An Electricity Trade Model for Microgrid Communities in Smart Grid


 Marvin Davidson
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1 An Electrcty Trade Model for Mcrogrd Countes n Sart Grd Tansong Cu, Yanzh Wang, Shahn Nazaran and Massoud Pedra Unversty of Southern Calforna Departent of Electrcal Engneerng Los Angeles, CA, USA {tcu, yanzhwa, shahn, Abstract Dstrbuted crogrd network s the ajor trend of future sart grd, whch contans varous knds of renewable power generaton centers and a sall group of energy users. In the dstrbuted power syste, each crogrd acts as a prosuer (producer and consuer) and axzes ts own socal welfare. In addton, dfferent crogrds can nteract aong each other through tradng over a arketplace. In ths paper, two odels are ntroduced for crogrds to deal wth the welfare axzaton probles. In the frst odel, a crogrd s consdered as a closed econoy group and decdes the optal power generaton dstrbuton n ters of te. In the second odel, each crogrd can trade wth ts neghborhoods and thus acheve a welfare ncrease fro akng use of ts coparatve advantage on power generaton durng a certan perod of te. For each odel, an effcent soluton s presented. Experental result shows the accuracy and effcency of our presented solutons. I. INTRODUCTION The current sart grd technology s undergong a transforaton fro a centralzed, producercontrolled network to one that s ore dstrbuted and consuernteractve [1][2]. Wth the ntroducton of the decentralzed network archtecture, the entre busness odel wll be changed together wth the relatonshp of all stakeholders, nvolvng and affectng utltes, regulators, energy servce provders, technology and autoaton vendors and all consuers of electrc power [2]. Aong all the changes n sart grd, the transforaton of the power generaton network s of the ost crtcal,.e., nstead of beng generated by a few faroff hghcapacty generators and transtted to end users, electrcal energy s ncreasngly beng produced by sallscale generators located closer to pontsofuse [3]. Ths dstrbuted power generaton center has ade t easer to ake use of all knds of renewable energy sources and sgnfcantly reduced the energy transsson cost. However, the nablty to control the ncreased nuber of dstrbuted energy sources can create huge dffcultes n operatng and controllng the dstrbuton network. To solve ths proble, the dea of crogrd s ntroduced [4]. A crogrd can be vewed as a prosuer (producer and consuer) [3]. It contans one or ultple knds of renewable power generaton centers and a sngle or a sall group of energy users, and offers the possblty of coordnatng the dstrbuted resources n a ore ntellgent way so that they can behave as a controlled entty. In ths way, dstrbuted resources can provde ther full advantages n a ore consstent way. Generally for each crogrd, as electrcty ust be used as t s beng generated, the usual practce s to atch supply to deand [5]. Ths s a challengng proble because of the followng reasons: frst, the power deand depends on exogenous factors and vares draatcally [6]; second, each energy user can have ts own utlty preference of energy usage at dfferent te slots; thrd, due to the varous types of power generaton centers, the power generaton cost wll also vary as a functon of te and weather factors, e.g., for a solar energy center, the power generaton cost wll be uch cheaper n the day te than durng the nght te. Under a certan resource constrant, t s necessary to decde the aount of energy generaton and consupton at each te so that the socal welfare of the crogrd can be axzed [7]. In addton, the arket dea has been n the heart of ajor roadaps for crogrd structures [36]. The nterconnect network enables the surplus generaton fro one crogrd to be used to eet the local deand n ts neghborhood. In the classc econoc study, trade s always benefcal to both sdes and thus should be encouraged [8]. By tradng wth each other, each crogrd can ake use of ts coparatve advantage and acheve an ncrease of socal welfare. Prevous papers such as [3] have studed structure of the energy tradng platfor n sart grd neghborhoods. However, they faled to provde an analyss on soe of the econoc factors such as what s the optal energy tradng volue or what deternes the relatve energy prce. To gve a detaled study on the above probles, two odels are ntroduced n ths paper. The frst odel deals wth the local welfare axzaton proble where a crogrd s consdered as a closed econoy group and decdes the optal power generaton dstrbuton n ters of te. In the second odel, each crogrd can trade wth ts neghborhoods and thus acheve a welfare ncrease fro akng use of ts coparatve advantage on power generaton durng a certan perod of te. A odfed Rcardan odel s used to study the econoc factors n the trade process [9]. The reander of ths paper s organzed as follows. In the next secton, we present our local welfare axzaton odel for each crogrd. In secton Secton III, the electrcty trade odel s ntroduced for crogrd neghborhoods. Secton IV reports the experental results and the paper s concluded n Secton V.
2 II. MODEL OF LOCAL WELFARE MAXIMIZATION In the frst odel, we start wth the assupton that a crogrd s a closed econoy group n ters of energy (.e., no energy tradng s allowed) and decdes the optal energy generaton dstrbuton n ters of te. A slotted te odel s assued,.e., all syste cost paraeters and constrants as well as energy generaton and consupton decsons are provded for dscrete te ntervals of constant length. We use C and P to represent the energy generaton and consupton levels (wth a unfed energy unt) at te slot where {1, 2,..., T }. T s the total nuber of equalszed te slots that we dvde one operatng perod nto. For exaple, f we set one operatng perod to be a day and T = 24, a day s dvded nto 24 te slots, each wth duraton of one hour. Econosts have agreed that users consue coonaltes (such as energy) at each te because ths energy consupton provdes satsfacton, or utlty, whch represents the level of a knd of socal welfare [9]. It s coonly odeled and verfed n econoc study that the relatonshp between the utlty derved fro the level of energy consupton at each te follows the for [9]:, (1) where s the preference factor at each te slot and we have 0 < < 1 for all. A hgher eans the users n ths crogrd prefer to consue ore energy at the correspondng te slot. Equaton (1) also reflects three characterstcs of the relatonshp between the utlty and energy consupton level: 1. As > 0, the utlty s an ncreasng functon of C, whch eans the overall satsfacton level wll be ncreased f users can consue ore energy at any te slot. 2. As < 1, the argnal utlty, U/ C = j Cαj j 1, s a decreasng functon of C. Ths eans f the energy consupton at one te slot s already at a very hgh level (.e., uch ore than necessary), users satsfacton level wll not ncrease too uch f we further ncrease the energy producton level at that te. As a result, to satsfy the energy users n the crogrd, t s better to ncrease the energy producton at the ore necessary te slots, whch s realstc. 3. The utlty functon drops to 0 f C = 0 for any 1 T, whch eans the level of satsfacton wll be very low f power outage occurs durng any perod of te. On the other hand, the generaton centers should decde the energy generaton dstrbuton n ters of te so that the total utlty of the crogrd s axzed. It s also coonly odeled n econocs that under a gven resource constrant, the energy generaton producton possbltes should follow the gven nequaton [9]: β P I, (2) where β s the nuber of resource unts that are needed to generate one unt of energy at each te slot and I s the total nuber of resource unts that s allowed to use durng one operatng perod. Notce that β s deterned by the type of energy generaton centers as well as the level of technology, and ght not be constant for dfferent values. Gven the odels for energy generaton and consupton, we need to deterne the relatonshp between the two. Unlke other coon fors of energy such as checal or knetc, electrcal energy should be used as t s beng generated. If storage s requred, electrcal energy wll typcally be converted edately nto another for of energy such as potental, knetc, or electrochecal[1][2]. In ost of the recent crogrd structures, energy storage s not used because of the hgh cost[3][4]. In addton, as stated at the begnnng of ths secton, the crogrd s assued to be a closed econoy group n ters of energy n ths odel. Therefore, no energy trade s consdered n ths odel. Based on the above assuptons, the energy generaton and consupton level should be the sae at every te nsde the crogrd,.e., C = P for each. Usng the above defntons and assuptons, the local welfare axzaton proble for one crogrd can be odeled as follows: Local Welfare Maxzaton Proble for a Mcrogrd Fnd the optal energy generaton P for 1 T. Maxze: β P I P 0 1 T C = P 1 T The geoetry optzaton dea [12] can be used to nuercally solve ths proble. Property 1: At the optal soluton pont of the above local welfare axzaton proble, we have: U/ C β P = I, (3) = U β C = D 1 T, (4) where D s the sae value for all, whch results n the optal soluton gven by: P = C = T β =1 α I 1 T. (5) Proof: It can be easly deterned that only when we ake full use of the energy generaton resources can we acheve the
3 axal utlty functon. In addton, assue at the optal soluton pont, there exsts a par of te slots {, j} that U/ C > U/ Cj I/ P j. We wll be able to fnd a new possble soluton wth C = C + σ/β and C j = C j σ/β j, where σ s a very sall value wth the sae unt of total resource I. As we have U/ C > U/ Cj I/ P j, t can be proven that U(C 1, C 2,..., C, C j,...) > U(C 1, C 2,..., C, C j,...), whch conflcts the optalty of the soluton. As a result, the optal soluton wll occur only when U/ C s the sae for every. It can be observed fro the soluton that for a closed crogrd, we need to spend a lot of resources to generate a certan aount of energy at the loweffcency power generaton te,.e., when β s large. In order to solve ths proble and also ake better use of ts hgheffcency te, a crogrd can choose to trade wth ts neghborhood, whch s dscussed n the next secton. III. MODEL OF WELFARE MAXIMIZATION WITH NEIGHBORHOOD TRADING In nternatonal econoc studes, countres engage n nternatonal trade because they are dfferent fro each other and both of the can beneft fro the dfferences by reachng an arrangeent n whch each does the thngs t does relatvely well [9]. Slarly, by buldng the nterconnect power network, each crogrd can perfor energy trade wth ts neghborhood n order to acheve a welfare ncrease. Bascally, an energy trade contract contans two steps: Frst, both crogrds get together to ake ther energy generaton decsons n order to acheve a axal overall utlty functon; Second, both sdes decde the dstrbuton of the beneft fro trade n a far way. The second step can be perfored usng a far beneft dstrbuton law that the rato between the utlty functon after trade and the axal utlty functon before trade (.e., U trade /U local,ax ) s the sae for both sdes, whch s relatvely easy and s not dscussed n detal n ths secton because of space lt (but s used and shown n experental result secton). In ths secton, we focus on the frst step to deal wth the total welfare axzaton proble for the two crogrds. A odfed Rcardan odel s used n ths secton [9]. In subsecton A, we start wth a splfed odel wth only two te slots n order to study the characterstcs of the optal soluton of the welfare axzaton proble. The odel s then extended to a ore general case n subsecton B. A. Tradng wth Two Te Slots In ths odel, we assue that the target crogrd and ts neghborhood has dfferent energy generaton cost as well as the resource. We denote the target crogrd s total resource by I s, the energy generaton at each te slot by P s, and the nuber of resource unts that are needed to generate one unt of energy at each te slot by β s,. Slarly, we have I n, P n, and β n, for ts neghborhood. As before, C denotes the total energy consupton at te slot and denotes the correspondng preference factor. The preference factor s assued to be the sae for both crogrds because they lve close to each other. Usng the above defntons and assuptons, the total welfare axzaton proble for these two crogrd can be odeled as follows: Total Welfare Maxzaton Proble for Two Mcrogrds Fnd the optal energy generaton P s, and P n, for 1 T. Maxze: β s, P s, = I s β n, P n, = I n P s, 0, P n, 0 1 T C = P s, + P n, 1 T Notce that we have sply used = nstead of n the frst two constrants because we have already proven that the axal utlty can be acheved only when each crogrd has ade full use of ts resources. In ths subsecton, we focus on a splfed stuaton that T = 2. Econosts have used the dea of coparatve advantage to analyze the otve of tradng. Coparatve advantage refers to the ablty of a party to produce a partcular good or servce at a lower argnal and opportunty cost over another. For energy generaton centers n a crogrd, the coparatve advantage coes fro the ablty of generatng energy at a partcular te slot at a lower opportunty cost over the energy generaton centers n another crogrd. For convenence, assue we have relabeled the te slots to ake β s,1 /β n,1 < β s,2 /β n,2. Based on the defnton, the target crogrd has coparatve advantage on energy generaton at te slot 1, and the neghborhood crogrd has coparatve advantage on energy generaton at te slot 2. Notce that coparatve advantage s deterned by the rato of β s, /β n, nstead of the absolute values, whch eans even though a crogrd has a hgher energy generaton cost at every te slot, t can stll have coparatve advantage at soe of the te slots. As we have stated before that energy tradng enables both crogrds to ake better use of ts coparatve advantage at a specal te slot, we are able to analyze the optal soluton. Property 2: At the optal soluton pont of the above total welfare axzaton proble wth T = 2, f β s,1 /β n,1 < β s,2 /β n,2, we have ether P n,1 = 0 or P s,2 = 0 (or both). Proof: Assue we have both P n,1 > 0 and P s,2 > 0 at the optal soluton pont, we wll be able to fnd a new possble
4 soluton wth P s,1 = P s,1 + σ s /β s,1, P s,2 = P s,2 σ s /β s,2, P n,1 = P n,1 σ n /β n,1 and P n,2 = P n,2 + σ n /β n,2, where σ s and σ n are a very sall values wth the sae unt of total resource I s or I n. As β s,1 /β n,1 < β s,2 /β n,2, there should exsts a par of {σ s, σ n } that β s,1 /β n,1 < σ s /σ n < β s,2 /β n,2. In ths case, we wll have both P s,1 + P n,1 > P s,1 + P n,1 and P s,2 +P n,2 > P s,2 +P n,2, whch leads to a hgher total utlty and conflcts the optalty of the soluton. Based on ths property, at least one crogrd wll be specfed n generatng energy at one te slot at the optal soluton pont. And thus the optal soluton of the total welfare axzaton proble wth T = 2 can be acheved by coparng the partal optal soluton at P n,1 = 0 or P s,2 = 0. Notce that there s another specal stuaton where β s,1 /β n,1 = β s,2 /β n,2. It can be proven that there are ultple optal soluton ponts n ths case and one of the optal soluton can be acheved when each of the two crogrds sply axzes ts own utlty as n secton II, whch eans none of the two crogrds can gan fro trade. Ths case s not dscussed n detal because of space lt but s verfed n the experental result secton. B. Tradng wth Multple Te Slots When we extend the proble to a ore general case wth T > 2, the dea of coparatve advantage can stll be used to analyze the characterstcs of the optal soluton. Slarly for convenence, assue we have already relabeled the te slots so that we have β s,1 /β n,1 < β s,2 /β n,2 < β s,3 /β n,3 <... < β s,t /β n,t. We can also coe up wth the property as follows: Property 3: At the optal soluton pont of the total welfare axzaton proble wth T > 2, f β s,1 /β n,1 < β s,2 /β n,2 < β s,3 /β n,3 <... < β s,t /β n,t, there exsts a te slot t that we have P n, = 0 for 1 < t and P s, = 0 for t < T. In other words, there s at ost one (probably zero) te slot n whch the energy s generated by both crogrds. Proof: When there already exsts a te slot t wth both P n,t > 0 and P s,t > 0, f there exsts another te slot 1 < t wth P n, > 0, we wll have β s, /β n, < β s,t /β n,t together wth P n, > 0 and P s,t > 0. We can fnd a better soluton accordng to the proof of Property 2. Slarly, there should not exst another te slot t < T wth P s, > 0. Assue we have already known the value of t, based on the analyss n secton II, we also have the followng property: Property 4: At the optal soluton pont of the above proble wth T > 2, U/ Ps, β s, s the sae for each 1 < t and U/ Pn, β n, s the sae for t < T. Proof: Based on Property 3, we have P n, = 0 for 1 < t and P s, = 0 for t < T, whch eans C = P s, for 1 < t and C = P n, for t < T. Thus Property 4 can be concluded accordng to the proof of Property 1. Property 3 and 4 can be used to deterne the energy generaton at all the other te slots when P s,t and P n,t are gven: P s, = t 1 β s, =1 α (I β s,t P s,t ) 1 < t.(6) P n, = T β n, =t+1 α (I β n,t P n,t ) t < T.(7) As a result, gven the value of t, the total welfare axzaton proble wth T > 2 can be splfed as follows: Total Welfare Maxzaton Proble wth a Gven t Fnd the optal energy generaton P s,t and P n,t. Maxze: =t+1 t 1 [ t 1 β s, =1 α (I β s,t P s,t )] α [ T β n, =t+1 α (I β n,t P n,t )] α (P s,t + P n,t ) αt 0 P s,t I s /β s,t 0 P n,t I n /β n,t There are only two varables n the above proble and they can be easly solved usng geoetry optzaton wth the help of atlab. The reanng proble s to deterne the value of t, and a straghtforward algorth s to swtch t fro 1 to T wth a coplexty of O(n). However, t can be proven that the axal utlty s a concave functon of t. In addton, t can be guaranteed that the optal t value s already acheved when we end up wth both P s,t > 0 and P n,t > 0 n solvng the aboveentoned optzaton proble. Detaled proof s otted n ths paper because of space lt. Usng these propertes, we can ternate our algorth once we have U(t) U(t 1) or both P s,t and P n,t are greater than zero. A specal case s that at the optal soluton pont, there s no te slot n whch the energy s generated by both copanes. Ths stuaton s already ncluded n our algorth and wll end up wth P s,t = 0 or P n,t = 0. IV. EXPERIMENTAL RESULTS To deonstrate the effectveness of the proposed solutons, cases correspondng to the aforesad odels are exaned. The proposed solutons have been pleented usng Matlab and tested for rando cases. In the frst sulaton, we focus on the case T = 2. The preference factor of each te slot s set to be 0.3 and 0.7. Before tradng, both the target crogrd and the neghborhood crogrd are consdered to be closed econoc groups and axze ther own utlty functons. When the crogrds open up to trade, they frst get together to decde the optal energy generaton at each te slot so that the total welfare can be axzed. After that, they dstrbute the total energy consupton n a far way, followng the rule that the rato
5 between the utlty functon after trade and the axal utlty functon before trade s the sae for both sdes. The odel s tested for rando cases wth dfferent energy generaton cost and total resource cobnatons. The detaled sulaton result s presented n Table I. TABLE I SIMULATION RESULTS FOR DIFFERENT CASES UNDER T = 2 before trade after trade U case grd β 1 β 2 I trade P 1 P 2 P 1 P 2 U local s n s n s n s n s n s n It can be observed fro the above table that n case 1, as we have β s,1 /β n,1 = β s,2 /β n,2, there s no coparatve advantage between the two crogrds. As a result, the optal soluton can be acheved when each of the two crogrds sply axzes ts own utlty as and none of the two crogrds can gan fro trade. Fro case 2 to case 6, as long as there s a dfference between β s,1 /β n,1 and β s,2 /β n,2, both crogrds can ake use of ts coparatve advantage and acheve a welfare ncrease fro tradng wth each other. The hgher ths dfference, the ore they can gan fro trade. In addton, copare case 2 and case 3, we can also observe that the total energy generaton resource wll also affect the cooperatve energy generaton decson for both crogrds. In case 2, the resource of the neghborhood crogrd s relatvely lted so the target crogrd needs to generate energy at both te slots. But n case 3, both crogrds have enough resources so that each of the turns out to be specfed n energy generaton at ts own advantageous te slot. In the second sulaton, we analyze the general case wth total te slot T = 6. We set β s,1 /β n,1 < β s,2 /β n,2 < β s,3 /β n,3 <... < β s,6 /β n,6 and test our odel under cases wth total resource cobnatons. The fnal result of cooperatve energy generaton at each te slot s shown n Table II TABLE II COOPERATIVE ENERGY GENERATION RESULTS UNDER T = 6 grd I P 1 P 2 P 3 P 4 P 5 P 6 U trade U local s n s n s n Our soluton s verfed n the above table. Case 1 and case 3 are the two extree cases n whch one crogrd s totally specfed n energy generaton at only one te slot, whle the other crogrd generates energy at every te slot. In case 2, as the two crogrds have relatvely the sae aount of resources, energy turns out to be generated by one certan crogrd wth a coparatve advantage at that te slot. No atter n whch case, both of the crogrds can acheve a welfare ncrease after openng up to trade. Another observaton fro the above experental results s that even f a crogrd has a uch better technology,.e., β s, s saller than ts neghborhood at any te slot, t wll stll beneft fro tradng wth other crogrds. And also, a crogrd wth less energy generaton resources s ore lkely to be specfed n generatng energy at a sngle te slot. V. CONCLUSION Two odels are presented n ths paper to deal wth the welfare axzaton probles for crogrds. In each odel, a crogrd s consdered to be a prosuer to have both energy generaton and consupton. In the frst odel, a closed econoy group s consdered for each crogrd and optal power generaton dstrbuton s solved. In the second odel, the crogrd cooperates wth ts neghborhood through energy tradng and a welfare ncrease s acheved for both sdes. The dea of coparatve advantage s used for crogrds n akng decsons on energy generaton. For each odel, an effcent soluton s presented. The accuracy and effcency of our presented solutons are verfed by experental results. REFERENCES [1] The Sart Grd: An Introducton, The US Departent of Energy, [2] L. Chen, S. Low, and J. Doyle, Two arket odels for deand response n power networks, Proc. of Sart Grd Councatons Conf., [3] D. Ilc, P.G. Da Slva, S.Karnouskos, M. Greseer, An energy arket for tradng electrcty n sart grd neghbourhoods, 6th IEEE Internatonal Conference on Dgtal Ecosystes Technologes (DEST), [4] MICROGRIDS: Large Scale Integraton of McroGeneraton to Low Voltage Grds, EU Contract ENK5CT , Techncal Annex, May [5] T. Cu, Y. Wang, S. Yue, S. Nazaran, and M. Pedra, A Gae Theoretc Prce Deternaton Algorth for Utlty Copanes Servng a County n Sart Grd, Proc. of IEEE PES Innovatve Sart Grd Technologes (ISGT) Conference, Feb [6] T. Saad, Technology Developents and R&D Challenges for Sart Grd Applcatons n Hoes, Buldngs, and Industry, Presentaton sldes avalable onlne. [7] P. Saad, H. MohsenanRad, R. Schober, V. Wong, and J. Jatskevch, Optal realte prcng algorth based on utlty axzaton for sart grd, Proc. of Sart Grd Councatons Conf., [8] N. Gregory Mankw, Prncple of Econocs, Dryden Press, [9] P. Krugan, M. Obstfeld, and M. Meltz, Internatonal Econocs: Theory and Polcy, Addson Wesley, [10] W. Saad, H. Zhu, H. V. Poor, and T. Basar, GaeTheoretc Methods for the Sart Grd: An Overvew of Mcrogrd Systes, DeandSde Manageent, and Sart Grd Councatons, IEEE Sgnal Processng Magazne, [11] Y. Narahar, D. Garg, R. Narayana, and H. Prakash, Gae Theoretc Probles n Network Econocs and Mechans Desgn Solutons, Advanced Inforaton and Knowledge Process, [12] S. P. Boyd and L. Vandenberghe, Convex optzaton, Cabrdge unversty press, 2004.
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