# Synergistic Optimization of Coal Power and Renewable Energy Based on Generalized Adequacy
## 3 Coordinated Optimization Model of Coal Power and Renewable Energy
### Constraints  
To meet the requirements of constructing a new power system under the \"dual carbon\" goals, this study considers not expanding coal-fired units. The retirement of units is performed directly based on existing installed capacities, and coal-fired units are clustered for decision-making on startup, shutdown, and output within the cluster. The following constraints are considered:  
1. Coal Unit Retirement Constraints: \[\alpha_{c}^{retite}=\frac{S_{c}-\overline{S}_{c}}{S_{c}}\] (24) \[\alpha_{c}^{retite,max}\geq\alpha_{c}^{retite}\geq\alpha_{c}^{retite,min}\] (25)  
The retirement of coal units should proceed in an orderly fashion based on local development, with maximum and minimum retirement rates of \(\alpha_{c}^{retite,max}\) and \(\alpha_{c}^{retite,min}\). 2. Output and Installed Capacity Constraints of Different Units:  
Due to the startup and shutdown costs, coal units cannot frequently start and stop, unlike wind and photovoltaic units, which can operate at their maximum output capacity. \[0\leq P_{c,t}\leq\overline{P}_{c,t}\leq\overline{S}_{c},\forall t \tag{26}\]  
\[\overline{P}_{c,t}=n_{c,t}\times\frac{S_{c}\times(1-\alpha_{c}^{retite})}{N_{c} }=n_{c,t}\times\frac{\overline{S}_{c}}{N_{c}} \tag{27}\]  
\[\lambda_{c}^{\min}\cdot\overline{P}_{c,t}\leq P_{c,t}\leq\lambda_{c}^{\max} \cdot\overline{P}_{c,t},\forall t \tag{28}\]  
\[P_{c,t}\leq\lambda_{c}^{\max}\cdot(\overline{P}_{c,t}-S_{c,t}^{SU}-S_{c,t+1}^ {SU})+SU_{c}\cdot S_{c,t}^{SU}+SD_{c}\cdot S_{c,t+1}^{SD},\forall t \tag{29}\]  
\[0\leq P_{w,t}\leq\overline{P}_{w,t}=\alpha_{w,t}\overline{S}_{w},\forall t, \forall w\in G_{w} \tag{30}\]  
\[\overline{S}_{w}=S_{w}+S_{w}^{new},\forall w\in G_{w} \tag{31}\]  
\[0\leq P_{PV,t}\leq\overline{P}_{PV,t}=\sigma_{PV,t}\cdot\overline{S}_{PV}, \forall t,\forall PV\in G_{PV} \tag{32}\]  
\[\overline{S}_{PV}=S_{PV}+S_{PV}^{new},\forall w\in G_{w} \tag{33}\]where \(P_{c,t}\) denotes the output of coal at time \(t\), \(\overline{P}_{c,t}\) denotes the online capacity of clustered units, \(n_{c,t}\) represents the number of online coal units at time \(t\), \(N_{c}\) is the total number of clustered units, \(\lambda_{c}^{\min}\) and \(\lambda_{c}^{\max}\) are the minimum and maximum output levels of the units, respectively, \(SU_{c}\) and \(SD_{c}\) represent the ramping levels for startup and shutdown, \(SU_{c,t}\), \(SD_{c,t}\) represent the startup and shutdown capacity at time \(t\), \(P_{w,t}\), \(P_{PV,t}\) are the wind and photovoltaic outputs at time \(t\), \(\sigma_{w,t}\), \(\sigma_{PV,t}\) represent the hourly output factors for wind and photovoltaic, \(\overline{P}_{w,t}\), \(\overline{P}_{PV,t}\) represent the hourly online capacities for wind and photovoltaic, and \(S_{w}\) and \(S_{PV}\) represent the initial installed capacities of wind and photovoltaic units, respectively. (3) System Balance Constraint  
\[ \begin{array}{l}P_{c,t}+\sum\limits_{w\in G_{ub}}P_{w,t}+\sum\limits_{PV\in G _{PV}}P_{PV,t}+\sum\limits_{bnt\in G_{bat}}(P_{bat,t}^{dis}-P_{bat,t}^{ch})+\\ \sum\limits_{cb\in G_{cb}}(P_{cb,t}^{dis}-P_{cb,t}^{ch})+P_{loss,t}=D_{t}+P_{C5, t},\forall t\end{array}  \tag{34}\]  
where \(D_{t}\) is the actual load demand of the system at time \(t\).