An introduction to the pricing of electric power transmission

Pergamon

P l h S0957-1787(97)00013-1

Utilities Policy, Vol. 6, No. 3, pp. 257-270, 1997 © 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0957-1787/97 $17.00 + 0.00

An introduction to the pricing of electric power transmission

Michael Hsu

The electric power transmission network plays an important role in the delivery of energy from outlying generation to demand centers located far away. Because power plants are interconnected via the transmission network, together they can provide improved reliability and lower overall generation costs. With the current thrust towards a competitive generation market with new independent entrants, the 'correct' spot pricing of electric power transmission service is crucial in providing signals to the marketplace for efficient short-run operations and long-term capital investments. The fundamental theory of transmission pricing is presented and three alternative market~pricing structures, contract network, financial replication and property rights, are discussed. © 1997 Elsevier Science Ltd. All rights reserved.

Keywords: Transmission pricing; Electric power; Market structure

Introduction

The electric power transmission network plays an important role in the delivery of energy from outlying generation to demand centers located far away. Because many power plants are interconnected via the transmission network, together they can provide improved reliability and lower overall generation costs. With the current thrust towards a competitive generation market with new independent entrants, the 'correct' spot pricing of electric power transmission is crucial in providing signals to the marketplace for efficient short-term operations and long-run capital investments.

The overall costs for a transmission network can be separated into the following four major components:

1. returns and depreciation of the capital equipment; 2. operation and maintenance to ensure that the network

is robust;

The author is with the Energy Modeling Forum, Terman Engineering Building, Room 402, Stanford University, Stanford, CA 94305, USA.

3. losses incurred in transmitting power; and 4. opportunity costs of system constraints.

Marginal cost pricing of transmission services defines the impact on the overall system costs when one additional megawatt is injected or withdrawn at some node. These costs include two major components: marginal losses throughout the network (i.e. item 3 above) and the opportunity costs of not being able to move cheaper power due to transmission line congestion (i.e. item 4 above). Under an ideal marketplace, transmission service charges should equal the short-run marginal costs of providing that service. However, because of the increasing returns to scale characteristic of an electrical network, marginal cost pricing may not generate sufficient revenues to cover the fixed capital investments. This makes implementing an efficient and equitable transmission pricing system difficult. For example, a usage independent connection charge may be necessary for full revenue reconciliation (i.e. embedded capital costs plus rate of return on investment). The remainder of this paper (1) covers the short-run marginal cost pricing of electric power transmission and (2) describes several proposed trading mechanisms for eliciting competitive short-run behavior from market participants while at the same time providing an avenue for hedging transmission cost risks. Other papers in the symposium discuss alternative revenue reconciliation mechanisms as implemented in individual countries.

What makes transmission pricing and investment decisions complicated are the physical laws that govern power flow in an interconnected network. For example, given two parallel paths between generation and demand center, electricity will flow according to the relative 'resistances' between the two lines. Unlike rail transportation where the contract path actually dictates movement of the commodity, electric power moves along all available paths between the supplier and the customer. Other industries such as gas pipelines face similar 'loopflow' transportation phenomena but electric power flow is unique in that supply and demand must be balanced at every instant in time without the help of

257

Electric power transmission pricing

cheap physical storage or flow control valves I. Since generators and customers are all connected to the same network, actions by one party can and will have significant consequences on other participants. Network

externality, which can be good (positive) or bad (negative), describes the benefit or cost imposed on other system users when one participant changes supply or demand. For example, a supplier increases generation by 10% at point A which causes congestion in a transmission line several hundred miles away to point B and those B endusers must now pay higher prices for more expensive local generation. In this case the supplier at node A should be charged some penalty for the externality it imposes on the B endusers.

Pioneering research on pricing transmission which incorporated the physical laws was done by Schweppe et

al. (1988). They concluded that the short-term value (i.e. price) of transmission services between any two locations is the difference of spot prices between those two points, if the spot prices reflect the system marginal generation costs, marginal losses, and congestion. Their results demonstrated that alternative geographic-based pricing methodologies such as the postage stamp rates do not provide the correct market signals. Hogan (1992) champions the concept of contract network which provides a mechanism for allocating long-term transmission capacity rights subject to maintaining short-run price efficiency. Oren et al. (1994) have demonstrated that 'virtual' transmission rights between any two nodes can be replicated using financial forwards available at those two nodes. Chao and Peck's (Chao and Peck, 1996) property rights approach explicitly incorporate the network externality impacts into the competitive trading mechanism. These alternative market structures all stem from a social welfare perspective and all achieve the socially optimal solution in short-run operations. Where these models differ is in the implementation: such as the definition, assignment, and trading of the capacity contract rights. This paper begins by reviewing the components which make up the spot price of electricity as developed by Schweppe et al. (1988). This basic model illustrates the correct short-term pricing of transmission services based on the marginal impact of changes in supply or demand at each node in the interconnected system. The salient features of the market structures proposed by Hogan (1992), Oren et al. (1994) and Chao and Peck (1996) will then be briefly covered.

General concepts

To develop the concepts of spot electricity prices we begin by formalizing the objectives of the policy makers: to maximize net social welfare (i.e. benefits minus costs) subject to physical constraints imposed by the electrical

258

transmission network. If the transmission grid had infinite capacity, power plants would be dispatched according to their marginal cost of generation with adjustments for losses. In reality, when the grid is heavily loaded and some lines have exceeded their NERC (North American Electric Reliability Council) specified safety limits, the control area operator must redispatch available resources to relieve the congestion. Some cheap plants will have to reduce output with the balance replaced by more expensive 'out of merit' generation units. Because of physical laws, a line constraint will impact power flow on all other transmission lines and subsequently, spot prices at nodes throughout the network.

There are two major sources of transmission constraints: thermal and voltage magnitude. Thermal constraints deal with real power transfers where the flow of electricity heats up the conducting wires. Voltage magnitude constraints deal with reactive power flows where the current is out of phase with the voltage, causing instability in the network. The stability constraints are altemating current (AC) in nature and are very difficult to model so most of the time these constraints are represented as interface limits and nomograms z. Ideally, transmission prices should be determined for both real and reactive power transfers (Hogan, 1992). Transmission limits (i.e. constraints) across links in the network are often computed based on a (n-I) contingency heuristic. A collection of (n-I) contingencies (i.e. a single line in the network is lost) is simulated and evaluated. The power transfer limits are then set such that in each of the simulated contingency the redistribution of power flows over the remaining lines do not violate thermal standards or create instability in the system (Harvey et al., 1996).

The analysis of spot pricing makes several key simplifying assumptions. Generation is assumed to be dispatched optimally such that marginal operating costs exceed average variable operating costs. Then, if prices are set equal to the marginal operating costs, operating revenues will be greater than total variable operating costs. The operating profits can be applied towards recouping the capital expenditures. The term 'dispatch' should be distinguished from 'commitment' of generation assets. Unit commitment looks forward to expected demand day ahead, week ahead, and month ahead. In the unit commitment problem we have to consider plant characteristics such as startup costs, ramp rates, and minimum downtime in addition to the variable cost of generation. Dispatch, on the other hand, considers only the next time interval (e.g. hour) with available online generation. The multiperiod coupling of supply costs and demand response greatly complicates the analysis. Consumer benefit and producer cost profiles are also assumed to be constant in the short-term. It is conceiv- able that consumer benefits will differ for various states

of the world, and the analysis should then be based on expectations.

Kirchoff's laws and power flow

The complexity of analyzing the electric network stems from the Kirchoff's laws which govern power flow. Since the power we use is based on the alternating current, a complete analysis of the network requires modeling both real and reactive power. Real power is the power used to do useful work such as running a refrigerator. Reactive power is the power an inductive or capacitive device consumes in one-half of the cycle but returns to the system in the second half of the cycle (i.e. behaving very much like storage). The reactive device consumes no actual energy since over one cycle the net reactive power usage is zero. However, reactive power requirements do impose real costs on the system because "it can cause unwanted local voltage changes on transmission networks, and it requires higher levels of current for a given amount of real power to be supplied than would be necessary in its absence" (EPRI, 1995). Reactive power makes system analysis much more difficult, but the simplified DC (direct current) flow model exemplifies many characteristics of the physical laws and is used here to avoid complicated calculations. For a complete derivation of the DC flow model (from an AC formulation) please refer to Appendix D of Spot Pricing of Electricity by Schweppe et al. (1988).


Consider the parallel interconnection between Node A and Node B presented in Figure 1. By the laws of physics, power generated from A will flow to B through lines 1 and 2 according to their relative line reactances. Reactance is defined to be the 'resistance' of individual components to the alternating current. This 'resistance' is frequency dependent for a given capacitor the higher the frequency the lower the reactance, and vice versa for an inductor (i.e. a resistor has no reactive component). Therefore, the lower the reactance of one line relative to other lines, the greater the power flow across that line. In the DC flow approximation, the reactance of a line is assumed to be proportional to its resistance. Suppose, for the Figure 1 example, the resistance of line I is half that

of line 2 (i.e. a = ~ ). So, given a one megawatt(s) (MW)

injection at node A, physical laws dictate that two-third megawatt will flow across line 1 and one-third megawatt across line 2. If the capacity of both lines is unlimited, the physical laws do not impact the ability to move power from A to B. However, if line 1 has a capacity limit and it becomes 'congested', then the total power transfer capability from A to B is limited even if line 2 has infinite capacity. If the capacity of line 1 is 10 MW then the maximum power that B can receive from A at any instant is 15 MW (10 MW through line 1 and 5 MW thorough line 2).

To analyze the infamous three node model we utilize the optimization technique of superposition, which

flow=13/(a+13) MW

A

+IMW

res is tance l ine 1 = tx

res is tance l ine 2 = 13

B

- 1 M W

Figure 1.

2 flow=o l( +13)

259


works for the DC flow model under a lossless assumption (i.e. no line losses; the losses can be estimated separately). In Figure 2, nodes A, B, and C are interconnected by three transmission lines. For sim- plicity, let A and B be generation nodes and C be the demand center.

In order to determine the flow path of power from A to C we pretend as if node B did not exist [Figure 3(a)]. The reduced network is the same as the one presented in Figure 1 and the flow across each line is computed.

We repeat the analysis by assuming that node A did not exist, and calculate the power flow from B to C [Figure 3(b)].

By superimposing power flows from the two separate solutions we derive the overall flow characteristics for the network (Figure 4). The direction of power flow between node A to node B depends on the line resistances as well as power injections.

The mathematics for spot pricing of electricity

Schweppe et al. (1988) derived the spot price charged to customer k and paid to generator j by formulating a social welfare maximization problem 3. We should stress that the spot price is computed without consideration of the intertemporal impacts of decisions made in the current hour on the costs of providing power in a future hour. The social welfare objective function is made up of three key components: customer benefits, generation fuel and maintenance costs, and transmission line congestion costs. The following is a simplified model of the one

presented by Schweppe et al. (1988)--hopefully, this reduction will illustrate the key results more clearly.

To begin we define the necessary variables:

d(t): vector of all demands, dk(t) is usage by customer k

g'(t): vector of all generation outputs, gj(t) is output from generator j

z~t): vector of all line power flows, zi(t) is flow across line i

g(t): total generation output= ~ gj(t)

d(t): total demand= E d~(t).

The time interval t can be set to any reasonably short period, say one-half hour or one hour. In this paper we will generally represent t on a hourly basis.

Total customer benefit is formulated as the sum of individual benefits:

B[d(t)]= ~ Bk[dk(t)] (1)

where dk(t) is the demand of customer k during hour t. Generation fuel and maintenance cost is unique to

individual generation units and written as:

GFM[g(t)] = Z GvMj[gj(t)] (2) J

where gj(t) is the output of the jth generation unit at time t.

Line losses are represented by:

L[z~t)] = ~. Li[zi(t) ] (3)

A

C~

5

C

B

Figure 2.

ct, [3, and 8 are the individual line resistances

260

where zi(t) is the power flow across line i. In the DC power flow approximation, the power loss

over some line i is a quadratic function of the power flow:L~[z~(t)] =R~z~ z where Ri is the resistance of line i.

The welfare maximization problem is computed subject to the following constraints:

1. energy balance: total generation equals total demand plus losses--represented as d(t)+L[~t)]=g(t). Energy flows and losses on a network are determined by physical laws. The network transfer admittance matrix, which characterizes power flow over each line i, is embedded in the energy balance equation;

2. generation limits: total demand in hour t cannot exceed the aggregate capacity of all available plants at hour t--represented as gcn,(t);

3. maximum individual generation: capacity of each generation unit--represented as gmaxj(t); and

(a) Suppose Node B did not exist

A

+x MW

(13+~)x/(a+l~)

ot

C

-x MW

ax/(,~+l~) ax/(a+l~)

(b) Suppose Node A did not exist

I ot+13

-Sy/(~+[]a4~)

~y/(~t+[~) C

-y MW

(~+13)y/(a+[l~)

Figure 3.

+y MW t B


By Superposition

A

+x MW

[(l~)x+Syl/(a+l~qS) C

--------7--- MW

\o (ax-~y)/(ct+l~) [~tx+(t~+13) y]/(et+13-~)

+y

Figure 4.

4. line flow limits: power flows over each transmission line i must be within specified operating limits-- represented by z .... i(t) •

The Lagrangian for our simplified social welfare model is:

MaxO(t) =B[d(t)] - Gig(t)] - N[z(t)]

-/~e(t) [d(t) + L[z-(t)] - g(t)]

where 4,

G[g(t) =~j~ GrM.flgj(t)]

(4)

[fuel and maintenance costs]

+ yos(t)[g(t) - g,,~,(t)] [total generation consraint]

+ ~ tz,~,.j(t)[&(t) - g,~,.j(t)] [individual generation constraint] " -7

N[~(t)] =ZlXos. i ( t )[zM)- zm . j [individual line capacity constraintp i (5)

zi(t) =zi[g*(t),d*(t)] is the power flow through line i given demand and generation at every node in the network except the swing node. The swing node, (g*(t),d*(t)), is any location selected to be the reference point. Typically, the swing node is chosen to be the node which contains the marginal generator. To be unambiguous, g'*(t) represents the vector of g'(t) less the swing node generation g'(t) and similarly, d(t) represents the vector ofd(t) sans the demand at the swing node.

First order necessary conditions from the Lagrangian are:

OG~,(t)] [ O L ~ ( t ) ] l ] Ogj(t) tZe(t) Ogj(t)

aZi(t) -- ~/ P"QS'/(/) ~g~.(t) = 0 Vgj(t) (6)

261


and

OB~[dk(t)] [ a L [ z ( t ) ] ] oak(t) /z~(t) 1+ adk(t~

. . Ozi(t) - E ~s . , to ~ =0 Vdk(t) (7)

Since individual consumer k will maximize benefits by setting marginal benefits equal to the price paid, we have:

OLIz'(t)] ] Z./~s,.(t) Oz,(t) aBk[dk(t)] =/x~(t) 1 + - - + Odk(t) Odk(t) " ' Odk(t)

=pk(t) Vdk(t) (8)

If generatorj and customer k are sharing the same bus, whether generatorj increases output by one megawatt or customer k reduces consumption by one megawatt, the marginal impact to the system should be identical; therefore, so should the price (i.e. pk(t)=pj(t)). This implies that the price paid to generator j should be the following:

O G [ g ( t ) ] [ a L [ z ( t ) ] ] Ogj(t~ - - tz~(t) ~ 1

OZi(t) -- ~. /~Qsj(t) ~ =pj(t) Vgj(t) (9)

or equivalently,

O GFMj[g j( t ) ] + TQS(t)+P.maxj(t)=pj(t) Vgj(t) (10)

o&(t)

Next we need to evaluate IXe(t ), the shadow price associated with the energy balance constraint. Taking the derivative of the Lagrangian with respect to swing node g*(t) we get:

1 OG[g(t)] 0N[z(t)] _ i.te(t ) 1 =p*(t) Og*(t) Og*(t) ag*(t)

(11)

By the definition of a swing node, it has no impact on the transmission network (i.e. because it is excluded

OZi(t) from the vector z(t)). So the component, 0 g ~ ' equals

zero for all lines z~. Given this, tZe(t) simplifies to:

OGFM[~t)] txe(t) - Og*(t) + yQs(t) +/Xmax'(t) (12)

Also, since g*(t) is the marginal unit it must not be

262

generating at its maximum capacity, so its Lagrange multiplier for capacity constraint, /x ..... (t), equals zero. By using the swing node as the reference we have:

p*= l~e(t)= OGFM,'[g*(t)] Og*(t) + Yos(t) (13)

If we let OGFM,[g*(t)] =,~(t), the equation becomes: Og*(t)

p*(t) = A(t) + yQs(t). So the price charged for consumption at any bus k

equals (substituting p*(t) = ,~(t) + 7Qs(t)):

aL[~t)] pk(t) =,~(t) + yQs(t) + [a(t) + yos(t)] Odk(t)

azi(t) (14) + Z. ~Qs,i(/) Odk(t)

Expanding the above equation we get customer k's market-based spot price for real power (i.e. from the DC flow model) is composed of four parts:

pk(t) = A(t) [marginal variable generation costs] + Yos(t) [system capacity premium]

Z ~L[zi(t)] ~zi(t) +[~(t)+ yos(t)] Ozi(t) Odk(t) [network marginal losses]

i

+ Z ~z~(t) I~Os'i Odk(t) [line congestion premium] (15)

While each pricing component has its physical and economic interpretation, they are interrelated. For example, network marginal losses depend on both generation marginal fuel and generation marginal maintenance costs. Individual customers will see different prices because of their spatial location in the network. As the mathematical results show, some users may impact the system significantly when they change their demand while other users impose minimal costs or even provide benefits.

Let us take each component of the spot price in turn:

1. A(t) is the marginal fuel and maintenance costs for the generator at the swing node. This value is typically called the system lambda. Since A(t) is composed of the derivative with respect to the power generated, only the variable portion of the production cost is relevant.

2. yQs(t) is the generation 'quality of supply' (i.e. hence the QS subscript) or capacity rent based on the critical system generation limits, gcrit(t). In a competitive generation market, short-run excess demand will bid up the spot price until the market clears; see Figure 5. In this case 7Qs(t) is the premium the market charges because there is overall capacity shortage in the system. Alternative approaches include setting YQs(t) to be the expected value of lost load (i.e. this method is used in the UK system) or the annualized cost of a

I


capacity rent

system lambda

Figure 5.

$/MWh

/ market clearing price and quantity

"excess" demand

Demand

MW

.

4.

peaking plant needed to maintain system stability. These three methods should be equivalent in an 'optimized' system.

cgLi[zi(t)] Ozi(t) [A(t)+yQs(t)] ~/ OZi(t) Odk(t) =r/Lk(t) (16)

is the network loss component. Since the system is integrated, a change in consumption by customer k will impact losses across the entire network. OL~[zi(t)] - - is the marginal energy losses across line i as

Ozi(t) Oz,(t)

flows change, is the change in flow across Odk(t)

each line i in the network when demand of customer k changes. In the DC power flow model, line losses is directly related to the square of the flows across that line so that the marginal losses is linear with respect to power flows. Note that the price of network loss component depends on system lambda and generation quality of supply.

OZi(t) ] = r/QS.k(t) E. /.~s,/(t) O~k(t) (17)

defines the short-run costs if the transmission lines are

congested. As defined in the optimization model, tZos.i(t) is the shadow price for line i if and when the line becomes congested and transfer capability limited. Since pQs.i(t) reflects the opportunity costs when line i is at its operating limits, it carries similar impact as the generation quality of supply and can be valued in the competitive marketplace. Suppose customer k wants additional power but is limited by transmission capacity in the network. Customer k will bid up the price of electricity, change the dispatch of the generation assets, and subsequently alter the power flow such that it receives the desired quantity. For the system capacity premium, all consumers see the same YQs(t), while for network congestion each customer sees a different ~Qs~(t) depending on their location in the grid.

The network externality concept is best captured by the two terms: r/QS.k(t) and r/L.k(t). From our formulation: if r/Qs.k(t) is negative then customer k actually lowers system congestion by increasing consumption. This is an instance of positive externality, where customer k helps other users by using more power. The UK system provides a real-world example: the marginal unit is located in an importing area (e.g. the southern region) such that increasing demand in the exporting area (e.g. the north) reduces the power flow between the two regions thereby relieving congestion.

In review, the spot price at some location k is equal

263


to:

pk(t) = A(t) + Yos(t) + r/Qs.k(t) + r/L.k(t). (18)

Locational prices

Consider the market clearing spot prices at two locations, node 1 and node 2. The spot price components for each node are:

pl( t )=A(t )+ Yos(t)+ rlm(t)+ rlOs.j(t) (19)

p2(t) = A(t) + yos(t) + rlL.2(t) + "qQs.2(t)

Taking the spot price difference between the two nodes, we have:

Apl2(t ) =p2(t) - pj(t)= "qos.2(t) + ~?L.z(t) - "qQS.l(t) -- r/m(t )

(20)

Ap~z(t) shows that the price difference between any two nodes depends on marginal losses and congestion throughout the entire network. Marginal losses can be thought of as the 'operating cost' of the transmission service while the congestion rentals reflect the opportunity cost of not being able to transfer additional cheap electricity. Therefore, Apjz(t) also represents the correct short-term wheeling rate or transmission charge to the contracting parties if the buyer withdraws power at node 2 and the seller injects power at node 1. Note that Ap~z(t) can imply a negative wheeling charge (when power is flowing from an expensive bus to a cheaper bus) and the transmission lines owner (e.g. the utility) must pay the contracting parties.

Capacity rights and financial replication

The designation and allocation of long-term capacity rights play a crucial role in short-term market clearing and long-run investment decisions. A capacity right defines how the rights holder can charge for transmission services, or sometimes equivalently, how the rights holder can inject and withdraw power at nodes in the network. A feasible capacity rights framework must satisfy the prerequisite that ownership of such rights will not alter the short-run behavior such that the equilibrium deviates from the competitive solution we derived in the previous sections. For example, it would seem inefficient if by owning some transmission rights, a generator would prefer to produce power even though its marginal cost of generation exceeds the market clearing price. On the other hand, the capacity rights must be useful to the market participants as a hedging instrument against locational price risks. There are three transmission capacity rights models: the basic contract path, the contract network (Hogan, 1992), and the property rights

264

or network externality model (Chao and Peck, 1996). The contract path definition includes the postage stamp or per-MW mile where capacity rights are assigned to individual lines. Contract path rights allow the rights holder to 'send' power through the contracted line. From our previous power flow analysis, we see that unless special trading rules are implemented, the basic contract path rights will not take into account the systemwide impacts of the power flow across an individual line. Therefore, the simple contract path approach fails our efficient short-term market clearing requirement.

Contract network rights

For the contract path rights model there is great difficulty in determining the initial number of contracts for allocation because transfer capacity across a particular line depends on the state of the system and often has nothing to do with the thermal or voltage limits. Hogan's contract network formulation solves these problems by looking at the 'transfer' capability between two nodes rather than the specific links that connect them. The contract network rights owner can choose either to be paid the dpo(t ) between two nodes i and j as they occur at each time t or inject power into and withdraw power out of the system. The contract network structure provides the mechanisms for transmission line owners to recoup their capital investments and for market participants to hedge their risk exposures.

Consider the three node network presented in Figure 6. Suppose, for this analysis, the line between node 1 and node 3 (line 1-3) has a 200 MW thermal capacity while lines connecting node 1 to node 2 and node 2 to node 3 have near unlimited capacity. Also assume that the demand center is at node 3 while generation assets are available at node 1 and node 2. To make the problem more descriptive, assume the resistances are: 1.5 for line 1-2, 0.5 for line 2-3, and 1 for line 1-3 (i.e. use the model presented in Figure 4 to compute power flows). The network contract approach simply defines a physically feasible rights assignment for power injection at each node. So, as shown in Figure 6, if node 1 were to generate all the power, the amount of power deliverable to node 3 is limited to 300 MW.

On the other hand, if node 2 were to generate all the power (Figure 7), the total deliverable power quadruples to 1200 MW.

Any combination of node 1 and node 2 injections that is physically feasible will also work. For amusement let us assign 120 MW of rights from node 1 and 720 MW of rights from node 2; illustrated in Figure 8.

Regardless of how the rights are assigned, the resulting transmission payments to the contract rights owners are equal. As we shall see shortly, 300 MW, 1200 MW, or 120/720 MW will all 'work' under the contract network framework. One more assumption

+ 3 0 0 M W rights

Node 1 200MW I ~

2 0 0 M W - max


Node 3

-300MW withdraw

100MW 100MW

Figure 6.

Node 2

before we present the results, let the marginal cost of generation be constant at: $10/MW for node 1 and $15/MW for node 2.

Suppose market clearing trading yields the dispatch and prices for some hour t as shown in Figure 9 (NB: we are ignoring losses in this particular model). Spot prices at node 2 and node 3 should reflect their respective marginal impacts on the system generation costs. A case in point, if demand at node 3 were to increase by one additional megawatt, to meet this increase generation at node 2 must be incremented by 1 1/3 MW while generation at node 1 is reduced by 1/3MW. Taking the marginal costs of generation ($15/MW for node 2 and

$10/MW for node 1) we have: 1 1 /3MWhX$15/ M W h - 1/3 MWh X $10/MWh=$16.67/MWh, which is the market clearing price at node 3 s.

Consumers at node 3 pay $16.67 for each megawatthour of energy consumed. Generators in node 2 are paid $15/MWh, while generators in node 1 receive $10/MWh.

The financial intermediary performs the payment accounting for hour t as follows:

1. collects $7639.86 ($16.67 x 458.3) from users at node 3;

2. pays $3166.50 ($15 x 211.1) to generators in node 2,

Node 1 200MW 1 ~

2 0 0 M W - max

Node 3

-1200MW withdraw

200MW 1000MW

Figure 7.

A + 1 2 0 0 M W ~ N o d e 2

rights /

265


+120MW rights

Node 1

I

200MW I ~

200MW - max

Node 3

-840MW withdraw

80MW 640MW

+720MW rights t Node 2

Figure 8.

and 3. pays $2472 ($10 x 247.2) to generators in node 1.

The total amount collected, $7639.86, less $3166.50+$2,472 payout yields approximately $2000 of congestion rental--which is to be distributed to the contract rights owners. In our example, the power transfer capacity to node 3 can be set at 300 MW from node 1 only, 1200 MW from node 2 only, or any feasible combination. Short-term market clearing efficiency is

preserved and payments to the rights owner for every feasible case is equal. Given the spot prices in Figure 9 we can compare the net transmission payments to the rights owners for some hour t6:

for 300 MW rights at node 1:

300 MWh X ($16.67 - $10)=@$2000

for 1200 MW rights at node 2:

1200 MWh × ($16.67 - $15)=@$2000

Node 1 $10/MWh

Node 3 $1& 6 7/MWh

+247.2MW 200MW

200MW - max -458.3MW

47.2.MW 258.3MW

+211.1MW

Figure 9.

Node 2 $15/MWh

266

for 120 MW fights at node 1 and 720 MW fights at node 2:

120 MWh x ($16.67 - $10)+720 MWh x ($16.67 - $15)= @$2000.

When there is congestion in the system, the congestion premium received from the network right equals the payoffs gained by injecting power at nodej and selling at node k. The key definition which makes the contract network approach desirable is that the contract network capacity rights allow the holder to either inject power at node k and withdraw the same amount a t j or just sit tight and collect the congestion premiums between nodes k and j as they occur, thereby decoupling power flows in the transmission network from the financial outcomes.

Market participants are interested in the usefulness of these contract rights for hedging transmission cost risks. We illustrate this result with an example. A generator holds 100 MW capacity rights from node 1 to node 3 and has previously signed a 100 MW fixed price contract at $20/MWh with an enduser in node 3. If the marginal cost of production for the generator is $8/MWh, we can write the following payoff equation for our example:

payoff= 100MWh[($10 - $8 at node 1]

+($20 - $16.67)[at node 3]]

+ 100MWh[$16.67 - $10] = 100MWh[$20 - $8]

=$1200 (21)

Notice that the $16.67 and the $10 both cancel, and the generator makes the margin between the contracted price and cost of generation. Transmission costs are exactly offset by the contract network payoff. Next suppose the rights are assigned as 1200 MW for power injection at


node 2. A generator at node 1 is contemplating a one year fixed price transaction of 50 megawatts (each hour) with a customer in node 3. The generator would like to hedge the transmission cost risks associated with this deal. What should the generator do since the contract network right is specified for node 2 only while the generator needs price protection based on injection at node 1? Since we know that the payout of the 300MW assignment at node 1 equals the payout for 1200 MW at node 2, implying an exchange ratio of 1:4. That is, one MW at node 1 is equivalent to four MW at node 2. Therefore, to hedge, the generator can purchase 200 MW of transmission rights from node 2 to node 3 for the course of year. This protection will be financially equivalent to 50 MW of transmission rights from node 1 to node 3.

It should be duly noted that once losses are included in the model the accounting may no longer balance. Figure 10 shows a simple two node loss accounting. Because energy lost in the transmission line is a second order relationship with respect to power flows (so marginal losses is linearly increasing with flow), the total amount collected based on the marginal losses will be greater than the average losses of the network. This operating surplus is usually credited towards recouping the capital investments.

While it resolves the hedging and transmission payment issues, the contract network structure generally requires an ex post settlement--that the transmission charges are computed after the dispatch has occurred. The definition and assignment of contract fights do not facilitate active trading. For example, if only 300 MW capacity rights is defined for injection at node 1 and withdraw at node 3, technically, market participants at

A

+(d+ctd 2) MWh at $p/MWh

Figure 10.

total losses = ctd 2 (MWh)

average losses = czd

marginal losses = 2¢xd

Accounting:

Collect from consumers: $(p+2¢xdp)d Pay to generators: $p(d+otd 2) Overcollection = +$otdZp

B

-d MWh at $(p+2o~dp)/MWh

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node 2 have no rights to inject power at that node. So in general, the economic dispatch is determined based on supply offers and demand bids, and the transmission charges are determined (by a computer program under the jurisdiction of the Independent System Operator) after the markets clear.

Financial replication

Since the actual transmission of power can be decoupled from the financial payments, the contract network capacity rights is replicable using financial instruments such as electricity forwards and options. (Oren et al. (1994)) First let us define the locational prices as random variables. Assume the current time period is 0, with t greater than or equal to 0 (i.e. time t is in the future).

Define ~(t) to be the uncertain market clearing price at node j at time t.

Let ~k(t) be the uncertain market clearing price at node k at time t.

The actual payoff structure o f the contract network right from node k to nodej for any hour t is equivalent to: payoff ( t ) = ~ / ( t ) - ~ ( t ) . To replicate this payoff scheme for each time period t, we simply buy one forward contract 7 at node j , sell one forward contract at node k, and buy some riskfree bonds. The payoff of the forward contracts are:

node j payoff= ~j(t) -fj(0,t) (22)

node k payoff=~(0,t) -/gk(t)

where f(0,t) and ~(0,t) are the forward contracts entered into now and expiring at time t for the respective nodes

j and k. At the same time purchase e -"~(0 , t ) - fk (0 , t ) ] amount o f riskfree bonds such that at time t the payoff is D')(0.t)-f~(0,t)]. The payoff o f the synthetic financial instrument at each time t equals:

Ibm(t) -fi(O,t)+fk(O,t) - h~.(t)+£(0, t) --f~.(0,t)] (23)

= [h/(t) -~b~(t)] =contract network right

A strip o f this synthetic contract, over some time interval s to t, exactly replicates the payoff structure o f the contract network rights; therefore, any market player or speculator can create virtual transmission rights for hedging purposes. This financial replication solution requires, a priori, a liquid forward market for each hourly market. The contract network right pays the rights holder the difference in locational prices each and every hour. That means to replicate the contract network right for individual hours, we must find counterparties willing to enter into forward contracts with settlement based on each hour separately. If these forward markets exist, there can be active trading of nodal prices ex ante to dispatch---an appealing proposition to power marketers and traders.

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Property rights

The contract network right internalizes the externality associated with the loopflow phenomenon. The property rights approach presented by Chao and Peck (Chao and Peck, 1996) allocates transmission rights based on contract paths but sets explicit trading rules which recognize the externality impacts. They define the following property rights and trading mechanism for the case with no transmission losses (we demonstrate their lossless example to illustrate the key points):

A transmission capacity right entitles its owner to the right to send a unit of power through a specific transmission line in a specific direction and to collect economic rents associated with the transmission line according to the trading rule. A fixed set of transmission capacity rights, P= {PJI <-i,j<-n }8, are issued for each link (i,j), and these rights are tradable...Without loss of generality, we may arbi- trarily designate a node, say n, as a base point and only need to define the trading rule that governs the transactions between the base point and every other node in the network. Chao and Peck, 1996

The trading rule consists o f a set o f coefficients B(t)= {/3~.(t)ll <-id,k<-n}, where the parameter/3k(t) represents the quantity o f transmission capacity rights on line (id') (i.e. line connecting node i and node j ) that a trader needs to acquire in order to inject one additional unit of power at node k. To transfer one unit o f power from node k to l, the trader procures for each link i-j, flkl(t) = [ilk(t) --/31~(t)] rights associated with flow from k to n and from n to l.

Let us use the following line resistances: 1.5 for line 1-2, 0.5 for line 2-3, and 1 for line 1-3. Assume node 2 to be the reference node n. As Figure 11 shows (i.e. the direction of arrow indicates positive power flow), to transfer one unit of power from node 1 to node 3 at time t, the trader needs to purchase transmission over the three links:

fll13(t)=fll12(t ) -- fl~2(t) for line 1 -- 2 (24)

13 l f123(t)=f123(t) - f133(t) for line 2 - 3

f l J l ~ ( t ) = f l l l 3 ( t ) - - fl~3(t) for line 1 - 3

We represent a concrete example in Table 1 for each MW transferred from node 1 to node 3.

The key benefit o f the Chao-Peck property rights approach is that both the transmission charges and the nodal electricity prices can be determined by a competitive market-based dynamic trading process so that the electricity marketplace does not necessarily have to be 'centralized' as suggested by the contract network model and can be separated into regional markets trading with each other. Because of the specialized trading rules, transmission rights owners participate actively in the marketplace, selling the rights to the highest bidders. A

inject

Node 1 ~1313

PI3 --- number of contracts assigned


Node 3

withdraw

~1213

PI2 P23

~2313

Node 2 reference node "n"

Figure 11.

competitive equilibrium solution is achieved and is socially optimal. The actual implementation may be complicated because to execute a transaction, the trader must simultaneously secure transmission rights on all links in the system.

Summary We have presented a simplified model of the transmission network in order to illuminate the key characteristics of transmission pricing. Actual system operations is considerably more complex; however, the DC flow model provide us with an understanding of the physical laws that moves power throughout the network. In the short-run two terms dominate transmission pricing: marginal losses and congestion opportunity costs. The transmission rights proposals presented here- with all satisfy the our criteria of short-term efficient market clearing. The property rights formulation by Chao--Peck makes bilateral trading a workable solution for the competitive market structure. Hogan's contract network rights internalizes the loopflow externality

Table I. Transmission contracts required for power injections

Inject at node 1 Inject at node 3 Net contracts

/3~ 2(0 1/2 1 I6 1/3 /~23(t) - 1/2 - 5/6 1/3 /3~3(t ) I/2 - I/6 2/3

impacts, thereby decoupling transmission from generation. With the development of efficient locational forward markets, virtual transmission rights can be synthesized to allow low cost hedging and speculation. One key issue remains to be resolved: how to organize long-term transmission expansion. Given economies of scale and investment externality it is likely that in the near future the independent system operator must continue to coordinate grid investments decisions.

The author would like to thank Richard Green, William Hogan, Hung-Po Chao, and Hill Huntington for their invaluable comments and suggestions; and the Energy Modeling Forum at Stanford University for its support. Any remaining errors are solely the responsibility of the author.

tExtensive research is currently being conducted in fuel cells (for storage) and FACTS devices (for flow control). 2Nomograms are graphical representation of tradeoff in transmission interface capacity across major tie lines given various operating conditions (i.e. increasing power flow across one tie line may reduce the capacity of power flow across a second tie line). 3For a more thorough explanation please see Schweppe et al. (1988), Chapter 7. 4Line maintenance costs are typically dominated by fixed components and are not flow driven. 5We should be careful with the units. Megawatt is a measure of power while megawatthour is in units of energy, where energy=power × time. Since we assume a time interval of one hour, one megawatthour=one megawatt × one hour. Therefore, a 200 megawatt power transfer limit implies a instantaneous capacity of 200 MW and a maximum deliverable energy of 200 MWh during any hour.

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6If we carry out the significant digits, the accounting will match exactly. 7A forward contract is an agreement today for delivery of and payment for a commodity at a future date. The price is determined at the present time but the exchange takes place in the future. gFor the DC power flow model P0 can be interpreted as the thermal constraint of the line linking node i and node j.

References

Chao, H. -R and Peck, S. (1996) A market mechanism for electric power transmission. Journal o f Regulatory Econom- ics 10, 25-59.

Electric Power Research Institute (1995)A Primer on Electric Power Flow for Economists and Utility Planners. EPRI TR- 104604, Project 2123-19.

Harvey, S., Hogan, W. and Pope, S. (1996) Transmission Capacity Reservations and Transmission Congestion Con- tracts. Putnam, Hayes & Bartlett.

Hogan, W. (1992) Contract networks for electric power transmission. Journal of Regulatory Economics 4, 211-242.

Oren, S., Spiller, P., Varaiya, P. and Wu, F. (1994) Nodal Prices and Transmission Rights: A Critical Appraisal. POWER.

Schweppe, F. C., Caramanis, M. C., Tabors, R. D. and Bohn, R. E. (1988) Spot Pricing of Electricity. Kluwer.

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Documents

An introduction to the pricing of electric power transmission