1192 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 2, MAY 2010 Extended Benders Decomposition for Two-Stage SCUC Cong Liu, Student Member, IEEE, Mohammad Shahidehpour, Fellow, IEEE, and Lei Wu, Member, IEEE Abstract—This letter presents the solution of a two-stage secu- rity-constrained unit commitment (SCUC) problem. The proposed SCUC model could include integer variables at the second stage. A framework of extended Benders decomposition with linear feasi- bility and optimality cuts is proposed for the solution of mixed-in- teger programming (MIP) problems at both stages. Test results show the effectiveness of the proposed methodology. Index Terms—L-shaped decomposition, MIP, SCUC. NOMENCLATURE Coefficients vector, matrix of MIP problem. Vector of ones and identity matrix. Index of nodes in branch and bound trees, iterations, and scenarios. Dimension of first/second stage problems. Dual price function of MIP problem. Dual price function of LP-relaxation of MIP. Number of scenarios and their probabilities. First stage, second stage, and slack variables. Spanning space of and variables. Variables represent objective function values. Ancillary binary variable. Dual variables. Predefined tolerance and a large number. I. PROPOSED MIP PROBLEM T HE security-constrained unit commitment (SCUC) problem shown in Table I is solved in two stages according to its specific structure. Generally, the first stage cor- responds to an optimal decision, while the second stage would examine the viability and optimality of first stage decisions. Benders cut applications are based on a prerequisite that the subproblem corresponding to the second stage is LP or convex NLP [1]. In practice, variables in the second stage can be integer or semi-continuous. Consider the following examples: 1) integer variables in power transmission control; 2) status of quick start units in post-contingency evaluation. Manuscript received August 19, 2009. First published January 26, 2010; cur- rent version published April 21, 2010. Paper no. PESL-00034-2009. The authors are with the Electrical and Computer Engineering Department, Illinois Institute of Technology, Chicago, IL 60616 USA (e-mail: cliu35@iit. edu; ms@iit.edu; lwu10@iit.edu). Digital Object Identifier 10.1109/TPWRS.2009.2038019 TABLE I SCUC DECOMPOSITION STRATEGIES We have developed an extended Benders decomposition method which can be applied to such SCUC problems. The SCUC problem is represented as a MIP formulation (1)–(4). Decision variables at the first stage and at the second stage in P could be either integer or continuous. The constraint structure is L-shaped and no coupling among subproblems: (1) (2) (3) (4) Nonlinear feasibility and optimality cuts were developed in [3] via the general duality theory and showed that an L-shaped decomposition can converge within a finite time. The formu- lation of optimality and feasibility cuts are given in (12) and (13). After introducing integer variables, the linear feasibility and optimality cuts (21)–(23) would transform the two-stage SCUC problem into a MIP set, which can be incorporated into the master problem and lead to the global optimal solution. As to the extended Benders decomposition, the mixed-integer master problem (BD-MP) and subproblems (BD-SP) are given in (5)–(8) and (9), respectively: (5) (6) (7) (8) (9) Here, is the current BD-MP solution. In addition, (7) and (8) represent optimality and feasibility cuts, respectively, which are formed by solving BD-SP. The BD-MP solution provides a lower bound while a feasible BD-SP solution leads to an upper 0885-8950/$26.00 © 2010 IEEE