Modelling CNECs and range actions

This page gathers core constraints, i.e constraints that need to be defined at the beginning of the optimization problem.

Used input data

Name	Symbol	Details
FlowCnecs	\(c \in \mathcal{C}\)	set of FlowCnecs[1]. Note that FlowCnecs are all the CBCO for which we compute the flow in the MILP, either: - because we are optimizing their flow (optimized flowCnec = CNEC) - because we are monitoring their flow, and ensuring it does not exceed its threshold (monitored flowCnec = MNEC) - or both
RangeActions	\(r,s \in \mathcal{RA}\)	set of RangeActions and state on which they are applied, could be PSTs, HVDCs, or injection range actions
RangeActions	\(r \in \mathcal{RA(s)}\)	set of RangeActions available at state \(s\), could be PSTs, HVDCs, or injection range actions
InjectionRangeActions	\(r \in \mathcal{IRA}(s)\)	set of InjectionRangeActions available at state \(s\)
Injection Distribution keys	\(d \in \mathcal{DK(r)}\)	set of distribution keys of InjectionRangeAction \(r\). Each distribution key is linked to a NetworkElement.
Iteration number	\(n\)	number of current iteration
ReferenceFlow	\(f_{n}(c)\)	reference flow, for FlowCnec \(c\). The reference flow is the flow at the beginning of the current iteration of the MILP, around which the sensitivities are computed
PrePerimeterSetpoints	\(\alpha _0(r)\)	set-point of RangeAction \(r\) at the beginning of the optimization
ReferenceSetpoints	\(\alpha _n(r)\)	set-point of RangeAction \(r\) at the beginning of the current iteration of the MILP, around which the sensitivities are computed
Sensitivities	\(\sigma _{n}(r,c,s)\)	sensitivity of RangeAction \(r\) on FlowCnec \(c\) for state \(s\)
Previous RA set-point	\(A_{n-1}(r,s)\)	optimal set-point of RangeAction \(r\) on state \(s\) in previous iteration (\(n-1\))
Activation cost of RA	\(c_{act}(r)\)	cost to spend to activate the range action
Upward variation cost of RA	\(c_{\Delta}^{+}(r)\)	cost to spend for each MW (resp. tap) changed in the upward direction for a standard range action (resp. PST range action)
Downward variation cost of RA	\(c_{\Delta}^{-}(r)\)	cost to spend for each MW (resp. tap) changed in the downward direction for a standard range action (resp. PST range action)

Used parameters

Name	Symbol	Details	Source
sensitivityThreshold		Set to zero the sensitivities of RangeActions below this threshold; thus avoiding the activation of RangeActions which have too small an impact on the flows (can also be achieved with penaltyCost). This simplifies & speeds up the resolution of the optimization problem (can be necessary when the problem contains integer variables). However, it also adds an approximation in the computation of the flows within the MILP, which can be tricky to handle when the MILP contains hard constraints on loop-flows or monitored FlowCnecs.	Equal to pst-sensitivity-threshold for PSTs, hvdc-sensitivity-threshold for HVDCs, and injection-ra-sensitivity-threshold for injection range actions
penaltyCost	\(c^{penalty}_{ra}\)	Supposedly a small penalization, in the use of the RangeActions. When several solutions are equivalent, this favours the one with the least change in the RangeActions’ set-points (compared to the initial situation). It also avoids the activation of RangeActions which have to small an impact on the objective function.	Equal to pst-penalty-cost for PSTs, hvdc-penalty-cost for HVDCs, and injection-ra-penalty-cost for injection range actions

Defined optimization variables

Name	Symbol	Details	Type	Index	Unit	Lower bound	Upper bound
Flow	\(F(c)\)	flow of FlowCnec \(c\)	Real value	One variable for every element of (FlowCnecs)	Flow unit	\(-\infty\)	\(+\infty\)
RA set-point	\(A(r,s)\)	set-point of RangeAction \(r\) on state \(s\)	Real value	One variable for every element of (RangeActions)	Degrees for PST range actions; MW for other range actions	Range lower bound[2]	Range upper bound[2]
RA set-point upward variation	\(\Delta^{+}(r,s)\)	The upward set-point variation of RangeAction \(r\) on state \(s\), from set-point on previous state to “RA set-point”	Real positive value	One variable for every element of (RangeActions)	Degrees for PST range actions; MW for other range actions	0	\(+\infty\)
RA set-point downward variation	\(\Delta^{-}(r,s)\)	The downward set-point variation of RangeAction \(r\) on state \(s\), from set-point on previous state to “RA set-point”	Real positive value	One variable for every element of (RangeActions)	Degrees for PST range actions; MW for other range actions	0	\(+\infty\)
RA activation	\(\delta(r,s)\)	Binary variable that indicates whether the range action \(r\) is activated at state \(s\)	Binary	One variable for every element of (RangeActions)	None	0	1

Defined constraints

Impact of rangeActions on FlowCnecs flows

The following equation is the RAO’s keystone, linking a FlowCnec’s flow to linear range actions’ set-points.

\[ \begin{equation} F(c) = f_{n}(c) + \sum_{r \in \mathcal{RA(s)}} \sigma_n (r,c,s) * [A(r,s) - \alpha_{n}(r,s)] , \forall (c) \in \mathcal{C} \end{equation} \]

with \(s\) the state on \(c\) which is evaluated

Definition of the set-point variations of the RangeActions

The following equation links a range action’s set-point to its upward/downward variations.

\[ \begin{equation} A(r,s) = A(r,s') + \Delta^{+}(r,s) - \Delta^{-}(r,s) , \forall (r,s) \in \mathcal{RA} \end{equation} \]

with \(A(r,s')\) the set-point of the last range action on the same element as \(r\) but a state preceding \(s\). If none such range actions exists, then \(A(r,s') = \alpha_{0}(r)\)

This equation always applies, except for PSTs when modeling them as APPROXIMATED_INTEGERS. If so, this constraint will be modeled directly via PST taps, see here as it gives better results [3].

Costly only- Range action activation variable

\[\left ( \alpha_{\max}(r, s) - \alpha_{\min}(r, s) \right ) \delta(r,s) \geq \Delta^{+}(r,s) + \Delta^{-}(r,s) , \forall (r,s) \in \mathcal{RA}\]

where \(\alpha_{\max}(r, s)\) and \(\alpha_{\min}(r, s)\) are respectively the maximum and minimum reachable set-points for range action \(r\) at state \(s\).

Shrinking the allowed range

The following equations are used to mitigate diverging behaviors that may occur due to non linearity.

If parameter ra-range-shrinking is enabled, the allowed range for range actions is shrunk after each iteration according to the following constraints:

\[ \begin{equation} t = UB(A(r,s)) - LB(A(r,s)) \end{equation} \]

\[ \begin{equation} A(r,s) \geq A_{n-1}(r,s) - (\frac{2}{3})^{n}*t \end{equation} \]

\[ \begin{equation} A(r,s) \leq A_{n-1}(r,s) + (\frac{2}{3})^{n}*t \end{equation} \]

The value \(\frac{2}{3}\) has been chosen to force the linear problem convergence while allowing the RA to go back to its initial solution if needed.

Costly only - Injection balance constraint

The network must remain balanced in terms of production and consumption after injection variations (redispatching):

\[\sum_{r \in \mathcal{IRA}(s)} \left ( \Delta^{+} (r, s) - \Delta^{-} (r, s) \right ) \times \sum_{d \in \mathcal{DK(r)}} d = 0, \forall s\]

Contribution of these constraints to the objective function

Margin optimization

Small penalisation for the use of RangeActions:

\[ \begin{equation} \min \sum_{r,s \in \mathcal{RA}} \left ( c^{penalty}_{ra}(r) \left ( \Delta^{+}(r,s) + \Delta^{-}(r,s) \right ) \right ) \end{equation} \]

Costly only - Remedial actions’ cost optimization

\[ \begin{equation} \min \sum_{r,s \in \mathcal{RA}} c^{act}(r) \delta(r,s) + c_{\Delta}^{+}(r) \Delta^{+}(r,s) + c_{\Delta}^{-}(r) \Delta^{-}(r,s) \end{equation} \]

• For PST range actions, the tap variation is used instead.

• If the variation costs are null, the penalties defined in the RAO parameters (\(c_{ra}^{penalty}\)) are used instead to force the RAO to change as few taps as possible (the value of the penalty depends on the type of range action: PST, injection of HVDC)

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[1]

CNECs that belong to a state for which sensitivity computations failed are ignored in the MILP

[2] (1,2)

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)

[3]

Relative bounds are originally in tap. When converting them in angle, we take the smallest angle step between two tap changes. Despite artificially tightening the bounds, this is the only way to ensure we respect the tap limits as the tap to angle map is non linear.
With PST modeled as APPROXIMATED_INTEGERS, we can directly use the taps and therefore avoid this approximation.