Modelling of energy service networks - Modelling Distributed Energy Resources in Energy Service Networks - page 33

k l

z kl

i lk

i kl

I k

I l

i lm

i km

y ko i ko y lo

i lo

i mk i ml

z lm

z km

m

i mo

I m

y mo

Figure 3.2 Electrical test network employed to develop the admittance matrix

relevant data regarding connectivity and line characteristics of the system. The con-
siderations taken if the admittance matrix is formed taking as a reference the three
node power system shown in Figure 3.2 are:

All nodal voltages are bus-to-ground values;

●

I k , I l and I m are the net injected nodal currents, respectively occurring, at nodes
k , l and m ;

●

Although node k is the slack bus, the nodal currents flow into the system
from multiple elements ( e.g. generators) but these are not shown for simplicity
purposes;

●

z kl , z lm and z km are the positive sequence impedances between nodes;

●

y ko , y lo and y ko are the positive sequence shunt admittances for the Pi-section.

●

The admittance value is the inverse of the impedance. If we apply KCL to the
network, the following sets of equations represent the system from Figure 3.2:

I k = i kl + i km + i ko = y kl ( V k − V l ) + y km ( V k − V m ) + y ko ( V k ) (3.5)

I l = i lk + i lm + i lo = y lk ( V l − V k ) + y lm ( V l − V m ) + y lo ( V l ) (3.6)

I m = i mk + i ml + i mo = y mk ( V m − V k ) + y ml ( V m − V l ) + y mo ( V m ) (3.7)

The above set of equations can be revised as:

I k = ( y kl + y km + y ko ) V k − y kl V l − y km V m (3.8)

I l =− y lk V k + ( y lk + y lm + y lo ) V l − y lm V m (3.9)

I m =− y mk V k − y ml V l + ( y mk + y ml + y mo ) V m (3.10)

Thus, the admittances for the matrix model (with capital letters) can be defined:

Y kk = y kl + y km + y ko (3.11)

Y ll = y lk + y lm + y lo (3.12)

Y mm = y mk + y ml + y mo (3.13)