1. First Stage
1.1. Sets
Name |
Dimension |
Sub index |
Description |
|---|---|---|---|
\(T\) |
1 |
time |
|
\(H\) |
1 |
Hydro powerplants |
|
\(CH\) |
1 |
Continous hydro powerplants |
|
\(DH\) |
1 |
On/off Hydro powerplants |
|
\(B\) |
1 |
Water basins |
|
\(S\) |
1 |
States |
|
\(S_H\) |
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 |
The sets \(T\), \(S_H\) and \(S_B\) are arranged in a specific order such that indexing a variable with the first or last element of a set (i.e \(t^{0}\) and \(t^{\text{END}} \in T\)) corresponds respectively to the lowest/first or highest/final element of that variable.
\(S_H\) and \(S_B\) include subset to specify the corresponding hydro powerplants and basins that the state is associated with, respectively. The sets are constructed as demonstrated in the following example. To collect all states associated with a basin \(b\) we can use the notation \(S_B\{b\}\)
\(S_B =\begin{cases} 1: \left[1, 2, 3\right] \\ 2: \left[4, 5\right] \\ 3: \left[6\right] \end{cases}\)
In a Pyomo model, it is not possible to directly index variables and parameters using sets that contain subsets, such as \(S\_B\). To handle this limitation, we need to create new sets (i.e. \(BS\)) which will explicitly represent the deployment of these subsets. These new sets will be structured to map the relationships required for indexing variables and parameters in the model effectively.
\(SB \in \{b, s\}=\{(1,~1),~(1,~2),~(2,~3),~(2,~4),~(2,~5),~(3,~6)\}\)
The set \(HBS\) defines the connections between each basin, its corresponding state, and the hydro powerplants. This link is established solely between a hydro powerplant and its associated upstream basin. It is assumed that the water level in downstream basins does not affect the behavior of turbined or pumped energy.
\(HBS \in \{b, ~h, ~s\}=\{(1, ~1, ~1),~(1, ~1, ~2),~(1, ~1, ~3,)\}\)
1.2. 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 |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}\) |
\(State^{t,~b,~s}\) |
The state of a basin derived from its volume: \(State^{t, b, s} = \begin{cases} 1 \text{ if } V_\text{BAS}^{t, b} \in \left[ V_\text{MIN}^{b,~s_b} ;V_\text{MAX}^{b,~s_b} \right] \\ 0 \text{ otherwise} \end{cases}\) |
\(\mathbb{B}\) |
|
\(P_\text{HYDRO}^{t,~h}\) |
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 |
\(\mathbb{R}^{+}\) |
\(\mathrm{m}^{3}/\mathrm{s}\) |
\(P_\text{ANC}^{t}\) |
Flexible power sell to the ancilliary market |
\(\mathbb{R}^{+}\) |
\(\mathrm{MW}\) |
1.3. Parameters
Name |
Description |
Type |
Default |
Units |
|---|---|---|---|---|
\(c_\text{DA}^{t}\) |
Day-ahead energy market price |
\(\mathbb{R}\) |
\(\mathrm{EUR}/\mathrm{MW}\) |
|
\(c_\text{FLEX}^{t}\) |
Primary frequency control 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 |
\(\mathbb{R}^{+}\) |
||
\(V_\text{START}^{b}\) |
Basins stating water volume |
\(\mathbb{R}^{+}\) |
0 |
\(\mathrm{m}^{3}\) |
\(V_\text{MAX}^{b,~s}\) |
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 |
\(\mathbb{R}^{+}\) |
0 |
\(\mathrm{m}^{3}/\mathrm{s}\) |
\(\alpha^{h,~s}\) |
Average factor for converting water flowing through a hydro power plant into power for each state (positive for turbines and negative for pumps). |
\(\mathbb{R}\) |
0 |
\(\mathrm{MW} \cdot \mathrm{s}/ \mathrm{m}^{3}\) |