The voltage variety rate, temperature variety rate, and temperature range are, respectively, in the following forms: k i ( v ) = v 2000 − v 1000 1000 , i = 1,2 , … , n , (2) k i ( T ) = T 2000 − T 1000 1000 , i = 1,2 , … , n , (3) Δ i ( T ) = T max − T min , i = 1,2 , … n , (4) where v 2000 and v 1000 are the voltage at the 2000s and 1000s in the i-th discharge process, respectively, T 2000 and T 1000 are the temperature at 2000 s and 1000 s in the i-th discharge process, respectively, T max and T min are the maximum and minimum temperature in the i-th discharge process, respectively, and i is the discharge cycle in the whole aging process. Therefore, the time series of indirect health factors can be expressed as follows: k ( v ) = { k 1 ( v ) , k 2 ( v ) , … , k n ( v ) } . (5) k ( T ) = { k 1 ( T ) , k 2 ( T ) , … , k n ( T ) } . (6) Δ ( T ) = { Δ 1 ( T ) , Δ 2 ( T ) , … , Δ n ( T ) } . (7)

Zhang et al. (2019) used the gray correlation method to analyze the correlation between candidate indirect health factor series and capacity series, which is cumbersome. This study uses the Pearson correlation coefficient method (Zhou et al., 2013) for research. r = ∑ i = 1 n ( x i − x ¯ ) ( y i − y ¯ ) ∑ i = 1 n ( x i − x ¯ ) 2 ∑ i = 1 n ( y i − y ¯ ) 2 . (8)

Thus, the correlation between candidate indirect health factor sequences and capacity sequences during discharge in NASA data sets B0005, B0006, B0007, and B0018 can be obtained.

It can be seen from Table 1 that the temperature change rate has the highest correlation and the visual effect is shown in Figure 2. Therefore, the temperature change rate is finally selected as an indirect health factor.

One-dimensional convolution in the TCN, such as two-dimensional convolution, still has the characteristics of weight sharing. Given a time series ( X t , Y t ) , the time series is transformed into the input and output of supervised learning. When the input X t and output Y t form a single channel, the output at any time t depends on the sub input sequence with the same convolution kernel length including the current time, that is, causal convolution, as shown in Figure 3 in the following part. When the convolution kernel length is k = 3 , then y t = x t − 2 ⋅ w 1 + x t − 1 ⋅ w 2 + x t ⋅ w 3 . In addition, TCN default step size s t r i d e = 1 , that is, the input sub time series moves one step at a time.

The deep network can be obtained by superposition based on a one-dimensional causal convolution network, but the increase of the receptive field is obtained at the expense of the depth of the network, so it is difficult to deal with long historical data. To solve this problem, TCN proponents refer to the dilatory causal convolution in the WaveNet model (Ding and Jia, 2019). The definition of extended causal convolution is given as follows: F ( s ) = ( x ∗ d f ) ( s ) = ∑ i = 0 k − 1 f ( i ) ⋅ x s − d ⋅ i , (9) d = b i , (10) where f : { 0 , … , k − 1 } is the convolution kernel and d is the dilation factor ( b is usually taken as 2).

The operation is realized using the dilation factor (default 2), which increases the receptive field w on the premise of fewer network layers. w = 1 + ∑ i = 0 n − 1 ( k − 1 ) ⋅ b i = 1 + ( k − 1 ) b n − 1 b − 1 , (11) where b is the base dilation factor, n − 1 is the number of network layers, and i is the number of network layers before the current layer.

It can be seen from the aforementioned text that through the selection of convolution kernel size and base expansion factor, a relatively small number of network layers can be used to achieve a large receptive field. However, despite the abovementioned operations, the network will be very deep. Therefore, TCN proponents introduced a residual block structure similar to Resnet. As shown in Figure 4, the residual block includes a two-layer convolution network and nonlinear mapping. To normalize the input of the hidden layer and offset the problem of gradient explosion, weight normalization is applied to each convolution layer. However, the aforementioned architecture can only realize complex linear regression, and nonlinearity needs to be introduced by adding the activation function. In addition, to prevent overfitting and gradient disappearance and accelerate model training, regularization is introduced by dropout after each convolution layer of each remaining block.

During the jump connection, because the number of channels in the input layer and the output layer may be inconsistent, the output and input cannot be added directly. To solve this problem, 1 can be introduced × 1 convolution to ensure that the two tensors have the same shape. o = A c t i v a t i o n ( x + F ( x ) ) , (12) where x is the input of the network and the function F ( x ) is the residual mapping to be learned. Through the residual connection, the problems of gradient disappearance and gradient explosion can be effectively alleviated, and the degradation of the model can be avoided.

The experimental analysis model in this study is based on Python 3 6. The experiment was carried out on the Dell Notebook. The relevant configurations are as follows: the system model is Inspiron 5488, the GPU is NVIDIA GeForce mx250, the CUDA version is 10.1, and the Pytorch version is 1.7.1.

The data set was provided by the NASA Ames prediction Excellence Center (Lin et al., 2021). On the accelerated life test platform developed by NASA, a 18650 lithium cobalate battery with a rated capacity of 2 Ah was used to carry out reference charge and discharge tests according to different test configuration files. First, for the reference charge test, the battery was charged to the charge cut-off voltage at constant current and then charged to the cut-off current at a constant voltage. For the reference discharge test, the battery should be discharged at a constant current until the voltage is below the discharge cut-off voltage. The dataset files in the mat format include two parts: one is the voltage, current, temperature, and time in the test; and the other is the capacity estimated using the Coulomb counting method (Lee et al., 2020). The discharge test conditions are shown in Table 2 below.

In this study, “charge–discharge” is regarded as a cycle. It should be noted that there are some wrong data in the NASA data set. For example, in the 310th and 313rd operations of B5, the charging record is missing, resulting in repeated discharge. For this phenomenon, the records of repeated charging or discharging operations are deleted (Zhao et al., 2021). Therefore, the cycle times of B5, B6, B7, and B18 are 167, 167, 167, and 132, respectively. Then, the time series comprising temperature change rate and SOH is transformed into samples required for supervised learning according to a certain window (is set to 8), and then the samples are divided into the training set, verification set, and test set according to different prediction starting points. If the starting point is set to 90, the training set is 73 samples generated from the data of the first 80 discharges. The validation set is 10 samples immediately after the training set, and the test set is the remaining samples. In addition, at the beginning of model training, the Z-score method should be used to standardize the data.

The model parameters are configured in Table 3, as follows:

In addition, to stop training in and obtain a better model, early stop technology is also adopted.

For the prediction effect of SOH, in addition to the average absolute error MAE (mean absolute error) and root mean error RMSE (root mean square error), the absolute error AE (absolute error) between the predicted value of EOL and the real value is used as the evaluation index. M A E = 1 N ∑ m = 1 N | y m − y m | , (13) R M S E = 1 N ∑ m = 1 N ( y m − y m ) 2 , (14) A E = | c y c l e E O L − c y c l e E O L | , (15) where y m and y m are the predicted and true values of the MTH discharge, respectively and c y c l e E O L and c y c l e E O L are the cycles from the predicted starting point to the predicted value or the true value reaching EOL, respectively.