Finn R. Førsund. Key words: Hydropower, electricity, reservoir, water value, market power

Transcription

1 1 andou 1 Auumn ECON 4925 YDROPOWER ECONOMICS by Finn R. Førsund Absrac: The key quesion in hydropower producion is he ime paern of he use of he waer in he reservoir. The waer used o produce elecriciy oday can alernaively be used omorrow. The analysis of he operaion of hydropower is herefore essenially a dynamic one. The paper inroduces some basic models for social allocaion of sored waer over discree ime periods using non-linear programming assuming capaciies of generaion and ransmission for given. The implicaions of consrains such as limied sorage capaciy and limied connecor capaciy for (inernaional) rade are sudied. Resuls are derived for waer allocaion and he developmen of he elecriciy price over ime. Graphical illusraions are provided in he wo- period case by means of he bahub diagram. Thermal capaciy is added o hydro and he opimal mix sudied. The walls of he hydro bahub are exended endogenously by hermal capaciies. Finally, he case of monopoly is sudied. Differen from sandard monopoly behaviour of conracing oupu, if oal available waer is o be used he sraegy of a monopolis is o redisribue he use of waer for elecriciy producion over periods compared wih he social opimal disribuion. Key words: ydropower, elecriciy, reservoir, waer value, marke power This maerial is inended for he Maser course ECON 4925 Resource Economics a he Deparmen of Economics, Universiy of Oslo

2 2 1. Inroducion Elecriciy Elecriciy is one of he key goods in a modern economy. The naure of elecriciy is such ha supply and demand mus be in a coninuous physical equilibrium. The sysem breaks down in a relaively shor ime if demand exceeds supply and vice versa. The sysem equilibrium is herefore demand-driven. The spaial configuraion of supply and demand is imporan for undersanding he elecriciy sysem. A nework for ranspor of elecriciy connecs he generaors and consumers. There is energy loss in he nework. Physical laws govern he flows hrough he neworks and he energy losses. The elecriciy is characerised by volage ( V) and he erz number (50 ± 2) for alernae curren and measured as effec (W), i.e. insananeous energy, and energy (kwh), i.e. he amoun of elecriciy during a ime period (he inegral of he effec over he ime period in quesion). The capaciy raing of he urbines of he generaors is in effec unis. The nework, or grid, has capaciy limis in effec unis for a given spaial configuraion of supply and demand nodes in he nework. The ime period used in a sudy of he elecriciy sysem is of crucial imporance for how he sysem is modelled. If he ime resoluion is one hour we can porray he demand by looking a he variaion in energy use hour by hour during a day. The demand varies ypically over he day wih he lowes energy consumpion during he nigh and peaks a breakfas ime and he sar of he working day, and again a peak round dinnerime. To see he need for effec capaciy i is common o look a he demand for one year and sor he 8760 hours according o he highes demand and hen decreasing. The hours wih he highes energy demand are called he peak load, and he hours wih he lowes demand are called he base load. In beween we have he shoulder. ydropower Elecriciy generaors can use waer, fossil fuels, bio-fuels, nuclear fuels, wind and geohermal energy as primary energy sources o run he urbines producing elecriciy. ydropower is based on waer driving he urbines. The primary energy is provided by graviy and he heigh he waer falls down on o he urbine. We will assume he exisence of a reservoir. The poenial for elecriciy generaion of one uni of waer (a cubic meer) is usually expressed by

3 3 he heigh from he dam level o he urbine level. The reservoir level will change when waer is released and hus influence he elecriciy producion. Elecriciy producion is also influenced by how processed waer is ranspored away from he urbine allowing fresh waer o ener he urbine. The urbine is consruced for an opimal flow of waer. Lower or higher inflow of waer will reduce he elecriciy per uni of waer. The key economic quesion in hydropower producion is he ime paern of he use of he waer in he reservoir. Given enough sorage capaciy he waer used oday can alernaively be used omorrow. The analysis of hydropower is herefore essenially a dynamic one. This is in conras wih a fossil fuel (e.g. coal) generaor. The quesion hen is how o uilize he given producion capaciy for each ime period. Assuming ha he marke for he primary energy source funcions smoohly his is no a dynamic problem, bu is a problem solved period for period (disregarding adjusmen cos going from a cold sae of no producing elecriciy o a ho sae producing). The economics of hydro producion wih reservoir was discussed early among economiss (see Lile (1955), Koopmans, 1957), bu he opic is a ypical engineering one. In Norway a cenral producion sysem was esablished afer he Second World War based on an undersanding of how he sysem was o be operaed (see veding (1967), 1968). This approach has been refined and developed ino a cenral model ool for Norway and laer he NordPool area (see augsad e al. (1990), Gjelsvik e al., 1992). The highly simplified approach aken in his paper is based on Førsund (1994) (see also Bushnell (2003), Crampes and Moreaux (2001), Johnsen (2001) and Sco and Read, 1996). The variables we are going o use are reservoir R, inflow of waer w and elecriciy Th producion, e, e, from hydro and hermal capaciies respecively. Flow variables in small leers are undersood o refer o he period, while sock variables in capial leers refer o he end of he period, i.e. waer inflow w akes place during period, while he conen of he reservoir R refers o he waer a he end of period. Release of waer during period is convered o elecriciy e measured in kwh according o a fixed ransformaion coefficien, reflecing he verical heigh from he cenre of graviy of he dam and o he urbines. Waer reservoir and inflow can also be measured in kwh using he same conversion. The reduced

4 4 elecriciy conversion efficiency due o a reduced heigh (head) he waer falls as he reservoir is used is disregarded. For he Norwegian sysem wih high differences in elevaions beween dams and urbine saions of mos of he dams, and having few river saions, his is an accepable simplificaion a our level of aggregaion. The ransformaion of waer ino elecriciy can be capured in he simples way by he producion funcion 1 e r (1) a where r is he release of waer from he reservoir during ime period and a is he fabricaion coefficien for waer. As menioned above he coefficien may vary wih he uilisaion of he reservoir, and also wih he release of waer due o he consrucion of he urbine giving maximal produciviy a a cerain waer flow. By assumpion here are no oher curren coss. This is a very realisic assumpion for hydropower. We will in he following assume ha he producion funcion (1) holds wih equaliy and herefore we can drop his relaion and measure waer in elecriciy unis. The dynamics of waer managemen is based on he filling and empying of he reservoir: R R + w e T (2) 1, Sric inequaliy means ha here is overflow. Some sudies of hydropower a a high level of aggregaion disregards he sorage process and specifies direcly he available waer wihin a yearly weaher cycle. The assumpions are hen no spill of waer and no binding upper reservoir consrain. The period concep may be as crude as wo periods (summer and winer season based on difference in inflow and/or release profile), and anyhing from monh, weeks, days and hours. A realisic modelling ( Samkjøringsmodellen, see augsad e al. (1990), Gjelsvik e al., 1992) may use a week as a period uni and involve 3 o 5 years. We will simplify and use a ime horizon for waer managemen problems following he yearly precipiaion cycle. Therefore we will also disregard discouning. Alhough i is obvious ha he world coninues afer one year, we will also simplify and no specify any erminal value of he reservoir or operae wih a scrap value for he reservoir conen of he las period.

5 5 2. The basic hydro model Social opimum We will assume ha here is unlimied ransferabiliy of waer beween he periods of he given oal amoun of waer available: e = R + w = W (3) T o T where W is he oal available inflows (including any waer R in he reservoir a he beginning of he firs period from he pas). The horizon, T, is assumed o be a seasonal cycle (one year) from spring o spring. The formulaion (3) requires for realism ha all waer is assumed o come in he firs period (i.e. w 1 > 0, w = 0 for = 2,..,T). In Norway he snow smeling during a few spring weeks fills he reservoirs wih abou 70 percen of he yearly oal. Waer is measures in energy unis, kwh, and no conversion from waer o elecriciy is shown. The energy consumpion in each period is evaluaed by uiliy funcions, which can be hough of as valid eiher for a represenaive consumer, or consiue a welfare funcion. There is no discouning (he horizon is oo shor for discouning o be of significance). The uiliy funcions represening he social value of elecriciy consumpion are: U ( e ), U '( e ) 0, U ''( e ) < 0, T (4) The uiliy funcions have he sandard propery of concaviy. We will define he marginal uiliy U ' measured in moneary unis as he marginal willingness o pay, p, i.e. he demand funcion for elecriciy: U '( e ) = p ( e ), (5) where p will also be referred o as he price of elecriciy for shor below. o The social opimisaion problem can be formulaed as follows: Max s.. T e T U ( e ) W The Lagrangian funcion is: (6)

6 6 (7) L= U ( e ) ν ( e W) T T Necessary condiions are 1 : L e = U '( e ) ν 0 e 0, T ν 0( = 0if e < W) T From he Kuhn Tucker condiions we know ha if we have an inerior soluion for he energy consumpion for period, e > 0, hen he shadow price on he energy consrain mus be posiive, ν > 0, and by complemenary slackness he energy consrain mus be binding. A sufficien condiion for a maximum is ha he Lagrangian (7) is concave, which is saisfied under our assumpions. (8) Resul 1: The law of one price. The price of elecriciy is consan and equal for all periods if marginal willingness o pay remains posiive for all periods. Demonsraion: According o (8) we have a binding consrain if he marginal willingness o pay, he price, remains posiive for all periods, and i is hen equal o he common shadow price on he energy consrain. By complemenary slackness he shadow price is posiive. Resul 1 is he oelling s rule for our model. We do no discoun, and by arbirage he price mus be he same for all periods. The ypical inerior soluion for boh periods is illusraed in Figure 1 in he case of wo periods by using a bahub diagram, showing oal available energy for he wo periods as he boom of he bahub and he wo marginal willingness o pay - funcions measured from each of he verical axes. The economic inerpreaion of he soluion o he allocaion problem is ha energy should be allocaed on he periods in such a way ha he shadow price of energy (i.e. he increase in he objecive funcion of a marginal increase in he given amoun of oal energy) is equal o he marginal uiliy of energy in each period, hus he marginal uiliies become equal. In he illusraion in Figure 1 if Period 1 is summer and Period 2 winer, he marginal uiliy should be equal. Alhough he marginal uiliy of energy 1 L L The use of is a shorhand noaion for he condiions 0, e i = 0 (Sydsæer, Srøm and Berck e e (1999), p. 100).

7 7 consumpion may be higher in he winer han in he summer for any level of consumpion, he marginal uiliy in he winer should no Period 1 Period 2 U 1 = ν U 2 = ν Period 1 Period 2 Toal available energy Figure 1. The bahub illusraion: Opimal allocaion of energy on wo periods become greaer han in he summer. Consrains in hydropower modelling There are many consrains on how o operae a dam. A fundamenal consrain is he maximal amoun of waer ha can be sored. This consrain will have a crucial imporance for how he dam can be operaed. Environmenal concerns may impose a lower limi on how much he dam can be empied. Empy dams creae eyesores in he landscape, and can creae bad smells from roing organic maerial along he exposed shores. Fish may have problems surviving or spawning a oo low waer levels. The environmenal lower consrain has a ime index, because he environmenal problems may vary wih season. In Norway where he dams are covered by ice in he winer season he lower level may be less hen han in he summer. The effec capaciy of a power saion may be consrained by he insalled urbines or he diameer of he pipe from he reservoir o he urbines. Such a consrain has no period subscrip. The effec concep will follow he period definiion. For example, if he period lengh is one hour he effec consrain is measured in kwh, by using he maximal kw for one hour.

8 8 In aggregaed analyses i is common no o specify he ransmission sysem. Bu a consrain on he ransmission can be represened he same way as for effec capaciy consrain, excep ha a ime index may be used on he consrain o indicae ha ransmission capaciy wihin some limis is an endogenous variable governed by physical laws of elecrical flows of acive and reacive power in a muli-link grid sysem beween inpu and oupu nodes. The loss may also vary wih emperaure: resisance is higher in winer han summer ime. Table 1. Consrains in he hydropower model Expression Consrain ype Max Reservoir R R Environmenal concerns, R R Min Reservoir Max Effec capaciy e e Max Transmission capaciy e e Waer flows, environmen e e e Ramping up 0 < e e 1 r Ramping down 0 < e 1 e r u d There may be environmenal concerns abou he size of he release from a reservoir. If he release occurs ino a river sysem here may be concerns boh abou he lower and he higher amoun of waer ha should be released due o impacs on he environmen downsream. Impacs on fishing and recreaional aciviy and pressure from ourism may be relevan. Erosion of riverbanks and emperaure change for agriculural aciviy nearby may also coun. Then here is concern abou navigaion and flood conrol. All hese effecs may also be presen when releases change, so upper consrains may be inroduced boh on ramping up and ramping down. These consrains are mos realisic for shorer ime periods.

9 9 3. ydro wih reservoir consrains Social opimum In he older lieraure on hydropower referred o in he Inroducion and in engineering lieraure he social objecive funcion is ofen expressed as minimising he oal coss of supplying a given amoun of elecriciy. In economics a sandard objecive funcion in empirical sudies is o maximise consumer plus producer surplus wih he produced quaniies as endogenous variables. The consumpion side is convenienly summarised by using demand funcions (defined in (5)) and he supply side by using cos funcions. This is a parial equilibrium approach because no ineracion wih he res of he economy is modelled. In he case of hydropower wih zero operaing marke coss he surplus is simplified o he area under he consumer demand funcion (since consumers expendiure is idenical o producers profi). The social planning problem is: Max x ( p ( z) dz ) T z= 0 s.. R R + w e, R R, T 1 The Lagrangian is: L= e T z= 0 λ R R 1 w e T ( p ( z) dz ) ( + ) T γ ( R R) (9) (10) Necessary firs order condiions are: 2 L e = p ( e ) λ 0 e 0 L = λ + λ+ 1 γ 0 R 0 R (11) 2 See Sydsæer, Srøm and Berck (1999, p. 51) for he derivaion of he firs relaion.

10 10 Assuming posiive producion in all periods yields p ( e ) = λ, λ + λ γ 0 R 0, T + 1 (12) The shadow price λ of he sored waer is ermed he waer value 3. According o Bellman s principle for solving dynamic programming problems wih discree ime we sar searching for he opimal soluion from he las period and hen work our way owards he firs period. Our horizon ends a T, we mus herefore have λ T + 1 = 0. For period T we have wo possibiliies as o he uilisaion of he waer in he reservoir, eiher i is empied, R T = 0, or some waer is remaining, R T > 0. Since he waer has no value from T+1 on he las siuaion can only be opimal if he marginal uiliy of elecriciy becomes zero before he boom of he reservoir is reached. We will adop he oher alernaive ha he marginal uiliy of elecriciy remains posiive o he las drop. This means ha we will have a siuaion of scarciy in he las period T. Scarciy gives economic value o he waer in he las period: λ 1 = 0, R = 0 λ γ < 0, γ = 0 λ > 0 (13) T+ T T T T T Since we canno have a siuaion of scarciy a he same ime as he upper limi on he reservoir is reached he shadow price on he upper consrain is se o zero. For all periods up o T we will hen have he waer values posiive if he upper consrain on he reservoir is no reached in any of he periods: Resul 2. The law of one price: The price of elecriciy remains posiive and consan for all periods including he las period T if he reservoir is empied only in he las period and he upper consrain on he reservoir is never reached. Demonsraion: The condiions above imply λ T > 0 (see (13), and we have by assumpion γ = 0, R > 0, u = 1,.., T. We hen have from he necessary condiions (11) u u 1 ha λu = λt > 0 u = 1,.., T. Scarciy during he period T Resul 3: The law of differen prices equalling he number of scarciy periods. 3 Bu remember ha in our simplified model waer is measured in elecriciy unis. We should really measure waer in m 3 o use he expression. By inroducing (1) ino he model assuming equaliy o hold in he producion funcion his can easily be done.

11 11 Assume he overflow or hrea of overflow never occurs. Then each period beween wo scarciy episodes will ge is separae price. The period beween he saring period and he firs period wih scarciy will ge he highes price, and hen he price will ypically fall for each scarciy period experienced up o he horizon. Demonsraion: Assume ha we have only one scarciy period in addiion o he erminal period scarciy. There is a scarciy in period, and 1 < < T, bu no overflow siuaion. Using condiion (11) and resul (13) we have: λ + λ+ 1 γ 0 R 0 λ+ 1 = λt > 0, R = 0, γ = 0, Ru > 0, γu = 0, u = 1,.., 1 (14) λ λ, u = 1,.., u T We have direcly from he firs order condiion in (11) ha he prices are equal o he waer values assuming ha a posiive amoun of elecriciy will be produced in every period. Overflow or hrea of overflow Assume overflow or hrea of overflow (reservoir compleely filled) for a period s <, where is he firs scarciy period afer s. We hen have from he necessary condiions (11): Rs > 0, λs+ 1 > 0, γ s > 0, s< λs + λs+ 1 γs = 0 (15) λ = λ γ 0 λ < λ = λ s s+ 1 s s s+ 1 The firs equaliy follows from he Kuhn-Tucker condiion in (11) when here is a posiive amoun of waer in he reservoir. The shadow price on waer λ s is zero if here is overflow. This follows from he complemenary slackness condiion for he second erm in (11). Then he shadow price on he reservoir consrain is equal o he waer value induced by he scarciy period, i.e. λ. This is he benefi of increasing he reservoir consrain wih one uni. If here is no spillage and he waer is jus mainained a he maximal level he waer value λ s will ypically be posiive. In any case he waer value λs is smaller han he waer value λ for laer periods up o he scarciy period. The firs condiion in (12) ells us ha a zero waer value can only go ogeher wih a zero value of he willingness o pay for elecriciy. A siuaion wih zero price, i.e. overflow, will ypically no be realisic in a world wihou uncerainy.

12 12 The siuaion wih wo periods is illusraed in Figure 2. In he firs period we have an inflow equal o AC, and in he second period an inflow equal o CD. The capaciy of he reservoir is BC. The opimal allocaion is o sore he maximal amoun BC in period 1 and consume wha Period 1 Period 2 λ 2 γ 1 λ 1 A B C D Figure 2. Social opimum wih reservoir consrain canno be sored, AB. In he second period he reservoir, conaining BC from he firs period and an inflow of CD in he second period, is empied. Using (15) above we have ha λ1 = λ2 γ1. Noice ha he waer allocaion will be he same for a wide range of period 1 demand curves keeping he same period 2 curve, or vice versa. (The period 1 curve can be shifed down o passing hrough B and shifed up o passing hrough he level indicaed for period 2 waer value, as indicaed by he broken lines.) Overflow or hrea of overflow beween wo periods wih scarciy Resul 4: Cycling prices beween periods wih overflow and periods wih scarciy. Assume ha he firs period of scarciy comes before he firs period of overflow or hrea of overflow, and afer his episode we have inerchanging scarciy and overflow periods in beween periods wih neiher scarciy nor overflow. The price will hen cycle from higher values in periods afer an overflow (or hrea of overflow) episode o he nex scarciy period o a lower price afer a scarciy period and unil he nex overflow (or hrea of overflow) period.

13 13 Demonsraion: Assume ha overflow hreaens in period s beween wo periods 1 and 2 wih scarciy; 1< 1 < s< 2. For he oher periods up o 2 we have neiher overflow nor scarciy. From earlier resuls we know he siuaion for he waer values up o 1, a s and from s+1 o 2. We need o esablish he siuaion from 1+1 o s -1: Ru > 0, γ u = 0 for u = 1+ 1,.., s 1, Rs > 0, γs> 0, λs + λs+ 1 γs = 0, λs+ 1 = λ2 (16) λ + λ γ = 0 λ = λ = λ γ 0 s 1 s s 1 s 1 s 2 s The marginal willingness o pay, or he price, is equal o he waer value according o he condiion (12) assuming ha elecriciy is produced in each period. Several producers There are over 600 hydropower producers in Norway, and a majoriy of hem have reservoirs. We will herefore briefly sudy he implicaions of several producers for he opimal allocaion of waer. Each plan is assigned one reservoir. The planning problem is he same as (9), bu now a foo index (j) for plan has o be inroduced. We will also need a relaion connecing he amoun consumed o he oal amoun produced. This is popularly ermed he energy balance: Max s.. x ( p ( z) dz ) T z= 0 R R + w e, R R, j j, 1 j j j j x e, T j J j (17) The energy balance for each period is he las resricion. I does no maer o he consumer who supplies he elecriciy. The Lagrangian is: L= x ( p ( z) dz ) T z= 0 λ ( R R w + e ) T j J T j J T j j j, 1 j j γ ( R R ) ν ( x e ) j j j j j J (18) The firs order condiions are:

14 14 L e L R j j = λ + ν 0 e 0 j j = λ + λ γ 0 R 0 j j, + 1 j j L = p( x) ν 0 x 0 x We assume ha elecriciy is consumed in all periods so he las condiion holds wih equaliy. The shadow price on he energy balance is hen equal o he marke price. The firs order condiions can be simplified o: λ j + p ( x ) 0 ej 0 (20) λ + λ γ 0 R 0, T, j J j j, + 1 j j Using he backward inducion principle assuming ha demand is no saiaed and reservoirs are empied in he erminal period T we ge: λ λ = p ( x ) > 0 (21) jt jt T T The equaliy follows from he assumpion ha all unis are producing elecriciy in he las period (a leas he inflows w jt ). Bu he condiion above is no specific o plan j, bu applies o all plans. In he opimal soluion all plans are assigned he same waer value in he las period and he consumpion of elecriciy is jt. R j J (19) For period T-1 he process is repeaed. Wihou any overflow a any plan or any plan empying is reservoir all plans are again facing he same waer values and he price mus be he same as for period T. To invesigae wheher i could be par of an opimal plan ha one plan has overflow in period T-1 le us ry wih overflow for plan j. Since overflow is a loss we will make he reasonable assumpion ha he plan has posiive producion implying from λ jt T T he firs condiion in (20) ha he waer value is equal o he marke price:, 1 = p 1( x 1). Assuming also posiive producion for all oher plans hey hen have he same waer value. Bu assuming overflow, or hrea of overflow for uni j implies ha γ jt, 1 > 0 and from he second condiion in (20) we have λ, 1 + λ γ, 1 = 0. Bu we hen have a conradicion jt jt jt since he las relaion implies pt 1( xt 1) < pt( xt) for plan j and pt 1( xt 1) = pt( xt) oher plans. We conclude ha opimaliy requires all plans o have hrea of overflow a he same ime. Any loss of waer is a social loss, so he opimal plan mus imply manoeuvring o

15 15 preven such losses. The price in period T-1 can only be lower han for period T if overflow hreaens all plans. The oher exreme siuaion is ha plan j empies is reservoir in period T-1, bu no he oher λ plans. The firs condiion in (20) again yields jt, 1 = pt 1( xt 1) since plan j has posiive producion. The second condiion in (20) now yields λ, 1+ λ 0. Assuming sric inequaliy we have ha for plan j i is required ha pt 1( xt 1) > pt( xt), while he condiion for he oher plans yield pt 1( xt 1) = pt( xt). Again we have a conradicion. We conclude ha in he regular case all reservoirs have o be empied a he same ime for he plan o be opimal. Bu noe ha he inequaliy involved is no sric, so i may be opimal for plans o empy heir reservoirs before ohers. The reasoning above leads o he following resul: jt jt Resul 5: veding s conjecure: In he case of many plans wih one limied reservoir each he plans can be regarded as one plan and he reservoirs can be regarded as one reservoir in he social soluion for operaing he hydropower sysem. The individual reservoirs will all be uilised in he same fashion, as if here is only one reservoir. This is a resul of imporan pracical value since i may simplify grealy he modelling effor. The resuls abou price movemen sudied in he previous secion for one plan and one reservoir are all valid also for he many plan case. Bu veding s conjecure may no hold if more of he consrains enered in Table 1 are inroduced. The consrains may be so demanding o fulfil, especially wih a deailed ime uni, ha some reservoirs may experience overflow before ohers, and some may be empied. Bu we may regard he conjecure as a firs bes soluion. This may hold even when exending he modelling o include sochasic inflows and demand. 4. Trade Social opimum wihou consrains on ransmission/rade Inroducing rade means ha we inroduce a second good, money, ino he sociey in addiion o elecriciy. We will simplify by jus adding (subracing) he expor (impor) sum o (from) he area under he demand curve for elecriciy, implying ha in he background we assume

16 16 uiliy funcions separable in elecriciy and money (aggregae for all oher goods). The objecive funcion will hen be he sum over he periods of consumer and producer surplus, which in our case for elecriciy will be he area under he demand curve since we have assumed zero producion cos (only waer value couns), and for money here is jus he amoun: posiive for expors and negaive for impors. There is no resricion of rade balance in elecriciy. The social planning problem is: x XI Max p( z) dz + p e T z = 0 s.. XI x = e e, e W, T T XI (22) The variable XI e is ne expor and is posiive if we have expor and negaive if we have impors. We assume ha in one period we can only have eiher expors or impors, or boh are zero. The Lagrangian is: x XI L= p( z) dz+ p e T z = 0 XI ν ( x e + e ) T λ( e W) T The necessary firs order condiions are: p ( x ) ν 0 x 0 ν λ 0 e 0 p XI ν = 0 XI (23) (24) Using he reasonable assumpion ha x > 0 Twe ge an adapaion of he foreign price regime. Resul 6: he law of adaping he foreign price regime. If elecriciy is used in all periods and here is no resricion on impor/expor capaciy, hen he foreign price regime will compleely deermine he home price. Demonsraion: We have direcly from combining he firs and he las condiion in (24) ha

17 17 XI p ( x ) = p T (25) Bu noice ha we do no necessarily use hydropower in all periods. If he shadow price on home consumpion is less han he shadow price on waer, no waer shall be used for hydropower producion in ha period, we jus impor. Wihou any consrain on he possibiliy o sore waer he model is oo exreme because we will only expor in one period, he period wih he highes expor price. The shadow price on waer will be se equal o his maximum price: XI { p } λ = max (26) T The oal expor will be: e = W p ( e ) (27) XI * * * where * is he period corresponding o he period wih he maximal expor price above. In all oher periods we will only impor o he going foreign price. Unlimied rade is herefore only of pracical ineres ogeher wih consrains on he possibiliy o sore waer (Førsund,1994). The social opimum wih consrains on ransmission/rade The social planning problem is: x XI Max p( z) dz + p e T z = 0 s.. XI x = e e, e W, T XI XI XI e e e, T XI The corresponding Lagrangian is x XI XI L= ( p ( z) dz+ p e ) T z= 0 XI ν ( x e + e ) T λ( e W) T T XI XI α ( e e ) XI XI β ( e e ) T The firs order condiions are: (28) (29)

18 18 p ( x ) ν 0 x 0 ν λ 0 e 0 p ν α + β = 0 XI (30) Resul 7: The law of one home price. If here is a limied impor/expor possibiliy hen he price will be consan for all periods and differen from he impor/ expor price. Demonsraion: Assume ha x, e > 0 T. Then from he firs condiion in (30) we have ha p( x) = ν and from he second consrain we have ha ν = λ. By subsiuion, he las condiion can be wrien: XI p ( x ) = λ = ν = p α + β T (31) The impor/expor capaciy will ypically be fully uilised, and we can divide he periods ino expor and impor periods: Expor periods a full capaciy XI { } EXP = : λ p < 0, T ( α > 0, β = 0) (32) Impor periods a full capaciy: XI { } IMP = : λ p > 0, T ( α = 0, β > 0) (33) We noe ha he home price can never be lower han he impor price in impor periods. A home price equal o he impor price implies ha impors will be less han he maximal capaciy. For expor periods he home price can never be higher han he expor price. A home price equal o he expor price implies ha expors will be less han he maximal capaciy. Noice ha he social soluion is independen of he values of he expor/impor price as long as he inequaliies in (32) and (33) are fulfilled. The soluion for he waer value mus obviously be posiive in a realisic case (exreme abundance of waer may resul in a home price of zero for all periods and maximal expors in all periods). The opimal waer value is se such ha he oal available waer, W, is jus used up on home consumpion and expors. An illusraion is provided in Figure 3. Two possible locaions of price of impors/expors a A and C compared wih waer value a B are enered:

19 19 (i) igher impor/expor price a A han he waer value a B: Opimal use of waer: Do maximal expors AA and use waer BB for home producion realising poin B on he demand curve. ome price is equal o he waer value. p A A B α b B C β C x Expor/impor capaciy Figure 3. Social opimum and expor/impor consrains (ii) Lower impor/expor price a C han waer value a B: Opimal use of waer: Impor he maximal elecriciy corresponding o Bb, use own waer in amoun bb realising poin B on he demand curve. ome price is equal o he waer value. Sum of oal use of waer mus obey he waer consrain. For wo periods wih idenical demand funcions we mus have 2BB = W wih expor price a A one period and impor price a C he oher period for he illusraion o be correc. 5. Thermal plans We inroduce plan specific variable cos funcions for he generaion of elecriciy based on hermal energy sources. Each plan has an upper capaciy ( e Th i ) for generaion ( e Th i ) ha can only be changed by invesmens. The cos funcions are no daed for simpliciy, bu he cos

20 20 funcion may change beween periods due o differen fuel prices (fuels may be more expensive in a high demand season): c = c ( e Th ), c ' > 0, e Th e Th, i N (34) i i i i i i The plan may be designed o have he smalles marginal cos a close o full capaciy uilisaion. We disregard coss of ramping up or down plans, and especially going from a cold o a spinning sae. (aving a phase of declining marginal coss may capure a sar-up effec.) The se of individual hermal plans can be aggregaed o a hermal secor by he following leas cos procedure saisfying a oal generaing requiremen of Min i N Th c ( e ) i i s.. Th Th Th Th e e, e e, i N i i i Th e for each period: (35) For each oal generaion level we ge a se of plans producing posiive oupu, and a se being idle according o he marginal cos levels. If he range of variaion in he marginal coss for each plan is sufficienly small all bu one plan will be uilised o full capaciy, and here will be a marginal uni parially uilised. We can perform a meri order ranking of he acive unis according o average coss a full capaciy uilisaion. Finally, he sequence of individual cos curves can be simplified or approximaed by a smooh funcion: Th Th Th Th c = c( e ), c' > 0, c'' > 0, e ei = e (36) i N Social soluion of mixed hydro and hermal capaciy The basic hydro model (6) wihou consrains on reservoirs, bu only on oal availabiliy of waer, is adoped. Insead of using he uiliy funcion as in (6) he demand funcion for elecriciy is used making explici he maximisaion of consumer plus producer surplus. We assume ha i does no maer how elecriciy is generaed, i.e. he willingness o pay is he same for he wo ypes of generaion (no green preference). The opimisaion problem faced by a sysem planner is:

21 21 x Th Max [ p( z) dz c( e )] s.. T z= 0 x = e + e, e W, e e Th Th Th T The Lagrangian funcion is: x Th L= [ p ( z) dz c( e )] T z= 0 Th ν ( x e e ) T T Th Th θ ( e e ) λ( e W) T The necessary condiions are: p ( x ) ν 0 x 0 ν λ 0 e 0 T T c'( e ) + ν θ 0 e 0 (37) (38) (39) Assuming ha elecriciy mus be produced in all periods we have: p ( x ) = ν (40) We mus hen in any period eiher acivae hydro or hermal, or boh. Thermal will no be used for periods where: c '(0) > ν = λ (41) ydro will no be used in periods where: c'( e T ) = ν < λ (42) For periods where boh hydro and hermal is used we have: T p ( x ) = λ = c'( e ) + θ (43) We have he basic resul of he law of one price if hydro is used in all periods. Resul 8: The law of one price wih mixed hydro and hermal capaciy. In a siuaion wih no reservoir consrains and assuming ha hydro will be used in every period he price will be consan for all periods. Demonsraion: The resul follows direcly from (43).

22 22 Resul 9: Thermal capaciy as consan base load. If he maximal hermal capaciy is no uilised in any period we will have a consan load of hermal for all periods. Demonsraion: Rearranging (43) wih he shadow price θ = 0 yields: c e = = p x e = c = cons (44) Th Th 1 '( ) λ ( ) ' ( λ). An illusraion for one period is shown in Figure 4. The marginal cos curve, c, for hermal Th capaciy sars a C and ends a he full capaciy value, e. Assuming b o be he available waer he opimal soluion is he price a level B equal o he shadow price of waer, and a hermal conribuion of Bb = e Th and a hydro conribuion of bb = e. If we assume ha he figure is represening jus one of many periods i is meaningful o inroduce wo alernaive waer values by he doed lines a C and A. For waer values from levels A o a he full capaciy of hermal unis will be uilised. For waer values higher han a he level a only hermal capaciy will be used. For waer values lower han a level C no hermal capaciy will be used. In a muli-period seing wih idenical demand funcions and average availabiliy of waer being bb he one period soluion shown in he figure will be repeaed each period. p a A c B b B C e Th ē Th e x Figure 4. ydro and hermal. Social opimum

23 23 For wo periods we may use he bahub diagram o illusrae he allocaion of he wo ypes of power on he wo periods. In Figure 5 he lengh of he bahub (bd) is exended a each end wih he hermal capaciy. Using he resul (44) we have ha he hermal exension of he bahub is equal a each end; wih (ab) in period 1 and (de) in period 2. We have ha (ab) = (de). The equilibrium allocaion is a poin c, resuling in an allocaion of (ab) hermal and (bc) hydro in period 1, and (cd) hydro and (de) hermal in period 2. Period 1 1 Period 2 2 p 1 (.) p 2 (.) c c a b c d e ydro energy Thermal exension Figure 5. Energy bahub wih hermal-exended walls of he hydro bahub Inroducing a reservoir consrain Inroducing a reservoir consrain ino problem (32) yields he following Lagrangian funcion:

24 24 x Th L= [ p ( z) dz c( e )] T z= 0 Th ν ( x e e ) T T Th Th θ ( e e ) λ ( R R w + e ) T T 1 γ ( R R) The oal energy supply condiion in (38) is replaced wih he wo las condiions in (45) showing he dynamics of waer sorage and he upper consrain on oal sorage. The necessary firs order condiions are: p ( x ) ν 0 x 0 ν λ 0 e 0 λ + λ γ 0 R 0, T + 1 T T c'( e ) + ν θ 0 e 0 Regarding combining hydro and hermal we will now have as a general rule ha he waer value is period specific in he second condiion, implying ha hermal capaciy may vary beween periods when boh hydro and hermal capaciies are used. A possible siuaion is illusraed in figure 6. Period 1 Period 2 (45) (46) c λ 1 p 1 (.) γ 1 p 2 (.) λ 2 c p 2 p 1 a A B C D ydro energy Thermal exension Figure 6.Thermal and hydro wih reservoir consrain d

25 25 The oal hydro capaciy is AD wih inflow AC in period 1 and CD in period 2 and sorage capaciy is BC. In period 1 he maximal amoun is sored for use in period 2. Since hermal capaciy is no uilised o is maximum in any of he wo periods he period waer value should be se equal o he marginal hermal coss. This implies ha less hermal capaciy, aa, is used in period 1 wih he lowes waer value, and more hermal capaciy, Dd, is aken ino use in he second period. Oher possible configuraions of he opimal social soluion follows he discussion in Secion 3. Resul 10: Thermal capaciy as peak load: When reservoir consrains are binding hen he hermal capaciy varies in uilisaion over he periods and serve as peak load. Demonsraion: From (44) we have ha he use of hermal capaciy is deermined by equalisaion of marginal coss and marke price. When he marke price varies due o reservoir consrains being binding, hen he peal load funcion follows. 6. Marke organisaions Free compeiion In Secion 3 we invesigaed he consequence for social planning of many hydropower producers, and found ha he sysem could be reaed as one aggregae uni (veding s conjecure). We now assume ha we are sudying one among several suppliers selling elecriciy in a spo marke for every period. There is no uncerainy, so he period prices p i are known. Given he capaciy of each producer and he size of his reservoir he will in he siuaion of no (acive) consrain on his reservoir (Secion 2) obviously choose o deliver all his elecriciy in he period wih he highes price in order o maximise profis. Therefore, in order o have posiive oal supply in all periods he prices mus be equal over periods in marke equilibrium. The allocaion over periods is hen compleely demand driven, and since he producers are indifferen abou when o produce some addiional rule has o be inroduced. In he case of a consrain on he reservoirs he profi maximisaion problem of a producer ( j ) is:

26 26 Max s.. T p e j R R + w e, R R, T j j, 1 j j j j (47) The Lagrangian for he problem is: L= T j λ ( R R w + e ) T T pe j j j, 1 j j γ ( R R ) j j j For noaional ease we have used he same symbols for shadow prices as in he social planning case wih a single producer. The shadow prices are plan specific. The necessary condiions are: L e j L R j = p λ 0 e 0 j j = λ + λ γ 0 R 0 j j, + 1 j j Le us assume ha here is a posiive marke price in every period. The producer will no supply any elecriciy if he waer value is higher han he marke price. For he periods he will supply a posiive amoun he marke price has o be equal o his waer value. In general he producer will srive o sell all his energy a he period wih he highes price, bu he is prevened o do his by he upper consrain on his reservoir. When overflow hreaens his waer value will be adjused downwards for ha period. e is willing o sell o a lower price now han a higher price in a laer period. Bu o he righ price he may sell in an even earlier period and preven an overflow siuaion happening. (48) (49) Comparing he privae condiions (49) wih he social condiions (20) we have ha if he prices faced by he producers are he same as in he social soluion, and provided he planning horizon is he same for all plans and equal o he social planning horizon, hen a compeiive marke will susain he social soluion. This is in accordance wih he exbook welfare heorems in economics. Bu remember he pifalls (exernal effecs, ec.), and noice ha we have no shown how such prices may be formed in privae markes. The reasoning of a hydropower producer deermining when o process his waer will follow he discussion se ou in Secion 3.

27 27 Monopoly wihou binding reservoir consrains We now urn o he case of all hydro producers being par of a monopoly, and simplifying furher by considering he monopolis as a single producion uni (i.e. he coordinaion problem expressed by veding s conjecure is solved by he monopolis). We assume ha he monopolis faces he demand funcions p = p ( e ), T. The opimisaion problem of he monopolis in he basic case of Secion 2 is: Max p ( e ) e s.. T e T W The Lagrangian is: p( e ) e ν ( e W) (51) T T The necessary firs order condiions are: L e = p '( e ) e + p ( e ) ν 0 e 0, T Assuming ha he monopolis will produce elecriciy in all periods he condiions may be wrien: p ( e )(1 + η ) = p ( e )(1 + η ),, ' T (53) ' ' ' In he expression for he marginal revenue we have inroduced he demand flexibiliy, η = p ' e / p, which is negaive (he inverse of he demand elasiciy). The condiion is ha he marginal revenue should be equal for all he periods and equal o he shadow price on sored waer. The absolue value of he demand flexibiliies mus be less han (or equal o) one. (50) (52) An illusraion in he case of wo periods is provided in Figure 7. The broken lines are he marginal revenue curves. We see ha in our case (he same demand curves as in Figure 1) he marginal revenue curves inersec for a posiive value, i.e. i will no be opimal for he monopolis o spill any waer. This value is he shadow value on waer. Bu his resul is depending on he form of he demand funcions. If we have spillage as an opimal soluion, hen he shadow waer value is zero. We see ha he waer value in general is smaller han he shadow value for waer in he social opimal case in Figure 1. Going up o he demand curves

28 28 Period 1 Period 2 p 2 M p 1 S p 1 M p 2 S Figure 7. The basic monopoly case gives us he monopoly prices for he wo periods. An imporan general resul is ha in he case of monopoly he marke prices become differen for he periods. For he period wih he mos inelasic demand he price becomes larger han he social opimal price, and for he mos elasic period he price becomes smaller. Thus we have a general shifing in he uilisaion of waer from periods wih relaive inelasic demand o periods wih relaive elasic demand. Resul 11: The law of differen prices. In he case of a monopolis wihou reservoir consrains he prices will differ beween periods according o differences in demand flexibiliies wih he highes/lowes price charged in inelasic/elasic periods. Demonsraion: Differing prices follow direcly from (53). Monopoly and reservoir consrains: The profi maximisaion problem is:

29 29 Max p ( e ) e T s.. (54) R R + w e, R R, T 1 The Lagrangian is: L= p ( e ) e T λ ( R R w + e ) T T 1 γ ( R R) The necessary firs order condiions are: L e = p '( e ) e + p ( e ) λ 0 e 0 L = λ + λ+ 1 γ 0 R 0 R (55) (56) Assuming elecriciy is always supplied and inroducing he demand flexibiliy η = p ' e / p : p '( e ) e + p ( e ) λ = p ( e )(1 + η ) λ = 0 λ + λ γ R 0 T + 1 The marginal willingness o pay (he price) is subsiued wih he marginal revenue. The discussion of he use of waer is parallel o he social opimum case. Bu will a monopolis choose he same ime profile for he same inflows, demand funcions, ec.? (57) Le us firs assume ha he monopolis will no find i profiable o spill any waer. In order for he sored waer o become scarce in any period i is necessary ha he consumers demand he oal amoun of sored waer o he price he monopolis is charging. Therefore, if he waer becomes scarce in he same period as in social opimum, he monopolis canno charge a higher price han in he social opimum. I is he shadow value of waer ha mus adjus downwards for his o be possible. Can he monopolis use more waer in elasic periods o achieve more periods wih scarciy? Can he monopolis choose o go empy in he period wih he highes price ha will empy he reservoir? Bu is i in his period ha he social soluion also goes empy? Maximising consumer surplus should lead o using as much waer as possible when demand is high.

30 30 If he monopolis reallocaes waer o elasic periods, his is no poin since price is deermined by waer value when reservoir is empied. Spilling waer is profiable if marginal revenue becomes zero before he reservoir is filled. Noice ha he monopolis does no have o empy he reservoir o creae value, jus demand a posiive price, i is only overflow ha will break he aemp o increase price in all periods. Period 1 Period 2 p 2 M p 1 M A B C D Figure 8. Monopoly wih reservoir consrain In he illusraion above he reservoir consrain is no binding, and we have no spillage. We ge he same ype of soluion as in Figure 2. Bu we noe ha he monopoly price in he period wih he relaively mos elasic demand becomes lower han he social opimal price wih a binding reservoir consrain, and he monopoly price in he period wih relaively inelasic demand becomes higher han in he social opimal case. This is he general shifing of waer from periods wih relaive inelasic demand o periods wih relaively elasic demand in he case of marke power. Resul 12. Increasing price differenials. A monopolis facing a reservoir consrain will coninue o ry o shif waer from inelasic periods o elasic periods and if he can do his he price differences will increase.

31 31 Demonsraion: The resul follows from (57). Monopoly wih hydro and hermal plans Le us assume ha a monopolis has full conrol over boh hydro and hermal capaciy. The demand funcions are p ( x ). The opimisaion problem is: Th Max [( p( x) x c( e )] s.. T x = e + e, e W, e e Th Th Th T (58) The Lagrangian is: Th L= ( p ( x ) x c( e )) T Th ν ( x e e ) T T Th Th θ ( e e ) λ( e W) T (59) The necessary condiions are: p '( x ) x + p ( x ) ν 0 x 0 ν λ 0 e 0 Th Th c'( e ) + ν θ 0 e 0 (60) Concenraing on periods where boh hydro and hermal are used he general resul is ha marginal revenue subsiues for he marginal willingness o pay in he social opimal soluion: Th p ( x )(1 + η ) = ν = λ = c'( e ) + θ (61) The monopoly soluion is illusraed in Figure 9. If he monopolis s waer value is OB in a period boh hermal and hydro capaciy will be used according o he marginal revenue condiion (60). The hermal capaciy will be Oe Th and he hydro capaciy (Oe - Oe Th ). For wo periods we may again use he bahub diagram o illusrae he allocaion of he wo ypes of power on he wo periods. In Figure 10 he lengh of he bahub (bd) is exended a

32 32 each end wih he hermal capaciy. Using he resul (57) we have ha he hermal exension of he bahub is equal a each end; wih (ab) in period 1 and (de) in period 2. We have ha (ab) = (de). The equilibrium allocaion is a poin c, resuling in an allocaion of (ab) hermal and (bc) hydro in period 1, and (cd) hydro and (de) hermal in period 2. p M p p(.) c B b B e Th e Max hermal x Figure 9. Monopoly. ydro and hermal capaciy Inroducing a reservoir consrain as in Figure 6 will no change he soluion for he case of an inersecion of he marginal revenue curves wihin he area delimied wih he lines from B and C in ha figure showing he sorage possibiliies. Resul 13. Monopoly and hydro - hermal: A monopolis will equae he waer value wih he marginal cos of hermal, and no he marke price. The use of hermal capaciy may be reduced in all periods and will be base load unless a hydro reservoir consrain is binding. For such periods hermal capaciy will also be used as peak. Demonsraion: The resuls follow from (61) and condiions similar o hose in (56) in case of limi on he reservoir.

33 33 Period 1 Period M p 2 M p 1 c c a b c d e ydro energy Thermal exension Figure 10. Two periods and monopoly, hydro and hermal 7. Furher opics There are, of course, many more ineresing aspecs of hydropower for economiss han he opics covered above. We will give an indicaion of he naure of some of he aspecs. Transmission Producion and consumpion of elecriciy akes place wihin a nework. The economic choice of ype and dimensions of a connecor from producion node o a consumpion node is a classic wihin economics, and is an example of an engineering producion funcion (see Førsund, 1999). Transmission lines have upper limis on how much elecriciy can be ransferred, and losses are incurred as a funcion of loads. The flow in neworks follow basic

34 34 physical laws. Economiss should noe ha here may be exernal effecs in neworks of significance for pracical policy (Borensein e al. (2000), ogan, 1997). There is an ineresing rade off beween increase in generaion capaciy and increase in ransmission capaciy, especially when he laer concerns inernaional connecors. (See Førsund (1994) for a general modelling of ransmission wihin he framework used in his paper.) Invesmens Physical capaciies of generaion and ransmission have been assumed given in he presenaion. We have focussed on decisions of operaion. A firs enaive invesmen analysis can be done wihin ha framework by inspecing he shadow prices on he capaciy consrains (Førsund, 1994). owever, his procedure is only valid for marginal invesmens. Large-scale invesmens mus be evaluaed by simulaing on he oal sysem wihin a longerm ime horizon and hen comparing values of he social objecive funcion including invesmen coss. Uncerainy For a realisic modelling of hydropower here is no way o escape he inroducion of sochasic variables. Inflows o he reservoirs are fundamenally sochasic variables. The household (general) demand for elecriciy in Norway is also dependen on ouside emperaure, especially due o he high share of elecriciy heaing of dwellings. This relaionship makes also par of demand sochasic. The objecive funcions used in he paper have o be reformulaed o expeced values. owever, here are some echnical difficulies of mahemaical naure o ge qualiaive insighs in he case of consrains, because he consrains have o be obeyed in a physical sense. I is no so ineresing for policy analysis o require ha a reservoir should no be empied in an expeced sense: empying is an absolue even. The mahemaical ool ha can be used is sochasic dynamic programming (see Wallace e al., 2002). Marke power The feaures of zero operaing coss, exremely quick regulaion of producion and sorage of huge amoun of waer compared wih wha is currenly used make hydropower a specially ineresing case for use of marke power. Alhough he share of hydropower was small in California in he crisis experienced in , hydropower producers was blamed for use of marke power (see Borensein e al. (1999), Borensein e al. (2002), Joskow and Kahn,