# 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  
#### 2.2.1. Carnot Battery Model  
The principle of the Carnot battery is based on thermal energy storage technology. It involves retrofitting existing decommissioned units by replacing the coal-fired boiler with a thermal storage tank, allowing the system to absorb excess renewable energy. The Carnot battery provides not only flexibility in energy storage but also inertia support for the system. Essentially, the Carnot battery remains a system primarily used for storing electrical energy. During the charging process, electrical energy is used to transfer heat from a low-temperature storage container to a high-temperature one, which can be achieved using traditional heat pumps, electric heaters, or other heating technologies. During discharge, heat can be cycled back into the thermoelectric generator through thermodynamic cycles such as the Rankine or Brayton cycle. The operational model of the Carnot battery is detailed in Equations (1)-(9) [30,31].  
\[P_{cb}^{\min}\leq P_{cb}\leq S_{c}-\overline{S}_{c} \tag{1}\]  
\[0\leq P_{cb,t}^{ch}\leq P_{cb,t}^{ch,\max} \tag{2}\]  
\[Q_{cb,t}^{ch}=\lambda_{cb}^{ch}\cdot P_{cb,t}^{ch} \tag{3}\]  
\[Q_{cb,t}^{dis}=\frac{P_{cb,t}^{dis}}{\lambda_{cb}^{dis}} \tag{4}\]  
\[Q_{cb,t+1}=Q_{cb,t}+\eta_{cb}^{ch}\cdot Q_{cb,t}^{ch}-\frac{Q_{cb,t}^{dis}}{ \eta_{cb}^{dis}}-\eta_{cb}^{self}\cdot Q_{cb,t} \tag{5}\]  
\[Q_{cb}^{plan}=b_{cb}\cdot Q_{cb}^{dis} \tag{6}\]  
\[Q_{cb}^{\min}\leq Q_{cb,t}\leq Q_{cb}^{plan} \tag{7}\]  
\[\lambda_{cb}^{\min}\cdot P_{cb}\leq P_{cb,t}^{dis}\leq P_{cb} \tag{8}\]  
\[-R_{cb}^{RD}\cdot\Delta t\leq P_{cb,t}^{dis}-P_{cb,t-1}^{dis}\leq R_{cb}^{RU} \cdot\Delta t \tag{9}\]  
where \(P_{cb}^{\min}\) represents the minimum transformation capacity of the Carnot battery, \(S_{c}\) and \(\overline{S}_{c}\) are the maximum installed capacities of coal-fired units before and during the  
Figure 1: Framework for coal power and renewable energy synergistic planning considering generalized adequacy.  
planning stage, \(P_{cb,t}^{ch}\) and \(P_{cb,t}^{dis}\) represent the charging and discharging power of the Carnot battery at time \(t\), \(P_{cb}^{ch,\max}\) and \(P_{cb}\) represent the maximum charging and discharging power of the Carnot battery, \(X_{cb}^{ch}\) and \(X_{cb}^{dis}\) are the efficiencies of the electrical-to-thermal and thermal-to-electrical conversions in the Carnot battery, \(Q_{cb,t}^{ch}\), \(Q_{cb,t}^{dis}\) and \(Q_{cb,t}\) represent the thermal storage and discharge power and the current thermal storage at time \(t\), \(\eta_{cb}^{ch}\), \(\eta_{cb}^{dis}\) and \(\eta_{cb}^{self}\) represent the thermal storage, discharge efficiencies, and self-discharge rate of the Carnot battery, \(Q_{cb}^{dis}\) and \(Q_{cb}^{plan}\) represent the maximum thermal discharge capacity and the planned thermal storage capacity of the Carnot battery, \(b_{cb}\) represents the maximum thermal discharge duration, \(\alpha_{cb}^{min}\) represents the minimum discharge coefficient of the Carnot battery, \(R_{cb}^{RD}\) and \(R_{cb}^{RU}\) represent the upward/downward ramp rates of the Carnot battery, and \(\Delta t\) represents the time interval.