Page 143 - Power Electronic Control in Electrical Systems

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Power electronic control in electrical systems 131

According to equations (4.53) and (4.54), the mismatch power equations are
(1) " #
net calc
P 1 P 1 P 1 ?
ˆ ˆ p:u:
P 2 P net P calc 1
2 2
(1) " # (4:58)
net calc
Q 1 Q 1 Q 1 ?
ˆ ˆ p:u:
Q 2 Q net Q calc 0:5
2 2
The Slack node plays no role during the iterative solution. Using equations (4.45)±
(4.52), and the numeric values P calc ˆ 0 and Q calc ˆ 0 calculated above, the Jacobian
2
2
elements for node two are
@P 2 @P 2 @Q 2 @Q 2
ˆ 10 ˆ 0 ˆ 0 ˆ 10
j
j
@y 2 @ V 2 j @y 2 @ V 2 j
The solution of the linearized system of equations is
(1) (1) (1)
P 2 10 0 y 2 y 2 0:10 1 0:1
ˆ ) ˆ ˆ (4:59)
j
j
Q 2 010 V 2 j V 2 j 00:1 0:5 0:05
Using equations (4.55) and (4.56) the voltage magnitude and phase angle at the end
of the first iteration are
(1) (0) (1)
y 2 0 0:1 0:1
ˆ ‡ ˆ p:u: (4:60)
j V 2 j 1 0:05 0:95
It should be noted that the voltage magnitude is in p.u. and the phase angle is in
radians.

Second iteration
Using updated voltage information, i.e. V 1 j ˆ 1, V 2 j ˆ 0:95, y 1 ˆ 0and y 2 ˆ 0:1,
j
j
in equations (4.42) and (4.43), new active and reactive powers are calculated, giving
the following result
P calc ˆ 0:94841 p:u:
2
Q calc ˆ 0:4275 p:u:
2
The mismatch power equations are

(2) " #
net calc
P 1 P 1 P 1 ? ? ?
ˆ ˆ ˆ p:u:
P 2 P net P calc 1 0:948417 0:051583
2 2
(2) " #
net calc
Q 1 Q 1 Q 1 ? ? ?
ˆ net ˆ ˆ p:u:
Q 2 Q 2 Q calc 0:5 0:42754 0:07246
2
(4:61)
It should be noted that the power mismatches have decreased by almost one order of
magnitude compared to the mismatch values obtained during the first iteration.