6.3.7 Example with Runga Kutta 4 th order method

Transcription

1 6.3.7 Example wth Runga Kutta 4 th order method Agan, as an example, 3 machne, 9 bus system shown n Fg. 6.4 s agan consdered. Intally, the dampng of the generators are neglected (.e. d = 0 for = 1, 2, 3). The load flow results and the ntal values of the magntudes and angles ( E and δ) of the nternal voltages of all the generators are same as those already gven n Table 6.2. As before, t s agan assumed that at t = 0.5 sec., a three phase to ground short crcut fault takes place at bus 7. The faulted generator termnal voltages and generator output powers are same as those shown n Table 6.3. Wth these values of P (o) e and ω o, the estmates dδ dt and dω dt ( = 1, 2, 3) are calculated from equatons (6.44) and (6.45) respectvely and are shown n columns 2 and 3 of Table 6.10 respectvely. Wth these frst estmates of the dervatves, the values of δ and ω ( = 1, 2, 3) are calculated from equatons (6.46) - (6.47) and are shown n columns 4 and 5 of Table 6.10 respectvely. Table 6.10: Calculatons wth RK method for frst estmate at t = 0.5 sec. (dampng = 0) dδ dt dω dt δ ω Wth the values of δ ( = 1, 2, 3) obtaned above, the generator termnal voltages and output powers are updated from equatons (6.23) - (6.25) and are shown n Table Please note that followng the notatons used earler, the output powers calculated at ths stage are denoted as P e ( = 1, 2, 3). Table 6.11: Caculatons wth RK method for second estmate at t = 0.5 sec. (dampng = 0) V Wth the values of ω and P e ( = 1, 2, 3) calculated above, the quanttes dδ dt and dω dt ( = 1, 2, 3) are calculated from equatons (6.48) - (6.49) and are shown n columns 2 and 3 of Table 6.12 respectvely. Subsequently, the values of δ and ω ( = 1, 2, 3) are calculated from equatons (6.50) - (6.51) and are shown n columns 4 and 5 of Table 6.12 respectvely. Agan, wth the values of δ ( = 1, 2, 3) obtaned above, the generator termnal voltages and output powers are updated from equatons (6.23) - (6.25) and are shown n Table Please note 272

2 Table 6.12: Calculatons wth RK method for second estmate at t = 0.5 sec. (dampng = 0) dδ dt dω dt δ ω that followng the notatons used earler, the output powers calculated at ths stage are denoted as P e ( = 1, 2, 3). Table 6.13: Caculatons wth RK method for thrd estmate at t = 0.5 sec. (dampng = 0) V Proceedng further, wth the values of ω and P e ( = 1, 2, 3) calculated above, the quanttes dδ dt and dω dt ( = 1, 2, 3) are calculated from equatons (6.52) - (6.53) and are shown n columns 2 and 3 of Table 6.14 respectvely. Subsequently, the values of δ and ω ( = 1, 2, 3) are calculated from equatons (6.54) - (6.55) and are shown n columns 4 and 5 of Table 6.14 respectvely. Table 6.14: Calculatons wth RK method for thrd estmate at t = 0.5 sec. (dampng = 0) dδ dt dω dt δ ω Agan, wth the values of δ ( = 1, 2, 3) obtaned above, the generator termnal voltages and output powers are updated from equatons (6.23) - (6.25) and are shown n Table Please note that followng the notatons used earler, the output powers calculated at ths stage are denoted as P e ( = 1, 2, 3). Lastly, wth the values of ω and P e and dω dt ( = 1, 2, 3) calculated above, the quanttes dδ dt ( = 1, 2, 3) are calculated from equatons (6.56) - (6.57) and are shown n columns 2 273

3 Table 6.15: Calculatons wth RK method for fnal estmate at t = 0.5 sec. (dampng = 0) V and 3 of Table 6.16 respectvely. Fnally, the values of δ and ω ( = 1, 2, 3) at the end of t = 0.5 sec. are calculated from equatons (6.58) - (6.59) and are shown n columns 4 and 5 of Table 6.16 respectvely. Table 6.16: Calculatons wth RK method for fnal estmate at t = 0.5 sec. (dampng = 0) dδ dt dω dt δ ω After the fnal values of δ and ω ( = 1, 2, 3) correspondng to t = 0.5 sec. are obtaned, we ncrement the tme by t (= sec.) and repeat the calculatons for t = sec. Towards ths goal, the quanttes δ and ω ( = 1, 2, 3) shown n Table 6.16 are substtuted for δ o and ω o ( = 1, 2, 3) n equatons (6.44) - (6.59). Wth these values of δ o ( = 1, 2, 3), equatons (6.23) - (6.25) are solved to calculate the ntal values of P e ( = 1, 2, 3) at t = sec. The results are shown n Table Table 6.17: Calculatons wth RK method for ntal estmate at t = sec. (dampng = 0) V (o) Wth the values of P (o) e ( = 1, 2, 3) thus obtaned, calculatons pertanng to the frst estmate are carred out by usng equatons (6.44) - (6.47) and the results are shown n Table Subsequently, the values of P e ( = 1, 2, 3) are calculated and the results are shown n Table Wth the values of P e, δ and ω ( = 1, 2, 3) obtaned as above, the calculatons pertanng to second estmate are performed to obtan P e, δ and ω ( = 1, 2, 3). The results are shown 274

4 Table 6.18: Calculatons wth RK method for frst estmate at t = sec. (dampng = 0) dδ dt dω dt δ ω Table 6.19: Calculatons wth RK method for frst estmate at t = sec. (dampng = 0) V n Tables 6.20 and Table 6.20: Calculatons wth RK method for second estmate at t = sec. (dampng = 0) dδ dt dω dt δ ω Table 6.21: Calculatons wth RK method for second estmate at t = sec. (dampng = 0) V Proceedng further, usng the values of P e, δ and ω ( = 1, 2, 3), the calculatons pertanng to thrd estmate are performed to obtan P e, δ and ω ( = 1, 2, 3). The results are shown n Tables 6.22 and Usng the values of P e, δ and ω ( = 1, 2, 3), the fourth estmates of the dervatves are computed and subsequently, the fnal values of δ and ω ( = 1, 2, 3) correspondng to t = sec. are obtaned. The calculatons are shown n Table

5 Table 6.22: Calculatons wth RK method for thrd estmate at t = sec. (dampng = 0) dδ dt dω dt δ ω Table 6.23: Calculatons wth RK method for thrd estmate at t = sec. (dampng = 0) V Table 6.24: Calculatons wth RK method for fnal estmate at t = sec. (dampng = 0) dδ dt dω dt δ ω For subsequent tme nstants, the calculatons proceed exactly n the same way as descrbed above. As n the case wth Euler s method, n ths case also, the fault s assumed to be cleared at t = 0.6 sec. and fnally, the smulaton study s stopped at t = 5.0 sec. The varatons of δ ( = 1, 2, 3) wth respect to the center of nerta (COI) are shown n Fg. 6.7 below. Please note that n ths fgure, no dampng of the generators has been consdered. The smulaton studes have also been carred out by consderng the dampng of the generators. The varatons of δ ( = 1, 2, 3) wth respect to the center of nerta (COI) for ths case are shown n Fg. 6.8 below. Comaprson of Fgs wth Fgs reveals that the responses obtaned wth these two methods are almost dentcal to each other. Wth ths example, we are now at the end of dscusson of transent stablty analyss. From the next lecture, we wll start the dscusson of small sgnal stablty analyss. 276

6 Fgure 6.7: Varaton of δ COI 1 (wth no dampng) obtaned wth Runga-Kutta method Fgure 6.8: Varaton of δ COI 1 (wth dampng) obtaned wth Runga-Kutta method 277