Declaration of the Newton-Raphson load flow solver for power system analysis. More...

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

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Classes
class	NewtonRaphson
	Newton-Raphson power flow solver implementing the ISolver interface. More...

Detailed Description

Declaration of the Newton-Raphson load flow solver for power system analysis.

The Newton-Raphson method is a robust and widely-used iterative algorithm for solving the nonlinear power flow equations in electrical power systems.

It uses the bus admittance matrix $$Y_{bus}$$ and iteratively updates the bus voltages $$V$$ by solving the set of nonlinear equations:

$$\Delta x^{(k)} = -[J(x^{(k)})]^{-1} F(x^{(k)})$$

where:

$$x$$ is the state vector (bus voltage magnitudes and angles),
$$F(x)$$ is the vector of power mismatch equations,
$$J(x)$$ is the Jacobian matrix of partial derivatives with respect to $$x$$,
$$\Delta x$$ is the update to the state vector at iteration $$k$$.

The power injection equations at each bus are:

Active power (P):

$$P_i = \sum_{j=1}^{n} |V_i||V_j|(G_{ij}\cos(\theta_{ij}) + B_{ij}\sin(\theta_{ij}))$$

Reactive power (Q):

$$Q_i = \sum_{j=1}^{n} |V_i||V_j|(G_{ij}\sin(\theta_{ij}) - B_{ij}\cos(\theta_{ij}))$$

where:

$$G_{ij} + jB_{ij}$$ is the element of the admittance matrix $$Y_{bus}$$,
$$\theta_{ij} = \delta_i - \delta_j$$ is the voltage angle difference between buses $$i$$ and $$j$$,
$$V_i, V_j$$ are the voltage magnitudes at buses $$i$$ and $$j$$.

The Jacobian matrix has the general block structure:

$$J = \begin{bmatrix} \frac{\partial P}{\partial \delta} & \frac{\partial P}{\partial |V|} \\ \frac{\partial Q}{\partial \delta} & \frac{\partial Q}{\partial |V|} \end{bmatrix}$$

with submatrices defined as follows:

$$J_1(kn) = \frac{\partial P_k}{\partial \delta_n} = \begin{cases} -V_k \sum_{n=1}^N V_n |Y_{kn}| \sin(\delta_k - \delta_n - \theta_{kn}) & \text{if } k = n \ V_k V_n |Y_{kn}| \sin(\delta_k - \delta_n - \theta_{kn}) & \text{if } k \ne n \end{cases}$$

$$J_2(kn) = \frac{\partial P_k}{\partial |V_n|} = \begin{cases} V_k |Y_{kk}| \cos(\theta_{kk}) + \sum_{n \ne k} V_n |Y_{kn}| \cos(\delta_k - \delta_n - \theta_{kn}) & \text{if } k = n \ V_k |Y_{kn}| \cos(\delta_k - \delta_n - \theta_{kn}) & \text{if } k \ne n \end{cases}$$

$$J_3(kn) = \frac{\partial Q_k}{\partial \delta_n} = \begin{cases} V_k \sum_{n=1}^N V_n |Y_{kn}| \cos(\delta_k - \delta_n - \theta_{kn}) & \text{if } k = n \ -V_k V_n |Y_{kn}| \cos(\delta_k - \delta_n - \theta_{kn}) & \text{if } k \ne n \end{cases}$$

$$J_4(kn) = \frac{\partial Q_k}{\partial |V_n|} = \begin{cases} -V_k |Y_{kk}| \sin(\theta_{kk}) + \sum_{n \ne k} V_n |Y_{kn}| \sin(\delta_k - \delta_n - \theta_{kn}) & \text{if } k = n \ V_k |Y_{kn}| \sin(\delta_k - \delta_n - \theta_{kn}) & \text{if } k \ne n \end{cases}$$

The process continues until all mismatches are within a specified tolerance or the maximum number of iterations is reached.

Definition in file NewtonRaphson.H.

Classes

Detailed Description