### Application of Singular Perturbation Method to Two Parameter Discrete Power System Model

#### Abstract

The singularly perturbed systems are ill-conditioned systems exhibiting slow fast phenomena with computational stiffness. Hence exact solution of these systems requires special numerical methods to overcome this stiffness. Singular perturbation theory of continuous-time control systems has reached its peak. Whereas singular perturbation theory of discrete-time control systems is still in developing stage. Singular perturbation methodology (SPM) is being developed for one- and multi parameter problems in discrete control systems. The SPM removes the system’s stiffness and satisfies all the specified boundary conditions thereby giving an approximate solution very close to the exact solution.

Here two parameter SPM is presented for linear time-invariant discrete control system for both initial and boundary value problems. Then these are applied to a fifth-order steam power system model up to second order approximation as a case study. These results are compared with the exact solution. This comparison justifies the application and SPM presented.

#### Full Text:

PDF#### References

[1]. Bjurel G. (1970) “Survey of stiff ordinary differential equations”. Royal Inst. Of Tech. Report NA 70.11, Stockholm, Sweden.6

Comstock C. and Hsiao G. C. (1976) “Singular perturbations for difference equations”. Rocky Mountain J. Math., 6, pp. 561-567.8

Dmitriev M. G. and Kurina G. (2006) “Singular perturbation method in control problems”. Automation and Remote Control, 67, pp. 1-43.13

Haller M. A. (1983) “A multiple-scale approach in the study of communication phenomena: application to a chopper supplying a step motor”. Elect. Machines & Power Syst., 8, pp.113-122.26

Krishnarayalu M. S. (2009) “Generalized Singular Perturbation Methodology for Boundary Value Problems of Digital Control Systems”. IJEEE, Vol. 3, No. 4, pp 104-121, 2009.29

Krishnarayalu M. S.(2004) “Singular perturbation methods for a class of initial and boundary value problems in multi-parameter classical digital control systems”. ANZIAM J. 46, 67-77 31

Krishnarayalu M. S.(1999) "Singular perturbation methods for one-point, two-point and multi-point boundary value problems in multiparameter digital control systems". Journal of Electrical and Electronics Engineering, Australia, Vol. 19, No. 3, PP 97-110, Sep 1999. 32

Krishnarayalu M. S.(1994) “Singular perturbation method for a class of initial and boundary value problems in digital control systems”. IE(I) Journal-EL, Vol. 75, PP 41-45, Aug 1994.34

Krishnarayalu M. S. and Naidu D. S. (1985) “Discrete modelling of singularly perturbed continuous systems”. International Journal of Modelling and Simulation.

Kishore Babu G. and Krishnarayalu M. S. (2015) “Suboptimal Control of Singularly Perturbed Multiparameter Discrete Control System”. PICC-2015 (IEEE Conference), Dec 9-11, 2015, Trichur, India.42

Kishore Babu G. and Krishnarayalu M. S. (2014) “Some Applications of Discrete One Parameter Singular Perturbation Method”. Journal of Control Engineering and Technology, Vol. 4, Issue 1, January 2014, Pages 76-81.46

Naidu D. S. (1987) “Singular perturbations and time scale in discrete control systems-an overview”. In Proceedings of the 26th IEEE Conference on Decision and Control, pages 2096–2103, Los Angeles, CA. 71

Naidu D. S. (1985) Authors reply to “Application of singular perturbation method to a steam power systems”. Elect. Power Syst. Res., 8, pp. 219-226.75

Naidu D. S. and Calise A. J. (1989) “Singular perturbations and time scales in control theory and applications: survey 1983-1989”. In IFAC Workshop on Singular Perturbations and Asymptotic Methods in Systems and Control, Boston, MA, August 17-18.77

Naidu D. S. and Rao A. K. (1985) “Singular Perturbation Analysis of Discrete Control Systems”. Volume 1154 of Lecture Notes in Mathematics. Springer-Verlag, New York.79

Phillips (1980) “Reduced order modelling and control of two-time scale discrete systems”, Int. J. Control, 31, pp.765-780 83

Rajagopalan and Naidu D. S. (1980) “A Singular perturbations method for discrete control systems”. Int. J. Control, 32, pp.925-936.85

Sannuti P. and Kokotovi´c P. V. (1969) “Near optimum design of linear systems by singular perturbation method”. IEEE Transactions on Automatic Control, AC-14:15–22, 1969.97

Yong Chen and Yongqiang Liu (2005) “Summary of Singular Perturbation Modelling of Multi-time Scale Power Systems”. Transmission and Distribution Conference and Exhibition: Asia and Pacific, IEEE/PES, pp 1-4. 100.

B.M. Patre and B. Bandyopadhyay (1999) “Near Optimal Control for Discrete Two-time-Scale Systems with Poles in Specified Circular Region”. European Control Conference (ECC).