Comparison of congestion management techniques: Nodal

Document technical information

Format pdf
Size 473.3 kB
First found Feb 4, 2016

Document content analisys

Language
English
Type
not defined
Concepts
no text concepts found

Persons

R. H. Boyd
R. H. Boyd

wikipedia, lookup

John W. Hogan
John W. Hogan

wikipedia, lookup

W. W. Hiltz
W. W. Hiltz

wikipedia, lookup

Andrew C. Weber
Andrew C. Weber

wikipedia, lookup

Theodore S. Peck
Theodore S. Peck

wikipedia, lookup

Organizations

Places

Transcript

Comparison of congestion management techniques: Nodal, zonal
and discriminatory pricing
Pär Holmberg* and Ewa Lazarczyk**
Wholesale electricity markets use different market designs to handle congestion in the
transmission network. We compare nodal, zonal and discriminatory pricing in general networks
with transmission constraints and loop flows. We conclude that in large games with many
producers and certain information, the three market designs result in the same efficient dispatch.
However, zonal pricing with counter-trading results in additional payments to producers in
export-constrained nodes, which leads to inefficient investments in the long-run.
Keywords: Congestion management, wholesale electricity market, transmission network, nodal
pricing, zonal pricing, counter-trading, discriminatory pricing, large game
* Corresponding author. Research Institute of Industrial Economics (IFN). P.O. Box 55665, SE-102 15 Stockholm,
Sweden, phone +46 8 665 45 59, fax + 46 8 665 4599. E-mail: [email protected] Associate Researcher of
Electricity Policy Research Group (EPRG), University of Cambridge.
** Research Institute of Industrial Economics (IFN). P.O. Box 55665, SE-102 15 Stockholm, Sweden, fax + 46 8
665 4599, Stockholm School of Economics, Department of Economics, Sveavägen 65, Stockholm, Sweden. Email: [email protected]
We are grateful for very helpful comments from Richard Friberg, Sven-Olof Fridolfsson, Jenny Fridström, Håkan
Pihl, Thomas Tangerås, Bert Willems, anonymous referees, seminar participants at Stockholm School of Economics,
Research Institute of Industrial Economics (IFN) and the Swedish Ministry of Enterprise, Energy and
Communications, and conference participants at IAEE 2011 in Stockholm, EWGCFM 2012 in London and EPRG
Online Symposium on Electricity Transmission Pricing and Congestion Management. We also want to thank Erik
Lundin for research assistance, and Christina Lönnblad and Dina Neiman for proof-reading our paper. The work has
been financially supported by the Research Program The Economics of Electricity Markets and the Torsten
Söderberg foundation.
1
1.
INTRODUCTION
Storage possibilities are negligible in most electric power networks, so demand and supply
must be instantly balanced. One consequence is that transmission constraints and the way they
are managed can have a large influence on market prices. The European Union’s regulation
1228/2003 (amended in 2006) sets out guidelines for how congestion should be managed in
Europe. System operators should coordinate their decisions and choose designs that are secure,
efficient, transparent and market based.
In this paper, we compare the efficiency and welfare distribution of three market designs
that are in operation in real-time electricity markets: nodal, zonal and discriminatory pricing.
Characteristics of the three designs are summarized in Table 1. The zonal market is special in that
it has two stages: a zonal clearing and a redispatch. We show that in competitive markets without
uncertainties the three designs result in the same efficient dispatch. However, zonal pricing with a
market based redispatch (counter-trading) results in additional payments to producers in exportconstrained nodes, as they can make an arbitrage profit from price differences between the zonal
market and the redispatch stage. This strategy is often referred to as the increase-decrease (incdec) game. This is the first paper that proves these results for general networks with general
production costs. Dijk and Willems (2011) are closest to our study. However, their analysis is
limited to two-node networks and linear production costs. The parallel study by Ruderer and Zöttl
(2012) is also analyzing similar issues, but the redispatch of the zonal market that they consider is
not market based, thus their model does not capture the increase-decrease game.
Table 1: Summary of the three congestion management techniques.
Congestion
Considered
Auction format
management
transmission
Uniform-price
technique
constraints
Nodal
All
Discriminatory
All
Zonal –stage 1
Inter-zonal
Redispatch –
Intra-zonal
stage 2
Pay-as-bid
X
X
X
X
2
1.1
Congestion management techniques
Producers submit offers to real-time markets just before electricity is going to be produced
and delivered to consumers. During the delivery period, the system operator accepts offers in
order to clear the real-time market, taking transmission constraints into account. The auction
design decides upon accepted offers and their payments. Nodal pricing or locational marginal
pricing (LMP) acknowledges that location is an important aspect of electricity which should be
reflected in its price, so all accepted offers are paid a local uniform-price associated with each
node of the electricity network (Schweppe et al., 1988; Hogan, 1992; Chao and Peck, 1996; Hsu,
1997). This design is used in Argentina, Chile, New Zealand, Russia, Singapore and in several
U.S. states, e.g. Southwest Power Pool (SPP), California, New England, New York, PJM1 and
Texas. Nodal pricing is not yet used inside the European Union. However, Poland has serious
discussions about implementing this design.
Under discriminatory pricing, where accepted offers are paid as bid, there is no uniform
market price. Still, the system operator considers all transmission constraints when accepting
offers, so there is locational pricing in the sense that production in import-constrained nodes can
bid higher than production in export constrained nodes and still be accepted. Discriminatory
pricing is used in Iran, in the British real-time market, and Italy has decided to implement it as
well. A consequence of the pay-as-bid format is that accepted production is paid its stated
production cost. Thus one (somewhat naïve) motivation for this auction format is that if
producers would bid their true cost, then this format would increase consumers’ and/or the
auctioneer’s welfare at producers’ expense.
The third type of congestion management is zonal pricing. Markets which use this design
consider inter-zonal congestion, but have a uniform market price inside each region, typically a
country (continental Europe) or a state (Australia), regardless of transmission congestion inside
the region. Denmark, Norway and Sweden2 are also divided into several zones, but this division
is motivated by properties of the network rather than by borders of administrative regions. 3
Britain is one zone in its day-ahead market, but uses discriminatory pricing in the real-time
market. Initially the zonal design was thought to minimize the complexity of the pricing
1
PJM is the largest deregulated wholesale electricity market, covering all or parts of 13 U.S. states and the District of Columbia. The Swedish government introduced four zones in Sweden from November 2011, as a result of an antitrust settlement between the European
commission and the Swedish network operator (Sadowska and Willems, 2012). 3
The optimal definition of zones for a given network is studied by e.g. Stoft (1997), Bjørndal and Jörnsten (2001) and Ehrenmann and Smeers
(2005). 2
3
settlement and politically it is sometimes more acceptable to have just one price in a
country/state. 4 Originally, zonal pricing was also used in the deregulated electricity markets of
the U.S., but they have now switched to nodal pricing, at least for generation. One reason for this
change in the U.S. is that zonal pricing is, contrary to its purpose, actually quite complex and the
pricing system is not very transparent under the hood. The main problem with the zonal design is
that after the zones of the real-time market have been cleared the system operator needs to order
redispatches if transmission lines inside a zone would otherwise be overloaded. Such a redispatch
increases accepted supply in import constrained nodes and reduces it in export constrained nodes
in order to relax intra-zonal congestion. There are alternative ways of compensating producers for
their costs associated with these adjustments. The compensation schemes have no direct influence
on the cleared zonal prices, but indirectly the details of the design may influence how producers
make their offers.
The simplest redispatch is exercised as a command and control scheme: the system operator
orders adjustments without referring to the market and all agents are compensated for the
estimated cost associated with their adjustments (Krause, 2005). In this paper we instead consider
a market oriented redispatch, also called counter-trading. This zonal design is used in Britain, in
the Nordic countries and it was used in the old Texas design.5 In these markets a producer’s
adjustments are compensated in accordance with his stated costs as under discriminatory pricing.
Thus the market has a zonal price in the first stage and pay-as-bid pricing in the second stage.
We consider two cases: a single shot game where the same bid curve is used in both the first and
second stage, and a dynamic game where firms are allowed to submit new bid curves in the
second stage. The dynamic model is appropriate if, for example, the first stage represents the dayahead market and the second stage represents the real-time market.
1.2
Comparison of the three market designs
Our analysis considers a general electricity network, which could be meshed, where nodes
are connected by capacity constrained transmission lines. We study an idealized market where
4
Policy makers’ and the industries’ critique of the nodal pricing design is summarized, for example, by Alaywan et al. (2004), de Vries et al.
(2009), Leuthold et al. (2008), Oggioni and Smeers (2012) and Stoft (1997). 5
Note that Britain is different in that it has pay-as-bid pricing for all accepted bids in the real-time market. The Nordic real-time markets only use
discriminatory pricing for redispatches; all other accepted bids are paid a zonal real-time price. 4
producers’ costs are common knowledge, and demand is certain and inelastic. There is a
continuum of infinitesimally small producers that choose their offers in order to maximize their
individual payoffs.6 Subject to the transmission constraints, the system operator accepts offers to
minimize total stated production costs, i.e. it clears the market under the assumption that offers
reflect true costs. We characterize the Nash equilibrium (NE) of each market design and compare
prices, payoffs and efficiencies for the three designs.
In the nodal pricing design, we show that producers maximize their payoffs by simply
bidding their marginal costs. Thus, in this case, the accepted offers do in fact maximize short-run
social welfare. We refer to these accepted equilibrium offers as the efficient dispatch and we call
the clearing prices the network’s competitive nodal prices. We compare this outcome with
equilibria in the alternative market designs.
For fixed offers, the system operator would increase its profit at producers’ expense by
switching from nodal to discriminatory pricing. But we show that even if there are infinitely
many producers in the market, discriminatory pricing encourages strategic bidding among
inframarginal production units. They can increase their offer prices up to the marginal price in
their node and still be accepted. 7 In the Nash equilibrium of the pay-as-bid design, accepted
production is the same as in the efficient dispatch and all accepted offers are at the network’s
competitive nodal prices. Thus, market efficiency and payoffs to producers and the system
operator are the same as for nodal pricing. As payoffs are identical in all circumstances, this also
implies that the long-run effects are the same in terms of investment incentives.
Under our idealized assumptions, the zonal market with counter-trading has the same
efficient dispatch as in the two other market designs. We also show that producers buy and sell at
the competitive nodal price in the counter-trading stage. Still producers’ payoffs are larger under
zonal pricing at consumers’ and the system operator’s expense. The reason is that the two-stage
clearing gives producers the opportunity to either sell at the zonal price or at the discriminatory
equilibrium price in the second stage, whichever is higher. In addition, even when they are not
producing any energy, production units in export-constrained nodes can make money by selling at
the uniform zonal price and buying back the same amount at the discriminatory price, which is
6
The idea to calculate Nash equilibria for a continuum of agents was first introduced by Aumann (1964). The theory was further developed by
Green (1984). 7
Related results have been found for theoretical and empirical studies of discriminatory auctions (Holmberg and Newbery, 2010; Evans and
Green, 2004). However, previous studies of discriminatory pricing have not taken the network into account.
5
lower, in the second stage. This increase-decrease game has been observed during the California
electricity crisis (Alaywan et al., 2004), it destroyed the initial PJM zonal design, and is present
in the UK in the form of large payments to Scottish generators (Neuhoff, Hobbs and Newbery,
2011). Our results show that inc-dec gaming is an arbitrage strategy, which cannot be removed by
improving competition in the market. If it is a serious problem, it is necessary to change the
market design as in the U.S. We show how producers’ profits from the inc-dec game can be
calculated for general networks, including meshed networks. Our results for the zonal market are
the same for the static game, where the same offer is used in the two stages, and in the dynamic
game, where firms are allowed to make new offers in the counter-trading stage.
Additional payments to producers in the zonal market cause long-run inefficiencies;
producers overinvest in export-constrained nodes (Dijk and Willems, 2011).8 Zonal pricing also
leads to inefficiencies in the operation of inflexible plants with long ramp-rates, which are not
allowed to trade in the real-time market. Related issues are analyzed by Green (2007). In practice
nodal pricing is considered superior to the other designs, as it ensures efficient allocation in a
competitive market also for uncertain demand and intermittent wind power production; an
advantage which is stressed by Green (2010).
The organization of the paper is as follows. In Section 2 we present a simple two–node
example illustrating the equilibrium under the nodal pricing. Section 3 discusses our model and in
Section 4 we present an analysis of the three congestion management designs. In section 5,
market equilibria for the discriminatory and zonal pricing designs are discussed with the means
of a simple example. The paper is concluded in section 6, which also briefly discusses how more
realistic assumptions would change our results. Three technical lemmas and all proofs are placed
in the Appendix.
2.
EXAMPLE – NODAL PRICING
In the following section we describe a simple example of bidding under nodal pricing and
the equilibrium outcome of this design. We consider a two-node network with one constrained
transmission-line in-between. In both nodes producers are infinitesimally small and demand is
8
Ruderer and Zöttl (2012) show that zonal pricing in addition leads to inefficient investments in transmission-lines, at least if the zonal market is
regulated such that redispatches are compensated according to producers’ true costs. 6
perfectlyy inelastic. For simpllicity, we m
make the ffollowing aassumptionss for each node: the
marginaal cost is equual to local output
o
and the
t productiion capacityy is 15 MW. In node 1, demand is
5 MW; in node 2 ddemand is 18
1 MW. Thee transmissiion line bettween thesee nodes is constrained
and can carry only 4 MW. Deemand in noode 2 exceedds its generration possibbilities so thhe missing
i
froom the otheer node.
electriciity must be imported
Figure 11. Equilibriium for nod
dal pricing..
NODE 1
NO
ODE 2
price
price
D1 D1+export
+
D2-impoort
=15
114
D2
N
p2 =14
N
p 1 =9
’
C1 =o1(qq)
’
C2 =o2(q)
5
0
5
N
0
q1 =9
MW
N
q2 =
=14 15
18
MW
W
With nodal ppricing, the equilibrium
m offers willl be as shoown in Fig. 1. In the first node
infinitessimally smaall producerss make noddal offers o((q) at their m
marginal coost. In orderr to satisfy
local deemand and export, 9 MW are going to be dispatchedd. Out of thhese, 5 MW
W will be
consumeed locally and 4 MW
W will be exported; tthe highest possible eexport leveel that the
transmisssion line alllows for. T
The marginaal cost and nnodal price is equal to 9, which coorresponds
to the tootal productiion of this nnode. In thee second nodde, the nodaal price is 144 as there arre 14 MW
that havve to be prooduced in the
t second node in ordder to satisffy demand and the traansmission
constrainnt. Productiion above tthose margiinal costs (99 in node 1 and 14 inn node 2) w
will not be
7
dispatched. All accepted production will be paid the nodal price of the node. The dispatch leads
to a socially efficient outcome. We use the superscript N to designate this outcome. We call nodal
production and nodal prices of competitive and socially efficient outcomes, the network’s
efficient dispatch and the network’s competitive nodal prices, respectively.
As our analysis will show, the offers in Figure 1 cannot constitute NE in the other two
designs. For discriminatory pricing it will be profitable for inframarginal offers to increase their
price up to the marginal offer of the node. For zonal pricing, the average demand in the two zones
would be 11.5 MW, so 11.5 MW would be accepted in each node at the zonal price 11.5 for the
offers in Figure 1. Production would be adjusted in the redispatch stage. However, as it applies
discriminatory pricing, it would not influence the payoff of producers that bid their true marginal
cost. Thus producers in the export-constrained node 1 would find it profitable to change their
offers downwards. They would like to sell as much as possible at the zonal price and then buy it
back at a lower price in the redispatch stage. Producers in the import constrained node 2 would
shift their offers upwards as in the pay-as-bid design, so that all production that is dispatched in
the redispatch stage is accepted at the marginal offer of the import constrained node.
3.
MODEL
The model described in this section is used to evaluate and compare three market oriented
congestion management techniques: nodal pricing, pay-as-bid and zonal pricing with countertrading. We study a general electricity network (possibly meshed) with n nodes that are connected
by capacity constrained transmission lines. Demand in a node i  1, , n is given by Di, which
is certain and inelastic up to a reservation price p . C’i(qi) is the marginal cost of producing qi
units of electricity in node i. We assume that the marginal cost is common knowledge, continuous
and strictly increasing up to (and beyond) the reservation price. 9 We let q i >0 be the relevant
total production capacity in node i, which has a marginal cost at the reservation price or lower.
Thus we have by construction that p  C 'i q i  . Capacity with a marginal cost above the
reservation price will not submit any offers.
9
Note that it is possible for a producer to generate beyond the rated power of a production unit. However, it heats up the unit and shortens its
lifespan. Thus the marginal cost increases continuously beyond the rated power towards a very high number (above the reservation price) where
the unit is certain to be permanently destroyed during the delivery period. Edin (2007) uses a similar marginal cost curve with a similar motivation. 8
In each node there is a continuum of infinitesimally small producers. Each producer in the
continuum of node i is indexed by the variable g i  0,1 . For simplicity, we assume that each
producer is only active in one node. Without loss of generality, we also assume that producers are
sorted with respect to their marginal cost in each node, such that a producer with a higher gi value
than another producer in the same node also has a higher marginal cost. The relevant total
production capacity q i in a node i is divided between the continuum of producers, such that firm
gi in node i has the marginal cost C'i g i q i  . Similarly, we let oˆi gi q i  represent the offer price of
firm gi in node i.
The system operator’s clearing of the real-time market must be such that local net-supply
equals local net-exports in each node and such that the physical constraints of the transmission
network are not violated. Any set qi in1 of nodal production that satisfies these feasibility
constraints is referred to as a feasible dispatch. We say that a dispatch is locally efficient if it
minimizes the local production cost in each node for given nodal outputs qi in1 , i.e. production
units in node i are running if and only if they have a marginal cost at or below C'i qi  . We
consider a set of demand outcomes Di i 1 , such that there is at least one feasible dispatch. In
n
principle the network could be a non-linear AC system with resistive losses. But to ensure a
unique cost efficient dispatch we restrict the analysis to cases where the feasible set of dispatches
is convex. Hence, if two dispatches are possible, then any weighted combination of the two
dispatches is also feasible. The feasible set of dispatches is for example convex under the DC
load flow approximation of general networks with alternating current (Chao and Peck, 1996). 10
The system operator sorts offers in ascending order in case a nodal offer curve oˆi qi  would
be locally decreasing. We denote the sorted nodal offer curve by oi qi  . The system operator then
chooses a feasible dispatch in order to minimize the stated production cost or equivalently to
maximize
10
Alternating currents (AC) result in a non-linear model of the network. Hence, in economic studies this model is often simplified by a linear
approximation called the direct current (DC) load flow approximation. In addition to Chao and Peck (1996), it is used, for example, by Schweppe
et al. (1988), Hogan (1992), Bjørndal and Jörnsten (2001, 2005, 2007), Glachant and Pignon (2005), Green (2007) and Adler et al. (2008). 9
n qi
W    oi  y dy ,
i 1 0


(1)
Stated cost
which maximizes social welfare if offers would reflect the true costs. Thus, we say that the
system operator acts in order to maximize the stated social welfare subject to the feasibility
constraints.
In a market with nodal pricing the system operator first chooses the optimal dispatch as
explained above. All accepted offers in the same node are paid the same nodal price. The nodal
price is determined by the node’s marginal price, i.e. the highest accepted offer price in the node.
We say that marginal prices or nodal prices are locally competitive if the dispatch is locally
efficient and the marginal price in each node equals the highest marginal cost for units that are
running in the node. An offer at the marginal price of its node is referred to as a marginal offer.
In the discriminatory pricing design all accepted offers are paid according to their offer price.
This gives producers incentives to change their offers and thereby state their costs differently.
Still, the dispatch is determined in the same way; by minimizing stated production cost. In the
zonal pricing design with counter-trading, the market is cleared in two stages. First the system
operator clears the market disregarding the intra-zonal transmission constraints (constraints inside
zones). Next, in case intra-zonal transmission lines are overloaded after the first clearing, there is
a redispatch where the system operator increases accepted production in import constrained
nodes and reduces it in export constrained nodes. Section 4.3 explains our zonal pricing model in
greater detail.
4.
ANALYSIS
We start our game-theoretical analysis of the three market designs by means of three
technical results that we will use in the proofs that follow.
Lemma 1. Assume that offers are shifted upwards (more expensive) in some nodes and shifted
downwards (cheaper) in others, then the dispatched production is weakly lower in at least one
node with more expensive offers or weakly higher in at least one node with cheaper supply.
One immediate implication of this lemma is that:
10
Corollary 1 (Non-increasing residual demand) If one producer unilaterally increases/decreases
its offer price, then accepted sales in its node cannot increase/decrease.
The system operator accepts offers in order to minimize stated production costs. Thus for a
given acceptance volume in a node, a firm cannot increase its chances of being dispatched by
increasing its offer price. Thus Corollary 1 implies that a producer’s residual demand is nonincreasing. The next lemma outlines necessary properties of a Nash equilibrium.
Lemma 2. Consider a market where an accepted offer is never paid more than the marginal price
of its node and never less than its own bid price. In Nash equilibrium, the dispatch must be
locally efficient and marginal prices of the nodes are locally competitive.
4.1
Nodal pricing
Below we prove that the nodal pricing design has at least one NE and that all NE results in
the same competitive outcome.11 It is only offers above and below the marginal prices of nodes
that can differ between equilibria.
Proposition 1 A market with nodal pricing has one NE where producers offer at their marginal
 
cost. All NE result in the same locally efficient dispatch qiN
n
i 1 and
the same competitive nodal
prices piN  CiqiN  .
As the system operator clears the market in order to maximize social welfare when offers
reveal true costs, we note that the equilibrium dispatch must be efficient. We use the superscript
N to designate this socially efficient outcome. We refer to the unique equilibrium outcome as the
11
Existence of pure-strategy NE in networks with a finite number of producers is less straightforward. The reason is that a producer in an importing node can find it profitable to deviate from a locally optimal profit maximum by withholding production in order to congest imports and push
up the nodal price (Borenstein et al., 2000; Willems, 2002; Downward et al., 2010; Holmberg and Philpott, 2012). Such unilateral deviations are
not feasible in a network with infinitesimally small producers, which makes existence of pure-strategy NE more straightforward. Escobar and
Jofré (2008) show that networks with a finite number of producers and non-existing pure-strategy NE normally have mixed-strategy NE.
Existence of NE in large games with continuous payoffs has been analyzed by Carmona et al. (2009).
11
 
network’s efficient dispatch qiN
n
i 1
 
and the network’s competitive nodal prices piN
n
i 1 .
Note
that as the dispatch is locally efficient, the unique equilibrium outcome exactly specifies which
units are running; production units in node i are running if and only if they have a marginal cost
at or below C'i qiN . Schweppe et al. (1988), Chao and Peck (1996) and Hsu (1997) and others
 
outline methods that can be used to calculate efficient dispatches qiN
n
i 1
for general networks.
Existence of the competitive outcome also indirectly establishes existence of a Walrasian
equilibrium, which has previously been proven for radial (Cho, 2003) and meshed networks
(Escobar and Jofré, 2008). Proposition 1 proves that all of our NE correspond to the Walrasian
equilibrium, so in this sense our NE is equivalent to the Walrasian equilibrium in a market with
nodal pricing. The reason is that the infinitesimal producers that we consider are price takers in
nodal markets, where all agents in the same node are paid the same market price.
4.2
Discriminatory pricing
Discriminatory pricing is different to nodal pricing in that each agent is then paid its
individual offer price rather than a uniform nodal price. Thus, even if agents are infinitesimally
small, inframarginal producers can still influence how much they are paid, so they are no longer
price takers. This means that the Walrasian equilibrium is not a useful equilibrium concept when
studying discriminatory pricing. This is the reason why we instead consider a large game with a
continuum of small producers in this paper.
Proposition 2. There exist Nash equilibria in a network with discriminatory pricing. All such NE
have the following properties:
1)
The dispatched production is identical to the network’s efficient dispatch in each node.
2)
All production in node i with a marginal cost at or below Ci qiN is offered at the
 
network’s competitive nodal price piN  CiqiN  .
3)
Other offers are not accepted and are not uniquely determined in equilibrium. However, it
can, for example, be assumed that they offer at their marginal cost.
12
Thus, the discriminatory auction is identical to nodal pricing in terms of payoffs, efficiency,
social welfare and the dispatch. As payoffs are identical for all circumstances, this also implies
that the long-run effects are the same in terms of investment incentives etc. Note that it is not
necessary that rejected offers bid at marginal cost to ensure an equilibrium. As producers are
infinitesimally small, it is enough to have a small finite amount of rejected bids at or just above
the marginal offer in each node to avoid deviations.
Finally we analyze how contracts influence the equilibrium outcome. We consider forward
contracts with physical delivery in a specific node at a predetermined price. For simplicity, we
consider cases where each infinitesimally small producer either has no forward sales at all or sells
all of its capacity in the forward market for physical delivery in its own node to consumers. In
the real-time market, consumers announce how much more power they want to buy in each node,
in addition to what they have already bought with contracts, and producers make offers for
changes relative to their contractual obligations. The system operator accepts changes in
production in order to achieve a feasible dispatch at the lowest possible net-increase in the stated
production costs.
Proposition 3. In a real-time market with nodal or discriminatory pricing, the equilibrium
dispatch is identical to the network’s efficient dispatch and marginal prices of the nodes are
competitive, for any set of forward contracts that producers have sold with physical delivery in
their own node.
We will use this result in our analysis of the zonal pricing design, where the first-stage clearing of
the zonal market can be regarded as physical forward sales.
4.3
Zonal pricing with counter-trading
4.3.1. Notation and assumptions
Zonal pricing with counter-trading is more complicated than the other two designs and we
need to introduce some additional notation before we start to analyze it. The network is divided
into zones, such that each node belongs to some zone k. We let Zk be a set with all nodes
belonging to zone k. To simplify our equations, we number the nodes in a special order. We start
with all nodes in zone 1, and then proceed with all nodes in zone 2 etc. Thus, for each zone k,
13
nodes are given numbers in some range n k to n k . Moreover, inside each zone, nodes are sorted
with respect to the network’s competitive nodal prices piN , which can be calculated for the nodal
pricing design, as discussed in Section 4.1. Thus, the cheapest node in zone k is assigned the
number n k and the most expensive node in zone k is assigned the number n k .
Counter-trading in the second-stage only changes intra-zonal flows. Thus it is important for
a benevolent system operator to ensure that the inter-zonal flows are as efficient as possible
already after the first clearing. In the Nordic multi-zonal market, system operators achieve this by
announcing a narrow range of inter-zonal flows before the day-ahead market opens. In particular,
flows in the “wrong direction”, from zones with high prices to zones with low prices, due to loop
flows, are predetermined by the system operator. We simplify the zonal clearing further by letting
the well-informed system operator set all inter-zonal flows before offers are submitted. Total netimports to zone k are denoted by I kN . We make the following assumption for these flows, as our
analysis shows that it leads to an efficient outcome:
Assumption 1: The system operator sets inter-zonal flows equal to the inter-zonal flows that
 
would occur for the network’s efficient dispatch qiN
n
i 1 .
These inter-zonal flows are announced
by the system operator before offers are submitted.
Assumption 1 sets all inter-zonal flows. Thus offers to each zonal market can be cleared
separately at a price where zonal net-supply equals zonal net-exports. We assume that the highest
potential clearing price is chosen in case there are multiple prices where zonal net-supply equals
zonal net-exports.12 The clearing price Πk in zone k is paid to all production in the zone that is
accepted in the zonal clearing. In case intra-zonal transmission-lines are overloaded after the first
clearing, there is a redispatch where the system operator increases accepted production in import
constrained nodes and reduces it in export constrained nodes. We consider a market oriented
redispatch (counter-trading), so all deviations from the first-clearing are settled on a pay-as-bid
basis. In the counter-trading stage, the system operator makes changes relative to the zonal
12
Normally this choice does not matter for our equilibria. However, it ensures existence of equilibria for degenerate cases when exogenous zonal
demand and exogenous net-exports happen to coincide with production capacities in one or several nodes for some zone. 14
clearing in order to achieve a feasible dispatch at the lowest possible net-increase in stated
production costs.
We consider two versions of the zonal design: a one shot game where the same offers are
used in the two clearing stages of the market and a dynamic game where agents are allowed to
make new offers in the counter-trading stage. The first model corresponds to the old pool in
England and Wales, while the latter model could for example be representative of the reformed
British market, where producers can first sell power at a uniform zonal price in the day-ahead
market and then submit a new bid to the real-time market with discriminatory pricing.13
4.3.2. Analysis
The equilibrium in a zonal market with counter-trading has some similarities with the
discriminatory auction. But the zonal case is more complicated, as the two clearing stages imply
that in equilibrium some producers can arbitrage between their zonal and individual
(discriminatory) counter-trading prices. Thus producers in nodes with low marginal prices will
play the inc-dec game, i.e. sell all their capacity at the higher zonal price and then buy back the
capacity at a lower price in the counter-trading stage or produce if the marginal cost is even
lower. We consider physical markets. This prevents producers from buying power or selling more
than their production capacity in the zonal market. Thus a producer in a node with a marginal
price above its zonal price cannot make an arbitrage profit. To maximize their profit in the
redispatch stage, bids of dispatched production in such import constrained nodes are shifted
upwards to the node’s competitive nodal price, similar to the case with discriminatory pricing.
First we consider a static game where producers cannot make new offers to the countertrading stage; the same offers are used in the two stages of the zonal market.
13
The dynamic model could also represent congestion management in the Nordic market, where the system operator does not accept offers in the
zonal clearing of the real-time market if these offers will cause intra-zonal congestion that needs to be countertraded in the second-stage. This is to
avoid unnecessary costs for the system operator and unnecessary payments to producers. In our model where there is no uncertainty, the zonal
day-ahead market then takes the role of the first-stage of the real-time market. The zonal real-time market becomes obsolete as without uncertainty, the day-ahead market has already cleared the zones. In this case offers to the real-time market, which are allowed to differ from day-ahead
offers, are only used in the discriminatory counter-trading stage. Proposition 5 shows that under our idealized assumptions switching to the Nordic
version of zonal congestion management is in vain, producers still get the same payoffs and the system operator’s counter-trading costs are unchanged. 15
Proposition 4. Under Assumption 1 there exists Nash equilibria in a zonal market with countertrading and the same offers in the zonal and countertrading stages. All of them have the following
properties:
1)
The zonal price in zone k is given by Π k*  p mNk  , where:
n 1
nk
n

N
N
if
n  n k ,, n k : I k   q i   Di  I k   q i

i nk
i nk
i n k
mk   
nk
nk

n k if  Di  I kN   q i

i n k
i nk
2)
nk
D
i nk
i
nk
 I kN   q i
i nk
(2)
As in the nodal pricing and pay-as-bid designs, the dispatched production in each node is
given by the network’s efficient dispatch, qiN .
3)
In strictly export-constrained nodes i  Z k , such that piN < Π k* , production with marginal
 
costs at or above piN are offered at the network’s competitive nodal price piN  Ci qiN . For
strictly import-constrained nodes in zone k where piN > Π k* , all production with a marginal cost
 
 
at or below Ci qiN is offered at piN  Ci qiN .
4)
Other offers are not uniquely determined in equilibrium. However, it can be assumed that
they offer at their marginal cost.
Equation (2) defines a marginal node, where the competitive nodal price equals the zonal
price. Next we show that the equilibrium outcome does not change in the dynamic game, where
agents are allowed to up-date their offers in the counter-trading stage.
Proposition 5. Under Assumption 1, it does not matter for payoffs or the equilibrium outcome of
the zonal market whether producers are allowed to up-date their offers in the counter-trading
stage.
We can now conclude that the dispatch for zonal pricing with counter-trading is the same as
for nodal pricing and discriminatory pricing. Thus, in the short run, the designs’ efficiencies are
equivalent. This also confirms that the system operator should set inter-zonal flows equal to the
corresponding flows in the competitive nodal market, as assumed in Assumption 1, if it wants to
maximize social welfare. However, it directly follows from Equation (2) and Propositions 4 and 5
16
that producers in strictly export-constrained nodes receive unnecessarily high payments in a zonal
pricing design:
Corollary 2. In comparison to nodal pricing, the total extra payoff from the system operator to
mk 1
producers in zone k equals:

i  nk
p
N
m k 

 piN q i under Assumption 1.
Even if zonal pricing is as efficient as nodal pricing in the short run, the extra payoffs will
cause welfare losses in the long run. Production investments will be too high in strictly exportconstrained nodes where piN < Πk. In addition, inflexible production that cannot take part in the
real-time market are paid the zonal price in the day-ahead market. Thus, the accepted inflexible
supply in this market is going to be too high in strictly export-constrained nodes and too low in
strictly import-constrained nodes.
5.
EXAMPLE – DISCRIMINATORY AND ZONAL PRICING
In the following section, we illustrate the equilibria for the discriminatory and zonal pricing
designs. The example that we use has an identical structure as the nodal pricing case that we
described in section 2. Again, we consider a two-node network with one constrained
transmission-line in-between. In both nodes producers are infinitesimally small and demand is
perfectly inelastic. In each node the marginal cost is equal to local output and the production
capacity is 15 MW. In node 1, demand is 5 MW; in node 2 demand is 18 MW. The transmission
line between these nodes is constrained and can carry only 4 MW. Demand in node 2 exceeds its
generation possibilities so the missing electricity must be imported from the other node.
The discriminatory design will result in the equilibrium offers presented in Fig. 2. In this
design, generators are paid according to their bid. Knowing this and having perfect information,
producers who want to be dispatched will bid the competitive nodal price of their node, to ensure
that they will be dispatched at the highest possible price. Thus, in node 1, they will bid 9 and in
node 2 they will bid 14. Producers who do not want to be dispatched may, for example, bid their
marginal costs, which are higher than the nodal prices of the respective nodes. The dispatch will
be the same as under nodal pricing design. Although producers will have different bidding
17
strategiees in both deesigns, the overall
o
resuult will be thhe same. Acccepted prodduction willl be paid 9
in node 1 and 14 in node 2.
Figure 22: Equilibrrium for disscriminatorry pricing.
NODE
E1
NOD
DE 2
price
price
D1+eexport
D2-im
mport
=15
144
14
o2(q)
o1(q)
9
C1
5
0
5
C2
’
9
15
’
0
MW
14 15
18
MW
In the zoonal design with counteer-trading, pproducers w
will offer as ffollows:
18
Figure 33: Zonal offfer in equillibrium for zonal priciing with coounter-trading.
price
D=D1+
+D2
o2(q))
144
9
o1(q)
0
9
15
23
29 330
M
MW
Figure 44: Nodal offfers in equilibrium for zonal priccing with counter-trad
ding.
NOD
DE 1
NO
ODE 2
pricce
pprice
D1+eexport
144
o2(q))
14
C1
D2-import
’
C2
9
’
o1(q)
0
15
9
MW
0
8
14 15
MW
W
19
Node 1:
Due to transmission constraints, producers in node 1 know that after the two stages, the
system operator can accept a maximum of 9 MW in their node. Therefore, producers with a
marginal cost at or below the competitive nodal price, offer at or below the competitive nodal
price as they will, in any case, be accepted and paid the zonal price, which is 14. The remaining 6
units in node 1 have a marginal cost above the competitive nodal price. They will bid low in
order to be accepted in the first stage and be paid the zonal price of 14. But due to the
transmission constraint, they will have to buy back their supply at their own bidding price in the
second round. As they are interested in maximizing their profit, they want this price difference to
be as large as possible, as long as they will not be chosen to produce. Therefore, they bid the
competitive nodal price 9 so that they will be “paid” not to produce and get 14 – 9 =5 (the
rectangle area in the figure 4). There are no profitable deviations from
these bids for
producers from node 1. In particular, we note that no infinitesimally small
producer
in
node 1 can unilaterally increase the zonal price at stage 1 above 14, as there are 6 units (in node
2) that offer their production at the price 14 without being accepted in the zonal market.
Node 2:
Due to the transmission constraint, producers in node 2 know that the system operator
needs to dispatch at least 14 units of electricity in their node after the two stages. Thus, all lowcost generators who want to be dispatched know that all offers at or below 14, the competitive
nodal price of node 2, will be accepted. 8 units are accepted in the zonal clearing and another 6
units are accepted in the counter-trade stage. The latter units are paid as bid and accordingly, they
maximize their profit by offering their supply at 14, the highest possible price for which they are
going to be accepted. Producers that do not want to be dispatched at all will bid above 14, for
example their marginal cost. In this way, 14 units will be produced in node 2. There are no
profitable deviations from these strategies for producers in node 2.
A comparison of these two examples and the nodal pricing example in Section 2 illustrates
that although the bidding strategies are different, the dispatch is the same in all scenarios.
However, the last design – zonal pricing with counter-trading – results in additional payments
that affect the long-term investment incentives.
It is interesting to note that the zonal price in our example is weakly higher than the nodal
prices in both nodes. This is always the outcome in two-node networks where the production
20
capacity in the cheapest node is not sufficient to meet the total demand, so that it is the marginal
cost in the most expensive node that sets the zonal price. The system operator will typically use
tariffs to pass its counter-trading cost on to the market participants, so it is actually quite plausible
that switching to nodal pricing will lower the cost for all electricity consumers, including the ones
in the high cost node.
6.
CONCLUSIONS AND DISCUSSION
We consider a general electricity network (possibly meshed), where nodes are connected by
capacity constrained transmission lines. In our game-theoretical model producers are
infinitesimally small and demand is certain and inelastic. We find that the three designs, nodal,
zonal with countertrading and discriminatory pricing, lead to the same socially efficient dispatch.
In addition, payoffs are identical in the pay-as-bid and nodal pricing designs. However, in the
design with zonal pricing and countertrading, there are additional payments from the system
operator to producers who can make money by playing the infamous inc-dec game. It does not
matter for our results whether we consider a static game where producers’ bids are the same in
the zonal and counter-trading stages or a dynamic game where producers are allowed to update
their offer curves in the counter-trading stage.
Similar to Dijk and Willems’ (2011) two-node model, our results for the zonal market imply
that producers overinvest in export-constrained nodes. While zonal pricing is good for producers,
consumers would gain overall from a switch from zonal to nodal pricing. In two-node markets, it
is normally the case that all consumers (also the ones in the most expensive node) would gain
from a switch to nodal pricing. In addition to the inefficiencies implied by our model, zonal
pricing also leads to inefficiencies in the operation of inflexible plants with long ramp-rates. They
are not allowed to trade in the real-time market, so they have to sell at the zonal price in the dayahead market. The consequence is that too much inflexible production is switched on in export
constrained nodes, where the competitive nodal price is below the zonal price, and too little in
import constrained nodes, where the competitive nodal price is above the zonal price. Related
issues are analyzed by Green (2007).
Another result from our analysis is that there is a significant number of firms that make
offers exactly at the marginal prices of the nodes in the zonal and pay-as-bid designs, which is not
necessarily the case under nodal pricing. This supports the common view that the zonal design is
21
more liquid. Although, the standard motivation for this is that the zonal design has less market
prices and thus fewer products to trade, and hence liquidity can be concentrated on these. Still it
is known from PJM that it is also possible to have a liquid market with nodal pricing (Neuhoff
and Boyd, 2011).
However increased liquidity can have more drawbacks than advantages. As illustrated by
Anderson et al. (2009), the elastic offers, especially in the pay-as-bid design but also in the zonal
design, mean that getting its offer slightly wrong can have a huge effect on a firm’s dispatch. This
increases the chances of getting inefficient dispatches when demand or competitors’ output is
uncertain, while the efficiency of the nodal pricing design is more robust to these uncertainties.
Similarly, Green (2010) stresses the importance of having designs that can accommodate
uncertainties from intermittent power.
There are other drawbacks with the zonal design.
We consider a benevolent system
operator that uses counter-trading to find the socially optimal dispatch. However, even if countertrading is socially efficient, it is costly for the system operator itself. Thus strategic system
operators have incentives to find the feasible dispatch that minimizes counter-trading costs. In
practice, counter-trading is therefore likely to be minimalistic and less efficient than in our
framework. Moreover, Bjørndal et al. (2003) and Glachant and Pignon (2005) show that network
operators have incentives to manipulate inter-zonal flows in order to lower the counter-trading
cost (and market efficiency) further. In our analysis we assume that the system-operator has full
control of the system and that it can set inter-zonal flow as efficiently as under nodal pricing, but
in practice market uncertainty, coordination problems and imperfect regulations lead to
significantly less efficient cross-border flows (Leuthold, 2008; Neuhoff, et al., 2011; Ogionni and
Smeers 2012). Studies by Hogan (1999), Harvey and Hogan (2000), and Green (2007) indicate
that nodal pricing is also better suited to prevent market power.
REFERENCES
Adler, I., S. Oren, J. Yao (2008). “Modeling and Computing Two-Settlement Oligopolistic
Equilibrium in a Congested Electricity Network.” Operations Research 56(1): 34 – 47.
22
Alaywan, Z., T. Wu, and A.D. Papalexopoulos (2004). “Transitioning the California Market from
a Zonal to a Nodal Framework: An Operational Perspective.” Power Systems and Exposition
2: 862-867.
Anderson, E.J., P. Holmberg, and A.B. Philpott, (2013). “Mixed Strategies in Discriminatory
Divisible good Auctions.” Rand Journal of Economics 44(1): 1–32.
Aumann, R.J., (1964). “Markets with a Continuum of Traders.” Econometrica 32 (1/2): 39-50.
Bernard, J. T., and C. Guertin (2002). “Nodal Pricing and Transmission Losses: An Application to
a Hydroelectric Power System.” Discussion Paper 02-34, Resources for the Future,
Washington DC.
Bjørndal, M., and K. Jörnsten (2001). “Zonal Pricing in a Deregulated Electricity Market.” The
Energy Journal 22 (1): 51-73.
Bjørndal, M., K. Jörnsten, and V. Pignon (2003). “Congestion management in the Nordic power
market: counter purchases.” Journal of Network Industries 4: 273 – 296.
Bjørndal, M., and K. Jörnsten (2005). “The Deregulated Electricity Market Viewed as a Bilevel
Programming Problem.” Journal of Global Optimization 33: 465 – 475.
Bjørndal, M. and K. Jörnsten (2007). “Benefits from coordinating congestion management – The
Nordic power market.” Energy Policy 35: 1978 – 1991.
Borenstein, S., J. Bushnell, J. and S. Stoft (2000). “The competitive effects of transmission
capacity in a deregulated electricity industry.” RAND Journal of Economics 31(2): 294 – 325.
Brunekreeft, G., K. Neuhoff, and D. Newbery (2005). “Electricity transmission: An overview of
the current debate.” Utilities Policy 13: 73 – 93.
Carmona, G., and K. Podczeck (2009). “On the Existence of Pure Strategy Nash Equilibria in
Large Games.” Journal of Economic Theory 144: 1300-1319.
Chao, H-P and S. Peck (1996). “A Market Mechanism For Electric Power Transmission.” Journal
of Regulatory Economics 10: 25-59.
Cho, In – Koo (2003). “Competitive equilibrium in a radial network.” The RAND Journal of
Economics 34(3): 438-460.
de Vries, L.J., J. de Joode, and R. Hakvoort (2009). “The regulation of electricity transmission
networks and its impact on governance.” European Review of Energy Markets 3(3): 13-37.
Dijk, J., and B. Willems (2011). “The effect of counter-trading on competition in the Dutch
electricity market.” Energy Policy 39(3): 1764-1773.
23
Downward, A., G. Zakeri, and A.B. Philpott (2010). “On Cournot Equilibria in Electricity
Transmission Networks”, Operations Research 58(4): 1194 – 1209.
Ehrenmann, A., and Y. Smeers (2005). “Inefficiencies in European congestion management
proposals.” Utilities Policy 13: 135 – 152.
Edin, K.A. (2007). “Hydro-Thermal Auction Markets.” Mimeo, Tentum, Stockholm.
Escobar, J.F., and A. Jofré (2008). “Equilibrium Analysis of Electricity Auctions.”, Department of
Economics, Stanford University.
Evans, J.E., and R.J. Green (2005). “Why Did British Electricity Prices Fall After 1998?”
Working paper 05-13, Department of Economics, University of Birmingham.
Glachant, J. M., and V. Pignon (2005). “Nordic congestion’s arrangement as a model for Europe?
Physical constraints vs. economic incentives.” Utilities Policy 13: 153 – 162.
Gravelle, H. and R. Rees (1992). Microeconomics. London:Longman.
Green, E.J. (1984). “Continuum and Finite-Player Noncooperative Models of Competition.”
Econometrica 52 (4): 975-993.
Green, R. (2007). “Nodal pricing of electricity: how much does it cost to get it wrong?” Journal
of Regulatory Economics 31: 125 – 149.
Green, R. (2010). “Are the British trading and transmission arrangements future-proof?” Utilities
Policy 18: 186 – 194.
Harvey, S. M., W.W Hogan (2000). “Nodal and Zonal Congestion Management and the Exercise
of Market Power”, John F. Kennedy School of Government, Harvard University.
Hogan, W. W. (1999). “Transmission Congestion: The Nodal-Zonal Debate Revisited.” John F.
Kennedy School of Government, Harvard University.
Hogan, W. W. (1992). “Contract Networks for Electric Power Transmission.” Journal of
Regulatory Economics 4: 211 – 242.
Holmberg, P., and D. Newbery (2010). “The supply function equilibrium and its policy
implications for wholesale electricity auctions.” Utilities Policy 18(4): 209-226.
Holmberg, P., and A. Philpott (2012). “Supply Function Equilibria in Networks with Transport
Constraints.”, IFN Working Paper 945, Research Institute of Industrial Economics, Stockholm.
Hsu, M. (1997). “An introduction to the pricing of electric power transmission.” Utilities Policy
6(3): 257 – 270.
24
Krause, T. (2005). “Congestion Management in Liberalized Electricity Markets – Theoretical
Concepts and International Application”, EEH – Power Systems Laboratory, Eidgenössische
Technische Hochschule Zürich.
Leuthold, F., H. Weight, and C. von Hirschhausen (2008). “Efficient pricing for European
electricity networks – The theory of nodal pricing applied to feeding-in wind in Germany.”
Utilities Policy 16: 284 – 291.
Neuhoff, K., and R. Boyd (2011). International Experiences of Nodal Pricing Implementation,
Working document, Climate Policy Initiative, Berlin.
Neuhoff, K., R. Boyd, T. Grau, J. Barquin, F. Echabarren, J. Bialek, C. Dent, C. von
Hirschhausen, B. F. Hobbs, F. Kunz, H. Weigt, C. Nabe, G. Papaefthymiou and C. Weber
(2011). “Renewable Electric Energy Integration: Quantifying the Value of Design of Markets
for International Transmission Capacity.” Working document, Climate Policy Initiative,
Berlin.
Neuhoff, K., B. F. Hobbs, and D. Newbery (2011). “Congestion Management in European Power
Networks, Criteria to Asses the Available Options.” Discussion Papers 1161, DIW Berlin.
Oggioni, G. and Y. Smeers (2012). “Degrees of Coordination in Market Coupling and CounterTrading.” The Energy Journal 33 (3):39 – 90.
Ruderer, D., and G. Zöttl (2012). “The Impact of Transmission Pricing in Network
Industries.” EPRG Working Paper 1214, University of Cambridge, UK.
Sadowska, M., and B. Willems (2012). “Power Markets Shaped by Antitrust.” TILEC Discussion
Paper 2012-043, University of Tilburg, Netherlands.
Schweppe, F. C., M.C. Caramanis, R.D. Tabors, and R.E. Bohn (1988). Spot Pricing of
Electricity. London: Kluwer academic publishers.
Stoft, S. (1997). “Transmission pricing zones: simple or complex?” The Electricity Journal 10
(1): 24-31.
Willems, B. (2002). “Modeling Cournot Competition in an Electricity Market with Transmission
Constraints.” The Energy Journal 23(3): 95– 126.
25
APPENDIX A: TECHNICAL LEMMAS
Lemma 3. m(k) is uniquely defined by Equation (2).
Proof:
We first note that the network’s efficient dispatch is feasible as the inter-zonal flows are efficient,
nk
nk
i n k
i nk
i.e.  Di  I kN   qiN . Thus
nk
nk
i nk
i nk
 Di  I kN   q i . We have m(k) = nk if
nk
nk
i nk
ink
n k 1
nk
nk
n 1
i nk
i nk
i nk
i n k
 Di  I kN   q i .
Otherwise we have I kN   q i   Di  I kN   q i . Moreover, I kN   q i is strictly increasing
in n, because q i  0 . Thus Equation (2) always has a unique solution. ■
The following two technical lemmas are used to prove that all Nash equilibria must result in the
same dispatch.


Lemma 4. If there is a set of nodal offer functions oˆi* q  i1 (not necessarily increasing) that
n
results in a locally efficient dispatch with the nodal output qi* i 1 and locally competitive
n


marginal prices, then any set of strictly increasing nodal offer functions oˆi q  i1 , such that
 
n
 
oˆ i qi*  oˆ i* qi* i  1,, n , will result in the same dispatch.


Proof: First, consider the case when offers oˆi* q  i1 are also strictly increasing in output. In this
n
case, the objective function (stated welfare) is strictly concave in the supply, qi. Moreover, the set
of feasible dispatches is by assumption convex in our model. Thus, it follows that the objective
function has a unique local extremum, which is a global maximum (Gravelle and Rees, 1992).
Thus the system operator’s dispatch can be uniquely determined. It follows from the necessary
Lagrange condition that the unique optimum is not influenced by changes in node i’s offers
below and above the quantity qi* , as long as offers are strictly increasing in output. Thus the
26


dispatch must be the same for any set of strictly increasing nodal offer functions oˆi q  i 1 , such
n
that oˆ i qi*   oˆ i* qi*  i  1,, n .


With perfectly elastic segments in the offer curves oi* q  i 1 there are output levels, for
n

which oi* q   0 in some node i. This means that the objective function is no longer strictly
concave in the supply. However, one can always construct strictly increasing curves that are
arbitrarily close to curves with perfectly elastic segments. Moreover, the system operator’s
objective function is continuous in offers. Thus, we can use the same argument as above with the
difference that the system operator may sometimes have multiple optimal dispatches, in addition


to the dispatch above, for a given set of offer curves oˆi* q  i1 .14 However, the same dispatch as
n
above is pinned down by the additional conditions that the dispatch is locally efficient and
marginal prices locally competitive.


Finally, we realize that there could be cases with non-monotonic offers oˆi* q  i1 . However,
n
the dispatch is locally efficient and marginal prices locally competitive, so such offers would
have to satisfy the following properties oˆi* q   oˆi* qi*  for q  qi* and oˆi* q   oˆi* qi*  for q  qi*
i  1,, n . Thus as the system operator sorts offers into ascending order, we can go through
the arguments above for sorted offers and conclude that the statement must hold for such cases as
well. ■
Lemma 5. If two sets of nodal offer functions both result in a locally efficient dispatch with
locally competitive marginal prices, then the two resulting dispatches must be identical.
Proof: Make the contradictory assumption that there are two pairs of offer functions with a
corresponding dispatch,
properties, except that
oˆ q
n
*
i
i 1
* n
i i 1
 qi
q 
 
, qi*
 
n
i 1
n
i 1

and
oˆ q

i
n
i 1
 
, qi
n
i 1
,
that satisfy the stated
. Lemma 4 states how these offers can be adjusted into
strictly increasing offer curves without changing the dispatch. We make such adjustments to get
14
Multiple optimal dispatches for example occur if several units in a node have the same stated marginal cost and some but not all of these units
are accepted in a dispatch that minimizes stated production costs. 27




two sets of adjusted nodal offer functions, oˆ i* q  i 1 and oˆ i q  i 1 that are identical in nodes with
n
n
the same dispatch and non-crossing in the other nodes.
By assumption we have oˆ i* q i*   C i q i*  and oˆ i q i   C i q i . The node’s marginal cost


curve is strictly increasing in output. Thus adjusted offers oˆ i* q  i 1 must be above (more

n

expensive) compared to adjusted offers oˆ i q  i 1 in all nodes where q i*  q i . Similarly, adjusted
n
offers oˆ i* q i 1 must be below (cheaper) compared to adjusted offers oˆ i q i 1 in all nodes
n
n
where q i*  q i . However, this would violate Lemma 1. Thus, the dispatches q i* i 1 and q i i 1
n
n
must be identical. ■
APPENDIX B: OTHER PROOFS
Proof of Lemma 1
We let the old dispatch refer to the feasible dispatch qiold i 1 that maximized stated social welfare
n
at old offers when supply in node i is given by oi qi  . Let oi qi  denote the shift of the supply
curve, so that oi qi   oi qi  is the new supply curve in node i. The new dispatch refers to the
q 
new n
i
i 1
feasible dispatch
that maximizes stated social welfare for new offers. Thus for new
offers, oi qi   oi qi  , the new dispatch qinew i 1 should result in a weakly higher social welfare
n
 
than the old dispatch qiold
n
i 1
, i.e.
new
n qi

i 1
old
n qi
 o x   o x dx    o x   o x dx.
i
0
i
i 1
i
i
(3)
0
Now, make the contradictory assumption that in comparison to the old dispatch, the new dispatch
has strictly more production in all nodes where offers have been shifted upwards (more
expensive) and strictly less production in all nodes where offers have been shifted downwards
(cheaper). Thus qinew > q iold when oi qi  ≥0 with strict inequality for some qi  0, qinew  , and qinew <
q iold when oi qi  ≤0 with strict inequality for some qi  0, qiold  , so that
28
new
n qi
old
n qi
  o x dx    o x dx .
i 1
i
0
i 1
(4)
i
0
But summing Equation (3) and Equation (4) yields
new
n qi

i 1
old
n qi
 o x dx    o x dx,
i
0
i 1
(5)
i
0
which is a contradiction since, by definition, the old dispatch qiold i 1 is supposed to maximize
n
stated welfare at old offers. ■
Proof of Lemma 2
The statement follows from that: 1) offers cannot be dispatched at a price below their marginal
cost in equilibrium, and that 2) all offers from production units with a marginal cost at or below
the marginal price of a node must be accepted in equilibrium. If 1) did not hold for some firm
then it would be a profitable deviation for the firm to increase its offer price to its marginal cost.
2) follows from that there would otherwise exist some infinitesimally small producer in the node
with a marginal cost below the marginal price, but whose offer is not dispatched. Thus, it would
be a profitable deviation for such a producer to slightly undercut the marginal price and we know
from Corollary 1 that such a deviation will not decrease the dispatched production in its node, so
the revised offer will be accepted. ■
Proof of Proposition 1
We note that the objective function (stated welfare) in Equation (1) is continuous in the nodal
output qi when offers are at the marginal cost. Moreover, the feasible set (the set of possible
dispatches) is closed, bounded (because of capacity constraints) and non-empty. Thus, it follows
from Weierstrass’ theorem that there always exists an optimal feasible dispatch when offers
reflect true costs (Gravelle and Rees, 1992).
Next, we note that no producer has a profitable deviation from the competitive outcome.
Marginal costs are continuous and strictly increasing. Hence, it follows from Corollary 1 that no
producer with an accepted offer can increase its offer price above the marginal price of the node
and still be accepted, as its offer price would then be above one of the previously rejected offers
29
in the same node.15 No producer with a rejected offer would gain by undercutting the marginal
price, as the changed offer would then be accepted at a price below its marginal cost. Thus, there
must exist an NE where all firms offer to produce at their marginal cost. Offers above and below
the marginal price of a node can differ between equilibria. But it follows from Lemma 2 and
Lemma 5 in Appendix A that all NE must have the same locally efficient dispatch and the same
locally competitive marginal prices, so nodal prices, which are set by marginal prices, must also
be the same. ■
Proof of Proposition 2
Proposition 1 ensures existence of the network’s efficient dispatch and competitive nodal prices.
Both nodal and discriminatory pricing are markets where an accepted offer is never paid more
than its node’s marginal price and never less than its own bid price, so in both cases the
equilibrium dispatch must be locally efficient and marginal prices of the nodes are competitive in
equilibrium, because of Lemma 2. Thus statement 1) follows from Lemma 5 in Appendix A. In
a discriminatory market it is profitable for a producer to increase the price of an accepted offer
until it reaches the marginal price of its node, which gives statement 2). Finally, we realise that
there are no profitable deviations from the stated equilibrium if rejected offers are at their
marginal cost.■
Proof of Proposition 3
We note that the stated production cost of contracted sales is a constant. Thus we can add it to the
objective function of the system operator’s optimization problem without influencing the optimal
dispatch. The set of feasible dispatches is not influenced by producers’ forward sales. Thus to
solve for the optimal dispatch we can add producers’ forward sales to their offered quantities, so
that offers include contracted quantities instead of being net of contracts, and then solve for the
feasible dispatch that minimizes the total stated production costs as defined by Equation (1).
Rewriting the dispatch problem in this way, implies that Lemma 1, Corollary 1, Lemma 4 and
Lemma 5 in Appendix A also apply to situations with contracts. Thus the stated result would
follow if we can prove that the dispatch must be locally efficient and marginal prices of the nodes
15
 
Also note that the last unit in a node cannot increase its offer above its marginal cost due to the reservation price p  C ' q i . 30
are competitive in equilibrium, also for contracts. Similar to the proof of Lemma 2, this follows
from that: 1) a production unit cannot be dispatched at a real-time price below its marginal cost in
equilibrium, and that 2) all production units with a marginal cost at or below the marginal realtime price of its node must be dispatched in equilibrium. The proof of Lemma 2 explains why 1)
must hold for uncontracted firms. If 1) would not hold for a contracted firm, then it would be a
profitable deviation for the firm to increase its offer price (to buy back the contract and avoid
being dispatched) to a price above the marginal real-time price and below its marginal cost. It
follows from Corollary 1 that such a unilateral deviation cannot increase the nodal production in
the contracted firm’s node. Thus its offer to buy back the contract is accepted at a price below its
marginal cost, which is cheaper than to follow the contracted obligation and produce at marginal
cost. 2) follows from that there would otherwise exist some infinitesimally small producer in the
node with a marginal cost below the marginal price, but whose offer is not dispatched. We
already know from the proof of Lemma 2 that such a producer would find a profitable deviation
if it was uncontracted. We also realize that a producer that has sold its production forward and
that has a marginal cost below the marginal price would lose from bidding above the marginal
price (to buy back the contract), so that its unit is not dispatched. It would be a profitable
deviation for such a producer to lower its bid to its marginal cost. It follows from Corollary 1 that
such a change would not decrease accepted production. Thus it increases its payoff by at least the
difference between its nodal marginal price and its marginal cost. ■
Proof of Proposition 4
Existence of a competitive equilibrium in the nodal design follows from Proposition 1.
Assumption 1 restricts inter-zonal flows to be efficient. However, we realize from the proof of
Proposition 3 that this extra constraint does not change the statement in Proposition 3. A
producer’s accepted offer in the zonal market is equivalent to a forward position with physical
delivery in its node. Thus it follows from Proposition 3 that, independent of the zonal clearing,
the equilibrium dispatch is identical to the network’s efficient dispatch and marginal prices of the
 
nodes are competitive in the counter-trading stage. This gives the unique dispatch qiN
n
i 1
as
stated in 2). The counter-trading stage uses discriminatory pricing, but all agents want to trade at
31
the best price possible, so all accepted offers in the counter-trading stage are marginal offers at
the network’s competitive nodal prices.
Consider a zone k with its associated nodes n  Z k or equivalently n  n k , , n k . A node
inside zone k where the network’s competitive nodal price piN is strictly below the zonal price
Πk is referred to as a strictly export constrained node. Price-taking producers in such nodes want
to sell as much production as they can at the zonal price, and then buy back production in the
discriminatory counter-trading stage at the lower price piN or produce at an even lower marginal
cost. Thus all capacity in a strictly export constrained node i is offered at or below piN < Πk. As
the real-time market is physical, producers in strictly import-constrained nodes of zone k (where
the network’s competitive nodal price piN is strictly above the zonal price Πk) are not allowed to
first buy power at a low price in the zonal market and then sell power at piN in the countertrading stage. Thus they neither buy nor sell any power in the zonal market, so they make offers
above Πk. We can conclude from the above reasoning that a marginal offer at the zonal price
cannot come from a production unit that is located in a node that is strictly export or import
constrained. In equilibrium there must be at least one marginal node m with p mN  Π k . Recall that
nodes have been sorted with respect to competitive nodal prices and that that the highest clearing
price is chosen in case there are multiple prices where zonal net-supply equals zonal net-exports.
Thus we can define one marginal node by Equation (2). 16 It follows from Lemma 3 that this
equation uniquely sets the zonal price Π k  p mNk  .
Offers in strictly import constrained nodes, which are above the zonal price, are never
accepted in the first stage of the zonal market. For these nodes, it is the rules of the countertrading stage that determine optimal offer strategies. Thus, the auction works as a discriminatory
auction, and we can use the same arguments as in Proposition 2 and Proposition 3 to prove the
second part of statement 3).
16
It is possible that nodes with numbers adjacent to m(k) have the same competitive nodal prices as node m(k), but it will not change the analysis.
It is enough to find one marginal node to determine the zonal price. As an example, it follows from Proposition 1 and our cost assumptions that in
the special case when zonal demand equals the zonal production capacity plus efficient imports, then the competitive nodal price equals the price
cap in all nodes. Thus any node could be chosen to be the marginal node, but n k is the most natural extension of the first part of Equation (2). 32
Production units in a strictly export-constrained node that have a higher marginal cost than
their competitive nodal price can sell their power in the zonal market at the zonal price and then
buy it back at a lower offer price in the counter-trade stage. Thus, to maximize profits this power
is offered at the lowest possible price, for which offers are not dispatched, i.e. at the marginal
price of the node. This gives the first part of statement 3). Non-dispatched
production
units
would not gain by undercutting the marginal price. Offers that are dispatched in strictly exportconstrained nodes are paid the zonal price. It is not possible for one of these units to increase its
offer price above piN < Πk and still be dispatched, as non-dispatched units in such nodes offer at
piN . Moreover, it is weakly cheaper for dispatched units to produce instead of buying back
power at piN . Thus, they do not have any profitable deviations. Accordingly, the stated offers
must constitute a Nash equilibrium. ■
Proof of Proposition 5
We solve the two-stage game by backward induction. Thus we start by analysing the
countertrading stage. A producer’s accepted offer in the zonal market is equivalent to a forward
position with physical delivery in its node. Thus it follows from Proposition 3 that, independent
of the zonal clearing, the equilibrium dispatch is identical to the network’s efficient dispatch and
marginal prices of the nodes are competitive in the counter-trading stage. The counter-trading
stage uses discriminatory pricing, but all agents want to trade at the best price possible, so all
accepted offers in the counter-trading stage are marginal offers at the network’s competitive nodal
prices.
We calculate a subgame perfect Nash equilibrium of the game, so rational agents realise
what the outcome of the second-stage is going to be, and make offers to the zonal market in order
to maximize profits. Thus, similar to the one-stage game, all production capacity in strictly
export-constrained nodes i  Z k , such that piN < Πk, is sold at the zonal price. As before,
production capacity in strictly import constrained nodes maximize their payoff by selling no
power in the zonal market; all production that is dispatched in strictly import constrained nodes is
accepted in the counter-trading stage. As in the one-stage game, the zonal price in zone k must be
set by the marginal price of some marginal node m as defined in Equation (2). Otherwise there
33
must be some offer to the zonal market from a production unit in a strictly export constrained
node (with piN < Πk ) that is rejected, and which would find it profitable to slightly undercut the
zonal price. All production units that are dispatched in marginal nodes are sold at the zonal price.
As in the one-stage game, there are always rejected offers from units in marginal nodes that can
be placed at or just above the zonal price. This rules out that profitable deviations for production
units in marginal nodes. Thus all agents get the same payoffs as the game in Proposition 4, where
the same offers were used in the zonal and countertrading stages. ■
34
×

Report this document