Pb1 y13 =-j10 Pb5. Pb4. y34 =-j10

Transcription

1 EE 55, Exam, Take-home. Due Monday, April, 06, 5:00pm. You may use class notes or any reference materials (e.g., books, etc.) that you like; however, you must work alone, i.e., you should not be communicating with anyone else about how to work the problems on this exam.. Generation planning (5 pts): Develop the complete set of equations for generation expansion plan formulation -b (GEP-b), based on the following data and assumptions. a. Plan for 30 years, at 5 year intervals (thus you should have 6 investment periods (at years 5, 0, 5, 0, 5, and 30). b. Use annual discount rate of 9%. c. Maintain a reserve of 5% above peak load. d. Assume a CO cost of $5/MT (you will need to add a term to the objective function to account for this). e. Transmission topology and data: Two comments follow Note the definitions of positive branch flow for each P bk, k=,,5. The admittances y ij are given on a 00 MVA power base. Pg Pg Pb3 y =-j0 Pb3 y4 =-j0 Pb y3 =-j0 Pb5 Pd y3 =-j0 4 Pb4 y34 =-j0 3 Pg4 Pd3 f. Line flow constraints: -500MW P bk 5000MW, k=,5. The per-unit expressions of these constraints, on a 00 MVA power base, are -500 P bk 500. Given the level of loads (see item (j) below), this implies the transmission has infinite capacity. g. Existing generation data: Bus Bus Bus 4 Technology PC (j=) NGCC (j=3) CT (j=4) Capacity (MW) h. Technology characteristics PC (j=) PC (j=) NGCC(j=3) CT (j=4) Wind (j=5) Capacity credit Investment cost (000$/MW) Fixed O&M costs (000$/MW/yr) Var O&M ($/MWhr) Full-load heat rate (MBTU/MWhr) Fuel cost ($/MBTU) CO emissions (lbs/mwhr) Capacity factor 0.35 Salvage value (000$/MW)

2 i. Load duration curve and corresponding blocks. Assume a load growth rate of %/year, applied to each value d, d, d 3, for each of the two loads in the system, with the durations h, h, and h 3 remaining the same. Therefore, for example, for the load at bus, d would be.0,.084,.95,.395,.4568,.6084,.7758 pu for years, 5, 0, 5, 0, 5, and 30, respectively. You will need to compute the values of d and d 3 for these years as well. Then you will need to do the same for the load at bus 3. d d L (MW) d3 h h+h h+h+h3 Number of hours that Load > L Initial load at bus (Pg) Block Block Block 3 Load (pu) d = d =0.8 d 3 =0.6 Duration (hrs) h =000 h =5000 h 3 =760 Initial load at bus 3 (Pg3) Block Block Block 3 Load (pu) d =. d =0.9 d 3 =0.7 Duration (hrs) h =900 h =500 h 3 =660. Production costing ( pts): a. (4 pts) A load duration curve characterizes the next year for a power system is given below. The power system is supplied by one 3 MW unit. Identify the minimum load, the loss of load probability, the loss of load expectation, and the expected unserved energy, assuming the 3 MW unit is perfectly reliable F Dr ( d e ) 0. d e (MW) b. (5 pts) Now consider that the power system is supplied with one 4 MW unit having a forced outage rate of 0.5. The load is still characterized by the load duration curve above. Compute loss of load probability, loss of load expectation, and expected unserved energy.

3 c. (3 pts) Consider the two situations you have assessed in parts (a) and (b) of this problem. Complete the following sentence: Although situation has the larger LOLP; situation has the larger EUE because. d. (4 pts) Compute the energy expected to be produced by the 4 MW unit of the year. e. (3 pts) Assume the 4 MW unit is a thermal unit. What is the minimum additional information you need to know to compute the cost incurred by unit 4 in producing the amount of energy developed in part (d)? f. (3 pts) After reviewing a production cost program which computes annual production costs in a way that is similar to the procedure suggested in parts (a)-(c) of this problem, an engineer decides to write a production cost simulation program that will allow representation of the power system with much greater operational fidelity. What is the single most important change that the engineer needs to make, relative to that suggested in parts (a)-(e)? 3. Benders (7 pts): The relation between primal and dual optimizations can be stated as: Primal subproblem : max z s. t. x 0 c A x b T A w * x Consider the following problem: max z subject to: 4x 5w x w 0 x 3w 8 w 0, w, x 0, w Dual subproblem : min z s. t. A integer T 0 c b A w* a. (4 pts) Express this problem in the form of Benders, i.e., express an appropriate master problem and an appropriate (primal) subproblem. b. (3 pts) Identify an initial solution to the master. c. (4 pts) Formulate the subproblem dual in terms of w*, and based on the dual formulation, and the solution to the master problem found in part (b), explain why you know that the primal subproblem is infeasible. T

4 d. (3 pts) Given the sub-problem primal is infeasible, explain why it is possible that we may still find a solution to this problem (and thus, that this solution will be feasible in the subproblem primal). e. (3 pts) Find a constraint on w which will make the subproblem primal feasible. 4. Planning principles (0 pts): A good paper on planning (you should read in its entirety - it can be found in IEEE XPlore) is: M. Awad, S. Broad, K. Casey, J. Chen, A. Geevarghese, J. Miller, A. Perez, A. Sheffrin, M. Zhang, E. Toolson, G. Drayton, A. Rahimi, and B. Hobbs, The California ISO transmission economic assessment methodology (TEAM): principles and application to Path 6, Power Engineering Society General Meeting, 006. This paper lists five principles to economic evaluation of proposed transmission upgrades. I have chosen from the paper an excerpt for each principle. For each one, identify how you would implement this principle. Where appropriate, base your answer on information, concepts, and/or methods we have covered in EE 55. I am expecting from you at minimum a -3 sentence paragraph for each principle. First key principle: Benefit Framework: Second key principle: Network Representation:

5 Third key principle: Market Prices: Fourth key principle: Uncertainty: Fifth key principle: Resource alternatives to transmission expansion: Fifth key principle (continued):

6 5. Transmission line parameters (6 pts): A 765 kv transmission line has phase separation of 36 feet. Each phase is a 4-conductor bundle with 4-inch diameter. The conductor type is Tern (795 kcmil). The following data is obtained from tables: Ind reactance ft spacing for 4-conductor bundle at 4 : X a =0.08Ω/mile Ind reactance spacing factor for 36 phase separation: X d =0.4348Ω/mile Cap reactance at ft spacing for 4 conductor bundle at 4 : X a =0.005MΩ-mile Cap reactance spacing factor for 36 phase separation: X d =0.06MΩ-mile a. (6 pts) Compute the SIL for this line configuration with 4 conductors per bundle. b. (4 pts) A designer considers reducing the number of conductors per bundle from 4 to 3, while maintaining conductor spacing and phase spacing at 4 inches and 36 feet, respectively. Would you expect this change to increase or decrease the surge impedance loading? Support your answer. c. (6 pts) The designer desires to maximize the power transfer capability of the line. Relative to the design of part (a) If the line is 30 miles long, what changes would you consider? If the line is 300 miles long, what changes would you consider?