Modelling the Power Gradient Constraints#

Used input data#

Name	Symbol	Details
Power gradient constraints set	\(\Gamma\)	Set of power gradient constraints
Upper power gradient constraint	\(\nabla p^{+}(g)\)	Maximum upward power variation between two consecutive timestamps for generator \(g\). This value must be non-negative.
Lower power gradient constraint	\(\nabla p^{-}(g)\)	Maximum downward power variation (in absolute value) between two consecutive timestamps for generator \(g\). This value must be non-positive.
Initial generator power	\(p_{0}(g,s,t)\)	Initial power of generator \(g\) at state \(s\) of timestamp \(t\), as defined in the network.
Timestamps	\(\mathcal{T}\)	Set of all timestamps on which the optimization is performed.
Time gap	\(\Delta_t(t, t + 1)\)	Time gap between consecutive timestamps \(t\) and \(t + 1\).

Defined optimization variables#

Name	Symbol	Details	Type	Index	Unit	Lower bound	Upper bound
Generator power	\(P(g,s,t)\)	the power of generator \(g\) at state at \(s\) of timestamp \(t\)	Real value	one per generator defined in \(\Gamma\), per state and per timestamp of \(\mathcal{T}\)	MW	\(-\infty\)	\(+\infty\)

Used optimization variables#

Name	Symbol	Defined in
Range Action upward set-point variation	\(\Delta^{+}(r,s,t)\)	CoreProblemFiller
Range Action downward set-point variation	\(\Delta^{-}(r,s,t)\)	CoreProblemFiller

Defined constraints#

Define the impact of injections on generator#

For a given generator \(g\) at state \(s\) of timestamp \(t \in \mathcal{T}\), we denote \(\mathcal{I}(g,s,t)\) the set of injection range actions defined at state \(s\) for timestamp \(t\) that act on \(g\). For each such injection range action \(i\), \(d_{i}(g)\) denotes the distribution key of \(g\) for this very remedial action. The power of \(g\) at state \(s\) is:

\[\forall g, t \quad P(g,s,t) = p_{0}(g,s,t) + \sum_{i \in \mathcal{I}(g,s,t)} d_i(g) \left [ \Delta^{+}(r,s,t) - \Delta^{-}(r,s,t) \right ]\]

This is only implemented for preventive injection range actions for the moment

Define the power gradient constraint#

\[\nabla p^{-}(g) * \Delta_t(t, t + 1) \leq P(g,s,t + 1) - P(g,s',t)\leq \nabla p^{+}(g) * \Delta_t(t, t + 1)\]

If the constraint is only defined with an upper (resp. lower) gradient then the lower (resp. upper) bound of the constraint is \(-\infty\) (resp. \(\infty\)).

\(s'\) is always the preventive state of timestamp \(t\) as curative states of different timestamps are independent