For the switching converter shown in the following figure, assume steady-state operation. Also assume that the components are ideal, the inductor current is always positive and continuous and switching period is $${T_s}.$$ If the voltage $${V_L}$$ is as shown, the duty cycle of the switch $$S$$ is __________.

The incremental costs (in rupees/$$MWh$$) of operating two generating units are functions of their respective powers $${P_1}$$ and $${P_2}$$ in $$MW,$$ and are given by $$${{d{C_1}} \over {d{P_1}}} = 0.2{P_1} + 50$$$ $$${{d{C_2}} \over {d{P_2}}} = 0.24{P_2} + 40$$$
Where, $$$\eqalign{ & 20\,MW \le {P_1} \le 150\,MW \cr & 20\,MW \le {P_2} \le 150MW. \cr} $$$
For a certain load demand, $${P_1}$$ and $${P_2}$$ have been chosen such that $$\,\,d{C_1}/d{P_1} = 76\,Rs/MWh\,\,$$ and $$\,d{C_2}/d{P_2} = 68.8\,Rs/MWh.\,\,$$ If the generations are rescheduled to minimize the total cost, then $${P_2}$$ is ____________.

A $$3$$-phase transformer rated for $$33 kV/11kV$$ is connected in delta/star as shown in figure. The current transformers $$\left( {C{T_s}} \right)$$ on low and high voltage sides have a ratio of $$500/5.$$ Find the currents $${i_1}$$ and $${i_2}$$, if the fault current is $$300$$ $$A$$ as shown in figure.

A $$3$$-bus power system network consists of $$3$$ transmission lines. The bus admittance matrix of the uncompensated system is
$$\left[ {\matrix{ { - j6} & {j3} & {j4} \cr {j3} & { - j7} & {j5} \cr {j4} & {j5} & { - j8} \cr } } \right]\,pu$$
If the shunt capacitance of all transmission lines is $$50$$% compensated, the imaginary part of the $$3$$^rd row $$3$$^rd column element (in $$pu$$) of the bus admittance matrix after compensation is