Case Problem $5$ Duke Energy Coal Allocation ${?}^2$

Duke Energy manufactures and distributes electricity to customers in the United States and

Latin America. Duke Energy acquired Cinergy Corporation in $2005,$ which had generating

facilities and energy customers in Indiana, Kentucky, and Ohio. For these customers

Cinergy has been spending $$725$ to $$750$ million each year for the fuel needed to operate

its coal$-$fired and gas$-$fired power plants; $92\%$ to $95\%$ of the fuel used is coal. In this region,

Duke Energy uses $10$ coal$-$burning generating plants: $5$ located inland and $5$ located on the

Ohio River. Some plants have more than one generating unit. Duke Energy uses $28-29$ mil$-$

lion tons of coal per year at a cost of approximately $$2$ million every day in this region.

Duke Energy purchases coal using fixed$-$tonnage or variable$-$tonnage contracts from

mines in Indiana $(49\%),$ West Virginia $(20\%),$ Ohio $(12\%),$ Kentucky $(11\%),$ Illinois $(5\%),$

and Pennsylvania $(3\%).$ Duke Energy must purchase all of the coal contracted for on fixed$-$

tonnage contracts, but on variable$-$tonnage contracts it can purchase varying amounts up to

the limit specified in the contract. The coal is shipped from the mines to Duke Energy's gen$-$

erating facilities in Ohio, Kentucky, and Indiana. The cost of coal varies from $$19$ to $$35$

per ton, and transportation$/$delivery charges range from $$1\mathrm{.}50$ to $$5\mathrm{.}00$ per ton.

A model is used to determine the megawatt$-$hours $(mWh)$ of electricity that each

generating unit is expected to produce and to provide a measure of each generating unit's

efficiency, referred to as the heat rate. The heat rate is the total British thermal units $[$BTUs$]$

required to produce $1$ kilowatt$-$hour $(kWh)$ of electrical power.

Coal Allocation Model

Duke Energy uses a linear programming model, called the coal allocation model, to allo$-$

cate coal to its generating facilities. The objective of the coal allocation model is to deter$-$

mine the lowest$-$cost method for purchasing and distributing coal to the generating units.

The supply$/$availability of the coal is determined by the contracts with the various mines,

and the demand for coal at the generating units is determined indirectly by the megawatt$-$

hours of electricity each unit must produce.

The cost to process coal, called the add$-$on cost$,$ depends upon the characteristics of the

coal $($moisture content, ash content, BTU content, sulfur content, and grindability$)$ and the

efficiency of the generating unit. The add$-$on cost plus the transportation cost are added to

the purchase cost of the coal to determine the total cost to purchase and use the coal.

Current Problem

Duke Energy signed three fixed$-$tonnage contracts and four variable$-$tonnage contracts. The

company would like to determine the least$-$cost way to allocate the coal available through these contracts to five generating units. The relevant data for the three fixed$-$tonnage con$-$

tracts are as follows:

For example, the contract signed with RAG requires Duke Energy to purchase $350,000$ tons of

coal at a price of $$22$ per Prepare a report that summarizes your recommendations regarding Duke Energy's coal al$-$

location problem. Be sure to include information and analysis for the following issues:

Determine how much coal to purchase from each of the mining companies and how

it should be allocated to the generating units. What is the cost to purchase, deliver,

and process the coal?

Compute the average cost of coal in cents per million BTUs for each generating unit

$($a measure of the cost of fuel for the generating units$).$

Compute the average number of BTUs per pound of coal received at each generat$-$

ing unit $($a measure of the energy efficiency of the coal received at each unit$).$

Suppose that Duke Energy can purchase an additional $80,000$ tons of coal from

American Coal Sales as an "all or nothing deal" for $$30$ per ton. Should Duke En$-$

ergy purchase the additional $80,000$ tons of coal?

Suppose that Duke Energy leams that the energy content of the coal from Cyprus

Amax is actually $13,000$ BTUs per pound. Should Duke Energy revise its procure$-$

ment plan?

Duke Energy has leamed from its trading group that Duke Energy can sell $50,000$

megawatt$-$hours of electricity over the grid $($to other electricity suppliers$)$ at a price

of $$30$ per megawatt$-$hour. Should Duke Energy sell the electricity? If so$,$ which

generating units should produce the additional electricity?