Three identical steam$-$turbine generators G$1,$ G$2,$ and G$3$ are operating in parallel. Each generator is: $7200$ V$,12\mathrm{.}5$ MVA, $50$ Hz$,0\mathrm{.}85$ PF lagging, $4-$poles, Y$-$connected, with synchronous reactance of $1$ and an armature resistance of $0\mathrm{.}1.$ The characteristic power$-$frequency slopes of generators G$1,$ G$2,$ and G$3$ are $4\mathrm{.}5$ MW$/$Hz$,5$ MW$/$Hz$,$ and $6$ MW$/$Hz$,$ respectively. $(1)$ If the no$-$load frequency of each of the three generators is set to $51\mathrm{.}5$ Hz$.$ How much power is supplied by each generator when the actual frequency is $50$ Hz$?$ What is the total power supplied by all generators? $(2)$ Without changing the no$-$load frequencies, the load is increased until one of the generators reaches its ratings before the other two generators. Which generator is that and why? What is the new system frequency? What is the total power supplied at this point? $(3)$ The operator decided to run each of the three generators at its maximum rated power, and at a system frequency of $50$ Hz$.$ This will be done by new settings of the no$-$load frequency of all generators. What is the no$-$load frequency of each generator? What is the total power supplied at this point? $(4)$ If the three generators are operating at their rated power, what is the internal generated voltage and the torque angle of each of the three generators?