By Kankesu Jayanthakumaran (Editor)
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1993). 672, m0 = 60, mi = 7 × 10 −3 i, j = 0, , N i = 1, , N Initial data: x0 (0 ) = y0 (0 ) = x0′ (0 ) = y0′ (0 ) = 0 . 8 cos +1 100i 100i N = 101 , 0 ≤ t ≤ b , b = end of the interval. , 2000). The performance of the sequential and parallel execution times for every problem is shown in Table 1 – 5 while Table 6 shows the speed up and efficiency performance for the problems. The notations are defined as follows: TOL MTD TS FS FCN MAXE Tolerances Method employed Total number of steps Failure steps Total function calls Magnitude of the global error (max yn − y (xn ) TIME(min) TIME(sec) S2PFDIR P2PFDIR The execution time in minutes The execution time in seconds Sequential implementation of the two point implicit block method Parallel implementation of the two point implicit block method In the code, we iterate the corrector to convergence.
For each partial derivative of line flow, 2 trigonometric functions are to be evaluated. Meanwhile, to compute calculated powers, 4 numbers of line flows, shown in Eq. 6 through Eq. 9 are to be computed. For each calculated power, 2 trigonometric functions are to be evaluated. Therefore the number of trigonometric functions that are to be evaluated is equal to 32 Nbr + 8 Nbr. 4 N, total number of trigonometric functions that are to be evaluated = 56 N (38) Based on Eq. (37) and Eq. (38), we can conclude that the proposed method uses significantly less number of mathematical operations and hence takes less CPU time particularly for larger networks compared to the NRSE method.
For such a general transmission network element, the real and reactive power flows are given by the following expressions. Newton-Raphson State Estimation Solution Employing Systematically Constructed Jacobian Matrix δ3 δ2 V1 V1 2 V2 3 V2 2 VN 3 VN 2 H= 3 pij i p ji i qij i qji i P1 2 P2 2 PN 2 P1 3 P2 3 PN 3 Q 1 2 Q 2 2 Q N 2 Q 1 3 Q 2 3 Q N 3 δN V1 V1 V1 N V2 pij j p ji j qij j qji j V1 V2 VN V1 V2 pij p ji VN VN VN pij Vi Hpij,δ Hpij,V p ji Hpji,δ Hpji,V qij Vj qji qji Q 1 N Q 2 N Q N N Vj Vi HV,V qij HV,δ Vj Vi VN V2 Vi V1 V2 V2 V1 VN VN P1 N P2 N PN N V2 V1 V2 N VN N 25 Vj (5) Hqij,δ Hqij,V Hqji,δ Hqji,V P1 V1 P2 V1 PN V1 P1 V2 P2 V2 PN V2 Q 1 V1 Q 2 V1 Q N V1 Q 1 V2 Q 2 V2 Q N V2 P1 VN P2 VN PN VN Q 1 VN Q 2 VN Q N VN HP,δ HP,V HQ,δ HQ,V 26 Advanced Technologies pi j Vi2 ( gi j a 2 gs h i ) p j i Vj2 ( gi j gs h j ) where ViVj a ( gi j cos i j bi j sin i j ) ViVj ( gi j cos i j bi j sin i j ) a ViVj bi j ( gi j sin i j bi j cos i j ) qi j Vi2 ( 2 bs h i ) a a ViVj q j i Vj2 (bi j bs h j ) ( gi j sin i j bi j cos i j ) a i j i j (6) (7) (8) (9) (10) All the line flows computed from Eq.
Advanced Technologies by Kankesu Jayanthakumaran (Editor)