## 4 Algorithm Analysis
### Comprehensive Performance Analysis of the Strategies  
The performance of energy management control strategies can be analyzed with respect to the following aspects: fuel economy, real-time performance, computational time, computational burden, and degree of realization at present. According to the space complexity of the algorithm, the computational burden of various strategies can be calculated, while computational time corresponds to the time complexity of the algorithms. Simultaneously, the complexity of an algorithm can be analyzed by combining the time complexity with the space complexity. According to Table 2, the computational time of the various strategies sorted from low to high is as follows: RB \(<\) ECMS \(<\) ADP/PMP \(<\) NN-DP/RL \(<\) MPC/SDP \(<\) DDP. Similarly, the computational burden of the various strategies, sorted from lowest to highest, is as follows: RB \(<\) ADP/MPC \(<\) RL/ECMS/NN-DP \(<\) SDP \(<\) PMP/DDP. For a given driving cycle, the fuel economy of the various strategies can be compared according to the simulation results under the same vehicle configuration and the same driving cycle. The DP-based strategy is able to obtain a theoretical global optimal fuel economy for HEVs, which provides a benchmark for assessing the optimality of other energy management strategies, despite the off-line optimization. If the performance of the minimum fuel consumption of the various strategies is divided into five levels from high to low, the DP-based strategy can be regarded as the first level, while the rule-based strategy can be regarded as the last level, due to the fact that it cannot obtain optimal fuel economy. The PMP-based strategy can obtain the global optimum solution by using the minimum principle when the obtained local optimal trajectory is a unique trajectory. Similarly, the ADP-based strategy (RL as a special structure of ADP) can obtain the approximation results for DP by approximating the performance index function and control strategy based on the function approximation structure to satisfy the optimality principle. The accuracy of the velocity prediction will influence the optimal results of the MPC-based strategy and the SDP-based strategy, and robustness and accuracy cannot be guaranteed when the actual driving cycle is quite different from the training driving cycle. The results of ECMS are very sensitive to the equivalent factor, while the results of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Time Complexity (\(n\)1)** & **Space Complexity** \\ \hline RB & \(O(1)=O(1)+O(1)\) & \(m\cdot O(n)\) \\ ECMS & \(O(n)\) & \(m\cdot O(n^{2})\) \\ MPC & \(O\left(n^{3}\right)\sim DP\) or \(O\left(n^{2}\right)\sim GA/SA\) & \(O\left(n^{2}\right)\) \\ PMP & \(O\left(n^{2}\right)\) & \(m\cdot O(n^{3})\) \\ DDP & \(O\left(n^{3}\right)=O(n(n+1)(n-1)/6)\) & \(m\cdot O(n^{3})\) \\ SDP & \(O\left(n^{3}\right)=O\left(n^{3}\right)+O\left(n^{2}\right)\) & \(O\left(n^{3}\right)\) \\ NN-DP & \(O\left(n^{2}\right)=O\left(m\cdot n^{2}\right)\) & \(m\cdot O\left(n^{2}\right)\sim train\) \\ RL & \(O\left(n^{2}\right)\) & \(m\cdot O\left(n^{2}\right)\sim train\) \\ ADP & \(O\left(n^{2}\right)=O\left(m\cdot n^{2}\right)\) & \(O\left(n^{2}\right)\) \\ \hline \hline \end{tabular}  
* \(n\) represents the scale of the problem, which can be regarded as the dimension of input vector; \(m\) represents the number of samples.  
\end{table} 
Table 2: The complexity of the above-mentioned algorithms. NN-DP strategy are influenced by the comprehensiveness of the training samples. According to the approximate degree between the results of each control strategy and the DP results, the fuel economy of the various strategies, sorted from lowest to highest, is as follows: RB \(<\) ECMS/NN-DP \(<\) SDP/MPC \(<\) ADP/RL \(<\) PMP/DDP.