潮流不同排序方案的比较文献翻译中英文对照

A Comparison of Power Flow by Different Ordering Schemes

Abstract—Node ordering algorithms, aiming at keeping sparsity as far as possible, are widely used today. In such algorithms, their influence on the accuracy of the solution is neglected because i t won’t make significant difference in normal systems. While, along with the development 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 ordering algorithm which could make reasonable compromising between memory and accuracy. Three schemes of node ordering for different purpose are proposed to compare the performance of the power flow calculation and an example of simple six-node network is discussed detailed.

Keywords—power flow calculation; node ordering; sparsity; accuracy; Newton-Raphson method ; linear equations

I. INTRODUCTION

Power flow is the most basic 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 numerical 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 linear equations directly influences the performance of Newton-Raphson power flow algorithm. Based on numerical mathematics and physical characteristics of power system in power flow calculation, scholars dedicated to the research to improve the computational efficiency of linear equations by reordering nodes’ number and received a lot of success which laid a solid foundation for power system analysis.

Jacobian matrix in power flow calculation, similar with the admittance matrix, has

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symmetrical structure and a high degree of sparsity. During the factorization 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 stores only nonzero terms, the reduction of fill-in reflects a great reduction of memory requirement and the number of operations needed to perform the factorization. So many extensive studies have been concerned with the minimization of the fill-ins. While it is hard to find efficient algorithm for determining the absolute optimal order, several effective strategies for determining near-optimal orders have been devised for actual applications [2, 3]. Each of the strategies is a trade-off between results and speed of execution and they have been adopted by much of industry. The sparsity-programmed ordered elimination mentioned above, which is a breakthrough in power system network computation, dramatically improving the computing speed and storage requirements of Newton’s method [4].

After sparse matrix methods, sparse vector methods [5], which extend sparsity 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 advantage, 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. Countering this problem, several node ordering algorithms [6, 7] were proposed to enhance sparse vector methods by minimizing the length of the paths while preserving the sparsity of the matrix.

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 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 accuracy of the solution. Nevertheless, if there are junctions of

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very high and low series impedances, long EHV lines, series and shunt compensation in the model of power flow problem, diagonal dominance will be weaken [8] and the assumption may not be tenable invariably. Furthermore, along with the development of modern power systems, various new models with parameters under various orders of magnitude appear in the model of power flow. The promotion of distributed generation also encourage us to regard the distribution networks and transmission systems as a whole in power flow calculation, and it will cause more serious numerical problem. All those things mentioned above will turn the problem into ill-condition. So it is necessary to discuss the effect of the node numbering to the accuracy of the solution.

Based on the existing node ordering algorithm mentioned above, this paper focus attention on the contradiction between memory and accuracy 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 follows. The contradiction between memory and accuracy in node ordering algorithm is introduced in section II. Next a simple DC power flow is showed to illustrate that node ordering could affect the accuracy of the solution in section III. Then, taking a 6-node network as an example, the effect of node ordering on the performance of power flow is analyzed detailed in section IV. Conclusion is given in section VI.

II.CONTRADICTION BETWEEN MEMORY AND ACCURACY

IN NODE ORDERING ALGORITHM

According to numerical mathematics, complete pivoting is numerically preferable to partial pivoting for systems of liner algebraic equations by Gaussian 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 pivoting affect the sensitivity of the LU factorization. Reference [10] proposes an effective and inexpensive test to recognize

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numerical difficulties during partial pivoting requires. Once the assessment criterion can not be met, complete pivoting will be adopted to get better numerical stability. In power flow calculations, partial pivoting is realized automatically without any row-interchanges and column-interchanges because of the diagonally dominant features of the Jacobin matrix, which could guarantee numerical stability in floating point computation in most cases. While due to rounding errors, the partial pivoting does not provide the solution accurate enough in some ill-conditionings. If complete pivoting is performed, at each step of the process, the element of largest module is chosen as the pivotal element. It is equivalent to adjust the node ordering in power flow calculation. So the node relate to the element of largest module is tend to arrange in front for the purpose of improving accuracy.

The node reordering algorithms guided by sparse matrix technology 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 admittance 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. So the results obtained form such algorithms will just deviate form the principle follow which the accuracy of the solution will be enhance. That is what we say there is contradiction between node ordering guided by memory and accuracy.

III. DIFFERENCE PRECISION OF THE SOLUTION USING PARTICAL PIVOTING AND COMPLETE PIVOTING

It is said that complete pivoting is numerically preferable to partial pivoting 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 of the ordering on the accuracy of the solution into consideration.

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Fig.1 DC model of Sample 4-node network

As an example, consider the DC model of sample 4-node system shown 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:

To simulate computer numerical calculation 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 exact 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. So this is a more reasonable ordering scheme for the purpose of getting more high accuracy.

IV. THE INFLUENCE OF NODE REODERING ON THE PERFORMANCE OF NEWTON-RAPHSON POWER FLOW METHOD

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Fig.2 Sample 6-node network

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 Newton-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 influence of them on the accuracy of the solution, the convergence rate, the calculated amount and the memory needed in power flow computation. The detail of the performance is shown in table IV.

A.Puropse 1 Saving Memory as far as possible

At present, there are various schemes widely used for node numbering 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 reduction 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 paper, what we concern about is how the result of the numbering affects the computational performance. The programming efficiency is beyond the scope of the present work. To save memory, a dynamic node ordering scheme similar

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to the third scheme presented in [2] is adopted in this section. Execution steps of the algorithm are as follows.

Scheme I

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 not sucn nodes any more, start with step b);

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 the one with the smallest original number. If we can not start with step a) or step b), turn to step c);

c) Number the node so that the fewest branches will be introduced when this node is eliminated. If not only node could introduce fewest branches, number the one with the largest degree.

Once certain node is numbered in the step above, update the degree of relevant nodes and topological information. Until all the nodes are numbered, the process of node numbering ends up.

TABLE I. REORDERED NODES USING SCHEME ONE

Following the steps of scheme I, the sequence 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 identical structure. So the memory requirement for the table of factors is

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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 Newton-Raphson method. While, the number of iterations required for this example is thirty-three because of the ill-conditioned caused by the small impedance branch. 123 multiply operations will be performed during forward substitution and backward substitution for each iteration, and 7456 multiply operations will be performed throughout the whole process of solving.

B. Puropse 2: Improving Accuracy Using Complete Pivoting

Considering that complete pivoting is numerically preferable to partial pivoting, in this section 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 arrange 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 diagonal of the Jacobian matrix. For convenience, we make use of admittance matrix to determine the order of numbers.

Scheme II

a) Form the nodal admittance matrix;

b) Factorize the nodal admittance matrix with complete pivoting. Record the changes on the position of the nodes;

c) Determine the new number of the node according to the positong of node in the end of the factorization;

TABLE II. REORDERED NODES USING SCHEME TWO

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Executing scheme II, 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 nodes, 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 of the node numbered for 6-node network is list in table II. Six fill-ins will be produced, so more memory (0.488Kb) and more operations (321 multiply operations) are spent in the procedure of forward and backward substitution during once iteration. The total number of iterations required reduces to thirteen, which suggests that the calculation accuracy for linear equations could be raised by complete pivoting. Finally, the number of multiply operations reduces to 5573 thanks to smaller number of iterations.

C. Puropse 3: Improving Accuracy while preserving the sparsity

Only one small impedance branch exists in the system, so only four 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 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, then numbered other nodes following the method used for purpose 1. That is what we called scheme III for the 6-node network. The sequence of the node numbered for the 6-node network is given in table III.

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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 requirements 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 reduction on the number of iterations indicates that more exact solutions for the linear equations could be got using scheme III. After analysis and comparison, the reasons are as follows:

? The diagonal element related to node 4 is just a little smaller than the one related to node 6, so eliminate node 4 first will not decrease accuracy. The scheme could meet complete pivoting approximately.

? Fewer operations in scheme III reduce the rounding error of calculator floating-point numbers. Especially, 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 element of node 6, producing a value in the normal range.

TABLE III. REORDERED NODES USING SCHEME THREE

TABLE IV. PERFORMACNE OF NEWTON POWER FLOW USING DIFFERENT SCHMEMS OF NODE ORDERING

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V. CONCLUSION

Theoretical analysis and the result of numerical calculating suggest that it is necessary to consider the influence of node ordering 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 conclusions of this paper are as follows:

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 our attention on keeping the sparsity to save memory requirement and compute operations.

For the ill-conditioning power system, the accuracy must be considered in node ordering algorithm to speed up 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.

VI. REFERENCES

[1] Allen J. Wood and Bruce F. Wollenberg, “Power Generation, Operation and Cotrol (Second Edition),” Tsinghuo University Press, 2003.

[2] W. F. Tinney and J. W. Walker. “Direct solutions of sparse network equations by optimally ordered triangular facto rization,” Proceedings of the IEEE, vol. 55, No.11, pp. 1801-1809, November 1967.

[3] K. M. Sambarapu and S. M. Halpin, “Sparse matrix techniques in power

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systems,” Thirty-Ninth Southeastern Symposium on System Theory, March 2007. [4] W. F. Tinney and C. E. Hart, “Power flow solution by Newton's Method,” IEEE Transactions on Power Apparatus and Systems, V ol. PAS-86, No. 11, pp. 1449-1460, November 1967.

[5] W. F. Tinney, V. Brandwajn, and S. M. Chan, “Sparse vector methods,” IEEE Transactions on Power Apparatus and Systems, V ol. PAS-104, No.2, pp. 295-301, February 1985.

[6] R. Betancourt, “An efficient heuristic ordering algorithm for partial matrix refactorization,” IEEE Transactions on Power Systems, V ol. 3, No. 3, pp. 1181-1187, August 1988.

[7] A. Gomez and L.G. Franquelo. “An efficient ordering algorithm to improve sparse vector methods,” IEEE Transactions on Power Systems, V ol. 3, No. 4, pp. 1538-1544, November 1988.

[8] B. Stott, “Review of load-flow calculation methods,” Proceedings of th e IEEE, V ol. 62, No. 7, pp. 916-929, July 1974.

[9] X. W. Chang and C. C. Paige, “On the sensitivity of the LU factorization,” BIT, V ol. 38, No. 3, pp. 486-501, 1998.

[10] P.A. Businger, “Monitoring the numerical stability of Gaussian elimination,” Numer. Math, V ol. 16, pp. 360-361, 1971.

[11] Paola Favati, Mauro Leoncini, and Angeles Martinez, “On the robustness of gaussian elimination with partial pivoting,” BIT, V ol. 40, No.1, pp.062-073, 2000

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潮流不同排序方案的比较

摘要：今天被广泛应用的节点排序算法，旨在尽可能地保证电力系统的稀疏性。在这些算法中，因为在正常的系统中算法对每种解决方案的精确度不会有显著的差异，所以它的影响通常被忽略。然而随着现代电力系统的发展，这个问题会变得更加严重，并且在节点排序过程中必须要把计数精度考虑在内。在本文中，我们试图为更多合理性排序算法奠定了基础，这样可以使内存和准确性之间进行合理的比较。本文列举出了三种不同目的的排序方案，旨在比较潮流计算的形式，并且以一个六节点网络为例进行具体讨论。

关键词：潮流计算，节点排序，稀疏性，精确度，牛顿—拉夫逊算法，线性方程组

1引言

潮流是在电力系统的分析中最基本和最重要的概念，而潮流计算则是进行电力系统规划，运行，调度和控制的基础。从数学上来讲，潮流问题是要找到一个非线性方程组的数值解。牛顿—拉夫逊算法是解决这个问题最常用的方法，它涉及到一系列线性方程组重复的直接求解。线性方程组求解的效率和精度直接影响了牛顿- 拉夫逊潮流算法的性能。在潮流计算中，电力系统的数值和物理特性，学者们通过重新安排节点的数目，致力于研究以便改善线性方程组的计算效率，并获得了很大的成功从而为电力系统的分析奠定了坚实的基础。

在潮流计算中的雅可比矩阵，类似于导纳矩阵，有着对称的结构和高度的稀疏性。在分解过程中，内存中的位置可以产生非零输入，从而在原始的雅可比矩阵中产生零输入。这一行动被称为最小填充。如果用只能处理和存储非零输入的编程术语，最小填充的减少反映了内存需求和执行分解所需的操作数量的大大减小。所以广泛的研究与最小填充的极小值有关。虽然很难找到为确定绝对的最佳排序的有效的算法，但是有着接近最好效果的一些有效算法已经得到了实际应用。每种策略是在结果和执行速度两者之间的折中，并且它们都被大部分工业所采纳。上面提到的稀疏性的编程排序消除，在电力系统网络计算中这是一个突破，这使得牛顿法的计算速度和存储需求显著提高。

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在稀疏矩阵的方法之后，稀疏向量扩展到向量的稀疏性探索的方法，当右手边的向量是稀疏的或在未知向量元素少数想用于求解线性方程组时，这种方法对求解线性方程是有用的。为了充分利用稀疏向量方法的优点，通过节点排序加强L-1的稀疏性是十分必要的。这相当于减少路径的长度，但它可能会导致更多的最小填充，更大的复杂性和费用。为了解决这个问题，提出了一些节点排序算法，这种算法试图通过减少路径的长度，同时保持矩阵的稀疏性来增强稀疏向量方法。

到目前为止，在任意一个节点的次序不会对数值精度产生负面影响的假设的基础上，大多数节点排序算法通常会采取单一迭代解决线性方程组作为研究对象的方法，旨在减少内存需求和计算操作。许多在网络问题中的强大对角线矩阵满足上述假设，并且为了保证稀疏性的排序方法增加了解决方案的准确性。然而，如果在潮流系统模型中存在一系列非常高或低的阻抗，长的超高压线路，串联和并联补偿等问题，对角占优将被削弱和假设可能并不总是站不住脚的。此外，随着现代电力系统的发展，不同数量级参数下的新模型出现在潮流模型中。分布式发电的推广也使我们坚定地把分布网络和传输系统融入到整个电力系统潮流计算中，当然它会造成更严重的数值问题。上面提到的所有这些事情会使问题变得更加糟糕。因此，有必要讨论节点编号对计算精度的影响。

基于上述提出的节点排序算法，本文重点关注这种节点排序在内存和准确性之间的矛盾，研究节点排序算法如何能影响的电力系统潮流计算的性能，从而为更理性的排序算法奠定基础。本文安排如下：在第二部分介绍了节点排序算法的内存和准确性之间的矛盾。接下来的第三部分通过一个简单的直流潮流来说明节点的顺序可能会影响算法的精度。然后在第四部分以6个节点的网络作为一个例子，对于节点排序对潮流性能的影响进行了详细分析。在第六部分给出了结论。

2 节点排序算法中内存和精确度之间的矛盾

根据计算数学，对于用高斯消元法求解的系统的线性代数方程组，完全消元法在数值上比部分消元法更可取。许多数学论文[9-11]都会关注高斯消元法的完全消元法与部分消元法的区别。参考文献[ 9 ]表明部分消元法和完全消元法

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是如何影响LU分解的灵敏度。参考文献[ 10 ]提出了一种有效而廉价的测试，从而找到在部分消元法在使用时的数学难题。一旦不能满足评估标准，就会采用完全消元法，以获得更好的数值稳定性。在潮流计算中，部分消元法可以再没有任何行交汇的情况下自动实现，因为在大多数情况下，雅可比矩阵的对角占优的功能可以保证在浮点运算的数值稳定性的。虽然由于舍入误差，部分消元法在有些极限点附近不能提供准确的解决方法。如果采用完全消元法，上面执行过程中的每一步，关键因素通常选择最大的模块元素。这相当于调整潮流计算的节点排序。因此，与最大的模块元素有关的节点往往安排在前面以达到提高精度的目的。

以稀疏矩阵技术为导向的节点重新排序算法已广泛应用于电力系统计算中，旨在最大限度地减少内存需求。在这些算法中，有着较少相邻节点的节点往往首先被编号。其结果是在节点导纳矩阵的对角线项往往根据自己的模块被安排从最小到最大排列。类似地，每一个涉及到一个节点的对角线子矩阵，往往根据他们的行列式按照从最小到最大的顺序进行排列。这样从这些算法形式中的获得的结果只会偏离形成的原则，但是后续的解决方案的精度将提高。这是我们所说的按照内存原则进行节点排序和精确度之间是有矛盾的。

3 使用部分消元法和完全消元法所产生的精确度差异

对于解决系统的线性代数方程组，完全消元法在数值上比部分消元法更可取。当系统系数变广时，解的精度几乎不可能受舍入误差的影响，因此把排序对于解决方案的准确性的顺序考虑在内是必要的。

图1有着四个节点的网络样本的直流模型

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以图1所示的有着四个节点的网络样本的直流模型为例。节点1是已知电压相角摆动节点;节点2-4负荷节点。按照原来的节点数量，直流潮流方程是：

为了模拟计算机数值计算操作，我们用四个有效数字来解决这个问题。没有消元地对公式（1）执行高斯消元得到的解为[ θ2,θ3,θ4]T=[-0.3036,-0.3239,-0.3249]T，其与精确解[θ2, θ3,θ4]T=[-0.3,-0.32,-0.321]的部分元素不同，通过完全消元法可以得到一个更加精确的解：[θ2,θ3, θ4]T=[-0.3007,-0.3207,-0.3217]，并且行和列的交汇处的节点的排序是3,2,4 。所以这是一个为了获得更高精确度的一个更加合理的方案。

4 节点排序对牛顿-拉夫逊潮流计算方法的表现形式的影响

图2有着六个节点的网络样本的直流模型

在上述分析的基础上，对节点重新排序的方案将不仅影响到内存的要求，

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17 而且影响到求解线性方程组时解的精度。因此，牛顿 – 拉夫逊潮流方法的性能将随着节点排序的变化而不同。在本节中将把三种不同的排序方案应用到如图2所示的6个节点的网络，以便对它们对潮流计算中解的精度、收敛速度、计算量和内存需求量进行比较。表四所示的是性能的细节。

A 目的一：尽可能地节省内存

目前，以减少最小优化和节省内存节点，有各种各样的方案应用于近优化的节点排序。这种方案所需要的唯一信息是描述网络节点分支连接模式的一个表。对减少网络的导纳矩阵有着最佳效果的排序也是相关的雅可比矩阵表的最优的因素。在编程的复杂性和最优性之间不同的方案可以达成不同的妥协。在本文中，我们关注的编号的结果是如何影响计算性能。编程效率是超出了目前的工作范围。为了节省内存，在这一部分中，提出了与[2]中提出的第三种方案类似一个动态节点排序方案。该算法的执行步骤如下。

方案一

a 定义其中一个节点度为一。如果一个以上的节点符合这个标准，

选择最原始的节点。如果没有任何节点符合要求，启动步骤b ；

b 当这个节点被淘汰，编号那些没有等效的分支节点可以被引入的节点。如果一个以上的节点符合这个标准，选择最原始的节点。如果我们不能启动步骤a 和步骤b ，打开步骤

c ；

c 当这个节点被淘汰，编号那些有最少分支的节点。如果不止一个节点可以引入最少的分支节点，给那个最大节点度的节点编号。

一旦在上述步骤中某个节点被编号，更新相关节点度和拓扑信息。直到所有的节点都编上号，节点编号就完成了。

表1 用方案一给节点再排序

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与节点4有关的对角线元素比节点6小一点，因此首先消除节点4不会降低精确度。该方案能够满足完全消元法。

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方案三的更少的操作次数减少计算器浮点数的舍入误差。尤其注意的是，如果首先消除节点6，非常小的值可能被添加到节点2和节点5的节点元素，这将导致严重的舍入误差。然而，如果首先消除节点4，一个相当大的值将被添加到节点6的对角线元素上，产生的新值在正常范围内。

表3用方案二给节点再排序

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