Evolutionary Techniques for Model Order Reduction of Large Scale Linear Systems
Recently, genetic algorithms (GA) and particle swarm optimization (PSO) technique have attracted considerable attention among various modern heuristic optimization techniques. The GA has been popular in academia and the industry mainly because of its intuitiveness, ease of implementation, and the ability to effectively solve highly non-linear, mixed integer optimization problems that are typical of complex engineering systems. PSO technique is a relatively recent heuristic search method whose mechanics are inspired by the swarming or collaborative behavior of biological populations. In this paper both PSO and GA optimization are employed for finding stable reduced order models of single-input- single-output large-scale linear systems. Both the techniques guarantee stability of reduced order model if the original high order model is stable. PSO method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. Both the methods are illustrated through numerical example from literature and the results are compared with recently published conventional model reduction technique.
[1] M.J.Bosley and F.P.Lees, "A survey of simple transfer function
derivations from high order state variable models", Automatica, Vol. 8,
pp. 765-775, !978.
[2] M.F.Hutton and B. Fried land, "Routh approximations for reducing
order of linear time- invariant systems", IEEE Trans. Auto. Control, Vol.
20, pp 329-337, 1975.
[3] R.K.Appiah, "Linear model reduction using Hurwitz polynomial
approximation", Int. J. Control, Vol. 28, no. 3, pp 477-488, 1978.
[4] T. C. Chen, C.Y.Chang and K.W.Han, "Reduction of transfer functions
by the stability equation method", Journal of Franklin Institute, Vol.
308, pp 389-404, 1979.
[5] Y.Shamash, "Truncation method of reduction: a viable alternative",
Electronics Letters, Vol. 17, pp 97-99, 1981.
[6] P.O.Gutman, C.F.Mannerfelt and P.Molander, "Contributions to the
model reduction problem", IEEE Trans. Auto. Control, Vol. 27, pp 454-
455, 1982.
[7] Y. Shamash, "Model reduction using the Routh stability criterion and
the Pade approximation technique", Int. J. Control, Vol. 21, pp 475-484,
1975.
[8] T.C.Chen, C.Y.Chang and K.W.Han, "Model Reduction using the
stability-equation method and the Pade approximation method", Journal
of Franklin Institute, Vol. 309, pp 473-490, 1980.
[9] Bai-Wu Wan, "Linear model reduction using Mihailov criterion and
Pade approximation technique", Int. J. Control, Vol. 33, pp 1073-1089,
1981.
[10] V.Singh, D.Chandra and H.Kar, "Improved Routh-Pade Approximants:
A Computer-Aided Approach", IEEE Trans. Auto. Control, Vol. 49. No.
2, pp292-296, 2004.
[11] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and
Machine Learning, Addison-Wesley, 1989.
[12] S.Panda and N.P.Padhy, "Comparison of Particle Swarm Optimization
and Genetic Algorithm for FACTS-based Controller Design", Applied
Soft Computing. Vol. 8, Issue 4, pp. 1418-1427, 2008.
[13] S.Panda and N.P.Padhy, "Application of Genetic Algorithm for PSS and
FACTS based Controller Design", International Journal of
Computational Methods, Vol. 5, Issue 4, pp. 607-620, 2008.
[14] S.Panda and R.N.Patel, "Transient Stability Improvement by Optimally
Located STATCOMs Employing Genetic Algorithm" International
Journal of Energy Technology and Policy, Vol. 5, No. 4, pp. 404-421,
2007.
[15] S.Panda and R.N.Patel, "Damping Power System Oscillations by
Genetically Optimized PSS and TCSC Controller" International Journal
of Energy Technology and Policy, Inderscience, Vol. 5, No. 4, pp. 457-
474, 2007.
[16] S.Panda and R.N.Patel, "Optimal Location of Shunt FACTS Controllers
for Transient Stability Improvement Employing Genetic Algorithm",
Electric Power Components and Systems, Taylor and Francis, Vol. 35,
No. 2, pp. 189-203, 2007.
[17] J. Kennedy and R.C.Eberhart, "Particle swarm optimization", IEEE
Int.Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ, 1995.
[18] S.Panda, N.P.Padhy, R.N.Patel, "Power System Stability Improvement
by PSO Optimized SSSC-based Damping Controller", Electric Power
Components & Systems, Taylor and Francis, Vol. 36, No. 5, pp. 468-
490, 2008.
[19] S.Panda and N.P.Padhy, "Optimal location and controller design of
STATCOM using particle swarm optimization", Journal of the Franklin
Institute, Elsevier, Vol.345, pp. 166-181, 2008.
[20] S.Panda, N.P.Padhy and R.N.Patel, "Robust Coordinated Design of PSS
and TCSC using PSO Technique for Power System Stability
Enhancement", Journal of Electrical Systems, Vol. 3, No. 2, pp. 109-
123, 2007.
[21] C. B. Vishwakarma and R.Prasad, "Clustering Method for Reducing
Order of Linear System using Pade Approximation", IETE Journal of
Research, Vol. 54, Issue 5, pp. 326-330, 2008.
[22] S.Mukherjee, and R.N.Mishra, Order reduction of linear systems using
an error minimization technique, Journal of Franklin Inst. Vol. 323, No.
1, pp. 23-32, 1987.
[23] S.Panda, S.K.Tomar, R.Prasad, C.Ardil, "Reduction of Linear Time-
Invariant Systems Using Routh-Approximation and PSO", International
Journal of Applied Mathematics and Computer Sciences, Vol. 5, No. 2,
pp. 82-89, 2009.
[24] S.Panda, S.K.Tomar, R.Prasad, C.Ardil, "Model Reduction of Linear
Systems by Conventional and Evolutionary Techniques", International
Journal of Computational and Mathematical Sciences, Vol. 3, No. 1, pp.
28-34, 2009.
[25] R.Parthasarathy, and K. N. Jayasimha, System reduction using stability
equation method and modified Cauer continued fraction, Proc. IEEE
Vol. 70, No. 10, pp. 1234-1236, Oct. 1982.
[26] L.S. hieh, and Y.J.Wei, A mixed method for multivariable system
reduction, IEEE Trans. Autom. Control, Vol. AC-20, pp. 429-432, 1975.
[27] R.Prasad, and J.Pal, Stable reduction of linear systems by continued
fractions, J. Inst. Eng. India, IE (I) J.EL, Vol. 72, pp. 113-116, Oct.
1991.
[28] J. Pal, Stable reduced -order Pade approximants using the Routh-
Hurwitz array, Electronics letters, Vol. 15, No. 8, pp. 225-226, April
1979.
[1] M.J.Bosley and F.P.Lees, "A survey of simple transfer function
derivations from high order state variable models", Automatica, Vol. 8,
pp. 765-775, !978.
[2] M.F.Hutton and B. Fried land, "Routh approximations for reducing
order of linear time- invariant systems", IEEE Trans. Auto. Control, Vol.
20, pp 329-337, 1975.
[3] R.K.Appiah, "Linear model reduction using Hurwitz polynomial
approximation", Int. J. Control, Vol. 28, no. 3, pp 477-488, 1978.
[4] T. C. Chen, C.Y.Chang and K.W.Han, "Reduction of transfer functions
by the stability equation method", Journal of Franklin Institute, Vol.
308, pp 389-404, 1979.
[5] Y.Shamash, "Truncation method of reduction: a viable alternative",
Electronics Letters, Vol. 17, pp 97-99, 1981.
[6] P.O.Gutman, C.F.Mannerfelt and P.Molander, "Contributions to the
model reduction problem", IEEE Trans. Auto. Control, Vol. 27, pp 454-
455, 1982.
[7] Y. Shamash, "Model reduction using the Routh stability criterion and
the Pade approximation technique", Int. J. Control, Vol. 21, pp 475-484,
1975.
[8] T.C.Chen, C.Y.Chang and K.W.Han, "Model Reduction using the
stability-equation method and the Pade approximation method", Journal
of Franklin Institute, Vol. 309, pp 473-490, 1980.
[9] Bai-Wu Wan, "Linear model reduction using Mihailov criterion and
Pade approximation technique", Int. J. Control, Vol. 33, pp 1073-1089,
1981.
[10] V.Singh, D.Chandra and H.Kar, "Improved Routh-Pade Approximants:
A Computer-Aided Approach", IEEE Trans. Auto. Control, Vol. 49. No.
2, pp292-296, 2004.
[11] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and
Machine Learning, Addison-Wesley, 1989.
[12] S.Panda and N.P.Padhy, "Comparison of Particle Swarm Optimization
and Genetic Algorithm for FACTS-based Controller Design", Applied
Soft Computing. Vol. 8, Issue 4, pp. 1418-1427, 2008.
[13] S.Panda and N.P.Padhy, "Application of Genetic Algorithm for PSS and
FACTS based Controller Design", International Journal of
Computational Methods, Vol. 5, Issue 4, pp. 607-620, 2008.
[14] S.Panda and R.N.Patel, "Transient Stability Improvement by Optimally
Located STATCOMs Employing Genetic Algorithm" International
Journal of Energy Technology and Policy, Vol. 5, No. 4, pp. 404-421,
2007.
[15] S.Panda and R.N.Patel, "Damping Power System Oscillations by
Genetically Optimized PSS and TCSC Controller" International Journal
of Energy Technology and Policy, Inderscience, Vol. 5, No. 4, pp. 457-
474, 2007.
[16] S.Panda and R.N.Patel, "Optimal Location of Shunt FACTS Controllers
for Transient Stability Improvement Employing Genetic Algorithm",
Electric Power Components and Systems, Taylor and Francis, Vol. 35,
No. 2, pp. 189-203, 2007.
[17] J. Kennedy and R.C.Eberhart, "Particle swarm optimization", IEEE
Int.Conf. on Neural Networks, IV, 1942-1948, Piscataway, NJ, 1995.
[18] S.Panda, N.P.Padhy, R.N.Patel, "Power System Stability Improvement
by PSO Optimized SSSC-based Damping Controller", Electric Power
Components & Systems, Taylor and Francis, Vol. 36, No. 5, pp. 468-
490, 2008.
[19] S.Panda and N.P.Padhy, "Optimal location and controller design of
STATCOM using particle swarm optimization", Journal of the Franklin
Institute, Elsevier, Vol.345, pp. 166-181, 2008.
[20] S.Panda, N.P.Padhy and R.N.Patel, "Robust Coordinated Design of PSS
and TCSC using PSO Technique for Power System Stability
Enhancement", Journal of Electrical Systems, Vol. 3, No. 2, pp. 109-
123, 2007.
[21] C. B. Vishwakarma and R.Prasad, "Clustering Method for Reducing
Order of Linear System using Pade Approximation", IETE Journal of
Research, Vol. 54, Issue 5, pp. 326-330, 2008.
[22] S.Mukherjee, and R.N.Mishra, Order reduction of linear systems using
an error minimization technique, Journal of Franklin Inst. Vol. 323, No.
1, pp. 23-32, 1987.
[23] S.Panda, S.K.Tomar, R.Prasad, C.Ardil, "Reduction of Linear Time-
Invariant Systems Using Routh-Approximation and PSO", International
Journal of Applied Mathematics and Computer Sciences, Vol. 5, No. 2,
pp. 82-89, 2009.
[24] S.Panda, S.K.Tomar, R.Prasad, C.Ardil, "Model Reduction of Linear
Systems by Conventional and Evolutionary Techniques", International
Journal of Computational and Mathematical Sciences, Vol. 3, No. 1, pp.
28-34, 2009.
[25] R.Parthasarathy, and K. N. Jayasimha, System reduction using stability
equation method and modified Cauer continued fraction, Proc. IEEE
Vol. 70, No. 10, pp. 1234-1236, Oct. 1982.
[26] L.S. hieh, and Y.J.Wei, A mixed method for multivariable system
reduction, IEEE Trans. Autom. Control, Vol. AC-20, pp. 429-432, 1975.
[27] R.Prasad, and J.Pal, Stable reduction of linear systems by continued
fractions, J. Inst. Eng. India, IE (I) J.EL, Vol. 72, pp. 113-116, Oct.
1991.
[28] J. Pal, Stable reduced -order Pade approximants using the Routh-
Hurwitz array, Electronics letters, Vol. 15, No. 8, pp. 225-226, April
1979.
@article{"International Journal of Electrical, Electronic and Communication Sciences:54853", author = "S. Panda and J. S. Yadav and N. P. Patidar and C. Ardil", title = "Evolutionary Techniques for Model Order Reduction of Large Scale Linear Systems", abstract = "Recently, genetic algorithms (GA) and particle swarm optimization (PSO) technique have attracted considerable attention among various modern heuristic optimization techniques. The GA has been popular in academia and the industry mainly because of its intuitiveness, ease of implementation, and the ability to effectively solve highly non-linear, mixed integer optimization problems that are typical of complex engineering systems. PSO technique is a relatively recent heuristic search method whose mechanics are inspired by the swarming or collaborative behavior of biological populations. In this paper both PSO and GA optimization are employed for finding stable reduced order models of single-input- single-output large-scale linear systems. Both the techniques guarantee stability of reduced order model if the original high order model is stable. PSO method is based on the minimization of the Integral Squared Error (ISE) between the transient responses of original higher order model and the reduced order model pertaining to a unit step input. Both the methods are illustrated through numerical example from literature and the results are compared with recently published conventional model reduction technique.
", keywords = "Genetic Algorithm, Particle Swarm Optimization, Order Reduction, Stability, Transfer Function, Integral Squared Error.", volume = "6", number = "9", pages = "947-7", }
