## 4 Results
### Using Artificial Neural Networks to Estimate Remaining Capacity  
The CE variation under variable working conditions can be estimated by Equations (12) and (13). When a battery discharges under various environmental conditions during certain cycles, the capacity fades. Without correction, the estimation errors of Equation (1) increase gradually with the operating cycle. Therefore, by introducing the correction of the CE and considering the declination of SoH, the estimation error is effectively reduced. Considering Equations (2) and (3), \(Q_{available}\) represents the available capacity of a battery, which is influenced by the CE. \(C_{max,p}\) represents the remaining capacity of a battery, which is not influenced by the CE. In order to obtain the true remaining capacity of a battery, the data of Tables 2-4 were normalized through Equation (15). \[C_{max,p}=\frac{Q_{available}}{\eta_{c(T)}} \tag{15}\]  
The estimation of \(SoH_{N}\) would normally be based on the relationship between charging/discharging cycles and remaining capacity. However, considering actual usage scenarios, such as in electric vehicles and other energy storage devices, the discharge C rate would vary sharply in a short period. Moreover, it is difficult for full charging and discharging to take place. Compound effects under different temperatures and discharge rates on capacity fade are difficult to estimate. Thus, this method cannot be direct applied to scenarios where large variations in environmental parameters exist. Therefore, the charging/discharging cycle numbers of the battery were normalized under different usage environments as discharging times in seconds. Through this method, the accumulated degradation of the batteries under various environments was analyzed. When the battery had undergone various rates of discharge under different temperatures, the remaining capacity was calculated by the ANN. The learning ability of ANN was realized by means of model training. The common training algorithm is a back-propagation algorithm, which means that the network error will be propagated back if it does not reach an expected value in the model training process; meanwhile, the network weights and biases values are adjusted constantly to obtain the minimum error [60]. This research utilized a multi-layered BPNN to study the model for estimating the remaining capacity (\(C_{max,p}\)). The network was trained using battery datasets collected under varying temperatures and discharging currents. To build the BPNN model, the complete data was split into training (70%) and testing (30%) data. The detailed operation flow of the BPNN is described below. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Parameters**} \\
**C Rate** & \(a_{1}\) & \(b_{1}\) & \(a_{2}\) & \(b_{2}\) \\ \hline
0.5 C & \(7.2\times 10^{3}\) & \(2.799\times 10^{-3}\) & \(-3.130\times 10^{2}\) & \(-9.669\times 10^{-2}\) \\
1 C & \(3.6\times 10^{3}\) & \(2.328\times 10^{-3}\) & \(-1.342\times 10^{2}\) & \(-8.570\times 10^{-2}\) \\
2 C & \(1.8\times 10^{3}\) & \(2.241\times 10^{-3}\) & \(-7.881\times 10^{1}\) & \(-6.820\times 10^{-2}\) \\
3 C & \(1.2\times 10^{3}\) & \(2.178\times 10^{-3}\) & \(-4.992\times 10^{1}\) & \(-4.990\times 10^{-2}\) \\ \hline \hline \multicolumn{5}{c}{**Parameters**} \\ \hline  & \(p_{1}\) & \(p_{2}\) & \(p_{3}\) & \(p_{4}\) & \(k_{1}\) & \(k_{2}\) & \(k_{3}\) & \(k_{4}\) \\ Value & \(3.6\times 10^{4}\) & \(3.9\times 10^{-3}\) & \(-1.628\times 10^{3}\) & \(-1.45\times 10^{-1}\) & \(-1.887\times 10^{-1}\) &The BPNN is one of the most commonly used neural network models. It is a multi-layer feedforward network that can learn and store base on the training of error back propagation. Its rule of learning makes use of gradient descent. Through continual adjustments of back propagation weights and thresholds of the whole network, the network's error square summation (cost) is minimized [60]. Moreover, BPNN can get many input-output models without the knowledge of mathematical equations of the mapping relationship [61].