For the three-bus power system shown in the figure, the trip signals to the circuit breakers B$$_1$$ to B$$_9$$ are provided by overcurrent relays R$$_1$$ to R$$_9$$, respectively, some of which have directional properties also. The necessary condition for the system to be protected for short circuit fault at any part of the system between bus 1 and the R-L loads with isolation of minimum portion of the network using minimum number of directional relays is

The expressions of fuel cost of two thermal generating units as a function of the respective power generation $${P_{G1}}$$ and $${P_{G2}}$$ are given as

$$\matrix{ {{F_1}({P_{G1}}) = 0.1aP_{G1}^2 + 40{P_{G1}} + 120\,Rs/hour} & {0\,MW \le {P_{G1}} \le 350\,MW} \cr {{F_2}({P_{G2}}) = 0.2P_{G2}^2 + 30{P_{G2}} + 100\,Rs/hour} & {0\,MW \le {P_{G2}} \le 300\,MW} \cr } $$

where $$a$$ is a constant. For a given value of $$a$$, optimal dispatch requires the total load of 290 MW to be shared as $${P_{G1}} = 175\,MW$$ and $${P_{G2}} = 115\,MW$$. With the load remaining unchanged, the value of $$a$$ is increased by 10% and optimal dispatch is carried out. The changes in $${P_{G1}}$$ and the total cost of generation, $$F( = {F_1} + {F_2})$$ in Rs/hour will be as follows

The bus admittance ($$Y_{bus}$$) matrix of a 3-bus power system is given below.

$$\quad\quad$$$$\matrix{ 1 & \quad\quad\quad2\quad\quad & 3 \cr } $$

$$\matrix{ 1 \cr 2 \cr 3 \cr } \left[ {\matrix{ { - j15} & {j10} & {j5} \cr {j10} & { - j13.5} & {j4} \cr {j5} & {j4} & { - j8} \cr } } \right]$$

Considering that there is no shunt inductor connected to any of the buses, which of the following can NOT be true?

A 50 Hz, 275 kV line of length 400 km has the following parameters:

Resistance, R = 0.035 $$\Omega$$/km;

Inductance, L = 1 mH/km;

Capacitance, C = 0.01 $$\mu$$F/km;

The line is represented by the nominal-$$\pi$$ model. With the magnitudes of the sending end and the receiving end voltages of the line (denoted by $$V_S$$ and $$V_R$$, respectively) maintained at 275 kV, the phase angle difference ($$\theta$$) between $$V_S$$ and $$V_R$$ required for maximum possible active power to be delivered to the receiving end, in degree is ___________ (Round off to 2 decimal places).