Analysis of Bifurcation Phenomenon in a Grid-Connected Electric Arc Furnace

Subject Areas : Renewable energy

Mohammad Behzad Es-haghi ¹ , Mehran Zamanifar ²

1 - 1 Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran 2 Smart Microgrid Research Center, Najafabad Branch, Islamic Azad University, Isfahan, Iran
2 - Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Isfahan, Iran Smart Microgrid Research Center, Najafabad Branch, Islamic Azad University, Isfahan, Iran

Received: 2017-11-13 Accepted : 2018-03-05 Published : 2018-05-05

Keywords: Electric Arc Furnace, Bifurcation theory, projection method, center manifold,

Abstract :

This paper aims to study the stability and dynamic behavior of a grid-connected electricarc furnace system, using bifurcation theory. This theory introduces a systematic method for stability analysis of dynamic systems, under changes in the system parameters. In fact, a parameter is constantly changed in each step, using MATLAB and/or AUTO software, and system eigenvalues are monitored simultaneously. Considering how the eigenvalues approach the system’s imaginary axis on S plain in accordance with the changes in the targeted parameter, the occurred saddle-node and/or Hopf bifurcation of the system is extracted. In this paper, at first, electric arc furnace is modeled by the differential-algebraic equations and then based on the projection method, two-dimensional center manifold at the Hopf bifurcation happens in the electric arc furnace system is achieved analytically. Projection method is discussed in the bifurcation theory and is presented totally in this paper. Finally, the results are compared by the computer simulations.

References:

[1] E. Acha, R.N. Semlyen, “A harmonic domain computational package for nonlinear problems and its applications to electrics arcs”, IEEE Trans. on Power Delivery, Vol. 5, No. 3, pp. 1390–1397, July 1990.

[2] G. Carpinelli, F. Iacovone, A. Russo, P. Varilone, “Chaos-based modeling of DC arc furnaces for power quality issues”, IEEE Trans. on Power Delivery, Vol. 19, pp. 1869–1876, 1990.

[3] A. Douglas, T.B. Martin, J.F. Witte, “Harmonic measurement, analysis, and power factor correction in a modern steel manufacturing facility”, IEEE Trans. on Power Delivery, Vol. 32, pp. 617–624, 1996.

[4] R.C. Dungan, “Simulation of arc furnace power systems”, IEEE Trans. on Industrial Application, Vol. 16, pp. 813–818, 1980.

[5] A. Medina, N. Garcia, “Newton methods for the fast computation of the periodic steady state solutions of systems with nonlinear and time-varying components”, Proceeding of the IEEE/PESS, pp. 36-42.  Edmonton, Alta., Canada, Canada, pp. 664-669, 1999.

[6] I.A. Hiskens, “Analysis tools for power systems-contending with nonlinearities”, Proceeding of the IEEE, Vol. 83, No. 11, pp. 1573–1587, Nov. 1995.

[7] V. Ajjarapu, B. Lee, “Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system”, IEEE Trans. on Power System, Vol. 7, pp. 312–319, 1992.

[8] H. Wang, E. Abed, A.M.A. Hamdan, “Bifurcations, chaos, and crises in voltage collapse of a model power system”, IEEE Trans. on Circuits System I: Fund. Theory Application, Vol. 41, pp. 294–302, 1994.

[9] V. Ajjarapu, B. Lee, “Bifurcation, theory and its application to nonlinear dynamical phenomena in an electrical power system”, IEEE Trans. on Power System, Vol. 7, pp. 424–431, 1992.

[10] W. Ji, V. Venkatasubramanian, “Dynamics of a minimal power system: invariant tori and quasi-periodic motions”, IEEE Trans. on Circuits System I: Fund. Theory Application, Vol. 42, pp. 981–1000, 1995.

[11] C.A. Canizares, Voltage stability assessment: concepts, practices and tools, Technical Report IEEE/PES Power System Stability Subcommittee, Final Document, available at http://www.power.uwaterloo.ca, Aug. 2002.

[12] N. Koppel, R B. Washburn, “Chaotic motions in the two-degree-of-freedom swing equations”, IEEE Trans. on Circuits System, Vol. 29, pp. 738–746, 1982.

[13] I. Dobson, H.D. Chiang, “Toward a theory of voltage collapse in electric power systems”, IEEE Trans. on Power System, Vol. 13, pp. 253–262, 1999.

[14] M. Varghese, F.F. Wu, P.Varaiya, “Bifurcations associated with subsynchronous resonance”, IEEE Trans. on Power System, Vol. 13, pp. 139–144, 1998.

[15] C. Kieny, “Application of the bifurcation theory in studying and understanding the global behavior of a ferroresonant electric power circuit”, IEEE Trans. on Power Delivery, Vol. 6, pp. 866–872, 1991.

[16] H.O. Wang, E.H. Abed, A.M.A. Hamdan, “Bifurcation, chaos and crisis in voltage collapse of a model power system”, IEEE Trans. on Circuits System, Vol. 41, pp. 294–302, 1994.

[17] S.H. Lee, J.K. Park, B.H. Lee, “A study on the nonlinear controller to prevent unstable Hopf bifurcation”, Proceeding of the IEEE/PESS, pp. 978–982, Vancouver, BC, Canada, Canada, July 2001.

[18] W.D. Rosehart, C.A. Cañizares, “Bifurcation analysis of various power system models”, International Journal of Electric Power Energy System, Vol. 21, pp. 171–182, 1999.

[19] C.A. Cañizares, “On bifurcations, voltage collapse and load modeling”, IEEE Trans. on Power System, Vol. 10, pp. 512–522, 1995.

[20] M.A. Pai, P.W. Sauer, B.C. Lesieutre, “Structural stability in power systems-effect of load models”, IEEE Trans. on Power System, Vol. 10, pp. 609–615., 1995.

[21] T.K. Vu, C.C. Liu, “Analysis of tap-changer dynamics and construction of voltage stability regions”, IEEE Trans. on Circuits System, Vol. 36, pp. 575–590, 1989.

[22] N. Mithulananthan, C.A. Cañizares, J. Reeve,  G.J. Rogers, “Comparison of PSSS, SVC and STATCOM controllers for damping power system oscillations”, IEEE Trans. on Power System, Vol. 18, pp. 786–792, 2003.

[23] R. Garcia-Kasusky, C.R. Fuerte-Esquivel, D. Torres-Lucio, “Assessment of the SVC effect on nonlinear instabilities and voltage collapse in electric power systems”, Proceeding of the IEEE/PES,  pp. 2659–2666, Toronto, Ont., Canada, July 2003.

[24] Y.A. Kuznetsov, “Elements of applied bifurcation theory”, Second edition, Springer, 1998.

[25] S. Wiggins, “Introduction to applied nonlinear dynamical systems”, Second edition, Springer, 2003.

[26] Y. Ma, H. Wen, X. Zhou, J. Li, H. Yang, “Bifurcation analysis on power system voltage stability”, Proceeding of the IEEE/ICICTA, pp. 26-29, Oct. 2009.

[27] Y. Ma, H. Wen, X. Zhou, J. Li, H. Yang,”Calculation and study of two-dimensional parameter local bifurcation boundary in wind power system stability model based on continuation method”, Proceeding of the IEEE/WNWEC, pp.10-16, Nanjing, China, Sep.2009 .

[28] Y.Ma, X. Li, X. Zhou, J. Li, “The comments on dynamic bifurcation of voltage stability in power system”, Proceeding of the IEEE/ICIE, pp. 272-275, Beidaihe, Hebei, China, Aug. 2010.

[29] K. Skandarama, R.C. Mala, N. Prabhu, “Control of bifurcation in a VSC based STATCOM”, Electrical Power and Energy Systems, Vol. 21, pp. 187-195, 2015.

[30] A. Medina, M.A Gómez-Martínez, C.R. Fuerte-Esquivel, “Application of bifurcations theory to assess nonlinear oscillations produced by ac electric arc furnaces”, IEEE Trans. on Power Delivery, Vol. 20, pp. 801-806, 2005.

[31] M.A. Gómez-Martínez, A. Medina, C.R. Fuerte-Esquivel, “AC arc furnace stability analysis based on bifurcation theory”, IEE Proceeding on Generation, Transmission and Distribution, Vol. 153, No. 4, pp. 463-468, July 2006.

[32] A. Khakpour, S. Franke, S. Gortschakow, D. Uhrlandt, R. Methling, K. D. Weltmann, “An improved arc model based on the arc dimeter”, IEEE Trans. on Power Delivery, Vol. 31, pp. 1335-1341, 2016.