An Application of the Ramsey Number in the Electricity Pricing

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1 Journal of Computer and Communcatons, 2016, 4, ISSN Onlne: ISSN Prnt: An Applcaton of the Ramsey Number n the Electrcty Prcng Hamng L, Ja He School of Computer and Informaton Engneerng, Shangha Unversty of Electrc Power, Shangha, Chna How to cte ths paper: L, H.M. and He, J. (2016) An Applcaton of the Ramsey Number n the Electrcty Prcng. Journal of Computer and Communcatons, 4, Receved: October 18, 2016 Accepted: November 25, 2016 Publshed: November 28, 2016 Copyrght 2016 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). Open Access Abstract The Ramsey number s a foundatonal result n combnatorcs. Ths artcle wll ntroduce Ramsey number wth the method of graph theory, and the Ramsey prcng theory s appled to the sales prce and study of cross subsdy. Based on the status of our sales prce and cross subsdy, Ramsey prcng methods theoretcally gude adjustment thoughts of sales prce and solve the practcal problems n our lfe. Keywords Ramsey Number, Graph Theory, Ramsey Prcng Theory 1. Introducton 1.1. The Background of Ramsey Number The defnton of Ramsey number was frst proposed by the Brtsh mathematcan Ramsey n Snce 1930, a large number of scholars made a thorough study of the Ramsey number, but now the exact value of the Ramsey number havng been developed by people remans low. It s descrbed n any dscrete structure, as long as the structure s suffcently large, there must be a partcular sub-secton. Ths defnton was subsequently developed nto Ramsey theory by Graham, Rothschld and Spencer [1]. Ramsey number of graphs s the promoton of Ramsey number theory. It has been much attenton as a well-known problem n graph theory, and ts research n graph theory has been rapdly developed. Graphc Ramsey numbers R( G1, G 2) refer to that for a gven graph G1, G 2, there s a mnmum postve nteger n, such that for an arbtrary order n n graph G, G1 G, or the complement graph of G contans G 2 [2]. Ramsey theory reveals an mportant phlosophcal thought n mathematcal: the dsorder s relatve, to a certan extent, some orderly structure must appear. Ramsey number s defned as the mnmum value of the amount n order to ensure that such a DOI: /jcc November 28, 2016

2 H. M. L, J. He structure must appear. Some people thnk that mathematcal nducton and Ramsey theory are the two branches of mathematcs. Mathematcal nducton s an approach that nfnty scenaro can be nferred from the lmted case, whle Ramsey theory s the study that a lmted number of exceptons s found from the nfnte case. Ramsey number corresponds to the mnmum groundbreakng number of mathematcal nducton. The groundbreakng number, whch was needed by mathematcal nducton, s usually easy to fnd, but the exact Ramsey number s dffcult to determne generally [3] Basc Concepts of Graph Theory [4] Graph: Graph G refers to an ordered trple ( V ( G), E( G), ϕ G ), ( ) vod set of vertces, E( G ) s the lnk between the edge set of vertces, V G s non- ϕ G s the correlaton functon, whch enables each edge of G corresponds to the dsordered pars of vertex of G (no dssmlarty). The number of vertces and edges of G respectvely represented by the symbol V ( G ) and dg ( v ), sde wth the ev (, v j). In the fgure G the degree of vertex v and dg ( v ) refers to the number of edges where G s assocated wth v and n the graph G maxmum and mnmum of vertces are denoted as ( G) and δ ( G). Smple Graph: A smple graph s a dagram what has nether rngs nor two lnks lnkng to the same par of vertces. Complete Graph: A complete graph s a smple undrected graph n whch every par of dstnct vertces s connected by a unque edge. The complete graph on n vertces s denoted by K n. Sub-graph: For a graph G and H, f V ( H) V ( G), E( H) E( G), and ϕ H s lmt of ϕ G n E( H ),then graph H s called sub-graph of graph G. If H G, H G, H s called the proper Sub-graph of G. Supposed that V s a non-vod set of V, the sub-graph s composed of edge set that s based on V as a vertex set and the whole edges wth ther two ponts n the V, and t s named sub-graph exported by V of G, whch s descrbed as GV [ ]. The sub-graph GVV [ ] s descrbed as G V. Supposed that E s non-vod set of E, the sub-graph s composed of vertex set that s based on E as a edge set and the endpont of E as vertex set, and t s called sub-graph exported by E of G, whch s descrbed as G[ E ]. The sub-graph GEE [ ] exported s descrbed as G E. 2. The Ramsey Number 2.1. The Introducton of Ramsey Problem Let s lead to Ramsey problem by a classc lfe example: sx people together, there are at least three people ether knowng each other or unknowng each other. If usng graph theory to prove ths problem, then we can represent these sx ndvduals wth sx vertces A, B, C, D, E, F. Over these sx vertces for the complete graph, f two people know each other connect the two vertces wth a sold lne, else wth a dotted lne. Ramsey problem s equvalent to prove the edge of ths complete graph wth sx vertces. Wth sold and broken lnes be represented, there must be at least ether one equ- 90

3 H. M. L, J. He lateral trangle wth sold lne, or one equlateral trangle wth dotted lne. Take the pont A for nstance, f we connect pont A and the other fve vertces, there are fve edges, each edge or as a sold lne or a dotted lne. Accordng to the pgeonhole prncple, at least three edges are the same lne. Assume there are three edges are sold lnes n Fgure 1. As shown, f set pont A to pont B, F as dotted lnes, to pont C, D, E as the sold lne, regardless of the connecton lnes of three endponts C, D, E are dotted or sold lne, then t wll meet Ramsey problem. If three connecton lnes are dfferent, accordng to the pgeonhole prncple, at least one s a sold lne, assumng that a sold lne of connecton C and E, the ACE can consttute an equlateral trangle wth the sold lne, and meet the Ramsey problem as well The Basc Theory of Ramsey Number In the same way we can prove 9 or 10 ndvduals, ether three people unknowng each other or four ndvduals knowng each other certanly. By dscussng these examples, we can fnd some numbers are specal, they can meet the Ramsey problem, whch are Ramsey numbers a number of scholars have studed so far. It has not been able to accurately calculate Ramsey numbers wth a formula, but we can stll draw a concluson: a par of constants a and b, correspondng to an nteger r, such that among r ndvduals ether a people know each other, or b ndvduals un-know each other; ether a strangers, or b acquantances, the mnmum number of r s denoted as R( ab, ), that s the complete graph wth R( ab, ) vertces. Wth dotted and sold lnes ndcate, n any case there s at least: 1) a complete sub-graph wth a vertces and sold lnes, or a complete sub-graph wth b vertces and dotted lnes; 2) there s at least one complete graph wth a vertces and dotted lnes or a complete graph b vertces and sold lnes. The two must have one set up, and R can be expressed as Ramsey number. A F B E C Fgure 1. Ramsey problem. D 91

4 H. M. L, J. He The above example can be transformed nto the equaton of the complete graph: K6, K3, K 3 ; By usng Ramsey number 9, 3, 4 R 3,3 = 6 ; R 3, 4 = 9. The Ramsey number whch has been proven s shown n Table 1 [5]. ( ) 3. The Ramsey Prcng Theory K K K s denoted as ( ) Theory s to solve practcal problems n our lves, then we wll nduce an example about how to apply Ramsey prcng method to solve the cross-subsdes sales prce problem The Introducton of the Sales Prce Cross-Subsdes Problem In the 21st century, the economc constructon n Chna has entered nto an unprecedented perod of rapd growth. Rapd economc development has led to a contnued ncrease n the demand for electrcty, whch undoubtedly put forward newer and hgher requrements to the power ndustry of our country. Wth the contnuous advance of the power system reform, our country has acheved ntal success n separaton of plant and network, constructon of power generaton sde market and other aspects. It has become a common concern that how to deepen the reform of the electrcty power market. As we all know, a reasonable electrcty prce mechansm s one of the key factors to ensure the smooth progress of Chna s electrc power market reform and to promote the sustanable and rapd development of power ndustry. Reasonable sales prce can reflect the power supply and demand tmely and effectvely, at the same tme t s advantageous to gude the user to use the electrc power reasonably, and promote the optmzaton of electrc power resources. By the mpact of the past planned economy, Chna s exstng electrcty prce system has been unable to meet the requrements of the electrcty market reform. In the tradtonal vertcally ntegrated electrc power ndustry management system, for economc development, socal stablty and other aspects, the government mplement preferental Table 1. Example of Ramsey numbers. b a ,

5 H. M. L, J. He tarff for some types of users, then the current sales prce cross-subsdes problem appears. From the perspectve of long-term development, cross-subsdes would not only serously hnder the effectve competton n the electrcty market of constructon, but not conducve to the overall coordnaton development of the natonal economy and the realzaton of the whole socal resources optmzaton allocaton. At present, Chna s electrc power system reform has entered the stage of lberalzng the user s choce and constructng the sale market. The power system reform program the government promulgated has been clear to develop consumers drect-purchasng electrcty plot, and gradually open the user s opton from the begnnng of large users. Wth the mplementaton of ths program, the cross-subsdes that exst n the current sale prce wll lead to a seres of problems n the electrcty market reform process. Therefore, at ths stage how to deal wth cross-subsdes of sales prce s an mportant ssue to be solved n the process of Chna s power ndustry development The Purpose and Sgnfcance of the Sales Prce Cross-Subsdes Problem All along, the power ndustry has been regarded as welfare, devatng from ts ntrnsc value to provde users wth low-cost power supply servce. In order to compensate for the loss caused by the power supply enterprses to provde low-cost power supply servces, governments have adopted dfferent ways to gve subsdes to the power enterprses. Among them, the cross-subsdes s one of the ways that the varous regons of Chna generally adopted. Because sales prce wth cross subsdes was generally unaffected by the supply-demand relatonshp and the cost fluctuaton, the prce n the polcy factor s much greater than ts operatng factors, resultng n an operatng loss of power enterprses and the decrease of supply servce levels, formng a vcous cycle of Loss-Cross-subsdze-Further loss. Gradually, the power companes cannot mprove ther economc fundamentally, makng t dffcult to be responsble for provdng excellent power servces. Takng nto account the actual stuaton that our country develops drect power-purchase plot for large consumers, the exstng sales prce cross-subsdes may cause many problems. On the one hand, large customers enjoy cheap electrcty servces by drect power-purchase, and power companes lost cross-subsdes revenue once provded by the bg users. At the same tme, power companes stll undertake ts oblgatons to provde resdental customers wth unversal electrcty servce, whch may cause serous fnancal losses to grd companes. On the other hand, to some extent, the exstng sales prce cross-subsdes hndered the smooth progress of the reform of the electrcty market. Reasonable soluton to the problem of cross-subsdes sales prce and mprovng the level of sellng prces for varous types of users, not only ensure healthy and sustanable development of the power companes, but also promote a smooth transton of Chna s power ndustry to the electrcty market of free competton. For such countres, Chna s rapd economc development, n the electrcty market reform process, t s a complex ssue that how to take nto account all the factors, a rea- 93

6 H. M. L, J. He sonable soluton to the sales prce cross-subsdes for varous types of users. Government mplements sales prce cross-subsdes, whch has two purposes n the fnal analyss, one s the polcy subsdes for economc development, the other s the unversal servce of the power. At the moment, the basc dea to solve ths problem s to reduce or abolsh unjustfed sales prce cross-subsdes, and gradually mprove the sales prce system for varous users, promptng prce levels more reasonable The Study of the Sales Prce Cross-Subsdes Cross-subsdy s a very mportant concept n the prcng theory, whch reflects the relatonshp between revenues and costs attrbutable to a sngle commodty or commodty combnatons. In the real prce-makng, cross-subsdy s often between the products and servces, also among the dfferent consumer groups, and electrcty commodty prcng s no excepton. Wth the rse of the reform of the electrcty market boom, countres around the world have very hgh levels of concern for the sales prce crosssubsdes, and scholars studed the sales prce cross-subsdes from dfferent angles. Paper [6] analyzed the relatve rato n the our current sales prce system among resdental, commercal, ndustral users, and comparng wth other countres, revealed the exstence of serous cross-subsdes from ndustral and commercal electrcty to resdental electrcty prces. At present, domestc scholars manly study the cross-subsdes faced by our country n the mplementaton of drect power-purchase plot for large consumers. Paper [7] proposed a new program that bg users enter the market to buy the electrcty, the program stpulates that large users accessed to the tradng center must bear some of the cost of cross-subsdes, and n accordance wth certan crtera, cross-subsdes are apportoned accordng to each kwh. Paper [8] proposed a phased soluton of sales prce cross-subsdes n the course of our country developng large users and power companes to deal drectly. In addton, because there s a close connecton between the sale prce cross-subsdes and unversal servce, many scholars are conductng research n unversal servce wth the sales prce cross-subsdes. All n all, domestc scholars have reached a prelmnary agreement n the sales prce cross-subsdes, and have made some progress, but solutons and specfc measures for the cross-subsdes has not been studed The Ramsey Prcng Method Mathematcal model of Ramsey prcng s expressed as follows. Suppose a publc nsttuton producng n types of products (or provde n types of customers wth servce), = 1,, n, and the demand for all knds of products (or groups of users) are ndependent. Product consumpton s q = ( q,, 1 qn ), when the prce vector s p ( p1,, pn) the demand functon s q (,, = D q1 qn). n Enterprses Revenues: R( q) = pq, 1 Cost Functon: C( q,, 1 qn ), n Total Consumer Surplus: S ( q) = D( q,, qn) dq pq, 1 Producer Surplus: R( q) C( q q ), 1, n =, 94

7 H. M. L, J. He The Ramsey prcng problem s denoted as max. { S( q) R( q) C( q, 1 qn )} ( ) ( ). +, S. t. R q C q, 0 1 q n The ntroducton of the Lagrange multpler λ makes the above maxmzaton problem nto: (, ) (, ) λ ( ) (, ) W = D q q dq C q q + R q C q q 1 n 1 n 1 n The formula of q were seekng a frst-order partal dervatves, Among them, class I user), f R η W p = ( p MC) + λ q + p MC = 0 q q p ( 1+ λ)( p MC) = λ q q p MC λ 1 = p 1+ λ n q = p λ 1 λ p q s the prce elastcty of demand of class I products (or = s denoted as the Ramsey ndex, then p = MC ( 1 R η ) + The Equaton shows that the prce of each type of product (user) on the margnal cost of an addton ( ) p MC p s nverse proporton to the prce elastcty of the product demand, ths s the famous rally rules. Ramsey prcng consder the mpact of user demand prce elastcty to product prces, greater elastcty of demand for the product, f the prce s too hgh, t wll cause a sgnfcant decrease n demand, thus resultng n the decrease of consumer surplus and the loss of socal welfare. If the prce elastcty of demand for products wth a lower prce, due to less demand for change, the mpact on consumer surplus and socal welfare s relatvely small [9] The Analyss of the Ramsey Prcng Method In order to better solve the problem of sales prce cross-subsdes, we can apply the Ramsey prcng theory to determne the level of sales prce of classfed users. Approprate adjustments to the exstng sales prce level wth reference to Ramsey prcng, to reduce or cancel the purpose of cross subsdes. Ramsey prcng s a sub optmal prcng method whch s determned by the margnal cost of the enterprse and the prce elastcty of dfferent knds of users n order to ensure the balance of power grd enterprses. The power cost of the resdents s hgher than that of the ndustral users, and the prce elastcty of the resdents demand s less than that of the ndustral users. In accordance wth the Ramsey prcng theory, the sales prce of the resdents should be hgher than the ndustral customers, whch s n lne wth the general stuaton of foregn resdents n the electrcty prce s hgher than the prce of electrcty [10]. The man steps of Ramsey prcng are shown n Fgure 2. The dffculty les n the determnaton of the margnal cost of power supply MC and the prce elastcty of demand for varous types of users. 95

8 H. M. L, J. He Calculate the prce elastcty of varous types of user power demand Determne the margnal cost of power supply for all knds of users Calculated Ramsey ndex Calculate the sales prce level of all knds of users under the Ramsey prcng method Fgure 2. A outlne of the Ramsey prcng method. Under normal crcumstances, the prce elastcty of electrcty demand ncreases wth the ncrease of user power consumpton. In accordance wth the Ramsey prcng theory to determne the user s electrcty prce, ndustral users demand prce elastcty s hgher and ts power supply margnal cost s low, so the Ramsey tarff for ndustral users should be low; n contrast, the demand prce elastcty of resdental users s small and ts power supply margnal cost s hgher, so t should be hgher than the prce of ndustral electrcty. To a certan extent, Ramsey prce reduces the cross-subsdes n the past, and ensure the balance of corporate earnngs under the premse to acheve the suboptmal goal of ncreasng socal welfare, but t s not satsfactory n the socal welfare dstrbuton. At present, Chna s current resdental electrcty prce s lower than ndustral tarffs. If you take Ramsey prcng method to determne the sales prce of users, resdental electrcty prces wll rse substantally, whch s obvously unfar. The power of human lfe s an ndspensable necessty, f substantal ncrease n resdental electrcty prces wll lead to poor areas and low-ncome users not affordng the hgh electrcty blls, the mpact of basc electrcty needs of these users daly. To solve the above problem, consder the lfelne tarff for resdental users combned wth Ramsey prcng, determne the lfelne of resdental users per capta electrcty consumpton standards, wthn ths standard resdental electrcty demand have greater rgdty. When resdents of the user s wthn ths standard, accordng to the lfelne electrcty tarff to be settled; f resdents lfelne electrcty consumpton exceeds the standard, the standard prce of electrcty by lfelne settlement, the excess electrcty tarff by Ramsey settlement. Ths approach not only ensures the poor areas, low-ncome resdents basc daly demand for electrcty, but also encourages users to conserve electrcty, to acheve optmal allocaton of resources. In the total welfare functon, dfferent users can set dfferent welfare weghts, and ther mathematcal expresson s as follow: ( p MC ) p λη( α λ) = +, α s socal welfare weght factor. Government related departments can set or adjust the value α of resdents, ndustry and other users, so as to acheve a reasonable adjustment of dfferent knds of user sales prce, and to meet the purpose of dfferent types of users of electrcty demand. 96

9 H. M. L, J. He 4. Conclusons From the ntroducton of the classc example of sx people, we found that there s a necessty that there must be 3 people knowng each other or not knowng each other. In the same way, we can fnd 9 people are bound to meet that ether 3 people not knowng each other or 4 people knowng each other. From the above, we found some partcularty of the number, and the number s called Ramsey number, but Ramsey number can t be obtaned wth the rght formula so far. The charm of Ramsey number les n ts uncertanty and practcalty, so a large number of scholars wll be nterested n t. I beleve that n the near future we wll come up wth more Ramsey number. Fnally, we apply the Ramsey number to real lfe, and ntroduce the Ramsey prcng method. Ramsey prcng theory focuses on solvng the ssues of the sales prce crosssubsdes, but the theory stll nvolves lttle n the feld of electrcty prcng, whose ratonalty and practcablty of the prcng should be further studed and tested; on the other hand, because the soluton to the sale prce cross-subsdes n practce mght lead to the problems n dfferent development stages of Chna s electrc power ndustry, such as the dffculty of the low ncome resdents, the mplementaton of the power unversal servce and so on, we need further study. References [1] Zhao, W.F. (2010) The Constructon Method and Random Graph of Ramsey Theory. Master s Degree Thess, Natonal Unversty of Defense Technology, Changsha. [2] Bao, Q.Q. (2009) Some Generalzed Ramsey Number Calculaton. Master s Degree Thess, Huazhong Unversty of Scence and Technology, Wuhan. [3] L, T. (2006) Ramsey Number on the Even-numbered Rounds. Master s Degree Thess, Dalan Unversty of Technology, Dalan. [4] Ba, L.F. (2004) Research on Some Problems n Generalzed Ramsey Numbers. Master s Degree Thess, HoHa Unversty, Nanjng. [5] Xu, X.D. (2002) Classcal Ramsey Theory and Its Applcatons. Master s Degree Thess, Natonal Unversty of Defense Technology, Changsha. [6] Que, G.H. (2003) Cross-Subsdzaton, Internatonal Comparson and Reform. Electrc Power Technologc Economc, No. 2, [7] Zeng, M., Yu, J.R. and Wang, H. (2007) Research on a New Large User Access to Electrcty Market Electrcty Purchasng. North Chna Electrc Power Technology, No. 4, [8] State Electrcty Regulatory Commsson, the State Grd Corporaton, State Grd Bejng Economc and Technology Research Insttute (2007) Research on Drect Tradng between Large Users and Power Companes. Bejng Economc Research Insttute State Grd. [9] Bgerna, S. and Bollno, C.A. (2016) Ramsey Prces n the Italan Electrcty Market. Energy Polcy, 88, [10] Bgerna, S. and Bollno, C.A. (2016) Optmal Prce Desgn n the Wholesale Electrcty Market. Energy Journal, 37, No. SI

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