In this paper we present a formulation of the unit commitment problem with AC power flow constraints. It is solved by a Benders decomposition in which the unit commitment master problem is formulated as a mixed-integer problem with linearization of the power generation constraints for improved convergence. Semidefinite programming relaxation of the rectangular AC optimal power flow is used in the subproblem, providing somewhat conservative cuts. Numerical case studies, including a 6-bus and the IEEE 118-bus network, are provided to test the effectiveness of our proposal. We show in our numerical experiments that the use of such strategy improves the quality of feasibility and optimality cuts generated by the solution of the convex relaxation of the subproblem, therefore reducing the number of iterations required for algorithm convergence.
In Lagrangian relaxation (LR) applications to UC it was common to relax reserve and demand coupling constraints in order to create individual subproblems for every unit. Other common approaches include the representation of individual unit subproblems as mixed-integer problems @cite_23 with techniques to select the Lagrange multipliers that maximize lower bounds produced by the relaxation, as well as the identification of identical solutions @cite_8 that cause dual solutions to be far from the optimum by means of successive subproblems.
@cite_23 @cite_8
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Lagrangian relaxation (LR) and general mixed integer programming (MIP) are two main approaches for solving unit commitment (UC) problems. This paper compares the LR and the state of art general MIP method for solving UC problems based on performance analysis and numerical testing. In this paper we have rigorously proved that UC is indeed an NP complete problem, and therefore it is impossible to develop an algorithm with polynomial computation time to solve it. In comparison with the general MIP methods, the LR methodology is more scaleable and efficient to obtain near optimal schedules for large scale and hard UC problems at the cost of a small percentage of deviation from the optimal solution. In particular, solving hydro generation subproblems within the LR framework can take advantages of both LR and general MIP methods and provide a synergetic combination of both approaches. When the Lagrangian relaxation based methods are applied to solve power system unit commitment, the identical solutions to the subproblems associated with identical units may cause the dual solution to be far away from the optimal solution and serious solution oscillations. As a result, the quality of the feasible solution obtained may be very unsatisfactory. This issue has been long recognized as an inherent disadvantage of Lagrangian relaxation based methods. In this paper, the homogeneous solution issue is identified and analyzed through a simple example. Based on this analysis, a successive subproblem solving method is developed. The new method combines the concepts of augmented Lagrangian relaxation and surrogate subgradient to produce a good search direction at the high level. The low level subproblems including those corresponding to the identical units are solved successively so that the commitments of the identical units may not be homogeneous in the dual solution. Compared with the standard Lagrangian relaxation method, the new method can obtain better dual solutions and avoid the solution oscillations. Numerical testing shows the new method is efficient and the quality of the feasible solution is greatly improved. We present a novel dual decomposition approach to MAP inference with highly connected discrete graphical models. Decompositions into cyclic k-fan structured subproblems are shown to significantly tighten the Lagrangian relaxation relative to the standard local polytope relaxation, while enabling efficient integer programming for solving the subproblems. Additionally, we introduce modified update rules for maximizing the dual function that avoid oscillations and converge faster to an optimum of the relaxed problem, and never get stuck in nonoptimal fixed points. For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods solve a sequence of subproblems of the form "minimize an augmented Lagrangian function subject to linearized constraints." Such methods converge rapidly near a solution but may not be reliable from arbitrary starting points. Nevertheless, the well-known software package MINOS has proved effective on many large problems. Its success motivates us to derive a related LCL algorithm that possesses three important properties: it is globally convergent, the subproblem constraints are always feasible, and the subproblems may be solved inexactly. The new algorithm has been implemented in MATLAB, with an option to use either MINOS or SNOPT (Fortran codes) to solve the linearly constrained subproblems. Only first derivatives are required. We present numerical results on a subset of the COPS, HS, and CUTE test problems, which include many large examples. The results demonstrate the robustness and efficiency of the stabilized LCL procedure.