Pricing and operation in deregulated electricity market by ...

Electric Power Systems Research 57 (2001) 133–139www.elsevier.com/locate/epsr**Pric ing**

134L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139as far as each participant **in** **operation**s of a powersystem is completely cooperative [14]. However, for the**deregulated** power systems, this rule may not holdbecause participants **in**tend to pursue their ownbenefits, there**by** may not always take cooperative attitude.In this paper **in** order to compare with theequaliz**in**g **in**cremental cost rule, we ma**in**ly use theStackelberg strategy **and** the Nash equilibrium solutionto analyze the negotiation process between utilities **and**IPPs s**in**ce they are widely recognized as rational decisionsfor competitive **market**s **in** terms of axiom [5,10].In Section 2, we first derive a common price of**electricity** for purchases **and** sales between a utility **and**IPPs when assum**in**g that all of IPPs construct a coalitionwho negotiates with the utility for barga**in****in**g. Bythe Stackelberg strategy [6], the Section 2 also providesvery clear **and** simple **operation** rules, which are similarto the well-known equaliz**in**g **in**cremental cost [8] butdifferent due to the fact of competitive environment. InSection 3, we give the prices of **electricity**, as well as the**operation** decisions for the situation when each IPPnegotiates with the utility **in**dependently, which meansthat IPPs sell their **electricity** at different prices depend**in**gon their negotiations with the utility. In Section 4,a Nash equilibrium [5] is used to analyze the negotiationbetween IPPs **and** the utility, which shows that theNash equilibrium **and** the Stackelberg strategy have thedifferent solutions when assum**in**g that the purchasedelectric energy depends on its price (or is a function ofthe price). Section 5 extends the results of the earliersections **by** tak**in**g account of transmission network. Aswill be shown **in** this paper, all obta**in**ed results can beviewed as an extension of the equaliz**in**g **in**crementalcost from regulated power systems to competitivepower **market**s. In Section 6, we derive an optimalsolution **by** maximiz**in**g total welfare of electric power**in**dustry society, for which the **operation** rule obviouslyco**in**cides with that of the conventional equaliz**in**g **in**crementalcost. To demonstrate the proposed model, severalexamples are simulated **and** used to compare withthe conventional approaches [3] **in** Section 7. F**in**ally,we give several general remarks to conclude this paper**in** Section 8.2. Negotiation between a utility **and** a coalition ofIPPsThis section derives the price of **electricity** **and** **operation**decisions on the basis of the noncooperative game[6] **by** suppos**in**g that all of IPPs form a coalition whojo**in**tly negotiates with a utility. That means all of the**electricity** sells at the same price from IPPs to theutility. S**in**ce the price of **electricity** for sales directly**in**fluences the payoffs of IPPs, it is reasonable for theutility to assume that the quantity of the purchased**electricity** energy from IPPs depends on its price. Thebasic structure of power **market** **in** this paper is thatthere are a utility with transmission network **and** severalIPPs without the transmission network. Note thatall results of this paper can straightforwardly be extendedto the multi-utility negotiation game, althoughonly one utility is analyzed **in** this paper for the sake ofsimplification [3].Let Z=(z 1 ,…,z m ) be supply power vector of theutility, where z j ,forj=1,…,m st**and**s for real poweroutput (MWH) of generator-j, **and**m is total number ofgenerators for the utility. Def**in**e g j (z j ) for j=1,…,m asthe **in**cremental cost ($ per MWH) of generator-j of theutility. Then z j0g j (z)dz is the cost function ($) for generator-jof the utility. Assume that there are n IPPs **and**def**in**e X i =(x i1 ,…,x **in**i ) to be supply power of IPP-i=1,…,n, where z ij for j=1,…,n i represents real poweroutput (MWH) of generator-j **and** n i is the total numberof generators for the IPP-i. Letf ij (x ij )fori=1,…,n;j=1,…,n i be the **in**cremental cost ($ per MWH) ofgenerator-j for the IPP-i, **and** f ij /x ij 0. Then x ij0 f ij (x)dx is the cost function ($) for generator-j of theIPP-i. S**in**ce this section analyses, the negotiation betweena coalition of IPPs **and** a utility, all of **electricity**is assumed to be sold at the same price from IPPs to theutility. Then let y be the price ($ per MWH) (or the**in**cremental cost) at which the utility purchases the**electricity** from IPPs. Let d be total dem**and** (MWH) ofconsumers **and** h(d) is earn**in**g ($) of the utility fromsell**in**g **electricity** to customers **and** is not related withZ,X i . Then IPPs will sell electric power to utility **by**maximiz**in**g their profits, which can be formulated asthe follow**in**g problems [3,15]:IPP−i:(i=1,…,n)nMaximize y i nj=1 x ij − ij=1x ij ;j=1,…,n ix ij0f ij (x)dx (1)where x ij0 f ij (x)dx is the cost function ($) for generator-jof the IPP-i, **and** IPP-i is assumed to decide x ij , j=1,…,n.On the other h**and**, the utility **in**tends to maximize itspayoffs **by** utiliz**in**g exist**in**g generators **and** lower**in**g thepurchas**in**g price of **electricity** from IPPs, which can bedescribed as the follow**in**g problem:Utility:Maximizey,z j ; j=1,…mn nh(d)−y i=1 i mj=1 x ij (y)− j=1z jj=1g j (z)dz(2)n ns.t. i=1 ij=1 x ij (y)+ mj=1 z j =d (3)where Eq. (3) means that the utility has the obligationto serve all customers’ load, **and** the utility is assumedto decide y, z i , j=1,...,m. Note that we use a presumedcondition **in** Eqs. (2) **and** (3), i.e. the purchased electricpower x ij (y) is supposed to be a function of y. S**in**ce the

L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139 135price of **electricity** for sales directly **in**fluences the payoffsof IPPs, it is reasonable for the utility to assumethat the quantity of the purchased **electricity** energyfrom IPPs depends on its price. Notice that x ij (y) isnota predeterm**in**ed function but an implicit function of y,which is decided **by** the game at the equilibrium.The optimality conditions for Eq. (1) is straightforward**by** differentiat**in**g the objective function with respectto x ij , i.e.y=f ij (x ij ) (i=1,…,n;j=1,…,n i ) (4)Accord**in**g to the Implicit Function Theorem **and** Eq.(4), there exists a differentiable function x ij (y) attheneighborhood of x 0 when df/dx0. Differentiat**in**gEq. (4) aga**in**, we have the follow**in**g relationsdx ijdy = 1(i=1,…,n;j=1,…,ndf ij (x ij )/dxi ) (5)ijOn the other h**and**, for Eqs. (2) **and** (3) of the utility,we establish a Lagrangian functionnL(Z,y,)=h(d)−y n+ n ii=1j=1mx ij (y)+ n ii=1j=1j=1mx ij (y)− j=1z j0g j (z)dzz j −d n (6)Note that h(d) is a constant, then the optimalityconditions for Eqs. (2) **and** (3) can be obta**in**ed **by**L/z j for j=1,…,m, L/y **and** L/=g j (z j ) (j=1,…,m) (7)ni=1ni=1n ij=1n ij=1nx ij (y)+(y−) mx ij (y)+ j=1n ii=1j=1dx ij=0 (8)dyz j =d (9)Substitut**in**g Eq. (5) **in**to Eq. (8) **and** comb**in****in**g Eq. (4)with Eq. (9), we have the solution [6]y=f ij (x ij ) (i=1,…,n i ) (10)=g j (z j ) (j=1,…,m) (11)−y=ni=1n ij=1n n i=1 ij=1 x ijn n i=1 i j=1 1/df ij /dx ij(12)mx ij + j=1z j =d (13)which are actually conditions of the equilibrium of thegame.Observ**in**g Eqs. (10) **and** (11), the generators for IPPs**and** the utility obviously have the same **in**crementalcost y **and** , respectively. As shown **in** Eq. (12),however, their **in**cremental costs are different due tocompetition between the utility **and** the coalition ofIPPs. S**in**ce df ij /dx ij is assumed to be positive, the**in**cremental cost for utility is generally larger than thatfor IPPs accord**in**g to Eq. (12). That implies that thegenerators of IPPs have to supply power economicallyat the **operation** po**in**t, compar**in**g to those of the utilitybecause the utility owns the transmission network **and**is generally burdened with the obligation to serve thecustomers’ loads. The price y of **electricity** for purchases**and** sales is taken as the same value of theequaliz**in**g **in**cremental cost for IPPs as **in**dicated **in** Eq.(10). It is evident that the discrepancy between the two**in**cremental costs narrows if df ij /dx ij have small value,which means that the IPPs can benefit more if theyhave generators with smaller f ij /x ij (i.e. flat cost function).Therefore, Eqs. (10)–(13) can be viewed as anextended equaliz**in**g **in**cremental cost rule for the **deregulated**power systems.Note that we can have similar equations as Eqs.(10)–(13) when tak**in**g the upper **and** lower limits ofeach variable **in**to consideration, even though moreLagrangan multipliers have to be **in**troduced. In thispaper, we assume that the utility decides the purchasedprices based on the outputs of IPPs while IPPs determ**in**etheir outputs depend**in**g on the purchased prices.Therefore, this model ma**in**ly targets the system wherethe utility is large enough compar**in**g to each IPP **and**almost dom**in**ates the **market**.3. Negotiation between a utility **and** **in**dividual IPPsThis section gives a noncooperative solution **by** suppos**in**gthat each IPP **in**dividually negotiates with theutility [3]. That means each IPP sells its **electricity** to theutility at the price different from those of other IPPs.Let Y=(y 1 ,…,y m ) where y i for i=1,...,m represents theprice at which the utility purchases the **electricity** fromIPP-i. All other def**in**itions are the same as those **in**Section 2.Assum**in**g that IPP-i **and** the utility decide x ij ( j=1,…,n j )**and**Y,Z, respectively, then we have the follow**in**gproblems for IPPs:IPP−i:(i=1,…,n)**and** for the utility:Utility: MaximizeY,Zm− j=1z j0Maximizex ij ;j=1,…n iy i j=1g j (z)dzn i nx ij − ij=1x ijn nh(d)− i=1 y i ij=1 x ij (y i )n ns.t. i=1 ij=1 x ij (y i )+ mj=1 z j =d0f ij (x)dxHence **in** the same way as the derivations of theSection 2, we have the follow**in**g conditions for anoncooperative solution

136y i f ij (x ij ) (i=1,…,n;j=1,…,n i ) (14)=g j (z j ) (j=1,…,m) (15)−y i =ni=1n ij=1n ij=1 x ijL. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139(i=1, …,n) (16)n i j=1 1/df ij /dx ijmx ij + j=1z j =d. (17)Observ**in**g Eq. (15), it is evident that the generatorsfor the utility have the same **in**cremental cost , whichco**in**cides with the conventional results. However, accord**in**gto Eq. (14), the generators for IPPs have different**in**cremental costs because IPPs **in**dividuallynegotiate with the utility **by** us**in**g their own strategies,while the equaliz**in**g **in**cremental cost is still hold for thegenerators with**in** an IPP. The price y i of **electricity** forpurchases **and** sales are also different for each IPP fromEq. (14). Eq. (16) **in**dicates that the flatter the fuel costfunctions of the generators for IPP-i is (i.e. the smallerdf ij /dx ij is), the less the discrepancy of the **in**crementalcosts between the utility **and** IPP-i is liable to be. Onthe other h**and**, Eqs. (14) **and** (16) also show that thecheaper the **in**cremental cost of generators for the IPPis, the more an IPP can sell power. Note that we c**and**erive a similar non cooperative solution when assum**in**gevery generator **in** IPPs sells power at its price y ij ,analogue to the derivation of Eqs. (14)–(17). Therefore,Eqs. (14)–(17) can also be viewed as an extendedequaliz**in**g **in**cremental cost rule for the **deregulated**power systems.4. Nash equilibriumIn this section, we only analyze the negotiation betweena utility **and** **in**dividual IPPs **by** us**in**g Nashequilibrium [5,3]. Therefore, we use the same def**in**itiondescribed **in** Section 3.IPP-i assumes that the price y i is a function of theelectric power X i sell**in**g to the utility. Assum**in**g thatIPP-i decides x ij for j=1,…,n j , then the strategy ofIPP-i can be formulated as:IPP−i:(i=1,…,n)ni− j=1x ij0nMaximize y i (X i ) ij=1x ij ;j=1,…,n if ij (x)dxOn the other h**and**, the utility assumes that thepurchased electric power x ij , from IPP-i is a function ofy i .Then the strategy of the utility can be described as:Utility:MaximizeY,Zx ijn nh(d)− i=1 y i ij=1 x ij (y i )zm− j=1 j0g j (z)dzn ns.t. i=1 ij=1 x ij (y i )+ mj=1 z j =dwhere the utility decides Y **and** Z.Supports y i /x ij 1/dx ij /dy i , we have the follow**in**gNash equilibrium solution [5].y i = 1n i j=1 x ijx ij0f ij (x)dx+n iji x ijy i (0)n i j=1 x ij(i=1,…,n;j=1,…n i ) (18)=g j (z j )(j=1,…,m) (19)−y i =ni=1n ij=1n ij=1 x ij(i=1,…,n) (20)n i nj=1 i j=1 x ij /f ij −y imx ij + j=1z j =d (21)where y i (0) is the **electricity** price when X i =0.The equations above are obviously different from theStackelberg strategy shown **in** Eqs. (14)–(17). For theStackelberg strategy, as **in**dicated **in** Eq. (14), eachgenerator **in** IPPs operates at equaliz**in**g **in**crementalcost, which is also taken as the purchased price, whilethe purchased prices **in** the Nash game are determ**in**ed**by** not only the **in**cremental costs but also outputs ofgenerators as shown **in** Eq. (18). Besides, compar**in**gEq. (16) with Eq. (20), the discrepancies of the **in**crementalcosts between the utility **and** IPPs are alsodifferent for the Stackelberg strategy **and** the Nashgame. Although this section only considers the negotiationbetween the utility **and** **in**dividual IPPs, the Nashequilibrium solution can be derived as the same way forthe negotiation between the utility **and** a coalition ofIPPs **by** assum**in**g y/x ij 1/dz ij /dy.5. Consideration of transmission networkThis section only deals with the negotiation betweenthe utility **and** **in**dividual IPPs. Assume that only theutility owns transmission network. Let H(Z,X 1 ,...,X n )be a transmission loss function (MWH), then we havethe follow**in**g noncooperative game [3,15]:IPP−i: Maximize y i j=1(i=1,…,n) x ij ;=1,…,n iUtility:m− j=1z j0MaximizeY,Zg j (z)dzn i nx ij − ij=1x ijn nh(d)− i=1 y i ij=1 x ij (y i )0f ij (x)dx

L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139 137Table 1Coefficients of generators for 8-generator power systemGenerator numberab c Capacity (MW)Utility Unit 1 0.002 11 105 660Unit 2 0.002 11 105 660Unit 3 0.0019 10 100 960Unit 4 0.0019 10 100 960IPP IPP 1 0.0026 9 100 435IPP 2 0.0027 9 100 420IPP 3 0.0028 10 100 310IPP 4 0.0029 10 100 3006. Socially optimal behaviorNext, we will show that the conventional equaliz**in**g**in**cremental cost rule optimizes the social welfare **and** isalso a Pareto solution. Consider a power **in**dustry society**in**clud**in**g a utility **and** several IPPs. Then the problemto maximize the welfare of the society can beformulated as follows [2]:Society:x ij0MaximizeX i ,Z;i=1,…,nxm jf ij (x)dx− j=10n nh(d)− i=1 ij=1g j (x)dzn ns.t. t=1 ij=1 x ij (y i )+ mj=1 x j −H(Z,X 1 ,…,X n )=d.n ns.t. i=1 ij=1 x ij (y i )+ mj=1 z j −H=dwhere H=H(z,X 1 (Y),...,X n (Y)). Then **in** the same wayas the derivations of the Section 3, we have a noncooperativesolution, which can be considered as an extendedequaliz**in**g **in**cremental cost ruley i f ij (x ij )(i=1,…,n;j=1,…,n i ) (22)= g j(z j )( j=1,…,m) (23)1−H/z j1−n ij=1H/x ij /df ij /dx ij −y i =n i 1/df /dx ij ij j=1n ij=1 x ijn i j=1 1/df ij /dx ij(i=1,…,n) (24)ni=1n ij=1mx ij + j=1z j −H(Z,X 1 ,…,X n )=d. (25)Obviously, for the generators of the utility, Eq. (23)co**in**cides with the conventional economic dispatch withthe consideration of the network loss. Generally, thetransmission loss H, which is expressed **in** quadraticforms of the generation powers can be estimated **by** theB-coefficient method [13].Thus the optimality conditions satisfy= g k(z k )= f ij(x ij )1−H/z k 1−H/x ij(i=1,…,n, j=1,…,n i ,k=1,…,m) (26)ni=1n ij=1mx ij + j=1z j −H(Z,X 1 ,…,X n )=d. (27)Eq. (26) is the equaliz**in**g **in**cremental cost with theconsideration of the transmission loss for all of thegenerators **in** the society, which means that the bestsolution to the economic dispatch problem is the equaliz**in**g**in**cremental cost if there is no competition butco**operation**.7. Numerical simulation7.1. An 8-generator power systemA power system with 8 thermal generators shown **in**Table 1 is used to illustrate the application, where theutility owns four thermal generators **and** four IPPs haveone generator, respectively. This paper assumes that allof cost function for generators, i.e. z 0g j (z)dz=a j z 2 +b j z+c j or x 0f ij (x)dx=a ij x 2 +b ij x+c ij , has thequadratic forms, as shown **in** Tables 1 **and** 2. TheTable 2Coefficients of generators for 40-generator power systemGenerator numberabcCapacity (MW)Utility Unit 15 0.00211 105 325Unit 6150.0019 10100 610Unit 1620 0.0018 10 100 360Unit 2125 0.0029 9 100 635Unit 2630 0.0039100885IPP IPP 120.00269100235IPP 35 0.0027 9 100 370IPP610 0.0028 9 100 500

138L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139Table 3Simulation results for 8-generator power system aDem**and**CoalitionIndividual4500 (MW) Output (MW) Cost ($ per h) Benefit I ($ per Output (MW) Cost ($ per h) Benefit IC ($ per($ per h) MWH)($ per h) MWH)Utility 3206.5 38 918 4722 13.6 3051 36 813 4253 13.46IPP-1 434.1 4496.9 391.1 11.26 428.84437.7 378.1 11.23IPP-2 4184334 372.9 11.26 412.9 4277.1 360.5 11.23IPP-3 224.5 2486.5 41.7 11.26 308.8 3456.5 167.2 11.73IPP-4 216.8 2404.2 36.9 11.26 298.33340.7 157.9 11.73Total 4500 52 639.6 5564.6 450052 325 5316.7a IC: **in**cremental cost.Table 4Simulation results for 40-generator power system aDem**and** Coalition Individual21 000 (MW)Output(MW)Cost($ per h)Benefit IC ($ per Output Cost Benefit($ per h) MWH) (MW) ($ per h) ($ per h)IC ($ perMWH)Utility Unit 15 1601.8 19 120.5 19 669.5 12.28 1648.3 19 743 20 307 12.32Unit 615 6003.3 67 885.4 73 725.4 12.28 6101.7 69 091 75 172.9 12.32Unit 1620 1779.8 21 272.5 21 855.5 12.28 1831.5 21 908.5 22 563.5 12.32Unit 2125 3155.2 34 073.5 38 746 12.28 3191 34 514 39 313 12.32Unit 2630 4401.2 42 780.5 54 046.5 12.28 4432.2 43 162 54 604.5 12.32IPP IPP 12 460.8 4623.4 4700.4 10.2638.2 6473.2 6803.2 10.66IPP 35 1099.1 10210 11 210.3 10.2 1079.6 10 018 10 969 10.16IPP 610 2498.5 21 834.5 25 484.7 10.2 2077.6 17 655.7 20 069.5 9.66Total21 000 221 850.3 249 438.3 21000 222 560.4 249 802.6a IC: **in**cremental cost.simulation results are summarized **in** Table 3. Accord**in**gto Table 3, the **electricity** prices of IPPs are lowerthan the **operation** cost of the utility because the utilityhas the obligation to serve all customers load. Whilethe total outputs of generators for IPPs **and** the utilityare the same for both coalition **and** **in**dividual. Actually,it can be proven that total outputs of IPPs are thesame for the negotiations of Sections 2 **and** 3 as far asthe cost functions x j0 f ij (z)dx **and** z j0g j (z)dz arequadratic. For the sake of the simplification, let h(d)=d where =80 ($ per MWH) is the **electricity** rate toconsumers.S**in**ce the df ij /dx ij IPP-1 are smaller than those ofIPP-2, IPP-3, IPP-4, the IPP-1 supplies more power tothe utility, especially for the case of the ‘**in**dividual’accord**in**g to Table 3. The results also show that thetotal benefit for the case of the coalition is larger thanthe case of the **in**dividual due to their jo**in**tly barga**in****in**gwith the utility.7.2. A 40-generator power systemThe proposed method also applied to a 40-generatorpower system with one utility **and** 10 IPPs. The utilityhas 30 generators **and** 10 IPPs own one generator,respectively. Table 4 shows the simulation results whereh(d)=80d is used. In addition to the price y, obviouslythe earn**in**g of the utility also depends on the h(d) toconsumers. Accord**in**g to Table 4, the total outputs ofgenerators for IPPs **and** the utility are the same forboth coalition **and** **in**dividual. The **in**cremental costs ofthe utility (12.8 **and** 12.32$ per h) are higher than the**in**cremental costs of IPPs (10.2 **and** 10.66$ per h). Asshown **in** Table 4, total benefit for the case of the**in**dividual is actually more than that for the case of thecoalition, but the benefit of each is not all **in**creased dueto their different **in**cremental costs.8. ConclusionsGame theory is a useful tool for the analysis ofelectric power **market**. The derived **operation** rulesreflect**in**g the competition can be viewed as an extensionof the conventional equaliz**in**g **in**cremental cost method.Besides, we also provided the prices of **electricity** forpurchases **and** sales, **and** as **in**dicated **in** this paper,these prices are generally cheaper than the **in**cremental

L. Geerli et al. / Electric Power Systems Research 57 (2001) 133–139 139costs of the generators belong**in**g to the utility, whenpresum**in**g that the purchased electric energy is a functionof the prices. To exam**in**e the proposed approach,two test systems were used as the demonstrated examples.As further research work, the pric**in**g structures**and** the wheel**in**g for power systems with a variety ofpractical constra**in**ts will be **in**vestigated **by** the gametheory.References[1] ‘Game Theory Applications **in** Electric Power **market**s’, IEEECatalog Number: 99TP136-0.[2] L. Chen, J. Toyoda, ‘Optimal pric**in**g policy **and** **operation**decisions for future power system’, International Conference onPower **and** Energy **in** Ch**in**a, Beij**in**g, 1990.[3] L. Chen, et al., ‘**Pric ing** structure of