Page 158 - Power Electronic Control in Electrical Systems

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146 Power flows in compensation and control studies

The active and reactive powers for the series converter are derived as follows:

S cR ˆ P cR ‡ jQ cR ˆ V cR I ˆ V cR Y V ‡ Y mm V ‡ Y mm V cR (4:92)

m
l
ml
m
2
j
P cR ˆ V cR j G mm ‡ V cR kV l j G ml cos (y cR y l ) ‡ B ml sin (y cR y l )f g
j
(4:93)
‡ V cR kV m j G mm cos (y cR y m ) ‡ B mm sin (y cR y m )g
j
f
2
j
f
Q cR ˆ jV cR j B mm ‡ V cR kV l j G ml sin (y cR y l ) B ml cos (y cR y l )g
(4:94)
j
‡ V cR kV m j G mm sin (y cR y m ) B mm cos (y cR y m )g
f
The active and reactive powers for the shunt converter are derived as follows

S vR ˆ P vR ‡ jQ vR ˆ V vR I vR ˆ V vR Y vR V vR V l (4:95)
2
P vR ˆ jV vR j G l0 ‡ V vR kV l j G l0 cos (y vR y l ) ‡ B l0 sin (y vR y l ) (4:96)
j
f
2
Q vR ˆ V vR j B l0 ‡ V vR kV l j G l0 sin (y vR y l ) B l0 cos (y vR y l )g (4:97)
j
f
j
Assuming lossless converters, the UPFC neither absorbs nor injects active power
with respect to the AC system. Hence, the following constraint equation must be
satisfied
P vR ‡ P cR ˆ 0 (4:98)
This is a complex model, which imposes severe demands on the numerical algorithms
used for its solution. Since in power systems planning and operation reliability
towards the convergence is the main concern, it is recommended that the Newton±
Raphson algorithm be used for its solution (Fuerte-Esquivel et al., 2000). In this
method the UPFC state variables are combined with the network nodal voltage
magnitudes and phase angles in a single frame-of-reference for a unified, iterative
solution. The UPFC state variables are adjusted automatically in order to satisfy
specified power flows and voltage magnitudes.
Following the general principles laid out in Section 4.4.4, the relevant equations in
(4.88)±(4.98) are derived with respect to the UPFC state variables. Equation (4.44) is
suitably modified to incorporate the linearized equation representing the UPFC
contribution. The UPFC is a very flexible controller and its linearized system of
equations may take several possible forms. For instance, if nodes l and m are the
nodes where the UPFC and the power network join together and the UPFC is set to
control voltage magnitude at node l, active power flowing from node m to node l and
reactive power injected at node m, then the following linearized equation shows the
relevant portion of the overall system of equations
2 3
@P l @P l @P l @P l @P l @P l @P l
@y l @y m @jV vR j @jV m j @y cR @jV cR j
2 3 @y vR 2 3
P l 6 @P m @P m 0 @P m @P m @P m 7 y l
j
6 0 7
@y l @y m @jV m j @y cR @ V cR j
6 P m 7 6 76 y m 7
@Q l @Q l @Q l @Q l @Q l @Q l
6 7 6 @Q l 76 7
j
j
j
Q l @y l @y m @ V vR j @ V m j @y cR @ V cR j @y vR
j
6 7 6 76 V vR j 7
6 7 6 76 7
@Q m @Q m @Q m @Q m @Q m
j
Q m 0 76
6 7 ˆ 6 0 V m j 7 (4:99)
j
j
@y l @y m @ V m j @y cR @ V cR j
6 7 6 76 7
0
6 7 6 @P ml @P ml @P ml @P ml @P ml 76 7
@y l @y m @ V m j @y cR @ V cR j
6 P ml 7 6 0 76 y cR 7
j
j
j
Q ml
4 5 6 74 V cR j 5
@Q ml @Q ml @Q ml @Q ml @Q ml
6 0 0 7
j
j
P bb 4 @y l @y m @ V m j @y cR @ V cR j 5 y vR
@P bb @P bb @P bb @P bb @P bb @P bb @P bb
@y l @y m @ V vR j @ V m j @y cR @ V cR j @y vR
j
j
j