Overview
The Linear RAO optimizes the Range Actions (for now, only PSTs) according to a configurable objective function (e.g. maximization of the minimum margin). The solving algorithm approximates the impact of the Remedial Actions on the network flows with linear sensitivity coefficients. It therefore solves linear optimisation problems to find the optimal application of the Remedial Actions. Moreover, it can iterate over several reference points in order to mitigate the linear approximation inherent to its optimisation problem.
In particular, the Linear RAO module is called from CASTOR.
Inputs
The main inputs of the algorithm are:
 the network of the “initial situation”, where the remedial actions are supposed to be at their “standard position”,
 an extract of the original Crac, containing only the range actions (network actions actions are removed).
Outputs
The Crac is updated by the linear RAO, as most of the results of the RAO are stored in Crac results extensions.
Algorithm
Let’s focus on PST to clarify how linear remedial actions are optimized. PST optimization relies on 3 types of computations:

A sensitivity analysis, which answers the question: how much does each PST shift angle impact the flow on each CNEC? These sensitivities (or Phase Shifter Distribution Factors, PSDF) are recomputed for each and every CNEC that are part of the studied perimeter, for each RA candidate during remedial action optimisation, and for each step of optimization.

A linear optimization: based on those sensitivities, what is the combination of phase shift angles which maximizes the objective function value?

A standard security analysis, checks that the new PST taps actually increase the objective function value. Indeed, the linear optimization remains an approximation (valid in DC), and transforming angles to discrete PST taps can also cause differences between the anticipated margins and the actual computed margin.
Hereunder, the specific workflow related to linear optimization.
Forcing tap positions of a defined set of PSTs to be equal
CASTOR can be also configured to define sets of PSTs whose taps should be maintained equal during optimisation. To see how this is done, you can check the details of the linear optimization problem.
Minimum impact of PST in the linear optimization
In order to control the usage of PST in the optimization, it is possible to set a constraint in the PST optimization problem: the change in phase shifting angle value of one particular PST should not have an impact on the objective function smaller than a configurable value pstsensitivitythreshold.
When computing the PST sensitivities, only the ones which are higher than pstsensitivitythreshold are considered and the others are considered null. This allows us to filter out the PSTs which don’t have a big enough impact on the CNECs.
Also, while solving the linear optimisation problem, the usage of PSTs is penalised. This is done by adding a configurable cost pstpenaltycost for moving the PSTs in the linear problem’s objective function:
\[\begin{equation} \max MM  \sum_{p \in \mathcal{P}} \Delta_{p} c^{PST} \end{equation}\]with \(MM\) the minimum margin, \(\mathcal{P}\) the set of PSTs, \(\Delta_p\) the variation of angle of the PST \(p\), and \(c^{PST}\) equal to pstpenaltycost.
This way, if two solutions are very close to each other, the problem will prefer the one that changes the PST taps the least.