Question

Duke Energy manufactures and distributes electricity to customers in the United States and Latin America. Duke recently purchased Cinergy Corporation, which has generating facilities and energy customers in Indiana, Kentucky, and Ohio. For these customers Cin- ergy 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: five located inland and five located on the Ohio River. Some plants have more than one generating unit. Duke Energy uses 28–29 million tons of coal per year at a cost of approximately $2 million every day in this region.

The company 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%). The company 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 generating facilities in Ohio, Kentucky, and Indiana. The cost of coal varies from $19 to $35 per ton and transportation/delivery charges range from $1.50 to

$5.00 per ton.

A model is used to determine the megawatt-hours (mWh) of electricity that each gen- erating unit is expected to produce and to provide a measure of each generating unit’s effi- ciency, referred to as the heat rate. The heat rate is the total 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:

 

 

Supplier	Number of Tons Contracted For	Cost ($/ton)	BTUs/lb
RAG	350,000	22	13,000
Peabody Coal Sales	300,000	26	13,300
American Coal Sales	275,000	22	12,600

 

 

For example, the contract signed with RAG requires Duke Energy to purchase 350,000 tons of coal at a price of $22 per ton; each pound of this particular coal provides 13,000 BTUs.

The data for the four variable-tonnage contracts follow:

 

 

Supplier	Number of Tons Available	Cost ($/ton)	BTUs/lb
Consol, Inc.	200,000	32	12,250
Cyprus Amax	175,000	35	12,000
Addington Mining	200,000	31	12,000
Waterloo	180,000	33	11,300

 

 

For example, the contract with Consol, Inc., enables Duke Energy to purchase up to 200,000 tons of coal at a cost of $32 per ton; each pound of this coal provides 12,250 BTUs.

The number of megawatt-hours of electricity that each generating unit must produce and the heat rate provided are as follows:

 

 

Generating Unit	Electricity Produced (mWh)	Heat Rate (BTUs per kWh)
Miami Fort Unit 5	550,000	10,500
Miami Fort Unit 7	500,000	10,200
Beckjord Unit 1	650,000	10,100
East Bend Unit 2	750,000	10,000
Zimmer Unit 1	1,100,000	10,000

 

 

For example, Miami Fort Unit 5 must produce 550,000 megawatt-hours of electricity, and 10,500 BTUs are needed to produce each kilowatt-hour.

 

 

The transportation cost and the add-on cost in dollars per ton are shown here:

 

 

 

Supplier	Miami Fort Unit 5	Miami Fort Unit 7	Beckjord Unit 1	East Bend Unit 2	Zimmer Unit 1
RAG	5.00	5.00	4.75	5.00	4.75
Peabody	3.75	3.75	3.50	3.75	3.50
American	3.00	3.00	2.75	3.00	2.75
Consol	3.25	3.25	2.85	3.25	2.85
Cyprus	5.00	5.00	4.75	5.00	4.75
Addington	2.25	2.25	2.00	2.25	2.00
Waterloo	2.00	2.00	1.60	2.00	1.60

 

 

 

Supplier	Miami Fort Unit 5	Miami Fort Unit 7	Beckjord Unit 1	East Bend Unit 2	Zimmer Unit 1
RAG	10.00	10.00	10.00	5.00	6.00
Peabody	10.00	10.00	11.00	6.00	7.00
American	13.00	13.00	15.00	9.00	9.00
Consol	10.00	10.00	11.00	7.00	7.00
Cyprus	10.00	10.00	10.00	5.00	6.00
Addington	5.00	5.00	6.00	4.00	4.00
Waterloo	11.00	11.00	11.00	7.00	9.00

 

Managerial Report

Prepare a report that summarizes your recommendations regarding Duke Energy’s coal allocation problem. Be sure to include information and analysis for the  following  issues:

1.           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?

2.           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).

3.           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).

4.           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 Energy purchase the additional 80,000 tons of coal?

5.           Suppose that Duke Energy learns 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?

6.           Duke Energy has learned 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 Ener

Answer 1

Step 1: First, for each generating unit, we calculate the requirement in terms of BTUs (million). To get this, Covert Electricity produced in mWh to Kwh by multiplying it by 1000, then multiply it by Heat Rate (BTUs per kWh) and divide it by 10^6 to get BTUs (million)

Step 2: Now we set up the linear programming model in Excel, in order to determine the optimal coal allocation plan from supplier to generating unit. The total tonnage allocated from the fixed contract supplier must be equal to the contracted quantity (so the constraint in solver will be set to =)

Step 3: The column total of the coal allocation table will be in Tons of Coal. We need to convert this into BTUs (million) in order to match the requirement of each generating unit (as calculated in the Step 1) This is calculated by multiplying the quantity of coal allocated from each supplier to the generating unit by 2000 (to convert tons into lb) and by the number of BTUs/lb corresponding to the coal supplied by that supplier. We divide this number by 10^6 to get the BTUs (million)

For example, For Miami Fort Unit 5, the formula used in cell C58 to get the BTU (millions) =SUMPRODUCT(C50:C56,$E$3:$E$9)*2000/10^6

Same formula is copied in C58:G58

1) The optimal coal allocation plan and the total cost of the same is determined by Linear programming model as below .

BTUs/lb 13,000 Number of Tons Contracted For 3,50,000 3,00,000 2,75,000 2,00,000 1,75,000 2,00,000 1,80,000 Cost (S/ton) 3 Fixed RAG 4 contract Peabody Coal Sales American Coal Sales 12,600 12,250 6 Variable Consol, Inc. 7 contract Cyprus Amax Addington Mining Waterloo 12,000 11,300 Generating Unit Electricity Produced (mWh) Heat Rate (BTUs per kwh) BTUs (million Miami Fort Unit 5 5,50,000 10,500 57,75,000 Miami Fort Unit 7 5,00,000 10,200 51,00,000 Beckjord Unit 1 6,50,000 10,100 65,65,000 East Bend Unit 2 Zimmer Unit 1 12 7,50,000 10,000 75,00,000 11,00,000 10,000 1,10,00,000 15 Transportation cost Add-On Cost (Ston) 39 Miami Fort Unit 5 15 13.75 Miami Fort Unit 7 15 13.75 Beckjord Unit 1 14.75 14.5 East Bend Unit 2 Zimmer Unit 1 10.75 10.5 11.75 American 13.25 10.25 15 Waterloo Optimal Coal Allocation Plan (Decision Variables) Miami Fort Unit 5 Miami Fort Unit 7 Beckjord Unit 1 61,538 71,617 East Bend Unit 2 Zimmer Unit 1 Row Total (Tons 50 Fixed RAG 51 contract Peabody 2,88,462 3,50,000 3,00,000 2,75,000 2,00,000 36,654 1,91,729 75,000 1,66,122 American 53 Variable Consol 54 contract Cyprus 33,878 2,00,000 2,00,000 98,673 Waterloo Column Total (Tons Column Total BTUs (million 98,673 2,65,705 65,65,000 115 2,36,654 57,75,000 1,91,729 51,00,000 2,88,462 75,00,000 4,41,122 1,10,00,000 Average Cost of Coal (cents per million BTUs Average number of BTUs und of Coal 12,201 13,300 13,000 12,468 Objective function Transportation cost +Add-On Cost Cost of Coal Total Cost 1,60,01,055 3,74,06,195 5,34,07,249

Solver Parameters Set Objective To: O Max By Changing Variable Cells: SC$65 OMin OY Value Of: $C$50:$G$56 Subject to the Constraints: $C$58:$G$58 $C$14:$G$14 SH$50:$H$52 $C$3:$C$5 SH$53:$H$56$C$6:$C$9 Add Change Delete Reset All Load/Save Make Unconstrained Variables Non-Negative Select a Solving Method Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Help

2) Average cost of coal in cents per million in computed in cells C59:G59 . The formula used in C59 =SUMPRODUCT(C50:C56,$D$3:$D$9)/C58*100

3) Average number of BTUs per pound of coal is computed in cells C60:G60. The formula used in C60 =C58/C57/2000*10^6

Answer 2

1)

Using the given data for the three-fixed tonnage contracts determine how much coal should be allocated to the generating units and the cost to purchase, deliver and process the coal.

Tonnage of coal to be acquired from each mining entity and destinations are noted as follows:

	City F # 5	City F # 7	City B	City E	City Z
Supplier R	0	0	61,538	288,462	0
Supplier P	217,105	11,278	71,617	0	0
Supplier AM	0	0	0	0	275,000
Supplier CO	0	0	33,878	0	166,122
Supplier CA	0	0	0	0	0
Supplier AD	0	200,000	0	0	0
Supplier W	0	0	98,673	0	0

The total cost to purchase, deliver, and manage the coal is $53,407,243.

2.)

Using the given data for the three-fixed tonnage contracts compute the average costs in coal in cents per million BTU’s.

The cost of the coal in cents per million BTUs for each producing unit is as follows:

City F # 5	City F # 7	City B	City E	City Z
111.84	136.97	127.24	103.85	114.51

3.)

Using the given data for the three-fixed tonnage contracts compute the average costs in coal in cents per million BTU’s.

Measure the energy efficiency of the coal received at each unit.

The average number of BTUs per pound of coal received at each producing unit is shown as follows:

City F # 5	City F # 7	City B	City E	City Z
13,300	12,069	12,354	13,000	12,468

4.)

A sensitivity analysis shows how the coefficients of a linear programming problem affect

the solution.

Review the data and determine if Energy Company D should purchase 80,000 tons of

coal.

The given data indicates the price per ton of coal purchased from AM coal is -$13

per ton with an allowable increase to 88,492 tons. The first set of given data listed as the

Number of Tons Contracted For shows that Energy Company D can purchase additional

amounts at the price of $22 per ton.

If the purchase occurs at $30 per ton the company could reduce cost by $5 per ton.

Purchasing an additional 80,000 tons could save the company $5 (80,000) = $400,000.

5)

Calculating and projecting that the energy content of the CA coal turns out to be

13,000 BTUs per ton the procurement plan changes are listed below:

	City F # 5	City F # 7	City B	City E	City Z
Supplier R	0	0	61,538	288,462	0
Supplier P	36,654	191,729	71,617	0	0
Supplier AM	0	0	0	0	275,000
Supplier CO	0	0	33,878	0	166,122
Supplier CA	0	0	0	0	0
Supplier AD	200,000	0	0	0	0
Supplier W	0	0	98,673	0	0

6.)

The concept of a shadow price is closely related to the concept of a dual price. The shadow price associated with a constraint in the change in the value of the optimal solution per-unit increase in the right hand side of the constraint.

The shadow prices for the demand constraints are listed below:

City F # 5	City F # 7	City B	City E	City Z
21	20	20	18	19

City E unit is the lowest cost producer at the margin ($18 per mWh), and an allowable increase is 160,000 mWh. Energy Company D should sell the 50,000 mWh over the grid.

The additional electricity should be produced at City E’s generating unit. Company D profit will be $12 per mWh.

An Excel model used in solving Company D’s coal allocation problem is shown below:

Fe Home Forma Duta Review Insert Page Layout Calibri B . View e Energy Company D.xlsx - Microsoft Excel Acrobat rt General . Center $ . RX Cut Atun Copy A.EE Con Cal e nt DeletePorno . Total Sourcing Costs (Ton] E Supplier $39.7 $39.7 $36.75 SA SO $39.75 $35.7 $32.75 536 50 $33.73 Supplier AM Supplier B $50.00 $38.25 Sad.00 $50.00 $38.25 Sad.00 $49.73 $39.00 $45.00 $37.25 $42.00 $37.00 * ** Tong Shipped 61.538 21.617 11278 41 $11.492,308 42 $11.97712.00 43 $9.281.250,00 44 $8.505,510.00 33.878 Supplier Supplier AM Supplierco Supplier CA Supplier AD Supplier 46 47 $ 7,650,000.00 $4,499.459.00 98,673 $%XXS TU Provided 5,775,000,000,000 5.775.000.000.000 5,100,000,000,000 5.100.000.000.000 6.565,000,000,000 6,565.000.000.000 City E 7,500,000,000,000 7.500.000.000.000 11.000.000.000.000 1.000.000.000.000