A Novel Approach for State of Health Estimation of Lithium-Ion Batteries Based on Improved PSO Neural Network Model

Abstract

The operation and maintenance of futuristic electric vehicles need accurate estimation of the state of health (SOH) of lithium-ion batteries (LIBs). To address this issue, a robust neural network framework is proposed to estimate the SOH. This article developed a novel approach that combines improved particle swarm optimization (IPSO) with bidirectional long short-term memory (Bi-LSTM) to effectively address the issue of precisely estimating SOH. The proposed IPSO-Bi-LSTM model is more effective than the other models for SOH estimation. This is because Bi-LSTM can capture both past and future appropriate information, making it more suitable for modeling complicated temporal sequences. The IPSO main objective is to optimize the model hyperparameters. To increase the model’s accuracy, the IPSO improves the parameters. The PSO-Bi-LSTM model performed better than the other approaches, according to experimental findings based on the NASA-PCOE battery dataset, and all of the SOH estimated outcomes, such as root mean square errors, were less than 0.50%. This result suggests that the proposed PSO-Bi-LSTM model has the ability to robustly estimate the SOH with a high accuracy.

1. Introduction

Electric vehicle economy and environmental sustainability are directly impacted by the performance of the power battery, which is also frequently known as the vehicles heart [1]. LIBs are the most widely used power battery type in the field of electric vehicles (EVs) because of their high energy density, long lifetime, low self-discharge rate, and limited environmental effects [2,3]. However, several safety issues concerning electric vehicles have increased frequently due to the EV industry’s rapid development [4]. The fundamental factors behind these accidents may be attributed to the inadequate management and maintenance of power batteries. These accidents not only cause significant financial losses but also have negative effects on society, seriously delaying the growth of the EV industry. Due to this, the roles of the battery management system (BMS) have become more noticeable [5]. BMS is responsible for both monitoring and controlling the operation of LIBs, as well as for executing the SOH estimation. By precisely measuring the SOH of a power battery, it is possible to effectively recognize the warning signs of battery aging and performance degradation. This allows us to take necessary actions to extend the battery lifetime and improve the dependability and safety of the EVs. It is essential to accurately estimate the SOH in order to ensure the safety, effectiveness, and durability of battery systems, especially in vital sectors such as electric vehicles, healthcare equipment, and energy storage. It improves the prevention of dangerous malfunctions, enhances performance, and prolongs battery lifespan, resulting in cost-effectiveness and lower environmental impact. An accurate SOH improves warranty management and user trust by safeguarding the predictable and dependable performance of battery-powered technologies.

There are three primary types of SOH estimation techniques for LIBs: direct measurement methods, model-based methods, and data-driven methods [6]. Direct measurement methods include the Coulomb counting method [7], the open circuit voltage (OCV) approach [8], and the estimation method depending on the internal resistance. The Coulomb counting technique is a quick and easy approach for evaluating the SOH of LIBs. The process involves determining the discharge capacity by estimating the charge that flows through the battery during discharge and then dividing it by the battery rated capacity. Nevertheless, a drawback of this approach is its dependence on the entire charge and discharge of the battery in a single cycle, which may be inconvenient in real-world scenarios. The OCV approach, which calculates the capacity by analyzing the correlation between the OCV and state-of-charge (SOC), needs several hours of inactivity to achieve an equilibrium battery condition. The internal resistance-based estimation method is a precise methodology for measuring SOH and can be examined using a hybrid pulse power characterization approach. This process is often executed offline but has challenges when executed online. Direct measurement techniques are highly precise; however, they are only applicable under laboratory conditions. Typically, batteries are not entirely drained to maintain the regular functioning of electric vehicles in everyday life; hence, this approach is not feasible for real-world implementations.

Model-based techniques typically involve the integration of measurable battery data, such as voltage, current, and temperature, with electrochemical models or state observers. These techniques estimate the battery SOH by considering it as a process of estimating parameters based on a battery model that describes its dynamic features. To address this limitation, researchers have developed several SOH observers using electrochemical models. Pang et al. [9] used the mathematical models developed for battery SOH together with the constant phase element parameters in the designed equivalent circuit model of electrochemical impedance spectroscopy to achieve a quick SOH estimation for retired power batteries. An online capacity estimation technique based on recursive least squares (RLS) and enhanced particle filtering (PF) was proposed by Haung et al. [10]. This technique uses the characteristic voltage (CV) as a measure of the health status from the discharge curve and constructs a correlation model between the CV and cycle capacity. Subsequently, the enhanced PF-RLS method was used to estimate the CV accurately in real time for SOH estimation. Fang et al. [11] used the forgetting-factor recursive least squares (FFRLS) technique to accomplish online model parameter identification. The double-extended Kalman filter (DEKF) algorithm was presented to estimate the SOH based on the link between the ohmic internal resistance and SOH. The model-based approach has difficulties, even if it exhibits good precision. These challenges involve the task of choosing a suitable model that achieves a balance between precision and computational demands. The large computational burden associated with parameter identification and the complexities of extending its usage due to the constant necessity for model adjustments to adjust various LIBs models and operating conditions. Furthermore, proficient expertise in the battery’s fundamental mechanisms is required to implement model-based approaches. It also relies on comprehensive physical and chemistry concepts, which include electrochemical phenomena, properties of materials, and solid-phase diffusion. These theories involve complex computational equations and functional relationships [12].

Data-driven methods are categorized as black-box models, and a key feature of these approaches is their investigation and use of high-performance algorithms. Data-driven procedures reduce the need for a comprehensive examination of the battery’s fundamental mechanism [13]. These methods are very useful and have become popular in recent years for SOH estimation. The fundamental concepts of their approach include examining data from battery performance tests to determine the correlation between external factors and the reduction in battery capacity [14]. Chang et al. [15] introduced an online technique for SOH estimation using incremental capacity (IC) and combining wavelet neural network and genetic algorithm (GA-WNN). The wavelet neural network initial connection weights, conversion factor, and scaling factor were optimized using a genetic algorithm. The GA-WNN model was utilized to determine the battery SOH, and the Pearson correlation coefficient method was employed to extract the health characteristic variables from the IC curves. Tian et al. [16] presented SOH estimation from the KF-joined particle filter. An attenuation framework was established using exceptional health indicators (HI), including vehicle loads and seasonal temperature. The battery model parameters were determined using the variable forgetting-factor recursive least squares. However, the estimated SOH error was high, up to 1.2%. Liu et al. [17] estimated the SOH by combining the Long Short-Term Memory (LSTM) and support vector regression (SVR) models. The SVR model and LSTM model were used as the main and secondary models, to assess the SOH, respectively. In addition to other cutting-edge models, this model was verified using the NASA database. Although the proposed model achieved a root mean square error (RMSE) of 0.60%, the information of battery parameters ignored, like temperature and impedance, which might improve accuracy. A Convolutional Neural Network (CNN) with an LSTM model-based SOH prediction method was developed by Xu et al. [18]. The CNN-LSTM model was trained using various ratios of training to testing. The model performed well, with an RMSE of 0.004%, although it was computationally laborious and challenging. Extracting relevant data distinctive features for training the model can mitigate this problem. Tang et al. [19] developed a CNN-LSTM- convolutional block attention module (CBAM) hybrid neural network model for LIBs SOH estimation. The CBAM module improves the sequential attention framework, minimizes the impact of noise on the input information, and boosts the ability of the CNN to extract features. Utilizing two public datasets and transfer learning with fine-tuning procedures, the SOH estimate had an RMSE of 0.17%.

Zhao et al. [20] introduced a novel regression-generative adversarial network that employs a generator to automatically produce auxiliary training samples with distributions that are similar yet distinct from the real samples. Additionally, they developed a comprehensive model to determine the optimal correlation between SOH and health features. Zhang et al. [21] introduced an enhanced method for estimating the SOH using a reference voltage-based IC and LSTM network. The health feature variables were derived from the IC curves, and the LSTM network was used to achieve SOH estimation. However, by evaluating a significant quantity of data using an efficient processor, data-driven state estimation methods can be developed without detailed knowledge of the battery background mechanism [22]. Furthermore, a data-driven strategy may predict multiple states with less knowledge of detailed chemical interactions and battery substance features and with a faster development time [23]. However, a hybrid framework can be built by merging data-driven and model-based approaches [24]. Whenever the models are hybridization, it may be not performed properly, it leads to a problematic configuration that provides mathematical complexities and delivers poor results [19]. Therefore, further research is required to address practical problems, and limitations must be considered when establishing an efficient hybrid framework. Estimating battery states requires the use of deep and machine learning algorithms. Nevertheless, the variability in the estimation performance of a single hybrid model can limit its precision [25]. To address this problem, it is essential to develop a hybrid deep learning approach that integrates two or more models to enhance the precision of the SOH estimation of LIBs.

This article developed a novel approach that combines improved particle swarm optimization (IPSO) with bidirectional long short-term memory (Bi-LSTM) to effectively deal with the issue of accurate SOH estimation. The proposed PSO-Bi-LSTM model was more successful than the PSO-LSTM for SOH estimation. This is because Bi-LSTM can capture both past and future appropriate information, making it more suited for modeling complicated temporal sequences. In addition, Bi-LSTM is more susceptible to the vanishing gradient issue, a phenomenon that may arise in LSTM when the gradients diminish significantly, leading to the termination of the network’s process of learning. Ma et al. [26] used improved PSO for hyperparameters selection with the backpropagation neural network model for remaining useful life and SOH estimation. The estimated performances showed good accuracy in the RMSE for RUL and SOH, at 6.05% and 0.78%, respectively. Zhi et al. [27] developed a SVR approach to predict the SOH after identifying important health parameters for lithium-ion batteries using the random forest technique. SVR parameter values are optimized using the GA-PSO technique, which increases the estimation accuracy and convergence speed. Using the four batteries, the suggested SOH estimation approach yields a 0.40% RMSE and 0.56% MAPE. Hu et al. [28] present a fractional-order calculus-based co-estimation technique for LIBs SOC and SOH estimation. The voltage response was predicted by the model using the Hybrid GA/PSO approach. The dual fractional-order extended Kalman filter estimates simultaneously. The study shows a very high computational burden but rapid convergence and good precision, with errors of less than 1% across both the SOC and SOH. Ye et al. [29] developed a model-based LIBs SOC estimation method employing enhanced adaptive PF and PSO. By updating the model parameters, implementing the adaptive PF technique achieved enhanced durability and fast convergence. In addition, the PSO algorithm was used to reduce the inconsistency of the battery in the presence of flat noise with a zero average value. The model that was suggested achieved SOC estimation error below 1%. Lai et al. [30] proposed a hybrid approach that combines ECM and extended Kalman filter (EKF) to estimate SOC. The optimal model parameters for the SOC estimation were chosen using the PSO approach. To estimate the SOC and SOP, combined with an estimated error of 2.95%. Ren et al. [31] presented a PSO algorithm and an LSTM model-based SOC estimation. The main objective of the PSO algorithm is to determine the optimized hyperparameters of the LSTM model. Ansari et al. [25] presented a hybrid framework that combines the PSO and RNN models to predict the remaining useful life (RUL) of LIBs. The framework was built using a systematic sampling method keeping consideration of the important battery statistics. PSO-GA method was used in [32] to identify the ideal battery features. The suggested co-estimator showed SOC estimation error of 1.2%. Reference [33], the SOE and SOP were determined at each macro time using the estimated SOC. PSO and the UKF were used in real time to determine the accurate battery parameters and estimate the SOC. The SOC error achieved an RMSE of less than 0.08%. How SOH affects SOE and SOP estimations has not yet been explored. In this research, the authors developed a PSO to improve the parameters of a Bi-LSTM model. In the available literature, no research has been found to attempt to apply an IPSO-Bi-LSTM network to estimate the SOH of a battery system.

This research proposes an improved version of the PSO approach to improve the optimization potential and rate of convergence. In addition, nonlinear variable inertial weights are included in the PSO method. The proposed approach was motivated by investigations carried out by Hu et al. [28], Zhi et al. [27], and Ma et al. [26]. Researchers have demonstrated a significant improvement in the precision of SOH estimation. The Bi-LSTM model efficiently captures the temporal dependencies and fluctuations in the LIBs dataset. The model can be improved by capturing such patterns and merging them with PSO, leading to estimates of SOH that are highly precise. Moreover, the SOH of the LIBs features is estimated well by combining PSO and Bi-LSTM. The PSO algorithm can contribute to reducing overfitting in the Bi-LSTM approach by identifying the most suitable combination of hyperparameters to achieve a balance between computational complexity and accurate estimation. For instance, this article distinguishes itself from others in terms of its approach to addressing the issue, organization of the neural network, parameters used in PSO and Bi-LSTM, and performance metrics used to assess the outcomes. The authors first divided the collected LIBs dataset into training, cross-validation, and testing sets. Subsequently, the authors proceeded to develop a Bi-LSTM neural network. After that, the IPSO approach is used to adjust the hyperparameters of the Bi-LSTM and, at last, the estimation of SOH. This article demonstrates the approach used for SOH estimation, as shown in Figure 1.

2. Proposed Methodology

The SOH is a metric used to quantify the present maximum dischargeable capacity of a power battery [34]. It specifically relates to the battery’s energy storage capacity. This is a crucial determinant in deciding when the batteries should be changed. Usually, the battery capacity and internal resistance are used as indicators to approximate its SOH, which may be described as follows [35]:

$$\mathrm{S}\mathrm{O}\mathrm{H}={\displaystyle \frac{{\mathrm{C}}_{\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}}}{{\mathrm{C}}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{c}\mathrm{a}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}}}}\times 100\%$$

(1)

$$\mathrm{S}\mathrm{O}\mathrm{H}={\displaystyle \frac{{\mathrm{R}}_{\mathrm{E}}{-\mathrm{R}}_{\mathrm{n}}}{{\mathrm{R}}_{\mathrm{E}}{-\mathrm{R}}_0}}\times 100\%$$

(2)

The C_{rated capacity,} the rated capacity, and C_{max capacity}, the current maximum usable capacity of a lithium-ion battery have the same specifications. Meanwhile, $R_n$, $R_E$, and $R_0$ denote the present internal resistance, the internal resistance at the end of its lifetime, and the original internal resistance of the lithium-ion battery, respectively. This study mainly aims to assess the SOH of a battery, with a particular focus on the degradation of its capacity. When a battery is first manufactured, its SOH is typically 100%. Over time, the SOH slowly declines due to both cycling and chronological deterioration caused by continual operation. Typically, a power battery is believed to have reached its end-of-life threshold when its capacity is reduced to around 80% to 70% of its rated capacity, and the internal resistance of the battery doubles compared to a new one. The proposed algorithms are discussed in this section.

2.1. Fundamental PSO Algorithm

Bird foraging involves swarming and clustering behaviors, which are represented by the PSO approach. This algorithm is widely utilized in various applications, leveraging swarm intelligence and a global random search approach. The direction and distance traveled by the particle during its search for the solution space are mostly determined by its change in speed. Simultaneously, it thoroughly assimilates its past knowledge and collective wisdom of the group to continuously fine-tune its local search capabilities. The purpose of the optimal solution and the global optimal solution currently found by the entire population is to continuously explore the solution space until the end of the iteration. Within the n-dimensional search space, a group of particles is initialized and dispersed throughout the area. The position of each particle’s represents a potential solution to a specific problem. Through their exploration, each particle can identify the most optimal position within the desired search space. Suppose the search space of a particle swarm consisting of N particles is a D-dimensional vector $xid=\left(xi1,xi2,\dots \dots ··xiD\right),$ the flight speed is represented [36].

$$vi=\left(vi1,vi2,\dots ,viD\right),where,i=1,2,3,\dots ,N)$$

The best solution obtained by the searched particles is $pi=(pi1,pi2,\dots ,piD)$. One of the best solutions for each particle is represented by $pg=\left(pg1,pg2,\dots ,pgD\right),$ which mimics the evolutionary, iterative calculation method. The PSO algorithm needs to adjust the speed and position of its flight in each iteration. The position update formula of the speed $v_{id}^{k+1}andk+1$ alterations of the $ith$ particle is:

$$v_{id}^{k+1}={wv}_{id}^k+c1r1\left(P_i^k-x_{id}^k\right)+c2r2\left(P_g^k-x_{id}^k\right)$$

(3)

$$x_{id}^{i+1}=x_{id}^k+v_i^{k+1}$$

(4)

The new location update formula is:

$$x_{id}^{k+1}=x_{id}^i\times wij+v_{id}^i\times w^{i}ij+rand\left(\right)\times gbes{t}^t$$

(5)

Within this scenario, $wij$ and $w^{i}ij$ are variables that dynamically regulate the influence of the current $x_{id}^i\mathrm{a}\mathrm{n}\mathrm{d}{w}^{i}ij$, respectively. The $gbes{t}^t$ denotes the most favorable position of the particle. The inertia weight, w, plays a crucial role in determining the extent to which the particle’s current velocity is inherited, thereby ensuring a balance between the detection and development capabilities of the PSO algorithm. Increasing the value of w results in more particle displacement in the original direction, enhancing exploration capabilities, but at a slower convergence rate. Conversely, decreasing the value of w improves the particle development abilities but increases the likelihood of being trapped in local optima. The learning factors c₁ and c₂ represent the proximity of each particle to its global best and historical merit, respectively. r₁ and r₂ are randomly generated integers that fall within the range of 0 to 1. To minimize the likelihood of particles exiting the search space during the iteration, $vij$ is often constrained within a certain range, $vij\in \left[-vamx,vmax\right]$.

Improvement in PSO Algorithm

To achieve optimal particle identification in the early stages and efficient particle growth in the latter stages, the parameter ‘$w$’ must be continually modified during the iteration process. This research employs a linear declining law, which is represented by the following formula [37]:

$$w_i=w_{max}-{\displaystyle \frac{i\left(w_{max}-w_{min}\right)}{I_{max}}}$$

(6)

The variable $w_i$ represents the inertia weight for the $i^{th}$ iteration. $I_{max}$ denotes the maximum number of iterations. $w_{min}and{w}_{max}$ correspond to the lowest and maximum inertia weights, respectively. The fundamental particle swarm technique suffers from the drawbacks of local extrema and sluggish convergence due to the limited variety of particle positions in the latter stages of the search. Experimental results indicate that during the first search phase, the value of c1 is rather high, while the value of $c_2$ is relatively low. Additionally, particles can freely explore and split inside the search space. As the number of iterations rises, the value of $c_1$ drops linearly, while the value of $c_2$ grows linearly. This leads to an improvement in the overall convergence of the particles toward the global optimum. This formula is represented by the following expression [31]:

$$c_{1i}=c_{1s}+{\displaystyle \frac{i(c_{1e}-c_{1s})}{I_{max}}}$$

(7)

$$c_{2i}=c_{2s}+{\displaystyle \frac{i(c_{2e}-c_{2s})}{I_{max}}}$$

(8)

The values of $c_1$ and $c_2$ for the $ith$ iteration are denoted as $c_{1i}$ and $c_{2i}$ respectively. The starting values of $c_1$ and $c_2$ are represented by $c_{1s}and{c}_{2s}$, while the final values are denoted as $c_{1e}and{c}_{2e}$. $I_{max}$ refers to the maximum number of iterations. The IPSO algorithm is similar to the PSO algorithm. The detailed algorithm process is as follows as shown in Figure 2:

Step 1: Generate the position and speed of N particles randomly and provide the necessary parameters such as the maximum number of repetitions, M.

Step 2: Determine the fitness value of each particle. Assign the $ith$ particle to its current best position and the $Nth$ particle to the global best position.

Step 3: Adjust the speed and position of the particles based on Equations (5) and (8).

Step 4: Compute the particle’s fitness value and modify the particle’s local and global extreme values based on the computed result.

Step 5: Evaluate whether the provided termination condition has been met. Upon satisfaction, terminate the search and produce the outcome; otherwise, proceed to Step 3 to resume the iteration.

2.2. LSTM

Recurrent neural networks (RNNs) of the LSTM network type are often used for processing sequential data, including time series signals and natural language. They are intended to solve the issue with standard RNNs vanishing gradient of standard RNNs, which may occur when the error signals gradient is too small, and the network is unable to identify long-term relationships in the data. The memory cell is a crucial part of an LSTM network because it enables the network to selectively forget or recall data from earlier time steps. Information enters and exits the memory cell via several gates that constitute the cell. The gates are realized as hyperbolic tangent activation functions or sigmoid activation functions, which have respective ranges of values between −1 and 1 and 0 and 1 [38]. Figure 3 depicts the overall layout of the LSTM cell.

The following are the mathematical formulas that characterize the LSTM network: The input at time step t should be represented by Xt, the hidden state by ℎt at time step t, the cell state by ct at time step t, and the sigmoid and hyperbolic tangent functions by σ and f, respectively. The LSTM formulas are expressed as follows [39]:

$$\left\{\begin{array}{c}{f}_t=\mathrm{sigmoid}\left(W_f{X}_t+D_f\left[{X_t,h}_{t-1}\right]+b_f\right)\\ i_t=\mathrm{sigmoid}\left(W_i{X}_t+D_i\left[{X_t,h}_{t-1}\right]+b_i\right)\\ o_t=\mathrm{sigmoid}\left(W_o{X}_t+D_o\left[{X_t,h}_{t-1}\right]+b_o\right)\\ c_t^{\mathrm{*}}=\tanh \left(W_c{X}_t+D_c\left[{X_t,h}_{t-1}\right]+b_c\right)\\ c_t=i_t·c_t^{\mathrm{*}}+f_t·c_{t-1}\\ h_t=o_t·f_t(c_t)\end{array}\right.$$

(9)

where, $c_t^{\mathrm{*}}$ is the candidate cell state, and it, ft, and ot are the input, forget, and output gates, respectively. During training, the biases $b_{i,f,o,c}$ as well as the weights $W_{i,f,o,c}$ are learned with the input of current, neuron, and gates it, ft, and ot. The weights $D_{i,f,o,c}$ are learned with the gates and hidden states. Any nonlinear activation function, such as the ReLU or the hyperbolic tangent, may be used as the function (ct). For time series regression applications, LSTM networks provide several benefits [40]. First, they may capture long-term dependencies in the data that typical regression models may find challenging to capture. This is accomplished by using the memory cell and its gates to selectively remember or erase data from earlier time steps. Second, LSTM networks function well with time series data that have irregular sampling intervals because they can accommodate input sequences of varied lengths. In addition, they may identify intricate temporal linkages and patterns in the data, which can result in better performance than with conventional regression models.

2.3. Bidirectional LSTM

Schuster and Paliwal [41] developed bidirectional RNN a kind of RNN that uses past and future input data sequences to train the network. Two connected layers are used to process the input data, and each layer processes the data in the forward and reverse time step directions. The BiLSTM uses two layers to process input data in both directions. The BiLSTM shows more accurate results than unidirectional LSTM in some applications, including battery management systems [39]. The network topology of BiLSTM is illustrated in Figure 4.

BiLSTM consists of two hidden layers, the forward and backward LSTM layers, which are connected to the same output layer. The forward LSTM layer output ${\overrightarrow{h}}_t$ were evaluated by BiLSTM independently after that $\overleftarrow{h_t}$ every time step t is likewise concatenated through BiLSTM. The following Equations (10)–(12) expresses the BiLSTM process [42].

$$\overrightarrow{h_t}=f\left(X_t,{\overrightarrow{h}}_{(t-1)};{\overrightarrow{\Theta }}_{BiLSTM}\right)=\left\{\begin{array}{c}\overrightarrow{f_t}=sigmoid(\overrightarrow{W_f{X}_t}{+\overrightarrow{D_f}[X}_t,\overrightarrow{h_{(t-1)}}]+\overrightarrow{b_t)}\\ \overrightarrow{i_t}=sigmoid\left(\overrightarrow{W_t{X}_t}{+\overrightarrow{D_t}[X}_t,\overrightarrow{h_{\left(t-1\right)}}\right]+\overrightarrow{b_i})\\ c_t^{*}=\tanh \left(\overrightarrow{W_c{X}_t}{+\overrightarrow{D_c}[X}_t,\overrightarrow{h_{\left(t-1\right)}}\right]+\overrightarrow{b_c})\\ \overrightarrow{o_t}=sigmoid\left(\overrightarrow{W_o{X}_t}{+\overrightarrow{D_o}[X}_t,\overrightarrow{h_{\left(t-1\right)}}\right]+\overrightarrow{b_o})\\ \overrightarrow{c_t}=\overrightarrow{f_t}ʘC_{\left(t-1\right)}+c_t^{*}ʘ\overrightarrow{i_t}\\ \overrightarrow{h_t}=\overrightarrow{o_t}ʘ\tanh \left(\overrightarrow{c_t}\right)\\ \end{array}\right.$$

(10)

$$\overleftarrow{h_t}=f\left(X_t,{\overleftarrow{h_t}}_{-1};{\overleftarrow{\Theta }}_{BiLSTM}\right)=\left\{\begin{array}{c}\overleftarrow{f_t}=sigmoid({\overleftarrow{W_f{X}_t}+\overleftarrow{D_f}[X}_t,\overleftarrow{h_{(t-1)}}]+\overleftarrow{b_f})\\ \overleftarrow{i_t}=sigmoid({\overleftarrow{W_i{X}_t}+\overleftarrow{D_i}[X}_t,\overleftarrow{h_{(t-1)}}]+\overleftarrow{b_i})\\ c_t^{*}=tanh({\overleftarrow{W_c{X}_t}+\overleftarrow{D_c}[X}_t,\overleftarrow{h_{\left(t-1\right)}}]+\overleftarrow{b_c})\\ \overleftarrow{o_t}=sigmoid({\overleftarrow{W_o{X}_t}+\overleftarrow{D_o}[X}_t,\overleftarrow{h_{\left(t-1\right)}}]+\overleftarrow{b_o})\\ \overleftarrow{c_t}=\overleftarrow{f_t}ʘC_{\left(t-1\right)}+c_t^{*}ʘ\overleftarrow{i_t}\\ \overrightarrow{h_t}=\overleftarrow{o_t}ʘ\tanh \left(\overleftarrow{c_t}\right)\\ \end{array}\right.$$

(11)

$$y_t=W_{\overrightarrow{h_y}}\overrightarrow{h_t}+W_{\overleftarrow{h_y}}\overleftarrow{h_t}+b_y$$

(12)

The weight from the forward and backward LSTM layer to the output layer is represented by $\overrightarrow{W_f}$ and $\overleftarrow{W_f}$, respectively. The bias $\overrightarrow{b_{z,r,h}}$ and $\overleftarrow{b_{z,r,h}}$ at output, layers are clearly stated.

2.4. IPSO-Bi-LSTM

This research proposed an IPSO-Bi-LSTM estimation framework that combines the PSO algorithm with the Bi-LSTM model and modifies the network configuration to the LIBs dataset. PSO algorithms are simple and exhibit rapid convergence. It can improve parameter accuracy by learning from the characteristics of the dataset. The Bi-LSTM model uses PSO to rapidly and effectively optimize parameters based on LIBs dataset characteristics. Consequently, the network configuration of the Bi-LSTM model and the features of the LIBs dataset are incorporated more efficiently. Figure 5 shows the proposed IPSO-Bi-LSTM framework.

The following steps should be followed to implement the proposed SOH estimation framework, as shown in Figure 1. First, the original data must be extracted and then normalized. Secondly, prepare the training, cross-validation, and testing portions of the dataset. Thirdly, population size, spatial dimension, maximum number of iterations, learning factor, inertia weight, speed, and location of each individual particle are initialized. Fourth, the fitness function is determined by calculating the difference between the estimated and reference values. The fitness value is then used to modify the local and global optimal values. Filth and save the model when halting requirements are achieved. Sixth, after integrating the parameter values into the suggested model, the test samples are utilized for estimation, and the estimated result is validated.

3. Experimental Details and Dataset Description

This research utilizes the NASA batteries public dataset from the NASA Ames Centre of Excellence for Estimation database [43]. The batteries are listed as B5, B6, B7, and B18. Choose a dataset that has four 2 Ah rated 18,650 LIBs in it. The batteries in this group are subjected to identical external conditions and measurement procedures. During the battery cycle aging test, the battery is maintained at an ambient temperature of 24 °C. The charging procedure starts with a period of constant current (CC) charging, followed by constant voltage (CV) charging. The charging procedure is as follows: first, charge with a constant current (CC) of 1.5 A until the voltage reaches the cut-off voltage of 4.2 V; thereafter, charge with a constant voltage (CV) of 4.2 V until the current decreases to the cut-off current of 20 mA. CC mode discharges the battery at 2 A until it reaches the cut-off voltage. Temperature, voltage, and current are among the variables that are monitored and saved throughout every charge/discharge operation.

The cycle life and capacity of LIBs continuously decrease because of the progressive depletion of the active material caused by an increase in charge/discharge cycles. The complete capacity decay curves for the four LIBs used in this experiment, labeled B5, B6, B7, and B18, are shown in Figure 6. The graph depicts a nonlinear degradation process in the capacity of these batteries, with an overall dropping pattern punctuated by instances of partial regeneration, in which the capacity momentarily improves before declining again. This brief regeneration highlights the intricacy of the battery aging process, which makes it more difficult to estimate the remaining capacity. Each battery begins with an individual’s initial capacity and health status, reflecting differences in its initial capacity and health status, and reflecting differences in its original situations and use history. The cycle life of these batteries is deemed to terminate whenever their capacity drops below 80% of the specified capacity. The graphs indicate that B18 exceeds the limit at approximately 80 cycles, whereas B5, B6, and B7 approach it between approximately 120 and 150 cycles. The variety observed emphasizes the difficulties in accurately estimating battery performance and durability [44,45,46,47,48,49,50].

Table 1. The hyperparameters information of the Bi-LSTM model optimized by IPSO.

Model Structure	Parameter
Layers	3
Learning rate	0.001
Activation function	ReLU
Batch size	23
Maximum iteration	700

. The operational information of hyperparameters of the Bi-LSTM model optimized by IPSO.

4. Result and Analysis

To evaluate the effectiveness of the IPSO-Bi-LSTM model in estimating the SOH of LIBs, the authors used error assessment measures, including the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Square Error (MSE). The LSTM, Bi-LSTM, and LSTM-Bi-LSTM are popular neural network models that will be included for comparison to validate the efficiency and superiority of the developed IPSO-Bi-LSTM model for the estimation of SOH of LIBs. The data for each battery include the voltage, current, and capacity. The model was trained on 70% of the data and tested on 30% of the data for SOH estimation. Figure 7 shows the estimated outcomes for battery B5. However, the estimated SOH of the LIBs at the 117th cycle was taken as an initial point. The LSTM model estimates a specific real value while estimating SOH; however, the highest estimation error and computation time were 8.79% and 77 s (sec), which cannot properly pre-estimate the LIBs SOH operation to the final stage, as shown in Figure 7a,b. The Bi-LSTM model has a maximum estimation error and computational time of 4.59% and 50 s. The Bi-LSTM model exhibits a reduction in the highest estimate error and computational time of 4.20% and 27 s. Nevertheless, when the number of cycles increases, the estimates of the Bi-LSTM model diverge from the actual value and exhibit a visible delay, which is unsuitable. The LSTM-Bi-LSTM model has a maximum estimate error and computational time of 4.37% and 39 s. The LSTM-Bi-LSTM model achieved a 0.22% reduction in the maximum estimation error and a computational time of 11 s compared to the Bi-LSTM model. This improvement leads to a considerable enhancement in the estimation performance. However, the model’s estimation accuracy remains insufficient. The IPSO-Bi-LSTM model has a maximum estimation error of 4.10%, and with a computational time of 17 s, the estimation curve steadily fits the actual value. After comparing to the other three models, there was a reduction in estimation error of 4.85%, while in computational time, there was a 22 s improvement in estimation precision, making it excellent.

Figure 8 demonstrates that the IPSO-Bi-LSTM model precisely estimated the SOH in the B6 battery, which is excellent in performance as compared to the other three models. Using the 119th cycle as the initial point for estimating the SOH of LIBs, both the LSTM and Bi-LSTM models show significant instabilities in their estimations. The maximum estimate errors and computational time for the LSTM and Bi-LSTM models are 2.15%, 1.84%, 69, and 43 s, respectively. The estimation error and computational time reduction achieved by BiLSTM compared to LSTM is 0.31% and 26 s. The highest estimation error of the LSTM-Bi-LSTM model is 1.51%, with a computational time of 24 s, which indicates greater stability throughout the middle and early stages of estimation. Compared to the Bi-LSTM model, the LSTM-Bi-LSTM model decreases its maximum estimation error and computational time by 0.33% and 19 s. However, as the number of cycles increases, the estimated outcomes of the LSTM-Bi-LSTM model frequently differ from the actual values, demonstrating that accuracy in measurement remains insufficient. The proposed IPSO-BiLSTM model achieves a maximum estimate accuracy of 1.5% while having a computational time of 13 s, demonstrating improved fitting, lower error fluctuations, and computationally efficient. This significant advancement significantly improves the estimated accuracy of the SOH for LIBs.

The estimation approach for the SOH of the B7 battery starts with the process of estimating the SOH of the LIBs at the 117th cycle, as illustrated in Figure 9. At the 117th cycle, the LSTM and Bi-LSTM models precisely estimated the initial fluctuation, with maximum estimation errors of 5.82% and 5.07%, respectively. However, the computational times are 71 s and 53 s. The Bi-LSTM maximum estimate error reduction of 0.75% is smaller than that of LSTM and is computationally efficient for 18 s. The maximum estimation error of the LSTM-Bi-LSTM model is 5%, and the computational time was 21 s; the error is reduced by 0.07% when compared to the Bi-LSTM model. However, it was computationally efficient for 32 s. The estimation error of the IPSO-Bi-LSTM model is reasonably stable and approaches the actual value, and the maximum estimated error that was observed is 4.91% with a computational time of 11 s. It obtained a reduction in error and computational efficiency compared to the previous model of 0.9% and 10 s. The IPSO-Bi-LSTM model estimates the LIB SOH more precisely and efficiently than the LSTM, Bi-LSTM, and LSTM-Bi-LSTM models when employed on the B7 battery.

Figure 9 shows that the B18 battery SOH estimation initiates at the 87th cycle. The LSTM model estimates poorly, with a maximum error of 6.70%, and requires a computational time of 38 s. Bi-LSTM estimates have a maximum error of 6.30% with a computational time of 17 s. Bi-LSTM reduces the maximum estimate error by 0.40% and enhances estimation results compared to LSTM, which is computationally efficient 18 s. It has unpredictable violent changes within the actual value range. The LSTM-Bi-LSTM model has a lower maximum estimate error of 6.58% with a computational time of 14 s than that of the Bi-LSTM model. A maximum estimation error and computational time lower than 0.12% and 24 s has been achieved. Although the LSTM-Bi-LSTM model improves the estimation performance, its accuracy is still high. The IPSO-Bi-LSTM model reduces the maximum estimation error by 6.17% compared to the LSTM-Bi-LSTM model. The maximum estimate error drops by 0.41%, reducing error fluctuations. The computational time is 7 s, and 7 s is efficient compared to LSTM-Bi-LSTM. The model’s estimation accuracy improves when the estimation curves eventually match the actual values. The IPSO-Bi-LSTM model estimates the LIB of SOH more efficiently than the LSTM, Bi-LSTM, and LSTM-Bi-LSTM models on the B18 battery dataset. Additionally, the LIB SOH estimations are significantly more accurate. To visualize the estimated SOH inaccuracy for each modeling approach and highlight the IPSO-Bi-LSTM model’s advantages.

Figure 10 displays the estimated error of the SOH estimation at each cycle, as evaluated by MAE, MSE, RMSE, and R² based on the experimental data. The compared outcomes of the estimated evaluation of the performance indicators for all four models on the B5, B6, B07, and B18 batteries are shown in

Table 2. The statistical performances of SOH estimation.

Battery	Approaches	MAE	MSE	RMSE	R2	Max_ Error	Computational Time (s)
B5	LSTM	4.0592	5.8309	4.2227	0.7954	8.79014	77
	BiLSTM	0.5584	0.5200	0.7211	0.9940	4.59323	50
	LSTM-BiLSTM	0.6772	0.8109	0.9005	0.9949	4.3757	39
	IPSO-Bi-LSTM	0.3302	0.4406	0.6638	0.9973	4.1032	17
B6	LSTM	0.5512	0.4128	0.6425	0.9981	2.1558	69
	BiLSTM	0.4820	0.3544	0.5953	0.9984	1.8461	43
	LSTM-BiLSTM	0.4772	0.3158	0.5620	0.9986	1.5144	24
	IPSO-Bi-LSTM	0.3804	0.2533	0.5032	0.9989	1.5003	13
B7	LSTM	1.8662	6.7284	2.5939	0.8912	5.8640	71
	BiLSTM	0.6324	0.7976	0.8931	0.9871	5.0765	53
	LSTM-BiLSTM	0.6324	0.7976	0.8931	0.9871	5.0001	21
	IPSO-Bi-LSTM	0.3986	0.4904	0.7003	0.9921	4.9157	11
B18	LSTM	1.5614	5.7007	2.3876	0.8938	6.7006	38
	BiLSTM	0.8311	1.5186	1.2323	0.9717	6.3000	17
	LSTM-BiLSTM	0.6007	1.3041	1.1420	0.9757	6.58172	14
	IPSO-Bi-LSTM	0.5193	1.2050	1.0977	0.9776	6.1794	7

. IPSO-Bi-LSTM is superior to LSTM-Bi-LSTM, Bi-LSTM, and LSTM in estimating efficiency under similar conditions, as shown by the outcome figures. The IPSO-Bi-LSTM model demonstrates excellent fitting and varying degrees of decrease in the MAE, MSE, and RMSE. Among the four cells, the IPSO-Bi-LSTM model has the highest values for MAPE, MAE, and RMSE, which are 0.3302, 0.2533, and 0.5032%, respectively. The IPSO-Bi-LSTM model demonstrated a decrease in the MAE, MSE, and RMSE in comparison to the LSTM-Bi-LSTM model, with average reductions of 0.1897%, 0.2102%, and 0.1331%, respectively.

The IPSO-Bi-LSTM model demonstrated a mean decrease of 0.2188% in MAE, 0.2003% in MSE, and 0.1192% in RMSE when compared to the Bi-LSTM model. The IPSO-Bi-LSTM model demonstrated an average reduction of 1.6023% in MAE, 4.5708% in MSE, and 1.7204% in RMSE when compared to the LSTM model. The average computational time taken by the LSTM, BiLSTM, LSTM-Bi-LSTM, and IPSO-Bi-LSTM models was 64, 41, 25, and 13 s, respectively. The IPSO-Bi-LSTM model demonstrated a decrease in computational time in comparison to the three models, with average reductions of 52, 29, and 13 s, respectively. The Bi-LSTM model increases the algorithm’s accuracy while incorporating IPSO strengthens the model’s estimating capabilities. This advantage improves the feasibility of implementing the proposed method in real-world scenarios [45,46].

4.1. Discussion

The Bi-LSTM architecture is very effective in capturing temporal relationships in sequential data by analyzing information in both the forward and backward directions. The bidirectional processing enables the model to include information from both previous and future time steps, which is essential for effectively modeling complicated time series data, such as estimating the SOH of a battery. Conventional models may have difficulties in capturing distant connections, particularly when the correlation between input characteristics extends across numerous time intervals. Bi-LSTM, on the other hand, overcomes this constraint by preserving pertinent information from both forward and backward directions, resulting in more precise and resilient predictions.

However, IPSO improves the performance of the model by improving the hyperparameters more efficiently than traditional optimization approaches. IPSO enhances the traditional PSO by including methods that avoid premature convergence and promote a more comprehensive exploration of the solution space. As a result, this improves the fine-tuning of the hyperparameters, which is crucial for optimizing the accuracy and stability of the Bi-LSTM model. IPSO optimizes the model’s performance on different datasets by adjusting important parameters, such as the learning rate and number of hidden units. The integration of Bi-LSTM’s capacity to capture complicated temporal patterns with IPSO’s effective hyperparameter optimization results in a robust framework that surpasses other approaches, especially in tasks related to complex and dynamic systems such as SOH estimate. This combination provides the proposed approach with an asset in the domain of battery management by enabling it to provide forecasts that are more precise, reliable, and broadly applicable.

4.2. Practical Implementation and Advantages

The BMS monitors and manages battery charging and discharging. Therefore, accurate SOH calculation is essential. The BMS may change the charge rates or operating settings to avoid overcharging or deep discharging using accurate SOH data. This degree of management decreases the battery stress and failure risk and ensures safe and optimum operation. Additionally, an accurate SOH estimate extends the battery life. The technology can forecast when a battery will decline to a level where it no longer meets the performance criteria by precisely measuring its health. Predictive maintenance or replacement prevents abrupt failures and extends battery life. Maintaining batteries under optimum operating conditions based on correct SOH data slows deterioration and extends battery life.

Another advantage of precise SOH estimates is cost savings. Battery lifetime extension decreases the need for frequent replacements and saves money, particularly in businesses like EVs, renewable energy storage, and UPS that use massive battery deployments. Accurate SOH monitoring reduces warranty claims and ensures that batteries are only replaced when needed, saving money. As well as economic advantages, precise SOH estimate enhances battery-powered system dependability, which is crucial in medical devices, aerospace, and military operations. Reliable SOH data keeps these systems running, establishing user confidence and encouraging battery-powered technology adoption.

An accurate SOH estimate reduces battery usage’s environmental effect. Reduced battery replacements reduce the environmental impact of battery waste and recycling. This method reduces electrical waste and conserves battery manufacturing resources by maximizing battery usage before replacement. It also improves energy reserve management in energy storage systems like renewable energy installations. Knowing the exact health of each storage array battery improves the energy dispatch, grid stability, and storage system performance. These practical consequences demonstrate the method’s potential to improve battery-powered system performance, dependability, and sustainability in a variety of applications.

5. Conclusions

In this work, a significant contribution by developing a novel IPSO-Bi-LSTM model for accurate estimation of SOH of LIBs for electrified transportation. This article proposes an IPSO approach to enhance the accuracy of estimating the SOH for LIBs. The proposed approach involves using a neural network model called Bi-LSTM to estimate the SOH based on the IPSO. The IPSO-Bi-LSTM model uses the Bi-LSTM to capture the relationship between the feature indicators and SOH in both directions. The IPSO is used to optimize the structural hyperparameters of the model to achieve a precise and robust estimation of the SOH of LIBs. The proposed IPSO-Bi-LSTM model is superior to the LSTM, Bi-LSTM, and LSTM-Bi-LSTM models based on the validated experimental outcomes using different types of battery datasets. The IPSO-Bi-LSTM model achieves an MAE lower than 0.33%, an MSE lower than 0.25%, and an RMSE lower than 0.50% compared to other approaches.

Future Works

The proposed IPSO-Bi-LSTM model for the SOH estimation of LIBs has shown potential, but further investigation is needed to improve its performance and application. To increase model robustness and accuracy across varied situations, environmental parameters, and real-time operating circumstances should be included. Hybrid models that integrate IPSO-Bi-LSTM with other artificial intelligence methods may capture more complicated battery behavior patterns and relationships. Additionally, applying this concept to different batteries or energy storage devices may reveal its generalizability and adaptation. These future studies will improve SOH estimation methods and improve battery management system reliability and efficiency.