Using integer variables for PST taps
Contents
- Used input data
- Used parameters
- Defined optimization variables
- Used optimization variables
- Defined constraints
Used input data
| Name | Symbol | Details |
|---|---|---|
| PstRangeActions | \(r \in \mathcal{RA}^{PST}\) | Set of PST RangeActions |
| reference angle | \(\alpha _n(r)\) | angle of PstRangeAction \(r\) at the beginning of the current iteration of the MILP |
| reference tap position | \(t_{n}(r)\) | tap of PstRangeAction \(r\) at the beginning of the current iteration of the MILP |
| PstRangeAction angle bounds | \(\underline{\alpha(r)} \: , \: \overline{\alpha(r)}\) | min and max angle1 of PstRangeAction \(r\) |
| PstRangeAction tap bounds | \(\underline{t(r)} \: , \: \overline{t(r)}\) | min and max tap1 of PstRangeAction \(r\) |
| tap-to-angle conversion function | \(f_r(t) = \alpha\) | Discrete function \(f\), which gives, for a given tap of the PstRangeAction \(r\), its associated angle value |
types of their ranges: ABSOLUTE, RELATIVE_TO_INITIAL_NETWORK, RELATIVE_TO_PREVIOUS_INSTANT (more information here)
Used parameters
| Name | Details |
|---|---|
| pst-optimization-approximation | This filler is used only if this parameters is set to APPROXIMATED_INTEGERS |
Defined optimization variables
| Name | Symbol | Details | Type | Index | Unit | Lower bound | Upper bound |
|---|---|---|---|---|---|---|---|
| PstRangeAction tap upward variation | \(\Delta t^{+} (r)\) | upward tap variation of PstRangeAction \(r\), between two iterations of the optimisation | Integer | One variable for every element of PstRangeActions | No unit (number of taps) | \(-\infty\) | \(+\infty\) |
| PstRangeAction tap downward variation | \(\Delta t^{-} (r)\) | downward tap variation of PstRangeAction \(r\), between two iterations of the optimisation | Integer | One variable for every element of PstRangeActions | No unit (number of taps) | \(-\infty\) | \(+\infty\) |
| PstRangeAction tap upward variation binary | \(\delta ^{+} (r)\) | indicates whether the tap of PstRangeAction \(r\) has increased, between two iterations of the optimisation | Binary | One variable for every element of PstRangeActions | No unit | 0 | 1 |
| PstRangeAction tap downward variation binary | \(\delta ^{-} (r)\) | indicates whether the tap of PstRangeAction \(r\) has decreased, between two iterations of the optimisation | Binary | One variable for every element of PstRangeActions | No unit | 0 | 1 |
Used optimization variables
| Name | Symbol | Defined in |
|---|---|---|
| RA setpoint | \(A(r)\) | CoreProblemFiller |
Defined constraints
Tap to angle conversion constraint
\[\begin{equation} A(r) = \alpha_{n}(r) + c^{+}_{tap \rightarrow a}(r) * \Delta t^{+} (r) - c^{-}_{tap \rightarrow a}(r) * \Delta t^{-} (r), \forall r \in \mathcal{RA}^{PST} \end{equation}\]Where the computation of the conversion depends from the context in which the optimization problem is solved.
For the first solve, the coefficients are calibrated on the maximum possible variations of the PST:
\[\begin{equation} c^{+}_{tap \rightarrow a}(r) = \frac{f_r(\overline{t(r)}) - f_r(t_{n}(r))}{\overline{t(r)} - t_{n}(r)} \end{equation}\] \[\begin{equation} c^{-}_{tap \rightarrow a}(r) = \frac{f_r(t_{n}(r)) - f_r(\underline{t(r)})}{t_{n}(r) - \underline{t(r)}} \end{equation}\]For the second and next solves (during the iteration of the linear optimization), the coefficients are calibrated on a small variation of 1 tap:
\[\begin{equation} c^{+}_{tap \rightarrow a}(r) = f_r(t_{n}(r) + 1) - f_r(t_{n}(r)) \end{equation}\] \[\begin{equation} c^{-}_{tap \rightarrow a}(r) = f_r(t_{n}(r)) - f_r(t_{n}(r) - 1) \end{equation}\]Note that if \(t_n(r)\) is equal to its bound \(\overline{t(r)}\) (resp. \(\underline{t(r)}\)), then the coefficient \(c^{+}_{tap \rightarrow a}(r)\) (resp. \(c^{-}_{tap \rightarrow a}(r)\)) is set equal to 0 instead.
Tap variation can only be in one direction, upward or downward
\[\begin{equation} \Delta t^{+} (r) \leq \delta ^{+} (r) [\overline{t(r)} - t_{n}(r)] , \forall r \in \mathcal{RA}^{PST} \end{equation}\] \[\begin{equation} \Delta t^{-} (r) \leq \delta ^{-} (r) [t_{n}(r) - \underline{t(r)}] , \forall r \in \mathcal{RA}^{PST} \end{equation}\] \[\begin{equation} \delta ^{+} (r) + \delta ^{-} (r) \leq 1 , \forall r \in \mathcal{RA}^{PST} \end{equation}\]See also