+ + +First Model Baseline# + +Sets# + + +Name +Dimension +Sub index +Description + + + +\(T\) +1 + +time + +\(H\) +1 + +Hydro powerplants + +\(B\) +1 + +Water basins + +\(S\_H\) +2 +\(H\) +State of each hydro powerplants + +\(S\_B\) +2 +\(B\) +State of each basins + +\(S\_BH\) +4 +\(\{ H; ~B; ~S\_H \;~S\_B \}\) +Index to make the correspondence between the states, basins and hydro powerplants + + + + + + +Variables# + + +Name +Description +Type +Units + + + +\(V_\text{BAS}^{t,~b}\) +Basins actual water volume +\(\mathbb{R}^{+}\) +\(\mathrm{m}^{3}\) + +\(V_\text{SPIL}^{t,~b}\) +Spilled water volumes when basins are full +\(\mathbb{R}^{+}\) +\(\mathrm{m}^{3}\) + +\(S_\text{BAS}^{t,~b,~s\_b}\) +The state of a basin derived from its volume: \(S_\text{BAS}^{t, b, s\_b} = \begin{cases} 1 \text{ if } V_\text{BAS}^{t, b} \in \left[ V_\text{BAS, MIN}^{b,~s\_b} ;V_\text{BAS, MAX}^{b,~s\_b} \right] \\ 0 \text{ otherwise} \end{cases}\) +\(\mathbb{B}\) + + +\(P_\text{TUR}^{t,~h}\) +Electrical power produced by a turbine +\(\mathbb{R}^{+}\) +\(\mathrm{MW}\) + +\(Q_\text{TUR}^{t,~h}\) +Amount of water that flows through a turbine +\(\mathbb{R}^{+}\) +\(\mathrm{m}^{3}/\mathrm{s}\) + +\(P_\text{PUM}^{t,~h}\) +Electrical power produced by a pump +\(\mathbb{R}^{+}\) +\(\mathrm{MW}\) + +\(Q_\text{PUM}^{t,~h}\) +Amount of water that flows through a pump +\(\mathbb{R}^{+}\) +\(\mathrm{m}^{3}/\mathrm{s}\) + +\(V_\text{BAS, S}^{t,~b,~s\_b}\) +Basin actual volumes for each state +\(\mathbb{R}^{+}\) +\(\mathrm{m}^{3}\) + +\(Q_\text{TUR, S}^{t,~h,~s\_h}\) +Amount of water that flows through a turbine for each state +\(\mathbb{R}^{+}\) +\(\mathrm{m}^{3}/\mathrm{s}\) + +\(Q_\text{PUM, S}^{t,~h,~s\_h}\) +Amount of water that flows through a pump for each state +\(\mathbb{R}^{+}\) +\(\mathrm{m}^{3}/\mathrm{s}\) + + + + + + +Parameters# + + +Name +Description +Type +Default value +Units + + + +\(c_\text{DA}^{t}\) +Day-ahead energy market price +\(\mathbb{R}\) + +\(\mathrm{EUR}/\mathrm{MW}\) + +\(nb_\text{SEC}\) +Number of seconds in one hour +\(\mathbb{R}^{+}\) +3600 + + +\(nb_\text{HOUR}^{t}\) +Number of hours for each time-step +\(\mathbb{R}^{+}\) + + + +\(V_\text{BAS, START}^{b}\) +Basins stating water volume +\(\mathbb{R}^{+}\) +0 +\(\mathrm{m}^{3}\) + +\(V_\text{BAS, MAX}^{b,~s\_b}\) +Basins water volume upper boundary for each state +\(\mathbb{R}^{+}\) +0 +\(\mathrm{m}^{3}\) + +\(V_\text{BAS, MIN}^{b,~s\_b}\) +Basins water volume lower boundary for each state +\(\mathbb{R}^{+}\) +0 +\(\mathrm{m}^{3}\) + +\(V_\text{BAS, DIS}^{t,~b}\) +Water volume entering a basin due to runoff and snowmelt +\(\mathbb{R}^{+}\) +0 +\(\mathrm{m}^{3}\) + +\(F_\text{TUR}^{b,~h}\) +Factor to specify whether the turbined water of a plant is added into a basin or removed +\(\in \{-1; ~0; ~1\}\) +0 +\(\mathrm{m}^{3}/\mathrm{s}\) + +\(F_\text{PUM}^{b,~h}\) +Factor to specify whether the pumped water of a plant is added into a basin or removed +\(\in \{-1; ~0; ~1\}\) +0 +\(\mathrm{m}^{3}/\mathrm{s}\) + +\(Q_\text{TUR, MAX}^{h,~s\_h}\) +Max water flows that can be turbined for each state +\(\mathbb{R}^{+}\) +0 +\(\mathrm{m}^{3}/\mathrm{s}\) + +\(Q_\text{PUM, MAX}^{h,~s\_h}\) +Max water flows that can be pumped for each state +\(\mathbb{R}^{+}\) +0 +\(\mathrm{m}^{3}/\mathrm{s}\) + +\(\alpha_\text{TUR, AVG}^{h,~s\_h}\) +Average factor for converting turbined water flow into generated electrical power for each state. +\(\mathbb{R}^{+}\) +0 +\(\mathrm{MW} \cdot \mathrm{s}/ \mathrm{m}^{3}\) + +\(\alpha_\text{PUM, AVG}^{h,~s\_h}\) +Average factor for converting pumped water flow into consumed electrical power for each state. +\(\mathbb{R}^{+}\) +0 +\(\mathrm{MW} \cdot \mathrm{s}/\mathrm{m}^{3}\) + + + + + + +Objective# + + +Constraints# + +Water basin volume evolution# + +(1)#\[\begin{split}\begin{align} + V_\text{BAS}^{t,~b} = + \begin{cases} + &V_\text{BAS, START}^{b} & \text{if } t = t_0 \\ + &V_\text{BAS}^{t - 1,~b} - V_\text{SPIL}^{t - 1,~b} + nb_\text{SEC} \cdot nb_\text{HOUR}^{t-1} \cdot + \sum_{h \in H} \left( + F_\text{TUR}^{b,~h} \cdot Q_\text{TUR}^{t-1,~h} + + F_\text{PUM}^{b,~h} \cdot Q_\text{PUM}^{t-1,~h} + \right) \quad & \text{if } t \neq t_0 + \end{cases} +\end{align}\end{split}\] + +(2)#\[V_\text{BAS, START}^{b} = V_\text{BAS}^{t_{end},~b} - V_\text{SPIL}^{t_{end},~b} + +nb_\text{SEC} \cdot nb_\text{HOUR}^{t_{end}} \cdot + \sum_{h \in H} \left( + F_\text{TUR}^{b,~h} \cdot Q_\text{TUR}^{t_{end},~h} + + F_\text{PUM}^{b,~h} \cdot Q_\text{PUM}^{t_{end},~h} + \right)\] + + +Water basin state# + +(3)#\[V_\text{BAS}^{t,~b} \leq V_\text{BAS, MAX}^{b,~s\_b} + V_\text{BAS, MAX}^{b,~S\_B_\text{MAX}\{b\}} +\cdot \left(1 -S_\text{BAS}^{t,~b,~s\_b} \right)\] + +(4)#\[V_\text{BAS}^{t,~b} \geq V_\text{BAS, MIN}^{b,~s\_b} \cdot S_\text{BAS}^{t,~b,~s\_b}\] + +(5)#\[\sum_{s \in S\_B\{b\}} S_\text{BAS}^{t,~b,~s} = 1\] + +(6)#\[V_\text{BAS, S}^{t,~b,~s\_b} = V_\text{BAS}^{t,~b} \cdot S_\text{BAS}^{t,~b,~s\_b}\] +The constraint (6) involves the multiplication of a floating-point variable and a binary variable. To +address this, the constraint requires the application of a Big M Linearization technique. + + +Water turbined# + +(7)#\[Q_\text{TUR, S}^{t,~h,~s\_h} \leq Q_\text{TUR, MAX}^{h,~s\_h} \cdot S_\text{BAS}^{t,~b,~s\_b}\] + +(8)#\[Q_\text{TUR}^{t,~h} = \sum_{s \in S\_H\{h\}} Q_\text{TUR, S}^{t,~h,~s}\] + +(9)#\[P_\text{TUR}^{t,~h} = \sum_{s \in S\_H\{h\}} \alpha_\text{TUR, AVG}^{h,~s} \cdot Q_\text{TUR, S}^{t,~h,~s}\] +The constraint (7) takes the set \(S\_BH\) as argument, enabling the connection between the +basin set \(B\) and the hydro powerplant set \(H\). + + + + + +
+ + +SmallFlex documentation# + +User Manual# +This section outlines the steps a user can follow to generate a new database with updated data. + + +First Model Baseline +Sets +Variables +Parameters +Objective +Constraints + + + + + + + + +