# Modelling the maximum minimum margin objective

Contents

## 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 $$c$$

## Used parameters #

Name Details
objective-function Used to set the unit (AMPERE/MW) of the objective function

## Defined optimization variables #

Name Symbol Details Type Index Unit Lower bound Upper bound
Minimum margin $$MM$$ the minimum margin over all OptimizedFlowCnecs Real value one scalar variable for the whole problem MW or AMPERE (depending on objective-function unit) $$-\infty$$ $$+\infty$$

## Used optimization variables #

Name Symbol Defined in
Flow $$F(c)$$ CoreProblemFiller

## Defined constraints #

### Define the minimum margin variable #

#### If objective-function is in MW #

$$$MM \leq f^{+}_{threshold} (c) - F(c), \forall c \in \mathcal{C} ^{o}$$$ $$$MM \leq F(c) - f^{-}_{threshold} (c), \forall c \in \mathcal{C} ^{o}$$$

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

#### If objective-function is in AMPERE #

$$$MM \leq \frac{f^{+}_{threshold} (c) - F(c)}{c^{A->MW}(c)}, \forall c \in \mathcal{C} ^{o}$$$ $$$MM \leq \frac{F(c) - f^{-}_{threshold} (c)}{c^{A->MW}(c)}, \forall c \in \mathcal{C} ^{o}$$$

where $$c^{A->MW}(c)$$ is the conversion factor from AMPERE to MW of the FlowCnec c, calculated as below:

$$$c^{A->MW}(c) = \frac{U_{nom}(c) \sqrt{3}}{1000}$$$

There is a real difference between the two objective functions, as the most limiting OptimizedFlowCnec (i.e. the one which defines the minimum margin), can be different in MW and in A, depending on the nominal voltages Unom.

## Contribution to the objective function #

The minimum margin should be maximised:

$$$\min (-MM)$$$