Page 140 - Power Electronic Control in Electrical Systems

//SYS21/F:/PEC/REVISES_10-11-01/075065126-CH004.3D ± 128 ± [106±152/47] 17.11.2001 9:54AM

128 Power flows in compensation and control studies

y l and jV l j are the incremental changes in nodal voltage magnitude and phase
angle at node l,
(r) represents the r-th iterative step and l ˆ 2, 3, 4, ... n.
The elements of the Jacobian matrix can be found by differentiating equations
(4.42) and (4.43) with respect to y l , y m , jV l j and jV m j.
For the case when l ˆ m:
n
@P l X 2 2
ˆjV l j jV m jf G lm sin (y l y m ) ‡ B lm cos (y l y m )g jV l j B ll ˆ Q l jV l j B ll
@y l
m ˆ l
(4:45)
n
@P l X P l
ˆ jV m jfG lm cos(y l y m ) ‡ B lm sin(y l y m )g‡jV l jG ll ˆ ‡jV l jG ll (4:46)
@jV l j jV l j
mˆl
n
@Q l X 2 2
ˆjV l j jV m jfG lm cos (y l y m ) ‡ B lm sin (y l y m )g jV l j G ll ˆ P l jV l j G ll
@y l
m ˆ l
(4:47)
n
@Q l X Q l
ˆ jV m jfG lm sin (y l y m ) B lm cos (y l y m )g jV l jB ll ˆ jV l jB ll
@jV l j jV l j
m ˆ l
(4:48)
For the case when l 6ˆ m
@P l
ˆjV l jjV m jfG lm sin (y l y m ) B lm cos (y l y m )g (4:49)
@y m
@P l 1 @Q l
ˆjV l jfG lm cos (y l y m ) ‡ B lm sin (y l y m )gˆ (4:50)
@jV m j jV m j @y m
@Q l
ˆ jV l jjV m jfG lm cos (y l y m ) ‡ B lm sin (y l y m )g (4:51)
@y m
@Q l 1 @P l
ˆjV l jfG lm sin (y l y m ) B lm cos (y l y m )gˆ (4:52)
@jV m j jV m j @y m
To start the iterative solution, initial estimates of the nodal voltage magnitudes and
phase angles at all the PQ nodes and voltage phase angles at all the PV nodes are
given to calculate the active and reactive power injections using equations (4.42±
4.43). Since it is unlikely that the initial estimated voltages will agree with the voltages
at the solution point, the calculated power injections will not agree with the known
specified powers.
The mismatch power vectors may be defined as

DP (r) ˆ (P gen P load ) P calc,(r) ˆ P net P calc,(r) (4:53)
DQ (r) ˆ (Q gen Q load ) Q calc,(r) ˆ Q net Q calc,(r) (4:54)
The Jacobian elements are then calculated and the linearized equation (4.44) is solved
to obtain the vectors of voltage updates
q (r ‡ 1) ˆ q (r) ‡ Dq (r) (4:55)

(r ‡ 1) (r) (r)
jVj ˆjVj ‡ DjVj (4:56)