Question 7.AE.19: Apply the coupled (Section 7.4) and decoupled (Section 7.5.1......
Apply the coupled (Section 7.4) and decoupled (Section 7.5.1) harmonic power flow algorithm to the distorted 18-bus system of Fig. E7.19.1 and compare simulation results including the fundamental and rms (including harmonics) voltages, distortion factor, and THD_v. The nonlinear load (installed at bus 5) is a 3 MW, 5 MVAr six-pulse converter. Line and load parameters are provided in [29]. For the DHPF calculations, the converter may be modeled with harmonic current sources (Table E7.19.1).
The Newton-based HPF (e.g., using an accurate converter model that includes harmonic couplings [5,6]) and the DHPF (modeling the converter with harmonic current sources of Table E7.19.1) are applied to the 18 bus system. Simulation results are compared in Table E7.19.2. Small levels of the maximum and average deviations (last two rows of Table E7.19.2) show the acceptable accuracy of the decoupled approach.
Table E7.19.1 Magnitude of Harmonic Currents (as a Percentage of the Fundamental) Used to Model Six-Pulse Converters^a.
\begin{array}{l|l|l|l|l|l} \begin{array}{l} \text { Harmonic } \\ \text { order } \end{array} & \begin{array}{l} \text { Magnitude } \\ (\%) \end{array} & \begin{array}{l} \text { Harmonic } \\ \text { order } \end{array} & \begin{array}{l} \text { Magnitude } \\ (\%) \end{array} & \begin{array}{l} \text { Harmonic } \\ \text { order } \end{array} & \begin{array}{l} \text { Magnitude } \\ (\%) \end{array} \\ \hline 1 & 100 & 19 & 2.4 & 37 & 0.5 \\ 5 & 19.1 & 23 & 1.2 & 41 & 0.5 \\ 7 & 13.1 & 25 & 0.8 & 43 & 0.5 \\ 11 & 7.2 & 29 & 0.2 & 47 & 0.4 \\ 13 & 5.6 & 31 & 0.2 & 49 & 0.4 \\ 17 & 3.3 & 35 & 0.4 & & \\ \hline \end{array}
All harmonic phase angles are assumed to be zero.
Table E7.19.2 Comparison of Simulation Results for the 18 Bus System (Fig. E7.19.1) Using Coupled (Section 7.4) and Decoupled (Section 7.5.1) Harmonic Power Flow
\begin{aligned}\text{fundamental voltage (pu)}&\text{rms voltage (pu)} &distortion\ factor\ D (\%)&THD\ of\ voltage\ (\%)\end{aligned}\\\begin{array}{c|l|l|l|l|l|l|l|l} \text { bus } & \text { HPF } & {\text { HPF }} & {\text { HPF }} & \text { HPF } & \text { HPF } & \text { HPF } & \text { HPF } & {\text { HPF }} \\ \hline 1 & 1.0545 & 1.0545 & 1.0549 & 1.0548 & 0.9996 & 0.9997 & 2.7159 & 2.3828 \\ 2 & 1.0511 & 1.0510 & 1.0516 & 1.0516 & 0.9995 & 0.9995 & 3.1353 & 3.1942 \\ 3 & 1.0456 & 1.0455 & 1.0463 & 1.0465 & 0.9993 & 0.9991 & 3.6591 & 4.2593 \\ 4 & 1.0425 & 1.0424 & 1.0433 & 1.0436 & 0.9992 & 0.9989 & 3.9386 & 4.7737 \\ 5 & 1.0359 & 1.0358 & 1.0370 & 1.0377 & 0.9989 & 0.9982 & 4.7701 & 6.0515 \\ 6 & 1.0348 & 1.0347 & 1.0360 & 1.0367 & 0.9988 & 0.9981 & 4.8692 & 6.1886 \\ 7 & 1.0326 & 1.0325 & 1.0339 & 1.0347 & 0.9987 & 0.9979 & 5.1238 & 6.5404 \\ 8 & 1.0268 & 1.0267 & 1.0281 & 1.0289 & 0.9987 & 0.9978 & 5.1528 & 6.5775 \\ 9 & 1.0496 & 1.0496 & 1.0501 & 1.0501 & 0.9995 & 0.9995 & 3.1397 & 3.1985 \\ 20 & 1.0505 & 1.0504 & 1.0509 & 1.0509 & 0.9996 & 0.9996 & 2.9329 & 2.9248 \\ 21 & 1.0496 & 1.0495 & 1.0505 & 1.0506 & 0.9991 & 0.9990 & 4.1782 & 4.4609 \\ 22 & 1.0479 & 1.0479 & 1.0488 & 1.0489 & 0.9991 & 0.9990 & 4.1847 & 4.4682 \\ 23 & 1.0451 & 1.0451 & 1.0474 & 1.0477 & 0.9978 & 0.9974 & 6.6676 & 7.1620 \\ 24 & 1.0485 & 1.0485 & 1.0519 & 1.0523 & 0.9968 & 0.9964 & 7.9886 & 8.4865 \\ 25 & 1.0419 & 1.0418 & 1.0449 & 1.0452 & 0.9971 & 0.9967 & 7.6002 & 8.1187 \\ 26 & 1.0415 & 1.0414 & 1.0445 & 1.0449 & 0.9971 & 0.9967 & 7.6029 & 8.1220 \\ 50 & 1.0501 & 1.0501 & 1.0501 & 1.0501 & 1.0000 & 1.0000 & 0.2482 & 0.1217 \\ 51 & 1.0500 & 1.0500 & 1.0500 & 1.0500 & 1.0000 & 1.0000 & 0.2482 & 0.0035 \\ \text { MD }^a & 0.0097 & – & 0.0778 & – & 0.0009 & – & 1.42473 & – \\ \text { AD }^b & 0.0059 & – & 0.0278 & – & 0.0003 & – & 0.57242 & – \\ \hline \end{array}
^a Maximum Deviation with respect to the coupled HPF (%).
^b Average Deviation with respect to the coupled HPF (%).