The two parallel lines in Example 3.13 supply a balanced load with a load current of $1.0\angle -{30}^{\circ }$ per unit. Determine the real and reactive power supplied to the load bus from each parallel line with

(a) no regulating transformer,

(b) the voltage-magnitude-regulating transformer in Example 3.13(a), and

(c) the phase-angle-regulating transformer in Example 3.13(b). Assume that the voltage at bus $abc$ is adjusted so that the voltage at bus $a^{\prime }{b}^{\prime }{c}^{\prime }$ remains constant at $1.0/0^{\circ }$ per unit. Also assume positive sequence. Comment on the effects of the regulating transformers.

Example 3.13

Two buses $abc$ and $a^{\prime }{b}^{\prime }{c}^{\prime }$ are connected by two parallel lines L1 and L2 with positive-sequence series reactances $X_{L1}=0.25$ and $X_{L2}=0.20$ per unit. A regulating transformer is placed in series with line L1 at bus $a^{\prime }{b}^{\prime }{c}^{\prime }$. Determine the $2\times 2$ bus admittance matrix when the regulating transformer.