880 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 19, NO. 2, MAY 2004Marginal Loss Modeling in LMP CalculationEugene Litvinov, Member, IEEE, Tongxin Zheng, Member, IEEE, Gary Rosenwald, andPayman Shamsollahi, Senior Member, IEEEAbstract—This paper discusses the pricing of marginal trans-mission network losses in the locational marginal pricing approachrecently deployed in the ISO New England (ISO-NE) standardmarket design (SMD) project implemented by ALSTOM’s T&DEnergy Automation and Information (EAI) Business. The tradi-tional loss model is studied and a new model is proposed. The newmodel achieves more defendable and predictable market-clearingresults by introducing loss distribution factors to explicitly bal-ance the consumed losses in the lossless dc power system model.The distributed market slack reference is also introduced anddiscussed. The LMP components produced by the two modelsare studied and compared under changes in slack reference.Numerical examples are presented to further compare the twomodels.Index Terms—Electricity market, marginal pricing, networklosses, optimization, power systems.NOMENCLATUREGeneration bid price vector.Loss distribution factor vector.Vector whose elements are 1.Market demand vector.Loss sensitivity vector, whose elements are calcu-lated with respect to the distributed slack referencerepresented as vector.Loss System losses variable.offsetSystem loss linearization offset, which is dependenton the slack weight.Market energy supply output vector.Minimum generator output limit.Maximum generator output limit.Line flow limit vector.Constraint sensitivity matrix, whose elements arecalculated with respect to the same distributed slackweight.Distributed slack reference weight vector.Shadow price for the energy balance equation.Shadow price for the system losses equation.Shadow prices vector for the transmission con-straints.Shadow prices for the maximum generator outputconstraints.Shadow prices for the minimum generator outputconstraints.Manuscript received August 12, 2003.E. Litvinov and T. Zheng are with ISO New England, Inc., Holyoke, MA01040 USA (e-mail: [email protected]; [email protected]).G. Rosenwald and P. Shamsollahi are with ALSTOM EAI Corp., Bellevue,WA 98004 USA (e-mail: [email protected]; [email protected]).Digital Object Identifier 10.1109/TPWRS.2004.825894I. INTRODUCTIONAcongestion management system (CMS) is a major part ofthe current competitive electricity markets. A 2002 No-tice of Public Ruling (NOPR) from Federal Energy Regula-tory Commission (FERC) [1] proposes location-based marginalpricing (LMP) together with firm transmission rights (FTR) asa mechanism to build efficient energy markets. LMP is part ofthe standard market design (SMD) promoted by FERC and isa fundamental principle in the majority of electricity markets[2]–[4].LMP at a location is defined as a cost of supplying an incre-ment of load at this location. This price reflects not only the costof producing energy, but also its delivery. Losses and/or trans-mission network congestion may make delivery of energy fromthe least expensive resource to a different location impossible oruneconomic. The resulting “out of merit” dispatch to accommo-date the system constraints using more expensive energy will, inturn, cause price separation at the various network locations, soit is very important to include the effect of congestion and lossesin both economic dispatch and price calculation.All CMS implementations in the current markets utilizelinear-programming (LP) techniques in the market-clearingalgorithms. LP requires linear network models that are nor-mally based on the dc idealization of power-flow equations[2]–[5], [9]. The dc model has no transmission losses by defi-nition, which makes it very challenging to correctly model themarginal effect of losses for economic dispatch and electricitypricing.Marginal loss modeling affects dispatch and LMPs in thesystem. It is shown in the new approach described below thatconsistent modeling of the marginal losses cannot only preservemajor LMP properties with dc model idealization, but also pro-mote a transparent and liquid electricity market by producingmore defendable and predictable results.II. LMP COMPONENTSOne of the important SMD features is the ability to hedgecongestion costs through the use of FTRs. In a system withoutlosses, the value of an FTR can be defined as the differencebetween the LMP at the sink of the FTR and the LMP at thesource [2]. However, in a system with losses, the LMPs at twolocations can be different even in the noncongested case. Thiscould be easily seen from the following expression for the LMPat node[4], [6], [7](1)whereis the constraint ’s power flow sensitivity to theinjection at nodewith respect to the slack reference , and0885-8950/04$20.00 © 2004 IEEELITVINOV et al.: MARGINAL LOSS MODELING IN LMP CALCULATION 881is the number of constraints. Even in the case of no con-gestion, when, the prices would be different at dif-ferent locations due to the variation in the loss sensitivity fac-tors. This makes it impossible to calculate the value of FTRssimply by subtracting the sink LMP from the source LMP forFTRs that hedge only congestion costs. The LMP in (1) couldbe split into three components [3], [4]: reference energy compo-nent, loss component , and congestioncomponent. In the above, although the refer-ence energy component is called “energy component,” in gen-eral, it does not represent the price for energy without loss andcongestion. Because loss factors and all shift factors are equal tozero at the slack reference (i.e., reference bus), the energy com-ponent is equal to the LMP at the reference. The energy com-ponent is the same for all locations in the system. In the modelwith losses, the value of a FTR must be measured by the differ-ence in only the congestion components of LMP. In a losslesssystem, the loss component of LMP in (1) is zero at all loca-tions and the difference in LMPs defining the value of FTRs isthe same as the difference in the congestion components.With the introduction of transmission losses, the LMP mustbe decomposed. Although the LMP is not dependent on theslack reference, the split into LMP components is dependent onthe selection of the slack reference. This dependency indicatesthat importance should not be placed on the absolute value ofeach component by itself. However, the differences in conges-tion components between locations are not