Question 12.4: Consider the network of Figure 12.5a. Let bus 1 be a swing b......
Consider the network of Figure 12.5a. Let bus 1 be a swing bus, buses 2 and 3 PQ buses, and bus 4 a PV bus. The loads at buses 2 and 3 are specified as is the voltage magnitude at bus 4. Construct P–θ and Q–V matrices.
P–θ Network
First construct the P–θ network shown in Figure 12.5b. Here the elements are calculated as follows:
B_{sr}^{\theta }= – \frac{1}{X_{sr}}=-B_{sr}(1+(\frac{G_{sr}}{B_{sr}})^2)
B_{sr}^{\theta }= \sum\limits_{r} ( \frac{1}{X_{sr}})=\sum\limits_{r}B_{sr}(1+(\frac{G_{sr}}{B_{sr}})^2)
All shunt susceptances are neglected and the swing bus connected to ground. The associated matrix is
\overline{B}^{\theta }=\left|\begin{matrix} 6.553 & -2.22 & -0.333 \\ -2.22 & 5.886 & -3.333 \\ -0.333 & -3.333 & 3.666 \end{matrix} \right|\left|\begin{matrix} \Delta \theta _{2} \\ \Delta \theta _{3} \\ \Delta \theta _{4} \end{matrix}\right| = \left|\begin{matrix}\Delta P_{2}/V_{2} \\ \Delta P_{3}/V_{3} \\ \Delta P_{4}/V_{4} \end{matrix} \right|Q–V Network
This has the structure of the original model, but voltage-specified buses, that is, swing bus and PV bus are directly connected to ground.
The Q–V network is shown in Figure 12.5c. The associated matrix is
\overline{B}^{v}=\left|\begin{matrix} 9.345 & -2.17 \\ -2.17 & 9.470 \end{matrix} \right|\left|\begin{matrix} \Delta V _{2} \\ \Delta V _{3} \end{matrix}\right| = \left|\begin{matrix}\Delta Q_{2}/V_{2} \\ \Delta Q_{3}/V_{3} \end{matrix} \right|The power flow equations can be written as in Example 12.3.