly or indirectly calculates SOH through experimentally measured data, such as OCV and impedance parameters. Although this method has a small amount of calculation, it requires experimental research on battery life, which takes a long time. It is difficult to implement in on-board BMS and is not suitable for online SOH estimation.

Model-based methods include the Kalman filter, particle filter, least squares method, and others. Model-based methods typically employ two alternately operating filters to perform joint estimation of SOC and SOH.

Data-driven SOH estimation methods mainly include artificial neural networks [31], Support Vector Machines (SVM) [32], Gaussian process regression [33], deep learning [34,35], etc.

Currently, the estimation of SOC and SOH primarily faces key challenges, such as insufficient model accuracy and parameter time-variability, which impact the reliability and precision of battery management systems. To further improve the estimation accuracy of battery SOC and SOH based on existing research, it is necessary to establish a more precise battery model with model parameters that can be updated online to adapt to various operating conditions and battery life stages. The future development trends mainly focus on aspects such as the integration of multiple methods, online update and adaptive algorithms, the application of big data and artificial intelligence, and multi-scale and multi-physics field coupling modeling [36,37,38,39,40].

To improve the accuracy and stability of SOC estimation, this paper, based on the traditional Unscented Kalman Filter (UKF), adopts Singular Value Decomposition (SVD) to replace Cholesky Decomposition, thereby developing the Singular Value Decomposition-Unscented Kalman Filter (SVDUKF).

Considering that battery capacity fading and parameter changes occur on a relatively long time scale, the Extended Kalman Filter (EKF) is used on the macro-time scale to estimate battery parameters that change slowly, including internal resistance, dual polarization impedance parameters, and current maximum available capacity. Considering that the change in SOC occurs on a short time scale, the Singular Value Decomposition Unscented Kalman Filter (SVDUKF) is used on the micro-time scale to estimate the SOC, which changes more rapidly. The two filters operate alternately, with the output of each serving as an input to the other.

2. Methods

This paper takes ternary lithium-ion power batteries for vehicles as the research object and conducts research on the joint estimation of battery SOC and SOH. Firstly, static capacity tests, aging cycle tests, and Hybrid Pulse Power Characteristic (HPPC) tests are carried out on the batteries through a lithium-ion battery test platform to establish the variation law of OCV with SOC and SOH. Secondly, a second-order RC model of the battery is established, and the Adaptive Forgetting Factor Recursive Least Square (AFFRLS) method is used to identify the battery parameters. Finally, a multi-time-scale joint estimation method for SOC and SOH based on the SVDUKF-EKF is proposed.

2.1. Lithium-Ion Battery Testing

This paper uses the ternary EVE-INR-18650/33V vehicle battery cell as the research object and employs the CT-4008Tn-5V12A charge–discharge tester to build a test platform. To study battery performance and establish a battery test database, static capacity tests, aging tests, and Hybrid Pulse Power Characteristic (HPPC) tests were conducted on the batteries under different aging degrees, providing data support for subsequent parameter identification and state estimation algorithms.

2.1.1. Static Capacity Test

Under a constant temperature of 25 °C, the battery was discharged at a 0.5 C rate until it reached the cut-off voltage. The discharge capacity of the single battery cell during the discharge process was recorded, and the average value of three discharge capacities was taken as the maximum available capacity of the single battery cell. Through the static capacity test, the initial maximum available capacity of the single battery cell was measured to be 3.1 Ah.

2.1.2. Battery Aging Cycle Test

To study the battery’s SOH and establish the dataset required for SOH estimation, the research team conducted an aging cycle test covering the full SOC range from 100% SOC to 0% SOC. Figure 1 shows the battery aging curve obtained from the test. The battery’s discharge capacity gradually decreases as the number of cycles increases. This phenomenon is mainly caused by the increase in internal resistance due to battery aging. The capacity degradation of a battery is not a smooth process. Local fluctuations in the curve occur due to capacity regeneration resulting from battery rest.

2.1.3. Hybrid Pulse Power Characteristic (HPPC) Test

To evaluate the dynamic performance of the battery, the research team conducted HPPC tests on lithium-ion batteries and obtained the external characteristics of the batteries under different depths of discharge.

By analyzing the HPPC test data at different battery aging stages, the corresponding SOC–OCV relationships at each aging stage were extracted. A 5th-order polynomial fitting method was adopted to generate the SOC–OCV curves under different aging degrees, as shown in Figure 2a. It can be seen from Figure 2a that there are significant differences in the SOC–OCV curves under different aging degrees. The maximum OCV difference at the same SOC between the SOH = 100% and SOH = 80% can reach 248 mV. This indicates that the impact of aging on the battery’s OCV cannot be ignored.

The voltage characteristic curves from HPPC tests under different aging degrees were analyzed, and a 6th-order polynomial surface fitting method was used to establish the relationship between OCV, SOC, and SOH, as shown in Figure 2b. The root mean square error (RMSE) of the fit is 0.0146 V.

A three-dimensional surface describing the variation of battery OCV with SOC and SOH was constructed using HPPC test data under different SOH conditions, laying a foundation for the joint estimation of battery SOC and SOH.

2.2. Mathematical Model of Lithium-Ion Batteries

Figure 3 shows the second-order RC equivalent circuit model of lithium-ion batteries. The two RC components reflect the electrochemical polarization and concentration polarization inside the battery, respectively, and the corresponding mathematical model of this circuit model is given by Equation (4).

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}{U}_1(t)}{\mathrm{d}t}}=-{\displaystyle \frac{1}{R_1{C}_1}}{U}_1(t)+{\displaystyle \frac{1}{C_1}}I(t)\\ \begin{array}{c}{\displaystyle \frac{\mathrm{d}{U}_2(t)}{\mathrm{d}t}}=-{\displaystyle \frac{1}{R_2{C}_2}}{U}_2(t)+{\displaystyle \frac{1}{C_2}}I(t)\\ U(t)=U_{\mathrm{OCV}}(t)-U_1(t)-U_2(t)-R_{0}I(t)\end{array}\end{array}\right.$$

(4)

In the equation, $U_{\mathrm{OCV}}$ is the nonlinear voltage source; $R_0$ is the ohmic internal resistance; $R_1$ and $R_2$ are the internal resistances of the two RC components, respectively; $C_1$ and $C_2$ are the capacitances of the two RC components, respectively; $U$ is the terminal voltage of the model; $U_1$ is the voltage across $R_1{C}_1$; and $U_2$ is the voltage across $R_2{C}_2$.

When using the SVDUKF method for battery SOC estimation, the most crucial step is to establish the state-space equations of the system. According to the Coulomb counting method (Ah Integration Method):

$$SO{C}_k=SO{C}_{k-1}-{\displaystyle \frac{I_{k}T}{Q_n}}$$

(5)

In the equation, $I_k$ is the current at time $k$ (with the current direction defined as negative for charging and positive for discharging), and $Q_n$ is the current maximum available capacity of the battery. Combined with the second-order RC model of the battery used in this study, its state-space equation can be expressed as Equation (6):

$$\left\{\begin{array}{l}\left[\begin{array}{c}SO{C}_k\\ U_{1,k}\\ U_{2,k}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{-T/{\tau }_1}& 0\\ 0& 0& e^{-T/{\tau }_2}\end{array}\right]\left[\begin{array}{c}SO{C}_{k-1}\\ U_{1,k-1}\\ U_{2,k-1}\end{array}\right]+\left[\begin{array}{c}-T/Q_N\\ R_1\left(1-e^{-T/{\tau }_1}\right)\\ R_2\left(1-e^{-T/{\tau }_2}\right)\end{array}\right]I_k\\ U_k=U_{OCV,k}(SOC)-U_{1,k}-U_{2,k}-R_0{I}_k\end{array}\right.$$

(6)

In the equation, $T$ is the sampling period, $R_0$ is the ohmic internal resistance, $R_1$ is the electrochemical polarization resistance, ${\tau }_1$ is the time constant, $R_2$ is the concentration polarization resistance, ${\tau }_2$ is the time constant, $U$ is the terminal voltage, $U_{OCV}$ is the open-circuit voltage, $U_1$ is the voltage across $R_1{C}_1$, and $U_2$ is the voltage across $R_2{C}_2$.

2.3. SVDUKF-EKF Joint Estimation Algorithm

The idea of joint estimation is to consider the influence of one state quantity in the estimation process of another, thereby enabling the two state quantities to iteratively operate alternately. Since battery capacity fading and parameter changes occur over a relatively long time scale, while SOC changes over a short time scale, parameters and capacity can be estimated in one filter, and SOC, which changes more rapidly, can be estimated in another filter. The two filters operate alternately, with the output of each serving as an input for the other.

The changes in battery parameters and capacity are relatively slow, so they do not need to be updated as frequently as SOC. To reduce computational complexity, a multi-time-scale joint estimation method for SOC and SOH is adopted, which reduces the computational load while ensuring high estimation accuracy. The principle of joint estimation is illustrated in Figure 4, where $\widehat{x}$ represents the state, and $\widehat{\theta }$ represents the parameter. The state filter and the parameter filter operate on time scales $l$ and $k$, respectively, with $l$ < $k$.

In a relatively short period of time, battery parameters change slowly, so the parameter estimate at the next moment can be considered as the estimate at the previous moment plus a small disturbance. To ensure the stability and convergence of the algorithm, the parameter estimator and the SOC estimator should share key information (e.g., terminal voltage, charge–discharge current, and prior estimation results). The state-space equation for the joint estimation of battery parameters and SOC can be expressed as (7):

$$\left\{\begin{array}{l}{x}_{k,l}=f(x_{k,l-1},{\theta }_k,u_{k,l-1})+w_{k,l-1}^x\\ {\theta }_{k+1}={\theta }_k+w_k^{\theta }\\ y_{k,l-1}=g(x_{k,l-1},{\theta }_k,u_{k,l-1})+v_{k,l-1}\end{array}\right.$$

(7)

In the equation, the subscripts $k$ and $l$ are the long time scale for Estimation $\theta$ and the short time scale for Estimation $x$, which satisfy $t_{k,0}=t_{k-1,L_Z}$, where $L_Z$ is the time conversion scale. $x_{k,l}$ is the state at time $t_{k,l}=t_{k,0}+l\times \mathsf{\Delta }t,(1\le l\le L_Z)$, ${\theta }_k$ is the parameter state at time $t_k$, and ${\theta }_k$ is constant within a micro-time scale $t_{k,0}\le t_{k,l}\le t_{k,L_Z-1}$; $u_{k,l-1}$ is the system input at time $t_{k,l-1}$; $y_{k,l-1}$ is the system observation output at time $t_{k,l-1}$; $w_{k,l-1}^x$ and $w_k^{\theta }$ are the process noises with a mean of 0 for state estimation and parameter estimation, and their covariances are $Q_{k,l-1}^x$ and $Q_{k,l-1}^x$, respectively; $v_{k,l-1}$ is the measurement noise of the system with a mean of 0, and its covariance is $R_{k,l-1}$.

Combined with the nonlinear discrete system of power batteries established in Equation (6), Equation (7) can be rewritten as Equation (8).

$$\left\{\begin{array}{l}{x}_{k,l}=f(x_{k,l-1},{\theta }_k,u_{k,l-1})=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{-T/{\tau }_{1,k}}& 0\\ 0& 0& e^{-T/{\tau }_{2,k}}\end{array}\right]x_{k,l-1}+\left[\begin{array}{c}-T/Q_{n,k}\\ R_{1,k,l}\left(1-e^{-T/{\tau }_{1,k}}\right)\\ R_{2,k,l}\left(1-e^{-T/{\tau }_{2,k}}\right)\end{array}\right]I_{k,l}\\ U_{k,l}=g(x_{k,l-1},{\theta }_k,u_{k,l-1})=U_{OCV,k,l}(SO{C}_{k,l},Q_{n,k})-U_{1,k,l}-U_{2,k,l}-R_{0,k}{I}_{k,l}\\ {\theta }_{k+1}={\theta }_k\end{array}\right.$$

(8)

In the equation, $I_{k,l}$ is the input current, $U_{k,l}$ is the output terminal voltage, $Q_n$ is the maximum available capacity at the current moment, $x_{k,l}={\left[\begin{array}{ccc}SO{C}_{k,l}& U_{1,k,l}& U_{2,k,l}\end{array}\right]}^{\mathrm{T}}$, ${\theta }_k={[\begin{array}{cccccc}{R}_{0,k}& R_{1,k}& C_{1,k}& R_{2,k}& C_{2,k}& Q_{n,k}\end{array}]}^{\mathrm{T}}$, $U^{\mathrm{OCV},k,l}$ is the open-circuit voltage at time $t_{k,l}$, $U_{1,k,l}$ is the voltage across the $R_1{C}_1$ link, $U_{2,k,l}$ is the voltage across the $R_2{C}_2$ link, ${\tau }_{1,k}$ and ${\tau }_{2,k}$ are the time constants of the two links $R_1{C}_1$ and $R_2{C}_2$, ${\tau }_{1,k}=R_{1,k}{C}_{1,k}$, ${\tau }_{2,k}=R_{2,k}{C}_{2,k}$.

From this, the state transition matrix and measurement matrix of the SVDUKF-EKF algorithm are obtained:

$$\left\{\begin{array}{l}{A}_{k,l-1}^x=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{-T/{\tau }_{1,k,l}}& 0\\ 0& 0& e^{-T/{\tau }_{2,k.l}}\end{array}\right]\\ B_{k,l-1}^x=\left[\begin{array}{c}-T/Q_{n,k}\\ R_{1,k}\left(1-e^{-T/{\tau }_{1,k}}\right)\\ R_{2,k}\left(1-e^{-T/{\tau }_{2,k}}\right)\end{array}\right]\\ C_{k,l-1}^x=\left[\begin{array}{ccc}{\left.{\displaystyle \frac{\mathsf{\partial }{U}_{OCV,k,l}(SOC,Q_{n,k})}{\mathsf{\partial }SOC}}\right|}_{SOC={\widehat{SOC}}_{k,l}^{-}}& -1& -1\end{array}\right]\end{array}\right.$$

(9)

$$\left\{\begin{array}{l}{A}_k^{\theta }=1\\ B_k^{\theta }=0\\ C_k^{\theta }={\left.{\displaystyle \frac{\mathrm{d}g(x_{k,0},\theta ,u_{k,0})}{\mathrm{d}\theta }}\right|}_{\theta ={\widehat{\theta }}_k^{-}}={\displaystyle \frac{\mathsf{\partial }g(x_{k,0},{\widehat{\theta }}_k^{-},u_{k,0})}{\mathsf{\partial }{\widehat{\theta }}_k^{-}}}+{\displaystyle \frac{\mathsf{\partial }g(x_{k,0},{\widehat{\theta }}_k^{-},u_{k,0})}{\mathsf{\partial }{x}_{k,0}}}{\displaystyle \frac{\mathrm{d}{x}_{k,0}}{\mathrm{d}{\widehat{\theta }}_k^{-}}}\end{array}\right.$$

(10)

Among them:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathsf{\partial }g(x_{k,0},{\widehat{\theta }}_k^{-},u_{k,0})}{\mathsf{\partial }{\widehat{\theta }}_k^{-}}}=\left[\begin{array}{cccccc}-I_{k,0}& 0& 0& 0& 0& {\displaystyle \frac{\mathsf{\partial }{U}_{OCV}(SOC,Q_n)}{\mathsf{\partial }SO{C}_{k,0}}}{\displaystyle \frac{\mathsf{\partial }SO{C}_{k,0}}{\mathsf{\partial }{\widehat{Q}}_{n,k}^{-}}}\end{array}\right]\\ \begin{array}{l}{\displaystyle \frac{\mathsf{\partial }g(x_{k,0},{\widehat{\theta }}_k^{-},u_{k,0})}{\mathsf{\partial }{x}_{k,0}}}=C_{k,0}^x\\ {\displaystyle \frac{\mathrm{d}{x}_{k,0}}{\mathrm{d}{\widehat{\theta }}_k^{-}}}=\left[\begin{array}{cccccc}0& a_1& a_2& 0& 0& 0\\ 0& 0& 0& a_3& a_4& 0\\ 0& 0& 0& 0& 0& a_5\end{array}\right]+A_{k-1}^x{\displaystyle \frac{\mathrm{d}{x}_{k-1,L_Z-1}}{\mathrm{d}{\widehat{\theta }}_k^{-}}}\end{array}\end{array}\right.$$

(11)

Among them:

$$\left\{\begin{array}{l}{a}_1={\displaystyle \frac{U_{1,k-1,L_Z-1}T}{R_{1,k-1}^2{C}_{1,k-1}}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_1}}}-{\displaystyle \frac{I_{k-1,L_Z-1}T}{R_{1,k-1}{C}_{1,k-1}}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_1}}}-I_{k-1,L_Z-1}({\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_1}}}-1)\\ a_2={\displaystyle \frac{U_{1,k-1,L_Z-1}T}{R_{1,k-1}{C}_{1,k-1}^2}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_1}}}-{\displaystyle \frac{I_{k-1,L_Z-1}T}{C_{1,k-1}^2}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_1}}}\\ a_3={\displaystyle \frac{U_{2,k-1,L_Z-1}T}{R_{2,k-1}^2{C}_{2,k-1}}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_2}}}-{\displaystyle \frac{I_{k-1,L_Z-1}T}{R_{2,k-1}{C}_{2,k-1}}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_2}}}-I_{k-1,L_Z-1}({\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_2}}}-1)\\ a_4={\displaystyle \frac{U_{2,k-1,L_Z-1}T}{R_{2,k-1}{C}_{2,k-1}^2}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_2}}}-{\displaystyle \frac{I_{k-1,L_Z-1}T}{C_{1,k-1}^2}}{\mathrm{e}}^{-{\displaystyle \frac{T}{{\tau }_2}}}\\ a_5={\displaystyle \frac{I_{k-1,L_Z-1}T}{{({\widehat{Q}}_{n,k}^{-})}^2}}\end{array}\right.$$

(12)

Based on the above derivation, a joint estimation method for SOC and SOH based on the SVDUKF-EKF under a multi-time scale is established. The steps within one cycle are as follows:

(1)

Initialize the states, parameters, and their corresponding error covariances:

$$\left\{\begin{array}{l}{\widehat{\theta }}_0=E({\theta }_0),P_0^{\theta }=E[({\theta }_0-{\widehat{\theta }}_0){({\theta }_0-{\widehat{\theta }}_0)}^T]\\ {\widehat{x}}_{0,0}=E(x_{0,0}),P_{0,0}^x=E[(x_{0,0}-{\widehat{x}}_{0,0}){(x_{0,0}-{\widehat{x}}_{0,0})}^T]\end{array}\right.$$

(13)

(2)

At the macro-time scale, perform the one-step prediction for the parameters and their error covariance matrix using the EKF algorithm:

$$\left\{\begin{array}{l}{\widehat{\theta }}_{k|k-1}={\widehat{\theta }}_{k-1}\\ P_{k|k-1}^{\theta }=P_{k-1}^{\theta }+Q_{k-1}^{\theta }\end{array}\right.$$

(14)

(3)

At the micro-time scale, perform the one-step prediction for the state and its error covariance matrix using the SVD-UKF algorithm:

(a)

Perform UT transformation using SVD decomposition to construct (2n + 1) Sigma points:

$$P_{k-1,l-1}^x=U_{k-1,l-1}\left[\begin{array}{cc}{S}_{k-1,l-1}& 0\\ 0& 0\end{array}\right]V_{k-1,l-1}^{\mathrm{T}}$$

(15)

$$\left\{\begin{array}{l}{\chi }_{k-1,l-1}^0={\widehat{x}}_{k-1,l-1}\\ {\chi }_{k-1,l-1}^i={\widehat{x}}_{k-1,l-1}+{\left[\sqrt{\left(n+\lambda \right)}{U}_{k-1,l-1}\sqrt{S_{k-1,l-1}}\right]}_i,i=1,2,\dots ,n\\ {\chi }_{k-1,l-1}^i={\widehat{x}}_{k-1,l-1}-{\left[\sqrt{\left(n+\lambda \right)}{U}_{k-1,l-1}\sqrt{S_{k-1,l-1}}\right]}_i,i=n+1,n+2,\dots ,2n\end{array}\right.$$

(16)

In the equation, n is the dimension of the state quantity, and λ is the scaling factor.

(b)

Calculate the mean and covariance of the one-step prediction of the state variables:

$${\chi }_{k-1,l|l-1}^i=f\left({\chi }_{k-1,l-1}^i,u_{k-1,l}\right)$$

(17)

$$\left\{\begin{array}{l}{\widehat{x}}_{k-1,l|l-1}={\displaystyle {\displaystyle \sum }_{i=0}^{2n}}{\omega }_m^i{\chi }_{k-1,l|l-1}^i\\ P_{k-1,l|l-1}^x={\displaystyle {\displaystyle \sum }_{i=0}^{2n}}{\omega }_c^i{\left({\chi }_{k-1,l|l-1}^i-{\widehat{x}}_{k-1,l|l-1}\right)}^{\mathrm{T}}+Q_{k-1,l-1}^x\end{array}\right.$$

(18)

(4)

Measurement update of the SVD-UKF algorithm under the micro-time scale:

(a)

Perform UT transformation again on the predicted mean and covariance using SVD decomposition to generate new (2n + 1) Sigma points:

$$P_{k-1,l|l-1}^x=U_{k-1,l|l-1}\left[\begin{array}{cc}{S}_{k-1,l|l-1}& 0\\ 0& 0\end{array}\right]V_{k-1,l|l-1}^{\mathrm{T}}$$

(19)

$$\left\{\begin{array}{l}{\chi }_{k-1,l|l-1}^0={\widehat{x}}_{k-1,l|l-1}\\ {\chi }_{k-1,l|l-1}^i={\widehat{x}}_{k-1,l|l-1}+{\left[\sqrt{\left(n+\lambda \right)}{U}_{k-1,l|l-1}\sqrt{S_{k-1,l|l-1}}\right]}_i,i=1,2,\dots ,n\\ {\chi }_{k-1,l|l-1}^i={\widehat{x}}_{k-1,l|l-1}-{\left[\sqrt{\left(n+\lambda \right)}{U}_{k-1,l|l-1}\sqrt{S_{k-1,l|l-1}}\right]}_i,i=n+1,n+2,\dots ,2n\end{array}\right.$$

(20)

(b)

Calculate the mean of the observation variables based on the (2n + 1) Sigma points obtained in step (a), and update the variance matrix:

$$\left\{\begin{array}{c}{y}_{k-1,l|l-1}^i=g\left({\chi }_{k-1,l|l-1}^i,u_{k-1,l}\right)\\ {\widehat{y}}_{k-1,l|l-1}={\displaystyle {\displaystyle \sum }_{i=0}^{2n}}{\omega }_m^i{y}_{k-1,l|l-1}^i\end{array}\right.$$

(21)

$$\left\{\begin{array}{l}{P}_{yy}^x={\displaystyle {\displaystyle \sum }_{i=0}^{2n}}{\omega }_c^i\left(y_{k-1,l|l-1}^i-{\widehat{y}}_{k-1,l|l-1}\right){\left(y_{k-1,l|l-1}^i-{\widehat{y}}_{k-1,l|l-1}\right)}^{\mathrm{T}}+R_{k-1,l-1}^x\\ P_{xy}^x={\displaystyle {\displaystyle \sum }_{i=0}^{2n}}{\omega }_c^i\left({\chi }_{k-1,l|l-1}^i-{\widehat{x}}_{k-1,l|l-1}\right){\left(y_{k-1,l|l-1}^i-{\widehat{y}}_{k-1,l|l-1}\right)}^{\mathrm{T}}\end{array}\right.$$

(22)

(c)

Calculate the Kalman gain, and update the system state and error covariance:

$$K_{k-1,l}^x={\displaystyle \frac{P_{xy}^x}{P_{yy}^x}}$$

(23)

$$\left\{\begin{array}{l}{\widehat{x}}_{k-1,l}={\widehat{x}}_{k-1,l|l-1}+K_{k-1,l}^x\left(y_{k-1,l}-{\widehat{y}}_{k-1,l|l-1}\right)\\ P_{k-1,l}^x=P_{k-1,l|l-1}^x-K_{k-1,l}^x{P}_{yy}^x{K}_{k-1,l}^x\end{array}\right.$$

(24)

(d)

Repeat steps (a)–(c) in (4) until $l=L_Z$, then exit the micro-time scale state estimation and set:

$${\widehat{y}}_{k,0}={\widehat{y}}_{k-1,L_Z},{\widehat{x}}_{k,0}={\widehat{x}}_{k-1,L_Z}$$

(25)

(5)

Measurement update of the EKF algorithm parameters and their error covariances under the macro-time scale:

$${\widehat{y}}_{k,0}=g({\widehat{x}}_{k,0},{\widehat{\theta }}_{k|k-1},u_{k,0})$$

(26)

$$K_k^{\theta }=P_{k|k-1}^{\theta }{(C_k^{\theta })}^T(C_k^{\theta }{P}_{k|k-1}^{\theta }{(C_k^{\theta })}^T+R_{k-1}^{\theta })$$

(27)

$${\widehat{\theta }}_k={\widehat{\theta }}_{k|k-1}+K_k^{\theta }(y_{k,0}-{\widehat{y}}_{k,0})$$

(28)

$$P_k^{\theta }=(I-K_k^{\theta }{C}_k^{\theta })P_{k|k-1}^{\theta }$$

(29)

(6)

Set $k=k+1$, then continue the iteration starting from step (2) until completion.

3. Results

Based on the established lithium-ion battery model and the aforementioned multi-time-scale SVDUKF-EKF joint estimation method, a joint estimation model was built in Matlab/Simulink (R2021a) to estimate SOC and SOH. The overall flow of the algorithm is shown in Figure 5. The joint estimation uses the SVDUKF on a micro-time scale to estimate the battery SOC, and the EKF on a macro-time scale to estimate battery parameters, including internal resistance, dual polarization impedance parameters, and current maximum available capacity. The two filters serve as inputs for each other and operate alternately, forming the SVDUKF-EKF algorithm.

Under the Dynamic Stress Test (DST) and Federal Urban Driving Schedule (FUDS) conditions, the proposed SVDUKF-EKF joint estimation algorithm was compared with the dual Extended Kalman Filter (EKF-EKF) method. The time conversion scale of both the EKF-EKF algorithm and the SVDUKF-EKF algorithm was set to 60 s.

3.1. DST Operating Condition Verification

To verify the multi-time-scale SVDUKF-EKF joint estimation algorithm under the DST condition, the initial SOC of the state estimators in both the SVDUKF-EKF and EKF-EKF algorithms is set to 80% (true SOC value: 100%), and the initial values of the parameter estimators are as follows: ${\theta }_0={\left[\begin{array}{cccccc}0.028& 0.049& 3527.2& 0.0058& 255.1508& 2.9\end{array}\right]}^{\mathrm{T}}$. The impedance parameters are obtained through the AFFRLS method, with the initial value of the maximum available capacity set to 2.9 Ah (actual value: 2.9588 Ah). The simulation results are shown in Figure 6 and Figure 7.

Figure 6 shows the terminal voltage estimation and its error of the multi-time-scale SVDUKF-EKF. It can be seen from the figure that under the DST condition, the algorithm can converge within 100 s, and the terminal voltage estimation error is basically within 50 mV. After 12,000 s, the terminal voltage error increases because the internal chemical reaction of the battery becomes intense when the SOC is lower than 20%.

Figure 7 presents a comparison of SOC estimation between the multi-time-scale EKF-EKF and SVDUKF-EKF. After convergence, the Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and maximum absolute error of SOC estimation for the two methods are listed in

Table 1. Comparison of SOC Estimation Errors Between the Two Algorithms Under DST Condition.

	EKF-EKF	SVDUKF-EKF
RMSE (%)	1.9007	1.0531
MAE (%)	1.7167	0.9710
Maximum Absolute Error (%)	2.9513	1.7862

.

As can be seen from Figure 7 and Table 1, both the multi-time-scale SVDUKF-EKF and EKF-EKF can converge within 100 s. However, the SOC estimation accuracy of the SVDUKF-EKF is higher than that of the EKF-EKF, and its RMSE, MAE, and maximum absolute error are all lower than those of the EKF-EKF.

Figure 8 and Figure 9 show the parameter estimation and SOH estimation results of the SVDUKF-EKF algorithm when the initial capacity is set to 2.9 Ah. The SOH is obtained by dividing the estimated capacity by the initial capacity. It can be seen from Figure 9b that the SOH estimation error gradually decreases. After 12,000 s, due to the large deviation of the battery model caused by the low SOC, the SOH estimation error increases. However, the overall error remains within 2%, which confirms the effectiveness of the algorithm.

3.2. Validation Under FUDS Operating Condition

Similar to Section 3.1, to verify the accuracy of the algorithm under the FUDS operating condition, the initial SOC of the state estimator and the initial values of the parameter estimator are set to be consistent with those under the DST operating condition (with a true capacity of 3.0993 Ah). The simulation results are shown in Figure 10 and Figure 11.

Figure 10 shows the terminal voltage estimation and estimation error of the SVDUKF-EKF algorithm. It can be seen from the figure that under the FUDS operating condition, the terminal voltage estimation value of the multi-time-scale SVDUKF-EKF algorithm can converge within 100 s, and the terminal voltage estimation error generally does not exceed 40 mV, indicating high accuracy and a fast convergence speed. Figure 11 presents a comparison of SOC estimation results between the EKF-EKF and SVDUKF-EKF methods under the multi-time scale. After convergence, the Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and maximum absolute error of the SOC estimation for the two methods are listed in

Table 2. Comparison of SOC Estimation Error Indicators Between the Two Algorithms Under FUDS Operating Condition.

	EKF-EKF	SVDUKF-EKF
RMSE (%)	1.6789	1.5611
MAE (%)	1.3723	1.4899
Maximum Absolute Error (%)	4.3017	2.4738

. It can be concluded from Figure 11 and Table 2 that both the multi-time-scale SVDUKF-EKF and EKF-EKF algorithms can achieve SOC estimation convergence within 100 s. Compared with the EKF-EKF algorithm, the RMSE of the SOC estimation of the SVDUKF-EKF algorithm is reduced by 7.02%.

Figure 12 shows the SOH estimation results of the SVDUKF-EKF algorithm. At this time, the reference SOH is 100%, and the initial SOH is 93.57%. It can be seen from the figure that the SOH estimation error gradually decreases and can quickly converge to near the true value within 400 s. This indicates that the algorithm can quickly correct the initial SOH error, and the stabilized SOH estimation error is within 1%, which proves the effectiveness of the algorithm under the FUDS operating condition.

3.3. Robustness Verification

To verify the robustness of the SVDUKF-EKF algorithm, different initial SOC values were set, and a comparison was conducted between the SVDUKF-EKF algorithm and the EKF-EKF algorithm. The SOC estimation results of the two algorithms under the DST condition are shown in Figure 13 and Figure 14. Under the DST condition, when there is an error in the SOC, both algorithms can converge within 400 s. However, in general, the SVDUKF-EKF algorithm has a faster convergence speed and better robustness than the EKF-EKF algorithm.

4. Conclusions

During the battery aging process, changes in model parameters and capacity affect SOC estimation over the long term. Inaccurate estimation of SOC can lead to overcharging and over-discharging of the battery pack, thereby affecting the battery’s SOH. This paper establishes a joint estimation method for SOC and SOH based on the SVDUKF-EKF under a multi-time scale. It uses a SVDUKF on a micro-time scale to estimate the battery’s SOC and an EKF on a macro-time scale to estimate impedance parameters and capacity. The two filters operate alternately, with the output of each serving as an input to the other. The joint estimation method directly handles the dynamic relationship between SOC and SOH simultaneously at the model or algorithm level, enabling real-time interaction and update between the two. The joint estimation method was validated under DST and FUDS conditions, and the results show that it has high estimation accuracy and stability under both conditions.