Question 7.AE.13: For the system of Fig. E7.8.1, compute the correction voltag......
For the system of Fig. E7.8.1, compute the correction voltage vector \Delta \bar{x}^0, updating the mismatch power vector, and check the convergence of the load flow algorithm (using a convergence tolerance of 0.0001).
Based on the results of Application Examples 7.9 to 7.11, the correction voltage vector \Delta \bar{x}^0 can be computed from
\Delta \vec{x}^0=\left(J^0\right)^{-1} \Delta W\left(\bar{x}^0\right)=(2.68 \mathrm{E}-4,2.3 \mathrm{E}-3,2.79 \mathrm{E}-3,3.57 \mathrm{E}-3,3.39 \mathrm{E}-3,4.03 \mathrm{E}-3)^tThe updated mismatch power vector is
\Delta \bar{W}^1=(1.41 \mathrm{E}-4,3.01 \mathrm{E}-4,-9.63 \mathrm{E}-6,7.62 \mathrm{E}-5,1.20 \mathrm{E}-3,5.96 \mathrm{E}-4)^tSince the components of \Delta \bar{W}^1 are larger than the specified convergence tolerance (e.g.,0.0001), the Newton–Raphson solution has not converged. However, it will converge in two iterations:
Fundamental power flow iteration summary (Fig. E7.8.1):
Fundamental power flow output solution (Fig. E7.8.1), where δ angles are rounded (approximate) values:
“Active and reactive powers in the line between “from bus” and “to bus.”
| Absolute real power mismatch | Absolute reactive power mismatch | |||||
| Iteration | Average (%) | Worst (%) | Worst bus | Average (%) | Worst (%) | Worst bus |
| 0 | 11.67 | 25 | 4 | 6.67 | 10.00 | 4 |
| 1 | 0.05 | 0.12 | 4 | 0.03 | 0.06 | 4 |
| 2 | 0.00 | 0.00 | 4 | 0.00 | 0.00 | 4 |
| Bus Voltage | Bus generation (G) and load (L) powers | Line power ^a | |||||||
| From bus | |v| (%) | \delta\ (^{\circ}) | P_G (%) | Q_G (%) | P_L (%) | Q_L (%) | To bus | P_{line} (%) |
Q_{line} (%)
|
| 1 | 100.0 | 0.00 | 35.09 | 20.15 | 0.00 | 0.00 | 2 | 13.35 | 10.69 |
| 4 | 21.73 | 9.47 | |||||||
| 2 | 99.76 | -0.02 | 0.00 | 0.00 | 10.00 | 10.00 | 1 | -13.32 | -10.66 |
| 3 | 3.32 | 0.66 | |||||||
| 3 | 99.64 | -0.16 | 0.00 | 0.00 | 0.00 | 0.00 | 2 | -3.32 | -0.65 |
| 4 | 3.32 | 0.65 | |||||||
| 4 | 99.59 | -0.20 | 0.00 | 0.00 | 25.00 | 10.00 | 3 | -3.32 | -0.65 |
| 1 | -21.68 | -9.35 | |||||||
^a Active and reactive powers in the line between “from bus” and “to bus.”