The CS2 dataset, consisting of CS2_35, CS2_36, CS2_37, and CS2_38, was chosen from the CACLE public data. The dataset was identified to contain trace amounts of Mn and had a chemical composition of LiCoO2 cathode, as indicated by the EDS analysis. Apply a steady current of 0.5 C until the voltage between the terminals reaches a stable value of 4.2 V. The charging process stopped when the current fell below 50 mA. During the process, a current of 1.1 A was used to deplete the battery until the terminal voltage reached the predefined cut off value of 2.7 V. The initial capacity of these four batteries was at least 1.1 Ah.

LiNi_0.8Co_0.15Al_0.05O₂ (NCA) cells (B0005, B0006, B0007, and B0018) were obtained by NASA from one of their databases. At 24°C, both charging and discharging took place at room temperature. A current of 1.5 A was employed to discharge the battery to 4.2 V during the charging process. In constant voltage (CV) mode, the charging procedure was discontinued when the charging current decreased to 20 mA or less. A constant current of 2 A was applied during discharge until the predetermined voltages of B0006 and B0018 were attained at 2.5 V, and B0005 and B0007 attained the predetermined voltages of 2.7 V and 2.2 V, respectively. The experiment was terminated when the battery achieved the end-of-life (EOL) criterion, which stipulated a reduction in rated capacity by 30%, from 2 to 1.6 Ah.

A selection of cells (Cell1-Cell8) was made from the Oxford dataset that featured an anode composed of graphite and a cathode composed of Li and LiNi-Co oxide. A hot cell underwent both charge and discharging processes at 40°C. At the same current level, the battery was charged to 4.2 V at 740 mA and discharged to 2.7 V.

The Equation 1 were applied to the data obtained from the three databases in order to ascertain the state of health (SOH): S O H = C c u r C n o m (1) where: C n o m denotes the initial designated capability, while C c u r indicates the actual capacity.

A SOH ranging from 1 to 0.8 distinguished primary batteries. In contrast, there were divergent viewpoints among experts regarding the secondary battery’s end of life (EOL) standard. Mussi et al. (2022), Liu et al. (2022) proposed decommissioning of the secondary battery at SOH 0.6, whereas others Braco et al. (2022) suggested retirement at SOH 0.2. Unsurprisingly, the secondary battery is ceased operation at the SOH is 0.2, a broad range.

This study is primarily concerned with estimating the SOH of a secondary battery. The SOH statistics for the battery ranged from 0.2 to 0.8. Figure 1 depicts the batteries consisting of the Oxford dataset and the NASA dataset, exhibiting SOH values ranging from 0.6 to 1.0. After meeting the necessary requirements, we have chosen the CACLE dataset as the subject of our studies. It is worth mentioning that the highest value of battery SOH in any of the three datasets was not equal to 1, as the rated capacity was only an approximation.

The experimental data were obtained from the CS2_35-CS2_38 battery datasets maintained by the CACLE group. The datasets pertaining to the secondary battery section comprise data on the complete discharge cycle of each battery, including time, voltage, current, temperature, and capacity. In particular, the secondary batteries CS2_35, CS2_36, CS2_37, and CS2_38 have respective cycle periods of 330, 423, 419, and 476.

Since the features were obtained from the discharge curve, it was necessary to analyse the discharge curve of the battery. The greater SOH range of the CALCE No. CS2_36 battery (Figure 2) could be ascertained through an examination of the battery’s discharge curve. The discharge trajectories of this battery at SOH values of 0.25, 0.5, and 0.8, respectively, are illustrated in Figure 2. The CS2_36 battery’s discharging process was observed to be divided into three phases, with point A delineating the initial stage from stage two, and point B distinguishing the second stage from stage three.

Point A signifies the maximum U-chord curvature, while point B denotes the minimum U-chord curvature. Over the course of the first and third phases, the voltage dropped precipitously. However, as time progressed through the second stage, the voltage changes diminished. This interval was investigated as it was referred to as the voltage plateau region. The secondary battery’s discharge curve exhibited a more rapid transition and the duration of the voltage plateau zone decreased as the battery aged.

In order to identify the voltage plateau region of the discharge curve, the coordinates A and B that divide the discharge curve must be determined. The identification of the characteristic points of the curve was predicated on the curvature of the profile. Curvature is a parametric measure that characterises the localised nature of a curve; more specifically, it quantifies the localised shape of a given point on the curve. Curve C represents the rate of rotation of the arc length of the point in the tangential direction. A point’s curvature decreases in proportion to the degree of curvature of the curve.

As a result of the discrete nature of the discharge curve and the lack of explicit function expressions, computing the curvature of individual points on the curve presents a technical challenge. Hence, it was necessary to develop a method for calculating curvature that takes into consideration discrete properties and the lack of explicit function expressions. The differential curvature method was typically employed for the computation of discrete curvature. With the advancement of mathematics, the U-chord curvature method exhibited superior performance in terms of noise resistance and rotation compared to earlier methodologies.

As shown in Figure 2, the location of nodes A and B shifted proportionally with the number of battery cycles. In order to derive the features, the data from the two sites and the voltage plateau region between them were utilised. In this study, the plateau discharge capacity of the battery was utilised as an feature. SOH can be accurately predicted by identifying representative features and associating them with the SOH of a battery. The present article utilised four features denoted by Eqs 8–13.

1) F 1 : Voltage at point B

The correlation between the extracted features and the cell SOH is investigated in this section. Equation 14 uses the Pearson correlation coefficient which is commonly employed to quantify the extent of linear correlation between two variables. ρ = ∑ i = 1 n F i − F ¯ C i − C ¯ ∑ i = 1 n F i − F ¯ 2 ∑ i = 1 n C i − C ¯ 2 (14) where: ρ ∈ 1 − , 1 denotes the Pearson correlation coefficient, F i and C i denote the two variables for which correlation is required, F i ¯ and C i ¯ denote their means.

When there is correlation coefficient ρ = ± 1 , there is a perfect linear relationship between the two variables, while when the correlation is 0, there is no correlation. The Pearson coefficients of the SOH and features of the CS2_35-CS2_38 cells are listed in Table 1. The obtained features had an excellent linear connection with the Pearson coefficient of battery SOH, which was close to 1, and could be used to estimate the battery health life.

By utilising the sum of squared errors as an empirical loss and converting the inequality constraints present in traditional SVM to equality constraints, LSSVM reduced the convex quadratic programming problem that SVM was originally designed to solve to a system of linear equations (Suykens et al., 1999). LSSVM is applicable for both regression curve fitting and classification. For predicting the health life of secondary batteries, its regression curve fitting function was utilised extensively. The fundamental principle is expounded upon as Eq. 15: min w , b , e ⁡ J w , e = 1 2 w T w + γ 2 ∑ i = 1 N e i 2 s . t .     y i = w T φ x i + b + e i   i = 1 , … , N (15) where: w is weight vector, b is error, while e and γ are slack variable and penalty factor, and φ x i is the kernel function. By defining the Lagrange multiplier as α , the Lagrange function can be constructed as Eq. 16. L w , b , e ; α = J w , e − ∑ i = 1 N α i w T φ x i + b + e i − y i (16)

By taking partial derivative of Eq. 16 and simplifying it to 0, Eq. 17 can be deduced. ∂ L ∂ w = 0 → w = ∑ i = 1 N α i ⋅ φ x i ∂ L ∂ b = 0 → ∑ i = 1 N α i = 0 ∂ L ∂ e i = 0 → α i = γ e i ∂ L ∂ α i = 0 → ω T ⋅ φ x i + b + e i − y i = 0 (17)

The Eq. 18 is the matrix form of Eq. 17: 0 Y T Y β + γ − 1 I b α = 0 y (18)

Eqs 19, 20 provide further explanation of the parameters in Eqs 18, 21 defines gamma, another key parameter of the LSSVM. β = φ x i T φ x j = K x i , x j   i , j = 1 , 2 , … , N (19) K x i , x j = ⁡ exp − d x i , x j 2 2 ⋅ σ 2 (20) g a m m a = 1 2 ⋅ σ 2 (21) where: Y T = y 1 , y 2 ⋯ , y N , β is the radial basis kernel function of the prediction model, while g a m m a and σ are the kernel function parameter and kernel function width factor.

When g a m m a setting was too large, there were few support vectors. While the training effect was reasonable, the generalization ability was low. The smaller g a m m a is, the more support vectors are, the slower the training speed, and hence, the accuracy was affected.

By solving the system of Eq. 18 the LSSVM regression function in Eq. 22 can be obtained as follows: y x = ∑ i = 1 N α i K x , x i + b (22)

The key parameters of LSSVM include kernel function parameter g a m m a and penalty factor γ .To find the optimal parameters, this work introduced PSO and SSA as the two optimization algorithms.

The algorithm known as particle swarm optimization (PSO) was suggested in consideration of avian behaviour (Kennedy and Eberhart, 1995). During the search process, each particle recorded its previous optimal results and parameters while proceeding in its own specific trajectory. Concurrently, every particle contributed its own outcomes and parameters. The particle swarm was thus informed of the present optimal parameters. With the passage of time and in consideration of their own and the group’s experience, the particles modified their subsequent search path and eventually discovered the global optimal solution.

A metaheuristic algorithm, the sparrow search algorithm (SSA) is designed to emulate the predation and anti-predation strategies employed by sparrows (Xue and Shen, 2020). Sparrows were classified as either producers or observers. Producers have larger energy reserves, and the amount of energy reserves depends on individual fitness. Each sparrow has the potential to develop into a producer, but the total number of producers remains unchanged. Identifying regions abundant in food resources and defending against predation, wherein individuals vied for resources, fell under the purview of producers. The producer led the followers as the sparrows at the periphery swiftly relocated to a secure location upon detecting impending danger. Those with diminished energy reserves, meanwhile, sought sustenance elsewhere.

Both PSO and SSA exhibited remarkable optimization capabilities. This work employed two optimization algorithms to optimise the hyperparameters of the LSSVM. Figure 5 displays a flowchart illustrating the process of creating the model.

The flowchart for the complete experiment is depicted in Figure 6. This study began by screening the battery SOH data between 0.2 and 0.8, then extracted the characteristic values of secondary batteries using U-chord curvature technology, and finally estimated the secondary battery’s health life using the following five models: GPR, CNN, LSTM, PSO-LSSVM, and SSA-LSSVM. Mean absolute error (MAE), root mean square error (RMSE), and absolute percentage error (MAPE) were subsequently employed as evaluation metrics to quantify the discrepancy between the anticipated and realised secondary battery health times. Eqs. 23–25 represented their respective calculation methodologies. M A E = 1 N ∑ i = 1 N y i − y ^ i (23) R M S E = 1 N ∑ i = 1 N y i − y ^ i 2 (24) M A P E = 1 N ∑ i = 1 N y i − y ^ i y i (25) where: y i is actual data, y ^ i is prediction data, while N is the number of prediction samples.

In order to assess the efficacy of the suggested approach, two experiments were conducted utilising three error indices MAE, MAPE, and RMSE for this evaluation. Several models in Experiment 1 exhibited significant errors. Nevertheless, with the expansion of the training data in Experiments 2.1 and 2.2, the prediction error exhibited a progressive decrease. This developed a methodology to enhance the precision of forecasting second-life batteries, particularly in situations involving discharge power uncertainties.

This section provides a detailed account of two experiments in which the precision and effectiveness of various machine algorithms and U-chord length curvature feature extraction techniques were contrasted and validated. A Li-battery discharge history cycle record was utilised to train the model in Experiment 1. Following that, the model that had undergone training was employed to predict the SOH of its own remaining cycles. Similar to the primary battery, this experiment validated the capability of the secondary battery to deduce its SOH for subsequent cycles despite the scarcity of data.

The remaining cycle expectancies of secondary batteries CS2_35-CS2_38 were predicted in Experiment 1 using their historical cycle records. In order to train the model, additional numbered secondary battery discharge recordings were utilised in Experiment 2. The model that had undergone training was subsequently applied to predict the battery’s SOH. The objective of this experiment was to evaluate the suitability of the model, augment the dataset, and ascertain whether it could overcome the difficult predictive characteristics of secondary batteries, which have restricted cycles and variability.

Experiment 2.1 and Experiment 2.2 constituted the two discrete components that comprised Experiment 2. Experiment 2.1 involved training the regression model with CS2_35-CS2_38 in order to generate predictions pertaining to the remaining three batteries’ health. To illustrate, the model was trained utilising CS2_35 in order to generate forecasts regarding the anticipated lifetimes of CS2_36, CS2_37, and CS2_38 batteries, respectively.

Experiment 2.2. involved training the model with three secondary batteries in order to generate predictions regarding the residual lifespan of the secondary batteries. In light of this rationale, the model underwent training utilising three secondary batteries, namely, CS2_35, CS2_36, and CS2_37, with the purpose of predicting the remaining useful life of CS2_38.

70% of the data from each secondary battery was initially designated as the training set in Experiment 1, with the remainder of the data being allocated as the test set. It is evident that the magnitude of the voltage plateau region of the secondary battery increased in tandem with the number of cycles performed. Notwithstanding this, an abrupt decline in length occurred, characterised by a swift progression of degradation. The behaviour of this conduct was in direct contrast to that of the primary battery. The work’s predictive capabilities were dramatically tested by the rapid rate of change and substantial degree of instability.

The positive prediction outcomes of GPR, CNN, LSTM, and SSA-LSSVM were comparable to the actual values, as shown in Figure 7A. The preponderance of the various prediction approaches, however, predicted subpar results, as shown in Figures 7B–D. Figure 7B did not display the GPR prediction method, however, because the GPR prediction results exceeded the axes’ range and were therefore irrelevant to the reference. Conversely, in comparison to alternative prediction methods, it was evident that the SSA-LSSVM prediction method exhibited a markedly superior performance.

On the other hand, as illustrated in Figure 7D, the LSTM prediction method demonstrated an inadequate initial performance. Conversely, the SSA-LSSVM prediction approach approached the actual value with a more pronounced degree of convergence (Figure 7C). In addition to improved CS2_35 cell prediction outcomes, GPR performance was substantially diminished. Failure of GPR was observed in the remaining cell predictions. The significant variations in the voltage plateau region of the secondary battery’s discharge curve with increasing cycle count, in conjunction with the non-representative nature of the historical data, may account for the suboptimal results observed with the various prediction methods. Experiment 1’s results indicate that alternative algorithms, excluding SSA-LSSVM, do not perform optimally when applied to historical data in order to estimate the remaining service life of secondary batteries.

Figure 8 displays the error plots of Experiment 1, which employed self-historical data to predict its own healthy life expectancy. Figure 8 employs a logarithmic coordinate axis to accommodate the significant fluctuation in inaccuracy resulting from the use of various techniques. The logarithmic coordinate axis was more effective than the traditional coordinate axis in accurately representing errors of different magnitudes. Figure 8 demonstrates that SSA-LSSVM outperformed the other four prediction methods when the experimental dataset was limited in size. The hybrid approaches have a benefit when there is less data, as they require more appropriate hyperparameters in such cases. The sparrow optimization algorithm enhances its ability to discover the global optimal solution more efficiently by adapting its search strategy and search speed. The particle swarm optimization technique exhibits superior global search capabilities during the first phase, but it is susceptible to converging towards a local optimal solution.

For each experiment, the error values are detailed in Table 2. More precisely, the error in forecasting the secondary battery (CS2_35) was assessed in Experiment 1 (EXP_1). According to the results, the MAE achieved with SSA-LSSVM was 1.24%, 2.16%, 2.86%, and 27.68% less than with LSTM, PSO-LSSVM, CNN, and GPR, respectively.

However, it is important to highlight that the MAE linked to the GPR prediction approach in Experiment 1, particularly in regards to the forecast of CS2_36 (EXP_1_2), was considerably elevated at 68.38%. The observed disparity in performance between the GPR’s exceptional predictive capabilities for primary batteries and its below average performance for secondary batteries implies that specific prediction techniques might not produce acceptable outcomes for the latter. The heightened prediction error that resulted from the accelerated decrease in voltage plateau and increased individualization of secondary batteries relative to primary batteries can be attributed to this.

Experiment 2.1 (EXP_2.1) employed five prediction approaches to estimate the remaining lifespan of three additional batteries. Assertions were produced using the CS2_35 training model. The computation of mean errors (EXP_2.1_1) revealed that the maximum MAE decreased to 9.51% with the implementation of several prediction algorithms. Conversely, the MAE obtained a value of 1.21%. Figure 12 exhibits the precise discrepancies between the predicted values and the actual values that were observed in Experiment 2.1.

Experiment 2.2 (EXP_2.2) consisted of training the model to predict the healthy life expectancy of CS2_35 (EXP_2.2_1) using data from CS2_36 to CS2_38. The minimal MAE remained at 0.83%, while the maximum MAE for the different prediction approaches reduced to 2.03%. The MAE in EXP_2.2_2 ranged from 1.1% to 0.6%. The MAE in EXP_2.2_3 ranged from 1.82% to 0.64%. The MAE in experiment EXP_2.2_4 reduced to a range of 1.90% to 0.64%.

In order to facilitate a thorough assessment of the effectiveness of these models in estimating the longevity of the secondary batteries, the mean errors generated by the five prediction methods throughout the experiment are displayed in Figure 13. From Experiment 1 to Experiment 2, as the volume of data increased, the predictive accuracy of GPR demonstrated a progressive enhancement. This implies that by increasing the quantity of experimental data, the proposed prediction model for secondary batteries could be improved in terms of effectiveness. Errors were reduced in Experiment 1 using the LSTM, PSO-LSSVM, and SSA-LSSVM prediction methodologies. Nevertheless, the SSA-LSSVM method exhibited the highest level of predictive efficacy in Experiment 2.1. In general, the predictive algorithms PSO-LSSVM and SSA-LSSVM produced smaller errors than alternative approaches. GPR is the most efficient in terms of computational load. Artificial neural networks, such as LSTM and CNN, exhibit sluggish performance, however hybrid technologies, including SSA-LSSVM and PSO-LSSVM, incorporate parameters such as optimization range, number of iterations, and optimization step length in the optimization method. Their computation time and workload varies from that of the optimization algorithm. Regarding parameter configuration.