## Problem Description
### Unit Commitment Problem Formulation  
To obtain the power generation costs of the units, the cost of each unit is represented by a function. For example, the fuel cost varies along a curve according to the load of the unit at various time periods; the function of fuel cost is shown in Equation (1) [18].  
Relevant constraints are specified as follows:  
\[FC_{i,t}(P_{i,t})=\Big{(}\alpha_{i}+\beta_{i}P_{i,t}+\gamma_{i}P_{i,t}^{2} \Big{)}. \tag{1}\]  
\(i\): Unit number  
\(t\): Time period of electricity use  
\(FC_{i}\): Fuel cost for unit i ($/hour)  
\(\alpha_{i},\ \beta_{i},\gamma_{i}\): Parameters for the fuel cost equation  
\(P_{i,t}\): Power of the unit \(i\) at time \(t\)  
* Generation Limit  
All the units have a power output limit, and their power supply must be within the tolerance range to prevent overload or shutdown.  
\[P_{MIN,i,t}\leq P_{i,t}\leq P_{MAX,i,t} \tag{2}\]  
\(P_{MIN,i,t}\): Minimum Power of the unit \(i\) at time \(t\)  
\(P_{MAX,i,t}\): Maximum Power of the unit \(i\) at time \(t\)  
* Power Balance  
The total power output of all the units of the system must satisfy the load demands and system losses to prevent insufficient power supply.  
\[\big{(}\sum_{i=1}^{\text{m}}u_{i,t}P_{i,t}\big{)}-P_{D,t}=0 \tag{3}\]  
\(P_{D,t}\): Total system load capacity at time \(t\)  
\(u_{i,t}\): Start-up and shut-down situations of the unit \(i\) at time \(t\)  
* Reserve Capacity  
The power capacity reserve available for the system refers to the remaining available power capacity after load demands and losses have been subtracted from the maximum power output available for the system at a specific time. The power system must have sufficient spinning reserve to respond to insufficient power supply due to emergency incidents (such as tripping or power outage).  
\[R_{t}-\sum_{i=1}^{\text{m}}u_{i,t}P_{MAX,i,t}\leq 0 \tag{4}\]  
\(R_{t}\): Spinning reserve at time \(t\)  
* Startup Cost  
Each unit has a distinct startup cost according to its start-up and shut-down characteristics. A hot startup means that the unit is started up within an acceptable time interval since the last shutdown. In a hot startup situation, the unit can be started up again rapidly, leading to low startup cost. A cold startup is a startup that occurs after the acceptable time interval since the last shutdown. In a cold startup situation, the unit requires more time to be restarted, leading to high startup cost. The hot and cold startup costs are defined in Equations (5) and (6).  
\[SC_{i,t}=\left\{\begin{array}{c}h-cost_{i,t}:\ T_{i}^{off}\leq X_{i,t}^{off} \leq H_{i}^{off}\\ c-cost_{i,t}:\ X_{i,t}^{off}>H_{i}^{off}\end{array}\right.. \tag{5}\]  
\[H_{i}^{off}=\ T_{i,t}^{off}+cs\_hour \tag{6}\]  
\(SC_{i,t}\): Startup cost of the unit \(i\) at time \(t\)  
\(h\)-\(cost_{i,t}\): Hot startup cost of the unit \(i\) at time \(t\)  
\(c\)-\(cost_{i,t}\): Cold startup cost of the unit \(i\) at time \(t\)  
\(cs\_hour\): Cold startup hours(h)  
\(T_{i}^{off}\): Minimum shutdown hours of the unit \(i\)  
\(X_{i,t}^{off}\): Continuous shutdown hours of the unit \(i\)  
\(H_{i}^{off}\): Number of hours required for cold startup  
Accordingly, the objective function can be presented as Equation (7).  
\[\min\text{TC}=\sum_{t=1}^{m}\sum_{i=1}^{n}u_{i,t}(FC_{i,t}+SC_{i,t}). \tag{7}\]  
TC: Total Cost (including generation cost and startup cost)  
\(\succcuit\): Unit Ramp-up and Ramp-down Limit  
Each unit features different ranges of ramp up. To satisfy load changes, the maximum ramp-up (R\({}_{\text{iup}}\)) of a unit within 2 consecutive hours must be considered, as presented in Equation (8).  
\[P_{i,t}\leq P_{i,t-1}+\text{R}_{\text{iup}} \tag{8}\]  
R\({}_{\text{iup}}\): Ramp-up rate of unit i