Equilibrium Prices Supported by Dual Price Functions in Markets with Non-Convexities

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1 Equilibrium Prices Suppored by Dual Price Funcions in Markes wih Non-Convexiies Mee Børndal Kur Jörnsen Deparmen of Finance and Managemen Science Norwegian School of Economics and Business Adminisraion (NHH) Helleveien N-545 Bergen Norway Absrac The issue of finding marke clearing prices in markes wih non-convexiies has had a renewed ineres due o he deregulaion of he elecriciy secor. In he day-ahead elecriciy marke, equilibrium prices are calculaed based on bids from generaors and consumers. In mos of he exising markes, several generaion echnologies are presen, some of which have considerable non-convexiies, such as capaciy limiaions and large sar up coss. In his paper we presen equilibrium prices composed of a commodiy price and an uplif charge. The prices are based on he generaion of a separaing valid inequaliy ha suppors he opimal resource allocaion. In he case when he sub-problem generaed as he ineger variables are held fixed o heir opimal values possess he inegraliy propery, he generaed prices are also suppored by non-linear price-funcions ha are he basis for ineger programming dualiy. Keywords Elecriciy Markes, Pricing, Ineger Programming Dualiy. Inroducion A reason for he renewed ineres in obaining marke clearing prices in markes wih nonconvexiies is he deregulaion of he elecriciy markes ha has aken place recenly. In such markes, he non-convexiies arise from sar-up and shu-down coss of power plans, and minimum oupu requiremens ha mus be fulfilled in order o run cerain plans (see for insance O Neill e al. (2, 22) or Hogan and Ring (2)). Moreover, non-convexiies in a marke wih uniform linear prices may conribue o and legiimae sraegies like hockey sick bidding, which may have considerable redisribuion effecs (Oren (2)). This may also be regarded as an argumen in favor of searching for alernaive price mechanisms o deal wih he underlying complicaions ha severe non-convexiies may creae in a marke.

2 As so clearly described by Scarf (994), The maor problem presened o economic heory in he presence of indivisibiliies in producion is he impossibiliy of deecing opimaliy a he level of he firm, or for he economy as a whole, using he crierion of profiabiliy based on compeiive prices. Scarf illusraes he imporance of he exisence of compeiive prices and heir use in he economic evaluaion of alernaives by considering an economic equilibrium (Walras). I is assumed ha he producion possibiliy se exhibis consan reurns o scale, so ha here is a profi of ero a he equilibrium prices. Moreover, each consumer evaluaes personal income a hese prices and marke demand funcions are obained by he aggregaion of individual uiliy maximiing demands. Since he sysem is assumed o be in equilibrium, supply equals demand for each of he goods and services in he economy. A new manufacuring echnology, ha also possesses consan reurns o scale, is o be considered, and, he quesion is if his new aciviy is o be used a a posiive level or no. In his seing, he answer is simple and sraighforward, as formulaed in Scarf (994): If he new aciviy is profiable a he old equilibrium prices, hen here is a way o use he aciviy a a posiive level so ha wih suiable income redisribuions, he welfare of every member of sociey will increase. Furhermore, if he new aciviy makes a negaive profi a he old equilibrium prices, hen here is no way in which i can be used o improve he uiliy of all consumers, even allowing he mos exraordinary scheme for income redisribuion. This shows he srengh of he pricing es in evaluaing alernaives. However, he opporuniy of using he pricing es relies on he assumpions made, i.e. ha he producion possibiliy se displays consan or decreasing reurns o scale. When we on he oher hand, have increasing reurns o scale he pricing es for opimaliy migh fail, and i is easy o consruc examples showing his, for insance by inroducing aciviies wih sar up coss (a very simple example is provided in O Neill e al. 22). Wih he lack of a pricing es, Scarf inroduces as an alernaive a quaniy es for opimaliy. Over ime several suggesions have been made o address he problem of finding prices for problems wih non-convexiies, especially for he case where he non-convexiies are modeled using discree variables. The obecive has been o find dual prices and inerpreaion of dual prices for ineger programming problems and mixed ineger programming problems. The decisive work of Gomory and Baumol (96) is addressing his issue, he ideas presened here has laer been used o creae a dualiy heory for ineger programming problems. Wolsey 2

3 (98) gives a good descripion of his heory, and shows ha in he ineger programming case we need o expand our view of prices o price-funcions in order o achieve inerpreable and compuable duals (refer also for insance, Alcaly e al. (966), Scarf (99), Williams (996), and Surmfels (2)). However, hese dual price-funcions are raher complex, and oher researchers have suggesed approximae alernaives, bu none of hese suggesions have, o our knowledge, been used successfully o analye equilibrium prices in marke wih non-convexiies. Recenly, O Neill e al. (22) presened an innovaive mehod for calculaing discriminaory prices, referred o as IP-prices, based on a reformulaion of he non-convex opimiaion problem. The mehod assumes ha he opimal producion plan is known and aims a generaing marke clearing prices, using a non-linear pricing scheme, pricing commodiies, sar-ups and capaciy. Due o he fac ha he commodiy prices generaed by O Neill e al. s procedure may possess some unwaned properies, Hogan and Ring (2) developed a very ineresing minimum uplif paymen scheme for markes wih non-convexiies. Their mehod is similar in spiri o he pricing rules used in he New York and PJM elecriciy markes. However, he minimum uplif pricing rule is us a simple adusmen of linear pricing o a marke in which non-convexiies are presen, and hence, no in line wih ineger programming dualiy heory. In his paper hen, we sugges he use of modified IP-prices as equilibrium prices in markes wih non-convexiies. The prices are derived using a minor modificaion of he idea of O Neill e al. and are based on he generaion of a valid inequaliy ha suppors he opimal soluion and can be viewed as an ineger programming version of he separaing hyperplane ha suppors he linear price sysem in he convex case. The modified IP-prices generaed, are based on ineger programming dualiy heory, and i can be shown ha here exiss a nonlinear non-discriminaory price-funcion, ha suppors he modified IP-prices. All hree pricing approaches assume ha he bidding forma is modified in order for he generaors o be able o use a muli-par bidding scheme. This is no cusomary in mos of oday s exising elecriciy markes, and he incenive problems ha migh appear as a resul of muli-par bidding proocols, have o be furher analyed.

4 The paper is organied as follows. In secion 2, he non-convex problem a hand is inroduced, and he reformulaion rendering he IP-prices derived by O Neill e al. (22) is presened. We coninue wih he parial equilibrium formulaion of he problem due o Hogan and Ring (2), and presen heir minimum uplif pricing scheme. In secion, we presen he basis for he modified IP-prices and how hey can be calculaed. In secion 4, we compare he hree pricing mehods on he modified Scarf example used by Hogan and Ring, and also illusrae he corresponding non-linear price funcions for a few levels of demand. Finally, some concluding remarks and issues for fuure research are presened in secion An Opimiaion Model for he Uni Commimen and Dispach Problem Hogan and Ring (2) presen an iner-emporal opimiaion model for economic uni commimen and dispach. The model allows consideraion of boh dynamics, i.e. muliple period issues, and ransmission consrains. The model is as follows: Max T Bi di C g ( ) ( ) = i subec o L ( y ) ι y = + S y = d g g m, g M, R ( g, g ), K ( y ) u and ineger for all where B i ( d i ) denoes a bid-based, well-behaved concave benefi funcion of demand for cusomer i in ime period, C ( g ) denoes he bid-based, well-behaved convex cos funcion for oupu of generaor ype in ime period, and S is he bid-based sar-up cos for generaor ype. Noe he sar-up coss could be ime-dependen in a more general version of he model. Also, i is possible o include a consrain ha guaranees ha he sar-up cos is only effecive in he period ha he generaor is sared up and hence, he generaor can be run wihou incurring addiional sar-up cos as long as he generaor is acive. Shu down cos is 4

5 also easy o include in he model. Furhermore, m and M are bid-based minimum and maximum oupus for each generaor of ype ha is commied, is an ineger variable showing he number of generaors of ype commied, u is an upper bound on he number of generaors of ype, and y is he vecor of ne load a each locaion. For locaion ϕ, y = d g. L y ) are he losses in period wih ne demand ι y where ϕ i ϕ i ϕ ( ι =(,,.,). K y ) represens he ransmission consrains for ne load y in period, and ( finally, R g, g ) are ramping limis or oher dynamic limis for generaor. ( The model presened is non-convex due o he ineger variables used for modeling he fixed sar-up coss and he aached minimum and maximum oupu consrains. Apar from his, he model is a convex mahemaical programming problem, in which we have exernaliies arising from he ransmission consrains. In his paper we will focus on he non-convexiies ha are presen due o he ineger variables. Hence, we will use a single period model wihou ransmission consrains and wihou losses. Also he model ha we use in his paper, has all demand locaed a a single locaion. The effecs of muliple periods, ransmission consrains and losses on he pricing scheme we sugges, will be he focus of fuure research. 2. The Simplified Single-Period Model In a given period, wih demand fixed a d, and no losses, ramping or ransmission consrains, he problem sudied is: Min subec o + C ( g ) S g g = d g m g M u and ineger for all g 5

6 The simplified model is a leas-cos uni commimen and dispach problem. Our main purpose of sudying his simplified model is o presen various alernaives for finding equilibrium prices in a marke wih non-convexiies, such as he day-ahead elecriciy marke, in which several differen producion echnologies are presen and are compeing. The simplified model is of he same ype as he allocaion model used by Scarf (994) in his seminal paper on resource allocaion and pricing in he presence of non-convexiies. Scarf presened, in he absence of usable prices, a quaniy es ha can be used o verify opimaliy or as a means o making exchanges ha lead o opimaliy. Recenly, O Neill e al. (2) presened an ineresing reformulaion ha generaes IP-prices ha are inerpreable as marke clearing prices, and can be viewed as a Walrasian decenralied price-direced equilibrium in he presence of indivisibiliies. The reformulaion used by O Neill e al. is accomplished by using knowledge of he opimal soluion values for he ineger variables, and adding consrains o he model ha guaranee ha hese opimal values are aained. Hence, he procedure and reformulaion is no inended o be used in order o calculae he opimal soluion, bu raher o use he knowledge o obain inerpreable prices ha can be used in order o creae viable conracs in he marke. 2.2 O Neill e al. s Reformulaion The single-period problem wih he ineger commimen variables,, fixed a heir known opimal values,, is he following: Min subec o + C ( g ) S g = d g m g M u g = 6

7 The reformulaed problem is a convex programming problem and has an associaed dual problem, wih dual variables p, α, β, γ and π. O Neill e al. inerpre hese dual variables as equilibrium prices, where p is he commodiy price, γ are he capaciy prices, and α, β, π are discriminaory uplif prices. Based on his, O Neill e al. deermine a se of marke clearing conracs, T, ha can be offered from he aucioneer o he marke paricipans, and ha are such ha hey clear he marke. They also show ha he IP-prices are opimal soluions o he decenralied opimiaion problems faced by each generaor ype, and ha in he case where he cos funcions ake he form funcions, each generaor s opimal profi is exacly equal o ero. C ( g ) = C g, i.e. linear cos However, as Hogan and Ring (2) poin ou, he marke clearing prices, or IP-prices, have some properies ha are quesionable. Firs, he commodiy IP-prices, p, can be volaile, i.e. a small change in demand can lead o a large change in he IP-commodiy price. Secondly, he IP-uplif charges can be posiive or negaive and also show a volaile behavior. This will be illusraed in he example in secion 4. Hogan and Ring hen make use of he reformulaion of O Neill e al. and inerpre he IP-equilibrium prices as prices generaed from a parial equilibrium model. This is done in order o highligh he imporance of he IP-uplif paymens ha is par of he marke selemen sysem. 2. Hogan and Ring s Parial Equilibrium Formulaion Hogan and Ring s parial equilibrium formulaion includes he following problems: A) The problem faced by he sysem operaor, i.e. he marke coordinaor: Max p( d i g ) π subec o i g = d i i u and ineger for all where p, g and are variables. I should be poined ou ha he sysem operaor only acs as an inermediary marke maker ha rades elecriciy wih generaors and loads and purchases / sells commimen ickes. The assumpion is ha he sysem operaor is a fully 7

8 regulaed monopolis ha hrough he regulaion acs as if i was a price aking profi maximier. B) The problem faced by each ype of generaor: Max pg + π C ( g ) S subec o g g m M u and ineger C) The problem for each consumer: Max B ( d ) + c i i i subec o pd i + ci ω i + > s i Π wih variables p, di, and ci. Here, s i, i si = are he share holdings of profis for each consumer and ω i is he iniial wealh endowmen for he numeraire good for consumer i wih consumpion level c i. Hogan and Ring show ha hese hree problems form a parial equilibrium model, in which he se of prices and quaniies ( p, π, d, g, ), consiues a compeiive parial equilibrium. However, he problem faced by he sysem operaor mus be reformulaed in he same way as is done by O Neill e al., in order for he se of problems o be well-behaved. I.e. he sysem operaor s problem is: Max p( d i g ) π subec o i g = d i i u = 8

9 The IP-prices and quaniies suppor he compeiive parial equilibrium. Wih he equilibrium profi for he sysem operaor = p d g π, which in he simplified Π ( ) i model, wih no ransmission effecs, reduces o he las erm. This las erm is equal o he ne paymen o he generaors for commimen ickes. From he consruc of he revised formulaion, including he equaliy consrains for he ineger variables, i is clear ha he dual prices π can be boh posiive and negaive. This also means ha he oal profi for he sysem operaor can eiher be posiive or negaive. Since he prices are equivalen wih he IPprices in O Neill e al. s model, he parial equilibrium inerpreaion only serves as a guide o how hese equilibrium quaniies and prices can be inerpreed. As Hogan and Ring make clear, he ne paymens for he commimen ickes mus be colleced in form of fixed charges from consumers and/or producers. They are independen of he level of consumpion and found in he parial equilibrium model based on he consumers ownership shares of he sysem operaor s loss or profi. In he day-ahead elecriciy marke, such fixed ransfers are no available wihin he marke designs used. However, here are markes in which ne paymens are colleced in he form of an uplif charge, allocaed beween consumers according o some a priori allocaion rule. Because of he volailiy of he IP-prices and he fac ha he uplif charge can be boh posiive and negaive, Hogan and Ring sugges he use of minimum uplif paymens and resuling equilibrium prices. The basis for his is ha hey would like he uplif paymens o be nonnegaive and he commodiy prices less volaile. 2.4 Hogan and Ring s Minimum Uplif Equilibrium Model Ring (995) sudied approaches for analying deviaions from he simple perfecly convex equilibrium model. One such deviaion of ineres is he non-convexiies generaed from he need o include ineger variables in he model formulaion. As an approximaion, which is close o he ideal siuaion in he linear case in which a linear price sysem suffices, Ring suggess an approach in which he necessary oal uplif paymen is kep as low as possible. The minimum uplif equilibrium model is based on he idea ha given he opimal soluion ( g, ), define he nonnegaive commimen paymens in erms of he uplif requiremens needed o make commied and uncommied generaors a leas indifferen. This means ha for a marke clearing price, p, he commimen paymen o each generaor is 9

10 π ( p) = Max(, Π + Π ) + where Π = pg C g ) S and is he opimal value of he problem ( Π Max pg C ( g ) S subec o g g m M u and ineger g The minimum uplif paying rule resuling from his is o pay each generaor an amoun π ( p) in addiion o he paymen received for he commodiies. The paymen π ( p) is condiional on acceping he commimen and dispach soluion. By consrucion, he uplif paymen is non-negaive and he oal charge ha has o be colleced from he cusomers/consumers is π. For each commodiy price, here exiss a corresponding uplif paymen. All hese suppor he compeiive marke soluion. Among all he possible pricing schemes (p, π ( p) ), he minimum uplif equilibrium pricing scheme is he commodiy price for which he oal uplif paymen is minimal. As is illusraed in he example in secion 4, he minimum uplif pricing scheme yields relaively sable commodiy prices as a funcion of demand however, he uplif paymens are raher volaile.. Modified IP-prices Along wih he minimum uplif pricing of Hogan and Ring and he IP-prices suggesed by O Neill e al., we sugges a hird alernaive. This alernaive is more relaed o he original uplif prices, bu yields more sable commodiy prices and uplif charges. The modified IP prices are based on he connecion beween he IP-prices and he dual informaion generaed when Benders decomposiion (Benders (962)) is used o solve he resource allocaion problem.

11 Benders decomposiion is a compuaional mehod which is ofen used o solve models in which a cerain se of variables, he complicaing variables, are fixed a cerain values and he remaining problem and is dual is solved in order o ge bounds for he opimal value of he problem and also generae informaion on how o fix he complicaing variables in order o obain a new soluion wih a poenially beer obecive funcion value. When Benders decomposiion is used o solve a nonlinear ineger programming problem, as he problem sudied in his paper, he problem is pariioned ino solving a sequence of easy convex opimiaion problems and heir duals, where he complicaing ineger variables are held fixed. These problems, ofen called he Benders sub-problems, are generaing lower bounds on he opimal obecive funcion value, and yield informaion ha is added o he Benders maser problem, a problem involving only he complicaing variables. The informaion generaed akes he form of cuing planes. Here, we are no ineresed in using Benders decomposiion o solve he opimiaion problem since we assume ha we already know he opimal soluion. However, viewing he reformulaion used by O Neill e al. as solving a Benders sub-problem in which he complicaing ineger variables are held fixed o heir opimal values reveals useful informaion concerning he IP-prices generaed in he reformulaion. The reformulaion viewed as a Benders sub-problem is he following: Min subec o + C g S g = d g m g M g where are he known opimal values for he ineger variables. Here we are using he linear version of he problem in order o simplify. However, he resuls derived are rue also for he case when he producion funcion is a general convex funcion, alhough in his case nonlinear programming dualiy has o be used. The dual o he Benders sub-problem wih he ineger variables fixed a heir opimal values is

12 Max pd + α m β M + subec o S p + α β C α, β The dual of he reformulaion used by O Neill e al. is Max pd + π subec o p + α β C α m + β M + π S α, β I is obvious ha he opimal dual soluion ( p, β ), α yields a feasible soluion o he dual of he reformulaion and ha π = S + α m β M. This means ha he opimal dual variable values of he equaliy consrains, =, are in fac he coefficiens in he Benders cu ha is generaed when solving he Benders sub-problem wih he ineger variables held fixed a heir opimal values. The Benders cu migh incorporae coefficiens in a valid inequaliy. The Benders obecive funcion cu has he form ( α m β M S ) p d + + From his we can generae an inequaliy of he form ( α ) ( ) m β M + S α m β M S + This inequaliy is a valid inequaliy for some problems, whereas for oher problems a differen se of variables, including boh ineger-consrained variables and coninuous variables, needs o be held fixed in order o generae a supporing valid inequaliy. From his, i is obvious ha 2

13 he dual informaion generaed in O Neill e al. s reformulaion and he dual informaion generaed from he Benders sub-problem saed above, is he same. In linear and convex programming he exisence of a linear price vecor is based on he use of he Separaing Hyperplane Theorem. Based on he convexiy assumpion, he equilibrium prices are he dual variables or Lagrange mulipliers for he marke clearing consrains. In he non-convex case i is well known ha no every efficien oupu can be achieved by simple cenralied pricing decisions or by linear compeiive marke prices. If we could find a supporing separaing valid inequaliy ha could serve he same purpose as he separaing hyperplane in he convex case, we would have a way o consruc equilibrium marke clearing prices in he non-convex case. However, in order o do so, we need o use a reformulaion of he original problem in which he coefficiens derived from he Benders sub-problem are coefficiens in valid inequaliy. The problem is ha his is no always he case when we fix he ineger variables o heir opimal values. Based on his observaion, i is easy o see why he IP commodiy and uplif prices are volaile. The reason for he volailiy comes from he fac ha for some problems he coefficiens in he Benders cu, derived when he ineger variables are held fixed o heir opimal values, are in fac coefficiens from a separaing valid inequaliy for he mixed ineger programming problem sudied. Hence, he separaing supporing valid inequaliy is he replacemen of he supporing hyperplane ha is he basis for linear pricing in he convex case. However, for oher problems his is no he case. This means ha if we are looking a a class of problems for which a supporing valid inequaliy only includes ineger variables, he IP-prices generaed will yield a supporing valid inequaliy in he sense ha he inequaliy suppors he opimal soluion and is a separaing hyperplane. Tha is, appending his valid inequaliy o he original problem does no cu off any oher feasible soluion. The original Scarf example, wih wo producion echnologies, Smokesack and High-Tech, falls ino his class of problems. However, Hogan and Ring s modified Scarf problem does no. Even if he problem under sudy is no wihin he class of problems in which a supporing valid inequaliy only includes ineger variables, i migh be he case ha for cerain demand values he coefficiens for he ineger variables generaed from he opimal dual variable values are coefficiens for hese variables in a supporing valid cuing plane, whereas for

14 oher demand values hey are no. For problems in his class, he IP-commodiy price and IPuplif prices will be volaile. Hence, in order o derive IP-prices ha are suppored by a separaing valid inequaliy, i is necessary o regard also some of he coninuous variables as complicaing variables ha are o be held fixed in he reformulaed problem. This means ha when using he pariioning idea in Benders decomposiion in order o calculae he opimal soluion, i is clear ha only he ineger variables are complicaing, when complicaing is inerpreed as complicaing from a compuaional poin of view. However, when our aim is o generae inerpreable prices, anoher se of variables should be regarded as complicaing. We need o find ou which variables should be held fixed a heir opimal values in order for he Benders cuing plane, derived by solving he Benders sub-problem and i s dual, o be a valid inequaliy ha suppors he opimal soluion. When hese variables are held fixed a heir opimal values, his cuing plane will, when added o he problem, generae commodiy prices and uplif charges ha have he properies we are looking for, i.e. marke clearing and non-volaile. We call he prices derived his way he modified IP-prices. In he example presened below, we will show how hese prices can be calculaed. If he problem sudied is such ha he sub-problem generaed when he ineger variables are held fixed o heir opimal values has he inegraliy propery, i is also possible o derive he corresponding nonlinear price funcion. The exisence of such non-linear price funcions and how hey can be generaed can be found in Wolsey (98). I should be noed ha since we know wha we are looking for, i.e. a valid inequaliy having cerain specific coefficiens, he generaion of he non-linear price funcion migh be easier. Also, we are no searching for a supporing separaing inequaliy ha when appended o he original problem yields an ineger feasible soluion since his is no necessary for our purposes when only inerpreable dual informaion is searched for. In he general mixed ineger case, he approach presened by Wolsey o generae he nonlinear price-funcion compaible wih sub-addiive ineger programming dualiy, canno be used. However, if we can generae a valid inequaliy ha suppors he mixed ineger opimal soluion, he prices derived from he original problem wih his valid inequaliy appended, can be inerpreed as equilibrium marke clearing prices. I is an ineresing research quesion o find ou if i is possible o generae a non-linear price funcion also for his general case. 4

15 I should be clear ha boh he Walrasian inerpreaion ha suppors O Neill e al. s marke clearing conracs and he parial equilibrium model ha is used by Hogan and Ring can be reformulaed for our modified IP-prices. 4. Hogan and Ring s Adaped Scarf s Example We use Hogan and Ring s example o illusrae he generaion of he modified IP prices ha are suppored by a nonlinear price funcion. The example consiss of hree echnologies, Smokesack, High Tech and Med Tech, wih he following producion coss: Smokesack High Tech Med Tech Capaciy Minimum Oupu 2 Consrucion Cos 5 Marginal Cos 2 7 Average Cos Maximum number We can formulae Hogan and Ring s allocaion problem as a mixed ineger programming problem as follows, denoed problem P. (P) Min q + 2q2 + 7q subec o q q + q = D q 72 q2 6 q 2 + q q, q2, q, 2, and ineger 5

16 Here, D denoes he demand and he consrucion variables for Smokesack, High Tech and 2 q q2 Med Tech are, and, respecively, whereas, and q denoe he level of producion using he corresponding Smokesack, High Tech and Med Tech echnologies. Noe 2 ha for fixed ineger values of,, and, he remaining problem in he coninuous variables have he inegraliy propery. Hence, he allocaion problem is in fac a pure ineger programming problem. The reason for his is he special form of he consrain marix for his example. The opimal soluions for demand levels ranging from o 6 is given in able in he appendix. The difficuly of finding a linear price srucure for his problem is obvious from he graph of he oal cos as a funcion of demand, illusraed in figure. The figure only shows he oal cos funcion for demand ranging from o 25. Obviously, as demand grows, he oal cos funcion becomes more linear, and from demand equal o, he increase is linear in demand. Figure : Toal Cos for Demands Ranging from o 25 oal cos Toal cos Demand Series In able 2 in he appendix he commodiy and uplif prices generaed from he hree approaches discussed in secions 2 and in his paper, are displayed (IP corresponding o O Neill e al., Minup o he Hogan and Ring approach, and ModIP is he modified IP prices we sugges in secion ). The hree mehods presened use hree differen reformulaions o calculae he prices. 6

17 O Neill e al. s reformulaion of problem P is: Min q + 2q2 + 7q subec o q q + q = D q 72 q2 6 q 2 + q 2 = = 2 = q, q2, q, 2, and ineger 2 where, and represen he opimal ineger soluion for he specified demand in he range o 6. For some of he demands, his reformulaion generaes negaive uplif prices, which causes he commodiy prices and uplif prices o be very volaile as a funcion of demand. This volailiy is illusraed in he wo graphs of figures 2 and. 7

18 Figure 2: Commodiy Price, IP IP Commodiy Price Commodiy Price Demand Series Figure : Uplif Price, IP IP Uplif Price 5 4 IP Uplif 2 Series Demand Hogan and Ring solve he following problem o calculae minimum-uplif commodiy and π is uplif prices: find commodiy price, p, for which he oal uplif paymen ( p) + minimal, where he uplif paymen o each generaor-class is π ( p) = Max(, Π Π ), and where Π = pq C q S, he uni cos vecor C = (,2,7), and he sar up cos vecor S Π + = (5,,). The s are he opimal obecive funcion values of he problems: 8

19 Π + = Max pq C q S subec o m q M u and ineger From he consrucion, i is clear ha he min uplif commodiy prices are less volaile as compared wih he IP prices. This is seen in figure 4 for he example. However, he min uplif paymens are volaile a a low level, which can be seen from figure 5. Figure 4: Commodiy Price, Minup Miin Uplif Commodiy Price 8 7 Commodiy Price Series Demand Figure 5: Uplif Price, Minup Min Uplif 25 Min Uplif 2 5 Series Demand 9

20 For he modified IP prices, he reformulaed problem o be solved is: Min q + 2q2 + 7q subec o q q + q = D q 72 q2 6 q 2 + q 2 = = 2 q = q q, q2, q, 2, and ineger The reason for requiring producion of unis of he hird ype o be held fixed ( q = q ) is ha by doing so, he coefficiens in he corresponding Benders cu will be coefficiens in a valid inequaliy, a separaing hyperplane, ha suppors he opimal soluion for all levels of demand. Hence, by consruc he modified IP prices are heoreically consisen wih mixed ineger programming dualiy heory. As seen in he graphs of figures 6 and 7, he modified IP-prices are no volaile, and he modified IP uplif prices form, as expeced, an increasing sepwise funcion of demand up o he poin where he problem s coninuous relaxaion has an ineger soluion. From ha level of demand on, in his example D =, linear pricing is sufficien. 2

21 Figure 6: Commodiy Price, ModIP Modified IP Commodiy Price Commodiy Price Demand Series Figure 7: Uplif Price, ModIP Modified IP Uplif Price Modified IP Uplif Series Demand When he modified IP-prices are derived, he following valid inequaliies are used: For demand ranging from -, he inequaliy 5 + q MIP is a price supporing i valid inequaliy. Here, MIP i is he modified IP-uplif seen in he able above. For demand ranging from 4-2, he valid inequaliy ha suppors he modified IP-prices, is q 4 MIP i, whereas for demands -6, he linear programming relaxaion of he problem has ineger soluion, and hence, a linear price is feasible. 2

22 In fac, he inequaliy q 4 MIP i is modified IP-price supporing for all values of demand in he range -2, excep for demands equal o, 8 and, given ha he righ hand sides for he demand values 2,, 4, 5, 6, 7, 9,,, and 2 are changed o 8, 2, 6, 2, 24, 2,, 5, 9, and 4, respecively. However, using his alernaive supporing valid inequaliy o generae he prices, leads o an increased volailiy in commodiy and uplif prices as can be seen in figures 8 and 9 below, for he demand values beween and 2. Figure 8: Uplif Price, ModIP Modified IP-uplif normal case IP-uplif Demand Series Figure 9: Alernaive Uplif Price, ModIP Modified IP-uplif alernaive IP-uplif Demand Series 22

23 The modified IP commodiy price is sable a he value 2 for demand ranging from -, and hen umps o he value (figure 6). This is when he supporing valid inequaliy q 5 MIP i is used for demands -. However, if he alernaive is used, he commodiy price will be for all demand values ranging from -2 excep he demands, 8 and, for which he commodiy price is 2. The modified IP-prices are suppored by a non-linear price funcion and hence, are nondiscriminaory. Here, we presen he shape of he nonlinear price funcion for a few demandvalues. The non-linear price funcion for demand equal o 55 can be derived as follows. Adding q q + q = D q 72 q2 yields he consrain q 55. Adding 7 imes he consrain 2 5, dividing by 6, and rounding up o he neares ineger value, gives us he consrain + q 2. Taking imes he inequaliy q 55 and adding he inequaliy + q 2 o his resul, we ge he nex inequaliy by dividing he resul by 7, and rounding up. The resuling inequaliy is + + q 24. Finally, he valid inequaliy q 82 is generaed by muliplying he inequaliy q 55 by 9, adding he inequaliy + q 2 o his inermediae resul, and finally, adding 2 imes he inequaliy q 24 o his. The price supporing valid inequaliy is hen derived by dividing by and rounding up o he neares ineger. The nonlinear price funcion ha suppors he modified IP prices for he demand level 55 is hen given by he following expression: 2

24 F + () 9 = () + () (() + (2) + ()) () + (2) + () + 7(7) (() + (2) + ()) () + (2) + () + 7(7) Here () denoes he coefficien or righ hand side of he consrain wih he respecive number in he formulaion of he original mixed ineger program. For demand equal o 56 he nonlinear price funcion becomes a lile bi more complicaed. The reason for his is ha for his demand level we need o use he echnology wih he high marginal cos. The opimal uplif for he demand equal o 56 is 87, bu he procedure used o consruc he price funcion for demand equal o 55 only generae an uplif of 86 for demand equal o 56. Hence, we have sill no generaed he price supporing valid inequaliy. In order o generae he price supporing inequaliy q 87 we need o derive a number of addiional valid inequaliies. Firs we generae he wo inequaliies q 6 and q + 6. These are generaed by adding he inequaliies q 86 and q 25, and o he resul add one of he inequaliies + q, q, dividing by wo and rounding up. Also 2 6 he inequaliy q 49 is easily generaed by adding he inequaliies q 86 and q 56, dividing he resul by five and rounding up. Anoher inequaliy needed o generae he price supporing valid inequaliy is he inequaliy q + 2. This inequaliy comes from adding hree of he previously generaed inequaliies namely, q 6, q + 6 and q 49, dividing he resul by wo and rounding up. The nex inequaliy ha needs o be generaed, is he inequaliy which sems from 2 imes he inequaliy q + 2 and imes he inequaliy q 49, added ogeher 24

25 wih he inequaliy 2 + q. Using division by wo and rounding up, his yields he inequaliy q 26. Finally, he price supporing inequaliy is generaed by aking hree imes he inequaliy q 56, adding o he previously generaed inequaliy q 26, and dividing by wo. In order o illusrae he non-linear price funcion supporing he demand levels -, we choose o generae he non-linear funcion supporing he equilibrium for demand equal o. The following hree inequaliies are easily derived, + +, + + q, and 2 2 q + q +, using he fac ha q + q + q, 6 q, 72 q2, and 2 6 q 2 =, adding, dividing, and rounding up. Wih hese hree inequaliies added, he opimiaion problem has he obecive funcion value 2 and he soluion is inegral. The dual variables can be used o generae he inequaliy q + 2q2 + 25q 5, subracing 2 imes he equaliy consrain yields q, which by adding 2 imes he inequaliy, generaes he desired valid inequaliy. I should be noed ha we do no sugges ha he non-linear price funcion supporing he opimal soluion and he modified IP prices should be generaed in pracical applicaions. However, he prices generaed using his idea, are due o he exisence of a non-linear nondiscriminaory price funcion, and heoreically sound. 5. Conclusions and Issues for Fuure Research In his paper we have shown ha he modified IP-prices yield commodiy and uplif charges ha have some nice properies. The use of he opimal soluion o generae a reformulaion ha generaes a supporing valid inequaliy o he resource allocaion opimiaion problem is also presened. This means ha he modified IP-prices, for problems which can be regarded as pure ineger programming problems, are consisen wih he basis in ineger programming dualiy. Knowing he coefficiens of a supporing valid inequaliy means ha we know ha, in his case, here exiss a non-discriminaory non-linear price-funcion ha ogeher wih he affine capaciy and produc prices yield equilibrium prices in markes wih non-convexiies. Also, knowing he coefficiens of he supporing valid inequaliy makes i easier o consruc he non-linear price-funcion as compared wih he case where hese coefficiens are unknown. Hence, he modified IP-prices have a non-linear non-discriminaory price-funcion 25

26 supporing hem, and alhough hey seem o be discriminaory, hey can also be viewed as a compac represenaion of he supporing non-linear price-funcion, and hence, are in fac nondiscriminaory. For he more general mixed ineger programming case, he modified IP-prices are suppored by a separaing valid inequaliy. However, he exisence of a non-linear price funcion and how i can be generaed is in his case more complicaed and would be an ineresing quesion for furher research. How he knowledge of he exisence of a separaing supporing valid inequaliy ha can be used o derive equilibrium prices in markes wih non-convexiies should be used o suppor a bidding forma and a conrac mechanism ha are incenive compaible, remains anoher ineresing quesion for fuure research. Oher research quesions ha should be addressed, are how elasic demand will effec and can be deal wih in markes wih non-convexiies, if and how he uplif charges should be colleced among he cusomers, and how his cos allocaion should influence he design of marke clearing conracs. If i is saisfacory us o have a heoreical foundaion for he use of modified IP-prices, or if procedures ha acually generaes a corresponding non-discriminaory price-funcion needs o be furher invesigaed References Alcaly, Roger E., and Alvin K. Klevorick A Noe on he Dual Prices of Ineger Programs. Economerica 4(): Benders, J.F Pariioning Procedures for Solving Mixed-Variables Programming Problems. Numerische Mahemaik 4: Børndal, Mee, and Kur Jörnsen. 24. Allocaion of Resources in he Presence of Indivisibiliies: Scarf s Problem Revisied. Discussion Paper, Deparmen of Finance and Managemen Science, Norwegian School of Economics and Business Adminisraion (NHH). Gomory, Ralph E., and William J. Baumol. 96. Ineger Programming and Pricing. Economerica 28(): Hogan, William W., and Brendan J. Ring. 2. On Minimum-Uplif Pricing for Elecriciy Markes. Working Paper, John F. Kennedy School of Governmen, Harvard Universiy. O Neill, Richard P., Paul M. Sokiewic, Benamin F. Hobbs, Michael H. Rohkopf, and William R. Sewar Jr. 22. Efficien Marke-Clearing Prices in Markes wih 26

27 Nonconvexiies. Working Paper (Forhcoming in European Journal of Operaional Research). O Neill, Richard P., Udi Helman, Paul M. Sokiewic, Michael H. Rohkopf, and William R. Sewar Jr. 2. Regulaory Evoluion, Marke Design and Uni Commimen. In The Nex Generaion of Elecric Power Uni Commimen Models, edied by Benamin F. Hobbs, Michael H. Rohkopf, Richard P. O Neill, and Hung-po Chao, Boson MA: Kluwer Academic Press. Oren, Shmuel S. 2. Marke Design and Gaming in Compeiive Elecriciy Sysems. Presened a The Arne Ryde Symposium on he Nordic Elecriciy Marke. Ring, Brendan J. 995, Dispach Based Pricing in Decenralied Power Sysems. Ph. D. Thesis, Deparmen of Managemen, Universiy of Canerbury, Chrischurch, New Zealand (Available a hp://ksgwww.harvard.edu/hepg/). Scarf, Herber E. 99. Mahemaical Programming and Economic Theory. Operaions Research 8(): Scarf, Herber E The Allocaion of Resources in he Presence of Indivisibiliies. Journal of Economic Perspecives 8(4): -28. Surmfels, Bernd. 2. Algebraic Recipes for Ineger Programming. Proceedings of Symposia in Applied Mahemaics. Williams, H. P Dualiy in Mahemaics and Linear and Ineger Programming. Journal of Opimiaion Theory and Applicaions 9(2): Wolsey, Laurence A. 98. Ineger Programming Dualiy: Price Funcions and Sensiiviy Analysis. Mahemaical Programming 2: Appendix Table : Opimal Soluions Demand Smoke Smoke High High Med Med Toal Sack Sack Tech Tech Tech Tech Cos Number Oupu Number Oupu Number Oupu

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29

30 Table 2: Commodiy and Uplif Charges Demand Commodiy prices Uplif charges IP Minup ModIP IP Minup ModIP

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