In the figure shows on period of the output voltage of an inverter. $$\alpha $$ should be chosen such that $${60^0} < \alpha < {90^0}.$$ If $$rms$$ value of the fundamental component is $$50$$ $$V$$, then $$\alpha $$ in degree is _________.

In an unbalanced three phase system phase current $${{\rm I}_a} = 1\angle \left( { - {{90}^0}} \right)\,\,pu,\,\,$$ negative sequence current $$\,{{\rm I}_{b2}} = 4\angle \left( { - {{150}^0}} \right)\,\,pu,\,\,$$ zero sequence current $$\,\,{{\rm I}_{c0}} = 3\angle {90^0}\,\,pu.\,\,\,$$ The magnitude of phase current $${{\rm I}_b}$$ in $$pu$ is

The fuel cost functions of two power plants are
Plant $${P_1}:\,{C_1} = 0.05\,Pg_1^2 + AP{g_1} + B$$
Plant $${P_2}:\,{C_2} = 0.10\,Pg_2^2 + 3AP{g_2} + 2B$$
Where, $$P{g_1}$$ and $$P{g_2}$$ are the generator powers of two plants and $$A$$ and $$B$$ are the constants. If the two plants optimally share $$1000$$ $$MW$$ load at incremental fuel cost of $$100$$ $$Rs/MWh,$$ the ratio of load shared by plants $${P_1}$$ and $${P_2}$$ is

The over current relays for the line protection and loads connected at the buses are shown in the figure.

The relays are IDMT is natural having the characteristic
$${t_{op}} = {{0.14\, \times \,time\,\,multiplier\,\,setting} \over {{{(plug\,\,setting\,\,multiplier)}^{0.02}} - 1}}$$
The maximum and minimum fault current at bus $$B$$ are $$2000A$$ and $$500$$ $$A$$ respectively. Assuming the time multiplier setting and plug setting for relay $${R_B}$$ to be $$0.1$$ and $$5$$ $$A$$ respectively, the operating time of $${R_B}$$ (in seconds) is __________.