Editing Power-flow study (section)

==Newton–Raphson solution method==
There are several different methods of solving the resulting nonlinear system of equations. The most popular{{according to whom|date=November 2023}} is a variation of the [[Newton–Raphson method]]. The Newton-Raphson method is an [[iterative method]] which begins with initial guesses of all unknown variables (voltage magnitude and angles at Load Buses and voltage angles at Generator Buses). Next, a [[Taylor Series]] is written, with the higher order terms ignored, for each of the power balance equations included in the system of equations. The result is a linear system of equations that can be expressed as:

<math display=block>\begin{bmatrix}\Delta \theta \\ \Delta |V|\end{bmatrix} = -J^{-1} \begin{bmatrix}\Delta P \\ \Delta Q \end{bmatrix} </math>

where <math>\Delta P</math> and <math>\Delta Q</math> are called the mismatch equations:

<math display=block>\Delta P_i = -P_i + \sum_{k=1}^N |V_i||V_k|(G_{ik}\cos\theta_{ik}+B_{ik}\sin \theta_{ik})</math>
<math display=block>\Delta Q_{i} =  -Q_{i} + \sum_{k=1}^N |V_i||V_k|(G_{ik}\sin\theta_{ik}-B_{ik}\cos\theta_{ik})</math>

and <math>J</math> is a matrix of partial derivatives known as a [[Jacobian matrix and determinant|Jacobian]]:
<math>J=\begin{bmatrix} \dfrac{\partial \Delta P}{\partial\theta} & \dfrac{\partial \Delta P}{\partial |V|} \\ \dfrac{\partial \Delta Q}{\partial \theta}& \dfrac{\partial \Delta Q}{\partial |V|}\end{bmatrix}</math>.

The linearized system of equations is solved to determine the next guess (''m'' + 1) of voltage magnitude and angles based on:

<math display=block>\theta_{m+1} = \theta_m + \Delta \theta\,</math>
<math display=block >|V|_{m+1} = |V|_m + \Delta |V|\,</math>

The process continues until a stopping condition is met. A common stopping condition is to terminate if the [[Matrix norm|norm]] of the mismatch equations is below a specified tolerance.

A rough outline of solution of the power-flow problem is:
# Make an initial guess of all unknown voltage magnitudes and angles. It is common to use a "flat start" in which all voltage angles are set to zero and all voltage magnitudes are set to 1.0 p.u.
# Solve the power balance equations using the most recent voltage angle and magnitude values.
# Linearize the system around the most recent voltage angle and magnitude values
# Solve for the change in voltage angle and magnitude
# Update the voltage magnitude and angles
# Check the stopping conditions, if met then terminate, else go to step 2.