Documentation for the TIMES Model PART I

Transcription

1 Energy Technology Systems Analyss Programme Documentaton for the TIMES Model PART I Aprl 2005 Authors: Rchard Loulou Uwe Remne Amt Kanuda Antt Lehtla Gary Goldsten 1

2 General Introducton Ths documentaton s composed of thee Parts. Part I comprses eght chapters consttutng a general descrpton of the TIMES paradgm, wth emphass on the model s general structure and ts economc sgnfcance. Part I also ncludes a smplfed mathematcal formulaton of TIMES, a chapter comparng t to the MARKAL model, pontng to smlartes and dfferences, and chapters descrbng new model optons. Part II s a comprehensve reference manual ntended for the techncally mnded modeler or programmer lookng for an n-depth understandng of the complete model detals, n partcular the relatonshp between the nput data and the model mathematcs, or contemplatng makng changes to the model s equatons. Part II ncludes a full descrpton of the sets, attrbutes, varables, and equatons of the TIMES model. Part III descrbes the GAMS control statements requred to run the TIMES model. GAMS s a modelng language that translates a TIMES database nto the Lnear Programmng matrx, and then submts ths LP to an optmzer and generates the result fles. In addton to the GAMS program, two model nterfaces (VEDA-FE and VEDA- BE) are used to create, browse, and modfy the nput data, and to explore and further process the model s results. The two VEDA nterfaces are descrbed n detal n ther own user s gudes. 2

3 PART I: TIMES CONCEPTS AND THEORY 3

4 TABLE OF CONTENTS FOR PART I 1 Introducton to the TIMES model A bref summary Usng the TIMES model The Demand Component of a TIMES scenaro The Supply Component of a TIMES Scenaro The Polcy Component of a TIMES Scenaro The Techno-economc component of a TIMES Scenaro The basc structure of the TIMES model Tme horzon Decouplng of data and model horzon The RES concept Overvew of the TIMES attrbutes Parameters assocated wth processes Parameters assocated wth commodtes Parameters attached to commodty flows nto and out of processes Parameters attached to the entre RES Process and Commodty classfcaton Economc ratonale of the TIMES modelng approach A bref classfcaton of energy models Top-Down Models Bottom-Up Models Recent Modelng Advances The TIMES paradgm A technology explct model Mult-regonal feature Partal equlbrum propertes Lnearty Maxmzaton of total surplus: Prce equals Margnal value Compettve energy markets wth perfect foresght Margnal value prcng Proft maxmzaton: the Invsble Hand A smplfed descrpton of the TIMES optmzaton program Indces Decson Varables TIMES objectve functon: dscounted total system cost Constrants Capacty Transfer (conservaton of nvestments) Defnton of process actvty varables

5 4.4.3 Use of capacty Commodty Balance Equaton: Defnng flow relatonshps n a process Lmtng flow shares n flexble processes Peakng Reserve Constrant (tme-slced commodtes only) Constrants on commodtes User Constrants Representaton of ol refnng n MARKAL New sets and parameters New varables New blendng constrants Lnear Programmng complements A bref prmer on Lnear Programmng and Dualty Theory Basc defntons Dualty Theory Senstvty analyss and the economc nterpretaton of dual varables Economc Interpretaton of the Dual Varables Reduced Surplus and Reduced Cost A comparson of the TIMES and MARKAL models Smlartes TIMES features not n MARKAL Varable length tme perods Data decouplng Flexble tme slces and storage processes Process generalty Flexble processes Investment and dsmantlng lead-tmes and costs Vntaged processes and age-dependent parameters Commodty related varables More accurate and realstc depcton of nvestment cost payments Clmate equatons MARKAL features not n TIMES Elastc demands and the computaton of the supply-demand equlbrum Theoretcal consderatons: the Equvalence Theorem Mathematcs of the TIMES equlbrum Defnng demand functons Formulatng the TIMES equlbrum Lnearzaton of the Mathematcal Program Calbraton of the demand functons Computatonal consderatons Interpretng TIMES costs, surplus, and prces The Lumpy Investment opton Formulaton and Soluton of the Mxed Integer Lnear Program Important remark on the MIP dual soluton (shadow prces) Endogenous Technologcal Learnng (ETL) The basc ETL challenge

6 8.2 The TIMES formulaton of ETL The Cumulatve Investment Cost Calculaton of break ponts and segment lengths New varables New constrants Objectve functon terms Addtonal (optonal) constrants Clustered learnng Learnng n a Multregonal TIMES Model Endogenous vs. Exogenous Learnng: Dscusson

7 1 Introducton to the TIMES model 1.1 A bref summary TIMES (an acronym for The Integrated MARKAL-EFOM 1 System) s an economc model generator for local, natonal or mult-regonal energy systems, whch provdes a technology-rch bass for estmatng energy dynamcs over a long-term, mult-perod tme horzon. It s usually appled to the analyss of the entre energy sector, but may also appled to study n detal sngle sectors (e.g. the electrcty and dstrct heat sector). Reference case estmates of end-use energy servce demands (e.g., car road travel; resdental lghtng; steam heat requrements n the paper ndustry; etc.) are provded by the user for each regon. In addton, the user provdes estmates of the exstng stock of energy related equpment n all sectors, and the characterstcs of avalable future technologes, as well as present and future sources of prmary energy supply and ther potentals. Usng these as nputs, the TIMES model ams to supply energy servces at mnmum global cost (more accurately at mnmum loss of surplus) by smultaneously makng equpment nvestment and operatng, prmary energy supply, and energy trade decsons, by regon. For example, f there s an ncrease n resdental lghtng energy servce relatve to the reference scenaro (perhaps due to a declne n the cost of resdental lghtng, or due to a dfferent assumpton on GDP growth), ether exstng generaton equpment must be used more ntensvely or new possbly more effcent equpment must be nstalled. The choce by the model of the generaton equpment (type and fuel) s based on the analyss of the characterstcs of alternatve generaton technologes, on the economcs of the energy supply, and on envronmental crtera. TIMES s thus a vertcally ntegrated model of the entre extended energy system. The scope of the model extends beyond purely energy orented ssues, to the representaton of envronmental emssons, and perhaps materals, related to the energy system. In addton, the model s admrably suted to the analyss of energyenvronmental polces, whch may be represented wth accuracy thanks to the explctness of the representaton of technologes and fuels n all sectors. In TIMES lke n ts MARKAL forebear the quanttes and prces of the varous commodtes are n equlbrum,.e. ther prces and quanttes n each tme perod are such that the supplers produce exactly the quanttes demanded by the consumers. Ths equlbrum has the property that the total surplus s maxmzed. 1.2 Usng the TIMES model The TIMES model s partcularly suted to the exploraton of possble energy futures based on contrasted scenaros. Gven the long horzons smulated wth TIMES, the 1 MARKAL (MARket ALlocaton model, Fshbone et al, 1981, 1983, Berger et al. 1992) and EFOM (Van Voort et al, 1984) are two bottom-up energy models whch nspred the structure of TIMES. 7

8 scenaro approach s really the only choce (whereas for the shorter term, econometrc methods may provde useful projectons). Scenaros, unlke forecasts, do not pre-suppose knowledge of the man drvers of the energy system. Instead, a scenaro conssts of a set of coherent assumptons about the future trajectores of these drvers, leadng to a coherent organzaton of the system under study. A scenaro bulder must therefore carefully test the assumptons made for nternal coherence, va a credble storylne. In TIMES, a complete scenaro conssts of four types of nputs: energy servce demands, prmary resource potentals, a polcy settng, and the descrptons of a set of technologes. We now present a few comments on each of these four components The Demand Component of a TIMES scenaro In the case of the TIMES model demand drvers (populaton, GDP, famly unts, etc.) are obtaned externally, va other models or from accepted other sources. As one example, the TIMES global model constructed for the EFDA 2 used the GEM-E3 3 general equlbrum model to generate a set of coherent (total and sectoral) GDP growth rates n the varous regons. Note that GEM-E3 tself uses other drvers as nputs n order to derve GDP trajectores. These GEM-E3 drvers consst of measures of technologcal progress, populaton, degree of market compettveness, and a few other perhaps qualtatve assumptons. For populaton and household projectons, both GEM-E3 and TIMES used the same exogenous sources (IPCC, Nakcenovc 2000, Moomaw and Morera, 2001). Other approaches may be used to derve TIMES drvers, whether va models or other means. For the EFDA model, the man drvers were: Populaton, GDP, GDP per capta, number of households, and sector GDP. For sectoral TIMES models, the demand drvers may be dfferent dependng on the system boundares. Once the drvers for a TIMES model are determned and quantfed the constructon of the reference demand scenaro requres computng a set of energy servce demands over the horzon. Ths s done by choosng elastctes of demands to ther respectve drvers, n each regon, usng the followng general formula: Elastcty Demand = Drver As mentoned above, the demands are provded for the reference scenaro. However, when the model s run for alternate scenaros (for nstance for an emsson constraned case, or for a set of alternate technologcal assumptons), t s lkely that the demands wll be affected. TIMES has the capablty of estmatng the response of the demands to the changng condtons of an alternate scenaro. To do ths, the model requres stll another set of nputs, namely the assumed elastctes of the demands to ther own prces. TIMES 2 EFDA: European Fuson Development Agreement 3 GEM-E3 General Equlbrum Model for Economy, Energy and Envronment 8

9 s then able to endogenously adjust the demands to the alternate cases wthout exogenous nterventon. In fact, the TIMES model s drven not by demands but by demand curves. To summarze: the TIMES demand scenaro components consst n a set of assumptons on the drvers (GDP, populaton, households) and on the elastctes of the demands to the drvers and to ther own prces The Supply Component of a TIMES Scenaro The second consttuent of a scenaro s a set of supply curves for prmary energy and materal resources. Mult-stepped supply curves can be easly modeled n TIMES, each step representng a certan potental of the resource avalable at a partcular cost. In some cases, the potental may be expressed as a cumulatve potental over the model horzon (e.g. reserves of gas, crude ol, etc), as a cumulatve potental over the ressource base (e.g. avalable areas for wnd converters dfferentated by veloctes, avalable farmland for bocrops, roof areas for PV nstallatons) and n others as an annual potental (e.g. maxmum extracton rates, or for renewable resources the avalable wnd, bomass, or hydro potentals). Note that the supply component also ncludes the dentfcaton of tradng possbltes, where the amounts and prces of the traded commodtes are determned endogenously (wthn any mposed lmts) The Polcy Component of a TIMES Scenaro Insofar as some polces mpact on the energy system, they may become an ntegral part of the scenaro defnton. For nstance, a No-Polcy scenaro may perfectly gnore emssons of varous pollutants, whle alternate polcy scenaros may enforce emsson restrctons, or emsson taxes, etc. The detaled technologcal nature of TIMES allows the smulaton of a wde varety of both mcro measures (e.g. technology portfolos, or targeted subsdes to groups of technologes), and broader polcy targets (such as general carbon tax, or permt tradng system on ar contamnants). A smpler example mght be a nuclear polcy that lmts the future capacty nuclear plants. Another example mght be the mposton of fuel taxes, or of ndustral subsdes, etc The Techno-economc component of a TIMES Scenaro The fourth and last consttuent of a scenaro s the set of techncal and economc parameters assumed for the transformaton of prmary resources nto energy servces. In TIMES, these techno-economc parameters are descrbed n the form of technologes (or processes) that transform some commodtes nto others (fuels, materals, energy servces, emssons). In TIMES, some technologes may be mposed and others may smply be avalable for the model to choose. The qualty of a TIMES model rests on a rch, well developed set of technologes, both current and future, for the model to choose from. The emphass put on the technologcal database s one of the man dstngushng factors of the class of Bottom-up models, to whch TIMES belongs. Other classes of models wll tend to emphasze other aspects of the system (e.g. nteractons wth the rest of the 9

10 economy) and treat the techncal system n a more succnct manner va aggregate producton functons. Remark: two scenaros may dffer n all or n only some of ther components. For nstance, the same demand scenaro may very well lead to multple scenaros by varyng the prmary resource potentals and/or technologes and/or polces, nsofar as the alternatve scenaro assumptons do not alter the basc demand nputs (Drvers and Elastctes). The scenaro bulder must always be careful about the overall coherence of the varous assumptons made on the four components of a scenaro. Organzaton of PART I Chapter 2 provdes a general overvew of the representaton n TIMES of the Reference Energy System (RES) of a typcal regon or country, focusng on ts basc elements, namely technologes and commodtes. Chapter 3 dscusses the economc ratonale of the model, and Chapter 4 presents a streamlned representaton of the Lnear Programmng problem used by TIMES to compute the equlbrum. Chapter 5 contans a comparson of the respectve features of TIMES and MARKAL, ntended prmarly for users already famlar wth MARKAL, whle Chapter 6 descrbes n detal the elastc demand feature and other economc and mathematcal propertes of the TIMES equlbrum. Chapters 7 and 8, respectvely descrbe two model optons: Lumpy Investments (LI), and Endogenous Technologcal Learnng (ETL). 10

11 2 The basc structure of the TIMES model It s useful to dstngush between a model s structure and a partcular nstance of ts mplementaton. A model s structure exemplfes ts fundamental approach for representng and analyzng a problem t does not change from one mplementaton to the next. All TIMES models explot an dentcal mathematcal structure. However, because TIMES s data 4 drven, each (regonal) model wll vary accordng to the data nputs. For example, n a mult-regon model one regon may, as a matter of user data nput, have undscovered domestc ol reserves. Accordngly, TIMES generates technologes and processes that account for the cost of dscovery and feld development. If, alternatvely, user suppled data ndcate that a regon does not have undscovered ol reserves no such technologes and processes would be ncluded n the representaton of that regon s Reference Energy System (RES, see sectons 2.3 and 2.4). Due to ths property TIMES can also be called a model generator that, based on the nput nformaton provded by the modeler, generates an nstance of a model. In the followng, f not stated otherwse, the expresson model s used wth two meanngs: the nstance of a TIMES model or more generally the model generator TIMES. The structure of TIMES s ultmately defned by varables and equatons determned from the data nput provded by the user. Ths nformaton collectvely defnes each TIMES regonal model database, and therefore the resultng mathematcal representaton of the RES for each regon. The database tself contans both qualtatve and quanttatve data. The qualtatve data ncludes, for example, lsts of energy carrers, the technologes that the modeler feels are applcable (to each regon) over a specfed tme horzon, as well as the envronmental emssons that are to be tracked. Ths nformaton may be further classfed nto subgroups, for example energy carrers may be splt by type (e.g., fossl, nuclear, renewable, etc). Quanttatve data, n contrast, contans the technologcal and economc parameter assumptons specfc to each technology, regon, and tme perod. When constructng mult-regon models t s often the case that a technology may be avalable for use n two dstnct regons; however, cost and performance assumptons may be qute dfferent (.e., consder a resdental heat pump n Canada versus the same pece of equpment n Chna). Ths chapter dscusses both qualtatve and quanttatve assumptons n the TIMES modelng system. The TIMES energy economy s made up of producers and consumers of commodtes such as energy carrers, materals, energy servces, and emssons. TIMES, lke most equlbrum models, assumes compettve markets for all commodtes. The result s a supply-demand equlbrum that maxmzes the net total surplus (.e. the sum of producers and consumers surpluses) as wll be fully dscussed n chapters 3 and 6. TIMES may, however, depart from perfectly compettve market assumptons by the ntroducton of user-defned explct constrants, such as lmts to technologcal penetraton, constrants on emssons, exogenous ol prce, etc. Market mperfectons can also be ntroduced n the form of taxes, subsdes and hurdle rates. 4 Data n ths context refers to parameter assumptons, technology characterstcs, projectons of energy servce demands, etc. It does not refer to hstorcal data seres. 11

12 Operatonally, a TIMES run confgures the energy system (of a set of regons) over a certan tme horzon n such a way as to mnmze the net total cost (or equvalently maxmze the net total surplus) of the system, whle satsfyng a number of constrants. TIMES s run n a dynamc manner, whch s to say that all nvestment decsons are made n each perod wth full knowledge of future events. The model s sad to have perfect foresght (or to be clarvoyant). In addton to tme-perods (whch may be of varable length), there are tme dvsons wthn a year, also called tme-slces, whch may be defned at wll by the user (see Fgure 2.1). For nstance, the user may want to defne seasons, day/nght, and/or weekdays/weekends. Tme-slces are especally mportant whenever the mode and cost of producton of an energy carrer at dfferent tmes of the year are sgnfcantly dfferent. Ths s the case for nstance when the demand for an energy form fluctuates across the year and a varety of technologes may be chosen for ts producton. The producton technologes may themselves have dfferent characterstcs dependng on the tme of year (e.g. wnd turbnes or run-of-the-rver hydro plants). In such cases, the matchng of supply and demand requres that the actvtes of the technologes producng and consumng the commodty be tracked at each tme slce. Examples of commodtes requrng tme-slcng may nclude electrcty, dstrct heat, natural gas, ndustral steam, and hydrogen. Two addtonal reasons for defnng sub yearly tme slces are a) the fact that the commodty s expensve (or even mpossble) to store (thus requrng that producton technologes be sutably actvated n each tme slce to match the demand), and b) the exstence of an expensve nfrastructure whose capacty should be suffcent to bear the peak demand for the commodty. The net result of these chracterstcs s that the deployment n tme of the varous producton technologes may be very dfferent n dfferent tme-slces, and furthermore that specfc nvestment decsons be taken to nsure adequate reserve capacty at peak. Model horzon Perod 1 Perod 2 Perod 3 Perod 4 Annual SP SU FA WI Seasons SP_WD SP_WE SU_WD SU_WE FA_WD FA_WE WI_WD WI_WE Weekly SP_WD_D SP_WD_N SP_WE_D SP_WE_N SU_WD_D SU_WD_N SU_WE_D SU_WE_N FA_WD_D FA_WD_N FA_WE_D FA_WE_N WI_WD_D WI_WD_N WI_WE_D WI_WE_N Daynte Fgure 2.1: Example of a tmeslce tree 12

13 2.1 Tme horzon The tme horzon s dvded nto a (user-chosen) number of tme-perods, each model perod contanng a (possbly dfferent) number of years. For TIMES each year n a gven perod s consdered dentcal, except for the cost objectve functon whch dfferentates between payments n each year of a perod. For all other quanttes (capactes, commodty flows, operatng levels, etc) any model nput or output related to perod t apples to each of the years n that perod, wth the excepton of nvestment varables, whch are usually made only once n a perod 5,. In ths respect, TIMES s smlar to MARKAL but dffers from the approach used n EFOM, where capactes and flows were assumed to evolve lnearly between so-called mlestone years. The ntal perod s usually consdered a past perod, over whch the model has no freedom, and for whch the quanttes of nterest are all fxed by the user at ther hstorcal values. It s often advsed to choose an ntal perod consstng of a sngle year, n order to facltate calbraton to standard energy statstcs. Calbraton to the ntal perod s one of the more mportant tasks requred when settng up a TIMES model. The man varables to be calbrated are: the capactes and operatng levels of all technologes, as well as the extracted, exported, mported, produced, and consumed quanttes for all energy carrers, and the emssons f modeled. In TIMES years precedng the frst perod also play a role. Although no explct varables are defned for these years, data may be provded by the modeler on past nvestments. Note carefully that the specfcaton of past nvestments nfluences not only the ntal perod s calbraton, but also the model s behavor over several future perods, snce the past nvestments provde resdual capacty n several years wthn the modelng horzon proper. 2.2 Decouplng of data and model horzon In TIMES, specal efforts have been made to de-couple the specfcaton of data from the defnton of the tme perods for whch a model s run. Two TIMES features facltate ths decouplng. Frst, the fact that nvestments made n past years are recognzed by TIMES makes t much easer to modfy the choce of the ntal and subsequent perods wthout major revsons of the database. Second, the specfcaton of process and demand nput data n TIMES s made by specfyng the years when the data apply, and the model takes care of nterpolatng and extrapolatng the data to represent the partcular perods chosen by the modeler for a partcular model run. 5 There are exceptonal cases when an nvestment must be repeated more than once n a perod, namely when the perod s so long that t exceeds the techncal lfe of the nvestment. These cases are descrbed n detal n secton 5.2 of PART II. 13

14 These two features combne to make a change n the defnton of perods qute easy and error-free. For nstance, f a modeler decdes to change the ntal year from 1995 to 2005, and perhaps change the number and duratons of all other perods as well, only one type of data change s needed, namely to defne the nvestments made from 1995 to 2004 as past nvestments. All other data specfcatons need not be altered 6. Ths feature represents a great smplfcaton of the modeler s work. In partcular, t enables the user to defne tme perods that have varyng lengths, wthout changng the nput data. 2.3 The RES concept The TIMES energy economy conssts of three types of enttes: Technologes (also called processes) are representatons of physcal devces that transform commodtes nto other commodtes. Processes may be prmary sources of commodtes (e.g. mnng processes, mport processes), or transformaton actvtes such as converson plants that produce electrcty, energy-processng plants such as refneres, end-use demand devces such as cars and heatng systems, etc, Commodtes consstng of energy carrers, energy servces, materals, monetary flows, and emssons. A commodty s generally produced by some process(es) and/or consumed by other process(es), and Commodty flows, that are the lnks between processes and commodtes. A flow s of the same nature as a commodty but s attached to a partcular process, and represents one nput or one output of that process. It s helpful to pcture the relatonshps among these varous enttes usng a network dagram, referred to as a Reference Energy System (RES). In TIMES, the RES processes are represented as boxes and commodtes as vertcal lnes. Commodty flows are represented as lnks between process boxes and commodty lnes. Usng graph theory termnology, a RES s an orented graph, where both the processes and the commodtes are the nodes of the graph. They are nterconnected by the flows, whch are the arcs of the graph. Each arc (flow) s orented and lnks exactly one process node wth one commodty node. Such a graph s called b-partte, snce ts set of nodes may be parttoned nto two subsets and there are no arcs drectly lnkng two nodes n the same subset. Fgure 2.2 depcts a small porton of a hypothetcal RES contanng a sngle energy servce demand, namely resdental space heatng. There are three end-use space heatng technologes usng the gas, electrcty, and heatng ol energy carrers (commodtes), respectvely. These energy carrers n turn are produced by other technologes, represented n the dagram by one gas plant, three electrcty-generatng plants (gas fred, coal fred, ol fred), and one ol refnery. To complete the producton chan on the prmary energy sde, the dagram also represents an extracton source for natural gas, an 6 However, f the horzon has been lengthened, addtonal data for the new years at the end of the horzon must of course be provded, unless the orgnal database horzon already covers the new model horzon. 14

15 extracton source for coal, and two sources of crude ol (one extracted domestcally and then transported by ppelne, and the other one mported). Ths smple RES has a total of 13 commodtes and 13 processes. These elements form Note that n the RES every tme a commodty enters/leaves a process (va a partcular flow) ts name s changed (e.g., wet gas becomes dry gas, crude becomes ppelne crude). Ths smple rule enables the nterconnectons between the processes to be properly mantaned throughout the network. Gas n ground Coal n ground Ol n ground Imported Ol Gas extracton Wet Gas Coal Crude ol Gas Plant Dry Gas Electrcty Gas fred Power plant Gas Furnace Home space Heatng Coal extracton Coal fred Power plant Electrc Heater Ol extracton Ol Import Ppelne Delvered Crude Ol refnery HFO LFO Ol fred Power plant Ol Furnace Fgure 2.2. Partal vew of a smple Reference Energy System (all arcs are orented left to rght) To organze the RES, and nform the modelng system of the nature of ts components, the varous technologes, commodtes, and flows may be classfed nto sets. Each TIMES set regroups components of a smlar nature. The enttes belongng to a set are referred to as members, tems or elements of that set. The same tem may appear n multple technology or commodty sets. Whle the topology of the RES can be represented by a mult-dmensonal network, whch maps the flow of the commodtes to the varous technologes, the set membershp conveys the nature of the ndvdual components and s often most relevant to post-processng (reportng) than nfluencng the model structure tself. 15

16 Contrary to MARKAL, TIMES has relatvely few sets for formng process or commodty groups. In MARKAL the processes are dfferentated dependng on whether they are sources, converson processes, end-use devces, etc., and processes n each set have ther own specalzed attrbutes. In TIMES most processes are endowed wth essentally the same attrbutes (wth the exceptons of storage and nter-regonal exchange processes), and unless the user decdes otherwse (e.g. by provdng values for some attrbutes and gnorng others), they have the same varables attached to them, and must obey smlar constrants. Therefore, the dfferentaton between the varous speces of processes or commodtes s made through data specfcaton only, thus elmnatng the need to defne specalzed membershp sets (unless desred for processng results). Most of the TIMES features (e.g. sub-annual tme-slce resoluton, vntagng) are avalable for all processes and the modeler chooses the features beng assgned to a partcular process by specfyng a correspondng ndcator set (e.g. PRC_TSL, PRC_VINT). However, the TIMES commodtes are stll classfed nto several Major Groups. There are fve such groups: energy carrers, materals, energy servces, emssons, and monetary flows. The use of these groups s essental n the defnton of some TIMES constrants, as dscussed n chapter Overvew of the TIMES attrbutes TIMES has some attrbutes that were not avalable n MARKAL. More mportantly, some attrbutes correspond to powerful new features that confer to TIMES addtonal flexblty. The complete lst of attrbutes s shown n PART II, and we provde below only succnct comments on the types of attrbute attached to each entty of the RES or to the RES as a whole. Attrbutes may be cardnal (e.g. numbers) or ordnal (e.g. sets). For example, some of ordnal attrbutes are defned for process to descrbe subsets of flows that are then used to construct specfc flow constrants. Subsecton 4.4 descrbes such flow constrants,a nd Chapter 2 of PART II gves the complete lst of TIMES sets. The cardnal attrbutes are usually called parameters. We gve below a bref dea of the types of parameters avalable n the TIMES model generator Parameters assocated wth processes TIMES process-orented parameters fall nto three general categores. Frst are techncal parameters that nclude effcency, avalablty factor(s), commodty consumptons per unt of actvty, shares of fuels per unt actvty, techncal lfe of the process, constructon lead tme, dsmantlng lead-tme and duraton, amounts of the commodtes consumed (respectvely released) by the constructon (respectvely dsmantlng) of one unt of the process, and contrbuton to the peak equatons. The effcency, avalablty factors, and 16

17 commodty nputs and outputs of a process may be defned n several flexble ways dependng on the desred process flexblty, on the tme-slce resoluton chosen for the process and on the tme-slce resoluton of the commodtes nvolved. Certan parameters are only relevant to specal processes, such as storage processes or processes that mplement trade between regons. The other class of process parameters s economc and polcy parameters that nclude a varety of costs attached to the nvestment, dsmantlng, mantenance, and operaton of a process. In addton, taxes and subsdes may be defned n a very flexble manner. Other economc parameters are the economc lfe of a process (whch s the tme durng whch the nvestment cost of a process s amortzed, whch may dffer from the operatonal lfetme) and the process specfc dscount rate, also called hurdle rate, both of whch serve to calculate the annualzed payments on the process nvestment cost. Fnally, the modeler may mpose a varety of bounds (upper, lower, equalty) on the nvestment, capacty, and actvty of a process. Note that many process parameters may be vntaged (.e. dependent upon the date of nstallaton of new capacty), and furthermore may be defned as beng dependent on the age of the technology. The latter feature s mplemented by means of specal data grouped under the SHAPE parameter, whch ntroduces user-defned shapng ndexes that can be appled to age-dependent parameters. For nstance, the annual mantenance cost of an automoble could be defned to reman constant for say 3 years and then ncrease n a lnear manner each year after the thrd year Parameters assocated wth commodtes Commodty-orented parameters fall nto three categores. Techncal parameters assocated wth commodtes nclude overall effcency (for nstance grd effcency), and the tme-slces over whch that commodty s to be tracked. For demand commodtes, n addton the annual projected demand and load curves (f the commodty has a subannual tme-slce resoluton) can be specfed. Economc parameters nclude addtonal costs, taxes, and subsdes on the overall or net producton of a commodty.. These cost elements are then added to all other (mplct) costs of that commodty. In the case of a demand servce, addtonal parameters defne the demand curve (.e. the relatonshp between the quantty of demand and ts prce). These parameters are: the demand s own-prce elastcty, the total allowed range of varaton of the demand value, and the number of steps to use for the dscrete approxmaton of the curve. Polcy based parameters nclude bounds (at each perod or cumulatve) on the overall or net producton of a commodty, or on the mports or exports of a commodty by a regon. 17

18 In TIMES the net or the total producton of each commodty may be explctly represented by a varable, f needed for mposng a bound or a tax. No such drect possblty was avalable n MARKAL, although the same result could be acheved va clever modelng Parameters attached to commodty flows nto and out of processes A commodty flow (more smply, a flow) s an amount of a gven commodty produced or consumed by a gven process. Some processes have several flows enterng or leavng t, perhaps of dfferent types (fuels, materals, demands, or emssons). In TIMES, unlke n MARKAL, each flow has a varable attached to t, as well as several attrbutes (parameters or sets) Techncal parameters (along wth some set attrbutes), permt full control over the maxmum and/or mnmum share a gven nput or output flow may take wthn the same commodty group. For nstance, a flexble turbne may accept ol or gas as nput, and the modeler may use a parameter to lmt the share of ol to at most 40% of the total fuel nput. Other parameters and sets defne the amount of certan outflows n relaton to certan nflows (e.g., effcency, emsson rate by fuel). For nstance, n an ol refnery a parameter may be used to set the total amount of refned products equal to 92% of the total amount of crude ols (s) enterng the refnery, or to calculate certan emssons as a fxed proporton of the amount of ol consumed. If a flow has a sub-annual tme-slce resoluton, a load curve can be specfed for the flow 7. Economc parameters nclude delvery and other varable costs, taxes and subsdes attached to an ndvdual process flow Parameters attached to the entre RES These parameters nclude currency converson factors (n a mult-regonal model), regon-specfc tme-slce defntons, a regon-specfc general dscount rate, and reference year for calculatng the dscounted total cost (objectve functon). In addton, certan swtches control the actvaton of the data nterpolaton procedure as well as specal model features to be employed (e.g., run wth ETL, see chapter 8). 2.5 Process and Commodty classfcaton Although TIMES does not explctly dfferentate processes or commodtes that belong to dfferent portons of the RES (wth the excepton of storage and tradng processes), there are three ways n whch some dfferentaton does occur. 7 It s possble to defne not only load curves for a flow, but also bounds on the share of a flow n a specfc tme-slce relatve to the annual flow, e.g. the flow n the tme-slce WnterDay has to be at least 10 % of the total annual flow. 18

19 Frst, TIMES does requre the defnton of Prmary Commodty Groups (pcg),.e. subsets of commodtes of the same nature enterng or leavng a process. For each gven process, the modeler defnes a pcg as a subset of commodtes of the same nature, etehr enterng or leavng the process as flows. TIMES uses the pcg to defne the actvty of the process, and also ts capacty. Besdes establshng the process actvty and capacty, these groups are convenent ads for defnng certan complex quanttes related to process flows, as dscussed n secton 4.4 and n PART II. As noted prevously TIMES does not requre that the user provde many set membershps. However, the TIMES report step does pass some set declaratons to the VEDA-BE result-processng system to facltate constructon of results analyss tables. These nclude process subsets to dstngush demand devces, energy processes, materal processes (by weght or volume), refneres, electrc producton plants, coupled heat and power plants, heatng plants, storage technologes and dstrbuton (lnk) technologes; and commodty subsets for energy, useful energy demands (splt nto sx aggregate subsectors), envronmental ndcators, and materals. The thrd nstance of commodty or process dfferentaton s not embedded n TIMES, but rests on the modeler. A modeler may well want to choose process and commodty names n a judcous manner so as to more easly dentfy them when browsng through the nput database or when examnng results. As an example, the World Mult-regonal TIMES model developed wthn ETSAP adopts a namng conventon whereby the frst three characters denote the sector and the next three the fuel (e.g., lght fuel ol used n the resdental sector s denoted RESLFO). Smlarly, process names are chosen so as to dentfy the sub-sector or end-use (frst three characters), the man fuel used (next three), and the specfc technology (last four). For nstance, a standard (001) resdental water heater (RHW) usng electrcty (ELC) s named RWHELC001. Namng conventons may thus play a crtcal role n allowng the easy dentfcaton of an element s poston n the RES. Smlarly, energy servces may be labeled so that they are more easly recognzed. For nstance, the frst letter may ndcate the broad sector (e.g. T for transports) and the second letter desgnate any homogenous sub-sectors (e.g. R for road transport), the thrd character beng free. In the same fashon, fuels, materals, and emssons are dentfed so as to mmedately desgnate the sector and sub-sector where they are produced or consumed. To acheve ths some fuels have to change names when they change sectors, whch s accomplshed va processes whose prmary role s to change the name of a fuel. In addton, such a process may serve as a bearer of sector wde parameters such as dstrbuton cost, prce markup, tax, that are specfc to that sector and fuel. For nstance, a tax may be leved on ndustral dstllate use but not on agrcultural dstllate use, even though the two commodtes are physcally dentcal. 19

20 3 Economc ratonale of the TIMES modelng approach Ths chapter provdes a detaled economc nterpretaton of the TIMES and other partal equlbrum models based on maxmzng total surplus. Partal equlbrum models have one common feature they smultaneously confgure the producton and consumpton of commodtes (.e. fuels, materals, and energy servces) and ther prces. The prce of producng a commodty affects the demand for that commodty, whle at the same tme the demand affects the commodty s prce. A market s sad to have reached an equlbrum at prces p* and quanttes q* when no consumer wshes to purchase less than q* and no producer wshes to produce more than q* at prce p*. Both p* and q* are vectors whose dmenson s equal to the number of dfferent commodtes beng modeled. As wll be explaned below, when all markets are n equlbrum the total economc surplus s maxmzed. The concept of total surplus maxmzaton extends the drect cost mnmzaton approach upon whch earler bottom-up energy system models were based. These smpler models had fxed energy servce demands, and thus were lmted to mnmzng the cost of supplyng these demands. In contrast, the TIMES demands for energy servces are themselves elastc to ther own prces, thus allowng the model to compute a bona fde supply-demand equlbrum. Ths feature s a fundamental step toward capturng the man feedback from the economy to the energy system. Secton 3.1 provdes a bref revew of dfferent types of energy models. Secton 3.2 dscusses the economc ratonale of the TIMES model wth emphass on the features that dstngush TIMES from other bottom-up models (such as the early ncarnatons of MARKAL, see Fshbone and Ablock, 1981, Berger et al., 1992, though MARKAL has snce been extended beyond these early versons). Secton 3.3 descrbes the detals of how prce elastc demands are modeled n TIMES, and secton 3.4 provdes addtonal dscusson of the economc propertes of the model A bref classfcaton of energy models Many energy models are n current use around the world, each desgned to emphasze a partcular facet of nterest. Dfferences nclude: economc ratonale, level of dsaggregaton of the varables, tme horzon over whch decsons are made (and whch s closely related to the type of decsons,.e. only operatonal plannng or also nvestment decsons), and geographc scope. One of the most sgnfcant dfferentatng features among energy models s the degree of detal wth whch commodtes and technologes are represented, whch wll gude our classfcaton of models n two major classes Top-Down Models At one end of the spectrum are aggregated General Equlbrum (GE) models. In these each sector s represented by a producton functon desgned to smulate the potental substtutons between the man factors of producton (also hghly aggregated nto a few varables such as: energy, captal, and labor) n the producton of each sector s output. In ths model category are found a number of models of natonal or global energy systems. 20

21 These models are usually called Top-Down, because they represent an entre economy va a relatvely small number of aggregate varables and equatons. In these models, producton functon parameters are calculated for each sector such that nputs and outputs reproduce a sngle base hstorcal year. 8 In polcy runs, the mx of nputs 9 requred to produce one unt of a sector s output s allowed to vary accordng to user-selected elastctes of substtuton. Sectoral producton functons most typcally have the followng general form: X s ρ ρ ρ 1/ ρ ( B K + B L + B E ) = A (3-1) 0 K s L S E S where X S s the output of sector S, K S, L S, and E S are the nputs of captal, labor and energy needed to produce one unt of output n sector S, s the elastcty of substtuton parameter, A 0 and the B s are scalng coeffcents. The choce of determnes the ease or dffculty wth whch one producton factor may be substtuted for another: the smaller s (but stll greater than or equal to 1), the easer t s to substtute the factors to produce the same amount of output from sector S. Also note that the degree of factor substtutablty does not vary among the factors of producton the ease wth whch captal can be substtuted for labor s equal to the ease wth whch captal can be substtuted for energy, whle mantanng the same level of output. GE models may also use alternate forms of producton functon (3-1), but retan the basc dea of an explct substtutablty of producton factors Bottom-Up Models At the other end of the spectrum are the very detaled, technology explct models that focus prmarly on the energy sector of an economy. In these models, each mportant energy-usng technology s dentfed by a detaled descrpton of ts nputs, outputs, unt costs, and several other techncal and economc characterstcs. In these so-called Bottom-Up models, a sector s consttuted by a (usually large) number of logcally arranged technologes, lnked together by ther nputs and outputs (commodtes, whch may be energy forms or carrers, materals, emssons and/or demand servces). Some bottom-up models compute a partal equlbrum va maxmzaton of the total net (consumer and producer) surplus, whle others smulate other types of behavor by economc agents, as wll be dscussed below. In bottom-up models, one unt of sectoral output (e.g., a bllon vehcle klometers, one bllon tonnes transported by heavy trucks or one Petajoule of resdental coolng servce) s produced usng a mx of ndvdual technologes outputs. Thus the producton functon of a sector s mplctly constructed, 8 These models assume that the relatonshps (as defned by the form of the producton functons as well as the calculated parameters) between sector level nputs and outputs are n equlbrum n the base year. 9 Most models use nputs such as labor, energy, and captal, but other nput factors may concevably be added, such as arable land, water, or even techncal know-how. Smlarly, labor may be further subdvded nto several categores. 21

22 rather than explctly specfed as n more aggregated models. Such mplct producton functons may be qute complex, dependng on the complexty of the reference energy system of each sector (sub-res) Recent Modelng Advances Whle the above dchotomy appled farly well to earler models, these dstnctons now tend to be somewhat blurred by recent advances n both categores of model. In the case of aggregate top-down models, several general equlbrum models now nclude a far amount of fuel and technology dsaggregaton n the key energy producng sectors (for nstance: electrcty producton, ol and gas supply). Ths s the case wth MERGE 10 and SGM 11, for nstance. In the other drecton, the more advanced bottom-up models are reachng up to capture some of the effects of the entre economy on the energy system. For nstance, the TIMES model has end-use demands (ncludng demands for ndustral output) that are senstve to ther own prces, and thus capture the mpact of rsng energy prces on economc output and vce versa. Recent ncarnatons of technology-rch models are mult-regonal, and thus are able to consder the mpacts of energy-related decsons on trade. It s worth notng that whle the mult-regonal top-down models have always represented trade, they have done so wth a very lmted set of traded commodtes typcally one or two, whereas there may be qute a number of traded energy forms and materals n mult-regonal bottom-up models. MARKAL-MACRO (see [9]) s a hybrd model combnng the technologcal detal of MARKAL wth a succnct representaton of the macro-economy consstng of a sngle producng sector. Because of ts succnct sngle-sector producton functon, MARKAL- MACRO s able to compute a general equlbrum n a sngle optmzaton step. The NEMS 12 model s another example of a full lnkage between several technology rch modules of the varous energy subsectors and a set of macro-economc equatons, although the lnkage here s not as tght as n MARKAL-MACRO, and thus requres teratve resoluton methods. The TIMES model ntroduces further enhancements over and above those of MARKAL. In TIMES, the horzon may be dvded nto perods of unequal lengths, thus permttng a more flexble modelng of long horzons: typcally, one may adopt short perods n the near-term (the ntal perod often conssts of a sngle base year), and longer ones n the out years; TIMES ncludes both technology related varables (as n MARKAL) and flow related varables (as n the EFOM model,van der Voort et. al., 1984), thus allowng the easy creaton of more flexble processes and constrants; the expresson of the TIMES objectve functon (total system cost) tracks the payments of nvestments and other costs much more precsely that n other bottom-up models; and, several other new features of TIMES that are fully dscussed n chapters 4 and 5, and n PART II of ths documentaton. 10 Model for Evaluatng Regonal and Global Effects (Manne et al., 1995) 11 Second Generaton Model (Edmonds et al., 1991) 12 Natonal Energy Modelng System, a detaled ntegrated equlbrum model of an energy system lnked to the economy at large, US Dept of Energy, Energy Informaton Admnstraton (2000) 22

23 In spte of these advances n both classes of models, there reman mportant dfferences. Specfcally: Top-down models encompass macroeconomc varables beyond the energy sector proper, such as wages, consumpton, and nterest rates, and Bottom-up models have a rch representaton of the varety of technologes (exstng and/or future) avalable to meet energy needs, and, they often have the capablty to track a wde varety of traded commodtes. The Top-down vs. Bottom-up approach s not the only relevant dfference among energy models. Among Top-down models, the so-called Computable General Equlbrum models (CGE) descrbed above dffer markedly from the macro econometrc models. The latter do not compute equlbrum solutons, but rather smulate the flows of captal and other monetzed quanttes between sectors (see e.g. Meade, 1996 on the LIFT model). They use econometrcally derved nput-output coeffcents to compute the mpacts of these flows on the man sectoral ndcators, ncludng economc output (GDP) and other varables (labor, nvestments). The sectoral varables are then aggregated nto natonal ndcators of consumpton, nterest rate, GDP, labor, and wages. Among technology explct models also, two man classes are usually dstngushed: the frst class s that of the partal equlbrum models such as MARKAL and TIMES, that use optmzaton technques to compute a least cost (or maxmum surplus) path for the energy system. The second class s that of smulaton models, where the emphass s on representng a system not governed purely by fnancal costs and profts. In these smulaton models (e.g. CIMS, Jaccard et al. 2003), nvestment decsons taken by a representatve agent (frm or consumer) are only partally based on proft maxmzaton, and technologes may capture a share of the market even though ther lfe-cycle cost may be hgher than that of other technologes. Smulaton models use market-sharng formulas that preclude the easy computaton of equlbrum at least not n a sngle pass. The SAGE ncarnaton of the MARKAL model possesses a market sharng mechansm that allows t to reproduce certan behavoral characterstcs of observed markets. The next secton focuses on the TIMES model, the most recent advanced partal equlbrum model. 3.2 The TIMES paradgm Snce certan portons of ths and the next sectons requre an understandng of the concepts and termnology of Lnear Programmng, the reader requrng a brush-up on ths topc may frst read secton 4.5, and then, f needed, some standard textbook on LP, such as Hller and Leberman (1990) or Chvàtal (1983). The applcaton of Lnear Programmng to mcroeconomc theory s covered n Gale (1960), and n Dorfman, Samuelson, and Solow (1958, and subsequent edtons). A bref descrpton of TIMES would express that t s a: Technology explct; Mult-regonal; 23

24 Partal equlbrum model; that assumes: Prce elastc demands; and Compettve markets: wth Perfect foresght (resultng n Margnal value Prcng) We now proceed to flesh out each of these propertes A technology explct model As already presented n chapter 2 (and descrbed n much more detal n Part II), each technology s descrbed n TIMES by a number of techncal and economc parameters. Thus each technology s explctly dentfed (gven a unque name) and dstngushed from all others n the model. A mature TIMES model may nclude several thousand technologes n all sectors of the energy system (energy procurement, converson, processng, transmsson, and end-uses) n each regon. Thus TIMES s not only technology explct, t s technology rch as well. Furthermore, the number of technologes and ther relatve topology may be changed at wll, purely va data nput specfcaton, wthout the user ever havng to modfy the model s equatons. The model s thus to a large extent data drven Mult-regonal feature Some exstng TIMES models coverng the entre energy system nclude up to 15 regonal modules, whle some exstng sectoral models consst of up to 27 regons. The number of regons n a model s lmted only by the dffculty of solvng LP s of very large sze. The ndvdual regonal modules are lnked by energy and materal tradng varables, and by emsson permt tradng varables, f desred. The trade varables transform the set of regonal modules nto a sngle mult-regonal (possbly global) energy model, where actons taken n one regon may affect all other regons. Ths feature s of course essental when global as well as regonal energy and emsson polces are beng smulated. Thus a mult-regonal TIMES modeled s geographcally ntegrated Partal equlbrum propertes TIMES computes a partal equlbrum on energy markets. Ths means that the model computes both the flows of energy forms and materals as well as ther prces, n such a way that, at the prces computed by the model, the supplers of energy produce exactly the amounts that the consumers are wllng to buy. Ths equlbrum feature s present at every stage of the energy system: prmary energy forms, secondary energy forms, and energy servces. A supply-demand equlbrum model has as economc ratonale the maxmzaton of the total surplus, defned as the sum of supplers and consumers surpluses. The mathematcal method used to maxmze the surplus must be adapted to the 24

25 partcular mathematcal propertes of the model. In TIMES, these propertes are as follows: Outputs of a technology are lnear functons of ts nputs (subsecton ); Total economc surplus s maxmzed over the entre horzon ( ), and Energy markets are compettve, wth perfect foresght ( ). As a result of these assumptons the followng addtonal propertes hold: The market prce of each commodty s equal to ts margnal value n the overall system ( ), and Each economc agent maxmzes ts own proft or utlty ( ) Lnearty A lnear nput-to-output relatonshp frst means that each technology represented may be mplemented at any capacty, from zero to some upper lmt, wthout economes or dseconomes of scale. In a real economy, a gven technology s usually avalable n dscrete szes, rather than on a contnuum. In partcular, for some real lfe technologes, there may be a mnmum sze below whch the technology cannot be mplemented (or else at a prohbtve cost), as for nstance a nuclear power plant, or a hydroelectrc project. In such cases, because TIMES assumes that all technologes may be mplemented n any sze, t may happen that the model s soluton shows some technology s capacty at an unrealstcally small sze. However, n most applcatons, such a stuaton s relatvely nfrequent and often nnocuous, snce the scope of applcaton s at the country or regon s level, and thus large enough so that small capactes are unlkely to occur. On the other hand, there may be stuatons where plant sze matters, for nstance when the regon beng modeled s very small. In such cases, t s possble to enforce a rule by whch certan capactes are allowed only n multples of a gven sze (e.g., buld or not a gas ppelne), by ntroducng nteger varables. Ths opton, referred to as Lumpy Investment (LI) s avalable n TIMES and s dscussed n chapter 7. Ths approach should, however, be used sparngly because t greatly ncreases soluton tme. Alternatvely and more smply, a user may add user-defned constrants to force to zero any capactes that are clearly too small. It s the lnearty property that allows the TIMES equlbrum to be computed usng Lnear Programmng technques. In the case where economes of scale or some other non-convex relatonshp s mportant to the problem beng nvestgated, the optmzaton program would no longer be lnear or even convex. We shall examne such a case n chapter 8 when dscussng Endogenous Technology Learnng. The fact that TIMES s equatons are lnear, however, does not mean that producton functons behave n a lnear fashon. Indeed, the TIMES producton functons are usually hghly non-lnear (although convex), representng non-lnear functons as a stepped sequence of lnear functons. As a smple example, a supply of some resource may be represented as a sequence of segments, each wth rsng (but constant wthn ts nterval) unt cost. The modeler defnes the wdth of each nterval so that the resultng supply 25

26 curve may smulate any non-lnear convex functon. In bref, dseconomes of scale are usually present at the sectoral level Maxmzaton of total surplus: Prce equals Margnal value The total surplus of an economy s the sum of the supplers and the consumers surpluses. The term suppler desgnates any economc agent that produces (and sells) one or more commodtes (.e., n TIMES, an energy form, a materal, an emsson permt, and/or an energy servce). A consumer s a buyer of one or more commodtes. In TIMES, the supplers of a commodty are technologes that procure a gven commodty, and the consumers of a commodty are technologes or demands that consume a gven commodty. Some technologes may be both supplers and consumers, but not of the same commodty (snce a technology never has the same commodty as nput and output, wth the excepton of storage technologes 13 ). Therefore, for each commodty the RES defnes a set of supplers and a set of consumers. It s customary n mcroeconomcs to represent the set of supplers of a commodty by ther nverse producton functon, that plots the margnal producton cost of the commodty (vertcal axs) as a functon of the quantty suppled (horzontal axs). In TIMES, as n other lnear optmzaton models, the supply curve of a commodty s not explctly expressed as a functon of aggregate factor nputs such as captal, labor and energy (as they would n typcal producton functons used n the economc lterature). However, t s a standard result of Lnear Programmng theory that the nverse supply functon s step-wse constant and ncreasng n each factor (see Fgures 2 and 3. for the case of a sngle commodty 14 ). Each horzontal step of the nverse supply functon ndcates that the commodty s produced by a certan technology or set of technologes n a strctly lnear fashon. As the quantty produced ncreases, one or more resources n the mx (ether a technologcal potental or some resource s avalablty) s exhausted, and therefore the system must start usng a dfferent (more expensve) technology or set of technologes n order to produce addtonal unts of the commodty, albet at hgher unt cost. Thus, each change n producton mx generates one step of the starcase producton functon wth a value hgher than the precedng step. The wdth of any partcular step depends upon the technologcal potental and/or resource avalablty assocated wth the set of technologes represented by that step. 13 Even here, a storage process consumes a gven commodty at a certan tme-slce or perod, and resttutes t at a later tme-slce or perod. Therefore, the output commodty s not dentcal to the nput commodty. 14 Ths s so because n Lnear Programmng the shadow prce of a constrant remans constant over a certan nterval, and then changes abruptly, gvng rse to a stepwse constant functonal shape. 26

27 Prce C C Supply Curve P E S S Equlbrum Demand Curve Q Q E Quantty Fgure 3.1. Equlbrum n the case of an energy form: the model mplctly constructs both the supply and the demand curves In a smlar manner, each TIMES nstance defnes a seres of nverse demand functons. In the case of demands, two cases are dstngushed. Frst, f the commodty n queston s an energy carrer whose producton and consumpton are endogenous to the model, then ts demand functon s mplctly constructed wthn TIMES, and s a stepwse constant, decreasng functon of the quantty demanded, as llustrated n Fgure 3.1 for a sngle commodty. If on the other hand the commodty s a demand for an energy servce, then ts demand curve s defned by the user va the spcfcaton of the own-prce elastcty of that demand, and the curve s n ths nstance a smoothly decreasng curve as llustrated n Fgure In both cases, the supply-demand equlbrum s at the ntersecton of the supply functon and the demand functon, and corresponds to an equlbrum quantty Q E and an equlbrum prce P E 16. At prce P E, supplers are wllng to supply the quantty Q E and consumers are wllng to buy exactly that same quantty Q E. Of course, the TIMES equlbrum concerns many commodtes, and the equlbrum s a mult-dmensonal analog of the above, where Q E and P E are now vectors rather than scalars. The above descrpton of the TIMES equlbrum s vald for any energy form that s entrely endogenous to TIMES,.e. an energy carrer, materal, or emsson permt. In the case of an energy servce, TIMES does not mplctly construct the demand functon. Rather, the user explctly defnes the demand functon by specfyng ts own prce 15 Ths smooth curve wll be dscretzed later for computatonal purposes, as descrbed n chapter 6 16 As may be seen n fgures 2 and 3, the equlbrum s not necessarly unque. In the case shown n Fgure 2, any pont on the vertcal segment contanng the equlbrum s also an equlbrum, wth the same Q E but a dfferent P E. In other cases, the multple equlbra may have the same prce and dfferent quanttes. 27

28 elastcty. Each energy servce demand s assumed to have a constant own prce elastcty functon of the form (see Fgure 3.2): D/D 0 = (P/P 0 ) E (3.3-1) Where {D 0,P 0 } s a reference par of demand and prce values for that energy servce over the forecast horzon, and E s the (negatve) own prce elastcty of that energy servce demand, as chosen by the user (note that though not shown by the notaton, ths prce elastcty may vary over tme). The par {D 0, P 0 } s obtaned by solvng TIMES for a reference scenaro. More precsely, D 0 s the demand projecton estmated by the user n the reference case based upon explctly defned relatonshps to economc and demographc drvers, and P 0 s the shadow prce of that energy servce demand obtaned by runnng the reference case scenaro of TIMES. Usng Fgure 3.1 as an example, the defnton of the supplers surplus correspondng to a certan pont S on the nverse supply curve s the dfference between the total revenue and the total cost of supplyng a commodty,.e. the gross proft. In Fgure 3.1, the surplus s thus the area between the horzontal segment SS and the nverse supply curve. Smlarly, the consumers surplus for a pont C on the nverse demand curve, s defned as the area between the segment CC and the nverse demand curve. Ths area s a consumer s analog to a producer s proft; more precsely t s the cumulatve opportunty gan of all consumers who purchase the commodty at a prce lower than the prce they would have been wllng to pay. For a gven quantty Q, the total surplus (supplers plus consumers ) s smply the area between the two nverse curves stuated at the left of Q. It should be clear from Fgure 3.1 that the total surplus s maxmzed exactly when Q s equal to the equlbrum quantty Q E. Therefore, we may state (n the sngle commodty case) the followng Equvalence Prncple: The supply-demand equlbrum s reached when the total surplus s maxmzed Ths s a remarkably useful result, as t leads to a method for computng the equlbrum, as wll be see n much detal n Chapter 6. In the mult-dmensonal case, the proof of the above statement s less obvous, and requres a certan qualfyng property (called the ntegrablty property) to hold (Samuelson, 1952, Takayama and Judge, 1972). One suffcent condton for the ntegrablty property to be satsfed s realzed when the cross-prce elastctes of any two energy forms are equal, vz. P / Q = P / Q for all j j j, In the case of commodtes that are energy servces, these condtons are trvally satsfed n TIMES because we have assumed zero cross prce elastctes. In the case of an energy carrer, where the demand curve s mplctly derved, t s also easy to show that the ntegrablty property s always satsfed 17. Thus the equvalence prncple s vald n all cases. 17 Ths results from the fact that n TIMES each prce P s the shadow prce of a balance constrant (see secton 4.5), and may thus be (loosely) expressed as the dervatve of the objectve functon F wth respect to the rght-hand-sde of a balance constrant,.e. F / Q. When that prce s further dfferentated wth 28

29 In summary, the equvalence prncple guarantees that the TIMES supply-demand equlbrum maxmzes total surplus.the total surplus concept has long been a manstay of socal welfare economcs because t takes nto account both the surpluses of consumers and of producers. 18 Prce Demand curve Supply Curve P E Equlbrum Q E Quantty Fgure 3.2. Equlbrum n the case of an energy servce: the user, explctly provdes the demand curve, usually usng a smple functonal form Remark: In older versons of MARKAL, and n several other least-cost bottom-up models, energy servce demands are exogenously specfed by the modeler, and only the cost of supplyng these energy servces s mnmzed. Such a case s llustrated n Fgure 3.3 where the nverse demand curve s a vertcal lne. The objectve of such models was smply the mnmzaton of the total cost of meetng exogenously specfed levels of energy servce. respect to another quantty Q j, one gets to 2 F / Q j Q, as desred. 2 F / Q Q 18 See e.g. Samuelson, P, and W. Nordhaus (1977) j, whch, under mld condtons s always equal 29

30 Prce Supply Curve P E Equlbrum Q E Quantty Fgure 3.3. Equlbrum when an energy servce demand s fxed Compettve energy markets wth perfect foresght Compettve energy markets are characterzed by perfect nformaton and atomc economc agents, whch together preclude any of them from exercsng market power. That s, nether the level any ndvdual producer supples, nor the level any ndvdual consumer demands, affects the equlbrum market prce (because there are many other buyers and sellers to replace them). It s a standard result of mcroeconomc theory that the assumpton of compettve markets entals that the market prce of a commodty s equal to ts margnal value n the economy. Ths s of course also verfed n the TIMES economy, as dscussed n the next subsecton. In TIMES, the perfect nformaton assumpton extends to the entre plannng horzon, so that each agent has perfect foresght,.e. complete knowledge of the market s parameters, present and future. Hence, the equlbrum s computed by maxmzng total surplus n one pass for the entre set of perods. Such a farsghted equlbrum s also called an nter-temporal, dynamc or clarvoyant equlbrum. Note that there are at least two ways n whch the perfect foresght assumpton may be relaxed: n one varant, agents are assumed to have foresght over a lmted porton of the horzon, say one perod. Such an assumpton of lmted foresght s emboded n the SAGE varant of MARKAL. There s for the tme beng no such varant of the TIMES model. In another varant, foresght s assumed to be mperfect, meanng that agents may only assume probabltes for certan key future events. Ths assumpton s at the bass of the Stochastc Programmng opton n Standard MARKAL. A Stochastc verson of TIMES s at the plannng stage. 30

31 Margnal value prcng We have seen n the precedng subsectons that the TIMES equlbrum occurs at the ntersecton of the nverse supply and nverse demand curves. It follows that the equlbrum prces are equal to the margnal system values of the varous commodtes. From a dfferent angle, the dualty theory of Lnear Programmng (secton 4.5) ndcates that for each constrant of the TIMES lnear program there s a dual varable. Ths dual varable (when an optmal soluton s reached) s also called the constrant s shadow prce 19, and s equal to the margnal change of the objectve functon per unt ncrease of the constrant s rght-hand-sde. For nstance (secton 4.5), the shadow prce of the balance constrant of a commodty (whether t be an energy form, materal, a servce demand, or an emsson) represents the compettve market prce of the commodty. The fact that the prce of a commodty s equal to ts margnal value s an mportant feature of compettve markets. Dualty theory does not necessarly ndcate that the margnal value of a commodty s equal to the margnal cost of producng that commodty. For nstance, n the equlbrum shown n Fgure 3.4 the prce does not correspond to any margnal supply cost, snce t s stuated at a dscontnuty of the nverse supply curve. In ths case, the prce s precsely determned by demand rather than by supply, and the term margnal cost prcng (so often used n the context of optmzng models) s ncorrect. The term margnal value prcng s a more approprate term to use. It s mportant to note that margnal value prcng does not mply that supplers have zero proft. Proft s exactly equal to the supplers surplus, and Fgures 2 through 5 show that t s generally postve. Only the last few unts produced may have zero proft, f, and when, ther producton cost equals the equlbrum prce, and even n ths case zero proft s not automatc as exemplfed n Fgure 3.3. In TIMES the shadow prces of commodtes play a very mportant dagnostc role. If some shadow prce s clearly out of lne (.e. f t seems much too small or too large compared to the antcpated market prces), ths ndcates that the model s database may contan some errors. The examnaton of shadow prces s just as mportant as the analyss of the quanttes produced and consumed of each commodty and of the technologcal nvestments. 19 The term Shadow Prce s often used n the mathematcal economcs lterature, whenever the prce s derved from the margnal value of a commodty. The qualfer shadow s used to dstngush the compettve market prce from the prce observed n the real world, whch may be dfferent, as s the case n regulated ndustres or n sectors where ether consumers or producers exercse market power. When the equlbrum s computed usng LP optmzaton, as s the case for MARKAL, the shadow prce of each commodty s computed as the dual varable of that commodty s balance constrant, as wll be further developed n secton

32 Prce Demand curve Supply Curve P E Equlbrum Q E Quantty Fgure 3.4. Case where the equlbrum prce s not equal to any margnal supply cost Proft maxmzaton: the Invsble Hand An nterestng property may be derved from the assumptons of compettveness. Whle the avowed objectve of the TIMES model s to maxmze the overall surplus, t s also true that each economc agent n TIMES maxmzes ts own proft. Ths property s akn to the famous nvsble hand property of compettve markets, and may be establshed rgorously by the followng theorem that we state n an nformal manner: Theorem: Let (p*,q*) be the par of equlbrum vectors. If we now replace the orgnal TIMES lnear program by one where the commodty prces are fxed at value p*, and we let each agent maxmze ts own proft, there exsts a vector of optmal quanttes produced or purchased by the agents that s equal to q* 20. Ths property s mportant nasmuch as t provdes an alternatve justfcaton for the class of equlbra based on the maxmzaton of total surplus. It s now possble to shft the model s ratonale from a global, socetal one (surplus maxmzaton), to a local, decentralzed one (ndvdual utlty maxmzaton). Of course, the equvalence suggested by the theorem s vald only nsofar as the margnal value prcng mechansm s strctly enforced that s, nether ndvdual producers nor ndvdual consumers behavors affect market prces both groups are prce takers. Clearly, many markets are not 20 However, the resultng Lnear Program has multple optmal solutons. Therefore, although q* s an optmal soluton, t s not necessarly the one found when the modfed LP s solved. 32

33 compettve n the sense the term has been used here. For example, the behavor of a few, state-owned ol producers has a dramatc mpact on world ol prces, that then depart from ther margnal system value. Market power 21 may also exst n cases where a few consumers domnate a market. The entre annual crop of a gven regon s supply of coffee beans may, for example, be purchased by a handful of purchasers who may then greatly nfluence prces. 21 An agent has market power f ts decsons, all other thngs beng equal, have an mpact on the market prce. Monopoles and Olgopoles are example of markets where one or several agents have market power. 33

34 4 A smplfed descrpton of the TIMES optmzaton program Ths chapter contans a smplfed formulaton of the TIMES Lnear Program. A Lnear Programmng problem conssts n the mnmzaton (or maxmzaton) of an objectve functon defned as a mathematcal expresson of decson varables, subject to constrants (also called equatons 22 ) also expressed mathematcally. Very large nstances of Lnear Programs nvolvng hundreds of thousands of constrants and varables may be formulated n the GAMS language, and solved va powerful Lnear Programmng optmzers 23. The Lnear Program descrbed n ths chapter s much smplfed, snce t gnores many exceptons and complextes that are not essental to a basc understandng of the prncples of the model. Secton 4.5 gves addtonal detals on general Lnear Programmng concepts. The full detals of the parameters, varables, objectve functon, and constrants of TIMES are gven n Part II of ths documentaton. An optmzaton problem formulaton conssts of three types of enttes: the decson varables:.e. the unknowns, or endogenous quanttes, to be determned by the optmzaton, the objectve functon: expressng the crteron to be mnmzed or maxmzed, and constrants: equatons or nequaltes nvolvng the decson varables that must be satsfed by the optmal soluton. 4.1 Indces The model data structures (sets and parameters), varables and equatons use the followng ndexes: r: ndcates the regon t or v: tme perod; t corresponds to the current perod, and v s used to ndcate the vntage year of an nvestment When a process s not vntaged then v = t. p: process (technology); s: tme-slce; ths ndex s relevant only for user-desgnated commodtes and processes that are tracked at fner than annual level (e.g. electrcty, lowtemperature heat, and perhaps run-of-rver hydro or natural gas, etc.). Tme-slce defaults to ANNUAL, ndcatng that a commodty s tracked only annually. c: commodty (energy, materal, emsson, demand); 4.2 Decson Varables 22 Ths rather mproper term ncludes equalty as well as nequalty relatonshps between mathematcal expressons. 23 For more nformaton on optmzers see The Solver Manual of the GAMS A USER'S GUIDE, Anthony Brooke, Davd Kendrck, Alexander Meeraus, and Ramesh Raman, December

35 The decson varables represent the choces to be made by the model,.e. the unknowns. The varous knds of decson varables n a TIMES model are: NCAP(r,v,p): new capacty addton (nvestment) for technology p, n perod v and regon r. For all technologes the v value corresponds to the vntage of the process,.e. year n whch t s nvested n. For vntaged technologes (declared as such by the user) the vntage (v) nformaton s reflected n other process varables, dscussed below. Typcal unts are PJ/year for most energy technologes, Mllon tonnes per year (for steel, alumnum, and paper ndustres), Bllon vehcle-klometers per year (B-vkms/year) or mllon cars for road vehcles and GW for electrcty equpment (1GW= PJ/year), etc. CAP(r,v,t,p): nstalled capacty of process p, n regon r and perod t (optonally wth vntage v). It represents the total capacty n place n perod t, consderng the resdual capacty at the begnnng of the modelng horzon and new nvestments made pror to and ncludng perod t that have not reached ther techncal lfetme. Typcal unts: same as nvestments. The CAP varables are actually not explctly defned n the model, but are derved from the NCAP varables and data on past nvestments and plant lfetmes. CAPT(r,t,p): total nstalled capacty of technology p, n regon r and perod t, all vntages together. The CAPT varables are only defned when some bound or user-constrant are specfed for them. They do not enter any other equaton. ACT(r,v,t,p,s): actvty level of technology p, n regon r and perod t (optonally vntage v and tme-slce s). Typcal unts: PJ for all energy technologes. The s ndex s relevant only for processes that produce or consume commodtes specfcally declared as tmeslced. Moreover, t s the process that determnes whch tme slces preval, va a specal attrbute. By default, only annual actvtys tracked. FLOW(r,v,t,p,c,s): the quantty of commodty c consumed or produced by process p, n regon r and perod t (optonally wth vntage v and tme-slce s). Typcal unts: PJ for all energy technologes. The FLOW varables confer consderable flexblty to the processes modeled n TIMES, as they allow the user to defne flexble processes for whch nput and/or output flows are not rgdly lnked to the process actvty. SIN(r,v,t,p,c,s)/SOUT(r,v,t,p,c,s): the quantty of commodty c stored or dscharged by storage process p, n tme-slce s, perod t (optonally wth vntage v), and regon r. TRADE(r,t,p,c,s,mp) and TRADE(r,t,p,c,s,exp):) quantty of commodty c (PJ per year) sold (exp) or purchased (mp) by regon r through export (resp. mport) process p n perod t (optonally n tme-slce s). Note that the topology defned for the exchange process p specfes the traded commodty c, the regon r, and the regons r wth whch regon r s tradng commodty c. In the case of b-lateral tradng, f t s desred that regon r trade wth several other regons, then each such trade requres the defnton of a separate b-lateral exchange process. Note that t s also possble to defne mult-lateral tradng relatonshps between regon r and several other regons r by defnng one of the 35

36 regons as the common market for trade n commodty c. In ths case, the commodty s put on the market and may be bought by any other regon partcpatng n the market. Ths case s convenent for global commodtes such as emsson permts or crude ol. Fnally, exogenous tradng may also be modeled by specfyng the r regon as an external regon. Exogenous tradng s requred for models that are not global, snce exchanges wth non modeled regons cannot be consdered endogenous. D(r,t,d): demand for end-use energy servce d n regon r and perod t. In the reference scenaro, ths varable s fxed by the user. In non-reference scenaros D(r,t,d) may dffer from the reference case demand due to the responsveness of demands to prces (based on each servce demand s own-prce elastcty). Note that n ths smplfed formulaton, we do not show the varables used to decompose D(r,t,d) nto a sum of step-wse quanttes (see chapter 6 and Part II for detals). Other varables: TIMES has a number of commodty related varables that are not strctly needed but are convenent for reportng purposes and/or for applyng certan bounds to them. Examples of such varables are: the total amount produced of a commodty (COMPRD), or the total amount consumed of a commodty (COMCON) TIMES objectve functon: dscounted total system cost The Surplus Maxmzaton objectve s frst transformed nto an equvalent Cost Mnmzaton objectve by takng the negatve of the surplus, and callng ths the total system cost. Ths practce s n part nspred from hstorcal custom from the days of the fxed demand MARKAL model. The TIMES objectve s therefore to mnmze the total cost of the system, properly augmented by the cost of lost demand. All cost elements are approprately dscounted to a selected year. Whle the TIMES constrants and varables are lnked to a perod, the components of the system cost are expressed for each year of the horzon (and even for some years outsde the horzon). Ths choce s meant to provde a smoother, more realstc rendton of the stream of payments n the energy system, as wll be dscussed below. Each year, the total cost ncludes the followng elements: Captal Costs ncurred for nvestng nto and/or dsmantlng processes; Fxed and varable annual Operaton and Mantenance (O&M) Costs, and other annual costs occurrng durng the dsmantlng of technologes; Costs ncurred for exogenous mports and for domestc resource producton; Revenues from exogenous exports; Delvery costs for requred commodtes consumed by processes; 24 These extra varables, as well as the flow varables, add only a moderate computatonal burden to the optmzaton process thanks to the use of a reducton algorthm to detect and elmnate redundant varables and constrants before solvng the LP. These varables and constrants are later renstated n the soluton fle for reportng purposes. 36

37 Taxes and subsdes assocated wth commodty flows and process actvtes or nvestments; Revenues from recuperaton of embedded commodtes, accrued when a process s dsmantlng releases some valuable commodtes; Salvage value of processes and embedded commodtes at the end of the plannng horzon; Welfare loss resultng from reduced end-use demands. Chapter 6 presents the mathematcal dervaton of ths quantty. As already mentoned, n TIMES, specal care s taken to precsely track the monetary flows related to process nvestments and dsmantlng n each year of the horzon. Such trackng s made complex by several factors: Frst, TIMES recognzes that there may be a lead-tme between the begnnng and the end of the constructon of some large processes, thus spreadng the nvestment nstallments over several years; Second, TIMES also recognzes that for some other processes (e.g. new cars), the nvestments n new capacty occur progressvely over several years of a tme perod, rather than n one lump amount (as n MARKAL); Thrd, there s the possblty that a certan nvestment decson made at perod t wll have to be repeated more than once durng that same perod (ths wll occur f the t th perod s long compared to the process lfe); Fourth, TIMES recognzes that there may be dsmantlng captal costs at the endof-lfe of some processes (e.g. a nuclear plant), and these costs, whle attached to the nvestment varable ndexed by perod t, are actually ncurred much later, and Fnally, TIMES assumes that the payment of any captal cost s spread over an economc lfe that may be dfferent from the techncal lfe of the process, and annualzed at a dfferent rate than the overall dscount rate. These varous TIMES features, whle addng precson and realsm to the cost profle, also ntroduce complex mathematcal expressons n the objectve functon. In ths smplfed formulaton, we do not provde much detal on these complex expressons, whch are fully descrbed n secton 5.1 of Part II. We lmt our descrpton to gvng general ndcatons on the cost elements composng the objectve functon, as follows: The nvestment and dsmantlng costs are transformed nto streams of annual payments, computed for each year of the horzon (and beyond, n the case of dsmantlng costs and recyclng revenues), along the lnes suggested above; A salvage value of all nvestments stll actve at the end of the horzon (EOH) s calculated and ts value s assgned to the (sngle) year followng the EOH; The other costs lsted above, whch are all annual costs, are added to the annualzed captal cost payments, mnus salvage value, to form the ANNCOST quantty (below), and TIMES then computes for each regon a total net present value of the stream of annual costs, dscounted to a user selected reference year. These regonal dscounted costs are then aggregated nto a sngle total cost, whch consttutes the objectve functon to be mnmzed by the model n ts equlbrum computaton. 37

38 R REFYR y NPV = ( 1+ d r, y ) ANNCOST ( r, y) r= 1 y YEARS where: NPV s the net present value of the total cost for all regons (the TIMES objectve functon); ANNCOST(r,y) s the total annual cost n regon r and year y; d r,y s the general dscount rate; REFYR s the reference year for dscountng; YEARS s the set of years for whch there are costs, ncludng all years n the horzon, plus past years (before the ntal perod) f costs have been defned for past nvestments, plus a number of years after EOH where some nvestment and dsmantlng costs are stll beng ncurred, as well as the Salvage Value; R s the set of regons n the area of study, and As already mentoned, the exact computaton of ANNCOST s qute complex and s postponed untl PART II, secton Constrants Whle mnmzng total dscounted cost the TIMES model must satsfy a large number of constrants (the so-called equatons of the model) whch express the physcal and logcal relatonshps that must be satsfed n order to properly depct the assocated energy system. TIMES constrants are of several knds. We lst and brefly dscuss the man types of constrants. A full descrpton s gven n Part II. If any constrant s not satsfed, the model s sad to be nfeasble, a condton caused by a data error or an overspecfcaton of some requrement. In the descrptons of the equatons that follow, the equaton and varable names (and ther ndexes) are n bold talc type, and the parameters (and ther ndexes), correspondng to the nput data, are n regular talc typeset. Furthermore, some parameter ndexes have been omtted n order to provde a streamlned presentaton Capacty Transfer (conservaton of nvestments) Investng n a partcular technology ncreases ts nstalled capacty for the duraton of the physcal lfe of the technology. At the end of that lfe, the total capacty for ths technology s decreased by the same amount. When computng the avalable capacty n 38

39 some tme perod, the model takes nto account the capacty resultng from all nvestments up to that perod, some of whch may have been made pror to the ntal perod but are stll n operatng condton (emboded by the resdual capacty of the technology), and others that have been decded by the model at, or after, the ntal perod, up to and ncludng the perod n queston. The total avalable capacty for each technology p, n regon r, n perod t (all vntages), s equal to the sum of nvestments made by the model at past and current perods, and whose physcal lfe has not yet ended, plus capacty n place pror to the modelng horzon that s stll avalable. The exact formulaton of ths constrant s made qute complex by the fact that TIMES accepts varable tme perods, and therefore the end of lfe of an nvestment may well fall n the mddle of a future tme perod. We gnore here these complextes and provde a streamlned verson of ths constrant. Full detals are shown n Part II. EQ_CPT(r,t,p) - Capacty transfer CAPT(r,t,p) = Sum {over all perods t precedng or equal to t such that t-t <LIFE(r,t,p) of NCAP(r,t,p)} + RESID(r,t,p) (4-1) where RESID(r,t,p) s the (exogenously provded) capacty of technology p due to nvestments that were made pror to the ntal model perod and stll exst n regon r at tme t Defnton of process actvty varables Snce TIMES recognzes actvty varables as well as flow varables, t s necessary to relate these two types of varables. Ths s done by ntroducng a constrant that equates an overall actvty varable, ACT(r,v,t,p,s), wth the approprate set of flow varables, FLOW(r,v,t,p,c,s), properly weghted Ths s accomplshed by frst dentfyng the group of commodtes that defnes the actvty (and thereby the capacty as well) of the process. In a smple process, one consumng a sngle commodty and producng a sngle commodty, the modeler smply chooses one of these two flows to defne the actvty, and thereby the process normalzaton (nput or output). In more complex processes, wth several commodtes (perhaps of dfferent types) as nputs and/or outputs, the defnton of the actvty varable requres frst to choose the prmary commodty group (pcg) that wll serve as the actvty-defnng group. For nstance, the pcg may be the group of energy carrers, or the group of materals of a gven type, or the group of GHG emssons, etc. The modeler then dentfes whether the actvty s defned va nputs or va outputs that beong to the selected pcg. Conceptually, ths leads to the followng relatonshp: EQ_ACTFLO(r,v,t,p,s) Actvty defnton ACT(r,v,t,p,s) = SUM{c n pcg of FLOW(r,v,t,p,c,s) / ACTFLO(r,v, p,c)} (4-2) 39

40 where ACTFLO(r,v, p,c) s a converson factor (often equal to 1) from the actvty of the process to the flow of a partcular commodty Use of capacty In each tme perod the model may use some or all of the nstalled capacty accordng to the Avalablty Factor (AF) of that technology. Note that the model may decde to use less than the avalable capacty durng certan tme-slces, or even throughout one or more whole perods, f such a decson contrbutes to mnmzng the overall cost. Optonally, there s a provson for the modeler to force specfc technologes to use ther capacty to ther full potental. For each technology p, perod t, vntage v, regon r, and tme-slce s, the actvty of the technology may not exceed ts avalable capacty, as specfed by a user defned avalablty factor. EQ_CAPACT (r,v,t,p,s) - Use of capacty ACT (r,v,t,p,s) or = AF(r,v,t,p,s)* CAPUNIT(r,p))*FR(r,s)* CAP(r,v,t,p) (4-3) Here CAPUNIT(r,p) s the converson factor between unts of capacty and actvty (often equal to 1, except for power plants). The FR(r,s) parameter s equal to the duraton of tmeslce s. The avalablty factor AF also serves to ndcate the nature of the constrant as an nequalty or an equalty. In the latter case the capacty s forced to be fully utlzed,. Note that the CAP(r,v,t,p) varable s not explctly defned n TIMES. Instead t s replaced n (4-3) by a fracton (less than or equal to 1) of the nvestment varable NCAP(r,v,p) 25 a sum of past nvestments that are stll operatng, as n equaton (4-1). Example: a coal fred power plant s actvty n any tme-slce s bounded above by 80% of ts capacty,.e. ACT (r,v,t,p,s) 0.8* * CAP(r,v,t,p), where CAPUNIT(r,p) = s the converson factor between the unts of the capacty varable (GW) and the actvty-based capacty unt (PJ/a) The actvty-based capacty unt s obtaned from the actvty unt(pj) by dvson by a denomnator of one year. The s ndex of the AF coeffcent n equaton (4-3) ndcates that the user may specfy tme-slced dependency on the avalabltyof the nstalled capacty of some technologes, f desrable. Ths s especally needed when the operaton of the equpment depends on the avalablty of a resource that cannot be stored, such as wnd and sun, or that can be only partally stored, such as water n a reservor. In other cases, the user may provde an AF factor that does not depend on s, whch s then appled to the entre year. The operaton profle of a technology wthn a year, f the technology has a sub-annual 25 That fracton s equal to 1 f the techncal lfe of the nvestment made n perod v fully covers perod t. It s less than 1 (perhaps 0) otherwse. Chapter 5 of PART II provdes detals of the computaton of that fracton. 40

41 process resoluton, s determned by the optmzaton routne. The number of EQ_CAPACT constrants s at least equal to the number of tme-slces at whch the equpment operates. For technologes wth only an annual characterzaton the number of constrants s reduced to one per perod (where s= ANNUAL ) Commodty Balance Equaton: In each tme perod, the producton by a regon plus mports from other regons of each commodty must balance the amount consumed n the regon or exported to other regons. In TIMES, the sense of each balance constrant ( or =) s user controlled, va a specal parameter attached to each commodty. However, the constrant defaults to an equalty n the case of materals (.e. the quantty produced and mported s exactly equal to that consumed and exported), and to an nequalty n the case of energy carrers, emssons and demands (thus allowng some surplus producton). For those commodtes for whch tme-slces have been defned, the balance constrant must be satsfed n each tme-slce. The balance constrant s very complex, due to the many terms nvolvng producton or consumpton of a commodty. We present a much smplfed verson below, to smply ndcate the basc meanng of ths equaton. For each commodty c, tme perod t (vntage v), regon r, and tme-slce s (f necessary or ANNUAL f not), ths constrant requres that the dsposton of each commodty balances ts procurement. The dsposton ncludes consumpton n the regon plus exports; the procurement ncludes producton n the regon plus mports. EQ_COMBAL(r,t,c,s) - Commodty Balance [ Sum {over all p,c TOP(r,p,c, out ) of: [FLOW(r,v,t,p,c,s) + SOUT(r,v,t,p,c,s)*STG_EFF(r,v,p)] } + Sum {over all p,c RPC_IRE(r,p,c, mp ) of: TRADE(r,t,p,c,s, mp )} + Sum {over all p of: Release(r,t,p,c)*NCAP(r,t,p,c)} ] * COM_IE(r,t,c,s) or = (4-4) Sum {over all p,c TOP(r,p,c, n ) of: FLOW(r,v,t,p,c,s) + SIN(r,v,t,p,c,s)} + Sum {over all p,c RPC_IRE(r,p,c, exp )} of: TRADE(r,t,p,c,s, exp ) + Sum {over all p of: Snk(r,t,p,c)*NCAP(r,t,p,c)} + 41

42 FR(c,s) * DM(c,t) where: The constrant s for energy forms and = for materals and emssons (unless these defaults are overrdden by the user). TOP(r,p,c, n/out ) dentfes that there s an nput/output flow of commodty c nto/from process p n regon r; RPC_IRE(r,p,c, mp/exp ) dentfes that there s an mport/export flow nto/from regon r of commodty c va process p; STG_EFF(r,v,p) s the effcency of storage process p; COM_IE(r,t,c) s the nfrastructure effcency of commodty c; Release(r,t,p,c) s the amount of commodty c recuperated per unt of capacty of process p dsmantled (useful to represent some materals or fuels that are recuperated whle dsmantlng a faclty); Snk(r,t,p,c) s the quantty of commodty c requred per unt of new capacty of process p (useful to represent some materals or fuels consumed for the constructon of a faclty); FR(s) s the fracton of the year covered by tme-slce s (equal to 1 for non- tme-slced commodtes). Example: Gasolne consumed by vehcles plus gasolne exported to other regons must not exceed gasolne produced from refneres plus gasolne mported from other regons Defnng flow relatonshps n a process A process wth one or more (perhaps heterogeneous) commodty flows s essentally defned by one or more ndependent nput and output flow varables. In the absence of relatonshps between these flows, the process would be completely undetermned,.e. ts outputs would be ndependent form ts nputs.we therefore need one or more constrants statng that the rato of the sum of some of ts output flows to the sum of some of ts nput flows s equal to a constant (whch s akn to an effcency). In the case of a sngle commodty n, and a sngle commodty out of a process, ths equaton defnes the tradtonal effcency of the process. Wth several commodtes, ths constrant may leave some freedom to ndvdual output (or nput) flows, as long as ther sum s n fxed proporton to the sum of nput (or output) flows. An mportant rule for ths constrant s that each sum must be taken over commodtes of the same type (.e. n the same group, say: energy carrers, or emssons, etc.). In TIMES, for each process the modeler dentfes the nput commodty group cg1, and the output commodty group cg2, and chooses a value for the effcency rato, named FLOFUNC(p,cg1,cg2). The followng equaton embodes ths: 42

43 EQ_PTRANS(r,v,t,p,cg1,cg2,s) Effcency defnton SUM {c n cg2 of : FLOW(r,v,t,p,c,s ) }= FLOFUNC(r,v,cg1,cg2,s) * SUM {c wthn cg1 of: COEFF(r,v,p,cg1,c,cg2,s)* FLOW(r,v,t,p,c,s)} (4-5) Where COEFF(r,v,p,cg1,c,cg2,s) takes nto account the harmonzaton of dfferent tme-slce resoluton of the flow varables, whch have been omtted here for smplcty, as well as commodty-dependent transformaton effcences Lmtng flow shares n flexble processes When ether of the commodty groups cg1 or cg2 contans more than one element, the prevous constrant allows a lot of freedom on the values of flows. The process s therefore qute flexble. The flow share constrant s ntended to lmt the flexblty, by constranng the share of each flow wthn ts own group. For nstance, a refnery output mght consst of three refned products: c1=lght, c2=medum, and c3=heavy dstllate. If losses are 9% of the nput, then the user must specfy FLOFUNC = 0.91 to defne the overall effcency. The user may then want to lmt the flexblty of the slate of outputs by means of three FLOSHAR(c) coeffcents, say 0.4, 0.5, 0.6, resultng n three flow share constrants as follows: FLOW(c1) 0.4*[FLOW(c1) + FLOW(c2) + FLOW(c3)], so that c1 s at most 40% of the total output, FLOW(c2) 0.5*[FLOW(c1) + FLOW(c2) + FLOW(c3)], so that c2 s at most 50% of the total output, FLOW(c3) 0.6*[FLOW(c1) + FLOW(c2) + FLOW(c3)], so that c3 s at most 60% of the total output, The general form of ths constrant s: EQ_INSHR(c,cg,p,r,t,s) and EQ_OUTSHR(c,cg,p,r,t,s) FLOW(c),, = FLOSHAR(c) * Sum {over all c n cg of: FLOW(c ) } (4-6) The commodty group cg may be on the nput or output sde of the process Peakng Reserve Constrant (tme-slced commodtes only) Ths constrant mposes that the total capacty of all processes producng a commodty at each tme perod and n each regon must exceed the average demand n the tme-slce where peakng occurs by a certan percentage. Ths percentage s the Peak Reserve 43

44 Factor, RESERV(r,t,c), and s chosen to nsure aganst several contngences, such as: possble commodty shortfall due to uncertanty regardng ts supply (e.g. water avalablty n a reservor); unplanned equpment down tme; and random peak demand that exceeds the average demand durng the tme-slce when the peak occurs. Ths constrant s therefore akn to a safety margn to protect aganst random events not explctly represented n the model. In a typcal cold country the peakng tme-slce for electrcty (or natural gas) wll be Wnter-Day, and the total electrc plant generatng capacty (or gas supply plant) must exceed the Wnter-Day demand load by a certan percentage. In a warm country the peakng tme-slce may be Summer-Day for electrcty (due to heavy ar condtonng demand). The user controls for whch tme-slces a peakng equaton s mantaned. For each tme perod t and for regon r, there must be enough nstalled capacty to exceed the requred capacty n the season wth largest demand for commodty c by a safety factor E called the peak reserve factor. EQ_PEAK(r,t,c,s) - commodty peak requrements Sum {over all p producng c wth c=pcg of CAPUNIT(r,p) * Peak(r,v,p,c,s) * FR(s) *CAP(r,v,t,p) * ACTFLO(r,v,p,c) } + Sum {over all p producng c wth c#pcg of Peak(r,v,p,c,s) *FLOW(r,v,t,p,c,s) + TRADE(r,t,p,c,s,) (4-7) [1+RESERVE(r,t,c,s)] * [ Sum {over all p consumng c of FLOW(r,v,t,p,c,s) + TRADE(r,t,p,c,s,e) } ] where: RESERV(r,t,c,s) s the regon-specfc reserve coeffcent for commodty c n tme-slce s, whch allows for unexpected down tme of equpment, for demand at peak, and for uncertan resource avalablty, and Peak(r,v,p,c,s) (never larger than 1) specfes the fracton of technology p s capacty n a regon r for a perod t and commodty c (electrcty or heat only) that s allowed to contrbute to the peak load n slce s; many types of supply processes are predctably avalable durng the peak and thus have a peak coeffcent equal to 1, whereas others (such as wnd turbnes or solar plants n the case of electrcty) are attrbuted a peak coeffcent less than 1, snce they are on average only fractonally avalable at peak (e.g., a wnd turbne typcally has a peak coeffcent of.25 or.3, whereas a hydroelectrc plant, a gas plant, or a nuclear plant typcally has a peak coeffcent equal to 1). 44

45 For smplcty t has been assumed n (4-7) that the tme-slce resoluton of the peakng commodty and the tme-slce resoluton of the commodty flows (FLOW, TRADE) are the same. In practce, ths s not the case and addtonal converson factors or summaton operatons are necessary to match dfferent tmeslce levels. Remark: to establsh the peak capacty, two cases must be dstngushed n equaton EQ_PEAK. For producton processes where the peakng commodty s the only commodty n the prmary commodty group (denoted c=pcg), the capacty of the process may be assumed to contrbute to the peak. For processes where the peakng commodty s not the only member of the pcg, there are several commodtes ncluded n the pcg. Therefore, the capacty as such cannot be used n the equaton. In ths case, the actual producton s taken nto account n the contrbuton to the peak, nstead of the capacty. For example, n the case of CHP only the producton of electrcty contrbutes to the peak electrcty supply, not the entre capacty of the plant, because the actvty of the process conssts of both electrcty and heat generaton n ether fxed or flexble proportons, and, dependng on the modeler's choce, the capacty may represent ether the electrc power of the turbne n condensng or back-pressure mode, or the sum of power and heat capactes n back-pressure mode. Note also that n the peak equaton (4-7), t s assumed that mports of the commodty are contrbutng to the peak of the mportng regon (thus, exports are of the frm power type) Constrants on commodtes In TIMES varables are optonally attached to varous quanttes related to commodtes, such as total quantty produced. Therefore t s qute easy to put constrants on these quanttes, by smply boundng the commodty varables at each perod. It s also possble to mpose cumulatve bounds on commodtes over more than one perod, a partcularly useful feature for cumulatvely boundng emssons or modelng reserves of fossl fuels. By ntroducng sutably namng conventons for emssons the user may constran emssons from specfc sectors. Furthermore, the user may also mpose global emsson constrants that apply to several regons taken together, by allowng emssons to be traded across regons. Alternatvely or concurrently a tax or penalty may be appled to each produced (or consumed) unt of a commodty (energy form, emsson), va specfc parameters. A specfc type of constrant my be defned to lmt the share of process (p) n the total producton of commodty (c). The constrant ndcates that the flow of commodty (c) from/to process (p) s bounded by a gven fracton of the total producton of commodty (c). In the present mplementaton, the same gven fracton s appled to all tme slces. 45

46 4.4.9 User Constrants In addton to the standard TIMES constrants dscussed above, the user nterested n developng reference case projectons of energy market behavor typcally ntroduces addtonal constrants to express these specal condtons. For example, there may a userdefned constrant lmtng nvestment n new nuclear capacty (regardless of the type of reactor), or dctatng that a certan percentage of new electrcty generaton capacty must be powered by renewable energy sources. User constrants may be employed across tme perods, to for example model optons for retrofttng and lfe extenson for processes. In order to facltate the creaton of a new user constrant TIMES provdes a template for ndcatng a) the set of varables nvolved n the constrant, and b) the user-defned coeffcents needed n the constrant Representaton of ol refnng n MARKAL Two alternatve approaches are avalable n MARKAL to represent ol refnng. Under the smplfed approach that s adopted n the majorty of MARKAL models, the refnery s treated as a set of one or more standard MARKAL technologes. But a more sophstcated approach s avalable where the modeler wshes to specfy bona fde qualty requrement constrants for each refned product, such as: Octane Ratng, Sulfur Content, Flash Index, Densty, Cetane Number, Vscosty, Red Vapor Pressure, etc. Ths approach s emboded n the specal ol-refnng module of MARKAL whose man features are outlned n ths secton and requres addtonal parameters, varables, and constrants New sets and parameters Frst, n order to properly apply the specfc constrants to ths sector, the model requres that several sets and other parameters that are unque to ths sector be defned, as follows: - Constant REFUNIT: specfes n what unts the refnery streams are defned (volume, weght, or energy) - Set REF: contans the lst of refnng processes (a subset of PRC) - Set OPR: contans the ntermedate energy carrers (refnery streams) that are produced by the members of REF, plus the avalable crude ols. Each stream wll enter the producton of one or more RPP. The members of OPR are expressed n unts specfed by REFUNIT (volume, weght, or energy). - Set BLE: contans the product commodty of the blendng actvty - Parameter CONVERT: contans the densty and energy content (by weght or by volume) of each blendng stream. These parameters are used as coeffcents of the blendng equatons to convert them to correct unts (see below). - Set SPE: contans the names of the specfcatons that must be mposed on refned petroleum products (RPPs), such as octane ratng, sulfur content, etc. 46

47 - Parameter BL_COM: contans the values of the blendng specfcatons SPE for the blendng streams OPR - Parameter BL_SPEC: contans the value of the specfcaton SPE of the blendng product BLE New varables Once these constants, sets and tables are defned by the user, the model automatcally creates blendng varables as follows: BLND t,,ble,opr : s the amount of blendng stock OPR enterng the producton of the refned product ble at tme perod t New blendng constrants These varables and nput parameters are fnally used to express the followng two types of blendng constrants: Blendng by volume (e.g. octane ratng) opr OPR ( oct BLND,, ) = oct BLND,, opr t ble opr ble opr OPR t ble opr where - oct opr s the octane content by volume of one unt of stream opr (tself expressed n unts REFUNIT. If REFUNIT s not equal to volume, some converson coeffcents (specfed n table CONVERT) must be appled to the varables of the equaton. - Oct ble s the requred volume octane ratng of the refned product enc - BLND varables are expressed n volume unts. Blendng by weght (e.g. sulfur content) opr OPR ( sulf opr BLNDt, ble, opr ) = sulf ble BLNDt, ble, opr OPR where - sulf opr s the sulfur content by weght of one unt of stream opr (tself expressed n unts REFUNIT. If REFUNIT s not equal to weght, then converson coeffcents (specfed n table CONVERT) must be appled to the varables of the equaton. - sulf blec s the weght sulfur content of the refned product enc - BLND varables are expressed n weght unts. opr 47

48 4.5 Lnear Programmng complements Ths secton s not strctly needed for a basc understandng of the TIMES model and may be skpped on a frst readng. However, t provdes addtonal nsghts nto the mcroeconomcs of the TIMES equlbrum. In partcular, t contans a revew of the theoretcal foundaton of Lnear Programmng and Dualty Theory. Ths knowledge may help the user to better understand the central role shadow prces and reduced costs play n the economcs of the TIMES model. More complete treatments of Lnear Programmng and Dualty Theory may be found n several standard textbooks such as Chvátal (1983) or Hller and Leberman (1990 and subsequent edtons). Samuelson and Nordhaus (1977) contans a treatment of mcro-economcs based on mathematcal programmng A bref prmer on Lnear Programmng and Dualty Theory Basc defntons In ths subsecton, the superscrpt t followng a vector or matrx represents the transpose of that vector or matrx. A Lnear Program may always be represented as the followng Prmal Problem n canoncal form: Max c t x (4-7) s.t. Ax b (4-8) x 0 (4-9) where x s a vector of decson varables, c t x s a lnear functon representng the objectve to maxmze, and Ax b s a set of nequalty constrants. Assume that the LP has a fnte optmal soluton, x*. Then each decson varable, x* j falls nto one of three categores. x * j may be: equal to ts lower bound (as defned n a constrant), or equal to ts upper bound, or strctly between the two bounds. In the last case, the varable x* j s called basc. Otherwse t s non-basc. For each prmal problem, there corresponds a Dual problem derved as follows: Mn b t y (4-10) s.t. A t y c (4-11) y 0 (4-12) 48

49 Note that the number of dual varables equals the number of constrants n the prmal problem. In fact, each dual varable y may be assgned to ts correspondng prmal constrant, whch we represent as: A x b, where A s the th row of matrx A Dualty Theory Dualty theory conssts manly of three theorems 26 : weak dualty, strong dualty, and complmentary slackness. Weak Dualty Theorem If x s any feasble soluton to the prmal problem and y s any feasble soluton to the dual, then the followng nequalty holds: c t x b t y (4-13) The weak dualty theorem states that the value of a feasble dual objectve s never smaller than the value of a feasble prmal objectve. The dfference between the two s called the dualty gap for the par of feasble prmal and dual solutons (x,y). Strong dualty theorem If the prmal problem has a fnte, optmal soluton x*, then so does the dual problem (y*), and both problems have the same optmal objectve value (ther dualty gap s zero): c t x* = b t y* (4-14) Note that the optmal values of the dual varables are also called the shadow prces of the prmal constrants. Complementary Slackness theorem At an optmal soluton to an LP problem: If y* s > 0 then the correspondng prmal constrant s satsfed at equalty (.e. A x*=b and the th prmal constrant s called tght. Conversely, f the th prmal constrant s slack (not tght), then y* = 0, If x* j s basc, then the correspondng dual constrant s satsfed at equalty, (.e. A t j*y =c j, where A t j s the j th row of A t,.e. the j th column of A. Conversely, f the j th dual constrant s slack, then x* j s equal to one of ts bounds. Remark: Note however that a prmal constrant may have zero slack and yet have a dual equal to 0. And, a prmal varable may be non basc (.e. be equal to one of ts bounds), and yet the correspondng dual slack be stll equal to 0. These stuatons are dfferentcases of the so-called degeneracy of the LP. They ofetn occur when constrants are over specfed (a trval case occurs f a constrant s repeated twce n the LP) 26 Ther proofs may be found n most textbooks on Lnear Programmng, such as Chvatal (1983) or Hller and Leberman (1990). 49

50 4.5.2 Senstvty analyss and the economc nterpretaton of dual varables It may be shown that f the j th RHS b j of the prmal s changed by an nfntesmal amount d, and f the prmal LP s solved agan, then ts new optmal objectve value s equal to the old optmal value plus the quantty y j * d, where y j * s the optmal dual varable value. Loosely speakng 27, one may say that the partal dervatve of the optmal prmal objectve functon s value wth respect to the RHS of the th prmal constrant s equal to the optmal shadow prce of that constrant Economc Interpretaton of the Dual Varables If the prmal problem conssts of maxmzng the surplus (objectve functon c t x), by choosng an actvty vector x, subject to upper lmts on several resources (the b vector) then: Each a j coeffcent of the dual problem matrx, A, then represents the consumpton of resource b j by actvty x ; The optmal dual varable value y* j s the unt prce of resource j, and The total optmal surplus derved from the optmal actvty vector, x*, s equal to the total value of all resources, b, prced at the optmal dual values y* (strong dualty theorem). Furthermore, each dual constrant A t j*y c j has an mportant economc nterpretaton. Based on the Complementary Slackness theorem, f an LP soluton x* s optmal, then for each x* j that s not equal to ts upper or lower bound (.e. each basc varable x* j), there corresponds a tght dual constrant y*a j = c j, whch means that the revenue coeffcent c j must be exactly equal to the cost of purchasng the resources needed to produce one unt of x j. In economsts terms, margnal cost equals margnal revenue, and both are equal to the market prce of x* j. If a varable s not basc, then by defnton t s equal to ts lower bound or to ts upper bound. In both cases, the unt revenue c j need not be equal to the cost of the requred resources. The technology s then ether non-compettve (f t s at ts lower bound) or t s super compettve and makes a surplus (f t s at ts upper bound). Example: The optmal dual value attached to the balance constrant of commodty c represents the change n objectve functon value resultng from one addtonal unt of the commodty. Ths s precsely the nternal prce of that commodty Reduced Surplus and Reduced Cost In a maxmzaton problem, the dfference y*a j - c j s called the reduced surplus of technology j, and s avalable from the soluton of a TIMES problem. It s a useful ndcator of the compettveness of a technology, as follows: 27 Strctly speakng, the partal dervatve may not exst for some values of the RHS, and may then be replaced by a drectonal dervatve. 50

51 f x* j s at ts lower bound, ts unt revenue c j s less than the resource cost (.e. ts reduced surplus s postve). The technology s not compettve (and stays at ts lower bound n the equlbrum); f x* j s at ts upper bound, revenue c j s larger than the cost of resources (.e. ts reduced surplus s negatve). The technology s super compettve and produces a surplus; and f x* j s basc, ts reduced surplus s equal to 0. The technology s compettve but does not produce a surplus. We now restate the above summary n the case of a Lnear Program that mnmzes cost subject to constrants: s.t. Mn c t x Ax b x 0 In a mnmzaton problem (such as the usual formulaton of TIMES), the dfference c j - y*a j s called the reduced cost of technology j. The followng holds: f x* j s at ts lower bound, ts unt cost c j s larger than the value created (.e. ts reduced cost s postve). The technology s not compettve (and stays at ts lower bound n the equlbrum); f x* j s at ts upper bound, ts cost c j s less than the value created (.e. ts reduced cost s negatve). The technology s super compettve and produces a proft; and f x* j s basc, ts reduced cost s equal to 0. The technology s compettve but does not produce a proft The reduced costs/surpluses may thus be used to rank all technologes, ncludng those that are not selected by the model. 51

52 5 A comparson of the TIMES and MARKAL models Ths chapter contans a pont-by-pont comparson of the TIMES and MARKAL models. It s of nterest prmarly to modelers already famlar wth MARKAL, and to modelers who are consderng adopton of ether model. The descrptons of the features gven below are not detaled, snce they are repeated elsewhere n the documentaton. Rather, the functon of ths chapter s to gude the reader, by mentonng the features that are present n one model and not n the other. 5.1 Smlartes The TIMES and the MARKAL models share the same basc modelng paradgm. Both models are technology explct, dynamc partal equlbrum models of energy markets. In both cases the equlbrum s obtaned by maxmzng the total surplus of consumers and supplers va Lnear Programmng. The two models also share the mult-regonal feature, whch allows the modeler to construct geographcally ntegrated (even global) nstances. These fundamental features were descrbed n chapter 3 of ths documentaton, and Secton 1.3, PART I of the MARKAL documentaton, and consttute the backbone of the common paradgm. However, there are also sgnfcant dfferences n the two models, whch we now outlne. These dfferences do not affect the basc paradgm common to the two models, but rather some of ther techncal features and propertes. 5.2 TIMES features not n MARKAL Varable length tme perods MARKAL has fxed length tme perods. However TIMES allows the user to defne perod lengths n a completely flexble way. Ths s a major model dfference, whch ndeed requred a complete re-defnton of the mathematcs of most TIMES constrants and of the TIMES objectve functon. The varable perod length feature s very useful n two nstances: frst f the user wshes to use a sngle year as ntal perod (handy for calbraton purposes), and second when the user contemplates long horzons, where the frst few perods may be descrbed n some detal by relatvely short perods (say 5 years), whle the longer term may be regrouped nto a few perods wth long duratons (perhaps 20 or more years) Data decouplng Ths somewhat msunderstood feature does not confer addtonal power to TIMES, but t greatly smplfes the mantenance of the model database and allows the user great flexblty n modfyng the new defnton of the plannng horzon. In TIMES all nput data are specfed by the user ndependently from the defnton of the tme perods employed for a partcular model run. All tme-dependent nput data are specfed by the year n whch the data apples. The model then takes care of matchng the data wth the 52

53 perods, wherever requred. If necessary the data s nterpolated (or extrapolated) by the model preprocessor code to provde data ponts at those tme perods requred for the current model run. In addton, the user has control over the nterpolaton and extrapolaton of each tme seres. The general rule of data decouplng apples also to past data: whereas n MARKAL the user had to provde the resdual capacty profles for all exstng technologes n the ntal perod, and over the perods n whch the capacty remans avalable, n TIMES the user provdes techncal and cost data at those past years when the nvestments actually took place, and the model takes care of calculatng how much capacty remans n the varous modelng perods. Thus, past and future data are treated essentally n the same manner n TIMES. One nstance when the data decouplng feature mmensely smplfes model management s when the user wshes to change the ntal perod, and/or the lengths of the perods. In TIMES, there s essentally nothng to do, except declarng the dates of the new perods. In MARKAL, such a change represents a much larger effort requrng a substantve revson of the database Flexble tme slces and storage processes In MARKAL, only two commodtes have tme-slces: electrcty and low temperature heat, and ther tme slces are rgdly defned (sx tme-slces for electrcty and three for heat). In TIMES, any commodty and process may have ts own, user-chosen tme-slces. These flexble tme-slces are segregated nto three groups, seasonal (or monthly), weekly (weekday vs weekend), and daly (day/nght), where any level may be expanded (contracted) or omtted. The flexble nature of the TIMES tme-slces s supported by storage processes that consume commodtes at one tme-slce and release them at another. MARKAL only supports nght-to-day (electrcty) storage. Note that many TIMES parameters may be tme-slce dependent (such as avalablty factor (AF), basc effcency (FLO_FUNC), etc) Process generalty In MARKAL processes n dfferent RES sectors are endowed wth dfferent (data and mathematcal) propertes. For nstance, end-use processes do not have actvty varables (actvty s then equated to capacty), and source processes have no nvestment varables. In TIMES, every process has the same basc features, whch are actvated or not solely va data specfcaton Flexble processes 53

54 In MARKAL processes are by defnton rgd, except for some specalzed processes whch permt flexble output (such as lmt refneres or pass-out turbne CHPs), and thus outputs and nputs are n fxed proportons wth one another. In TIMES, the stuaton s reversed, and each process starts by beng entrely flexble, unless the user specfes certan coeffcents to rgdly lnk nputs to outputs. Ths feature permts better modelng of many real-lfe processes as a sngle technology, where MARKAL requres several technologes (as well as dummy commodtes) to acheve the same result. A typcal example s that of a boler that accepts any of 3 lqud fuels as nput, but whose effcency depends on the fuel used. In MARKAL, to model ths stuaton requres four processes (one per possble fuel plus one that carres the nvestment cost and other parameters), plus one dummy fuel. In TIMES one process s suffcent, and no dummy fuel s requred. Note also that TIMES has a number of parameters that lmt the nput share of each fuel, whereas n MARKAL, mposng such lmts requres that the user defne several user constrants Investment and dsmantlng lead-tmes and costs New TIMES parameters allow the user to model the constructon phase and dsmantlng of facltes that have reached ther end-of-lfe. These are: lead tmes attached to the constructon or to the dsmantlng of facltes, captal cost for dsmantlng, survellance costs durng the dsmantlng lead-tme. Lke n MARKAL, there s also the possblty to defne flows of commodtes consumed at constructon tme, or released at dsmantlng tmes, thus allowng the representaton of lfe-cycle energy and emsson accountng Vntaged processes and age-dependent parameters The varables assocated wth user declared vntaged processes employ both the tme perod p and vntage perod v (n whch new nvestments are made and assocated nput data s obtaned). The user ndcates that a process s to be modeled as a vntaged process by usng a specal vntage parameter. Note that n MARKAL vntagng s possble only for demand devces (for whch there s no actvty varable) or va the defnton of several replcas of a process, each replca beng a dfferent vntage. In TIMES, the same process name s used for all vntages of the same process. 29 In addton, some parameters can be specfed to have dfferent values accordng to the age of the process. In the current verson of TIMES, these parameters nclude the avalablty factors, the n/out flow ratos (equvalent to effcences), and the fxed cost 28 In the end the two models use equvalent mathematcal expressons to represent a flexble process. Only TIMES reduces the user s effort to a mnmum, whle MARKAL requres the user to manually defne the multple processes, dummy fuels and user constrants. 29 The representaton of vntage as a separate ndex helps elmnate a common confuson that exsted n MARKAL, namely the confuson of vntage wth the age of a process. For nstance, f the user defnes an annual cost for a car equal to 10 n 2005 and only 8 n 2010, the decrease would not only apply to cars purchased n 2010, but also to cars purchased n 2005 and earler when they reach the 2010 perod. 54

55 parameters only. Several other parameters could, n prncple, be defned to be agedependent, but such extensons have not been mplemented yet Commodty related varables MARKAL has very few commodty related varables, namely exports/mports, and emssons. TIMES has a large number of commodty-related varables such as: total producton, total consumpton, but also specfc varables representng the flows of commodtes enterng or extng each process. Ths allows the user many handles to put lmts, and costs on commodtes More accurate and realstc depcton of nvestment cost payments In MARKAL each nvestment s assumed to be pad n ts entrety at the begnnng of some tme perod. In TIMES the tmng of nvestment payments s qute detaled. For large facltes (e.g. a nuclear plant), captal s progressvely lad out n yearly ncrements over the faclty s constructon tme, and furthermore, the payment of each ncrement s made n nstallments spread over the economc lfe of the faclty. For small processes (e.g. a car) the capacty expanson s assumed to occur regularly each year rather than n one large lump, and the payments are therefore also spread over tme. Furthermore, when a tme perod s qute long (.e. longer that the lfe of the nvestment), TIMES has an automatc mechansm to repeat the nvestment more than once over the perod. These features allow for a much smoother (and more realstc) representaton of the stream of captal outlays n TIMES than n MARKAL. Moreover, n TIMES all dscount rates can be defned to be tme-dependent, whereas n MARKAL both the general and technology-specfc dscount rates are constant over tme Clmate equatons TIMES now possesses a set of varables and equatons that endogenze the concentraton of CO2 and also calculate the radatve forcng and global temperature change resultng from GHG emssons and accumulaton n the atmosphere. Ths new feature s descrbed n chapter 7 of PART II. 55

56 5.3 MARKAL features not n TIMES Over the years, several extensons were added to the MARKAL model. Most of these extensons have been mplemented n TIMES, for nstance elastc demands, multregonal tradng, lumpy nvestments, and endogenous technologcal learnng. Four extensons of MARKAL have not (yet) been mplemented n TIMES, namely: Stochastc programmng, Integraton of damage costs n the objectve functon, Myopc energy markets (SAGE), and MARKAL-MACRO. There are plans to mplement most of these mssng optons n the future. The nterested reader s referred to the MARKAL documentaton 30 for the descrptons of these four optons, and the full descrpton of the model wth lnks to Standard MARKAL, MARKAL-MACRO and SAGE wrte-ups. 56

57 6 Elastc demands and the computaton of the supply-demand equlbrum In the precedng chapters, we have seen that TIMES does more than mnmze the cost of supplyng energy servces. Instead, t computes a supply-demand equlbrum where both the supply optons and the energy servce demands are computed by the model. The equlbrum s drven by the user-defned specfcaton of demand functons, whch determne how each energy servce demand vares as a functon of the market prce of that energy servce. The TIMES code assumes that each demand has constant own-prce elastcty n a gven tme perod, and that cross prce elastctes are zero. Economc theory establshes that the equlbrum thus computed corresponds to the maxmzaton of the net total surplus, defned as the sum of the supplers and of the consumers surpluses (Samuelson, 1952, Takayama and Judge, 1972). The total net surplus has been often consdered a vald metrc of socetal welfare n mcroeconomc lterature, and ths fact confers strong valdty to the equlbrum computed by TIMES. The TIMES model s normally run n two contrasted modes: frst to smulate some reference case, and then to smulate alternate scenaros, each of whch departs n some way from the reference case assumptons and parameters. For nstance, an alternate scenaro may make dfferent assumptons on the avalablty or the cost of some new technologes. Or, t may assume that certan energy or envronmental polces are beng mplemented (e.g. emsson taxes, or portfolo standards, or effcency mprovements). Or agan, a scenaro may assume that a certan goal must be reached (such as a cap on emssons), leavng the model free to acheve that goal at least cost. In almost any such alternate scenaro, some stran s put on some sectors, resultng n ncreases n the margnal values of at least some energy servces (for example, severe emsson reductons may ncrease the prce of auto transportaton). In TIMES demands self-adjust n reacton to changes (relatve to the reference case) of ther own prce, and therefore the model goes beyond the optmzaton of the energy sector only. Thus although TIMES falls short of computng a general equlbrum, t does capture a major element 31 of the feedback effects not prevously accounted for n bottom-up energy models. In ths chapter, we explan how Lnear Programmng computes the equlbrum. Addtonal techncal detals may be found n Tosato (1980) and n Loulou and Lavgne (1995). One of the frst large scale applcaton of these methods was realzed n the Project Independence Energy System (PIES, Hogan, 1975), although n the context of demands for fnal energy rather than for energy servces as n TIMES or MARKAL. 6.1 Theoretcal consderatons: the Equvalence Theorem The computatonal method s based on the equvalence theorem presented n chapter 3, whch we restate here: 31 It has been argued, based on strong crcumstantal evdence, that the change n demands for energy servces s ndeed the man feedback economc effect of energy system polces (Loulou and Kanuda, 2002) 57

58 "A supply/demand economc equlbrum s reached when the sum of the producers and the consumers surpluses s maxmzed" Fgure 3.2 of Chapter 3 provdes a graphcal llustraton of ths theorem n a case where only one commodty s consdered. 6.2 Mathematcs of the TIMES equlbrum Defnng demand functons For each demand category, defne a demand curve,.e. a functon determnng demand as a functon of prce. In TIMES, a constant elastcty relatonshp s used, represented as: E DM ( p ) = K p (6 1) where DM s the th demand, p s ts prce, taken to be the margnal cost of procurng the th commodty, and E s the own prce elastcty of that demand. Note that although the regon and tme ndexes r, t have been omtted n ths notaton, all quanttes n Equaton (1.6-1), ncludng the elastctes are regon (f approprate) and tme dependent. Constant K may be obtaned f one pont (p 0,DM 0 ) of the curve s known (the reference case). Thus Equaton (1.6-1) may be rewrtten as: DM / DM 0 0 E = ( p / p ) (6 2) Or ts nverse: p = p 0 ( DM / DM 0 1/ ) E where the superscrpt 0 ndcates the reference case, and the elastcty E s negatve. Note also that the elastcty may have two dfferent values, one for upward changes n demand, the other for downward changes Formulatng the TIMES equlbrum Wth nelastc demands, the TIMES model may be wrtten as the followng Lnear Program Mn s. t. and c X k CAP k, B X b ( t) DM ( t) = 1,2,.., I; t = 1,.., T (6 3) (6 4) (6 5) where X s the vector of all varables and I s the number of demand categores. In words: 58

59 (6-3) expresses the total dscounted cost to be mnmzed. (6-4) s the set of demand satsfacton constrants (where the CAP varables are the capactes of end-use technologes, and the DM rght-hand-sdes are the exogenous demands to satsfy). (6-5) s the set of all other constrants. Wth elastc demands the role of TIMES s to compute a supply/demand equlbrum among equatons (1.6-3) through (1.6-5) where both the supply sde and the demand sde adjust to changes n prces, and the prces charged by the supply sde are the margnal costs of the demand categores (.e. p s the margnal cost of producng demand DM ). A pror ths seems to be a dffcult task, because the prces used on the demand sde are computed as part of the soluton to equatons (1.6-3), (1.6-4), and (1.6-5). The Equvalence Theorem, however, states that such an equlbrum s reached as the soluton of the followng mathematcal program, where the objectve s to maxmze the net total surplus: s. t. and Max k CAP B X b t p k, 0 ( t) ( t) DM 0 [ DM ( t) ] ( t) 0 1 / E DM ( t ) 1 / E a q dq c X = 1,.., I ; t = 1,.., T (6 6) (6 7) (6 8) Where X s the vector of all TIMES varables wth assocated cost vector c, (6-6) expresses the total net surplus, and DM s now a vector of varables n (6-7), rather than fxed demands. The ntegral n (6-6) s easly computed, yeldng the followng maxmzaton program: Max s. t / E 1+ 1/ E ( p ( t) [ DM ( t) ] DM ( t) /(1 + 1/ E ) ) t k CAP k, B X b ( t) DM ( t) = 1,.., I; t = 1,.., T c X (6 6)' (6 7)' (6 8)' Lnearzaton of the Mathematcal Program The Mathematcal Program emboded n (6-6), (6-7) and (6-8) has a non-lnear objectve functon. Because the latter s separable (.e. does not nclude cross terms) and concave n the DM varables, each of ts terms s easly lnearzed by pece-wse lnear functons whch approxmate the ntegrals n (6-6). Ths s the same as sayng that the nverse demand curves are approxmated by starcase functons, as llustrated n fgure 6.1. By so dong, the resultng optmzaton problem becomes lnear agan. The lnearzaton proceeds as follows. 59

60 and a) For each demand category, the user selects a range wthn whch t s estmated that the demand value DM (t) wll always reman, even after adjustment for prce effects (for nstance the range could be equal to the reference demand DM o (t) plus or mnus 50%). The smallest range value s denoted DM(t) mn. b) Select a grd that dvdes each range nto a number n of equal wdth ntervals. Let ß (t) be the resultng common wdth of the grd, ß (t)= R (t)/n. See Fgure 6.1 for a sketch of the non-lnear expresson and of ts step-wse constant approxmaton. The number of steps, n, should be chosen so that the step-wse constant approxmaton remans close to the exact value of the functon. c) For each demand segment DM (t) defne n step-varables (one per grd nterval), denoted s 1, (t), s 2, (t),, s n, (t). Each s varable s bounded below by 0 and above by ß (t). One may now replace n equatons (6-6) and (6-7) each DM (t) varable by the sum of the n-step varables, and each non-lnear term n the objectve functon by a weghted sum of the n step-varables, as follows: DM ( t) = DM ( t) DM ( t) 1+ 1/ E DM ( t) n mn + s j, ( t) 6 9 j= / E mn ( t) s ( t) / β ( t) The resultng Mathematcal Program s now fully lnearzed. + n j= 1 A 6 10 Remark: nstead of maxmzng the lnearzed objectve functon, TIMES mnmzes ts negatve, whch then has the dmenson of a cost. The porton of that cost representng the negatve of the consumer surplus s akn to a welfare loss. j, s, j, 60

61 F(x) = x (1/E) A,j (t) β (t) x=dm (t) Range = R (t) Fgure 6.1. Step-wse approxmaton of the non-lnear terms n the objectve functon Calbraton of the demand functons Besdes selectng elastctes for the varous demand categores, the user must evaluate each constant K (t). To do so, we have seen that one needs to know one pont on each demand functon n each tme perod, { p 0 (t),dm 0 (t) }. To determne such a pont, we perform a sngle prelmnary run of the nelastc TIMES model (wth exogenous DM 0 (t)), and use the resultng shadow prces p 0 (t) for all demand constrants, n all tme perods for each regon Computatonal consderatons Each demand segment that s elastc to ts own prce requres the defnton of as many varables as there are steps n the dscrete representaton of the demand curve (both upward and down f desred), for each perod and regon. Each such varable has an upper bound, but s otherwse nvolved n no new constrant. Therefore, the lnear program s augmented 61

62 by a number of varables, but does not have more constrants than the ntal nelastc LP (wth the excepton of the upper bounds). It s well known that wth the modern LP codes the number of varables has lttle or no mpact on computatonal tme n Lnear Programmng, whether the varables are upper bounded or not. Therefore, the ncluson n TIMES of elastc demands has a very mnor mpact on computatonal tme or on the tractablty of the resultng LP. Ths s an mportant observaton n vew of the very large LP s that result from representng mult-regonal and global models n TIMES Interpretng TIMES costs, surplus, and prces It s mportant to note that, nstead of maxmzng the net total surplus, TIMES mnmzes ts negatve (plus a constant), obtaned by changng the sgns n expresson (6-6). For ths and other reasons, t s napproprate to pay too much attenton to the meanng of the absolute objectve functon values. Rather, examnng the dfference between the objectve functon values of two scenaros s a far more useful exercse. That dfference s of course, the negatve of the dfference between the net total surpluses of the two scenaro runs. Note agan that the popular nterpretaton of shadow prces as the margnal costs of model constrants s naccurate. Rather, the shadow prce of a constrant s, by defnton, the ncremental value of the objectve functon per unt of that constrant s rght hand sde (RHS). The nterpretaton s that of an amount of surplus loss per unt of the constrant s RHS. The dfference s subtle but nevertheless mportant. For nstance, the shadow prce of the electrcty balance constrant s not necessarly the margnal cost of producng electrcty. Indeed, when the RHS of the balanced constrant s ncreased by one unt, one of two thngs may occur: ether the system produces one more unt of electrcty, or else the system consumes one unt less of electrcty (perhaps by choosng more effcent end-use devces or by reducng an electrcty-ntensve energy servce, etc.). It s therefore correct to speak of shadow prces as the margnal system value of a resource, rather than the margnal cost of procurng that resource. 62

63 7 The Lumpy Investment opton In some cases, the lnearty property of the TIMES model may become a drawback for the accurate modelng of certan nvestment decsons. Consder for example a TIMES model for a relatvely small communty such as a cty. For such a scope the granularty of some nvestments may have to be taken nto account. For nstance, the sze of an electrcty generaton plant proposed by the model would have to conform to an mplementable mnmum sze (t would make no sense to decde to construct a 50 MW nuclear plant). Another example for mult-regon modellng mght be whether or not to buld cross-regon electrc grd(s) or gas ppelne(s) n dscrete sze ncrements. Processes subject to nvestments of only specfc sze ncrements are descrbed as lumpy nvestments. For other types of nvestments, sze does not matter: for nstance the model may decde to purchase 10, electrc cars, whch s easly rounded to 10,950 wthout any serous nconvenence. The stuaton s smlar for a number of resdental or commercal heatng devces, or for the capacty of wnd turbnes or ndustral bolers, or for any technologes wth relatvely small mnmum feasble szes. Such technologes would not be canddates for treatment as lumpy nvestments. It s the user s responsblty to decde that certan technologes should (or should not) respect the mnmum sze constrant, weghng the pros and cons of so dong. Ths chapter explans how the TIMES LP s transformed nto a Mxed Integer Program (MIP) to accommodate mnmum or multple sze constrants, and states the consequences of so dong on computatonal tme and on the nterpretaton of dualty results. The lumpy nvestment opton avalable n TIMES s slghtly more general than the one descrbed above. It nsures that nvestment n technology k s equal to one of a fnte number N of pre-determned szes: 0, S 1 (t), S 2 (t),,s N (t). As mpled by the notaton, these dscrete szes may be dfferent at dfferent tme perods. Note that by choosng the N szes as the successve multples of a fxed number S, t s possble to nvest (perhaps many tmes) n a technology wth fxed standard sze. Imposng such a constrant on an nvestment s unfortunately mpossble to formulate usng standard LP constrants and varables. It requres the ntroducton of nteger varables n the formulaton. The optmzaton problem resultng from the ntroducton of nteger varables nto a Lnear Program s called a Mxed Integer Program (MIP). 7.1 Formulaton and Soluton of the Mxed Integer Lnear Program Typcally, the modelng of a lumpy nvestment nvolves Integer Varables,.e. varables whose values may only be non-negatve ntegers (0, 1, 2, ). The mathematcal formulaton s as follows 63

64 NCAP( p, t) = wth N = 1 S ( p, t) Z ( p, t) each t = 1,.., T Z ( p, t) = 0 or 1 and N = 1 Z ( p, t) 1 The second and thrd constrants mply that at most one of the Z varables s equal to 1. Thereforew, the frst constrant now means that NCAP s equal to one of the preset szes or s equal to 0, whch s the desred result. Although the formulaton of lumpy nvestments looks smple, t has a profound effect on the resultng optmzaton program. Indeed, MIP problems are notorously more dffcult to solve than LPs, and n fact many of the propertes of lnear programs dscussed n the precedng chapters do not hold for MIPs, ncludng dualty theory, complementary slackness, etc. Note that the constrant that Z(p,t) should be 0 or 1 departs from the dvsblty property of lnear programs. Ths means that the feasblty doman of nteger varables (and therefore of some nvestment varables) s no longer contguous, thus makng t vastly more dffcult to apply purely algebrac methods to solve MIP s. In fact, practcally all MIP soluton algorthms make use (at least to some degree) of partal enumeratve schemes, whch tend to be tme consumng and less relable 32 than the algebrac methods used n LP. The reader nterested n more techncal detals on the soluton of LPs and of MIPs s referred to references (Hller and Leberman, 1990, Nemhauser et al. 1989). In the next secton we shall be content to state one mportant remark on the nterpretaton of the dual results from MIP optmzaton. 7.2 Important remark on the MIP dual soluton (shadow prces) Usng MIP rather than LP has an mportant mpact on the nterpretaton of the TIMES shadow prces. Once the optmal MIP soluton has been found, t s customary for MIP solvers to fx all nteger varables at ther optmal (nteger) values, and to perform one 32 A TIMES LP program of a gven sze tends to have farly constant soluton tme, even f the database s modfed. In contrast, a TIMES MIP may show some erratc soluton tmes. One may observe reasonable soluton tmes (although sgnfcantly longer than LP soluton tmes) for most nstances, wth an occasonal very long soluton tme for some nstances. Ths phenomenon s predcted by the theory of complexty as appled to MIP, see Papadmtrou and Stegltz (1982) 64

65 addtonal teraton of the LP algorthm, so as to obtan the dual soluton (.e. the shadow prces of all constrants). However, the nterpretaton of these prces s dfferent from that of a LP. Consder for nstance the shadow prce of the natural gas balance constrant: n a pure LP, ths value represents the prce of natural gas. In MIP, ths value represents the prce of gas condtonal on havng fxed the lumpy nvestments at ther optmal nteger values. What does ths mean? We shall attempt an explanaton va one example: suppose that one lumpy nvestment was the nvestment n a gas ppelne; then, the gas shadow prce wll not nclude the nvestment cost of the ppelne, snce that nvestment was fxed when the dual soluton was computed. In concluson, when usng MIP, only the prmal soluton s fully relable. In spte of ths major caveat, modelng lumpy nvestments may be of paramount mportance n some nstances, and may thus justfy the extra computng tme and the partal loss of dual nformaton. 65

66 8 Endogenous Technologcal Learnng (ETL) In a long-term dynamc model such as TIMES the characterstcs of many of the future technologes are almost nevtably changng over the sequence of future perods due to technologcal learnng. In some cases t s possble to forecast such changes n characterstcs as a functon of tme, and thus to defne a tme-seres of values for each parameter (e.g. unt nvestment cost, or effcency). In such cases, technologcal learnng s exogenous snce t depends only on tme elapsed and may thus be establshed outsde the model. In other cases there s evdence that the pace at whch some technologcal parameters change s dependent on the experence acqured wth ths technology. Such experence s not solely a functon of tme elapsed, but typcally depends on the cumulatve nvestment (often global) n the technology. In such a stuaton, technologcal learnng s endogenous, snce the future values of the parameters are no longer a functon of tme elapsed alone, but depend on the cumulatve nvestment decsons taken by the model (whch are unknown). In other words, the evoluton of technologcal parameters may no longer be establshed outsde the model, snce t depends on the model s results. ETL s also named Learnng-By-Dong (LBD) by some authors. Whereas exogenous technologcal learnng does not requre any addtonal modelng, endogenous technologcal learnng (ETL) presents a tough challenge n terms of modelng ngenuty and of soluton tme. In TIMES, there s a provson to represent the effects of endogenous learnng on the unt nvestment cost of technologes. Other parameters (such as effcency) are not treated, at ths tme. 8.1 The basc ETL challenge Emprcal studes of unt nvestment costs of several technologes have been undertaken n several countres. Many of these studes fnd an emprcal relatonshp between the unt nvestment cost of a technology at tme t, INVCOST t, and the cumulatve nvestment n that technology up to tme t, Ct = VAR _ INV j. t j= 1 A typcal relatonshp between unt nvestment cost and cumulatve nvestments s of the form: INVCOST = a b t C t ( 8 1) where a s the ntal unt nvestment cost (when C t s equal to 1) and b s the learnng ndex, representng the speed of learnng 33. As experence bulds up, the unt nvestment 33 It s usual to defne, nstead of b, another parameter, pr called the progress rato, whch s related to b va the followng relatonshp: pr = 2 b 66

67 cost decreases, and thus may make nvestments n the technology more attractve. It should be clear that near-sghted nvestors wll not be able to detect the advantage of nvestng early n learnng technologes, snce they wll only observe the hgh ntal nvestment cost and, beng near-sghted, wll not antcpate the future drop n nvestment cost resultng from early nvestments. In other words, tappng the full potental of technologcal learnng requres far-sghted agents who accept makng ntally nonproftable nvestments n order to later beneft from the nvestment cost reducton. Wth regard to actual mplementaton, smply usng (8-1) as the objectve functon coeffcent of VAR_INV t wll yeld a non-lnear, non-convex expresson. Therefore, the resultng mathematcal optmzaton s no longer lnear, and requres specal technques for ts soluton. In TIMES, a Mxed Integer Programmng (MIP) formulaton s used, that we now descrbe. 8.2 The TIMES formulaton of ETL The Cumulatve Investment Cost We follow the basc approach descrbed n Barreto, 2001 The frst step of the formulaton s to express the total nvestment cost,.e. the quantty that should appear n the objectve functon for the nvestment cost of a learnng technology n perod t. TC t s obtaned by ntegratng expresson (8-1): TC t = C 0 t a y b a * dy = C 1 b b+ 1 t (8 2) TC t s a concave functon of C t, wth a shape as shown n fgure 8.1 Hence, 1-pr s the cost reducton ncurred when cumulatve nvestment s doubled. Typcal observed pr values are n a range of.75 to.95 67

68 1600 Cumulatve nvestment cost (B$) Cumulatve nvestment (TW) Fgure 8.1. Example of a cumulatve learnng curve Wth the Mxed Integer Programmng approach mplemented n TIMES, the cumulatve learnng curve s approxmated by lnear segments, and bnary varables are used to represent some logcal condtons. Fgure 8.2 shows a possble pecewse lnear approxmaton of the curve of Fgure 8.1. The choce of the number of steps and of ther respectve lengths s carefully made so as to provde a good approxmaton of the smooth cumulatve learnng curve. In partcular, the steps must be smaller for small values than for larger values, snce the curvature of the curve dmnshes as total nvestment ncreases. The formulaton of the ETL varables and constrants proceeds as follows (we omt the perod, regon, and technology ndexes for notatonal clarty): 1. The user specfes the set of learnng technologes (TEG). 2. For each learnng technology, the user provdes: a) The progress rato pr (from whch the learnng ndex b may be nferred) b) One ntal pont on the learnng curve, denoted (C 0, TC 0 ) c) The maxmum allowed cumulatve nvestment C max (from whch the maxmum total nvestment cost TC max may be nferred) d) The number N of segments for approxmatng the cumulatve learnng curve over the (C 0, C max ) nterval (note that N may be dfferent for dfferent technologes). 3. The model automatcally selects approprate values for the N step lengths, and then proceeds to generate the requred new varables and constrants, and the new objectve functon coeffcents for each learnng technology. The detaled formulae are shown and brefly commented on below. 68

69 TC max Cumulatve nvestment cost C max TC 0 Fgure 8-2. C 0 Cumulatve nvestment Example of a 4-segment approxmaton of the cumulatve cost curve Calculaton of break ponts and segment lengths The successve nterval lengths on the vertcal axs are chosen to be n geometrc progresson, each nterval beng twce at wde as the precedng one. In ths fashon, the ntervals near the low values of the curve are smaller so as to better approxmate the curve n ts hgh curvature zone. Let {TC -1, TC } be the th nterval on the vertcal axs, for = 1,, N-1. Then: N 1 N TC = TC ( TCmax TCo ) / (1 0.5 ), = 1,2,, N Note that TC max s equal to TC N. The break ponts on the horzontal axs are obtaned by pluggng the TC s nto expresson (1.10-2), yeldng: ( 1 b) ( TC ) 1 b, = 1, 2, N C =..., a 1 69

70 8.2.3 New varables Once ntervals are chosen, standard approaches are avalable to represent a concave functon by means of nteger (0-1) varables. We descrbe the approach used n TIMES. Frst, we defne N contnuous varables x, = 1,,N. Each x represents the porton of cumulatve nvestments lyng n the th nterval. Therefore, the followng holds: C N = x = We now defne N nteger (0-1) varables z that serve as ndcators of whether or not the value of C les n the th nterval. We may now wrte the expresson for TC, as follows: TC N = = 1 a z + b x 8 4 where b s the slope of the th lne segment, and a s the value of the ntercept of that segment wth the vertcal axs, as shown n fgure 8.3. The precse expressons for a and b are: b a TC = C = TC TC C b C 1 = 1,2,..., N = 1,2,..., N New constrants For (1.8-4) to be vald we must make sure that exactly one z s equal to 1, and the others equal to 0. Ths s done (recallng that the z varables are 0-1) va: N = 1 z = 1 We also need to make sure that each x les wthn the th nterval whenever z s equal to 1 and s equal to 0 otherwse. Ths s done va two constrants: C 1 z x C z 70

71 8.2.5 Objectve functon terms Re-establshng the perod ndex, we see that the objectve functon term at perod t, for a learnng technology s thus equal to TC t - TC t-1, whch needs to be dscounted lke all other nvestment costs. TC b TC -1 a C -1 C Fgure 8.3. The th segment of the step-wse approxmaton Addtonal (optonal) constrants Solvng nteger programmng problems s facltated f the doman of feasblty of the nteger varables s reduced. Ths may be done va addtonal constrants, that are not strctly needed but that are guaranteed to hold. In our applcaton we know that experence (.e. cumulatve nvestment) s always ncreasng as tme goes on. Therefore, f the cumulatve nvestment n perod t les n segment, t s certan that t wll not le n segments -1, -2,.., 1 n tme perod t+1. Ths leads to two new constrants (reestablshng the perod ndex t for the z varables): 71

72 j = 1 z j, t j = 1 z j, t + 1 = 1,2,..., N 1, t = 1,2,..., T 1 N j = z j, t N j = z j, t + 1 Summarzng the above formulaton, we observe that each learnng technology requres the ntroducton of N*T nteger (0-1) varables. For example, f the model has 10 perods and a 5-segment approxmaton s selected, 50 nteger (0-1) varables are created for that learnng technology, assumng that the technology s avalable n the frst perod of the model. Thus, the formulaton may become very onerous n terms of soluton tme, f many learnng technologes are envsoned, and f the model s of large sze to begn wth. In secton 8.5 we provde some comments on ETL, as well as a word of warnng. 8.3 Clustered learnng An nterestng varaton of ETL s also avalable n TIMES, namely the case where several technologes use the same key technology (or component), tself subject to learnng. For nstance, table 8.1 lsts 11 technologes usng the key Gas Turbne technology. As experence bulds up for gas the turbne, each of the 11 technologes n the cluster benefts. The phenomenon of clustered learnng s modeled n TIMES va the followng modfcaton of the formulaton of the prevous secton. Let k desgnate the key technology and let l = 1, 2,, L desgnate the set of clustered technologes attached to k. The approach conssts of three steps: ) Step 1: desgnate k as a learnng technology, and wrte for t the formulaton of the prevous secton; ) Step 2: subtract from each INVCOST l the ntal nvestment cost of technology k (ths wll avod double countng the nvestment cost of k); ) Step 3: add the followng constrant to the model, n each tme perod. Ths ensures that learnng on k spreads to all members of ts cluster: VAR _ INVk VAR _ INV l L l = 1 = 0 72

73 Table 8-1: Cluster of gas turbne technologes (from A. Sebregts and K. Smekens, unpublshed report, 2002) Descrpton Integrated Coal gasfcaton power plant Integrated Coal Gasfcaton Fuel Cell plant Gas turbne peakng plant Exstng gas Combned Cycle power plant New gas Combned Cycle power plant Combned cycle Fuel Cell power plant Exstng gas turbne CHP plant Exstng Combned Cycle CHP plant Bomass gasfcaton: small ndustral cog. Bomass gasfcaton: Combned Cycle power plant Bomass gasfcaton: ISTIG+reheat 8.4 Learnng n a Multregonal TIMES Model Technologcal learnng may be acqured va global or local experence, dependng on the technology consdered. There are examples of technques that were developed and perfected n certan regons of the World, but have tended to reman regonal, never fully spreadng globally. Examples are found n land management, rrgaton, and n household heatng and cookng devces. Other technologes are truly global n the sense that the same (or close to the same) technology becomes rather rapdly commercally avalable globally. In the latter case, global experence benefts users of the technology world wde. Learnng s sad to spllover globally. Examples are found n large electrcty plants, n steel producton, wnd turbnes, and many other sectors. The frst and obvous mplcaton of these observatons s that the approprate model scope must be used to study ether type of technology learnng. The formulaton descrbed n the prevous sectons s adequate n two cases: a) learnng n a sngle regon model, and b) regonal learnng n a multregonal model. It does not drectly apply to global learnng n a multregonal global model, where the cumulatve nvestment varable must represent the sum of all cumulatve nvestments n all regons together. We now descrbe an approach to global learnng that may be mplemented n TIMES, usng only standard TIMES enttes. The frst step n modelng multregonal ETL (MRETL) s to create one addtonal regon, regon 0, whch wll play the role of the Manufacturng Regon. Ths regon s RES conssts only of the set of (global) learnng technologes (LT s). Each such LT has the followng specfcatons: a) The LT has no commodty nput b) The LT has only one output, a new commodty c representng the learnng. Ths output s precsely equal to the nvestment level n the LT n each perod. c) Commodty c may be exported to all other regons 73

74 Fnally, n each real regon, the LT s represented wth all ts attrbutes except the nvestment cost NCAP_COST Furthermore, the constructon of one unt of the LT requres an nput of one unt of the learnng commodty c (usng the NCAP_ICOM parameter see chapter 3 of PART II). Ths ensures that the sum of all nvestments n the LT n the real regons s exactly equal to the nvestment n the LT n regon 0, as desred. 8.5 Endogenous vs. Exogenous Learnng: Dscusson In ths secton, we formulate a few comments and warnngs that may be useful to potental users of the ETL feature. We start by statng a very mportant caveat to the ETL formulaton descrbed n the prevous sectons: f a model s run wth such a formulaton, t s very lkely that the model wll select some technologes, and wll nvest massvely at some early perod n these technologes unless t s prevented from dong so by addtonal constrants. Why ths s lkely to happen may be qualtatvely explaned by the fact that once a learnng technology s selected for nvestng, two opposng forces are at play n decdng the optmal tmng of the nvestments. On the one hand, the dscountng provdes an ncentve for postponng nvestments. On the other hand, nvestng early allows the unt nvestment cost to drop mmedately, and thus allows much cheaper nvestments n the learnng technologes n the current and all future perods. Gven the consderable cost reducton that s usually nduced by learnng, the frst factor (dscountng) s hghly unlkely to predomnate, and hence the model wll tend to nvest massvely and early n such technologes, or not at all. Of course, what we mean by massvely depends on the other constrants of the problem (such as the extent to whch the commodty produced by the learnng technology s n demand, the presence of exstng technologes that compete wth the learnng technology, etc.). However, there s a clear danger that we may observe unrealstcally large nvestments n some learnng technologes. ETL modelers are well aware of ths phenomenon, and they use addtonal constrants to control the penetraton trajectory of learnng technologes. These constrants may take the form of upper bounds on the capacty of or the nvestment n the learnng technologes n each tme perod, reflectng what s consdered by the user to be realstc penetratons. These upper bounds play a determnng role n the soluton of the problem, and t s most often observed that the capacty of a learnng technology s ether equal to 0 or to the upper bound. Ths last observaton ndcates that the selecton of upper bounds (or capacty/nvestment growth rates) by the modeler s the predomnant factor n controllng the penetraton of successful learnng technologes. In vew of the precedng dscusson, a fundamental queston arses: s t worthwhle for the modeler to go to the trouble of modelng endogenous learnng (wth all the attendant computatonal burdens) when the results are to a large extent condtoned by exogenous upper bounds? We do not have a clear and unambguous answer to ths queston; that s left for each modeler to evaluate. However, gven the above caveat, a possble alternatve to ETL would consst n usng exogenous learnng trajectores. To do so, the same sequence of realstc upper bounds on capacty would be selected by the modeler, and the values of the unt nvestment costs (INVCOST) would be externally computed by pluggng these upper 74

75 bounds nto the learnng formula (8-1). Ths approach makes use of the same exogenous upper bounds as the ETL approach, but avods the MIP computatonal burden of ETL. Of course, the runnng of exogenous learnng scenaros s not entrely foolsafe, snce there s no absolute guarantee that the capacty of a learnng technology wll turn out to be exactly equal to ts exogenous upper bound. If that were not the case, a modfed scenaro would have to be run, wth upper bounds adjusted downward. Ths tral-and-error approach may seem nelegant, but t should be remembered that t (or some other heurstc approach) mght prove to be necessary n those cases where the number of learnng technologes and the model sze are both large (thus makng the rgorous ETL formulaton computatonally ntractable). 75

76 References for Part I Altdorfer, F., "Introducton of prce elastctes on energy demand n MARKAL", Memorandum No 345, KFA, Julch, July Barreto, L. (2001), Technologcal Learnng n Energy Optmsaton Models and the Deployment of Emergng Technologes, Ph.D. Thess No 14151, Swss Federal Insttute of Technology Zurch (ETHZ). Zurch, Swtzerland. Berger, C., R. Dubos, A. Haure, E. Lessard, R. Loulou, and J.-Ph. Waaub, "Canadan MARKAL: an Advanced Lnear Programmng System for Energy and Envronment Modellng", INFOR, Vol. 20, , Brge, J. R., and Rosa, C. H. (1996), Incorporatng Investment Uncertanty nto Greenhouse Polcy Models, The Energy Journal, 17/1, Chvátal, V., Lnear Programmng, Freeman and Co, New-York, 1983 Dantzg, G.B., Lnear programmng and Extensons, Prnceton Unversty Press, Prnceton, New-Jersey, Department of Energy/Energy Informaton Admnstraton, US Government, "The Natonal Energy Modelng System : An overvew (2000)", DOE/IEA-0581, Washngton. DC, March 2000 Dorfman, R., P.A. Samuelson, and R.M. Solow, Lnear programmng and Economc Analyss, McGraw-Hll, New-York, Edmonds, J.A., H.M. Ptcher, D. Barns, R. Baron, and M.A. Wse, "Modellng future greenhouse gas emssons: The second generaton model descrpton", n Modellng global change, L.R. Klen and Fu-Chen Lo eds, Unted Natons Unversty Press, Tokyo-New-York-Pars, 1991, Fankhauser, S. (1994), "The socal cost of GHG emssons: an expected value approach", Energy Journal 15/2, Fshbone L G, Gesen G, Hymmen H A, Stocks M, Vos H, Wlde, D, Zoelcher R, Balzer C, and Ablock H. (1983), Users Gude for MARKAL: A Mult-perod, Lnear Programmng Model for Energy Systems Analyss, BNL Upton, NY, and KFA, Julch, Germany, BNL Fshbone, L.G., and H. Ablock, (1981), "MARKAL, A Lnear Programmng Model for Energy Systems Analyss: Techncal Descrpton of the BNL Verson", Internatonal Journal of Energy Research, Vol. 5, Fragnère, E. and Haure, A. (1996), "MARKAL-Geneva: A Model to Assess Energy- Envronment Choces for a Swss Canton", n C. Carraro and A. Haure (eds.), Operatons Research and Envronmental Management, Kluwer Academc Books. Gale, D., The Theory of Lnear Economc Models, McGraw-Hll, New-York, Gerkng. H., and Voss, A. (1986), An Approach on How to Handle Incomplete Foresght n Lnear Programmng Models, Internatonal Conference on Models and Uncertanty n the Energy Sector, Rsö Natonal Laboratory, Denmark, February, 1986, p Gorenstn, B. G. et al. (1993), Power System Expanson Plannng Under Uncertanty, IEEE Transactons on Power Systems, 8, 1, pp Grubb, M., (1993), "Polcy Modellng for Clmate Change: The Mssng Models," Energy Polcy, 21, 3, pp

77 Hller, F. S, and G. L. Lerberman, Introducton to Operatons Research, Ffth Edton, McGraw-Hll, New-York, 1990 Hogan, W.W., (1975), "Energy Polcy Models for Project Independence", Computers and Operatons Research, 2, Jaccard, M., J. Nyboer, C. Batalle, and B. Sadownk (2003). "Modellng the Cost of Clmate Polcy: Dstngushng between alternatve cost defntons and long-run cost dynamcs." Energy Journal 24(1): Kanuda, A., and R. Loulou, Robust Responses to Clmate Change va Stochastc MARKAL: the case of Québec, European Journal of Operatons Research, vol. 106, pp , 1998 Kunsch, P. L. and Teghem, J., Jr. (1987), "Nuclear Fuel Cycle Optmzaton Usng Mult-Objectve Stochastc Lnear Programmng," European Journal of Operatons Research 31/2, Larsson, T. and Wene, C.-O. (1993), "Developng Strateges for Robust Energy Systems. I: Methodology," Internatonal Journal of Energy Research 17, Larsson, T. (1993), "Developng Strateges for Robust Energy Systems. II: Applcaton to CO 2 Rsk Management," Internatonal Journal of Energy Research 17, Loulou, R., and A. Kanuda, Mnmax Regret Strateges for Greenhouse Gas Abatement: Methodology and Applcaton, Operatons Research Letters, 25, , Loulou, R. and A. Kanuda, Usng Advanced Technology-rch models for Regonal and Global Economc Analyss of GHG Mtgaton, n Decson and Control: Essays n honor of Alan Haure, G. Zaccour ed., Kluwer Academc Publshers, Norwell, USA, pp , A condensed verson of ths artcle was publshed electroncally n the Proceedngs of the Internatonal Energy Agency Internatonal conference on Clmate Change Modellng and Analyss, held n Washngton DC, June15-17, Loulou, R., and Lavgne, D., MARKAL Model wth Elastc Demands: Applcaton to GHG Emsson Control, n Operatons Research and Envronmental Engneerng, C. Carraro and A. Haure eds., Kluwer Academc Publshers, Dordrecht, Boston, London, 1996, pp Manne, A., Mendelsohn, R. and Rchels, R. (1995), "MERGE: a Model for Evaluatng Regonal and Global Effects of GHG Reducton Polces", Energy Polcy 23 (1), Manne, A.S., and Wene, C-O., "MARKAL-MACRO: A Lnked Model for Energy- Economy Analyss", BNL report, Brookhaven Natonal Laboratory, Upton, New-York, February Meade, D. (1996) LIFT User s Gude, Techncal report, INFORUM, IERF, College Park, Md (July 1996). Messner, S. and Strubegger, M. (1995), User s Gude for MESSAGE III, WP-95-69, IIASA, Luxembourg, Austra. Moomaw, W.R. and J.R. Morera «Technologcal and Economc Potental of Greenhouse Greenhouse Gas Emssons Reducton». In Clmate Change 2001: Mtgaton, edted by Metz, B., Davdson, O., Swart, R. and J. Pan, Intergovernmental Panel on Clmate Change (IPCC), Thrd Assessment Report, Workng Group III, p Cambrdge : Cambrdge Unversty Press. 77

78 Nakcenovc, N. (ed.) Specal Report on Emssons Scenaros. A Specal Report of Workng III of the Intergovernmental Panel on Clmate Change. Cambrdge : Cambrdge Unversty Press, p.599. Nordhaus, W. (1993), "Rollng the DICE: an optmal transton path for controllng GHG's", Resources and Energy Economcs 15/1, Nemhauser, G.L., A.H.G. Rnnooy Kan, and M.J. Todd (edtors), Handbooks n Operatons Research and Management Scence, Vol I: Optmzaton, North-Holland, Papadmtrou, C.H., and K. Stegltz, Combnatoral Optmzaton -- Algorthms and Complexty, Prentce-Hall, New-Jersey, Peck, S. C., and Tesberg,T. J. (1995), Optmal CO 2 Control Polcy wth Stochastc Losses from Temperature Rse, Clmatc change, 31/1, Raffa, H., Decson Analyss, Addson-Wesley, Readng, Mass., 1968 Samuelson, P.A., "Spatal Prce Equlbrum and Lnear Programmng", Amercan Economc Revew, 42, , Samuelson, P.A., and W. Nordhaus, Economcs (17th edton), John Wley, 1977 Takayama, T., and Judge G.G., Spatal and Temporal Prce and Allocaton Models, North Holland, Amsterdam, Tosato, G.C., "Extreme Scenaros n MARKAL LP Model: use of Demand Elastcty", presented at the 5th Italan-Polsh Symposum on Applcatons of Systems Theory to Economcs and Technology, Torun, June Voort, E. van der, Donn, E., Thonet, C., Bos d'enghen, E., Dechamps, C. & Gulmot, J.F Energy Supply Modellng Package EFOM-12C Mark I, Mathematcal descrpton. Louvan-la-Neuve, Cabay: Commsson of the European Communtes, EUR