State of health estimation and remaining useful life prediction for lithium-ion batteries using FBELNN and RCMNN

Abstract

This study proposes the novel method of lithium-ion battery state of health (SoH) estimation and remaining useful life (RUL) prediction to ensure the safety and reliability of the energy storage system. A fuzzy brain emotional learning neural network (FBELNN) is employed to estimate SoH and a recurrent cerebellar model neural network (RCMNN) is used for the RUL prediction. The inputs of FBELNN are extracted features from the monitoring curve of the constant voltage and current, because the lithium-ion battery is seldom completely discharged and the discharging situation in actual operating process is complex. The FBELNN learns the battery aging features that are extracted and selected by discrete wavelet transform and principal component analysis, respectively. The SoH estimation results from the FBELNN are accurate due to the special structure and parameters adaptive laws. The RCMNN and online training again can improve the performance of RUL prediction, because recurrent units can capture the dynamic features. Experimental data are performed by using NASA Prognostics Center of Excellence battery datasets to verify the effectiveness of the proposed method. The results show that the root mean square error of SoH estimation is smaller by the FBELNN and the prediction accuracy of RUL is higher by RCMNN under the different starting points.

Keywords

Fuzzy brain emotional learning neural network recurrent cerebellar model neural network lithium-ion battery remaining useful life state of health

1 Introduction

Hybrid electric vehicles (HEV) and microgrids were developed rapidly and they can promote the adoption of renewable energy sources with global energy crisis and global climate change [1]. Lithium-ion batteries have been played an important role in many electrification fields such as HEV, microgrids, and other consumer electronics, due to their high energy density, long cycle life and low self-discharge rate [2]. In actual work, the lithium-ion batteries are regarded as complex chemical and physical structure. The electrode material will be changed with accumulated operation, resulting in the capacity decrease and internal resistance increase [3]. On the one hand, state of health (SoH) is general defined as a percentage that reflects capacity decrease. When the capacity of the lithium-ion battery has decreased to 70% or 80% of the rated capacity, the end of life (EoL) of the battery also reaches in different applied scenario [4]. The remaining useful life (RUL) is general defined as the number of cycle from present time to the EoL, it is the measure of reflecting the aging conditions of the battery [5]. Hence, SoH estimation and RUL prediction for the lithium-ion battery play the important roles in the safety and economic analysis of the energy storage system (ESS) [6].

In recent years, extensive researches on SoH estimation and RUL prediction have been conducted. Many methods have been studied to SoH estimation that can be generally divided into three categories: electrochemical/physical measurement, model-based method and data-driven method [7]. There are two main methods of the approaches of RUL prediction: data-driven method and model-based method [8]. The methods of SoH estimation and RUL prediction are together introduced in this section. The coulomb counting method is easily implemented by calculating discharged battery capacity. This method is one of the most accurate measurement, but it is time-consuming and has only limited application [9]. Electrochemical impedance spectroscopy (EIS) is adopted for extracting the internal parameters that show varying aging processes of the battery [10]. But the applications of EIS are restricted for size limitations, complex experimental procedures and high cost. The model-based methods generally combine mathematical models with filtering techniques to establish the model of battery degradation by specifically designed experiments and cycling data [11]. Saxena et al. found the empirical model to establish the relationship between SoH with the number of cycle under different state of charge (SoC) [12]. Singh et al. presented the electrochemistry model to determine the initial parameters of the semi-empirical model that is used to estimate the SoH [13]. Zhang et al. used an improved particle filter to establish an empirical model for RUL prediction, and the Markov Chain Monte Carlo technique was used to improve the problem of inaccurate RUL prediction that caused by the issue of particle impoverishment [14]. Qiu et al. proposed the multiscale hybrid Kalman filter to estimate SoC and SoH, and RUL prediction is achieved by cuckoo search particle filter. The effect of cuckoo search algorithm is that transfer the particles to the maximum likelihood region [15]. But the application of model-based methods is affected by the high quality of experiment data, reliable equivalent model and high computational complexity [16]. Data-driven methods are a promising route of SoH estimation and RUL prediction that regard the lithium-ion battery as a black box. It is less complex than high-level empirical model and only depends on enough degradation data [17]. However, the main weakness of these methods is that the experiments usually take several months to obtain complete data. Li et al. presented a novel hybrid Elman and long short-term memory method for RUL prediction by combining the empirical model decomposition [18]. Dong et al. utilized the adaptive extended Kalman filter to predict RUL that combines with Brownian motion model to adjust the parameters under different aging speeds and this proposed method can ensure the flexibility, reliability of RUL prediction results [19]. But these methods used the known capacities at the past cycles and have not discussed how to get SoH information. Guo et al. presented SoH estimation of relevance vector machine, in which the input features are extracted by the rational analysis and principal component analysis (PCA) under charging conditions [20]. Wei et al. proposed a novel method for SoH estimation and RUL prediction, that is established by SVR to simulate the lithium-ion battery aging; and they introduced an impedance aging parameter model to overcome estimation errors [21].

The cerebellar model neural network (CMNN) is an imitation of the cerebellum learning structure and is regarded as a non-fully connected perceptron-like associative memory network with overlapping receptive-fields [22]. The CMNN has the structure of multiple layers and local learning characteristics, and is widely used in the control of complex dynamic systems with its advantages of simple computation, good generalization capability and fast learning speed [23]. One disadvantage is that the output of the CMNN is constant within each quantized state and the derivative information is not preserved [24]. Hence, the delayed recurrent units are introduced into the recurrent CMNN (RCMNN) of the association memory space and the weight memory space to capture the dynamic features in this study. Lin et al. proposed a novel adaptive filter using the RCMNN to a nonlinear channel equalization system. The results demonstrate that the performance of the proposed filter is better than the adaptive fuzzy filter or the neural fuzzy filter [25]. Guan et al. used genetic particle swarm optimization to seek for the optimal learning rates of the RCMNN so that it has well performance in trajectory tracking due to the dynamic characteristics of the recurrent units. The proposed control method is reliable that tested by comparing with other control schemes [26]. Huynh et al. designed a new intelligent RCMNN that have a dynamic deletion threshold for retaining or deleting the hypercubes automatically to improve the structure of the RCMNN [27]. This method can get well tracking performance for uncertain nonlinear systems under external disturbances. Since, the RCMNN has recurrent units in association memory space and weight memory space, it can improve the performance of RUL prediction by learning the nonlinear aging trend of lithium-ion battery in this study.

On the basis of emotional stimuli theory, the brain emotional learning neural network (BELNN) was created by Moren and Balkenius in 2001 [28]. The BELNN has two parts: the amygdala, which is involved in emotional judgment, and the orbitofrontal cortex corresponds to emotional control; these two parts affect each other, and they determine the output of the BELNN [29]. The BELNN have features of fast self-learning, simple implementation and good robustness so it is applied in power systems, motor drives, nonlinear systems and image analysis [29 –32]. Lin et al. proposed a novel self-organizing fuzzy BELNN (FBELNN) to apply in a biped robot, and the self-organization algorithm changes the sensory cortex space to give the most efficient structure [30]. Zhao et al. proposed wavelet FBELNN to apply in chaotic system and medical image analyzing; the proposed method can get better performance in approximation ability or accuracy than other methods [31, 32]. The FBELNN can fast and accurately get satisfactory results due to the special structure and parameters adaptive laws.

In this study, a data-driven method is proposed using the FBELNN and RCMNN for SoH estimation and RUL prediction. The proposed method uses discrete wavelet transform (DWT) and PCA to extract features from the part curve of charging voltage and current. These features are the input of FBELNN for SoH estimation, and then the results of SoH estimation are the input of RCMNN for RUL prediction. The public datasets of NASA Prognostics Center of Excellence (PCoE) are verified to illustrate the effectiveness of the proposed method [33].

The remainder of this paper is organized as follows. The datasets analysis and the method of feature extraction are presented in Section 2. In Section 3, the structure of FBELNN and RCMNN are introduced. The processes and results of SoH estimation and RUL prediction are detailed in Section 4. Conclusions are drawn in the final section.

2 Public datasets analysis and features extraction

In this section, the datasets of PCoE are introduced for showing the curve of voltage and current and the phenomenon of capacity degradation. The processes of feature extraction and selection are illustrated in section 2.2.

2.1 Battery aging datasets of PCoE

A set of three lithium-ion batteries labeled #5, #6 and #7 are collected from PCoE. During the processes of cycling test, the parameters and conditions of the lithium-ion batteries are presented in Table 1.

Table 1
The parameters and conditions of the lithium-ion batteries

Parameter Value Parameter Value

Rated capacity 2 Ah Charging current (CC) 1.5 A

Charging voltage (CV) 4.2 V Cut-off current (CV) 20 mA

Discharging current 2 A Temperature 24 °C

Parameter	Value	Parameter	Value
Rated capacity	2 Ah	Charging current (CC)	1.5 A
Charging voltage (CV)	4.2 V	Cut-off current (CV)	20 mA
Discharging current	2 A	Temperature	24 °C

As shown in Fig. 1, the lithium-ion batteries undergo three different operational phases including constant current (CC) charge, constant voltage (CV) charge, CC discharge and impedance measurement. At the phase of CC charge, the charging current was 1.5 A until the voltage reached to 4.2 V, and then the charging current will drop to 20 mA at the phase of CV charge phase. The discharging current was 2A until the lithium-ion batteries voltage reached to 2.7 V, 2.5 V and 2.2 V for #5, #6 and #7, respectively.

Fig. 1

The curve of voltage and current at the 2nd cycle (#5).

Figure 2 shows the relation between the capacity degradation and the cycle number with #5, #6 and #7. The value of failure threshold is set as 70% of the rated capacity (1.4 Ah) in this study. The curves of capacity degradation are not monotone decreasing due to some experiment intervals. During the aging process, the phenomenon of capacity regeneration has an important influence on RUL prediction due to highly dynamic and nonlinear of degradation curves. So it is necessary that not only need to accurately estimate SoH but also need improve the problem of capacity regeneration.

Fig. 2

Capacity degradation curves.

In Fig. 3, the lithium-ion battery charging curves are provided at the different cycles containing the 40th, 80th, 120th and 160th cycles. With the increase of cycles, the shape of the charging curves are influenced by the lithium-ion battery aging. For example, the CC charging time of 40th cycle is shortest in Fig. 3(a). So the changing shape of voltage and current curves is the basis of feature extraction in next part.

Fig. 3

Charging curve evolution at different cycles (#5).

2.2 Feature extraction based on DWT-PCA

Because of the output of the ESS is affected by the various conditions, the discharging process is unpredictable and incomplete in practice. The processes of charging controlled by ESS, and the aging features are extracted from the curves that the voltage of the last 500 s under CC charge and the current of the first 500 s under CV charge. The energy from discrete wavelet transform (DWT) is applied to extract features in this study. It can characterizes the energy distribution of the signal in the domain of time and frequency [34]. The energy will change with the shape of the charging curves. The finite energy signal f(t) in terms of both the scaling and the wavelet functions is defined by the formula: $f (t) = \sum_{k = 0}^{2^{N - m} - 1} h_{M, k} Ψ_{M, k} (t) + \sum_{m = 1}^{M} \sum_{k = 0}^{2^{N - m} - 1} g_{m, k} ψ_{m, k} (t)$ (1) where M is the decomposition level, 2^N is the length of the sampling data, where N is a positive integer. And h_{M, k} is the scaling coefficient, which is also known as the approximation coefficients, g_{m, k} is the wavelet coefficient, which is also known as the detail coefficients, Ψ (t) and ψ (t) are the scaling function and wavelet function, respectively. Wavelet energy is the index to reflect the energy concentration of wavelet coefficients on certain scales [35]. The energy can be calculated by: $E_{h_{M}} = \sum_{l = 1}^{q_{M}} {| h_{Ml} |}^{2}, E_{g_{m}} = \sum_{l = 1}^{q_{m}} {| g_{ml} |}^{2}$ (2) where l stands for the point in wavelet coefficients, q_M is the maximum length of scaling coefficient at scale is M and q_m is the length of wavelet coefficients at scale m. It is well known that PCA is a statistical method that can help to decrease the number of redundant features [37]. The process of DWT-PCA is shown in Fig. 4 when the decomposition level is 9. The scaling energy and the wavelet energy optimized by PCA are the input of SoH estimator in the end. The principal components are the new features when P_l≥95% and F = [F₁, F₂, …, F_q] conclude the q-th principal component, where P_l is the contribution rate based on variance accumulation.

Fig. 4

The feature extraction process of DWT-PCA.

3 The structure of FBELNN And RCMNN

The structure and learning algorithm for the FBELNN and the RCMNN are detailed in this section, FBELNN is used as SoH estimator and RCMNN is used as RUL predictor.

3.1 The structure and learning algorithm of FBELNN

As mentioned before, the FBELNN contains two parts so two fuzzy inference rules are defined as: $If I_{1} is S_{1 j} and I_{2} is S_{2 j}, \dots, I_{n_{i}} is S_{nij}, then A = V_{ij}$ (3) $If I_{1} is S_{1 j} and I_{2} is S_{2 j}, \dots, I_{n_{i}} is S_{nij}, then O = W_{ij}$ (4) $for j = 1, 2, \dots, n_{j}$ where n_i is the input dimension, S_ij is the fuzzy set for the i-th input and j-th layer, V_ij is the output weight of the amygdala, W_ij is the output weight of the orbitofrontal cortex, A is the output of the amygdala and O is the output of the orbitofrontal cortex. The structure of FBELNN is shown in Fig. 5.

Fig. 5

The structure of FBELNN.

3.1.1 Forward computation

Layer 1. Input space: According to a given problem, I_i ∈ Rⁱ, i = 1, 2, … , n_i, where n_i is considered to be the feature dimension. I_i is the feature extracted by DWT-PCA.

Layer 2. Sensory cortex space: This space performs the fuzzification operation and transmit the input to the amygdala and orbitofrontal cortex space. It is different from the sensory input of a normal BELNN. The Gaussian function is adopted as the basis function, which is represented as:

$S_{ij} = \exp (\frac{- (I_{i} - m_{ij})^{2}}{σ_{ij}^{2}}), \begin{matrix} i = 1, 2, \dots, n_{i} \\ j = 1, 2, \dots, n_{j} \end{matrix}$ (5) where S_ij represents a Gaussian function with the mean m_ij and variance σ_ij. Meanwhile, the thalamus affects the output of amygdala, and the output of thalamus S_th is determined by the maximum value of S_ij: $S_{th} = \max (S_{ij})$ (6)

Layer 3. Amygdala and orbitofrontal cortex space: This layer is the algebraic sum of the amygdala and orbitofrontal cortex weights with the sensory cortex value S_ij. The amygdala space output is:

$A = \frac{\sum_{i = 1}^{n_{i}} \sum_{j = 1}^{n_{j}} S_{ij} * V_{ij}}{\sum_{i = 1}^{n_{i}} \sum_{j = 1}^{n_{j}} S_{ij}} + S_{th} V_{th}$ (7) where V_th is the weight of thalamus. The output of the thalamus S_th is multiplied by the weight V_th and submitted to the amygdala [38]. Similarly, the orbitofrontal cortex space output is: $O = \frac{\sum_{i = 1}^{n_{i}} \sum_{j = 1}^{n_{j}} S_{ij} * W_{ij}}{\sum_{i = 1}^{n_{i}} \sum_{j = 1}^{n_{j}} S_{ij}}$ (8)

The weights of V_ij and W_ij are adjusted by specific learning rules.

Layer 4. Output space: The output of the FBELNN is the result of interaction between the amygdala space and the orbitofrontal cortex space. The output of the FBELNN is represented as:

$U = A - O$ (9)

3.1.2 Learning algorithm of FBELNN

The learning rules for the amygdala and the orbitofrontal cortex involve updating the weights of V_ij and W_ij. The updating laws for the weights space are represented as: $V_{ij} (k + 1) = V_{ij} (k) + Δ V_{ij} (k)$ (10) $W_{ij} (k + 1) = W_{ij} (k) + Δ W_{ij} (k)$ (11) where the parameters adjustment laws for the weight for the amygdala and the orbitofrontal cortex are respectively defined as [33]: $Δ V_{ij} (k) = λ_{V} (S_{ij} \max (0, Rw (k) - A (k)))$ (12) $Δ W_{ij} (k) = λ_{W} (S_{ij} (U (k) - S_{th} V_{th} - Rw (k)))$ (13) where λ_V and λ_W are the learning rate for the amygdala and the orbitofrontal cortex, respectively. The parameter Rw is the reward signal. According to the nerve physiology, the output of amygdala is correspond to the specific emotional hints and the output of the orbitofrontal cortex is adjusted to minimize the difference in the amygdala output [27]. For this study, Rw is: $Rw = \sum_{i = 1}^{n_{i}} α_{i} I_{i} + β U$ (14) where α and β are both the signal constant gains. From (12), ΔV_ij cannot be decreased because the function of max adjusts the weight monotonically, it represents the ability of forgetting the previous emotion signals [28]. And ΔW_ij can increase or decrease in order to speed up the learning process and achieve the desired output. The sensory cortex space has the learning rule because the mean m_ij and variance σ _ij need to be updated, which the gradient descent algorithm is used to adjust the parameters of Gaussian function. A cost function is defined as: $E (k) = \frac{1}{2} {(Ut (k) - U (k))}^{2}$ (15) where Ut is the desired output. Then the chain rule can be used to express the adjustment law of mean and variance, that are represented as: $\begin{matrix} Δ m_{ij} = - λ_{m} \frac{\partial E}{\partial m_{ij}} = - λ_{m} \frac{\partial E}{\partial U} \frac{\partial U}{\partial S_{ij}} \frac{\partial S_{ij}}{\partial m_{ij}} \\ = λ_{m} (Ut - U) (V_{ij} - W_{ij}) S_{ij} \frac{2 (I_{i} - m_{ij})}{σ_{ij}^{2}} \end{matrix}$ (16) $\begin{matrix} Δ σ_{ij} = - λ_{σ} \frac{\partial E}{\partial σ_{ij}} = - λ_{σ} \frac{\partial E}{\partial U} \frac{\partial U}{\partial S_{ij}} \frac{\partial S_{ij}}{\partial σ_{ij}} \\ = λ_{σ} (Ut - U) (V_{ij} - W_{ij}) S_{ij} \frac{2 {(I_{i} - m_{ij})}^{2}}{σ_{ij}^{3}} \end{matrix}$ (17) where λ_m and λ_σ are the learning rates of mean and variance, respectively. So Equations (10)-(11) and Equations (16)-(17) is the learning rules of the FBELNN.

3.2 The structure and learning algorithm of RCMNN

In this part, the structure of the RCMNN is shown in Fig. 6, which recurrent units insert in association memory space and weight memory space. And in the association memory space and receptive-field space, the fuzzy inference rules follow as: $\begin{matrix} If p_{1} is φ_{1 bc} and p_{2} is φ_{2 bc}, \dots, p_{n_{a}} is φ_{n_{a} bc}, \\ then y = w_{b} \\ for j = 1, 2, \dots, n_{b}, k = 1, 2, \dots, n_{c} \end{matrix}$ (18)

where n_a is the input dimension, φ_abc is the fuzzy set for the a-th input b-th layer and c-th block, and w_b is a output weight for the b-th layer.

Fig. 6

The structure of RCMNN.

3.2.1 Forward computation

Layer 1. Input space: p_a ∈ R^a, a = 1, 2, … , n_a, p_a is the lithium-ion battery of SoH in RUL prediction, all variables of the input p_a can be quantized into discrete states.

Layer 2. Association memory space: In this space, one input simultaneously activate different fuzzy sets for different layers. The mechanism for this mapping is shown in Fig. 7 that several states are accumulated into one block that adopted Gaussian function as a receptive-field basis function. It can be represented as:

$φ_{abc} = \exp (\frac{- ({pr}_{a} - χ_{abc})^{2}}{v_{abc}^{2}}), \begin{matrix} b = 1, 2, \dots, n_{b} \\ c = 1, 2, \dots, n_{c} \end{matrix}$ (19)

Fig. 7

Two-dimensional RCMNN with n_b = 4 and n_c = 3.

where φ_abc denotes the output of the b-th layer and c-th block for the a-th input of pr_a with the mean χ_abc and variance v_abc. In the RCMNN, the input of this block can be described as: ${pr}_{a} (t) = p (t) + r_{abc} * φ_{abc} (t - T)$ (20) where r_abc is the recurrent weight in this space, and φ_abc (t-T) indicates the value of φ_abc through time delay T. The input of this block contains the memory term φ_abc (t-T), that saves all previous information and captures dynamic characteristics. When an input state is (p₁, p₂) = (6, 6), according Fig. 7 the combined cubes (φ ₁₁₃ φ ₂₁₃ , φ ₁₂₂ φ ₂₂₂ , φ ₁₃₂ φ ₂₃₂ , φ ₁₄₂ φ ₂₄₂ ) are simultaneously activated for different layers. And each block has three adjustable parameters, named χ _abc , v_abc and r_abc.

Layer 3. Receptive field space: Areas formed by blocks are called receptive field space. The b-th receptive-field function is defined as:

$R_{b} = \prod_{a = 1}^{n_{a}} φ_{abc}$ (21)

In the fuzzy rules in (18), the product is used as the “and” computation in the antecedent part of the fuzzy rules. The multidimensional receptive field functions can be expressed in a vector form: $R = {[R_{1}, \dots, R_{b}, \dots, R_{n_{b}}]}^{T}$ (22)

Layer 4. Weight memory space: In this space, each location of R_b to a particular adjustable value can be expressed as:

$w = {[w_{1}, \dots, w_{b}, . ., w_{n_{b}}]}^{T}$ (23) where w_b denotes the connecting weight value of the b-th receptive-field.

Layer 5. Output space: In this paper, the output of RCMNN is expressed as:

$y = \sum_{b = 1}^{n_{b}} w_{b} * {Rr}_{b}$ (24)

Due to the effect of the recurrent units in the weight memory space, Rr_b is represent as: ${Rr}_{b} (t) = R_{b} (t) + {wr}_{b} * w_{b} * {Rr}_{b} (t - T)$ (25) where wr_b is the recurrent weight of weight memory space. The role of the recurrent units is to consider the past value in the association memory space and weight memory space, so this RCMNN have the advantage of simple structure with dynamic characteristics.

3.2.2 Learning algorithm of RCMNN

The parameters of the RCMNN are adjusted by gradient descent algorithm. A cost function is defined as: $L (k) = \frac{1}{2} {(yt (k) - y (k))}^{2}$ (26) where yt is the desired output. To minimize the error of cost function, the chain rule is used to calculate each parameters as below: $\begin{matrix} Δ {wr}_{b} = - η_{wr} \frac{\partial E}{\partial {wr}_{b}} = - η_{wr} \frac{\partial E}{\partial y} \frac{\partial y}{\partial {Rr}_{b}} \frac{\partial {Rr}_{b}}{\partial {wr}_{b}} \\ = η_{wr} (yt - y) w_{b} * {Rr}_{b} (t - T) \end{matrix}$ (27) $Δ w_{b} = - η_{w} \frac{\partial E}{\partial w_{b}} = - η_{w} \frac{\partial E}{\partial y} \frac{\partial y}{\partial w_{b}} = η_{w} (yt - y) {Rr}_{b}$ (28) $\begin{matrix} Δ r_{abc} = - η_{r} \frac{\partial E}{\partial r_{abc}} = - η_{r} \frac{\partial E}{\partial y} \frac{\partial y}{\partial R_{b}} \frac{\partial R_{b}}{\partial φ_{abc}} \frac{\partial φ_{abc}}{\partial {pr}_{a}} \frac{\partial {pr}_{a}}{\partial r_{abc}} \\ = η_{r} (yt - y) w_{b} * R_{b} * \frac{2 (χ_{abc} - {pr}_{a})}{v_{abc}^{2}} * φ_{abc} (t - T) \end{matrix}$ (29) $\begin{matrix} Δ χ_{abc} = - η_{m} \frac{\partial E}{\partial χ_{abc}} = - η_{χ} \frac{\partial E}{\partial y} \frac{\partial y}{\partial R_{b}} \frac{\partial R_{b}}{\partial φ_{abc}} \frac{\partial φ_{abc}}{\partial χ_{abc}} \\ = η_{χ} (yt - y) w_{b} * R_{b} * \frac{2 ({pr}_{a} - χ_{abc})}{v_{abc}^{2}} \end{matrix}$ (30) $\begin{matrix} Δ v_{abc} = - η_{v} \frac{\partial E}{\partial v_{abc}} = - η_{v} \frac{\partial E}{\partial y} \frac{\partial y}{\partial R_{b}} \frac{\partial R_{b}}{\partial φ_{abc}} \frac{\partial φ_{abc}}{\partial v_{abc}} \\ = η_{v} (yt - y) w_{b} * R_{b} * \frac{2 {({pr}_{a} - χ_{abc})}^{2}}{v_{abc}^{3}} \end{matrix}$ (31) }}where η_wr, η_w, η_r, η_χ and η_ν are the learning rates of wr_b, w_b, r_abc, χ _abc and v_abc, respectively.

4 Processes and results analysis

According to the above illustration, the processes and results of SoH estimation and RUL prediction are further analyzed and compared in this section.

4.1 The processes of SoH estimation and RUL prediction

The flow chart of SoH estimation and RUL prediction is shown in Fig. 8. This flow chart not only includes the offline training and the online monitoring, but also illustrates the processes of SoH estimation and RUL prediction.

Fig. 8

The flow chart of SoH estimation and RUL prediction.

In the part of offline training, the method of feature extraction is showed in the section of 2.2, and the results of feature extraction will be analyzed in next section. The parameters of FBELNN are updated based on the features and the residual capacity from other lithium-ion battery for building SoH estimator. The RCMNN learns the curve of capacity degradation of other lithium-ion battery for building RUL predictor, and the role of the recurrent units is better catch the fluctuation of capacity degradation. In the part of online monitoring, the charging curve of the lithium-ion battery in use is collected, and then feature extraction and SoH estimation are carried out by trained FBELNN. The results of SoH estimation are shown to the operator to ensure the safety of the ESS, and are used as the basis of RUL prediction of the lithium-ion battery in use. And the SoH results estimated under online monitoring can be used to train RCMNN again to improve the accuracy of RUL prediction, that can help the operator to prepare for lithium-ion battery replacements. But limited by experimental conditions in the lab, this study uses the datasets of PCoE as the basis of offline training and online monitoring. And the analysis and results will be detailed in the following two sections.

4.2 Feature extraction and results of SoH estimation

The first step of DWT is the selection of decomposition level and wavelet function. In this study, the Daubechies wavelet family (DbN) be chosen because they have shown excellent performance in analyzing the change of the signals, where N is the order of the wavelet function [34]. It is important to select a suitable decomposition level and the order of the wavelet function that can ensure to get enough detail of the signals. To explain that the reason of selection, the wavelet energy of the 40th cycle curve (#5 lithium-ion battery) extracted by Db2, Db3 and Db4 wavelet function are shown in Fig. 9. When the decomposition level is increased to 9, the value of normalized Db3 wavelet energy is larger than other. Therefore, Db3 and decomposition level of 9 can make the effect of feature extraction better during the process of DWT.

Fig. 9

Wavelet energy from 40th cycle with the different decomposition level.

According to section 2.2, the number of feature from DWT is 10 including the wavelet energy and the scaling energy, when the decomposition level is 9. The PCA can uses orthogonal transformation to convert the possibly correlated variables into a set of uncorrelated linearly variables [37], so wavelet energy be optimized by the PCA to improve the performance of SoH estimation. The first principal component of wavelet energy explains 99.68% and 99.85% for the current curve and voltage curve, respectively. It can be seen that the contribution rate of first principal component already reach above 99%, so these results illustrate the performance of features optimization based on the PCA. Thus, the scaling energy and the first principal component from the wavelet energy are used as the input of FBELNN for SoH estimation in this study.

Furthermore, the correlation between the capacity and features is analyzed in order to show the effectiveness of the feature extraction. As shown in Fig. 10, this scatter plot presents the correlation between the capacity and features that are the first principal component from the voltage and current wavelet energy. Spearman rank correlation analysis is a measure of whether the two variables are strictly monotonic. The range of Spearman rank correlation coefficient is r_S ∈ [-1, + 1]. The two variables are strictly monotone when r_S = ±1 [39]. The calculation formula is shown as in the following function:

Fig. 10

Linear correlation between capacity and features.

$r_{s} = \frac{\sum_{i = 1}^{N} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sqrt{\sum_{i = 1}^{N} {(x_{i} - \bar{x})}^{2} \sum_{i = 1}^{N} {(y_{i} - \bar{y})}^{2}}}$ (32) where x_i is the rank of features of voltage or current, y_i is the rank of capacity, and $\bar{x}$ and $\bar{y}$ are the means of variables features and capacity, respectively. From this function, the Spearman rank correlation coefficient of current features is 0.9793 and that of voltage features is -0.9709. These results illustrate the effectiveness of feature extraction, but it must be noted that features are not completely linear related to capacity due to the characteristic of the battery, the capacity regeneration and sampling frequency. So it is necessary to further establish the model of the relationship between input features and capacity.

According to Fig. 8, the features extracted from #7 lithium-ion battery are used as the input of the FBELNN in the offline training. The number of layer of sensory cortex space is 30. The features extracted from #5 and #6 lithium-ion batteries are tested in online monitoring. The results of SoH estimation for #5 and #6 lithium-ion batteries are respectively shown in Fig. 11 (a) and (b). According to these datasets, #5 and #6 lithium-ion batteries reach the EoL at the 125th and 109th cycle, respectively.

Fig. 11

The results of SoH estimation used by the FBELNN.

These results of the FBELNN are further compared with the RCMNN, the CMNN and the fuzzy neural network (FNN). The number of layer and block of association memory space are respectively set as 8 and 3 for the RCMNN and the CMNN. The number layer of membership function layer is 90 for the FNN. The absolute output errors of different methods are shown in Fig. 12 (a) and (b). In Fig. 12, the absolute output errors sometimes abruptly increase which may be caused by the phenomenon of capacity regeneration. The advantages of different methods have not been directly compared in Fig. 12, so the root mean square error (RMSE) and the coefficient of determination (R²) are used in evaluating the performance of different methods. according to function (33) and (34), respectively.

Fig. 12

Absolute output errors of different methods.

$RMSE = \sqrt{\frac{\sum_{i = 1}^{n} {(C_{i} - {\hat{C}}_{i})}^{2}}{n}}$ (33) $R^{2} = 1 - \frac{\sum_{i = 1}^{n} {(C_{i} - {\hat{C}}_{i})}^{2}}{\sum_{i = 1}^{n} {(C_{i} - {\bar{C}}_{i})}^{2}}$ (34) where C is the actual capacity, $\hat{C}$ is the estimated capacity, $\bar{C}$ is the average of the actual capacity. It can be seen from the Table 2 that the FBELNN is superior to other methods.

Table 2

The performance analysis results of SoH estimation

Methods	RMSE(#5)	R²(#5)	RMSE(#6)	R²(#6)
FBELNN	0.0138	0.9915	0.0283	0.9796
RCMNN	0.0178	0.9857	0.0355	0.9679
CMNN	0.0193	0.9832	0.0364	0.9664
FNN	0.0195	0.983	0.0367	0.9658

4.3 The comparison and results of RUL prediction

The lithium-ion battery of RUL also need to be predicted for making the ESS more intelligent. According to Fig. 8, the capacity of #7 lithium-ion battery is used as the training set of the RCMNN in the offline training. Then the results of SoH estimation by the FBELNN are used to train again in online monitoring. These processes can not only reduce the calculation cost by offline training in online monitoring, but also make the RCMNN learn the capacity degradation of the lithium-ion battery better for RUL prediction. The number of iterations for offline training is 5000 and the number of iterations for online training is 100. In the processes of RUL prediction, the predicted output is used as the one of input of the RCMNN in the next moment because the SoH of the lithium-ion battery can’t be known in the future. The RCMNN is compared with the CMNN and the recurrent neural network (RNN) under the same processes. Multiple step prediction is started at the different starting points, and the results of RUL prediction for the different lithium-ion batteries are respectively shown in Figs. 13 and 14. Figure 13 (a) and (b) respectively show the results when the starting points of #5 lithium-ion battery are the 63rd and 82nd cycle. Similarly, Fig. 14 (a) and (b) respectively show the results when the starting points of #6 lithium-ion battery are the 54th and 69th cycle.

Fig. 13

The results of RUL prediction for #5 battery.

Fig. 14

The results of RUL prediction for #6 battery.

The number of inputs to these neural network is 3. The number of layer and block of association memory space are respectively set as 5 and 6, the initial means are the median of each block, the initial variances are 1, and the initial weights are 0.1 for the RCMNN and the CMNN. The number of hidden layer neurons of the RNN is 30. The number of layer of sensory cortex space is 10 for the FBELNN. Function (35) is the prediction error (PE_r) between the real value and the predicted value. The results of RUL prediction errors are shown in Table 3.

${PE}_{r} = {RUL}_{R} - {RUL}_{P}$ (35) where RUL_R is the real RUL value and RUL_P is the predicted RUL value. It can be seen from the Table 3 that the prediction error decreases as the starting points close to EoL. The special structure of association memory space can better learn the capacity degradation with not monotone decreasing, but the recurrent units have more advantages in the prediction problem. So the RCMNN that have the recurrent units in association memory space and weight memory space can get the better results for RUL prediction.

Table 3

The performance analysis results of SoH estimation

Methods	Starting Points at 1/2		Starting Points at 2/3
	PE_r(#5)	PE_r(#6)	PE_r(#5)	PE_r(#6)
RCMNN	8	5	2	3
RNN	21	8	7	–5
CMNN	22	12	12	4
FBELNN	22	20	13	7

5 Conclusion

This study proposes a method to ensure the safety and stability of the ESS by online monitoring the charging curve for estimating SoH and predicting RUL. The features extracted from the charging curve in different charging phases can guarantee this method to be applied to different discharging conditions. The correlation analysis demonstrates a strong linear relationship between the extracted features and the lithium-ion battery capacity, which is important for the lithium-ion battery. The proposed method of SoH estimation can provide accurate result using DWT-PCA and the FBELNN. The special structure and parameters adaptive laws from the FBELNN can get more accurate results of SoH estimation than other neural networks. The structure of the RCMNN and the processes of again training can better predict the ageing trend of the lithium-ion battery, and the result of RUL prediction using the RCMNN is more accurate than other methods. Finally, these results show that the proposed method have good accuracy to help operators to analyze the working state of the lithium-ion battery.

In the next step, the Fujian Special Equipment Inspection and Research Institute will work with us for creating the new aging datasets of the lithium-ion battery. The discharging voltage and discharging current need to be further considered at the different temperatures and running conditions. And the electrochemical characteristics of the lithium-ion battery need to be also analyzed for improving the method of feature extraction. The proposed method can be used to provide important references for the workers to analyze and predict the SoH of a lithium-ion battery. The results of this study also allow more intelligent microgrids for energy management system. Moreover, the structure or parameter adaptive laws of the FBELNN and the RCMNN need to be amended for engineering application.

Footnotes

Acknowledgments

The authors appreciate the financial support in part from the Ministry of Science and Technology of Republic of China under grant MOST 109-2811-E-155-504-MY3, and from Technology Innovation Fund Support Project by the company of Powerchina Fujian Electric Power Engineering Co., LTD. under Grant DJ-ZDXM-2020-15.

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State of health estimation and remaining useful life prediction for lithium-ion batteries using FBELNN and RCMNN

Abstract

Keywords

1 Introduction

2 Public datasets analysis and features extraction

2.1 Battery aging datasets of PCoE

Table 1 The parameters and conditions of the lithium-ion batteries Parameter Value Parameter Value Rated capacity 2 Ah Charging current (CC) 1.5 A Charging voltage (CV) 4.2 V Cut-off current (CV) 20 mA Discharging current 2 A Temperature 24 °C

3.1 The structure and learning algorithm of FBELNN

4.1 The processes of SoH estimation and RUL prediction

Footnotes

Acknowledgments

References

Table 1
The parameters and conditions of the lithium-ion batteries

Parameter Value Parameter Value

Rated capacity 2 Ah Charging current (CC) 1.5 A

Charging voltage (CV) 4.2 V Cut-off current (CV) 20 mA

Discharging current 2 A Temperature 24 °C