A transmission-constrained unit commitment method in ... - CiteSeerX

More documents

Recommendations

Info

302 ( ) C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310 corresponding to the most unsatisfied capacity constraint and increasing only the multiplier mt of this hour. This approach can be modified to simultaneously updating the multipliers m t’s of it all hours that the spinning capacity requirement is violated. Such a modification can speed up the feasibility phase at the cost of overcommitment w5,14 x. Under the three-phase scheme, this overcommitment effect can nevertheless be corrected in the final unit decommitment phase to be addressed in a later section. The feasibility phase presented in this paper contains two parts. Part 1 seeks a reserve-feasible commitment by increasing the multipliers mt of all hours with insufficient spinning capacities. Starting from this reserve- feasible commitment obtained, Part 2 solves a linear program Ž LP. for each hour to verify if this commitment is also dispatchable at this hour. If not, information can be extracted from the solution of the LP to determine how much additional spinning capacity at this hour is required so that a dispatchable commitment exists. The spinning reserve requirement of this hour is thus uplift. Part 1 is then rerun with respect to the new Ž uplift. spinning reserve requirements. Suppose Part 1 of the feasibility phase terminates at iteration k with an obtained commitment u k 4 it . Let qbt be the variable representing the total generation to be dispatched to bus b in time t. k 4 min min k max max k Given a commitment uit let qbt sÝig L pi uit and qbt sÝig L pi u it, which form an immediate b b lower bound and upper bound for q bt. ŽLPŽ t.. Ž . min Ý B y yq max Ž 19. q bt, ybt bs1 bt bt s.t. qbt min FqbtFy bt, ;b Ž 20a. q max Fy FÝ p max , ;b Ž 20b. bt bt ig L b i Ýbs1 B qbtsDt Ž 20c. yFlFÝbs1 B G l bŽ qbtyDbt. FF l , ; l Ž 20d. q Gr D , ;b Ž 20e. bt bt bt Ž Ž .. max In LP t , y is a variable upper bound of both q and q . Let y ) be the solution of ŽLPŽ t ...Ifu k 4 bt bt bt bt it is max dispatchable in time t, y )sq solves ŽLPŽ t.. and the optimal objective value of ŽLPŽ t.. is zero. If u k 4 bt bt it is not dispatchable in time t, ŽLPŽ t.. yields a positive objective value, which indicates that to be dispatchable, more units should be committed in time t, and the target value of the new bus capacity requirement is now increased to y bt) for each bus b in time t. That is, an additional bus spinning capacity requirement is imposed to each bus subproblem d Ž P .: Ý i it bt b p max u Gy ). Ž 21. igL b Ž . Part 1 of the feasibility phase is therefore rerun with respect to the updated capacity requirement 21 . The phase 2 algorithm is summarized below. 3.2.1. Phase 2 algorithm Step 0: k§0; m 0 and r 0 and other multipliers are from Phase 1. k k Ž . Ž k k . Step 1: Given m and r and other multipliers, solve the dual subproblems 14 to obtain u , p . Ž. Ž. Step 2: If 3 and 6 are satisfied, go to Step 5 if Step 5 has not been visited, otherwise stop. kq1 k k Ž B max k . kq1 k k Step 3: For ;b,t, m §m qs Pmin 0, R yÝ Ý p u ; r §r qs PminŽ t t t bs1 i g L i it bt bt 0,sbtRbty b max Ý p u k .. i g L b i it
Step 4: k§kq1, go to Step 1. ( ) C.-L. Tseng et al.rDecision Support Systems 24 1999 297–310 303 k Step 5: Solve ŽLPŽ t.. for all t. If the optimal value of ŽLPŽ t.. is 0 for all t, stop and u is reserve-feasible and dispatchable. Otherwise, commit more units in any bus b and time t such that y bt))qbt max to satisfy Eq. Ž 21 ., where y ) solves ŽLPŽ t .. bt . In the algorithm, the loop between Step 1 and Step 4 corresponds to the Part 1 of the feasibility phase, and Step 5 corresponds to Part 2. The feasibility phase is nevertheless heuristic. An important issue is how robust the feasibility phase of a transmission-constrained unit commitment method is. To test the robustness of the feasibility phase, one can apply it to an instance in which each single bus has enough capacity to handle its own load over the planning horizon. Then the transmission line capacities are gradually reduced to zero. In such a problem, a feasible solution obviously exists, and it is equivalent to solving many single-area unit commitment problems. It can be easily verified that our feasibility phase reduces to solving many single-area problems in such an instance. 3.3. Economic dispatch With the DC power flow model, the transmission constraints are formulated as linear constraints. If the fuel cost is represented by a quadratic function of the power generation, the transmission-constrained economic dispatch is a quadratic programming problem. Many methods can be used to solve this type of problem, e.g., the reduced gradient method. In our implementation, the transmission-constrained economic dispatch is solved by the commercial software MINOS, which does general nonlinear programming. 3.4. Phase 3: unit decommitment A Ž single area. unit decommitment Ž UD. method was developed as a postprocessing method for the Ž LR. approach for solving the unit commitment problem in Ref. w15 x. Given the demand requirement D t, reserve requirement R and a feasible solution Ž u, p. Ž satisfying D, R and all physical constraints . t ˜ ˜ t t , the unit decommitment method improves Ž u, ˜ p˜. while maintaining feasibility. Thus, the unit decommitment can be regarded as a primal approach. A transmission-constrained unit decommitment algorithm can be devised by repeatedly solving the Ž single area. unit decommitment to each bus. Suppose at iteration k, we have a feasible schedule Žu, ˜ k p ˜ k .. The schedule of bus b can be improved by the unit decommitment with respect to the it current load assigned to be the bus demand at this iteration, i.e., Ýi g L p˜it k u˜it k . The unit decommitment method b can obtain a more efficient schedule, if possible, which generates the same load Ýi g L p˜it k u˜it k . After the b commitments of all buses are refined, a transmission-constrained economic dispatch is performed to redispatch the units in all buses. The decommitment procedure can thus proceed until no improvement in the total cost is made. Since at each iteration the decommitment procedure improves the commitment without affecting the aggregated bus generation, the transmission feasibility is retained at each iteration. The algorithm is as follows. 3.4.1. Transmission-constrained UD algorithm Step 0: A feasible schedule Žu ˜ 0 , p˜ k . is given, k§0. Step 1: Given Žu,p ˜ k ˜ k . each bus performs the unit decommitment presented in Ref. w15x independently with the current load assigned to be the demand requirement Ýi g L p˜it k u˜it k also subject to the satisfaction of Eq. b k Ž. 6 . The resultant commitment is denoted by u ˆ . Step 2: Apply the transmission-constrained economic dispatch with respect to uˆ k to obtain a dispatch pˆ k . Let Žu ˜ kq1 , p ˜ kq1 .§ Žu,p ˜ k ˜ k ..
Page 1 and 2: Ž . Decision Support Systems 24 19
Page 3 and 4: ( ) C.-L. Tseng et al.rDecision Sup
Page 5: is a constant term given the multip

A transmission-constrained unit commitment method in ... - CiteSeerX

Create successful ePaper yourself

Delete template?

Save as template?