Finding a stable solution of a system of nonlinear
equations arising from dynamic systems

Carl. T. Kelley; Liqun  Qi; Xiaojiao Tong; Hongxia Yin

doi:10.3934/jimo.2011.7.497

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April 2011, 7(2): 497-521. doi: 10.3934/jimo.2011.7.497

Finding a stable solution of a system of nonlinear equations arising from dynamic systems

Carl. T. Kelley ^1, , Liqun Qi ^2, , Xiaojiao Tong ^3, and Hongxia Yin ^4,

1.	Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, United States
2.	Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
3.	Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421002, China
4.	Department of Mathematics and Statistics, Minnesota State University Mankato, Mankato, MN 56001, United States

Received August 2009 Revised March 2011 Published April 2011

Abstract
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In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of constrained semismooth equations. The advantage of the new models is twofold. Firstly, the stability requirement of dynamical systems is controlled by nonlinear inequalities. Secondly, the semismoothness property of the stability functions makes the models solvable by efficient numerical methods. We introduce smoothing functions for the stability functions and present a smoothing Newton method for solving the problems. Global and local quadratic convergence of the algorithm is established. Numerical examples from dynamical systems are also given to illustrate the efficiency of the new approach.

Keywords: stable solutions, saddle-node bifurcation, Hopf bifurcation, System of nonlinear equations, stability functions, smoothing Newton method..

Mathematics Subject Classification: Primary: 65H10, 65K10; Secondary: 90C3.

Citation: Carl. T. Kelley, Liqun Qi, Xiaojiao Tong, Hongxia Yin. Finding a stable solution of a system of nonlinear equations arising from dynamic systems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 497-521. doi: 10.3934/jimo.2011.7.497

References:

[1]	F. Alvarado, I. Dobson and Y. Hu, Computation of closest bifurcations in power systems, IEEE Trans. Power System, 9 (1994), 918-928. doi: 10.1109/59.317655. Google Scholar
[2]	C. A. Cañizares, Calculating optimal system parameters to maximize the distance to saddle-node bifurcation points, IEEE Trans. Circuits and System, 45 (1998), 225-237. doi: 10.1109/81.662696. Google Scholar
[3]	X. Chen, H. Qi, L. Qi and K.-L. Teo, Smooth convex approximation to the maximum eigenvalue function, J. of Global Optimization, 30 (2004), 253-270. doi: 10.1007/s10898-004-8271-2. Google Scholar
[4]	H. D. Chiang, I. Dobson and R. J. Thomas, On voltage in electric power systems, IEEE Trans. Power Systems, 5 (1990), 601-611. doi: 10.1109/59.54571. Google Scholar
[5]	T. Coffey, C. T. Kelley and D. E. Keyes, Pseudo-transient continuation and differential-algebraic equations, SIAM J. Sci. Comp., 25 (2003), 553-569. doi: 10.1137/S106482750241044X. Google Scholar
[6]	F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley, New York, 1983. Google Scholar
[7]	H. Dan, N. Yamashita and M. Fukushima, Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions, Optimization Methods and Software, 17 (2002), 605-626. doi: 10.1080/1055678021000049345. Google Scholar
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity, J. Math. Anal. Appl., 338 (2008), 101-110. doi: 10.1016/j.jmaa.2007.05.011. Google Scholar
[9]	J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations," SIAM, Philadelphia, 1996. Google Scholar
[10]	P. Deuflhard, Adaptive pseudo-transient continuation for nonlinear steady state problems, ZIP-Report02-12(March 2002), in "Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms," Springer Series in Computational Mathematics, 35, Springer (2004). Google Scholar
[11]	I. Dobson, An iterative method to compute the closest saddle node or Hopf bifurcation in multidimensional parameter space, in "Proceedings of the IEEE International Symposium on Circuits and Systems," San Diego, 1992, 2513-2516. Google Scholar
[12]	I. Dobson and L. Lu, Computing an optimum direction in control space to avoid saddle node bifurcation and voltage collapse in electric power systems, IEEE Trans. Automatic Control, 37 (1992), 1616-1620. doi: 10.1109/9.256397. Google Scholar
[13]	K. R. Fowler and C. T. Kelley, Pseudo-transient continuation for nonsmooth nonlinear equations, SIAM J. Numer. Anal., 43 (2005), 1385-1406. doi: 10.1137/S0036142903431298. Google Scholar
[14]	C. P. Gupta, R. K. Varma and S. C. Srivastava, A method to determine closest Hopf bifurcation in power systems considering exciter and load dynamics, in "Proceedings of 'Energy Management and Power Delivery Conference 1998 (EMPD'98)," Singapore, 1998, 293-297. Google Scholar
[15]	M. Hintermüller and M. Hinze, A SQP-semi-smooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints, SIAM J. Optim, 16 (2006), 1177-1200. doi: 10.1137/030601259. Google Scholar
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," in "Frontiers in Applied Mathematics, 16," SIAM, Philadelphia, 1995. Google Scholar
[17]	C. T. Kelley, "Iterative Methods for Optimization," SIAM, Philadelphia, 1999. Google Scholar
[18]	C. T. Kelley, "Solving Nonlinear Equations with Newton's Method," in "Fundamentals of Algorithms, 1," SIAM, Philadelphia, 2003. Google Scholar
[19]	C. T. Kelley and D. E. Keyes, Convergence analysis of pseudo-transient continuation, SIAM J. Numer. Anal., 35 (1998), 508-523. doi: 10.1137/S0036142996304796. Google Scholar
[20]	C. T. Kelley, Li-Zhi Liao, Liqun Qi, Moody T. Chu, J. P. Reese and C. Winton, Projected Pseudotransient continuation, SIAM J. Numer. Nanl, 46 (2008), 3071-3083. doi: 10.1137/07069866X. Google Scholar
[21]	P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. V. Cutsem and V. Vittal, Definition and classification of power system stability, IEEE Transaction on Power Systems, 19 (2004), 1387-1401. doi: 10.1109/TPWRS.2004.825981. Google Scholar
[22]	A. S. Lewis, Nonsmooth analysis of eigenvalues, Math. Program., 84 (1999), 1-24. Google Scholar
[23]	A. S. Lewis and M. Overton, "Eigenvalue Optimization," Acta Numerica, 5 (1996), 149-190. doi: 10.1017/S0962492900002646. Google Scholar
[24]	Y. Ma, H. Kawakami and C. K. Tse, Bifurcation analysis of switched dynamical systems with periodically moving borders, IEEE Transactions on Circuits and Systems, 51 (2004), 1184-1193. doi: 10.1109/TCSI.2004.829240. Google Scholar
[25]	Y. V. Makarov, Z. Y. Dong and D. J. Hill, A general method for small signal stability analysis, IEEE Transaction on Power Systems, 13 (1998), 979-985. doi: 10.1109/59.709086. Google Scholar
[26]	R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 957-972 doi: 10.1137/0315061. Google Scholar
[27]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solutions of Nonlinear Equations in Several Variables," Academic Press, New York, 1970. Google Scholar
[28]	F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function, Math. Program, 89 (2000), 1-33. doi: 10.1007/PL00011388. Google Scholar
[29]	J. S. Pang, D. F. Sun and J. Sun, Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Cone Complementarity Problems, Math. Oper. Res., 28 (2003), 39-63. doi: 10.1287/moor.28.1.39.14258. Google Scholar
[30]	H. Qi and L. Liao, A smoothing Newton method for extended vertical linear complementarity problems, SIAM J. Matrix Anal. Appl., 21 (1999), 45-66. doi: 10.1137/S0895479897329837. Google Scholar
[31]	H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM J. Matrix Anal. Appl., 28 (2006), 360-385. doi: 10.1137/050624509. Google Scholar
[32]	H. Qi and X. Yang, Semismoothness of spectral functions, SIAM J. Matrix Anal. Appl., 25 (2004), 766-783. doi: 10.1137/S0895479802417921. Google Scholar
[33]	L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), 227-244. doi: 10.1287/moor.18.1.227. Google Scholar
[34]	L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35. Google Scholar
[35]	L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367. doi: 10.1007/BF01581275. Google Scholar
[36]	D. Sun and J. Sun, Semismooth matrix valued functions, Math. Oper. Res., 27 (2002), 150-169. doi: 10.1287/moor.27.1.150.342. Google Scholar
[37]	D. Sun and J. Sun, Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems, SIAM J. Numer Anal., 40 (2002), 2352-2367. doi: 10.1137/S0036142901393814. Google Scholar
[38]	M. D. Smooke and R. M. Mattheij, On the solution of nonlinear two-point boundary value problem on successively refined grids, Appl. Numer. Math., 1 (1985), 463-487. doi: i:10.1016/0168-9274(85)90032-7. Google Scholar
[39]	A. Shestakov and J. Milovich, "Applications of Pseudo-Transient Continuation and Newton-Krylov Methods of the Poisson-Boltzmann and Radiation Diffusion Equation," Tech. report UCRL-JC-139339, Lawrence Livermore National Laboratory, Livermore, CA, 2000. Google Scholar
[40]	X. J. Tong and S. Zhou, A smoothing projected Newton-type method for semismooth equations with bound constraints, Journal of Industrial Management Optimization, 1 (2005), 235-250. Google Scholar
[41]	V. Venkatasubramanian, H. Schattler and J. Zaborsky, Dynamics of large constrained nonlinear systems - a taxonomy theory, Proc. IEEE, 83 (1995), 1530-1561. doi: 10.1109/5.481633. Google Scholar
[42]	M. Ulbrich, Semismooth Newton Methods for operator equations in function spaces, SIAM J. Optim., 13 (2002), 805-842. doi: 10.1137/S1052623400371569. Google Scholar
[43]	X. Wu, C. K. Tse, O. Dranga and J. Lu, Fast-scale instability of single-stage power-factor-correction of power supplies, IEEE Transactions on Circuits and Systems, 53 (2006), 204-213. doi: 10.1109/TCSI.2005.854293. Google Scholar

show all references

References:

[1]	F. Alvarado, I. Dobson and Y. Hu, Computation of closest bifurcations in power systems, IEEE Trans. Power System, 9 (1994), 918-928. doi: 10.1109/59.317655. Google Scholar
[2]	C. A. Cañizares, Calculating optimal system parameters to maximize the distance to saddle-node bifurcation points, IEEE Trans. Circuits and System, 45 (1998), 225-237. doi: 10.1109/81.662696. Google Scholar
[3]	X. Chen, H. Qi, L. Qi and K.-L. Teo, Smooth convex approximation to the maximum eigenvalue function, J. of Global Optimization, 30 (2004), 253-270. doi: 10.1007/s10898-004-8271-2. Google Scholar
[4]	H. D. Chiang, I. Dobson and R. J. Thomas, On voltage in electric power systems, IEEE Trans. Power Systems, 5 (1990), 601-611. doi: 10.1109/59.54571. Google Scholar
[5]	T. Coffey, C. T. Kelley and D. E. Keyes, Pseudo-transient continuation and differential-algebraic equations, SIAM J. Sci. Comp., 25 (2003), 553-569. doi: 10.1137/S106482750241044X. Google Scholar
[6]	F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley, New York, 1983. Google Scholar
[7]	H. Dan, N. Yamashita and M. Fukushima, Convergence Properties of the Inexact Levenberg-Marquardt Method under Local Error Bound Conditions, Optimization Methods and Software, 17 (2002), 605-626. doi: 10.1080/1055678021000049345. Google Scholar
[8]	G. Degla, An overview of semi-continuity results on the spectral radius and positivity, J. Math. Anal. Appl., 338 (2008), 101-110. doi: 10.1016/j.jmaa.2007.05.011. Google Scholar
[9]	J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations," SIAM, Philadelphia, 1996. Google Scholar
[10]	P. Deuflhard, Adaptive pseudo-transient continuation for nonlinear steady state problems, ZIP-Report02-12(March 2002), in "Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms," Springer Series in Computational Mathematics, 35, Springer (2004). Google Scholar
[11]	I. Dobson, An iterative method to compute the closest saddle node or Hopf bifurcation in multidimensional parameter space, in "Proceedings of the IEEE International Symposium on Circuits and Systems," San Diego, 1992, 2513-2516. Google Scholar
[12]	I. Dobson and L. Lu, Computing an optimum direction in control space to avoid saddle node bifurcation and voltage collapse in electric power systems, IEEE Trans. Automatic Control, 37 (1992), 1616-1620. doi: 10.1109/9.256397. Google Scholar
[13]	K. R. Fowler and C. T. Kelley, Pseudo-transient continuation for nonsmooth nonlinear equations, SIAM J. Numer. Anal., 43 (2005), 1385-1406. doi: 10.1137/S0036142903431298. Google Scholar
[14]	C. P. Gupta, R. K. Varma and S. C. Srivastava, A method to determine closest Hopf bifurcation in power systems considering exciter and load dynamics, in "Proceedings of 'Energy Management and Power Delivery Conference 1998 (EMPD'98)," Singapore, 1998, 293-297. Google Scholar
[15]	M. Hintermüller and M. Hinze, A SQP-semi-smooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints, SIAM J. Optim, 16 (2006), 1177-1200. doi: 10.1137/030601259. Google Scholar
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," in "Frontiers in Applied Mathematics, 16," SIAM, Philadelphia, 1995. Google Scholar
[17]	C. T. Kelley, "Iterative Methods for Optimization," SIAM, Philadelphia, 1999. Google Scholar
[18]	C. T. Kelley, "Solving Nonlinear Equations with Newton's Method," in "Fundamentals of Algorithms, 1," SIAM, Philadelphia, 2003. Google Scholar
[19]	C. T. Kelley and D. E. Keyes, Convergence analysis of pseudo-transient continuation, SIAM J. Numer. Anal., 35 (1998), 508-523. doi: 10.1137/S0036142996304796. Google Scholar
[20]	C. T. Kelley, Li-Zhi Liao, Liqun Qi, Moody T. Chu, J. P. Reese and C. Winton, Projected Pseudotransient continuation, SIAM J. Numer. Nanl, 46 (2008), 3071-3083. doi: 10.1137/07069866X. Google Scholar
[21]	P. Kundur, J. Paserba, V. Ajjarapu, G. Andersson, A. Bose, C. Canizares, N. Hatziargyriou, D. Hill, A. Stankovic, C. Taylor, T. V. Cutsem and V. Vittal, Definition and classification of power system stability, IEEE Transaction on Power Systems, 19 (2004), 1387-1401. doi: 10.1109/TPWRS.2004.825981. Google Scholar
[22]	A. S. Lewis, Nonsmooth analysis of eigenvalues, Math. Program., 84 (1999), 1-24. Google Scholar
[23]	A. S. Lewis and M. Overton, "Eigenvalue Optimization," Acta Numerica, 5 (1996), 149-190. doi: 10.1017/S0962492900002646. Google Scholar
[24]	Y. Ma, H. Kawakami and C. K. Tse, Bifurcation analysis of switched dynamical systems with periodically moving borders, IEEE Transactions on Circuits and Systems, 51 (2004), 1184-1193. doi: 10.1109/TCSI.2004.829240. Google Scholar
[25]	Y. V. Makarov, Z. Y. Dong and D. J. Hill, A general method for small signal stability analysis, IEEE Transaction on Power Systems, 13 (1998), 979-985. doi: 10.1109/59.709086. Google Scholar
[26]	R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 957-972 doi: 10.1137/0315061. Google Scholar
[27]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solutions of Nonlinear Equations in Several Variables," Academic Press, New York, 1970. Google Scholar
[28]	F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function, Math. Program, 89 (2000), 1-33. doi: 10.1007/PL00011388. Google Scholar
[29]	J. S. Pang, D. F. Sun and J. Sun, Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Cone Complementarity Problems, Math. Oper. Res., 28 (2003), 39-63. doi: 10.1287/moor.28.1.39.14258. Google Scholar
[30]	H. Qi and L. Liao, A smoothing Newton method for extended vertical linear complementarity problems, SIAM J. Matrix Anal. Appl., 21 (1999), 45-66. doi: 10.1137/S0895479897329837. Google Scholar
[31]	H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM J. Matrix Anal. Appl., 28 (2006), 360-385. doi: 10.1137/050624509. Google Scholar
[32]	H. Qi and X. Yang, Semismoothness of spectral functions, SIAM J. Matrix Anal. Appl., 25 (2004), 766-783. doi: 10.1137/S0895479802417921. Google Scholar
[33]	L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), 227-244. doi: 10.1287/moor.18.1.227. Google Scholar
[34]	L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35. Google Scholar
[35]	L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367. doi: 10.1007/BF01581275. Google Scholar
[36]	D. Sun and J. Sun, Semismooth matrix valued functions, Math. Oper. Res., 27 (2002), 150-169. doi: 10.1287/moor.27.1.150.342. Google Scholar
[37]	D. Sun and J. Sun, Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems, SIAM J. Numer Anal., 40 (2002), 2352-2367. doi: 10.1137/S0036142901393814. Google Scholar
[38]	M. D. Smooke and R. M. Mattheij, On the solution of nonlinear two-point boundary value problem on successively refined grids, Appl. Numer. Math., 1 (1985), 463-487. doi: i:10.1016/0168-9274(85)90032-7. Google Scholar
[39]	A. Shestakov and J. Milovich, "Applications of Pseudo-Transient Continuation and Newton-Krylov Methods of the Poisson-Boltzmann and Radiation Diffusion Equation," Tech. report UCRL-JC-139339, Lawrence Livermore National Laboratory, Livermore, CA, 2000. Google Scholar
[40]	X. J. Tong and S. Zhou, A smoothing projected Newton-type method for semismooth equations with bound constraints, Journal of Industrial Management Optimization, 1 (2005), 235-250. Google Scholar
[41]	V. Venkatasubramanian, H. Schattler and J. Zaborsky, Dynamics of large constrained nonlinear systems - a taxonomy theory, Proc. IEEE, 83 (1995), 1530-1561. doi: 10.1109/5.481633. Google Scholar
[42]	M. Ulbrich, Semismooth Newton Methods for operator equations in function spaces, SIAM J. Optim., 13 (2002), 805-842. doi: 10.1137/S1052623400371569. Google Scholar
[43]	X. Wu, C. K. Tse, O. Dranga and J. Lu, Fast-scale instability of single-stage power-factor-correction of power supplies, IEEE Transactions on Circuits and Systems, 53 (2006), 204-213. doi: 10.1109/TCSI.2005.854293. Google Scholar

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Finding a stable solution of a system of nonlinear equations arising from dynamic systems

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