## 2 Preliminaries
### Three-Binary UCP Formulations  
In this part, we introduce the classical three-binary (3-bin) formulation of the UCP from [9], i.e., commitment, startup, and shutdown variables.  
1. Objective Function: The objective function represents the total cost of the power system, which is shown as \[min{\sum_{g=1}^{N}\sum_{t=1}^{T}[\alpha_{g}u_{g,t}+\beta_{g}P_{g,t}+S_{g,t}]}\] (1) which comprises the production cost \(\alpha_{g}u_{g,t}+\beta_{g}P_{g,t}\) and the startup cost \(S_{g,t}\).
2. Startup cost: The startup cost \(S_{g,t}\) can be expressed as an MIP formulation in a 3-bin formulation, with \(s_{g,t}\) and \(d_{g,t}\). The cost is determined by comparing the time since the unit was last shut down to a threshold representing the cold-start duration. It is formulated as \[\left\{\begin{aligned} S_{g,t}\geq C_{hot,g}s_{g,t}\\ S_{g,t}\geq C_{cold,g}\Big{[}s_{g,t}-\sum_{t=max(t-\sum_{t,g,-}T_{ cold,g},1)}^{t-1}d_{i,\tau}-f_{init,g,t}\Big{]}\end{aligned}\right.\] (2) Since the UCP must account for the initial status of generating units, the classification of hot and cold starts in the early time periods may depend on residual states from the previous day. As a result, the standard rules for determining startup types cannot be uniformly applied across all time periods. In particular, when the following two conditions are satisfied, \(t-\underline{T}_{off,g}-T_{cold,g}\leq 0\) and \(max\big{(}-T_{g,0},0\big{)}<\left|t-\underline{T}_{off,g}-T_{cold,g}-1\right|+1\), we set \(f_{init,g,t}=1\), indicating that the startup at time \(t\) is considered a hot start due to the influence of the unit's initial state. Otherwise, \(f_{init,g,t}=0\), and the classification of the startup should instead be determined based on the summation term.
3. Unit generation limits: \[u_{g,t}P_{g}\leq P_{g,t}\] (3) \[P_{g,t}\leq u_{g,t}\overline{P}_{g}\] (4)
4. Power balance constraint: \[\sum_{g=1}^{N}P_{g,t}-P_{D,t}=0\] (5)
5. System spinning reserve requirement: \[\sum_{g=1}^{N}u_{g,t}\overline{P}_{g}\geq P_{D,t}+R_{t}\] (6)
6. Ramp rate limits: \[P_{g,t}-P_{g,t-1}\leq u_{g,t}\Big{(}P_{up,g}+\underline{P}_{g}\Big{)}-u_{g,t -1}\underline{P}_{g}+s_{g,t}\Big{(}P_{start,g}-P_{up,g}-\underline{P}_{g} \Big{)}\] (7)\[P_{g,t-1}-P_{g,t}\leq u_{g,t-1}\Big{(}P_{down,g}+\underline{P}_{g}\Big{)}-u_{g,t }\underline{P}_{g}+d_{g,t}\Big{(}P_{\text{shut},g}-P_{\text{down},g}-\underline {P}_{g}\Big{)} \tag{8}\]
7. \[\text{Minimum up/downtime constraints:}\] \[\sum_{\omega=\max(t-\sum_{m,g,0})+1}^{t}s_{g,\omega}\leq u_{g,t},t\in\big{[} U_{g}+1,\ldots,T\big{]}\] (9)  
\[\sum_{\omega=max(t-\sum_{m,g,0})+1}^{t}d_{g,\omega}\leq 1-u_{g,t},t\in\big{[} L_{g}+1,\ldots,T\big{]} \tag{10}\]  
Here, \(\omega\) is an index used to enforce the minimum up/downtime constraints. Taking Equation (9) as an example, if the unit is in the off state at time \(t\) (i.e., \(u_{g,t}=0\)), then any startup operation occurring at time \(\omega\) would imply that the unit shuts down before completing its required minimum uptime, thereby violating the constraint.
8. Initial status of units: \[u_{g,t}=u_{g,0},t\in\big{[}1,\ldots,U_{g}+L_{g}\big{]}\] (11)
9. State constraints: State constraints were introduced in the 3-bin formulation to link the startup, shutdown, and state variables: \[s_{g,t}-d_{g,t}=u_{g,t}-u_{g,t-1}\] (12)