Solvers

General information

The goal is to compute the voltages at each bus and the currents and powers flow in each branch of the network. The computation must respect the Kirchhoff’s laws and the constraints of the network.

More formally, this is done by solving a system of \(n\) nonlinear equations with \(n\) variables:

\[F: \mathbb{R}^n \to \mathbb{R}^n\]

The goal is then to find \(x\) such that:

\[F(x) = 0\]

Computationally, this translates to finding a solution \(x\) such that:

\[||F(x)||_{\infty} < \varepsilon\]

With \(\varepsilon\) being a small tolerance. In code, \(\varepsilon\) can be set with en.solve_load_flow(tolerance=...) (by default 1e-6).

There are several solvers to solve this kind of problems. In Roseau Load Flow, the following solvers are available:

Newton-Raphson

This is the classical Newton-Raphson method.

First, an initial solution \(x_0\) is chosen by initializing the voltages either by propagating the voltage of the sources or by re-using the results from the last successful run. The choice of either option depends on the warm_start argument to the en.solve_load_flow() method.

Then, multiple iterations are made with:

(1)\[x_{k+1} = x_k - J_F^{-1}(x_k)F(x_k)\]

with \(J_F\) being the jacobian of \(F\).

The algorithm stops when it finds a solution \(x_k\) such that \(||F(x_k)||_{\infty} < \varepsilon\) within a maximum number of iterations (modify with en.solve_load_flow(max_iterations=...)). If the maximum number of iterations is exceeded, the solver did not converge and the execution fails.

Parameters

The Newton-Raphson solver doesn’t accept any parameter.

Goldstein and Price

This is a variant of the classical Newton-Raphson solver with a linear search.

At each iteration, \(x_{k+1}\) is calculated using:

(2)\[x_{k+1} = x_k + t d(x_k)\]

with \(d = -J_F^{-1}F\)

For the classical Newton-Raphson solver, \(t=1\) is chosen for the next iterate. The idea of the linear searches, in this case the Goldstein and Price variant, is to find a “better” \(t\) that improves the convergence of the solver.

Let \(g\) be a function to be minimized:

\[\begin{split}g &: \mathbb{R}^n \to \mathbb{R} \\ g(x) &:= \frac{1}{2} ||F(x)||_2\end{split}\]

Let \(q\) be the function \(g\) in the direction \(d\):

\[\begin{split}q &: \mathbb{R} \to \mathbb{R} \\ q(t) &:= g(x_k + t d(x_k))\end{split}\]

A search is made to find \(t\) such that:

(3)\[m_2q'(0) \leqslant \frac{q(t) - q(0)}{t} \leqslant m_1q'(0)\]

In the figure above, any \(t\) such that \(a < t < b\) is satisfactory.

This \(t\) is found by dichotomy with multiple iterations, but in most cases only one iteration is needed. This is especially true when there are no flexible loads in the network.

\(t\) is then used to compute \(x_{k+1} = x_k + t d(x_k)\)

The Goldstein and Price variant is thus as fast as the classical Newton-Raphson while being more robust.

Parameters

The Goldstein and Price solver accepts the following parameters:

• "m1" the first constant of the Goldstein and Price variant. By default: 0.1.

• "m2" the second constant of the Goldstein and Price variant. By default: 0.9. Note that the constraint \(m_1 < m_2\) must be met.