Outline Natural gas, market, players and roles GAMS 1. A stochastic capacity expansion and equilibrium model for the global natural gas market

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1 Thu March th,, F A stochastc capact expanson and eulbrum model for the global natural gas market Ruud Eggng, Wnterschool, March Ruud Eggng March, page Outlne Natural gas, market, plaers and roles GAMS Capact expansons GAMS BREAK 5 mn Uncertant GAMS 3 Stoch MCP Benders for Stoch MCP BREAK/ GAMS 4 Ruud Eggng March, page Man sources Natural Gas Eggng, PhD Thess, IEA 5 Energ Statstcs Manual, BP Statstcal Revew of World Energ, / Gabrel, Fuller,. A Benders Decomposton Method for Solvng Stochastc Complementart Problems wth an Applcaton n Energ. Computatonal Economcs 35 (more on natural gas: Ruud Eggng March, page 3 Ruud Eggng March, page 4 Natural gas producton & processng Natural gas flows Ruud Eggng March, page 5 Source: IEA 5, fg. 3.3, p 58 Ruud Eggng March, page 6 (C) Ruud Eggng,

2 Thu March th, World energ consumpton NG ~% Developments and challenges Globall rsng demand Locall depletng reserves Increasng number of plaers nvolved Increasng complext of market and trade relatons Increasng uncertantes and rsks Interpla wth envronmental ssues Conflctng objectves wthn and between countres BP Ruud Eggng March, page 8 In sum Complext Uncertant A Stochastc Global Gas Market Model Mult perod stochastc MCP wth endogenous expansons for transport and storage 98% of worldwde consumpton and producton Market power Ppelne and LNG suppl chans Contracts Demand seasonalt Decson varables: varous operatng levels and capact expansons Ruud Eggng March, page 9 Ruud Eggng March, page Producton Trade State owned vs. prvatel owned ExxonMobl, RD Shell, GazProm, Qatargas, Vertcal/Horzontal ntegraton Ol, electrct,.. Trade, storage, Representaton: Geographcal regons Produce and sell gas Varng ownershp and ntegraton GazExport, Pepco, RWE, GDF SUEZ, Washngton Gas Representaton: Geographcal regons Purchase gas from producers Sell gas to marketers Storage and transportaton servces Market power Ruud Eggng March, page Ruud Eggng March, page (C) Ruud Eggng,

3 Thu March th, Standard Trader Representaton Optonal Cartel Representaton Trader Cartel Trader Trader Trader Trader Ruud Eggng March, page 3 Ruud Eggng March, page 4 Luefacton and regasfcaton 6F: m 3 LNG ~ 6 m 3 NG Cheaper or onl opton Luefers E.g., GDF Suez, Qatargas Regasfers E.g., GDF Suez, Chenere Energ Ruud Eggng March, page 5 Storage Operators Ownershp Access regmes Roles: Demand seasonalt, strategc storage, speculaton,... Representaton: regulated servce provder Injecton Aprl Ma June Jul August September Extracton October November December Januar Februar March Ruud Eggng March, page 6 Transport network Suppl chan Internatonal/domestc Ownershp Access regmes Trader Network arcs Marketer Representaton Trans Sstem Operator All transportaton arcs Regulated servce provder Regulated fee + congeston charge Storage faclt Ruud Eggng March, page 7 Ruud Eggng March, page 8 (C) Ruud Eggng, 3

4 Thu March th, Gas market structure Countr Countr 3 C C3 Ppelne Transt countres Trader T Sectors K,,3 S Storage M Marketer T LNG Node L T LNG trade T Regas Node R3 T K,,3 M3 T3 S3 Tme to pla The assgnment at the end of Monda s lecture Ruud Eggng March, page Olgopol Suppler : Inverse demand : Ppelne operator : max s. t. s. t. f cap c prod INT SLP max ppe f ppe ppe f ppe cap Market clearng for ppelne capact : ppe ppe prod ppe ppe Soluton olgopol Suppler : Ppelne operator : Market clearng : f f ppe prod ppe ppe INT SLP SLP prod ppe c cap cap prod, ppe ppe ppe f ppe ppe free n sgn Ruud Eggng March, page Ruud Eggng March, page Olgopol n GAMS Olgopol n GAMS GAMS Fle Thu...gms Walk through Fll out the statonart condtons (see slde ) ~5 mnutes Ruud Eggng March, page 3 Ruud Eggng March, page 4 (C) Ruud Eggng, 4

5 Thu March th, Some assumptons Perfect foresght? Perfect nformaton! Rsk neutralt Open loop: p all decsons for all stages at start Vs. closed loop: feedback strateges : at ever stage former decsons and outcomes taken nto account when choosng a course of acton Assumptons open loop more restrctve but generall mathematcall tractable Investments n MCP outlne Agent wth perfect foresght Decde on sales n ear SALES capact expansons Sellng prce dscount rate (both exogenous) Convex cost curve c( SALES ) Intal capact CAP Investment costs per unt b Upper bounds on expansons Ruud Eggng March, page 5 Ruud Eggng March, page 6 Investments Formulaton L SALES, lm max SALES c ( SALES ) b Y s.. t SALES CAP SALES ' ' Investments KKT c( SALES) L SALES SALES b ' ' CAP ' SALES ' Ruud Eggng March, page 7 Ruud Eggng March, page 8 GAMS Capact expanson Code up the model on slde 8, for a two perod model, monopol plaer, current capact 5, nvestment costs /unt, Inverse Demand curve When ou re done, tme for a break ~5 mnutes, ncludng break Ruud Eggng March, page 9 Ruud Eggng March, page 3 (C) Ruud Eggng, 5

6 Thu March th, Demand Uncertant Smple Stochastc Investment Problem : c()=. Current Cap = 5 Two perods Perod : Cap expanson I: $/unt Perod : 5% 5%: p= OR p= Invest how much to maxmze exp proft? p =-,? p =-, Ruud Eggng March, page 3 Demand Uncertant Smple Stochastc Investment Problem max,, s. t. 5 5 Ruud Eggng March, page 3 Stochastc Investments KKT 5 5 Mnmzaton! GAMS! Code up the model on slde 33, usng a set of scenaros ~5 mnutes (see former exercse!) Ruud Eggng March, page 33 Ruud Eggng March, page 34 GAMS tpe t all up GAMS, usng sets and varables Ruud Eggng March, page 35 Ruud Eggng March, page 36 (C) Ruud Eggng, 6

7 Thu March th, Stochastc Natural Gas Market Model Opt problem Extensve form stochastc MCP Consder multple futures when makng capact expanson decsons Addtonal assumpton: rsk neutralt Maxmze expected profts Include all consdered futures and assgn probabltes Ruud Eggng March, page 37 Ruud Eggng March, page 38 KKT Arc operator Opt problem Ruud Eggng March, page 39 Ruud Eggng March, page 4 Arc operator KKT Plaers n the model V Arc network operator V Trader See Thess Storage Operator See Thess Market clearng condtons, next Et cetera Ruud Eggng March, page 4 Ruud Eggng March, page 4 (C) Ruud Eggng, 7

8 Thu March th, Market clearng condtons Scalablt ssues Scenaro tree wth four scenaros Ruud Eggng March, page 43 Capact addtons n 5: four futures; -: two futures Ruud Eggng March, page 44 Stochastc gas market model Remarks Two uncertan events s not so much Doublng of model sze, calc tme: 5 tmes as bg When uncertant s n far future, the mpact s largel dscounted awa Hedgng affects tmng and szes, but man results close to averages Some detaled developments and results dffer due to nterpla of tmng, hedgng and game theoretc approach Ruud Eggng March, page 45 Benders Decomposton Outlne Some varables make the problem hard. Fxng these: remanng problem eas Decompose problem n two parts Dffcult varables Master Problem (MP) Remanng varables Sub Problems (SP) Iteratvel MP and SP are solved MP: fxng values SP: feasble soluton + nfo to mprove MP Ruud Eggng March, page 46 Org problem: block structure -Q >= -CAP -Q >= -CAP -Q 3 >= -CAP + 3 -Q 4 >= -CAP Org problem: complcatng var -Q >= -CAP -Q >= -CAP -Q 3 >= -CAP + 3 -Q 4 >= -CAP Ruud Eggng March, page 47 Complcatng varables Ruud Eggng March, page 48 (C) Ruud Eggng, 8

9 Thu March th, Sub problems n BD -Q >= -CAP -Q >= -CAP - -Q 3 >= -CAP - - -Q 4 >= -CAP Subproblems separate b block Benders Decomposton Loop INIT: Convergence Gap = +INF WHILE(ConvGap >Treshold) DO: Solve MP Fnd suggested values for fxng varables Solve SPs Fx vars accordng to last MP Soln Collect nfo to update MP Update ConvGap Pass nfo from SPs to MP b Addng a Benders Cut Ruud Eggng March, page 49 Ruud Eggng March, page 5 BD for Smple Stochastc Investment Problem Defne MP: Defne SP(): Defne SP(): INIT LB= INF, UB=+INF MP: I=, = nf SP: (,z, ) =(5, ½, ) (,z, ) =(5, 37½, 5) Add Cut to MP mn I, st.. I I, mn z = ( -), st.. 5 I, I I k ( ) mn z = ( k - ), st.. 5 I, I I ( ) Ruud Eggng March, page 5 Benders Cuts Ever cut lmts MP soluton space, cuttng off nfeasble solutons Uses Dual Prces (, ) of the SPs the fxed MP soluton I k Provde lnear approxmaton how aggregate SP objectve would change from current fxed soluton Approxmaton overestmate k k k k z z I I 5 5I Ruud Eggng March, page 5 Addng Frst Benders Cut # Z I -nf -nf #: (I,,z) Second MP soluton # Z I -nf -nf : (,-nf,-nf) Cap expans : (,-nf,-nf) Cap expans : (,-,-8) Ruud Eggng March, page 53 Ruud Eggng March, page 54 (C) Ruud Eggng, 9

10 Thu March th, Second teraton SP solutons SP(,): SP(,): mn z = ( -), st.. 5 (,z, ) =( 5, ½,) (,z, )=(, 5,) Add Cut to MP: 6½ mn z = ( - ), st.. 5 Addng Second Benders Cut -4 # Z I -nf -nf : (,-nf,-nf) : (, --8,) Ruud Eggng March, page 55 Ruud Eggng March, page 56 Thrd MP soluton -4 # Z I -nf -nf : (.5,-6.5,-57.5) : (,-nf,-nf) : (,-,-8) And so on -4 9: (3.,-6.5,-54.5) cut cut cut 3 cut 4 cut 5 cut 6 cut 7 cut 8 cut Capact expanson # Z I -nf -nf Ruud Eggng March, page 57 Ruud Eggng March, page 58 Approach BD and cuts orgnall for optmzaton BD for Stoch MCP (Gabrel & Fuller, ) general, two stage, electrct market (stlzed) small number plaers, man scenaros market power n SP: MP are LCP, SP are LCP G&F derve alternatve cut to be used for mperfectl compettve lower level problems Benders Decomposton Benders (96) Orgnall Mxed Integer Problems Fuller and Chung (7): VIs Gabrel and Fuller (8): Stoch MCP two stage small#plaers man scenaros Ruud Eggng March, page 59 Ruud Eggng March, page 6 (C) Ruud Eggng,

11 Thu March th, Notaton Tradtonal Benders Optmalt Cuts Ruud Eggng March, page 6 Ruud Eggng March, page 6 Cut accordng to Gabrel and Fuller () Cut APPLIED Frst stage varables and Approxmated mpact approxmated mpact second stage varables - Ruud Eggng March, page 63 Ruud Eggng March, page 64 Implementaton Mult stage GAMS, GB RAM, dual core x. GHz Convergence analss A B Model perods 4 6 Scenaros 4 4 Scenaro nodes 9 Num capact expanson Total num varables 7, 47,373 Full MCP calc tme (seconds) 63,5 VI-MCP Net calc tme (seconds) 67,36 Num teratons Ruud Eggng March, page 65 Ruud Eggng March, page 66 (C) Ruud Eggng,

12 Thu March th, Convergence analss contnued C (*) D (*) E (*) Model perods Scenaros Scenaro nodes Num expanson var,87,87,35 Total num varables 77,77 67,55 7,48 Full MCP calc tme 3,853 3,5 8,679 Num teratons feasble MP calc tme & nfeasble MP calc tme % 96 3 feasble SP calc tme & 4,934 4,576 4,373 nfeasble SP calc tme % 7 6 Num nfeasble MP Num nfeasble SP Ruud Eggng March, page 67 A B C (*) D (*) E (*) Model perods Scenaros Convergence analss Scenaro nodes Num expanson var ,87,87,35 Total num varables 7, 47,373 77,77 67,55 7,48 Full MCP calc tme 63,5 3,853 3,5 8,679 VI-MCP Net calc tme^ 67,36 5,57 5, 5,3 Num teratons VI-MCP Gross calc tme 5 3,684 5,7 5,7 3,5 feasble MP calc tme & nfeasble MP calc tme % feasble SP calc tme & 59,847 4,934 4,576 4,373 nfeasble SP calc tme % 7 6 Num nfeasble MP Num nfeasble SP Convergence crteron Expans Expans MP nfeas MP nfeas MP nfeas Ruud Eggng March, page 68 Benders n GAMS Same problem wth monopol suppler facng a hgh and a low demand scenaro GAMS small BD Note: numercal devatons, but no complcatons et Conclusons Makng use of Benders cuts from (Gabrel and Fuller, ) allows easer dervaton MP and SP GAMS needs much tme for model generaton (unless t would be possble to keep models n memor) Implementatons should use software that allows effcent fle processng and model generaton Large potental to reduce soluton tmes of large scale stochastc MCP (consderng parallel processng) Numercal complcatons (!!!) Ruud Eggng March, page 69 Ruud Eggng March, page 7 Benders for Stoch MCP? GAMS Speculaton: Fewer tme perods Smaller frst stage problem Ruud Eggng March, page 7 Package: AssgnmentsOppdal.docx Thu_4_... _ Gms Open t. Two data fles, the man fle, cournot parameter Dan& Frtz! Optonal: maxmum expanson constrant (derve and adjust KKT and mplement n GAMS) Untl dnner or when our are done Ruud Eggng March, page 7 (C) Ruud Eggng,