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Locational Marginal Pricing in North American Power SystemsMr. Henry Louie, University of Washington, USA

Prof. Kai Strunz, Technische Universitt Berlin, Deutschland

Abstract

The methodology used in the pricing of electrical energy is a fundamental characteristic of electricity market design.

In deregulated North American power systems the utilization of locational marginal pricing is the dominant approach

to pricing electrical energy. Locational marginal prices (LMPs), which are spatially and temporally distinguished nodal

pricesbased uponshort-run marginal costs, reflect economicandphysical realities of thepower systemas well as operating

constraints. In addition, LMPs can be used to ascertain transmission congestion costs and are often included in ancillary

service market clearing or settlement calculations. It is therefore requisite to understand the fundamentals of LMPs to be

able to analyze deregulated North American power system economics. In this paper, the concept, calculation, utilization

and practical application of LMPs as well as a thorough educational illustration are provided.

1 IntroductionThe North American power system served approximately

4,926 TWh of electrical energy to customers in the U.S.,

Canada and Mexico in 2006 [1]. In the U.S. and Canada,

the majority of the load is procured through market trans-

actions in the geographic footprint of Regional Trans-

mission Organizations (RTOs), which are independent,

revenue-neutral entities charged with, among other respon-

sibilities, maintaining system reliability and market over-

sight [2]. Market designs differ amongst RTOs; however,

a characteristic common to many is the utilization of lo-

cational marginal prices (LMPs) for electrical energy pric-ing [3].

Locational marginal pricing, which is based on the short-

run marginal cost of supplying energy, was developed in

the 1980s [4, 5, 6] and has grown to be the dominant

method of pricing energy in electricity markets in North

America that operate under the auspices of an RTO [7, 3,

8, 9, 2]. The use of LMPs has grown because the phys-

ical constraints of the system and economic realities are

accurately represented. The resulting LMPs can be readily

utilized to price transmission and to determine congestion

costs. This paper formulates the concept of LMPs in an

educational manner and provides details on their practicalcalculation and utilization in North American power sys-

tems.

Currently, nodal LMPs are utilized in four North America

RTOs: Midwest Independent System Operator (MISO),

New England RTO, New York ISO (NYISO) and the

Pennsylvania-Jersey-Maryland Interconnection (PJM) [2].

In addition, LMPs are included in the new market designs

to be fully implemented in the California ISO (CAISO) in2008 and in the Electric Reliability Council of Texas (ER-

COT) in 2009 [8]. Other markets rely on bilateral transac-

tions, zonal pricing, or the last cleared generation offer to

set electricity prices. The impetuses for the adoption of the

LMP-basedpricing are to achieve greater marketefficiency

and to more accurately represent the price of transmission

and the cost of congestion [8, 9].

The LMPs are computed for each node and market period

in both the forward and balancing energy markets. The

forward energy market is typically for a period one day

ahead, commonly called a day-ahead market. The LMPs

are utilized in the market settlement processes to determinegenerator payments and load charges by multiplying the

amount of energy produced or consumed at a node by the

LMP at that node. In addition, LMPs are used in ancillary

service calculations and to price transmission and manage

congestion and therefore the understanding of their cal-

culation is crucial in North American power system eco-

nomic analyses.

The paper is arranged as follows. In Section 2, the concept

of locational marginal pricing is introduced. In Section 3,

the practical calculation of LMPs is formulated. In Sec-

tion 4, an educational illustration of LMP calculation is

demonstrated on a four-bus study system considering theeffects of congestion. Conclusions are drawn in Section 5.

2 Concept

The concept of locational marginal pricing is to assign the

price of energy based on the short-run marginal cost of


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supply. In so doing, economic signals for system opera-

tion are provided, including costs of congestion. An anal-

ogous formulation of LMP calculation for reactive power

is possible; however, real power is the focus of this paper

as reactive power and other component costs of supplying

energy are compensated in ancillary service markets and

market settlement procedures [3, 6].

The short-run marginal cost at a node is equal to the change

in optimal economic cost associated with supplying an

additional increment of load at that node. This value is

readily available from the Lagrangianmultiplier associated

with the nodal real power balance equation at the solution

to the N-node Optimal Power Flow (OPF) problem:

min {f(PG,PD)} (1)

where

f(PG,PD) =Ni=1

CG,i(PG,i) Ni=1

CD,i(PD,i) (2)

and CG,i(PG,i) is the generator cost as a function of the

real power output of a generator at node i, PG,i, and

CD,i(PG,i) is the dispatchable load cost as a function of

the real load at node i, PD,i. Nodes without generation or

without load have the respective PG,i or PD,i values set

equal to zero. This optimization is subject to:

g(x) = 0 (3)

h(x) 0 (4)

where x is a vector of the optimization variables, which

include generator power output and dispatchable load; and

g(x) and h(x) are equality and inequality constraints,

respectively. The equality constraints are composed of

the nodal power balance equations, whereas the inequality

constraints are the voltage, branch power flow and genera-

tor maximum and minimum output limitations.

The definition of the LMPthe cost of serving an ad-

ditional increment of load at a nodeis equivalent in

meaning to that of the Lagrangian multiplier of the real

power balance equationat the node at the optimal solution.

Therefore, the LMP for node j is:

LMPj =

p,j (5)

wherep,j is the Lagrangianmultiplier at the optimal solu-

tion to (1) associated with the real power balance equation

at node j. Obtaining the LMP from the OPF implicitly

includes voltage, transmission and generation constraints

that affect the economics of the delivery of energy in a

transparent fashion.

3 Practical Calculation

Solving (1) for each market period is not practical in most

market applications due to computational and implemen-

tation requirements. Therefore, in practical application

the simplified dc system model is used in LMP calcula-

tions [7]. Differences in specific LMP calculation method-

ologies between the RTOs dictate that a specific method

be delineated hereafter, though it is noted that similar ap-

proaches are used in other markets.

In the following formulation, theLMPcalculation is reflec-

tive of the method utilized in the PJM market, which is se-

lected due to its prominence as the worlds largest electric-

ity market and its lengthy experience with LMPs [7]. The

most notable influenceof this method is the omission of the

lossesand theutilizationof a dc systemmodel [10]. Losses

can be ignored without significant inaccuracy if they are

small in value, as in the case of tightly meshed systems in-

dicative of certain regions of North America. In addition,

generation offers and load bids and not costs are utilized in

the calculation.

The LMP at a given node can be decomposed into the

marginal cost of generation based upon generator offers,

congestion costs and marginal loss costs [6]. Consistent

with the dc model, marginal loss costs are ignored and the

LMP for node j can then be expressed as:

LMPj = s Kk=1

kk,j (6)

where s is the marginal price of generation at the refer-

ence node, K is the number of branches, and k is the

shadow price of the congestion of branch k. Conceptually,

the shadow price, k, is the change in cost due to an in-

cremental relaxation of the constraint on branch k. The In-

jection Shift Factor (ISF) [11], k,j , is the fraction of real

power that flows on branch k due to a unit production of

powerat nodej and consumptionof power at the reference

node under dc assumptions. A derivation of ISF calcula-tion is reserved in the Appendix. The sum of products of

k and k,j is the congestion cost.

The variables s and k are Lagrangian multipliers whose

values are determined from solving an incremental linear

optimization around a given operating point. The oper-

ating point in terms of real power produced, PG,j , and

consumed, PD,j , at each node j is based upon the sched-

uled dispatch and load in the day-ahead market and is in-

put from a state-estimator in balancing markets. Nodes

without generation or without load have the corresponding

PG,j and PD,j values set equal to zero.

The objective of the incremental optimization in the prac-tical LMP calculation of an N-node system is:

min

Ni=1

OG,i(PG,i) Ni=1

OD,i(PD,i)

(7)

where PG,i and PD,i are the changes in real power

production and consumption from the operating point, and


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OG,i(PG,i) and OD,i(PD,i) are the offers and bids as-

sociated with the changes, respectively. The hypothetical

changes to power production and consumption must re-

spect the power balance equation, generator and dispatch-

able load limits:

Ni=1

PG,i Ni=1

PD,i = 0 (8)

PminG,j PG,j PmaxG,j j (1, . . . ,N ) (9)

PminD,j PD,j P

maxD,j j (1, . . . ,N ) (10)

where the superscript max and min refer to the maximum

and minimum changes to real power production and con-

sumption at each node, respectively. In addition to the con-

straints in (8)(10), the limitations on branch flow must be

obeyed:

Fmaxk Ni=1

{(PG,i + PG,i)k,i}

N

i=1

{(PD,i + PD,i)k,i} Fmaxk k (1, . . . , K )

(11)

where Fmaxk is the magnitude of the maximum power flow

limit on branch k. In certain markets, only branches at or

approaching their limits are included in (11).

Solving (7) produces the Lagrangianmultiplierss and k.

The Lagrangianmultiplier,s, is associated with the power

balance equality constraint (8), whereas k is associated

with branch power flow constraint. There is a k for each

constraint in (11).

A consequence of utilizing the dc system model is that

voltage, stability and reactive power constraints cannot beexplicitly modeled. In practical application, these con-

straints are approximately represented and enforced in

LMP calculation as real power branch flow constraints.

4 Educational Illustration

A four-bus study system is used to demonstrate the prac-

tical calculation of LMP under two scenarios: Uncon-

strained Case and Constrained Case. It is shown that in the

Unconstrained Case, the only non-zero component com-

prising the LMP is the marginal generationpriceand hencethe LMPs are uniform. In the Constrained Case, conges-

tion in the network results in non-zero and non-uniform

congestions costs and there is differentiation in LMP. The

illustration also shows that the definitionof the LMP as the

cost to serve an additional increment of demand at each

node is equivalent to (6) if linear generator energy offer

functions are utilized.

4.1 Study System Parameters

The study system is comprised of four buses and four

branches with equal reactances of 0.10 p.u. The corre-

sponding ISFs for a unit of power produced at each bus

with Bus 1 arbitrarily selected as the reference is shown in

Table 1.

Table 1: Injection Shift Factors

Injection Branch Terminus Buses

Bus 12 23 24 43

1 0 0 0 0

2 1 0 0 0

3 1 0.667 0.333 0.333

4 1 0.333 0.667 0.333

Table 1 is interpreted as follows. Concerning the second

row, if1 MW of power is injected in Bus 2 and withdrawn

from the reference bus, Bus 1, then 1 MW would flow from

Bus 2 to Bus 1. No power would flow on any other branchdue to this injection. Therefore, since the ISF are direc-

tional, there is a 1 in the second column and zeros in the

third through fifth columns. For the third row, the power

flow resulting from an injection of 1 MW of power into

Bus 3 divides between the branches in accordancewith the

branch impedances enroute to the reference bus. Account-

ing for the direction of the ISFs defined in Table 1 and due

to the equal branch impedances, one third the power flows

from Bus 3 to Bus 4 and then to Bus 2; two thirds of the

power flows from Bus 3 to Bus 2. The entire unit of power

flows from Bus 2 to Bus 1. The elements in Table 1 for the

other bus injections are calculated in a similar fashion.

The system has three generators, whose technical and eco-

nomic data are shown in Table 2. In this educational il-

lustration, it is assumed that generator offers are linear,

as described by the coefficient c1 and that the load is not

dispatchable. Differentiation of the generator offer with

respect to power for each generator gives the offer-based

marginal generation cost which is used in LMP calcula-

tion. It is also assumed that the market period is one hour.

For the purposes of clarity in illustration, voltage limits and

losses are ignored.

Table 2: Generator Data

Gen. c1 Pmax Pmin

Bus ($/MW) (MW) (MW)

1 20 500 0

3 25 200 0

4 30 200 0


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4.2 Unconstrained Network

The calculation of the LMP is intuitively demonstrated by

determining the change in total cost of supplying energy

when the demand is independently increased at each bus

by 1 MW for one hour. The dispatch, load and resulting

branch flows are shown in Fig. 1 with a resulting total pro-

duction cost of $8, 000, which is found by evaluating the

generator offer functions at the given dispatch.

1

4

32

100 MW 300 MW

0 MW

200 MW

100 MW100 MW

0 MW

0 MW400 MW

Figure 1: Illustrative 4-bus system for the unconstrained

case.

The LMP at Bus 1 is calculated by assuming a hypotheti-

cal load of1 MW is present at Bus 1. The least cost hy-

pothetical dispatch that serves the load is to increase the

power output of Generator 1 by 1 MW to 401 MW, thus

increasing the operating costs to $8,020an increase of

$20. Therefore the LMP at Bus 1 is $20/MWh. A visual-

ization of this LMP calculation is provided in Fig. 2.

1

4

32

100 MW 300 MW

0 MW

200 MW

100 MW100 MW

0 MW

0 MW401 MW

1 MW

Figure 2: Conceptualization of LMP calculation for Bus 1.

For Bus 2, an increase of load to 101 MW can be economi-

cally met by increasing the power output of Generator 1 to

401 MW, resulting in an increase in operatingcosts by $20.Therefore the resulting LMP of Bus 2 is also $20/MWh. In

fact, since there are no constraints and marginal losses are

ignored, the LMP at each node is equal to $20/MWh. This

result is applicable to any system with like conditions.

This intuitive result is now compared with (6). Solving (7)

yields a s value equal to $20 and k equal to zero for

all branches k. Therefore, from (6) the LMP is equal to

$20/MWh at all buses.

The results for the Unconstrained Case are summarized in

Table 3. The absence of congestion or other constraints

has lead to uniform LMPs equal to the marginal generation

cost of $20/MWh. From the last two columns of Table 3, it

is seen that money charged to the loads equals the money

credited to the generators. Therefore, the RTO remains

revenue neutral and there are no congestion costs.

Table 3: Unconstrained Case Results

LMP Generation Load Credit Charge

Bus ($/MWh) (MW) (MW) ($) ($)

1 20 400 8,000 0

2 20 100 0 2,000

3 20 0 300 0 6,000

4 20 0 0 0 0

Total 8,000 8,000

4.3 Constrained NetworkThe next scenario examined includes a branch power flow

constraint. Assume that the real power flow on branch 43

is limited to 50 MW and that the dispatch, load and result-

ing branch flows of the system are shown in Fig. 3. The

production cost for this dispatch is $8,750.

1

4

32

100 MW 300 MW

0 MW

100 MW

50 MW50 MW

0 MW

150 MW250 MW

Figure 3: Illustrative 4-bus system for the constrainedcase.

The LMP for Bus 1 and 2 are the same as in the uncon-

strained case as Generator1 cansupplyoneadditional MW

of power without violating the branch limit. However, if

the load at Bus 3 is increased to 301 MW, Generator 1 can-

not supply additional power since it would cause the powerflow on branch 43 to increase past its limit to 50.333

MW. Instead, power must come from the next cheapest

power sourceGenerator 3whose increased power out-

putto 151 MW does not affect the power flow on branch 4

3 if the load is concomitantly increased to 301 MW. There-

fore, the production cost increases to $8,775an increase

of $25, and hence the LMP at Bus 3 becomes $25. The vi-


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sualization for this LMP calculation is provided in Fig. 4.

1

4

32

100 MW 301 MW

0 MW

100 MW

50 MW50 MW

0 MW

151 MW250 MW


Finally, in order to serve an additional MW of load at Bus

4 at the lowest production cost, Generator 1 increases its

power output by 2 MW to 252 MW and Generator 3 de-

creases its power output by 1 MW to 149 MW, as shown in

Fig. 5. The resulting power flow on branch 43 in this case

is 50 MW and the net result is an increase in production

cost by $15. Therefore the LMP at Bus 4 is $15/MWh.

Note that this value is less than the offer of any generator.

1

4

32

100 MW 300 MW

1 MW

101 MW

50 MW51 MW

0 MW

149 MW252 MW


Solution of (7) yields s equal to $20 and a non-zero kfor branch 43 of $15. Since the ISFs for Bus 1 and 2 onto

branch 43 are zero, the LMP is equal to $20/MWh. For

Bus 3, the ISF is equal to 0.333 so that the LMP becomes

$25/MWh in accordance with (6). In similar fashion, the

LMP for Bus 4 is computed to be $15/MWh. Therefore,

the results are identical to the approach of increasing the

demand by 1 MW at each bus.

The results for the Constrained Case are summarized in

Table 4. The presence of congestion has caused differen-tiated LMPs equal to the marginal generation cost as de-

termined from generation offers plus the congestion costs.

From the last two columns of Table 4, it is seen that money

charged to the loads is greater than the money credited

to the generators. The congestion costs are equal to the

difference between the money charged and credited and

is equal to $750. Quantifying the congestion cost in this

manner is a significant advantage of LMPs and allows

for loads to hedge against high energy prices through fi-

nancial instruments such as Financial Transmission Rights

(FTRs) [12, 7].

Table 4: Constrained Case Results

LMP Generation Load Credit Charge

Bus ($/MWh) (MW) (MW) ($) ($)

1 20 250 5,000 0

2 20 100 0 2,000

3 25 150 300 3,750 7,500

4 15 0 0 0 0

Total 8,750 9,500

5 Conclusions

Locational marginal pricing is a powerful and elegant tool

that is commonly used in electricity markets in the U. S.

and Canada. The method of assigning nodal prices based

upon short-run marginal costs, which is interpreted as the

cost of serving the next increment of demand at each nodein the system, is intuitive and adequately reflects system

constraints and economics. Based upon this definition, the

LMP at a node is equal to the Lagrangian multiplier asso-

ciated with the real power balance equation at that node.

In practice however, a simplified dc model is employed

and the LMP can be determined from an incremental lin-

ear program by adding the offer-based marginal generation

cost at the reference node with congestion costs, as shown

in this paper.

The calculation of LMPs for a four-bus study system for

constrained and unconstrained operation illustrated an in-

tuitive method for calculating LMPs and demonstrated theutility in LMPs to determine congestion costs. The con-

gestion costs are determined from the differences in LMPs

between two nodes and are used to economically manage

congestion through the use of financial instruments such as

financial transmission rights.

The use of Injection Shift Factors (ISFs) is a computation-

ally efficient method of determining the impact of changes

in generation or load on branch power flow under dc as-

sumptions.

In accordance with the linear problem formulation, the to-

tal real power flowing on any branch under dc assumptions

is found by:

F = H (12)

where F is a vector of branch power flow, H is a matrix

of coefficients that relate the node voltage angles,, across

a branch to the power flowing on that branch. The ele-

ments ofH are determined from the bus impedance ma-

trix. Since there is a linear relationship between the bus


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voltage angles and the net power injection at a node, it is

possible to calculate the amount that a particular nodes

net power injection contributes to the flow on any branch.

Therefore (12) can be expressed as:

F = Pnet (13)

wherePnet is a vectorof net power injections at each node.

The net power injected is equal to PG minus PD at each

node. For an N-node system with Kbranches the matrix,

, is a concatenation of ISFs arranged as:F1...

FK

=

1,1 1,N

......

1,N K,N

Pnet,1

...

Pnet,N

(14)

wherek,j is the injection shift factor from a nodej onto a

branch k and Fk is the real power flow on branch k whose

terminal nodes are p and q. The ISF is computed as:

k,j =zpj zqj

xk(15)

where xk is the simple reactance of branch k and zpj and

zqj are the elements of the node impedance matrix under

dc assumptions [11].

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