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Doctoral Thesis

Analysis of flow-based market coupling in oligopolistic powermarkets

Author(s): Kurzidem, Martin

Publication Date: 2010

Permanent Link: https://doi.org/10.3929/ethz-a-006097327

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Diss. ETH No. 19007

Analysis of Flow-based Market Coupling inOligopolistic Power Markets

A dissertation submitted to the

ETH Zurich

for the degree of

Doctor of Sciences ETH Zurich

presented by

Martin Johann Kurzidem

Dipl.-Ing. RWTH Aachen

born April 24, 1979

citizen of Germany

accepted on the recommendation of:

Prof. Dr. Goran Andersson, examiner

Prof. Dr. Julian Barquin, co-examiner

Dr. Martin Eschle, co-examiner

2010

Acknowledgments

I would like to express my gratitude to Prof. Dr. Goran Andersson for his excellent

supervision of my PhD thesis. Most of all I appreciated his understanding of

leadership, which gave me the flexibility and the freedom to pursue my research in

different directions and in personal responsibility.

Special thanks go to Dr. Martin Eschle and Prof. Dr. Julian Barquin for their

co-examination. Their remarks and comments improved the quality of the thesis

substantially.

Furthermore, I would like to thank my colleagues and friends at the Power System

Laboratory. Our discussions helped me to understand the complex interrelation of

different areas of power system economics.

Many thanks to my parents, Renate and Johann Kurzidem, for their continuous

support before and during the time as an assistant at ETH Zurich.

Finally and most of all, I would like to thank Ruth, the better part of me. Her

efforts and contribution to my PhD thesis cannot be expressed in one line.

Zurich, March 2010

Martin Kurzidem

iii

Abstract

The liberalisation of the power supply sector has led to a higher interdependency

of the European electricity markets as cross-border trades increased significantly.

Different levels of electricity prices arise due to frequently congested transmission

capacity between the price areas. This limited transmission capacity is allocated to

the users by the application of various congestion management methods whereby

the explicit auction of the Net Transfer Capacity (NTC) is most widely spread. A

more sophisticated congestion management approach, term flow-based allocation,

has been proposed which overcomes the drawbacks of bilateral NTC allocation by

introducing a mechanism for efficiently allocating transmission capacity involving

multiple borders.

However, the power supply sector is predominantly oligopolistic, thus, it is often

exposed to the power producers’ exercise of their market power as electricity and

transmission prices frequently do not reflect the true cost levels. In this thesis, a

case study focuses on the comparison of the NTC-based with the flow-based con-

gestion management method based on imperfectly competitive electricity markets

in the central European region.

Another topic studied is the modelling of strategic behaviour by generating com-

panies by assuming the companies to affect electricity and transmission prices by

conjectured response parameters. Those parameters allow a flexible modelling of

electricity and transmission market competition, ranging from perfectly competi-

tive to Cournot competitive price levels. Furthermore, a closed-form formulation

of the conjectured response parameters for electricity and transmission markets

based on multiple price areas is derived. An estimation procedure is presented

based on an implicit parameter fitting by using historical market data. This leads

to a more accurate price generation process, i.e. it allows the modelling of power

markets in the vicinity of historical electricity prices. Based on that approach,

the comparison between the NTC-based and flow-based congestion management

method is performed.

v

Kurzfassung

Seit Anfang der 90er Jahre wird die monopolistische und teils staatsregulierte

Marktstruktur der europaischen Elektrizitatsmarkte fortgehend durch wesentliche

Grundzuge der Liberalisierung ersetzt. Zu diesen zahlen unter anderem die Re-

duzierung politischer Regulierungen und Eingriffe in das Marktgeschehen, der nicht

diskriminierende Zugang Dritter zum Stromnetz und somit die Einfuhrung von

Wettbewerb unter den Stromproduzenten, sowie die Offnung der Markte fur neue

Stromanbieter. Die mit der Strommarktliberalisierung in vielen Landern einherge-

hende Grundung von Stromhandelsplattformen (Stromborsen) ermoglicht einen

geregelten Handel standardisierter Stromprodukte mit dem Ziel marktgerechte,

wettbewerbsfahige Strompreise auf Basis der Stromerzeugungsgrenzkosten zu erzie-

len.

Die Europaische Union legt mit ihrer Zielsetzung die nationalen Strommarkte

der Mitgliedstaaten zu einem gemeinsamen, internen Strombinnenmarkt zusam-

menzufuhren den Grundstein fur einen intensiveren Austausch und Handel von

elektrischer Energie fest. Damit sollen Handelsliquiditat und Wettbewerb bei

effi-zienterer Nutzung der elektrischen Infrastruktur (UCTE Ubertragungsnetz)

und bei stetiger Gewahrleistung der Versorgungssicherheit weiter gefordert wer-

den. Resultierend aus der wirtschaftlichen Kopplung der Markte und unter opti-

maler Ausnutzung der grenzuberschreitenden, physikalischen Transferkapazitaten

im Ubertragungsnetz ware somit die Moglichkeit fur ein einheitliches Preisniveau

der nationalen Strommarkte geschaffen. Jedoch fuhrt die grosstenteils unzurei-

chende Ubertragungskapazitat zwischen zwei benachbarten Preiszonen aufgrund

physikalischer Netzengpasse zur Einschrankung bishin zur Verhinderung weiterer

grenzuberschreitender Handelsaktivitaten, wodurch eine vollstandige okonomische

Kopplung der Strommarkte und ein einheitliches Preisniveau nicht erreicht wer-

den konnen. Andererseits sind die teils betrachtlich voneinander abweichenden

Strompreisniveaus an den Europaischen Stromborsen auf die massgeblich oligopolis-

tische Struktur der nationalen Stromerzeugungsmarkte zuruckzufuhren. Deren

Anfalligkeit fur die Ausubung von Marktmacht ist besonders ausgepragt.

vii

Ein von der ENTSO-E vorgesehenes Engpassmanagementkonzept sieht vor maxi-

male Ubertragungskapazitaten (Net Transfer Capacities, NTCs) beim grenzuber-

schreitenden Stromhandel zu definieren, die nach marktbasierten Prinzipien an

die Teilnehmer auktioniert werden (NTC Auktionsverfahren). Ein entscheiden-

der Nachteil des NTC Verfahrens liegt in der beim Allokationsprozess entkoppel-

ten Betrachtungsweise von kommerziellen Transaktionen und den daraus resul-

tierenden physikalischen Lastflussen auf den Grenzleitungen. Im Hinblick auf neue

und effizientere Engpassmanagementsysteme entwickelt sich die Methode der last-

flussbasierten Kapazitatsallokation (Flow-based Allocation, FBA) als wahrschein-

licher zukunftiger Standard heraus, in der die physikalischen Lastflusse zwischen

mehreren Landern zur Optimierung verwendet werden. Durch die zusatzlichen

Freiheitsgrade des FBA Verfahrens konnen zusatzliche Handelspotentiale als Folge

einer effizienteren Nutzung der physikalischen Grenzkapazitaten entstehen.

Ein wesentlicher Beitrag dieser Dissertation liegt in der Herleitung und Formulier-

ung eines Modells zur Untersuchung des strategischen Marktverhaltens in Strom-

markten und in Markten fur grenzuberschreitende Transferkapazitaten basierend

auf dem FBA Verfahren. Die Einfuhrung des als ”conjectural variation” bekan-

nten Konzepts erlaubt anhand der Spezifikation modelleigener Parameter die Simu-

lation unterschiedlicher Wettbewerbsgrade in Strom- und Grenzkapazitatsmarkten.

Verglichen mit den haufig angewandten spieltheoretischen Ansatzen des perfekten

Wettbewerbs und des Cournot Wettbewerbs, ermoglicht eine Abschatzung dieser

Parameter anhand von historischen Marktdaten genauere Strompreisanalysen.

Ziel dieser Dissertation ist eine Vergleichsanalyse der NTC- und der FBA-basierten

Engpassmanagementmethode hinsichtlich ihrer Auswirkung auf Strommarktpreise,

auf das grenzuberschreitende Stromhandelsvolumen und auf die fur den Erwerb von

Ubertragungskapazitaten auftretenden Kosten.

Contents

Acknowledgments iii

Abstract v

Kurzfassung vii

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Publications and Conference Papers . . . . . . . . . . . . . . . . . . 5

2 Aspects of Power System Design 7

2.1 Complementarity-Based Equilibrium Models . . . . . . . . . . . . . 7

2.1.1 Definition of Complementarity . . . . . . . . . . . . . . . . . 8

2.1.2 Derivation of Complementarity Conditions . . . . . . . . . . 9

2.1.3 Modelling Software . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 The Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Network Representation . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Standard Market Clearing Mechanisms . . . . . . . . . . . . . . . . 11

ix

2.4.1 Bilateral Markets . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 Pool-Based Markets . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Time Horizon of Market Analysis . . . . . . . . . . . . . . . . . . . 13

3 Flow-based Allocation of Cross-Border Capacities 15

3.1 Congestion Management . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 General Principles of Flow-based Allocation . . . . . . . . . . . . . 16

3.2.1 The Simplified Transmission System Model . . . . . . . . . . 18

3.2.2 Modelling Comparison: Critical Branches vs. Flowgates . . . 19

3.3 Power Transfer Distribution Factors . . . . . . . . . . . . . . . . . . 20

3.3.1 Definition of PTDFs . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Computation of PTDFs . . . . . . . . . . . . . . . . . . . . 22

3.4 Flowgate Capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.1 Transfer Capacity Definitions . . . . . . . . . . . . . . . . . 30

3.4.2 Determination of the Total Flowgate Capacity . . . . . . . . 31

3.4.3 Determination of the Net Flowgate Capacity . . . . . . . . . 37

3.4.4 Contingency Analysis and Capacity Harmonisation . . . . . 39

3.4.5 Case Study - Capacity Calculation . . . . . . . . . . . . . . 40

3.5 Time Dependencies of PTDFs and Flowgate Capacities . . . . . . . 41

3.5.1 Physical Capacity Definitions . . . . . . . . . . . . . . . . . 42

3.5.2 Planning Phase . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5.3 Allocation Phase . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Concepts of Flow-based Auctions . . . . . . . . . . . . . . . . . . . 44

3.7 Concepts of Congestion Management: NTC vs. FBA . . . . . . . . 45

3.7.1 NTC-based Auctions . . . . . . . . . . . . . . . . . . . . . . 46

3.7.2 FBA Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.8 Equilibrium Model of a Flow-based Auction . . . . . . . . . . . . . 49

3.9 Case Study - Explicit Flow-based Auction . . . . . . . . . . . . . . 51

3.9.1 Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.9.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Modelling Strategic Generator Behaviour 59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Classical Concepts from Game-Theory . . . . . . . . . . . . . . . . 61

4.2.1 Perfect Competition . . . . . . . . . . . . . . . . . . . . . . 63

4.2.2 Cournot Competition . . . . . . . . . . . . . . . . . . . . . . 64

4.3 The Concept of Conjectural Variation . . . . . . . . . . . . . . . . . 65

4.3.1 Conjectural Variations in the Electricity Market . . . . . . . 67

4.3.2 Conjectural Variations in the Transmission Market . . . . . 68

4.4 Setting up the Equilibrium Model . . . . . . . . . . . . . . . . . . . 69

4.5 Case Study - Simple System . . . . . . . . . . . . . . . . . . . . . . 72

4.5.1 Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5.2 Configuration of Conjectural Variations Parameters . . . . . 74

4.5.3 Case Study I: Perfect Competition . . . . . . . . . . . . . . 76

4.5.4 Case Study II: Imperfect Competition . . . . . . . . . . . . 78


5 Conjectural Variation Based Price Forecasting 93

5.1 Estimation of Conjectural Variation Parameters . . . . . . . . . . . 93

5.1.1 Explicit Fitting Procedure . . . . . . . . . . . . . . . . . . . 95

5.1.2 Implicit Fitting Procedure . . . . . . . . . . . . . . . . . . . 97

5.2 Case Study - Central European Region . . . . . . . . . . . . . . . . 101

5.2.1 Modelling Assumptions . . . . . . . . . . . . . . . . . . . . . 102

5.2.2 Availability of Market Data . . . . . . . . . . . . . . . . . . 105

5.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 107


6 Conclusions and Outlook 113

A Aspects of Power System Design 117

A.1 Elasticity of Electricity Demand . . . . . . . . . . . . . . . . . . . . 117

A.2 Elasticity of Transmission Capacity Demand . . . . . . . . . . . . . 117

A.3 Definition of Social Welfare . . . . . . . . . . . . . . . . . . . . . . 118

A.4 DC Power Flow Approximation . . . . . . . . . . . . . . . . . . . . 121

B Flow Based Allocation of Cross-Border Capacities 123

B.1 Complementarity Conditions of Flow-based Auction . . . . . . . . . 123

B.2 Case Study: AC vs. DC PTDF . . . . . . . . . . . . . . . . . . . . 123

B.3 Derivation of the Net Flowgate Capacity . . . . . . . . . . . . . . . 124

C Modelling Strategic Generator Behaviour 127

C.1 Classical Concepts from Game-Theory . . . . . . . . . . . . . . . . 127

C.1.1 Cournot Competition . . . . . . . . . . . . . . . . . . . . . . 127

C.2 The Concept of Conjectural Variation . . . . . . . . . . . . . . . . . 128

C.2.1 CV in the Electricity Market . . . . . . . . . . . . . . . . . . 128

C.2.2 CV in the Transmission Market . . . . . . . . . . . . . . . . 128

C.3 Setting up the Equilibrium Model . . . . . . . . . . . . . . . . . . . 128

List of Abbreviations 131

List of Symbols 133

Bibliography 137

Curriculum Vitae 143

1 Introduction

1.1 Background and Motivation

The introduction of liberalisation in many European electricity markets has led

to a fundamental reformation aiming at the implementation of competition in the

power supply industry in order to increase efficiency in the industry and to intro-

duce market-based electricity prices. In several European countries, this has given

rise to the establishment of organised market places, known as the Power Exchanges

(PX), where electricity can be traded by means of different structured products

and within different time scales ranging from several years to a few hours prior to

physical delivery. Another key issue of electricity market liberalisation is the un-

bundling of transmission and generation in order to provide a non-discriminatory

third-party access to the transmission networks. By separating transmission and

generation, discrimination in the competitive power supply sector can be reduced.

However, the most efficient approach is to eliminate discrimination by an ownership

unbundling or divestiture. Focussing on the demand side, another major element

of liberalisation is the free choice of consumers for their supplier.

Inefficiencies of the old regulatory framework such as costly planning errors have

led to excess generating capacity in many countries. Thus, one of the goals with

the introduction of competition in the power supply sector is to put pressure on

the profit margins of power suppliers in order to minimise the cost of electric-

ity supply and to ensure that electricity prices are truly cost-based. However, a

major concern with the introduction of competition is the risk of strategically be-

having companies due to the predominantly oligopolistic structure of the power

supply market, i.e. power supply is shared by only a few generating companies.

Common features of liberalised electricity markets are a low short-term demand

responsiveness to wholesale electricity prices and the separation of power markets

1

2 1 Introduction

by congested transmission lines. Based on that power suppliers are able to use their

market power by restricting their generation output (production inefficiencies) and

adjusting their sales characteristic in order to hold or further raise electricity prices

above competitive levels (price distortions). This leads to a redistribution of wel-

fare among producers, consumers and the Transmission System Operators (TSOs).

Furthermore, the opening of national electricity markets allows access for interna-

tional power suppliers and arbitragers, thus, resulting in an increase of cross-border

trading activities. However, as seen throughout northwestern Europe, cross-border

capacity is limited due to the congested transmission infrastructure. In an effort

to promote the international exchange of electricity, market-based congestion ma-

nagement methods are frequently implemented for the allocation of limited cross-

border transmission capacity. Similar to electricity markets, strategically behaving

companies play a major role in the price generation process of the transmission

market. Depending on the location of those companies’ generating units, strategic

companies are likely to manipulate the congestion price to their favour in order to

increase their benefit margin.

In order to help analyse market designs and regulatory policies, the implementation

of reliable models of liberalised electricity markets is particulary important. Sev-

eral aspects such as the choice of the market design, the congestion management

scheme, the generating companies’ bidding behaviour and the network representa-

tion may have a major impact on the resulting market outcomes and thus need to

be modelled adequately. In the literature a wide range of oligopolistic power mar-

ket models exists aiming at studying strategic generator behaviour. Different game

theoretical approaches are being used to provide strategic generators with the es-

sential knowledge and information to model the existing imperfect competition in

order to achieve reasonably market prices. However, most frequently those models

are focussed only on imperfectly competitive electricity markets as the power sup-

ply market is particulary exposed to the exercise of market power by large supply

companies. Additionally, cross-border trades gain increasing importance due to

the opening of national electricity markets, thus, models of imperfect competition

need to account for both, the electricity and the transmission markets.

The introduction of liberalisation in the European electricity market sector and

the opening of national power markets for international market participants have

also led to a substantial increase of cross-border electricity exchange. Several con-

gestion management approaches can be applied in order to guarantee an efficient

allocation of scarce transmission capacity. An overview of the currently applied

European cross-border congestion management methods is given in [1]. In general,

1.1 Background and Motivation 3

two designs for the allocation of limited transmission capacity are used worldwide

differing from each other by means of the sequence of energy and transmission

markets. In the separated transmission and energy markets design, transmission

capacity is first allocated in an auction, and then local energy spot markets are

cleared. In contrast, the integrated market design implies that transmission capac-

ity is allocated in accordance with the energy bids of strategic generators. Both

designs are also known as explicit and implicit auctioning mechanisms of cross-

border transmission capacity.

In the highly meshed European transmission network congestion is mostly man-

aged through explicit auctions of cross-border transmission capacity [2]. The value

of that capacity, also known as the Net Transfer Capacity (NTC), is given as an

exchange capacity rather in terms of a physical capacity between two neighbouring

countries. Although each cross-border trade induces a power flow in the trans-

mission network, a physical allocation does not take place during the capacity

allocation phase. Additionally, as a result of the highly meshed European trans-

mission network, any cross-border trades between third-party countries have an

effect on the power flow on all cross-border tie lines. However, NTC values are

of bilateral nature, which do not account for the physical impact resulting from

the capacity allocation by other NTC auctions. Therefore, those TSOs involved

in the NTC calculation need to include an increasing security margin in order to

prevent any network security problems in their control area. However, this leads

to a reduced capacity to be used for inter-regional commercial activities.

The European Network For Transmission System Operators for Electricity (ENTSO-

E) has launched a proposal for an innovative market-based congestion management

mechanism, introduced as flow-based allocation (FBA) to be applied across Euro-

pean borders. Its major innovation is its multinational approach. By the establish-

ment of a centralised auction office, physical cross-border capacity is allocated in a

coordinated way within a specified region involving several countries and multiple

borders. Furthermore, the FBA mechanism transfer all inter-regional commercial

transfer into the resulting physical power flow on all interconnectors in that region.

By that method, the physical infrastructure is to be used more efficiently while fa-

cilitating cross-border trades at the same time. Furthermore, the FBA mechanism

supports the integration of regional power markets towards the goal of a single

European electricity market. The FBA method is currently in a dry-run in the

South-East European and Central-West European regions [3], [4].

The motivation for this thesis is twofold: In a first step, the strategic market

behaviour of supply companies should be analysed in electricity and transmission

4 1 Introduction

markets based on multiple areas. In the second step, the concept of imperfect

competition is to be applied to the central European region in order to study the

benefits from the FBA mechanism compared to the NTC-based approach.

1.2 Contributions

In terms of strategic generator behaviour, the major contributions of this thesis are

based on the flow-based congestion management. They are summarised as follows:

• Introduction of the concept of conjectural variations simultaneously in elec-

tricity and transmission markets

• Study of strategic behaviour of generating companies in the electricity and

transmission market

• Introduction of the explicit and the implicit fitting procedure for conjectural

variation parameters according to the electricity and transmission market

based on multiple control areas / price zones and historical market data

As for the flow-based allocation of transmission capacity, the major contributions

are:

• Provision of a detailed documentation of power transfer distribution factors

and net flowgate capacity calculation

• Implementation of the explicit flow-based coordinated auction

• Quantitative comparison of the FBA and the NTC-based mechanism based on

a real-sized system in an imperfectly competitive market environment. Con-

jectural variation parameters for electricity markets are estimated by means

of the implicit fitting procedure

1.3 Thesis Outline 5

1.3 Thesis Outline

The thesis structure and content is as follows:

Chapter 2

A complementarity-based formulation of power markets as equilibrium models and

basis aspects of power market modelling are reviewed.

Chapter 3

The flow-based allocation mechanism of transmission capacity is introduced in-

cluding a detailed documentation of power transfer distribution factors and net

flowgate capacity calculation. Then, the implementation of the explicit flow-based

coordinated auction is presented.

Chapter 4

A complementarity-based model of strategic behaviour of generating companies in

multiple electricity and transmission markets is formulated. The issue of strategic

market behaviour is addressed by means of the conjectural variations approach.

Chapter 5

A formulation of the explicit and implicit fitting procedure of conjectural variation

parameters for electricity and transmission markets is given. Then, the NTC-based

and the FBA congestion management method are compared to each other based on

the estimation of conjectural variation parameters by means of historical market

data. The geographical scope of this study is limited to the central European

Region.

Chapter 6

A summary of qualitative and quantitative benefits from the flow-based congestion

management method and an outlook for future work are given.

1.4 Publications and Conference Papers

The work has been presented in the following papers and documents:

1. M.Kurzidem, T.Krause, E.Beck:

Strategic Bidding and Congestion Management in Electricity Markets

Latsis Symposium 2006, Research Frontiers in Energy Science and Technol-

ogy, Zurich

6 1 Introduction

2. M.Kurzidem, G.Andersson:

A Study of the Transmission Price Conjecture in an Oligopolistic Power Mar-

ket

IEEE Power Tech Conference, Lausanne, 1st − 5th July, 2007.


An Application of the Energy and Transmission Price Conjecture in an Oligopolis-

tic Power Market

16th Power Systems Computation Conference (PSCC), Glasgow, 14th − 18th

July, 2008.

4. C.Duthaler, M.Kurzidem, M.Emery, G.Andersson:

Analysis of the Use of PTDF in the UCTE Transmission Grid

16th Power Systems Computation Conference (PSCC), Glasgow, 14th − 18th

July, 2008.


Comparison of Different Congestion Management Methods

Advisory Report based on co-operation with Bundesamt fur Energie (BFE),

2009.

6. M.Kurzidem, G.Andersson, Axpo Group:

An Application of PTDF to Determine Commercially Used Cross-Border

Transfer Capacities Resulting From Physical Bottlenecks in Electrical Net-

works

Project with Nordostschweizerische Kraftwerke (NOK), 2008/2009.

2 Aspects of Power System Design

This chapter addresses some basic aspects of power market modelling. A mathemat-

ical formulation of complementarity-based equilibrium models is provided followed

by the introduction of the Nash equilibrium as a market equilibrium under imperfect

competition. Subsequent to that, a widely used approximation of the transmission

network is described based on that all case studies in this work are carried out.

Finally, the two commonly applied market clearing mechanisms are presented fol-

lowed by a general discussion on the validity of the market analysis with respect to

the time horizon.

2.1 Complementarity-Based Equilibrium Models

The formulation of a market equilibrium by means of complementarity conditions

allows the modelling of several market participants’ profit maximisation problems,

e.g. that of generating companies, arbitragers and Transmission System Operators

(TSOs), by deriving their first-order optimality conditions, known as the Karush-

Kuhn Tucker (KKT) conditions. By adding market clearing conditions to the set

of KKT conditions, the market participants’ profit maximisation problems are cor-

related, which is necessary to find a market equilibrium. In the most general form,

such model formulation is termed a mixed complementarity problem [5] - [9].

Basically, the use of KKT conditions to find a unique market equilibrium implies

a convex feasible region of each market participant’s optimisation problem. In

case the optimisation problem is nonconvex, there is no guarantee that an optimal

solution can be found by deriving the KKT conditions. Thus, a unique market

equilibrium does not exist in general. By way of example, if strategic generating

companies include the TSO’s optimality conditions into their set of constraints

in order to correctly anticipate their impact on congestion prices, the feasible re-

7

8 2 Aspects of Power System Design

gion of their optimisation problem becomes nonconvex [10]. Likewise applies for a

pool-based market design, where strategic companies include the TSO’s optimality

conditions for the determination of Locational Marginal Prices (LMP) [7].

2.1.1 Definition of Complementarity

In the following, the complementarity problem is derived according to [9]. Given a

non-negative variable χi of a vector of variables χ = {χi} and a non-positive valued

function ξi(χ) of a set of functions ξ(χ) = {ξi(χ)}. Then, a complementarity

condition is defined as

χi ≥ 0; ξi(χ) ≤ 0; χi · ξi(χ) = 0 (2.1)

which can also be formulated more compactly as

0 ≤ χi ⊥ ξi(χ) ≤ 0. (2.2)

Since complementarity conditions can have inequalities in either direction (e.g.

ξi(χ) ≥ 0), the complementarity problem is stated as:

Find vector χ that satisfies the conditions

0 ≤ χ ⊥ ξ(χ) ≤ 0 (2.3)

where 0 ≤ χ ⊥ ξ(χ) ≤ 0 is the compact notation of χ ≥ 0; ξ(χ) ≤ 0; χ·ξ(χ) = 0.

There should be as many individual conditions ξ as variables χ. If all functions

ξi(χ) are affine, then, the complementarity problem is named a Linear Comple-

mentarity Problem (LCP).

Assuming there is another vector of variables called ψ = {ψj}, and ζ(χ,ψ) =

{ζj(χ,ψ)} is a set of vector-valued functions of dimension ψ. Furthermore, each

individual function ξi is to be a function of χ and ψ. Then, the Mixed Comple-

mentarity Problem (MCP) is defined as:

Find vectors χ, ψ that satisfy the conditions

0 ≤ χ ⊥ ξ(χ,ψ) ≤ 0 and ζ(χ,ψ) = 0. (2.4)

If ξ(χ,ψ) and ζ(χ,ψ) are affine, then this is a Mixed Linear Complementarity

Problem (MLCP).

2.1 Complementarity-Based Equilibrium Models 9

2.1.2 Derivation of Complementarity Conditions

There are two constrained optimisation problems formulated as

max Π1(χ,φ) (2.5)

s.t.: ξ(χ,φ) ≤ 0

χ ≥ 0

and

max Π2(ψ,φ) (2.6)

s.t.: ζ(ψ,φ) ≤ 0

ψ ≥ 0

where χ and ψ are vectors of decision variables, φ is supposed to be a free variable,

Π1(χ, φ) and Π2(ψ, φ) are the objective functions to be maximised, ξ(χ, φ) ≤ 0

and ζ(χ, φ) ≤ 0 are the sets of constraints. Π1, Π2, ξ and ζ are assumed to

be smooth functions, while Π1 and Π2 being scalar and concave, ξ and ζ being

vector-valued and convex. Furthermore, both optimisation problems are coupled

by free variable φ which is determined by the following market clearing condition

σ(χ,ψ,φ) = 0 (2.7)

where σ is assumed to be a smooth and convex function. By these assumptions,

the feasible region of both optimisation problems is convex. Thus, any local opti-

mum is a global optimum.

Assuming λ and ν to be the vectors of LaGrange multipliers for the constraints

ξ(χ, φ) ≤ 0 and ζ(χ, φ) ≤ 0, then, the set of KKT conditions for both optimisa-

tion problems and the market clearing condition are:

0 ≤ χi ⊥ ∂Π1

∂χi−

∑

m

λm∂ξm∂χi

≤ 0 (∀i) (2.8)

0 ≤ λm ⊥ ξm ≤ 0 (∀m) (2.9)

0 ≤ ψj ⊥ ∂Π2

∂ψj−

∑

n

νn∂ζn∂ψj

≤ 0 (∀j) (2.10)

0 ≤ νn ⊥ ζn ≤ 0 (∀n) (2.11)

φk ⊥ σk = 0 (∀k) (2.12)

According to Definition (2.4), the set of KKT conditions (2.8) - (2.12) is a MCP.

Each element χi and ψj of decision variables χ and ψ and each element λm and νn

of dual variables λ and ν is associated with one inequality constraint in the set of

KKT conditions. By including the market clearing condition (2.12), the number of

conditions (equations) and variables (unknowns) is equal by which the necessary

condition for optimality in order to find a market equilibrium are satisfied.


2.1.3 Modelling Software

Complementarity-based market equilibrium model can be directly solved by us-

ing the PATH solver within GAMS [11]. It allows the application of equilibrium

models to large power system with thousands of generating unit and hundreds of

constrained interfaces such as the PJM Interconnection [12].

2.2 The Nash Equilibrium

A widely spread modelling approach of strategically behaving generating compa-

nies in oligopolistic power markets is based on defining a market equilibrium as

a set of electricity and transmission prices, generation and demand levels, etc. It

has the property, that each market participant’s first-order conditions for profit

maximisation are fulfilled while accounting for the market clearing conditions at

the same time. Thus, no market participant will change its decision unilaterally.

When competition is not perfect some companies might be able to unilaterally

manipulate electricity prices through their actions in order to raise their profits.

However, in order to optimise its profits it must consider the other companies’

reactions to its actions. Let us assume the total profit Πf of company f is defined

as

Πf = π ·Gf − Cf (Gf ) (2.13)

where π denotes the market price, Gf is company f ’s total generation output and

Cf (Gf) the total generation cost for producing Gf amount of power. Because the

market price depends on the output decisions of all companies, each market partic-

ipant can only optimise its profits when considering the actions of its competitors.

Hence

π = π(Gf +G−f ) (2.14)

where the total generation output of all other companies than f is denoted by G−f .

Thus, company f ’s profit is a function of its own actions Xf and the actions X−f

of its competitors:

Πf = Πf (Xf , X−f ). (2.15)

The exchange of information between strategic companies is rather unrealistic,

however, it is reasonable to assume that all companies behave in a rational manner,

2.3 Network Representation 11

i.e. they are trying to maximise their profits. Therefore, each company f has to

adapt its actions Xf such that

Πf (Xoptf , Xopt

−f ) ≥ Πf (Xf , Xopt−f ) (2.16)

where Xoptf represents company f ’s optimal actions and Xopt

−f those of f ’s competi-

tors. Such interacting optimisation problems form what is called in game theory

a noncooperative game. The solution of such a game, if it exists, is called a Nash

equilibrium and represents a market equilibrium under imperfect competition.

2.3 Network Representation

The network representation is of particular importance. In several equilibrium

models of imperfect competition the DC load flow approximation is used in order to

account for physical realities of the transmission network (see also Appendix A.4).

Because of its linear properties, no further computational burdens are imposed

on the tractability of such models. Moreover, if the entire equilibrium models is

convex, large systems with as many as thousands of power plants and transmission

lines can be modelled [5].

2.4 Standard Market Clearing Mechanisms

Different types of power markets can be classified by means of the market clearing

mechanism, whose outcome are the electricity price, the quantity of supply and

demand and the selection of power producers. In general, electricity can either be

sold in a bilateral way or via a central power auction [7].

2.4.1 Bilateral Markets

In the OTC (Over-The-Counter) or bilateral market, customers are directly con-

tracted by power suppliers as price and quantity for a well-defined amount of

power are determined bilaterally between both parties. Customers are frequently

modelled by a single linear decreasing demand curve [7], [13] which implies the

consumers mid- to long-term market behaviour with respect to electricity price

variations. Competition only exists on the supply side, as producers try to max-

imise their profits by optimising their generation and sales output while satisfying


the price-responsive electricity demand. When accounting for the transmission

network, consumers and producers might be located at different nodes. Then, pro-

ducers choose their sales at each node, followed by a request for transmission service

from the owner of the transmission network (the TSO). Any transaction between

a producer and a consumer, when both are located at different nodes, is charged

a fixed transmission fee and/or a market-based transmission price, depending on

the applied (intra zonal) congestion management scheme. If transmission conges-

tion is based on a market-based pricing method, the clearing mechanism yields to

location-specific electricity prices.

In this context, power suppliers are able to act strategically in the electricity mar-

ket by adjusting their sales output. More theoretically, the bilateral market design

allows for the modelling of competition in the transmission market provided that

a market-based congestion management method is implemented. However, since

electricity and transmission prices are nodal-based, the liquidity and competition

at each individual node is rather low. Based on that, producers are frequently

modelled as prices takers with respect to transmission as the transmission market

is assumed to be perfectly competitive in several works.

In the absence of arbitragers, differences in the electricity prices between nodes

can diverge from the transmission price to be paid for a transaction between those

nodes. This effect occurs when electricity markets are not perfectly competitive

and is frequently referred as spatial price discrimination [14]. By including arbi-

tragers, which are modelled as price takers, non-cost price differences over space

are eliminated. Arbitragers recognise opportunities arising from electricity price

differences by expanding and redistributing their sales until each transmission price

reflects the difference of corresponding electricity prices.

2.4.2 Pool-Based Markets

In pool-based power auctions electricity is sold to a single central entity, known as

the Power Exchange (PX), which determines quantity and price within a prede-

fined area by matching aggregated supply and demand bids. In most cases, demand

side bidding is replaced by a single strictly linear decreasing demand curve, rep-

resenting the consumers’ willingness to buy electricity. Power suppliers do not

directly contract consumers but rather sell their generation anonymously to the

PX. If the transmission network is considered, the PX takes over the grid owner’s

responsibility for managing transmission congestion by choosing location-specific

demand and supply bids to maximise its net profits. Electricity prices are deter-

2.5 Time Horizon of Market Analysis 13

mined on a nodal base and differ from each other in case transmission congestion

occurs. Thus, power producers maximise their net profits based on the electricity

price at their location as only their generation output and not the individual sales

to consumers matters. Moreover, the representation of the network by means of

Power Transfer Distribution Factors (PTDF) allows the formulation of congestion-

based transmission prices for each single node with respect to an arbitrary hub

node. In pool-based power markets, the transmission price between each pair of

nodes is equal to the difference of electricity prices at those nodes because generat-

ing companies recognise price arbitrage in their set of constraints. The scheme by

which electricity prices are determined is also known in the literature as Locational

Marginal Pricing (LMP). This market concept allows generating companies to act

strategically by holding back their generating output in order to raise the electric-

ity price at its node. A market for transmission capacity does not exist, however,

transmission prices are implicitly influenced by the generating companies’ strate-

gies to manipulate electricity prices.

2.5 Time Horizon of Market Analysis

An important modelling issue of liberalised power markets concerns the time hori-

zon of their validity. Since market participants face different strategic decisions in

the short-, medium- and long-term, power market models usually differ from each

other by means of implementation and design in order to make credible statements

within the desired time frame.

In long-term market analysis, generating companies’ decisions comprise the time

frame of several years. Their decisions usually deal with the planning and construc-

tion of new power plants. Furthermore, maximum production capacity is fixed by

implicitly accounting for seasonal hydro operation and fuel purchases or energy

sales under long-term contracts [9]. In order to allow for future changes in the

electricity demand, demand is assumed to be price responsive. In most case, the

correlation between electricity price and demand is linear. Regarding the TSO, its

responsibility is to forecast the state of the network by estimating a future gener-

ation and load pattern. By that, major congestion problems can be detected and

handled by the planning and building of new transmission lines.

The generating companies’ short-term decisions are in the range of several days to

a few weeks. Those include the forecast of demand a generating company wishes

to supply in the short-run by submitting bidding curves in power auctions that are

based on a pre-established hourly production schedule. Here, the marginal cost


of generation are close to the variable cost of the most expensive generation unit

that has been assigned for production. Several pieces of information needed for

the unit commitment problem in the short-term horizon, such as the quantity and

the marginal cost of hydro resources and the maximum production capacity, are

previously provided by the medium-term market analysis. The goal is to utilise the

available generation units in an optimal way in order to comply with these referen-

tial values. A common approach to satisfy this objective is to build the company’s

hourly residual demand curve, which can be obtained by subtracting aggregated

supply bids by the company’s competitors from aggregated demand bids. By that,

the companies are able to evaluate how alternative bidding sets modify the mar-

ket clearing price on an hourly basis. Residual demand curves can be estimated

by several techniques that are based on available historic bidding information [9].

As a result, the company’s optimal offer and bid curves to be submitted to the

market can be obtained. Additionally, the company receives some information on

the availability and the marginal cost of reserve capacity which can be used for the

secondary reserve market.

Generating companies make some of their most important operational decisions in

the medium-term, which roughly comprises the horizon from a few months to one

year. The company’s objective is to make decisions on fuel purchases, physical and

financial contracting, emission allowances management, yearly market share objec-

tives and the allocation of hydro resources [15]. Thus, the medium-term market

analysis is very suitable for the forecasting of electricity and transmission prices

and associated costs and revenues.

It is widely accepted that the medium-term market analysis can be addressed ade-

quately by market equilibrium conditions [16], [17]. Those conditions characterise

an equilibrium point which is defined by the market participants’ output decisions

and by the values of overall market variables such as prices and consumptions. No

market participant can change its decisions unilaterally without making less net

profits. The market equilibrium can be considered as a good estimation of the

average behaviour of all market variables.

3 Flow-based Allocation of Cross-

Border Capacities

This chapter introduces the Flow-based Allocation (FBA) of cross-border transmis-

sion capacity as a congestion management method to be applied in an highly meshed

transmission system. After introducing the need for a congestion management con-

cept that respects the physical realities of the transmission network in a better way,

the general principles of the FBA approach are summarised followed by a descrip-

tion of its major innovations: power transfer distribution factors and net flowgate

capacities. Both are determined based on a power system state and updated when

the allocation horizon approaches. Thus, their time dependency will be highlighted.

Although, the FBA is not yet in operation in Europe, three different implementation

concepts of FBA are currently under investigation and will be presented: flow-based

coordinated explicit auctions, flow-based market coupling and open-market coupling.

Subsequently, the FBA approach will be compared to the widely spread concept of

the bilateral Net Transfer Capacity (NTC) based mechanism. The chapter closes

with the formulation of an equilibrium-based complementarity model of a flow-based

auction and a case study based on a four area power system.

3.1 Congestion Management

The electricity market liberalisation has led to an increase of cross-border power

trading and thus to a higher capacity utilisation of the transmission network. Of-

tentimes, the transmission capacity is not adequate in order to keep up with the

rising trading activities. Several physical bottlenecks occur in the transmission

system and put a limitation on the power exchange between the countries.

In 1999, the former ETSO (since 2009 called ENTSO-E) introduced transfer ca-

pacities to be used by neighbouring Transmission System Operators (TSOs) for

15

16 3 Flow-based Allocation of Cross-Border Capacities

the allocation of interconnection capacity between their control areas. In order to

conform with the guidelines of a market-based congestion management method,

the NTC is allocated in cross-border power auctions in a non-discriminatory and

market-based approach. Henceforth, NTC values are important indicators for the

market participants to plan their inter-regional transactions and for the TSOs to

manage the power flow resulting from cross-border exchanges.

A very specific feature of electric power is that the flow resulting from a power

exchange is governed by physical laws (Kirchhoff’s and Ohm’s Laws), i.e. electric

power takes several parallel paths from source to destination of the underlying

power exchange and not the direct contract path. For instance, in the highly

meshed European electricity network, a single transaction between Germany and

France will partly flow directly between the two countries but also through the

Netherlands-Belgium-France, Switzerland-France and Switzerland-Italy-France. An

adequate congestion management method needs to account for the capacity allo-

cation in complex networks by integrating the impact of cross-border trades on

the physical power flow. Thus, bilateral contract path mechanisms, like the NTC

models, rather fail in highly meshed power networks since they do not account for

the physical reality of electric power transmission.

The ENTSO-E has launched a proposal for an innovative market-based congestion

management mechanism, introduced as flow-based allocation, to be applied across

European borders within a specified region, denoted as the FBA region in the fol-

lowing. The FBA mechanism links the physical power flow on all inter-regional

interconnectors resulting from cross-border trades between any two control areas

within the FBA region. By that method, the physical infrastructure is to be used

more efficiently while facilitating cross-border trades at the same time. In terms

of the practical implementation of FBA, ENTSO-E and EuroPEX agree on the es-

tablishment of a series of regional initiatives in order to achieve efficiency benefits

in the medium term. Furthermore, the FBA mechanism supports the integration

of regional power markets towards the goal of a single European electricity mar-

ket. The flow-based allocation method is currently in a dry-run in the South-East

European and in the Central-West European regions [3], [4].

3.2 General Principles of Flow-based Allocation

Based on the laws of physics of the interconnected transmission network, any com-

mercial exchange between adjacent regional power markets results in a physical

power flow partly on the direct inter-regional transmission lines and partly across

3.2 General Principles of Flow-based Allocation 17

all available parallel paths in the network. The power flow on those parallel paths,

also known as loop flow, is ignored in a bilateral capacity allocation and might

cause network security problems in the control area of each TSO.

The objective of the FBA mechanism is to facilitate cross-border trades by provid-

ing maximum available inter-regional transmission capacity without compromising

the system security. More specifically, the FBA mechanism describes the interde-

pendency of commercial cross-border transactions between regional power markets

and the (physical) power flow those transactions cause on all cross-border inter-

connections in the FBA region. In case of transmission congestion, a commercial

exchange between any two power markets within that region has an effect on all

congested interconnectors. Besides, this allocation procedure also accounts for the

netting of the induced power flow on each single interconnector between neighbour-

ing control areas by evaluating the impact of all cross-border transactions on that

transmission link. Ultimately, this yields to a more efficient utilisation of inter-

regional cross-border capacities.

A further characteristic of the FBA mechanism is its inter-regional feature. By the

establishment of a flow-based coordinated auction in a regional setting, a central

entity, the auction office, collects all bids for energy and/or cross-border capacity

which are specified by a source and a destination area and the demanded quantity.

In fact, since cross-border transactions are no longer limited to the interconnec-

tions where they are reported, all bids compete with each other by means of their

contribution to the congestion. By converting all regional commercial transactions

into the physical power on the flowgates, the auction office chooses the most eco-

nomically efficient transactions subject to transmission capacity constraints, i.e. in

case of transmission congestion only those bids are selected with the highest mar-

ket value regarding the congested interconnection. Furthermore, the auction office

identifies the location of the congestion in the transmission network and provides

price incentives for the investment in the transmission infrastructure as a result of

the regional economic optimisation.

Moreover, the FBA approach facilitates the integration of regional electricity mar-

kets to achieve a higher economic efficiency across regions. An underlying assump-

tion is the establishment of a day-ahead power market in each single price region

that enables market participants to trade electricity between regional markets as

long as it is economically worthwhile and technically feasible. Thus, by imple-

menting the coordinated transmission auction at the day-ahead stage, electricity

and transmission markets operate at the same time interval, which facilitates the

planning of cross-border trades. Several supporting processes that go along with

the operation of day-ahead markets are explicitly described in [18].


3.2.1 The Simplified Transmission System Model

The ENTSO-E proposes the implementation of the FBA mechanism within some

specified regions of the European transmission system, also known as the regional

initiatives, in order to accelerate the process of establishing a single integrated Eu-

ropean electricity market. Each of those regional initiatives is treated separately

but has to account for the physical and commercial interactions with the surround-

ing transmission system. In order to enhance market transparency by means of

facilitating commercial trading activities, the complexity of the physical transmis-

sion system is reduced. In detail, the implementation of the FBA mechanism goes

along with the transformation of the highly-meshed transmission system of the

FBA region into a simplified transmission system model.

Control Area / Price Region

TransmissionTransmissionLine

Figure 3.1: Complete transmission system

By example, Figure 3.1 illustrates a transmission system which is divided into four

control areas, each of which is controlled by a single TSO. Furthermore, each control

area is assumed to coincide with a single electricity price region. Two approaches

are developed by the way single control areas are modelled: By allowing multiple

nodes per control area, a set of critical branches is defined, containing those inter-

and intra-regional transmission lines that can be restrictive in the cross-border al-

location (left Figure 3.2). In the second approach, price regions are symbolised

by single regional price nodes in the simplified transmission system model. Those

price nodes are interconnected by single cross-border links, called flowgates, which

represent the set of transmission lines between two price regions in the complete

transmission system (right Figure 3.2).

The flow properties of both simplified transmission system models are characterised

by sensitivity parameters, called power transfer distribution factors. They describe

the induced physical power flow on a given critical branch / flowgate caused by com-

mercial transactions between regional markets. The maximum power flow between

3.2 General Principles of Flow-based Allocation 19

Regional Price N d

Price Region

Flowgate

Node

CriticalBranch

Figure 3.2: Simplified transmission system models: critical branches (left) and

flowgates (right)

two price regions without violating the system security is given by the capacity of

the critical branches or by the flowgate capacity.

3.2.2 Modelling Comparison: Critical Branches vs. Flowgates

Both approaches account for physical network capacities as the limiting factor

of cross-border power trades. However, in the flowgate approach, the underlying

transmission network is represented by a simplified transmission system model,

where the trading volume is limited by the flowgate capacities. By comparison,

the concept of critical branches identifies single critical transmission lines to be

relevant in the allocation process. Both approaches give a price incentive for the in-

vestment in the limited element of the network infrastructure, whereas the concept

of critical branches is more accurate since it identifies the individual transmission

lines. However, the complexity increases by the amount of critical transmission

lines to consider in the allocation process. Additionally, the lack of transparency is

demonstrated by the composition of the congestion price. By definition, the price

according to any given congested transmission line is weighted by the impact of

each trade on the power flow on that line. Though, for an increasing number of

congested transmission lines, the number of contributions rise at the expense of

a transparent price derivation. By comparison of these approaches, there are far

less flowgate capacities than critical branches, in general. Hence, the allocation

of scarce transmission capacity and the process of transmission price generation is

more clearly represented in the flowgate concept.


Figure 3.3: Cross-border power flows resulting from a power exchange of 100MW

between Germany and France [19]

3.3 Power Transfer Distribution Factors

Generally, the power flow modelling in transmission networks is based on the eval-

uation of the commonly known AC load flow equations, which determine an AC

power flow on each single branch from the voltage magnitudes and angles. If the

capacity limit of any equipment in the power system is violated, TSOs usually

influence the power flow by re-dispatching generation. Empirically, they choose

generation units based on their production characteristics and their impact on the

critical equipment. By introducing Power Transfer Distribution Factors (PTDFs),

the sensitivity of nodal power injections with respect to the power flow on any

given transmission line is determined. Thus, PTDFs can be applied to intra-zonal

congestion management by providing nodal congestion prices that represent the

generators impact on the congested transmission lines.

Furthermore, PTDFs are a central feature of the FBA mechanism to be applied

in inter-zonal congestion management by accounting for the impact of commercial

transactions on cross-border transmission capacities. For that matter, Figure 3.3 il-

lustrates the power flow on each border between two neighbouring control areas fol-

lowing a commercial exchange of 100MW from Germany to France. Relatively large

power flows take the path Netherlands-Belgium-France and Switzerland-France.

Only about 36% of the traded power goes through the direct border between Ger-

many and France (the contract path). The relation between the power exchange

program of two zones and the induced physical power flow on the borders between

neighbouring control areas is explicitly formulated by the introduction of PTDFs.

3.3 Power Transfer Distribution Factors 21

c c‘

i‘

i Ti,i‘

Ti,i‘

Tc,c‘ Tc,c'

i

Figure 3.4: Transaction between two nodes (left) and between two zones (right)

This section is structured as follows: Subsection 3.3.1 introduces PTDFs in a

nodal-based and in a zonal-based representation of the transmission network. A

closed-form definition of PTDFs is given followed by a description of the general

determination procedure in the context of flow-based modelling. Several impact

factors in the calculation of PTDFs are discussed in Subsection 3.3.2 including

some useful simplifications resulting from the DC load flow approximation.

3.3.1 Definition of PTDFs

PTDFs can either be applied in a nodal-based or in a zonal-based representation of

the transmission network. In the former one, PTDFs are transmission line depen-

dant parameters that describe the percentage power flow induced on each single

transmission line based on a power transfer between two nodes. The definition of

PTDFs according to the nodal-based approach yields to

ΔPn

ΔTi,i′

∣∣∣∣BC

= PTDFi,i′,n, (3.1)

where ΔTi,i′ denotes the shift in power injections nodes i and i′ and ΔPn the

induced power flow on transmission line n as seen according to the transmission

system on the left hand side of Figure 3.4. Then, an additional transfer between

nodes i and i′ leads to a PTDF for each transmission line n. In fact, the load

flow equations can be replaced by a set of PTDFs which describes the power flow

in the vicinity of the current state of the power system. Then, PTDFs allow

TSOs to identify the impact location-specific power injections have on the loading

of transmission lines or transformers which can be very useful for re-dispatching

generation in order to handle local transmission congestion more transparently.

Additionally, PTDFs are one of the main features in the context of flow-based

modelling in inter-zonal congestion management. Unlike the nodal-based approach,

a commercial transaction between two regional markets is transformed into the

expected physical power flow on each flowgate, i.e. PTDFs are determined for the


aggregated transmission lines per each bilateral border. The definition of a PTDF

in the zonal-based approach is

ΔPm

ΔTc,c′

∣∣∣∣BC

= PTDFc,c′,m (3.2)

where ΔPm denotes the change in power flow on flowgatem resulting from a change

in the transaction from zone c to zone c′ denoted by ΔTc,c′. The amount by which

the transaction is changed is not treated as a real increase of the exchange program

between zones c and c′. In fact, the objective is to highlight the impact of ΔTc,c′ on

the power flow in the vicinity of the forecasted base case. According to the right

hand side of Figure 3.4, a generation shift between zones c and c′ leads to a PTDF

related to the interconnecting flowgate m.

Furthermore, the PTDF calculation is performed based on the expected state of

the power system for a certain future time horizon. That base case (BC) is defined

by the expected generation and load pattern, the transformers’ tap positions, the

network topology and the assumed exchange programs at all borders of the power

system. The composition of a single regional load flow model includes the merging

of all control areas in that region. As each single TSO provides an equivalent model

of its control area, the underlying harmonisation procedure between all TSOs is

organised by the auction office.

3.3.2 Computation of PTDFs

In the following, several aspects concerning the calculation of PTDFs are studied.

Depending on the generation units’ specific location in control zone c (source) and

control zone c′ (sink), an increase of the exchange program by ΔTc,c′ raises the

question about the underlying generation shift method. Since power flow calcula-

tions are usually performed in the AC mode in order to most accurately represent

the current state of the power system, volume and direction of generation shift

have to be specified. Finally, the DC load flow is under consideration in order to

find out in how far simplifications in the PTDF calculation are justified without

causing too much inaccuracy.

Additionally, the influence of each aspect on the PTDF calculation is explained

based on the four area example shown in Figure 3.5, where the FBA region con-

sists of areas A, B and C. Furthermore, the power system is assumed to be in a

base state, which is described by power injections g∗1 to g∗6 and power withdrawals

l∗1 to l∗4 caused by local generators and loads. The power flow on the flowgates

between areas A, B and C is denoted as P ∗AB, P ∗AC and P ∗CB.


g1*

* g2*

l2* g3*

*

PAB*

Area A Area B

FBA region

l1*g2 g4*

PAC* PCB

*

*

l3* g5*Area C Area Dl4*

g6*

Figure 3.5: Expected base case scenario of the power system

Generation Shift Method

Generally, the generation shift is performed in the PV nodes of the source and

the destination area, proportionally to the engagement of those generators in the

base case scenario. Thus, generators that are out-of-service in the base case are

not taken into account in the generation shift process. According to Figure 3.6, a

generation shift of ΔTA,B between areas A and B leads to local generation increases

Δgi (i ∈ {1, 2}) in source area A according to

Δgi =g∗i∑

k∈{1,2} g∗k

· ΔTA,B (3.3)

and local generation decreases Δgj (j ∈ {3, 4}) in destination area B according to

Δgj =g∗j∑

k∈{3,4} g∗k

· ΔTA,B (3.4)

Any other generation shift based on the same amount ΔTA,B leads to locally differ-

ent power injections Δg′i and withdrawals Δg′j. Consequently, the induced inter-

regional power flows ΔP ′AB,ΔP′AC and ΔP ′CB do not coincide with ΔPAB,ΔPAC

and ΔPBC. Generally, the sets of PTDFs based on two different generation shift

methods differ from each other:

ΔPm

ΔTA,B

∣∣∣∣BC

=ΔPm

Δg1 + Δg2

∣∣∣∣BC

= PTDFA,B,m

�= PTDF ′A,B,m =ΔP ′m

Δg′1 + Δg′2

∣∣∣∣BC

=ΔP ′mΔTA,B

∣∣∣∣BC


PAB

+ g1

+- g3

Area A Area B

TA,B TA,B

PCBPAC

+ g2 - g4

*

FBA region

l3* g5* l4*

g6*

Area C Area D

Figure 3.6: Generation shift between zones A and B

Volume of Generation Shift

By definition, a PTDF describes the impact of a commercial cross-border trans-

action on the power flow on a given flowgate within the FBA region based on a

forecasted future state of the power system. If the future state of the power system

approximately corresponds to the forecasted one, PTDFs are most accurately de-

termined by an infinitesimal amount of generation shift. However, a small amount

of generation shift leads to unreasonable values of PTDFs due to network losses in

the AC load flow calculation. Instead, a comparatively large amount of generation

shift does not describe the power system in the vicinity of the assumed future state,

thus, leading to unprecise PTDF values.

According to the power system shown in Figure 3.6, if the amount of generation

shift changes from ΔTA,B to ΔT ′A,B, the additional power flows ΔP ′AB,ΔP′AC and

ΔP ′BC do not linearly change related to ΔPAB,ΔPAC and ΔPBC. Due to the non-

linear feature of AC power flow calculation, two sets of PTDFs that are based

on the same generation shift method but differ in terms of the amount of their

generation shift are not necessarily identical:

ΔPm

ΔTA,B

∣∣∣∣BC

= PTDFA,B,m �= PTDF ′A,B,m =ΔP ′mΔT ′A,B

∣∣∣∣∣BC

Direction of Generation Shift

PTDFs are calculated based on an increase of generation in the source area and

a decrease of generation by the same amount in the destination area. A PTDF

calculation in the opposite direction takes into account the same set of generation


units in both areas. However, due to network losses resulting from the AC load

flow calculation, PTDFs on the same flowgate differ from each other, not only by

sign but also by their absolute value.

Based on the power system illustrated in Figure 3.6, the set of PTDFs resulting

from a generation shift from A to B does not correspond to that from B to A, by

sign and absolute value, because of the directional feature of PTDFs. Thus, based

on the same generation shift method and the same amount of generation shift:

ΔPm

ΔTA,B

∣∣∣∣BC

= PTDFA,B,m �= PTDF ′A,B,m =ΔP ′mΔTA,B

∣∣∣∣BC

Underlying Base Case

In the context of flow-based modelling, PTDFs are flowgate-related parameters

which are based on the forecasted state of the power system (base case). More

specifically, the calculation of PTDFs is related to the network topology, to the load

flow on all inter- and intra-regional transmission lines and to the set of scheduled

generation units and loads in the whole region. Thus, any change of that base case

leads to a different load flow pattern in the power system, which itself defines a

new base case. Ultimately, PTDFs based on different base case scenarios are not

identical.

The state of the power system shown in Figure 3.6 is assumed to a base case.

Any change of that base case scenario by means of transmission line outages or

different power injections/withdrawals has an impact on the power flow in the whole

transmission system. Thus, two different base cases based on the same generation

shift method and amount lead to different sets of PTDFs according to

ΔPm

ΔTA,B

∣∣∣∣BC

= PTDFA,B,m �= PTDF ′A,B,m =ΔP ′mΔTA,B

∣∣∣∣BC′

DC Load Flow

The DC load flow is a linear approximation of the AC power flow (see Appendix

A.4). By assuming a flat voltage profile and small voltage angles, the AC load

flow equations are reduced such that the active power flow on a given line linearly

depends on the transmission line reactances and the difference of the voltage angles

at each end of that line. Additionally, transmission line losses are omitted. The


DC load flow approximation is widely used in the calculation of PTDFs since it

allows for some very useful simplifications regarding the process of calculation.

Assuming the same generation shift method, the lossless linear DC load flow allows

for an arbitrary choice of the generation shift volume between two areas c and c′

without changing the set of PTDFs on a given flowgate m:

ΔPm

ΔTc,c′

∣∣∣∣BC

=ΔP ′mΔT ′c,c′

∣∣∣∣∣BC

= PTDFc,c′,m (3.5)

Equation (3.5) states that any kind of generation shift volume leads to the same

set of PTDFs. In the SEE-Region [3], for instance, PTDFs are calculated based

on a standardised generation shift volume of 100MW for each pair of zones.

Furthermore, the following equations are fully satisfied:

∑

m∈M(c,l)

PTDFc,c′,m = 1 (3.6)

∑

m∈M(l′,c′)

PTDFc,c′,m = −1 (3.7)

∑

m∈M(k,k′)

PTDFc,c′,m = 0 (3.8)

where M(c, l) denotes that set of flowgates m connecting source area c with its

neighbouring areas l, M(l′, c′) those flowgates connecting destination area c′ with

its neighbouring areas l′ and M(k, k′) those flowgates connecting any other area k

with its neighbouring areas k′. Equations (3.6) and (3.7) state that the amount

by which generation is shifted (ΔTc,c′) is fully converted into the additional physi-

cal power flow exporting from source area c and importing to destination area c′.Furthermore, the balance of additional importing and exporting power regarding

any other areas is always guaranteed as shown by Equation (3.8).

However, the main advantage of the DC load flow approximation is the associative

property of DC PTDFs, which reduces the amount of simulations when calculating

the PTDF matrix of a given power system. Let assume the power system consists

of N areas and M flowgates. According to Definition (3.2), a single generation shift

between two areas c, c′ ∈ N specifies a set of PTDFs according to the number of

flowgates m. In AC load flow, the PTDF matrix of the entire system is determined

by computing a generation shift between each pair of areas in both directions. In

total, N · (N −1) generation shifts are to be simulated resulting in a PTDF matrix

of dimension N · (N − 1) ×M .


Due to the linear properties of the DC load flow, the following equations are fully

satisfied for all c, c′, k ∈ N and m ∈M :

PTDFc,c′,m = −PTDFc′,c,m (3.9)

PTDFc,c′,m = PTDFc,k,m + PTDFk,c′,m (3.10)

By Equation (3.9), the changing of source area c and destination area c′ only modi-

fies the sign of DC-based PTDFs, while Equation (3.10) constitutes the associative

feature of DC-based PTDFs. By choosing an arbitrary area k, also known as the

hub-area, only generation shifts towards that hub need to be determined. Based

on the set of PTDFs resulting from that calculation, all the remaining PTDFs can

be determined by means of Equation (3.10). Compared with AC-based PTDFs,

only N − 1 generation shifts are to be computed. Table 3.1 presents the amount

of simulations based on AC and DC load flow and the dimension of the resulting

PTDF matrix for a couple of combinations (N,M). Basically, the calculation of

Table 3.1: Amount of simulation for AC- and DC- load flow based on several

regional models

Description of Dimension of � Simulations

Power System Model N M PTDF Matrix AC-based DC-based

Power System (Fig. 3.4) 3 3 6 x 3 6 2

CWE-Region 4 4 12 x 4 12 3

CEE-Region 7 10 42 x 10 42 6

SEE-Region 8 12 56 x 12 56 7

Europe 22 30 462 x 30 462 21

the PTDF matrix based on DC load flow reduces the amount of simulations by

N when compared with AC load flow. Focussing on the European power system

where flow-based market coupling is assumed to be applied, the computational ef-

fort of PTDF calculation becomes increasingly important. A comparison between

AC- and DC-based PTDFs based on a winter planning data set of the European

transmission system for a typical peak hour is described in Appendix B.2. Similar

to [20], the results reveal that DC-based PTDFs are a good approximation of AC-

based PTDFs as around 90% of the calculated DC PTDFs lead to less than 1MW

difference in power flow on any flowgate. Thus, the use of DC PTDFs is justified.


„Exchange programs“ „Active power flows“[MW]

maxE

TRMFRM

NF+OFE

TTC

NTC

maxF

TFC

NF+OF

Transfer direction

BCE

N C

NTF

NFC

max

TRM : Transmission Reliability MarginE : Maximum Generation Shift

TTC : Total Transfer Capacity

max

FRM: Flow Reliability MarginF : Maximum Additional Flow

TFC : Total Flowgate CapacityNTC : Net Transfer CapacityBCE : Base Case Exchange

NFC : Net Flowgate CapacityNTF : Notified Transmission FlowNF : Natural FlowOF : Outside FlowOF : Outside Flow

Figure 3.7: Transfer capacity definitions: left commercial, right physical values

3.4 Flowgate Capacities

In the existing explicit auctions of cross-border transmission capacity throughout

Europe, the Net Transfer Capacity (NTC) defines the maximum power exchange

program between two neighbouring areas compatible with security standards ap-

plicable in both areas and taking into account the technical uncertainties on future

network conditions [2]. NTC values are calculated twice a year based on a typical

winter and summer scenario of the power system and offered to the auctioning of

transmission rights.

A central feature of the FBA mechanism in cross-border congestion management is

the transformation of all cross-border trades within the specified FBA region into

the physical power flow on all the transmission lines between two neighbouring

areas in that region. Then, the aggregated set of transmission lines per each bilat-

eral border is defined as a flowgate. For each flowgate within the FBA region, the

neighbouring TSOs calculate the limit of the physical power flow which is based on

technical constraints in their control areas. That is calculated in both directions

of the flowgate and called the flowgate capacity.

Figure 3.7 illustrates the transfer capacity definitions and other corresponding val-

3.4 Flowgate Capacities 29

NTFAB

Area A Area BBCEA,B

NTF

BCEA,C BCEC,B

BCED,B

FBA regionNTFAC

NTFCB

Area C Area DBCEC,D

Figure 3.8: Power system in base case scenario

ues based on the NTC and the FBA model by comparison. In the NTC model

all capacities and values are in terms of power exchange programs between two

neighbouring areas. Their definitions are documented in [2]. By contrast, in the

FBA model all capacities and other corresponding values are given in terms of

active power flows per each bilateral border within a specified region, called the

FBA region. However, there are also several common relations between the two

models. Both, the NTC and the Net Flowgate Capacity (NFC) are directional and

considered as the relevant capacities to be offered to the market for transmission

rights. Prior to their determination, the Total Transfer Capacity (TTC) and the

Total Flowgate Capacity (TFC) have to be calculated. Those are based on a base

case situation of the underlying transmission network described by the Base Case

Exchange (BCE) values between area and the flowgate-related Notified Transmis-

sion Flow (NTF) values. Upon the base case situation in both models, a generation

shift is performed between two neighbouring areas until the (n-1) security criterion

is breached in either of the two areas. Furthermore, all transfer capacities and other

values, wether defined in the NTC or in the FBA model, are time-dependent since

all of them are determined based on a given state of the power system. However,

for simplicity, the time index will be omitted in the following.

The network shown in Figure 3.8 serves to clarify the modus operandi of TFC and

NFC calculation. It illustrates a power system with areas A, B, C and D. The

FBA region consists of areas A, B and C, thus, the NFCs are to be determined in

both directions of flowgates m ∈ {AB, AC, CB} while transmission congestion on

the borders of area D is assumed to be handled by a different congestion manage-

ment scheme. Besides, the power system is assumed to be in a base case which is

described by the NTF and the BCE programs between each pair of neighbouring


areas.

This section is structured as follows: In Subsection 3.4.1 the transfer capacity def-

initions in the FBA model are formulated according to [3] followed by an explicit

description of the TFC and the NFC calculation in Subsections 3.4.2 and 3.4.3.

Subsequently, Subsection 3.4.4 focuses on the contingency analysis in the process of

capacity calculation and on the capacity harmonisation on the borders within the

FBA region. The section closes with an example on NFC calculation in Subsection

3.4.5.

3.4.1 Transfer Capacity Definitions

Outside Flow

The Outside Flow (OF) is a physical active power flow on the flowgate between two

areas applying the FBA mechanism for transmission capacity, caused by the elec-

tricity transactions with and between the areas not participating in such allocation

mechanism, but synchronous connection with them.

Natural Flow

The Natural Flow (NF) is a physical active power flow on the flowgate between two

areas which would exist between the two areas if all systems in a synchronously

interconnected network would be balanced, i.e. when their exchange total is zero.

Notified Transmission Flow

The Notified Transmission Flow (NTF) is the physical active power flow on the

flowgate between two areas resulting from the Base Case Exchanges (BCE) and

from the loop flows (NF+OF) prior to any generation shift between those areas.

Maximum Additional Physical Flow

The Maximum Additional Physical Flow (ΔFmax) is the maximum additional physi-

cal power flow on the flowgate between two areas, compatible with security standards

at each system.

Total Flowgate Capacity

The Total Flowgate Capacity (TFC) is the maximum physical active power flow

on the flowgate between two areas, compatible with security standards, applicable at

each system and interdependent neighbouring systems, if future network conditions,

generation and load pattern were perfectly known in advance.


Flow Reliability Margin

The Flow Reliability Margin (FRM) is a physical active power flow on the flowgate

between two areas taken as a security margin that copes with uncertainties on the

computed TFC.

Net Flowgate Capacity

The Net Flowgate Capacity (NFC) is the maximum physical active power flow on

the flowgate between two areas, compatible with security standards applicable at each

system and interdependent neighbouring systems, resulting from the commercial

transactions among the systems participating in the FBA auction, and taking into

account the technical uncertainties on future network conditions.

3.4.2 Determination of the Total Flowgate Capacity

According to the left hand side of Figure 3.7, the Total Transfer Capacity (TTC)

from area c to neighbouring area c′ is defined as

TTCc,c′ = BCEc,c′ + ΔEmaxc,c′ (3.11)

where BCEc,c′ denotes the base case exchange program and ΔEmaxc,c′ the maximum

generation shift between the two areas without comprising the system security.

According to the FBA model, the TFC is calculated between each two neighbour-

ing areas that belong to the FBA region. Similar to the TTC calculation, the

determination of the TFC is related to a given power system scenario, known as

the base case, which is defined by the network topology, the generation schedule

and the consumption pattern. Thus, as shown on the right hand side of Figure

3.7, the TFC on flowgate m, linking areas c and c′ within the FBA region, is

mathematically stated as follows:

TFCm = NTFm + ΔFmaxm (3.12)

Based on the NTF on flowgate m, the cross-border power exchange between neigh-

bouring areas c and c′ is stepwise increased, until a network constraint based on

the (n-1) security criterion in either of the two systems is violated. This leads to a

change of the power flow on the considered flowgate by ΔFmaxm . Then, the aggre-

gated pre-contingency power flow on flowgate m determines the value of TFCm.

However, the specification of the NTF and the ΔFmax can be realised in various

ways. Hence, the TFC calculation is based on the agreed methods used by the

TSOs in areas c and c′ or by a central entity that is in charge of the capacity


calculation within the FBA region.

In [3] two frequently used approaches are presented regarding the determination

of the NTF. Since the TFC is directional and highly dependent on the base case

power flow on the considered flowgate, both methods differ from each other in

terms of the setting of the underlying base case condition of the transmission net-

work to avoid very small or even negative TFC values. In addition to that, the

TFC of a given flowgate is determined by the generation shift method that causes

the maximum additional power flow on the considered flowgate. Two widely used

generation shift methods will be presented that differ from each other by the dis-

tribution and contribution of scheduled generating units during the shift process.

Specification of NTF

By the first approach (Approach I), the TFC determination is similar to the con-

ventional advance to be used in bilateral TTC calculation according to [2]. Based

on a base case situation of the transmission network, which is described by the

most probable exchange programs between each two neighbouring areas and the

power flow on each single transmission line, the NTF is the aggregated power flow

on a flowgate within the FBA region. According to Figure 3.8, NTFm accounts for

all loop flows on flowgate m created by the exchange programs between the FBA

region and external area D, i.e. BCEBD and BCECD, also known as the Outside

Flow (OF). Additionally, the power flow on each flowgate m is affected by those

power exchanges between each pair of areas within the FBA region, i.e. BCEAB,

BCEAC and BCEBC. A third contribution to the loop flows in the transmission

network of the FBA region is based on the Natural Flow (NF). This is due to the

characteristics of synchronously interconnected electric networks, where there is a

power flow on the transmission lines even though each single area is balanced.

Based on the NTF, the TFC is calculated in both directions for each flowgate in

the FBA region by increasing the power exchange between the two areas at both

ends of the considered flowgate. A drawback by this approach is the dependency

of the calculated TFCs on the underlying base case situation of the network, which

can lead to rather unrealistic TFC values. If, for instance, a large BCE from area

A to area B results in a comparatively high NTF on flowgate AB, the TFC in the

opposite direction of the flowgate might be very small or even negative, if ΔFmaxB,A

from B to A is less that NTFAB in direction from A to B.

The algorithm of the second approach (Approach II) does not change from the first

one. However, it addresses the problem of unrealistically small or even negative

TFC values in one direction of a flowgate by assuming zero base case power ex-


NTF‘AB

Area A Area B

NTF‘ BCED,B

FBA regionNTF AC

NTF‘CB

Area C Area DBCEC,D

Figure 3.9: NTF Approach II: Eliminating BCE programs within FBA region

changes between among all areas in the FBA region. As a result of that, the NTF

on flowgate AB tend to be lower than in Approach I, hence, allowing for a higher

TFC value in direction from B to A. However, changing the setting of the base case

scenario by means of the BCE programs in the FBA region, the TFC values are not

calculated based on the most probable future exchange scenario anymore. Thus,

capacities might need to be adjusted in subsequent allocation phases when network

problems occur. Furthermore, the existing NTF values need to be re-calculated in

order to eliminate the impact of the BCE programs on the power flow within the

FBA region.

By example, let assume the BCE program from area A to area B in the power

system depicted in Figure 3.8 to be comparatively high, thus, resulting in a large

NTF on flowgate AB in direction from A to B. When calculating the TFC in op-

posed direction of NTFAB, the maximum additional power flow ΔFmax induced on

that flowgate in direction from B to A is less than NTFAB. Hence, according to

Equation (3.12), the TFC from B to A is negative. In order to address that prob-

lem, all BCE programs within the FBA region are set to zero, i.e. BCEA,B = 0,

BCEA,C = 0 and BCEB,C = 0. Figure 3.9 shows the new state of the power sys-

tem.

By means of the calculated PTDF matrix for the FBA region, as documented in

Section 3.3, the impact of all BCE programs within the FBA region on the power

flow on the flowgates can be calculated and eliminated in order to obtain the base

case situation according to Approach II. Then, the power flow Xm caused on flow-

gate m ∈ M = {AB,AC,CB} by BCEc,c′, (c, c′) ∈ F = {(A,B), (A,C), (C,B)} is


obtained as follows:

Xm =∑

(c,c′)∈F

BCEc,c′ · PTDFc,c′,m (3.13)

Hence, the adjusted NTF values on each flowgate m within the FBA region are

NTF ′m = NTFm −Xm (3.14)

Focussing on the power flow on flowgate AB, each BCE program within the FBA

region contributes to the NTF on flowgate AB in direction from A to B. By setting

those BCE programs to zero, NTFAB will be reduced. Ultimately, the TFC in the

opposite direction of NTFAB tends to be higher by Approach II than by Approach

I. In fact, the adjusted power flow NTF ′m is partly composed of the NF and the

induced by power flow by external power exchange programs, i.e. by BCEC,D and

BCED,B. Thus, NTF ′m can also be formulated as

NTF ′m = NFm +OFm (3.15)

Specification of ΔFmax

As shown by Equation (3.12), the TFC calculation of a given flowgate between two

neighbouring areas includes the determination of the maximum additional power

flow ΔFmax on that flowgate. Starting from an agreed base case situation of the

power system between the participating TSOs, a generation increase is performed

in the exporting area and an equivalent generation decrease in the importing area.

During the generation shift process the loads in both areas are constant. The gen-

eration shift is to be made gradually and causes a change of the power flow on

the flowgate linking those areas. This approach is carried out until a network con-

straint based on the (n-1) security rule is violated in either of the two areas. Then,

the maximum generation shift ΔEmax causes the maximum additional power flow

ΔFmax on the considered flowgate, which is compatible with the network security

rules.

There are a variety of generation shift methods that differ from each other in terms

of the choice of the participating generating units in the exporting and importing

areas and by which the generation shift is distributed over the scheduled gener-

ating units. Generally, the choice among the different shift approaches is left to

the responsibility of the TSOs in the exporting and importing areas. A common

feature of all methods is that only scheduled power plants in the base case scenario

are used in the generation shift method.


NTFAB

Area A Area Bg1*

l1*g2*

l2* g3*

g4*

+ FmaxArea A Area B

+ g1

l1* + g2

l2* g3

g4Emax Emax

A,BA,B

A,B

Figure 3.10: Transmission system in base case situation based on Method II

In the following, three commonly used generation shift methods are introduced

and explained based on a generation shift between neighbouring areas A and B as

depicted in Figure 3.8. Generators and loads are distributed over the two areas ac-

cording to Figure 3.10. The generators’ upper and lower capacity limits are known

and denoted by Gmax1 to Gmax

4 and Gmin1 to Gmin

4 , respectively.

By Method I, the generation increase and decrease is performed proportionally to

the power reserve of all scheduled generating units, hence, all units simultaneously

reach their generation capacity limits. This method implicitly assumes the up-

per and lower generation capacity limits for each scheduled generation unit to be

known. According to Figure 3.10, each generating unit i ∈ {1, 2} in exporting area

A is increased by

Δgi =Gmax

i − g∗i∑k∈{1,2}(G

maxk − g∗k)

· ΔEmaxA,B (3.16)

subject to

Gmini ≤ g∗i + Δgi ≤ Gmax

i (3.17)

Accordingly, each generating unit j ∈ {3, 4} in importing area B is decreased by

Δgj =g∗j −Gmin

j∑k∈{3,4}(g

∗k −Gmin

k )· ΔEmax

A,B (3.18)

subject to

Gminj ≤ g∗j − Δgj ≤ Gmax

j (3.19)

If generating units in either of the two areas reach their lower or upper generation

capacity limit, they remain at the level of the last incrementation and decremen-

tation, respectively, of the generation shift process. Another stopping criterion is

fulfilled if all the generators reach their capacity limit although network constraints


are not yet violated. Then, the generation shift process can be continued by further

shifting generation proportionally to the levels in the base case scenario in order

to determine the ΔFmax.

Alternatively, the generation shift can be realised by increasing the generating units

in the exporting area and decreasing those in the importing area proportionally to

their engagement in the base case scenario. Then, based on the power system in

Figure 3.10, each generating unit i ∈ {1, 2} on the exporting side is increased by

Δgi =g∗i∑

k∈{1,2} g∗k

· ΔEmaxA,B (3.20)

Accordingly, each generating unit j ∈ {3, 4} on the importing side is decreased by

Δgj =g∗j∑

k∈{3,4} g∗k

· ΔEmaxA,B (3.21)

Again, the generating units’ operational constraints, formulated in Equations (3.17)

and (3.19), are to be fulfilled at each step of the generation shift process. If one

generating unit reaches its capacity limits during that process, that unit will remain

at its maximum or minimum level in all subsequent steps of generation shifting.

Henceforth, this method is denoted by Method II.

By Method III, those TSOs participating in the generation shift process, make a

list of the power plants they will use to increment and decrement. That list may

represent a specific order of the scheduled power plants that are characterised by

their generation capacity limits.

In TTC calculation the maximum generation shift ΔEmax between two neighbour-

ing areas is equivalent to the additional power exchange program between those

areas. According to the TFC calculation, the maximum additional power flow

ΔFmax on a certain flowgate is induced by a generation shift between the areas

on both ends of that flowgate. If the generation shift methods in the TTC and

TFC model are identical, then, a relation between ΔEmax and ΔFmax can be es-

tablished. The maximum generation shift ΔEmax from area A to area B results in

the maximum additional power flow ΔFmax on flowgate m, that links both areas,

according to

ΔFmaxm = PTDFA,B,m · ΔEmax

A,B (3.22)

where the determination of the PTDF is based on the same methodology as ΔEmax

and ΔFmax. Thus, by inserting Equation (3.22) into Equation (3.12) the TFC on

flowgate m is calculated as follows:

TFCm = NTFm + ΔEmaxA,B · PTDFA,B,m (3.23)


3.4.3 Determination of the Net Flowgate Capacity

Subsequent to the calculation of the TTC, the NTC between two neighbouring

areas is determined by accounting for the Transmission Reliability Margin (TRM),

which is a security margin dealing with on the computed TTC value [2]. Thus, the

NTC from area c to area c′ is defined as

NTCc,c′ = TTCc,c′ − TRMc,c′ (3.24)

By contrast, the NFC is related to a specific flowgate m within the FBA region.

Then, the NFC is calculated by subtracting the Flow Reliability Margin (FRM)

and the sum of the NF and the OF from the prior computed TFC according to

NFCm = TFCm − FRMm − (NFm +OFm) (3.25)

In the following, the focus will be on the determination of the FRM and the sum

of NF and OF.

Determination of FRM

Similar to the TRM, a FRM is defined in order to account for different kinds of

uncertainties in the TFC calculation. In [3], the FRM is defined as a percentage

of the TFC according to

FRMm = 10% · TFCm (3.26)

Alternatively, the FRM can be computed by means of the TRM and PTDFs, only

if the underlying generation shift methods for ΔEmax and ΔFmax computation are

identical. Furthermore, the method for PTDF calculation must comply with the

method for capacity calculation. Then, based on the TRM for an exchange program

from area c to area c′, the FRM related to flowgate m between those areas can be

determined approximately as follows

FRMm ≈ TRMc,c′ · PTDFc,c′,m (3.27)

Basically, a relation between the FRM and the TRM is not evident, since both secu-

rity margins comply with uncertainties either related to a power exchange (TTC) or

to a physical power flow (TFC). However, the effect of the generation shift between

two adjacent areas on the considered flowgate can be described by means of the

PTDF on that flowgate, thus, the underlying assumption of Equation (3.27) is the

conversion of both security margins among each other by means of the same PTDF.


Determination of NF and OF

A specific feature of the FBA model is the validity of the computed physical ca-

pacities within a spatially limited area, the FBA region, of a larger power system.

The determination of those capacities is quite sensitive to the underlying base case

condition of the power system in that region. That state of the power system, how-

ever, is not only influenced by the regional generation and load pattern and the

inter-regional power exchange programs. To some extent, the power flow within

the FBA region is affected, e.g. by power exchanges between areas beyond the

FBA region, which can have an restrictive effect on the NFC values. Thus, those

induced power flows are integrated in the NFC calculation by means of the NF and

the OF. As shown by Equation (3.25), they are to be considered in the determi-

nation of the NFC. By inserting Equations (3.14) and (3.15) into Equation (3.13),

the sum of NF and OF on a given flowgate m can be computed as follows

NFm +OFm = NTFm −∑

(c,c′)∈F

BCEc,c′ · PTDFc,c′,m (3.28)

Usually, values for BCE and NTF are given by the base case scenario of the power

system. Thus, the sum of NF and OF is obtained by subtracting the impact of all

BCE programs between neighbouring areas c and c′ within the FBA region from

NTF in the base case.

The NFC is determined by computing all capacities and corresponding values ac-

cording to Equation (3.25). Alternatively, the NFC on a given flowgate can be

approximated in both directions by the NTC values between the areas on both

ends of the flowgate, known from the base case scenario of the power system. By

inserting Equations (3.23), (3.26) and (3.28) into Equation (3.25) and after rear-

ranging, the NFC for flowgate m can be computed approximately by

NFCm ≈ NTCc,c′ · PTDFc,c′,m +∑

(k,k′)∈F,(k,k′) �=(c,c′)

BCEk,k′ · PTDFk,k′,m (3.29)

The full derivation of (3.29) is given in Appendix B. The main advantage of this

approach is the dependency of the NFC on NTC and BCE, both of which are

known from the base case scenario.


DE

620 MW1048 MW

DE

343 MW1885 MW

ATCH

620 MW

236 MW

1048 MW

ATCH

343 MW

860 MW

1885 MW

236 MW

Figure 3.11: Reference case scenario (W 2008/2009): BCE (left) and NTF (right)

3.4.4 Contingency Analysis and Capacity Harmonisation

According to the TTC calculation, the generation shift process is carried out on

a bilateral basis by the TSOs in each of the two participating neighbouring areas.

By the definition of an exporting and an importing area, the maximum generation

shift ΔFmax becomes directional, thus, each TSO calculates an exporting TTC and

an importing TTC with regard to its control area. Furthermore, the TSO who is

performing the generation shift is only responsible for the contingency analysis in

its own transmission system including the cross-border transmission lines without

taking into account the other area or any third systems that are affected by the

generation shift. Thus, based on the same base case scenario two TTCs are cal-

culated by the participating TSOs per each direction on the considered border.

During the NTC calculation, a harmonisation process between them results into

the agreement of one NTC per direction on each bilateral border.

The TFC calculation for any flowgate within the FBA region can also be per-

formed by the TSOs in the participating exporting and importing area. Similar

to the NTC model, a harmonisation of NFCs per each flowgate would be needed

in order to agree on one valid set of NFCs in both directions of a given flowgate

within the FBA region. In [3] it is stated, that this approach is justified in a first

step of FBA implementation. A more sophisticated approach focusses on a single

entity to calculate all the TFCs in the FBA region based on a commonly agreed

network model by all TSOs in that region. Then, the contingency analysis during

the generation shift process includes all areas within the FBA region.


Table 3.2: PTDF matrix and NTC values for FBA region in Fig. 3.11c, c′ \ m DE-AT DE-CH AT-CH c c′ NTCc,c′ NTCc′,c

AT CH -0.379 0.413 0.210 AT CH 470 MW 1200 MW

DE CH 0.120 0.515 0.132 DE CH 1500 MW 3200 MW

DE AT 0.499 0.102 -0.078 DE AT 2000MW 1800 MW

Table 3.3: NTF and the sum of NF and OFm NTFm NFm +OFm

AT-CH 860 MW 761 MW

DE-CH 1885 MW 806 MW

DE-AT 343 MW -276 MW

3.4.5 Case Study - Capacity Calculation

Let assume cross-border transmission congestion between neighbouring countries

AT, CH and DE to be managed by the FBA mechanism. The state of the power

system in the FBA region is defined by BCE and NTF values published by the

ENTSO-E for the winter scenario 2008/2009 [21] as depicted in Figure 3.11. Based

on that base case scenario, the PTDF matrix is calculated and shown together

with the NTC values in Table 3.2. Furthermore, the FBA region is embedded in

the UCTE network, thus, there is a physical and commercial exchange between the

FBA region and the remaining UCTE power system. The NFC values are to be

determined on the common borders of the countries in the FBA region according to

Equation (3.29). By comparison, their values will be determined when the impact

of NF and OF is not eliminated.

The calculated sum of NF and OF and the NTF values from the base case scenario

are shown in Table 3.3 for each flowgate m. The interconnection of the FBA region

with the remaining UCTE power system induces a substantial power flow on the

flowgates within the FBA region, particulary on flowgate AT-CH in direction from

AT to CH. Besides, the base case flowgate flow NTFDE-AT is opposed to the power

flow induced byNFDE-AT+OFDE-AT. As expected, the impact of the BCE programs

within the FBA region (left hand side of Figure 3.11) leads to an increased power

flow in direction of the flowgates.

Table 3.4 shows the calculated NFC values when the sum of NF and OF is and

is not eliminated in the NFC calculation. Generally, the NFC values are shifted

in both directions according to the values of NF and OF. Focussing on flowgate

AT-CH, the calculation of the NFC from CH to AT is negative when the impact

of NF and OF is not discounted. This is due to the substantial contribution of the

3.5 Time Dependencies of PTDFs and Flowgate Capacities 41

Table 3.4: NFC values with elimination (left) without elimination (right) of NF

and OFm NFC−→m NFC←−m m NFC−→m NFC←−m

AT-CH 148 MW 202 MW AT-CH 909 MW -559 MW

DE-CH 1312 MW 1108 MW DE-CH 2118 MW 303 MW

DE-AT 1307 MW 589 MW DE-AT 1031 MW 864 MW

induced power flow by means of NF and OF on flowgate AT-CH. In other words,

without eliminating the effects of NF and OF, the additional power flow generated

on flowgate AT-CH, based on the generation shift from CH to AT, does not yield

to a positive NFC in direction from CH to AT.

3.5 Time Dependencies of PTDFs and Flowgate Ca-

pacities

The determination of transfer capacities in the bilateral NTC model is based on a

given state of the power system defined by the network topology, the generation

and load pattern and the exchange programs between neighbouring control zones.

More specifically, that system state is related to a base case scenario at a certain

time in the past and adapted to an agreed set of forecasted system conditions typ-

ical for a certain time in the future. Transfer capacities calculated in that time

frame, also known as the planning phase, are best estimates of the future power

system state that should support the planning of the market participants’ cross-

border power trading. Thus, these transfer capacities, the NTC values, are rather

indicative and non-binding [2]. They are calculated twice a year by the ENTSO-E

based on the expected power system states for a typical winter and summer peak

hour, also known as the ”snapshots” of the power system.

Subsequent to the planning phase, the transfer capacities are assigned to the market

participants in different consecutive time frames, known as the allocation phases,

in order to provide a flexible scope of trading, i.e. from longer term (year-ahead)

to shorter term (day-ahead or even hour-ahead) trading. For each allocation time

frame, the Available Transfer Capacities (ATCs) are re-calculated in order to ac-

count for the exchange programs already allocated in previous time frames, known

as the Already Allocated Capacities (AACs). Additionally, any assumptions ac-

cording to the network topology, the generation and load pattern are updated in

order to most accurately determine the expected state of the power system in the


Planning phase Allocation phases[MW]

„Exchange programs“

Planning phase Allocation phases[MW]

„Active power flows“

[MW]

ATC

[MW]

NTC

ATC

Transfer directionNFC

AFC

AAC ANF

[t]

AAC: Already Allocated CapacityATC : Available Transfer Capacity

yearly monthly dailyyearly [t]

ANF: Already Nominated FlowAFC : Available Flowgate Capacity

yearly monthly dailyyearly

ATC : Available Transfer Capacity C : v b e owg e C p c y

Figure 3.12: Planning and allocation phases in the NTC and FBA model

subsequent allocation phase. The picture on the left hand side of Figure 3.12 illus-

trates the planning phase and the different allocation phases of the NTC and the

ATC, respectively.

According to the FBA mode, the definition of the physical capacities is also as-

sociated with a single planning and several allocations rounds. This section is

structured as follows: After giving the definitions of the physical capacities in

Subsection 3.5.1 according to the allocation phases, Subsections 3.5.2 and 3.5.3

highlight the activities in the planning and allocation phases in the FBA model.

3.5.1 Physical Capacity Definitions

Already Nominated Flow

The Already Nominated Flow (ANF) is the physical active power flow on the flow-

gate between two neighbouring areas resulting from the transmission rights already

allocated at some of the previous allocations rounds, and nominated prior to the

current auctions round.

Available Flowgate Capacity

The Available Flowgate Capacity (AFC) is the part of the NFC that remains avail-

able, after the previous phases of the allocation, for current allocation rounds.

3.5 Time Dependencies of PTDFs and Flowgate Capacities 43

3.5.2 Planning Phase

In the planning phase, the NFC values and the PTDF matrix are calculated in

order to provide a signal to the market participants to plan their long-term trading

strategies. As in the NTC model, both are related to a given reference scenario

of the power system in the past, which is adjusted according to the agreed power

system state in the future. Thus, NFC and PTDF values are time dependent and

have to be adapted when approaching the horizon of programme execution.

If NFC and PTDF values are calculated on a seasonal time basis (half year ahead),

for instance, the system state on the 3rd Wednesday of January for the winter

scenario and on the 3rd Wednesday of July for the summer scenario, each at 10:30h,

can be chosen to comply with the calculation method of bilateral NTC models [2].

3.5.3 Allocation Phase

The NFC and PTDF values are based on an expected power system state in a cer-

tain hour to be representative for a certain future time frame. Generally, the NFC

and PTDF values vary when approaching the horizon of allocation due to unforseen

changes of the power flow resulting from different base case scenarios than the one

used in the planning phase. Moreover, the larger the allocation time frame, the

more challenging is the derivation of an expected power system state upon which

NFC and PTDF values are calculated to be representative for the whole allocation

time frame. Any distortion of NFC and PTDF values may result in system security

problems due to inaccuracies in the allocation of physical cross-border capacities.

Furthermore, any remaining physical capacities subsequent to the allocation stage

or an any additional physical capacities resulting from capacity calculation based

on base case scenarios closer to the allocation period cannot be used for further

commercial activities.

Thus, the calculated NFC values are allocated in subsequent auction rounds that

are based on different time frames. Commonly known time frames for auction

rounds are year-ahead, month-ahead, day-ahead. This procedure results in a va-

riety of capacity products to be used by the market participants to secure their

longer term trading and to optimise their trading in the short term. For each

subsequent auction round, the Available Flowgate Capacity (AFC) to be used for

further capacity allocation is calculated based on the expected power system state

for a closer time frame by integrating the results of the previous allocation phases


Allocationrounds

Capacityallocation

rd3 WedYearly

rounds allocation

NFC

NFC

rd3 Wed rd3 Wed rd3 Wedrd3 Wedrd3 Wedrd3 WedMonthly

NFC

ANF ANF ANF ANF ANF ANF

AFC AFC AFC AFC AFC AFC

Winter scenario

Daily

ANF ANF

NFC

... ... ... ... ... ...

... ... ... ... ... ...

... ... ... ... ... ...

Oct Nov Dec Jan Feb Mar

Octst st1 ... 31 st th1 ... 30

Novst st1 ... 31Dec

st st1 ... 31Jan

st th1 ... 28Feb

st st1 ... 31Mar Allocation

horizonOctst st1 ... 31 st th1 ... 30

Novst st1 ... 31Dec

st st1 ... 31Jan

st th1 ... 28Feb

st st1 ... 31Mar Allocation

horizon

Figure 3.13: Allocation rounds and horizon

according to

AFCm = NFCm − ANFm (3.30)

where ANF denotes the Already Nominated Flow on flowgate m. The ANF in-

cludes the allocation results of each previous auction round and the unforeseen

power flow resulting from the difference of the prevailing and forecasted system

conditions. Figure 3.13 illustrates the splitting of the allocation phases starting

from a yearly planning phase and the physical allocation after each auction round.

In day-ahead auctions, the forecasted system state provides the best approxima-

tion of prevailing system state in the sequence of auction rounds. Unused flowgate

capacities (AFC) can be offered to the market for further cross-border transactions

without compromising the system security.

3.6 Concepts of Flow-based Auctions

There are three major approaches of FBA implementation currently under investi-

gation, which can be classified by means of the commonly known market clearing

concepts of explicit and implicit auctions.

In flow-based coordinated explicit auctions, the transmission market is timely sep-

arated from the electricity market. Market participants submit their bids for

transmission capacity prior to the market clearing of regional power exchanges

by specifying source- and sink area and the demanded quantity. A central entity,

the auction office, collects all regional commercial transactions by converting them

3.7 Concepts of Congestion Management: NTC vs. FBA 45

into physical power flows on the flowgates. In case of congestion, the auction office

determines the most valuable bids per MW of flow on the congested flowgates to

be accepted while the lowest ones are to be cancelled. Subsequent to the market

clearing of the transmission market, market participants take part in the regional

power exchanges. The flow-based coordinated explicit auction is currently under

investigation in the SEE region [3].

Another approach of FBA implementation is known as Flow-based Market Cou-

pling (FMC) [18]. It is also based on the concept of decentralised market coupling,

pioneered by the Association of European Power Exchanges (EuroPEX) [22], that

provokes the integration of separated regional power markets by an inter-regional

coupling process. More specifically, regional power markets are coupled by an

implicit auction, i.e. transmission rights do not have to be acquired explicitly

since energy and the use of flowgate capacities are traded simultaneously. In this

approach, all flowgate capacities are made available to a central entity, which opti-

mises the use of that capacities by matching energy bids and offers in the respective

regional power markets.

Furthermore, the FBA mechanism is integrated in the concept of Open Market

Coupling (OMC), which is a hybrid model of explicit and implicit auctions [23].

In addition to a day-ahead implicit auction, an explicit auction based on a day-,

month- and year-ahead stage works in parallel. Besides, bids for transmission ca-

pacity between two zones can also be submitted to the auction office. Those bids

are also taken into account in the price determination process.

3.7 Concepts of Congestion Management: NTC vs.

FBA

The most widely used congestion management method in the European transmis-

sion system is the auctioning of the bilateral NTC between almost each pair of

neighbouring countries. However, the NTC value only defines the contract path

along which transmission capacity is allocated between two countries, hence, it

is an exchange transfer capacity between adjacent areas that does not represent

the physical capacity reservation of the transmission system. Exchange and phys-

ical border capacity correspond approximately only in longitudinal systems, e.g.

the U.S. West Coast, or in two-party peninsular systems, e.g. France-Spain and

Sweden-Finland. However, in the highly meshed European transmission network,

physical transmission capacity is reserved as a result of the cross-border trades


A BA B

C D

Figure 3.14: Transaction from A to D: Acquisition of transmission rights in different

NTC-based auctions

between all interconnected countries. It is a common resource that has to be op-

erated jointly and in a coordinated way [24]. Thus, contract path mechanisms are

inappropriate for the application to multiple borders in highly interconnected net-

works because transmission capacity cannot be portioned. The FBA mechanism

is an extension of the bilateral NTC concept as a congestion management method

to be introduced in complex systems with multiple borders and control areas.

3.7.1 NTC-based Auctions

A major drawback of the NTC mechanism is the complexity of multiple indepen-

dent cross-border auctions. According to Figure 3.14, a market participant, which

is located in area A, wants to have access to the power market in area D. Then, that

market actor needs to acquire transmission rights by participating in each single

NTC auction along an arbitrarily chosen path from the source to the destination

area. However, there are several possible ways how to accomplish the contract

path from area A to area D. Obviously, direct paths A-B-D (black coloured) and

A-C-D (blue coloured) include the participation in only two cross-border auctions.

Another possibility is path A-B-C-D (green coloured). In this case, the market par-

ticipant assumes a higher price level in auction A-C than in auctions A-B and B-C

together. Similar considerations hold for path A-C-B-D (red coloured). Besides,

the market participant needs to synchronise its cross-border trades by accounting

for the clearing time of each single cross-border auction along the chosen contract

path. Furthermore, the participation in two or even more cross-border auctions

comes along with an increased risk of transmission costs because the market par-

ticipant has to acquire all transmission rights in order to have access to the power

3.7 Concepts of Congestion Management: NTC vs. FBA 47

market in destination area D. In total, the strictly bilateral and independent fea-

ture of NTC auctions yields to an increased complexity regarding the settlement

of cross-border trades in multiple borders and control zones.

Moreover, the separation of NTC-based auctions only provides very vague economic

signals for the market participants although the underlying network is strongly in-

terconnected. Basically, every cross-border transaction contributes to the physical

congestion by either relieving or aggravating the loading of the critical branch in

the transmission network. The NTC value, however, only represents an exchange

capacity between adjacent areas without providing a direct link to the underlying

transmission network. Thus, there is no price incentive for market participants

to relieve congestion by adapting their trades since the contribution to the con-

gested element cannot be assigned to a specific auction. From the TSOs point

of view, the congestion price does not provide an efficient investment foundation

in the infrastructure of the transmission network. As a result of the missing link

between commercial transactions and the induced physical power flow, the NTC

on a given border can be auctioned in both directions, yielding to two prices at a

single cross-border simultaneously.

3.7.2 FBA Auctions

The FBA mechanism is an extension of the bilateral NTC concept as a congestion

management method to be applied to complex systems with multiple borders and

control areas. It is to be implemented within a specific region, the FBA region,

by representing that region through a simplified transmission system model as

shown in Section 3.2. The simplified transmission system model evidently reduces

the complexity of the complete transmission system, which has an impact on the

accuracy of this model as a congestion management tool. However, it provides

significant improvements over the existing NTC-based model by a better represen-

tation of the physical capacity limits in the transmission system while maintaining

adequate market liquidity at each regional node.

The major improvement of the FBA mechanism is the clear distinction between

trade and physical flow. Traders only need to specify the source and destination

area corresponding to their trade within the FBA region unlike the NTC mech-

anism where transmission rights at independent cross-border auctions have to be

acquired. All trades, however, are transferred into the resulting power flow on the

flowgates between neighbouring control areas. Thus, they are coupled with each

other by the underlying interconnected transmission network. Consequently, all


A B

A B A,B

B,A

C DxA,B

xB,A

Figure 3.15: Efficient transmission pricing

bids for transmission capacity within the FBA region compete with each other.

Any bid that contributes to saturate one or more physical bottlenecks in that

region is charged a fee according to its impact on the congestion. Transmission

capacity is allocated only to the most highly valued bids with respect to the con-

gested flowgates.

Moreover, the FBA mechanism implies an efficient congestion pricing scheme,

which includes the netting of the power flow on the same flowgate. According

to the example depicted in Figure 3.15, there are two trades between areas A and

B, that are equal by quantity but different by means of the source and the desti-

nation area. Assuming there is congestion on flowgate CD, the amount by which

trade τA,B aggravates congestion is denoted by xA,B. Consequently, trade τA,B pays

a congestion price which is equal to the congestion price that trade τB,A receives

for alleviating congestion by xB,A which is equal to −xA,B.

Compared with the bilateral NTC mechanism, the FBA concept leads to an in-

crease of the social welfare (see Appendix A.3) as a result of an economic optimisa-

tion of the most highly valued bids within the FBA region subject to transmission

capacity limits. Then, this optimisation also implies an optimal utilisation of the

physical infrastructure within the FBA region, which leads to an increased trad-

ing volume when compared to independent, bilateral NTC auctions. As a result

of the exact location of the transmission congestion, it additionally provides an

efficient investment foundation of the transmission infrastructure. However, the

implementation of the FBA mechanism within a limited region requires a high

level of cooperation and coordination among the participating TSO in the joined

auction. Quite some administrative procedures have to be carried out prior to the

bidding phase of the FBA auction, i.e. the exchange and the harmonisation of the

network model of each participating TSO, the merging of the network models and

the calculation of the technical parameters (PTDF matrix and NFC values).

3.8 Equilibrium Model of a Flow-based Auction 49

3.8 Equilibrium Model of a Flow-based Auction

An equilibrium model of a flow-based coordinated explicit auction is presented in

this section. The model is formulated by the optimisation problems of auction

participants and the auction office. Their complementarity conditions are derived

from their optimisation problem according to the procedure described in Chapter

2.1 and shown in Appendix B.1.

The bidders’ goal is to submit their bids for transmission capacity in order to ac-

quire transmission rights between regional power markets. The independent auc-

tion office collects all regional bids, settles a location-specific transmission price in

case of flowgate congestion and allocates the limited transmission capacity to those

bidders with the most highly-valued bids. Market clearing conditions are needed to

link the complementarity conditions of auction participants and the auction office

in order to find a unique market equilibrium.

The Bidders Model

In explicit auctions each bidder submits its bids for transmission capacity to the

auction office prior to the clearing of regional power markets. Based on the bidders’

electricity price forecasts in those markets, they specify their bid b in terms of the

requested quantity qbidb,c,c′, the bid price pbid

b,c,c′ and the desired direction, i.e. from

source area c to destination area c′. If congestion occurs on a given flowgate m

within the FBA region, a location-specific transmission price τb,c,c′ has to be paid

for the allocated quantity qb,c,c′ for bid b. That price is determined by the flowgate-

related congestion price ωm which is weighted by the impact of that bid on all

congested flowgates m ∈M according to

τb,c,c′ =∑

m∈M

ωm · PTDFc,c′,m (3.31)

where PTDFc,c′,m defines the flow on flowgate m resulting from a 1MW trans-

action from area c to area c′. As seen by Equation (3.32), the bidder’s profits

related to bid b then is the difference of its willingness to pay the price pbidb,c,c′ and

the location-specific transmission price τb,c,c′ times the allocated quantity qb,c,c′. If

there is no transmission congestion on any flowgate m ∈M , ωm equals to zero and

the bidder’s requested quantity qbidb,c,c′ equals to the allocated quantity qb,c,c′.


The profit maximisation problem for a given bid b can be stated as follows:

max∑

c,c′∈C

(pbidb,c,c′ · qb,c,c′ − (

∑

m∈M

ωm · PTDFc,c′,m) · qb,c,c′) (3.32)

s.t.: qb,c,c′ ≤ qbidb,c,c′ (νb,c,c′) (3.33)

qb,c,c′ ≥ 0. (3.34)

The Model of the Auction Office

The objective of the auction office is the efficient allocation of scarce transmission

capacity to the most highly valued transmission service subject to physical capacity

constraints on each flowgate. Each flowgate m is described by the net flowgate

capacity in both directions (NFC+m, NFC

−m) which limits the physical power flow

on that interconnection. If congestion occurs on a given flowgate m, i.e. the power

flow on that flowgate zm equals the flowgate capacity, a flowgate-related congestion

price ωm occurs. That price represents the system value of the congested flowgate.

By comparison with other congested flowgates, the level of ωm indicates the need

for an investment in that specific interconnection capacity.

The profit maximisation problem for the auction office yields to:

max∑

m∈M

ωm · zm (3.35)

s.t.: zm ≤ NFC+m (λ+

m) (3.36)

−zm ≤ NFC−m (λ−m) (3.37)

zm free.

The Market Clearing Condition

By the market clearing condition stated in Equation (3.38) the market participants’

profit maximisation problems are linked to each other in order to find a global

market equilibrium. All allocated bid quantities qb,c,c′ are transferred into the

physical power flow zm on flowgate m.

zm =∑

c,c′∈C

PTCDc,c′,m ·∑

b∈B

qb,c,c′ (ωm) (3.38)

3.9 Case Study - Explicit Flow-based Auction 51

A B

C D

Figure 3.16: Power system model

Table 3.5: PTDF matrix and NFC valuesc, c′ \ m AB AC BC BD CD m NFCm

A B 0.667 0.333 -0.167 -0.167 0.167 AB 100 MW

A C 0.333 0.667 0.167 0.167 -0.167 AC 200 MW

A D 0.500 0.500 0.000 0.500 0.500 BC 200 MW

B C -0.333 0.333 0.333 0.333 -0.333 BD 200 MW

B D -0.167 0.167 0.167 0.667 0.333 CD 60 MW

C D 0.167 -0.167 -0.167 0.333 0.667

3.9 Case Study - Explicit Flow-based Auction

The following example focusses on the application of the FBA mechanism within

a flow-based coordinated explicit auction which is modelled based on the comple-

mentarity conditions stated in Section 3.8. Special features in the application of

the FBA mechanism such as the bid characteristics, the marginal congestion pric-

ing, location-specific transmission prices and the rewarding of counterflows should

be highlighted.

3.9.1 Test System

Figure 3.16 displays the simplified transmission system model of a region where

cross-border transmission capacity is supposed to be allocated in a flow-based coor-

dinated explicit auction. That region consists of four areas c ∈ {A, B, C, D} that

are interconnected by flowgates m ∈ {AB, AC, BC, BD, CD}. Each area is con-

sidered to operate as a marketplace for electrical energy. Thus, for any inter-areal

transaction, transmission rights have to be acquired beforehand by submitting bids


Table 3.6: First set of bids and market clearing resultsb c c′ qbid

b,c,c′ pbidb,c,c′ qb,c,c′ τb,c,c′

b1 A B 40 3.00 40 0.0

b2 A B 40 2.50 40 0.0

b3 C B 50 2.00 50 0.0

b4 C B 50 1.50 50 0.0

to a central auction office, that are specified by price, quantity and direction. Ta-

ble 3.5 illustrates the PTDF matrix and the flowgate capacities by means of NFC

values. Those hold for both directions on a given flowgate.

3.9.2 Results

Table 3.6 presents a set of bids b ∈ {b1, b2, b3, b4} that is assumed to be submit-

ted to the auction office. According to Equation (3.38), all requested bid quantities

are transferred into the physical power flow on each flowgate by means of the corre-

sponding PTDFs. Here, all bids are accepted since they do not lead to a violation

of any flowgate capacity limit. Thus, the requested quantity qbidb,c,c′ corresponds to

the allocated quantity qb,c,c′ and there is no transmission price τb,c,c′ to be paid for

the acquisition of transmission rights.

Another bid (b5) is added to the existing set of bids as shown in Table 3.7. By

that additional bid, the capacity limit of flowgate AB is violated, hence, the re-

quested bid quantities cannot be allocated. As each single bid has an impact on

the congested flowgate AB, those with the highest market value of that flowgate

are accepted. The determination of each bid’s market value according to flowgate

AB is as follows:

kABb,c,c′ =

pbidb,c,c′

PTDFc,c′,AB(3.39)

Table 3.7 illustrates the values of all bids related to the congested flowgate. Since

bid b2 has the lowest value, the requested quantity of that bid is the first one to

be reduced (marginal bid). Figure 3.17 illustrates the power flow on each single

flowgate according to the requested and the allocated bid quantities. A limitation

of bid b2 at 30MW is sufficient to keep the power flow on the flowgates within

their capacity limits.

As for the determination of the location-specific transmission price τb,c,c′, the price


106.7 MW

A B 51 7 MW 51 7 MW

100.0 MW

A B 50 0 MW 50 0 MW

C D

3.3 MW

51.7 MW 51.7 MW

51.7 MWC D

0.0 MW

50.0 MW 50.0 MW

50.0 MW

Figure 3.17: Power flow for requested (left) and allocated (right) bid quantities

according to Table 3.7

Table 3.7: Second set of bids and market clearing resultsb c c′ qbid

b,c,c′ pbidb,c,c′ kAB

b,c,c′ qb,c,c′ τb,c,c′

b1 A B 40 3.00 4.50 40 2.50

b2 A B 40 2.50 3.75 30 2.50

b3 C B 50 2.00 6.00 50 1.25

b4 C B 50 1.50 4.50 50 1.25

b5 A B 30 3.00 4.50 30 2.50

of the marginal bid b2 sets the marginal congestion price related to flowgate AB:

τb2,A,B = pbidb2,A,B = 2.50 AC/MW

The remaining τb,c,c′, that are related to bids b′ ∈ {b1, b3, b4, b5}, are determined

based on those bids’ influence on the power flow on flowgate AB according to:

τb′,c,c′ = kABb2,A,B · PTDFc,c′,AB

Thus, the location specific transmission price of bid b2 also applies to bids b1 and

b5 since they have the same influence on the power flow on the congested flowgate.

Bids b3 and b4 have an influence which is half of the marginal bid, hence, their

transmission prices are half the price of bid b2.

A sixth bid (b6) requests a transmission right from area C to area B and is added to

the existing set of bids as depicted in Table 3.8. That additional bid increases the

loading of the transmission network and leads to a violation of the capacity limits

of flowgates AB and CD as shown in Figure 3.18. According to Table 3.8, bids

b2 and b4 are identified as the least-valued bids related to flowgates AB and CD,

respectively. Thus, their bid quantities are reduced in order to relieve transmission


123.3 MW

A B 100.0 MW

A B

C D

13.3 MW

68.3 MW 68.3 MW

C D

20.0 MW

60.0 MW 60.0 MW

C D68.3 MW

C D60.0 MW



Table 3.8: Third set of bids and market clearing resultsb c c′ qbid


b,c,c′ kCDb,c,c′ qb,c,c′ τb,c,c′

b1 A B 40 3.00 4.50 17.96 40 2.50

b2 A B 40 2.50 3.75 14.97 10 2.50

b3 C B 50 2.00 6.00 6.00 50 1.50

b4 C B 50 1.50 4.50 4.50 40 1.50

b5 A B 30 3.00 4.50 17.96 30 2.50

b6 C B 50 2.00 6.00 6.00 50 1.50

congestion on those flowgates. Since bids b2 and b4 are marginal bids, they define

the marginal congestion prices on flowgates AB and CD according to

τb2,A,B = pbidb2,A,B = 2.50 AC/MW

τb4,C,B = pbidb4,C,B = 1.50 AC/MW

Bids b1 and b5 have the same impact on the power flow on flowgate AB as marginal

bid b2, thus, their transmission prices correspond to τb2,C,B. Instead, the influence

of bids b3 and b6 on flowgate CD equals to that of marginal bid b4, so, the trans-

mission price they are charged is equal to τb4,C,B.

Alternatively, τb,c,c′ can be calculated by means of the flowgate-related congestion

price ωm and the PTDF of each bid related to the critical flowgates. That conges-

tion price describes the system value of the congested flowgate and is determined

by simulation. Here, congestion occurs on flowgates AB and CD and the market

simulation reveals ωAB = 3.50 AC/MW and ωCD = 1.00 AC/MW. For each bid b,

τb,c,c′ is determined according to Equation (3.31) as follows

τb,c,c′ = 3.50 AC/MW · PTDFc,c′,AB + 1.00 AC/MW · PTDFc,c′,CD


103 3 MW 100 0 MW103.3 MW

A B

33.3 MW

68.3 MW 88.3 MW

100.0 MW

A B

35.0 MW

67.5 MW 87.5 MW

C D48.3 MW

C D47.5 MW



Table 3.9: Fourth set of bids and market clearing resultsb c c′ qbid


b,c,c′ qb,c,c′ τb,c,c′

b1 A B 40 3.00 4.50 40 2.50

b2 A B 40 2.50 3.75 35 2.50

b3 C B 50 2.00 6.00 50 1.25

b4 C B 50 1.50 4.50 50 1.25

b5 A B 30 3.00 4.50 30 2.50

b6 C B 50 2.00 6.00 50 1.25

b7 D A 40 2.00 -4.00 40 -1.87

By that formulation, the total transmission price τb,c,c′, to be paid by bid b, is

decomposed in two portions each of which results from that bid’s impact on the

congested flowgates AB and CD. Basically, the transmission price for any trans-

action between two areas consists of several portions of costs resulting from that

transaction’s impact on the power flow on all congested flowgates anywhere located

in the FBA region.

Another bid (b7) is added to the existing set of bids according to Table 3.9. That

bid is specified by a request for transmission service from area D to area A. As

illustrated by Figure 3.19, the requested quantities cannot be allocated since their

combination violate the capacity limit of flowgate AB. Again, the requested quan-

tity of marginal bid b2 is reduced until that flowgate reaches its limit. Then, bid

b2 also sets the marginal congestion price related to flowgate AB.

While bids b1 to b6 lead to a power flow on flowgate AB in direction of the conges-

tion, bid b7 induces a power flow in the opposite direction of that flowgate. Thus,

according to its impact on that flowgate, the transmission price to be received by


bid b7 for relieving congestion on flowgate AB is calculated as

τb7,D,A = ωAB · PTDFD,A,AB

The ex ante determination of ωAB yields to 3.75 AC/MW.

3.10 Concluding Remarks

The FBA mechanism is an extension of the bilateral allocation of exchange capacity,

currently applied in NTC auctions, that allocates physical transmission capacity of

multiple interconnected cross-borders within a specified region. That region, also

denoted as the FBA region, is represented by a simplified model of the transmis-

sion system. There two approaches of capacity models for flow-based allocation

currently under investigation: One approach takes into account a set of individual

critical network branches (critical branch approach) while the other determines

physical cross-border capacities between neighbouring control areas (flowgate ap-

proach). While the former one provides a better representation of the physical

transmission network, the latter one leads to more transparency in terms of trad-

ing and the derivation of the location-specific transmission prices.

A key feature of flow-based auctions is the transformation of commercial trans-

actions between two areas into the physical power flow induced on the flowgates

within the FBA region. The central element, by which this transformation is per-

formed, is the PTDF matrix, which contains the flowgate-related sensitivities of

power flow changes based on changes in the power exchange from a source to a

destination area. Several factors have an impact on the accuracy of AC-based

PTDFs: the method, the volume, the direction of the generation shift and the un-

derlying base case scenario of the transmission system. By applying the DC load

flow approximation, a part of the PTDF matrix can be calculated algebraically.

Compared to the AC-based PTDF matrix, the number of simulation runs for de-

termining the complete PTDF matrix within the FBA region can be reduced by

the amount of areas in that region.

Regarding the conceptual derivation of flowgate capacities, there are several par-

allels to the bilateral NTC approach. Both are based on a base case scenario of

the power system upon which a maximum additional generation shift is performed

between two neighbouring areas while accounting for the (n-1) system security cri-

terion. Furthermore, the NTC and NFC are allocated in different time horizons

by accounting for already allocated capacities. The major difference between these

two approaches lies in the definition of the capacities. While the NTC defines a

3.10 Concluding Remarks 57

power exchange capacity, the NFC is a flowgate-related physical transmission ca-

pacity between neighbouring control areas.

An equilibrium model of the flow-based coordinated explicit auction is presented.

All bids for cross-border transmission capacity between different areas in the FBA

region compete with each other. Bids are defined by quantity, price and the spec-

ification of source and destination area unlike the NTC approach where bids need

to be specified for each single cross-border. A central entity, the auction office,

collects all area-wide bids and allocates scarce cross-border transmission capacity

in an economic optimisation by identifying the value each single bid has on the con-

gested flowgates subject to flowgate capacity limits. Those bids with the lowest

values are first to be reduced when the capacity limit on any flowgate is violated.

If the bid is accepted, a location-specific transmission price is to be paid based on

the bid’s impact on the power flows on all congested flowgates. Those bids which

lead to a flow in the opposite direction of a congested flowgate receive a payment

for relieving transmission congestion.

4 Modelling Strategic Generator

Behaviour

The predominantly oligopolistic structure of the power supply market provides the

basis for strategically behaving power producers to hold or further increase electric-

ity and transmission prices above competitive levels by exerting their market power

in the markets for electrical energy and cross-border transmission capacity. In this

chapter, the concept of conjectural variation is introduced as a more flexible mod-

elling approach of strategic market behaviour by supplying companies than the most

widely applied concepts of perfect and Cournot competition. It is derived for the

electricity and transmission market. Subsequent to that, a formulation of the power

market model is presented by means of complementarity-based equilibrium condi-

tions of power producers, arbitragers and a centralised auction office. The chapter

closes with a case study on strategically behaving power producers in electricity and

transmission markets based on a simplified transmission-constrained power market

model.

4.1 Introduction

One of the goals with the introduction of liberalisation in the electricity mar-

ket sector is to enhance competition among generating companies in order to im-

prove social welfare (see Appendix A.3) and market efficiency. In this new market

environment, power producers face different operation strategies and investment

decision risks in their medium and long-term planning than in the previously mo-

nopolistic electricity market as they have to consider competitive behaviour from

power producers. Meanwhile, new power suppliers face a high barrier to enter the

electricity market as the construction of new power plants is costly, both in terms

of capital and time. Thus, the supply side is more akin to an oligopoly as the

59

60 4 Modelling Strategic Generator Behaviour

market is often divided among a few generating companies each with large market

shares. The oligopolistic market structure provides a basis for those companies

to exert their market power in order to affect electricity and transmission prices

profitably.

In addition to bilaterally contract consumers of electrical energy, like municipal

utilities and industries, strategic companies may participate in markets for electri-

cal energy known as power exchanges. Those are often based on an uniform price

auction, where the market clearing price is determined by the supply bid of the

marginal unit. In order to gain higher profits, strategically behaving companies

can adjust their bid characteristics. Furthermore, they have an incentive to restrict

their generation output in order to raise the offer price on marginal units. Since

supply has to meet the demand at any time, electricity prices rise above competitive

levels. According to the transmission market, strategic companies take advantage

of their location towards the congested transmission line, or flowgate, and redis-

tribute their trades in order to influence the demand for transmission capacity and

thus the transmission price. In both cases large generating companies exploit their

market dominance to influence market prices to their favour.

In order to understand the complexities of competition and to help analyse mar-

ket designs and regulatory policies, computationally tractable models of strategic

interactions among generating companies are becoming increasingly important.

Frequently used modelling technique of oligopolistic electricity markets are equi-

librium models which are based on the profit maximisation of each market actor.

As discussed in Chapter 2, a market equilibrium is defined as a set of market

outcomes, which no market participant would modify unilaterally by his actions

without decreasing its profits. Many oligopolistic price equilibrium models appear

in the literature targeting at studying strategic behaviour among competing gen-

erating companies. In general, these models differ in terms of market design, the

type of oligopoly game and the solution methodology.

Focussing on the interaction between market participants, different game theoret-

ical approaches are being used to provide strategic generators with the essential

knowledge and information to influence the market equilibrium to their favour.

Assuming the company’s strategic variable to be the generation quantity rather

than the price bid (Bertrand model, [7], [13]), the most classical concepts are per-

fect and Cournot competition. Both are widely used in the literature for analysing

competitive large-scale market models while retaining the mathematical convex-

ity. However, both the concepts tend to either underestimate (perfect competi-

tion) or overestimate (Cournot competition) the electricity price. A more sophis-

ticated modelling approach of strategic interaction among generating companies is

4.2 Classical Concepts from Game-Theory 61

achieved by supply function equilibrium models [25]. Each company finds its opti-

mal supply function, while embedding the KKT optimality conditions of the ISO

in its set of constraints. This approach, however, comes along with an increased

computational complexity, which puts a constraint on its application to large-scale

systems. By introducing the concept of conjectural variation, each strategic com-

pany estimates its competitors’ responses to its own strategic actions in the energy

and transmission market. Thus, different levels of competition can be simulated

among generating companies by parametrically changing the companies conjecture

on its competitors’ responses. Additionally, some formulations show how to esti-

mate that parameter bases on historical market data. Ultimately, this approach

leads to more credible electricity and transmission prices than the concepts of per-

fect and Cournot competition do.

As for the market design, pool- or bilaterally-based electricity market models are

most common. In pool-based markets companies sell power to the ISO at the lo-

cational price of their bus, while in bilateral market models customers or power

exchanges are individually contracted. If transmission constraints are included, a

linearised DC load flow is adopted in most cases.

This chapter is structured as follows: Section 4.2 focusses on the classical game

theoretical concepts of perfect and Cournot competition while the subsequent Sec-

tion 4.3 presents the integration of the concept of conjectural variations in the

energy and transmission market. In Section 4.4 the oligopolistic equilibrium model

is presented. The chapter closes with a numerical example presented in Section

4.5.

4.2 Classical Concepts from Game-Theory

The modelling of electricity markets based on either perfect or Cournot competi-

tion is widely spread in the literature. Both concepts are based on rather simplified

assumptions with respect to the integration of competitors’ responses when a power

supplier decides its optimal generation output. Hence, they fail at providing cred-

ible market prices.

In perfectly competitive markets, suppliers believe they cannot affect prices, e.g.

by holding back generation from the market, because generation and sales profits

would be lost at the favour of their competitors. Hence, electricity prices are based

on the producers’ marginal generation costs while transmission is priced according

to the difference of corresponding electricity prices. Cournot competition, however,


Gf

Node BNode A

~~

f

G-fDA

Figure 4.1: Transmission constrained network

leads to comparatively high electricity prices as producers neglect their competi-

tors’ response to their strategic actions.

In order to provide a better insight into the basic game-theoretical concepts, their

impact on market results will be derived based on the simplified power system

shown in Figure 4.1. It represents a lossless transmission constrained network with

electricity demand located at node A to be served through a congested transmis-

sion line by generation located at node B. Total generation (Gf + G−f ) is shared

by company f and its competitors −f and is assumed to have no generation capac-

ity limits. Following the nomenclature in Chapter 2.2, company f ’s only decision

variable (Xf ) is its generation output:

Xf = Gf . (4.1)

Furthermore, assuming a linear decreasing demand function with intercept (α > 0)

and slope (β > 0), the electricity price at node A (πA) can be expressed as follows:

πA = πA(Gf , G−f ) = α− β · (Gf +G−f ). (4.2)

If there is no congestion on the transmission line, no transmission costs occur for

any company in order to satisfy the demand. In case the transmission line is

congested, a transmission price ω−→BA

has to be paid, which itself is assumed to be

a function of total transferred generation according to:

ω−→BA

= ω−→BA

(Gf , G−f ) = δ · (Gf +G−f ). (4.3)

By introducing parameter δ (> 0), ω−→BA

and Gf + G−f are linearly correlated.

Strictly speaking, this correlation only holds in case of transmission congestion.

The validity of Equation (4.3) can also be extended to complex meshed systems,

where no company is likely to anticipate the impact of its own decisions on the

transmission price. In order to account for transmission costs, company f ’s profit

function, as formulated in Equation (2.13), changes to

Πf = πA ·Gf −MCf ·Gf − ω−→BA

·Gf , (4.4)

4.2 Classical Concepts from Game-Theory 63

where marginal generation cost MCf are assumed to be constant resulting in linear

total cost.

The focus of the following studies is on the impact of company f ’s generation

output on electricity and transmission prices while assuming perfect competition

on the one hand (see Subsection 4.2.1) and Cournot competition on the other hand

(see Subsection 4.2.2).

4.2.1 Perfect Competition

The implicit assumption of perfectly competitive electricity markets is that there

are an infinite number of competitors so that market outcomes cannot be affected

by the modification of any market participants’ decision. Thus, according to the

power system in Figure 4.1, each company acts as a price taker with respect to the

electricity and transmission price:

πA = π∗A (4.5)

ω−→BA

= ω∗−→BA. (4.6)

Both equations are basic assumptions when designing perfectly competitive mar-

kets. Each company takes the electricity and transmission price as fixed, which is

indicated by the asterisk (∗). When calculating company f ’s optimal generation,

its profit function, described in Equation (4.4), is derived by decision variable Gf ,

set to zero and rearranged:

∂Πf

∂Gf= π∗A −MCf − ω∗−→

BA= 0

⇐⇒ π∗A − ω∗−→BA

= MCf . (4.7)

In this model, each company should increase its production up to the point where

its marginal generation cost is equal to the difference of electricity and transmission

price. If transmission is not constrained, the electricity price equals the marginal

generation cost. When each market participant acts in a perfectly competitive

manner, the most efficient operation of production capacities is guaranteed, which

ultimately results in competitive market prices.

In reality, there are often just a few supplying companies with a substantial pro-

portion of market share. Thus, market participants can abuse their market power

by strategic behaviour in order to raise electricity and transmission prices above

competitive levels. Therefore, the design of perfectly competitive markets does not

result in realistic electricity prices. In fact, electricity prices are underestimated in

many cases.


4.2.2 Cournot Competition

Assuming that generators are not able to change market prices results in com-

petitive and rather unrealistic electricity and transmission prices. Applying the

Cournot strategy gives market players the possibility to influence market prices

through their actions. However, they believe that other companies will not alter

their decisions. As long as demand functions are monotonously decreasing, a mar-

ket equilibrium can always be found [26].

Based on the power system shown in Figure 4.1, generating companies should now

act in a Cournot-like manner with respect to the electricity and the transmission

market. In order to distinguish the impact of Cournot behaviour on electricity and

transmission prices in a better way, both markets will be studied separately.

Focus on the electricity market

Each company plays a Cournot game in the electricity market while assuming that

it cannot alter the transmission price by its actions according to Equation (4.6).

Thus, based on Equation (4.4) company f ’s equilibrium condition for its optimal

production output is (see Appendix C.1.1 for full derivation):

∂Πf

∂Gf=

∂πA(Gf , G∗−f )

∂Gf·Gf + πA −MCf − ω∗−→

BA= 0

⇐⇒ πA − ω∗−→BA

= MCf + β ·Gf . (4.8)

Equation (4.8) shows that the difference of electricity and transmission price does

not correspond to company f ’s marginal generation cost. Moreover, as company

f holds back generation from the market, electricity price πA rises above MCf by

β · Gf . Additionally, πA might further be boosted due to an increase of demand

slope β. As there is no correlation among strategic companies, because companies

assume that withdrawing their generation does not have an effect on their com-

petitors’ generation output (∂G∗

−f

∂Gf= 0), Cournot models of strategic interaction

tend to overestimate electricity prices.

Focus on the transmission market

Now, all companies behave in a Cournot-like manner with respect to transmis-

sion while acting as a price taker in the electricity market as seen in Equation

(4.5). Based on Equation (4.4), the optimality condition according to company f ’s

4.3 The Concept of Conjectural Variation 65

generation output is (see Appendix C.1.1 for full derivation):

∂Πf

∂Gf= π∗A −MCf − ∂ω−→

BA(Gf , G

∗−f )

∂Gf·Gf − ω−→

BA= 0

⇐⇒ π∗A − ω−→BA

= MCf + δ ·Gf . (4.9)

Each company f naively assumes that changes in its generation output do not affect

electricity prices, thus, transmission price ω−→BA

drops as a result of holding back

generation output. However, company f ’s optimality condition does not include

any response of its competitors, thus, the company’s strategies might have a too

high impact on ω−→BA

.

4.3 The Concept of Conjectural Variation

The ability of strategic companies to correctly anticipate the impact of their gen-

eration output on the electricity price, requires the KKT optimality conditions of

social welfare maximisation to be imbedded in each company’s set of constraints.

Then, the optimisation problem for each company becomes a Mathematical Pro-

gram with Equilibrium Constraints (MPEC), which has a non-convex objective

function. As each strategic company has to solve an MPEC, the overall problem

becomes an Equilibrium Problem with Equilibrium Constraints (EPEC), which is

generally difficult to compute due to its non-convex feasible region [10], [27].

Although the application of classical concepts such as the widely known perfect or

Cournot competition leads to market models that are solvable for realistically large

systems, those models lack of providing credible electricity price. The conjectural

variations approach is gaining increasing attention in the literature as it allows for

a flexible representation of competing generating companies’ interactions in the

electricity market while being numerically tractable for large systems [5]. Basi-

cally, when a company decides its optimal production, the concept of conjectural

variation considers the response of its competitors by anticipating how they change

their generation as a result of a change in the company’s generation output. This

improvement allows a more accurate price generation process than the more com-

monly applied concepts of perfect and Cournot competition.

In the literature, there are several concepts of conjectural variation, which actually

have their origin in the field of microeconomics [6]. They all have in common that

competitor companies’ reactions are integrated in each strategic company’s profit

maximisation problem. Those concepts mainly differ related to the time scale, they

are applied to and in terms of the definition of competitors’ responses. In [5] the


linear conjectured supply function approach is introduced to oligopolistic power

markets. This approach can be taken as an approximation of the supply function

equilibrium model by assumed linear supply functions around the market equilib-

rium that represent each strategic company’s belief how total competitor supply

varies due to price changes. The results show that such additional responsiveness

leads to a reduction of electricity prices. In [28], [29] the competitors’ reactions are

modelled by means of residual demand functions, which are obtained by subtract-

ing the competitors’ supply functions from the electricity demand curve. Basically,

the residual demand functions represents each strategic company’s potential to ma-

nipulate the market price. This leads to more accurate market prices compared

with Cournot competition. While the previous works mainly refer to model each

strategic company’s decision in the medium-term horizon (see Chapter 2.5), the

subsequent ones are related to short-term strategies, as individual supply functions

are determined for bidding in spot markets.

In [30] a conjectured supply function equilibrium model is derived from the sup-

ply function equilibrium theory. It determines a general supply function for each

strategic company based on the companies’ conjecture on its competitors’ change

of supply due to price changes. If the conjectures are constant, then supply func-

tions will be linear and a unique equilibrium exists. Furthermore, the conjectured

supply function equilibrium model can be applied when demand is inelastic. In

[31], [32] it is stated that the companies’ profits in electricity spot markets can

be maximised when their bidding strategy is based upon conjectural variation of

competitors’ change in generation output. Furthermore, it is shown that classi-

cal game-theoretic concepts such as perfect and Cournot competition, Stackelberg

(leader and follower) and the case of monopoly are covered by the conjectural vari-

ations approach. In [33], [34] a closed-form formulation of the conjectural variation

based on the competitors’ reactions to changes in the generation output is derived.

The conjectural variation parameters are calculated based on available historical

market data, such as marginal generation cost, market share, price elasticity of

demand and market clearing prices.

Just a few market equilibrium models account for some kind of interaction of

strategically behaving companies in the transmission market while, in many cases,

the transmission network is not considered at all [28], [29], [33]. In [5], [13], [35]

the transmission market is assumed to be perfectly competitive, as each company

beliefs that its actions will not affect the transmission price. In contrast, mod-

els in which a company correctly anticipates the influence of its decisions on the

transmission price, require the first-order conditions for the TSO to be imbedded

in the companies’ constraint set [10]. In this way, strategic generators foresee the

4.3 The Concept of Conjectural Variation 67

impact of their actions on the transmission price. As in energy markets, the re-

sulting profit maximisation problem for each generator becomes an MPEC. Then,

the overall market model is formulated as an EPEC and for that reason such a

model is generally difficult to compute for large systems. Thus, the assumption

of a perfectly competitive transmission market is a compromise between compu-

tational tractability and the representation of the transmission network. In [36]

a hybrid Bertrand-Cournot model with respect to transmission decisions by the

ISO is presented, in which, depending on the congestion pattern within individual

subnetworks, the choice for one approach is more defensible than for the other.

Alternatively, smooth functions for modelling the manipulation of the transmis-

sion prices can be included. This has been done by introducing the conjectured

transmission price response function in [27], [37]. It allows strategic companies to

behave in a more sophisticated manner, such that they anticipate how the prices of

transmission will change as a result of the companies’ demand for transmission ser-

vices. Furthermore, it overcomes the difficulties of solving MPEC/EPEC models

of strategic interaction.

4.3.1 Conjectural Variations in the Electricity Market

Firstly, the conjectural variations approach is applied to the electricity markets

(CVE) while assuming the transmission price to be fixed. Equation (4.10) yields

company f ’s optimality condition for its generation output (see Appendix C.2.1

for full derivation)

∂Πf

∂Gf=

∂πA(Gf , G−f )


BA= 0

⇐⇒ πA − ω∗−→BA

= MCf + β · (1 + CV Ef) ·Gf . (4.10)

while CV Ef is defined as

CV Ef =∂G−f (Gf )

∂Gf. (4.11)

Equation (4.11) represents the variation of the competitors’ total generation when

company f ’s generation output changes. In that sense, a conjectural variation

is the conjecture company f holds on competitors’ total generation which corre-

sponds to a first-order Taylor approximation of the local response of other suppliers

around the market equilibrium [5], [9]. Meanwhile, CV Ef = −1 yields the per-

fectly competitive case, described in Equation (4.7), while CV Ef = 0 corresponds

to Cournot competition as shown in Equation (4.8).


However, both these concepts lead to either under- or overestimate electricity prices

as each of them defines only one specific level of strategic interaction. The true

value of conjectural variations is its flexibility to model various levels of competition

among strategic companies, ranging from perfect competition to no competition

(Cournot). By parametrically varying the conjecture about total competitors’ sup-

ply response according to

−1 ≤ CV Ef ≤ 0, (4.12)

a range of electricity prices can be modelled as indicated in Equation (4.10). Fur-

thermore, the impact of a high demand slope on the electricity price increase is of

less significance due to being scaled down by 1 + CV Ef .

4.3.2 Conjectural Variations in the Transmission Market

In a very similar way, the concept of conjectural variations can be applied to the

transmission market (CVT). For simplification, the electricity price is assumed to

be fixed according to Equation (4.5). Thus, company f ’s optimality condition for

its generation yields to (see Appendix C.2.2 for full derivation):

∂Πf


BA(Gf , G−f )


BA= 0

⇐⇒ π∗A − ω−→BA

= MCf + δ · (1 + CV Tf) ·Gf . (4.13)

while CV Tf is defined as

CV Tf =∂G−f (Gf )

∂Gf. (4.14)

Equation (4.13) indicates that company f ’s impact on the transmission price is

less than in Cournot competition when respecting other companies’ responses in

the transmission market. Similar to CV Ef , perfect and Cournot competition are

modelled by CV Tf = −1 and CV Tf = 0, respectively. Different intensities of

strategic interactions can be simulated by varying the CV Tf parameter within its

definition range:

−1 ≤ CV Tf ≤ 0. (4.15)

It is important to note that the conceptual derivation of CV Ef and CV Tf leads

to an identical definition of both concepts as Equations (4.11) and (4.14) demon-

strate. However, in a more realistic scenario of electricity markets, several cus-

tomers/power exchanges and several strategic companies exist that can be located

4.4 Setting up the Equilibrium Model 69

at different places in the network. Those companies usually have strategic vari-

ables such as generation output, energy sales and demand for transmission capacity.

Thus, the companies’ energy sales might be a better fit in terms of the definition of

CV Ef than the companies’ generation output. Similarly, the demand for transmis-

sion capacity replaces the companies’ generation output according to the definition

of CV Tf .

4.4 Setting up the Equilibrium Model

According to the definition in Chapter 2.1, the network-constrained market equi-

librium model is a MLCP as a result of using linear marginal generation costs and

linear demand functions. It simulates a bilateral market with perfect arbitrage

based on a flow-based allocation of transmission capacity. The major market par-

ticipants are generating companies and arbitragers while an Independent System

Operator (ISO) represents the grid owner and the market operator together. Each

of them solves a profit maximisation problem based on a set of constraints in order

to find the optimal values of their decision variables. The market clearing con-

ditions couple the equilibrium conditions of all participants in order to enforce a

market equilibrium. Consumers are modelled by means of a linear price-demand

curve and included in the companies’ and arbitragers’ optimisation problem. Based

on a reference price for electricity (πrefc ), a reference electricity demand level (dref

c )

and a price elasticity of demand (εelc ) (see Appendix A.1), the electricity price (πc)

for each area c ∈ C is given by

πc = πrefc · (1 − 1

εelc) +

πrefc

drefc

· 1

εelc· (

∑

f∈F

sf,c + asc − apc), (4.16)

where sf,c denotes the energy sales by company f in area c while asc and apc are

the arbitragers’ sales and purchases in the same area. In a very similar way, the

congestion price (ωm) can be formulated by means of the price elasticity of demand

for transmission capacity (εtrm) (see Appendix A.2) for each flowgate m ∈M

ωm =ωref

m

NFCm· 1

εtrm· (

∑

f∈F

tfgf,m + atfgm), (4.17)

where tfgf,m denotes company f ’s and atfgm the arbitragers’ allocated physical trans-

mission capacity. By definition, congestion price ωm only arises from physical

congestion on flowgate m. Thus, if flowgate m is fully congested (NFCm =


∑f∈F t

fgf,m + atfgm), then congestion price ωm is determined by Equation (4.17).

Otherwise (NFCm ≥ ∑f∈F t

fgf,m + atfgm), ωm is equal to zero.

In the following, each market participant is described by its optimisation problem

and the set of constraints. The KKT optimality conditions are presented in Ap-

pendix C.3.

The Power Producer Model

The optimisation problem of a generating company f can be formulated as follows:

max∑

c∈C

πc · sf,c −∑

c′∈C

∑

h∈H(f,c′)

MCf,h,c′ · gf,h,c′ −∑

m∈M

ωm · tfgf,m (4.18)

s.t.: gf,h,c ≤ Gf,h,c (μf,h,c) (4.19)

∑

c′∈C

tf,c,c′ −∑

h∈H(f,c)

gf,h,c = 0 (θgf,c) (4.20)

tfgf,m −∑

c,c′∈C

tf,c,c′ · PTDFc,c′,m = 0 (ηf,m) (4.21)

sf,c −∑

c′∈C

tf,c′,c = 0 (θsf,c) (4.22)

gf,h,c, tf,c,c′, sf,c ≥ 0. (4.23)

Each generating company f may own one or several generation units h ∈ H(f, c) to

be located in area c ∈ C that are described by generation output gf,h,c and marginal

generation cost MCf,h,c. According to Constraint (4.19), generation is limited by

the maximum capacity Gf,h,c. Constraint (4.20) accounts for the energy balance

between company f ’s total generation in area c and its total trades tf,c,c′ from that

area to areas c′. The physical impact of company f ’s total trades on flowgate m is

given by tfgf,m which results from multiplying total commercial trades tf,c,c′ by the

power transfer distribution factor PTDFc,c′,m as indicated by Constraint (4.21).

If there is congestion on flowgate m, power producer f ’s induced physical power

flow tfgf,m on that flowgate is charged against the congestion price ωm. Depending

wether their total trades aggravate or relieve congestion on flowgate m, company f

makes or receives a payment from the auction office according to ωm. Furthermore,

power producer f ’s total trades to area c must be consistent with its energy sales

sf,c to that area as stated in Constraint (4.22).

A strategic company has to decide on its generation gf,h,c, trades tf,c,c′ and sales

sf,c output in order to maximise its net profits which is formulated by the optimi-

sation function (4.18). Company f ’s net profits (NPf ) can also be stated as the

4.4 Setting up the Equilibrium Model 71

company’s generation plus transmission cost (GCf + TCf ) to be subtracted from

sales profits (SPf ) as indicated by the optimisation function.

The Arbitrager Model

The arbitragers’ optimisation problem is formulated as follows:

max∑

c∈C

π∗c · asc −∑

c′∈C

π∗c′ · apc′ −∑

m∈M

ω∗m · atfgm (4.24)

s.t.:∑

c∈C

(−asc + apc) = 0 (ρa) (4.25)

∑

c′∈C

atc,c′ − apc = 0 (θapc ) (4.26)

atfgm −∑

c,c′∈C

atc,c′ · PTDFc,c′,m = 0 (ηam) (4.27)

asc −∑

c′∈C

atc′,c = 0 (θasc ) (4.28)

apc, atc,c′, asc ≥ 0. (4.29)

Arbitragers erase any non-cost based price differences by trading electricity between

two areas until the price for transmission meets the difference of corresponding elec-

tricity prices. They do not own any generation capacities, thus, total electricity is

purchased in one area (apc) and sold in another area (asc′). The balance of total

purchased and sold energy is given by Constraint (4.25). Electricity purchased in

area c must meet total trades atc,c′ from that area to areas c′ (Constraint (4.26)).

The consistency condition relating to the arbitragers’ sales and total trades to area

c is stated in Constraint (4.28).

Since arbitragers are assumed to act perfectly competitive, electricity and transmis-

sion price are fixed (indicated by the asterisk ∗) as formulated in the optimisation

function (4.24). Consequently, the arbitragers’ sales profit always meet its total

cost of generation and transmission.

The Model of the Auction Office

The optimisation problem of the auction office can be formulated as follows:

max∑

m∈M

ω∗m · zm (4.30)

s.t.: zm ≤ NFC+m (λ+

m) (4.31)


−zm ≤ NFC−m (λ−m) (4.32)

zm free.

The objective of the auction office is the efficient allocation of scarce transmission

capacity to the most highly valued transmission service subject to physical capacity

constraints on each flowgate. Each flowgate m is described by flowgate capacities

in both direction (NFC+m, NFC

−m) which limit the physical power flow on that

flowgate. The auction office behaves like a price taker with respect to transmission

prices (ω∗m) while deciding on the most valuable set of transmission services (zm)

to provide in order to maximise its profits.

The Market Clearing Conditions

The market clearing conditions are stated as follows:

πc = πrefc · (1 − 1

εelc) +

πrefc

drefc

· 1

εelc· (

∑

f∈F

sf,c + asc − apc) (π∗c ) (4.33)

zm =∑

c,c′∈C

PTDFc,c′,m · (∑

f∈F

tf,c,c′ + atc,c′) (ω∗m) (4.34)

Market clearing conditions are needed to correlate the market participants’ profit

maximisation problem in order to find a global market equilibrium. Condition

(4.33) presents the electricity price in area c as a function of all companies’ total

sales and the arbitragers’ sales and purchases in that area. Basically, all supplying

companies and arbitragers contribute to the process of price formation. By the

second market clearing condition (4.34), all commercial transactions by producers

and arbitragers are transferred into their physical power flow on flowgate m. It

represents one of the key items of flow-based allocation of transmission capacity.

4.5 Case Study - Simple System

The following case study presents a power market when the supplying industry is

oligopolistic. As generating companies own substantial proportions of total gen-

eration, they are able to abuse their market power through strategic behaviour

in the electricity and transmission market. There is no accounting for arbitragers

since market prices are assumed to be subject mainly to the generating companies’

market behaviour. The purpose of the study is twofold: Firstly, as there are just

4.5 Case Study - Simple System 73

a few power suppliers to satisfy system-wide electricity demand, electricity prices

are mainly subject to their strategic decisions. The aim is to make a statement on

the overall impact of imperfect competition in oligopolistic electricity markets on

the system economy. Secondly, as strategic companies make cross-border transac-

tions, they are able to manipulate the congestion price by changing their demand

for transmission capacity. However, due to their location with respect to the con-

gested flowgate, strategic companies apply different strategies in the transmission

market. This approach aims at providing some insight into the individual sup-

pliers’ strategies to beneficially affect the location-specific transmission price they

have to pay.

Since there are no arbitragers, only supplying companies are considered to satisfy

system-wide electricity demand. Thus, their generation, trade and sales policies

are the main drivers to influence electricity and transmission price levels. In terms

of strategic behaviour, each power supplier makes conjectures on the change of

competitors’ sales of electricity and demand for transmission service to their own

decisions. Starting from the perfectly competitive case, the intensity of competi-

tion between the companies will be decreased until the least competitive case is

reached, i.e. the Cournot competition. The whole spectrum between perfect and

Cournot competition is achieved by changing the companies’ conjectures in the

electricity and transmission market parametrically. The power market is modelled

in terms of market equilibrium conditions which are formulated as a MLCP as

described in Section 4.4. A bilateral market is simulated based on a DC load flow

approximation. See Chapter 2.3 for more details.

4.5.1 Test System

Figure 4.2 displays three areas c ∈ {A,B,C} interconnected by flowgates m ∈{AB,AC,BC}. Each area can be viewed as a zone overlapping with country bor-

ders, which is controlled by the national TSO. Furthermore, areas are considered to

operate as marketplaces for electrical energy, which is equivalent to the operation

of power exchanges, resulting in area-specific electricity prices, demand and supply

levels. For simplification, electricity prices are linearly correlated with demand by

specifying a price elasticity of demand (εelc ). Reference values in terms of electricity

price (πrefc ) and demand (dref

c ) are given on the left hand side of Table 4.1.

There are three generating companies f ∈ {fA, fB, fC}, with company fA located

in area A, company fB in area B and company fC in area C. Each of them has

constant marginal generation cost and an unlimited generation capacity accord-


Area B Area C

Area A

Figure 4.2: Power system model

Table 4.1: Electricity demand and supplying companies’ generation parameterc πref

c drefc εelc f MCf Gf

A 15.00 400 -0.3 fA 13.00 ∞B 15.00 400 -0.3 fB 17.00 ∞C 15.00 400 -0.3 fC 15.00 ∞

ing to the right hand side of Table 4.1. Since power suppliers may sell power

to areas where they have no generation units, they have to be charged for the

transmission service they use. Thus, as far as the congestion management mecha-

nism is concerned, markets are coupled through a PTDF-based implicit auctioning

of transmission capacity. Transmission capacity within a zone is assumed to be

adequate, so that intra-zonal transactions will not cause congestion on any trans-

mission line. In terms of inter-zonal transactions only flowgate AB is assumed to

have limited capacity. Table 4.2 gives the parameters assumed for the application

of flow-based market coupling.

4.5.2 Configuration of Conjectural Variations Parameters

The generating companies’ intensity of competitive market behaviour can be de-

scribed by specifying their conjectured parameters according to the electricity and

transmission market. As to electricity markets, they make conjectures about their

competitors’ change in sales in area c due to their own change in sales output in

the same area according to

CV Ef,c =∂s−f,c(sf,c)

∂sf,c. (4.35)


Table 4.2: PTDF matrix and NFC valuesc, c′ \ m AB AC BC m NFCm

A B 0.667 0.333 -0.333 AB 25

A C 0.333 0.667 0.333 AC ∞B C -0.333 0.333 0.667 BC ∞

Based on the configuration of the power system shown in Figure 4.2, there are

f ∈ {fA, fB, fC} power producers and c ∈ {A,B,C} areas, thus, CV Ef,c is a

3×3 matrix. In the transmission market, strategic companies conjecture how their

competitors change their total trades that affect flowgate m as a result of their

own variation of trade output

CV Tf,m =∂tfg−f,m(tfgf,m)

∂tfgf,m

. (4.36)

As there are m ∈ {AB,AC,BC} flowgates, the dimension of CV Tf,m is equal to

3 × 3. However, those CVT parameters according to flowgates m ∈ {AC,BC} do

not have to be determined, since transmission congestion does not occur on these

flowgates. Thus, only CV Tf,AB is significant. For simplification, the following

assumptions are made according to the specification of CV Ef,c:

1. Power producers play the same level of competition among each other

2. Competition is equal in all areas

The first assumption implicitly reflects the supplying companies’ behaviour in an

oligopolistic market. As the supply side is shared by a few large generating com-

panies, it is believed that each company’s market share is large enough, such that

the impact of the company’s strategy on electricity prices is equal among them.

Mathematically speaking, the suppliers’ conjecture on their competitors’ change of

sales in area c is equal among all suppliers:

CV Ef,c = CV Ec ∀f (4.37)

As there is no discrimination among the companies concerning the level of compe-

tition, each company holds the same market share as in the perfectly competitive

case. The second assumption permits the interpretation of the results in a more

condensed form since market imperfections are assumed to be equal in each area:

CV Ef,c = CV Ef ∀c (4.38)


Table 4.3: Results for perfect competitionA B C AB AC BC

πc [AC/MWh] 13.02 17.01 15.02 ωm [AC/MWh] 6.00 0.00 0.00

dc [MW] 416 384 400 zm [MW] 25 13 -13

gc [MW] 453 346 400

MSfA[%] 38 38 38 fA fB fC

MSfB[%] 29 29 29 SP [AC] 6787 5180 5984

MSfC[%] 33 33 33 GC [AC] 5894 5888 5998

TC [AC] 884 -713 21

PS [AC] CS [AC] AR [AC] SW [AC] NP [AC] 9 5 7

0 30032 150 30182

Both assumptions conform with the overall goal to examine the economic impact

of the supplying companies’ strategy in oligopolistic electricity markets. Thus,

CV Ef,c is equally changed for all producers and in all areas.

Starting from the perfectly competitive case, parameters CV Ef,c and CV Tf,m will

be changed according to the following procedure:

CV E0 = −1.0

CV Ev = CV Ev−1 + 0.1 (for v = 1...10)(4.39)

andCV T 0 = −1.0

CV Tw = CV Tw−1 + 0.1 (for w = 1...10)(4.40)

These procedures assure that both parameters are within their domain of definition

between -1.0 and 0.0.

4.5.3 Case Study I: Perfect Competition

In perfectly competitive markets power producers believe they cannot change elec-

tricity and transmission prices unilaterally. They implicitly assume there are a

number of competitors, so that any strategy to affect the supply for electricity or

the demand for transmission capacity profitably fails in favour of its competitors.

Mathematically speaking, each company’s conjecture on competitors adjustment

in energy sales and demand for transmission capacity equals to -1.0.

Table 4.3 summarises the results for CV E0 = −1.0 and CV T 0 = −1.0. Three

different price zones evolve due to congestion on flowgate AB (zAB = 25 MW).

Electricity prices (πc) correspond to the marginal generation cost of the local sup-

ply company resulting in ”low-price” area A, ”high-price” area B and price area


C, whose price level is in between the former ones. Electricity generation at low

marginal cost exceeds more expensive generation resulting in an exporting power

balance of area A and an importing power balance of area B. As local electricity

demand is covered by all power suppliers’ sales, those suppliers located in adja-

cent areas additionally make/receive a payment to/from the auction office as a

result of their request for transmission service. The transmission service is valued

most efficiently as the area-to-area transmission price (τc,c′) reflects the difference

of corresponding electricity prices in each area c, c′ ∈ {A,B,C}:

πc′ − πc =∑

m∈M

PTDFc,c′,m · ωm = τc,c′, (4.41)

where PTDFc,c′,m denotes the percentage flow on flowgate m induced by a 1 MW

transfer from area c to area c′.In perfectly competitive electricity markets none of the companies makes any

strategic adjustments of their generation, trades and sales. Since electricity prices

are competitive, the demand for electricity is comparably high. Thus, the major

beneficiaries are consumers as the consumer surplus (CS) accounts for the main

contribution to overall social welfare (SW). The auction revenue (AR) represents

the sum of all power suppliers’ trading costs between the areas as a result of the

congestion on flowgate AB.

Focussing on local electricity demand (dc), each company’s market share, denoted

by MSf , is equal in all areas as there is no strategic behaviour underlying the

companies’ sales policy. In fact, each company’s single market share corresponds

to the company’s share of total electricity generation in the system.

Power supplier fA’s location in a low-price area results from its low marginal gener-

ation cost when compared with its competitors. However, that company increases

its sales profits (SP) by additional sales of a part of its generation output in higher

price areas B and C. Ultimately, its sales profits exceed that supplying company’s

generation cost (GC). However, company fA is located upstream of the congestion,

and hence, has to account for transaction costs (TC) for any cross-border trade

to area B or C. At the same time, any transaction made by company fB, relieves

congestion on flowgate AB, as its location is downstream of the congestion. Hence,

company fB receives a payment from the auction office, which is indicated by the

negative sign of its transaction cost. However, due to more costly generation in

area B, its sales profits are lower than its generation cost. Finally, none of the

suppliers makes any net profits (NP) as electricity is sold at marginal generation

cost.


4.5.4 Case Study II: Imperfect Competition

In the absence of arbitragers the electricity demand is covered exclusively by the

supplying companies’ energy sales. Thus, electricity prices are totally exposed

to the companies’ sales policy. Similar considerations hold for the transmission

market as supplying companies influence the congestion price by adjusting their

demand for transmission capacity through their trading behaviour.

Three case studies are presented aiming at providing some information on the eco-

nomic impact of imperfect competition in the electricity and transmission market.

Case Study II.A explores the companies’ strategic behaviour in the electricity mar-

ket, while the transmission market is assumed to be perfectly competitive. Starting

from the perfectly competitive case, competition in the electricity market is being

decreased gradually according to the procedure formulated in Equation (4.39). As

all suppliers make identical conjectures on their competitors in all areas (see Equa-

tions (4.37) and (4.38)), this approach aims at stating the overall economic impact

resulting from imperfectly competitive electricity markets. Case Studies II.B and

II.C additionally account for the companies’ strategic behaviour in the transmis-

sion market in terms of the variation of their demand for transmission capacity.

Here, the focus is on the individual supplier’s potential to manipulate congestion

prices profitably as a result of their specific location with respect to the congested

flowgate. As company fA is located upstream and company fB downstream of the

congestion, their strategies in the transmission market are opposed to each other.

Case Studies II.B. and II.C deal with strategic behaviour by company fA and by

company fB in the transmission market, however focussing on one company at a

time while all remaining power suppliers behave perfectly competitive.

II.A: CVEf , f ∈ {fA, fB, fC}

Market observations : In order to influence market outcomes most profitably, strate-

gic power producers are able to adjust their decision variables such as generation,

trades and sales by making conjectures on their competitors’ total response in the

sales output. Perfectly competitive results are obtained for CV Ev = −1.0. Any

change towards CV Ev = 0.0 represents an increase of imperfect competition in the

electricity markets. At CV Ev = 0.0 (Cournot competition) companies believe they

can unilaterally change market prices, thus, competition does not exist. Figure 4.3

displays the development of electricity prices and demand levels as a function of

CV Ev. As competition deteriorates in each single area, the demand for electricity

declines while electricity prices rise and peak at CV Ev = 0.0. Furthermore, prices


10.00

15.00

20.00

25.00

30.00

350

400

450

500

550

ectr

icity

Dem

and

(dc)

Area A Area B Area C[MW] [€/MW]

Ele

ctri

city

Pri

ce (

c)

0.00

5.00

10.00

15.00

20.00

25.00

30.00

250

300

350

400

450

500

550

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Ele

ctri

city

Dem

and

(dc)

CVEv

Area A Area B Area C[MW] [€/MW]

Ele

ctri

city

Pri

ce (

c)

Perfect Cournot

Figure 4.3: Electricity prices and demand as a function of CV Ev

35'000

201897

150 134 119 105 92 80 68 57 46 36 2630'000

35'000Consumer Surplus Producer Surplus Auction Revenue[€]

201897

35595018

63027433

8430

150 134 119 105 92 80 68 57 46 36 26

25'000

30'000

35'000

Wel

fare

Consumer Surplus Producer Surplus Auction Revenue[€]

3001228120

26385

201897

35595018

63027433

84309310 10088 10775 11381

150 134 119 105 92 80 68 57 46 36 26

25'000

30'000

35'000

Soci

al W

elfa

re


3001228120

2638524806

23366 22047 20838 19725 18700

201897

35595018

63027433

84309310 10088 10775 11381

150 134 119 105 92 80 68 57 46 36 26

20'000

25'000

30'000

35'000

Soci

al W

elfa

re


3001228120

2638524806

23366 22047 20838 19725 18700 17753 16876

201897

35595018

63027433

84309310 10088 10775 11381

150 134 119 105 92 80 68 57 46 36 26

15'000

20'000

25'000

30'000

35'000

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Soci

al W

elfa

re


3001228120

2638524806

23366 22047 20838 19725 18700 17753 16876

201897

35595018

63027433

84309310 10088 10775 11381

150 134 119 105 92 80 68 57 46 36 26

15'000

20'000

25'000

30'000

35'000

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Soci

al W

elfa

re

CVEv


Perfect Cournot

3001228120

2638524806

23366 22047 20838 19725 18700 17753 16876

201897

35595018

63027433

84309310 10088 10775 11381

150 134 119 105 92 80 68 57 46 36 26

15'000

20'000

25'000

30'000

35'000

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Soci

al W

elfa

re

CVEv


Perfect Cournot

Figure 4.4: Social welfare as a function of CV Ev

346 336 326 316 307 298 290 282 274 267 260

400 387 375 364 353 343 333 324 316 308 300

600

900

1200

1500

Toto

al G

ener

atio

n

Producer A Producer B Producer C[MW]

453 438 424 411 399 388 377 367 357 348 340

346 336 326 316 307 298 290 282 274 267 260

400 387 375 364 353 343 333 324 316 308 300

0

300

600

900

1200

1500

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Toto

al G

ener

atio

n

CVEv

Producer A Producer B Producer C[MW]

Perfect Cournot

Figure 4.5: Total generation as a function of CV Ev


in all areas converge towards one single electricity price. Figure 4.4 illustrates that

producers gain increasing profits (PS) while consumer surplus (CS) drops drasti-

cally. Social welfare (SW), auction revenue (AR) and consumer surplus are at their

minimum when Cournot competition is reached.

Table 4.4: Average rates of change of electricity prices and demand[AC/MWh] ΔπA ΔπB ΔπC [MW] ΔdA ΔdB ΔdC

ΔCV Ev 1.42 1.07 1.25 ΔCV Ev -11.38 -8.60 -9.99

Market analysis : Regarding the power producers’ generation in Figure 4.5, gener-

ating companies withdraw an increasing part of their generation output as com-

petition deteriorates. Hence, they reduce their sales and trading activities. Con-

sequently, total electricity demand decreases, which gives rise to higher electricity

prices. Table 4.4 demonstrates the average rates of change of electricity prices and

demand levels based on a stepwise change of CV Ev by ΔCV Ev = 0.1. The results

reveal that price and demand in low-price area A are most influenced on average

by changes in the competition level, followed by higher-price areas C and B. Thus,

as competition in electricity markets is shifting from perfect to Cournot, strategic

companies reduce their sales in low-price areas, such that the demand for electricity

in those areas decreases noticeably. Since low-price areas are more exposed to price

increases than high-price areas, electricity prices converge in all areas, as market

competition deteriorates.

It is known that strategic behaviour in electricity markets affects producer and con-

sumer surpluses, auction revenues and, thus, the overall social welfare. In perfectly

competitive markets the price for electricity is based on the marginal generation

cost curve. Consumer benefit and social welfare peak while producers’ total costs

and their sales profits are balanced. Any level of competition other than the per-

fectly competitive one, leads to a redistribution of surpluses and to a reduction of

social welfare. As a result of increasing electricity prices, producers make profits

at the expense of consumers and the auction office.

Basically, supplying companies have two major sources of income. Firstly, as they

hold back generation capacity, generating companies are able to reduce part of

their generation cost. Secondly, although the producers’ total sales decline, they

are able to increase their sales profits by selling power at higher electricity prices.

The saving of trading cost is considered to be a third source of income. In this

case, however, it is of less significance. As illustrated in Figure 4.6, producers are

able to increase their total sales profits while reducing their generation cost at

the same time. Thus, total producer surplus rises as competition deteriorates. In


1795019245

20355 21296 22092 22762 23324 23792 24179 24495 24749

17780 17214 16677 16173 15697 15249 14826 14425 14045 13684 1334110'000

15'000

20'000

25'000

30'000Sales Profits (SP) Generation Cost (GC) Trading Cost (TC)[€]

1795019245

20355 21296 22092 22762 23324 23792 24179 24495 24749

17780 17214 16677 16173 15697 15249 14826 14425 14045 13684 13341

150 134 119 105 92 80 68 57 46 36 260

5'000

10'000

15'000

20'000

25'000

30'000

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

CVEv

Sales Profits (SP) Generation Cost (GC) Trading Cost (TC)[€]

Perfect Cournot

Figure 4.6: Producers’ sources of income as a function of CV Ev

Table 4.5: Average rates of change of different income sources[AC] ΔSP ΔGC ΔTC

ΔCV Ev 680 -444 -12

Table 4.5 the producers’ sales profits are identified to be the major contribution

to increasing total producer surplus as their average impact on producer surplus

exceeds the other two.

II.B: CVEf , f ∈ {fA, fB, fC}∧

CVTf , f ∈ {fA}

In Case Study II.A all producers are treated equally in terms of their competitive

behaviour in the electricity market. The producers’ conjectures on their competi-

tors’ change of sales output are identical among them at each simulated level of

competition. Thus, a supplying company faces the same response function like each

of its competitors when deciding on its optimal strategy in the electricity market.

At the same time, the transmission market is assumed to be perfectly competitive.

Now, assumptions according to the transmission market are changed. As the topol-

ogy in Figure 4.2 shows, company fA is located in low-price area A, which is up-

stream of the congested flowgate AB. Thus, any sales by that company to area B

or C results in a payment of a location-specific transmission price τc,c′ according

to Equation (4.41). In order to better highlight company fA’s individual impact

on congestion price ωAB, only that supplier is assumed to act strategically in the

transmission market while its competitors implicitly assume perfect competition.

Apart from that, the same assumptions are made as in Case Study II.A.


2 00

3.00

4.00

5.00

6.00

7.00

nges

tion

Pric

e(

AB)

-1.0 -0.9-0.8 -0.7-0.6 -0.5-0.4 -0.3-0.2 -0.10.0

[€/MW]

CVTfAw

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Con

gest

ion

Pric

e(

AB)

CVEv

-1.0 -0.9-0.8 -0.7-0.6 -0.5-0.4 -0.3-0.2 -0.10.0

[€/MW]

Perfect Cournot

CVTfAw

Figure 4.7: Congestion price as a function of CV Ev and CV TwfA

0.50

1.00

1.50

2.00

(B

A)

A

,B

-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3

[€/MW]CVTf A

w

-0.50

0.00

0.50

1.00

1.50

2.00

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

(B

A)

A

,B

CVEv

-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.0

[€/MW]CVTf A

w

Perfect Cournot

Figure 4.8: Price divergence for a transaction from area A to area B as a function

of CV Ev and CV TwfA

2.0%

4.0%

6.0%

8.0%

10.0%-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

CVTfAw

-4.0%

-2.0%

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

CVTfAw

Perfect Cournot

Figure 4.9: Company fA’s profit gains from divergence of electricity and transmis-

sion prices as a function of CV Ev and CV TwfA


Market observations : Figure 4.7 displays the congestion price as a function of

CV Ev and CV TwfA

. At CV TwfA

= −1.0 (red coloured line) the transmission mar-

ket is perfectly competitive as producers believe they cannot change the congestion

price through their trading strategy. Though, ωAB decreases as a function of CV Ev

due to the overall convergence of rising electricity prices caused by local electricity

price increases (see Case Study II.A). At CV Ev = 0.0, the congestion price hits

its lowest mark as electricity prices almost match each other (Figure 4.3). As soon

as producer fA behaves strategically in the transmission market, ωAB drops be-

low its perfectly competitive outcome as indicated by the blue coloured lines. For

−0.7 < CV Ev ≤ 0.0, the congestion price even falls to zero for certain levels of

imperfect competition in the transmission market.

Moreover, the transmission price to be paid for any transaction between two areas

does not conform with the difference of electricity prices between those areas, which

is also known as spatial price discrimination (see Chapter 2.4). By way of exam-

ple, Figure 4.8 illustrates the difference of the left and right hand side of Equation

(4.41) based on a transaction from area A to area B. Even for a perfectly competi-

tive transmission market (red coloured line), πB −πA does not match transmission

price τA,B. In fact, as competition in electricity markets decreases (CV Ev → 0.0),

the transmission price is slightly higher than indicated by πB − πA. However, by

acting strategically in the transmission market (blue coloured lines), the divergence

between the electricity price difference and the transmission price increases for the

benefit of company fA. In fact, an increasing imperfect competition in the trans-

mission market, caused by producer fA, leads to a noticeably lower transmission

price to be paid for any transaction from area A to area B when compared with

the difference of corresponding electricity prices. The same considerations hold

for transactions from area A to area C. Company fA’s additional gains resulting

from the spatial price discrimination amount up to 9% of its total net profits as

illustrated in Figure 4.9.

Market analysis : As company fA is located upstream of congested flowgate AB,

any of its transactions other than to the area of its location aggravates conges-

tion. Thus, company fA pays a location-specific transmission price for any of

its sales to area B or area C. Its best strategy is to influence the demand for

transmission capacity such that the congestion price declines. As depicted in Fig-

ure 4.10, producer fA decreases its total trades for each CV Ev as a function of

CV TwfA

by holding back additional generation from the market shown in Figure

4.11. Since its competitors implicitly assume a perfectly competitive transmis-

sion market, they believe they cannot change the congestion price by adjusting


150

200

250

300

350

Trad

es b

y Pr

oduc

er A

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[MW] CVTf A

w

0

50

100

150

200

250

300

350

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Tota

l Tra

des b

y Pr

oduc

er A

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[MW] CVTf A

w

Perfect Cournot

Figure 4.10: Producer fA’s total trades as a function of CV Ev and CV TwfA

350

400

450

500

ion

by P

rodu

cer A

(gA)

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[MW]CVTf A

w

250

300

350

400

450

500

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Gen

erat

ion

by P

rodu

cer A

(gA)

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[MW]CVTf A

w

Perfect Cournot

Figure 4.11: Producer fA’s generation as a function of CV Ev and CV TwfA

2'000

3'000

4'000

5'000

Prof

its b

y Pr

oduc

er A

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[€]

CVTfAw

0

1'000

2'000

3'000

4'000

5'000

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Net

Pro

fits b

y Pr

oduc

er A

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[€]

CVTfAw

Perfect Cournot

Figure 4.12: Producer fA’s profits as a function of CV Ev and CV TwfA


Table 4.6: Producer fA’s profit maximising combinations of CV Ev and CV Tw,peakfA

CV Ev -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

CV Tw,peakfA

- 0.0 0.0 0.0 -0.2 -0.4 -0.5 -0.6 -0.6 -0.7 -0.8

their demand for transmission capacity. Thus, the congestion price is only sub-

ject to company fA’s strategic decisions in the transmission market. Ultimately,

ωAB converges towards zero. According to Figure 4.12, company fA gains some

additional profits by manipulating the congestion price when compared to its net

profits based on a perfectly competitive transmission market (red coloured bars).

Within −1.0 ≤ CV Ev ≤ −0.7, producer fA’s maximum net profits are reached

at each CV Tw,peakfA

= 0.0, i.e. when acting in a Cournot-like manner with respect

to transmission. In the interval of −0.7 < CV Ev ≤ 0.0, however, producer fA’s

net profits peak at CV Tw,peakfA

< 0.0. Any further decrease of competition beyond

CV Tw,peakfA

towards CV TwfA

= 0.0 leads to a reduction of company fA’s net profits.

In some cases, they are even lower than in the perfectly competitive case. Supply-

ing company fA’s profit maximising combinations of CV Ev and CV Tw,peakfA

values

are displayed in Table 4.6. Two intervals can be identified relating to company

fA’s change of its net profits as a function of CV TwfA

. For each CV Ev company

fA’s net profits

increase for −1.0 ≤ CV TwfA

≤ CV Tw,peakfA

(Interval 1) and

decrease for CV Tw,peakfA

< CV TwfA

≤ 0.0 (Interval 2).

This raises the question in how far electricity and transmission prices are benefi-

cially manipulated by company fA’s strategic decisions in the transmission market.

By comparison with Figure 4.7, the combinations of CV Ev and CV Tw,peakfA

also

give rise to producer fA’s largest impact on the congestion price without overes-

timating ωAB. Furthermore, as indicated by Figure 4.8, producer fA’s strategy in

the transmission market affects the transmission price by a higher rate than the

difference of relating electricity prices. Ultimately, this results in lower transmis-

sion costs for that company. However, at certain levels of imperfect competition

in the transmission market, company fA’s net profits decrease.

As described in Case Study II.A, the company’s net profits are composed of its

sales profits, generation and transmission costs. Figure 4.13 illustrates the average

increase rate of company fA’s net profits for each CV Ev based on a decrease of

competition in the transmission market within Interval 1. As company fA holds

back generation output, it reduces its total trades in order to affect the demand

for transmission capacity to its favour. Hence, since company fA is able to cut


-100

0

100

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

-300

-200

-100

0

100

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

CVEv

Sales Profits (SP) Generation Costs (GC)

Trading Costs (TC) Net Profits (NP)[€]

Perfect Cournot

Figure 4.13: Producer fA’s average increase rate of profits as a function of CV Ev

-100

0-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

-300

-200

-100

0-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

CVEv

Sales Profits (SP) Generation Costs (GC)

Trading Costs (TC) Net Profits (NP)[€]

Perfect Cournot

Figure 4.14: Producer fA’s average decrease rate of profits as a function of CV Ev

down some generation and transmission costs while making less sales profits in the

same way, its additional benefits can only be based on the saving of its costs. For

−1.0 ≤ CV Ev ≤ −0.7 the saving of generation costs is the major contribution

to company fA’s increasing net profits (black coloured bars), while the saving of

transmission costs makes the main part within the interval of −0.7 < CV Ev ≤ 0.0.

Figure 4.14 displays the average decrease rate of company fA’s net profits for each

CV Ev based on a decrease of competition in the transmission market within Inter-

val 2. It accounts for exactly those CV TwfA

yielding to overestimate the congestion

price such that company fA’s decline of its trades is higher than needed in order to

level the congestion price. For each −0.7 < CV Ev ≤ 0.0, company fA’s net profits

decrease on average. Transmission costs are equal to zero, as further savings of

generation costs does not come up with substantial losses of sales profits.


4.00

6.00

8.00

10.00

nges

tion

Pric

e(

AB)

-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3

[€/MW]CVTfB

w

0.00

2.00

4.00

6.00

8.00

10.00

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Con

gest

ion

Pric

e(

AB)

CVEv

-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.0

[€/MW]CVTfB

w

Perfect Cournot

Figure 4.15: Congestion price as a function of CV Ev and CV TwfB

-1.50

-1.00

-0.50

0.00

(B

A)

A

,B

-1.0-0.9-0.8-0.7-0.6-0.5-0.4

[€/MW]

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

(B

A)

A

,B

CVEv

-1.0-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.0

[€/MW]

CVTfBw

Perfect Cournot

Figure 4.16: Price divergence for a transaction from area A to area B as a function

of CV Ev and CV TwfB

6.0%

8.0%

10.0%

12.0%

14.0%

16.0%-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

CVTfBw

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

16.0%

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

CVTfBw

Perfect Cournot

Figure 4.17: Company fB’s profit gains from divergence of electricity and trans-

mission prices as a function of CV Ev and CV TwfB


II.C: CVEf , f ∈ {fA, fB, fC}∧

CVTf , f ∈ {fB}

If flow-based allocation of cross-border transmission capacity is the underlying con-

gestion management method within a predefined region, any cross-border trades

in that region affect the flowgate capacity utilisation. Then, congestion can ei-

ther be aggravated or relieved by commercial transactions that involve two areas.

Unlike the situation in Case Study II.B, supplying company fB, which is located

downstream of the congested flowgate AB, acts strategically in the transmission

market in this study while its competitors’ decisions with respect to transmission

are based on perfect competition. As that company’s location is in high-price area

B, any cross-border trades by supplier fB are only profitable because of receiving

a payment for relieving congestion on flowgate AB.

Market observations : Figure 4.15 shows congestion price ωAB to rise for each CV Ev

above its competitive level (red coloured line) as a result of increasing imperfect

competition in the transmission market caused by producer fB’s strategy. As soon

as company fB behaves in a Cournot-like manner with respect to transmission

(CV TwfB

= 0.0), the congestion price peaks for any given CV Ev.

The difference of electricity and transmission prices is illustrated in Figure 4.16 by

means of a transaction from area A to area B. As in Case Study II.B, the divergence

between πB − πA and τA,B increases as a result of decreasing competition in the

transmission market. Unlike Case Study II.B, the difference of electricity prices

provides an underestimated indication of the transmission price. Producer fB is

able to profitably raise τA,B above πB − πA, thus, increasing the payment from

relieving transmission congestion. These additional profits amount up to 15% of

its total net profits as shown in Figure 4.17.

Market analysis : Producer fB’s willingness to trade electricity to lower-price areas

A or C is based on the amount of payment for relieving transmission congestion

on flowgate AB. In perfectly competitive electricity and transmission markets, the

transmission price for any of producer fB’s transactions reflects the difference of

relating electricity prices according to Equation (4.41). Thus, since electricity and

transmission prices are cost-based, producer fB’s benefit from relieving transmis-

sion congestion matches its diminished profits from selling power to lower price

areas. However, as shown in Subsection 4.5.3, supplier fB’s does not realise any

profits. Similar to supplier fA’s behaviour in Case Study II.B, company fB holds

back generation output (Figure 4.19) in order to reduces its total trades as illus-

trated in Figure 4.18. Thus, producer fB’s strategy in the transmission market is

to decrease its demand for transmission capacity in order to charge their trans-


100

150

200

250

300

Trad

es b

y Pr

oduc

er B

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[MW]CVTfB

w

0

50

100

150

200

250

300

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Tota

l Tra

des b

y Pr

oduc

er B

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[MW]

Perfect Cournot

CVTfBw

Figure 4.18: Producer fB’s total trades as a function of CV Ev and CV TwfB

300

350

400

ratio

n by

Pro

duce

r B

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[MW]

CVTfBw

200

250

300

350

400

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Gen

erat

ion

by P

rodu

cer B

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

Perfect Cournot

[MW]

CVTfBw

Figure 4.19: Producer fB’s generation as a function of CV Ev and CV TwfB

2'000

3'000

4'000

Prof

its b

y Pr

oduc

er B

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[€]

CVTfBw

0

1'000

2'000

3'000

4'000

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Net

Pro

fits b

y Pr

oduc

er B

CVEv

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5-0.4 -0.3 -0.2 -0.1 0.0

[€]

Perfect Cournot

CVTfBw

Figure 4.20: Producer fB’s profits as a function of CV Ev and CV TwfB


-50

0

50

100

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

[€]

-200

-150

-100

-50

0

50

100

-1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

CVEv

Sales Profits (SP) Generation Costs (GC) Trading Costs (TC) Net Profits (NP)

[€]

Perfect Cournot

Figure 4.21: Producer fB’s average increase rate of its profits as a function of

CV Ev

actions against a higher congestion price. By that, as illustrated in Figure 4.20,

company fB’s net profits rise above competitive levels and peak for each CV Ev at

CV Tw,peakfB

= 0.0, i.e. when Cournot competition is reached.

Figure 4.21 illustrates the increase rate of company fB’s net profits for each CV Ev

based on a decrease of competition in the transmission market. Company fB’s

increase in its net profits is mainly driven by substantial savings of its genera-

tion costs, as for −1.0 > CV Ev ≥ −0.8 those surpass the supplier’s losses in its

sales profits noticeably. Moreover, in that range company fB’s gains from relieving

congestion decrease on average in spite of acting strategically in the transmission

market. For CV Ev > −0.8 an increasing contribution arises from benefits relating

to the transmission costs, which is then added to the power supplier’s net profits.


The generating companies’ market behaviour based on perfect and imperfect com-

petition has been analysed in an oligopolistic market environment when trans-

mission capacity is implicitly allocated by a flow-based congestion management

method. Three case studies of strategically behaving generating companies are

presented by introducing the concept of conjectural variations in the electricity

and transmission market. By parametrically changing each supplying company’s

conjecture on its competitors’ market behaviour in each of both markets, different

intensities of competition can be modelled.


In a region of perfectly competitive electricity and transmission markets, the power

supply is based on marginal generation costs. Hence, cheapest generation units are

scheduled in order to satisfy overall electricity demand in that region. Electricity

and transmission prices are perfectly competitive. In fact, congestion-based differ-

ences of electricity prices between any two power markets reflect the transmission

price to be paid for a transaction between those markets. Within this economic

framework, the power suppliers’ generation and transaction costs and their sales

profits cancel each other out while consumers are the biggest beneficiaries.

However, when the supplying industry is oligopolistic, power suppliers are able

to exploit their market dominance in order to maintain or raise electricity prices

above competitive levels. Assuming there are no arbitragers, the electricity de-

mand is only subject to the power suppliers’ sales policy. Thus, electricity prices

are exposed to major manipulations by the strategic companies’ market decisions

such that non-cost based prices for electricity arise. In fact, as shown by Case

Study II.A, electricity prices in low price areas are increased by a higher rate than

in higher price areas since holding back cheap generation from the market has a

higher impact on electricity prices than holding back more expensive generation.

As a result of increasing electricity prices, consumer benefit and social welfare

decline as power suppliers are able to gain higher sales profits while decreasing

their generation costs at the same time. Finally, the results identify the generat-

ing companies’ increase of their sales profits to be the major contribution in their

additionals net profits.

Assuming generating companies to act strategically in the transmission market,

their location with respect to the congested flowgate is significant in order to de-

termine their strategy. In Case Study II.B, the company located upstream of the

congestion reduces its transaction costs by decreasing its demand for transmission

capacity in order to bring down the congestion price. Consequently, transmission

prices are no longer cost-based which gives rise to spatial price discrimination, i.e.

when the difference of electricity prices in any two market does not correspond

to the transmission price to pay for any transaction between those markets. This

effect is typical for imperfectly competitive markets when arbitrage is low or even

completely neglected. The results reveal that this supplier’s benefits resulting from

spatial price discrimination amount up to 9% of its total net profits.

The power producer’s increasing net profits are mainly based on cost savings as the

reduction of both generation and transmission costs exceeds the supplier’s losses in

their sales profits. The producer’s net profits peak, when their strategic decisions

have the biggest impact on the congestion price. Any further decrease of gener-

ation output, ultimately leading to less demand for transmission capacity, results


in declining net profits for that producer as the congestion price is implicitly over-

rated.

In Case Study II.C the company located downstream of the congestion is assumed

to behave in a strategic manner in the transmission market. As any of its trans-

actions to adjacent power markets induce a power flow in the opposite direction

of the congested flowgate, that company’s strategy is to increase the congestion

price in order to get most highly rewarded for relieving transmission congestion.

By decreasing their demand for transmission capacity, the congestion price rises

which results in increasing net profits for that company. As in Case Study II.B,

the producer additionally benefits from creating spatial price discrimination, which

amounts up to 15% of its net profits. Focussing on the producer’s increasing net

profits, the savings of generation cost are identified as its main source of income.

5 Conjectural Variation Based

Price Forecasting

This chapter presents a quantitative comparison of the NTC-based and the FBA

congestion management method in the central European power system by introduc-

ing CVE parameters. The application of those parameters allows the comparison

of both approaches in the vicinity of historical electricity prices and demand levels.

In a first phase, termed as the fitting period, the CVE parameters are estimated by

means of historical electricity prices and demand levels. Then, in the second phase,

termed as the forecasted period, the electricity demand is increased in each country

by a typical yearly rate for five consecutive time periods. At the end of each time

period, both models are compared by means of the simulated market results.

So far, an estimation of conjectural variation parameters has been a challenging

task. A closed-form mathematical expression is derived for CVE and CVT pa-

rameters for the case of multiple power markets in FBA congestion management.

Based on that, CVE and CVT parameters can be fitted explicitly by historically

known market data, however, the availability of the data is not always provided.

By a more sophisticated approach, the implicit fitting procedure, CVE and CVT

parameters are estimated iteratively by means of publicly available electricity and

transmission prices. Subsequent to that, a comparison between the NTC-based and

the FBA approach is made followed by concluding remarks.

5.1 Estimation of Conjectural Variation Parameters

A variety of power market models exists in the literature. Their validity can be

evaluated by means of the modelled output variables to be under observation. If

electricity prices are relevant, models of strategic interaction among power suppli-

ers play a major role in the price generation process. The concepts of perfect and

93

94 5 Conjectural Variation Based Price Forecasting

Cournot-based competition in power markets are most widely spread. However,

they fail at providing credible electricity prices due to rather unrealistic assump-

tions by strategic power suppliers regarding their competitors’ market behaviour

as described in Chapter 4. Ultimately, electricity prices tend to be either perfectly

competitive or highly overrated.

The introduction of the CVE and CVT concepts allows to model a range of differ-

ent competitive levels in the electricity and transmission markets. By this flexible

modelling approach, electricity and transmission prices can be obtained within the

range of perfectly competitive and Cournot-based price levels. A common way to

estimate CVE and CVT parameters is based on historical market data. Then, CVE

values represent the power suppliers’ market behaviour in a past market scenario

and could be applied for forecasting a future state of the power market, e.g. by

increasing the historical load by a typical rate. The assumption of the same fu-

ture market behaviour by all individual companies leads to a more realistic market

outcome when compared with perfectly competitive or Cournot-based competitive

future market behaviour of power suppliers since it allows to study the power mar-

ket in the vicinity of its past state.

However, the estimation of those parameters remains a challenging task which has

been addressed by several previous works. In [28] [33], a mathematical expression

is derived for the CVE parameters based on the profit maximisation problem of

each power producer. It requires the knowledge of publicly available market data

in order to specify the CVE parameters. This approach is termed an explicit fitting

procedure. However, since the complete market data set is rarely obtainable due

to its specific characteristic and sensitivity, e.g. providing the companies’ marginal

generation cost, a more sophisticated method has been developed which determines

the CVE values iteratively subject to historically available electricity prices. This

procedure is introduced in [29] and termed implicit fitting procedure. So far, both

estimation methods have been strictly applied to the modelling of a single power

market without accounting for the transmission network. The estimation of CVT

parameters has not been addressed so far.

This work provides an extension of the implicit fitting procedure to the modelling

of multiple power markets while cross-border trades are assumed to be managed by

the FBA mechanism introduced in Chapter 3. Furthermore, a closed-form math-

ematical expression is derived according to the CVT parameters, followed by a

discussion of the explicit fitting procedure and an application of the implicit fitting

procedure.

5.1 Estimation of Conjectural Variation Parameters 95

5.1.1 Explicit Fitting Procedure

The calculation of CVE and CVT parameters based on historical market data

requires the derivation of a mathematical closed-form solution for both parameters.

By definition, CVE parameters represent each company’s belief of how competitors

change their sales in a certain power market as a result of the firm’s own change

in sales (see Chapter 4). CVT parameters, however, represent that company’s

assumption on its competitors’ change of allocated trades resulting from its own

change in allocated trades on a certain flowgate. Starting from the company f ’s

profit maximisation function, which is stated in Equation (4.18), the optimality

conditions for its sales sf,c in market c, its trade tf,c,c′ from market c to c′ and the

allocated quantity tfgf,m on flowgate m are given as follows (see also KKT conditions

(C.7), (C.8) and (C.12) in Appendix C)

−πc − βc · (1 + CV Ef,c) · sf,c + θsf,c = 0 (5.1)

−θsf,c + θg

f,c′ −∑

m∈M (ηf,m · PTDFc′,c,m) = 0 (5.2)

ωm + δm · (1 + CV Tf,m) · tfgf,m + ηf,m = 0 (5.3)

In the following, the explicit fitting procedure is derived separately for CVE and

CVT parameters based on a modification of Equations (5.1) - (5.3).

Focus on CVE Parameters

For simplification, the estimation of CVE parameters is based on a perfectly com-

petitive transmission market (CV Tf,m = −1.0), i.e. congestion price ωm is assumed

to be given in the process of CVE estimation. Furthermore, demand slope βc is

replaced by the expression for the price elasticity of electricity demand shown in

Equation (A.1) in Appendix A. By inserting Equation (5.3) into (5.2), the CVE

parameter is formulated as

CV Ef,c =εelc · dc

πc· θ

sf,c − πc

sf,c− 1 (5.4)

while θsf,c is given by

θsf,c = θg

f,c′ +∑

m∈M

(ωm · PTDFc′,c,m) (5.5)

Equation (5.4) presents a closed-form solution of the CVE parameter for the case

of multiple power markets while Equation (5.5) gives a formulation of company f ’s

marginal cost involved in its sales to market c. Those are composed of the marginal


generation cost of their production units located in zone c′ and the transmission

price to be paid from zone c′ to zone c. In particular, this is the case when zones

of power generation and sales do not coincide. Then, company f determines its

optimal sales output in market c by trading off its sales benefit against the cost of

generation and transmission of the same amount of power from zone c′. In case,

zones of power generation and sales do coincide, there is no payment for transmis-

sion and θsf,c equals to company f ’s marginal generation cost denoted by θg

f,c

The basic idea of the explicit fitting procedure is the calculation of CVE parameters

by means of publicly available market data according to Equation (5.4). However,

the specification of CVE parameters reveals to be a challenging task since it re-

quires the knowledge of a huge amount of market data. While electricity prices

(πc), demand levels (dc), the price elasticities of demand (εelc ) and congestion prices

(ωm) are often publicly available, the companies’ marginal generation cost (θsf,c,

θgf,c′) and their sales characteristics (sf,c) are not known precisely or even at all.

The modelling of large electricity market models with multiple price zones and

strategic companies is almost impossible due to enormous amount of sensitive data

needed.

Even though the companies’ historical marginal generation cost and their sales are

known, there is no guarantee the calculated CVE parameters are within their defi-

nition range of −1 ≤ CV Ef,c ≤ 0. This is due to the variety of different sources the

market data is extracted from and the fact that the companies’ historical genera-

tion and sales outputs are driven by short and long term decisions, which cannot

be distinguished at the time of CVE calculation. In fact, it is hardly possible to

find the needed market data based on a single historical reference scenario.

Furthermore, it is very unlikely the calculated CVE parameters represent equi-

librium values of the power market model. Thus, applying the CVE parameters

to the same scenario of the power market their calculation is based on, leads to

completely different market results, e.g. in terms of electricity and transmission

prices, generation output etc. Ultimately, this does not lead to a forecast of a

future power market state in the vicinity of the historical one.

Focus on CVT Parameters

Similarly, the estimation of CVT parameters is based on perfectly competitive

electricity markets (CV Ef,m = −1.0), i.e. the electricity price is not assumed to

be altered in the process of CVT calculation. For simplification, only flowgate

m is assumed to be congested. Then, replacing the capacity demand slope δm of

Equation (5.3) by the formulation of the price elasticity of transmission capacity


demand (see Equation (A.2) in Appendix A) and inserting Equation (5.1) into

(5.2) yields to

CV Tf,m =εtrm ·NFCm

ωm· −ηf,m − ωm

tfgf,m

− 1 (5.6)

while ηf,m is given by

ηf,m =θgf,c − πf,c′

PTDFc′,c,m(5.7)

According to Equation (5.6), a closed-form solution of the CVT parameter is pre-

sented while Equation (5.7) provides a definition of company f ’s marginal value of

the congested flowgate m for a transfer from zone c′ to zone c. In case of perfectly

competitive transmission markets, ηf,m equals to the flowgate-related congestion

price ωm. Otherwise its marginal evaluation depends that company’s competitive

behaviour in the transmission market.

However, as the congestion price (ωm), the electricity price (πf,c′), the flowgate

capacity (NFCm) and the price elasticity of transmission capacity demand (εtrm)

are often publicly available, the companies’ marginal value of the congested flow-

gate (ηf,m) and their contribution to the congestion on that flowgate (tfgf,m) are not

easily obtained.

5.1.2 Implicit Fitting Procedure

The explicit fitting procedure for conjectural variation parameters has several draw-

backs mainly due to the unavailability of the required market data. In order to

fit those parameters based on historical market data, which is publicly available,

an iterative estimation method for CVE and CVT parameters has been developed,

also known as the implicit fitting procedure. The CVE and CVT parameters are

updated iteratively until the simulated market data meets the historical one. As

for the electricity market, the fitting of CVE parameters is based on previously

known electricity prices and demand levels. In terms of the transmission market,

transmission prices and the capacity demand are assumed to be obtainable for the

fitting of CVT parameters.

Furthermore, if the implicit fitting procedure converges, then, the CVE and CVT

parameters are market equilibrium-based, i.e. the simulated market data resulting

from the application of those parameters is equal to the historical one, which is used

for the parameter estimation. This feature allows to make studies in the vicinity

of the reference scenario by varying the market model’s parameters while keeping

the power suppliers’ market behaviour in the forecasted scenario unchanged.


Focus on CVE Parameters

Based on Equation (5.4), the CVE parameter at iteration k is determined as follows

CV Ekf,c = Kel · θ

s,kf,c − πk

c

skf,c

− 1 (5.8)

where Kel is a constant according to

Kel =εelc · dref

c

πrefc

(5.9)

Then, assuming the change of θs,kf,c and sk

f,c to be much lower in subsequent iterations

than the change of CV Ekf,c yields to

ΔCV Ekf,c = CV Ek

f,c − CV Ek−1f,c = Kel · π

k−1c − πk

c

skf,c

(5.10)

If the electricity price πk−1c at iteration k-1 corresponds to the publicly available

historical electricity price πrefc and if there is a difference to the simulated price πk

c ,

then, ΔCV Ekf,c indicates the direction of the update of the CVE parameter in the

subsequent iteration according to

CV Ek+1f,c = CV Ek

f,c + γkc · ΔCV Ek

f,c (5.11)

where

ΔCV Ekf,c = Kel · π

refc − πk

c

skf,c

(5.12)

By introducing the coefficient γkc , the step size for each subsequent iteration is

reduced in order to achieve faster convergence. Since the convergence criterion

is formulated in terms of electricity prices, the price error εel,kc is defined by the

difference of historical and simulated electricity prices

εel,kc = πref

c − πkc (5.13)

It is important to point out that the convergence criterion according to Equation

(5.13) must hold for each single power market c ∈ C. In general, the number of

iterations increases with the number of involved power markets C. The parameter

γkc is proposed to be adjusted separately for each power market in order to reduce

the number of total iterations.

Figure 5.1 illustrates the steps of the implicit fitting procedure. It is important to


Initialisation:k = 0; CVEf,c

0

Solve:Market Equilibrium

Update:CVEf

k+1Storage:

f c, sf c, cs,k k k

| c - c |< c ?ref kno

CVEf,c f,c, sf,c, c

el,k

yes

E dEnd

Figure 5.1: Implicit fitting procedure for CVE parameters

note that the initial values of the CVE parameter have an effect on the conver-

gence and on the total number of iterations. Here, best initial values are found

when starting from the perfectly competitive case.

Furthermore, since the CVE parameter fitting is only subject to historical elec-

tricity prices, the simulation results provide some information on the market data

which is historically not available, e.g. the companies’ marginal generation cost and

their sales output in each power market. These results, however, might be different

from the real market data as this fitting approach does not account for the com-

panies’ short or long term strategies during the time of decision making. Another

consequence of the unavailability of some market data leads to CVE values that

are not necessarily unique. As shown by Equation (5.8), the convergence criterion

is of dimension c, however, the dimension of CVE values is f × c. Depending on

the initialisation of CVE parameters, different sets of CVE values may satisfy the

convergence criterion, however, leading to slightly different simulated values of the

market model’s variables, e.g. marginal generation cost and sales characteristics.

An interesting aspect of the convergence criterion in terms of electricity price is

shown by means of the market clearing condition formulated in Equation (4.33)


and stated here again at iteration step k

πkc = πref

c · (1 − 1

εelc) +

πrefc

drefc · εelc

· (∑

f∈F

skf,c + ask

c − apkc) (5.14)

If the convergence criterion is reached at iteration k, i.e. πrefc ≈ πk

c , then, after

rearranging Equation (5.14) the simulated demand is equal to the reference demand

as shown in the following

drefc =

∑

f∈F

skf,c + ask

c − apkc

In fact, the convergence criterion in terms of electricity prices is also a criterion in

terms of the electricity demand. Then, price elasticity of demand εelc has a major

impact on the sensitivity of simulated electricity prices with respect to the changes

of the CVE value. As shown by the company’s KKT condition with respect to its

sales

−πkc −

πrefc

drefc · εelc

· (1 + CV Ekf,c) · sk

f,c + θs,kf,c ≥ 0 (5.15)

if εelc is close to zero, which is equal to an approximately constant electricity de-

mand, then, small changes of the CVE value have a relatively large impact on the

simulated electricity prices. Thus the range of modelled electricity prices can be

enlarged for decreasing εelc .

Focus on CVT Parameters

For CVT parameter estimation, changes of ηkf,m and tfg,k

f,m are assumed to be much

lower in consecutive iterations than the change of CV T kf,m. Thus, the increment

ΔCV T kf,m is formulated as follows

ΔCV T kf,m = Ktr · ω

refm − ωk

m

tfg,kf,m

(5.16)

where

Ktr =εtrm ·NFCref

m

ωrefm

(5.17)

Similar to the CVE parameter fitting, the convergence criterion is given in term

of transmission prices, where εtr,kc defines the difference of historical and simulated

transmission prices

εtr,kc = ωref

m − ωkm (5.18)

Then, the sequence of the implicit fitting procedure is comparable to that of CVE

parameters.

5.2 Case Study - Central European Region 101

DE

FR CH AT

IT

Figure 5.2: Central European Region

5.2 Case Study - Central European Region

The aim of the following study is a quantitative comparison of the NTC-based

and the flow-based congestion management method based on the European trans-

mission system. The study is limited to the geographical scope of the Central

European Region (CE-Region) consisting of TSO-controlled areas Austria (AT),

Switzerland (CH), Germany (DE), France (FR) and Italy (IT) as depicted in Fig-

ure (5.2). Each control area also coincides with a single price zone where electricity

is assumed to be sold in the corresponding national power exchange (PX). Further-

more, countries are able to participate in any power market within the CE-Region.

Then, transmission rights need to be acquired according to the underlying con-

gestion management scheme. In the NTC-based model, cross-border trades are

limited by the NTC values between each pair of neighbouring price zones. As for

the FBA mechanism, the physical transmission capacity on each flowgate is given

by the corresponding NFC values (see Chapter 3.4). In both congestion manage-

ment methods, transmission congestion leads to a transmission price to be paid by

the party which requests transmission services.

Both congestion management approaches are integrated into a power market model

and simulated separately. While the implementation of the FBA mechanism is ex-

plicitly documented in Chapter 4.4, some minor adjustments of the power market

formulation are made, particulary in terms of the functioning of the transmission

market, in order to conform with the rules of the NTC-based congestion manage-


ment method 1. In both models, the main market components can be identified as

countries (equivalent to power producers), arbitragers and an inter-regional auction

office. They all optimise their decision variables in order to maximise their prof-

its.2 The CVE parameters are estimated for each company in each power market

in the CE-Region by the implicit fitting procedure based on peak-hourly electricity

prices and demand levels according to the winter period 2008/2009, also denoted

as the reference scenario. The estimation of CVE parameters is carried out on a

monthly basis, whereby each month within the reference scenario is represented

by its monthly averaged peak-hourly electricity price and demand level. This ap-

proach does not only reduce the computational time. Furthermore, it avoids a

CVE parameter estimation based on unexpectedly high or low daily electricity

prices and allows a more generalised statement of the simulation results. Accord-

ing to Figure 5.3, after the monthly fitting period at time y, a forecast of the power

market is given separately for each month in the subsequent winter scenario y+1

based on a yearly increase rate of each country load by 2%. Although, the load

increase seems to be slightly higher than what has been observed in recent years, it

better illustrates the effect of both congestion management methods on forecasted

electricity prices. Then, the CVE parameters are estimated based on the power

market state at time y+1 and applied to the winter scenario at time y+2 etc. This

approach continues until the winter scenario of 2013/2014. The electricity prices

resulting from both congestion management method are compared with each other

based on each forecasted scenario.

5.2.1 Modelling Assumptions

In the following, some modelling assumptions are listed. They are based on the

inadequacy of the power market model to represent the observed reality. Fur-

thermore, they are needed to provide a common basis for the comparison of both

congestion management methods and to prevent unrealistic market results.

• A major modelling assumption concerns the chronological sequence of the

market clearing in capacity and energy auctions. Today, cross-border trans-

mission capacity in the CE-Region is allocated in explicit auctions. Any

1As for the NTC model there is no transformation of cross-border trades into the physical power flow,thus, PTDFs are not defined. Moreover, each party is looking for its optimal set of cross-border exchangetrades.

2The model contains around 600 generating units that are shared among the five countries. Further-more, there are about 1400 single variables and about 1400 single equations to be solved.


Forecasting PeriodHistorical Market Data

W 13/14W 08/09 W 12/13W 10/11 W 11/12W 09/10

y+1 y+2 y+3 y+4 y+5y

Oct

Nov

Oct

Nov

Oct

Nov

Oct

Nov

Oct

Nov

Oct

Nov

Dec

Jan

Dec

Jan

Dec

Jan

Dec

Jan

Dec

Jan

Dec

Jan

Feb

Mar

Feb

Mar

Feb

Mar

Feb

Mar

Feb

Mar

Feb

Mar

CVEy CVEy+1 CVEy+2 CVEy+3 CVEy+4

Fitting Period Fitting Period Fitting Period Fitting Period Fitting Period

Figure 5.3: Fitting and forecasting periods

party participating in the neighbouring countries’ power market or power ex-

changes, needs to acquire transmission rights beforehand. From the modelling

perspective, the time gap between the clearing of cross-border capacity auc-

tions and the clearing of power exchanges is difficult to tackle. Consequently,

in both congestion management approaches the allocation of transmission ca-

pacity is integrated in the market clearing process of the energy market, i.e.

NTC-based implicit auctioning is compared to flow-based implicit auctioning.

• According to the NTC-based model, two NTC auctions are assumed to be

implemented at each border of the CE-Region in both directions except for

the case of the France-Switzerland interconnection. There, the NTC from FR

to CH is not auctioned since it is usually fully blocked by long-term bilateral

(power supply) contracts between those two countries. As for the flow-based

congestion management approach, the bilateral contracts are disposed be-

cause they reveal to be not conform with the FBA mechanism. Simulation

results show that the physical transmission capacity of flowgate FR → CH

is fully used after the allocation of the bilateral contract. Then, any export

trades from France and any cross-border trades by other countries that lead to

a physical power flow in direction from FR to CH are blocked. Furthermore,

the background of introducing the FBA mechanism is to study an optimal

allocation of physical transmission capacity by including all physical borders

within the CE-Region.


• The total amount of cross-border trades from AT, DE and FR to CH is limited

by a value also known as the Swiss roof. This is due to restricting the transit

power flow through Switzerland coming over its northern borders in direction

to the importing area IT. In the context of FBA modelling, the Swiss roof

is defined as the maximum physical power flow from AT, DE and FR to CH

and assumed to be represented by the same value as in the NTC context.

• A key feature of FBA modelling is the netting of power flow on a given

flowgate by accounting for all inter-regional cross-border trades in the capacity

allocation phase. However, this can cause unrealistically high cross-border

trades in both directions between neighbouring source and destination areas

since the induced power flows on the flowgate between those areas cancel each

other out. Thus, in terms of cross-border trading import and export limits

are defined for each area according to the sum of importing and exporting

NTC values.

• As for NFC determination, the approximation by Equation (3.29) is applied,

which is repeated here for convenience:


(k,k′)∈F,(k,k′)�=(c,c′)

BCEk,k′ · PTDFk,k′,m

The reasoning behind this approach is twofold: Firstly, the provided winter

base case to be used for PTDF and NFC calculation is different from the

one used by ENTSO-E for the calculation of NTC values. However, the DC

PTDF matrix can be assumed to change only marginally in case of minor

differences between the network topologies of the two base case scenarios,

which is due to the huge amount of transmission lines in the European trans-

mission network. Thus, in order to find a common basis for the NTC, NFC

and PTDF calculation, Equation (3.29) can be applied which determines the

NFC by combining PTDFs and NTC values. Secondly, the performed NFC

calculations based on the provided base case of the transmission system, has

led to extremely low NFC values in one direction of some flowgates which is

due to the relatively large impact of the base case power flow on the capac-

ity calculation. Although the impact of the remaining UCTE power system

on the power flow in the CE-Region has been eliminated, some NFC values

remained quite small (see more on this problem in [3]).


UCTE

NOK

UCTE

NOK

ENTSO-E

NOK

NOK / ENTSO-E

NOK

X

XMarginalCost

Exchangewith UCTE

X

Source

X

Network

NTC

Hourly

X

X

Load

Spot Market Prices

PTDF

NFC

X

Snapshot Half-Yearly Monthly

X

Figure 5.4: Availability of market data

5.2.2 Availability of Market Data

Some general information is given on the availability of the market and the network

data. If possible all data are obtained on an peak-hourly basis for the whole winter

scenario. Otherwise it has to be converted into an hourly basis, if possible peak-

hourly basis. As the simulations are performed for each single month of the winter

period, the peak-hourly values for a specific data set are averaged over each month

in order to provide a representative monthly peak-hourly value. Some market data

is given only in a single specific hour (”snapshot”). This data set is assumed to be

valid for each single month in the winter scenario. Table 5.4 gives a compilation

of the market data and its availability for different time intervals.

Power Exchange between CE-Region and UCTE

The geographical scope of the case study is limited to the countries in the CE-

Region as shown in Figure 5.2. However, there is an interaction between the

CE-Region and its neighbouring countries of the UCTE power system in terms

of cross-border power exchanges. Those are included into the reference scenario

of winter 2008/2009 and assumed to be constant at each forecasted scenario of

the power market. The corresponding data is available on a monthly basis on the

ENTSO-E platform [41].

Electricity Prices


Electricity prices for all countries are represented by the spot market prices of the

corresponding power exchanges for the whole reference period. They are publicly

available on an hourly basis on the homepages of the power exchanges EEX [43],

Powernext [44], Swissix [43], EXAA [45] and IPEX [46].

Electricity Demand

The electricity demand in each country of the CE-Region is represented by the

load values available on an hourly basis on the ENTSO-E platform [41]. This is

due to the marginal generation cost curves, which are based on the total annual

electricity consumption.

Price Elasticity of Electricity Demand

The price elasticity of demand (definition in Appendix A.1) is set to -0.01 in each

power market, which means that demand is rather price independent.

Marginal Generation Cost Curves

The marginal generation cost curves for each country are based on 17th January

2007, at 10:00h and provided by NOK [42]. They are related to the total annual

consumption in each country and adjusted to the year 2009. It is assumed that

the marginal generation cost curves will not alter in the forecasted periods.

NTC, Swiss Roof

NTC and Swiss roof values are obtained for the reference scenario according to the

published NTC matrix on the ENTSO-E platform [41].

Transmission Network, PTDF, NFC

A winter base case of the UCTE transmission system is provided by the NOK [42]

for a single peak-hour in order to calculate the PTDF matrix and the NFCs needed

for the FBA approach. The data set is adjusted by NOK in order to account for

future network conditions in a typical winter scenario.

As for the FBA mechanism, the DC PTDF matrix is calculated for the CE-Region

based on the network topology of the entire UCTE transmission system. The


103.14

109.82

89.74

100.53

95.86

106.85

90.00

100.00

110.00

120.00

CH AT DE FR IT FBA NTC[€/MWh]

103.14

109.82

70.67

81.56

78.08

89.74

80.59

100.53

95.86

106.85

70.00

80.00

90.00

100.00

110.00

120.00

08/09 09/10 10/11 11/12 12/13 13/14

CH AT DE FR IT FBA NTC[€/MWh]

Figure 5.5: Development of electricity prices in NTC-based and FBA congestion

management

underlying calculation method is documented in Chapter 3.3. The DC PTDF

matrix is assumed to be valid for all forecasted periods since DC PTDFs are less

dependant on the underlying power system state than AC PTDFs.

NFC values are determined by applying the approximation method introduced in

Chapter 3.4. As with DC PTDFs, they are assumed to be constant in all forecasted

scenarios.

5.2.3 Simulation Results

The fitting period is based on monthly electricity prices and demand levels of

winter scenario W08/09 whereas the forecasting period comprises the following

future winter scenarios: W09/10, W10/11, W11/12, W12/13 and W13/14. In

order to present the results in a more condensed form, the monthly simulated

peak-hourly market results are averaged over each corresponding winter half-year.

Observation: Figure 5.5 illustrates the electricity price development of each country

in the CE-Region for the NTC-based (dotted lines) and the FBA (solid lines) con-

gestion management methods. Both are based on the reference scenario W08/09,

where the average peak-hourly electricity price in Italy is the highest within the

CE-Region followed by the Swiss, the French, the German and the Austrian price


Figure 5.6: Left: Difference (”FBA - NTC”) of total trading volume; Right: Dif-

ference of total trading cost

levels as a result . As for the forecasted periods, electricity prices in AT, DE and FR

according to the FBA model exceed those in the NTC-based one. The Italian and

the Swiss electricity prices, however, do not differ due to the underlying congestion

management method. In fact, electricity prices resulting from the FBA simulation

are closer to each other than those from the NTC-based modelling approach.

Furthermore, concerning the FBA model, the total trading volume, i.e. the sum of

all cross-border trades by the countries and arbitragers located in the CE-Region,

exceeds the trading volume for the NTC-based approach for all simulated winter

periods by an average value of 2500 MWh/h. The corresponding total trading

cost, however, are still lower for the FBA approach by an average of 175.000 AC/h

as illustrated in Figure 5.6.

Figure 5.7 shows the difference between the FBA and the NTC-based modelling

approach for total exporting (left hand side) and for total importing trades (rights

hand side). According to the FBA model, any cross-border trades to AT and FR

are substantially higher while imports to IT decrease noticeably. Regarding the

exporting trades for both models, there is an increase of cross-border trades from

CH, DE and IT.

Analysis: According to the FBA approach, Figure 5.8 gives some information on

frequency and location of all congested flowgates for each winter scenario (denoted

by the bars). Since the simulations are carried out on a monthly time basis, the

resulting monthly congestion pattern is summed up over the half-year of the cor-

responding simulated winter scenario, e.g. based on the winter scenario W08/09,

flowgate AT → CH is congested in four out of six simulated months. In addition

to that, the difference of electricity prices between those countries on both ends

of each flowgate is illustrated for each winter scenario (red crosses). Obviously,


Figure 5.7: Left: Difference (”FBA - NTC”) of exporting trades; Right: Difference

of importing trades

those flowgates between countries with the highest difference in electricity prices,

i.e. flowgates AT → CH and AT → IT, are frequently congested. It implies that

the major flow of trades goes from low-price areas AT, DE and FR to high-price

areas CH and IT which provides the basis of the price convergence as depicted in

Figure 5.5. Furthermore, as a result of the implicit feature of the FBA mechanism

to benefit those cross-border trades that relieve congestion, total trading cost are

reduced on average as illustrated by Figure 5.6. Then, the increase of total trades

implies that the flow-based allocation scheme creates additional possibilities for

price arbitrage between the power markets in the CE-Region. Thus, the conver-

gence of electricity prices is more evident for the FBA than for the NTC-based

approach.

According to the flow-based allocation scheme, each single cross-border trade is

transferred into the induced physical power flow on all flowgates within the CE-

Region. In case of congestion, a transmission price has to be paid based on the

impact of the inter-regional trade on the congested flowgates. Figure 5.9 visualises

the change of the FBA and the NTC-based approach by means of exporting and

importing trades. Additionally, the most frequent congested flowgates are shown.

While maximum import and export trades for each country are given by the same

physical limits for both models, the main reason of the additional trading volume

and reduced trading cost is based on the increasing export trades by CH, DE and

IT, combined with rising imports to AT, CH and FR. In fact, that change of export

and import balance reduces the physical loading of frequently congested flowgates

AT → CH and AT → IT, thus, decreasing the congestion price in direction of the

power flow on the congested flowgate. Consequently, due to the netting of power

flow, transmission cost are lower while total cross-border trades can be increased

according to the FBA approach.


15.00

20.00

25.00

30.00

35.00

2

3

4

5

6qu

ency

of C

onge

stio

nTransmission Congestion W08/09Transmission Congestion W13/14Electricity Price Difference

[Months] [€/MWh]

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0

1

2

3

4

5

6

CH FR CH-DE AT CH CH-IT DE FR DE AT FR IT AT IT

Freq

uenc

y of

Con

gest

ion

Flowgates

Transmission Congestion W08/09Transmission Congestion W13/14Electricity Price Difference

[Months] [€/MWh]

Figure 5.8: Frequency of congested flowgates and electricity price differences based

on the FBA approach

Frequency ofCongestion

Ex Im

Ex: Export TradesIm: Import Trades

Ex Im

Im Im

Ex Im

Figure 5.9: Difference (”FBA-NTC”) of export trades and of import trades



The main goal of finding adequate estimations of CVE and CVT parameters is

to study the power market in the vicinity of its reference scenario. A closed-form

mathematical expression has been derived for CVE and CVT parameter estimation

for the case of multiple power markets. While the explicit fitting procedure features

major drawbacks such as the incomplete availability of the needed market data,

the implicit fitting procedure overcomes this problem by an iterative adjustment

of CVE and CVT parameters by means of historical electricity and transmission

prices, respectively.

The CVE parameter estimation is applied in the context of a power market study

in the central European region which aims at comparing the FBA and the NTC-

based congestion management scheme. The results show that according to the FBA

approach, transmission congestion is shifted to those flowgates between countries

with high electricity price differences, i.e. flowgates AT → CH and AT → IT, since

the main flow of trades goes to high-price areas CH and IT. Furthermore, total

trading costs are reduced as a result of the financial compensation for those cross-

border trades that relieve transmission congestion. More specifically, the export

trades of DE and IT rise while a substantial increase of import trades of AT and

FR can be observed. Thus, the total trading volume is increased which leads to

additional possibilities of inter-regional price arbitrage. As for the FBA approach,

this results in the more evident convergence of electricity prices between low-price

areas AT, DE, FR and high-price areas CH, IT when compared to NTC-based

modelling.

6 Conclusions and Outlook

The oligopolistic structure of the power supply market is particulary exposed to the

exercise of market power by large generating companies, which can result in imper-

fectly competitive electricity and transmission prices, production inefficiencies and

a redistribution of benefits among power producers and consumers. Besides, cross-

border trades are gaining increasing importance due to the opening of electricity

markets for national and international market participants. A new congestion ma-

nagement method has been proposed by the organisation European Networks of

transmission System Operators for Electricity (ENTSO-E), term flow-based alloca-

tion of transmission capacity, which better accounts for the power flows induced by

cross-border transactions. In order to understand the complexities of competition

in the liberalised electricity and transmission market and to help analyse market

designs, regulatory policies and congestion management methods, computationally

tractable models of imperfect competition are becoming increasingly important.

In the first part of this thesis, the concept of conjectural variations has been in-

troduced to the electricity and transmission market in order to study the effect

of imperfect competition under the flow-based allocation scheme. By paramet-

rically changing each supplying company’s conjecture on its competitors’ market

behaviour in each of both markets, different intensities of competition can be mod-

elled. As shown in the case study (see Section 4.5.4), electricity prices are exposed

to the power producers’ strategic market decisions as they hold back cheap gener-

ation from the market in order to raise electricity prices above competitive levels.

Furthermore, strategic companies are able to manipulate the congestion price to

their favour by changing their demand for transmission capacity. This gives rise to

spatial price discrimination, i.e. when the difference of electricity prices in any two

markets does not correspond to the transmission price to pay for a transaction be-

tween those markets. The case studies illustrate that strategic power producers ex-

ploit favourably the weaknesses of the market design such as the lack of arbitragers

113

114 6 Conclusions and Outlook

by additionally profiting from self-induced spatial price discrimination. Perfectly

competitive arbitragers exploit any non-cost based differences of electricity prices

by purchasing electricity in lower-price areas and selling it to higher-price areas.

Thus, by allowing arbitragers to have access to all power markets, the transmis-

sion price between two power markets should always correspond to the difference

of electricity prices.

The implementation of reliable models of liberalised electricity markets in order to

achieve reasonable market prices remains a challenging task. The second part of

this thesis presents the implicit estimation of the conjectural variation parameters

related to electricity (CVE) and transmission markets (CVT) based on multiple

power markets by means of historical market data. An application is shown to the

power market model of the central European region, which is used for comparing

the existing Net Transfer Capacity (NTC) based mechanism with the proposed

Flow-based Allocation (FBA) scheme. By fitting the CVE parameters with histor-

ical market data, both congestion management methods are studied in the vicinity

of historical electricity prices and demand levels. The results reveal additional

benefits of the FBA approach by means of a higher total trading volume and lower

total trading costs in that region. This is due to the financial compensation for

those cross-border trades that relieve the power flow on congested flowgates. Fur-

thermore, regarding the FBA mechanism, the increase of total cross-border trades

leads to an additional inter-regional price arbitrage in the central European region.

As a result of that, the convergence of electricity prices between low-price areas

Austria, Germany, France and high price areas Switzerland, Italy is more evident

when compared with the NTC-based approach.

A major assumption of this case study is the validity of the available market and

network data, i.e. network data, marginal generation cost, flowgate capacities,

NTC values etc., for future scenarios of the power system. The ”snapshot” of the

underlying European transmission network, however, only represents the state of

the power system in a single hour. In order to provide more accurate simulation

results over a given time horizon, the amount of available market and network data

must be extensive.

Another focus of future work concerns the modelling of the power supply market.

In this work, marginal generation cost curves are given in an aggregated form

per each country in the central European region. In fact, the presented power

market model is qualified for the modelling of several power supply companies

with different generation characteristics. This approach allows the modelling of

competition among supplying companies within each country, thus, resulting in a


more accurate price generation process.

Furthermore, the oligopolistic power market model can be extended by including

the market for financial transmission rights and the market for ancillary services

as documented in [40]. Then, the introduction of conjectural variations addresses

the issue of imperfect competition in those two markets in a very similar way as

in the electricity and the transmission market.

116 6 Conclusions and Outlook

A Aspects of Power System Design

A.1 Elasticity of Electricity Demand

The (electricity) price elasticity of demand represents the response of electricity

demand to changes in the electricity price. Based on reference values for electricity

price (πref) and demand (dref), a small increase of the electricity price yields to a

decrease of the electricity demand according to

εel =πref

dref· Δd

Δπ(−1 ≤ εel ≤ 0). (A.1)

If εel < 0 the demand is said to be elastic. At εel = −1 the demand is unit elastic

while at εel = 0 the demand is constant since changes in the electricity price do

not affect the demand level.

A.2 Elasticity of Transmission Capacity Demand

The (transmission) price elasticity of demand represents the response of trans-

mission capacity demand to changes in the transmission price. Based on reference

values for transmission price (ωref) and flowgate capacity (NFCref), a small increase

of the transmission price yields to a decrease of the capacity demand according to

εtr =ωref

NFCref· Δtfg

Δω(0 ≤ εtr ≤ 1). (A.2)

If εtr > 0 the transmission capacity demand is said to be elastic. At εtr = 1 the

capacity demand is unit elastic while at εtr = 0 the capacity demand is constant

since changes in the transmission price do not affect the transmission capacity

demand.

117

118 A Aspects of Power System Design

Area A Area B

A

CSACS

CSBCS

DA

DB

PSAPS

PSBPSB

SA SB

qqAq qqBq

Figure A.1: Social welfare in isolated areas

A.3 Definition of Social Welfare

In liberalised power markets suppliers and consumers of electricity are the main

participants. Their offer and ask bids for a certain amount of electricity define the

market’s aggregated supply and demand curves. The intersection of both curves

gives the market clearing price and the volume. Basically, the aggregated supply

curve describes a price/quantity characteristic of the suppliers’ willingness to offer

electricity, while the willingness of consumers to buy a certain amount of electricity

for a certain price is described by the aggregated demand curve.

In the following, supply curve Sc and demand curve Dc in area c are assumed to

be linear, i.e. the relation between offer price πc(qSc ) and offer quantity qS

c and the

relation between ask price πc(qDc ) and ask quantity qD

c is linear as follows

Si : πc(qSc ) = αc + βc · qS

c (A.3)

Di : πc(qDc ) = γc − δc · qD

c (A.4)

Intercept and slope of the supply function are denoted by αc (> 0), βc (> 0) while

γc (> 0), δc (> 0) are intercept and slope of the demand function.

In the following, the social welfare will be introduced which is widely used as a

measure to evaluate the efficiency of liberalised power markets. Assuming there are

multiple power markets, three cases are studied based on different specifications

on the cross-border trade of electricity.

No Cross-Border Trades

Figure A.1 shows a region of two areas A and B assuming there is no cross-border

trade of electricity between those areas. In this case, the intersection of supply and

A.3 Definition of Social Welfare 119

Area A Area BCross Border Trade

DA

PSPS‘

CSACS‘

A‘A

B‘ CSBCS‘

R

IDA

DB

PSAPS‘

‘ ‘ ‘

B PSBPS‘ESBSA SB

q qqqAq qABq qBqqA qBq

Figure A.2: Social welfare with cross-border trade

demand function in each area leads to the market clearing price π′c, c ∈ {A,B} and

the quantity q′c = qS

c = qDc , c ∈ {A,B}. By definition, the social welfare of area c is

the sum of producer and consumer surplus according to

SWc = PSc + CSc (A.5)

Assuming there is no cross-border trade between the areas, the social welfare for

the entire region becomes

SWtot = SWA + SWB

=∑

c∈{A,B}(PSc + CSc)

Cross-Border Trades, no Transfer Capacity Limit

Let assume cross-border trades of electricity between areas A and B as illustrated in

Figure A.2. If there is sufficient transfer capacity between those areas, electricity is

shifted from low-price area B to high-price area A until a single regional electricity

price is reached. That price can be determined by means of the import-demand

curve IDA of high-price area A and the export-supply curve ESB of low-price area

B. Based on Equations (A.3) and (A.4), both are defined as follows

IDA : qImpA = qD

A(πA) − qSA(πA)

⇒ πA(qImpA ) =

αAδA + βAγA

βA + δA− βAδAβA + δA

· qImpA (A.6)

ESB : qExpB = qS

B(πB) − qDB (πB)

⇒ πB(qExpB ) =

αBδB + βBγB

βB + δB+

βBδBβB + δB

· qExpB (A.7)


Area A Area BCross Border Trade

CS‘‘CSA IDA

DA

DB

PS‘‘PSA

A

‘‘A

A

‘‘B

B PS‘‘PSB

CS‘‘CSB

A

ESBSARAB

q qqq‘‘qA qABq‘‘ qBq‘

B

qA qBq

ESBSA SB

Figure A.3: Social welfare with cross-border trade and transfer capacity limit

where IDA is linear decreasing while ESB is linear increasing. As indicated in Figure

A.2, regional price πR and transferred quantity q′AB result from the intersection of

IDA and ESB. In that case, πR is equal to π′A and π

′B due to the transfer of

electricity from low-price area B to high-price area A. Then, the social welfare of

the region becomes

SW ′tot = SW ′

A + SW ′B

=∑

c∈{A,B}(PS ′c + CS′c)

while

SW ′tot > SWtot (A.8)

when compared to Figure A.1.

Cross-Border Trades, Transfer Capacity Limit

Now, a capacity limit is imposed on the cross-border between areas A and B as

shown in Figure A.3 which limits the maximum transfer to q′′AB. In that case, elec-

tricity prices in area A and B are not equal and a transmission prices has to be paid

to shift electricity from B to A according to π′′A−π′′B. Then, in case of transmission

congestion, the auction revenue is added to the producer and consumer surplus in

order to determine the social welfare for the entire region:

SW ′′tot = SW ′′

A + SW ′′B + ARAB

=∑

c∈{A,B}(PS ′′c + CS′′c ) + ARAB

A.4 DC Power Flow Approximation 121

while

SW ′tot > SW ′′

tot > SWtot (A.9)

when compared to Figures A.1 and A.2.

A.4 DC Power Flow Approximation

The assumption of a linear DC power flow is often used in optimisation prob-

lems of power markets when accounting for the effect of the transmission network.

However, in most of these models the focus is on power economics rather than on

the exact modelling of the power flow. Instead of using non-linear AC power flow

equations, the most critical cases in the transmission network can also be identi-

fied by the DC power flow approximation. Furthermore, the linear DC power flow

equations retain the convexity of optimisation problems and are fast to be solved

which is of great value in the operation and planning of electric power systems.

The π - model of a transmission lines between nodes k andm is shown in Figure A.4

(left). It is characterised by the series impedance Zk,m and the shunt admittance

Y shk,m

Zk,m = Rk,m + jXk,m

Yk,m = Gshk,m + jBsh

k,m

where Rk,m and Xk,m denote the series resistance and reactance while Gshk,m and

Bshk,m are the shunt conductance and susceptance. The formulation of the network

equations requires the series admittance Yk,m of the transmission line

Yk,m = Z−1k,m = Gk,m + jBk,m

=Rk,m

R2k,m +X2

k,m

− jXk,m

R2k,m +X2

k,m

In many cases Gshk,m can be neglected (”short lines”), thus, the active and reactive

power flows from node k to node m are obtained as follows

Pk,m = U2kGk,m − UkUmGk,mcos(θk,m) − UkUmBk,msin(θk,m) (A.10)

Qk,m = −U2k (Bk,m +Bsh

k,m) + UkUmBk,mcos(θk,m) − UkUmGk,msin(θk,m)

where voltage magnitudes and angles are denoted by Um, Uk and θm, θk. Besides,

the notation θk,m = θk − θm holds.


k

Pk,m ,Qk,mUk , k Um , m

Pk,m

Xk,m

k m

Zk,m

Uk , k Um , m

Yk,msh Yk,m

sh

Figure A.4: π - model of transmission line and DC power flow model

In DC load flow, the following approximations are valid:

Uk ≈ Um

sin(θk,m) ≈ θk,m

cos(θk,m) ≈ 1

These equations are implicitly based on the assumptions of a flat voltage profile,

which particulary is the case for light load conditions. Additionally, θk,m is assumed

to be small. Furthermore, since Bk,m = −1/Xk,m, the expression for the active

power flow, stated in Equation (A.10), can be simplified to

Pk,m = θk,m/Xk,m =θk − θm

Xk,m(A.11)

Figure A.4 (right) illustrates the simplified transmission line model. It is analogous

to Ohm’s law applied to a resistor with a DC current, where Pk,m denotes the DC

current, θk and θm are the DC voltages at both end of the resistor and Xk,m is the

resistance [39].

B Flow Based Allocation of Cross-

Border Capacities

B.1 Complementarity Conditions of Flow-based Auc-

tion

Bidders Model KKTs

For each qb,c,c′ (b ∈ B, (c, c′) ∈ C):

0 ≤ qb,c,c′⊥− pbidb,c,c′ +

∑

m∈M

ωm · PTDFc,c′,m + νb,c,c′ ≥ 0 (B.1)

For each νb,c,c′ (b ∈ B, (c, c′) ∈ C):

0 ≤ νb,c,c′⊥qbidb,c,c′ − qb,c,c′ ≥ 0 (B.2)

The Auction Office Model KKTs

For each λ+m (m ∈M):

0 ≤ λ+m⊥NFC+

m − zm ≥ 0 (B.3)

For each λ−m (m ∈M):

0 ≤ λ−m⊥NFC−m + zm ≥ 0 (B.4)

For each zm (m ∈M):

0 ≤ −ωm + λ+m − λ−m = 0 (B.5)

B.2 Case Study: AC vs. DC PTDF

The comparison between AC and DC PTDFs is based on a planning data set of the

UCTE power system for a typical winter peak-hour provided by [42]. The scope of

123

124 B Flow Based Allocation of Cross-Border Capacities

100

60%

70%

40%

50%

60%

TDFs

[%]

45

20%

30%

40%

ount

of

PT

12

310%

20%

Am

o

0%0.5 1.0 1.5 2.0

Change of power flow [MW]

0.0 0.5 0.5 1.0 1.0 1.5 1.5 2.0

Figure B.1: Comparison of AC and DC PTDF

the study is confined to countries AT, CH, DE, FR and IT, thus, PTDFs are only

related to the flowgates between those countries. By applying the same generation

shift method for both modes, AC and DC PTDFs are compared with each other

on each single flowgate according to a generation shift of 100MW based on the

operating point of the power system as follows

ΔPTDF100MW = |PTDFAC100MW − PTDFDC

100MW|

The results are illustrated in Figure B.1, where the horizontal axis represents in-

tervals of absolute changes of power flow on any given flowgate. The vertical axis

depicts the percentage of all PTDFs within a certain interval, e.g. slightly less

than 30% of all evaluated ΔPTDF100MW leads to a change of power flows on all

flowgates in the interval of 0.5 MW < ΔPTDF100 MW ≤ 1.0 MW.

B.3 Derivation of the Net Flowgate Capacity

Inserting Equations (3.23), (3.26) and (3.28) into Equation (3.25) yields to

NFCm ≈ NTFm + ΔEmaxc,c′ · PTDFc,c′,m −

TRMc,c′ · PTDFc,c′,m −NTFm +

∑

(c,c′)∈F

BCEc,c′ · PTDFc,c′,m

B.3 Derivation of the Net Flowgate Capacity 125

And after rearranging

NFCm ≈ ΔEmaxc,c′ · PTDFc,c′,m −

TRMc,c′ · PTDFc,c′,m +

BCEc,c′ · PTDFc,c′,m +∑

(k,k′)∈F,(k,k′) �=(c,c′)


⇐⇒

NFCm ≈ (BCEc,c′ + ΔEmaxc,c′ − TRMc,c′) · PTDFc,c′,m +

∑

(k,k′)∈F,(k,k′)�=(c,c′)


=⇒


(k,k′)∈F,(k,k′) �=(c,c′)


the NFC can be formulated as a function of the BCE and NTC.

126 B Flow Based Allocation of Cross-Border Capacities

C Modelling Strategic Generator

Behaviour

C.1 Classical Concepts from Game-Theory

C.1.1 Cournot Competition

Focus on the electricity market

Inserting Equation (4.2) into Equation (4.4), then taking the derivative, yields the

following optimality condition for firm f :

∂Πf

∂Gf=

∂πA(Gf , G∗−f )


BA= 0

⇐⇒ ∂(α− β · (Gf +G∗−f ))


BA= 0

⇐⇒ −β ·Gf + πA −MCf − ω∗−→BA

= 0

⇐⇒ πA − ω∗−→BA

= MCf + β ·Gf . (C.1)

Focus on the transmission market

Inserting Equation (4.3) into Equation (4.4), then taking the derivative, yields the

following optimality condition for firm f :

∂Πf


BA(Gf , G

∗−f )


BA= 0

⇐⇒ π∗A −MCf − ∂(δ · (Gf +G∗−f ))


BA= 0

⇐⇒ π∗A −MCf − δ ·Gf − ω−→BA

= 0

⇐⇒ π∗A − ω−→BA

= MCf − δ ·Gf . (C.2)

127

128 C Modelling Strategic Generator Behaviour

C.2 The Concept of Conjectural Variation

C.2.1 CV in the Electricity Market

∂Πf

∂Gf=

∂πA(Gf , G−f )


BA= 0

⇐⇒ ∂(α− β · (Gf +G−f ))


BA= 0

⇐⇒ −β · (1 +∂G−f

∂Gf) ·Gf + πA −MCf − ω∗−→

BA= 0

⇐⇒ −β · (1 + CV Ef) ·Gf + πA −MCf − ω∗−→BA

= 0

⇐⇒ πA − ω∗−→BA

= MCf + β · (1 + CV Ef) ·Gf . (C.3)

C.2.2 CV in the Transmission Market

∂Πf


BA(Gf , G−f )


BA= 0

⇐⇒ π∗A −MCf − ∂(δ · (Gf +G−f ))


BA= 0

⇐⇒ π∗A −MCf − δ · (1 +∂G−f

∂Gf) ·Gf − ω−→

BA= 0

⇐⇒ π∗A −MCf − δ · (1 + CV Tf) ·Gf − ω−→BA

= 0 (C.4)

⇐⇒ π∗A − ω−→BA

= MCf − δ · (1 + CV Tf) ·Gf . (C.5)

C.3 Setting up the Equilibrium Model

Company Model KKTs

For each gf,h,c (f ∈ F , h ∈ H(f, c), c ∈ C):

0 ≤ gf,h,c⊥MCf,h,c − θgf,c + μf,h,c ≥ 0 (C.6)

For each sf,c (f ∈ F , c ∈ C):

0 ≤ sf,c⊥− π∗c +πref

c

drefc · εelc

· (1 + CV Ef,c) · sf,c + θsf,c ≥ 0 (C.7)

C.3 Setting up the Equilibrium Model 129

For each tf,c,c′ (f ∈ F , c, c′ ∈ C):

0 ≤ tf,c,c′⊥− θsf,c′ + θg

f,c −∑

m∈M

(ηf,m · PTDFc,c′,m) ≥ 0 (C.8)

For each μf,h,c (f ∈ F , h ∈ H(f, c), c ∈ C):

0 ≤ μf,h,c⊥Gf,h,c − gf,h,c ≥ 0 (C.9)

For each θsf,c (f ∈ F , c ∈ C):

θsf,c⊥sf,c −

∑

c′∈C

tf,c′,c = 0 (C.10)

For each θgf,c (f ∈ F , c ∈ C):

θgf,c⊥

∑

c′∈C

tf,c,c′ −∑

h∈H(f,c)

gf,h,c = 0 (C.11)

For each tfgf,m (f ∈ F , m ∈M):

tfgf,m⊥ω∗m +ωref

m

NFCc · εtrc· (1 + CV Tf,m) · tfgf,m + ηf,m = 0 (C.12)

For each ηf,m (f ∈ F , m ∈M):

ηf,m⊥tfgf,m −∑

c,c′∈C

(tf,c,c′ · PTCUc,c′,m) = 0 (C.13)

Arbitrager Model KKTs

For each apc (c ∈ C):

0 ≤ apc⊥π∗c − θapc + ρa ≥ 0 (C.14)

For each asc (c ∈ C):

0 ≤ sf,c⊥− π∗c + θasc − ρa ≥ 0 (C.15)

For each atc,c′ (c, c′ ∈ C):

0 ≤ atc,c′⊥− θasc′ + θap

c −∑

m∈M

(ηam · PTDFc,c′,m) ≥ 0 (C.16)

For each θasc (c ∈ C):

θasc ⊥asc −

∑

c′∈C

atc′,c = 0 (C.17)

130 C Modelling Strategic Generator Behaviour

For each θapc (c ∈ C):

θapc ⊥

∑

c′∈C

atc,c′ − apc = 0 (C.18)

For each atfgm (m ∈M):

atfgm⊥ω∗m + ηf,m = 0 (C.19)

For each ηam (m ∈M):

ηam⊥atfgm −

∑

c,c′∈C

(atc,c′ · PTCUc,c′,m) = 0 (C.20)

The Auction Office Model KKTs

For each λ+m (m ∈M):

0 ≤ λ+m⊥NFC+

m − zm ≥ 0 (C.21)

For each λ−m (m ∈M):

0 ≤ λ−m⊥NFC−m + zm ≥ 0 (C.22)

For each zm (m ∈M):

0 ≤ −ω∗m + λ+m − λ−m = 0 (C.23)

Market Clearing Conditions

For each πc (c ∈ C):

πc⊥αc + βc · (∑

f∈F

sf,c + asc − apc) − πc (C.24)

For each ωm (m ∈M):

∑

c,c′∈C

PTDFc,c′,m · (∑

f∈F

tf,c,c′ + atc,c′) − zm (C.25)

List of Abbreviations

AAC Already Allocated Capacity

AFC Available Flowgate Capacity

ANF Already Nominated Flow

ATC Available Transfer Capacity

BCE Base Case Exchange

CVE Conjectural Variations for Electricity Markets

CTE Conjectural Variations for Transmission Markets

ENTSO-E European Networks of Transmission System Operators for Electricity

EPEC Equilibrium Program with Equilibrium Constraints

EuroPEX Association of European Power Exchanges

FBA Flow-based Allocation

FMC Flow-based Market Coupling

ISO Independent System Operator

KKT Karush-Kuhn Tucker Conditions

LCP Linear Complementarity Problem

LMP Locational Marginal Prices

MCP Mixed Complementarity Problem

MLCP Mixed Linear Complementarity Problem

MPEC Mathematical Program with Equilibrium Constraints

NFC Net Flowgate Capacity

NTC Net Transfer Capacity

OMC Open Market Coupling

OTC Over-The-Counter

PTDF Power Transfer Distribution Factor

PX Power Exchange

SEE South East Europe

TSO Transmission System Operator

131

List of Symbols

Indices and Sets

c ∈ C Set of power markets/price areas c

b ∈ B Set of bids b

f ∈ F Set of generating companies f

h ∈ H Set of generating units h

H(f, c) Set of generating units owned by company f ,

located in price area c

m ∈M Set of flowgates m

Primal Variables

apc Arbitragers’ purchases in power market c, [MW]

asc Arbitragers’ sales in power market c, [MW]

atc,c′ Arbitragers’ cross-border trades from market c to c′, [MW]

atfgm Allocated physical transmission capacity on flowgate m

resulting from the arbitragers’ total trades, [MW]

dc Electricity demand in power market c, [MW]

gf,h,c Generation output by unit h, owned by company f

and located in price area c, [MW]

qb,c,c′ Allocated quantity for bid b from market c to c′, [MW]

sf,c Sales by company f in power market c, [MW]

tf,c,c′ Company f ’s cross-border trades from market c to c′, [MW]

tfgf,m Allocated physical transmission capacity on flowgate m

resulting from company f ’s total trades, [MW]

zm Power flow on flowgate m, [MW]

πc Electricity price in power market c, [AC/MWh]

ωm Congestion price related to flowgate m, [AC/MWh]

133

Dual Variables

ηf,m Dual variable for allocated capacity - total trades constraint

of company f according to flowgate m, [AC/MWh]

ηam Dual variable for arbitragers’ allocated capacity - total trades

constraint according to flowgate m, [AC/MWh]

λ+m Dual variable for net flowgate capacity constraint

in direction +, [AC/MWh]

λ−m Dual variable for net flowgate capacity constraint

in direction −, [AC/MWh]

μf,h,c Dual variable for generation capacity constraint of company f

in price zone c, [AC/MWh]

νb,c,c′ Dual variable for allocated quantity limit for bid b

from market c to c′, [AC/MWh]

ρa Dual variable for arbitragers’ total sales - purchase

constraint m, [AC/MWh]

θapc Dual variable for arbitragers’ purchase - trades balance


θasc Dual variable for arbitragers’ trades - sales balance


θgf,c Dual variable for generation - trades balance of company f


θsf,c Dual variable for trades - sales balance of company f


Coefficients and Functions

drefc Reference electricity demand in power market c, [MW]

Gf,h,c Generation capacity limit of unit h, owned by company f

and located in price area c [MW]

GCf Company f ’s generation cost, [AC]

MCf,h,c Marginal generation cost of unit h, owned by

company f , located in price area c, [AC/MW]

NPf Company f ’s net profits, [AC]

NFC+m Net flowgate capacity related to flowgate m in direction +, [MW]

NFC−m Net flowgate capacity related to flowgate m in direction −, [MW]

pbidb,c,c′ Price for bid b from market c to c′, [AC/MW]

PTDFc,c′,m Power transfer distribution factor related to flowgate m,

based on a 1MW transaction from market c to c′, [p.u.]

qbidb,c,c′ Requested quantity for bid b from market c to c′, [MW]

SPf Company f ’s sales profits, [AC]

TCf Company f ’s transmission cost, [AC]

εelc Price elasticity of electricity demand in power market c, [p.u.]

εtrm Price elasticity of transmission capacity demand

related to flowgate m, [p.u.]

πrefc Reference electricity price in power market c, [AC/MWh]

τb,c,c′ Transmission price for bid b from market c to c′, [AC/MWh]

ωrefm Reference congestion price related to flowgate m, [AC/MWh]

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Curriculum Vitae

2005 − 2010 Assistant at the Power Systems Laboratory,

ETH Zurich, Switzerland;

PhD thesis under supervision of Prof. Dr. G. Andersson

1999 − 2005 Study of Electrical Engineering and Information Technology,

RWTH Aachen, Germany

1998 − 1999 Military service, Karlsruhe, Germany

1989 − 1998 Secondary school, Aloisiuskolleg, Bonn, Germany

1985 − 1989 Primary school, Donatusschule, Bonn, Germany

143

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