Problems with Iterative Load Flow

To illustrate the issues with Newton-Raphson iterative mathematics when performing load flow calculations, it's convenient to use a simple two-bus network consisting of one load bus and one swing bus. Since there are only two buses in the network, the mathematics are vastly simplified.

See a complete description of the mathematical formulae.

The swing bus by definition has a fixed voltage value (V) of 1 per unit (pu) and a phase angle (δ) of 0 degrees regardless of the power flowing to the load bus. The complex impedance of the line connecting the two buses is R + jX. The load flow calculation challenge is to determine the value of V and δ at the load bus as we vary real power (P) and reactive power (Q) at the load bus.

A Java-based program that accomplishes this using Newton-Raphson mathematics has been developed and was used to produce the screen seen on the right.

In the new diagram, we've run the Newton-Raphson load flow using values of P=-0.8 and Q=-0.2. The calculation is performed many times, using each possible starting point in the voltage-phase space once. The color of that starting point is then changed to reflect the outcome.

The point is colored GREEN if the calculation converges to the correct physical solution.

The point is colored PURPLE if the calculation converges to an incorrect solution.

The point is colored RED if the calculation converges to a spurious solution.

The point is colored BLACK if the Newton-Raphson calculation won't converge with that starting point.

Note that a spurious solution is mathematically correct, but it cannot be realized in a physical system. An incorrect solution is neither spurious or correct.

We can summarize that in this example, Newton-Raphson converges to the correct solution reliably if the starting point is close to the solution, and that it will provide an incorrect answer otherwise.

Now we've changed the values of P and Q in the load bus to -0.8 and -0.8, respectively. This moves the spurious solution closer to the correct solution (if they were in the same spot, the system would be at voltage collapse, so in effect, we're moving the system closer to voltage collapse). Note that in general, we still get pretty good convergence to the correct solution if we use a starting point in the vicinity of the correct solution, and that we'll converge to an incorrect solution otherwise.

Additionally, Newton-Raphson will now give spurious answers for some of the starting points for the values of P and Q we've chosen.

The P and Q values have now been changed to -0.8 and -1.05, respectively. Again, the correct solution and the spurious solution are closer together than before, and the system is closer to voltage collapse.

Interestingly enough, for these values a randomly-selected point would more often produce a correct result, though to get a reliably correct answer you must choose a starting point very close to the correct answer.

Notice also that for the first time, the Newton-Raphson calculation won't converge at all for certain starting points (the black area in the graph).

In our final graph, we've changed the values of P and Q to -0.8 and -1.115, respectively. We've essentially increased the VAR flow so that the system is almost at voltage collapse.

Note that for the most part, the Newton-Raphson methodology of calculating load flow won't work at all in the vicinity of voltage collapse.