Time-Frequency Image Analysis and Transfer Learning for Capacity Prediction of Lithium-Ion Batteries

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Motivation
Lithium-ion batteries (LIBs) are recognised as a key future form of technology for renewable energy and electric vehicles (EVs) due to their high power and energy densities, low maintenance cost, long lifetime, and low self-discharge rate [1]. However, to optimise the energy performance of the LIBs, prolong their life cycle, and reduce their cost, it is thus critical to monitor the internal state of the battery, such as state-of-charge (SoC), state-of-health (SoH), and remaining useful life (RUL) [2,3]. During operation, the continuous determination of the LIB's internal state is achieved through the battery management system (BMS), which guarantees the reliability and efficiency of LIBs. LIB. The above study investigated the diagnostic indicators under predetermined operation conditions, such as certain current or voltage ranges, which involve CC charge and cycling discharge. Nonetheless, in real-world applications like EV applications, the LIB never operates in a static scenario.
Few studies have evaluated a battery's capacity estimation under a dynamic current profile. Venugopal et al. [29] extracted 18 time-domain diagnostic features which affect the battery capacity from the measured voltage, current, and temperature of an LIB operated under a dynamic charging/discharging current profile. The selected features were fed into an independently recurrent NN (IndRNN) for capacity evaluation. The authors in [30] developed a data-driven LIB health diagnosis method based on time and frequency domain features such as mean, covariance, and kurtosis from the time domain features; from the frequency domain features, the authors selected median frequency, band-power, signal-to-noise ratio (SNR), and total harmonic distortion (THD) as the features to determine the battery's capacity. The extracted time and frequency features are then evaluated and ordered based on trendability and monotonicity metrics to select the most features that provide high accuracy and lower computational burden to estimate the battery's capacity. A Gaussian process regression (GPR) model was then designed to capture the correlation between the selected features and the capacity of the LIB.
To enhance the quality of the extracted diagnostic characteristics under a variable current profile, Jonghoon Kim [31] proposed an advanced signal processing method known as discrete wavelet transform (DWT) with multiresolution analysis (MRA) to analyse the non-linearity and non-stationarity behaviour of the battery terminal voltage under dynamic load profile. More useful diagnostic features were extracted for capacity evaluation in the time and frequency domains, respectively. Similarly, the authors in [32] proposed a fast wavelet transform to extract the dynamic features of voltage and current. Then, an xD-Markov machine learning model was established to obtain the battery's capacity from the extracted features. All of the above mentioned studies considered the time or frequency domain for the extraction of the most useful diagnostic features to describe the degradation process inside the LIB. Analyses are usually used to extract indicators from the measured signal and the time and frequency domains, known as conventional feature extraction techniques. However, conventional feature extraction and selection are time-consuming and labour-intensive processes requiring detailed knowledge of the relevant features of the system. This selection process can introduce uncertainty and biased results [33]. Moreover, the performance of machine learning techniques is based on the quality of the features extracted from the measured data. That is, if the extracted characteristic features are weak, this can negatively impact the performance of these approaches [34,35]. Another study [33] reported the difficulty of identifying suitable features with the relevant information required for capacity evaluation.

Contributions
To overcome the aforementioned issues, a time-frequency image (TFI) analysis-based approach has been proposed to analyse the non-stationary and non-linear behaviour of the LIB terminal voltage in both time and frequency domains and for online prediction of LIB capacity. Moreover, in this paper, we concentrate on the operation of the battery under the variable current load profiles of different battery cycles. The key benefit of the proposed solution is that the battery's terminal voltage is presented in a two-dimensional (2D) representation (time-frequency domain), which provides more helpful characteristics information related to the battery degradation than a one-dimensional (1D) representation (time or frequency domain). Wavelet transforms combined with a deep learning method have been developed for the analysis of the characteristics of LIBs under different operating conditions. The proposed method utilises a deep learning featureless methodology to learn the features of the data automatically, unlike traditional machine learning methods. This avoids manual feature extraction, which relies heavily on human knowledge and experience. Finally, the proposed scheme is a non-parametric estimation method, and, thus, offline testing and modelling are not required. To validate the proposed technique, experimental data on LIB cells from the experimental datasets published by the Prognostics Center of Excellence (PCoE) NASA were used. The results reveal that the proposed scheme improves over competing baseline schemes and achieves 95.60% prediction accuracy. This indicates good potential for the design of improved battery management systems based upon this method.

Structure
The remainder of the paper is organised as follows: Section 2 introduces the randomised battery dataset analysis. Section 3 presents the proposed capacity imaging analysis scheme. Section 4 presents the results and discussion of the obtained results. Conclusions and future work directions are presented in Section 5.

Randomised Battery Dataset
The practical operating environment of a real-world EV battery includes a dynamic and partial driving pattern. Most of the literature utilises a battery dataset with limited assumptions; for example, the battery is cycled using only a constant current profile and specific voltage limit. Nevertheless, these assumptions do not cover the real operating situation of EV batteries. In this study, a randomised battery usage dataset was adopted from the NASA Ames Prognostics Center of Excellence [36,37], to ascertain the impact of actual, dynamic EV driving cycles. This dataset contains the ageing results of four LIBs named RW9, RW10, RW11, and RW12, acquired at room temperature. The general properties of the battery are summarised in Table 1. These four LIBs were cycled using two cycling protocols known as random walk cycling mode and reference charge and discharge cycling mode. The cycle process is shown in Figure 1 [36]. prediction accuracy. This indicates good potential for the design of improved battery management systems based upon this method.

Structure
The remainder of the paper is organised as follows: Section 2 introduces the randomised battery dataset analysis. Section 3 presents the proposed capacity imaging analysis scheme. Section 4 presents the results and discussion of the obtained results. Conclusions and future work directions are presented in Section 5.

Randomised Battery Dataset
The practical operating environment of a real-world EV battery includes a dynamic and partial driving pattern. Most of the literature utilises a battery dataset with limited assumptions; for example, the battery is cycled using only a constant current profile and specific voltage limit. Nevertheless, these assumptions do not cover the real operating situation of EV batteries. In this study, a randomised battery usage dataset was adopted from the NASA Ames Prognostics Center of Excellence [36,37], to ascertain the impact of actual, dynamic EV driving cycles. This dataset contains the ageing results of four LIBs named RW9, RW10, RW11, and RW12, acquired at room temperature. The general properties of the battery are summarised in Table 1. These four LIBs were cycled using two cycling protocols known as random walk cycling mode and reference charge and discharge cycling mode. The cycle process is shown in Figure 1 [36].

Random Walk Cycling Mode
A sequence of currents set varying between (−4.5 A, −3.75 A, −3 A, −2.25 A, −1.5 A, −0.75 A, 0.75 A, 1.5 A, 2.25 A, 3 A, 3.75 A, 4.5 A) is used for charging and discharging; the four LIBs were continuously cycled. Hence, negative currents are related to the charging operation, while positive currents denote the discharge operation of the batteries. This mode of charging and discharging protocol is known as random walk (RW) operation mode. The aforementioned current sequence is randomly applied to an LIB for five minutes, which is identified as a step in the dataset. It is important to mention that a single random walk (RW) cycle consist of 1500 RW steps and 1500 rests; each RW profile consists of numerous RW cycles. After every RW profile, the battery undergoes several reference charging and discharging tests to measure its capacity and calculate its SoH value. Figures  2 and 3 show the measured voltage, current, and temperature of the first and last 100 RW charging and discharging steps of battery RW9, respectively [29].

Random Walk Cycling Mode
A sequence of currents set varying between (−4.5 A, −3.75 A, −3 A, −2.25 A, −1.5 A, −0.75 A, 0.75 A, 1.5 A, 2.25 A, 3 A, 3.75 A, 4.5 A) is used for charging and discharging; the four LIBs were continuously cycled. Hence, negative currents are related to the charging operation, while positive currents denote the discharge operation of the batteries. This mode of charging and discharging protocol is known as random walk (RW) operation mode. The aforementioned current sequence is randomly applied to an LIB for five minutes, which is identified as a step in the dataset. It is important to mention that a single random walk (RW) cycle consist of 1500 RW steps and 1500 rests; each RW profile consists of numerous RW cycles. After every RW profile, the battery undergoes several reference charging and discharging tests to measure its capacity and calculate its SoH value. Figures 2 and 3 show the measured voltage, current, and temperature of the first and last 100 RW charging and discharging steps of battery RW9, respectively [29].

Reference Charge and Discharge Cycle Mode
After the start of each RW cycle, a sequence of reference charging and discharge current profiles is implemented to set a standard benchmark for the battery's SoH. Initially, by applying a 2 A constant current profile, the LIB is charged to its maximum voltage and then a 4.2 V constant voltage is sustained until its current falls to 0.01 A, then it rests for 20 min. Afterwards, 2 A discharge constant current profile is applied to the LIB until its voltage reduces to its minimum voltage, 3.2 V. This procedure is known as reference charging and discharging cycle mode [38]. To measure the LIB's capacity and calculate its SoH value, after every RW cycle, the current cell capacity ( ) is calculated using the Coulomb counting method as follows: where is the total discharge current of the battery cell. Once the battery capacity is calculated, the SoH of the battery can be calculated, as shown in (2).

=
. 100% where is the measured capacity after every RW cycle and is the measured capacity of the battery at the beginning of its life. The capacities of the measured cells of the four batteries are shown in Figure 4.   Figure 2. Voltage, current, and temperature of the first 100 steps measured for battery RW9.

Reference Charge and Discharge Cycle Mode
After the start of each RW cycle, a sequence of reference charging and discharge current profiles is implemented to set a standard benchmark for the battery's SoH. Initially, by applying a 2 A constant current profile, the LIB is charged to its maximum voltage and then a 4.2 V constant voltage is sustained until its current falls to 0.01 A, then it rests for 20 min. Afterwards, 2 A discharge constant current profile is applied to the LIB until its voltage reduces to its minimum voltage, 3.2 V. This procedure is known as reference charging and discharging cycle mode [38]. To measure the LIB's capacity and calculate its SoH value, after every RW cycle, the current cell capacity ( ) is calculated using the Coulomb counting method as follows: where is the total discharge current of the battery cell. Once the battery capacity is calculated, the SoH of the battery can be calculated, as shown in (2).

=
. 100% where is the measured capacity after every RW cycle and is the measured capacity of the battery at the beginning of its life. The capacities of the measured cells of the four batteries are shown in Figure 4.   Figure 3. Voltage, current, and temperature of the last 100 steps measured for battery RW9.

Reference Charge and Discharge Cycle Mode
After the start of each RW cycle, a sequence of reference charging and discharge current profiles is implemented to set a standard benchmark for the battery's SoH. Initially, by applying a 2 A constant current profile, the LIB is charged to its maximum voltage and then a 4.2 V constant voltage is sustained until its current falls to 0.01 A, then it rests for 20 min. Afterwards, 2 A discharge constant current profile is applied to the LIB until its voltage reduces to its minimum voltage, 3.2 V. This procedure is known as reference charging and discharging cycle mode [38]. To measure the LIB's capacity and calculate its SoH value, after every RW cycle, the current cell capacity (Q current ) is calculated using the Coulomb counting method as follows: where I d is the total discharge current of the battery cell. Once the battery capacity is calculated, the SoH of the battery can be calculated, as shown in (2).
Energies 2020, 13, 5447 where Q Current is the measured capacity after every RW cycle and Q Fresh is the measured capacity of the battery at the beginning of its life. The capacities of the measured cells of the four batteries are shown in Figure 4. As illustrated in Figure 4, the capacity fade is a non-linear and non-homogeneous process since each cell degrades in different ways, even under the same test conditions. Therefore, the main aim of this study is to investigate if the time-frequency image analysis method can capture the nonstationary behaviour of the battery during cycling. Figure 5 shows the block diagram of the proposed TFI paradigm. Here, the terminal voltage of the cycled battery using randomised current profile data is utilised to predict the capacity of the battery. Initially, the terminal voltage of the battery is measured at different capacities, and then the measured voltage is transformed from 1D raw data to a 2D time-frequency image (TFI) by applying a CWT algorithm. The raw data of the measured voltage for the LIB at different capacities contains only the time-domain information, but the converted 2D TFI features of the raw data clearly represent the time and frequency domain information at once. Finally, the generated TFIs for different battery capacities are fed into a deep learning convolutional neural network for feature extraction and image classification. The following subsections explain the capacity prediction steps in detail.  As illustrated in Figure 4, the capacity fade is a non-linear and non-homogeneous process since each cell degrades in different ways, even under the same test conditions. Therefore, the main aim of this study is to investigate if the time-frequency image analysis method can capture the non-stationary behaviour of the battery during cycling. Figure 5 shows the block diagram of the proposed TFI paradigm. Here, the terminal voltage of the cycled battery using randomised current profile data is utilised to predict the capacity of the battery. Initially, the terminal voltage of the battery is measured at different capacities, and then the measured voltage is transformed from 1D raw data to a 2D time-frequency image (TFI) by applying a CWT algorithm. The raw data of the measured voltage for the LIB at different capacities contains only the time-domain information, but the converted 2D TFI features of the raw data clearly represent the time and frequency domain information at once. Finally, the generated TFIs for different battery capacities are fed into a deep learning convolutional neural network for feature extraction and image classification. The following subsections explain the capacity prediction steps in detail.

The Proposed Capacity Imaging Analysis Scheme
battery. Initially, the terminal voltage of the battery is measured at different capacities, and then the measured voltage is transformed from 1D raw data to a 2D time-frequency image (TFI) by applying a CWT algorithm. The raw data of the measured voltage for the LIB at different capacities contains only the time-domain information, but the converted 2D TFI features of the raw data clearly represent the time and frequency domain information at once. Finally, the generated TFIs for different battery capacities are fed into a deep learning convolutional neural network for feature extraction and image classification. The following subsections explain the capacity prediction steps in detail.

Time-Frequency Image (TFI) Analysis
The main aim of this research was the analysis and classification of the measured voltage of LIBs at different capacities using TFI analysis. As stated in Section 1, most studies have used traditional methods of time or frequency domain to analyse the LIB's measured signals and extract diagnostic characteristics to describe the degradation inside the battery cell. Like most real-life signals, the measured LIB terminal voltage can exhibit non-stationary characteristics [31,32]. Thus, analysing the LIB's measured voltage using traditional time or frequency domains is insufficient to extract comprehensive features about the degradation process inside the battery [39]. Nevertheless, time-frequency approaches can extract several domain features in order to assess the measured signal for the time and frequency domains, precisely extracting better features from the measured signal [40]. The main goal of a time-frequency signal processing analysis is to extract useful information features from the measured battery terminal voltage by converting the time series signal into the time-frequency domain [41,42].
Various methods can be used for time-frequency representation, such as CWT, Wigner-Ville distribution, and short-time Fourier transform (STFT). Of these, CWT is employed to transform the measured signal from the time domain to a TFI, whose energy concentrations visualise changes in frequency components over time as the battery degrades [43]. In the proposed prediction scheme, shown in Figure 5, a CWT was applied to the measured terminal voltage, to transform it from a time domain signal into a time-frequency domain scalogram image. The aim of this conversion process was to extract more useful information about the degradation process inside the battery during a randomised charging/discharging current profile. Through a set of wavelet basis functions, the wavelet transforms (WT) degraded the LIB's terminal voltage in the time-frequency domain. This transform employed a wavelet function of a finite bandwidth in terms of the time and frequency domains. Via the scaling and translation of the wavelet basis function, the signal was degraded with various resolutions at differing scales of time and frequency [44]. Equation (3) describes the mathematical scaling and translation of basic wavelet function: where ψ a, b (t) is an analysis wavelet and is called a child wavelet, a ∈ (0, +∞) is the scale parameter of the wavelet transform, and b ∈ (−∞, +∞) is the translation or shift parameter in time. The function ψ(t) is known as the mother wavelet with finite length and fast depletion. Two limit conditions must be satisfied in the mother wavelet signal, which are ( given time series signal x(t), the wavelet coefficients wt(a, b) are obtained by the convolution integral of the mother wavelet ψ(t) and the given signal x(t), as presented in (4): where ψ * means the complex conjugate of function ψ. Throughout this procedure, the signal x(t) is divided into a sequence of scaled and shifted wavelet coefficients, in which the wavelet set is the basis function. After this, the signal x(t) is altered by CWT and sent to the 2D time-frequency domains [45]. With this approach, the 1D time domain signals are transformed into TFI. Figure 6 shows an example schematic of the battery measured voltage at three different capacities and the transformed time-frequency domain representation features for each capacity. It can be observed that there are clear discriminative variations in the CWT coefficients at each capacity in the TFI. Therefore, battery degradation is well reflected by the multi-domain TFI features, which can thus be used as input for the deep learning convolutional neural network (CNN) to classify the battery's capacity during the degradation.

Time-Frequency Image Analysis and Classification Using Deep Learning Algorithm
The obtained TFIs from the battery measured voltage at different capacities are fed into the deep learning predictive model to extract the relationship between the TFIs and corresponding capacity. In this research, a data-driven method based on transfer deep learning model is adopted to perform capacity classification in LIB.
Generally, a DL-CNN is built from three types of layers, namely (1) convolution layers, (2) pooling layers, and (3) fully connected layers [44]. The convolution and pooling layers are linked to form convolution blocks, and many of these blocks are stacked to create a deep architecture. The fully connected layer has generally been used as the final layer in the classification or regression. The general architecture of the DL-CNN is shown in Figure 7 [46].

Time-Frequency Image Analysis and Classification Using Deep Learning Algorithm
The obtained TFIs from the battery measured voltage at different capacities are fed into the deep learning predictive model to extract the relationship between the TFIs and corresponding capacity. In this research, a data-driven method based on transfer deep learning model is adopted to perform capacity classification in LIB.
Generally, a DL-CNN is built from three types of layers, namely (1) convolution layers, (2) pooling layers, and (3) fully connected layers [44]. The convolution and pooling layers are linked to form convolution blocks, and many of these blocks are stacked to create a deep architecture. The fully connected layer has generally been used as the final layer in the classification or regression. The general architecture of the DL-CNN is shown in Figure 7 [46]. capacity classification in LIB.
Generally, a DL-CNN is built from three types of layers, namely (1) convolution layers, (2) pooling layers, and (3) fully connected layers [44]. The convolution and pooling layers are linked to form convolution blocks, and many of these blocks are stacked to create a deep architecture. The fully connected layer has generally been used as the final layer in the classification or regression. The general architecture of the DL-CNN is shown in Figure 7 [46].

Pooling
Convolution + ReLU Convolution + ReLU Feature extraction Classification The convolutional layer combines the input image from the initial layer with several filters, known as kernels, and these are then fed to the activation function to create a features map [47]. The output features map is the convolutional result of the input maps and can be calculated as in (5)  The convolutional layer combines the input image from the initial layer with several filters, known as kernels, and these are then fed to the activation function to create a features map [47]. The output features map is the convolutional result of the input maps and can be calculated as in (5) [48]: Here, * is the convolutional operation, x i is ith input map, k is a F × F convolutional filter, b j is the additive bias, M j is the feature map of the convolutional layer, and l is the lth layers in the neural network. Lastly, the obtained results from the convolutional layers are fed to the activation function. The common used activation function is the rectified linear unit (ReLU), defined as in (6) [47][48][49].
For the pooling layer, which follows the convolutional layer, low-resolution maps are created from the most significant local information. The maximum value is derived from each region by using the max pooling layer, as shown in Figure 8 [48]. In the fully connected layer, a 1D vector is utilized to represent all the feature maps, which is fully connected to the output layer. The output of the fully connected layer is described as given in (7): where O i is the output layer, x j F is the j th neuron in the fully connected layer, w ij is the weight related to O i and x j F , b i is the bias corresponding to O i , and f is the activation function [48,50].
Energies 2020, 13, x FOR PEER REVIEW 9 of 18 Here, * is the convolutional operation, is ith input map, is a F × F convolutional filter, is the additive bias, is the feature map of the convolutional layer, and is the lth layers in the neural network. Lastly, the obtained results from the convolutional layers are fed to the activation function. The common used activation function is the rectified linear unit (ReLU), defined as in (6) [47][48][49].
For the pooling layer, which follows the convolutional layer, low-resolution maps are created from the most significant local information. The maximum value is derived from each region by using the max pooling layer, as shown in Figure 8 [48]. In the fully connected layer, a 1D vector is utilized to represent all the feature maps, which is fully connected to the output layer. The output of the fully connected layer is described as given in (7): where is the output layer, is the neuron in the fully connected layer, is the weight related to and , is the bias corresponding to , and is the activation function [48,50]. In this study, two deep transfer learning algorithms were applied to classify the generated results from TFI. The classification results are presented in the Results and Discussion section.

Results and Discussion
The proposed scheme (see Figure 5) was implemented using the MATLAB environment. First, the results of TFI estimation using complex Morlet mother wavelet transform (CMMW) will be presented in this section, and then the results of image analysis and classification using deep learning algorithms will be demonstrated. In this study, two deep transfer learning algorithms were applied to classify the generated results from TFI. The classification results are presented in the Results and Discussion section.

Results and Discussion
The proposed scheme (see Figure 5) was implemented using the MATLAB environment. First, the results of TFI estimation using complex Morlet mother wavelet transform (CMMW) will be presented in this section, and then the results of image analysis and classification using deep learning algorithms will be demonstrated. Figure 9a shows the measured voltage signal of the first cycle of RW9 at the beginning of the life of the LIB, indicating that no degradation had occurred at this point and the battery's capacity was full (here, C = 2.1 Ah). Figure 9c presents the measured voltage at the end of the LIB's life and shows that the battery degraded, and its capacity decreased to its minimum value (C = 0.8 Ah). From both results, it can be clearly observed that the battery had aged and that the time period of the measured voltage (31.07 h) at the end of life was less than that for the first cycle of the battery (93.18 h). This is because the battery has reached its threshold voltage level very frequently due to an increase in the internal resistance of the LIB, and this inherently reduces the LIB capacity.

Time-Frequency Image (TFI) Results
Energies 2020, 13, x FOR PEER REVIEW 10 of 18 and varies from 1 to 2 Hz [51]. In contrast, Figure 9d shows the energy distributed in the middle and high frequency bands (up to 100 Hz) of the image, meaning that the amplitude of the wavelet coefficients increased to cover the rapid changes between the upper and lower voltage range. A comparison of the TFIs for the battery at the beginning and end of its life clearly shows a significant difference in the energy density distribution. Therefore, the capacity degradation of the battery is well reflected by the TFI information and multiple domain features can be calculated using TFI. The datasets for the four test batteries were processed by the proposed CMMW. Figure 10 presents a sample of the results of the time-frequency images of the measured LIB terminal voltage at different capacities. They show the energy concentration distribution of the battery terminal voltage according to the level of degradation in the TFI. As shown in Figure 10, as the battery's capacity decreases, the distribution of the energy in the TFI changes, clearly illustrating the degradation process inside the battery cell. From the extracted wavelet coefficients using CMMW, the energy concentrations for wavelet coefficients are calculated in the form of TFIs, which are then fed to the pre-trained CNN for To extract appropriate features capable of describing the degradation phenomena inside the battery from the measured voltage, CMMW was applied using Equations (3) and (4)  voltage, as illustrated in Figure 9b,d. In the TFI, the energy concentration of the measured voltage of the LIB is shown at different capacities, with the horizontal and vertical axes representing time and frequency, respectively. The amplitudes of the wavelet coefficients are represented in different colours on the TFI [47]; red colour indicates that the level of energy density is high. In Figure 9b, when the LIB is at the beginning of its life, the energy distribution is concentrated in the middle frequency range of the image and varies from 1 to 2 Hz [51]. In contrast, Figure 9d shows the energy distributed in the middle and high frequency bands (up to 100 Hz) of the image, meaning that the amplitude of the wavelet coefficients increased to cover the rapid changes between the upper and lower voltage range. A comparison of the TFIs for the battery at the beginning and end of its life clearly shows a significant difference in the energy density distribution. Therefore, the capacity degradation of the battery is well reflected by the TFI information and multiple domain features can be calculated using TFI.
The datasets for the four test batteries were processed by the proposed CMMW. Figure 10 presents a sample of the results of the time-frequency images of the measured LIB terminal voltage at different capacities. They show the energy concentration distribution of the battery terminal voltage according to the level of degradation in the TFI. As shown in Figure 10, as the battery's capacity decreases, the distribution of the energy in the TFI changes, clearly illustrating the degradation process inside the battery cell. From the extracted wavelet coefficients using CMMW, the energy concentrations for wavelet coefficients are calculated in the form of TFIs, which are then fed to the pre-trained CNN for image pattern recognition and classification.   To further validate the proposed technique, the experimental datasets of the LIBs RW10, RW11, and RW12 at different capacity ranges from 2.1 to 1.2 Ah have been tested. Table 2 demonstrates the effectiveness of the proposed TFI method to predict the capacity of these cells. Ongoing research is mainly attempting to apply the proposed technique to quantify which type of degradation mode most affects the battery's capacity.

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent

DL-CNN Results
In this subsection, we demonstrate how the generated TFIs from the measured terminal voltage at different capacities through CWT were fed into AlexNet and VGG-16 DL-CNNs for capacity classification, after they had been trained for TFI classification. The Stochastic gradient descent (SGDM) and adaptive moment estimation (ADAM) optimisation algorithms were used to train the generated TFIs.

AlexNet Neural Network
The AlexNet neural network was created in 2012 by Krizhevsky et al. [52] and is built from eight layers (five convolutional and three fully connected layers). In this study, we assumed that the battery's capacity is classified into five classes from the first measured capacity value, which is 2.1 Ah, to the measured capacity value of 1.20 Ah in the experimental dataset-similar to the work presented in [53,54]. The measured voltage of each battery per RW cycle was labelled to show its corresponding capacity. The datasets of LIBs RW9-12 were split into training data (70% of all TFI in the four folds) and 30% for testing data. Therefore, the LIB capacity estimation can be considered a 5-class classification. The size of the generated TFIs is 227 × 227 × 3, which is suitable for the pretrained AlexNet network. The training settings for modelling the network and the AlexNet architecture are given in Tables 3 and 4. The AlexNet output layer was replaced with a new output layer with five neurons corresponding to five classes of capacity values. Since the AlexNet network is not trained to classify TFIs, only the weights of the last convolution layer and fully connected layer were trained, while the first four convolution layers were not. Training only the last layers of the AlexNet network reduced the training time of the model [44].  these classification results demonstrate that the proposed method achieved accurate predictions of battery capacity for each battery cell. The second pretrained model tested in this study is a convolutional neural network from the Oxford Visual Geometry Group (VGG) which is a 16-layer network [55]. VGG-16 has achieved classification accuracy performance on the ImageNet dataset and Table 6 shows the layer structure of VGG-16. With the same hyperparameter settings, the VGG-16 model is compared with the AlexNet model and the prediction accuracy results are presented in Table 7. The comparison classification results from Tables 5 and 7 for AlexNet and the VGG-16 models show that as the DL-CNN becomes deeper, the accuracy of the model will increase. However, as the DL-CNN becomes deeper, the training time will increase, and more computational complexity will be added to the model.

Conclusions
This paper proposed a new capacity evaluation method for LIBs using multi-domain features obtained from a TFI algorithm. The terminal voltage of the battery was transformed into a 2D image feature using CWT instead of 1D raw data of terminal voltage or the extraction of multiple statistical features. The proposed method was applied CWT to produce a 2D multi-domain features time-frequency representation of the battery terminal voltage, known as a TFI. To this end, experimental data on four LIBs cells published by the Prognostics Center of Excellence (PCoE) NASA were adopted to validate the effectiveness of the proposed method. The generated TF images clearly demonstrated the degradation process of the battery throughout the distribution of the energy concentration of the measured voltage for the battery at different capacity values. Two pre-trained DL-CNN were utilised to classify the generated TFIs at various capacities into five classes. The classification results achieved 95.69% accuracy using the AlexNet network and 95.52% accuracy from the VGG-16 network. The accuracy of the proposed method indicates that the proposed technique can be an effective health prognostic tool for managing LIBs for various applications such as electric vehicles and grid applications. However, the proposed method should be tested under different operating conditions (i.e., different temperatures and high-discharge current profile) to increase the accuracy of prediction and to develop a general prediction model. It is important to emphasise that the TFIs generated can be improved since the TFIs were generated by the use of a classical CWT. The resolution of the obtained energy concentration spectra affects the performance of the deep learning algorithm for capacity estimation. Our future work will therefore explore the applicability of the multi-domain features using energy concentration in time-frequency image analysis to provide robust results under different operating and temperature conditions. In addition, other time-frequency techniques such as STFT and Winger-Ville distribution will be investigated and compared to each other in terms of time-frequency resolution and computational complexity for online capacity prediction. Finally, future work will also compare computation load for the proposed algorithms.