Question

A coal-fired power plant can produce electricity at a variable cost of $0.07 per kilowatt-hour when running at its full capacity of 30 megawatts per hour, $0.16 per kilowatt-hour when running at 20 megawatts per hour, and $0.24 per kilowatt-hour when running at 10 megawatts per hour. A gas-fired power plant can produce electricity at a variable cost of $0.12 per kilowatt-hour at any capacity from 1 megawatt per hour to its full capacity of 5 megawatts per hour. The cost of constructing a coal-fired plant is $70 million, but it costs only $14 million to build a gas-fired plant.

Required:
a. Consider a city that has a peak afternoon demand of 80 megawatts of electricity. If it wants all plants to operate at full capacity, what combination of coal-fired plants and gas-fired plants would minimize construction costs?
b. How much will the city spend on building that combination of plants?
c. What will the average cost per kilowatt-hour be if you average over all 80 megawatts that are produced by that combination of plants?

Answer 1

Answer:

a-The construction of 2 coal-fired plants and 4 gas-fired plants will have minimum construction costs.

b-Total construction cost is $196 Million.

c-The average cost is $0.0825 per killowatt-hour.

Explanation:

a

In order to estimate the best combination of the two, consider the following linear programming model

$30X+5Y\leq80\\X\geq0\\Y\geq0$

with minimizing function as $70X+14Y$

This yeilds in the optimum solution of 2 coal fired plants and 4 gas fired plants with minimum construction costs.

The construction of 2 coal-fired plants and 4 gas-fired plants will have minimum construction costs.

b

The construction cost is as follows

Number of coal-fired plants=2

Number of gas-fired plants=4

Total Cost=(Cost of 1 Coal-Fired Plant*Number of coal-fired plants)+(Cost of 1 Gas-Fired Plant*Number of gas-fired plants)

$\text{Total Cost}=(70\times 2)+(14\times4)\\\text{Total Cost}=140+56\\\text{Total Cost}=\ 196 \text{Million}$

Total construction cost is $196 Million.

c

Average Cost of Electrici$\text{Average Cost}=\text{Fraction of Coal-Fired}\times\text{Cost of Coal-Fired}+\text{Fraction of Gas-Fired}\times\text{Cost of Gas-Fired}\\\text{Average Cost}=\text{0.75}\times\text{0.07}+\text{0.25}\times\text{0.12}\\\text{Average Cost}=\ 0.0525+\ 0.03\\\text{Average Cost}=\ 0.0825$ty production is given by estimating the share of electricity produced by coal-fired plants and gas-fired plants

Total energy=80 MW

Energy produced by Coal-Fired Plants at full capacity=2*30=60 MW

Energy produced by Gas-Fired Plants at full capacity=4*5=20 MW

Fraction of Coal-Fired Plants is given as

                                  $\dfrac{\text{Coal-Fired Share}}{\text{Total Energy}}=\dfrac{60}{80}=0.75$

Fraction of Gas-Fired Plants is given as

$\dfrac{\text{Gas-Fired Share}}{\text{Total Energy}}=\dfrac{20}{80}=0.25$

Cost of Producing KW-hr by Coal-Fired Plant is $0.07

Cost of Producing KW-hr by Gas-Fired Plant is $0.12

So

$\text{Average Cost}=[\text{Fraction}_{Coal-Fired}\times\text{Cost per KW-hr}_{Coal-Fired}]+[\text{Fraction}_{Gas-Fired}\times\text{Cost per KW-hr}_{Gas-Fired}]\\\text{Average Cost}=(0.75\times0.07)+(0.25\times0.12)\\\text{Average Cost}=(0.0525)+(0.03)\\\text{Average Cost}=\ 0.0825$The average cost is $0.0825 per killowatt-hour.