Using integer variables for PST taps

Used input data

Name	Symbol	Details
PstRangeActions	\(r \in \mathcal{RA}^{PST}\)	Set of PST RangeActions
PstRangeActions	\(r,s \in \mathcal{RA}^{PST}\)	set of PST RangeActions and state on which they are applied
reference angle	\(\alpha _n(r, s)\)	angle of PstRangeAction \(r\) at state \(s\), at the beginning of the current iteration of the MILP
reference tap position	\(t_{n}(r, s)\)	tap of PstRangeAction \(r\) at state \(s\), at the beginning of the current iteration of the MILP
PstRangeAction tap bounds	\(t^-(r) \: , \: t^+(r)\)	min and max tap[1] 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

Used parameters

Name	Details
pst-model	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	\(\Delta t^{+} (r, s)\)	upward tap variation of PstRangeAction \(r\), at state \(s\), between two iterations of the optimisation	Integer	One variable for every element of PstRangeActions and for evey state in which it is optimized	No unit (number of taps)	0	\(+\infty\)
PstRangeAction tap downward variation	\(\Delta t^{-} (r, s)\)	downward tap variation of PstRangeAction \(r\), at state \(s\), between two iterations of the optimisation	Integer	One variable for every element of PstRangeActions and for evey state in which it is optimized	No unit (number of taps)	0	\(+\infty\)
PstRangeAction tap upward variation binary	\(\delta ^{+} (r, s)\)	indicates whether the tap of PstRangeAction \(r\) has increased, at state \(s\), between two iterations of the optimisation	Binary	One variable for every element of PstRangeActions and for evey state in which it is optimized	No unit	0	1
PstRangeAction tap downward variation binary	\(\delta ^{-} (r, s)\)	indicates whether the tap of PstRangeAction \(r\) has decreased, at state \(s\), between two iterations of the optimisation	Binary	One variable for every element of PstRangeActions and for evey state in which it is optimized	No unit	0	1

Used optimization variables

Name	Symbol	Defined in
RA setpoint	\(A(r, s)\)	CoreProblemFiller

Defined constraints

Tap to angle conversion constraint

\[ \begin{equation} A(r, s) = \alpha_{n}(r, s) + c^{+}_{tap \rightarrow a}(r, s) * \Delta t^{+} (r, s) - c^{-}_{tap \rightarrow a}(r, s) * \Delta t^{-} (r, s) , \forall (r,s) \in \mathcal{RA}^{PST} \end{equation} \]

Where the computation of the conversion depends on 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:

\[ \begin{equation} c^{+}_{tap \rightarrow a}(r, s) = \frac{f_r(t^+(r)) - f_r(t_{n}(r, s))}{t^+(r) - t_{n}(r, s)} \end{equation} \]

\[ \begin{equation} c^{-}_{tap \rightarrow a}(r, s) = \frac{f_r(t_{n}(r, s)) - f_r(t^-(r))}{t_{n}(r, s) - t^-(r)} \end{equation} \]

For the second and next solves (during the iteration of the linear optimization), the coefficients are calibrated on a small variation of 1 tap:

\[ \begin{equation} c^{+}_{tap \rightarrow a}(r, s) = f_r(t_{n}(r, s) + 1) - f_r(t_{n}(r, s)) \end{equation} \]

\[ \begin{equation} c^{-}_{tap \rightarrow a}(r, s) = f_r(t_{n}(r, s)) - f_r(t_{n}(r, s) - 1) \end{equation} \]

Note that if \(t_n(r, s)\) is equal to its bound \(t^+(r)\) (resp. \(t^-(r)\)), then the coefficient \(c^{+}_{tap \rightarrow a}(r, s)\) (resp. \(c^{-}_{tap \rightarrow a}(r, s)\)) is set equal to 0 instead.

Tap variation can only be in one direction, upward or downward

\[ \begin{equation} \Delta t^{+} (r) \leq \delta ^{+} (r, s) [t^+(r) - t_{n}(r, s)] , \forall (r, s) \in \mathcal{RA}^{PST} \end{equation} \]

\[ \begin{equation} \Delta t^{-} (r) \leq \delta ^{-} (r, s) [t_{n}(r, s) - t^-(r)] , \forall (r, s) \in \mathcal{RA}^{PST} \end{equation} \]

\[ \begin{equation} \delta ^{+} (r, s) + \delta ^{-} (r, s) \leq 1 , \forall (r, s) \in \mathcal{RA}^{PST} \end{equation} \]

RangeActions relative tap variations

If PST \(r\) has a RELATIVE_TO_PREVIOUS_INSTANT range constraints, between \(t^-_{rel}(r)\) and \(t^+_{rel}(r)\), the following constraint is added:

\[ \begin{equation} t^-_{rel}(r) \leq (t_{n}(r, s) + \Delta t^+(r, s) - \Delta t^-(r, s)) - (t_{n}(r, s') + \Delta t^+(r, s') - \Delta t^-(r, s')) \leq t^+_{rel}(r) \end{equation} \]

where \(s'\) is the last state preceding \(s\) where \(r\) is also optimized.

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[1]

PST range actions’ lower & upper bounds are computed using CRAC + network + previous RAO results, depending on the types of their ranges: ABSOLUTE, RELATIVE_TO_INITIAL_NETWORK, RELATIVE_TO_PREVIOUS_INSTANT (more information here)