Article A Graph-Based Power Flow Method for Balanced Distribution Systems 3, ID

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1 energe Artcle A Graph-Baed Power Flow Method for Balanced Dtrbuton Sytem Tao Shen 1,,, Yanjun L 1, *, ID and J Xang 3, ID 1 School of Informaton and Electrcal Engneerng, Zhejang Unverty Cty College, Hangzhou , Chna; @zju.edu.cn College of Control Scence and Engneerng, Zhejang Unverty, Hangzhou 31007, Chna 3 College of Electrcal Engneerng, Zhejang Unverty, Hangzhou 31007, Chna; jxang@zju.edu.cn * Correpondence: lyanjun@zucc.edu.cn Thee author contrbuted equally to th work. Receved: 14 December 017; Accepted: 1 February 018; Publhed: 7 February 018 Abtract: A power flow method baed on graph theory preented for three-phae balanced dtrbuton ytem. The graph theory ued to decrbe the power network and facltate the dervaton of the relatonhp between bu Current and the bu Voltage Ba from the feeder bu (the CVB equaton). A dtnctve feature of the CVB equaton t unfed form for both radal and mehed network. The method requre nether a trcky numberng and layerng of node nor breakng mehe and loop-analy, whch are both neceary n prevou work for mehed network. The convergence of the propoed method proven ung the Banach fxed-pont theorem. Keyword: dtrbuton ytem; power flow; graph theory; radal network; mehed network; backward/forward weep 1. Introducton Power flow calculaton the mot fundamental numercal problem for power ytem analy. A fat and general power flow method wll be requred by dtrbuton ytem a the development of mart grd and mut be a effcent a poble n the future [1]. Method on tranmon network are well developed uch a Gau Sedel, Newton Raphon [] and Fat-Decoupled method [3]. Dtrbuton network have ome pecal charactertc uch a radal/weakly mehed tructure, hgh R/X rato of mpedance, large number of branche and node, etc. Thee feature may caue problem when the algorthm for power flow of tranmon network are appled to dtrbuton ytem [4]. Power flow calculaton may be executed every fve mnute on tradtonal network, but a for mcrogrd, t may not meet the requrement. Mcrogrd have the nature of uncertanty and volatlty, o they need real-tme montorng to guarantee ther relablty, and requre a fater power flow method. Power flow method alo a very mportant tool for mprovng the relablty and effcency of fault analy [5], and t can alo provde evdence for protecton for power dtrbuton ytem. Backward/forward weep (BFS) method,whch ntended to olve unbalanced radal dtrbuton network, ha a very good performance, where all node are labeled nto dfferent layer accordng to dtance from the feeder node. The branch current are calculated n the backward weep, whle the bu voltage then calculated n the forward weep [6]. However, the BFS method cannot be appled drectly to network even wth weakly mehed tructure becaue the dtance from the feeder node are not unque n the preence of loop. Shrmohammad et al. have propoed a compenaton-baed power flow method for olvng weakly mehed network by ung the mult-port compenaton technque and bac formulaton of Krchhoff law [7]. Teng ha propoed a drect method for both radal and mehed network by developng the bu-njecton to branch-current Energe 018, 11, 511; do: /en

2 Energe 018, 11, 511 of 11 (BIBC) matrx and the branch-current to bu-voltage (BCBV) matrx. However, when olvng mehed network, the method ha to apply ome prelmnary operaton, ncludng Kron Reducton and modfyng the two matrce by loop-analy [8]. Wu and Zhang developed a power flow method for dealng wth mehed network baed on compenaton and loop-analy [9]. Thee method all need extra proceng for mehed network. Th paper propoe a graph-baed power flow method for dtrbuton ytem, whch ha a unfed formulaton for both radal and mehed network. Compared wth prevou work, the method ue graph theory to drectly buld the CVB equaton, a map to the bu current from the bu voltage ba from the feeder node. It requre nether a trcky numberng and layerng of node nor breakng mehe and loop-analy, whch are both neceary n prevou work for mehed network. Although graph theory ha been ued for power ytem n many apect [10 1], there are a few n power flow calculaton. The mot relevant the work publhed more recently [13], where graph theory ued for buldng the Z-bu matrx, and the reult obtaned are only for radal dtrbuton ytem. The other contrbuton of th paper that the convergence of the graph-baed method addreed by ung the Banach fxed-pont theorem, aocated wth the convergence rate of a clear phycal meanng. Notaton: The followng notaton are ued n th paper: CVB: The equaton between the bu current and the bu voltage ba from the feeder bu. BFS: The backward/forward weep method. BIBC, BCBV: The bu-njecton to branch-current matrx and the branch-current to bu-voltage matrx n [8]. R, C denote the ratonal, complex number et. denote the conjugate operator of complex number. I, U: The nject current and voltage vector of all node ncludng the feeder node. I, U : The nject current and voltage vector of all node except the feeder node. U L, I L : The voltage drop, mpedance, current vector of all branche. Z l, S : Matrce of all mpedance of branche/complex power of node except feeder node. H, H : The ncdence matrx wth/wthout the row of feeder node. U d : The voltage dfference vector between the feeder node and other node. Φ, Z d : The mappng between U d and I. 1 n : A n order vector wth all element beng 1.. Problem Formulaton.1. Topologcal Decrpton of the Network A dtrbuton network ha a typcal tree tructure, the root of whch the feeder node wth a known voltage. Sometme there are ome extra branche between node o a to form a mehed tructure. We ue undrected graph G = {N, E} to depct the topology tructure of a gven dtrbuton network where N = {1,, n} and E = {L 1,, L m } are node et and branch et, repectvely. If there a branch between two node n a graph, the two node are connected. Although ncdence matrx ha been ued n prevou tude [14,15], t ha to be numbered from front to back n equence. By ung of graph theory, we can gve an arbtrary numberng to node and branche. Fgure 1 how two mple typcal dtrbuton network contanng 4 node, where the feeder node node 3, not the frt node. Wthout lo of generalty, the potve drecton of branch current defned to be alway flowng out of the node wth the lower number. In th ettng, the ncdence matrx H = (h j ) of the graph G defned a

3 Energe 018, 11, of 11 +1, the branch current L j tart at node h j = 1, the branch current L j end at node. L4 0, otherwe L4 L The ncdence matrce of network n Fgure 1, for ntance, are gven a L 3 1 L1 1 L 1 L L 3 L5 L5 L 1 L L 3 L 4 L 5 L L , H = 3L Specally, m = n 1 for radal network. It ratonal to aume that the condered network connected, whch mple that the rank of H n L L1 1 L1 L3 (a) L L L1 L4 L4 L L3 (b) L L1 1 3 L3 4 4 L5 L5 Fgure 1. A typcal radal network and mehed network: (a) radal tructure (n = 4, m = 3); and (b) mehed tructure (n = 4, m = 5)... Bac Formulaton of Krchhoff Law L The power flow related to the teady-tate behavor L 3 3of the power ytem, 1 where all the voltage L1 1 and current are nuodal gnal wth the ame frequency. For a three-phae balanced dtrbuton ytem, each gnal can be repreented by a complex value. Wthout L1lo of generalty, all the electrcal varable are complex number n th paper f not pecfcally tated. L3 4 4 In th ene, the current, voltage and complex power of node k are L3denoted by complex number k, u k and k, repectvely. The potve drecton of k nject to node k, a llutrated n Fgure 1. The concatenated current and voltage vector are denoted by I = [ 1,,, n ] T C n and U = [u 1, u,, u n ] T C n. Let f be the number of feeder node. The feedng power of dtrbuton network can be gven by f = u f f, where denote the conjugate operator. Let Lk, u Lk and z Lk be the current, voltage and mpedance of branch k, repectvely. The potve drecton of Lk and u Lk follow that of h j,.e., from the node wth lower number to the node wth larger number. Smlarly I L = [ ] T L1, L,, Lm C m and U L = [ ] T u L1, u L,, u Lm C m. The dagonal mpedance matrx defned by Z L = dag ( ) z L1, z L,, z Lm C m m. Clearly one ha [ ] T I = 1, u,..., n u, (1) n and u 1 U L = Z L I L. () The Krchhoff Current and Voltage Law can be convenently decrbed by ue of ncdence matrx, repectvely,

4 Energe 018, 11, of 11 I = HI L, U L = H T U, (3) and they are vald for both radal and mehed network. Combnng wth Equaton (), t follow that.3. Reformulaton of Power Flow Equaton I = HZ 1 L HT U. (4) Equaton (1) and (4) form the power flow equaton. However, matrx (HZL 1 HT ) ngular, whch hamper contructng an dentty map from them (an dentty map a functon that alway return the ame value that wa ued a t argument.). For a dtrbuton network, the feeder node a lack node, whoe voltage fxed a the bae value V 0 R. Th paper addree the cae that all the other bue except for the feeder bu are modeled a P, Q bu. Such a contant power cae ncreangly common n the modern dtrbuton ytem becaue more and more power electronc devce are ued. Gven the feeder node voltage V 0 and the complex power k for other node k N \ f, where notaton N \ f denote the ubet of N deletng the element f, the goal of power flow to calculate the current and voltage of node n the et N \ f, whch we ue I, U C n 1 to denote repectvely, that, I = [ 1,, f 1, f +1,, n ] T, U = [u 1,, u f 1, u f +1,, u n ] T. (5) Correpondngly, let H R (n 1) m be the matrx removng the f -th row of H, wth whch A for the example n Fgure 1, node 3 the feeder node, then I = H I L. (6) H = 1 4 L 1 L L , L L 3 L 4 L L (7) Denote by U d C n 1 the voltage dfference between the feeder node and other node, namely, U d = U V 0 1 n 1. (8) Throughout of th paper, notaton 1 n denote a n order vector wth all element beng 1. Due to H T 1 n = 0, the followng can be obtaned, 3. Man Reult 3.1. Revew BFS Method U L = H T U = H T U V 0 H T 1 n = H T U d. (9) Tradtonal BFS method generally take advantage of the radal topology. It tart wth numberng and layerng from the feeder node to termnal node. The backward weep, tartng from the termnal layer and endng at the frt layer, to calculate the branch current I L by a current ummaton wth a poble voltage update. The forward weep operatng n an oppote drecton to calculate the voltage drop of node U d wth the branch current obtaned n the backward proce. Note that H nonngular n a connected radal network, we can drectly obtan branch current I L from I by Equaton (6), ntead of by current ummaton n the backward weep. In our tudy,

5 Energe 018, 11, of 11 the backward and forward procee can be decrbed, baed on graph theory, mply wthout layerng a: (1) () (3) (4) (5) (6) I (k) L L d U (k+1) [ ] I (k) T = 1 u1,..., f 1 u, f +1 f 1 u,... n f +1 u, n = H 1 I (k), = Z L I (k) = H T L, L, = V 0 1 n 1 + d, repeat (1) to (5) untl U (k+1) < ɛ, where upercrpt (k) denote the value at the k-th teraton and ɛ the convergence tolerance. 3.. Unfed Method for both Radal and Mehed Network The above graph-baed proce no longer applcable for mehed network becaue H not a quare matrx for mehed tructure. The tradtonal method cannot apply ether n that the preence of crculatng current prohbt layerng node. Generally, radal and mehed network are dealt wth eparately when we conder the power flow for dtrbuton ytem. For dealng wth mehed network, breakng mehe or loop-analy were needed n prevou tude. Below, a unform method for both radal and mehed network preented. Combnng Equaton (6), (), and (9) yeld, I = H Z 1 L HT U d = ΦU d. (10) Here, Φ = H Z 1 L HT nothng but the Laplacan matrx weghted by branch admttance of G removng the row and column correpondng to the feeder node. For a connected network, Φ alway nonngular no matter f H a quare matrx. Defne Z d := Φ 1. (11) Equaton (10) buld a bjectve mappng between I and U d, by whch the functon of the tep () (4) n BFS method can be compactly rewrtten a d = Z d I (k), (1) whch nothng but the CVB equaton. Equaton (10) and (1) look gnfcant by themelve nce they mean that the njected current of node could be drectly related not to node power but to the node voltage ba from the feeder node. Now, our unform graph-baed method now can be delvered a follow: (g1) (g) (g3) (g4) d U (k+1) [ ] I (k) T = 1 u1,..., f 1 u, f +1 f 1 u,... n f +1 u, n = Z d I (k), = V 0 1 n 1 + d, Repeat (g1) to (g3) untl U (k+1) < ɛ. The ntal value U (0) generally et a V 0 1 n 1. Note that the nveron of Φ,.e., Z d doe not need to calculate durng the teraton. The flowchart of graph-baed method hown n Fgure.

6 Energe 018, 11, of 11 S, H, Z L, ϵ U 0 = V 0 1 n 1 Z d =(H Z 1 L HT ) 1 I (k) = [ 1 u 1 ] T,..., f 1 u, f +1 f 1 u,... n f +1 u n U (k+1) = V 0 1 n 1 + Z d I (k) U (k+1) < ϵ U Fgure. The flowchart of graph-baed method Convergence of Method Snce the above algorthm explct about the nvolved electrcal varable, t convergence can be analyzed by ung Banach fxed-pont theorem. ) Let S = dag ( 1, f 1, f +1, n, the dagonal matrx of njected power for node et N \ f. Denote by [x ] N \ f a vector contng of all x ndexed by N except for the f th one. Let V be the oluton of power flow,.e., the teady tate of the algorthm, and v be the element of V wth the mnmal magntude,.e., v = mn N \ f v where v the th element of V. Theorem 1. Algorthm (g1) (g4) table for all ntal value U (0) atfyng U (0) V < R, (13) where R = v 1 v Z ds. (14)

7 Energe 018, 11, of 11 Proof. The propoed algorthm a mappng from U to telf whch eentally a knd of fxed-pont teraton and can be rewrtten a U (k+1) ( = g ) = V 0 1 n 1 Z d [ u (k) ] T N \ f. (15) Baed onbanach fxed-pont theorem, the fxed-pont V atfyng V = g(v ) ext and unque f g(u ) a contracton mappng on U. It can be een that U (k+1) ( V = g [ (k) Z d S u v u (k) v ] T ) g(v ) = N \ f where L (k) = Z ds v u (k) wth v u (k) = mn N \ f v u (k). [ Z ds 1 v L(k) U (k) V, ] T 1 u (k) Due to U (k) V < R, one ha u (k) v < R and u (k) > v R for all N \ f. Subequently v u (k) > ( v R) v for one N \ f. Furthermore, ( v R) v v(v R) for all N \ f. Therefore, due to Equaton (13), N \ f (16) L (k) < Z ds 1, (17) v(v R) whch mple that U (k+1) V < R. It together wth the ntal condton U (0) V < R how that f (U ) a contracton mappng on U. Remark 1. The condton n Equaton (14) mple that v > R > 0 and v > Z d S. Remark. A maller Z d S wll lead to a larger R. Th mple roughly that a trong network (a mall rato between tranmtted power and the branch admttance, /(z L ) 1 ) allow a large permble regon for drect approache of power flow. Meanwhle a mall Z d S mean a mall Lpchtz contant L (k) and ubequently a fat convergence Comparon to the Drect Approach It can be een that our Algorthm (g1) (g3) mlar to that n [8]. Th not urprng, n that both are baed on Equaton (1), the mappng from I to U d. The dfference how to obtan Equaton (1). The dtnctve feature of our graph-baed method to preent a much mpler way than the drect approach n [8]. Recall the CVB equaton obtaned by the drect approach [8] a: for radal network, and for mehed network, [ U d 0 U d = (BCBV)(BIBC)I, ] = (BCBV)(BIBC) followed by a Kron Reducton. The followng comparon tated. In the radal network, the drect approach need: (1) equentally numberng node and edge from layer to layer begnnng at the feeder node; () performng a x-tep algorthmto buld the matrce (BCBV) and (BIBC) ; and (3) obtanng Z d by multplng (BCBV) by (BIBC). Our method [ I B new ],

8 Energe 018, 11, of 11 need: (1) an arbtrary numberng node and edge; () drectly wrtng matrce H, H, and Z L ; and (3) calculatng Z d = H T Z L H 1. In the mehed network, the drect approach need an extra drawng the correpondng radal veron of the mehed network, addng two tep for every extra branch that make the network mehed to buld matrce (BCBV) and (BIBC), and a Kron reducton. Thu, the complexty of the drect approach would ncreae largely a the degree of meh ncreae, whle our method ha the ame procedure for the mehed network a that for the radal network. In fact, our method can apply to any mehed network rather than to the weakly-mehed network. Moreover, our method ha a clearer phycal meanng becaue no network reducton ha to be made. The above content are ummarzed n Table 1 for a clear nght. Table 1. Comparon wth Drect Approach. Drect Approach Radal Network Propoed Method Numberng Sequental Arbtrary Matrce BIBC, BCBV H, Z L Operaton Z d = INV(BIBC BCBV) Z d = H T Z L H 1 Drect Approach Mehed Network Propoed Method Mehe Need Recognton No Need Numberng Sequental for radal tructure and place mehe to the end Arbtrary Matrce Modfed BIBC, BCBV H, Z L Operaton Modfyng BIBC, BCBV and Applyng Kron Reducton Z d = (H ZL 1 ) 1 4. Tet Reult The propoed method teted and compared on both radal and mehed network on MATLAB. Table how the dtrbuton ytem of 14-, 33-, 69-, 84-, 119-, 135-, and 874-node radal network and ther mehed edton, whch are from paper [16 0]. Table. Network Confguraton. No. of Node No. of Branch (Radal) No. of Branch (Mehed) Four method are teted. Method I the Gau Sedel Method, Method II the Newton Raphon method, Method III the drect approach propoed n [8] and Method IV our Graph-baed method. The convergence tolerance et at p.u. Table 3 and 4 how the performance of thee four method for radal and mehed network, where Tme and IT denote the teraton tme and teraton number, repectvely, and L denote the approxmate Lpchtz contant, whch decrbe the convergence rate of Method IV. Accordng to the table, Method I, the Gau Sedel method, cot much more tme and teraton tep than the three other method, nce t ha a very low convergence rate o that even f t doe not need much tme at each teraton, t tll cot much tme. Method II, the Newton Raphon method, need fewer teraton tep than other method nce t follow the drecton of gradent decent at every tep.

9 Energe 018, 11, of 11 However, the Newton Raphon method tll cot more tme than Method III and IV, becaue t requre calculatng the Jacoban matrx and t nveron matrx at each teraton, whch cot majorty of tme. Therefore, the tme conumpton of Newton Raphon more related to the number of node compared wth Method III and IV. No. of Node Table 3. Radal Network Tet. Method I Method II Method III Method IV Tme IT Tme IT Tme IT Tme IT L No. of Node Table 4. Mehed Network Tet. Method I Method II Method III Method IV Tme IT Tme IT Tme IT Tme IT L A for Method III and Method IV, the reult how the drect approach approxmately equvalent to the propoed method, whch content wth the theoretcal analy. However, the advantage of our method the proce to get the CVB equaton. A mentoned above, the drect approach requre loop-analy and Kron Reducton for mehe network, whle our method doe not need any extra proceng. Table 5 provde a comparon of tme pent on gettng the CVB equaton, and the reult how that the propoed method take le tme than the drect approach to get the CVB equaton. Moreover, for the example of the 14-node network, we only ncreae the number of meh, and the reult how that the tme conumpton of propoed method are almot the ame when the number of mehe ncreaed, unlke the ncreang tme conumpton of drect approach, manly due to the Kron Reducton. Note that, n the cae that the node numberng, the radal tructure drawng and the mehed branch dentfyng have been made n advance, t can be een that the graph-baed method much better f the tme pent on thee pretreatment are contaned. Lmtaton The propoed method ha hown obvou advantage compared to prevou work. However, t ha ome lmtaton. Frt, a for the mpact of dtrbuted generator, DG can be condered a PQ node wth contant actve/reactve power a well a PV node wth contant actve power and voltage magntude. If DG are condered a PQ node, our method can deal wth t. Alternatvely, f DG are condered a PV node, our method not applcable. Bede, a for the applcablty to unbalanced network, we have a prelmnary outlne to extend our method for unbalanced network, but the work tll need delberate dcuon and proof. Fnally, the mpact of FACTS devce not dcued n th paper nce FACTS devce cannot be mply modeled a PQ node and our method am at dtrbuton network wth PQ node.

10 Energe 018, 11, of 11 Table 5. Tme to get the CVB equaton. No. of Node No. of Branch Method IV Tme Method III Tme No. of Node No. of Branch Method IV Tme Method III Tme Concluon Th paper ha propoed a graph-baed power flow method for three-phae balanced dtrbuton ytem wth PQ node. For the nature of dtrbuton ytem uch a radal/weakly mehed tructure, and large number of branche and node, tradtonal power flow method may fal or cannot meet the requrement. Wth regard to th, we have made ome progre. The propoed method provde a unform formulaton for both radal and mehed network. The unform formulaton much mpler than before, requrng nether a trcky numberng and layerng of node nor breakng mehe and loop-analy, whch are both neceary n prevou work for mehed network. The convergence of the propoed method ha been hown by ung the Banach fxed-pont theorem. The comparon tet reult how the effcency of our compettve method. Acknowledgment: The author would lke to thank anonymou revewer for ther valuable comment and nght. Th reearch wa upported by the Natonal key reearch and development program of Chna (016YFB ), the Natonal Natural Scence Foundaton of Chna( , ), the cence and technology project of SGCC(511SX16000J). Author Contrbuton: Tao Shen conducted analy and mulaton. Yanjun L provded gudance, concepton and gave fnal approval of the veron to be ubmtted. J Xang provded gudance, concepton and plan. Conflct of Interet: The author declare no conflct of nteret. Reference 1. Balamurugan, K.; Srnvaan, D. Revew of power flow tude on dtrbuton network wth dtrbuted generaton. In Proceedng of the 011 IEEE Nnth Internatonal Conference on Power Electronc and Drve Sytem (PEDS), Sngapore, 5 8 December 011; pp Nguyen, H.L. Newton-Raphon Method n Complex Form. IEEE Tran. Power Syt. 1997, do: /tdc Zmmerman, R.D.; Chang, H.D. Fat decoupled power flow for unbalanced radal dtrbuton ytem. IEEE Tran. Power Syt. 1995, 10, Luo, G.X.; Semlyen, A. Effcent load flow for large weakly mehed network. IEEE Tran. Power Syt. 1990, 5, Ou, T.C. A novel unymmetrcal fault analy for mcrogrd dtrbuton ytem. Int. J. Electr. Power Energy Syt. 01, 43, Kertng, W.H. A Method to Teach the Degn and Operaton of a Dtrbuton Sytem. IEEE Tran. Power Appar. Syt. 1984, 7,

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