This section presents prior work on RUL prediction. The approaches to predicting RUL can be categorized into model-based methods and data-driven methods. Model-based approaches utilize electrochemical principles, dynamic models, and equivalent circuit models. (Xiong et al., 2017) utilized electrochemical impedance spectroscopy (EIS) and proposed a method to identify degradation behavior. (Xu and Chen. 2017) proposed a state-space model utilizing expectation-maximization (EM) and extended Kalman filter (EKF) for RUL prediction. (Pinson and Bazant, 2012) introduced a single particle model to explain degradation in capacity, focusing on the growth of the SEI layer. (Hu et al., 2017a) utilized a reduced-form electrochemical model by proposing a moving horizon estimation framework (MHE). Even though model-based approaches give us a basic understanding of the dynamics of electrochemical reactions inside a battery, they are computationally expensive. Moreover, the parameters depend on operating conditions, so the model has limited applicability.

Because of the drawbacks and limitations of model-based approaches, data-driven approaches have received substantial attention recently. The advancements in machine and deep learning techniques and their non-reliance on inherent electrochemical dynamics make them an attractive alternative. Numerous machine learning and deep learning algorithms have been investigated such as the auto-regression model (Song et al., 2017), relevance vector machine (RVM) (Yuchen et al., 2018), Gaussian process regression (GP-regression) (Liu et al., 2013) and support vector machine (SVM) (Dong et al., 2014). Neural networks have proved efficient because of their nonlinear modeling capability (Wu et al., 2016; Tarar et al., 2022). Sequential dependency networks, such as recurrent neural networks (RNNs), are widely used (Liu et al., 2010; Sbarufatti et al., 2017; Zhang et al., 2022) proposed a prediction framework based on a combination of offline global models developed by different machine learning methods and cell-individualized models that were online adapted. (Hu et al., 2020) provided a timely and comprehensive review of the battery lifetime prognostic technologies, focusing on recent advances in model-based, data-driven, and hybrid approaches. These approaches’ details, advantages, and limitations were presented, analyzed, and compared.

A novel hybrid method is proposed (Xu et al., 2022) to predict battery capacity degradation trajectory, which combines physics-based and data-driven approaches in three steps: hybrid feature extraction, clustering with data augmentation, and training a deep neural network. The method provides accurate predictions with only 20% training data and is robust to noisy input. Validation results show mean absolute percentage errors below 2.5% for capacity degradation trajectory and 6.5% for remaining useable cycle life under different aging conditions. (Deng et al., 2022) use battery aging data to recognize degradation patterns and improve the state of health (SOH) estimation accuracy using transfer learning. Four discharge capacity curve features are extracted, with two distinct degradation patterns and two used for SOH estimation. Long short-term memory (LSTM) network achieves the best SOH estimation accuracy compared to other algorithms, and degradation pattern recognition and transfer learning methods further improve accuracy to mean MAE and RMSE values of 0.94% and 1.13%.

Jiang et al. (2021) proposed a reliable cycling aging prediction based on a data-driven model to address the urgent issue of adaptive and early prediction of lithium-ion batteries’ remaining useful life. First, a multi-kernel relevance vector machine (RVM) model with two alternative kernel functions was built to improve the learning and generalization capabilities of the traditional RVM. Next, the kernel and weight parameters were determined using the particle swarm optimization approach. A similarity criterion of the battery capacity curves is then provided to filter offline battery data for model training to accomplish early life prediction. For model training and verification, battery cycling aging data from two different types of batteries under various aging conditions were employed. Ren et al. (2018) proposed an integrated deep learning approach in which 21-dimensional features extracted from the National Aeronautics and Space Administration (NASA) dataset are fused by an auto-encoder that reduces their dimensionality to 15. The reduced 15-dimensional features are then fed to a deep neural network (DNN) model for RUL prediction.

The capacity of the battery degrades over charge-discharge cycles. In data-driven approaches, various features extracted from the measured data exhibit valuable patterns that can be exploited to predict the battery’s RUL. These trends can be exploited in a time series analysis using features spanning multiple cycles. However, these memory-based features that span multiple cycles have rarely been used.

With a growing trend of using deep neural networks in such problems, domain knowledge-based features are also declining, and researchers bank on the power of neural networks to extract features from the data. Feature engineering is an important aspect where domain knowledge plays an important role. A careful presentation of the data can help identify the trends in the data and can be an important step in validating the dataset. Furthermore, the identified features are used with memory to predict RUL by introducing a window spanning multiple cycles capturing features of the previous n cycles. The main contributions of this research are as follows

1. This paper developed a novel framework—DNN with memory features (DNNwMF)—to accurately predict the RUL of Li-ion batteries using features of current and previous n cycles.

Incorporating memory in features in such a way for batteries has not been found in the literature. Memory-based features in a simple deep neural network are used rather than RNN or LSTM. RNN and LSTM networks are more complex and take longer to train than DNN. LSTM networks also require more memory to train and can easily overfit. Moreover, dropout is much harder to implement in LSTM. In addition, LSTM is sensitive to different random weight initializations. Also, LSTM processes time-dependent data, so using the extracted features might not be a good way to feed to an LSTM network. Instead, providing sequential raw information is a better approach with the LSTM network. These limitations make it challenging to find the optimal value of n, for which the proposed method is used. Hence, the proposed method allows for finding an optimal window to incorporate memory, which has not been done for Li-on battery datasets.

The generalizability of the results is another crucial aspect often overlooked in data-driven approaches. The design of experiments for dataset creation is constrained because only a few attributes, like charging profiles, discharging profiles, temperature, and humidity, are changed to build a dataset. Therefore, datasets capture different degradation mechanisms; hence, a data-driven technique optimized for a dataset may not work if the dataset is changed. In order to confirm the generalizability of the results, two different datasets, a) from the Prognostic Center of excellence of National Aeronautics and Space Administration (NASA) and b) the Center for Advanced Life Cycle Engineering (CALCE) dataset provided by the University of Maryland College Park are used to validate the results. For these datasets, the tests have been performed in different conditions. For instance, all the tests for NASA’s dataset are run at room temperature, whereas the CALCE dataset has 22 different test conditions that vary in temperature, C-rate, and the time interval between cycles.

The next section describes the proposed framework and problem formulation. Section 3 presents and analyzes the experimental results and Section 4 concludes this paper.

Figure 2 shows one cycle of the most widely used constant current constant voltage (CC-CV) charge and discharge process.

(a) Charging process: In the charging process, the first step is constant current (CC) charging, in which current is set to a constant value while voltage is raised to the maximum upper limit. After that, the voltage is kept at this constant value, called the constant voltage (CV) charging step. While the voltage is kept at the CV, the current drops to a certain threshold.

(b) Discharging process: This process consists entirely of CC discharging in which the voltage drops to the lower limit.

In typical experimental data, the charging and discharging processes described above are repeated until the battery reaches its end of life. In this paper, the NASA dataset (Saha and Goebel, 2019) and the Center for Advanced Life Cycle Engineering (CALCE) dataset provided by the University of Maryland (UMD) College Park have been used for RUL estimation. Features are extracted from the charging and discharging voltage and current profile from the two datasets.

When the battery goes through charge-discharge cycles, several degradation mechanisms occur, which include loss of lithium inventory, lithium plating, and metallic dendrites, negative electrode degradation, degradation in composite electrode, metal dissolution, positive electrode degradation in bulk, loss of active material, and increased impedance. These degradation mechanisms reduce the battery capacity, manifested in the shortening of the time it takes to reach upper and lower cutoff voltage during charging and discharging. Similarly, constant current charging time and constant current discharging time decrease over the cycles. Hence, these features can help extract helpful degradation patterns to help predict RUL.Figures 3, 4 shows the battery voltage and battery output current over time during multiple charge and discharge cycles of battery B0005 from the NASA data set.

The figures visibly capture various trends as the battery ages and the number of cycles increases. This kind of plotting shows the variation of all these features over the charge-discharge process and how the trend varies over the number of cycles. For example, Figure 3 shows how the battery voltage varies over time during the charge-discharge cycle. During the CC phase in the charging cycle, as shown in Figure 3A, the battery voltage increases to its cut-off voltage, where it stays constant for the rest of the cycle; however, the time required to reach the cut-off voltage decreases with the number of cycles. Figure 3B demonstrates that the time to reach the lower cut-off voltage during the discharge phase decreases with the increasing number of cycles. Similarly, the CC charging and discharging time decrease as the number of cycles increases, as shown in Figure 4. Thus, clear patterns emerge as the battery ages, benefiting battery prognostics.

However, as the battery ages and its capacity fades, the collected samples in the charging and discharging process are different: 4,500–5,000 for newer and 700–800 for old batteries. Therefore, it is not feasible to use these data samples directly from the charge and discharge cycle into the prognostic framework without data pre-processing or a mechanism for feature extraction. To learn from this data, features are extracted from the charge and discharge cycle that can be used in RUL prognostics. The extracted features are as follows:

Upper Cut-off Voltage Time: During the charging process, the battery terminal voltage features are t min i , V i , V i ⩾ upper cut - off voltage , i = 1,2,3 , … , m , (1)

where t _min(i) represents the time when the battery terminal voltage reaches the maximum value first, and V _i represents the value of the output voltage in the i _th cycle.

Constant Current Charging Time (CCCT): One of the output current features is CCCT which represents the time when the output current of the battery starts to drop, t min i , A i , A i ⩽ constant current value , i = 1,2,3 , … , m , (2)

where A _i represents the current value in the i _th cycle. Maximum Charging Voltage Time: The time required to reach the maximum battery measured voltage is given as t max , V max = t i , V i | max V i , i = 1,2,3 , … , m , (3)

where t _max is the time stamp for which the maximum voltage V _max is measured in the i _th cycle.

Constant Voltage Charging Time: The time for which the battery is charged at constant voltage t min i , V i , V i ⩽ constant voltage value , i = 1,2,3 , … , m , (4)

where V _i represents the voltage value in the i _th cycle.

Minimum Discharging Voltage time: During the discharging process, the time for which the battery terminal voltage reaches its minimum voltage is given as t min , V min = t i , V i | min V i , i = 1,2,3 , … , m , (5)

where t _min is the time to reach the minimum battery terminal voltage value V _min in the i _th cycle.

Constant Current Discharging Time: The time when the output current during the discharging phase starts to rise again is the constant current charging time given as t min i , A i , A i ⩾ - constant current value , i = 1,2,3 , … , m , (6)

where t _min(i) represents the time and A _i represents the value of current in the i _th cycle.

Battery Capacity: Battery capacity, C, represents the discharge capacity available at each cycle. The discharge capacity feature is the product of the last two features (t _min(i), A _i ), so it may seem unnecessary to include it. However, the experiments show accuracy loss with no significant reduction in the number of optimization parameters. Moreover, the optimal values of n obtained by excluding this feature are the same when including this feature. Also, because an autoencoder using the optimal value of n is employed, the redundancy of features is being taken care of.

Prediction of RUL requires the true labels of RUL for the training phase. For this, the total number of battery charge and discharge cycles are calculated and assigned as the total cycle life L. Then, for the i ^th charge and discharge cycle (0⩽i⩽n), RUL (y _i ) is calculated as y i = L + 1 − i . (7) Finally, x _i and y _i are combined to get the supervised data (x _i , y _i ) where x _i is the input feature with label y _i . To mitigate any negative effects of different data ranges, data normalization is carried out to map each feature in an interval [0, 1] using the minimum-maximum normalization method on the input x _i .

A DNN is employed with memory features that use the features of the last n cycles at the input. The DNN is a supervised learning model. The input is fed to the network along with the ground truth. The network processes input assigns weights to different inputs, and passes it through a nonlinear function to generate the mapping for the next layer. For instance, for the i ^th hidden layer, its input is mapped as h i = f W h i − 1 + b , (8)

where f is non-linear activation function, W is the weight matrix, b is the bias, and h ⁱ⁻¹ is the output of the previous hidden layer. The output of the last layer is compared with the ground truth to form a loss function. During the training phase, the layers’ weights are optimized so that the loss function is minimized. The framework uses a DNN model consisting of four fully connected layers; the number of neurons in each layer is 10, 7, 4, and 1. The current DNN structure was picked based on the results of tuning the model. The number of hidden layers and the number of neurons in each hidden layer were varied, and the best results were used. The rectified linear unit (ReLu) function is used as an activation function for the intermediate hidden layers, and the sigmoid function is used as an activation function for the output layer. The network is then trained to minimize the root mean square error (RMSE) as the error (loss) metric given by R M S E = 1 N t ∑ i = 1 N t y ̂ i − y i 2 , (9)

where N _t is the size of the training set. y ̂ i and y _i are the predicted value and the true value of sample i respectively.

As the number of features in the DNNwMF framework increases, the number of optimization parameters also increases. Redundancies in the features also creep into the system. In such a scenario, an auto-encoder is used to reduce the dimensionality of the input features. An autoencoder is used instead of other compression algorithms, such as PCA, because the features have a nonlinear relationship to the cycle life. An autoencoder can compress the information better into low dimensional latent space, leveraging its capability to model complex nonlinear functions. Moreover, the autoencoder is better at feature fusion than PCA. PCA simply learns a linear transformation that projects the data into a different space, with the projection vectors determined by the data’s variance. Autoencoder is a neural network that tries to mimic the input and hence tries to learn an identity system (Gondara, 2016; Tarar and Khalid, 2021). It consists of an encoder-decoder pair. A fully connected autoencoder encodes the input vector x into a latent space representation as vector z using a transformation of the form z = f W e x + b e , (10)

where f is non-linear activation function, W _e is the weight matrix, and b _e is the bias at the encoder. Latent space representation z is then decoded into an output space x ̃ having the same dimensionality as x. This transformation also uses a similar mapping, that is, x ̃ = f W d z + b d . (11)

(W _e , W _d , b _e , b _d ) are the weights optimized using a back propagation algorithm so that the error in the reconstruction is minimized.

Autoencoders are used for many applications, including feature extraction, feature fusion, dimensionality reduction, and denoising, to name a few. Here, the task is two-fold; dimensionality reduction and feature fusion. When the parameters above are optimized, the hidden neurons can be used to express the input, albeit with a reduced dimension. In this paper, autoencoders compress the high dimensional highly correlated features to low dimensional complex features as opposed to (Ren et al., 2018), which used autoencoder only to reduce the 21-dimensional features to 15-dimensional features. The autoencoder contains one hidden layer containing nine neurons. So the features were compressed to nine features. The input and output layers contain ((n _opt + 1) × 13) number of neurons.

This paper used the Li-ion batteries data set from NASA’s Prognostic Center of excellence (PCoE) (Saha et al., 2009; Saha and Goebel, 2019) and the Center for Advanced Life Cycle Engineering (CALCE) dataset provided by the University of Maryland (UMD) College Park. The NASA dataset was collected under room temperature and rated conditions. B0005, B0006, B0007, and B0018 batteries were used to generate results. The total number of cycles, i.e., m, was 168 for batteries B0005, B0006, and B0007, while m = 132 for battery B0018. The CALCE dataset contains data from six vendors under twenty-two different test conditions of temperature, discharge C-rate, and rest time. Although the cycle datasets used in the paper contain batteries that only last a few hundred cycles, the developed method will work for batteries with longer cycle life ( > 1000 cycles). Also, it is pertinent to mention that the CALCE dataset contains battery life cycles closer to 1,000 cycles. Therefore, the developed method can be applied to batteries with any life cycle. Table 1 presents CALCE dataset test conditions used in the paper. For all test conditions of the CALCE dataset, the charging took place at 0.8 C.

All the features explained in Section 2 were extracted for batteries from two datasets. Batteries B0005, B0006, and B0007 were used for training, and battery B0018 was used for testing from the NASA dataset. For the CALCE dataset, as each test condition contained three battery samples, two samples were used for training, and the remaining sample was used for testing. Thus, for nine test conditions shown in Table 1, nine separate models were trained using two samples, with the third sample used for testing. As there are 168 cycles for the batteries B0005, B0006, and B0007, RUL corresponding to i ^th cycle is calculated as 169 − i. For battery B0018, i ^th cycle RUL was calculated as 133 − i as it contains 132 cycles. The paper used a similar process for the CALCE dataset. The data were normalized using minimum-maximum normalization. As indicated previously, there were separately trained models for two datasets. In fact, for the CALCE dataset, there were nine trained models for the nine test conditions used in the paper. Training so many models may raise the question of why not a universal model is trained for both datasets. However, the two datasets contain batteries of different chemistries. Furthermore, the CALCE dataset comes from six different vendors with 22 different testing conditions. Each test condition triggers different degradation mechanisms. Hence, all these factors make it challenging to make a universal model even for the same dataset with different test conditions, let alone for different datasets. Thus, as will be seen from the results, the optimal value of n is different for different test conditions, indicating difficulty in obtaining a universal model.

This subsection shows that without using memory features, an auto-encoder is not required. This section also presents the results of DNNwMF with different values of n and finds the optimal value n _opt. The optimal value of n refers to the number of previous cycles used with the current cycle to predict the RUL. For each value of n, training data is used to train the model. Within the training data, validation data is used to test the model, and the model is selected based on the minimum validation loss. The testing data is then used on the trained model, and RMSE (%) is obtained. This process is repeated for all the values of n. The optimal value of n is chosen for which the change in RMSE (%) starts to diminish (0.01% in this case). In other words, for each n, the optimal model is chosen using validation loss, with the optimal value of n chosen using the testing loss. The proposed DNNwMF is compared with logistic regression and SVM models, and the effect of increasing C rate and temperature on the optimal value n _opt is explored. The SVM benchmark used Gaussian Kernel Radial Basis Function (RBF) and the hyperparameters were optimized within the Regression Learner app in MATLAB to automate the selection of hyperparameter values. Finally, the results with ADNNwMF show that an autoencoder can be used to reduce the number of optimization parameters considerably without compromising accuracy. Stochastic gradient descent with RMSE as a loss function is used for all these cases during the training phase. After optimizing parameters during the training, RUL prediction results are verified using the testing set. The results are presented using RMSE (%), which is calculated as follows. R M S E % = 1 N t ∑ i = 1 N t y ̂ i − y i y i 2 × 100 , (12) where N _t is the size of the training set. y ̂ i and y _i are the predicted and actual value of sample i, respectively.

Table 5 shows the optimal value n _opt for each test condition for the CALCE dataset. It can be seen that increasing discharge C-rate and temperature increases the n _opt. Figure 8 shows the RMSE (%) for each test number of the CALCE dataset at n _opt for DNNwMF, logistic regression and SVM. DNNwMF outperforms both logistic regression and SVM.

Table 3 also shows that the number of parameters to optimize increases with the increase in n. For instance, for n = 17, the number of optimization parameters is an order of magnitude higher compared to n = 0. Thus, improved accuracy is achieved at the cost of computational complexity.