Modelling of energy service networks - Modelling Distributed Energy Resources in Energy Service Networks - page 41

● K rnp is the reduced nodal-pipe incidence matrix since it does not consider the
slack node;
● F pipe is the vector of flows in pipes as a function of the upstream and downstream
pressures.
Since pipe flows are not usually known in gas networks, they need to be calcu-
lated from the pressure drop equations. Consequently, the pressure changes in the
pipe elements are related to the nodal pressure values, and for Figure 3.4 they can be
defined as:
p kl

= p k − p l (3.44)

p lm = p l − p m (3.45)

p km = p k − p m (3.46)

The pressure drop equations can be represented in matrix form as:
⎡

⎤ ⎡ ⎤ ⎡ ⎤

p kl
p lm
p km

− 1 + 10 p k

p l
p m

⎣ ⎦ =− ⎣ ⎦ ⎣ ⎦

0 − 1 + 1 (3.47)

− 10 + 1

As a result, formula (3.47) can have the following general form:
p pipe =− K np p node (3.48)

The terms from (3.48) can be described as:
● p pipe is the vector of pressure drops in the network pipes;
● K np is the transpose of the nodal-pipe incidence matrix;
● p node is the vector of nodal pressures in the network.
Taking the premises from (3.37), the flow in pipes can be defined as a set of
pressure drop functions. This fact permits us to link the vector of pressure drops
stated in expression (3.48) with the gas flowing in the pipes (3.37), hence resulting in:
F pipe =

ζ p pipe = ζ − K np p node

(3.49)

where ζ ( p pipe ) is the vector of pressure drop functions.
Subsequently, term (3.49) can be combined with expression (3.43), and as a
consequence establishes a relationship among the nodal pressures and the nodal
flow injections in the system. Thus, the net nodal flow injections at node k can be
determined as:
F Tk =

K np p node (3.50)
According to (3.34), expression (3.50) allows us to rewrite the mismatch flow
equation as:
F k =

K rnp ζ −

F node =

F Rk − K rnp ζ − K np p node = 0 (3.51)

Acceptable results from the gas load flow studies are achieved when the iterative
solver reduces the flow mismatch terms below a specified tolerance value ( e.g. ),
stated for node k as:
|

F k | ≤

F (3.52)