潮流不同排序方案的比較外文翻譯

1、<p><b>　　中文4897字</b></p><p>　　A Comparison of Power Flow by Different Ordering Schemes</p><p>　　Wenbo Li, Xueshan Han, Bo Zhang</p><p>　　The School of Electric Engi

2、neering</p><p>　　Shandong University</p><p>　　Jinan, China</p><p>　　Email:liwenbo_1984@foxmail.com</p><p>　　Abstract—Node ordering algorithms, aiming at keeping sparsit

3、y as far as possible, are widely used today. In such algorithms, their influence on the accuracy of the solution is neglected because it won’t make significant difference in normal systems. While, along with the developm

4、ent of modern power systems, the problem will become more ill-conditioned and it is necessary to take the accuracy into count during node ordering. In this paper we intend to lay groundwork for the more rationality order

5、</p><p>　　Keywords—power flow calculation; node ordering; sparsity; accuracy; Newton-Raphson method ; linear equations</p><p>　　I. INTRODUCTION</p><p>　　Power flow is the most b

6、asic and important concept in power system analysis and power flow calculation is the basis of power system planning, operation, scheduling and control [1].Mathematically speaking, power flow problem is to find a numeric

7、al solution of nonlinear equations. Newton method is the most commonly used to solve the problem and it involves repeated direct solutions of a system of linear equations. The solving efficiency and precision of the line

8、ar equations directly influences the</p><p>　　Jacobian matrix in power flow calculation, similar with the admittance matrix, has symmetrical structure and a high degree of sparsity. During the factorization

9、procedure, nonzero entries can be generated in memory positions that correspond to zero entries in the starting Jacobian matrix. This action is referred to as fill-in. If the programming terms is used which processed and

10、 stores only nonzero terms, the reduction of fill-in reflects a great reduction of memory requirement and the number of</p><p>　　After sparse matrix methods, sparse vector methods [5], which extend sparsity

11、exploitation to vectors, are useful for solving linear equations when the right-hand-side vector is sparse or a small number of elements in the unknown vector are wanted. To make full use of sparse vector methods advanta

12、ge, it is necessary to enhance the sparsity of L-1by ordering nodes. This is equivalent to decreasing the length of the paths, but it might cause more fill-ins, greater complexity and expense. Counterin</p><p&

13、gt;　　Up to now, on the basis of the assumption that an arbitrary order of nodes does not adversely affect numerical accuracy, most node ordering algorithms take solving linear equations in a single iteration as research

14、subject, aiming at the reduction of memory requirements and computing operations. Many matrices with a strong diagonal in network problems fulfill the above assumption, and ordering to conserve sparsity increased the acc

15、uracy of the solution. Nevertheless, if there are junctions of ve</p><p>　　Based on the existing node ordering algorithm mentioned above, this paper focus attention on the contradiction between memory and ac

16、curacy during node ordering, research how could node ordering algorithm affect the performance of power flow calculation, expecting to lay groundwork for the more rationality ordering algorithm. This paper is arranged as

17、 follows. The contradiction between memory and accuracy in node ordering algorithm is introduced in section II. Next a simple DC power flow is showe</p><p>　　CONTRADICTION BETWEEN MEMORY AND ACCURACY<

18、/p><p>　　IN NODE ORDERING ALGORITHM</p><p>　　According to numerical mathematics, complete pivoting is numerically preferable to partial pivoting for systems of liner algebraic equations by Gaus

19、sian Elimination Method (GEM). Many mathematical papers [9-11] focus their attention on the discrimination between complete pivoting and partial pivoting in (GEM). Reference [9] shows how partial pivoting and complete pi

20、voting affect the sensitivity of the LU factorization. Reference [10] proposes an effective and inexpensive test to recognize numeri</p><p>　　The node reordering algorithms guided by sparse matrix technology

21、 have wildly used in power system calculation, aiming at minimizing memory requirement. In these algorithms, the nodes with fewer adjacent nodes tend to be numbered first. The result is that diagonal entries in node admi

22、ttance matrix tend to be arranged from least to largest according to their module. Analogously, every diagonal submatrices relate to a node tend to be arranged from least to largest according to their determinants. </

23、p><p>　　III. DIFFERENCE PRECISION OF THE SOLUTION USING PARTICAL PIVOTING AND COMPLETE PIVOTING</p><p>　　It is said that complete pivoting is numerically preferable to partial p

24、ivoting for solving systems of linear algebraic equations. When the system coefficients are varying widely, the accuracy of the solution would be affect by rounding errors hardly and it is necessary to take the influence

25、 of the ordering on the accuracy of the solution into consideration. </p><p>　　Fig.1 DC model of Sample 4-node network</p><p>　　As an example, consider the DC model of sample 4-node system show

26、n in Figure 1. Node 1 is the swing node having known voltage angle; nodes 2-4 are load nodes. Following the original node number, the DC power flow equation is:</p><p>　　To simulate computer numerical calcul

27、ation operations, four significant figures will be used to solve the problem. Executing GEM without pivoting on (1) yields the solution[ θ2,θ3,θ4]T=[-0.3036,-0.3239,-0.3249]T, whose components differ from that of the exa

28、ct solution [θ2, θ3,θ4]T=[-0.3,-0.32,-0.321]T. A more exact solution could be obtained by complete pivoting: [θ2,θ3, θ4]T=[-0.3007,-0.3207,-0.3217]T, and the order of the node after row and column interchanges is 3,2,4.

29、So this is a more reaso</p><p>　　IV. THE INFLUENCE OF NODE REODERING ON THE PERFORMANCE OF NEWTON-RAPHSON POWER FLOW METHOD</p><p>　　Fig.2 Sample 6-node network</p><p>

30、　　On the basis of the above-mentioned analysis, the scheme for node reordering will not only affect memory requirement but also the accuracy of the solution in solving linear simultaneous equations. So performance of New

31、ton-Raphson power flow method will be different with various node ordering. In this section three schemes of ordering for different purpose will be applied to a sample 6-node network shown in Fig 2 to compare the influen

32、ce of them on the accuracy of the solution, the convergence ra</p><p>　　Puropse 1 Saving Memory as far as possible </p><p>　　At present, there are various schemes widely used for node numbering

33、 in near-optimal order to reduce fill-ins and save memory. The only information needed by the schemes is a table describing the node-branch connection pattern of the networks. An order that would be optimal for the reduc

34、tion of the admittance matrix of the network is also optimal for the table of factors related Jacobian matrix. Different schemes reach different compromise between programming complexity and optimality. In this p</p&g

35、t;<p><b>　　Scheme I </b></p><p>　　a) Number the node degree of which is one. If more than one node meet this criterion, number the node with the smallest original number. If there are no

36、t sucn nodes any more, start with step b); </p><p>　　b) Number the node so that no equivalent branches will be introduced when this node is eliminated. If more than one node meets this criterion, number th

37、e one with the smallest original number. If we can not start with step a) or step b), turn to step c); </p><p>　　c) Number the node so that the fewest branches will be introduced when this node is eliminat

38、ed. If not only node could introduce fewest branches, number the one with the largest degree.</p><p>　　Once certain node is numbered in the step above, update the degree of relevant nodes and topological inf

39、ormation. Until all the nodes are numbered, the process of node numbering ends up. </p><p>　　TABLE I. REORDERED NODES USING SCHEME ONE</p><p>　　Following the steps of scheme I, the sequenc

40、e of the node numbered for the 6-node network is given in table I. No fill-in will be introduced during the procedure of solving the linear equation, so the table of factors and the Jacobian matrix will have completely i

41、dentical structure. So the memory requirement for the table of factors is 0.256Kb, which is the same with that for the Jacobian matrix. Normally, an acceptable solution can be obtained in four or five iterations by Newto

42、n-Raphson method</p><p>　　B. Puropse 2: Improving Accuracy Using Complete Pivoting </p><p>　　Considering that complete pivoting is numerically preferable to partial pivoting, in this section

43、complete pivoting is adopted to improve accuracy of the solution of the linear equations, aiming at reducing the number of iterations. Here nodes relate to large determinant of the diagonal submatrices intend to be arran

44、ge in front. To some extern, the modulus of the entries on the main diagonal of the admittance matrix could indicate the magnitude of the determinant of the submatrices on the main d</p><p>　　Scheme II </

45、p><p>　　a) Form the nodal admittance matrix; </p><p>　　b) Factorize the nodal admittance matrix with complete pivoting. Record the changes on the position of the nodes; </p><p>　　c

46、) Determine the new number of the node according to the positong of node in the end of the factorization; </p><p>　　TABLE II. REORDERED NODES USING SCHEME TWO</p><p>　　Executing scheme I

47、I, complete pivoting might automatic performed without row and column exchanges. The module of entries on main diagonal corresponding to such node may become larger by summing more branch parameter, as a result, the node

48、s, degree of which is larger, tend to be numbered first. So the results of such scheme may depart form the principle of node numbering guided by sparse matrix methods and many fill-ins might be introduced. The sequence o

49、f the node numbered for 6-node network is </p><p>　　C. Puropse 3: Improving Accuracy while preserving the sparsity </p><p>　　Only one small impedance branch exists in the system, so only four

50、entries (submatrices) corresponding to node 4 and node 6 are very large in admittance matrix (Jacobin matrix). During the process of forward substitution, once node 4 or node 6 is elimination, the submatrix comprised of

51、rest elements could keep good numerical stability and numbering of rest nodes would not make a difference to the accuracy of the solution. To take both accuracy and sparsity into account, we numbered node 4 first,</p&

52、gt;<p>　　Since only one small impedance branch exists in the system and it connects to node 4, the degree of which is one. Scheme III will meet the request of purpose 1. So the number of fill-ins, memory requireme

53、nts and operations needed for factorization are all the same with scheme I. Only nine iterations will be needed to insure the convergence, result in a large save of calculation (only 2107 multiply operations). The reduct

54、ion on the number of iterations indicates that more exact solutions for the </p><p>　　? The diagonal element related to node 4 is just a little smaller than the one related to node 6, so eliminate node 4 fi

55、rst will not decrease accuracy. The scheme could meet complete pivoting approximately. </p><p>　　? Fewer operations in scheme III reduce the rounding error of calculator floating-point numbers. Especially,

56、if eliminate node 6 first, very small value might be added to diagonal element of node 2 and node 5, which would cause serious rounding error. While, if eliminate node 4 first, a sizable value will be added to diagonal e

57、lement of node 6, producing a value in the normal range. </p><p>　　TABLE III. REORDERED NODES USING SCHEME THREE</p><p>　　TABLE IV. PERFORMACNE OF NEWTON POWER FLOW USING DIFFERENT

58、 SCHMEMS OF NODE ORDERING</p><p>　　V. CONCLUSION</p><p>　　Theoretical analysis and the result of numerical calculating suggest that it is necessary to consider the influence of node ordering

59、 on the accuracy of the power flow calculation. If the node ordering algorithm takes both memory and accuracy into account reasonably, the performance of power flow calculation could be further improved. Elementary concl

60、usions of this paper are as follows:</p><p>　　For the well-conditioning power system, the influence of node ordering on the accuracy of power flow calculation could be neglect. It is more important to focus

61、our attention on keeping the sparsity to save memory requirement and compute operations.</p><p>　　For the ill-conditioning power system, the accuracy must be considered in node ordering algorithm to speed up

62、 the convergence rate. On this basis, if the sparsity is considered meanwhile, more accuracy might be obtained because of the reduction of float point computation.</p><p>　　VI. REFERENCES</p><p&g

63、t;　　[1] Allen J. Wood and Bruce F. Wollenberg, “Power Generation, Operation and Cotrol (Second Edition),” Tsinghuo University Press, 2003. </p><p>　　[2] W. F. Tinney and J. W. Walker. “Direct solutions of

64、sparse network equations by optimally ordered triangular factorization,” Proceedings of the IEEE, vol. 55, No.11, pp. 1801-1809, November 1967. </p><p>　　[3] K. M. Sambarapu and S. M. Halpin, “Sparse matrix

65、 techniques in power systems,” Thirty-Ninth Southeastern Symposium on System Theory, March 2007. </p><p>　　[4] W. F. Tinney and C. E. Hart, “Power flow solution by Newton's Method,” IEEE Transactions on

66、 Power Apparatus and Systems, Vol. PAS-86, No. 11, pp. 1449-1460, November 1967. </p><p>　　[5] W. F. Tinney, V. Brandwajn, and S. M. Chan, “Sparse vector methods,” IEEE Transactions on Power Apparatus and S

67、ystems, Vol. PAS-104, No.2, pp. 295-301, February 1985. </p><p>　　[6] R. Betancourt, “An efficient heuristic ordering algorithm for partial matrix refactorization,” IEEE Transactions on Power Systems, Vol.

68、3, No. 3, pp. 1181-1187, August 1988. </p><p>　　[7] A. Gomez and L.G. Franquelo. “An efficient ordering algorithm to improve sparse vector methods,” IEEE Transactions on Power Systems, Vol. 3, No. 4, pp. 15

69、38-1544, November 1988. </p><p>　　[8] B. Stott, “Review of load-flow calculation methods,” Proceedings of the IEEE, Vol. 62, No. 7, pp. 916-929, July 1974. </p><p>　　[9] X. W. Chang and C. C.

70、Paige, “On the sensitivity of the LU factorization,” BIT, Vol. 38, No. 3, pp. 486-501, 1998. </p><p>　　[10] P.A. Businger, “Monitoring the numerical stability of Gaussian elimination,” Numer. Math, Vol. 16

71、, pp. 360-361, 1971. </p><p>　　[11] Paola Favati, Mauro Leoncini, and Angeles Martinez, “On the robustness of gaussian elimination with partial pivoting,” BIT, Vol. 40, No.1, pp.062-073, 2000</p>&l

72、t;p>　　VII. BIOGRAPHIES</p><p>　　Wenbo Li was born in Shandong Province in P. R. China, 1984. He received his B. S. from Electrical Engineering Institute of Shandong University, China, in 2007. He is curr

73、ently pursuing the Ph.D. degree at Shandong University. His main interest is in power system analysis and control.</p><p>　　Xueshan Han was born in Liaoning Province in P. R. China, 1959. He received B. S. a

74、nd M .S. degree from Electrical Engineering Department of Northeast Institute of electrical Power, Jilin In 1990 and PhD from Harbin Institute of Technology, Harbin in 1994. Now he is a Professor of the School of Electri

75、cal Engineering, Shandong University, China. His interests focus on power system analysis and control. </p><p>　　Bo Zhang was born in Shandong Province, China, 1963. Now he is a Professor of the School of El

76、ectrical Engineering, Shandong University, China. His interests focus on power system analysis and control.</p><p>　　潮流不同排序方案的比較</p><p>　　李文博，韓學山，張波</p><p>　　山東大學電氣工程學院 濟南，中國</p&

77、gt;<p>　　郵箱：liwenbo_1984@foxmail.com</p><p>　　摘 要：今天被廣泛應(yīng)用的節(jié)點排序算法，旨在盡可能地保證電力系統(tǒng)的稀疏性。在這些算法中，因為在正常的系統(tǒng)中算法對每種解決方案的精確度不會有顯著的差異，所以它的影響通常被忽略。然而隨著現(xiàn)代電力系統(tǒng)的發(fā)展，這個問題會變得更加嚴重，并且在節(jié)點排序過程中必須要把計數(shù)精度考慮在內(nèi)。在本文中，我們試圖為更多合理性排序算法

78、奠定了基礎(chǔ)，這樣可以使內(nèi)存和準確性之間進行合理的比較。本文列舉出了三種不同目的的排序方案，旨在比較潮流計算的形式，并且以一個六節(jié)點網(wǎng)絡(luò)為例進行具體討論。</p><p>　　關(guān)鍵詞：潮流計算，節(jié)點排序，稀疏性，精確度，牛頓—拉夫遜算法，線性方程組</p><p><b>　　1引言</b></p><p>　　潮流是在電力系統(tǒng)的分析中最基本和最

79、重要的概念，而潮流計算則是進行電力系統(tǒng)規(guī)劃，運行，調(diào)度和控制的基礎(chǔ)。從數(shù)學上來講，潮流問題是要找到一個非線性方程組的數(shù)值解。牛頓—拉夫遜算法是解決這個問題最常用的方法，它涉及到一系列線性方程組重復的直接求解。線性方程組求解的效率和精度直接影響了牛頓 - 拉夫遜潮流算法的性能。在潮流計算中，電力系統(tǒng)的數(shù)值和物理特性，學者們通過重新安排節(jié)點的數(shù)目，致力于研究以便改善線性方程組的計算效率，并獲得了很大的成功從而為電力系統(tǒng)的分析奠定了堅實的基礎(chǔ)

80、。</p><p>　　在潮流計算中的雅可比矩陣，類似于導納矩陣，有著對稱的結(jié)構(gòu)和高度的稀疏性。在分解過程中，內(nèi)存中的位置可以產(chǎn)生非零輸入，從而在原始的雅可比矩陣中產(chǎn)生零輸入。這一行動被稱為最小填充。如果用只能處理和存儲非零輸入的編程術(shù)語，最小填充的減少反映了內(nèi)存需求和執(zhí)行分解所需的操作數(shù)量的大大減小。所以廣泛的研究與最小填充的極小值有關(guān)。雖然很難找到為確定絕對的最佳排序的有效的算法，但是有著接近最好效果的一些有

81、效算法已經(jīng)得到了實際應(yīng)用。每種策略是在結(jié)果和執(zhí)行速度兩者之間的折中，并且它們都被大部分工業(yè)所采納。上面提到的稀疏性的編程排序消除，在電力系統(tǒng)網(wǎng)絡(luò)計算中這是一個突破，這使得牛頓法的計算速度和存儲需求顯著提高。</p><p>　　在稀疏矩陣的方法之后，稀疏向量擴展到向量的稀疏性探索的方法，當右手邊的向量是稀疏的或在未知向量元素少數(shù)想用于求解線性方程組時，這種方法對求解線性方程是有用的。為了充分利用稀疏向量方法的優(yōu)點

82、，通過節(jié)點排序加強L-1的稀疏性是十分必要的。這相當于減少路徑的長度，但它可能會導致更多的最小填充，更大的復雜性和費用。為了解決這個問題，提出了一些節(jié)點排序算法，這種算法試圖通過減少路徑的長度，同時保持矩陣的稀疏性來增強稀疏向量方法。</p><p>　　到目前為止，在任意一個節(jié)點的次序不會對數(shù)值精度產(chǎn)生負面影響的假設(shè)的基礎(chǔ)上，大多數(shù)節(jié)點排序算法通常會采取單一迭代解決線性方程組作為研究對象的方法，旨在減少內(nèi)存需求

83、和計算操作。許多在網(wǎng)絡(luò)問題中的強大對角線矩陣滿足上述假設(shè)，并且為了保證稀疏性的排序方法增加了解決方案的準確性。然而，如果在潮流系統(tǒng)模型中存在一系列非常高或低的阻抗，長的超高壓線路，串聯(lián)和并聯(lián)補償?shù)葐栴}，對角占優(yōu)將被削弱和假設(shè)可能并不總是站不住腳的。此外，隨著現(xiàn)代電力系統(tǒng)的發(fā)展，不同數(shù)量級參數(shù)下的新模型出現(xiàn)在潮流模型中。分布式發(fā)電的推廣也使我們堅定地把分布網(wǎng)絡(luò)和傳輸系統(tǒng)融入到整個電力系統(tǒng)潮流計算中，當然它會造成更嚴重的數(shù)值問題。上面提到的

84、所有這些事情會使問題變得更加糟糕。因此，有必要討論節(jié)點編號對計算精度的影響。</p><p>　　基于上述提出的節(jié)點排序算法，本文重點關(guān)注這種節(jié)點排序在內(nèi)存和準確性之間的矛盾，研究節(jié)點排序算法如何能影響的電力系統(tǒng)潮流計算的性能，從而為更理性的排序算法奠定基礎(chǔ)。本文安排如下：在第二部分介紹了節(jié)點排序算法的內(nèi)存和準確性之間的矛盾。接下來的第三部分通過一個簡單的直流潮流來說明節(jié)點的順序可能會影響算法的精度。然后在第四部

85、分以6個節(jié)點的網(wǎng)絡(luò)作為一個例子，對于節(jié)點排序?qū)Τ绷餍阅艿挠绊戇M行了詳細分析。在第六部分給出了結(jié)論。</p><p>　　2 節(jié)點排序算法中內(nèi)存和精確度之間的矛盾</p><p>　　根據(jù)計算數(shù)學，對于用高斯消元法求解的系統(tǒng)的線性代數(shù)方程組，完全消元法在數(shù)值上比部分消元法更可取。許多數(shù)學論文[9-11]都會關(guān)注高斯消元法的完全消元法與部分消元法的區(qū)別。參考文獻[ 9 ]表明部分消元法和完全消

86、元法是如何影響LU分解的靈敏度。參考文獻[ 10 ]提出了一種有效而廉價的測試，從而找到在部分消元法在使用時的數(shù)學難題。一旦不能滿足評估標準，就會采用完全消元法，以獲得更好的數(shù)值穩(wěn)定性。在潮流計算中，部分消元法可以再沒有任何行交匯的情況下自動實現(xiàn)，因為在大多數(shù)情況下，雅可比矩陣的對角占優(yōu)的功能可以保證在浮點運算的數(shù)值穩(wěn)定性的。雖然由于舍入誤差，部分消元法在有些極限點附近不能提供準確的解決方法。如果采用完全消元法，上面執(zhí)行過程中的每一步，

87、關(guān)鍵因素通常選擇最大的模塊元素。這相當于調(diào)整潮流計算的節(jié)點排序。因此，與最大的模塊元素有關(guān)的節(jié)點往往安排在前面以達到提高精度的目的。</p><p>　　以稀疏矩陣技術(shù)為導向的節(jié)點重新排序算法已廣泛應(yīng)用于電力系統(tǒng)計算中，旨在最大限度地減少內(nèi)存需求。在這些算法中，有著較少相鄰節(jié)點的節(jié)點往往首先被編號。其結(jié)果是在節(jié)點導納矩陣的對角線項往往根據(jù)自己的模塊被安排從最小到最大排列。類似地，每一個涉及到一個節(jié)點的對角線子矩陣

88、，往往根據(jù)他們的行列式按照從最小到最大的順序進行排列。這樣從這些算法形式中的獲得的結(jié)果只會偏離形成的原則，但是后續(xù)的解決方案的精度將提高。這是我們所說的按照內(nèi)存原則進行節(jié)點排序和精確度之間是有矛盾的。</p><p>　　3 使用部分消元法和完全消元法所產(chǎn)生的精確度差異</p><p>　　對于解決系統(tǒng)的線性代數(shù)方程組，完全消元法在數(shù)值上比部分消元法更可取。當系統(tǒng)系數(shù)變廣時，解的精度幾乎不

89、可能受舍入誤差的影響，因此把排序?qū)τ诮鉀Q方案的準確性的順序考慮在內(nèi)是必要的。</p><p>　　圖1有著四個節(jié)點的網(wǎng)絡(luò)樣本的直流模型</p><p>　　以圖1所示的有著四個節(jié)點的網(wǎng)絡(luò)樣本的直流模型為例。節(jié)點1是已知電壓相角擺動節(jié)點;節(jié)點2-4負荷節(jié)點。按照原來的節(jié)點數(shù)量，直流潮流方程是：</p><p>　　為了模擬計算機數(shù)值計算操作，我們用四個有效數(shù)字來解決這

90、個問題。沒有消元地對公式（1）執(zhí)行高斯消元得到的解為[ θ2,θ3,θ4]T=[-0.3036,-0.3239,-0.3249]T，其與精確解[θ2, θ3,θ4]T=[-0.3,-0.32,-0.321]的部分元素不同，通過完全消元法可以得到一個更加精確的解：[θ2,θ3, θ4]T=[-0.3007,-0.3207,-0.3217]，并且行和列的交匯處的節(jié)點的排序是3,2,4 。所以這是一個為了獲得更高精確度的一個更加合理的方案。&

91、lt;/p><p>　　4 節(jié)點排序?qū)εｎD-拉夫遜潮流計算方法的表現(xiàn)形式的影響</p><p>　　圖2有著六個節(jié)點的網(wǎng)絡(luò)樣本的直流模型</p><p>　　在上述分析的基礎(chǔ)上，對節(jié)點重新排序的方案將不僅影響到內(nèi)存的要求，而且影響到求解線性方程組時解的精度。因此，牛頓 – 拉夫遜潮流方法的性能將隨著節(jié)點排序的變化而不同。在本節(jié)中將把三種不同的排序方案應(yīng)用到如圖2所示的

92、6個節(jié)點的網(wǎng)絡(luò)，以便對它們對潮流計算中解的精度、收斂速度、計算量和內(nèi)存需求量進行比較。表四所示的是性能的細節(jié)。</p><p>　　A 目的一：盡可能地節(jié)省內(nèi)存</p><p>　　目前，以減少最小優(yōu)化和節(jié)省內(nèi)存節(jié)點，有各種各樣的方案應(yīng)用于近優(yōu)化的節(jié)點排序。這種方案所需要的唯一信息是描述網(wǎng)絡(luò)節(jié)點分支連接模式的一個表。對減少網(wǎng)絡(luò)的導納矩陣有著最佳效果的排序也是相關(guān)的雅可比矩陣表的最優(yōu)的因素。

93、在編程的復雜性和最優(yōu)性之間不同的方案可以達成不同的妥協(xié)。在本文中，我們關(guān)注的編號的結(jié)果是如何影響計算性能。編程效率是超出了目前的工作范圍。為了節(jié)省內(nèi)存，在這一部分中，提出了與[2]中提出的第三種方案類似一個動態(tài)節(jié)點排序方案。該算法的執(zhí)行步驟如下。</p><p><b>　　方案一</b></p><p>　　a 定義其中一個節(jié)點度為一。如果一個以上的節(jié)點符合這個標

94、準，選擇最原始的節(jié)點。如果沒有任何節(jié)點符合要求，啟動步驟b ；</p><p>　　b 當這個節(jié)點被淘汰，編號那些沒有等效的分支節(jié)點可以被引入的節(jié)點。如果一個以上的節(jié)點符合這個標準，選擇最原始的節(jié)點。如果我們不能啟動步驟a和步驟b，打開步驟c ；</p><p>　　c 當這個節(jié)點被淘汰，編號那些有最少分支的節(jié)點。如果不止一個節(jié)點可以引入最少的分支節(jié)點，給那個最大節(jié)點度的節(jié)點編號。<

95、;/p><p>　　一旦在上述步驟中某個節(jié)點被編號，更新相關(guān)節(jié)點度和拓撲信息。直到所有的節(jié)點都編上號，節(jié)點編號就完成了。</p><p>　　表1 用方案一給節(jié)點再排序</p><p>　　緊跟著方案一之后，6節(jié)點網(wǎng)絡(luò)的節(jié)點編號次序如表1所示。在求解線性方程組的過程中，沒有引進最小填充，所以表格的因素和雅可比矩陣將有完全一致的結(jié)構(gòu)。所以表格的因素的內(nèi)存需求是0.256K

96、b的，這個與該雅可比矩陣相同。通常情況下，通過四五次牛頓-拉夫遜迭代方法就可以得到解。可是這個例子所需的迭代次數(shù)是三十三次，因為小阻抗分支所造成的病態(tài)性。在每次迭代期間前后替代的過程中需要123次乘法運算，整個解答過程需要7456次乘法運算。</p><p>　　B 目的二：用完全迭代法改善精確度</p><p>　　考慮到完全消元法在數(shù)值上比部分消元法更可取，在本節(jié)中，為了提高解決線性

97、方程組的準確性而采用完全消元法，旨在減少迭代次數(shù)。這里涉及到大量的對角線子矩陣行列式的節(jié)點以便安排在前面。在某種程度上，導納矩陣的主對角線上的入口模數(shù)可以表明雅可比矩陣的主對角線上的子矩陣的行列式的幅度。為方便起見，我們利用導納矩陣確定數(shù)字的順序。</p><p><b>　　方案二</b></p><p>　　a 形成節(jié)點導納矩陣；</p><

98、p>　　b 用完全消元法因式分解節(jié)點導納矩陣。記錄節(jié)點的當前位置上的變化；</p><p>　　c 根據(jù)因式分解的節(jié)點的最終位置確定節(jié)點的新編號；</p><p>　　表1 用方案二給節(jié)點再排序</p><p>　　執(zhí)行方案二，完整消元法可以再沒有行和列交匯的情況下自動進行。對應(yīng)這些節(jié)點的主對角線的入口模數(shù)通過總結(jié)更多的分支參數(shù)而變得更大，因此，節(jié)點度越大

99、的往往首先被編號。因此，該方案的的結(jié)果可能與稀疏矩陣方法和許多引入的最小填充下形成的節(jié)點編號的原則相異。6節(jié)點網(wǎng)絡(luò)的節(jié)點編號次序如表2所示。將產(chǎn)生6個最小填充，所以在一次迭代中前后替代過程中將花費更多的內(nèi)存（ 0.488Kb ）和更多的操作（ 321個乘法運算） ，所需的迭代總數(shù)減少到十三次，這表明線性方程組的計算精度通過完全消元法得以提高。最后，由于迭代次數(shù)的減少乘法運算的次數(shù)減少到5573次。</p><p>

100、;　　C 目的三：保持稀疏性的同時提高精確度</p><p>　　在系統(tǒng)中只存在一個小的阻抗分支，所以相應(yīng)于節(jié)點4和節(jié)點6的只有四個條目（子矩陣）是非常大的導納矩陣（雅可比矩陣）。在提出替代的過程中，一旦節(jié)點4和節(jié)點6被消除，其余元素組成的子矩陣能保持良好的數(shù)值穩(wěn)定性，并且其余節(jié)點的編號不會對解決方案的精確度產(chǎn)生影響。把準確性和稀疏性都考慮在內(nèi)，我們把4節(jié)點編為1號，然后按照目的一的方法給其他節(jié)點編號。這就是我