Sample Problems for Exam 1, EE Note:

Transcription

1 Sample Problems for Exam, EE 57 Note: Problem 3) Has been changed from what was handed out Friday, this is closer to what will come on the quiz. Also note that the hour time period length must be incorporated in the Lagrangian as shown in the solution. Problem 4) The handout on Friday morning had a fourth problem, but since it was a problem dealing with Hydro scheduling I eliminated it from this version we will not include hydro scheduling until the next exam. Prof Wollenberg

2 EE57, Exam, Problem Session Samples PROBLEM Given the three generating units below: UNIT : where UNIT : UNIT 3: 50 < P < 600 MW 300 < P < 875 MW 65 < P 3 < 400 MW F ( P ) = P P F ( P ) = P P F 3 ( P 3 ) = P P 3 Solve for the economic dispatch of these three units when they are to supply a total load of 500MW. Assume all generators are on line. WATCH GENERATOR LIMITS

3 Sample problem solution Unit Unit Min Incremental Cost Unit Max Incremental Cost Total Steps All units at min output: total = 75 Unit will load to its max giving a total of 90 Unit 3 now loads to its max, total now = 55 Back unit 3 down by 5 MW Solution MW Inc Cost Unit = at min Unit = at max Unit 3= Total 500

4 EE57, Exam, Problem Session Samples PROBLEM You are to find the optimal unit commitment schedule for the three generating units shown in the table below: (SEE NOTES NEXT PAGE) Unit Maximum Minimum Incremental No Load Start (MW) (MW) Cost Energy Input Up Cost ($/MWH) ($/HR) ($) The minimum up and down times for the units are: MINIMUM MINIMUM Up Time Down Time (HR) (HR) The load to be served is: (Each load period is hour long.) Period Load (MW) Use the following as the states to be searched in the DP: State Unit Unit Unit 3 Capacity of all units on 800 MW MW MW Assume that all four generators are off line prior to the first hour, and that they each have been off longer than their minimum down time. FIND: A) The optimal unit commitment schedule for the 4 hour time period B) The total production cost for the optimal schedule C) See part C at bottom of next page. Note: Please show all your work. If you skip some states in your scheduling, explain why.

5 EE57, Exam, Problem Session Samples The units in this example have linear F(P) functions : F(P) No Load Cost Pmin P Pmax The F(P) function is: F( P) = NoLoadcost + Inccos t * P Note however, that the unit must operate within its limits. To dispatch units with linear cost functions follow these steps: Step ) Sum the Pmax for all units on line - if this is less than the load the state is infeasible, quit. Step ) For each unit that is on line set its generation output at Pmin, subtract the sum of the Pmin for all the on line units from the total load to be supplied during the time period. Step 3) Raise the generation on the unit with the minimum incremental cost until either the total load is met or that unit hits its maximum. If at the maximum, then subtract that generations Pmax plus the sum of the Pmin of the other generator(s) from the load. Step 4) Raise the generation on the unit with the next higher incremental cost until either the total load is satisfied or that unit hit max. If at the maximum, then subtract that generations Pmax of the units at max plus the sum of the Pmin of the other generator from the load. Step 5) Take the highest incremental cost unit and start raising its output until load is satisfied. Part C) Show the equations you would use to set up a Lagrange Relaxation solution for this prob-.

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7 EE57, Exam, Problem Session Samples PROBLEM 3 A small municipal power company in northern Minnesota has two power plants. One burns coal and the other burns natural gas supplied from a pipeline coming from Canada. The power company has ample supplies of coal and it purchases gas as take or pay contracts for fixed periods of time. For the 4 hour time period shown below, the power company must burn.0*0 6 cuft of gas. The fuel costs to the paper company are: coal gas.0 $/MBTU the gas is rated at 000 BTU/cuft Input/Output characteristics of generators: Unit (coal unit) H ( P ) = 0.P MBTU/HR 50 < P < 500 Unit (gas unit) q( P ) =.5P cuft/hr 50 < P < 000 Load: (Both load periods are hours long) Period Load (MW) Assume both units are on line for the entire 4 hours FIND: The most economic operation of the power plants over the 4 hours which meets the gas consumption requirements. (You may stop when you are within + or - 0.5*0 6 cuft of the gas target.).

8 Solution to Prob 3: Notation, subscript is the generator number, superscript is the time step. Set starting γ = 0.0 Lagrangian = F ( P ) + F( ) + λ (500 P P ) + λ(800 P ) + P γ (q( P ) + q( P ) *0 P Note, in the lagrangian, the last part requires a multiplication by hours since the q(p) function is the cuft/hour and each time period is hours. Two sets of equations should be set up: Set : is used to solve P, P, andλ 6 ) F P λ = 0 q γ λ P = P P = 0 Set :is used to solve P, P, andλ F P λ = 0 q γ λ P = P P = 0 Solution algorithm: a) Pick a value for γ b) Solve set then set c) Using P and P calculate the total gas used and compare to *0^6 See next page for answers.

9 First iteration: Set solves to: P = 375. andp = 5. withλ = 75 Set : solves to: P = 600andP = 00. withλ = 0 6 Total gas is: q( P = 5) + q( P = 00) =.668* 0 cuft Go back to step a and reduce γ slightly. Etc.