Exercise $3$: A two$-$pole, three$-$phase, synchronous generator is capable of supplying $250$ kVA at $0\mathrm{.}8$ PF lagging at full load. The generator's prime mover has a no$-$load speed of $3650\frac{r}{m}in$ and a full$-$load speed of $3570\frac{r}{m}in.$ This generator is operating in parallel with a second generator that is a four$-$pole, three$-$phase, machine capable of supplying $250$ kVA at $0\mathrm{.}85$ PF lagging at full load. The prime mover of this second generator has a no$-$load speed of $1800\frac{r}{m}in$ and a full$-$load speed of $1780\frac{r}{m}in.$ The loads powered by the two generators consist of $300$ kW $.$

a$)$ Determine the slopes $s_p$ of the frequency$-$power characteristics for both the generators.

b$)$ Calculate the operating frequency of this power system with the two generators supplying the load.

c$)$ Find the real power that is supplied by each of the generators in this system.

d$)$ What should the no$-$load speed of the second generator be modified to in order to adjust the operating frequency to $60$ Hz for supplying power to the same load, while all other system characteristics remaining the same?