Load Flow or Power Flow Analysis

Load flow, or power flow, analysis calculates a power network’s steady-state operating point for specified generation, demand and network data. It is a static phasor calculation, not a transient-stability simulation.
Core load-flow facts:

1. A load flow study enforces real- and reactive-power balance at the buses of the network model.
2. It tests a specific operating case defined by loads, generator schedules, voltage setpoints, branch status and transformer taps.
3. In the standard AC formulation, nonlinear algebraic equations relate bus voltage magnitudes and phase angles to real and reactive power injections. Reference, PV and PQ bus types determine which quantities are specified and which are unknown.
4. Iterative methods such as Newton-Raphson, fast-decoupled power flow and Gauss-Seidel reduce the bus power mismatches until the chosen tolerance is met. Convergence must be checked; a numerical result is not valid merely because the solver stopped.
5. Typical results include bus voltage magnitudes and angles, generator reactive power, real and reactive flow at both ends of each branch, network losses and the balancing real power at the reference or slack bus.

Load Flow Steps

A reproducible load-flow study has three broad stages:

1. Build a consistent per-unit network model and identify the reference, PV, PQ and isolated buses.
2. Form the bus-admittance matrix and the real- and reactive-power balance equations for the selected formulation.
3. Solve the load flow equations, check convergence and limits, then calculate branch flows, losses and remaining generator injections.

Modeling of Power System Components

Generator

A generator is represented as a complex power injection at its bus. At a PV bus, scheduled real power and voltage magnitude are specified while reactive power and voltage angle are solved, subject to the model’s limits and control logic.

Load


A basic load-flow case treats demand as specified real and reactive power. More detailed tools may also use current- or impedance-dependent load models.Transmission Line
A Transmission line is commonly represented by a nominal π equivalent for steady-state analysis.

The series branch contains R + jX. The shunt charging admittance is divided between the two line ends as Y/2.

Off Nominal Tap Changing Transformer
A nominal-ratio transformer follows the voltage relation shown here:
An off-nominal transformer introduces a tap ratio into the branch model.

The transformation ratio a is defined by the convention shown below.

The tapped transformer and series branch can be represented by an equivalent network between its terminal buses.

Figure 2: line containing an off-nominal transformer.
The equivalent π representation between buses p and q is shown next.

Figure 3: equivalent π model of the branch.

The required admittances Y₁, Y₂ and Y₃ must give the same terminal current-voltage relationship in Figures 2 and 3.
For Figure 2, write the two terminal-current equations:


For Figure 3, write the equivalent equations:

Comparing the coefficients of E_p and E_q in equations I and III gives:

Comparing equations II and IV gives the remaining terms:

The resulting relations provide useful checks on the equivalent model:

The values of Y₂ and Y₃ can change sign with the off-nominal tap ratio.

Interpret the sign through the susceptance and the stated injection convention, not through a general complex admittance alone.
Negative shunt susceptance corresponds to inductive reactive-power absorption, as for an inductor.
Positive shunt susceptance corresponds to capacitive reactive-power injection, as for a capacitor.
Modeling of a Network

The figure shows a two-bus system used to form nodal balance equations.
Use one consistent sign convention for generation, demand and net injection.
Generated complex power at bus i is:

Demanded complex power at bus i is:


The net injected power is generation minus demand under this convention: