An Effective Approach for Distribution System Power Flow Solution

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1 World Academy of Scence, Engneerng and Technology nternatonal Journal of Electrcal and Computer Engneerng ol:, No:, 9 An Effectve Approach for Dstrbuton System Power Flow Soluton A. Alsaad, and. Gholam nternatonal Scence ndex, Electrcal and Computer Engneerng ol:, No:, 9 waset.org/publcaton/7 Abstract An effectve approach for unbalanced three-phase dstrbuton power flow solutons s proposed n ths paper. The specal topologcal characterstcs of dstrbuton networs have been fully utlzed to mae the drect soluton possble. Two matrces the bus-njecton to branch-current matrx and the branch-current to busvoltage matrx and a smple matrx multplcaton are used to obtan power flow solutons. Due to the dstnctve soluton technques of the proposed method, the tme-consumng LU decomposton and forward/bacward substtuton of the Jacoban matrx or admttance matrx requred n the tradtonal power flow methods are no longer necessary. Therefore, the proposed method s robust and tme-effcent. Test results demonstrate the valdty of the proposed method. The proposed method shows great potental to be used n dstrbuton automaton applcatons. Keywords Dstrbuton power flow, dstrbuton automaton system, radal networ, unbalanced networs.. NTRODUCTON S an mportant tool and the foundaton of Dstrbuton A Management System (DMS), power flow calculaton problem has been pad more and more attenton. Many programs of real-tme applcatons n the area of dstrbuton automaton, such as networ optmzaton plannng, swtchng, state estmaton, and so forth, requre a robust and effcent power flow method [] []. Such a power flow method must be able to model the specal features of dstrbuton systems n suffcent detal. The well-nown characterstcs of an electrc dstrbuton system are radal; multphase and unbalanced operaton; unbalanced dstrbuted load; extremely large number of branches and nodes; wderangng resstance and reactance values. Those features cause the tradtonal power flow methods used n transmsson systems, such as the Gauss-Sedel and Newton-Raphson technques, to fal to meet the requrements n both performance and robustness aspects n the dstrbuton system applcatons. n partcular, the assumptons necessary for the smplfcatons used n the standard fast-decoupled Newton- Raphson method [] often are not vald n dstrbuton systems. Therefore, a novel power flow algorthm for dstrbuton systems s desred. To qualfy for a good dstrbuton power flow algorthm, all of the characterstcs mentoned before, need to be consdered. Several power flow A. Alsaad and. Gholam are wth the slamc Azad Unversty, Kazeroon branch, ran. algorthms specally desgned for dstrbuton systems have been proposed n the lterature [] []. Some of these methods were developed based on the general meshed topology le transmsson systems [] [9]. From those methods, the Gauss mplct -matrx method [7] s one of the most commonly used methods; however, ths method does not explctly explot the radal networ structure of dstrbuton systems and, therefore, requres the soluton of a set of equatons whose sze s proportonal to the number of buses. Recent research proposed some new deas on how to deal wth the specal topologcal characterstcs of dstrbuton systems [] [], but these deas requre new data format or some data manpulatons. n [], the authors proposed a compensaton-based technque to solve dstrbuton power flow problems. ranch power flows rather than branch currents were later used n the mproved verson and presented n []. Snce the forward/bacward sweep technque was adopted n the soluton scheme of the compensaton-based algorthm, new data format and search procedure are necessary. Extenson of the method, whch emphaszed on modelng unbalanced loads and dspersed generators, was proposed n []. n [], the feeder lateral based model was adopted, whch requred the layer-lateral based data format. One of the man dsadvantages of the compensaton-based methods s that new databases have to be bult and mantaned. n addton, no drect mathematcal relatonshp between the system status and control varables can be found, whch maes the applcatons of the compensaton-based algorthm dffcult. The algorthm proposed n ths paper s a novel technque. The only nput data of ths algorthm s the conventonal busbranch orented data used by most utltes. The goal of ths paper s to develop a formulaton, whch taes advantages of the topologcal characterstcs of dstrbuton systems, and solve the dstrbuton power flow drectly. t means that the tme-consumng LU decomposton and forward/bacward substtuton of the Jacoban matrx or the Y admttance matrx, requred n the tradtonal Newton Raphson and Gauss mplct matrx algorthms, are not necessary n the new development. Two developed matrces, the bus-njecton to branch-current matrx and the branch-current to bus-voltage matrx, and a smple matrx multplcaton are utlzed to obtan power flow solutons. The proposed method s robust and very effcent compared to the conventonal methods. Test results demonstrate the feasblty and valdty of the proposed method. nternatonal Scholarly and Scentfc Research & nnovaton () 9 scholar.waset.org/7-89/7

2 World Academy of Scence, Engneerng and Technology nternatonal Journal of Electrcal and Computer Engneerng ol:, No:, 9. UNALANCED THREE-PHASE LNE MODEL Fg. shows a three-phase lne secton between bus and j. The lne parameters can be obtaned by the method developed by Carson and Lews []. A matrx, whch taes nto account the self and mutual couplng effects of the unbalanced threephase lne. r P + jq = ( ) + ( ) = j () Where and are the bus voltage and equvalent current r njecton of bus at the -th teraton, respectvely. and are the real and magnary parts of the equvalent current njecton of bus at the -th teraton, respectvely. * A. Relatonshp Matrx Developments nternatonal Scence ndex, Electrcal and Computer Engneerng ol:, No:, 9 waset.org/publcaton/7 Fg. Three-phase lne secton model A matrx, whch taes nto account the self and mutual couplng effects of the unbalanced threephase lne secton, can be expressed as: aa ab ac an = ba bb bc bn () abcn ca cb cc cn na nb nc nn [ ] After Kron s reducton s appled, the effects of the neutral or ground wre are stll ncluded n ths model as shown n () aa n ab n ac n = abc ba n bb n () bc n ca n cb n cc n [ ] The relatonshp between bus voltages and branch currents n Fg. can be expressed as a b c = A C aa n ba n ca n ab n bb n cb n ac n bc n cc n For any phases faled to present, the correspondng row and column n ths matrx wll contan null-entres.. ALGORTHM DEELOPMENT The proposed method s developed based on two derved matrces, the bus-njecton to branch-current matrx and the branchcurrent to bus-voltage matrx, and equvalent current njectons. n ths secton, the development procedure wll be descrbed n detal. For dstrbuton networs, the equvalentcurrent-njectonbased model s more practcal [] []. For bus, the complex load S s expressed by S = P + jq =... N () And the correspondng equvalent current njecton at the -th teraton of soluton s Aa b Cc () Fg. Smple dstrbuton system A smple dstrbuton system shown n Fg. s used as an example. The power njectons can be converted to the equvalent current njectons by (), and the relatonshp between the busmcurrent njectons and branch currents can be obtaned by applyng Krchhoff s Current Law (KCL) to the dstrbuton networ. The branch currents can then be formulated as functons of equvalent current njectons. For example, the branch currents, and, can be expressed by equvalent current njectons as = + = + = Therefore, the relatonshp between the bus current njectons and branch currents can be expressed as = Equaton (7a) can be expressed n general form as: [ ] [ C][ ] () (7a) = (7b) nternatonal Scholarly and Scentfc Research & nnovaton () 9 scholar.waset.org/7-89/7

3 World Academy of Scence, Engneerng and Technology nternatonal Journal of Electrcal and Computer Engneerng ol:, No:, 9 nternatonal Scence ndex, Electrcal and Computer Engneerng ol:, No:, 9 waset.org/publcaton/7 where C s the bus-njecton to branch-current (C) matrx. The constant C matrx s an upper trangular matrx and contans values of and only. The relatonshp between branch currents and bus voltages as shown n Fg. can be obtaned by (). For example, the voltages of bus,, and are = = = (8a) (8b) (8c) where s the voltage of bus, and j s the lne mpedance between bus and bus j. Substtutng (8a) and (8b) nto (8c), (8c) can be rewrtten as = (9) From (9), t can be seen that the bus voltage can be expressed as a functon of branch currents, lne parameters, and the substaton voltage. Smlar procedures can be performed on other buses; therefore, the relatonshp between branch currents and bus voltages can be expressed as = Equaton (a) can be rewrtten n general form as: [ Δ ] = [ C ][ ] (b) (a) where C s the branch-current to bus-voltage (C) matrx.. uldng Formulaton Development Observng (7), a buldng algorthm for C matrx can be developed as follows: Step ) For a dstrbuton system wth m-branch secton and n- bus, the dmenson of the C matrx s m (n-). Step ) f a lne secton ( ) s located between bus and bus j, copy the column of the -th bus of the C matrx to the column of the j-th bus and fll a to the poston of the -th row and the j-th bus column. Step ) Repeat procedure () untl all lne sectons are ncluded n the C matrx. From (), a buldng algorthm for C matrx can be developed as follows. Step ) For a dstrbuton system wth m-branch secton and n- bus, the dmenson of the C matrx s (n-) m. Step ) f a lne secton s located between bus and bus j, copy the row of the -th bus of the C matrx to the row of the j-th bus and fll the lne mpedance ( j ) to the poston of the j-th bus row and the -th column. Step ) Repeat procedure () untl all lne sectons are ncluded n the C matrx. The algorthm can easly be expanded to a multphase lne secton or bus. For example, f the lne secton between bus and bus s a three-phase lne secton, the correspondng branch current wll be a vector and the n the C matrx wll be a dentty matrx. Smlarly, f the lne secton between bus and bus j s a three-phase lne secton, the j n the C matrx s a mpedance matrx as shown n (). t can also be seen that the buldng algorthms of the C and C matrces are smlar. n fact, these two matrces were bult n the same subroutne of our test program. Therefore, the computaton resources needed can be saved. n addton, the buldng algorthms are developed based on the tradtonal bus-branch orented database; thus, the data preparaton tme can be reduced and the proposed method can be easly ntegrated nto the exstent DA. C. Soluton Technque Developments The C and C matrces are developed based on the topologcal structure of dstrbuton systems. The C matrx represents the relatonshp between bus current njectons and branch currents. The correspondng varatons at branch currents, generated by the varatons at bus current njectons, can be calculated drectly by the C matrx. The C matrx represents the relatonshp between branch currents and bus voltages. The correspondng varatons at bus voltages, generated by the varatons at branch currents, can be calculated drectly by the C matrx. Combnng (7b) and (b), the relatonshp between bus current njectons and bus voltages can be expressed as [ ] = [ C ][ C][ ] = [ DLF][ ] Δ () And the soluton for dstrbuton power flow can be obtaned by solvng () teratvely * r P + jq = ( ) + ( ) = j (a) Δ + = DLF (b) [ ] [ ][ ] + + [ ] = [ ] + [ Δ ] (c) Accordng to the research, the arthmetc operaton number of LU factorzaton s approxmately proportonal to N. For a large value of N, the LU factorzaton wll occupy a large porton of the computatonal tme. Therefore, f the LU factorzaton can be avoded, the power flow method can save tremendous computatonal resource. From the soluton technques descrbed before, the LU decomposton and forward/bacward substtuton of the Jacoban matrx or the Y admttance matrx are no longer necessary for the proposed method. Only the DLF matrx s necessary n solvng power flow problem. Therefore, the proposed method can save consderable computaton resources and ths feature maes the proposed method sutable for onlne operaton. nternatonal Scholarly and Scentfc Research & nnovaton () 9 scholar.waset.org/7-89/7

4 World Academy of Scence, Engneerng and Technology nternatonal Journal of Electrcal and Computer Engneerng ol:, No:, 9 nternatonal Scence ndex, Electrcal and Computer Engneerng ol:, No:, 9 waset.org/publcaton/7. TEST RESULTS The proposed three-phase power flow algorthm was mplemented usng MATLA. Two methods are used for tests and the convergence tolerance s set at. p.u. Method : The forward/bacward method []. Method : The proposed algorthm. A. Accuracy Comparson For any new method, t s mportant to mae sure that the fnal soluton of the new method s the same as the exstent method. An eght-bus system Fg. Eght-bus dstrbuton system TALE FNAL CONERGED OLTAGE SOLUTONS TALE TEST FEEDER TALE NUMER OF TERATON AND NORMALED EXECUTON TME (equvalent -node system), ncludng the three-phase, double-phase, and sngle-phase lne sectons and buses as shown n Fg. s used for comparsons. The fnal voltage solutons of method and method are shown n Table. From Table, the fnal converged voltage solutons of method are very close to the soluton of method. t means that the accuracy of the proposed method s almost the same as the commonly used forward/bacward method.. Performance Test A man feeder trun wth x9-phase buses, s used for ths test. The sngle and double-phase laterals have been lumped to form the unbalanced loads for testng purposes. Ths trun s then chopped nto varous szes for tests as shown n Table. The substaton s modeled as the slac bus. Table lsts the number of teratons and the normalzed executon tme for both methods. t can be seen that method s more effcent, especally when the networ sze ncreases, snce the tme-consumng processes such as LU factorzaton and forward/bacward substtuton of admttance matrx are not necessary for method. For a 7-node system, method s almost tmes faster than method. C. Robustness Test One of the major reasons, whch mae the power flow program dverge, s the ll-condton problem of the Jacoban matrx or admttance matrx. t usually occurs when the system contans some very short lnes or very long lnes. n order to prove that the proposed method can be utlzed n severe condtons, EEE 7-bus test feeder s used []. The test feeder s adjusted by changng the length of eght lne sectons. Four of them are multpled by ten, and the other four are dvded by ten. The test result shows the number of teratons for ths case s and the executon tme s.8 s. t means that the proposed method s robust and very sutable for onlne use.. DSCUSSON AND CONCLUSON n ths paper, a drect approach for dstrbuton power flow soluton was proposed. Two matrces, whch are developed from the topologcal characterstcs of dstrbuton systems, are used to solve power flow problem. The C matrx represents the relatonshp between bus current njectons and branch currents, and the C matrx represents the relatonshp between branch currents and bus voltages. These two matrces are combned to form a drect approach for solvng power flow problems. The tme-consumng procedures, such as forward/bacward substtuton of the Jacoban matrx or admttance matrx, are not necessary n the proposed method. The ll-condtoned problem that usually occurs durng the other tradtonal methods wll not occur n the proposed soluton technques. Therefore, the proposed method s both robust and effcent. Test results show that the proposed method s sutable for large-scale dstrbuton systems. nternatonal Scholarly and Scentfc Research & nnovaton () 9 scholar.waset.org/7-89/7

5 World Academy of Scence, Engneerng and Technology nternatonal Journal of Electrcal and Computer Engneerng ol:, No:, 9 nternatonal Scence ndex, Electrcal and Computer Engneerng ol:, No:, 9 waset.org/publcaton/7 REFERENCES [] EEE Tutoral Course on Dstrbuton Automaton. [] EEE Tutoral Course on Power Dstrbuton Plannng. [] W. M. Ln and M. S. Chen, An overall dstrbuton automaton structure, Elect. Power Syst. Res., vol., pp. 7 9, 98. []. Stott and O. Alsac, Fast decoupled load flow, EEE Trans. Power Apparat. Syst., vol. 9, pp , May/June 97. [] J. H. Teng andw. M. Ln, Current-based power flowsolutons for dstrbuton systems, n Proc. EEE nt. Conf. Power Syst. Technol., ejng, Chna, 99, pp. 8. [] T. S. Chen, M. S. Chen, T. noue, and E. A. Chebl, Three-phase cogenerator and transformer models for dstrbuton system analyss, EEE Trans. Power Delvery, vol., pp. 7 8., Oct. 99. [7] T.-H. Chen, M.-S. Chen, K.-J. Hwang, P. Kotas, and E. A. Chebl, Dstrbuton system power flow analyss A rgd approach, EEE Trans. Power Delvery, vol., pp., July 99. [8] T. H. Chen and J. D. Chang, Open wye-open delta and open delta-open delta transformer models for rgorous dstrbuton system analyss, n Proc. nst. Elect. Eng., vol. 9, 99, pp. 7. [9] K. A. rt, J. J. Graffy, J. D. McDonald, and A. H. El-Abad, Three phase load flow program, EEE Trans. Power Apparat. Syst., vol. PAS- 9, pp. 9, Jan./Feb. 97. [] D. Shrmohammad, H. W. Hong, A. Semlyen, and G. X. Luo, A compensaton- based power flow method for wealy meshed dstrbuton and transmsson networs, EEE Trans. Power Syst., vol., pp. 7 7, May 988. [] G. X. Luo and A. Semlyen, Effcent load flow for large wealy meshed networs, EEE Trans. Power Syst., vol., pp. 9, Nov. 99. [] C. S. Cheng and D. Shrmohammad, A three-phase power flowmethod for real-tme dstrbuton system analyss, EEE Trans. Power Syst., vol., pp. 7 79, May 99. [] R. D. mmerman and H. D. Chang, Fast decoupled power flow for unbalanced radal dstrbuton systems, EEE Trans. Power Syst., vol., pp., Nov. 99. [] W. M. Kerstng and L. Wlls, Radal Dstrbuton Test Systems, EEE Trans. Power Syst., vol., EEE Dstrbuton PlannngWorng Group Rep., Aug. 99. nternatonal Scholarly and Scentfc Research & nnovaton () 9 scholar.waset.org/7-89/7

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