Natural Gas Supply in Denmark - A Model of Natural Gas Transmission and the Liberalized Gas Market

Transcription

1 Natural Gas Supply n Denmark - A Model of Natural Gas Transmsson and the Lberalzed Gas Market A Masters Thess submtted to the department of Informatcs and Mathematcal Modelng at the Techncal Unversty of Denmark Author: Supervsors: Lars Bregnbæk Thomas K. Stdsen, Assstant Professor Hans Ravn, Dr. Techn. Submtted June 2005

2 Abstract In the wake of the lberalzaton of European energy markets a large area of research has spawned. Ths area ncludes the development of mathematcal models to analyze the mpact of lberalzaton wth respect to effcency, supply securty and envronment, to name but a few subjects. Ths project descrbes the development of such a model. In Denmark the parallel lberalzaton of the markets of natural gas and electrcty and the exstence of an abundance of de-centralzed combned heat and power generators of whch most are natural gas fred, leads to the natural assumpton that the future holds a greater deal of nterdependency for these markets. A model s developed descrbng network flows n the natural gas transmsson system, the man arteres of natural gas supply, from a techncal vewpont. Ths yelds a techncal boundng on the supply avalable n dfferent parts of the country. Addtonally the economc structure of the Dansh natural gas market s formulated mathematcally gvng a descrpton of the transmsson, dstrbuton and storage optons avalable to the market. The supply and demand of natural gas s put nto a partal equlbrum context by ntegratng the developed model wth the Balmorel model, whch descrbes the markets for electrcty and dstrct heat. Specfcally on the demand sde the consumpton of natural gas for heat and power generaton s emphaszed. General results and three demonstraton cases are presented to llustrate how the developed model can be used to analyze varous energy polcy ssues, and to dsclose the strengths and weaknesses n the formulaton.

3 Contents 1 Introducton Motvaton Objectve Structure Readng the Thess Acknowledgements Network Bound Energy Supply Energy Markets and Supply Systems Three Interconnected Markets Electrcty Supply Organzaton of the Electrcty Market Legal Foundaton for Lberalzed Electrcty Markets Natural Gas Supply Organzaton of the Market for Natural Gas Legal Foundaton for Gas Market Lberalzaton Dstrct Heatng Combned Heat and Power (CHP) Market Defnton and Regulaton Summary Model Structure Overvew Flow Model Economc Model Summary The Balmorel Model Top-Down - Bottom-Up Market Equlbrum Partal Equlbrum and Operatons Research Elements of the Balmorel Model Geography The Temporal Dmenson The Objectve Functon

4 CONTENTS Investments Energy Transformaton Transmsson and Dstrbuton Energy Demands Emsson Quotas More on the Balmorel Model Flow Model General One Dmensonal Flow Conservaton of Mass Momentum Energy Conservaton Thermodynamc State Means of Transent Flow Analyss Steady-State Analyss Head and Bernoull s Equaton Gas Networks Practcal Formulaton Quas Steady-State Model Conservaton of Mass Non-lnear Momentum Constrants Pressure Drop over Ppe-length From Non-lnear to Pecewse-lnear Bnary Flow Drecton Varables Lnkng Tme Segments The Flow Model Computatonal Intractablty A Soluton A Conc Varaton Summary Gas Market Model The Transmsson System Dstrbuton Gas Storage Gas Market Lnk wth the Balmorel Model Summary Model Executon Geography and Tme Fuel Prces Demands Demand for Electrcty and Dstrct Heat Natural Gas Demand Gas Producton Technology Data Model Complexty Summary

5 CONTENTS 1 8 Smulaton Results System Load Margnal Values Gas prces Electrcty prces Fuels for Energy Transformaton Transformaton Technologes Investments n Transformaton Technology Emssons from Transformaton The Natural Gas Transmsson System Area Dstrbuton Summary Demonstraton Cases De-centralzed Combned Heat and Power Fxed Tarff vs. the Spot Market Technologes Dstrbuton of Intal Capacty Model Results Prospectve Natural Gas Reserves Emssons of CO Implementaton Summary Concluson Evaluaton of the Product Soundness of Assumptons Perfect Informaton Perfect Competton Quas Steady-State Modelng Demand Inelastcty Outstandng Issues Tme Delay Exogenous Prces Heat Demand Resdual Natural Gas Demand Heurstc Qualty Contrbuton Summary

6 Nomenclature α β e β h β x χ E s,t s,t ɛ h a ɛ e r Γ s ι κ s EN κ Y EN κ s EX κ Y EX Λ A G s constant 1 for ppe flow equatons cost factor for electrcty dstrbuton cost factor for heat dstrbuton cost of transmsson export rate at node energy gan duraton of tme perod s, t duraton of tme segment s, t dstrbuton loss factor for heat dstrbuton loss factor for electrcty fracton of purchased storage capacty, whch must be n the storage faclty by season s njecton rate nto faclty monthly entry capacty booked n month s annual entry capacty booked monthly ext capacty booked n month s annual ext capacty booked stock of a faclty node-ppe ncdence matrx graph descrbng the transmsson network vector of source strengths n the network

7 CONTENTS 3 G M A C D E I L P R S T Y ν Ω ι σ ψ l e υ l e ε σ m c,m p l e V σ Φ( ) π s,exp π s,imp ρ ρ s ρ n σ set of generaton technologes set of emsson types set of areas set of countres set of dstrbuton areas set of edges subset of areas where a storage faclty s located. pont of lnearzaton subset of areas where producton of gas occurs or gas can be exported. set of regons set of seasons set of tme segments set of years supply rate at node added heat njecton allowance for storage product σ slope of lnearzaton plane n drecton of pressure slope of lnearzaton plane n drecton of flow rate extracton allowance for storage product σ lmt on emsson type m n country c ntersect of lnearzaton plane volume allowance for storage product σ functon representng the emssons resultng from the generaton profle export prce at n season s mport prce at n season s densty cost of monthly capacty contract as fracton of annual contract densty under normal condtons ndex of storage products 1 f transmsson factor

8 CONTENTS 4 τ ι τ ξ δ τen Y τex Y τ V ε Ξ ζ σ A c B c v D dh f E e d e s f f t njecton tarff dstrbuton tarff n dstrbuton area δ for prce step ξ tarff for annual entry capacty tarff for annual ext capacty volumetrc transmsson tarff extracton rate from faclty set of prce steps for the dstrbuton system operators product unts purchased of storage contract σ area of ppe cross-secton back-pressure rato between electrcty and heat producton specfc heat capacty at constant volume s the ppe dameter head loss due to frcton effcency factor of a ppe electrcty made avalable to the consumer after loses n transmsson, dstrbuton etc. generated electrcty frcton factor dependent on the ppe roughness and Reynolds number theoretcal frcton factor F x net forces actng n drecton x F s,t,g ( ) Functon descrbng the natural gas consumpton of technology g n area n tme segment s, t g g g h s K a p Q q Q n R acceleraton of gravty generaton constrants of technology g generated heat cost of heat and power generaton wthn a certan area pressure volumetrc flow rate added heat volumetrc flow rate under normal condtons the unversal gas constant

9 CONTENTS 5 R T t e t h u U e U h V W w wg ξ x r,ρ resdual demand for natural gas temperature energy tax rate on electrcty energy tax rate on heat mass flow rate utlty functon of electrcty utlty functon of heat volume performed work average flow velocty across over a cross secton of ppe weght assocatng dstrbuton prce steps wth generaton types transmsson of electrcty between regons X x(r,ρ) nvestment n electrcty transmsson capacty Z compressblty factor of natural gas

10 CHAPTER 1 Introducton Ths project encompasses the development of a techncal supply model of the Dansh natural gas transmsson system, and a mathematcal descrpton of the economc structures relevant for the supply of natural gas. The two models nteract to shed lght on the nterplay between the three markets of namely natural gas, electrcty, and dstrct heat. 1.1 Motvaton The lberalzaton of the Dansh energy markets, partcularly for electrcty and natural gas, has been accompaned by a wde range of nterestng challenges. The degree of ntegraton and market nteracton between network bound energy supply forms s partcularly nterestng n lght of the central energy polcy ssues of effcency, supply securty, and envronmental mpact. The nterplay between energy markets s perhaps most evdent when consderng a gas fred combned heat and power plant (CHP). Ths s a meetng pont between the electrcty, dstrct heatng markets and the natural gas market. Some de-centralzed CHPs have been producng power on market-lke terms snce January 1st 2005, whle stll under the oblgaton to supply dstrct heatng n accordance wth demand. Ths s a complex economcal producton node, where the decsons taken by the plant operator are nested n the development of three markets. It s an open queston how de-centralzed CHP plants wll react to the new market structure on a long term bass, yet ther potental for mpact on the above mentoned polcy ssues s consderable. In 2002, the total electrcty output from de-centralzed CHPs was 22 PJ out of a total 127 PJ of generated electrcty [12]. Thus around 17% of the electrcty producton, whch s mostly gas fred, has an uncertan future. The second motvatonal factor s that the development of a model for analyzng energy polcy requres a combned look at techncal as well as economc aspects of the energy supply system. Ths combnaton of analyzng techncal systems (.e. the natural gas transmsson network) wth respect to ther capablty for energy supply usng an economcally based market optmzaton, by means of mathematcal modelng, bascally sums up the scentfc nterest of the author of ths thess.

11 1.2. OBJECTIVE Objectve The objectve of the project s to develop a decson support tool for addressng challengng energy polcy ssues and ssues pertanng to energy systems analyss. The emphass s upon the development of a mathematcal descrpton of the natural gas sector from an ntegrated techncal and economcal vewpont. Ths descrpton s ntegrated wth the Balmorel model of electrcty and dstrct heatng ( hopng thereby to acheve a comprehensve representaton of the three network bound energy supply forms. Hence developng a tool for analyzng the nterplay between these markets. Requrements for the model nclude that t should descrbe the capacty and ncentves of relevant market players wth a sutable level of accuracy, and thus be able to predct market development. The model should take nto account the techncal capabltes of exstng systems and the organsatonal/economc structures under whch they are operated. 1.3 Structure Ths thess s structured as follows. Chapter 2 contans a presentaton of relevant energy supply systems and markets. A hstorc revew s combned wth a dscusson of current and future challenges n the sector. The lberalzaton process whch s ongong n Europe s dscussed wth specal emphass on the Dansh model. Chapter 3 outlnes the overall structure of the developed model and how t s ntended to be mplemented. Chapter 4 s a quck revew of the Balmorel model. A comprehensve look at theory of, and models for, compressble flow leads to the development of a transmssons model n chapter 5. The structure of the Dansh market for natural gas s outlned n chapter 6. Ths leads to the model formulaton whch encompasses the economcs of natural gas supply. Chapter 7 descrbes varous confguraton optons and the data set whch s mplemented. Sample results are presented n chapter 8 and n chapter 9 some nspratonal demonstraton cases are presented. Fnally chapter 10 sums up the project contents and presents concludng remarks. As the thess contans an abundance of symbolc terms, attenton s drawn to the nomenclature n the begnnng of the thess, for reference. 1.4 Readng the Thess An understandng of operatons research on an ntroductory level s necessary, as well as famlarty wth basc mechancs and thermodynamcs. As such, the use of lnear, mxednteger and bnary programmng wll not be addressed (see for example [20] and [21]). A bref nsght nto conc programmng s provded as ths s not commonly appled even n OR crcles. More emphass s placed on the feld of flud dynamcs. Wthout gvng a comprehensve revew of the entre feld, the theory necessary for gas flow calculatons and modelng s presented extensvely. 1.5 Acknowledgements The author s grateful to those who have provded assstance n connecton wth ths project, and would especally acknowledge and express apprecaton for the assstance provded by Gastra (Energnet Danmark), specfcally to Jess Bernt Jensen and Torben Brabo for takng the tme to provde the necessary nsght nto the busness of gas transmsson.

12 CHAPTER 2 Network Bound Energy Supply Ths chapter s a comprehensve descrpton of the Dansh network bound energy supply. The bascs of energy supply and markets are ntally dscussed. Four areas are descrbed n the context of nfrastructure, organzaton, and lberalzaton. These areas are: 1. Electrcty supply 2. Natural gas supply 3. Dstrct heatng 4. Combned heat and power (CHP) These areas are naturally nterdependent and n a post-lberalzed energy market the developments n one area have an ncreasng mpact on the other areas. Fgure 2.1 provdes an overvew of the system of energy supply as a whole, showng the dstrbuton of electrcty and dstrct heatng capacty, connectvty to the natural gas networks and avalablty of publc heatng supply. 2.1 Energy Markets and Supply Systems The basc objectve of an energy supply system s naturally supplyng consumers wth demanded energy commodtes. The objectve, when establshng a market structure for energy commodtes, s to ensure that producton and delvery s performed effcently, to make the consumer able to obtan the lowest possble prce, whle mantanng a focus on ssues such as supply securty and any envronmental mplcatons. The exstence of a market for varous energy commodtes, reles on the presence of nfrastructure to enable ther delvery from producer to consumer. The techncal systems enablng supply (e.g. the electrcty grd and natural gas transmsson and dstrbuton networks) are often consdered natural monopoles, snce the nvestment costs whch would be

13 2.1. ENERGY MARKETS AND SUPPLY SYSTEMS 9 Fgure 2.1: Dansh energy producton and supply. Heat and power generaton facltes are dstrbuted throughout the country. Where avalable most are connected to the natural gas network. The penetraton of bo-fuels especally n the publc heatng sector s also a notceable trat for the Dansh energy supply. (SOURCE: Dansh Energy Authorty)

14 2.2. THREE INTERCONNECTED MARKETS 10 nflcted on each market player to develop and mantan hs own techncal system, supersedes the potental for effcency gan by havng a perfectly compettve market. The avalablty of energy and the securty of supply s a publc commodty as a matter of polcy. The consumpton of energy unts (e.g. molecules of natural gas or MWh of electrcty) s a prvate commodty. For ths reason, and the natural monopoly consderaton, energy and energy supply s sold and purchased through two organzatonal structures; a publc servce structure and a market orented structure. The key to lberalzng energy markets s to separate the publc and the prvate commodtes n order to enable transparency for consumers wth respect to energy prces and to ensure that super-vsonal structures are able to asses the performance of companes n charge of supplyng publc commodtes. 2.2 Three Interconnected Markets In Denmark three energy systems form a very nterestng and nterconnected structure. The electrcty and dstrct heatng systems meet n combned heat and power (CHP) generaton facltes, of whch most are natural gas fred, and spread wdely over the country. As such the three networks nterface n the technology of co-generaton. The structure of today s Dansh energy markets s a product of the European sngle market project, the purpose of whch s to ncrease competton and effcency wth respect to natonal and European level concerns for supply securty and envronmental conservaton. As a result, the electrcty and natural gas sectors have been reformed n parallel. The purpose of the lberalzaton s as quoted from the Treaty to secure the free movement of goods, persons, servces and captal wthn the nternal market n ths case wth regard to the markets of electrcty and natural gas. The emphass s placed on ncreased transparency and access for market players to ensure an ntegrated, compettve and effcent market. Ths encompasses the establshment of general prncples for a framework at communty level, whle leavng the mplementaton of ths framework to the member-states, n recognton of the dfferences n the structures of the natonal energy systems. There s a certan degree of freedom for member-states to subsdze or otherwse prortze electrcty generated from renewable and co-generaton methods. Recently, n an effort to further ntegrate the energy sector, the three system companes bearng system responsblty for electrcty and natural gas supply (Eltra, Elkraft System and Gastra) have been merged nto one company bearng the full weght of system responsblty pertanng to electrcty and natural gas supply, Energynet.dk. Ths, along wth a large number of mergers between market players, some planned and others already performed, s a very obvous example of why the knd of research undertaken n ths project s hghly relevant, n lght of current developments. 2.3 Electrcty Supply Electrcty supply n Denmark s manly secured by three types of generaton. 1. Central plants 2. De-centralzed combned heat and power plants 3. Wnd power

15 2.3.1 Organzaton of the Electrcty Market 11 Fgure 2.2: Organzaton of the electrcty sector (SOURCE: Dansh Energy Authorty[28]) The central plants were orgnally large powerplants, manly ol fred, untl the energy crss n the 1970s. Almost all have snce been converted to combned producton of heat and electrcty, and they now supply Denmark s largest ctes wth dstrct heatng whle retanng a large share of the total electrcty producton. Most of them are today fred by ether coal, bomass or natural gas, n part to decrease dependency on nsecure ol supples. Each plant s located on one of the 15 central power plant locatons n the largest Dansh ctes. De-centralzed combned heat and power plants were orgnally local supplers of dstrct heatng, organzed at muncpal level or as consumer owned prvate companes. Many of these heat producers have through the 1990s been converted to co-generaton due to the ntroducton of a favorable subsdy on combned producton enablng decentralzed CHPs to sell electrcty at a feed-n tarff on prortzed terms. There are approxmately 600 decentralzed plants currently n operaton.[28] Wnd power s stll prortzed on the electrcty market. There are around 5,400 wnd turbnes spread out around the country. The total wnd generaton capacty accounts for 3118 MW by January 1st 2005 and n % of electrcty generaton was performed by wnd power.[28] Organzaton of the Electrcty Market Today, the Dansh electrcty supply structure s organzed as dsplayed on fgure 2.2, as a result of the market lberalzaton. The fgure (2.2) demonstrates how the roles of the dfferent actors are connected. The red arrows show the actual electrcty flows. Green arrows show how payment for delvered electrcty occurs. The black arrows show payments for the publc servce oblgatons, whch the dfferent companes serve the consumer. Fnally, the purple arrows demonstrate the flow of network tarffs from the consumer to the network companes.

16 2.3.2 Legal Foundaton for Lberalzed Electrcty Markets 12 A further descrpton of what the payments mean follows below: 1. Electrcty tradng takes place on market terms. Ether by blateral agreements or on one of the power exchanges (ether the Nordc exchange, Nord Pool, or the German exchange, EEX). The supply oblgated companes supply customers who do not wsh to take advantage of the free choce of electrcty suppler. These companes supply customers at regulated prces. 2. Network tarffs are payments to cover the expenses of the delvery of electrcty from producer to consumer. Ths covers expenses of the system responsble company, transmsson operators, and grd operators. 3. PSO payments cover the common nterests of the electrcty market. Ths ncludes supply securty, subsdes for envronmentally frendly producton, energy related research etc. Electrcty tradng on the Nord Pool power exchange s a bd-ask process between generators and electrcty traders. The system prce (spot-prce) s formed 24 tmes n the day-ahead market. All traders can take bds at the spot-prce assumng there s suffcent capacty for transmsson. If ths s not the case, the exchange forms area prces whch reflect the supply stuaton. Asde from the day-ahead market, Nord Pool also deals wth futures n electrcty.[31] Legal Foundaton for Lberalzed Electrcty Markets On a European level drectve 96/92/EC provded general defntons for actor roles wthn the electrcty systems of members states, and more mportantly the un-bundlng of accounts. Ths has later been replaced by drectve 2003/54/EC and Regulaton (EC No 1228/2003) governng condtons for cross-border trade n electrcty. The rules enforce the prncples of non-dscrmnant access to networks, as well as transparency. The rules dctate that efforts should be undertaken to ensure that system operators make avalable all necessary nformaton for obtanng access to the network, wth transparent and non-dscrmnant access prces. Also they dctate that system operators must preserve confdentalty of commercally senstve nformaton.[29] The latest Dansh mplementaton of these measures nto natonal legslaton are Law No 494 and 495 both of June 9th 2004.[28] 2.4 Natural Gas Supply The Dansh supply of natural gas orgnates n the off-shore ol and gas felds n the North Sea. Two hgh pressure ppelnes extend along the sea bed and make landfall n Jutland. They meet at the Nybro gas treatment plant near the western coast of Denmark (see Fgure 2.3), where up to 24 mllon cubc meters (energy content roughly equal to 1000 TJ) of gas can be treated daly. From Nybro two 30 nch transmsson lnes extend across Jutland towards the major juncton at Egtved. From here one connecton goes South to the Dansh- German border at Ellund. Another goes North to the gas storage faclty at Llle Torup and termnates n the cty of Aalborg. Fnally a transmsson lne runs all the way East across the country, passng Odense and crossng both Belts to arrve on the outskrts of Copenhagen near Karlslunde. From here one lne proceeds to the Stenllle storage faclty whle others

17 2.4.1 Organzaton of the Market for Natural Gas 13 Fgure 2.3: The Dansh natural gas transmsson system (SOURCE: Gastra) proceed to supply the area of Greater Copenhagen and the northern parts of Zealand. Ultmately a transmsson lne crosses Øresund to supply our Swedsh neghbors.[30] Most of these major transmsson lnes are nches n dameter and perform at a maxmum pressure of 80 bars. At no pont n the transmsson network s the pressure allowed to descend below 42 bars, n order to secure adequate pressure at the fnal delvery locatons. Meterng and regulaton statons (M/R statons) are located along the transmsson lnes. From here, natural gas s extracted from the transmsson system nto the underlyng dstrbuton networks. Here the responsblty for network operaton s also passed from the transmsson system operator Gastra [30] to one of the four dstrbuton system operators. These operators, along wth the storage system operator, are publc companes responsble for provdng the basc servces of natural gas supply. They develop products for capacty and volumetrc throughput n the system, and provde balancng servces. The model used n ths artcle s a reflecton of present and prevous structures, of servces avalable to the gas shpper Organzaton of the Market for Natural Gas The breakdown of nsttutons n connecton wth the lberalzaton process has resulted n the new, and unbundled, structure of the Dansh natural gas sector. The overall systemc responsble company for the natural gas system s Gastra (EnergNet Danmark), whch s also responsble for the operaton and development of the transmsson system. There are fve dstrbuton networks, two of whch are operated by DONG Dstrbuton, and the remanng three are operated by Naturgas Fyn, Naturgas Mdt-Nord and Hovedstadens Naturgas. The dstrbuton companes are also corporately assocated wth

18 2.4.2 Legal Foundaton for Gas Market Lberalzaton 14 Fgure 2.4: Dansh natural gas market structure the supply oblged companes and certan supplers on market terms, however, each of these act a as an ndvdual legal entty n accordance wth the requrement for unbundlng. The storage operator, DONG Lager, s also a separate entty and part of the DONG corporate structure. There s stll no exchange for natural gas. All gas s traded blaterally. There s, however, a possblty for traders to swap gas amongst each other. Gastra has developed a vrtual Gas Transfer Faclty (GTF), whch enables traders to swap gas n the network. Such a faclty s also avalable for capacty wthn the transmsson system, called the Capacty Transfer Faclty (CTF). These vrtual tradng ponts are the frst steps towards an actual tradng pont for natural gas. The next step, whch s currently beng nvestgated by Gastra n conference wth Nord Pool, s to establsh a vrtual tradng hub. Fnally, an actual natural gas exchange mght be developed n the not so dstant future.[23] Legal Foundaton for Gas Market Lberalzaton Drectve 98/30/EC of the European Parlament and the Councl of 22 June 1998 concernng common rules for the nternal market n natural gas, provdes the bass for the lberalzaton of the natural gas market. Proceedng ths drectve, two other drectves (90/377/EEC and 91/296/EEC) had been adopted n 1990 and 1991 respectvely. These drectves called for transparency and reportng of prces for EU statstcal purposes, and access rghts to natonal hgh pressure transmsson networks. The natural next step, n lght of 96/92/EC, was the call for a non-dscrmnant and transparent access to natural gas supply servces n the European Communty, as had been done wth regard to electrcty as explaned prevously.

19 2.5. DISTRICT HEATING 15 The contents of 98/30/EC s smlar to that of 96/92/EC, but the dfferences n the techncal/physcal propertes of electrcty and natural gas and ther connected systems set ther mark on the specfcs of the drectve. There s the addtonal functonal possblty of natural gas storage as opposed to electrcty, and the possblty of dealng n lquefed natural gas (LNG). Bascally the prncples regardng market structure n 96/92/EC are echoed n 98/30/EC, callng for the separaton (un-bundlng) of dfferent natural gas undertakngs (producton, transmsson, dstrbuton, supply, purchase and storage) as well as transparency and non-dscrmnaton. Drectve 98/30/EC was replaced on June 26th 2003 by 2003/55/EC. All the mentoned documents can be found on the EUR-Lex webste[29]. 2.5 Dstrct Heatng Publc heatng supply s extensve n Denmark havng connected 60% of all prvate homes to some form of publc heatng. Publc heat plannng was undertaken from 1979 and onwards and as a result a large number of muncpalty or prvate-consumer owned heat companes appeared. The central plannng of heatng supply was a reacton to the energy crss of the 1970s, and part of the larger project of natonal energy plannng as a whole. The projected ntroducton of natural gas was undertaken n the same year, and naturally there was a poltcal desre to utlze ths nvestment effcently. As part of ths plannng, muncpaltes where gven the authorty to oblge prvate propertes to be connected and suppled through the local publc heatng supply (dstrct heatng or ndvdual natural gas heatng), as ths was developed. Ths oblgaton stll stands today. The supply of dstrct heat s consdered a natural monopoly snce potental effcency gans from havng perfectly compettve markets, do not justfy nvestments nto parallel supply systems. Ths was also the case for electrcty and natural gas supply as descrbed n the prevous sectons. Dstrct heatng, however, s not effcently transported over great dstances and as such heat generaton s also most often consdered to be a natural monopoly. Therefore, the generaton of heat s a publc oblgaton and the prcng of heat s regulated by the supervsng authortes. The basc gudelne for regulaton s that the busness of producng and supplyng heat must be self sustanable, whch means that companes can only charge what s requred to secure operatng and nvestment costs. The desre for energy effcency and envronmentally frendly producton has spurred a reform of the dstrct heatng sector. Almost all dstrct heatng bolers have been replaced by ether co-generaton unts, prmarly fred by natural gas, or by bo-fuel heatng unts. 2.6 Combned Heat and Power (CHP) Co-generaton of heat and electrcty became a matter of natonal prorty wth the combned heat and power agreement of 1986[25]. Combned generaton of electrcty and heat results n a hgher total fuel effcency than for separate generaton. The process s bascally to use the heat waste product from electrcty generaton, and use ths n the local dstrct heatng network.

20 2.7. MARKET DEFINITION AND REGULATION 16 Almost all central power plants have been converted to co-generaton and the remanng central facltes serve only as back-up or peak load producers. De-centralzed CHP capacty has ncreased from roughly 200 MW n 1990 to nearly 2.5 GW n The ncentve for co-generaton has been the presence of a favorable feed-n tarff for unloadng electrcty nto the network. The lberalzaton of the electrcty market s now puttng an end to ths way of sellng electrcty. Now larger de-centralzed producers must unload electrcty n competton wth other producers. The current drecton of developments s that all de-centralzed electrcty producers wll soon operate on market terms. The subsdy of local co-generaton has not completely dsappeared. Were t so, t would be at the expense of the heat consumers, who are oblgated to take part n the local heat supply. The subsdy s now put not on the electrcty generaton sde, but on the heat supply sde of the equaton. Ths means that local CHPs now have to produce electrcty to sell on the market so as to be able to reduce prces for ts heat costumers. 2.7 Market Defnton and Regulaton The two grey boxes n fgures 2.2 and 2.4 descrbe defnng and regulatng natonal agences. The frst box contanng the Dansh Energy Authorty s responsble for defnng the rules of the energy markets n general. They nterpret how legslaton should be mplemented n praxs. They also support research and development projects deemed n the publc nterest. The other box contans the regulatory bodes of whch the Energy Regulatory Board montors the prces for PSOs and net-tarffs as well as the prces the supply oblged companes charge consumers. The Energy Board of Appeal handles cvl complants between consumers and electrcty companes, whereas the Energy Supples Complants Board deals wth complants aganst the decsons of the Energy Authorty and the Energy Regulatory Board. 2.8 Summary Energy supply systems have been put nto place all over Europe durng the last century. In most cases the development of energy markets have been a matter of natonal concern, causng a spawnng of publcly owned and managed energy supply companes, oblged to brng energy at a far prce to every corner of Europe. Many of these have snce been prvatzed n recognton of the tendency that publc monopoles are generally not economcally effcent. In lne wth the EU Drectves concernng rules for the nternal markets n electrcty and natural gas, the structure of the sectors n Denmark have been developed to ensure open and transparent access to transmsson and dstrbuton systems. Also, an unbundlng of accounts has occurred to ensure that the tarffs charged for transmsson and dstrbuton of energy commodtes reflect the nvestment and operatng costs of the transmsson/dstrbuton system. The mergng of the three system responsble companes (Eltra, Elkraft System and Gastra) to form the new company, Energnet.dk, s a development, whch has made the research undertaken n ths project more relevant than ntally expected. It can be expected that the merger wll serve to consoldate efforts between the electrcty and gas sectors when

21 2.8. SUMMARY 17 addressng future challenges for energy supply, market development, effcent energy utlzaton, and relevant envronmental concerns. One example of such actvty s the research program enttled A Model of and Analyses of an Integrated Gas and Electrcty System. undertaken by EnergNet Danmark n cooperaton wth relevant research nsttutons and supported by the Dansh Energy Authorty.

22 CHAPTER 3 Model Structure The model developed n ths project combnes the techncal and economc aspects of natural gas supply. The mplementaton s dvded nto two parts, whch are descrbed separately. However, n ths chapter the overall structure of the mplementaton s presented to gve a non-techncal overvew, whch does not demand experence wth mathematcal modelng and operatons research methods. The connecton between the developed model and the Balmorel model s also descrbed. The detals and mathematcal formulaton of the techncal model of natural gas supply s descrbed later n chapter 5 and subsequently n chapter 6 the economcs are formulated. 3.1 Overvew Fgure 3 provdes an overvew of the ntegrated model. The mportant thng to note s the partal equlbrum model of the two commodtes of electrcty and natural gas. Each market takes nput from and generates feed-back nto the other market. The nput taken by the electrcty market from the market for natural gas affects the supply functons of electrcty. Conversely output from the electrcty market affects the demand functon for natural gas. Ths s a reflecton of the fact that natural gas s a prmary energy source, whereas electrcty s a secondary energy commodty. The blue felds contan the fxed data and supply modelng of the natural gas model. The yellow felds contan fxed data and supply modelng of the electrcty and dstrct heatng. Green felds contan the data that reflects the top-down elements of the model. Fnally the red felds ndcate results of the model executon. Note that one result feld, namely the feld concernng nvestments gves feedback nto the model. Ths reflects the fact that decsons regardng nvestments are transferred to the followng years. 3.2 Flow Model The flow model s bascally a transportaton model for natural gas. The flow model ensures the techncal feasblty of the supply soluton. These techncal aspects are ncluded drectly

23 3.2. FLOW MODEL 19 Fgure 3.1: Overall model structure.

24 3.3. ECONOMIC MODEL 20 n the model n order to lure out the mpact of for nstance capacty shortfalls n economc terms.e. the value of addtonal capacty. The am s not to make an accurate operatonal model. Ths would be too complex to nclude n ths model. Rather, the ntenton s to be able to extract the economc mpact of techncal restrctons. It s concevable that later work could be able to nclude some elements of capacty nvestment etc., but ths s beyond the scope of ths project. Capacty n the transmsson system can be descrbed n terms of pressure and flow. There s a lmt to the stran one can subject ppelne components to, and therefore there are pressure defned operatonal lmts n the transmsson system. Pressure dfference s the drvng force, whch causes flow n the transmsson system. Therefore, the hgher the pressure s at the source, the more flow can be pushed through the network. Ths s also the case on a dstrbuton level, and therefore there are mnmum pressure levels at transmsson system outlets, n order to ensure that the dstrbuton systems are able to push adequate flow through to ther customers. Natural gas transmsson networks have the addtonal property of beng able to store gas n the ppelnes; a concept termed lne-pack. Ths s done by rasng the pressure n the transmsson system by feedng n more natural gas than s taken out. Ths gves a buffer, whch grants the operator a strong tool to react aganst outages, or can be used to compensate for short-term varatons n demand or producton. 3.3 Economc Model Part of the model concerns the structure of the natural gas market. Ths module has two components. One s the determnaton of whch contracts are purchased by ndustry from the system operators to gan the desred access to the system. Ths has regard for capacty and transmtted volumes n the transmsson system and subordnately n the dstrbuton networks. Also, t s a determnaton of whch contracts are made wth the storage system operator to ensure that storage capacty, njecton and extracton capacty are all payed for as well as the varable costs of njecton. The second component s the constructon of the optmzaton crtera, by the sum of costs nflcted upon the market. 3.4 Summary The project s structured around the development of the Natural Gas Supply System Model. Model emphass s placed on accurately descrbng the economc structure of the supply system, ensurng techncal feasblty, and lnkng ths wth the Balmorel model.

25 CHAPTER 4 The Balmorel Model The Balmorel model s a partal equlbrum model, whch descrbes jontly an nternatonal electrcty and heatng system. The model was orgnally developed to shed lght on nternatonal energy condtons n the Baltc Sea regon and was n part fnanced by the Dansh Energy Authorty s Energy Research Programme around the year Top-Down - Bottom-Up The Balmorel model combnes the approach of bottom-up modelng n a classc techncal modelng tradton wth top-down economc analyss, projectons and forecasts. By descrbng mathematcally the mechansms, whch defne acton and reacton to changes n the state of the system, the bottom-up part drves the model towards a stable state where, held up by boundary condtons descrbng the world outsde the model dynamcs, the model s able to produce results whch are both realstc and comprehensve n terms of what they descrbe. 4.2 Market Equlbrum The model s solved by optmzng the value of an objectve functon. The objectve s an expresson of dfference between consumer utlty and total cost of supply. As such the prce of commodtes s reflected n the cost that the fnal consumer s wllng to pay, where a producer s able to supply at the bded prce wthout generatng a loss. Equlbrum s ensured by constranng the amount of energy commodtes demanded by consumers at a pont of consumpton, to be equal to the amount suppled to that locaton. Market equlbrum s the state of a market where supply and demand are equal for all consdered commodtes. The theory of general equlbrum apples to an entre economy, encompassng all goods traded n the economy. Partal equlbrum theory states that developments n a descrbed market, or a group of related markets, have neglgble mpact on other markets where prces are fxed. Ths s an

26 4.3. PARTIAL EQUILIBRIUM AND OPERATIONS RESEARCH 22 attrbute whch makes t possble to add a great amount of detal to the descrpton of the examned market, usng only smple boundary condtons for descrbng the dependency on other markets. The development of partal equlbrum theory s attrbuted to Antone Augustn Cournot [6] and Alfred Marshall [7]. 4.3 Partal Equlbrum and Operatons Research Partal, and general, equlbrum theory reles on functons descrbng supply and demand. Operatons research by tradton uses optmzaton models capable of smultaneous dervaton of a massve number of varables accordng to some crterum, whle subjected to constrants. The supply and demand curves of partal equlbrum theory are thus constructed and formulated mathematcally, and the equlbra determned by mposng equalty between supply and demand. Ths makes t possble to smultaneously take nto account supply and demand condtons at all the market locatons at the modeled tme-steps, and determne equlbra for all these. By mantanng a lnear model, where non-lnear convex functons can be formulated by pecewse lnear approxmaton, the sze of the model,.e. the number of constrants and varables, can be very large and yet mantan computatonal tractablty. In the Balmorel model, the consdered commodtes are heat and electrcty. Fuel costs are exogenously fxed accordng to data and forecasts for developments n prces. Ths mples that the prce of natural gas s exogenous n the Balmorel. In ths project, an addtonal market s modeled n determnng the correlaton between electrcty, dstrct heat and natural gas, namely the market for natural gas. 4.4 Elements of the Balmorel Model Equlbra are reflected n the soluton of the model on a number of ssues. Equlbra between consumer margnal utlty and the margnal cost of supply by relevant geographcal dvson and for ever modeled tme-segment Equlbra between tme-segments caused by presence of storage optons. Equlbra between geographcal dvsons by transmsson optons. Equlbra of margnal utlty between traded commodtes. Equlbra between short-run and long run margnal costs mpled by nvestment optons. In order to understand the mplcatons of the above, consder the deal system wth nfnte capacty for transmsson, storage and dstrbuton wthout loss, wth no cost assocated to these operatons. One common prce would appear for all geographcal dvsons and tmesegments for whch would reflect both the margnal cost of supply everywhere at any tme as well as a global consumer utlty. The ntroducton of the aforementoned techncal and economc elements mpose lmtatons or costs on the transfer of resources between geographcal dvsons and tmes, and as such prces become geographcally and temporally dependent.

27 4.4.1 Geography Geography Geographcally, the Balmorel model s constructed on a three-level herarchy of countres, regons and areas. The country level features detal of natonal polcy wth regard to taxes and emsson control as well as provdes a logcal geographcal dstncton for aggregaton of results. Regons are subdvsons of a country at a level where the geography descrbed by the regon can be assumed to feature a fully connected electrcty dstrbuton network. At an nterregonal level the process of electrcty transmsson s handled. Transmsson bottlenecks appear at an nterregonal level. Electrcty demand s also ncurred on a regonal level and hereby are electrcty prces also determned at ths level. Areas are subdvsons of regons and can be assumed to feature a fully connected dstrct heatng network. Consumpton of heat and producton of both heat and power s assocated wth areas. Producton capacty s naturally also nstalled at area level. Ths has the postve sde effect of gvng more resoluton wth regardng to dstrct heatng. Snce dstrct heatng systems are unable to transmt heat over great dstances, ncreased resoluton on the producton and supply of dstrct heatng s also desrable. It was necessary to make new data for heatng demand, the process of whch s descrbed n secton Ths project concerns only Dansh network bound energy supply. As such the set of countres C contans only Denmark. It would be farly easy to nclude neghborng countres (at least wth regard to electrcty and heat), but ths would mpose addtonal requrements for computatonal power, and mnor data adjustments. The set of regons n Denmark contans two elements R = {DK W, DK E}, snce there are dfferent electrcal systems n Eastern and Western Denmark. As the focus s the natural gas transmsson system, the logcal choce of areas are areas suppled wth natural gas from a specfc meterng and regulaton staton. There are about 50 of such areas n the set A. Ths yelds the postve sde effect of greater resoluton wth regard to dstrct heat, whch s desrable snce dstrct heatng systems are unable to transmt heat over great dstances. Ths also makes t more lkely that the model wll use some of the smaller, and perhaps less effcent technologes whose man justfcaton are ther sutablty for small scale heat supply. These are lkely lost n overly aggregated models The Temporal Dmenson There are three temporal levels n Balmorel. The hghest level s years, and each year s smulated wthout foresght regardng condtons n the subsequent years. There are two subdvsons of the year, generally called seasons and tme perods. There s no restrcton as to how these are to be nterpreted, or to how many perods should be ncluded. When smulatng for one season and one tme segment for example, ths could correspond to smulaton usng annually averaged values. 12 seasons can correspond to months whle 168 tme-segments could ndcate hourly averages for a week wthn the gven month (season). The followng sets descrbe the segmentaton of the year nto tme perods: S = {s 1,..., s 12 } (4.1) T = {t 1,..., t 12 } (4.2)

28 4.5. THE OBJECTIVE FUNCTION 24 The elements of the S set naturally represent the months of the year. The elements of T represent hours of a typcal week n the approprate month. The hours have varyng weght (or duraton) as some hours of the day and week are more nterestng than others. As mentoned, the set of years, Y, controls the annual dynamcs. A separate lnear program s solved for each year and results are transferred to the succeedng years. Ths specfcally concerns results regardng nvestments n capacty. 4.5 The Objectve Functon The objectve s to maxmze the sum of consumer utlty and the negatve cost of producton and supply. Consumer utlty s formulated as follows: Utlty of electrcty: c C s S t T r R(c) U e,r,s,t (e r,s,t d ) Utlty of dstrct heat: c C s S t T a A(c) U h,a,s,t (e a,s,t d ) Here e r,s,t d s the electrcty made avalable to the consumer after loses n transmsson, dstrbuton etc. n the regon r n the set of regons R(c) pertanng to the country c and the tme segment (s, t). U e,r,s,t s the actual utlty functon of electrcty dependent on consumer preferences. The utlty of dstrct heat s analogous to electrcty, where a A(c) descrbes the and area a n the set of areas A(c) of country c. In ths project only non-elastc demands are employed, and as such the utlty functons are constants for each tme-segment and consumpton locaton. The assocated costs contrbuton to the objectve functon are defned as follows. { c C s S t T Energy taxes r R(c) te e r,s,t s (1 ɛ e r) a A(c) th h a,s,t s (1 ɛ h a) Generaton costs a A(c) Ks,t a (e r,s,t s, h a,s,t s ) Transmsson: operatons { and nvestments β x(r,ρ) x r,ρ,s,t + X x(r,ρ)} Dstrbuton costs (r,ρ) R(c) 2,r ρ r R(c) βe e r,s,t d r 1 ɛ e r a A(c) βh a } h a,s,t d 1 ɛ h a Here e s and h s represent the generated amount of electrcty and heat respectvely. ɛ e r, ɛ h a are the percentage loss n the dstrbuton process. t e and t h are the energy tax rates assocated wth electrcty and heat. Ka s,t (e r,s,t s, h a,s,t s ) s the cost functon assocated wth a certan generaton of heat and electrcty n a gven area. Ths functon ncludes fuel costs, fuel taxes, emsson taxes and operatng costs. β x(r,ρ) s the cost of transmsson between the regons r and ρ, x r,rho s the transmtted amount, whle X x(r,ρ) represents an nvestment n transmsson capacty. Fnally βr e and βa h represent cost factors for dstrbuton Investments The model features the possblty to nvest n both generaton capacty and electrcty transmsson capacty between regons. These nvestments can ether be endogenously performed at run-tme, or, dependng on the modeled scenaro, be gven as exogenous nput data. The nvestment opton s lmted by the shortsghtedness of the temporal resoluton. As annual plannng s quas-dynamc, the crtera for performng nvestments s a matter of the feasblty for the nvestment wthn the year n whch t s undertaken. Ths means

29 4.5.2 Energy Transformaton 25 the comparson made s between the fnancal cost wthn the frst year of operaton wth the effcency gan n the overall system. In the followng year the nvestment s treated as already exstng capacty, and the nvestment costs are consdered sunk costs Energy Transformaton In the Balmorel model varous forms of energy transformaton are supported. Technologes are descrbed n terms of transformaton potental, effcency, cleanlness as well as economc parameters such as varable producton costs, fxed annual costs and nvestment costs. In the followng the basc forms of transformaton supported by Balmorel are descrbed. The technology types are exemplfed wth specfc technologes, but may well be used to represent dfferent technologes wth smlar techncal characterstcs. Generaton constrants are formulated generally as a functon of the produced amount of heat and electrcty: gg s,t (e s,t s,g, h s,t s ) 0, g G s S, t T Sngle Energy Type Transformaton The two frst technologes, llustrated on fgure 4.1, represent technologes producng ether only electrcty or only heat. These can be exemplfed by tradtonal condensng power plants, where the heat wast product s cooled by an ntake of seawater, and tradtonal heat-only bolers, whch produce only heat respectvely. Fgure 4.1: Electrcty only and heat only producton technologes. CHP Technologes Combned heat and power facltes come n many shapes and forms, but overall they can be dvded nto two types. These are fxed-rato technologes, whch produce heat and electrcty at some near-constant rato, and varable rato technologes. Fxed rato unts are exemplfed by gas engnes or back-pressure gas turbnes. Generally the rato between electrcty and heat s termed the c B value of the technology. Varable rato technologes are for example extracton steam turbnes, where heat can be extracted at some pont along the

30 4.5.2 Energy Transformaton 26 turbne to be used for dstrct heatng, or t can run along the full length of the turbne from where ts temperature becomes too low to have practcal use n the dstrct heat network. Fgure 4.2 llustrates the feasble regon of these producton technologes. Fgure 4.2: Combned heat and power technology types. Storage Technologes Heat and electrcty storage facltes can also be descrbed. Heat storage s generally a large nsulated contaner wth hot water. Electrcty can be stored by hydrogen fuel cells or by pumpng water nto a reservor, from where t at a later tme can drop through a turbne releasng the energy potental. Fgure 4.3 llustrates the producton profles of storage facltes. Fgure 4.3: Heat and electrcty storage technologes

31 4.5.3 Transmsson and Dstrbuton 27 Wnd Power and Heat pumps Fnally a technology descrbes fxed electrcty producton unts. These are wnd or solar powered unts who s producton s fxed by the avalablty of wnd or sunlght. Thus these appear as a pont on the electrcty-heat chart n fgure 4.4. Ths technology opton s sketched alongsde heat pumps and smlar technologes (such as electrcty powered heat bolers), whch use electrcty to generate heat. Fgure 4.4: Heat and electrcty storage technologes Transmsson and Dstrbuton Transmsson of electrcty s possble between regons, but transmsson s lmted by exogenous and endogenous transmsson capacty. Transmsson capacty s thus constraned by a smple lnear flow model. Costs for transmsson and loss n the network are also ncurred. Regons serve as ponts of consumpton for electrcty. These are characterzed by a loss factor, costs etc., n representaton of a dstrbuton network. Transmsson of dstrct heat over great dstances s nfeasble, and thus heat demand must be suppled by generaton from wthn each area. Areas are assocated wth dstrbuton losses and costs wth respect to heat, as wth electrcty consumpton nodes above Energy Demands Energy demands are represented by a nomnal demand profle, whch vares over tmesegments. The bult n data contans a representaton of varatons over the day, week and between seasons. The profle s appled to an annual demand by consumpton node (regon for electrcty and area for heat). There s an opton to ntroduce own-prce elastctes, yet ths s not appled n ths project. The demand satsfacton constrants can be stated for electrcty as: g G(r) e s,t s,g + ρ R(c),r ρ x (r,ρ),s,t (1 ɛ x(ρ,r) ) = er,s,t d 1 ɛ e, r R, s S, t T r

32 4.5.5 Emsson Quotas 28 So demand for electrcty s suppled by local producton and net transmsson nto the regon, subject to loses n dstrbuton. For heat: g G(a) h s,t s = ha,s,t d 1 ɛ h, a A, s S, t T a Emsson Quotas Emssons can be lmted by taxes or quotas. Where taxes appear n the objectve functon, quotas naturally take the form of constrants. Ths sort of emsson polcy s descrbed by: Φ m (e s,t s,g, h s,t s ) m c,m, c C, m M g G(c) s S t T Where m M ndex varous emsson types (CO 2, SO 2, etc.), and the Φ( )-functon represents the emssons resultng from the generaton profle. The m c,m expresson s the emsson lmt of emsson type m n country c. The dual values of these constrants can be nterpreted as the margnal value of emsson allowances. Gven a value of an emsson allowance one can effectvely mplement quotas as an emsson tax, by assgnng a prce to tradable emsson allowances. Ths s dscussed further n secton More on the Balmorel Model For a more complete descrpton of the model one can refer to the followng documents: [1], [3], [2], [4], [5]. These and the Balmorel model tself can be downloaded from

33 CHAPTER 5 Flow Model Interacton between a techncal model descrbng flow and pressure wth an economc model descrbng the costs and utltes, makes t possble to address two ssues. The flow model ensures that commodtes bought and sold on the market can actually be delvered. If not, t mposes restrctons and sheds lght on the lost proft from such restrctons. Ths makes t possble to consder the effects of varous polces such as capacty ratonng and tarffs. Secondly t mples a value of addtonal capacty whch, when held aganst nvestment costs could be used as a sgnal that nvestment may be fnancally sustanable. The flow problem reflects on ssues of supply securty and effcency from a capacty perspectve, and s thus relevant n lght of the stated objectve of ths project. It s emphaszed that the purpose of the flow model s not to derve an accurate descrpton of exactly how natural gas s delvered to ndvdual consumpton ponts, or to be able to determne the precse pressure and flow rate n dfferent parts of the transmsson network. Rather, the purpose s to mpose restrctons on the soluton by modelng the flow n terms of the restrctons (mostly pressure related) of the transmsson ppelnes, and to gve ndcatons of capacty value. In short, the flow model ensures that the delvery whch occurs falls wthn the techncal lmtatons of the transmsson system. Ths chapter concerns the techncal aspects of flow modelng. Frst, a bref ntroducton to the way n whch fluds, and n partcular compressble fluds such as natural gas, respond to the forces relevant to natural gas transmsson. Next, a more practcal modelng approach s ntroduced whch has been the prme nspraton for the fnal model formulaton. Fnally, the flow model s formulated takng account of the specal consderatons and advantages of the Dansh transmsson network. 5.1 General One Dmensonal Flow There s the general agreement that flud flow s descrbed by four man condtons, expressed n a sngle spacal dmenson along the length of a ppe. (see for example [14]). Conservaton of Mass (ρw) x = ρ t (5.1)

34 5.1.1 Conservaton of Mass 30 Fgure 5.1: System for descrpton of general one-dmensonal flow. Momentum Equaton Conservaton of Energy State Equaton Fx = d (mw) (5.2) dt Ω W = E (5.3) ρ = f(p) (5.4) In the above, m represents mass, ρ s an expresson of densty, F x represents the net forces actng n drecton x whch s along the ppe, w s the flow velocty averaged over a crosssecton of the ppe, Ω s added heat, W s the performed work, E energy gan, p represents the pressure. Refer also to the nomenclature when necessary Conservaton of Mass The conservaton of mass, or contnuty equaton n a ppe flow context, states mass may nether be created or destroyed. Ths means that accumulaton of mass wthn a control volume must be equal to the net flow nto the control volume. In other words what comes n, ether goes out or stays n. The mass present wthn a control volume can be descrbed by: m = V ρdv (5.5) where V represents a control volume. Below two expressons are presented for the movement of mass nto and out of the control volume. dm = dt A dm = dt ρwda (5.6) V ρ dv (5.7) t The frst equaton (5.6) descrbes the change n mass by the mass-flux through the control surface. The second equaton (5.7) descrbed the change n densty wthn the control volume

35 5.1.2 Momentum 31 over tme n relaton to the mass leavng the control volume. These together form the equaton: V dt V [ (ρw) x A ρwda = dt (ρw) x dv = + ρ t V ρ dv V t (5.8) ρ dv t (5.9) ] dv = 0 (5.10) Snce the above must hold for any control volume, the ntally presented formulaton of the contnuty equaton s derved: (ρw) x = ρ t (5.11) Momentum Newton s second law of moton expressed n the drecton of x, along the length of the ppe, adequately descrbes momentum of gas flow n ppes [17]. The net force n the drecton of x on gas wthn the control volume s the algebrac sum of three ndvdual forces projected on x. These forces are: 1. pressure forces 2. shearng forces (frcton) 3. gravtatonal force Snce pressure s defned as force per area unt, the force nduced by pressure dfference over the ppe length dx s: ( F pressure = pa p + p ) x dx A = p Adx (5.12) x Ths of course n the drecton of moton, as flow runs from hgh to low pressure. Ths pressure force s the component, whch enables the transmsson of gas through ppes, and pressure s the man control wth whch a transmsson system operator s able to nfluence the rate of the flow at compressor statons or other pressure sources such as hgh pressure storage facltes. The shearng force s caused by frcton wth the ppe and vscd forces wthn the gas. F shear = Aw2 2 4f dx D (5.13) Ths s defned n the drecton of the flow, hence the negatve term. The orgn of the term s Darcy s equaton, whch defnes frcton nduced head loss. The concept of head s ntroduced n secton below. Darcy s equaton states that the change n head due to frcton can be descrbed by:

36 5.1.2 Momentum 32 dh f = 2fρw2 dx (5.14) gd The f term s a functon of the roughness of the ppe and the Reynolds number. Ths factor s dscussed further later n the chapter. The net body force on gas wthn the control volume can be formulated by: F gravty = gρadx sn α, (5.15) where α s the vertcal angel between the ppe s orentaton and the horzon, and g s the acceleraton of gravty. The rght hand sde of the momentum equaton empresses the flux of momentum through the control volume. Ths term can be reformulated by the followng consderatons. Fgure 5.2: Closed system of mass movng wthn a ppelne, through a control volume. By consderng a system of constant mass flowng through a control volume t s possble to reformulate the momentum flux expresson as follows: An ntal state of the system s gven at the tme t. At ths stage mass has entered the control volume, but no mass has yet left the control volume. As such the system s contaned wthn the control volume or s approachng t. In the second state at the tme t + t all mass has entered the control volume and some of the mass has also left the control volume. The momentum flux can be descrbed by the lmt for t 0 of the dfference n momentum between the two states. d(mw) dt d(mw) dt d(mw) dt [(mw) cv + (mw) 2 ] = lm t+ t [(mw) cv + (mw) 1 ] t t 0 t [(mw)] t+ t [(mw)] t = lm t 0 = d(mw) cv dt t + lm t 0 + lm t 0 [(mw) 2 ] t+ t [(mw) 1 ] t t [(mw) 2 ] t+ t [(mw) 1 ] t t (5.16) (5.17) (5.18) The momentum of mass n the volume labeled 1 n the frst state s n the lmt the momentum of mass leavng that volume, hence the flux nto the control area over the tme t. Ths can be expressed as:

37 5.1.3 Energy Conservaton 33 (mw) 1 lm t 0 t m 1 w 1 = lm t 0 t = ρ 1 Aw 2 1 (5.19) Identcally the momentum of the mass whch has left the control volume n the second state, labeled 2, can be descrbed at the lmt by the momentum flux out through the boundary of the control volume. Ths gves: (mw) 2 lm t 0 t m 2 w 2 = lm t 0 t When the x component of the control volume goes towards 0. = ρ 2 Aw 2 2 (5.20) dm dt w 2 dm dt w 1 = ρaw2 x Fnally, the momentum contrbuton of mass wthn the control volume: (5.21) d(mw) cv dt = d (ρawdx) (5.22) dt By combnng (5.21) and (5.22) wth (5.18) the followng expresson for momentum flux appears. d(mw) dt = t (ρawdx) + x (ρaw2 )dx (5.23) Now the rght hand sde (5.23) and left hand sde (5.12), (5.13), (5.15) of the momentum equaton (5.2) s combned yeldng the momentum equaton expressed as: p Aw2 Adx x 2 4f dx D gρadx sn α = t (ρawdx) + x (ρaw2 )dx (5.24) p x 2fρw2 (ρw) gρ sn α = + (ρw2 ) (5.25) D t x The fnal equaton s the general form of the momentum equaton for one-dmensonal flow [14] Energy Conservaton The energy conservaton equaton s the frst law of thermodynamcs. The E term n (5.3) represents the change n system energy. Ths energy can be dvded nto knetc energy, potental energy and nternal energy stored n the gas atoms and molecules (U). E = 1 2 mw2 + mgz + U (5.26) Usng the system concept of the prevous secton, the thermodynamcs can be descrbed n the lmt for t 0. The energy expresson s thus dvded nto the content of the control volume plus the enterng energy subtracted the leavng energy. On dfferental form, the energy conservaton equaton (5.3) can be formulated as:

38 5.1.3 Energy Conservaton 34 dω dt dw dt = de cv dt + dm out dt ( ) w gz 2 + u 2 dm ( ) n w 2 1 dt 2 + gz 1 + u 1 (5.27) The terms u are here the specfc nternal energes. The work can be dvded nto the flow work and every other knd (ths ncludes frcton). The flow work s performed at the up and down stream system boundares. Ths work s formulated as dw flow dt = dm out dt (p 2 v 2 ) dm n (p 2 v 2 ) (5.28) dt The resdual tme rate of work s termed dws dt. The energy equaton can be expressed as dω dt dw s dt = de cv dt + dm out dt ( ) w gz 2 + u 2 + p 2 v 2 dm ( ) n w 2 1 dt 2 + gz 1 + u 1 + p 1 v 1 (5.29) Introducng specfc enthalpy h smplfes the expresson as h = u + pv. dω dt dw s dt = de cv dt + dm out dt Fnally n the lmt of the length of the system x 0: ( ) w gz 2 + h 2 dm ( ) n w 2 1 dt 2 + gz 1 + h 1 (5.30) dω dt dw s dt dω dt dw s dt dω dt dw s dt (( = de cv + dm ) out w + w 2 x dx dt dt 2 ( +g z + z ) ( x dx + h + h ) ) x dx dm ( ) n w 2 dt 2 + gz + h (5.31) ( ) = de cv 1 w 2 z h + ρaw dx + g dx + dt 2 x x x dx (5.32) [ ( ) ] = w 2 ρaw + gz + u + pv dx (5.33) x 2 [ ( ) ] + w 2 ρaw + gz + u + pv dx (5.34) x 2 Wth respect to gas flow three cases are generally consdered: 1. sothermal processes - where gas temperature s constant throughout the system. 2. adabatc processes - f ppes are nsulated so the added heat from the surrounds s zero. 3. polytropc flow, whch s n essence the ntermedary between the two aforementoned.

39 5.1.4 Thermodynamc State 35 The adabatc flow would assume dω = 0. Isothermal flow smplfes the nternal energy statements consderably. How depends on a number of ssues, such as the type of gas and the state functon. In ths project all flow s assumed to be stablzed by ground temperature and thus temperature s constant throughout the system. As such, flow s generally assumed to be sothermal. Ths s a common assumpton for modelng natural gas transmsson systems, but t s a smplfcaton. In the artcle [17] Osadacz and Chaczykowsk conclude, unsurprsngly that sgnfcant errors occur when modelng a network where temperature fals to stablze usng an sothermal model. There s, however, no ndcaton that temperature varatons play a major role n the Dansh transmsson network Thermodynamc State Two state equatons were consdered for dfferent reasons. Lterature regardng sothermal flow suggests a state equaton where[14]: ρc 2 = p (5.35) However, elsewhere t s stated that the conventon wthn the natural gas ndustry s to use the followng state equaton: p = ρzrt (5.36) The Z-compressblty functon can often be consdered constant, and ths s generally the case for natural gas where the calculaton method s standardzed accordng to [15]. Ths yelds a compressblty of Ths fnal method was selected manly to avod ntroducng too many non-lnear terms Means of Transent Flow Analyss The equatons derved above for contnuty, momentum, energy and state can wth tedous effort and smplfcaton be combned to form one second order partal dfferental equaton. However, accordng to [14] ths s hghly mpractcal. Instead there are several other ways to reformulate the problem and by approprate smplfcaton devse models sutable for a varety of purposes. All usable transent formulatons are, however, n the form of dfferental equatons whch must be solved numercally, makng ther use mpractcal for the purpose of ths project. In the followng we proceed to dscuss how steady-state analyss can be used n an approxmaton effort to generate a set of flow constrants, whch can be appled to the overall model. 5.2 Steady-State Analyss A steady-state flow process descrbes a system state where the flow and pressure are constant n the ndvdual locatons n the system over tme. Steady state analyss s very useful when there are no large varatons n pressure and flow. Steady-state analyss s also generally used as a startng pont for numercal solutons usng transent flow analyss.

40 5.2.1 Head and Bernoull s Equaton 36 From the steady state condton we know that ρ t = 0. Hereby the contnuty can be expressed by: (ρw) x = u = 0, (5.37) x where u defnes mass flow-rate, whch s constant over the length of ppe under steady state condtons. When flow s expressed n terms of volumetrc flow rate under normalzed condtons t s often termed Q n n lterature. Here no dstncton s necessary between u and Q n as the appled unt of mass s Nm Head and Bernoull s Equaton In the steady state case energy conservaton can be formulated by the followng equaton whch s Bernoull s Equaton at two ponts along the ppe adjusted for shearng head loss: p ρg + w2 2g + z = p + dp (w + dw)2 + + (z + dz) + dh f (5.38) ρg 2g Bernoull s equaton gves rse to the concept of drvng head. Each component of Bernoull s equaton s an expresson of energy per unt of weght of the flud. The frst term represents the pressure head, whch s the ablty of per unt weght of the flud to perform work. The velocty head, the second term, s an expresson of knetc energy per unt weght. Fnally, the potental head, z, represents the potental energy per unt weght that can be transformed nto work. Head loss has been descrbed prevously n secton At ths pont the assumpton s made that potental head s neglgble as Denmark s a rather flat country. Hence Bernoull s equaton (5.38) s smplfed to the followng: dp ρg dp ρg = (dw2 ) 2g = (dw2 ) 2g + dh f (5.39) + 2fw2 dx (5.40) gd The change n knetc energy due to a change n velocty s also assumed neglgble. dp ρg dp ρg = dh f (5.41) = 2fw2 dx (5.42) gd dp = 2fρw2 dx (5.43) D The contnuty equaton (5.37) gves the followng usefull expressons: The state equaton yelds the followng as flow s sothermal: ρw = ρ 1 w 1 w = ρ 1 ρ w 1 (5.44) p ρ = ZRT p ρ = p 1 ρ 1 ρ = p 1 ρ ρ 1, w = p 1 p w 1 (5.45)

41 5.2.1 Head and Bernoull s Equaton 37 Substtuton nto (5.43) gves the followng equaton: dp = 2fp ρ 1 Dp 1 dp = 2f D ρ 1 ( p1 ( p1 p ) 2 w 2 p 1dx ) (5.46) w1dx 2 (5.47) pdp = 2f D ρ2 1w 2 1ZRT dx (5.48) pdp = 2f D ρ 2 nq 2 n (π(d/2) 2 2 ZRT dx (5.49) ) (5.50) The suffxed n ndcates normal condtons. When usng a mass flow rate n terms of Nm3/h per tme unt. ρ n becomes 1Nm 3 /m 3 and the equaton can be stated as: pdp = 32f u 2 π 2 ZRT dx (5.51) D5 Snce the flow s sothermal, the compressblty s constant, the frcton factor s assumed constant, and the mass flow rate s constant over the length of ppe, both sdes of the equaton can be ntegrated. pl L pdp = p=p 0 1 ( p p 2 32f L) = p 2 0 p 2 L = = 64f x=0 32f u 2 π 2 ZRT dx (5.52) D5 u 2 π 2 ZRT L D5 (5.53) u 2 π 2 ZRT L D5 (5.54) Note that ths s subject to the assumpton that the flow drecton s postve. For negatve flow drecton: 1 ( p 2 2 L p 2 32f u 0) = π 2 ZRT L D5 (5.55) p 2 L p 2 0 = = 64f u 2 π 2 ZRT L D5 (5.56) When these expressons are combned the followng equaton can be used for calculatng the relatonshp between flow and pressure at the end ponts of a ppelne secton: p 2 0 p 2 L = 64f u u π 2 ZRT L (5.57) D5 If mportant observaton s that f f s assumed, whch n essence mples that the Reynolds number s constant and the roughness of ppelnes are constant, a resstance coeffcent can be defned for each ppe e termed c e where:

42 5.2.1 Head and Bernoull s Equaton 38 c e = 64f π 2 D 5 ZRT L e (5.58) Ths also mples that f a resstance coeffcent s known for ppe e 1 the resstance of ppe e 2, whch has the same nternat roughness, can be found by: c e2 = c e1 D 5 e 1 L e1 Ths gves the smple equaton for the pressure flow relatonshp: L e2 D 5 e 2 (5.59) p 2 0 p 2 L e = c e u u (5.60) Ths relatonshp generally holds for low pressure networks, however, when dealng wth hgh pressure networks. Unfortunately the frcton factor f cannot always be consdered constant. The terms 1 f s often called the transmsson factor. For fully turbulent flow ths factor can be descrbed by the followng [14]: Where Re s Reynolds number whch agan s descrbed by: 1 f = 6.872(Re)0.073 (5.61) Re = 4Qρ µdπ = 4u µdπ (5.62) Where µ s the vscosty of the gas and s assumed constant. Ths s n accordance wth the so called smooth gas law[14] for fully turbulent flow. Ths can naturally by substtuted nto equaton (5.56), however frst a fnal consderaton s made. In praxs the nner surface of a gas ppe reduces the flow actually ncurred n comparson to that, whch can be calculated usng the smooth ppe law. The rough ppe law nduces the concept of ppe effcency. The effcency of a ppe s the share of the theoretcal flow whch actually occurs over a pressure drop. As such t s appled to the transmsson factor as follows: 1 1 f = E = 6.872( 4u f t µdπ )0.073 E (5.63) Consequently the constants of ths expresson can be ncluded n the dervaton of the resstance coeffcents c e and the u factor when squared, nverted and multpled to the other u s of (5.56) yelds the followng: p 2 0 p 2 L e = c e u u α (5.64) where α 0.854[14]. Ths s the pressure flow relatonshp appled n the model. Below the assumptons leadng to ths formulaton are collected:

43 5.3. GAS NETWORKS one-dmensonal flow 2. steady state flow 3. flow s sothermal 4. constant compressblty of gas 5. constant specfc gravty of gas 6. constant vscosty of gas 7. Darcy s head loss relatonshp s applcable 8. potental head s neglgble 9. neglgble change n knetc energy across the ppe 10. frcton s constant over the length of ppe 11. the rough ppe law holds 5.3 Gas Networks The prevous sectons outlned how the flow process can be descrbed generally and for the steady-state case n a sngle ppe. Ths must be extended to descrbe developments n the network as a whole. In the steady-state case, t s common to assume that flow-rate s constant throughout each ppe-leg and pressure s determned a both ends of the ppe. Ths mples that pressure s gven n terms of varables connected to junctons between ppes. By sustanng mass flow rate through nodes by a mass-balance equaton (lnear), and sustanng momentum over ppe legs through a relatonshp between upstream pressure, flow rate and downstream pressure (non-lnear). As such, the steady state flow can practcally be descrbed by two sets of equatons, one lnear and one non-lnear, gven that, the network s acyclc. The lnear property of the flow through ppelnes s actually a consequence of Krchoff s frst law. Gven that at one locaton n the network the pressure s fxed, and gven that all nflow and outflow are known (or controlled) the state of the network s unquely determed[18]. If, however, the network s not acyclc a thrd set of equatons come nto play. As a consequence of Krchoff s second law, the pressure drop over a full cycle s zero. Equatons must be added for cycles n the network Practcal Formulaton In representaton of a transmsson ppelne network, consder the graph G = (A, E) where A s vertex n the set of vertexes (nodes) and e E s an arc n the set of arcs (ppes) n the graph. A node-ppe ncdence matrx A can formulated as: 1, f ppe e comes out of node ; a,e = 1, f ppe e goes nto node ; 0, otherwse (5.65)

44 5.3.1 Practcal Formulaton 40 Gven that p descrbes the pressure at node and u e descrbes the flow n network arc e, t s possble to express the two frst sets of equatons as follows. Mass balance equatons: Au = s (5.66) The steady-state ppe flow equatons derved n the prevous secton: p 2 p 2 j = c e u e u e α, (5.67) where p and p j are up- and downstream pressure respectvely for the ppe wth end nodes and j. u e s the mass flow rate n the ppe leadng from to j, and α s a constant nearly equal to 1. c e are parameters descrbng the resstance of the specfc ppe, whch can be determned emprcally calculated accordng to [14]. The supply varables s are amounts comng nto, or leavng the system at (assumng s = 0). The network flow can be descrbed as: Au = s, (5.68) A T p 2 = φ(u), (5.69) where p 2 = (p 2 1,..., p2 V )T, and φ(u) = (φ 1 (u 1 ),..., φ l (u E )) T, and φ e = c e u e u e α. (5.70) It holds that f the s vector s gven (or otherwse determned) and pressure of a reference node s gven, the system has a unque soluton. The system can be reformulated by 1)usng the node-ppe ncdence matrx removng the reference node and 2) lettng B f be the reduced cycle matrx wth respect to some spannng tree. In essence, ths s removng the cycles of the system for separate treatment. The formulaton s as follows: A f u = s f, (5.71) B f φ(u) = 0, (5.72) A T p 2 = φ(u). (5.73) It can be shown[18] that the frst two sets of equatons have a unque soluton, and snce they are ndependent of the pressure ths can be calculated after solvng the optmzaton problem. Below a small example s presented to emphasze how ths concept works. Example 5.1 The followng example llustrates how the above descrbed method can be used to calculate the network state. Gven a drected graph G, whch s depcted on fgure 5.3, the node-arc ncdence matrx A s gven below. A =

45 5.3.1 Practcal Formulaton 41 a b 3 c e d f 6 5 Fgure 5.3: Small network example. Thck lnes represent a spannng tree. The rank of ths matrx can be shown to be n 1 where n s the number of rows. Ths means that one row s a lnear combnaton of the others, and can therefore be omtted wthout loss n determnng a unque soluton. If the pressure at one reference node s gven, and ths node s omtted from the model mass balance equatons, ths does not change the feasble regon of model. For the example t s addtonally assumed that the source vector s gven as s = [ ] T. Hence the reduced ncdent matrx s defned for our example as: A f = The reference node (1) has thus been omtted, and the mass flow balance equatons can be expressed as: A f u = s f where s f s the source vector wth the reference node omtted. A spannng tree T = {a, b, c, d, f} s emphaszed by the thck lnes on fgure 5.3. From ths a matrx contanng all fundamental cycles can be formed. Fundamental cycles are cycles whch appear when, gven a spannng tree of a graph, one edge s added to the graph. It can be shown that, gven all fundamental cycles of a graph, any addtonal cycles can be expressed as lnear combnatons of a number of fundamental cycles[16]. In the example there s only one cycle, whch then naturally s the fundamental cycle, whch appears when addng edge e to the spannng tree. It follows that: when selectng the cyclc drecton of edge b. B f = [ ], By theorem 2 of [16] t holds that B f A T = 0. Hence, consderng the pressure-flow relatonshp:

46 5.3.1 Practcal Formulaton 42 A T p 2 = φ(u) (5.74) B f A T p 2 = B f φ(u) (5.75) B f φ(u) = 0 (5.76) The orgnal formulaton of the problem s thus equvalent wth: A f u = s f, (5.77) B f φ(u) = 0, (5.78) A T p 2 = φ(u). (5.79) The two frst lnes contan only flow varables and t has been shown n [16] that these have a unque soluton. For the example these are expressed below [ ] φ(u a ) φ(u b ) φ(u c ) φ(u d ) φ(u e ) φ(u f ) u a u b u c u d u e u f = (5.80) = 0 (5.81) The φ( ) functon n the example s gven as φ(u) = cu u where c = [ By numercal soluton of the system (5.80)-(5.81) the flow values are calculated and presented n the table below: ]. Flow Varable Value u a 100 u b u c u d u e u f 50 Ths means that all flows follow ther defned drecton wth the excepton of the flow n arc d. Gven a reference pressure value p 1 = 200 and the flow values pressure at the remanng nodes can be calculated teratvely. Pressure at the nodes are presented n the table below:

47 5.4. QUASI STEADY-STATE MODEL 43 Pressure Varable Value p p p p p p The pressure value n node 5 s used to confrm the result snce ths can be calculated from both arc d and arc e. 5.4 Quas Steady-State Model The developed flow model reles on the hypothess that suffcently accurate results may be obtaned by consderng flow wthn a tme perod from a node to the center of a ppe-leg to be steady. Ths s dependent on the assumpton that varatons n nflow and outflow of the network are n practce gradual to some extent, or they can be modeled wth suffcent accuracy wth a mean consderaton near ther occurrence. The prncple s that by lftng the flow conservaton constrant on the ppe (whch s defned mplctly by usng one varable for mass flow rate per ppe-leg per tme perod) and ntroducng two flow varables per ppe-leg, the flow varables are nterpreted n/out flow rates from each endpont. Contnuty s mposed by ntroducng a storage varable descrbng the amount of gas present n the ppe at the begnnng of each tme perod. Pressure at the mdsecton of the ppe s derved mposng the state equaton on the mdsecton. Pressure-flow relatonshps are formulated from both sdes of the mdsecton of the ppes, so that the nodal pressure s bound usng the flow and mdsecton pressure at each ppe ncdent to the node. The network s assumed to by acyclc. The only cycles n the Dansh transmsson system are a result of lne duplcatons, where flow can be assumed to run n the same drecton n both parallel lnes. The possble mplementaton of a ppelne project such as the Nordc Gas Rng, would naturally requre a reconsderaton of ths assumpton. Algebrac formulaton of ths dea follows. Fgure 5.4: System descrbed by the developed model

48 5.4.1 Conservaton of Mass Conservaton of Mass Snce ppes are broken down nto two sectons t s requred that mass s conserved both wth respect to edges and wth respect to nodes. The edge mass balance constrant s formulated as follows: O s,t+1 = O s,t e + u s,t,e + us,t j,e, e, s, t, e, j e (5.82) The varable Oe s,t states the mass of gas present n the ppe at the begnnng of tme segment (s, t). Hence the net sum of mass enterng or leavng the ppe s present n the ppe n the followng tme perod. For nodes the mass balance constrant s analogous to the example 5.1. e u s,t,e = νs,t χ s,t d s,t (5.83) for nodes, where d s,t s the demand nto node, ν s,t s the nput from an adjacent supply source, and χ s,t s the rate of export Non-lnear Momentum Constrants Commencng wth the steady-state expresson of equaton (5.64), the pressure flow relatonshp s gven below: p 2 p 2 j = c e u e u e α, (5.84) Gven a network wth nodes and j and edges ncdent to nodes ndexed e. Consder a mean pressure of an edge (ppe) p e, whch n the steady state case s the average of p and p j, whch s the pressure at the end ponts. The ppe specfc parameters c e are derved. The flow u,e s the average flow-rate between node and the mdpont of edge e. Durng a tme perod t we assume that the flow between node and edge mdpont can be consdered near-steady (quas-steady). Hence the relatonshp: (p s,t ) 2 (p s,t e ) 2 = 1 2 c eu s,t,e us,t,e α e holds Pressure Drop over Ppe-length Consder a network wth pressure varables ncdent to nodes and flow varables ncdent to arcs. The network s acyclc. Assumng the flow s postve (runnng from an upstream node to a downstream node), the flow pressure relatonshp s as follows: p 2 p 2 j = c e u α+1 e Assumng that the upstream pressure p s known, ths means the downstream pressure p j s:

49 5.4.3 Pressure Drop over Ppe-length 45 p j = p 2 c eu α+1 e Wthout loss of generalty c e s set to 1. Ths functon s dsplayed n the 3-dmensonal plot n fgure pdown 50 pdown pup u u pup pdown 50 pdown pup u u 0 0 pup Fgure 5.5: The relatonshp between flow-rate and pressure at each end of a ppe shown from four vewponts. It s evdent that as flow ncrease, so does the rate at whch pressure drops. As hgh pressure s desrable n the system, and the downstream pressure s concave as a functon of both upstream pressure and flow-rates, ths can be lnearzed. However, as flow can move n both drectons n the ppe we return to the orgnal formulaton of flow and pressure relatonshps (rentroducng the absolute value for sgn preservaton on flows. p 2 p 2 j = c e u e α u e Assumng u e < 0 ths gves the followng expresson for the downstream varable (physcally the upstream varable). p j = p 2 + c eu α+1 e. The combned functon takes the form llustrated n fgure 5.6 Ths functon s nether convex nor concave over the entre span of feasble flow-rates. However, one can observe that when flow-rates become negatve the orgnal upstream down effectvely becomes a downstream node and downstream becomes upstream.

50 5.4.4 From Non-lnear to Pecewse-lnear pdown pup u u pup pdown pup u u pup Fgure 5.6: The relatonshp between flow rate and pressure at each end of a ppe. Allowng negatve flow-rates, also shown from 4 vews. It s apparent from the graph that at a flow rate of 0, the rate at whch pressure drops/rses at the unfxed end of the ppe changes sgn. Ths shows that the functon of flow at one end of a ppe, gven a fxed pressure at the other end, n relaton to the flow rate s none convex/concave From Non-lnear to Pecewse-lnear Now assume that for an edge-node ncdence par t holds that u,e 0. Ths means that the flow-drecton s from node to ppe, and therefore p t ps,t e. The energy relatonshp s then: (p s,t ) 2 (p s,t e ) 2 = 1 2 c e(u s,t,e )1+α e mplyng that: p s,t = (p s,t e ) c e(u s,t,e )1+α Obvously for u s,t,e = 0 we have p = p e. Wth fxed p e, ncreasng the value of u s,t,e the second term becomes domnant and the relatve dstance to p s,t 1 = 2 c eu s,t,e decreases. For ths soluton area of u s,t,e and for fxed ps,t e the functon s convex. Now assume u s,t,e 0. Ths leads to the energy relaton:

51 5.4.5 Bnary Flow Drecton Varables 47 (p s,t ) 2 (p s,t e ) 2 1 = c e 2 (us,t,e )1+α e Solvng for p s,t gves: p s,t = (p s,t e ) c e(u s,t,e )1+α Assumng agan that p s,t e s fxed, t s evdent agan that for u s,t,e = 0 t holds that ps,t = p s,t e. decreases as the numerc value of us,t,e ncreases and at Ths functon wth respect to u s,t,e an acceleratng rate. The functon s concave wth respect to u s,t,e. Snce the purpose s to restrct the throughput of the network n representaton of techncal lmtatons the two cases for flow drecton have ndvdually the correct functonal form. Gven s, ether a value of upstream or downstream pressure. When upstream pressure s gven, downstream pressure s a functon of flow rate s concave. In ths case a concave functon sets the upper lmt of the sustanable downstream pressure, gven flow and upstream pressure. Ths can be approxmated by a famly of pecewse-lnear functons. The other case, gven downstream pressure, the upstream pressure as a functon of flow rate s convex. Ths convex functon s nterpreted as a lower lmt on the upstream pressure requred to sustan the flow. Hence t can also be approxmated by a famly of pecewse lnear functons. Below the pecewse lnearzed versons of the energy relaton are presented n the two cases of u s,t,e. For u s,t,e 0: For u s,t,e 0: p s,t p l e + ψl e us,t,e + υl ep s,t e (5.85) p s,t p l e + ψ l eu s,t,e + υl ep s,t e (5.86) The ndex l L refers to a pont on surface of the orgnal pressure functon llustrated n fgure 5.6 where the functon s lnearzed. The parameters p l e, ψl e and υ e descrbe tangent planes to the orgnal functon, whch form a lower bound for the physcal upstream pressure whch enables the flow u s,t,e 0. Lkewse parameters pl e, ψ l e and υ l e descrbe the tangent planes to the orgnal functon, whch form an upper bound on the downstream pressure whch enable the flow u s,t,e Bnary Flow Drecton Varables As t s obvous that both sets of constrants wll not hold at once (except a 0-flow soluton), a set of bnary varables B s,t,e {0, 1} s ntroduced. By defnton: u t,e > 0 B s,t,e = 1 (5.87) u t,e < 0 B s,t,e = 0 (5.88)

52 5.4.6 Lnkng Tme Segments 48 Ths s ensured by the ntroducton of the followng constrants: where M s a suffcently large number. u s,t,e B s,t,e M (5.89) u s,t,e (B s,t,e 1)M (5.90) The bnary varables are also ntroduced n the energy equatons such that: Lnkng Tme Segments p s,t p l e + ψl e us,t,e + υl ep s,t e + (B s,t,e 1)M (5.91) p s,t p l e + ψ l eu s,t,e + υl ep e + B s,t,e M (5.92) It has been repeatedly stated that the pressure at the mdsecton of a ppe s known, but how s ths brought about? By contnuty what enters a ppe n one tme segment ether leaves the ppe or s present n the next tme segment. The state equaton descrbes the statc relatonshp between the present amount of natural gas and the pressure n the ppe. Ths property sustans the flow soluton of the network between tme segments. Assumng the state equaton s lnear n terms of pressure and amount (constant compressblty Z) and defnng O t e as the amount of gas n the ppe e at tme t. From the state equaton the followng s derved: p = ρrt Z (5.93) RZ = p (5.94) ρt p n RZ = (5.95) ρ n T n p T n ρ = ρ n p n T (5.96) The mass present wthn the ppe s expressed by the volume of the ppe and the average densty: O s,t e p s,t e T n = ρ n p n T V e e, t, Ths expresson s practcal as ρ n = 1Nm 3 by defnton and the normalzed condtons for pressure and temperature are p n = 1bar and T n = 273K. Ths system ensures that the value of p s,t e s fxed from the prevous tme perod. Hence the decsons taken at tme s, t are p and u s,t,e. Therefore f decsons are sequentally taken n order of t there are two unknowns to the energy relatons n each tme-step per edge-node ncdence par.

53 5.4.7 The Flow Model The Flow Model The complete model now appears as follows: p s,t e = K e Oe s,t e, s, t (5.97) O s,t+1 = O s,t + (u s,t ) s, t, e, s, t, e, j e (5.98) e e,e + us,t j,e u s,t,e = ν s,t χ s,t d s,t s, t, (5.99) p s,t p l e + ψl e us,t,e + υl ep s,t e + (B s,t,e 1)M s, t, (, e) e (5.100) p s,t p l e + ψ l eu s,t,e + υl ep s,t e + B s,t,em s, t, (, e) e (5.101) u s,t,e B s,t,em s, t, (, e) e (5.102) u s,t,e (B s,t,e 1)M s, t, (, e) e (5.103) The model has too much freedom wth regard to the node pressure, but snce the node pressure s only ndrectly connected across tme perods va the contnuty equatons for ppe-pressure, ths freedom should not be explotable. 5.5 Computatonal Intractablty Unfortunately the number of bnary varables makes the model computatonally ntractable. Havng around 50 edges wth two flow sectons each gves approxmately 100 bnary varables per tme perod. It s desrable to have at least around 144 tme perods thus leavng the model computatonally ntractable. Unfortunately t has been found mpossble to reformulate n terms of bnary varables wthout loosng ether accuracy or operatonal functonalty. It s therefore the logcal step to search for a model reducton wth least possble mpact on functonalty. Several deas are put forward below. All ppe-legs to whch all paths to source nodes are undrectonal should be locked n terms of flow drecton. Ths forces a modest functonalty reducton n order that lne-pack near the end of the graph can only be used to supply demand from farther out. Examples affected: Llle Torup Aalborg, Torslunde Lynge. Ppe-legs near a strong supply source can be drected. If one assumes that a supply source s always actve, and demand near ths source wll always be suppled from ths source, then flow drecton can be fxed as flowng away from the source node assumng no relevant crcuts are present. Example affected Nybro Egtved. All ppe-legs have only 1 flow drecton. Ths mples that flow cannot enter a ppe from both sdes durng a tme perod nor leave from both sdes durng a tme perod, thus forcng flow drecton to change on the nodes nstead of on ppes. Ths gve a reducton by 50% on bnary varables, and the functonalty reducton appears to be rrelevant. Ppe-legs can be grouped. Instead of havng full flexblty n terms of flow drecton between meterng statons one can group consecutve M/R statons makng these

54 5.6. A SOLUTION 50 dependent on the same bnary flow drecton varable. Ths mposes the lmtaton that all demand ponts wthn the group must be suppled from the same drecton. Examples groups: {Egtved, Llleballe, Taulov} or {Vborg, Karup, Hernng} etc. Unfortunately these restrctons nsuffcently reduce the problem complexty. There are stll multple bnary varables wthn a tme perod and many perods n the model. Seasonal flow drecton could also be mposed. By restrctng flow drecton to the season (e.g. the month) the number of bnary varables could be reduced drastcally. However, ths may also gve very bad results as the flow drecton s lkely to shft durng the day, as demand s hgher durng certan peak hours than durng nghttme. All the descrbed smplfcaton possbltes have been attempted and though the model can be solved for a steady state scenaro, the search tree of the branch and cut algorthm explodes exponentally once demand and supply varatons are mplemented. Multple solvers for both the mxed-nteger problem and the lnear subproblem have been tested. Tamperng wth the types of cuts generated by the MIP solvers has also been attempted. The concluson s that the problem s smply too large to solve as a mxed-nteger problem. 5.6 A Soluton In order to acheve results from the flow model a smple heurstcal approach s adopted. By frst determnng what s desred by the market, one can obtan a far dea concernng the drecton of flow n the ndvdual tme perods. Subsequently these drectons can be forced nto the model and the flow model can be solved lnearly. Ths naturally rules out some flexblty n the model, but f the guess of flow drecton makes sense, the results should be nterestng enough. The three types of model executon can be termed NO FLOW, DI- RECTION IMPOSED FLOW and FREE FLOW. The frst model excludes the flow model and nstead ncludes a supply=demand equaton for natural gas on the ndvdual meterng statons. The second (DIRECTION IMPOSED FLOW ) ncludes the flow model but wth the bnary flow drecton varables fxed a pror. Fnally, FREE FLOW s the computatonally ntractable mxed-nteger model. Usng the frst two models t s assumed that a farly good estmaton of the fnal model usng the followng teratve scheme: Soluton Scheme: 1: solve NO FLOW 2: determne drectons and fx bnary varables 3: solve DIRECTION IMPOSED FLOW Ths scheme could potentally be mproved consderably by makng teratve changes to flow drecton as result of the resoluton of the problem. Ths has not been made a prorty for the project and has been left for future consderaton. 5.7 A Conc Varaton Ths secton nvestgates the possblty of applyng conc programmng n the pressure flow model (5.98)-(5.103). The secton s manly of mathematcal nterest and s ncluded n the report to demonstrate an elegant alternatve to pecewse-lnear approxmaton.

55 5.7. A CONIC VARIATION 51 In short, usng conc programmng makes t possble to extend the range of applcable constrants from lnear programmng slghtly, by ntroducng conc constrants. Conc constrants are non-lnear, but t s stll possble to solve conc models usng an nteror pont method. Thus the soluton complexty s near that of a lnear programmng problem. Bascally two knds of conc constrants can be modeled. These are the quadratc cone and the rotated quadratc cone. C t = x Rnt : x 1 nt x 2 j (5.104) j C t = x Rnt : 2x 1 x 2 nt x 2 j, x 1, x 2 0 (5.105) Revst the non-lnear formulatons for the flow-pressure relatonshps from the past chapter. Here t s necessary to assume that α = 1. j p s,t = p s,t = (p s,t e ) c e(u s,t,e )2, (p s,t e ) c e(u s,t,e )2, u s,t,e 0 (5.106) u s,t,e 0 (5.107) As conc constrants can be lnked wth lnear constrants, consder the followng. p s,t = p s,t e = u s,t,e = (p s,t e ) 2 + (u s,t,e )2, u s,t,e 0 (5.108) (p s,t ) 2 + (u s,t,e )2, u s,t,e 0 (5.109) 1 2 c eu s,t,e (5.110) Followng the logc from the secton the equalty sgns can be replaced wth nequaltes, thus boundng the numerc values of the us. p s,t p s,t e u s,t,e = (p s,t e ) 2 + (u s,t,e )2, u s,t,e 0 (5.111) (p s,t ) 2 + (u s,t,e )2, u s,t,e 0 (5.112) 1 2 c eu s,t,e (5.113) The result s two cones whch effectvely bnd the flow-rate by the allowed pressure. These cones elmnate the need for a pecewse lnear approxmaton. Unfortunately the examned solver wth capacty for ntroducng conc constrants s not currently able to combne ths wth a mxed nteger programmng nterface, and as such the dea was abandoned. It has later become apparent that another solver may be able to combne these elements and ths would be an nterestng aspect to look nto.

56 5.8. SUMMARY Summary Wth offset n general theory of compressble flow, a model has been developed by smplfcaton and assumptons whch descrbes the relatonshp between ppelne flow and pressure on a network level. Unfortunately t was necessary to mplement a heurstcal soluton to the flow drecton subproblem, and as such optmalty cannot be guaranteed for the complete model as t s formulated mathematcally. The developed flow model serves the purpose of constranng flow n the transmsson system n-lne wth what s techncally feasble. Although the accuracy of the model can be questoned and the number of constrants whch are ntroduced s consderable, t s possble to attan a soluton wthn tmes whch are deemed acceptable n relaton to the modeled tme horzon. Fgures for tme and resource consumpton are presented n chapter 7, but frst the dervaton of the natural gas market model s descrbed n chapter 6

57 CHAPTER 6 Gas Market Model In ths chapter, the man propertes regardng the structure of the Dansh transmsson, dstrbuton and storage facltes for gas, are outlned. The publshed polces of the system operators are nterpreted to sut the model doman. From ths a number of model constrants are defned and a cost contrbuton to the objectve functon. Also the mplementaton of the market for natural gas s descrbed n terms of a domestc and an nternatonal market. 6.1 The Transmsson System The cost assocated wth the use of the transmsson system s organzed n a structure of tarffs. The current structure of natural gas transmsson tarffs, s a combnaton of a tarff on capacty n the system, and a tarff on the actual amounts transmtted through the system. For capacty there s a tarff for reservng entry capacty and a tarff for ext capacty. The capacty system s bascally an entry-ext system, whch means that a tarff s payed for entry at one of three entry ponts, whch are the gas treatment plant at Nybro, the Dansh-German border at Ellund, and the Dragør border wth Sweden. However, there s at present no actual entry possblty from Sweden. System ext tarffs are payed for ext n the ext zone, whch s bascally the whole country and the two nternatonal connectons to Germany and Sweden agan. The transmsson operator has developed a seres of capacty products some of whch, are mplemented n ths model. The basc product s an annual capacty contract, whch gves the customer the opton to use a certan amount of hourly entry or ext capacty for a year. Ths contract can be supplemented by monthly, weekly or daly contracts, whch are relatvely more expensve but provde flexblty. Snce gas consumpton s strongly correlated wth temperature, t s hard to forecast accurately the capacty requrements for a year n advance. In the begnnng of a gven month one would have a better forecast of consumpton n that month and one would perhaps choose to supplement one s annual capacty contract wth a monthly contract. Weekly and daly contracts become relevant when the week-ahead weather forecast becomes relable. In ths model t s nconvenent to ntroduce weekly and daly contracts. Ths factor s attrbuted to the temporal resoluton,

58 6.1. THE TRANSMISSION SYSTEM 54 January February March Aprl May June 35% 35% 30% 15% 8% 8% July August September October November December 8% 8% 8% 10% 15% 30% Table 6.1: Cost of monthly capacty as fractons of annual capacty contracts. (SOURCE: Gastra A/S) whch gves a pcture of average hours wthn a month. Thus all days appear alke n the vew of the model, and thus the more nexpensve monthly contracts would always be chosen. It s assumed that market players have perfect foresght wthn the year, for purposes of capacty bookng. Thus, fndng the correct combnaton of capacty products between annual and monthly products s a lnear subproblem. The followng varables are ntroduced: κ Y EN κ Y EX κ s EN κ s EX annual entry capacty booked annual ext capacty booked monthly entry capacty booked n month s monthly ext capacty booked n month s At present entry and ext capacty are dentcally prced at kr./kwh/h/year [22]. Symbolcally, the annual capacty tarffs are termed τen Y, τ EX Y, for entry and ext respectvely. Monthly contracts are gven as factonal annual contracts ρ s varyng monthly. Table 6.1 shows cost of monthly capacty as percentage of annual contracts. Introducng addtonally the subset of areas P A where producton of gas occurs or gas can be exported, the objectve functon contrbuton of transmsson tarffs can be formulated as: τ V P s t ν s,t s,t } {{ } volume ( + τen Y κ Y EN + ) ( 12 s 8760 ρs κ s EN + τex Y κ Y EX + s s } {{ } entry 12 s ) 8760 ρs κ s EX s } {{ } ext (6.1) The expresson 12 s 8760 ensures proper scalng when S 12. The followng constrants ensure that suffcent capacty s always booked: A d s,t P + P ν s,t κ Y EN + κ s EN, s, t (6.2) χ s,t κ Y EX + κ s EX, s, t (6.3) Equaton 6.2 states that for any gven tme perod, the sum of ordered annual and seasonal entry capacty relevant to that tme perod must be greater than or equal to supply of that perod. Equaton 6.3 states that for any gven tme perod, the sum of ordered annual and seasonal ext capacty, relevant to that tme perod, must exceed total delveres and exports combned.

59 6.2. DISTRIBUTION Dstrbuton There are fve dstrbuton network operators n Denmark. These are DONG Dstrbuton, Hovedstadens Naturgas, Naturgas Mdtnord and Naturgas Fyn. DONG Dstrbuton operated the dstrbuton networks of southern Jutland and most of Zealand (wth the excepton of the area of Greater Copenhagen). Mdt Nord operates n the central and northern parts of Jutland. Naturgas Fyn operates on the sland of Fynen and Hovedstadens Naturgas operates the network n Greater Copenhagen. All of the dstrbuton network operators have presently opted for a tarff based on volume. There s no cost of securng capacty n ths system. The tarff structure s arranged so that the greater the demand from a sngle customer, the lower the unt cost of dstrbuton becomes. Ths s unfortunate from a modelng perspectve. As the market s modeled as a sngle player there s no dstncton between ndvduals, and as such the mpact of dstrbuton tarffs cannot be dstrbuted approprately for a sngle plant. However, as generaton technologes typcally reflect a certan plant sze, technologes are assocated wth a step on the dstrbuton tarff ladder, or a lnear combnaton of steps. All operators have 8 prce steps. A set Ξ s defned to represent these. The tarffs assocated wth a specfc dstrbuton area, δ D, n a gven volume nterval, ξ Ξ, s thus defned as τ ξ δ. For each generaton technology type fueled by natural gas a weghtng s estmated accordng to how much of the producton capacty would normally fall wthn a certan dstrbuton volume nterval. Ths weghtng s termed wg. ξ As such the economc mpact of dstrbuton tarffs, and thus ther objectve functon contrbuton s formulated as: s t δ τ ξ δ wg ξ F s,t,g ( ) ξ Ξ g G δ s,t (6.4) Where F s,t,g ( ) s a functon descrbng the natural gas consumpton of technology g n area n tme segment s, t. 6.3 Gas Storage The Dansh gas storage operator, DONG Lager, s n charge of the two Dansh gas storage facltes at Stenllle and Llle Torup. These facltes have a combned workable storage capacty of 820 mllon Nm 3, roughly equvalent to 20% of the annual domestc consumpton. The functon of the storage facltes are two-fold. Frst, usng gas storage t s possble to compensate for the large seasonal varables n demand, and second they provde a reserve opton to sustan supply n case of a breakdown n supply. Ths reserve s scaled n order to mantan 60 days of non-nterruptble supply and 3 days of nterruptble supply. All customers who are not economcally compensated for nterruptblty are unnterruptble. The pressure n the caverns of the storage facltes s kept around bar and as such the gas must be compressed and chlled upon njecton, whle t must be heated upon extracton. Ths process lmts the rate at whch gas can be njected and extracted from the facltes. DONG Lager supples two dfferent storage products, whch may be combned at the user s dscreton. One s more flexble than the other wth regard to njecton/extracton capacty and s as such also more expensve. Table 6.3 descrbes the two man storage products. A

60 6.3. GAS STORAGE 56 Volume cap. Extracton cap. Injecton cap. Tarff Unt V σ ε σ ι σ τ σ kwh % of volume % of volume kr./kwh Product Product Table 6.2: Storage products from DONG Lager (SOURCE: DONG Lager) σ s used to ndex storage products. The subset I A ncludes those nodes whch contan a storage faclty. A lnear combnaton of the two storage products s assumed to be able to cover most users requrements, and for the purpose of ths project t s assumed to be the only storage opton. It s also possble to trade storage, njecton and extracton capacty amongst storage users. As the developed model consders the market to be a sngle actor, such tradng s assumed to be 100% effcent for practcal reasons. There s addtonally a varable cost of njecton, currently τ ι = kr./kwh. For supply securty reasons, storage contracts are subjected to a fllng requrement. Ths s to ensure that all players do not fll at the last mnute thus ensurng a smooth rate of fllng n the autumn months. The fllng requrements are dependent on the amount of storage purchased. Table 6.3 shows the percentage of total purchased capacty must be present by the frst of each month. Γ s s ntroduced to symbolcally descrbe the fracton of purchased storage capacty, whch must be n the storage faclty by season s. The followng varables are used to descrbe the utlzaton of the storage faclty. Λ s,t ι s,t ε s,t ζ σ stock of a faclty at a gven tme perod njecton rate nto faclty at gven tme perod extracton rate from faclty at gven tme perod product unts purchased of each storage contract The varables are subject to faclty specfc techncal upper bounds n terms of maxmum capacty, and rate of extracton and njecton. These are specfed n table 6.3. As a consequence the varables are bound as follows: 0 Λ s,t Λ s,t, I, s, t (6.5) October November December January February March Aprl Γ s 0% 20% 55% 60% 40% 10% 0% Table 6.3: Fllng requrements from DONG Lager (SOURCE: DONG Lager) Workng gas volume Extracton cap. Injecton cap. Λ s,t : mll. Nm 3 ε s,t : knm 3 /h ι s,t : knm 3 /h Llle Torup Stenllle Table 6.4: Techncal constrants on the storage facltes (SOURCE: DONG Naturgas A/S)

61 6.4. GAS MARKET 57 0 ε s,t ε s,t, I, s, t (6.6) 0 ι s,t ι s,t, I, s, t (6.7) The equatons relevant to the storage facltes are now presented. As mentoned, the market operates wth only one vrtual storage faclty and as such, the constrants n relaton to the products of table 6.3 are defned over the sums of the actual storage facltes. Energy unt converson factors are omtted for clarty. V σ ζ σ σ I ε σ ζ σ σ I ι σ ζ σ σ I Γ s σ Λ s,t, s, t (6.8) ε s,t, s, t (6.9) ι s,t, s, t (6.10) V σ ζ σ I Λ s, t 1 s (6.11) The frst three equatons (6.8)-(6.10) ensure that adequate volumetrc capacty, extracton capacty and njecton capacty s reserved respectvely. The forth (6.11) ensures that the fllng requrement s upheld. The storage system s contrbuton to the objectve functon s thus the cost of purchasng capacty products and the varable cost of njecton. τ σ ζ σ σ } {{ } capacty products + I τ ι ι s,t s,t s t } {{ } njecton (6.12) 6.4 Gas Market Two gas markets are mplemented n the model. The frst s the nternatonal market, where t s possble to buy and sell gas accordng to exogenous prces based on hstorcal developments at the Zeebrügge gas hub n Belgum. A prce dfference of 0.02 kr./mwh s lad between the mport and export prce to account for transportaton to and from the border. These prces are adjusted to follow the expectatons wth regard to developments n annual fuel prces. These expectatons are descrbed n chapter 7. Ths way of handlng mport and exports s selected as t s closely related to the method Gastra (now Energnet Danmark) uses to calculate a neutral gas prce, whch t employs n ts balancng operatons. The domestc market works as a partal equlbrum market where the supply rate from the North Sea s prce ndependent. Ths, by the assumpton that the varable costs of producton are neglgble when compared to fxed costs and nvestments n offshore extracton. Demand s gven by exogenous and nelastc demand from consumers wth the excepton of demand from heat and power generators. These consume the gas, whch s avalable to them to make a proft. They are also able trade on the nternatonal market. As such a prce ndcaton can be derved when heat and power generators consume some, but not all the gas whch s avalable to them.

62 6.5. LINK WITH THE BALMOREL MODEL 58 Imports and exports naturally also have an mpact on the objectve functon. When mport prces are defned as π s,imp and export prces are defned as π s,exp. Ther contrbuton s: ( s t P π s,imp ν s,t π s,exp 6.5 Lnk wth the Balmorel Model ) χ s,t s,t (6.13) The lnk between the orgnal Balmorel model and the gas flow model les n a set of equatons ensurng that the outtake of gas at the M/R statons matches the demand from the heat and power sectors and the resdual demand (prvate homes, ndustry etc.). The resdual demand s descrbed n chapter 7, but for now t s enough to know that ths demand s exogenous and defned as R s,t. For clarty the consumpton of gas by the electrcty and heatng sector s defned functonally as F s,t,g ( ). d s,t = R s,t + g G F s,t,g ( ), A, s S, t T (6.14) 6.6 Summary Presently the entre modelng effort has been descrbed. The model descrbed n the prevous chapter dealt wth the physcal restrctons of natural gas transmsson. Ths chapter outlned how consderatons of a both techncal and economc nature have been formulated nto operator polcy and prces/tarrfs, and these have been nterpreted to sut the modelng doman. The market for natural gas has also been descrbed brefly ntroducng both the nternatonal and domestc market functons. These wll be consdered further when dscussng results of the model executon.

63 CHAPTER 7 Model Executon Ths chapter concerns the confguraton and executon of the model. Varous nput alternatves are descrbed to gve an mpresson of how the model can be confgured to the needs of the user. The sources of standard data are presented, and the manner n whch data s ftted to the model doman s dsclosed. The mpact of nput alternatves n ther mplcaton on fnal results are dscussed. 7.1 Geography and Tme At present the geography s fxed to descrbe Denmark by means of two electrcty regons and a total of 50 areas wthn these regons for descrbng producton of power and dstrct heat and delvery of dstrct heat and natural gas. On the long term the ntenson s to be able to aggregate areas nto larger geographcal sectons accordng to requrements of ndvdual model applcatons. The years 2003, 2005, 2010, 2015 and 2025 are ncluded n the model. Other years may be supported, but data may not be avalable n all respects. Data s ncluded to support up to 12 seasons and 12 tme perods. There s full flexblty for down-scalng the resoluton as long as one s wary of the mplcatons. Seasons and tme varaton profles are mplemented so that seasons bascally represent months, whle tme perods represent dfferent representatve hours wthn the week. Table 7.1 shows the natve ntervals of for tme segments. The flexblty of the temporal resoluton gves rse to some nterestng nterpretatons of whch a few are lsted below: S = {s 1 }, T = {t 1 } Model s executed usng annualzed averages for all data. The flow model descrbes the steady-state of the annual net flows. S = {s 1, s 7 }, T = {t 1 } As above, but wth summer and wnter smulatons based on the varaton between January and June. S = {s 1,... s 12 }, T = {t 1 } All ndvdual months modeled wth the steady-state flow condtons.

64 7.2. FUEL PRICES 60 Tme Element Day Type Hours t 1 Weekday 00:00-09:00 t 2 09:00-11:00 t 3 11:00-11:30 t 4 11:30-15:00 t 5 15:00-18:00 t 6 18:00-18:30 t 7 18:30-21:00 t 8 21:00-00:00 t 9 Weekend 00:00-07:00 t 10 07:00-16:00 t 11 16:00-21:00 t 12 21:00-00:00 Table 7.1: The subdvson of an average week wthn a month nto tme segments. S = {s n }, T = {t 1,..., t 1 2} An annually averaged week wth the varaton profle of month n. Quas steady-state flow modelng descrbe transmsson. S = {s 1,... s 12 }, T = {t 1,..., t 12 } The full model wth twelve months and as much varaton as the data provdes. Numerous other combnatons are possble each wth ts own nterpretaton and area of usefulness. Table 7.1 shows the hours whch are represented by tme perods wthn a week. 7.2 Fuel Prces The latest publshed projecton for fuel prce developments by Dansh Energy Authorty of February 2003 [9] s appled n the model. The fuel prces appled are presented n fgure 7.1. There are no seasonal varatons n fuel prces, wth the excepton of natural gas whch s treated apart from other fuels. Prces for natural gas are not used drectly, as ths prce s determned by the model. The prce development for natural gas s, however, used to project the border prces as mentoned n chapter Demands Demands for network bound energy supply s derved for electrcty, dstrct heat and resdual natural gas (gas not used for energy transformaton purposes). It s assumed that the profle of these demands s sustaned across years, whle the annualzed demands, to whch the profles are appled change over the years. Thereby each energy form a profle s found or derved, and a forecast for annual demand s found or derved Demand for Electrcty and Dstrct Heat Annualzed demand for dstrct heatng s found through aggregaton of annual delveres of heat from heat producers wthn the respectve areas. A total annual demand estmaton can be made for each area n the model, whch s coherent wth the technology and capacty

65 7.3.1 Demand for Electrcty and Dstrct Heat 61 Fuel Prces Natural Gas Coal Fuel Ol Lght Ol Straw Wood 30 kr./gj Tme/Year Fgure 7.1: Fuel prce development forecasts Demand for Dstrct Heat and Power Heat demand DK Electrcty demand DK E Electrcty Demand DK W MWh/h Hours Natural Gas Demand 800 Total hstorcal deamand for natural gas Resdual NG demand profle knm 3 /h Hours Fgure 7.2: Exogenous nput data relatng to demands wth full temporal resoluton. data. Annual demand s assumed to be constant. The varaton profle for dstrct heat n the Balmorel data set s employed. Electrcty annual demand s derved analogously to dstrct heat demand. However, here an annual ncrease of 1.0% s assumed. Fgure 7.2 (top) shows the varaton profles for dstrct heatng and electrcty demand wth the full resoluton of 12 seasons and 12 tme perods. Fgure 7.3 (top) shows the same, but aggregated to 4 seasons and 4 tme perods.

66 7.3.2 Natural Gas Demand 62 Demand for Dstrct Heat and Power Heat demand DK Electrcty demand DK E Electrcty Demand DK W MWh/h knm 3 /h Hours Natural Gas Demand Total hstorcal deamand for natural gas Resdual NG demand profle Hours Fgure 7.3: Exogenous nput data relatng to demands wth four seasons and four tme perods Natural Gas Demand Demand data for natural gas s only avalable as total consumpton over tme. It s necessary to subtract that whch has been used for heat and power generaton. Ths data s, however, only avalable on an annualzed bass. Usng the profles for heat an electrcty producton however, a decent estmaton can be made. Ths estmaton s llustrated on fgure 7.2 (bottom) and n the tme aggregated form wth four seasons and four tme perods on fgure 7.3 (bottom). 7.4 Gas Producton Gas supply from the North Sea s assumed to be constant throughout the year. Ths reflects the economc advantage of operatng at maxmum capacty on the offshore rgs n order to reduce the long-run costs of sustanng the extracton operaton. Fgure 7.4 shows two possble trends n the North Sea producton over the modeled tme frame. One s the expected annual producton gven only presently confrmed reserves. The other shows and estmated contrbuton from future prospects. The bass scenaro, for whch results are presented n chapter 8, consders only the confrmed resources and thus follows the rather pessmstc producton curve. Secton 9.2 ncludes results usng the producton curve whch ncludes prospects. 7.5 Technology Data Technologes and exogenous capacty wth regard for energy transformaton has been updated n relaton to the data set dstrbuted wth Balmorel. The Balmorel transformaton capacty s not dstrbuted at a resoluton suffcent for ths project. It was necessary to generate a new capacty dstrbuton.

67 7.6. MODEL COMPLEXITY North Sea Annual Producton ndentfed resources prospectve contrbuton 7 6 Nm 3 /day Years Fgure 7.4: Natural gas producton from the North Sea (SOURCE: Dansh Energy Authorty) The root source of transformaton capacty data used for the model s the Energy Producer Census Census of all Dansh energy producton unts s performed annually by the Dansh Energy Authorty. Ths data set ncludes plant level nformaton on unt locaton, electrcty and heat generaton capacty, annual electrcty and heat producton, annual fuel consumpton and much more. The generaton capacty data set s aggregated by technology type wthn each area. Capacty s sorted by postal code and assocated wth the nearest meterng and regulaton staton where applcable, usng maps of the natural gas transmsson and dstrbuton networks n conjuncton wth a postal code map. Subsequently, generaton capacty s aggregated to match the technology ndex. A new technology ndex was also mplemented. The Energy Producer Census contans plant by plant technologcal data, but ths s too fne to be appled practcally. Also, t lacks data wth regard to operatng and nvestment costs among other shortfalls. Instead the technology ndex created by Eltra, Elkraft System and the Dansh Energy Authorty [8] s appled, and plants are assgned a best match to technology types from the ndex. It s mportant to note also that wnd powered technologes are omtted entrely from ths project. There has been no ntenton to make scenaros on ths front, and snce demand s generated by observng hstorc delveres from thermal technologes only, t s assumed that there s a demand layer, whch s beng suppled by wnd power. Snce wnd power s prortzed and producton s fxed to an exogenous wnd profle, the omsson of a matchng amount of supply and demand has no effect on the soluton or assocated shadow prces. 7.6 Model Complexty In the bass scenaro wth 12 seasons and 12 tme perods the model conssts of 5 teratve solutons of two lnear programmng problems and a fast heurstc (lnear tme). The lnear programmng problem, whch expresses the ntal desres of the market, s a lot smaller than the second, whch ncludes the techncal consderatons of the transmsson system.

68 7.7. SUMMARY 64 Excludng model generaton tme the soluton of the model takes n the neghborhood of 43 hours or just short of two full days. Table 7.6 contans more detal about the model sze and executon tme. One nterestng tendency s that the ntal soluton of both models takes notceably longer than the subsequent teratons. Ths s the case for both models. Ths tendency must be attrbuted to solvers ablty to warm start, meanng that the second teraton commences at the soluton to the frst teraton. Ths property results n the rapd re-optmzaton as long as there s not too bg a varaton n the nput data between the modeled years. The solver used s CPLEX 9.0 by ILOG. Model Complexty Model: Varables Constrants Non-Zeroes Executon Tme seconds (mn-max) Intal model 480, ,000 1,500, ,700 Physcal model 520, ,000 1,900,000 10,200-63,200 Total (5 years) 156,200 Table 7.2: Lnear program dmensons and executon tmes. 7.7 Summary The model s dependent on data from a number of sources, and the qualty of sources and the method of data fttng s crucal for attanng relable results. Although the generaton of a realstc data set was not the focus of ths report, even the generaton of the mperfect data used to demonstrate the applcaton of the model has been unreasonably tme consumng. In the followng chapter the strengths and weakness of the model, but also the appled data set wll be dsclosed.

69 CHAPTER 8 Smulaton Results The model produces a vast amount of data upon ts executon. In ths chapter t s shown how ths output can be nterpreted and llustrated n a context upon whch t can be used for analytcal purpose. Results are presented for a standard executon of the model as descrbed n the prevous chapter. Dfferent temporal resolutons are employed n some of the dfferent results. Ths s to demonstrate how effectve of the flexble descrpton of tme s, but also due to the rather lengthy executon tmes of the full resoluton model. 8.1 System Load One of the more nterestng results of a smulaton s load dstrbuton of the system. In the electrcty market for example, there s a strong correlaton between hgh load and hgh prces, as the generators wth hghest operatng costs produce only durng these perods. Due to these aforementoned operatng costs they are unable to provde compettve bds n low and medum load stuatons, hence at peak hours they must bd to cover long term margnal costs. These are naturally also the most nterestng hours wth respect to nvestments. For the transmsson system operator (TSO) the peak load hours are also the most nterestng. Gas transmsson s a busness where the cost share of nvestment s domnant over the cost of operatons. Therefore, the sgnal most relevant to the TSO s a sgnal ndcatng capacty shortfalls; sgnals whch could trgger nvestment. Load duraton s hghly ndcatve of how far the system s from runnng out of capacty. Fgure 8.1 (left) features the load duraton curves for electrcty producton n the modeled years. Snce electrcty demand n ths smulaton s nelastc, the results are hardly surprsng. On an annual bass a small ncrease n generaton s requred over tme to supply the annually ncreased demand. The demonstraton features maxmum resoluton wth respect to tme perods to gve a better mpresson of the shape of the load-duraton curve. The smulaton whch produced the graph ncdentally featured 12 seasons and 12 tme perods. Fgure 8.1 (rght) smlarly llustrates load duraton wth regard to ext from the natural gas transmsson system. That s the sum of hourly exports and ext nto the domestc market. The smulaton uses only confrmed reserves as explaned n secton 7.4. Notably

70 8.2. MARGINAL VALUES 66 the varaton n hourly values s less when there s plentful supply. Excess avalable gas s exported, whch means a more effcent use of the transmsson system, and a hgher utlzaton of capacty products. The low supply years reflect a dffculty n even provdng for resdual demand. Hence the load profle reflects manly the resdual profle, and not much gas s for other purposes (exports, dstrct heatng and power generaton). Load duraton Electrcty Producton Domestc Gas Consumpton GWh/h 5 4 Nm 3 /h Tme (hours) Tme (hours) Fgure 8.1: Load-duraton curves for electrcty demand (left) and natural gas transmsson system ext (rght) 8.2 Margnal Values Margnal values of nterestng nterpretatons n many areas of the model. The value of addtonal gas at some locaton n the model can be derved from the shadow prce of the nodal mass balance equaton (5.83) and can be nterpreted as an area prce consumers must be wllng to pay to ensure supply. That s of course f gas was sold at area specfc prces. Dual values to capacty constrants could be used to nterpret a cost of addtonal capacty f the capacty constrant s bndng. In the followng the nterpretaton of margnal value of the supply varable are used to determne a gas prce at the supply locaton. Followng ths, the dervaton of an electrcty prce s explaned Gas prces In secton 6.4 t was descrbed how two forms of prces appeared n the model. Import and export prces are gven by exogenous values wth approprate varaton over seasons and

71 8.2.2 Electrcty prces 67 tme perods. An endogenous prce appears from the domestc supply source n the North Sea. The dual varable µ s,t to the doman constrant for the supply varable ν s,t s nterpreted as the shadow prce for an ncrease of supply rate by 1kNm 3 /h for the duraton of tme segment s, t. Hence a gas prce at the entry pont s derved by: where s the Nybro network node. p = µs,t s, t (8.1) Fgure 8.2 demonstrates an example of ths for 144 tme perods. The gas prce s notably lower than the wholesale prce of gas on the market today, whch s currently around 1.8kr./Nm 3 dependent on the supplyng company. Interestngly enough ths same calculaton usng only 4 seasons and 4 tme perods gves the results on fgure 8.3. Here the average prce s defntely hgher and more equal to the prce observed n the market. Ths mght ndcate a weakness n the flow model as there s more varaton n demand on the one wth fner resoluton. Supply ssues resultng from ths flow model could result n gas not beng delvered to where t has the hghest value, thus affectng the smulated market value. Domestc Supply Prce kr./nm hours Fgure 8.2: Gas prce for the frst year of the smulaton wth 144 tme perods Electrcty prces The prce of electrcty n the model s reflected n the shadow prces of the electrcty balance constrants. At optmalty: π s,t r g G(r) e s,t s,g + ρ R(c),r ρ x (r,ρ),s,t (1 ɛ x(ρ,r) ) er,s,t d 1 ɛ e = 0, r R, s S, t T r (8.2)

72 8.2.2 Electrcty prces Domestc Supply Prce kr./nm hours Fgure 8.3: Gas prce for the frst year of the smulaton wth 16 tme perods. Here π s,t R reflects the dual varable of the equaton, whch when optmal values of the prmal varables are nserted, reflects the shadow prce of power generaton. In other words, the value of π s,t R s normalzed value of a margnal ncrease n avalable power, or the margnal generaton cost of a unt of power. From partal equlbrum theory t s known that ths reflects the market prce under perfect competton, gven that there s a market prce for every regon r n every tme segment (s, t). Fnally t s mportant to note that snce the energy balances are gven terms of power (MW), the electrcty energy prce s p e = πs,t R. s,t Electrcty Prce 600 DK East DK West kr./mwh hours Fgure 8.4: Electrcty prce for the frst year of the smulaton. The electrcty prce, depcted on fgure 8.4, remans farly constant throughout the smulated year. It s nterestng that the electrcty prce n the early months of the year ht the

73 8.3. FUELS FOR ENERGY TRANSFORMATION 69 bottom mark. Ths s most lkely due to the hgh heat demand makng fxed rato CHP generators of lmted use. Theoretcally ths would mply that electrcty to heat technologes such as heat pumps would possbly be an effcent nvestment. These are, however, not among the smulated technologes and therefore the prce of electrcty drops drastcally. 8.3 Fuels for Energy Transformaton The fuel mx used for dstrct heatng and power producton s another smulaton result. The most decdng factors wth respect to the fuel mx are fuel prces (exogenous and endogenous), transformaton capacty and technology, compettveness of nvestments and fuel potental wthn specfc areas/regons. For natural gas, the supply system plays a decdng role. If fuel s used n one place t wll naturally not be avalable for consumpton at another. If prces n Sweden and Germany are favorable, the avalable supply for the domestc market dwndles at least untl export capacty s reached. Bottlenecks n the transmsson system would naturally also have an mpact on the emphass on natural gas n the fuel mx. Results from a twelve season, twelve tme perod smulaton are shown on fgure 8.5 n annualzed terms, and fgure 8.6 shows the load duraton fuel consumpton. The smulatons uses only the confrmed natural gas reserves (as descrbed n secton 7.4). It s as evdent from the fgures, as t s ntutve that gas s well represented n the frst years where producton s hgh. The share declnes untl all domestc producton as well as mport capacty s used to ensure the nelastc demand of resdual customers. Ths s dscussed further n secton 8.4 below n context of transformaton technologes Fuel Consumpton for Transformaton Gas Coal Fuel ol Waste Straw Wood Fuel Consumpton GWh Years Fgure 8.5: Fuel consumpton by technologes n four smulated years.

74 8.4. TRANSFORMATION TECHNOLOGIES Fuel for Dstrct Heat and Power 5000 Fuel for Dstrct Heat and Power GWh/h GWh/h GWh/h Fuel for Dstrct Heat and Power GWh/h Fuel for Dstrct Heat and Power Gas Coal Fuel ol Waste Straw Wood Fgure 8.6: Fuel consumpton by technologes n four smulated years. The load duraton for fuels (fgure 8.6) also has some nterestng mplcatons. The shape of the duraton curves can reveal detals about whch fuels are used for base load generaton and whch prmarly are used n peak load hours, by consderng the slope and curvature of the graphs. In the frst two smulaton years llustrated, the natural gas consumpton has an almost constant slope and no curvature. In the two fnal years there s almost no gas fred generaton. It appears that coal and wood s used for hgh and peak load generaton ntally. As natural gas supples dsappear, coal takes over some of the base load and s supplemented by straw fred generaton. Wood fred generaton s stll manly used n the hgh/peak load hours. 8.4 Transformaton Technologes The model can demonstrate n dfferent ways, whch technologes are used to generate heat and power. One can vew results n terms of annual dstrbuton between technologes. How much producton s on new or old capacty etc. One can also see how producng technologes are geographcally dstrbuted. Naturally, generaton s separable n heat and power, and total fuel consumpton can also be llustrated. Each technology s dentfed on the fgures by a label. To facltate nterpretaton of the graphs these labels are descrbed n table 8.4. Fgure 8.7 demonstrates how ntally gas power s compettve n the smulated market. Especally wth respect to power generaton, the large scale combned cycle facltes are well represented. The specfc smulaton s run wth four seasons and four tme perods and the North Sea producton s lmted to confrmed reserves only, as was descrbed n secton 7.4. The

75 8.5. INVESTMENTS IN TRANSFORMATION TECHNOLOGY 71 Label Descrpton APF-E01-CO Large scale coal fred CHP wth extracton capablty APF-E01-NG Large scale natural gas fred CHP wth extracton capablty W2E-B05-MW Muncpal waste fred back pressure CHP unt GTLS-B06-NG Large sngle cycle gas turbne (fxed rato) GTMD-B06-NG Medum sngle cycle gas turbne (fxed rato) GTMN-B06-NG Mnature sngle cycle gas turbne (fxed rato) CCLA-E07-NG Large combned cycle faclty wth extracton capablty CCSM-B06-NG Small combned cycle faclty (fxed rato) GE-B08-NG Gas engne (fxed rato) ST-B09-WW Wood pellet or wood chp fred steam turbne (fxed rato) HO-51-WW Heat-only boler fred by wood HO-52-NG Heat-only boler fred by natural gas HO-54-MW Heat-only boler fred by muncpal waste ST-CO-FOsn Power plant fred by fuel ol wth de-no x and de-so 2 ST-CO-COsn Power plant fred by coal wth de-no x and de-so 2 HO-B0-FO Heat only boler fred by fuel ol HO-B0-ST Heat only boler fred by straw ST-B8-NG Steam turbne fred by natural gas (fxed rato) ST-B8-CO Steam turbne fred by coal (fxed rato) ST-B9-ST Steam turbne fred by straw (fxed rato) ST-B8-FO Steam turbne fred by fuel ol (fxed rato) G-HSTORE Heat storage faclty compettveness of gas power n the early years of the smulaton thus reflects the hgh capacty of supply. In later years, as reserves are depleted, producton s moved to straw and coal fred technologes, as can also be seen on fgure 8.7. Evdently these consttute the best alternatve when no cheap or techncally feasble opton for gas supply exsts. There s also a notceable tendency to use manly the large scale facltes, rather than the smaller facltes such as gas engnes. Ths s n spte of the fact that these are the most numerously nstalled faclty types n ths country. Ther ndvdual capacty s, however, lmted, so they were not expected to domnate the market, but one mght have expected a larger market share for these technologes. Ths s dscussed further n chapter 9. The extracton possblty s also a lkely reason for the technology choce. Added flexblty to produce a hgher share appears to be desrable. 8.5 Investments n Transformaton Technology There s a clear tendency towards nvestment n large-scale combned cycle facltes ntally. Ths can be seen from fgure 8.9. A few comments on ths tendency. The Balmorel model uses a quas-dynamc tme representaton. Perfect foresght s present for the duraton of the currently smulated year, yet no foresght s present wth regard for the followng years. Therefore, the decson n the frst year to nvest n gas powered technology also reflects the vew that the domestc producton of natural gas wll be sustaned for the economc lfetme of the nvestment. When ths s not the case t can result n a stuaton where an nvestment n gas fred technology wll produce for the frst couple of years, and then subsequently shut down due to supply shortage or a rse n gas prces before nvestment costs are covered.

76 8.6. EMISSIONS FROM TRANSFORMATION Electrcty Generaton 70 Dstrct Heat Generaton TWh TWh Years Years Fgure 8.7: Generaton of dstrct heat and power by dfferent technologes. A masters thess project, recently completed, nvestgated an alternatve way of handlng nvestments n Balmorel, and these deas are lkely to be mplemented n some form n the not so dstant future [27]. Ths frst smulaton was executed under the assumpton that no new non-confrmed gas reserves are brought to producton wthn the smulaton perod, as was stated n chapter 7. It s also nterestng to note that once the gas reserves are depleted, large nvestments are made n straw fred generaton. Ths s lkely due to the fact that straw s a CO 2 neutral fuel, and as such does not requre CO 2 allowances. 8.6 Emssons from Transformaton Sx graphs on fgure 8.10 show the development n emssons resultng from the smulaton. These are only from energy transformaton, so any emssons from gas consumpton n prvate homes etc. do not mpact the results. One mght add that these are by no means nterestng n the context of the model, as emssons from resdual consumpton can be assumed lnear wth respect to consumpton levels whch were gven exogenously. The frst fve graphs show emssons of CO 2, SO 2, NO x, CH 4 and N 2 O. The fnal fgure s the total energy generaton. Ths s ncluded n order to be able to see emssons developments n lght of developments n energy consumpton. It s evdent that the move towards more gas fred heat and power has a postve mpact on most emssons, the one excepton beng methane. Ths s as can be expected snce

77 8.7. THE NATURAL GAS TRANSMISSION SYSTEM Fuel Consumpton by Technologes TWh Years Fgure 8.8: Fuel consumpton by technologes. frng of natural gas results n fewer emssons per generated energy unt than other fossl fuels. In 2015 more producton s once agan coal fred, resultng n an ncrease n all emssons except methane. Fnally, n 2025 CO 2 emssons are reduced as a consequence of the aforementoned nvestment n bomass technology. In secton 9.3 addtonal analyss s performed on the mpact of CO 2 emssons allowances. 8.7 The Natural Gas Transmsson System Illustratng what happens n the natural gas transmsson system n a comprehensve manner s rather complex. For each tme perod there s an abundance of data avalable descrbng the flows and pressure n dfferent parts of the network. These s no fnal destnaton attached to gas flows. Ths makes t challengng to connect flow-values wth the market effects or otherwse draw the connecton for the broader pcture. Pressure values are even more dffcult to nterpret clearly, snce the pressure calculatons are relaxed n formulatng the pressure nduced flow constrants. A falsehood, whch s an otherwse ntutve nterpretaton, s that when pressure at the nput sources s the maxmally allowed, whle the pressure at the far end of the network s mnmal, then ths system must be runnng at full capacty. Ths would be the case f the actual pressure was calculated. However, due to the pecewse lnear approxmaton descrbed n secton 5.4.4, the only thng one can know for sure s that f the smulaton result s feasble, the resultng supply s possble. In theory, the pressure results of the smulaton are replcable f one were to use the lne-valves n the transmsson system to reduce pressure at certan locatons along the transmsson ppelne. Ths, however, has no practcal operatonal purpose.

78 8.7. THE NATURAL GAS TRANSMISSION SYSTEM Electrcty New Cap. 20 Dstrct Heat New Cap TWh TWh Years Years Fgure 8.9: Generaton on newly purchased technology. Emssons 2.4 x per Year CO Emssons per Year SO Emssons per Year NO x tonnes tonnes tonnes Years 20 Emssons per Year CH Years 1.6 Emssons per Year N 2 O Years Generaton of Electrcty and Heat 91 tonnes tonnes Generaton (TWh) Years Years Years Fgure 8.10: Emssons from transformaton n relaton to total transformaton.

79 8.8. AREA DISTRIBUTION 75 AALBORG ELLIDSHOEJ DK WR ural HAVERSLEV LITORUP 55 VIBORG KARUP HERNING 278 BRANDE NOERSKOV LYNGE MAALOEV HELLE EGTVED NYBRO VARDE BROENDBY HVIDOVRE 480 VALLENSBAEK 470 LILBALLE TORSLUNDE AMAGERF HCOERSTED SYDHAVNEN AVEDOERE VESTAMAGER TAULOV STENLILLE 424 DRAGOER LYNGSODDE KARLSLUNDE OERESUND 76 STANDST SKAERBAEK MIDDELFART KOEGE POTTEHUSE 418 SOROE BILLESBOELLE KONGSMARK SLAGELSE RINGSTED KOELBJERG 402 NYBORG LISELSKAER HOEJBY ULLERSLEV TERKELSBOEL 0 DK ER ural 5 0 ELLUND Fgure 8.11: Flows n the transmsson system. Fgure 8.11 shows the flows durng one tme perod of a smulaton. The fgure contans too much nformaton to be of practcal use n ths scale, but s ncluded to provde an overvew of how the flow soluton can be vsualzed. The teal and pnk labels are flow rates at the entry and ext of each ppe secton. The red labels ndcate producton, mport or extracton from storage facltes. The blue labels represent export or njecton nto storage facltes. Fgure 8.12 shows an enhanced mage of the transmsson system n Western Jutland. Here pressure values are also labeled. Green labels ndcate nodal-pressure whle yellow labels ndcate mean ppe pressure. On ths specfc smulaton t s nterestng to note the flow values n the parallel ppe sectons. If the system was runnng far from full capacty, one would expect the lnear program solver to suggest flow n only one of the parallel ppes besdes supplyng the few small M/R statons located on the northern ppelne. If, on the other hand, the system was at full capacty, the flow would be dvded evenly between the two ppes (agan takng the ntermedares nto account). As such, one s led to the assumpton that the flow soluton s not at full capacty, but one would not be able to supply the same amount wthout the redundant ppelne. 8.8 Area Dstrbuton Other results can also be specfed wth detaled geographcal resoluton. Ths capacty s very useful, f not analyzng results n detal, at least for gettng a sense of what s gong on. A complete absence of gas fred generaton, n a larger area for example, mght alert one to a capacty ssue, or modelng error, that one would msnterpret as a sgn of general reducton n gas fred transformaton, should one only consder the annually aggregated results. Only one addtonal example s demonstrated to avod cloudng the chapter wth results. Fgure 8.13 shows fuel consumpton by transformaton technology wth area level resoluton. Note that the scalng of abscssa are ndependent between areas, so nothng can be read from ths pcture concernng the scale of nvestments.

80 8.9. SUMMARY NOERSKOV gas producton flow values node pressure values NYBRO HELLE VARDE flow n parallel ppes EGTVED mean pressure n ppes Fgure 8.12: Flow and pressure n a porton of the transmsson system. AALBORG DK WR ural ELLIDSHOEJ HAVERSLEV LITORUP VIBORG KARUP HERNING BRANDE NOERSKOV LYNGE HELLE NYBRO VARDE EGTVED LILBALLE TAULOV LYNGSODDE STANDST SKAERBAEK POTTEHUSE MIDDELFART BILLESBOELLE KOELBJERG NYBORG LISELSKAER HOEJBY ULLERSLEV MAALOEV BROENDBY VALLENSBAEK HVIDOVRE TORSLUNDE AMAGERF AVEDOERE VESTAMAGER HCOERSTED SYDHAVNEN STENLILLE KARLSLUNDE DRAGOER OERESUND SOROE KOEGE KONGSMARK SLAGELSE RINGSTED TERKELSBOEL DK ural ER ELLUND Fgure 8.13: Investments dstrbuted by area. 8.9 Summary Hopefully ths chapter has succeeded n llustratng how nterestng results can be nterpreted from the model output. There s an abundance of output not presented, whch has equally nterestng potental for nterpretaton. The trck s not as much how one can fnd,

81 8.9. SUMMARY 77 but rather to determne what one s lookng for.

82 CHAPTER 9 Demonstraton Cases In ths chapter the model s used to analyze three demonstraton cases. The purpose s to llustrate how the model can be used to evaluate relevant energy polcy ssues and ssues pertanng to systems analyss. Input varatons are descrbed where appled, and a selecton of output deemed approprate s presented and dscussed. These nterpretatons are ntended to be nspratonal wth respect to model applcaton, rather than judgemental wth regard to the ssues descrbed n the demonstraton cases. The selected cases are: 1. De-centralzed CHP facltes operatng on market terms and ther mpact on the natural gas system. 2. Prospectve reserves n the North Sea and the mplcaton nvestments n heat and power. 3. Market prces for CO 2 allowances and ther mpact on emssons from the heat and power sectors. 9.1 De-centralzed Combned Heat and Power All electrcty generators not located at one of the 15 central plant locatons are by defnton de-centralzed. In 1986 t was decded to subsdze the converson of a number of dstrct heatng bolers nto CHP unts [25]. Subsequently, n 1990 the decson to convert all larger dstrct heatng bolers to ether CHPs or bomass fred bolers was taken [26]. The ncentve was that electrcty produced by de-centralzed co-generaton unts could be sold at a fxed and favorable feed-n tarff wth prorty over the large central plants. The feed-n tarff became known as the 3-stage tarff due to dfferental tarff levels for the three stages of low, hgh and peak load generaton. As of January 1st 2005, ths structure has been replaced by subsdy on the heat sde of the energy mx for larger de-centralzed producers. The rest are assumed to come on smlar terms wthn a short tme frame. Eventually all CHPs wll lkely compete on the electrcty market, wth the possble excepton of muncpal waste fred facltes.

83 9.1.1 Fxed Tarff vs. the Spot Market 79 Fgure 9.1: Left: The three stage tarff at dfferent transformer levels and for dfferent seasons. Rght: The varaton n spot prces for the frst week of January Fxed Tarff vs. the Spot Market There s a huge dfference between producng aganst known prces and bddng aganst an uncertan spot market. The fxed tarff case one can plan aganst an expected heat demand n the producton optmzaton problem becomes a farly determnstc problem nvolvng the decson of when to producton on the CHP unt, when to use a heat boler, and how to use the heat storage faclty, n order to generate a surplus for the heat customers. Now they are faced wth the challenge of bddng aganst a hghly volatle and uncertan market prce. Fgure 9.1 llustrate the 3-stage tarff sde by sde wth a sample week on the spot market. The uncertanty s passed on to the natural gas system, whch provdes the fuel for most de-centralzed CHPs. Here t s especally nterestng how the freely operatng CHPs wll respond to the spot market n the peak hours n the gas system. If the electrcty prce s hgh smultaneously wth heat demand, there s a potental that the last free capacty n the transmsson system, could come under pressure Technologes One unfortunate property of the model s that there s no dstncton between central and decentralzed electrcty generators. Ths results from an unfortunate lack of foresght durng the categorzaton and aggregaton process. Ths data generaton process was unfortunately ncredbly tme consumng and s only partly automated. Once all CHPs operate on market terms ths wll be less of a problem, as the need for dstncton s decreased. For now, the strategy s to consder the technologes most common for gas fred CHPs. These are descrbed n the followng sectons. Engne Drven Plants Engne drven plants are the most common form of de-centralzed CHP. Bascally a combuston engne powers a turbne to generate electrcty. Heat recovery systems enable the use of heat from coolng water, lubrcants and exhaust n the dstrct heatng network. Engnes are generally ether natural gas or desel fred. These plants account for about 85% n terms of number of nstalled plants, and 45% of the electrcty generaton capacty n Denmark [13].

84 9.1.3 Dstrbuton of Intal Capacty 80 Steam Turbnes A steam turbne s bascally a boler, whch heats water formng steam. Ths steam s pressurzed over the length of a turbne drvng a generator. Tradtonally ext steam has been cooled by an ntake of seawater, but by replacng ths wth a dstrct heatng condenser, the heat s transferred to the dstrct heatng system. Steam turbnes can ether be back pressure unts, whch supply electrcty and heat at a constant rato, or extracton unts gvng added flexblty from a full back-pressure mode to a full condensng. The extracton unt type of steam turbne combnes the seawater condenser wth the dstrct heatng condenser to get the desred rato between electrcty and heat. Approxmately 6% of de-centralzed CHP unts n Denmark are steam turbnes, but beng generally larger unts they cover near 20% of nstalled electrcty generaton capacty[13]. Gas Turbnes A gas turbne can be coupled wth a generator to generate electrcty. Agan heat can be recovered from exhaust gas. The advantage of gas turbnes s that heat can be recovered wthout reducng the electrcty producton effcency. However, gas turbnes have low regulatory ablty n that effcent s drastcally reduced when operatng below nomnal effect[13]. Combned Cycle Technology The combned cycle technology encompasses the combnaton of a gas turbne and a steam turbne. Instead of the steam turbne beng suppled wth steam from a boler, t s drven by the hgh pressure exhaust from the gas turbne. Combned cycle facltes are usually of a rather large sze and account for 6% of facltes n Denmark and an electrcty generaton capacty share of 35%[13] Dstrbuton of Intal Capacty Among the technologes whch make up the de-centralzed power generaton capacty the dstrbuton of ntal capacty s llustrated on fgure 9.2. Ths dstrbuton results from the aggregaton and categorzaton of heat and power generators lsted n the energy producers census. The capacty shares devate notably from those presented n secton Ths s unfortunately the result of naccurate methods for categorzaton, as some of the capacty llustrated here s not de-centralzed. The technologes were lsted n table Model Results An nterestng result of the ntal smulaton was the ncreased emphass on gas power. Ths mpled nvestments n large-scale combned cycle facltes. The relatvely large capacty share of gas engnes s not put to much use by the model. One could nterpret that these are beng replaced by more effcent technologes. Unfortunately the model does not accurately descrbe the advantages of gas engnes. Normally of small scale they are able to use the heat demand of a relatvely small dstrct heatng system, however, ths advantage s not evdent snce the geographcal resoluton does not acheve a dstrbuton level. An estmaton of the hstorc consumpton load of CHPs s generated by dstrbutng annualzed data for gas consumpton by de-centralzed CHPs across a load duraton varaton curve

85 9.1.4 Model Results 81 Fgure 9.2: Dstrbuton of electrcty generaton capacty among de-centralzed technologes as mplemented n the model. based on the electrcty generaton varaton for CHPs publshed on the Elkraft System web ste ( The load duraton curve of the natural gas fred technologes descrbed above s presented alongsde the hstorc combned CHP consumpton on fgure 9.3. It s clear to see that the two depctons are ncommensurable. The smulated results obvously nclude a larger than expected share of central generaton capacty. Hstorc Profle for CHP Gas Consumpton Smulated Load for De centralzed CHPs knm3/h knm3/h Hours hours Fgure 9.3: Left: Estmated hstorc consumpton of natural gas for de-centralzed CHPs. Rght: Smulated consumpton from de-centralzed CHPs An alternatve method of analyss whch attempts to salvage the case by consderng only the consumpton patterns of the ndvdual plants. A load duraton curve s generated for each technology ndvdually n relaton to the maxmum load generated by that technology. The load duraton profles are the normalzed n relaton to the peak load hour of that technology. For the hstorcal profle the hghest peak value s leveled as ths lkely represents a specfc extreme ncdent. Now the varaton s normalzed n relaton to the hghest remanng peak. Ths s compared wth the load varaton of the smulated technologes on fgure 9.4. The relatve load dstrbuton of the ndvdual technologes n relaton to the hstorc load of the de-centralzed CHPs can be vewed n lght the exstng capacty and share of the generaton mx.

86 9.2. PROSPECTIVE NATURAL GAS RESERVES CHP Consumpton Varaton Compared to Hstorc Varaton GTLS B06 NG GTMD B06 NG GTMN B06 NG CCLA E07 NG CCSM B07 NG GE B08 NG ST B8 NG 0.7 Share of peak hour load Hours Fgure 9.4: The hstorc varaton profle compared wth varaton of smulated technologes. Analyss of the demonstraton case s nconclusve. Gven a better aggregaton of capacty data, t would be possble to derve more relevant results. The mportant ssue of scalng s lost, and thus t s not possble to determne what happens from de-centralzed CHPs n the peak hours. Ths s essentally what s nterestng from the gas systems perspectve. The case does, however, demonstrate some of the possbltes for performng analyss and selectng amongst the model output n order to shed lght on a partcular ssue. 9.2 Prospectve Natural Gas Reserves The rapd depleton of domestc reserves for natural gas s not a certanty as mentoned n chapter 7. The Dansh Energy Authorty operates wth an expectaton of prospectve dscoveres, whch can extend natural gas producton some years nto the future. Ths was llustrated on fgure 7.4. It s now consdered how ths prospectve contrbuton would nfluence the energy stuaton. The results presented n chapter 8 ndcated the rather obvous fact that when natural gas supples were depleted and mport optons lmted, the market moved towards alternatve technologes for heat and power generaton. The dfference between the orgnal smulaton descrbed n chapter 8 and the followng s only the domestc supply. However, many other stuatons could be modeled. Other possble supply optons could be the mplementaton of the Nordc Gas Rng project. It remans a possblty that the BGI consortum wll construct a transmsson ppelne connectng Germany wth Denmark and Sweden. The ppe s expected to branch out to Skåne and Zealand and make landfall at the Avedøre plant. The ntenton of ths case s to demonstrate that alternatve supply stuatons can be descrbed. Not to provde thorough analyss of ther precs mpact. Therefore, only the prospectve ncrease n North Sea wll actually be mplemented n ths project (also due to the unfortunately rather long model executon tmes as mentoned n chapter 7). Fgure 9.5 shows the domestc fuel consumpton by heat and power technologes gven the new supply stuaton on the left and the orgnal settng on the rght. One can see when comparng wth the orgnal, where t was prevously necessary to convert to coal and straw towards the end of the smulaton perod. Now gas power sustans a large share n 2015 but subsequently s stll lost from heat and power generaton.

87 9.3. EMISSIONS OF CO Fuel Consumpton by Technologes 140 Fuel Consumpton by Technologes TWh TWh Years Years Fgure 9.5: Fuel consumpton by technologes. Left: Gven addtonal supply from the North Sea. Rght: Gven only confrmed reserves. 9.3 Emssons of CO 2 As a consequence of the adopton of the Kyoto Protocol for reducton of greenhouse emssons, quotas of CO 2 have been mplemented. Emsson allowances are tradable and ths can be done on the Nord Pool power exchange. The clearng prce of EUA (European Unon Allowances) s publshed on Currently EUAs trade at around EURO/ton. 2.4 x 104 Emssons per Year CO x 104 Emssons per Year CO tonnes tonnes Years Years Fgure 9.6: CO 2 emssons. Left: Wth prced CO 2 allowances. Left: Wthout restrctons on emssons Implementaton There are two ways to ncte the model to reduce emssons of CO 2. One can mplement mpose an emsson quota explctly. Alternatvely one can assgn a prce to emssons allowances. The last opton s selected. Ths s done as the mpact on soluton tme should be mnmal as addtonal costs n the objectve functon have less mpact than addng constrants contanng a summaton over a large number of varables. Quotas would naturally be attached to all generaton varables across all tme perods. Subsequently t wll be possble to evaluate whether the market prce for emsson allowances s reasonable. Actually the CO 2 prce has been ncluded n all scenaros so far. Consequently