## Used input data

Name | Symbol | Details |
---|---|---|

FlowCnecs | \(c \in \mathcal{C}\) | set of FlowCnecs. 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 on state \(s\), could be PSTs, HVDCs, or injection range actions |

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 | \(\alpha _0(r)\) | setpoint of RangeAction \(r\) at the beginning of the optimization |

ReferenceSetpoints | \(\alpha _n(r)\) | setpoint 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 \(s\) |

## 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’ setpoints (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) | MW | \(-\infty\) | \(+\infty\) |

RA setpoint | \(A(r,s)\) | setpoint 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^{1} |
Range upper bound^{1} |

RA setpoint absolute variation | \(\Delta A(r,s)\) | The absolute setpoint variation of RangeAction \(r\) on state \(s\), from setpoint on previous state to “RA setpoint” | Real positive value | One variable for every element of (RangeActions) | Degrees for PST range actions; MW for other range actions | 0 | \(+\infty\) |

## Defined constraints

### Impact of rangeActions on FlowCnecs flows

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

### Definition of the absolute setpoint variations of the RangeActions

\[\begin{equation} \Delta A(r,s) \geq A(r,s) - A(r,s') , \forall (r,s) \in \mathcal{RA} \end{equation}\] \[\begin{equation} \Delta A(r,s) \geq - A(r,s) + A(r,s') , \forall (r,s) \in \mathcal{RA} \end{equation}\]with \(A(r,s')\) the setpoint of the last range action on the same element as \(r\) but a state preceding \(s\). If none such range actions exists, then \(A(r,s') = \alpha_{0}(r)\)

## Contribution to the objective function

Small penalisation for the use of RangeActions:

\[\begin{equation} \min \sum_{r,s \in \mathcal{RA}} (c^{penalty}_{ra}(r) \Delta A(r,s)) \end{equation}\]Code reference: CoreProblemFiller