# American Institute of Mathematical Sciences

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

 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

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.
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 and 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. [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. [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. [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. [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. [6] F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley, New York, 1983. [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. [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. [9] J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations," SIAM, Philadelphia, 1996. [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). [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. [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. [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. [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. [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. [16] C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," in "Frontiers in Applied Mathematics, 16," SIAM, Philadelphia, 1995. [17] C. T. Kelley, "Iterative Methods for Optimization," SIAM, Philadelphia, 1999. [18] C. T. Kelley, "Solving Nonlinear Equations with Newton's Method," in "Fundamentals of Algorithms, 1," SIAM, Philadelphia, 2003. [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. [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. [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. [22] A. S. Lewis, Nonsmooth analysis of eigenvalues, Math. Program., 84 (1999), 1-24. [23] A. S. Lewis and M. Overton, "Eigenvalue Optimization," Acta Numerica, 5 (1996), 149-190. doi: 10.1017/S0962492900002646. [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. [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. [26] R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 957-972 doi: 10.1137/0315061. [27] J. M. Ortega and W. C. Rheinboldt, "Iterative Solutions of Nonlinear Equations in Several Variables," Academic Press, New York, 1970. [28] F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function, Math. Program, 89 (2000), 1-33. doi: 10.1007/PL00011388. [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. [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. [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. [32] H. Qi and X. Yang, Semismoothness of spectral functions, SIAM J. Matrix Anal. Appl., 25 (2004), 766-783. doi: 10.1137/S0895479802417921. [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. [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. [35] L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367. doi: 10.1007/BF01581275. [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. [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. [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. [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. [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. [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. [42] M. Ulbrich, Semismooth Newton Methods for operator equations in function spaces, SIAM J. Optim., 13 (2002), 805-842. doi: 10.1137/S1052623400371569. [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.

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##### 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. [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. [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. [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. [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. [6] F. H. Clarke, "Optimization and Nonsmooth Analysis," John Wiley, New York, 1983. [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. [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. [9] J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations," SIAM, Philadelphia, 1996. [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). [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. [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. [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. [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. [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. [16] C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," in "Frontiers in Applied Mathematics, 16," SIAM, Philadelphia, 1995. [17] C. T. Kelley, "Iterative Methods for Optimization," SIAM, Philadelphia, 1999. [18] C. T. Kelley, "Solving Nonlinear Equations with Newton's Method," in "Fundamentals of Algorithms, 1," SIAM, Philadelphia, 2003. [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. [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. [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. [22] A. S. Lewis, Nonsmooth analysis of eigenvalues, Math. Program., 84 (1999), 1-24. [23] A. S. Lewis and M. Overton, "Eigenvalue Optimization," Acta Numerica, 5 (1996), 149-190. doi: 10.1017/S0962492900002646. [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. [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. [26] R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 957-972 doi: 10.1137/0315061. [27] J. M. Ortega and W. C. Rheinboldt, "Iterative Solutions of Nonlinear Equations in Several Variables," Academic Press, New York, 1970. [28] F. Oustry, A second-order bundle method to minimize the maximum eigenvalue function, Math. Program, 89 (2000), 1-33. doi: 10.1007/PL00011388. [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. [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. [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. [32] H. Qi and X. Yang, Semismoothness of spectral functions, SIAM J. Matrix Anal. Appl., 25 (2004), 766-783. doi: 10.1137/S0895479802417921. [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. [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. [35] L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367. doi: 10.1007/BF01581275. [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. [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. [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. [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. [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. [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. [42] M. Ulbrich, Semismooth Newton Methods for operator equations in function spaces, SIAM J. Optim., 13 (2002), 805-842. doi: 10.1137/S1052623400371569. [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.
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