Health State Prediction of Lithium-Ion Battery Based on Improved Sparrow Search Algorithm and Support Vector Regression

Abstract

The state of health (SOH) prediction of lithium-ion batteries is a pivotal function within the battery management system (BMS), and achieving accurate SOH predictions is crucial for ensuring system safety and prolonging battery lifespan. To enhance prediction performance, this paper introduces an SOH prediction model based on an improved sparrow algorithm and support vector regression (ISSA-SVR). The model uses nonlinear weight reduction (NWDM) to enhance the search capability of the Sparrow algorithm and combines a mixed mutation strategy to reduce the risk of local optimal traps. Then, by analyzing the characteristics of different voltage ranges, the charging time from 3.8 V to 4.1 V, the discharge time of the battery, and the number of cycles are selected as the feature set of the model. The model’s prediction capabilities are validated across various working environments and training sample sizes, and its performance is benchmarked against other algorithms. Experimental findings indicate that the proposed model not only confines SOH prediction errors to within 1.5% but also demonstrates robust adaptability and widespread applicability.

1. Introduction

In recent years, in the context of the increasing energy crisis and environmental challenges, the automotive industry has focused intently on achieving low pollution and high efficiency. Lithium-ion batteries, known for their compact size, low weight, high energy density, substantial output power, and exceptional safety, have emerged as the preferred option for energy storage in electric vehicles [1,2,3]. However, with the charge and discharge behavior of lithium-ion batteries, battery attenuation will be inevitable. In real-world applications, it is necessary to maintain continuous surveillance of the State of Health (SOH) [4] of lithium batteries to forestall significant deterioration in their performance. Accurate predictions of a lithium battery’s SOH allow us to foresee its Remaining Useful Life (RUL) [5] beforehand.

Over the past few decades, extensive research has been conducted on assessing the SOH prediction for batteries. The primary methods for estimating SOH can be categorized into three groups: direct measurement techniques, model-based approaches [6], and data-driven strategies [7].

The degradation mechanistic model analyzes the degradation rule of lithium-ion battery performance from the essence of the internal chemical reaction. Junfu Li et al. proposed a simplified mechanistic model-based estimation method of lithium-ion battery SOC and established the functional relationship between mechanical parameters and battery. This model can extend from a single battery to a battery pack [8]. Issam Baghdadi et al. introduced a battery aging model utilizing the Dakin degradation approach, investigating how current and battery temperature influence the rate of aging [9].

The equivalent circuit model represents the battery as a fundamental circuit setup. Fang et al. constructed such a model and employed the Forgetting Factor Recursive Least Squares method for real-time identification of its parameters. Additionally, they utilized the Double Extended Kalman Filter algorithm to estimate SOC and SOH [10]. Shen et al. introduced a second-order Thevenin equivalent circuit model for determining the parameters of the lithium-ion battery model, incorporating an adaptive genetic algorithm for this purpose [11].

Due to its high applicability and capacity for real-time processing, the data-driven approach is better suited for prediction and modeling tasks. Among the most widely used data-driven techniques are statistical methods and machine learning algorithms. Robert R. Richardson et al. used Gaussian process regression to estimate SOH [12]. Lv Jiechao et al. proposed an adaptive unscented Kalman filter (AUKF) algorithm, utilizing residuals as inputs to accurately forecast. This AUKF algorithm fine-tunes the filter gain and attains optimal estimation by monitoring the variations in the filter residuals [13]. The data-driven approach avoids the need for intricate internal electrochemical assessments or model construction. Oji outlined the strengths and weaknesses of each method, taking into account the essential attributes necessary for precise SOH estimation in practical applications [14]. Guo et al. proposed a SOH estimation model based on the time correlation of charging data and Ensemble SVR and a new health indicator(HI) indicator includes voltage and capacity, A collaborative support vector regression (SVR) model, known as Ensemble SVR (ESVR) has been proposed to build the correlation between HI and SOH [15]. Lin et al. introduced an innovative approach for estimating the SOH by integrating the simulated annealing algorithm with SVR [16]. Wang et al. introduced an RUL prediction model utilizing the artificial bee colony (ABC) algorithm and SVR [17].

The basic intelligent optimization algorithm has many limitations, and many scholars use improved optimization algorithms to optimize SVR hyper-parameters. Li et al. introduced a model for estimating the SOH utilizing an enhanced ant lion optimization (IALO) algorithm. They chose four health indicators that are strongly associated with SOH degradation as the input variables for the SVR model [18]. Qin et al. optimized SVR hyper-parameters by improved particle swarm optimization (IPSO) [19]. Since the selection of SVR parameters directly affects the prediction performance, the optimization of SVR parameters is still a research issue.

This paper introduces a battery health forecast model that utilizes an improved Sparrow Search Algorithm. In this model, the feature set includes the charge time within a specified voltage range for each cycle, the discharge time, and cycles. Building on this, the Improved Sparrow Search Algorithm (ISSA) is employed to determine the optimal parameters for the SVR model. Ultimately, the output of this optimized model is utilized to predict the SOH of the lithium battery. The main contributions and new ideas presented in this paper are as follows: (1) Use NWDM to improve the searchability of the sparrow algorithm and adopt a hybrid mutation strategy to avoid the algorithm becoming stuck in a local optimum. (2) Select the charging time of battery voltage rising from 3.6 V to 4.2 V, the discharging time of the whole discharging process (voltage dropped from 4.2 V to 2.7 V), and the charge/discharge cycle number as the input of the model. (3) Optimize SVR parameters and propose an SOH prediction model ISSA-SVR.

The structure of the remainder of this paper is as follows: Section 2 and Section 3 introduce SVR and SSA. Section 4 illustrates the improved strategy of ISSA. Section 5 analyzes battery data, extracts feature sets, and proposes an SOH prediction model based on ISSA-SVR. Section 6 carries out different experiments to confirm the efficacy and practicality of the proposed model. Section 7 concludes the paper by summarizing the findings.

2. Support Vector Regression

As an extension of the support vector machine (SVM) tailored for regression tasks, SVR boasts accelerated convergence and heightened accuracy, particularly when addressing small sample sizes and nonlinear challenges. It accomplishes this task by utilizing nonlinear mappings to transform SOH feature data into a higher-dimensional space, enabling effective estimation of SOH through the application of linear regression in this transformed space.

T denotes the sample training set and is formulated as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill T=\left\{\left(x_1,y_1\right),\cdots ,\cdots ,\left(x_l,y_l\right)\right\}\in {\left(R^n\times y\right)}^l\end{array}\end{array}$$

(1)

where $x_i$ signifies the input and $y_i$ signifies the corresponding output for each sample within the training set.

The fundamental representation of the SVR hyperplane in a lower-dimensional space is depicted as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left(x_i\right)=w^T{x}_i+b\end{array}\end{array}$$

(2)

where w represents the weight vector of the hyperplane, and b serves as the bias term, determining the distance between the hyperplane and the origin.

During the actual data-fitting process, it is unrealistic for all sample points to lie within the interval band. To address this issue, a technique known as “soft margin” is employed to enhance fitting accuracy. Subsequently, the introduction of slack variables ${\xi }^{\left(\ast \right)}={\left({\xi }_1,{\xi }_1^{\ast },\cdots {\xi }_l,{\xi }_l^{\ast }\right)}^T$ and penalty factor C leads to the formulation of the optimization function as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}\underset{w,b,\xi ,{\xi }^{(\ast )}}{min}{\displaystyle \frac{1}{2}}∥w∥+C{\displaystyle \sum _{i=1}^1}({\xi }_i,{\xi }_i^{{}^{(\ast )}})\hfill \\ s.t.\left\{\begin{array}{c}((w·x_i)+b)-y_i\le \epsilon +{\xi }_i\hfill \\ y_i-((w·x_i)+b)\le \epsilon +{\xi }_i^{{}^{(\ast )}}\hfill \\ {\xi }_i^{\ast }\ge 0,{\xi }_i\ge 0\hfill \end{array}\right.\hfill \end{array}\end{array}\end{array}$$

(3)

where $\epsilon$ is the sample error.

Furthermore, introduce the Lagrangian multipliers and convert the optimizing function to:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}\begin{array}{cc}\hfill {\displaystyle \underset{{\alpha }^{(\ast )},{\eta }^{(\ast )}\in R^{2l}}{max}}=& -{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i,j=1}^l}\left({\alpha }_i^{\ast }-{\alpha }_i\right)\left({\alpha }_j^{\ast }-{\alpha }_j\right)\left(x_i·x_j\right)\hfill \\ \hfill & -\epsilon {\displaystyle \sum _{i=1}^l}\left({\alpha }_i^{\ast }+{\alpha }_i\right)+{\displaystyle \sum _{i=1}^l}{y}_i\left({\alpha }_i^{\ast }-{\alpha }_i\right)\hfill \end{array}\hfill \\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}{\displaystyle {\sum }_{i=1}^l}\left({\alpha }_i^{\ast }-{\alpha }_i\right)=0\hfill \\ C={\alpha }_i^{(\ast )}+{\eta }_i^{(\ast )}\hfill \\ {\alpha }_i^{(\ast )}\ge 0,{\eta }_i^{(\ast )}\ge 0\hfill \end{array}\right.\hfill \end{array}\end{array}\end{array}$$

(4)

where ${\alpha }_i^{\ast }$ and ${\alpha }_i$ are the Lagrange multipliers. Minimize the Lagrangian function and deduce the decision function as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left(x\right)={\displaystyle \sum _{i=1}^n}\left({\alpha }_i^{\ast }-{\alpha }_i\right)k\left(x_i,x_j\right)+b\end{array}\end{array}$$

(5)

where $k(x_i,y_i)$ is the kernel function. Among all kernel functions, the radial basis function (RBF) stands out as the most widely used in machine learning. It can be expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill k\left(x_i,x_j\right)=\exp (-{\displaystyle \frac{{∥x_i-x_j∥}^2}{2{\sigma }^2}})=\exp (-g{∥x_i-x_j∥}^2)\end{array}\end{array}$$

(6)

where $\sigma$ denotes the spread or range of the kernel function and $g={\displaystyle \frac{1}{2{\sigma }^2}}$.

3. Sparrow Search Algorithm

The sparrow search algorithm (SSA) draws primary inspiration from sparrows’ foraging behavior, including their tendency to follow leaders and evade predators while searching for food. The algorithm simulates the sparrow’s behavior to establish a mathematical model and then conducts the global search.

During the optimization process, setting up a hypothetical location for sparrows to explore for food becomes essential. And a matrix composed of sparrows’ position can be obtained as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{X}=\left[\begin{array}{c}{x}_{1,1},x_{1,2}\cdots ,\cdots ,\cdots x_{1,d}\\ x_{2,1},x_{2,2}\cdots ,\cdots ,\cdots x_{2,d}\\ ⋮,⋮,⋮\\ ⋮,⋮,⋮\\ ⋮,⋮,⋮\\ x_{n,1},x_{n,2}\cdots ,\cdots ,\cdots x_{n,d}\end{array}\right]\end{array}\end{array}$$

(7)

where d is the dimension and n is the size of sparrow. The fitness values associated with their positions can be formulated as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill F_{\mathrm{x}}=\left[\begin{array}{c}f\left(\left[x_{1,1},x_{1,2}\cdots ,\cdots ,\cdots x_{1,d}\right]\right)\\ f\left(\left[x_{2,1},x_{2,2}\cdots ,\cdots ,\cdots x_{2,d}\right]\right)\\ ⋮\\ ⋮\\ ⋮\\ f\left(\left[x_{n,1},x_{n,2}\cdots ,\cdots ,\cdots x_{n,d}\right]\right)\end{array}\right]\end{array}\end{array}$$

(8)

In the sparrow population, the sparrows with good fitness are called discoverers, and they give priority to gaining food in the process of foraging. The position update formula for discoverers is:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill X_{i,j}^{t+1}=\left\{\begin{array}{cc}{X}_{i,j}^t·exp\left(-{\displaystyle \frac{t}{\alpha ·T_{max}}}\right)\hfill & \mathrm{if}{R}_2<ST\hfill \\ X_{i,j}^t+Q·L\hfill & \mathrm{if}{R}_2\ge ST\hfill \end{array}\right.\end{array}\end{array}$$

(9)

where t denotes the iteration count at the present moment and $j\in \left(1,d\right)$. $T_{max}$ is a pre-determined, adjustable parameter representing the maximum iterations allowed. $X_{i,j}$ signifies the location of the ith sparrow in the jth dimension. Q is a randomly chosen step size obtained from a normal distribution with a mean of zero and a standard deviation of one. $\alpha$ is a random number and $\alpha \in \left(0,1\right)$. $R_2$ represents a safety value and $R_2\in \left(0,1\right)$. $ST$ stands for the security threshold, and both are random numbers. L is a matrix of 1×d, where every element is set to 1.

In the sparrow population, the sparrows that follow discoverers are called followers. The position update formula for followers is:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill X_{i,j}^{t+1}=\left\{\begin{array}{cc}Q·exp\left({\displaystyle \frac{X_{worst}-X_{x,j}^t}{i^2}}\right)\hfill & \mathrm{if}\mathrm{i}>{\displaystyle \frac{2}{n}}\hfill \\ X_p^{t+1}+\left|X_{x,j}^t-X_p^{t+1}\right|·A^{+}·L\hfill & \mathrm{if}\mathrm{i}\le {\displaystyle \frac{2}{n}}\hfill \end{array}\right.\end{array}\end{array}$$

(10)

where $X_{worst}$ and $X_p^{t+1}$ are the worst position of the current population and the best position of the discoverers. A is a matrix of 1×d with elements of 1 or −1 and $A^{+}=A^T{\left(A{A}^T\right)}^{-1}$.

Upon sensing danger, certain sparrows will exhibit behavior to evade predators. These sparrows are called scouters, and their positions would be updated by:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill X_{i,j}^{t+1}=\left\{\begin{array}{cc}{X}_{best}^t+\beta ·\left|X_{i,j}^t-X_{best}^t\right|\hfill & \mathrm{if}{f}_i>f_g\hfill \\ X_{i,j}^t+K·\left({\displaystyle \frac{\left|X_{i,j}^t-X_{worst}^t\right|}{\left(f_i-f_w\right)+\epsilon }}\right)\hfill & \mathrm{if}{f}_i=f_g\hfill \end{array}\right.\end{array}\end{array}$$

(11)

where $X_{best}$ represents the most favorable position within the current population. The step size $\beta$ is governed by a random number. K influences the direction and magnitude of movement for the sparrows, $K\in \left[-1,1\right]$. $f_g$ and $f_w$ represent the highest and lowest fitness values, respectively, $f_i$ denotes the fitness of the ith individual. To prevent the denominator from being zero, a tiny, constant value $\epsilon$ is also included.

4. Improved Sparrow Search Algorithm

4.1. Discoverer Position Update Strategy

The nonlinear weight-decreasing strategy (NWDS) is very prevalent in IPSO. And the weight increases or decreases adaptively between the maximum and minimum. Adding weight strategy into the search strategy can increase the scope of the initial search, improve the precision of local search in the later stage, and prevent falling into local optimum.

This paper introduces NWDS into the discoverers’ search mechanistic. The weight $\omega$ gradually decreases with the iteration process. During the initial iteration stage, it is beneficial to reduce the impact of random initialization of the sparrow population, enhance the convergence speed and accuracy, and achieve a good balance between global search and jumping out of local optimum. The NWDS in this paper can be expressed as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \omega =sin\left({\displaystyle \frac{\pi ·t}{2T_{max}}}+\pi \right)+1\end{array}\end{array}$$

(12)

Then, the position update formula of discoverers can be renewed as:

$$X_{i,j}^{t+1}=\{\begin{array}{cc}\left(\begin{array}{c}{X}_{ij}^t+\left(X_{best}^t-X_{ij}^t\right)·\omega \hfill \\ ·exp\left(-{\displaystyle \frac{i}{\alpha ·T_{max}}}\right)\hfill \end{array}\right)\hfill & \mathrm{if}{R}_2<{ST}_1\hfill \\ X_{ij}^t+Q·\mathrm{L}·\omega \hfill & \mathrm{if}{R}_2\ge {ST}_1\hfill \end{array}$$

(13)

4.2. Scouter Position Update Strategy

(1)

Gaussian mutation

Gaussian distribution (GD), also known as normal distribution, is a continuous probability distribution model. Its probability density function can be expressed as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left(x|\mu ,{\sigma }^2\right)={\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{e}^{-{\displaystyle \frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}}\end{array}\end{array}$$

(14)

where x is the random variable. $\mu$ and $\sigma$ represents the mean and variance.

This paper adopts a forward direction disturbances strategy based on Gaussian mutation to make the sparrow population more effectively escape from the local optimum. The position of some scouters will be updated as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill X_{i,j}^{t+1}=X_{best}^t+\left(X_{best}^t-X_{i,j}^t\right)\left|Gas\left(\delta \right)\right|\end{array}\end{array}$$

(15)

where $\left|Gas\left(\delta \right)\right|$ is the Gaussian mutation operator.

(2)

Cauchy mutation

Cauchy distribution (CD) is one of the common continuous distributions. Its probability density function can be expressed as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill f\left(x|\theta ,\alpha \right)={\displaystyle \frac{\alpha }{\pi \left({\alpha }^2+{\left(x-\theta \right)}^2\right)}}\end{array}\end{array}$$

(16)

where $\theta$ and $\alpha$ are location parameter and scale parameter.

The position of some scouters will be updated as follows:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill X_{i,j}^{t+1}=X_{i,j}^t+\left(X_{i,j}^t-X_{worst}^t\right)\left|C\left(\delta \right)\right|\end{array}\end{array}$$

(17)

where $\left|C\left(\delta \right)\right|$ is the Cauchy mutation operator.

Compared with GD, CD is more dispersed, the vertex of the axis of symmetry is gentler, and the tail at both ends of the distribution curve is narrower. Such distribution features make the Cauchy distribution more likely to generate large function values and make the range at both ends of the Cauchy distribution more uneven.

(3)

Gravity coefficient

The gravity coefficient based on inertia can prevent the algorithm from falling into local optimization to a certain extent. To balance the local and global performance of the algorithm, the gravity coefficient G is introduced and expressed as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill G=G_{max}+(G_{max}-G_{min})(1-\exp (-{\left({\displaystyle \frac{\alpha t}{T_{max}}}\right)}^2)\end{array}\end{array}$$

(18)

where $G_{max}$ and $G_{min}$ are the maximum and the minimum gravity coefficient, respectively. $\alpha$ is the nonlinear control parameter.

According to the Gauss mutation strategy, Cauchy mutation strategy, and gravity coefficient G, the position update formula of scouters can be renewed as:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill X_{i,j}^{t+1}=\left\{\begin{array}{cc}{X}_{best}^t+G\left(X_{best}^t-X_{i,j}^t\right)\mid Gas\left(\delta \right)\mid \hfill & \mathrm{if}{R}_2<{ST}_2\hfill \\ X_{i,j}^t+G\left(X_{i,j}^t-X_{worst}^t\right)\left|C\left(\delta \right)\right|\hfill & \mathrm{if}{R}_2\ge {ST}_2\hfill \end{array}\right.\end{array}\end{array}$$

(19)

In the optimization strategy, the Gaussian operator expands the ability of local searching, and the Cauchy operator increases the range of reverse solutions.

4.3. ISSA Flow

In ISSA, the NWDS is employed to enhance the discoverers’ ability to conduct an extensive search. This reduction in individuals exceeding boundaries accelerates the convergence rate in later stages. Additionally, Gaussian variation is utilized to refine the scouter’s local search, leveraging current position data to mitigate premature convergence and elevate positional accuracy. Furthermore, Cauchy variation and the gravity coefficient are incorporated to further improve the scouter’s wide-ranging search. These strategies expand the search domain, assisting the population in escaping local minima. The integration of these diverse approaches renders the algorithm more adaptable in optimization, promotes population diversity, augments its capacity to transcend local optima, balances global and local search capabilities, and facilitates the discovery of robust solutions. The workflow of ISSA is shown in Figure 1.

Based on ISSA, the algorithm proceeds through the following steps:

(1)

Determine the training data set $\left(x_i,y_i\right)$.

(2)

Set the main parameters such as P, M, B, $R_2$, ${ST}_1$, ${ST}_2$.

(3)

Set the initial positions of the sparrows, evaluate their fitness levels, and update $X_{best}^t$ and $X_{worst}^t$.

(4)

Update the sparrow position according to Equations (10), (13) and (19).

(5)

Update the positions and re-evaluate the fitness of the subsequent generation of sparrows. and update $X_{best}^t$ and $X_{worst}^t$.

(6)

If the iteration limit is reached, stop the process and output the optimal solution. If not, proceed back to step (4).

Figure 1. Flowchart of ISSA.

Figure 1. Flowchart of ISSA.

5. SOH Prediction Model Based on ISSA-SVR

5.1. Experimental Data

National Space Administration (NASA) provides the open battery data and the batteries B0005, B0006, B0007, and B0018 are selected to verify the effectiveness of the proposed model. Four batteries are charged and discharged in three different working modes at room temperature.

First, initiate charging with constant current (CC) mode at 1.5 A until the battery voltage attains 4.2 V. Following this, maintain constant voltage (CV) mode for charging until the current diminishes to 20 mA. Subsequently, discharge the battery using CC mode at 2 A until the voltages of B0005, B0006, B0007, and B0018 fall to 2.7 V, 2.5 V, 2.2 V, and 2.5 V, respectively. Figure 2 illustrates the decline in capacity over time for these four battery types.

5.2. Feature Extraction

In real-world scenarios, the discharge process of the battery cannot be directly controlled, and the charging process usually adopts the CC-CV approach. Figure 3 shows the charging voltage profiles of batteries with different SOH. As the battery’s capacity diminishes, polarization becomes more evident. A fresh battery requires 2587 s to attain the predefined voltage limit. In contrast, a battery at 75% SOH only needs about 1874 s, representing just 72.4% of the time taken by a new battery.

In general, the voltage of a lithium-ion battery is restricted to a specific range. Figure 3 demonstrates how the battery charging time changes as the voltage increases from 3.6 volts to 4.2 volts. It is noteworthy that the length of the selected voltage interval corresponds to the downward trend in the battery’s SOH. Therefore, this paper chooses the charging time of charging battery voltage from 3.6 V to 4.2 V as one feature set of the model. Discharging time of the whole discharging process (voltage dropped from 4.2 V to 2.7 V), and charge/discharge cycle number are selected as model input too.

5.3. Evaluation Criterion

Similar to many other papers, this paper adopts root mean square error (RMSE), mean square error (MSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) as the criteria to evaluate the performance of the algorithm.

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}}{\left({\mathrm{y}}_{\mathrm{t}}-{\mathrm{y}}_{\mathrm{p}}\right)}^2}\end{array}\end{array}$$

(20)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{MAPE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}}\left|{\displaystyle \frac{{\mathrm{y}}_{\mathrm{p}}-{\mathrm{y}}_{\mathrm{t}}}{{\mathrm{y}}_{\mathrm{t}}}}\right|\times 100\%\end{array}\end{array}$$

(21)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{MSE}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}}{\left({\mathrm{y}}_{\mathrm{t}}-{\mathrm{y}}_{\mathrm{p}}\right)}^2\end{array}\end{array}$$

(22)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{MAE}={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{n}}}\left|{\mathrm{y}}_{\mathrm{p}}-{\mathrm{y}}_{\mathrm{t}}\right|\end{array}\end{array}$$

(23)

where $y_t$ and $y_p$ are the actual and predicted maximum battery capacity, respectively.

In general, RMSE can effectively indicate the extent to which the predicted value deviates from the true value. However, during the experiment, there will always be a lot of data deviation from the true value. Even a small amount of data will have a significant impact on RMSE.

Therefore, MSE, MAE, and MAPE are added as additional evaluation criteria. The closer these indicators approach 0, the more accurate the algorithm’s prediction becomes.

5.4. Prediction Model

The battery health-prediction model based on ISSA-SVR was established to predict battery SOH. Initially, data on external parameters like voltage, current, and time for each aging cycle of four batteries were extracted from the NASA database. Subsequently, these data were split into training and testing datasets. Within each cycle, cycle number, discharge time, and charging time within the specified voltage range were selected to form the feature set. Subsequently, the ISSA-SVR model was constructed, and ISSA was employed to optimize the SVR parameters, therefore establishing an optimal prediction model. The proposed framework’s reliability and precision were confirmed through error analysis. Figure 4 shows the ISSA-SVR flow.

6. Experiments and Analysis

To assess the performance of the proposed ISSA-SVR model, the comparative experiments with GA-SVR, GWO-SVR, and SSA-SVR are carried out, and the main parameters are shown in

Table 1. Main parameters of different models.

Optimize	Setting Parameters
GA-SVR	M = 300, V = 2, B = [100, 100], P = 20, ${\mathrm{P}}_A$ = 0.9, ${\mathrm{P}}_B$ = 0.1, ${\mathrm{P}}_C$ = 0.5
GWO-SVR	M = 300, V = 2, B = [100, 100], P = 15
SSA-SVR	M = 300, V = 2, B = [100, 100], P = 20, ${\mathrm{P}}_p$ = 0.7, ST = 0.6
ISSA-SVR	M = 300, V = 2, B = [100, 100], P = 20, ${\mathrm{P}}_p$ = 0.7, $\mathrm{S}{T}_1=0.6$, $\mathrm{S}{T}_2=0.9$

.

6.1. Model Performance Comparison

To fairly evaluate model performance, the same feature set is adopted as the training set. Figure 5 shows the test results of battery B0005.

In Figure 5, the ISSA-SVR prediction results are closest to the actual results.

Table 2. Comparison of prediction performance.

Optimizer	SSA-SVR	ISSA-SVR	GWO-SVR	GA-SVR
MAE	0.12	0.08	0.13	0.09
MAPE (%)	0.17	0.11	0.18	0.12
MSE	0.03	0.02	0.03	0.04
RMSE	0.17	0.14	0.17	0.20

lists the prediction results, and it is obvious that ISSA-SVR provides the minimum MAE, MAPE, MSE, and RMSE. The MAE, MAPE, and MSE are 0.08, 0.11, 0.02, and 0.14, respectively. Specifically, the MAE of the ISSA-SVR model is only 68%, demonstrating its enhanced predictive accuracy. Moreover, the results also highlight the ISSA-SVR model’s effectiveness in tackling nonlinear issues with limited data samples.

6.2. Dependence of ISSA-SVR on Feature Set

To illustrate the dependence of ISSA-SVR on the training set, the training sets with different proportions are extracted from the original data, and the remaining proportion data are used to validate the downward trend of SOH. For battery B0005, Figure 6 shows the prediction results of three different training sets with 50%, 60%, and 70% proportions.

Table 3. Prediction performance with different training sets.

Proportion	50%	60%	70%
MAE	0.13	0.10	0.09
MAPE (%)	0.11	0.08	0.07
MSE	0.03	0.03	0.02
RMSE	0.17	0.17	0.14

gives the specific statistical results.

Obviously, the predicted SOH can track the true value throughout the operation cycle. Training sets with different proportions have the approximate estimation results. The MAE, MAPE, MSE, and RMSE decrease with the increase of feature set proportion, which means that the higher the proportion, the better the prediction performance can be obtained. However, the higher the proportion, the lower the proportion of prediction data. Hence, selecting a training set with an appropriate ratio based on the actual circumstances and the required accuracy is essential.

6.3. Universality Validation

To compare the universality of ISSA-SVR and SSA-SVR, batteries B0006, B0007, and B0018 are used to carry out tests. Figure 7, Figure 8 and Figure 9 show the capacity attenuation profiles of the three batteries. It is evident that the ISSA-SVR model offers higher prediction accuracy compared to the SSA-SVR model.

For B0006, the predicted SOH based on ISSA-SVR can accurately track the true value in the whole process with an error of less than 1%, as shown in Figure 9. In the case of B0006 and B0018, noticeable variations are detected during the mid to late phases of forecasting, likely attributed to differences in battery capacity and the influence of the experimental temperature on the results. Even so, the prediction accuracy of the three batteries is still acceptable, and the errors are all less than 1.5%.

Table 4. Prediction performance based on ISSA-SVR and SSA-SVR.

Optimizer	SSA-SVR	ISSA-SVR	SSA-SVR	ISSA-SVR	SSA-SVR	ISSA-SVR
Battery	B0006		B0007		B0018
MAE	0.23	0.16	0.18	0.11	0.22	0.15
MAPE (%)	0.40	0.31	0.26	0.15	0.37	0.28
MSE	0.29	0.09	0.28	0.05	0.29	0.07
RMSE	0.53	0.30	0.52	0.22	0.54	0.26

gives the MAE, MAPE, MSE, and RMSE based on ISSA-SVR and SSA-SVR.

In Table 4, all the evaluation indicators based on ISSA-SVR are better than those of SSA-SVR. The maximum RMSE and MAPE of the three batteries are 0.30 and 0.31, respectively. Furthermore, the values of indicators based on ISSA-SVR are all quite small, and it shows that ISSA-SVR has a favorable universality capability and can be adapted to predict the SOH of other batteries.

7. Conclusions

This paper introduces a lithium battery health-prediction model utilizing an improved Sparrow Search Algorithm. The model selects charging time within a specific voltage range (3.6 V to 4.2 V), discharging time (voltage dropping from 4.2 V to 2.7 V), and the number of charge/discharge cycles as its feature set. To enhance the search capabilities of the Sparrow algorithm, the paper employs the Nonlinear Weight Dimensionality Reduction (NWDM) method and adopts a hybrid mutation strategy to prevent premature convergence to local optimums. Additionally, the paper optimizes the parameters of the SVR and proposes an SOH prediction model based on ISSA-SVR.

Compared with other different algorithms, ISSA-SVR has a preferable estimation capability and is more effective in solving small-sample nonlinear problems. Meanwhile, the experiment results of dependence on feature set show that ISSA-SVR has good performance in different proportions of training sets. Furthermore, three batteries are used to carry out universality verification experiments. The prediction error base on ISSA-SVR is less than 1.5%.

In summary, the model demonstrates high accuracy, strong robustness, and wide applicability. Looking ahead, we plan to delve deeper into the impact of different input features on the prediction of SOH. Additionally, exploring a practical method to accurately reflect the evolution of battery pack health status is also a key focus of our future research work.