Electrical Power Systems

158 Electrical Power Systems
Q ik = |V i| 2 |Y ik |sin q ik – |V i||V k||Y ik|sin (q ik – d i + d k )
Similarly power flows from bus k to i can be written as:
–|V i| 2 |y o ik | ...(7.35)
P ki = –|V k | 2 |Y ik |cos q ik + |V i ||V k ||Y ik |cos (q ik – d k + d i ) ...(7.36)
Q ki = |V k | 2 |Y ik |sin q ik – |V i ||V k ||Y ik |sin (q ik – d k + d i )
–|Vk | 2 |yo ki | ...(7.37)
Now real power loss in the line (i ® k) is the sum of the real power flows determined from
eqn. (7.34) and (7.36)
\ PLoss ik = P ik + P ki
\ PLoss ik = –|V i | 2 |Y ik |cos q ik +|V i ||V k ||Y ik |cos (q ik – d i + d k )
–|V k | 2 |Y ik |cos q ik +|V i ||V k ||Y ik |cos (q ik – d k + d i )
= – (|V i | 2 +|V k | 2 )|Y ik |cos q ik
+ |V i||V k||Y ik|[ cos { q ik – (d i – d k )} + cos {q ik + (d i – d k )}]
= – (|V i| 2 +|V k| 2 )|Y ik|cos q ik + 2|V i||V k||Y ik|cosq ik cos (d i – d k )
\ PLoss ik = [2|V i||V k|cos (d i – d k ) –|V i| 2 – |V k| 2 ]|Y ik|cosq ik ...(7.38)
Let
Y ik = G ik + jB ik
G ik =|Y ik | cosq ik
B ik =|Y ik | sinq ik
\ PLoss ik = G ik [2|V i||V k|cos (d i – d k ) –|V i| 2 – |V k| 2 ] ...(7.39)
Reactive power loss in the line (i ® k) is the sum of the reactive power flows determined
from eqns. (7.35) and (7.37), i.e.
QLoss ik = Q ik + Q ki
\ QLoss ik = |V i| 2 |Y ik|sin q ik –|V i||V k||Y ik|sin(q ik – d i + d k )
–|V i| 2 |y o ik| +|V k| 2 |Y ik|sinq ik
– |V i||V k||Y ik|sin( q ik – d k + d i ) – |V k| 2 |y o ki|
\ QLoss ik = (|V i| 2 +|V k| 2 ) B ik – |V i||V k||Y ik|[ sin(q ik – d i + d k ) + sin(q ik – d k + d i)]
– (|V i| 2 |y o ik| +|V k| 2 |y o ki|)
\ QLoss ik = (|V i| 2 +|V k| 2 ) B ik – 2|V i||V k| B ik cos (d i – d k ) – (|V i| 2 |y o ik| +|V k| 2 |y o ki|)
\ QLoss ik = B ik [|V i| 2 + |V k| 2 –2|V i||V k|cos ( d i – d k)] – [|V i| 2 |y o ik| +|V k| 2 |y o ki|]
...(7.40)
7.10 ALGORITHM OR GAUSS-SEIDEL METHOD
Step-1: Initial Computation
With the load profile known at each bus (i.e. PL i and QL i are known), allocate Pg i and Qg i to all
generating units. While active and reactive generations are not allocated to the slack bus, these