2. Second Stage
2.1. Sets
Name |
Dimension |
Sub index |
Description |
|---|---|---|---|
\(T\) |
1 |
time |
|
\(F\) |
1 |
Ancillary services flex time |
|
\(H\) |
1 |
Hydro powerplants |
|
\(CH\) |
1 |
Continous hydro powerplants |
|
\(DH\) |
1 |
On/off Hydro powerplants |
|
\(B\) |
1 |
Water basins |
|
\(S\) |
1 |
States |
|
\(TF\) |
1 |
\(\{ T; ~F \}\) |
Index to make correspondance between time and flex time |
\(S_H\) |
2 |
\(H\) |
State of each hydro powerplants |
\(S_B\) |
2 |
\(B\) |
State of each basins |
\(HS\) |
2 |
\(\{ H; ~S \}\) |
Index to make the correspondence between the states and hydro powerplants |
\(BS\) |
2 |
\(\{ B; ~S \}\) |
Index to make the correspondence between the states and basins |
\(HBS\) |
3 |
\(\{ H; ~B; ~S \}\) |
Index to make the correspondence between the states, basins and hydro powerplants |
2.2. Variables
Name |
Description |
Type |
Units |
|---|---|---|---|
\(V_\text{BAS}^{t,~b}\) |
Basins actual water volume |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
\(V_\text{END}^{t,~b}\) |
Basins water volume at the end of the optimization |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
\(V_\text{SPIL}^{t,~b}\) |
Spilled water volumes when basins are full |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
\(State^{t,~b,~s}\) |
The state of a basin derived from its volume: \(State^{t, b, s} = \begin{cases} 1 \text{ if } V_\text{BAS}^{t, b} \in \left[ V_\text{MIN}^{b,~s_b} ;V_\text{MAX}^{b,~s_b} \right] \\ 0 \text{ otherwise} \end{cases}\) |
\(\mathbb{B}\) |
|
\(Run_\text{HYDRO}^{t,~h}\) |
Binary variable to indicates if ON/OFF hydro power plant is running or not |
\(\mathbb{B}\) |
|
\(P_\text{HYDRO}^{t,~h}\) |
Electrical power produced or consumed by a hydro power plant |
\(\mathbb{R}\) |
\(\mathrm{MW}\) |
\(Q^{t,~h}\) |
Amount of water that flows through an hydro power plant |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}/\mathrm{s}\) |
\(Q_\text{S}^{t,~h,~s}\) |
Amount of water that flows through a hydro power plant for each corresponding basin state |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}/\mathrm{s}\) |
\(P_\text{ANC}^{f}\) |
Flexible power sell to the ancilliary market |
\(\mathbb{R}^{+}\) |
\(\mathrm{MW}\) |
\(V_\text{QUOTA}^{h}\) |
Water volume quota that can flow through an hydro power plant (from first stage optimization) |
\(\mathbb{R}\) |
\(\mathrm{m}^{3}\) |
\(dV_\text{+}^{h}\) |
Overage water volume flowing through an hydro power plant compared to the first stage volume quota |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
\(dV_\text{-}^{h}\) |
Shortage water volume flowing through an hydro power plant compared to the first stage volume quota |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
2.3. Parameters
Name |
Description |
Type |
Default |
Units |
|---|---|---|---|---|
\(c_\text{DA}^{t}\) |
Day-ahead energy market price |
\(\mathbb{R}\) |
\(\mathrm{EUR}/\mathrm{MW}\) |
|
\(c_\text{FLEX}^{f}\) |
Primary frequency control market price |
\(\mathbb{R}\) |
\(\mathrm{EUR}/\mathrm{MW}\) |
|
\(nb_\text{HOUR}^{t}\) |
Number of hours for each time-step |
\(\mathbb{R}^{+}\) |
||
\(V_\text{START}^{b}\) |
Basins stating water volume |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
|
\(V_\text{MAX}^{b,~s}\) |
Basins water volume upper boundary for each state |
\(\mathbb{R}^{+}\) |
0 |
\(\mathrm{m}^{3}\) |
\(V_\text{MIN}^{b,~s}\) |
Basins water volume lower boundary for each state |
\(\mathbb{R}^{+}\) |
0 |
\(\mathrm{m}^{3}\) |
\(V_\text{DIS}^{t,~b}\) |
Water volume entering a basin due to runoff and snowmelt |
\(\mathbb{R}^{+}\) |
0 |
\(\mathrm{m}^{3}\) |
\(F_\text{HYDRO}^{b,~h}\) |
Factor to specify whether the water flowing through a hydro power plant is added into a basin or removed |
\(\{-1; ~0; ~1\}\) |
0 |
|
\(Q_\text{MAX}^{h,~s}\) |
Max water that can flows through an hydro power plant for each state |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}/\mathrm{s}\) |
|
\(\alpha^{h,~s}\) |
Average factor for converting water flowing through a hydro power plant into power for each state (positive for turbines and negative for pumps). |
\(\mathbb{R}\) |
\(\mathrm{MW} \cdot \mathrm{s}/ \mathrm{m}^{3}\) |
|
\(F_\text{dV +}^{h,~s}\) |
Factor multipling the overage water volume in the objective penalty (positive for turbine negative for pump) |
\(\mathbb{R}\) |
\(\mathrm{EUR} / \mathrm{m}^{3}\) |
|
\(F_\text{dV -}^{h,~s}\) |
Factor multipling the shortage water volume in the objective penalty (positive for turbine negative for pump) |
\(\mathbb{R}\) |
\(\mathrm{EUR} / \mathrm{m}^{3}\) |
|
\(V_\text{BUF +}^{h}\) |
Maximmum overage water volume flowing through an hydro power plant |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
|
\(V_\text{BUF -}^{h}\) |
Maximmum shortage water volume flowing through an hydro power plant |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
|
\(M\) |
Arbitrary big number for big_m decomposition |
\(\mathbb{R}^{+}\) |
1E6 |