GaussSeidel.H File Reference

Declaration of the Gauss-Seidel load flow solver for power system analysis. More...

#include <Eigen/Dense>
#include <utility>
#include <vector>
#include "ISolver.H"

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Classes
class	GaussSeidel
	Gauss-Seidel power flow solver implementing the ISolver interface. More...

Detailed Description

Declaration of the Gauss-Seidel load flow solver for power system analysis.

The Gauss-Seidel method is an iterative algorithm to solve the power flow equations for bus voltages in a power system. It uses the bus admittance matrix $$ Y_{bus} $$ and iteratively updates the bus voltages $$ V $$ to satisfy the power balance equations:

$$ S_i = V_i \sum_{j=1}^N Y_{ij}^* V_j^* $$

where:

• $$ S_i $$ is the specified complex power at bus $$ i $$
• $$ V_i $$ is the voltage at bus $$ i $$
• $$ Y_{ij} $$ is the element of the admittance matrix

The Gauss-Seidel voltage update for the k-th iteration is:

$$ V_i^{(k+1)} = \frac{1}{Y_{ii}} \left( \frac{S_i^*}{V_i^{*(k)}} - \sum_{j=1, j \neq i}^N Y_{ij} V_j^{(k+1 \text{ or } k)} \right) $$

For voltage-controlled buses (PV buses), the reactive power injection $$ Q_k $$ is unknown initially but can be computed at iteration i by:

$$ Q_k^{(i)} = V_k^{(i)} \sum_{n=1}^N |Y_{kn}| V_n^{(i)} \sin \left( \delta_k^{(i)} - \delta_n^{(i)} - \theta_{kn} \right) $$

The total reactive power generation at bus k is:

$$ Q_{Gk} = Q_k + Q_{Lk} $$

where $$( Q_{Lk} )$$ is the reactive load demand at the bus.

If the calculated $$( Q_{Gk} )$$ remains within its specified limits (e.g., $$ Q_{Gk}^{min} \leq Q_{Gk} \leq Q_{Gk}^{max} $$), the voltage magnitude $$ V_k $$ is fixed at its specified value, and only the voltage angle $$ \delta_k $$ is updated using the Gauss-Seidel formula. This means:

• The updated voltage magnitude $$ V_k^{(i+1)} $$ is set to the specified magnitude,
• Only the angle $$ \delta_k^{(i+1)} $$ changes according to the iterative update.

If $$( Q_{Gk} )$$ exceeds its limits during any iteration, the bus type is switched from voltage-controlled (PV) to load bus (PQ), with $$ Q_{Gk} $$ fixed at the violated limit (either $$ Q_{Gk}^{max} $$ or $$ Q_{Gk}^{min} $$). Under this condition, the voltage magnitude $$ V_k $$ is no longer fixed and is recalculated by the load flow program.

To accelerate convergence, a relaxation (or acceleration) coefficient $$ \omega $$ (typically $$(0 < \omega \leq 1) $$) can be applied to the voltage update:

$$ V_i^{(k+1)} = \omega \times V_i^{(k+1)} + (1 - \omega) \times V_i^{(k)} $$

The iteration proceeds until the maximum change in bus voltages between iterations is below a specified tolerance, or the maximum number of iterations is reached.

Definition in file GaussSeidel.H.