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126 Power flows in compensation and control studies

the Newton±Raphson algorithm and derived formulations have been used instead
(Tinney and Hart, 1967; Peterson and Scott-Meyer, 1971).
The issue of convergence has become even more critical today, with the wide range
of power systems controllers that need inclusion in power flow computer algorithms
(Fuerte-Esquivel, 1997; Ambriz-Perez, 1998). As outlined in Section 4.2, the new
controllers regulate not just voltage magnitude but also voltage phase angle, imped-
ance magnitude and active and reactive power flow. It is unlikely that the nodal
impedance based method would be able to cope with the very severe demands
imposed by the models of the new controllers on the power flow algorithms. Inten-
sive research work has shown the Newton±Raphson algorithm to be the most reliable
vehicle for solving FACTS upgraded networks (Acha et al., 2000; Ambriz-Perez
et al., 2000; Fuerte-Esquivel et al., 2000), with other formulations derived from the
Newton±Raphson method becoming viable but less desirable alternatives (Acha,
1993; Noroozian and Andersson, 1993).

4.4.4 Newton±Raphson power flow method
The equation describing the complex power injection at node l is the starting point
for deriving nodal active and reactive power flow equations suitable for the Newton±
Raphson power flow Algorithm (Tinney and Hart, 1967)

S l ˆ P l ‡ jQ l ˆ V l I (4:37)
l
where S l is the complex power injection at node l,
P l is the active power injection at node l,
Q l is the reactive power injection at node l,
V l is the complex voltage at node l and
I l is the complex current injection at node l.
The injected current I l may be expressed as a function of the currents flowing in the
n branches connected to node l,
n
X
I l ˆ Y lm V m (4:38)
m ˆ 1
where Y lm ˆ G lm ‡ jB lm and Y lm , G lm and B lm are the admittance, conductance and
susceptance of branch l±m, respectively.
Substitution of equation (4.38) into equation (4.37) gives the following inter-
mediate result
n
X

P l ‡ jQ l ˆ V l Y V m (4:39)
lm
m ˆ 1
Expressions for the active and reactive powers are obtained by representing the
complex voltages in polar form, V l ˆjV l je jy l and V m ˆjV m je jy m
n
X j(y l y m )
P l ‡ jQ l ˆjV l j jV m j(G lm jB lm )e (4:40)
m ˆ 1
n
X
P l ‡ jQ l ˆjV l j jV m j(G lm jB lm )fcos (y l y m ) ‡ j sin (y l y m )g (4:41)
m ˆ 1