## 2 Comprehensive Hydrogen-Integrated Energy System Model
### Tiered Carbon Trading Model  
The tiered carbon trading approach, similar to tiered electricity pricing, seeks to gradually motivate consumers to decrease carbon emissions by establishing varying levels of carbon costs. This approach utilizes a tiered carbon trading system that incorporates gratuitous quotas. If the system's carbon emissions fall below the assigned quota, any excess carbon allowances can be sold on the trading market to generate profit. If, on the other hand, the carbon emissions of the system surpass the assigned limit, the corresponding carbon allowances must be bought back. The formula for computing carbon allowances during a schedule cycle is as follows:  
\[E_{\rm C}=\sum_{t=1}^{T}\lambda_{\rm c2e}P_{\rm cchp}(t)+\lambda_{\rm c2h}H_{ \rm cchp}(t)+\lambda_{\rm GB}H_{\rm gb}(t)+\lambda_{\rm e}P_{\rm buy}(t)+ \lambda_{\rm q}Q_{\rm buy}(t) \tag{6}\]  
where \(E_{\rm C}\) represents the carbon quota within a scheduling cycle; \(\lambda_{\rm c2e}\) and \(\lambda_{\rm c2h}\) denote the allocated quotas for CCHP unit's unit electricity and thermal power, respectively; \(\lambda_{\rm GB}\) represents the allocated quota for the gas boiler's unit thermal power; \(\lambda_{\rm e}\) and \(\lambda_{\rm q}\) represent the allocated quotas for purchased unit electricity and unit mass of gray hydrogen, respectively; \(P_{\rm cchp}(t)\) and \(H_{\rm cchp}(t)\) represent the electricity and heat power generated by the CCHP unit at time \(t\); \(H_{\rm gb}(t)\) represents the heat power generated by the gas boiler during time interval \(t\); \(P_{\rm buy}(t)\) and \(Q_{\rm buy}(t)\) denote the purchased electricity power and the mass of purchased gray hydrogen at time \(t\), respectively.  
Actual Carbon Emission:  
\[M_{\rm real}(t)=M_{\rm total}(t)-M_{\rm CCS}(t) \tag{7}\]  
where \(M_{\rm real}(t)\) represents the actual carbon emissions of the system at time \(t\), and \(M_{\rm total}(t)\) represents the total carbon emissions generated by the system at time \(t\).  
The tiered carbon trading cost expression is as follows:  
\[C_{\text{p}}^{\text{CO}_{2}}(t)=\begin{cases}\chi[M_{\text{real}}(t)-E_{\text{C}} ]&M_{\text{real}}(t)-E_{\text{C}}\leq l\\ \chi(1+\varphi)[M_{\text{real}}(t)-E_{\text{C}}-l]+\chi l&l\leq M_{\text{real}}( t)-E_{\text{C}}\leq l\\ \chi(1+2\varphi)[M_{\text{real}}(t)-E_{\text{C}}-2l]+\chi(2+\varphi)l&2l\leq M _{\text{real}}(t)-E_{\text{C}}\leq 3l\\ \chi(1+3\varphi)[M_{\text{real}}(t)-E_{\text{C}}-3l]+\chi(3+3\varphi)l&3l\leq M _{\text{real}}(t)-E_{\text{C}}\leq 4l\\ \chi(1+4\varphi)[M_{\text{real}}(t)-E_{\text{C}}-4l]+\chi(4+6\varphi)l&M_{\text {real}}(t)-E_{\text{C}}\geq 4l\end{cases} \tag{8}\]  
where \(C_{\text{p}}^{\text{CO}_{2}}(t)\) represents the stepwise carbon trading cost at time \(t\); \(\chi\) represents the carbon trading base price; \(l\) represents the interval length; \(\varphi\) represents the rate of price increase after an increase in carbon emissions. This equation set represents the input values as the actual carbon emissions of the system. Based on the actual carbon emissions, it establishes five stages with incrementally increasing carbon emission prices. As the carbon emissions increase, the system costs also rise. Therefore, to enhance economic efficiency, the system automatically reduces carbon emissions, achieving a balance between economic viability and carbon emissions.