Electromagnetic Force Analysis of Transformer on the Ferroresonance due to Consecutive 3–Phase Short–Circuit Faults Using Finite Element Method (FEM)
Subject Areas : Renewable energy
Ali Ahmadpour
1
,
SeyedJalal Seyed Shenava
2
,
Abdolmajid Dejamkhooy
3
,
Elham Mokaramian
4
1 - Department of Electrical Engineering, Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran.
2 - Department of Electrical Engineering, Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran.
3 - Department of Electrical Engineering, Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran.
4 - Department of Electrical Engineering, Faculty of Engineering, University of Mohaghegh Ardabili, Ardabil, Iran.
Keywords: Finite Element Method, Ferroresonance, Transformer, Electromagnetic force analysis, Short–circuit,
Abstract :
In this paper, Finite Element Method (FEM) is used to study the ferroresonance phenomenon from the perspective of the electromagnetic forces introduced on the transformer windings. In order to simulate this phenomenon, a consecutive 3–phase short–circuit fault is considered to be on the primary side at a relatively distant from the transformer. Then, due to the capacitance of the line, between the error point and transformer, the single–phase short–circuit faults are generated, one–by–one. Because of the switching, for isolation feeder from the fault point, the ferroresonance appears in power system. Then, voltages and currents of the primary and secondary windings will be changed. The effect of current changes on the forces that transformer coils withstand, when the ferroresonance happens, is an interesting topic to this paper. With modeling of the transformer and feeder switching in the FLUX 12.2 software, the ferroresonance condition is simulated. Due to the dependence of the ferroresonance on the non-linearity of the magnetic characteristic of the transformer core, Jiles–Atherton vector hysteresis model is used to modeling the core hysteresis loops for enhancement the results accurate.
[1] R. D. Evans, A. C. Monteith, and R. L. Witzke, "Power-system transients caused by switching and faults", IEEE Electrical Engineering, vol. 58, no. 8, pp. 386–396, Aug. 1939 (doi: 10.1109/T-AIEE.1939.5057978).
[2] A. Tokić, and J. Smajić, "Modeling and simulations of ferroresonance by using bdf/ndf numerical methods", IEEE Trans. on Power Delivery, vol. 30, no. 1, pp. 342–350, Sep. 2015 (doi: 10.1109/TPWRD. 2014.2346766).
[3] W. Sima, M. Yang, Q. Yang, T. Yuan, and M. Zou, "Simulation and experiment on a flexible control method for ferroresonance", IET Generation, Transmission and Distribution, vol. 8, no. 10, pp. 1744–1753, May. 2014 (doi: 10.1049/iet-gtd.2014.0046).
[4] P. H. Odessey, and E. Weber, "Critical conditions in ferroresonance", Electrical Engineering, vol. 57, no. 8, pp. 444–452, Aug. 1938 (doi: 10.1109/EE.1938.6430867).
[5] J. T. Salihi, "Theory of ferroresonance", Trans. of the American Institute of Electrical Engineers, Part I: Communication and Electronics, vol. 78, no. 6, pp. 755–763, Jan. 1960 (doi: 10.1109/TCE.1960.6368465).
[6] J. C. Davidson, "The phenomenon of ferroresonance", Students’ Quarterly Journal, vol. 41, no. 161, pp. 172–175, Sep. 1970 (doi: 10.1049/sqj.1970.0067).
[7] E. F. Kratz, L. W. Manning, and M. Maxwell, "Ferroresonance in series capacitor–distribution transrormer applications", IEEE Trans. of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, vol. 78, no. 3, pp. 438–445, Apr. 1959 (doi: 10.1109/AIEEPAS.1959.4500349).
[8] R. F. Karlicek, and E. R. Taylor, "Ferroresonance of grounded potential transformers on ungrounded power systems", IEEE Trans. of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, vol. 78, no. 3, pp. 607–614, Apr. 1959 (doi: 10.1109/AIEEPAS.1959.4500391).
[9] R. H. Hopkinson, "Ferroresonance during single–phase switching of 3–phase distribution transformer banks", IEEE Trans. on Power Apparatus and Systems, vol. 84, no. 4, pp. 289–293, Apr. 1965 (doi: 10.1109/TPAS.1965.4766193).
[10] A. E. A. Araujo, A. C. Soudack, and J. R. Marti, "Ferroresonance in power systems: chaotic behavior", IEE Proceedings C–Generation, Transmission and Distribution, vol. 140, no. 3, pp. 237–240, May. 1993 (doi: 10.1049/ip-c.1993.0035).
[11] H. Abdi, S. Abbasi, and M. Moradi, "Analyzing the stochastic behavior of ferroresonance initiation regarding initial conditions and system parameters", Electrical Power and Energy Systems, vol. 63, pp. 134–139, Dec. 2016 (doi: 10.1016/j.ijepes.2016.04.016).
[12] M. Esmaeili, M. Rostami, G. B. Gharehpetian, and C. P. McInnis, "Ferroresonance after islanding of synchronous machine-based distributed generation", Canadian Journal of Electrical and Computer Engineering, vol. 38, no. 2, pp. 154–161, May. 2015 (doi: 10.1109/CJECE.2015.2411713 ).
[13] U. Karaagac, J. Mahseredjian, and L. Cai, "Ferroresonance conditions in wind parks", Electric Power Systems Research, vol. 138, pp. 41–49, Sep. 2016 (doi: 10.1016/j.epsr.2016.04.007).
[14] R. P. Pineda, R. Rodrigues, and A. A. Telléz, "Analysis and simulation of ferroresonance in power transformers using simulink", IEEE Latin America Trans., vol. 16, no. 2, pp. 460–466, Mar. 2018 (doi: 10.1109/TLA.2018.8327400).
[15] E. Cazacu, V. Ionita, and Lucian Petrescu, "An efficient method for investigating the ferroresonance of single-phase iron core devices", The 10th Intetnational Symposium on Advanced Topics in Electrical Engineering, Bucharest: Romania, pp. 363–368, 23-25 Mar. 2017 (doi: 10.1109/ATEE.2017.7905167).
[16] J. A. Corea–Araujo, F. González–Molina, J. A. Martínez, F. Castro–Aranda, J. A. Barrado–Rodrigo, and L. Guasch–Pesquer, "Single-phase transformer model validation for ferroresonance analysis including hysteresis", IEEE Power & Energy Society General Meeting, Denver: CO: USA, 26-30 Jul. 2015 (doi: 10.1109/PESGM.2015.7285872).
[17] P. S. Moses, M. A. S. Masoum, and H. A. Toliyat, "Impacts of hysteresis and magnetic couplings on the stability domain of ferroresonance in asymmetric three–phase three–leg transformers", IEEE Trans. on Energy Conversion, vol. 26, no. 2, pp. 581–592, Dec. 2011 (doi: 10.1109/TEC.2010.2088400).
[18] A. Rezaei-Zare, M. Sanaye-Pasand, H. Mohseni, S. Farhangi, and R. Iravani, "Analysis of ferroresonance modes in power transformers using preisach–type hysteretic magnetizing inductance", IEEE Trans. on Power Delivery, vol. 26, no. 2, pp. 919–929, Apr. 2007 (doi: 10.1109/TPWRD.2006.877078).
[19] M. Yang, W. Sima, L. Chen, P. Duan, P. Sun, and T. Yuan, "Suppressing ferroresonance in potential transformers using a model–free active–resistance controller", Electrical Power and Energy Systems, vol. 95, pp. 384–393, Feb. 2018 (doi: 10.1016/j.ijepes.2017.08.035).
[20] T. C. Akinci, N. Ekren, S. Seker, and S. Yildirim, "Continuous wavelet transform for ferroresonance phenomena in electric power systems", Electrical Power and Energy Systems, vol. 44, pp. 403–409, Jan. 2013 (doi: 10.1016/j.ijepes.2012.07.001).
[21] H. Radmanesh, and G. B. Gharehpetian, "Ferroresonance suppression in power transformers using chaos theory", Electrical Power and Energy Systems, vol. 45, pp. 1–9, Feb. 2013 (doi: 10.1016/j.ijepes.2012. 08.028).
[22] M. Yang et al., "Electromagnetic transient study on flexible control processes of ferroresonance", Electrical Power and Energy Systems, vol. 93, pp. 194–203, Dec. 2017 (doi: 10.1016/j.ijepes.2017.05.026).
[23] B. Rezaeealam, B. Norouzi, "Investigating ferroresonance phenomenon in a single–phase transformer with the effect of magnetic hysteresis", Indonesian Journal of Electrical Engineering and Computer Science, vol. 2, no. 2, pp. 248–258, May. 2016 (doi: 10.11591/ijeecs.v2.i2.pp248-258).
[24] C. A. Charalambous, Z. Wang, P. Jarman, and J. P. Sturgess, "Time-domain finite-element technique for quantifying the effect of sustained ferroresonance on power transformer core bolts", IET Electric Power Applications, vol. 8, no. 6, pp. 221–231, Mar. 2014 (doi: 10.1049/iet-epa.2013.0330 ).
[25] C. A. Charalambous, Z. D. Wang, P. Jarman, and M. Osborne, "2–D finite–element electromagnetic analysis of an autotransformer experiencing ferroresonance", IEEE Trans. on Power Delivery, vol. 24, no. 3, pp. 1275–1283, Apr. 2009 (doi: 10.1109/TPWRD.2009.2016629).
[26] A. Arroyo, R. Martinez, M. Manana, A. Pigazo, R. Minguez, "Detection of ferroresonance occurrence in inductive voltage transformers through vibration analysis", Electrical Power and Energy Systems, vol. 106, pp. 294–300, Mar. 2019 (doi: 10.1016/j.ijepes.2018.10.011).
[27] V. Valverde, G. Buigues, A. J. Mazón, I. Zamora, I. Albizu, "Ferroresonant configurations in power systems", International Conference on Renewable Energies and Power Quality (ICREPQ’12), Santiago de Compostela, Spain, 28–30 March, vol. 1, no. 10, pp. 474–479, Mar. 2012 (doi: 10.24084/repqj10.351).
[28] A. Djebli, F. Aboura, L. Roubache, O. Touhami, "Impact of the eddy current in the lamination on ferroresonance stability at critical points", Electrical Power and Energy Systems, vol. 106, pp. 311–319, Mar. 2019 (doi: 10.1016/j.ijepes.2018.10.008).
[29] M. M. Beyranvand, B. Rezaeealam, J. Faiz, A. R. Zare, "Impacts of ferroresonance and inrush current forces on transformer windings", IET Electric Power Applications, vol. 13, no. 7, pp. 914–921, Feb. 2019 (doi: 10.1049/iet-epa.2018.5193).
[30] Alexandre B. Nassif, M. Dong, S. Kumar, G. Vanderstar, "Managing ferroresonance overvoltages in distribution systems", IEEE Canadian Conference of Electrical and Computer Engineering (CCECE), Edmonton, AB, Canada, 5–8 May 2019 (doi: 10.1109/CCECE.2019.8861797).
[31] M. Navaei, A. A. Abdoos, M. Shahabi, "A new control unit for electronic ferroresonance suppression circuit in capacitor voltage transformers", Electrical Power and Energy Systems, vol. 99, pp. 281–289, Jul. 2018 (doi: 10.1016/j.ijepes.2018.01.021).
[32] F. Delinc, A. Nicolet, W. Legros, and A. Genon, "Analysis of ferroresonance with a finite element method taking hysteresis into account", Journal of Magnetism and Magnetic Materials, vol. 133, pp. 557–560, May. 1994 (doi: 10.1016/0304-8853(94)90621-1).
[33] A. G. MacPhee, S. McKee, and R. R. S. Simpson, "Ferroresonance in electrical systems", COMPEL–The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 21, no. 2, pp. 265–273, Jun. 2002 (doi: 10.1108/03321640210416359).
[34] W. Sima, M. Zou, M. Yang, D. Peng, and Y. Liu, "Saturable reactor hysteresis model based on Jiles–Atherton formulation for ferroresonance studies", Electrical Power and Energy Systems, vol. 101, pp. 482–490, Oct. 2018 (doi: 10.1016/j.ijepes.2018.04.003).
[35] M. M. Beyranvand, and B. Rezaeealam, "Finite element study of ferroresonance in single–phase transformers considering magnetic hysteresis", Journal of Magnetics, vol. 22, no. 2, pp. 196–202, Apr. 2017 (doi: 10.4283/JMAG.2017.22.2.196).
[36] J. Gyselinck, P. Dular, N. Sadowski, J. Leite and J. P. A. Bastos, "Incorporation of a Jiles–Atherton vector hysteresis model in 2D FE magnetic field computations, application of the Newton-Raphson method", COMPEL–The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 23, no. 3, pp. 685–693, Sep. 2004 (doi: 10.1108/03321640410540601).
[37] J. Gyselinck, L. Vandevelde, J. Melkebeek, and P. Dular, "Complementary two-dimensional finite element formulations with inclusion of a vectorized Jiles–Atherton model", COMPEL–The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 23, no. 4, pp. 959–967, Dec. 2004 (doi: 10.1108/03321640410553382).
[38] C. Guérin, K. Jacques, R. V. Sabariego, P. Dular, C. Geuzaine, and J. Gyselinck, "Using a Jiles–Atherton vector hysteresis model for isotropic magnetic materials with the finite element method, Newton–Raphson method, and relaxation procedure", International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 30, no. 5, pp. 1–12, Sep. 2017 (doi: 10.1002/jnm.2189).
[39] K. Chwastek, "The applications of fixed-point theorem in optimisation problems", Archives of Electrical Engineering, vol. 61, no. 2, pp. 189–198, Jun. 2012 (doi: 10.2478/v10171-012-0016-0).
_||_[1] R. D. Evans, A. C. Monteith, and R. L. Witzke, "Power-system transients caused by switching and faults", IEEE Electrical Engineering, vol. 58, no. 8, pp. 386–396, Aug. 1939 (doi: 10.1109/T-AIEE.1939.5057978).
[2] A. Tokić, and J. Smajić, "Modeling and simulations of ferroresonance by using bdf/ndf numerical methods", IEEE Trans. on Power Delivery, vol. 30, no. 1, pp. 342–350, Sep. 2015 (doi: 10.1109/TPWRD. 2014.2346766).
[3] W. Sima, M. Yang, Q. Yang, T. Yuan, and M. Zou, "Simulation and experiment on a flexible control method for ferroresonance", IET Generation, Transmission and Distribution, vol. 8, no. 10, pp. 1744–1753, May. 2014 (doi: 10.1049/iet-gtd.2014.0046).
[4] P. H. Odessey, and E. Weber, "Critical conditions in ferroresonance", Electrical Engineering, vol. 57, no. 8, pp. 444–452, Aug. 1938 (doi: 10.1109/EE.1938.6430867).
[5] J. T. Salihi, "Theory of ferroresonance", Trans. of the American Institute of Electrical Engineers, Part I: Communication and Electronics, vol. 78, no. 6, pp. 755–763, Jan. 1960 (doi: 10.1109/TCE.1960.6368465).
[6] J. C. Davidson, "The phenomenon of ferroresonance", Students’ Quarterly Journal, vol. 41, no. 161, pp. 172–175, Sep. 1970 (doi: 10.1049/sqj.1970.0067).
[7] E. F. Kratz, L. W. Manning, and M. Maxwell, "Ferroresonance in series capacitor–distribution transrormer applications", IEEE Trans. of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, vol. 78, no. 3, pp. 438–445, Apr. 1959 (doi: 10.1109/AIEEPAS.1959.4500349).
[8] R. F. Karlicek, and E. R. Taylor, "Ferroresonance of grounded potential transformers on ungrounded power systems", IEEE Trans. of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems, vol. 78, no. 3, pp. 607–614, Apr. 1959 (doi: 10.1109/AIEEPAS.1959.4500391).
[9] R. H. Hopkinson, "Ferroresonance during single–phase switching of 3–phase distribution transformer banks", IEEE Trans. on Power Apparatus and Systems, vol. 84, no. 4, pp. 289–293, Apr. 1965 (doi: 10.1109/TPAS.1965.4766193).
[10] A. E. A. Araujo, A. C. Soudack, and J. R. Marti, "Ferroresonance in power systems: chaotic behavior", IEE Proceedings C–Generation, Transmission and Distribution, vol. 140, no. 3, pp. 237–240, May. 1993 (doi: 10.1049/ip-c.1993.0035).
[11] H. Abdi, S. Abbasi, and M. Moradi, "Analyzing the stochastic behavior of ferroresonance initiation regarding initial conditions and system parameters", Electrical Power and Energy Systems, vol. 63, pp. 134–139, Dec. 2016 (doi: 10.1016/j.ijepes.2016.04.016).
[12] M. Esmaeili, M. Rostami, G. B. Gharehpetian, and C. P. McInnis, "Ferroresonance after islanding of synchronous machine-based distributed generation", Canadian Journal of Electrical and Computer Engineering, vol. 38, no. 2, pp. 154–161, May. 2015 (doi: 10.1109/CJECE.2015.2411713 ).
[13] U. Karaagac, J. Mahseredjian, and L. Cai, "Ferroresonance conditions in wind parks", Electric Power Systems Research, vol. 138, pp. 41–49, Sep. 2016 (doi: 10.1016/j.epsr.2016.04.007).
[14] R. P. Pineda, R. Rodrigues, and A. A. Telléz, "Analysis and simulation of ferroresonance in power transformers using simulink", IEEE Latin America Trans., vol. 16, no. 2, pp. 460–466, Mar. 2018 (doi: 10.1109/TLA.2018.8327400).
[15] E. Cazacu, V. Ionita, and Lucian Petrescu, "An efficient method for investigating the ferroresonance of single-phase iron core devices", The 10th Intetnational Symposium on Advanced Topics in Electrical Engineering, Bucharest: Romania, pp. 363–368, 23-25 Mar. 2017 (doi: 10.1109/ATEE.2017.7905167).
[16] J. A. Corea–Araujo, F. González–Molina, J. A. Martínez, F. Castro–Aranda, J. A. Barrado–Rodrigo, and L. Guasch–Pesquer, "Single-phase transformer model validation for ferroresonance analysis including hysteresis", IEEE Power & Energy Society General Meeting, Denver: CO: USA, 26-30 Jul. 2015 (doi: 10.1109/PESGM.2015.7285872).
[17] P. S. Moses, M. A. S. Masoum, and H. A. Toliyat, "Impacts of hysteresis and magnetic couplings on the stability domain of ferroresonance in asymmetric three–phase three–leg transformers", IEEE Trans. on Energy Conversion, vol. 26, no. 2, pp. 581–592, Dec. 2011 (doi: 10.1109/TEC.2010.2088400).
[18] A. Rezaei-Zare, M. Sanaye-Pasand, H. Mohseni, S. Farhangi, and R. Iravani, "Analysis of ferroresonance modes in power transformers using preisach–type hysteretic magnetizing inductance", IEEE Trans. on Power Delivery, vol. 26, no. 2, pp. 919–929, Apr. 2007 (doi: 10.1109/TPWRD.2006.877078).
[19] M. Yang, W. Sima, L. Chen, P. Duan, P. Sun, and T. Yuan, "Suppressing ferroresonance in potential transformers using a model–free active–resistance controller", Electrical Power and Energy Systems, vol. 95, pp. 384–393, Feb. 2018 (doi: 10.1016/j.ijepes.2017.08.035).
[20] T. C. Akinci, N. Ekren, S. Seker, and S. Yildirim, "Continuous wavelet transform for ferroresonance phenomena in electric power systems", Electrical Power and Energy Systems, vol. 44, pp. 403–409, Jan. 2013 (doi: 10.1016/j.ijepes.2012.07.001).
[21] H. Radmanesh, and G. B. Gharehpetian, "Ferroresonance suppression in power transformers using chaos theory", Electrical Power and Energy Systems, vol. 45, pp. 1–9, Feb. 2013 (doi: 10.1016/j.ijepes.2012. 08.028).
[22] M. Yang et al., "Electromagnetic transient study on flexible control processes of ferroresonance", Electrical Power and Energy Systems, vol. 93, pp. 194–203, Dec. 2017 (doi: 10.1016/j.ijepes.2017.05.026).
[23] B. Rezaeealam, B. Norouzi, "Investigating ferroresonance phenomenon in a single–phase transformer with the effect of magnetic hysteresis", Indonesian Journal of Electrical Engineering and Computer Science, vol. 2, no. 2, pp. 248–258, May. 2016 (doi: 10.11591/ijeecs.v2.i2.pp248-258).
[24] C. A. Charalambous, Z. Wang, P. Jarman, and J. P. Sturgess, "Time-domain finite-element technique for quantifying the effect of sustained ferroresonance on power transformer core bolts", IET Electric Power Applications, vol. 8, no. 6, pp. 221–231, Mar. 2014 (doi: 10.1049/iet-epa.2013.0330 ).
[25] C. A. Charalambous, Z. D. Wang, P. Jarman, and M. Osborne, "2–D finite–element electromagnetic analysis of an autotransformer experiencing ferroresonance", IEEE Trans. on Power Delivery, vol. 24, no. 3, pp. 1275–1283, Apr. 2009 (doi: 10.1109/TPWRD.2009.2016629).
[26] A. Arroyo, R. Martinez, M. Manana, A. Pigazo, R. Minguez, "Detection of ferroresonance occurrence in inductive voltage transformers through vibration analysis", Electrical Power and Energy Systems, vol. 106, pp. 294–300, Mar. 2019 (doi: 10.1016/j.ijepes.2018.10.011).
[27] V. Valverde, G. Buigues, A. J. Mazón, I. Zamora, I. Albizu, "Ferroresonant configurations in power systems", International Conference on Renewable Energies and Power Quality (ICREPQ’12), Santiago de Compostela, Spain, 28–30 March, vol. 1, no. 10, pp. 474–479, Mar. 2012 (doi: 10.24084/repqj10.351).
[28] A. Djebli, F. Aboura, L. Roubache, O. Touhami, "Impact of the eddy current in the lamination on ferroresonance stability at critical points", Electrical Power and Energy Systems, vol. 106, pp. 311–319, Mar. 2019 (doi: 10.1016/j.ijepes.2018.10.008).
[29] M. M. Beyranvand, B. Rezaeealam, J. Faiz, A. R. Zare, "Impacts of ferroresonance and inrush current forces on transformer windings", IET Electric Power Applications, vol. 13, no. 7, pp. 914–921, Feb. 2019 (doi: 10.1049/iet-epa.2018.5193).
[30] Alexandre B. Nassif, M. Dong, S. Kumar, G. Vanderstar, "Managing ferroresonance overvoltages in distribution systems", IEEE Canadian Conference of Electrical and Computer Engineering (CCECE), Edmonton, AB, Canada, 5–8 May 2019 (doi: 10.1109/CCECE.2019.8861797).
[31] M. Navaei, A. A. Abdoos, M. Shahabi, "A new control unit for electronic ferroresonance suppression circuit in capacitor voltage transformers", Electrical Power and Energy Systems, vol. 99, pp. 281–289, Jul. 2018 (doi: 10.1016/j.ijepes.2018.01.021).
[32] F. Delinc, A. Nicolet, W. Legros, and A. Genon, "Analysis of ferroresonance with a finite element method taking hysteresis into account", Journal of Magnetism and Magnetic Materials, vol. 133, pp. 557–560, May. 1994 (doi: 10.1016/0304-8853(94)90621-1).
[33] A. G. MacPhee, S. McKee, and R. R. S. Simpson, "Ferroresonance in electrical systems", COMPEL–The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 21, no. 2, pp. 265–273, Jun. 2002 (doi: 10.1108/03321640210416359).
[34] W. Sima, M. Zou, M. Yang, D. Peng, and Y. Liu, "Saturable reactor hysteresis model based on Jiles–Atherton formulation for ferroresonance studies", Electrical Power and Energy Systems, vol. 101, pp. 482–490, Oct. 2018 (doi: 10.1016/j.ijepes.2018.04.003).
[35] M. M. Beyranvand, and B. Rezaeealam, "Finite element study of ferroresonance in single–phase transformers considering magnetic hysteresis", Journal of Magnetics, vol. 22, no. 2, pp. 196–202, Apr. 2017 (doi: 10.4283/JMAG.2017.22.2.196).
[36] J. Gyselinck, P. Dular, N. Sadowski, J. Leite and J. P. A. Bastos, "Incorporation of a Jiles–Atherton vector hysteresis model in 2D FE magnetic field computations, application of the Newton-Raphson method", COMPEL–The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 23, no. 3, pp. 685–693, Sep. 2004 (doi: 10.1108/03321640410540601).
[37] J. Gyselinck, L. Vandevelde, J. Melkebeek, and P. Dular, "Complementary two-dimensional finite element formulations with inclusion of a vectorized Jiles–Atherton model", COMPEL–The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 23, no. 4, pp. 959–967, Dec. 2004 (doi: 10.1108/03321640410553382).
[38] C. Guérin, K. Jacques, R. V. Sabariego, P. Dular, C. Geuzaine, and J. Gyselinck, "Using a Jiles–Atherton vector hysteresis model for isotropic magnetic materials with the finite element method, Newton–Raphson method, and relaxation procedure", International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol. 30, no. 5, pp. 1–12, Sep. 2017 (doi: 10.1002/jnm.2189).
[39] K. Chwastek, "The applications of fixed-point theorem in optimisation problems", Archives of Electrical Engineering, vol. 61, no. 2, pp. 189–198, Jun. 2012 (doi: 10.2478/v10171-012-0016-0).
