A strong semidefinite programming relaxation of the unit commitment problem

The unit commitment (UC) problem aims to find an optimal schedule of generating units subject to the demand and operating constraints for an electricity grid. The majority of existing algorithms for the UC problem rely on solving a series of convex relaxations by means of branch-and-bound or cutting-planning methods. In this paper, we develop a strengthened semidefinite program (SDP) for the UC problem by first deriving certain valid quadratic constraints and then relaxing them to linear matrix inequalities. These valid inequalities are obtained by the multiplication of the linear constraints of the UC problem such as the flow constraints of two different lines. The performance of the proposed convex relaxation is evaluated on several instances of the UC problem. For most of the instances, globally optimal integer solutions are obtained by solving a single convex problem. Since the proposed technique leads to a large number of valid quadratic inequalities, an iterative procedure is devised to impose a small number of such valid inequalities. For the cases where the strengthened SDP does give a global integer solution, we incorporate other valid inequalities, including a set of Boolean quadric polytope constraints. The proposed relaxations are extensively tested on various IEEE power systems in simulations.

[1]  W. C. Andrews,et al.  THE AMERICAN INSTITUTE OF ELECTRICAL ENGINEERS. , 1901, Science.

[2]  L. L. Garver,et al.  Power Generation Scheduling by Integer Programming-Development of Theory , 1962, Transactions of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems.

[3]  John A. Muckstadt,et al.  An Application of Lagrangian Relaxation to Scheduling in Power-Generation Systems , 1977, Oper. Res..

[4]  A. Turgeon Optimal unit commitment , 1977 .

[5]  A. Turgeon Optimal scheduling of thermal generating units , 1978 .

[6]  F. Albuyeh,et al.  Evaluation of Dynamic Programming Based Methods and Multiple area Representation for Thermal Unit Commitments , 1981, IEEE Transactions on Power Apparatus and Systems.

[7]  Charles R. Johnson,et al.  Positive definite completions of partial Hermitian matrices , 1984 .

[8]  Hanif D. Sherali,et al.  A new reformulation-linearization technique for bilinear programming problems , 1992, J. Glob. Optim..

[9]  Hiroshi Sasaki,et al.  A solution method of unit commitment by artificial neural networks , 1992 .

[10]  Michel X. Goemans,et al.  Semideenite Programming in Combinatorial Optimization , 1999 .

[11]  H. A. Smolleck,et al.  A fuzzy logic approach to unit commitment , 1997 .

[12]  Chih-Chang Tseng,et al.  A unit decommitment method in power system scheduling , 1997 .

[13]  Warren P. Adams,et al.  A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems , 1998 .

[14]  Y. Nesterov Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

[15]  V. Quintana,et al.  An interior-point/cutting-plane method to solve unit commitment problems , 1999, Proceedings of the 21st International Conference on Power Industry Computer Applications. Connecting Utilities. PICA 99. To the Millennium and Beyond (Cat. No.99CH36351).

[16]  Chuan-Ping Cheng,et al.  Unit commitment by Lagrangian relaxation and genetic algorithms , 2000 .

[17]  O. SIAMJ. CONES OF MATRICES AND SUCCESSIVE CONVEX RELAXATIONS OF NONCONVEX SETS , 2000 .

[18]  S. Oren,et al.  Solving the Unit Commitment Problem by a Unit Decommitment Method , 2000 .

[19]  G. Purushothama,et al.  Simulated Annealing with Local Search: A Hybrid Algorithm for Unit Commitment , 2002, IEEE Power Engineering Review.

[20]  A. Papalexopoulos,et al.  Optimization based methods for unit commitment: Lagrangian relaxation versus general mixed integer programming , 2003, 2003 IEEE Power Engineering Society General Meeting (IEEE Cat. No.03CH37491).

[21]  W. Ongsakul,et al.  Unit commitment by enhanced adaptive Lagrangian relaxation , 2004, IEEE Transactions on Power Systems.

[22]  Deepak Rajan,et al.  IBM Research Report Minimum Up/Down Polytopes of the Unit Commitment Problem with Start-Up Costs , 2005 .

[23]  Xiaoqing Bai,et al.  Semi-definite programming-based method for security-constrained unit commitment with operational and optimal power flow constraints , 2009 .

[24]  C. Gentile,et al.  Tighter Approximated MILP Formulations for Unit Commitment Problems , 2009, IEEE Transactions on Power Systems.

[25]  S. Burer,et al.  OLD WINE IN A NEW BOTTLE : THE MILP ROAD TO MIQCP , 2009 .

[26]  Adam N. Letchford,et al.  On Nonconvex Quadratic Programming with Box Constraints , 2009, SIAM J. Optim..

[27]  Prateek Kumar Singhal,et al.  Dynamic Programming Approach for Large Scale Unit Commitment Problem , 2011, 2011 International Conference on Communication Systems and Network Technologies.

[28]  Kurt M. Anstreicher,et al.  On convex relaxations for quadratically constrained quadratic programming , 2012, Math. Program..

[29]  M. Anjos,et al.  Tight Mixed Integer Linear Programming Formulations for the Unit Commitment Problem , 2012, IEEE Transactions on Power Systems.

[30]  S. Low,et al.  Zero Duality Gap in Optimal Power Flow Problem , 2012, IEEE Transactions on Power Systems.

[31]  Benjamin F. Hobbs,et al.  The Next Generation of Electric Power Unit Commitment Models , 2013 .

[32]  Andres Ramos,et al.  Tight and Compact MILP Formulation of Start-Up and Shut-Down Ramping in Unit Commitment , 2013, IEEE Transactions on Power Systems.

[33]  Javad Lavaei,et al.  Efficient convex relaxation for stochastic optimal distributed control problem , 2014, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[34]  Hatim S. Madraswala,et al.  Genetic algorithm solution to unit commitment problem , 2016, 2016 IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES).