Consider the simple model shown in Figure 1, where a large stiff network supplies a constant power load through an impedance :
Figure 1. Simple power flow model (note that all quantities are in per-unit
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)
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.