Modelling of energy service networks - Modelling Distributed Energy Resources in Energy Service Networks - page 40

The pressure drop between nodes k and m can be stated as [171]:

K km F km p km = p k − p m = (3.36)

⎧
⎨

if low-pressure network, then q 2

if medium-pressure networks, then q

=

where = 1 . 848

⎩

if high-pressure networks, then q = 1 . 854

Since a change in the flow direction of the gas stream might take place if an
assumption is erroneous, (3.36) can be rearranged into:
F km = σ km σ km ( p k −

1 /q

p m )

(3.37)

K km

where if p k > p m , then σ km =

1
if p k < p m , then σ km =− 1

The connectivity data of the system is known in the literature as the nodal-pipe
incidence matrix, and for Figure 3.4 it has the following composition:

⎡ ⎤

− 10 − 1

⎣ ⎦

K np = + 1 − 10 (3.38)

0 + 1 + 1

In formulating the nodal equations for Figure 3.4, the convention employed
considers incoming flow injections into nodes as positive, and therefore yields:

− F k =− F kl − F km (3.39)

F l = F kl − F lm (3.40)

F m = F km + F lm (3.41)

From the above set of equations, F k has a negative sign because it is the slack
node and otherwise the nodal balance at k would not be possible. However, the slack
node term is eliminated due to the fact that summing all pipe flows from the three
terms yields 0. This creates the necessity of dismissing (3.39) and as a consequence
the remaining expressions (3.40) and (3.41) take the following matrix form:
F l
F m

⎡
⎣

⎤

+

F kl
F lm
F km

1 − 10

⎦

= (3.42)

+ +

0 1 1

This allows us to generalise the net nodal injection equations into the matrix
form:

F node = K rnp F pipe (3.43)

The terms from (3.43) can be described as:

F node is the vector of net nodal injections. Whenever flow moves towards the node
it is regarded as positive, while it is considered negative if the flow moves away
from the node;

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