2758 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 3, AUGUST 2013 The Subgradient Simplex Cutting Plane Method for Extended Locational Marginal Prices Congcong Wang, Student Member, IEEE, Tengshun Peng, Peter B. Luh, Fellow, IEEE, Paul Gribik, and Li Zhang, Member, IEEE Abstract—In current electricity markets of the USA, locational marginal prices (LMPs) are obtained from the economic dispatch process and cannot capture costs associated with commitment de- cisions. The extended LMPs (ELMPs) were established as the op- timal Lagrangian multiplier of the dual of the unit commitment and economic dispatch problem. Commitment related costs are in- cluded and uplift payments are minimized. To obtain ELMPs, the dual problem should be solved with multiplier optimality and com- putational efficiency. Subgradient methods suffer from the multi- plier zigzagging difficulty. Cutting plane methods encounter com- putational complexity issues in calculating query points. In this paper, a subgradient simplex cutting plane method is developed to obtain ELMPs. Transmission is not considered for simplicity, while key features of ELMPs are still captured. By innovatively using subgradients and simplex tableaus, query points are efficiently ob- tained through an adaptive three-level scheme. A query point along the subgradient is easily calculated at Level 1. As needed, Level 2 obtains Kelley’s query point and Level 3 obtains the Chebyshev center, both by pivoting simplex tableaus. Numerical results show that the optimal multiplier is efficiently obtained. Index Terms—Cutting plane methods, electricity markets, extended locational marginal prices (ELMPs), Lagrangian relax- ation. I. INTRODUCTION I N the current wholesale electricity markets of the USA, the auction mechanism selects generation offers and their cor- responding levels to minimize the total bid cost. The conges- tion-dependent locational marginal prices (LMPs) are then de- termined in the economic dispatch process with fixed unit com- mitment decisions. Start-up and no-load costs associated with unit commitment decisions cannot be included in LMPs, re- sulting in significant uplift payments [1]. To improve price sig- nals, extended LMPs (ELMPs) were developed in [2] as the op- timal Lagrangian multiplier of the dual of the unit commitment and economic dispatch (UCED) problem. The corresponding optimal dual values as a function of demand form the convex Manuscript received May 23, 2012; revised October 15, 2012 and December 12, 2012; accepted January 21, 2013. Date of publication March 07, 2013; date of current version July 18, 2013. This work was supported in part by the Na- tional Science Foundation under Grant ECCS-1028870 and a project funded by MISO. The preliminary results were presented in 2009 and 2010 IEEE Power and Energy Society General Meetings. Disclaimer: The views expressed in this paper are solely those of the authors and do not necessarily represent those of MISO or NSF. Paper no. TPWRS-00549-2012. C. Wang and P. B. Luh are with Department of Electrical and Computer En- gineering, University of Connecticut, Storrs, CT 06269-2517 USA. T. Peng, P. Gribik, and L. Zhang are with Midwest Independent Transmission System Operator, Inc., Carmel, IN 46032 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2013.2243173 hull of the total cost function. Commitment related costs are included in ELMPs, and uplift payments are minimized. Eco- nomic significance of ELMPs as compared with that of the cur- rent LMPs was presented in [2], and the following will focus on developing an efficient algorithm to obtain the optimal multi- plier as ELMPs. For simplicity but without loss of key features of ELMPs, transmission capacity constraints and ancillary ser- vices are not considered. The mixed-integer UCED problem has been well solved by branch and cut methods, however, in the primal space. To ob- tain ELMPs, multiplier optimality rather than the optimality or feasibility of the primal solution is required. Traditional subgra- dient methods suffer from the multiplier zigzagging difficulty [3], and are thus ineffective in obtaining the optimal multiplier. Central cutting plane methods have been used to solve non-dif- ferentiable optimization problems [4], [5]. Given an initial poly- hedron containing the optimal solution, non-optimal portions of the polyhedron are iteratively cut off by constructing cuts from query points [5]. A key issue is to select query points such that effective cuts can be constructed. Without knowing in advance which part to cut off, various centers deep inside the polyhedron have been investigated as query points. However, centers such as the center of gravity or the analytic center can be computa- tionally expensive to obtain. In this paper, a subgradient simplex cutting plane method (SSCPM) is developed to efficiently solve the dual problem. After the literature review in Section II, the UCED problem is formulated, and the dual problem is obtained by using La- grangian relaxation in Section III. The SSCPM is developed in Section IV to solve the dual problem by innovatively using sub- gradients and simplex tableaus. Given an initial polyhedron and query point, cuts are constructed from the query point. In the up- dated polyhedron, an adaptive three-level scheme is established to obtain the next query point. Rather than calculating centers of the polyhedron through an expensive procedure, a point along the subgradient is easily obtained as the query point at Level 1. However, this point may not always be deep inside the polyhe- dron, resulting in ineffective cuts. In this case, Kelley’s query point is calculated at Level 2. If effective cuts cannot be con- structed at Kelley’s point, then the Chebyshev center is ob- tained at Level 3, and the process repeats. By pivoting simplex tableaus, query points at Levels 2 and 3 are efficiently obtained. Numerical results are presented in Section V. A simple two-hour example illustrates the adaptive three-level process to cut off non-optimal multipliers. A 24-hour 32-unit example based on the IEEE Reliability Test System shows that our algo- rithm efficiently obtains the optimal multiplier. A MISO-sized problem for the day-ahead market and one hundred Monte Carlo simulation runs then demonstrate that our method solves large problems with robust performance. The preliminary results were presented at the IEEE Power and Energy Society General Meetings in 2009 [6] and 2010 U.S. Government work not protected by U.S. copyright.