Battery Lifetime Prediction via Neural Networks with Discharge Capacity and State of Health

Abstract

The market share of electric vehicles (EVs) has grown exponentially in recent years to reduce air pollution and greenhouse gas emissions. The principal part of an EV is the energy storage system, which is usually the batteries. Thus, the accurate estimation of the remaining useful life (RUL) of the batteries, for an optimal health management and a decision-making policy, still remains a challenge for automakers. In this paper, the problem of battery RUL prediction is studied from a new perspective. Unlike other estimation strategies existing in the literature, the proposed technique uses an intelligent prediction of the lifespan of lithium–iron–phosphate (LFP) batteries via a modified version of neural networks. It uses a data-driven life estimation approach and optimization method and does not require any prior comprehension and initialization of any parameters of the battery model. To validate and verify the proposed technique, we use LFP battery data sets, and the experimental results showed that the proposed methodology can well learn the characteristic relationship of battery discharge capacities as well as its state of health (SOH), where the battery life cycle changes as the battery ages with time and cycles.

1. Introduction

The harmful gas emissions created by the transportation sector are a threat to the health of humans due to increasing pollution and global warming. According to works by Leach et al. [1] and Palencia et al. [2], almost 99.8% of global transport is powered by internal combustion engines, and 95% of transport energy comes from liquid fuels made from petroleum. Definitely, transport is responsible for 71.7% of CO2 emissions coming from road transportation according to a report from the Annual Energy Outlook (AEO) [3]. Currently, the introduction of environment friendly efficient energy conversion systems are facing the challenges for decarbonizing energy in the transportation sector [4]. In this context, electricity is the fastest-growing energy in the transportation sector with an average projected growth of 7.4% per year by 2050. Moreover, electric vehicles, (EVs) have received massive attention and popularity around the world due to their capability to reduce environmental pollution and minimize carbon emissions and global warming [5]. In summary, EVs are the promising alternative to diesel vehicles not only in terms of emissions levels, but also in terms of simplicity, reliability, efficiency and comfort [6]. As an important component, the battery system has a great impact on the performance of electric vehicles. The authors in [7,8] show that lithium-ion batteries are widely used in many real applications of electric vehicles. This great interest in the lithium-ion battery is due to several advantages, such as high energy density, high power density, high cell voltage, long lifespan, light weight and environmental friendliness. They are the first choice for power sources for electric vehicles. Therefore, the efficiency of electric vehicles depends on the precise evaluation of some key parameters of the battery for proper operation and diagnosis of this energy storage system.

However, a poor battery storage system monitoring and safety policy can lead to critical issues. To address these concerns, the prognostic and health management (PHM) for battery packs plays a crucial role in enhancing battery performance, including precise monitoring of battery safety [9,10]. Accordingly, one of the most critical roles of the PHM is to accurately predict the remaining useful life of a battery, detail best practices for operation and maintenance (preventive maintenance), and ensure safe use [11,12]. Indeed, the state of health (SOH) is a key index for evaluating the health status of batteries and to determine the remaining useful life, plan maintenance while ensuring minimal downtime and ensure the safety of the assembly [13,14,15,16]. A wide range of methods and design approaches has been proposed in the literature to track battery lifetime prediction, which can be divided into three categories: physical models, empirical models and data-driven approaches [17,18]. Several works [19,20,21,22] show that the physical models, which are based on mathematical models describing the battery dynamics of internal reactions in the electrodes, often face challenges in practical applications due to model complexity and the identification of parameters. The empirical battery models, such as the equivalent circuit model [23] and the reduced electrochemical model [18], can easily be applied to online SoH monitoring, but the algorithm development often requires profound experimental validation and debugging to provide adequate accuracy. Recently, many data-based methods [24,25,26,27] have been applied for the prediction of battery life; these methods have some limitations due to the number of neurons in the hidden layer and the fact that the number of parameters of the regression vector are user-defined. That is why we developed a new method to predict the lifetime of a battery with high precision. Firstly, we developed a new algorithm for the modeling of discharge capacity. Secondly, an optimization neural network structure was applied. Finally, the model obtained can be used to predict LFP battery lifetime.

This paper is structured as follows: Section 2 describes the proposed method based on the Levenberg–Marquardt algorithm, and a comparative study is carried out with other learning algorithms existing in the literature. In addition, an optimization method is applied to the proposed adaptive algorithm. Finally, the proposed method is used to predict the discharge capacity (DC) and SOH of the battery in order to determine its remaining useful life.

2. Methodology

The Levenberg–Marquardt (LM) training algorithm is commonly used for nonlinear data and optimization problems [28]. LM is a supervised algorithm, and it is suitable for nonlinear time series data sets. Hence, the LM method is a combination of two numerical minimization algorithms: the gradient descent and the Gauss–Newton methods [29,30]. The gradient descent method ensures that the loss function decreases at each iteration, but the convergence is slow. In the Gauss–Newton, the sum of the squared errors is reduced by assuming the least squares function is locally quadratic in the parameters and finding the minimum of this quadratic. The Levenberg–Marquardt method acts more like a gradient-descent method when the parameters are far from their optimal value and acts more like the Gauss–Newton method when the parameters are close to their optimal value [31]. In order to ameliorate the Levenberg–Marquardt algorithm with respect to convergence time and approximation quality, we propose a new version of this algorithm, named the Levenberg–Marquardt backpropagation neural network (LM-BPNN).

2.1. Proposed Method

The proposed LM-BPNN algorithm minimizes the following cost function:

$$J=\sum _{i=1}^N\frac{1}{2}{\left(e_k\right)}^2+\sum _{i=1}^N\frac{1}{4}\beta {\left(e_k\right)}^4$$

(1)

and

$$e_k=y_k-{\widehat{y}}_k$$

(2)

where $e_k$ is the learning error, $y_k$ is the real output at time k, and ${\widehat{y}}_k$ is the predicted output of the system.

Using the structure of Figure 1, the i-th neuron of the output layer is expressed as:

$$y_{k_i}={o_{k_i}}^{\left(2\right)},=f^{\left(2\right)}\left({h_{k_i}}^{\left(2\right)}\right),i=1,2,...,n_2$$

(3)

where $f^{\left(2\right)}$ is the activation function of the output $h_{k_i}^{\left(2\right)}$. This latter is defined by:

$${h_{k_i}}^{\left(2\right)}=\sum _{i=1}^{n_2}\sum _{j=1}^{n_1}{w_{k_{ij}}}^{\left(2\right)}\ast {o_{k_j}}^{\left(1\right)}+{b_{k_i}}^{\left(2\right)}$$

(4)

where $b_i^{\left(2\right)}$ is the bias of the i-th neuron of the output layer, and the activation output of the j-th neuron of the hidden layer $o_{k_j}^{\left(1\right)}$ is:

$${o_{k_j}}^{\left(1\right)}=f^{\left(1\right)}\left({h_{k_j}}^{\left(1\right)}\right),j=1,2,...,n_1$$

(5)

The output $h_j^{\left(1\right)}$ of j-th neuron of the hidden layer is:

$${h_{k_j}}^{\left(1\right)}=\sum _{j=1}^{n_1}\sum _{p=1}^{n_o}{w_{k_{jp}}}^{\left(1\right)}\ast {x_p}_{\left(k\right)}+b_j^{\left(1\right)};$$

(6)

The first step is to correct the weights and the bias of the output layer. The weights of this layer are modified using the following equation:

$$\begin{array}{c}{w}_{{(k+1)}_{ij}}^{\left(2\right)}=w_{k_{ij}}^{\left(2\right)}+\Delta w_{k_{ij}}^{\left(2\right)}+\eta \left(l\right)\mu \left(l\right)\Delta w_{{(k-1)}_{ij}}^{\left(2\right)}\hfill \end{array}$$

(7)

with $\eta \left(l\right)$ being the learning rate, and the correction term $\Delta {w_{ij\left(k\right)}}^{\left(2\right)}$ is defined as:

$$\Delta w_{k_{ij}}^{\left(2\right)}=-\eta \left(l\right)\frac{\partial J}{\partial {w_{ij}}^{\left(2\right)}}$$

(8)

with:

$$\frac{\partial J_k}{\partial {w_{ij}}^{\left(2\right)}}=\frac{\partial J_k}{\partial o_{k_i}^{\left(2\right)}}\frac{\partial o_{k_i}^{\left(2\right)}}{\partial {h_{k_i}}^{\left(2\right)}}\frac{\partial {h_{k_i}}^{\left(2\right)}}{\partial w_{k_{ij}}^{\left(2\right)}}$$

(9)

For ${\delta }_i=\frac{\partial J_k}{\partial o_{k_i}^{\left(2\right)}}\frac{\partial o_{k_i}^{\left(2\right)}}{\partial h_{k_i}^{\left(2\right)}}$, Equation (9) becomes:

$$\frac{\partial J_k}{\partial {w_{ij}}^{\left(2\right)}}={\delta }_i\frac{\partial h_{k_i}^{\left(2\right)}}{\partial w_{k_{ij}}^{\left(2\right)}}$$

(10)

with $\frac{\partial h_{k_i}^{\left(2\right)}}{\partial w_{k_{ij}}^{\left(2\right)}}=o_{k_j}^{\left(1\right)}$, Equation (8) is approximated by:

$$\Delta w_{k_{ij}}^{\left(2\right)}=-\eta {\delta }_i{o}_{k_j}^{\left(1\right)}$$

(11)

and

$$\begin{array}{c}\frac{\partial J_k}{\partial o_{k_i}^{\left(2\right)}}=\frac{\partial J_k}{\partial \widehat{y_{k_i}}}=\hfill \\ -\left(y_{k_i}-\widehat{y_{k_i}}\right)-\beta {\left(y_{k_i})-{\widehat{y}}_{k_i}\right)}^3\hfill \end{array}$$

(12)

$$\frac{\partial o_{k_i}^{\left(2\right)}}{\partial h_{k_i}^{\left(2\right)}}=f^{\prime }\left(h_{k_i}^{\left(2\right)}\right)$$

(13)

$${\delta }_i=-f^{\prime }\left(h_{k_i}^{\left(2\right)}\right)\left(\left(y_{k_i}-\widehat{y_{k_i}}\right)-\beta {\left(y_{k_i}-\widehat{y_{k_i}}\right)}^3\right)$$

(14)

Using Equations (7), (11) and (14), we obtain:

$$\begin{array}{c}{w}_{{(k+1)}_{ij}}^{\left(2\right)}=w_{k_{ij}}^{\left(2\right)}+\eta \left(l\right)f^{\prime }\left(h_{k_i}^{\left(2\right)}\right)o_{k_j}^{\left(1\right)}\hfill \\ \times {\epsilon }_i\left(k\right)(1+\beta {\epsilon }_i{\left(k\right)}^2)+\eta \left(l\right)\mu \left(l\right)\Delta w_{{(k-1)}_{ij}}^{\left(2\right)}\hfill \end{array}$$

(15)

The output layer bias is updated using the following equation:

$$\begin{array}{c}{b}_{{(k+1)}_{ij}}^{\left(2\right)}=b_{k_{ij}}^{\left(2\right)}+\eta \left(l\right)f^{\prime }\left({h_{k_i}}^{\left(2\right)}\right){\epsilon }_{k_i}\hfill \\ \times (1+\beta {{\epsilon }_{k_i}}^2)+\eta \left(l\right)\mu \left(l\right)\Delta w_{{(k-1)}_{ij}}^{\left(2\right)}\hfill \end{array}$$

(16)

The second step is to correct the weights and bias of the hidden layer. The update equation of the hidden layer’s weights is:

$$\begin{array}{c}\hfill \begin{array}{c}{w}_{{(k+1)}_{jp}}^{\left(1\right)}=w_{k_{jp}}^{\left(1\right)}-\eta \left(l\right)\frac{\partial J\left(k\right)}{\partial w_{k_{jp}}^{\left(1\right)}+\eta \left(l\right)\mu \left(l\right)\Delta w_{{(k-1)}_{jp}}^{\left(1\right)}}\hfill \end{array}\end{array}$$

(17)

with:

$$\begin{array}{c}\hfill \Delta w_{k_{jp}}^{\left(1\right)}=-\eta \left(l\right)\frac{\partial J_k}{\partial w_{k_{jp}}^{\left(1\right)}}\end{array}$$

(18)

and $\frac{\partial J_k}{\partial w_{k_{jp}}^{\left(1\right)}}$ can be formulated as follows:

$$\begin{array}{c}\frac{\partial J_k}{\partial w_{k_{jp}}^{\left(1\right)}}=\frac{\partial J_k}{\partial o_{k_i}^{\left(2\right)}}\frac{\partial o_{k_i}^{\left(2\right)}}{\partial h_{k_i}^{\left(2\right)}}\frac{\partial h_{k_i}^{\left(2\right)}}{\partial o_{k_j}^{\left(1\right)}}\times \frac{\partial o_{k_j}^{\left(1\right)}}{\partial {h_{k_j}}^{\left(1\right)}}\frac{\partial {h_{k_j}}^{\left(1\right)}}{\partial w_{k_{jp}}^{\left(1\right)}}\hfill \end{array}$$

(19)

Given

$$\frac{\partial J_k}{\partial o_{k_i}^{\left(2\right)}}\frac{\partial o_{k_i}^{\left(2\right)}}{\partial h_{k_i}^{\left(2\right)}}={\delta }_i=-\left({\epsilon }_{k_i}(1+\beta {{\epsilon }_{k_i}}^2\right)f^{\prime }\left(h_{k_i}^{\left(2\right)}\right)$$

(20)

$$\frac{\partial h_{k_i}^{\left(2\right)}}{\partial o_{k_j}^{\left(1\right)}}\frac{\partial o_{k_j}^{\left(1\right)}}{\partial {h_{k_j}}^{\left(1\right)}}={\delta }_j=w_{k_{ij}}^{\left(2\right)}{f}^{\prime }\left({h_{k_j}}^{\left(1\right)}\right)$$

(21)

$$\frac{\partial h_{k_{jp}}^{\left(1\right)}}{\partial w_{k_{jp}}^{\left(1\right)}}={x_p}_{\left(k\right)}$$

(22)

and combining Equations (20)–(22) yields

$$\begin{array}{c}\frac{\partial J_k}{\partial w_{k_{jp}}^{\left(1\right)}}=-f^{\prime }\left(h_{k_i}^{\left(2\right)}\right)w_{k_{ij}}^{\left(2\right)}{f}^{\prime }\left({h_{k_j}}^{\left(1\right)}\right){x_p}_{\left(k\right)}\hfill \\ \times \left({\epsilon }_{k_i}(1+\beta {\epsilon }_{k_i}^2\right)\hfill \end{array}$$

(23)

Then, the complete form of Equation (17) is:

$$\begin{array}{c}{w}_{{(k+1)}_{jp}}^{\left(1\right)}=w_{k_{jp}}^{\left(1\right)}+\eta \left(l\right)f^{\prime }\left(h_{k_i}^{\left(2\right)}\right)\times {\epsilon }_{k_i}{w}_{k_{ij}}^{\left(2\right)}{f}^{\prime }\left({h_{k_j}}^{\left(1\right)}\right){x_p}_{\left(k\right)}\hfill \\ \begin{array}{c}=w_{k_{ij}}^{\left(2\right)}+\eta \left(l\right)f^{\prime }\left({h_{k_i}}^{\left(2\right)}\right){o_{k_j}}^{\left(1\right)}\hfill \\ \times {\epsilon }_{k_i}(1+\beta {{\epsilon }_{k_i}}^2+\eta \left(l\right)\mu \left(l\right)\Delta w_{{(k-1)}_{ij}}^{\left(2\right)}\hfill \end{array}\hfill \end{array}$$

(24)

The hidden layer bias is updated by:

$$\begin{array}{c}{h}_{{(k+1)}_{jp}}^{\left(1\right)}=h_{{\left(k\right)}_{jp}}^{\left(1\right)}+\eta f^{\prime }\left(h_{k_i}^{\left(2\right)}\right){\epsilon }_{k_i}\hfill \\ \times b_{{\left(k\right)}_{ij}}^{\left(2\right)}{f}^{\prime }\left({h_{k_j}}^{\left(1\right)}\right)\hfill \\ =b_{{\left(k\right)}_{ij}}^{\left(2\right)}+\eta \left(l\right)f^{\prime }\left({h_{k_i}}^{\left(2\right)}\right){\epsilon }_i\left(k\right)(1+\beta {{\epsilon }_{k_i}}^2\hfill \\ +\eta \left(l\right)\mu \left(l\right)\Delta w_{{(k-1)}_{ij}}^{\left(2\right)}\hfill \end{array}$$

(25)

In Equations (15) and (24), the momentum coefficient $\mu$ and the learning rate $\eta$ are modified as follows [32]:

$$\mu \left(l\right)=\left\{\begin{array}{ccc}1.2\times \mu (l-1)& \mathrm{if}& MSE\left(l\right)<MSE(l-1)\\ \mu \left(l\right)& \mathrm{if}& MSE\left(l\right)=MSE(l-1)\\ \frac{\mu (l-1)}{1.2}& \mathrm{if}& MSE\left(l\right)>MSE(l-1)\end{array}\right.$$

(26)

$$\left\{\begin{array}{ccc}\mu \left(l\right)=0.01& if& \mu \left(l\right)>1\end{array}\right.$$

(27)

with $\eta$ in the range [0,1]

$$\eta \left(l\right)=\left\{\begin{array}{ccc}1.2\times \eta (l-1)& \mathrm{if}& MSE\left(l\right)>MSE(l-1)\\ \eta \left(l\right)& \mathrm{if}& MSE\left(l\right)=MSE(l-1)\\ \frac{\eta (l-1)}{1.2}& \mathrm{if}& MSE\left(l\right)<MSE(l-1)\end{array}\right.$$

(28)

with $MSE$ is the mean square error and used as the performance index. It is defined as:

$$MSE=\frac{1}{N}\sum _{k=1}^N(y_k-\widehat{y_k}{)}^2$$

(29)

where N is the number of observations.

The specific steps of the proposed algorithm are summarized as follows:

Fix the number of iterations l, the number of hidden layers $n_1$ and $n_2$ output, initialize the matrices of the different weights randomly.

Fix the stopping criterion $ϵ>0$.

Repeat for

step 1. Via Equation (5), calculate the outputs of the hidden layer ${o_j}^{\left(1\right)}$.

step 2. Via Equation (3), calculate the outputs ${o_i}^{\left(2\right)}$ of the output layer.

step 3. Via Equation (8), calculate the local gradients of the output layer $\Delta {w_{ij}}^{\left(2\right)}$.

step 4. Via Equation (19), calculate local gradients $\Delta {w_{jp}}^{\left(1\right)}$ of the hidden layer.

step 5. Via Equation (15), (16), (24) and (25) update all weights and biases of different layers.

step 6. Calculate the error between the desired output and the estimated one.

Untilerr$err\le ϵ$, then stop. Otherwise $l=l+1$ and return to step 1.

To illustrate the performance of the proposed LM-BPNN algorithm with respect to the convergence time, a comparative study was carried out with other standard BPLM learning algorithms existing in the literature [33] and mixed least square and least fourth (MLSLF) [34].

The parameters adopted for the three algorithms are: l = 1000, $n_0$ = 4, $n_1$ = 9, $n_2$ = 1, $\eta$ = 0.4, $\mu$ = 0.1, $\beta$ = 0.2 et $\left\{y_{(k-1)},y_{(k-2)},y_{(k-3)},y_{(k-4)}\right\}$.

We have taken as the activation function for the hidden layer the hyperbolic tangent activation function and for the output layer the linear activation function. The simulation results of the three algorithms are plotted in Figure 2 and Figure 3. From these figures, we can see that the BPLM algorithm presents lower MSE values at the first 10 iterations than the other algorithms. This is also observed through the minimization of the criterion J in Figure 2. The analysis of the results in Figure 3 also shows the influence of the integration of the adaptive calculations of the $\mu$ and $\eta$ coefficients on the convergence speed of the proposed LM-BPNN algorithm.

The latter requires a reduced number of iterations to approximate the nonlinear system.

2.2. Structure Selection

The techniques used for parameter identification for nonlinear systems are generally based on minimizing an objective function of the difference between the estimated and the measured outputs of the actual system. These methods are based on iterative optimization techniques.

2.2.1. Model Structure Selection

The choice of a model structure must satisfy a compromise between simplicity and complexity. The model must be as simple as possible to use and as complex as possible to represent the process. According to [35,36], several models exist in the literature to identify nonlinear systems, such as the neural network autoregressive model with eXogenous inputs (NNARX), the neural network autoregressive moving average model with eXogenous inputs (NNARMAX) and the neural network output error model (NNOE). In our study, we use the standard representation model NNARX. The latter is described by the following expression:

$$y_k=F\left(y_{(k-1)},...,y_{(k-n)},u_{(k-1)},...,u_{(k-m)}\right)+e_k$$

(30)

Equation (30) can be rewritten in the following form:

$${\widehat{y}}_k=F_{NN}\left(\psi \left(k\right),\widehat{\theta }\left(k\right)\right)$$

(31)

where $\widehat{\theta }\left(k\right)$ is the parameters vector of the network,and the regression vector $\psi \left(k\right)$ is as follows:

$$\psi \left(k\right)={\left[y_{(k-1)},...,y_{(k-n)}{u}_{(k-1)},...,u_{(k-m)}\right]}^T$$

(32)

The NNARX model is a non-recurrent model. The regression vector consists of the past inputs of u and the past measured outputs of y.

2.2.2. Architecture of Neural Networks

After choosing the identification algorithm to model the battery, it remains to determine the number of hidden layers and the number of neurons in hidden layers (number of hidden layers) as well as to define the parameters of the regression vector.

Different model selection strategies exist in the literature [37,38]. Among them, we can cite the weighted information criteria (WIC). This criterion is shown below.

$$\begin{array}{c}WI{C}_t=0.1\left(AI{C}_t+BI{C}_t\right)+0.2\left(RMS{E}_t+MAP{E}_t\right)\\ +0.2\left(\left(1-\mathrm{D}{\mathrm{A}}_{\mathrm{t}}\right)+\mathrm{MD}{\mathrm{A}}_{\mathrm{t}}\right)\end{array}$$

(33)

with t representing the different possible architectures for an NNARX model. This criterion is calculated by adding several weighted selection criteria. It validates the accuracy of the predictive model using several performance measures. The Akaïke information criterion (AIC) is used to determine the complexity of the model by minimizing the following equation:

$$AIC=log\left(\frac{{\sum }_{k=1}^N{\left[y_k-{\widehat{y}}_k\right]}^2}{N}\right)+\frac{2m_1}{N}$$

(34)

where $m_1$ is the number of weights in the neural network. The Baysian information criterion (BIC), root mean square error (RMSE), mean absolute percentage error (MAPE) and direction accuracy criterion (DA) are defined by:

$$BIC=log\left(\frac{{\sum }_{k=1}^N{\left[y_k-{\widehat{y}}_k\right]}^2}{N}\right)+\frac{m_{1}log\left(N\right)}{N}$$

(35)

$$RMSE={\left(\frac{{\sum }_{k=1}^N{\left[y_k-{\widehat{y}}_k\right]}^2}{N}\right)}^{1/2}$$

(36)

$$MAPE=\frac{1}{N}\sum _{k=1}^N\left|\frac{y_k-{\widehat{y}}_k}{y_k}\right|$$

(37)

$$\begin{array}{c}DA=\frac{1}{N}\sum _{k=1}^N{a}_k,\hfill \\ \mathrm{with}{a}_k=\left\{\begin{array}{cc}1& \mathrm{if}\left(y_{(k+1)}-y_k\right)\left({\widehat{y}}_{(k+1)}-y_k\right)>0\\ 0& \mathrm{Otherwise}.\end{array}\right.\hfill \end{array}$$

(38)

In Equation (33), the modified direction accuracy (MDA) criterion can be calculated by:

$$\mathrm{MDA}=\frac{1}{N-1}\sum _{k=1}^{N-1}{\left(A_k-F_k\right)}^2$$

(39)

with $a_k$ and $F\left(k\right)$ selected as:

$$A_k=\left\{\begin{array}{c}1\mathrm{si}{y}_{(k+1)}-y_k<0\hfill \\ 0\mathrm{si}{y}_{(k+1)}-y_k>0\hfill \end{array}\right.$$

(40)

$$F_k=\left\{\begin{array}{c}1\mathrm{si}{\widehat{y}}_{(k+1)}-{\widehat{y}}_k<0\hfill \\ 0\mathrm{si}{\widehat{y}}_{(k+1)}-{\widehat{y}}_k>0\hfill \end{array}\right.$$

(41)

The different steps used to determine the optimal network architecture are as follows:

Step 1. Define the different possible architectures for an NNARX model.

Step 2. Calculate $\mathrm{AIC},\mathrm{BIC},\mathrm{RMSE},\mathrm{MAPE},\mathrm{DA}$ and $\mathrm{MDA}$ criteria.

Step 3. Standardize the AIC, BIC, RMSE, MAPE, DA and MDA criteria for each architecture. For example, for m1 values of AIC, the standardization is as follows:

$$AI{C}_t=\frac{\left(AI{C}_t-min\left(AIC\right)\right)}{\left(max\left(AIC\right)-min\left(AIC\right)\right)}$$

(42)

Step 4. Calculate the $WI{C}_t$ by Equation (33).

Step 5. Choose the architecture with the minimum value of $WI{C}_t$.

The selection of the optimal architecture of the proposed LM-BPNN algorithm is illustrated in Figure 4.

3. Experimental Results and Discussion

In this study, we used data sets from the Center for Advanced Life Cycle Engineering (CALCE) [39]. The characteristics of the LFP battery are listed in

Table 1. Characteristics of the LFP battery.

Capacity rating	2600 mAh
Cell chemistry	LiFePO4
Weight	80.5 g
Length	26.5 mm ± 0.2 mm
Height	65.2 mm ± 0.4 mm
Nominal voltage	3.2 V

.

The data sets are collected according to the following experimental procedure:

1. Discharge the battery for constant current at 2.6 A;

2. Discharge until voltage is 4.2 V;

3. Charge at constant voltage until current <0$.08\mathrm{A}$;

4. Rest for 2 min and measure the resistance;

5. Rest for 1 min.

Figure 5 illustrates the proposed methodology for battery lifetime prediction. We use the proposed LM-BPNN algorithm to predict the discharge capacity and to estimate the state of health of the LFP battery. In this context, we use the neuronal model selection algorithm presented in Figure 4 to select the optimal structure. Once the various optimal parameters are obtained, then they are used as input parameters for the LM-BPNN algorithm.

Several validity measure metrics are used to examine the prediction accuracy of DC, such as the mean squared error (MSE)), root mean square error (RMSE), normalized root mean square error (NRMSE) and root mean squared percentage error (RMSPE) with:

$$NRMSE={\frac{RMSE}{y_{k_{max}}-y_k}}_{min}$$

(43)

$$RMSPE=\sqrt{\frac{1}{N}\sum _{k=1}^N{\left(\frac{y_k-{\widehat{y}}_k}{y_k}\right)}^2}\ast 100$$

(44)

3.1. Comparative Analysis

Using the achitecture of Figure 5, the data sets are divided into three test groups: training data, validation data and test data. We used 80% of data sets for model learning, 15% for validation and the rest as the testing data set.

First, a structural learning phase is applied to the proposed LM-BPNN algorithm to determine optimal architecture using the flowchart of Figure 4. Using this technique, the optimal parameter values adopted for the three algorithms are: $n_0$ = 6, $n_1$ = 9, $n_2$ = 1, $\beta$ = 0.2, l = 5000, $\eta$ = 0.04, $\mu$ = 0.002 and the regression vector $\left\{y_{(k-1)},y_{(k-2)},y_{(k-3)},y_{(k-4)}\right\}$.

The hyperbolic tangent and the linear function are used as activation functions in the hidden layer and output layer, respectively. The results of training, validation and testing of the DC model using standard BPLM, MLSLF and LM-BPNN algorithms are shown in Figure 6, Figure 7 and Figure 8.We can see from the evolution of prediction errors in Figure 6, Figure 7 and Figure 8 that the proposed algorithm has little prediction error compared to those obtained using the other two algorithms (BPLM and MLSLF). Using the proposed LM-BPNN algorithm, the experimental results also show a good match between the actual and the estimated of DCs for LFP battery.

From Figure 6, Figure 7 and Figure 8, we can note that the performance of the other two algorithms is degraded, especially during the validation and testing phases of the model, while our LM-BPNN algorithm always keeps its best performance for all three parts of the data sets (training, validation and testing).

A similar analysis can also be observed in

Table 2. Performance indices of the BPLM algorithm.

	MSE	RMSE	NRMSE	RMSPE (%)
Training	4.5481 × 10${}^{-4}$	0.0213	0.0380	1.1871
Validation	0.0245	0.1564	2.5069	10.8574
Test	0.0413	0.2032	29.8817	14.4836

,

Table 3. Performance indices of the MLSLF algorithm.

	MSE	RMSE	NRMSE	RMSPE (%)
Training	3.5783 × 10${}^{-4}$	0.0189	0.0337	1.0486
Validation	0.1281	0.3579	5.7363	24.8415
Test	0.2130	0.4615	67.8708	32.8967

and

Table 4. Performance indices of the LM-BPNN algorithm.

	MSE	RMSE	NRMSE	RMSPE (%)
Training	3.2221 × 10${}^{-5}$	0.0057	0.0101	0.3176
Validation	1.1290 × 10${}^{-5}$	0.0034	0.0538	0.2385
Test	5.8381 × 10${}^{-6}$	0.0024	0.3553	0.1722

. As indicated in these tables, our method provides the best performance indices compared to those obtained by the two other techniques and in particular through performance indices of the testing sets for the discharge capacities obtained by the three algorithms.

The RMSPE performance index of the LM-BPNN algorithm is 0.1722%, while the BPLM and MLSLF algorithms have bad performance with RMSPEs of 14.4836% and 32.8967%.

This finding led to the conclusion that the adaptive calculations of the $\mu$ and $\eta$ coefficients have a bigger impact on improving the approximation quality of the proposed algorithm.

To confirm the validity of the models obtained by three algorithms, correlation tests were carried out to check whether the models are appropriate. As reported in [40,41,42], the model of a system is adequate if the following conditions are satisfied:

$$\left\{\begin{array}{cc}{\varphi }_{{\epsilon }^{\prime }{\epsilon }^{\prime }}\left(\tau \right)=\delta \left(\tau \right),& \forall \tau \\ {\varphi }_{{\epsilon }^{\prime }{\left({\epsilon }^2\right)}^{\prime }}\left(\tau \right)=0,& \forall \tau \\ {\varphi }_{{\left({\epsilon }^2\right)}^{\prime }{\left({\epsilon }^2\right)}^{\prime }}\left(\tau \right)=\delta \left(\tau \right),& \forall \tau \end{array}\right.$$

(45)

with

$$\left\{\begin{array}{c}{\epsilon }_k^{\prime }=e_k-\overline{\epsilon }=e_k-\frac{1}{N}{\displaystyle \sum _{k=1}^N}{e}_k\\ {\left({\epsilon }_k^2\right)}^{\prime }={\epsilon }_k^2-\overline{{\epsilon }^2}={\epsilon }_k^2-\frac{1}{N}{\displaystyle \sum _{k=1}^N}{\epsilon }_k^2\end{array}\right.$$

(46)

where $\delta$ is the Kronecker delta, $\varphi$ is the intercorrelation function, and $\epsilon$ is the residue (prediction error) of the model. The results of the correlation tests are shown in Figure 9, Figure 10 and Figure 11. The dashed lines in each plot denote the 95% confidence bands at $\pm 1.96\sqrt{N}$.

Figure 10 and Figure 11 show that many points for the estimated discharge capacity using the BPLM and MLSLF algorithms are outside the 95% confidence interval. These figures confirm that the BPLM and MLSLF algorithms are not statistically valid.

However, the DC model validity tests for the proposed LM-BPNN algorithm, shown in Figure 9, are well within 95% confidence bands showing the validity of the model. As a result, we can then judge that the model is suitable to represent the real behavior of the LFP battery.

3.2. Lifetime Prediction Based on Discharge Capacity and State of Health

The state of health (SOH) used in this work is defined as the ratio of the discharge capacity after stabilization to the maximum discharge capacity. It is expressed by:

$$SOH=\frac{C_d}{C_{max}}\ast 100$$

(47)

where $C_{max}$ and $C_d$ are the maximum (nominal) capacity and the discharge capacity.

The evolution plots of the discharge capacities and state of health for the LFP battery are given in Figure 12 and Figure 13. It can be seen from these figures that the CD or SOH values decrease with an increasing number of charge/discharge cycles. As in the work of [43,44], we considered the threshold capacity value of 1.58h for the LFP battery, which is in accordance woth a loss of 30% of the nominal capacity. As shown in Figure 12 and Figure 13, it is clear that the DC and SOH values of the battery decrease with increasing number of cycles, especially after the 1093 cycle, thus indicating the aging of the battery. Consequently, the battery becomes unusable from cycle 1093. Therefore, after this cycle, the battery must be changed.

4. Conclusions

This paper is a contribution relating to the methods of prognosis of the aging of batteries of electric vehicles using artificial intelligence. Firstly,a new version of the gradient backpropagation algorithm has been proposed. Indeed, the proposed algorithm is a mixed version of the BPLM and MLSLF algorithms. This new version was used for the prognosis of the aging of an LFP battery. Secondly, the LM-BPNN algorithm was combined with a structural optimization technique to predict the future behavior of the LFP battery. The performance of the proposed approach was evaluated through a comparative study on the real data collected from a LFP battery. The experimental results show that the proposed method has a good predictive precision that allows it to follow-up the evolution of the real behavior of the battery. It can also be a highly valuable method for other types of batteries. However, we have challenges to improve the method developed in this work for a more powerful one by considering the effect of temperature and depth of charge or discharge.