## 4 Case Study
 \tag{6}\]  
Equation (9) is a continuity equation, where \(V_{m,t+1}\) is storage for the \(m\)th reservoir at time \(t+1\), \(Q_{m,t}\) is the outflow of the \(m\)th reservoir, which includes the turbine release and spillway release, \(IQ_{m+1,t}\) is the internal inflow into the \(m+1\) reservoir, \(\tau_{m}\) is the routed outflow from the \(m\)th reservoir into the \(m+1\) reservoir downstream, \(\eta\) is the conversion coefficient, and \(I_{m+1,t}\) is the inflow into the \(m+1\) reservoir. Equation (7) is  
\[ \begin{array}{l}F=\max\sum\limits_{t-1}^{T}\sum\limits_{m-1}^{M}N_{m,t} \Delta T\\ N_{m,t}=k_{m}q_{m}h_{m,t}\end{array}  \tag{7}\]  
where \(F\) is an objective function (kwh), \(T\) is the number of operational periods, \(M\) is the number of hydropower plants, \(N_{m,t}\) is the output hydropower for the \(m\)th hydropower plant (MW), \(\Delta T\) is the time interval, \(k_{m}\) is the hydropower plant coefficient, \(q_{m}\) is the release passing turbines, and \(l_{m,t}\) is the hydraulic head. The following constraints are considered for this problem [26]:  
\[\underline{N}_{m,t}\leq N_{m,t}\leq\overline{N}_{m,t}. \tag{8}\]  
Equation (8) represents the constraint for each hydropower plant, where \(\underline{N}_{m,t}\) is the lower and \(\overline{N}_{m,t}\) is the upper output for the \(m\)th hydropower plant. Equation (9) is  
\[N_{-t}\leq N_{m,t}\leq\overline{N}_{t}. \tag{9}\]  
Equation (9) represents the constraint related to the total output of the system, where \(\underline{N}_{t}\) and \(\overline{N}_{t}\) represent the lower and upper outputs of the system, respectively. Equation (10) is  
\[Z_{-mt}\leq Z_{m,t}\leq\overline{Z}_{mt}. \tag{10}\]  
Equation (10) is related to the water level of each reservoir, where \(Z_{m,t}\) and \(\overline{Z}_{m,t}\) represent the lower and upper levels of the reservoirs, respectively. Equation (11) is  
\[Q_{-mt}\leq Q_{m,t}\leq\overline{Q}_{mt} \tag{11}\]  
Equation (11) is considered to be one constraint for the outflow, where \(\underline{Q}_{m,t}\) and \(\overline{Q}_{m,t}\) represent the lower and upper outflows of the reservoirs, respectively. Equation (12) is  
\[Z_{m,1}=Z_{m,b},Z_{m,T+1}=Z_{m,e} \tag{12}\]Equation (12) is considered to be the boundary constraint, where \(Z_{m,b}\) is the water level at the beginning of the operation period, and \(Z_{m,e}\) is the water level at the end of the operation period. If the constraints are not satisfied, penalty functions are added to the objective function:  
\[F=-\sum\limits_{t=1}^{T}\left(\sum\limits_{m=1}^{M}N_{m,t}-\delta\left(\frac{N _{t}}{N_{t}}-\sum\limits_{m=1}^{M}N_{m,t}\right)^{2}\right)\Delta t, \tag{13}\]  
where \(\delta=\left[ \begin{array}{c}1,\sum\limits_{m=1}^{M}N_{m,t}\leq\underline{N _{t}}\\ 0,\sum\limits_{m=1}^{M}N_{m,t}\geq\underline{N_{t}}\end{array} \right]\). Equation (14) is  
\[N_{m,t}=\left[ \begin{array}{c}p_{1}\Big{(}q_{m,t}-\underline{q_{m,t}}\Big{)},q_{m,t}\in\left(-\infty,\underline{q}_{m,t}\right)\\ k_{m}q_{m,t}h_{m,t},q_{m,t}\in\left(\underline{q}_{m,t},\overline{q}_{m,t} \right)\\ p_{2}\Big{(}\overline{q}_{m,t}-q_{m,t}\Big{)},q_{m,t}\in\left(\overline{q}_{m,t},+\infty\right)\end{array} \right]. \tag{14}\]  
\(N_{m,t}\) represents the hydropower output for each period, and \(p_{1}\) and \(p_{2}\) are penalty coefficients. The following indexes are used to evaluate the hydropower output based on the CA [6]:  
1. The temporal reliability index. This index shows the number of periods where the generated hydropower is greater than the hydropower plant capacity. A high percentage for this index is suitable: \[TR=\frac{N_{t=1}^{T}(N_{m,t}\geq PPC_{m})}{T}\times 100,\] (15) where \(TR\) is the temporal reliability index, \(N_{t=1}^{T}(N_{m,t}\geq PPC_{m})\) is the number of periods where the generated hydropower is greater than the hydropower plant capacity, \(PPC_{m}\) is the hydropower plant capacity, and \(T\) is the total number of operational periods;
2. Volumetric reliability index.