FARAO search-tree is initially designed to optimize non-costly remedial actions (including generation unit trip where applicable).

Unlike purely linear optimization, a search-tree algorithm does not neglect the non-linearity of efficient remedial actions such as curative and preventive topological actions. Incorrect modeling of remedial actions such as topological remedial actions could lead to over or underestimate their impact and then result in a biased network situation that could endanger the system security.

So far, the search-tree algorithm has proved for many years its relevance and efficiency on current daily operational process on CWE, Northern Italy and SWE Capacity calculation.

For each topological remedial action applied, the search-tree will systematically optimize PST taps/HVDC by applying a linear optimization. By considering both topological remedial actions and linear remedial actions at every step, instead of considering only one and then the other, FARAO results are better optimized.

The optimization problem is a non-convex and non-linear one, dealing with topology changes on the network, which represent discrete actions by definition. The problem treated by the optimizer is a combinatorial problem. Search-tree algorithms are commonly used for high complexity mathematical problems, containing combinatorial and discrete aspects.

Some remedial actions such as PSTs/HVDC are treated via a linear optimization. It has to be noticed that approximation of discrete PST taps treated as linear variable can be valid in DC but a security analysis is performed by FARAO after their optimization to ensure security of the grid despite of the approximation.

### Higher grid security

A search-tree algorithm will optimize all couples of critical branches and critical outages to define a set of remedial actions covering all these network states. This limits the dependency to net position forecast of which the accuracy is highly uncertain due to deviations of generation from renewable sources or unforeseen events (crisis,…).

For example, let’s assume there is no existing PRA for simplicity reasons and focus on the CNEC with the minimum relative margin (CNE1C1). The optimizer will search for CRA increasing margin of CNE1C1. However, considering only CNE1C1 could be relevant if and only if there is a high reliability and accuracy of net position forecast. As stated before, this could not be acceptable for the security of the system. That is why the search tree will also identify optimized remedial actions for CNE1C2, CNE1C3…

### Remedial actions considered

For Capacity calculation process, FARAO search-tree considers the following remedial actions :

- Preventive remedial actions (fully shared) hereafter “PRA”
- Curative remedial actions (shared after critical outage) hereafter “CRA” (not yet implemented…)

### Linear Remedial actions

The impact of some types of remedial actions on flows could be considered to be linear: optimization of HVDC setpoints and optimization of generation unit setpoints. In addition to these, FARAO search-tree also considers phase shifter transformers as linear remedial actions.

A phase shifter transformer (PST) is defined by its range of acceptable tap settings. This acceptable tap setting can be defined by three different ways:

- a range of tap positions relatively to the neutral tap position of the PST (e-g +6/-6 around the neutral tap position)
- a range of tap positions relatively to the tap position of the PST as described in the grid model
- in curative: a range of tap positions relatively to the optimised tap position of the PST in preventive (e-g +4/-4 around the tap position selected in preventive)

### Non-linear remedial action

Topological and other discrete remedial actions are considered without any approximation, and can be optimized in both instants (preventive or curative). This is a non-exhaustive list of discrete remedial actions considered by FARAO search-tree :

- Change of circuit breaker position (open/close): line opening or busbar coupler opening/closing
- Change of switch position

As a matter of clarification, connecting/disconnecting a generation unit can also be considered by FARAO search-tree for Capacity calculation (without element of costs).

### Objective function

#### For Flow-based Capacity calculation – minimum margin

The objective function is used to determine at each step of the search tree which remedial action is the best. A variant of it is also used when solving the linear optimization problem.

The active flow \(F_i\) on a CNEC \(i\) is:

\[\begin{equation} F_i = F_{ref_{0,i}} + ( \sum_{j \in J} \Delta\alpha_j * PSDF_{j,i}) + \Delta F_i \end{equation}\]With:

- \(F_{ref_{0,i}}\) the initial active transit on CNEC \(i\)
- \(J\) the set of all PSTs
- \(PSDF_{j,i}\) the sensitivity of PST \(j\) on CNEC \(i\)

And:

- \(\Delta\alpha_j\) variation angle value of PST \(j\) (between initial situation and final situation)
- \(\Delta F_i\) impact of a set of applied non-PST RAs on CNEC \(i\)

Optimization will try to maximize the objective function value, formulated as such:

\[\begin{equation} OF = min_i(\rho_i * F_{max_i}-F_i) \end{equation}\]With :

- For the sake of readability, we consider in the rest of the explanation that the FRM value is already considered in F_{max_i}.
- and \(\rho_i = 1 \ \ \forall i\quad\) if at least one CNEC is in overload

\(\rho_i = \frac{1}{\sum_{ma}|PTDF_{ma,i}|}\ \ \forall i\quad\) if all margins are positive, with \(PTDF_{ma,i}\) the PTDF of CNEC \(i\) relatively to a commercial border (zone-to-zone) of the capacity calculation region = one commercial border in the capacity calculation region (e-g FR-BE)

The objective function of FARAO search-tree is also configurable. The constraints taken into account within the optimization are also configurable in order to comply with process specificities (i.e limitation of loop flows required for Flow based CORE Capacity Calculation).

#### For NTC Capacity calculation/CEP Validation

The objective function is :

\[\begin{equation} OF = min_i(F_{max_i}-F_i) \end{equation}\]For the NTC Capacity calculation, only the positive margin stop criterion is activated. This is equivalent to an iterative security analysis at different values of commercial exchanges. It is also valid for the CEP Validation performed at national or regional level.

### Inputs

This algorithm acts on sets of states that share common remedial actions, also called **perimeters**.

The main inputs of the algorithm are:

- the network at the root of the perimeter,
- an extract of the original Crac, containing only the remedial actions that are available in the given perimeter (filtered on usage rules).

### Stop criterion

For Flow based Capacity calculation, considering that the objective function is the minimum margin (meaning the minimum of all flow margins of every CNEC), there are currently two stop criteria, ie two ways to stop the search tree algorithm:

- when the minimum margin is positive, meaning that the network is secured (all the CNEC flows are under line thresholds):
**positive margin**stop criterion. As mentioned above, for NTC Capacity calculation/ CEP Validation, this stop criterion is applied. - when the the minimal margin on every CNEC cannot be increased anymore:
**maximum margin**stop criterion.

These stop criteria only make sense for a minimum margin objective function (may it be absolute or relative).

On both stop criteria, additional constraints can be added, for example:

- the maximal number of consecutive chosen network actions, also called
**search tree depth**, - the minimal relative gain of objective function between two consecutive network actions (i.e. between two search tree depths).

### Algorithm

For each iteration/step (a level of depth in tree):

- Determination of available remedial actions.
- Once the list of available remedial actions is defined, candidates are created. Each candidate corresponds to a grid situation, where one (or more) remedial actions are applied.
- A skippable optimization of the linear remedial actions is done.
- A security analysis determines for each candidate the value of the objective function. The security analysis consists of a series of DC (or AC) load-flow computations (for each defined contingency).
- In order to maximise the objective function, values obtained for each candidate are compared: the candidate leading to the best increase of objective function value is selected. The remedial actions corresponding to this candidate are applied.

Note that if no candidate can increase the objective function value more than configurable value, the optimisation of the current studied perimeter (preventive/curative) stops. All candidates applied in previous steps/iterations of optimisation will be considered as final found remedial actions.

#### Usefulness of optimizing linear remedial actions at every step

By default, FARAO search-tree always studies the combination of PST and other linear remedial actions with each individual non-PST remedial action. This allows to better take into consideration the joint effect of non-PST remedial with available PST (in particular if the non-PST remedial action impacts significantly the PSDF of available PST, typically if both actions are located in a close “electrical vicinity”).

In this figure above, 3 non-PST remedial actions are available, as well as a list of PSTs. Each non-PST remedial action is applied on the grid situation. However, prior to an assessment of the objective function value, PSDF are assessed (with consideration of the applied non-PST action), and PSTs are optimised.

RA1 alone could have led to bigger objective function value than RA2 or RA3. However, when taking PSTs into account, RA2 with optimised PSTs is more efficient than other RA and optimised tap settings.