Projections

When the control algorithm is trying to find the best control for given voltage constraints, it could find a solution that is not “feasible” by the load. This means that the active and reactive powers \(P\) and \(Q\) that constitute the solution lie outside the feasible domain defined by a part of the disc of radius \(S^{\max}\) in the \((P, Q)\) space. In these cases, the solution has to be projected into the feasible domain. We can choose how the projection is performed using three available projection types: the Euclidean projection, the projection at Constant \(P\) and the projection at Constant \(Q\).

The projection accepts two approximation parameters: alpha and epsilon.

  • alpha is used to compute soft sign function and soft projection function. The higher alpha is, the better the approximations are.

  • epsilon is used to approximate a smooth square root function:

    \[\sqrt{S} \approx \sqrt{\varepsilon \times \exp\left(\frac{-{|S|}^2}{\varepsilon}\right) + {|S|}^2}\]

    The lower epsilon is, the better the approximations are.

Important

Please note that no projection is performed if the final \(\underline{S(U)}\) point lies in the disc of radius \(S^{\max}\).

Euclidean projection

A Euclidean projection on the feasible domain. This is the default value for projections when it is not specified.

../../../_images/Euclidean_Projection.svg 
import roseau.load_flow as rlf

projection = rlf.Projection(type="euclidean")  # alpha and epsilon can be provided

Important

Please note that using the Euclidean projection may reduce the provided \(P^{\mathrm{th.}}\) and \(Q^{\mathrm{th.}}\) of the load. See the Feasible Domain page for more details.

Constant \(P\)

Keep the value of \(P\) computed by the control and project \(Q\) on the feasible domain.

../../../_images/Constant_P_Projection.svg 
import roseau.load_flow as rlf

projection = rlf.Projection(type="keep_p")  # alpha and epsilon can be provided

Important

Please note that using the Constant \(P\) projection may reduce the provided \(Q^{\mathrm{th.}}\) of the load. See the Feasible Domain page for more details.

Constant \(Q\)

Keep the value of \(Q\) computed by the control and project \(P\) on the feasible domain.

../../../_images/Constant_Q_Projection.svg 
import roseau.load_flow as rlf

projection = rlf.Projection(type="keep_q")  # alpha and epsilon can be provided

Important

Please note that using the Constant \(Q\) projection may reduce the provided \(P^{\mathrm{th.}}\) of the load. See the Feasible Domain page for more details.