Iterative Methods and Applications

Journal of Applied Mathematics

Guest Editors: Giuseppe Marino, Filomena Cianciaruso, Luigi Muglia, Claudio H. Morales, and Daya Ram Sahu Iterative Methods and Applications JournalofAppliedMathematics

Iterative Methods and Applications

This is a special issue published in “Journal of Applied Mathematics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Saeid Abbasbandy, Iran Meng Fan, China Yongkun Li, China Mina B. Abd-El-Malek, Egypt Ya Ping Fang, China Wan-Tong Li, China Mohamed A. Abdou, Egypt A. M. Ferreira, Portugal J. Liang, China Subhas Abel, India Michel Fliess, France Ching-Jong Liao, Taiwan Mostafa Adimy, France M. A. Fontelos, Spain Chong Lin, China Carlos J. S. Alves, Portugal Huijun Gao, China Mingzhu Liu, China Mohamad Alwash, USA B. J. Geurts, The Netherlands Chein-Shan Liu, Taiwan Igor Andrianov, Germany Jamshid Ghaboussi, USA Kang Liu, USA Sabri Arik, Turkey Pablo Gonzalez-Vera,´ Spain Yansheng Liu, China Francis T.K. Au, Hong Kong Laurent Gosse, Italy Fawang Liu, Australia Olivier Bahn, Canada K. S. Govinder, South Africa Shutian Liu, China Roberto Barrio, Spain Jose L. Gracia, Spain Zhijun Liu, China Alfredo Bellen, Italy Yuantong Gu, Australia J. Lopez-G´ omez,´ Spain Jafar Biazar, Iran Zhihong GUAN, China Shiping Lu, China Hester Bijl, The Netherlands Nicola Guglielmi, Italy Gert Lube, Germany Anjan Biswas, Saudi Arabia F. G. Guimaraes,˜ Brazil Nazim I. Mahmudov, Turkey S. P.A. Bordas, USA Vijay Gupta, India O. D. Makinde, South Africa James Robert Buchanan, USA Bo Han, China F. J. Marcellan,´ Spain Alberto Cabada, Spain Maoan Han, China G. Mart´ın-Herran,´ Spain Xiao Chuan Cai, USA Pierre Hansen, Canada Nicola Mastronardi, Italy Jinde Cao, China Ferenc Hartung, Hungary M. McAleer, The Netherlands Alexandre Carvalho, Brazil Xiaoqiao He, Hong Kong Stephane Metens, France Song Cen, China Luis Javier Herrera, Spain Michael Meylan, Australia Q. S. Chang, China J. Hoenderkamp, The Netherlands Alain Miranville, France Tai-Ping Chang, Taiwan Ying Hu, France Ram N. Mohapatra, USA Shih-sen Chang, China Ning Hu, Japan JaimeE.M.Rivera,Brazil Rushan Chen, China Zhilong L. Huang, China Javier Murillo, Spain Xinfu Chen, USA Kazufumi Ito, USA Roberto Natalini, Italy Ke Chen, UK Takeshi Iwamoto, Japan Srinivasan Natesan, India Eric Cheng, Hong Kong George Jaiani, Georgia Jiri Nedoma, Czech Republic Francisco Chiclana, UK Zhongxiao Jia, China Jianlei Niu, Hong Kong J.-T. Chien, Taiwan Tarun Kant, India Roger Ohayon, France C. S. Chien, Taiwan Ido Kanter, Israel Javier Oliver, Spain Han H. Choi, Republic of Korea Abdul Hamid Kara, South Africa Donal O’Regan, Ireland Tin-Tai Chow, China Hamid Reza Karimi, Norway M. Ostoja-Starzewski, USA M. S. H. Chowdhury, Malaysia Jae-Wook Kim, UK Turgut Ozis¨ ¸, Turkey Carlos Conca, Chile Jong H. Kim, Republic of Korea Claudio Padra, Argentina Vitor Costa, Portugal Kazutake Komori, Japan R. M. Palhares, Brazil Livija Cveticanin, Serbia Fanrong Kong, USA Francesco Pellicano, Italy Eric de Sturler, USA Vadim . Krysko, Russia Juan M. Pena,˜ Spain Orazio Descalzi, Chile JinL.Kuang,Singapore Ricardo Perera, Spain Kai Diethelm, Germany Miroslaw Lachowicz, Poland Malgorzata Peszynska, USA Vit Dolejsi, Czech Republic Hak-Keung Lam, UK James F. Peters, Canada Bo-Qing Dong, China Tak-Wah Lam, Hong Kong M. A. Petersen, South Africa Magdy A. Ezzat, Egypt PGL Leach, Cyprus Miodrag Petkovic, Serbia Vu Ngoc Phat, Vietnam A.-M.A.Soliman,Egypt Qing-Wen Wang, China Andrew Pickering, Spain Xinyu Song, China Guangchen Wang, China Hector Pomares, Spain Qiankun Song, China Junjie Wei, China Maurizio Porfiri, USA Yuri N. Sotskov, Belarus Li Weili, China Mario Primicerio, Italy P. C . Spre ij , Th e Ne t h e r l a n d s Martin Weiser, Germany M. Rafei, The Netherlands Niclas Stromberg,¨ Sweden Frank Werner, Germany Roberto Reno,` Italy Ray KL Su, Hong Kong Shanhe Wu, China Jacek Rokicki, Poland Jitao Sun, China Dongmei Xiao, China Dirk Roose, Belgium Wenyu Sun, China Gongnan Xie, China Carla Roque, Portugal XianHua Tang, China Yuesheng Xu, USA Debasish Roy, India Alexander Timokha, Norway Suh-Yuh Yang, Taiwan Samir H. Saker, Egypt Mariano Torrisi, Italy Bo Yu, China Marcelo A. Savi, Brazil Jung-Fa Tsai, Taiwan Jinyun Yuan, Brazil Wolfgang Schmidt, Germany Ch Tsitouras, Greece Alejandro Zarzo, Spain Eckart Schnack, Germany K. Vajravelu, USA Guisheng Zhai, Japan Mehmet Sezer, Turkey Alvaro Valencia, Chile Jianming Zhan, China N. Shahzad, Saudi Arabia Erik Van Vleck, USA Zhihua Zhang, China Fatemeh Shakeri, Iran Ezio Venturino, Italy Jingxin Zhang, Australia Jian Hua Shen, China Jesus Vigo-Aguiar, Spain Shan Zhao, USA Hui-Shen Shen, China M. N. Vrahatis, Greece Chongbin Zhao, Australia F. Simoes,˜ Portugal Baolin Wang, China Renat Zhdanov, USA TheodoreE.Simos,Greece Mingxin Wang, China Hongping Zhu, China Contents

Iterative Methods and Applications, Giuseppe Marino, Filomena Cianciaruso, Luigi Muglia, Claudio H. Morales, and Daya Ram Sahu Volume2014,ArticleID827064,2pages

ConvergenceRegionofNewtonIterativePowerFlowMethod:NumericalStudies, Jiao-Jiao Deng and Hsiao-Dong Chiang Volume 2013, Article ID 509496, 12 pages

A Rank-Two Feasible Direction Algorithm for the Binary Quadratic Programming,XuewenMuand Yaling Zhang Volume 2013, Article ID 963563, 7 pages

Symmetric SOR Method for Absolute Complementarity Problems,JavedIqbalandMuhammadArif Volume 2013, Article ID 172060, 6 pages

Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems,L.Bergamaschiand A. Mart´ınez Volume 2013, Article ID 767042, 10 pages

General Iterative Methods for System of Equilibrium Problems and Constrained Convex Minimization Problem in Hilbert Spaces,PeichaoDuan Volume 2013, Article ID 957363, 11 pages

Log-Likelihood Ratio Calculation for Iterative Decoding on Rayleigh Fading Channels Using Pade´ Approximation, Gou Hosoya and Hiroyuki Yashima Volume 2013, Article ID 970126, 10 pages

Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings,M.DelaSen Volume 2013, Article ID 310106, 12 pages

A Novel Iterative Method for Solving Systems of Fractional Differential Equations, E. Hesameddini and A. Rahimi Volume 2013, Article ID 428090, 7 pages

Existence and Iterative Approximation Methods for Generalized Mixed Vector Equilibrium Problems with Relaxed Monotone Mappings,RabianWangkeereeandPanuYimmuang Volume 2013, Article ID 973408, 11 pages

An Iterative Method with Norm Convergence for a Class of Generalized Equilibrium Problems, Haixia Zhang and Fenghui Wang Volume 2013, Article ID 647524, 6 pages

Reconsiderations on the Equivalence of Convergence between Mann and Ishikawa Iterations for Asymptotically Pseudocontractive Mappings, Haizhen Sun and Zhiqun Xue Volume 2013, Article ID 274931, 5 pages Viscosity Approximation Methods and Strong Convergence Theorems for the Fixed Point of Pseudocontractive and Monotone Mappings in Banach Spaces,YanTang Volume 2013, Article ID 926078, 8 pages

Modified Preconditioned GAOR Methods for Systems of Linear Equations,Xue-FengZhang, Qun-Fa Cui, and Shi-Liang Wu Volume 2013, Article ID 850986, 7 pages

New Iterative Algorithm for Two Infinite Families of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces, Fang Zhang, Huan Zhang, and Yulong Zhang Volume 2013, Article ID 649537, 7 pages

Approximation Analysis for a Common Fixed Point of Finite Family of Mappings Which Are Asymptotically 𝑘-Strict Pseudocontractive in the Intermediate Sense,H.ZegeyeandN.Shahzad Volume 2013, Article ID 821737, 7 pages

Relaxed Viscosity Approximation Methods with Regularization for Constrained Minimization Problems, Lu-Chuan Ceng, Hong-Kun Xu, and Ching-Feng Wen Volume 2013, Article ID 531859, 19 pages

Developing a Series Solution Method of 𝑞-Difference Equations,Hsuan-KuLiu Volume 2013, Article ID 743973, 4 pages

The Split Common Fixed Point Problem for 󰜚-Strictly Pseudononspreading Mappings,ShuboCao Volume 2013, Article ID 241789, 9 pages

A Numerical Approach to Static Deflection Analysis of an Infinite Beam on a Nonlinear Elastic Foundation: One-Way Spring Model, Jinsoo Park, Hyeree Bai, and T. S. Jang Volume 2013, Article ID 136358, 10 pages

Viscosity Method for Hierarchical Fixed Point Problems with an Infinite Family of Nonexpansive Nonself-Mappings, Yaqin Wang Volume 2013, Article ID 574215, 8 pages

Parallel Rank of Two Sandpile Models of Signed Integer Partitions,G.Chiaselotti,T.Gentile,G.Marino, andP.A.Oliverio Volume 2013, Article ID 292143, 12 pages

On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces,ShinMinKang,ArifRafiq,andSunYoungCho Volume 2013, Article ID 284937, 5 pages

Comparison Theorems for Single and Double Splittings of Matrices, Cui-Xia Li, Qun-Fa Cui, and Shi-Liang Wu Volume 2013, Article ID 827826, 4 pages Contents

A Discrete Dynamical Model of Signed Partitions,G.Chiaselotti,G.Marino,P.A.Oliverio,andD.Petrassi Volume 2013, Article ID 973501, 10 pages

Strong Convergence for Hybrid 𝑆-Iteration Scheme, Shin Min Kang, Arif Rafiq, and Young Chel Kwun Volume 2013, Article ID 705814, 4 pages Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 827064, 2 pages http://dx.doi.org/10.1155/2014/827064

Editorial Iterative Methods and Applications

Giuseppe Marino,1 Filomena Cianciaruso,1 Luigi Muglia,1 Claudio H. Morales,2 and Daya Ram Sahu3

1 Dipartimento di Matematica, Universita´ della Calabria, 87036 Arcavacata di Rende, Italy 2 Department of Mathematics, University of Alabama in Huntsville, Huntsville, AL 35899, USA 3 Department of Mathematics, Banaras Hindu University, Varanasi 221005, India

Correspondence should be addressed to Giuseppe Marino; [email protected]

Received 10 December 2013; Accepted 10 December 2013; Published 9 March 2014

Copyright © 2014 Giuseppe Marino et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An important branch of nonlinear analysis theory, applied in We invited the authors to present their original articles that the study of nonlinear phenomena in engineering, physics, will stimulate the continuing efforts in developing new results and life sciences, is related to the existence of fixed points in the previous mentioned areas. of nonlinear mappings, the approximation of fixed points of F. Zhang et al. introduce a new iterative scheme for nonlinear operators, of zeros of nonlinear operators, and the finding a common fixed point of two countable families of approximation of solutions of variational inequalities. multivalued quasinonexpansive mappings and prove a weak Thisspecialissueisfocusedonthelatestachievementsin convergence theorem under the suitable control conditions these topics and the related applications. The aim is to present in a uniformly convex Banach space. the newest and extended coverage of the fundamental ideas, S. M. Kang et al. establish the strong convergence for the concepts, and important results of the topics below. Topics of hybrid S-iterative scheme associated with nonexpansive and interest include, but are not limited to the following. Lipschitz strongly pseudocontractive mappings in Banach spaces. (i) New iterative schemes to approximate fixed points of R. Wangkeeree and P. Yimmuang first consider an aux- nonlinear mappings, common fixed points of nonlin- iliary problem for the generalized mixed vector equilibrium ear mappings, or semigroups of nonlinear mappings. problem with a relaxed monotone mapping and prove the (ii) Iterative approximations of zeros of accretive-type existence and uniqueness of the solution for the auxiliary operators. problem; then they introduce a new iterative scheme for approximating a common element of the set of solutions of a (iii) Iterative approximations of solutions of variational generalized mixed vector equilibrium problem with a relaxed inequalities problems or split feasibility problems and monotone mapping and the set of common fixed points of a applications. countable family of nonexpansive mappings. (iv) Optimization problems and their algorithmic Y. Tang and D. Wen prove strong convergence theorems approaches. foracommonelementofthesetoffixedpointsofafinitefam- ily of pseudocontractive mappings and the set of solutions of (v) Methods for the global continuation of fixed point a finite family of monotone mappings. The common element curves in engineering problems. is the unique solution of a certain variational inequality. (vi) Fixed point of nonlinear operators in cone metric L.-C. Ceng et al. introduce a new relaxed viscosity spaces with applications and fixed points of nonlinear approximation method with regularization and prove the operators in ordered metric spaces with applications. strong convergence of the method to a common fixed 2 Journal of Applied Mathematics pointoffinitelymanynonexpansivemappingsandastrict M.DelaSeninvestigatesthelimitpropertiesofdistances pseudocontraction that also solves a convex minimization and the existence and uniqueness of fixed points, best prox- problem and a suitable equilibrium problem. imity points, and existence and uniqueness of limit cycles, to Recently,S.TakahashiandW.Takahashiproposedaniter- which the iterated sequences converge, of single-valued, and ative algorithm for finding common solutions of generalized so-called, contractive precyclic self-mappings. equilibrium problems governed by inverse strongly mono- X.-F. Zhang et al. propose three kinds of preconditioners tone mappings and of fixed points problems for nonexpansive to accelerate the generalized AOR (GAOR) method for the mappings. linear system from the generalized least squares problem. The H. Zhang and F. Wang provide a result that allows for the convergence and comparison results are also obtained. removal of one condition ensuring the strong convergence of L. Bergamaschi and A. Mart´ınez propose a parallel the algorithm. preconditioner for the Newton method in the computation of S. Cao introduces and analyzes the viscosity approxi- the leftmost eigenpairs of large and sparse symmetric positive mation algorithm for solving the split common fixed point definite matrices. problem for the strictly pseudononspreading mappings in J. Park et al. apply a numerical procedure introduced Hilbert spaces. by Choi and Jang in 2012 for the numerical analyzing of Y.Wang presents a viscosity method for hierarchical fixed static defection of an infinite beam on a nonlinear elastic point problems to solve variational inequalities, where the foundation. involved mappings are nonexpansive nonself-mappings. G. Chiaselotti et al. introduce the concept of fundamental H. Zegeye and N. Shahzad introduce an iterative process sequence for a finite graded poset X which is also a discrete which strongly converges to a common fixed point of finite dynamical model. family of uniformly continuous asymptotically strict pseudo- J. Iqbal and M. Arif study symmetric successive over- contractive mappings in the intermediate sense. relaxation (SSOR) method for absolute complementarity C.-X. Li et al. present some comparison theorems for the problems. spectral radius of double splittings of different matrices under suitable conditions, which are superior to the corresponding Acknowledgment results in the recent papers. P. Duan proposes an implicit iterative scheme and an The editors would like to thank the authors for their interest- explicit iterative scheme for finding a common element of ing contributions. the set of solutions of system of equilibrium problems and a constrained convex minimization problem by the general Giuseppe Marino iterative methods in Hilbert spaces. Filomena Cianciaruso H. Sun and Z. Xue demonstrate that the proof of Luigi Muglia main theorem of B. E. Rhoades and Stefan M. Soltuz in Claudio H. Morales J.Math.Anal.Appl.283(2003),681–688,isincorrect.The Daya Ram Sahu authors provide the correct version of this result concerning the equivalence between the convergences of Ishikawa and Mann iterations for uniformly L-Lipschitzian asymptotically pseudo-contractive maps. S. M. Kang et al. study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. G. Chiaselotti et al. use a discrete dynamical model with three evolution rules in order to analyze the structure of a partially ordered set of signed integer partitions whose main properties are actually not known. This model is related to the study of some extremal combinatorial sum problems. Applying the Taylor and multiplication rule of two generalized polynomials, H.-K. Liu develops a series solution of linear homogeneous q-difference equations. X. Mu and Y. Zhang, among the others, present a rank- two feasible direction algorithm based on the semide nite programming relaxation of the binary quadratic programming.Theproposedalgorithmrestrictstherankofmatrix variabletobetwointhesemidenite programming relaxation, and yields a quadratic objective function with simple quadratic constraints. Moreover a feasible direction algorithm is used to solve the nonlinear programming and the convergentanalysisandtimecomplexityofthemethodis given. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 509496, 12 pages http://dx.doi.org/10.1155/2013/509496

Research Article Convergence Region of Newton Iterative Power Flow Method: Numerical Studies

Jiao-Jiao Deng1 and Hsiao-Dong Chiang1,2

1 School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China 2 School of ECE, Cornell University, Ithaca, NY 14850, USA

Correspondence should be addressed to Jiao-Jiao Deng; [email protected]

Received 22 March 2013; Revised 10 October 2013; Accepted 14 October 2013

Academic Editor: Luigi Muglia

Copyright © 2013 J.-J. Deng and H.-D. Chiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Power flow study plays a fundamental role in the process of power system operation and planning. Of the several methodsin commercial power flow package, the Newton-Raphson (NR) method is the most popular one. In this paper, we numerically study the convergence region of each power flow solution under the NR method. This study of convergence region provides insights of the complexity of the NR method in finding power flow solutions. Our numerical studies confirm that the convergence region of NR method has a fractal boundary and find that this fractal boundary of convergence regions persists under different loading conditions. In addition, the convergence regions of NR method for power flow equations with different nonlinear load models are also fractal. This fractal property highlights the importance of choosing initial guesses since a small variation of an initial guess near the convergence boundary leads to two different power flow solutions. One vital variation of Newton method popular in power industry is the fast decoupled power flow method whose convergence region is also numerically studied on an IEEE 14-bus test system which is of 22-dimensional in state space.

1. Introduction (iv) the specified real power generation at each (generator) bus except one generator bus. Since power flow calculation started in 1956 [1–3], a variety of numerical methods have been developed for power flow Power flow study is performed extensively both for sys- study, such as Gauss method [1], Newton method [4], and tem planning purposes to analyze alternative plans of future fast decoupled method [5]. Over the past 20 years, Newton systems and for system operation purposes to evaluate differ- method and its variations are widespread, utilized, and ent operating conditions of existing systems. Indeed, power enhanced [6–11]. Power flow study (or load flow study) is the flow study is used in transmission planning to check for determination of steady-state conditions of a power system branch overloads, bus voltage problems. In static contingency for a specified power generation and load demand. The power analysis, power flow study is used to assess the effect of branch flow problem is the computation of voltage magnitude and and/or generator outages. In transfer capability analysis, phaseangleateachbusinapowersystemunderthefollowing (repetitive) power flow study is performed to calculate the conditions: power transfer limits. In the literature, there was little work on the study of (i) balanced three-phase steady-state conditions and the convergence regions of power flow solutions. The existing system is in sinusoidal steady-state; relevant papers that the authors were able to identify were (ii) the transmission network is composed of constant, mainly focused on the following aspects: power flow fractals linear, and lumped-parameter branches; and truncated fractals on a 3-bus system [12–17], different (iii) the specified real and reactive power demand at each convergence regions under polar and rectangular expresses (load) bus; Newton method [18]. This study of convergence regions 2 Journal of Applied Mathematics provides insights to explain the reasons behind power flow where divergence and may lead to improving the robustness of 𝑥 𝑘 Newton power flow via some parameters adjustment. 𝑘:statevariablesofthe th iteration; In this paper, we numerically study the convergence 𝑥𝑘+1:statevariablesofthe(𝑘+1)th iteration; region of each power flow solution under the Newton- Δ𝑥𝑘:errorbetween𝑥𝑘+1 and 𝑥𝑘; Raphson (NR) method. One important variation of Newton method popular in power industry is the fast decoupled 𝐽(𝑥𝑘): Jacobian matrix of 𝑥𝑘. power flow method. We also numerically study the conver- 𝑓(𝑥) gence region of fast decoupled power flow method. If the components of are differentiable, we define the The key results presented in this paper are summarized as Jacobian matrix in (2)as 󵄨 follows. 𝜕𝑓 󵄨 𝐽(𝑥 )=𝑓󸀠 (𝑥 )= 󵄨 . 𝑘 𝑘 𝜕𝑥󵄨 (3) (1) Our numerical studies on a larger system, IEEE 14-bus 󵄨𝑥𝑘 test system, as compared with the 3-bus power system, confirm that the convergence region of NR method ItiswellknownthatNewtonmethodislocallyconvergent has a fractal boundary. under the following assumptions [19]. (2) The convergence region shrinks with the increase ∗ Assumption 1. Equation (1)hasasolution𝑥 . in loading conditions. This property can be used to 󸀠 𝑛×𝑛 explain the difficulty of Newton method in finding a Assumption 2. 𝑓 :Ω → 𝑅 is Lipschitz continuous near ∗ power flow solution under heavy loading conditions, 𝑥 . in addition to the numerical ill-condition. 󸀠 ∗ (3) The fractal boundary of convergence regions persist Assumption 3. 𝑓 (𝑥 ) is nonsingular. under different loading conditions (the lightly loaded condition, the medium loaded condition, and the The classic convergence theorem is stated as follows. heavily loaded condition). Theorem 4 ((local convergence) [19]). If Assumption 1 (4) Convergence regions of NR method for power flow through Assumption 3 hold and the initial guess 𝑥0 is ∗ equations with different load models all reveal fractal. sufficiently near 𝑥 , then the Newton sequence exists (i.e., 󸀠 ∗ (5) Convergence region of fast decoupled power flow 𝑓 (𝑥𝑛) is nonsingular for all 𝑛>0) and converges to 𝑥 and method also has a fractal boundary. there is 𝐾>0such that (6) The required number of iterations is large (i.e., greater 󵄩 󵄩 󵄩 󵄩2 󵄩𝑒 󵄩 ≤𝐾󵄩𝑒 󵄩 (4) than 10) when the initial guesses are located near the 󵄩 𝑛+1󵄩 󵄩 𝑛󵄩 boundaries of convergence regions. It is interesting for n sufficiently large, where the mismatch errors decrease to note that the region of convergence with different quadratically. required number of iterations is disconnected. Assumption 3 assures that the singular Jacobian matrix This fractal property highlights the importance of choos- does not appear during iterations. By the same token, the ing initial guesses since a small variation of an initial guess initial point and the singularity of Jacobian matrix do make near the convergence boundary leads to two different power great effect on the convergence of Newton method. Starting flow solutions. This fractal property persists under different from an initial guess 𝑥0,thealgebraicequations(1)are loading conditions. This study of convergence region pro- iteratively solved. Generally, the Newton method converges vides insights of the complexity of the NR method as well as within a reasonable number. However, the Newton method the fast decoupled power flow method in finding power flow may diverge. To this end, the study of convergence region is solutions even for large-scale power systems. This study also relevant. sheds lights on the difficulty of finding a power flow solution of heavily-loaded power systems. 2.1. Fractal Convergence Boundaries. The convergence region of a power flow solution under the Newton method is the 2. Convergence Region of Newton Method region in the state space such that initial points starting from Newton method is a locally convergent iterative method for this region can nicely converge, under the Newton method, solving a set of nonlinear equations, which can be expressed to a power flow solution. It is known that the boundaries as of convergence regions of power flow solutions obtained by Newton method are a set that is analogous to the “Julia 𝑓 (𝑥) =0. (1) set” [15–18]. These boundaries are known to have fractal geometric features. Using initial points close to any of the 𝑛 𝑛 Here 𝑓:𝑅 →𝑅. solutions would result in a rapid convergence. However, using The Newton iteration can be expressed as initial points near the boundary of a convergence region, the Newton method becomes unpredictable. In this study, 𝑓(𝑥𝑘)=−𝐽(𝑥𝑘)Δ𝑥𝑘,Δ𝑥𝑘 =𝑥𝑘+1 −𝑥𝑘, (2) aregioninstatespaceinwhichtheinitialpointsdonot Journal of Applied Mathematics 3 converge (in a specified tolerance) to any power flow solution is termed the divergence region of Newton method. 𝑛 We consider a -bus power system with m generators. 𝜋 There exits three different types of buses: slack bus1, 𝑃-𝑉 bus PV (numbered from bus 2 to bus 𝑚), and 𝑃-𝑄 bus (numbered from bus 𝑚+1to bus 𝑛). The power flow equations defined 2 in polar coordination can be expressed as [2] 𝜋

Δ𝑃 𝑃 (𝜃, 𝑉) +𝑃𝑑 −𝑃𝑔 𝑓 (𝑥) =[ ]=[ ]=0, (5) PV Δ𝑄 𝑄 (𝜃, 𝑉) +𝑄𝑑 −𝑄𝑔 1 𝜋 where 𝑛 𝑃𝑖 (𝜃, 𝑉) =𝑉𝑖∑𝑉𝑗 (𝐺𝑖𝑗 cos 𝜃𝑖𝑗 +𝐵𝑖𝑗 sin 𝜃𝑖𝑗 ), 𝑖=2⋅⋅⋅𝑛, 3 𝑗=1 Figure 1: A modified 3-bus system. 𝑛 𝑄𝑖 (𝜃, 𝑉)=𝑉𝑖∑𝑉𝑗 (𝐺𝑖𝑗 sin 𝜃𝑖𝑗 −𝐵𝑖𝑗 cos 𝜃𝑖𝑗 ), 𝑖=𝑚+1⋅⋅⋅𝑛, 𝑗=1 (6) Table 1: Multiple power flow solutions. Initial and 𝑃𝑑 ={𝑃𝑑𝑖}, 𝑖 = 2⋅⋅⋅𝑛are the active load powers of the Equilibrium Color 𝜃2 𝜃3 Stability numbers buses; 𝑃𝑔 ={𝑃𝑔𝑖}, 𝑖=2⋅⋅⋅𝑚are the real power output of the 1 Blue 0.1191 0.1195 33703 Stable generators; 𝜃={𝜃𝑖}, 𝑖=2⋅⋅⋅𝑛are the voltage phase angles of − the buses; 𝑉={𝑉𝑖}, 𝑖=𝑚+1⋅⋅⋅𝑛arethevoltagemagnitudeof 2Green0.3151 3.0906 30430 Type-1 the buses. Equation (7) characterizes the real power balance 3Red−1.8206 2.3522 16463 Type-2 at 𝑃-𝑉 buses and the real and reactive power at 𝑃-𝑄 buses. 4 Purple 3.0919 −0.3137 30432 Type-1 The Newton iteration for solving the power flow equation 5Yellow2.3525−1.8220 16364 Type-2 (7) proceeds as follows: 6Sky-blue−3.0952 −3.0940 32587 Type-1 Δ𝑃 Δ𝜃 [ ]=−𝐽[ ]. Δ𝑄 Δ𝑉 (7) The power flow solutions obtained by using the Newton The Jacobian matrix in (8)is method are summarized in Table 1.Thecolorcolumnin 𝜕Δ𝑃 𝜕Δ𝑄 Table 1 denotes the colors in the convergence regions and its [ ] 2𝜋 − [ 𝜕𝜃 𝜕𝜃 ] fractal boundaries are highlighted in Figure 2,whichisa 𝐽= . (8) 𝑏𝑦 − 2𝜋 𝜃 𝜃 [𝜕Δ𝑃 𝜕Δ𝑄] grid of initial conditions in 2 and 3,representing [ 𝜕𝑉 𝜕𝑉 ] 160,000 power flow solutions. The scale in each voltage angle is 0 to 2𝜋 in radians. The blue region is the convergence To study the stability of solutions, a dynamic system is region of stable equilibrium point (SEP). The Newton power considered flow method starting from the initial conditions in this region all converges to the stable equilibrium point of (9). 𝑥=−𝑓̇ (𝑥) , (9) The green, purple, and sky-blue regions correspond to those initial conditions that converge to the three type-1 unstable ̇ 𝑇 where 𝑥=[̇ 𝜃 𝑉]̇ .Itisevidentthatthesolutionsof equilibrium points (UEP), at which one of the eigenvalues algebraic equation (7) correspond to the equilibrium points of the Jacobian is positive and the other is negative. Finally, of the ordinary differential equation defined9 in( ). A solution the red and yellow parts are the convergence regions of two of algebraic equation is exactly an equilibrium point of the type-2 equilibrium points, at which both eigenvalues are ordinary differential equation. For a power flow solution, it negative. The numerical difference between pixels is 0.157 in is a stable equilibrium point of (9) if all of its eigenvalues lie radians. in the open left half plane while a type-1 equilibrium point of Note that there are regions in the state space in which (9) has exactly one eigenvalue that lies in the open right half a small change in initial conditions results in a different plane. Type-2 equilibrium points of (9) are similarly defined. convergence region. The number of iterations required for We study the convergence region of Newton method on convergence is large in the fractal areas. Depending on a modified 3-bus system shown in Figure 1.Themodified3- themaximumiterationnumberusedinthepowerflow bus system contains 2 𝑃-𝑉 buses and 1 slack bus, with bus 1 program, these fractal areas may change. Note that due to being selected as the slack bus, and the remaining buses, bus the singularity of Jacobian, the Newton power flow will not 2andbus3,areboth𝑃-𝑉 buses. Hence, there are only two converge from any initial condition at which the Jacobian is real power equations to be solved with two unknowns (the singular or from a point that maps on to a point where the voltage angles 𝜃2 and 𝜃3). Jacobian is singular. 4 Journal of Applied Mathematics

6 Table 2: Required Iterations starting from different initials. Convergence iteration number Initial numbers 5 [0, 2]235 (2, 4] 60197 4 (4, 6] 69038 (6, 8] 21945 3 (8, 10] 6162 (10, 12] 1776 2 (12, 14] 472 (14, 16] 140 1 (16, 18] 25 (18, 20] 9 0 >20 1 0 1 2 3 4 5 6 —— Figure 2: Convergence region of each power flow solution of a modified 3-bus test system. 6 20 18 5 2.2. Required Number of Iterations for Initials in Region of 16 Convergence. The convergence of a Newton power flow has 14 4 strong relationship with the maximum iteration number and 12 the convergence threshold used in the computation. Newton power flow method usually converges within 10 iterations. 3 10 For the study of required iterations starting from different 8 initial points, we perform numerical studies on the modified 2 3-bus system (see Figure 1)withthefollowingparameters: 6 4 (i) threshold for iterations: 20; 1 −8 2 (ii)thresholdforconvergenceinpowermismatch:10 . 0 0 1 2 3 4 5 6 Table 2 shows the required number of iterations starting from different initial conditions. The initial points that Figure 3: The convergence region and the number of iterations converge within 10 iterations constitute about 98.5%. Also, needed for the Newton method on (𝜃2,𝜃3)plane. the required iterations from initials that lie in or near fractal boundaries are shown in Figure 3.Thecolor-mapinFigure 3 denotes the correspondence of colors with the required region of the modified 3-bus system is 2-dimensional. For the iterations. As can be seen from the figure, the initial points IEEE 14-bus system, it contains 4 𝑃-𝑉 buses, 9 𝑃-𝑄 buses, in convergence region can nicely converge with 10 iterations, and 1 slack bus. Hence the state variable is a 22-dimensional which are always less than 10 and the initial points which vector, which means that the convergence region of IEEE 14- require large iteration numbers all lie near the boundaries of bussystemis22-dimensional.Forthesakeofthevisualization convergence regions. of the convergence region, the same two-dimensional cross- sectionsareselectedtodisplayconvergenceregions.Thebasic 3. Convergence Regions under data for study is listed as follows: Different Conditions (i) horizon axis is the voltage angle 𝜃2 in bus 2, and the In this section, the convergence regions on several conditions vertical axis is the voltage angle 𝜃3 in bus 3; areinvestigatedviaamodified3-bussystemandtheIEEE14- (ii) the scale in each coordinate is from 0 to 2𝜋; bus system: (iii) each figure is 2𝜋 − 𝑏𝑦 −2𝜋 grid of initial conditions (i) base-case, on (𝜃2,𝜃3) plane, representing 160,000 power flow (ii) various loading conditions, solutions. (iii) different load models (models representing load demands). 3.1. Convergence Regions at Base-Case. Figure 4 shows the convergence region of IEEE 14-bus system of the base case on For the modified 3-bus system, there are 2 𝑃-𝑉 buses and the (𝜃2,𝜃3) plane. The blue region is the convergence region of 1slackbus.Hencethestatevariable𝑥 in power flow equations the SEP. The power flow starting from the initial conditions is a 2-dimensional vector, which means that the convergence in this region all converges to the stable equilibrium point. Journal of Applied Mathematics 5

Table 3: Number of power flow solutions and fractal properties at different loading conditions.

Case name Loading condition Number of solutions Number of initials converged to SEP Fractal 16 33703Yes 4.02 4 34743 Yes 3-bus Loading factor 52 90051Yes 9.1 0 0 — 12016274Yes 28 16538Yes 14-bus Loading factor 42 17125Yes 50 0 —

6 Table 4: Power flow solutions on different loading conditions.

Power flow solutions 5 Loading factor SEP UEP theta2 theta3 theta2 theta3 4 6 0.7663 0.7689 2.5373 2.5433 7 0.9318 0.9350 2.3713 2.3777 3 8 1.1377 1.1416 2.1650 2.1714 9 1.5089 1.5140 1.7935 1.7994 2

1 However,thenumberofpowerflowsolutionsdecreases whiletheconvergenceregionofSEPincreasesinsizewith 0 the increase of loading factor 𝜆,whichisshowninFigure 5. 0 1 2 3 4 5 6 In other words, the number of power flow solutions decreases when the loading conditions become heavier. During this Figure 4: Convergence region of power flow solutions of the IEEE process of load increases, saddle node bifurcations occur 14-bus system on (𝜃2,𝜃3)plane. which brings about the solutions that vanished in pair, and then at the end there are only two power flow solutions left The green region is the convergence region of one of the type- (one SEP and one type-1 UEP); finally these two power flow 1equilibrium.Theleftcolorizedregions,likegreen,purple, solutions come together when the loading factor 𝜆 reaches yellow, and so forth, are regions corresponding to those initial its bifurcation value at which these two power flow solutions conditions that converge to the correspond