Modelling the maximum minimum relative margin objective function

Used input data

Name	Symbol	Details
OptimisedFlowCnecs	\(c \in \mathcal{C} ^{o}\)	Set of FlowCnecs which are ‘optimised’. OptimisedFlowCnecs is a subset of FlowCnecs: \(\mathcal{C} ^{o} \subset \mathcal{C}\)
upper threshold	\(f^{+}_{threshold} (c)\)	Upper threshold of FlowCnec \(c\), in MW, as defined in the CRAC
lower threshold	\(f^{-}_{threshold} (c)\)	Lower threshold of FlowCnec \(c\), in MW, defined in the CRAC
nominal voltage	\(U_{nom}(c)\)	Nominal voltage of OptimizedFlowCnec \(c\)
Absolute PTDF sum	\(\sigma_{ptdf}(c)\)	Absolute zone to zone PTDF sum[1] of FlowCnec \(c\).
Highest threshold value	\(MaxRAM\)	A “bigM” which is computed (by OpenRAO) as the greatest absolute possible value of the CNEC threshold, among all CNECs in the CRAC. It represents the common greatest possible value for a given CNEC’s margin (exception made of CNECs only constrained in one direction, but this value should be high enough not to have any effect on those).

Used parameters

Name	Symbol	Details
type		This filler is only used if the objective function is MAX_MIN_MARGIN_IN_MEGAWATT, or MAX_MIN_MARGIN_IN_AMPERE. This parameter is also used to set the unit (AMPERE/MW) of the objective function
ptdf-sum-lower-bound	\(\varepsilon_{PTDF}\)	zToz PTDF sum below this value are lifted to the ptdf-sum-lower-bound, to avoid a bad conditionning of the problem where the value of relative margins are very high. Its impact on the accuracy of the problem is insignificant, as high relative margins do not usually define the min. relative margin.

Defined optimization variables

Name	Symbol	Details	Type	Index	Unit	Lower bound	Upper bound
Minimum relative margin	\(MRM\)	the minimum negative margin over all OptimizedFlowCnecs	Real value	one scalar variable for the whole problem	Relative MW or relative AMPERE (depending on objective-function	0	\(+\infty\)
Is minimum margin positive	\(P\)	binary variable, equal to 1 if the min margin is positive, 0 otherwise	Binary	one scalar variable for the whole problem	no unit	0	1

Used optimization variables

Name	Symbol	Defined in
Flow	\(F(c)\)	CoreProblemFiller
Minimum margin	\(MM\)	MaxMinMarginFiller

Defined constraints

Making the absolute minimum margin \(MM\) negative

The absolute minimum margin defined in MaxMinMarginFiller will now only be used for when the minimum margin is negative. So the following constraints are added:

\[ \begin{equation} MM \leq 0 \end{equation} \]

\[ \begin{equation} MM \geq -(1 - P) * m_{min}^{RAM} \end{equation} \]

where \(m_{min}^{RAM}\) represents the maximum (absolute) value of the margin when it is negative. It is computed as follows:

\[ m_{min}^{RAM} = MaxRAM * 5 \]

Defining the minimum relative margin

The following constraints define the new \(MRM\) variable:

\[ \begin{equation} MRM \leq \frac{f^{+}_{threshold} (c) - F(c)}{\sigma^{\prime}_{ptdf}(c) c^{unit}(c)} + (1 - P) * m_{min}^{relRAM}, \forall c \in \mathcal{C} ^{o} \end{equation} \]

\[ \begin{equation} MRM \leq \frac{F(c) - f^{-}_{threshold} (c)}{\sigma\prime_{ptdf}(c) c^{unit}(c)} + (1 - P) * m_{min}^{relRAM}, \forall c \in \mathcal{C} ^{o} \end{equation} \]

• where \(\sigma^{\prime}_{ptdf}(c)\) is a “safe” version of the zone-to-zone absolute PTDF sum, where small values are lifted to avoid bad conditioning of the MILP:

\[\sigma^{\prime}_{ptdf}(c) = \max{(\sigma_{ptdf}(c), \varepsilon_{PTDF})} \]

• the max possible positive relative RAM is:

\[m_{max}^{relRAM} = MaxRAM / \varepsilon_{PTDF}\]

• the max possible negative relative RAM is (in absolute value):

\[m_{min}^{relRAM} = m_{max}^{relRAM} * 5\]

• and the unit conversion coefficient is defined as follows:

  • If the objective-function is in MW: \(c^{unit}(c) = 1\)

  • If it is in AMPERE: \(c^{unit}(c) = \frac{U_{nom}(c) \sqrt{3}}{1000}\)

Note that an OptimizedFlowCnec might have only one threshold (upper or lower). In that case, only one of the two constraints above is defined.

Making \(MRM\) positive

When the MM is negative, P is forced to 0 (see above). The following constraint sets the MRM to 0:

\[ \begin{equation} MRM \leq P * m_{max}^{relRAM} \end{equation} \]

Contribution to the objective function

The sum of minimum absolute & relative margins should be maximised:

\[ \begin{equation} \min -(MM + MRM) \end{equation} \]

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[1]

Computed outside the linear optimization, as below:
\( \begin{equation} \sigma_{ptdf}(c) = \sum_{(z1, z2) \in zToz} \mid PTDF_{zTos}(z1, c) - PTDF_{zTos}(z2, c) \mid \end{equation}\)
With \(zToz\) the set of zone-to-zone PTDFs used for the relative margin computation, defined in the configuration parameter relative-margin-ptdf-boundaries.
And \(PTDF_{zTos}(z1, c)\), the zone-to-slack PTDF of bidding zone \(z1\) on CNEC \(c\).