OPTIMAL PURCHASING PROCESS FOR ELECTRICITY AND RENEWABLE ENERGY CREDITS WITH PRICE AND DEMAND UNCERT

J Syst Sei Syst Eng (Jun 2012) 21(2): 184-203ISSN: 1004-3756 (Paper) 1861-9576 (Online)DOI: 10. 1007/s11518-012-5192-3CN11-2983/NOPTIMAL PURCHASING PROCESS FOR ELECTRICITY ANDRENEWABLE ENERGY CREDITS WITH PRICE AND DEMANDUNCERTAINTYZhiyuan CHEN Yiwen SU2 Houmin YAN3'Department ofAutomation, Tsinghua Universit, Beijing, 100084. China'Department of Sytems Engineering and Engineering Management, The Chinese University of Hong Kong,Shatin, N.T, China'Department ofManagement Sciences, City University of Hong Kong Kowloon, Chinahoumin. yan@cityu. edu.hk (8)AbstractThis paper optimizes the electricity and renewable energy credit (REC) purchasing process forenergy distribution. Electricity is traded in deregulated time-sequential markets at fluctuating prices.Optimal electricity purchasing under price and demand uncertainty is a challenging task for electricitydistributors, and the recently implemented renewable portfolio standards (RPS) further complicate thepurchasing process. Goverment regulatory decisions concemning the RPS require distributors topurchase corresponding certificates, namely RECs, equivalent to a certain percentage of theirelectricity sales. This paper formulates and optimizes the joint purchasing process for electricity andRECs. It also analyzes the effect of RPS policy on electricity distributors.Keywords: Renewable energy credit, information update, dynamic programming1. Introductionemissions sources across the United States." ToCarbon dioxide (CO2) emissions are thereduce carbon emissions, many U.S. states havemain cause of climates change, such as risingadopted a mandatory renewable energy policysea levels, severity and frequency of extremenamed as renewable portfolio standard (RPS).weather, ecosystems imbalance, and otherThe RPS requires that the electricity purchasingenvironmentalproblems.The electricityprocess include the purchase of correspondinggeneration sector is an intensely high-carbongovermment certificates called renewable energyindustry with a heavy reliance on fossil fuels.credits (RECs) to support the generation ofAccording to the U.S. Environments Protectionelectricity from renewable energy sources (U.S.Agency (2010) “The process of generatingDepartment of Energy 2010). In Califormia, forelectricity is the single largest source of CO2example, electricity dstributors need to showemissions in the United States, representing 40%that中国煤化工of total CO2 emissions from all CO2equivTYHCNMH G◎Systems Engineering Society of China & Springer-Verlag Berlin Heidelberg 2012Chen et al: Optimal Purchasing Process for Electricity and Renewoble Energy CreditsJ Syst Sci Syst Eng185The U.S. electricity market is a rather specialelectricity it produces from such sources. Thecase. Power generation piants sell elecricity to apower plant can sell these RECs 10 ectricitypower pool, Electricity distributors thendistributors to gaincompensation for itpurchase power from the pool and sell it tcinvestment costs in exploring renewable energycustomers. In the electricity markets, thesources. These electricity distributors then passdistributors bid oa and purchase electricity moreon the corresponding RECs, based on their sales,than once in the facc of imperfect demand10 the govemment to demonstrate theiuinformation and uncertainty in both price ancompliance with their regulatory obligations.demand. Take the electricity markets irRECs are central to the implementation of anCalifomia as an example. The states electricityRPS. Such a credit is a tradable certificate ofdistributorsbavemultiplepurchasingproof that me kWh of elecincity has beenopportunities, and there are three typicalgenerated from a renewable fuel source. Creditstime-sequentialelectricitymarkets:theare denominated in kWh and are a separateday- ahead market, the hour-ahead market (alsocommodity from the power itself. Electricity andknown as day-of market), and the indcpendentRECs can be, and often are, sold separately.systcm operator (ISO) teal-time marke. As timeThere thus exists an open REC market ingocs by, ditributors bave two opportunities toconjunction with the corresponding electricityprocure electricity from the day-ahead andmarket. The price of RECs depends on thehour-ahead markets. In the ISO real time market,equilibrium of demand and supply. An RPSthe total quantities of electricity purchased fromrequires all electricity distributors (or electricitytbe day-ahead and hour-ahead markets anrctailers, depending on the policy design) tocompared to the realized demand. Electricitydemonstrate, through the ownersbip of credits,distributors are then forced to buy/sell thethat they have supported an amount ofdifference betweea the realized demand andreaewable energy generation tbat is equivaleatelectricity purcbased. The optimal purchasing切0 a given percentagc of their to1a) xWh sales'.problem in the electricity market has beenUnder a mandatory RPS policy, electricitysolved by Sethi et al. (2005) and Yan et al.distributors must cach day determine the order(2000a, 2000b).quantities of kWhs electricity that need to beThe electrons that make up commoditypurchased in the day-ahead or hour-aheadelectncity are physically the same and cannot bemarkets to meet demand and tbe requiredidentified as coming from traditional energyquantities of RECs. In this paper, we derive asources or renewable sources. Governmentsmodel that optimizes the purchasing processcreaced RPS to address this challenge. The RPSsubjcct to RPS policy and demand informationmechanism generally places 叨obligation nupdates in time- scquential markets for clectricitypower plants to produce a specified fraction ofelectricity from renewable energy sources, andI For example, if the RPS is set to 5% and anthe govermment issues coresponding RECs toelectricity, distributor sells 100,000 kWhs, then itncc中国煤化工the power plantsthe pawer plant based on the quantity ofor cfYHCNM HGChen et sa; Optimal Purchasing Process for Electricity and Renewable Energy Credits186J Syst Sci Syst Eng，distributors. This paper thus investigates thecan analyze the important role of demandoptimal purchasing problem under price andinformation updates in this problem.demanduncertaintyor’ distributors irThe remainder of the paper is organized astime- sequential electricity markets with thefollows. We review the related literature inobjective of highlighting a potentially importantSection 2 and present the notation in Section 3.emerging research area with societal impact andConsidering the REC constraint, we formulatea new set of managerial concemns.the electricity procurement process in aTo the best of our knowledge, this papertime- sequential market as a two-stage dynamicconstitutes the first attempt to study the effect ofprogramming problem. In Section 4, we employthe RPS policy on the operating decisions ofnumerical results to explain our solution andelectricity distributors in U.S. electricity markets.engage in managerial analysis. Finally, we drawOur objective is to minimize distributors totalconclusions and suggest ideas for future researchpurchasing cost subject to a given RECin Section 5. The appendix contains the proofs.purchasing constraint. The system dynamicsencompass both two parts: the electricity bought2. Literature Reviewin each market and demand informationThree streams of literature are related to ourrevisions.Informationrevisionsincluderesearch. First, our paper can be linked with theconsumer demand, This paper has three specialwork on the optimal purchasing problem in thecharacteristics. First, it models the mechanismselectricity market in which the challenge isof govermment regulation considering the effectfinding a way to determine the electricity orderof RECs in the electricity purchasing process.quantity under price and demand uncertainty.An electricity distributor purchases electricityYan et al. (2000b) and Sethi et al. (2005a)from the electricity markets and RECs fromconsider the electricity purchasing problem withpower plants or the corresponding open marketdemand information updates. Inspired by thesesimultaneously. Second, the procurements oftwo papers, which investigates the problem ofelectricity and RECs are linked by a linearoptimal electricity purchases in time-sequentialconstraint in the model. This assumption imarkets, we add a new element to incorporatenaturally derived from the govermment'sthe RECs purchasing process and the carbonannouncedtarget. Third,in electricityemissions constraint to match current industrydistributor has its own purchasing target, andpractice.can adjust the ratio of electricity and RECsSecond, our paper is also related todepending on its requirement. Two-stagenewsvendor models (Khouja 1999, Chen et al.dynamic programming is utilized to analyze this2006, Chen & Xu 2001, Zhang et al. 2011) thatproblem. Moreover, owing to the characteristicsconsider both demand information updates andof the electricity markets, our model ismultiple delivery modes. Our work constitutesconstructed using a three-layer time framework,an extension of the classical newsvendor model,that is able to reveal the purchasing process andmmodities withdemand information updates. Consequently, we dem中国煤化工These twoTYHCNM HGCben et al山: Optimal Purchasing Process for Eletricity and Renewable Energy CreditsJ Syst Sci Syst Eng187commodities are correlated via a linearRECs to be procured to support the developmentconstraint. Other extensions of the newsvendorof renewable energy sources.problem with certain constraints can bc found inThird, our research is also closely related tothe literature (Fisher & Raman 1996, Sethi et al.the recentiterature on carbon footprint2007, Bensoussan et al. 2007).problems and green supply chain management.Note that there is a fundamental differenceThe discussion of the carbon footprint problembetween the constraints in these studies. Inin the popular and trade press (Butner et al. 2009)Fisher & Raman (1996)， a total capacitybas aroused considerable interest amongconstraint is imposed to provide a balancedacademic researchers. Carbon emission concernsallocation of a limitedresource to variousare integ. s.ted into operational decision-makingproducts. Sethi et al. (2007) examine thevith regard to procurement, production, andsingl-period, two-stage supply chain withinventory management in Benjaafar et al. (2010).demand information updates under an aggregateBy associating carbon footprint parameters withor long-term service constraint. In their paper,various decision variables, they show that thethe buyer has two procurement opportunities,traditional models can be modified to supportwitb the second coming after observation of adecision-making that accounts for both cost andmarket signal under a service constraint.carbon footprint. Yu & Li (2010) consider theWhereas they consider only one product, wefuel replenishment problem in power plants withaddress problem of two commodities with acarbon emission constraints, whereas our paperlinear purchasing quantity constraint. The aim offocuses on the electricity and REC purchasingthe model in Bensoussan et al. (2010) is toproblems from the electricitydistributor'smaintain an overall customer satisfaction level atperspective.a low cost by balancing service levels underGreen supply chain management is defineddifferent demand conditions. A series of papersastheintegrationof environmentalbave dealt with evaluation of the serviceconsiderations into supply chain management,performance of replenishment decisions inincluding product design, distributor selectiondiferent contexts, for example, Sethi & Chengandmaterial sourcing,he manufacturing(1997), Gumani & Tang (1999), and Beyer et al.process, product packaging, product delivery to(2010). A thorough review of the rescarch onconsumers, and end-of-life management of thedemand information updates in the supply chainproduct after usc. For examples, Blumbergcan been found in Cboi et al. (2001). As in Sethi(2005) and Pochampally (2009) discuss theet al. (2001, 2003, 2005a), Liu et al. (2006)carbonemissionprobleminreverseapplystochasticdynamicprogramminglogitics/closed-loop supply chains. Ata (2010)techniques to solve the single-period, two-stagealso addresses this problem, and determines theproblem of procurement commitments over time.profit-maximizing operating strategy for aThe constraint imposed in our paper is that thewaste-to- energy firm. Moreover, the carbonenergy distributor must meet a govermentemisprtation call forregulation that requires a certain oumber of polic中国煤化工，regulationTYHCNMHGChen et aL: Optimal Purchasing Process for Electricity and Renewable Energy Credis188JSyst Sci Syst Engmechanisms. A recent paper by Hoen et al.has the option of purchasing an additional(2010) investigates bow two regulationvolume of electricity in this market. Based onmechanisms can affect the transportation modethe updated demand information i acumulatedselection decision. It analyzes a situation induring the first stage, the electricity distributorwhich a single tansportation mode is selectedplaces a supplemental post-update order ofby a decision-maker for all transportation needs.quantity x-x for electricity and y-y forAn overview of the green supply chainRECs in the second-stage, where x and y aremanagement literature appears in Srivastara et al.the total purchased quantities of electricity and(2007), Sasikumar (2009) and Gupta (2009). InRECs respectively. At the end of the secondthis paper, we examine the ectricity market,stage t , if the realized demand D exceeds theand focus on the purchasing decisions made bytotal purchased quantity x，then the shortfallelectricity distributors.will be satisfied at a penalty cost p in the .real-time market. If the realized demand D is3. Problem Formulationless than the total purchased quantity, then theWe formulate the electricity and RECdifference will be sold at a salvage price spurchasing process as a one-period, two-stageimmediately.model with demand information updates and anStage 1REC constraint. Before electricity is distributedto customers, the electricity distributor orders ittt3from the time-sequential market: the day-abead工1 ,4[叨market and the hour-ahead market. Meanwhile,h ,山1Stage 2he distributor purchases RECs from powerplants or the corresponding open market. Thetgsystem dynamics are ilustrated by the timeline出一工1,U[D啊y- yn,W2in Figure 1, which represents the time at whichFigure 1 Timeline of decions and informationpurchase decisions are made and the informationdynamicsstatus in the market. Let n, 2, and 5 denoteTo complete theformulation of thethe epochs representing the start of Stage I, thedistributor's problem, we introduce the followingstart of Stage 2, and the end of Stage 2. There isnotation and assumptions. Let (Q,F, P) denotea day-ahead market at t，in which thethe probability space and E denote thedistributor purchases η of electricity at oneexpectation operation. Let D denote randomunit cost η and y of RECs at one unit costdemand, and I denote the random signal at t2m. Its decisions are based on the availablethat updates the demand distribution. We assumeinformation about the uncertain demand D tothat the joint probability function of demand andbe realized at tz and the distribution of thethe signal at t is Pp./(z,i) or中(z,i). Thecosts for electricity and the RECs at t2. At 2,signshghilit Adomeity function g(i) ,the hour-ahead market opens, and the distributor wi中国煤化工0 for ech iYHCNMHGChen et al: Optimal Pwrchasing Process for Elecricity and Renewable Energy CreditsJ Syst Sci Syst Eng189At tr， after observing signal i, demand iswritten as:updated, and the conditional probability densityU(xy,巧,以2, w,i)= min'(x,y,)(2)function of demand,ziven1=i, iss.t. x2况，f(zli)=(z,i)1g(i) . Let the distributionsy2y,corresponding to the densities (z,i)， g(),where n'(x,y,i)=y2(x- q)+w2(v-y)and f(z|i) be denoted by中(z,i), G(I), and+E[p(D-x)*-s(x-D)*li]，Let L(a,i)=F(z\i), respectively. Also let E[D]= u,E[p(D-a)* -s(a- D)* |i. We can also writeE[/]=η and ED|I=i]= u(). To simplifyn'(x,y,I)= v2(x-x)+w2(y-yI)+ L(x,I).the notation, we use E[D|i] to represent theFor any given first-stage quantity ofconditional expectation of demand given thatelectricity order 气，REC quantity y，realizedI=i. We assume that the signal i exhibits aprice口for electricity, w2 for the RECs, andstocbasticorder,i.e, iz>片and >以>m because themin Il(q.y)(4)closer to the end the timeline is, the higher thes.t. η20,market clearing price.n≥0,Moreover, the distributor sets a mandatoryREC target that the order quantity of RECswhere N(x,h)=(4-吃)所+(m -w吃)y+E[U(x,2y,1)}.should account for δ percent of the totalordered electricity quantity. δ is assumed to beU(q,M,i)= min(x,y,I)(5)constant during tbe selling period and δ∈[0,1],s.t. x2前，y≥8x.(1)中国煤化工The optimization problem at Stage 2 can be wherMYHCNMHGChen et al: Optimal Purchasing Process for Electricity and Renewable Energy Credis190JSyst Sci Syst Eng3.1 Optimal Order Quantity at Stage 2function of x. The nonlinear function of x isIn this section, we solve the distributor'sthe standard newsvendor profit under an orderStage2 purchasingproblem, dynamicquantity of x and a demand of [D|i] withprogramming equation(5). Given q, letgiven information i . Both parts are convexfunctions. Therefore, the objective function is()= argmax, F(q1i)z(P约)p-sconvex, and the constraints are linear. We canandobtain the global 'optimal solution using thKKT conditions. Hence, we have the followingi(9)= arg max, F(q|i)2(E-以-0%).theorem.P-sIn general, for 0<θ≤1， let ig(q):=Theorem 3.3 At Stage 2. given the electricityargmax;F(q|i)2θ and qe()= F~(0li).and REC order. up-to levels ofStage1,(x, y)Then, for any given q and θ, ig(q) is thethe realized demand information signal i, andthe exogenous carbon emissions constraintcritical signal below wbich the conditionalprobability of the order quantity q satisfyingparameter δ, the optimal ordering policy forelectricity and REC levels arethe updated demand is greater than the required(x' (x,)y1),y(*x,),), which are stateprobability θ and above which the conditionaldependent order-up-to levels.probability of q satisfying tbe updated demandIf η≥8x and y> 8xc, thenis less than the required probability θ. For anygiven日and i, q(i) is the quantity that不，if x≤,ensures that the conditional probability ofsatisfying the updated demand is equal to θ.x'(所,y,1)={xu， if吊x, thenLemma 3.2 Given the electricity order-up-tolevels of the first-stage 吊and y1 and the{x (xy,y,)=小realized demand infomation signal i, the costvy'(x,y,i)=8x. .function of the second-stage, n(x,y,i), is a●If yμ≤δx and吊≤x, thenjoint convex function that correspondsto x andy.[x(,n,1)=xe,Clearly, m(x,y,I) can be separated into twoy (r,y,i)=8x。.parts: a linear function of y and a nonlinear中国煤化工YHCNMHGCben et L: Optimal Purchasing Process for Elecricity and Renewable Energy CreditsJ Syst Sci Syst Eng191x(i)= arg min{vzx+ (x,0}=F-(521), .3.2 Optimal Order Quantity at Stage 1p-s .Afer solving the purchasing problem atandStage 2, we solve the distributor's purchasingx() = argmin{v2x + m2δx+ L(x,i)}problem at Stage1， i.e.,the dynamic=F-l(P-2-OW2|n.programming equation (4).p-sLemma 3.4 (q,y) is joint convex andFor simplicity, we use x and x。 tocorresponds to芍and .represent xy() and x(). Obviously, we knowThis lemma supports the existence of thethat xg Sxu for any given i.optimal solutions of (x,y).In Figure 2, the optimal electricity orderingTheorem 3.5 The results of the electricity andpolicy is indicated by the broad-brush line.REC purchasing problem can be shown asOptimal ordering quantities x' and y' lie infollows.the boundary of the feasible region.1. At Stage I, optimal order quantities吊andyi satisfy the first-order condition. Then, 石and yi satisfv/y=x(dI(x.y)21;=0,dyyt2. At Stage 2, optimal order up-to-levels x andx,y' can be listed as follows.●When y≥δx,xn y;6x0，ifisi(x),(a) η≥δxy↑y=δXx，iti(q) 0, the optimal orders area worthless information case. Explaining thiscase mathematically, random variables I andx=x and yi=8x&.D are independent, i.e., F(z|i)=F(z) andWhen x <0< xu, the optimal orders are中(z,i)= F(z)G(i). The observed information atx'=x and yi'=0.h has no value for updating the demand in thisWhen x <0, the optimal orders arecase. With the assumption that the marketx =0 and yi=0.clearing price at r will be higher than that atNote that the threshold values in the斤，n0,x(i)= arg min{zx+ L(D,)}y -片>0,y" -δx° >0.=F-\(P=兰|i),p-sThe unconstrained optimal solutions belongx()= argmin{vzx + w28x + L(D,)}10 the feasible area if and only if w=0 .=F-(P-只-0|η.Therefore, this case is invalid.Case2: i">0, =0 and j=0,To simplify the notation, weuse x and xeto represent x(i) and x[(i). As F-(|i) isz-p+(p-s)F(xi |I)-名"=0, .an increasing function, we know that xc≤xw2=0,for any given i.The Lagrangian function can be witten asy' -片>0,follows.{y -8x° >0.A(x,y,n,2,对)= 2x+ W2J"+E[p(D-x)* -s(x- D)间to the_ feasihle- area_ if and nnly if 以=0.-q(x-x)-h(v-y)-k(y-8x),π中国煤化工YHCNMHGChen et l: Optimal Purchasing Process for Elecricity and Renewable Energy Credis198JSyst Sci Syst EngCase3:矿=0,名>0, and j=0,以- p+(p-s)F(x' |0-h +8ij =0,[2 -p+(p-s)F(x' |1)=0,w2-j=0,wr-i=0,x =气，y >y,y -yn=0,y" =8x'.y* -8x* >0.The optimal solutions areJx' =x，The optimal solutions belong to the crosspointof x =x and y, =外.[y =dx.Case7: h"=0,名>0,and j>0, .[x* =x，[y" =y小2-p+(p-s)F(x' |1)+82;* =0,m-z-δ=0,Case4:布>0, 名>0,and号=0，n-p+(p-s)F(x'|i)-q=0,| wr-名=0,ly* =8x.x =莉，The optimal solutions arcy =y,y" >8x*.ly*=yCase8:省>0,名>0,and 7>0,ly' =yp-p+(p-s)F(x |)-名+8h =0, .CaseS: i"=0,名=0,and j>0,w-h-ij=0,|口-p+(p-s)F(x' |i)+ 8' =0,以-j =0,>而，Uy* =8x'.y' =8x*.{y =y,y =8x.[y" =8xe.Therefore, the optimal policy can be shownas followe中国煤化工Case6:心>0, 石=0,and >0,TYHCNMHGChen et aL: Optimal Purchasing Process Jor Elecmicity and Renewable Energy CreditsJ Syst Sci Syst Eng199{m-p+(p-s)F(x)-H'+&%j=0,x <所，x =所，w-名-ζ=0,在=0，所≤x≤j小x =x，y =0;x0, 石=0,and李=0,x≥η,y22y},y28(x2+xy2)} is a convex[r-p+(p-)F(x)-A =0,set. T(x,y,i) is a joint convex function of与m=0,and y. According to Theorem A.4 in Porteusη =0,(2002), U(x,y,i) is a joint convex function ofη and y. The operator of the xpectation[i-8>0.does not change the convexity of所and y .Therefore, I(x,y) is joint convex in石andto the feasible area, if and only if m =0.Case3:矿=0,名>0,and矿=0,Proof of Proposition 3.6r-p+(p-s)F(;")=0,The Lagrangian function isw-石=0,A(q,y,不,右,④)=Y所+my>0,+E[p(D-x)+ -s(所-D)*]=0,-你- hn-%( -8x),(vi-8x*>0.VA'=「n-p+(p-s)F(x)-布+8I]The optimal solutions belong to the crossm-名-4pointof q=F~(二4)=x and yi=0.andp-sv2a_[(-5)/(F) 0].不=x%,o]lyi =0.The KKT conditions are中国煤化工d矿=0,YHCNMHGChen et aL: Optimal Purchasing Process for Electricity and Renewable Energy Credits200」Syst Sci Syst Eng[n-p+(p-s)F(q)-° =0,The optimal solutions arew-石=0,[xi=0,=0,lyi'=0.y =0,Case8:矿">0,石">0 and " >0,(i>8xj. .[n-p+(p-s)F(*)-i +8i' =0,w-h-j=0,」i=0,lyi=0.为=0,CaseS5:矿=0,名"=0,and矿>0,(i =8xi.[r-p+(p-s)F(x)+81^ =0,m-h =0,吊>0,yi'>0,yi =8x. .Therefore, the optimal policies for theworthless information case are as follows.|xi=Fr(B-n-0w)=x,When x'>0， the optimal orders arex°=x and r =8x'.lxi =8x'.When x <00,石=0,and >0,η=x and ji=0.(n-p+(p-s)F()-4^" +&lg =0,When x{<0 ，the optimal orders areη=0 and yi=0.u-名”=0,Proof of Proposition 3.7The first-order condition with respect to不lyi =δx°.for the perfect information case is as follows.{i=0,dn"(x,y)=n一%Case7: q =0,石>0,and矿>0,+C[以+w68-J]g()dim-p+(p-s)F(x)+8%" =0,=η-% +(% +w8-s)G(r ~'(x))=0.m-石一矿”=0,Then, at the first stage, the optimal orderxi>0,quantity x =t(G~'(-吃一片))， and the以+ w28-lyi =8xi.optim中国煤化工/ satisfiesMYHCNMHGChen et al: Optimal Pwrechasing Process for Electricity and Renewable Energy Credits201J Syst Sci Syst Engif i