# Simple Power Flow Example

Consider the simple model shown in Figure 1, where a large stiff network supplies a constant power load through an impedance :

Suppose the load power is known and we want to calculate the load bus voltage . Unfortunately, this cannot be computed in a straightforward manner because the load is of constant power and thus the load current and impedance are voltage dependent, i.e. the load draws more current as voltage decreases. As a result, the load bus voltage is non-linearly related to the load itself.

## Derivation of the Load Bus Voltage

Note that in this derivation, all quantities are in per-unit. Recall that the load complex power can be calculated from the voltage and current phasors as follows:

Suppose that we represent both the load power and impedance in polar coordinates, i.e.

Then the power equation can be re-written as:

Conjugating the terms in brackets and simplifying, we get:

Applying Euler's law, we get:

Separating the real and imaginary terms of the above equation:

Squaring both equations and summing them together, we get:

Simplifying and re-arranging the equation above, we can get the following homogenous equation:

This equation can be solved for the load bus voltage using the quadratic formula:

- ... Equ. (1)

Where

and

## Worked Example

Suppose that the source bus has a nominal voltage of 33kV and a short circuit level of 800MVA at X/R ratio of 10. What is the voltage at the load bus if it is supplying a constant 50MW load at 0.8pf (lagging)?

The source impedance is:

Converted to per-unit values (on a 100MVA base):

For an X/R ratio of 10, the source impedance is therefore:

- (angle in radians)

The 50MW load converted to per unit is (by convention, a load with lagging power factor has a negative reactive power):

- (angle in radians)

Plugging the parameters , , and into Equation (1), we get the load bus voltage:

## Intuition for General Power Flow Solutions

We saw in this simple example that the power flow problem for constant power loads is non-linear (i.e. quadratic) and cannot be solved with linear techniques. This intuition can be extended to more general power flow problems. While a closed form solution was found for this simple case, the general power flow problem is typically solved using iterative numerical methods, for example a Newton-Raphson algorithm.