# Synergistic Optimization of Coal Power and Renewable Energy Based on Generalized Adequacy
## 2 Generalized Adequacy Theory and Coal Power Transition Modeling
### Coal Power Transition Modeling
#### 2.2.2 Carbon Capture and Reduction Model  
Currently, coal-fired power generation remains dominant, and carbon reduction is imperative. However, it is not feasible to retire a large number of coal-fired plants in a short time, as they are still needed to ensure a reliable power supply for the safe operation of the power system. Therefore, technologies that help coal-fired power plants reduce carbon emissions have been developed. Carbon capture, utilization, and storage is a promising low-carbon technology crucial for the development of coal power. Additionally, captured high-concentration CO\({}_{2}\) can be processed and converted into methane for economic benefits. The operational model of the CCUS is detailed in Equations (10)-(16) [32].  
\[rt^{CCUS,\min}\leq rt^{CCUS}\leq rt^{CCUS,\max} \tag{10}\]  
where \(rt^{CCUS}\) represents the proportion of coal-fired units equipped with carbon capture devices, and \(rt^{CCUS,\min}\) and \(rt^{CCUS,\max}\) represent the minimum and maximum proportions of carbon capture device installations during the planning stage, which can be determined based on regional development levels and policy requirements.  
The maximum amount of CO\({}_{2}\) captured by the carbon capture device depends on the maximum capture capacity of the units equipped with carbon capture devices and the carbon emissions of coal-fired units in the system, which is expressed as:  
\[K_{tan,t}^{CCUS,\max}=\theta^{CCUS}\cdot\min\{P_{c,t}\cdot k_{c},rt^{CCUS} \cdot\overline{S}_{c}\cdot k_{c}\} \tag{11}\]  
\[0\leq K_{tan,t}^{CCUS}\leq K_{tan,t}^{CCUS,\max} \tag{12}\]  
where \(K_{tan,t}^{CCUS,\max}\) represents the maximum amount of CO\({}_{2}\) that can be captured by the carbon capture device, \(K_{tan,t}^{CCU}\) represents the amount of CO\({}_{2}\) captured by the carbon capture device at time \(t\), \(\theta^{CCUS}\) represents the maximum CO\({}_{2}\) capture rate of coal-fired units, \(P_{c,t}\) represents the output of the cluster units at time \(t\), and \(k_{c}\) is the average carbon emission coefficient of the cluster units.  
The energy consumption generated by installing carbon capture devices is as follows:  
\[P_{CS,t}=P_{CS}+e\cdot K_{tan,t}^{CCUS} \tag{13}\]  
where \(P_{CS,t}\) represents the power consumption of the carbon capture device at time \(t\), \(P_{CS}\) represents the fixed power consumption of the carbon capture device, and \(e\) represents the energy consumption per unit of captured CO\({}_{2}\).  
The resulting modification costs are as follows:  
\[C_{CCUS}=c_{CCUS}^{inv}\cdot rt^{CCUS}\cdot\overline{S}_{c} \tag{14}\]  
where \(c_{CCUS}^{inv}\) represents the unit investment price of the carbon capture, utilization, and storage device.  
The CO\({}_{2}\) generated can be treated using power-to-gas (P2G) technology, which converts CO\({}_{2}\) into CH\({}_{4}\), thereby generating economic benefits.  
\[K_{tun,t}^{P2G}=\psi_{CH_{4}}K_{tun,t}^{CCUs} \tag{15}\]  
\[C_{CH_{4}}=c_{CH_{4}}^{sale}\cdot\sum\limits_{t\in T}K_{tun,t}^{P2G} \tag{16}\]  
where \(K_{tun,t}^{P2G}\) denotes the amount of methane converted at time \(t\), \(\psi_{CH_{4}}\) represents the efficiency of the P2G technology in converting CO\({}_{2}\) to CH\({}_{4}\), \(C_{CH_{4}}\) signifies the economic benefits of converting CO\({}_{2}\) to methane, and \(c_{CH_{4}}^{sale}\) indicates the unit price of CH\({}_{4}\).