Non-linear algebraic equations and their solution

In the next lecture, we will compute the steady state frequency of a power system, given the load characteristics. We shall see that in general, this will require us to carry out a "loadflow". A loadlfow involves the solution of a set of non-linear algebraic equations. Therefore, in this lecture we revise the basic methods to solve non-linear algebraic equations.

We are aware that a transmission network in sinusoidal state state can be modelled by linear algebraic equations in the node voltage phasors(V) and the nodal current phasor injections (I):

where Ybus is the bus admittance matrix.

However, in power system studies, nodal injections are not specified as current phasors but as real and reactive power injections (nonlinear functions of V and I) , and/or voltage magnitudes of some nodes. We have also seen that real and reactive power can be a function of frequency. In such a situation, obtaining the steady state solution (i.e. node voltage phasors and frequency) will require us to solve a set of non-linear equations.

Therefore we take a silight diversion from the main theme and review why and how we use numerical techniques for solving non-linear algebraic equations.

Let us consider the "why" question first. If we wish to solve an equation of the form:

Perhaps, "simplifying" it will help us solve it ?

Perhaps, if we take the natrual logarithm of both sides we may be able to do something ?

But soon enough you will realize that we seem to be getting nowhere !

It is clear that some other way (guess work ?) may be required to get the solution.