Modelling of energy service networks - Modelling Distributed Energy Resources in Energy Service Networks - page 34

Y kl = Y lk =− y kl (3.14)

Y lm = Y ml =− y lm (3.15)

Y km = Y mk =− y km (3.16)

The sum of admittances further simplifies the expressions for nodal currents:

I k = Y kk V k + Y kl V l + Y km V m (3.17)

I l = Y lk V k + Y ll V l + Y lm V m (3.18)

I m = Y mk V k + Y ml V l + Y mm V m (3.19)

Furthermore, (3.17)-(3.19) can be represented with matrices as:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤

I k
I l
I m

Y kk Y kl Y km
Y lk Y ll Y lm
Y mk Y ml Y mm

V k
V l
V m

⎣ ⎦ = ⎣ ⎦ ⎣ ⎦

(3.20)

Hence, generic results from the i -th to the j -th node of a system establish that
nodal current equations can be presented in matrix form as:
⎡
⎣

⎤
⎦ =

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

I i
.
I j

Y ii ··· Y ij V i

.
V j

. .

. . .

(3.21)

Y ji ··· Y jj

The values of the admittance matrix from the i -th to the j -th element can be
calculated from the following algorithm:

⎧
⎨ j , then Y ii = y is the sum of all admittances connected to

node i
if i

if i =

j , and node i is not connected to node j , then the element
Y ij =

=

Y node =

⎩

0

if i j , and node i is connected to node j , then the element

Y ij =−

=

y ij

Consequently, the matrix from (3.21) can be generalised into:
I node = Y node V node

(3.22)

The terms from (3.22) can be described as:

I node is the vector of injected nodal currents. Whenever current is flowing towards
the node it is regarded as positive, while it is considered negative if the current
is moving away from the node.

●

Y node is the nodal admittance matrix and it comprises two sets of items:
-

●

The diagonal element known as the self-admittance ( i.e. Y ii ).

- The off-diagonal element called the mutual admittance ( i.e. Y ij ).

V node is the vector of nodal voltages measured with respect to the slack node.

●