Environmental Engineering Reference
In-Depth Information
The generators employ constant PQ models ( i.e. injected powers into the sys-
tem); therefore, their internal admittance does not enter the corresponding nodal
matrix formulation;
The load on each node of the system is assumed to be a three-phase balanced
load.
3.1.3 Nodal formulation and the admittance matrix
The nodal method to calculate load flows consists of detailing equations that estab-
lish the sum of all net power injections within each node in the system must be equal
to 0. These equations are commonly referred to as power 'mismatches' or 'residuals'
and at node k for both active and reactive power they take the following forms:
P k
=
P Gk
P Dk
P Tk =
P Rk
P Tk =
0
(3.1)
Q k
=
Q Gk
Q Dk
Q Tk =
Q Rk
Q Tk =
0
(3.2)
The terms P Gk and Q Gk represent, respectively, the generator injections of real
and reactive power at node k . In load flow settings it is assumed that these variables
are known beforehand by the network operator. This rule is only exempted when
referring to the slack node, in which case the slack generator produces sufficient
power to supply required load and losses, originally unknown variables of the problem.
Meanwhile, P Dk and Q Dk are, respectively, the input data for real and reactive power
load demands at node k . Since the generation and load can be measured by the
electric utility, their net values are the required power in which supply meets demand:
P Rk
=
P Gk
P Dk
(3.3)
Q Rk
=
Q Gk
Q Dk
(3.4)
Therefore, the electric load flow equations focus on computing the transmitted
real and reactive power injections, P Tk and Q Tk , as functions of nodal voltages and
network impedances. In order to effectively calculate the load flows in the lines,
the nodal voltages need to be known with a good degree of accuracy. A solution
to the problem is reached when the equality constraints from (3.1) and (3.2) are
satisfied. However, if the nodal voltage values are not precise, then the power flowing
through the lines will be inexact. Thus, the power mismatches will not be 0. This
is why load flow problems employ iterative numerical techniques in order to correct
and reduce the difference in the mismatch value. In modern load flow computer
programs, it is ordinary for mismatch expressions to satisfy a tolerance very close
to0( e.g. 1e-10) before the iterative solution can be recognised as a success. Once
a solution is reached, important data is obtained about the steady-state operating
conditions of the electrical system and are commonly referred to as state variables. If
the load flow is done for multiple periods, then multiple snap shots of how the system
performs can be determined.
A key step in solving electrical load flow problems is to classify all the nodes
of the system in order to build the nodal admittance matrix . This matrix contains
 
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