Environmental Engineering Reference
In-Depth Information
Once the mismatch (3.31) and (3.32) are met, the final variation of the state
variables for node k can be updated; for the i -th iteration this is analogous to term
(3.58):
1
i + 1
V k
V k
V i + 1
k
V k
=
+
(3.69)
θ i + 1
k
θ k + θ i + 1
=
(3.70)
k
After the voltage magnitudes and phase angles have been determined, subse-
quently active and reactive power flows are obtained. Finally, once the flows and
losses in power lines are computed, the net power injections provided by the slack
node can be calculated. Figure 3.5 presents a flow diagram of the basic load flow
algorithm for electrical systems.
3.3.2.2 The natural gas system Jacobian matrix
For the natural gas load flow, the state variables of the system represented by x 0 in
(3.56) are the nodal pressure magnitudes. Hence, the Newton-Raphson algorithm
can be presented by the following vector relationship:
δF
δp
1
p =−
F
(3.71)
The number of elements in the Jacobian matrix for natural gas systems consists
of the number of load nodes minus the slack node. Terms in the natural gas sys-
tem regarding flow and pressure drops in pipes are employed to build the Jacobian
matrix; this expression takes the following form:
K rnp DK rnp
J
=−
(3.72)
The products give the Jacobian matrix both square and symmetrical proper-
ties, where term D represents a diagonal matrix and for pipe km it can be expressed
as [171]:
diag 1
F
p
D
=
q ·
(3.73)
Two distinguished cases arise when building the first partial derivatives: the diag-
onal and non-diagonal elements. First, the diagonal elements of the matrix are related
to a specific load node and consist of the sum of the expressions F/p for all the pipes
connected to that node. Second, each off-diagonal element is associated to connectiv-
ity between nodes, consisting of the negative expression
F/p for the branch that
connects the two nodes. Henceforth, the diagonal components are always positive;
meanwhile, the non-diagonal components are always negative.
In general, once the mismatch term (3.52) is met, the final variation of the state
variables for node k can be updated; for the i -th iteration this is analogous to (3.58):
p i + 1
k
p i k +
p i + 1
k
=
(3.74)
 
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