## 3 Power Sizing and Energy Management Strategy
### Component Sizing Problem  
The sizing of energy sources in a vehicle's power system is a critical factor influencing its performance, efficiency, and overall cost. Selecting appropriate dimensions is essential, as undersized energy sources may fail to meet performance demands, while oversized ones result in increased manufacturing costs.  
In light-duty passenger vehicles where the FCS serves as the primary power source rather than functioning as a range extender and in plug-in hybrid configurations, it is imperative that the FCS satisfies the performance specifications of the powertrain. Predefining the FCS dimensions based on these operational requirements presents a pragmatic design strategy. This is particularly critical, as the concurrent optimization of multiple interdependent parameters--including fuel cell stack sizing, battery capacity sizing, economy, and system durability--introduces significant computational complexity. The proposed sizing methodology not only guarantees compliance with powertrain performance constraints, but also streamlines the multi-objective optimization process through design space reduction. Consequently, it enables an enhanced system-level trade-off analysis between energy economy and component durability.  
The proposed methodology focuses on meeting power requirements as the foundation for sizing. To determine the fuel cell parameters, a testing-based method is utilized, aiming to ensure that the performance criteria are met. Specifically, the maximum power outputs of the fuel cell and the battery capacity are determined according to the approach in [26].  
The fuel cell's maximum power output is evaluated using the following two key tests: (1) it must sustain the vehicle's maximum speed and (2) it must maintain speed under a constant road grade. The parameters used for the described tests are collected in Table 4.  
\[\begin{array}{c c c}\hline\text{Test 1: }P_{bat}=0,\,v=v_{s1},a=0,\alpha=0.\\ \\ P_{fc,1}=\frac{P_{req(t)}}{\eta_{fc_{d,ck}}}\\ \\ \text{Test 2: }P_{bat}=0,\,v=v_{s2},a=0,\alpha=\alpha_{s}.\\ \\ P_{fc,2}=\frac{P_{req(t)}}{\eta_{fc_{d,ck}}}\\ \\ \end{array} \tag{18}\]  
To ensure that the vehicle maintains a sufficient power reserve, a reserve power coefficient of 1.2 is incorporated into the calculations.  
\[P_{fc,max}=1.2*max\Big{\{}P_{fc,1},P_{fc,2}\Big{\}} \tag{20}\]  
Both \(P_{fc,max}\) and \(Q_{bat}\) are initially unknown when calculating the required power. As the energy source sizes increase, the required power correspondingly grows. To resolve this, the battery capacity is initially set to its minimum value and the maximum fuel cell power is iteratively updated until the difference between the assumed and final values converges to \(10^{-6}\).  
Through simulation experiments, the maximum power output of the fuel cell is determined to be 80.6 kW.  
The selection of battery capacity will be determined in conjunction with the EMS, as detailed in Section 3.2. It is important to note in advance that variations in battery capacity will be achieved by parallelizing battery cells. In this study, a single \(LiFePO_{4}\) battery cell with a capacity of \(11\) Ah is used.