## 2 Approach for Optimal Allocation of MFCS
### Approach for Optimal Allocation  
The selected target maximum demand power depends on the drive motor, which can be calculated to be 400 kW according to the vehicle balance equation. The MFCS power supply ratio and conversion efficiency are set to be 0.5 and 0.95, respectively. Based on this, the MFCS power rating can be calculated to be 210 kW. To allocate all stack power in the MFCS, it is necessary to determine the influencing factors and constraints, which can be converted into an optimization problem to solve. The \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Unladen mass (t) & 13 \\ Full load mass (t) & 30 \\ Rotating mass factor & 1.1 \\ Travelling resistance coefficient & 0.01 \\ Vehicle Parameters & Wheel rolling radius (m) & 0.39 \\ Number of wheels & 12 \\ Air resistance coefficient & 0.75 \\ Windward area (m\({}^{2}\)) & 7 \\ Deceleration ratio & 3.8 \\ \hline \multirow{2}{*}{Dynamics Indicators} & Maximum speed (km/h) & 100 \\ Maximum climbing gradient (\%) & \textgreater{}30 \\
0–50 km/h acceleration time (s) & \textless{}15 \\ \hline \end{tabular}
\end{table} 
Table 1: Target dynamics parameters. Figure 1: CWTVC condition. influencing factors and constraints considered in this paper include the number of stacks, the range of power ratings for each stack, the efficiency and lifetime of the stacks, and the power allocation strategy. The solution required for optimal allocation is an optimization model with a two-level objective function, which can be formulated as  
\[ \begin{split}& J_{o}^{\max}=\sum\limits_{i=1}^{\text{end}}\frac{t_ {i}}{t_{\text{end}}}(\alpha\eta_{\text{MFCS}}(\overline{P}_{\text{fc},j})+ \beta\text{RUL}_{\text{MFCS}}(\overline{P}_{\text{fc},j}))\\ &\text{s.t.}\left\{\begin{array}{l}\eta_{\text{MFCS}}(\overline{ P}_{\text{fc},j})=\frac{\overline{P}_{\text{fc},j}}{\sum\limits_{i=1}^{ \infty}\frac{\overline{P}_{\text{fc},j}}{\overline{\
u}(\overline{\
u}_{\text{ fc},j})}}\\ \alpha+\beta=1\\ P_{\text{fc},j}^{\text{fc}}\leq P_{\text{fc},j}\leq P_{\text{fc},j}\\ \sum\limits_{i=1}^{\infty}P_{\text{fc},j}=P_{\text{MFCS},d}\\ \Delta P_{\text{low}}\leq\sum\limits_{i=1}^{n}P_{\text{fc},j}-P_{\text{MFCS},d }^{\max}\leq\Delta P_{\text{high}}\end{array}\right.,\end{split}  \tag{1}\]  
where \(J_{o}^{\max}\) denotes the optimal solution objective; \(\overline{P}_{\text{fc},j}\) denotes the real-time output power of the \(i\)th stack; \(\alpha\) and \(\beta\) denote the weights of the MFCS efficiency and RUL, respectively; \(\eta_{\text{MFCS}}\) and \(\text{RUL}_{\text{MFCS}}\) denote the efficiency and remaining useful life of the MFCS, respectively; \(P_{\text{fc},j}\) denotes the demanded power of the \(i\)th stack; \(P_{\text{fc},j}^{\text{min}}\) denotes the minimum rated power; \(P_{\text{fc},j}^{\max}\) denotes the maximum rated power; \(\Delta P_{\text{high}}\) and \(\Delta P_{\text{low}}\) denote the upper and lower bounds of the MFCS rated power, respectively; and \(P_{\text{MFCS},d}^{\max}\) is the rated power of the MFCS. For the optimal allocation scheme shown in Figure 2, a two-layer objective function is solved. The outer-layer optimization model adopts a genetic algorithm (GA) to solve the optimal stacking scheme based on the power demand determined by the MFCS on the basis of the first-level energy management strategy. The inner-layer optimization model uses sequential quadratic programming (SQP) algorithm to determine the optimal solution for the power demand allocated to each PEM fuel cell. According to the above process, the number of PEM fuel cells is taken as 2 to 4, and the number of stack combinations is 100. When the number of PEM fuel cells is 2,  
Figure 2: Flowchart of the optimal allocation approach. there is no optimal solution; when the number of PEM fuel cells is 3, the optimal stack solution is 20/70/120 kW; and when the number of stacks is 4, the optimal stack solution is 10/10/70/120 kW.