## 2 Optimized Hybrid Energy System for High-Speed Rail Integrating Carbon-REC Trading
### Models of System Components
#### 2.2.3 Carbon Model
This accounting methodology conducts daily settlement of carbon emissions from diverse sources and aligns the temporal scale to facilitate the modeling of subsequent carbon trading mechanisms and the establishment of multi-objective functions. In this study, the daily carbon emissions of the entire high-speed rail power system are calculated, enabling the determination of carbon emission costs based on the carbon price. Carbon emission costs constitute a fundamental cost component, meaning that any carbon emissions inherently incur such costs. \(\text{Cost}_{\text{Carbon}}\) represents the carbon emission cost in $, while \(\text{Price}_{\text{Carbon}}\) denotes the carbon price in $/t. The carbon price in this study is set to \(13.2\,\mathrm{\SIUnitSymbolMicro}/\mathrm{t}\)[43]. Since carbon emissions in this work are measured in kg, unit conversion is required to ensure consistency for calculations. Equation (13) is presented below. \[Cost_{Carbon}=\frac{Emission_{Total}\times Price_{Carbon}}{1000} \tag{13}\]  
In this study, the daily total carbon emission allowance for the high-speed rail power system is set to \(7900\,\mathrm{kg}\), denoted as \(\text{Carbon}_{\text{Allowance}}\). This value is derived from the 95 th percentile of the daily carbon emission data for high-speed rail power systems relying solely on grid energy. When the total daily carbon emissions of the high-speed rail exceed the specified carbon emission allowance, additional carbon allowances must be purchased. Therefore, in addition to paying for the daily carbon emission costs, the high-speed rail must also incur a carbon transaction cost, referred to as \(\text{Cost}_{\text{CarbonT}}\). If the total daily carbon emissions do not exceed the carbon emission allowance, only the carbon emission costs are required. The price of carbon allowances is consistent with the carbon price, both set at 13.2 $/t, denoted as \(\text{Price}_{\text{Allowance}}\). Equation (14) is presented below. Figure 4:0  
If the total daily carbon emissions are less than the specified carbon emission allowance, the high-speed rail can trade the unused portion of its carbon allowance for the day, which is referred to as carbon incentivization or \(\text{Cost}_{\text{CarbonI}}\). In this study, carbon credits are treated as negative costs to simplify calculations. This study introduces a carbon penalty mechanism into the model to internalize environmental externalities, compelling the system to balance between economic optimality and emission compliance. In addition to daily carbon emission modeling, annual carbon emissions are also incorporated. When the total annual carbon emissions exceed the prescribed upper limit, the penalty term imposes a carbon penalty cost, thereby incentivizing the high-speed rail power system to prioritize low-carbon operations. Conversely, if the annual emissions remain below the threshold, the penalty value is 0. Equation (15) is presented below. Figure 4:1  
This study sets the annual carbon emission upper limit (\(\text{Carbon}_{\text{Limit}}\)) to 2,020,321 kg, derived from the 70 th percentile of the annual carbon emission data for high-speed rail power systems relying solely on grid energy. The function \(\sigma(x)\) represents the Sigmoid function, employed to smooth the penalty term. The parameter \(\tau\) denotes a normalization coefficient, set to 1000 in this study, to regulate the sensitivity of the penalty term. \(W_{\text{Carbon}}\) represents the penalty weight (set to \(10^{5}\) here), determined through grid search. A high penalty weight ensures strict penalties for excessive emissions, aligning with dual-carbon policy goals of China.