A small power system is supplied by three generators. The cost characteristics

of these generators are given by:

C$1($P$1)=300+11$P$1+0\mathrm{.}05$P $2$

$1$

C$2($P$2)=250+13$P$2+0\mathrm{.}06$P $2$

$2$

C$3($P$3)=150+16$P$3+0\mathrm{.}08$P $2$

$3$

$1$

where the cost characteristic of each generator is expressed in units of $$/$h and the power

provided by each generator is expressed in units of MW$.$

a$)$ Calculate the hourly cost of optimally supplying a load of $1300$ MW with these three

generators. What is the marginal cost of generation for this operating condition? $(10$

points$)$

b$)$ Assume now that Generator $1$ is located in the Northern region while Generators $2$

and $3$ are located in the Southern region. The load in the Northern region is $200$ MW

and the load in the Southern region is $1100$ MW$.$ The interconnection between the

Northern and Southern regions has a transmission capacity of $300$ MW$.$ Calculate the

minimal hourly operating cost and the marginal value of increasing the transmission

capacity of the interconnection. Discuss how your results differ from those obtained in

Part $($a$).(16$ points$)$