# Lmps for (Technically-Inclined) Dummies

By Savitski, David W. | Energy Law Journal, July 1, 2019 | Go to article overview

# Lmps for (Technically-Inclined) Dummies

Savitski, David W., Energy Law Journal

(ProQuest: ... denotes formulae omitted.)

Symbol Key

AC = alternating current

DC = direct current

A = change in variable (e.g., AQ = a change in Q)

df(x)/dx = the derivative (slope) of the function f(x)

ðf(x,z)/ðx = the partial derivative (slope) of the function f(x,z)

GEN1, GEN2, GEN3 = generator 1, 2, and 3

Jf(x)dx = integral, the area under the function f(x)

Kı, K2, etc. = the capacity of generators 1, 2, etc.

LMP = locational marginal price

...= the Lagrangian (the constrained objective function)

... = the Lagrangian multipliers (shadow prices) for energy at the reference bus, transmission, and generators 1, 2, and 3

MCı, MC2, МСз = the marginal (incremental) cost of one MWh of energy supplied by generators 1, 2, and 3

MW = megaWatt

MWh = megaW att hour

P1, P2, Рз = price of energy (the LMPs) at buses 1, 2, and 3

π1, π2, πз = the profit of generators 1, 2, and 3

Q = Total energy demanded (and supplied)

Q1, Q2, Q3 = the energy supplied by generators 1, 2, and 3

S1, S2 = flows of GENİ and GEN2 over the constrained line to the reference bus

Σ = summation

T = the maximum flow on the constrained transmission line

z = transmission line resistance

I. Introduction

Marke yts commonly use price to equate supply and demand, perhaps best illustrated by the stock market.1 Many wholesale electricity markets similarly use price to equate the supply and demand for energy (transmission prices are still regulated), including CAISO, MISO, and PJM, among others.2 Because the transmission of electricity is sometimes constrained, however, there is often not a single market-clearing price as in the stock market. Instead, prices often depend on location, hence "locational marginal prices" (LMPs). LMPs result from an optimization process whereby wholesale electricity prices in organized markets are set locally (at the bus level) based on least-cost dispatch, subject to generation and transmission constraints.3 These constraints play a central role in dispatch and greatly complicate analyses of LMPs.4 In such markets, load (electricity demand) pays the LMP at the bus where energy is taken and generators receive the LMP at the bus where energy is delivered.5

The Federal Energy Regulatory Commission (FERC) first approved LMPs for PJM in its November 25, 1997, order in Docket No. OA97-261.6 This seminal case involved debate over the merits of LMP, which FERC evaluated and summarized in this order (at pages 37-51).7 Proponents argued that LMPs send correct price signals for efficiency because the marginal benefit to load equals the marginal cost of delivering energy to each location, with transmission congestion costs reflected in LMP differences, unlike for average-cost pricing.8 The proponents also argued that LMPs send correct signals for investment in the industry (e.g., producers prefer to build generators in locations where LMP is high and large energy consumers prefer to locate their load where LMP is low) and for transmission use and planning, among other things.9 Opponents argued that LMPs overcompensate inframarginal baseload units,10 are based on offer prices rather than cost data, will not lower prices, are not known ex ante, and are overly complex.11 While the merits of these points are not explored here, an understanding of LMP derivation sheds light on them, as noted periodically below.12

The LMP literature varies greatly based on approach, depending on audience and application. A relatively simple approach describes LMPs in general as consisting of three cost components (energy, transmission congestion, and transmission losses) of delivering a MWh of electricity to specific locations.13 A more complicated approach explores efficiency, LMPs, market power, welfare effects, and so on using graphs and tables.14 A more abstract approach explores similar issues using advanced mathematics, often vector calculus. …

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