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MIP-Based Post-Contingency Corrective Action With Quick-Start Units Lei Wu, Member, IEEE, Mohammad Shahidehpour, Fellow, IEEE, and Cong Liu, Student Member, IEEE Abstract—This letter provides a new methodology for the postcontingency corrective action in security-constrained unit commitment (SCUC). It formulates a mixed integer programming (MIP) subproblem instead of linear programming (LP) when considering corrective actions based on quick-start units. Test results show that a better solution is achieved by applying MIP. Index Terms—Mixed integer programming (MIP), post-contingency action, quick-start units.
,
,
, , ,
, , , , , , , , , ,
NOMENCLATURE Bus-unit/bus-load/bus-branch incidence matrix. Sets of available/on-outage units in contingency . Index of contingencies/units/time horizon. Production cost function of unit . Non-quick-start units set committed in the base case. Sets of non-quick-start/quick-start units. Power generation/commitment of units. Minimum/maximum capacity. Power flow/reactance of transmission lines. Quick-start capacity of units. Slack variables. Spinning up/down capability of units. Startup/shutdown cost of units. Bus/phase shifter angle. Dual variables. Cut coefficients. Given values from the previous iteration.
and reserve limits, and network security constrains [1], [2]. Constraints for each contingency (3) include power balance, power generation (restricted by the base case solution), and network security constraints. The contribution of this letter is in calculating commitment decisions of quick-start units with post-contingency corrective actions. A practical SCUC problem is not solved in its entirety, which is due to a large number of possible contingencies. A proper decomposition using Benders cuts [3] is discussed next for the SCUC solution. A. Solution Methodology The decomposition procedure is as follows: Step 1) Solve the base case problem described as (1) and (2). The MIP gap threshold of 0.5% or 0.1% is used. Step 2) Solve an MIP subproblem for each contingency as in and derived from step 1. (4) with the base case solution With the relaxation of power balance constraint, (4) is always is zero, the base case is a viable solution feasible. If for post-contingency corrective actions when contingency occurs. Otherwise, a feasibility cut (5) is added to the base case problem (1), (2). Derivations of and in (4) and the feasibility cut in (5) are further discussed in the next subsection. Here, (4) becomes an LP problem if the system has no quick-start units, which is a simplified version of the proposed model. The third set of constraints in (4) includes emergency thermal ratings in contingencies with time constants shorter than the quick-start time. Step 3) Once all contingencies are considered, add all cuts to (1) and (2) and solve the revised base case problem again. The iteration will stop when all contingency violations are remedied within the base case solution:
Transpose of matrix. I. PROBLEM MODELING TATIC security addresses the ability of power systems to withstand disturbances by determining whether there is a new secure operation point following contingences. Securityconstrained unit commitment (SCUC) is formulated as a mixed integer programming (MIP) model in (1)–(3) for minimizing base case operation costs by considering base case and possible contingency states:
S
(1) (2) (3)
(4)
Constraints in the base case (2) include power balance, ramp up/down, minimum up/down time, power generation Manuscript received January 13, 2009; revised March 10, 2009. First published September 18, 2009; current version published October 21, 2009. Paper no. PESL-00158-2008. The authors are with the Electrical and Computer Engineering Department, Illinois Institute of Technology, Chicago, IL 60616 USA (e-mail: lwu10@iit. edu;
[email protected];
[email protected]). Digital Object Identifier 10.1109/TPWRS.2009.2030262 0885-8950/$26.00 © 2009 IEEE
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TABLE I GENERATOR DATA Fig. 1. One-line diagram of the two-bus system.
B. Contingency Subproblems Constraints in (4) are represented in (6) where all state-variables except and :
identifies (6)
The contingency solution is given as follows: . If all solutions Step 1) Solve (6) by relaxing are integer {0,1}, the solution is optimal. Otherwise, go to step 2. Step 2) Choose one non-integer value, , and generate two new LP problems (7) and (8) to eliminate the non-integer solution. By applying (9), an active cut for (7) and (8) is generated calculated in step in (10) which eliminates the non-integer 1. We assume (10) is an active cut for feasible solutions of (7) and (8). The choice of non-integer is based on the most fractional integer variable [4]. Other rules based on the idea of estimating the cost of forcing the variable to become integer may also apply: (7) (8)
(9) (10) Step 3) Add cut (10) to (6) and go back to step 1. The iteration will continue until all such solutions are eliminated. Here, (4) is equivalent to the LP given in (11). Using dual solutions of (11), which are expressed as and in (4), a feasibility cut (5) is generated. Since the number of available quick-start units, i.e., number of integer variables in (4), is limited, (4) is solved in a finite number of iterations using a disjunctive decomposition [5] described above. Iterations between the base case and contingences terminate in finite iterations because the number of feasibility cuts generated from each LP problem (11) is finite (each feasibility cut corresponds to an extreme ray of the convex polyhedral cone of the LP problem (11), and the number of extreme rays is finite [3]):
(11) If the cost of post-contingency corrective action is included in the objective, the proposed method may not be applicable since the optimality Benders cuts could eliminate potential solutions of feasible set.
TABLE II SOLUTION FOR THREE CASES
II. NUMERICAL EXAMPLES A two-bus system including four units and one load depicted in Fig. 1 is discussed. G2 and G4 are quick-start units with 30 MW quick-start capacities. The line capacity is 100 MW and the one-hour load is 90 MW. The contingency of G1 is studied. All other constraints are relaxed in the study. Generator data are shown in Table I. Three cases are studied to illustrate the effectiveness of the proposed method: Case 1) Linear subproblem, lower generation limit of each quick-start unit is set at zero. Case 2) LP-based subproblem, lower generation limit of each quick-start unit is set at its minimum capacity. Case 3) MIP-based subproblem proposed in the letter. Table II shows dispatch solutions and base case costs for the three cases. In Case 1, the post-contingency dispatch solution for G4 is 10 MW, which is lower than its minimum capacity of 20 MW. In fact since the 10-min spinning reserve of G3 is 5 MW, in the post-contingency state of Case 1, quick-start units G2 and G4 can only dispatch 15 MW [i.e., 90-(80-5)]. So by setting lower generation limits of quick-start units to zero, the postcontingency dispatch solution may be infeasible which would violate minimum capacity constraints. In Case 2, lower generation limits of quick-start units are set to minimum capacity. All quick-start units are committed in post-contingency which would lower the dispatch of cheap unit G3 and increase the base case operation cost. In Case 3, since an MIP subproblem is considered for the post-contingency corrective action, the MIPbased unit commitment of quick-start units G2 and G4 is optimized with a lower dispatch cost. REFERENCES [1] L. Wu, M. Shahidehpour, and T. Li, “Cost of reliability analysis based on stochastic unit commitment,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1364–1374, Aug. 2008. [2] A. J. Conejo, E. Castillo, R. Mínguez, and R. García-Bertrand, Decomposition Techniques in Mathematical Programming. New York: Springer, 2006. [3] M. Shahidehpour and Y. Fu, “Benders decomposition,” IEEE Power and Energy Mag., vol. 3, no. 2, pp. 20–21, Mar. 2005. [4] L. A. Wolsay, Integer Programming. New York: Wiley, 1998. [5] R. G. Jeroslow, “A cutting-plane game for facial disjunctive programs,” SIAM J. Control Opt., vol. 18, no. 3, pp. 264–281, May 1980.
