Scrambling on Electrical Power Grids

Transcription

1 Scrmbling on Electricl Power Grids Thoms V. Huynh Deprtment ofsystems Engineering The Nvl Postgrdute School Monterey, CA 93943, USA John S. Osmundson Deprtment ofnformtion Sciences The Nvl Postgrdute School Monterey, CA 93943, USA Abstrct - Deregulted, the U.S. electricl power grid system hs shown the emergence of behviors tht include scrmbling by buyers to purchse electricity nd unexpected incresing costs of electricity to consumers. To infer those behviors, in this pper, we dpt theoreticl frmelvork, developed by Nelvmn in his lvork on spred of epidemic disese on networks, to the propgtion of electricity buying nd selling in power grid system. Specificlly, lve represent the electricl power grid s rndom biprtite grph of electricity sellers nd buyers, lvhich is then projectedto one-mode nellvork of intercting buyers. The so-clled trnsction trnsferbility mong buyers is clculted s function of the offered buying prices. The verge number of buyers scrmbling to buy electricity is obtined s function of the trnsction trnsferbility. Together, the trnsction trnsferbility nd the verge number ofscrmbling buyers indicte tht scrmbles for electricity buying drive up costs to buyers. Keywords: Electricl power grid system, rndom biprtite grph, trnsction trnsferbility, nd scrmbling. 1 ntroduction Coopertion mong electricl utilities, which is criticl to mintin relible trnsmission of electricity, hs become even more importnt with deregultion of the electricl power industry in the u.s. [1], s trnsmission of electricity between regions increses nd requires more creful opertion in the bsence of regultion. The deregultion ws intended to induce competition in the mrket nd to reduce costs to consumers. However, the utilities hve different nd, sometimes competing, economic interests. Now, more thn one electricl power service provider cn be involved in selling electricity to customer, nd customer cn buy electricity from more thn one seller. As result, the electricl power grid system tht evolved under one set of operting principles is now dpting to new set of operting principles, with the emergence ofbehviors tht include scrmbling by buyers to purchse electricity, increse in grid problems nd filures, nd unexpected incresing costs of electricity to consumers [2]. These behviors re expected to be relted scrmble by buyers who compete with ech other to purchse electricity. This work provides theoreticl frmework to explin the behviors observed in [2]. t will show tht connectivity mong electricity buyers is incresed s result of this competition. The price sellinglbuying increse is implied by the incresed connectivity. Since electricity is the only commodity to buy nd sell here, the words 'buyers' nd 'sellers' men electrify buyers nd sellers, respectively, from now on. A trnsction is defined to be buying nd/or selling interction between seller nd buyer. A trnsction trnsfer between two buyers occurs when one buyer is outbid by the other buyer, resulting in the first buyer losing his opportunity to buy electricity from seller nd the second buyer succeeding in buying electricity from the seller. n other words, trnsction trnsfer is effected between buyer who hs been buying electricity from seller nd who now lowers his buying price, which is rejected by the seller, nd buyer who hs rised his bidding price, which is ccepted by the seller. The probbility of trnsction trnsfer from buyer with current trnsction to potentil buyer depends on both decrese in or no chnge in the former's buying price nd n increse in the ltter's offered buying price. The purpose of this work is to pply the theoreticl frmework employed in Newmn's work [4] on spred of epidemic disese on networks to infer some of the behviors observed in [2]. The theoreticl frmework includes: Modeling of the power grid system s biprtite grph; Modeling ofthe trnsction trnsferbility; Computing the verge number of buyers scrmbling to purchse power, using the theoreticl frmework in [4]; nd Combining the trnsction trnsferbility nd the verge number of scrmbling buyers to obtin sensitivity ofscrmbling to offered prices. Our gols in this pper re: Apply Newmn's work [4] on spred of epidemic disese on networks to the propgtion of trnsctions on the one-mode network of electricity buyers /08/$ EEE

2 Model nd compute the trnsction trnsferbility s function ofoffered buying prices. Discuss the implictions of the results: scrmbling by buyers to purchse electricity nd incresing costs of electricity to consumers observed in [2]. The rest of the pper is orgnized s follows. We discuss the modeling ofthe power grid system s biprtite grph. We then describe the modeling nd the computtion of the trnsction trnsferbility nd the computtion of the verge number of buyers scrmbling to buy electricity. We next discuss the results nd their implictions. We end with some concluding remrks. 2 Power Grid System s Biprtite Grph A power grid system treted here is simplified to be network ofelectricity sellers nd buyers. This network my be considered to belong to the clss of collbortion networks [3], which re known to hve biprtite structure nd strongly non-poisson degree distributions [4], [5]. This network of electricity sellers nd buyers cn thus be represented by biprtite grph, whose vertices re the sellers nd buyers nd whose rcs run between the sellers nd their buyers. Since sellers neither sell nor buy from ech other, there re no rcs connecting them. Biprtite grph model of the network of sellers nd buyers One-mode network of buyers ~Selle, o Buye, Figure 1. A biprtite grph model of network ofelectricity sellers nd buyers (left) nd the corresponding one-mode network ofbuyers (right). The exmple network in Fig. 1 shows three sellers (A, B, nd C in squres) nd five buyers (1,..., 5 in circles) forming biprtite grph. The biprtite grph is then projected onto the so-clled one-mode network (uniprtite grph) on the right in Fig. 1. n the one-mode network, the vertices represent the buyers nd the rcs connect the buyers buying from the sme sellers. An rc connects two buyers ifthere is trnsction trnsfer between them. Trnsctions propgte in this one-mode network s buyers re ble to buy nd others lose their buying. A scrmble to buy electricity occurs when the propgtion of trnsctions in the network occurs s n epidemic disese on humn network [4]. The percoltion of trnsctions through network depends on both the price bidding nd the structure of the network [4], [6]. The structure of the network is reflected by the verge degree of the network, which is obtined from degree probbility distribution. The degree of vertex indictes the number of vertices to which it is connected. As this type of network is known to dmit strongly non Poisson degree distributions, we ssume the probbility Pk of rndomly chosen vertex (buyer) hving degree k in the one-mode network is of the truncted power-lw form Pk=~k-e k K, k~1, in which > 1, K is the exponentil cutoff frequency, nd ~, the normliztion constnt, is given by where Lin (x) is the nth polylogrithm of x. As pointed out in [3], [5], nd [7], the exponentil cutoff is included becuse it ppers to exist in mny relworld grphs nd it mkes the distribution normlizble for ll. 3 Cluster in Network s Scrmble on Power Grid System Following Newmn [4], let rij denote fixed probbility of trnsction propgtion per unit time ssocited with the rc connecting the ith vertex nd the Jth vertex. t is the probbility tht the i th buyer, being in stte of buying, will trnsfer its buying to the Jth buyer in given time intervl. Assuming discrete time steps, if the i th buyer is trnsferring its buying for r time steps, then the probbility tht the Jth buyer will be buying from the sme seller is ~j = 1- (1- rij)1". We will consider single time step, i.e., T = 1; in this cse, ~j = rij' The electricity buying trnsferbility (trnsferbility, for short) is T = (rij), where ( ) denotes the expecttion vlue.

3 Clcultion oftrnsferbility Let ~i nd~j denote the frctionl offered price chnges from the jth buyer nd the jth buyer, respectively. We ssume tht both c;j > 0 nd c;i < 0 hppen in order to effect buying trnsfer between the jth buyer nd the jth buyer. Furthermore, we ssume tht ;; dmit norml distribution with men f.1i nd vrince (J"/ nd tht ~; dmit norml distribution with men f.1 j nd vrince o}. We define ~ij = ~j -~i. Then, ~ij dmits norml distribution h d 2 2 F Wt men f.1 = f.1 j - f.1i n vnnce (J"j + O"i. or. 1 f smp Cty 0 c cu ton, we ssume 0"j = (J"i =0". Thus, ~ij dmits norml distribution with men f.1 nd vrince 20" 2. The per unit time probbility of trnsction trnsferred from the jth buyer to the jth buyer is then, rij = p(c;ij > 0) or rij = J- p(c;ij ~ 0). From the norml distribution of ~ij we then hve rij =- t1jl-uji), nd, hence, the trnsction trnsferbility is given by (1) t/j is the cumultive distribution function of the stndrd norml distribution b 0.7 :: 0.6. ~ 0.5 i 0.4 ~ 0.3 Trnferbility vs. Rtio of Difference in Mens of Offered Prices to Stndrd Devition Ot--'-==T----,--,..---,--_r---,----,---; Rtio of Difference in Mens of Offered Prices Figure 2. Trnsferbility s function ofthe rtio, f.1,ofthe difference in the mens ofoffered (J" prices to the stndrd devition Cpturing (1), Fig. 2 displys the trnsferbility s function of the rtio, f.1, of the men to the (J" stndrd devition ofthe distribution of c;ij. Tht T increses with the rtio f.1 mens tht, on verge, 0" the higher is the difference ~ij between the offered prices of two buyers, the more likely there will be trnsction trnsfer between the two buyers. Clcultion ofthe Averge Number of Buyers Scrmbling to Buy Electricity Were interested in the size of the cluster of buyers tht re involved in trnsction, i.e., the number of vertices reched vi trnsction propgtion long the rcs in the network. The size of the cluster of buyers tht re involved in trnsction is indictive of the effect of the trnsction propgtion nd the instbility of the grid system. For fixed network, s in the cse of the power grid system, trnsction trnsfer rte threshold exists; only smll, finite-sized scrmbles re possible below the threshold, bove which scrmbles become untenbly controlled. Following Newmn et le [3], we introduce the generting function for the probbility distribution of the degree of rndomly chosen vertex, Go (x) = LPk Xk, k=o nd the generting function for the probbility distribution of the degree of vertex t the end of rndomly chosen rc, LPk Xk G](x)= k=o '(), Go J where G~(x) is the derivtive of Go (x) with respect to x. The generting function for the probbility distribution of m rcs of the occupied rcs from rndomly chosen vertex is [9] Similrly, the generting function for the probbility distribution of the number of occupied rcs from the vertex t the end of rndomly chosen rc is Now the generting function Hj(x;T) for the totl number of buyers buying electricity s result of single trnsction trnsfer long n rc in the /08/$ EEE

4 network must stisfy self-consistency condition of the form [3], [8] Hj(x; T) = xgj(h j(x; T); T). The generting function H 0 (x; T) for the distribution of the number of buyers ffected by scrmble strting with single electricity buying buyer is 10.~ 6 en & 5 l! ~ 4 The verge size (s) of n electricity buying scrmble is now given by Trnsferbility, T (2) According to (2), the verge size H 0 (x; T) becomes unbounded when T is equl to the criticl vlue T c = --!--(), which mrks the onset of G j 1 uncontrolled scrmbling (epidemic [4]) behvior. For the truncted power-lw network, (s) =1+ TG~(J) 1-TG~(1) (3) As in [9], we consider = 2 nd the three different vlues of the cutoff frequency K of 5, 10, nd 20. The results generted by (3) re cptured in Fig. 3, which displys the grphs of the verge scrmble size (s) s function of trnsferbility T, prmeterized by K Averge size vs. Trnsferbility for lph =2 1, 0L---_...L...-_...L.-_---l...-_----L.._----l J o 0.1 (kpp), for the fixed vlue of !--kpp = kpp = kpp = Figure 3. Averge scrmble size s function of trnsferbility.. Prentheticlly, Fig. 3 shows the sme results produced in [9]. The criticl vlue oftrnsferbility, t which the verge scrmble size (s) becomes unbounded, vries with the vlue of the cutoff frequency. The results contined in Fig. 2 nd Fig. 3 will be used to infer the behviors of the deregulted power grid system. 4 Discussion ofresults Focused on the curve corresponding to the cutofffrequency of 5 (K = 5 ) in Figure 3, we observe tht s T pproches the threshold vlue of 0.6, the scrmbling becomes chotic, in the sense tht ll buyers re ffected by single trnsction. Below this threshold the scrmble is mngeble. At T = 0.55, the verge number of scrmbling buyers is roughly 10 (Fig. 3). Figure 2 indictes tht the rtio J.1 is pproximtely equl to 0.18 t T = This mens tht, on verge, the bidding price is higher thn the current buying price nd tht trnsction tkes plce between two buyers. At T =0.3, shown by Figure 3, the verge number of scrmbling buyers is roughly 2, nd, indicted by ~ (-1). The negtive vlue of Figure 2, the rtio J.1 this rtio indictes three possibilities: (1) buyer who hs been buying electricity from seller nd who now rises his buying price tht is higher thn the bidding price of buyer who hs not been buying from the seller; (2) buyer who hs been buying electricity from seller nd who continues to mintin his buying price tht is higher thn the bidding price of buyer who hs not been buying from the seller; nd (3) the buying price of buyer who hs been buying electricity from seller is much higher thn the other offered prices. n either cse, there is no trnsction. n relity, the rtio J.1 increses nd, hence, trnsferbility increses; scrmbling then ugments nd the power grid problem grows. The observtion of some behviors of the power grid system reported in [2] re thus supported by the theoreticl frmework explined in this pper. 5 Conclusions n this work we pply Newmn's work [4] on spred of epidemic disese on networks to the propgtion of trnsctions on the one-mode network of electricity buyers. We lso model nd compute the trnsction trnsferbility s function of offered buying prices. The trnsction trnsferbility nd the verge number of scrmbling buyers indicte tht

5 scrmbles for electricity buying drive up costs to buyers. The theoreticl frmework explined in this pper supports some of the behviors of the power grid system observed in [2]. Future work will extend this theoreticl frmework to ssess the utiliztion of the grid nd the verge number of system filures resulting from buying scrmbles. 6 References [1] Federl Energy Regultory Commission (FERC) Order 888, April 24, 1996 [2] 1. S. Osmundson, T.V. Huynh, nd Gry O. Lngford, "Emergent Behvior in Systems of Systems," Proceedings of the Conference of Systems Engineering Reserch (CSER), Los Angeles, April 4-6, [3] M. E. J. Newmn, S. H. Strogtz, nd D. 1. Wtts, "Rndom grphs with rbitrry degree distributions nd their pplictions," Phys. Rev. E 64, (2001). [4] M. E. 1. Newmn, "Spred of epidemic disese on networks, Physicl Review E 66 (2002), no. 1, rt. no MR (2003e:60223) [5] L. A. N. Amrl, A. Scl, M. Brthelemy, nd H. E.Stnley, "Clsses of behvior of smll-world networks," cond-mt/ [6] L. A. Meyers, "Contct network epidemiology: bond percoltion pplied to infectious disese prediction nd control," Bulletin (New Series) of the Americn Mthemticl Society, Volume 44, Number 1, Jnury 2007, Pges [7] M. E. J. Newmn, "The structure of scientific collbortion networks," Snt Fe nstitute working pper [8] C. Moore nd M. E. 1. Newmn, "Exct solution of site nd bond percoltion on smll-world networks," Phys. Rev. E 62, (2000). [9] M. E. 1. Newmn, "Exct solutions of epidemic models on networks," (Dted: November 28, 2001) /08/$ EEE