# Switching

The various representations are shown in Figure 10.37. A switching operation changes the topology of the network and hence the [Y] matrix. If the [Y] matrix is formed with all the switches open, then the closure of a switch is obtained by the adding together of the two rows and columns of [Y] and the associated rows of [i].

**Figure 10.36 **The equivalent circuit of Figure 10.35

Another area where switching is used is to account for non-linear C — i characteristics of transformers, reactors, and so on. The representation is shown in Figure 10.38(a) and (b), in which

**Figure 10.37 **Representation of switch

and

where *bt* = incremental inductance.

If C is outside of the limits of segment K, the operation is switched to either k *—* 1 or k + 1. This changes the [Y] matrix.

Because of the random nature of certain events, for example switching time or lightning incidence, Monte Carlo (statistical) methods are sometimes used. Further information can be obtained from the references at the end of this book.

# Travelling-Wave Approach

Lines and cables would require a large number of n circuits for accurate representation. An alternative would be the use of the travelling-wave theory.

Consider Figure 10.39,

where *U =* speed of propagation and Z0 = characteristic impedance. At node k,
where

**Figure 10.40 **Equivalent circuit for single-phase line

and

The equivalent circuit is shown in Figure 10.40. An advantage of this method is that the two ends of the line are decoupled. The value of *I _{k}* depends on the current and voltage at the other end of the line t seconds previously, for example if

*t =*0.36 ms and At

*=*100 m

^{s}, storage of four previous times is required.

Detailed models for synchronous machines and h.v.d.c. converter systems are given in the references.