潮流不同排序方案的比较文献翻译中英文对照

1、 外文翻译（原文）A Comparison of Power Flow by Different Ordering SchemesAbstract Node ordering algorithms, aiming at keeping sparsity as far as possible,are widely used today.In such algorithmst,heirinfluenceon the accuracyof thesolution is neglected bectauwsoeni t make significant difference in normal sys

2、tems.While, along with the development of modern power systems,the problem willbecome more ill-conditioned and it is necessary to take the accuracy into count durinode ordering.In thispaper we intendto lay groundwork forthe more rationalityorderingalgorithmwhich could make reasonablecompromising bet

3、ween memoryand accuracy. Three schemes of node ordering for different purpose are proposed tocompare the performanceof the power flow calculatioannd an example of simplesix-node network is discussed detailed.Keywordspower flow calculation;node ordering; sparsity; accuracy;Newton-Raphson method ; lin

4、ear equationsPower flow is the most basic and important concept in power system analysis andpower flow calculation is the basis of power system planning, operation, schedulingand control 1.Mathematically speaking, power flow problem is to find a numericalsolution of nonlinear equations. Newton metho

5、d is the most commonly used to solvethe problem and it involves repeated direct solutions of a system of linear equationThe solvingefficiencaynd precisionof the linearequationsdirectliynfluencestheperformance of Newton-Raphson power flow algorithm.Based on numericalmathematics and physical character

6、istics of power system in power flow calculation,scholars dedicated to the research to improve the computational efficiency of linearequationsby reorderingnodesnumber and receiveda lotof successwhich laidasolid foundation for power system analysis.Jacobian matrix in power flow calculation, similar w

7、ith the admittance matrix, has1 外文翻译（原文）symmetrical structureand a high degree of sparsityD.uring the factorizationprocedure, nonzero entries can be generated in memory positions that correspond tozero entries in the starting Jacobian matrix. This action is referred to as gramming terms i

8、s used which processedand storesonly nonzero terms,thereduction of fill-in reflects a great reduction of memory requirement and the numberof operationsneeded to perform the factorizatioSno. many extensivestudieshavebeen concerned with the minimization of the fill-ins. While it is hard to find effica

9、lgorithm for determining the absolute optimal order, several effective strategies fdeterminingnear-optimalordershave been devisedfor actualapplications2,3.Each of the strategies is a trade-off between results and speed of execution and thehave been adopted by much of industry.The sparsity-programmed

10、orderedeliminationmentioned above, which isa breakthroughin power system networkcomputation, dramatically improving the computing speed and storage requirementsof Newtons method 4.Aftersparsematrixmethods,sparsevectormethods 5,which extend sparsityexploitationto vectors,are useful for solving linear

11、 equations when theright-hand-side vector is sparse or a small number of elements in the unknown vectorare wanted. To make full use of sparse vector methods advantage, it is necessary toenhance thesparsitoyf L-1by orderingnodes.This isequivalento decreasingthelength of the paths, but it might cause

12、more fill-ins, greater complexity and expensCounteringthisproblem,severalnode orderingalgorithms6,7 were proposedtoenhance sparse vector methods by minimizing the length of the paths whilepreserving the sparsity of the matrix.Up to now, on the basis of the assumption that an arbitrary order of nodes

13、 does noadverselyaffectnumericalaccuracy,most node orderingalgorithmstake solvinglinear equatioinnsa single iterataisornesearchsubjecta,iming at the reduction ofmemory requirementsand computing operations.Many matriceswith a strongdiagonal in network problems fulfill the above assumption, and orderi

14、ng to conservesparsity increased the accuracy of the solution. Nevertheless, if there are junction2 外文翻译（原文）very high and low series impedances, long EHV lines, series and shunt compensationin the model of power flow problem, diagonal dominance will be weaken 8 and theassumption may not be tenable i

15、nvariably. Furthermore, along with the developmentof modern power systems, various new models with parameters under various ordersof magnitude appear in the model of power flow.The promotion of distributedgenerationalsoencourageus to regardthe distributionnetworksand transmissionsystems as a whole i

16、n power flow calculationa,nd itwillcause more seriousnumericalproblem.All those thingsmentioned above willturn the problem intoill-conditioSno.itisnecessaryto discussthe effectof the node numbering to theaccuracy of the solution.Based on the existing node ordering algorithm mentioned above, this pap

17、er focusattention on the contradiction between memory and accuracy during node ordering,researchhow could node orderingalgorithmaffectthe performanceof power flowcalculation, expecting to lay groundwork for the more rationality ordering algorithmThis paper is arranged as follows. The contradiction b

18、etween memory and accuracyin node ordering algorithm is introduced in section II. Next a simple DC power flowis showed to illustrate that node ordering could affect the accuracy of the solutionsection III. Then, taking a 6-node network as an example, the effect of node orderinon the performance of p

19、ower flow is analyzed detailed i n .s eCcotnicolnu sIiVo n isgiven in section VI.II. CONTRADICTIONIN NODEMEMORYORDERINGALGORITHMAccording to numerical mathematics, complete pivoting is numerically preferableto partial pivoting for systems of liner algebraic equations by Gaussian EliminationMethod (G

20、EM). Many mathematicalpapers 9-11focus theirattentionon thediscrimination between complete pivoting and partial pivoting in (GEM). Reference9 shows how partial pivoting and complete pivoting affect the sensitivity of the Lfactorization. Reference 10 proposes an effective and inexpensive test to reco

21、gniz3 外文翻译（原文）numerical difficulties during partial pivoting requires. Once the assessment criterican not be met, complete pivoting will be adopted to get better numerical stability.power flow calculationsp,artialpivotingis realizedautomaticallywithout anyrow-interchangesand column-interchangesbecau

22、se of the diagonallydominantfeatures of the Jacobin matrix, which could guarantee numerical stability in floatinpoint computation mions t cases.While due torounding errors, the partial pivotingdoes not provide the solution accurate enough in some ill-conditionings. If completepivoting is performed,

23、at each step of the process, the element of largest module ischosen as the pivotal element. It is equivalent to adjust the node ordering in powerflow calculation. So the node relate to the element of largest module is tend to arrin front for the purpose of improving accuracy.The node reorderingalgor

24、ithmsguided by sparsematrixtechnologyhave wildlyused in power system calculationa,iming at minimizingmemory requirement.Inthese algorithms, the nodes with fewer adjacent nodes tend to be numbered first. Theresult is that diagonal entries in node admittance matrix tend to be arranged from lto largest

25、 according to their module. Analogously, every diagonal submatrices relateto a node tend to be arranged from least to largest according to their determinants.the results obtained form such algorithms will just deviate form the principle followhich the accuracyof the solutionwillbe enhance.That is wh

26、at we say thereiscontradiction between node ordering guided by memory and accuracy.COMPLETEItissaidthatcompletepivotingisnumericallypreferablteo partiaplivotingforsolvingsystems of linearalgebraicequations.When the system coefficientasrevarying widely, the accuracy of the solution would be affect by

27、 rounding errors hardand it is necessary to take the influence of the ordering on the accuracy of the solinto consideration.4 外文翻译（原文）Fig.1 DC model of Sample 4-node networkAs an example, consider the DC model of sample 4-node system shown in Figure 1.Node 1 isthe swing node having known voltageangl

28、e;nodes 2-4 are load nodes.Following the original node number, the DC power flow equation is:To simulate computer numerical calculation operations, four significant figures wibe used to solvethe problem.ExecutingGEM withoutpivotingon (1) yieldsthesolution 2,3,-04.3T0=36,-0.3239,-0.3249T, whose compo

29、nents differ from thatof the exact solution 2, -30,.3,4-0T.=32,-0.321T. A more exact solution couldbe obtained by complete pivoting: 2,-30,.30047,T-=0.3207,-0.3217T, and theorderof the node afterrow and column interchangesis 3,2,4.So thisis a morereasonable ordering scheme for the purpose of getting

30、 more high accuracy.IV. THE INFLUENCEPERFORMANCE OFOF NODE REODERINGPOWERON THEFLOW METHODNEWTON-RAPHSON5 外文翻译（原文）On the basis of the above-mentioned analysis, the scheme for node reordering willnot only affect memory requirement but also the accuracy of the solution in solvinglinearsimultaneousequa

31、tions.So performance of Newton-Raphson power flowmethod will be different with various node ordering. In this section three schemes oordering for different purpose will be applied to a sample 6-node network shown inFig 2 to compare the influenceof them on the accuracy of the solution,theconvergencer

32、ate,the calculatedamount and the memory needed in power flowcomputation. The detail of the performance is shown.in table IVAt present,there are variousschemes widely used for node numbering innear-optimal order to reduce fill-ins and save memory. The only information neededby the schemes isa tablede

33、scribingthe node-branchconnectionpatternof thenetworks. An order that would be optimal for the reduction of the admittance matrixof the network is also optimalfor the tableof factorsrelatedJacobianmatrix.Different schemes reach different compromise between programming complexity andoptimality. In th

34、is paper, what we concern about is how the result of the numberingaffectsthe computationalperformance.The programming efficienciys beyond thescope of the present work. To save memory, a dynamic node ordering scheme similar6 外文翻译（原文）to the third scheme presented in 2 is adopted in this section. Execu

35、tion steps ofalgorithm are as follows.a) Number the node degreeof which isone.Ifmore than one node meet thiscriterion,umber the node with the smallestoriginalnumber. Ifthereare not sucnnodes any more, start with step b);b) Number the node so that no equivalent branches will be introduced when thisno

36、de is eliminated. If more than one node meets this criterion, number the one withthe smallest original number. If we can not start with step a) or step b), turn to sc) Number the node so thatthe fewestbrancheswillbe introducedwhen thisnode is eliminated. If not only node could introduce fewest branc

37、hes, number the onewith the largest degree.Once certainnode isnumbered in the stepabove,updatethe degreeof relevantnodes and topological information. Until all the nodes are numbered, the process ofnode numbering ends up.Following the steps of scheme I, the sequence of the node numbered for the 6-no

38、denetwork is given in tableI.No fill-iwnillbe introducedduringthe procedureofsolving the linear equation, so the table of factors and the Jacobian matrix will hacompletely identical structure. So the memory requirement for the table of factors i7 外文翻译（原文）0.256Kb, which is the same with thatfor the J

39、acobian matrix.Normally, anacceptablesolutioncan be obtainedin fouror fiveiterationbsy Newton-Raphsonmethod. While, the number of iterationrsequiredfor thisexample is thirty-threebecause of the ill-conditioned caused by the small impedance branch. 123 multiplyoperations will be performed during forw

40、ard substitution and backward substitutionforeach iterationa,nd 7456 multiplyoperationswillbe performed throughoutthewhole process of solving.Considering that complete pivoting is numerically preferable to partial pivoting,this section complete pivoting is adopted to improve accuracy of the solution

41、 of thelinearequationsa,iming at reducingthe number of iterationHse.re nodes relatetolarge determinant of the diagonal submatrices intend to be arrange in front. To someextern,the modulus of the entrieson the main diagonalof the admittancematrixcould indicatethe magnitude of the determinantof the su

42、bmatriceson the maindiagonal of the Jacobian matrix. For convenience, we make use of admittance matrixto determine the order of numbers.b) Factorizethe nodal admittancematrixwith complete pivoting.Record thechanges on the position of the nodes;c) Determine the new number of the node according to the

43、 positong of node inthe end of the factorization;TABLE II. REORDEREDNODESUSING SCHEMETWO8 外文翻译（原文）Executingscheme II,complete pivotingmight automaticperformedwithoutrowand column exchanges.The module of entrieson main diagonalcorrespondingtosuch node may become largerby summing more branchparameter,

44、as a resultt,henodes,degreeof which islargert,end to be numbered firstS.o theresultsof suchscheme may departform theprinciploef node numbering guided by sparsematrixmethods and many fill-ins might be introduced. The sequence of the node numberedfor 6-node network is list in table II. Six fill-ins wi

45、ll be produced, so more memor(0.488Kb) and more operations (321 multiply operations) are spent in the procedureof forward and backward substitutiodnuringonce iterationT.he totalnumber ofiterationrsequiredreducesto thirteenw,hich suggeststhatthe calculatioanccuracyfor linearequationscould be raisedby

46、 complete pivoting.Finallyt,he number ofmultiply operations reduces to 5573 thanks to smaller number of iterations.C. Puropse 3: Improving Accuracy while preserving the sparsityOnly one small impedance branch existsin the system,so only four entries(submatrices) corresponding to node 4 and node 6 ar

47、e very large in admittance matrix(Jacobin matrix). During the process of forward substitution, once node 4 or node 6eliminationt,he submatrixcomprised of restelementscould keep good numericalstability and numbering of rest nodes would not make a difference to the accuracy ofthe solution. To take bot

48、h accuracy and sparsity into account, we numbered node 4firstt,hen numbered othernodes followingthe method used forpurpose1. That iswhat we calledscheme IIIfor the 6-node network.The sequence of the nodenumbered for the 6-node network is given in table III.9 外文翻译（原文）Since only one small impedance br

49、anch exists in the system and it connects to node4, the degree of which is one. Scheme III will meet the request of purpose 1. So thenumber of fill-ins, memory requirements and operations needed for factorization areallthe same with scheme I. Only nine iterationwsill be needed to insuretheconvergenc

50、e, result in a large save of calculation (only 2107 multiply operations).reduction on the number of iterations indicates that more exact solutions for the liequations could be got using scheme III. After analysis and comparison, the reasonsare as follows:The diagonalelement relatedto node 4 is justa

51、 littlsemallerthan the onerelated to node 6, so eliminate node 4 first will not decrease accuracy. The schemecould meet complete pivoting approximately.Fewer operationsin scheme IIIreduce the rounding errorof calculatorfloating-point numbers. Especially, if eliminate node 6 first, very small value m

52、igbe added to diagonalelementof node 2 and node 5, which would cause seriousroundingerror.While, ifeliminatenode 4 firsta, sizablevaluewillbe added todiagonal element of node 6, producing a value in the normal range.TABLE III. REORDEREDTABLE IV .DIFFERENT SCHMEMS OFNODE ORDERING10 外文翻译（原文）V. CONCLUS

53、IONTheoreticalanalysisand the resultof numericalcalculatinsguggestthatit isnecessaryto considerthe influenceof node orderingon the accuracyof the powerflow calculation. If the node ordering algorithm takes both memory and accuracy intoaccount reasonably,the performanceof power flow calculatiocnould

54、be furtherimproved. Elementary conclusions of this paper are as follows:For the well-conditioninpgower system,the influenceof node orderingon theaccuracy of power flow calculation could be neglect. It is more important to focus oattentionon keeping the sparsityto save memory requirementand computeop

55、erations.For the ill-conditionipnogwer system,the accuracymust be consideredin nodeordering algorithm to speed up the convergence rate. On this basis, if the sparsityconsidered meanwhile, more accuracy might be obtained because of the reduction offloat point computation.VI. REFERENCES1 Allen J. Wood

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