# Learning-Based Branching Acceleration for Unit Commitment with Few Training Samples
## 2 Formulation of Unit Commitment Problem
### Constraints  
This subsection describes the constraints that must be satisfied in minimizing the total cost of a UC problem.  
1. System power balance constraints: \[\sum\limits_{i=1}^{N}P_{i,t}-P_{t}^{\text{D}}=0,\] (2) where \(\sum\limits_{i=1}^{N}P_{i,t}\) is the total unit output power at the period \(t\), and \(P_{t}^{\text{D}}\) is the total load of the system at the period \(t\).
2. Generation capacity constraints: \[u_{i,t}\underline{P}_{i}\ \leq P_{i,t}\ \leq u_{i,t}\overline{P}_{i},\] (3) where \(\overline{P}_{i}\) is the maximum output power of the unit \(i\); \(\underline{P}_{i}\) is the minimum output power of the unit \(i\); \(u_{i,t}\) is the on/off status of the unit \(i\) at the period \(t\).
3. Ramp-up/down rate constraints: \[P_{i,t}-P_{i,(t-1)}\leq R_{i}^{\text{u}}(1-y_{i,t})+\overline{P}_{ i}y_{i,t},\] (4) \[-P_{i,t}+P_{i,(t-1)}\leq R_{i}^{\text{d}}(1-z_{i,t})+\overline{P} _{i}z_{i,t},\] (5)where \(R_{i}^{u}\) and \(R_{i}^{d}\) are the ramp-up rate and ramp-down rate of the unit \(i\), respectively. The startup/shutdown ramp rate only needs to meet the generation capacity constraints.  
4. Dependency of binary variables: \[u_{i,t}-u_{i,t-1}=y_{i,t}-z_{i,t}.\] (6)  
When the \(u_{i,t}\) changes, the correct values of \(y_{i,t}\) and \(z_{i,t}\) can be ensured. For example, when \(u_{i,t}=1\) and \(u_{i,t-1}=0\), \(y_{i,t}\) and \(z_{i,t}\) are forced to be set to 1 and 0, respectively.  
5. Minimum up/down time constraints: \[\sum_{k=\max(t-\underline{T}_{i}^{\text{off}}+1,1)}^{t}y_{i,k}\leq u_{i,t},t \in[G_{i}+1,T],\] (7) \[\sum_{k=\max(t-\underline{T}_{i}^{\text{off}}+1,1)}^{t}z_{i,k}\leq 1-u_{ i,t},t\in[L_{i}+1,T],\] (8)  
where \(G_{i}\) is the number of periods during which the unit \(i\) must continue to operate from the initial period, \(G_{i}=\max\{0,\min[T,u_{i,0}(\underline{T}_{i}^{\text{on}}-T_{i,0})]\}\); \(L_{i}\) is the number of periods during which the unit \(i\) must remain continuously shut down from the initial period, \(L_{i}=\max\{0,\min\left[T,(1-u_{i,0})\left(\underline{T}_{i}^{\text{off}}-T_ {i,0}\right)\right]\}\); \(u_{i,0}\) is the on/off status of the unit \(i\) at the initial period; \(T_{i,0}\) is the initial continuous startup time or continuous shutdown time of the unit \(i\); \(\underline{T}_{i}^{\text{on}}\) is the minimum continuous startup time of the unit \(i\), and \(\underline{T}_{i}^{\text{off}}\) is the minimum continuous shutdown time of the unit \(i\).  
6. Hydropower plant generation capacity constraints: \[\underline{P}_{n}^{\text{Hpp}}\leq\sum_{i=1}^{N_{n}^{\text{Hpp}}}P_{n,i,t}^ {\text{Hydro}}\leq\overline{P}_{n}^{\text{Hpp}},\] (9)  
where \(P_{n,i,t}^{\text{Hydro}}\) is the output power of the hydro unit \(i\) at the period \(t\) in the hydropower plant \(n\); \(\overline{P}_{n}^{\text{Hpp}}\) is the maximum output power of the hydropower plant \(n\) at each period; \(\underline{P}_{n}^{\text{HPP}}\) is the minimum output power of the hydropower plant at each period; \(N_{n}^{\text{HPP}}\) is the total number of hydropower units in the hydropower plant \(n\).  
7. Hydropower plant daily generation constraints: \[\Delta t\sum_{t=1}^{T}\sum_{i=1}^{N_{n}^{\text{Hpp}}}P_{n,i,t}^{\text{Hydro} }\leq\overline{W}_{n}^{\text{HPP}},\] (10)  
where \(\overline{W}_{n}^{\text{HPP}}\) is the maximum daily generation of the hydropower plant \(n\); \(\Delta t\) is the unit's generation period in hours.