## 5 Optimization and Scheduling Model of the Complementary System
### Equation Constraints  
1. Water balance: Considering the flow balance between reservoirs, the flow into the lower reservoirs includes the discharge of the upstream reservoirs and the natural inflow of the interval. The reservoir capacity of the \(i\)-th hydropower station at the end of the \(t\)-th period is as follows: \[ \begin{array}{l}v(i,t)=v(i,t-1)+I(i,t)-[s(i,t)+q(i,t)]\times\Delta T+q_{p}( i,t)\times\Delta T\\ \qquad\qquad\qquad\qquad\qquad+\sum\limits_{k=1}^{N_{u}(i)}[s(k,t)+q(k,t)] \times\Delta T\end{array} \] (18) where \(v(i,t)\) is the capacity of the \(i\)-th hydropower station at the end of the \(t\)-th period, \(10^{4}\) m\({}^{3}\); \(I(i,t)\) is the natural incoming water flow, m\({}^{3}\)/s; \(s(i,t)\) is the abandoned water flow, m\({}^{3}\)/s; \(q(i,t)\) is the power generation flow, m\({}^{3}\)/s; \(q_{p}(i,t)\) is the pumping water flow, m\({}^{3}\)/s; \(N_{u}(i)\) is the number of hydropower stations connected to the hydropower station upstream; \(s(k,t)\) is the abandoned water flow by upstream stations, m\({}^{3}\)/s; \(q(k,t)\) is the power generation flow by upstream stations, m\({}^{3}\)/s; \(\Delta T\) is the conversion factor of flow and capacity, \(\Delta T=0.36\). 2. Capacity constraints: \[ \begin{array}{l}v(i,0)=v(i)^{begin}\\ v(i,T)=v(i)^{end}\end{array} \] (19) where \(v(i)^{begin}\) is the initial capacity of the \(i\)-th hydropower station, \(10^{4}\) m\({}^{3}\); \(v(i)^{end}\) is the final capacity of the \(i\)-th hydropower station, \(10^{4}\) m\({}^{3}\). 3. Water head constraints: \[ \begin{array}{l}h(i,t)=h_{f}(i,t)-h_{b}(i,t)-\Delta h(i,t)\end{array} \] (20) where \(h(i,t)\) is the water head of the \(i\)-th hydropower station in the \(t\)-th period, m; \(h_{f}(i,t)\) is the pre-dam water level, m; \(h_{b}(i,t)\) is the tailwater level, m; \(\Delta h(i,t)\) is the head loss, m. The \(h_{f}(i,t)\) can be calculated using the following equation: \[ \begin{array}{l}h_{f}(i,t)=a_{0}+a_{1}v(i,t)+a_{2}v^{2}(i,t)\end{array} \] (21) The pre-dam water level-storage capacity coefficient of the reservoir is a constant, \(a_{i}\), \(i=0,1,2\), which is typically fitted using the corresponding data of actual hydropower station water levels and storage capacities. The \(h_{b}(i,t)\) can be calculated using the following equation: \[ \begin{array}{l}h_{b}(i,t)=b_{0}+b_{1}[s(i,t)+q(i,t)]+b_{2}[s(i,t)+q(i,t)]^{2 }\end{array} \] (22)The tailwater level-outflow coefficient of the reservoir is a constant, \(b_{i}\), \(i=0,1,2\), which is typically fitted using the corresponding data of actual hydropower station tailwater levels and outflow. 4. Generating power: \[P_{g}(i,t)=A\eta h(i,t)q_{g}(i,t)\] (23) where \(A\) is the conversion coefficient of power and water; \(\eta\) is the efficiency of hydropower units. 5. Pumping power: \[P_{p}(i,t)=\frac{q_{p}(i,t)gh_{p}(i,t)}{\eta_{p}(i)}\] (24) where \(h_{p}(i,t)\) is the pumping head of the \(i\)-th pumping unit in the \(t\)-th period, m; \(\eta_{p}(i)\) is the pumping operation efficiency of the \(i\)-th pumping unit, which typically ranges from 0.75 to 0.85. Since the value of \(\eta_{p}(i)\) is less than 1, it reflects the existence of energy losses during pumping operations. \(g\) is the local gravity acceleration. 6. Complementary constraints of pumping/power generation: The system cannot be in the pumping and power generation modes at the same time: \[q_{g}(i,t)q_{p}(i,t)=0\] (25)
7. The relationship between wind power and wind speed is expressed as follows: \[P_{w}(t)= \begin{cases}0&v\leq v_{in},v\geq v_{out}\\ P_{E}\frac{v-v_{in}}{v_{r}-v_{in}}&v_{in}\leq v\leq v_{r}\\ P_{E}&v_{r}\leq v\leq v_{out}\end{cases} \] (26) where \(P_{w}(t)\) is the output of the wind power in the \(t\)-th period, kW; \(P_{E}\) is the rated power output of the wind turbine generator, kW; \(v\) is the pmeasured wind speed, m/s; \(v_{in}\) is the cut-in wind speed, m/s; \(v_{out}\) is the cut-out wind speed, m/s; \(v_{r}\) is the rated wind speed, m/s. 8.