Elektrotehniški vestnik 80(1-2): 8-12, 2013 Original Scientific Paper Impact of the system parameters on the ferroresonant modes Marina Pejic, Amir Tokic University of Tuzla, Faculty of Electrical Engineering, Bosnia and Herzegovina E-mail: marina.benes@uuntz.ba, amir.tokic@untz.ba Abstract. The ferorresonance is a complicated and hard to predict phenomenon, and because of its harmful impact on the electrical equipment very interesting to be known in a greater detail. The ferroresonance occurs in several modes: fundamental, subharmonic, quasi-periodic and chaotic. This paper analyses the impact of the system parameters on obtaining the different ferroresonance modes. The analysis is carried out by simulating the behavior of one of the most common examples of the ferroresonance occurrence: unloaded single-phase transformer which is, when switched on, energized over the grading capacitor. The ferroresonant modes are indentified and represented by three different techniques: the spectral-density analysis, the phaseplane analysis and the Poincare map. Keywords: ferroresonance, analysis, modes, techiques. Vpliv sistemskih parametrov na feroresonančne pojave Feroresonanca je zapleten in težko predvidljiv pojav, ki je zaradi možnega škodljivega vpliva na električno opremo zelo zanimiv za analizo. Feroresonanca se pojavlja na različne načine: osnovni, podharmonski, kvazi-periodični in kaotični. V članku obravnavamo vpliv sistemskih parametrov na različne feroresonančne pojave. Analizo smo izvedli s simulacijo delovanja neobremenjenega enofaznega transformatorja, enega izmed najbolj pogostih primerov pojavov feroresonance. Feroresonančni vplivi so predstavljeni s tremi različnimi tehnikami: z analizo spektralne gostote, z analizo v fazni ravnini in s Poincarovo analizo. 1 Introduction The ferroresonance is a nonlinear phenomenon that is sensitive to the parameters and initial conditions of the system. The basic element of the ferroresonant circuit is a nonlinear inductance, but for the ferroresonance to occur, the electrical circuit must also contain a capacitor, voltage source (usually sinusoidal) and low losses [1], [2]. Due to the existence of many sources of capacitors and non-linear inductances, and a wide range of operating states, configurations under which the ferroresonance takes place are innumerable. One of these configurations is an unloaded single-phase transformer which is, when switched on, energized over the grading capacitor (Fig. 1), [3]-[5]. Figure 1. Equivalent scheme of the ferroresonant serial electrical circuit The waveforms of magnitudes occurring in a power system and experiments carried out on a reduced system model together with numerical simulations enable the ferroresonant modes to be divided into four types: fundamental, subharmonic, quasi-periodic and chaotic. In the fundamental mode, the voltage or current waveforms are distorted, but their period of oscillation is equal to the period of the source. The subharmonic mode is characterized by the periodic voltage or current signals, but the period of its oscillation is an integer multiple of the source period. In the quasi-periodic mode, the voltage or current signals are not periodic and in the chaotic mode, the voltage or current signals show an unpredictable behavior [6], [7]. The different ferroresonant modes can be obtained by changing the system parameters, which are: sizes that define the transformer magnetizing curve, grading capacitor, switching time, system initial states, amplitude of the voltage source, etc. [8]. By changing the grading capacitor Cg, the resistance of the transformer magnetizing branch Rm and the amplitude of the voltage source Um, the different ferroresonant modes can be obtained. The ferroresonant behavior of the dynamic systems can be analyzed on the basis of three different methods: the spectral-density analysis, the phase-plane analysis and the Poincare map [3], [6]. Our presentation of the ferroresonant modes will be made by using these techniques. 2 Ferrroresonant modes To simulate the single-phase transformer ferroresonance, the software ATP (Alternative Transient Program) [10] is used. It is a version of EMTP (Electromagnetic Transient Program) [11], the software for the analysis of the electromagnetic transient phenomena taking place in the power system. The program has a graphical user interface implemented in the graphic preprocessor ATPDraw [12], enabling a relatively simple construction of models of the electrical circuits. The single-phase transformer is represented by its equivalent scheme, where the magnetizing branch of the transformer is represented by a linear resistance Rm and nonlinear inductance. Resistance Rg and inductance Lg represent the network participation. The simulation duration is , while the step time is A t = 1 0 " 6 sec. The ATPDraw simulation scheme of the serial ferroresonant electrical circuit is shown in Fig. 2, [13]. Transformer voltage Table 1 : System parameter ferroresonance values fundamental Fundamental ferroresonance Rm [A] 2180,9 Cg [pF] 3,75 Um [V] 325,27 650,0487,5325,0162,50,0-162,5-325,0-487,5-650,00,70 0,75 0,80 (file jednofazni.pl4; x-var t) v:XX0005 Figure 3. Transformer voltage waveform - fundamental ferroresonance The harmonic components of the voltage and current signals are analyzed by using the spectral density method. This method is used to obtain the characteristic frequencies that are present in the signal. The presence of more than one characteristic frequency indicates the multiple periodicity, which is common in some ferroresonant states. The spectral analysis used for the transformer voltage waveform is shown in Fig. 4. Figure 2. ATPDraw simulation scheme of the serial ferroresonant electrical circuit 2.1 Fundamental ferroresonance The fundamental ferroresonance occurs when the values of the system parameters are the ones shown in Table 1. 220 200 - 180 - 160 - S 140 -(D I 120- o > 100 -CO 180 -60 - 40 - 20 - 0 Harmonic spectrum - fundamental ferroresonance 150 200 Frequency [Hz] The transformer voltage waveform of the fundamental ferroresonance is shown in Fig. 3. Figure 4. Spectral analysis of the transformer voltage waveform - fundamental ferroresonance Based on the results of the spectral analysis of the transformer voltage waveform, we can conclude that the voltage spectrum consists of a basic harmonic (/0 = 50 Hz), and its harmonics ( 3/0,5/0, etc.). The phase plane is a diagram which consists of two state variables: transformer voltage and current (Fig. 5). The result is a shift of the point in the time that follows the trajectory. The periodic solutions correspond to the closed trajectories. 0,85 0,90 0,95 1,00 50 100 250 300 Um [V] 325,27 400 300 200 „ 100 aj n CH 0 -S Ö > -100 -200 -300 -400 Poincare map (fundamental ferroresonance) -2 -1.5 -0.5 0 0.5 Current [A] 1.5 Figure 6. Poincare map - fundamental ferroresonance The Poincare map shows the point far away from the point representing the normal state [3], [6]. 2.2 Subharmonic ferroresonance The subharmonic ferroresonance occurs when the values of the system parameters have values are the ones shown in Table 2. The transformer voltage waveform of the subharmonic ferroresonance is shown in Fig. 7. Transformer voltage -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Current [A] Figure 5. Phase- plane analysis - fundamental ferroresonance The phase-plane is represented by a closed trajectory which tells us that this is a periodic voltage signal. The Poincare map is a diagram of the two state variables voltage and current, but the system period (frequency) is taken for a sampling period (frequency). Because of that, the Poincare map of the periodic solution consists of only one point (Fig. 6). The Poincare map for the case of the fundamental ferroresonance is shown in Fig. 6. 1000-[V] 750- 2500-250-500-750- 0,70 0,75 0,80 (file jednofazni.pl4; x-var t) v:XX0005 0,95 [s] 1,00 Figure 7. Transformer voltage waveform - subharmonic ferroresonance A spectral analysis of the transformer voltage waveform is shown in Fig. 8. Harmonic spectrum - subharmonic ferroresonance 220 200 180 160 Si 140 aj I 120 0 > 100 CO 1 80 60 40 20 0 5 10 15 20 25 30 35 40 45 50 Frequency [Hz] Figure 8. Spectral analysis of the transformer voltage waveform - subharmonic ferroresonance From results of the spectral analysis of the transformer voltage waveform we see that the voltage spectrum consists of a basic harmonic (/0 = 50 Hz) and its subharmonics, of which the most dominant is the third subharmonic component -j. The phase plane for the case of the subharmonic ferroresonance is shown in Fig. 9. Table 2: System parameter values ferroresonance subharmonic Subharmonic ferroresonance Rm [ß] 12500 Cg [^F] 10 0,85 0,90 -1 2 Phase plane (subharmonic ferroresonance) Transformer voltage > 0 n Ü3 0 Ö > -200 -400 -600 -800 — -15 -5 0 5 Current [A] Figure 9. Phase plane analysis - subharmonic ferroresonance The phase plane is represented by a close trajectory with three sizes and a period of 3T or 60 ms. The Poincare map for the case of the subharmonic ferroresonance is shown in Fig. 10. 500 400 300 200 — 100 "(D CB 0 -S 0 > -100 -200 -300 -400 -500 Poincare map (subharmonic ferroresonance) 6 8 Current [A] 10 12 14 Chaotic ferroresonance Rm [ß] 12500 Cg [^F] 48 Um [V] 600 20001500-1000-5000 -500-1000-1500-20000,70 0,75 0,80 (file jednofazni.pl4; x-var t) v:XX0005 Figure 11. Transformer voltage waveform - chaotic ferroresonance A spectral analysis of the transformer voltage waveform is shown in Fig. 12. 220 Harmonic spectrum - chaotic ferroresonance 200 180 160 > » 140 cn f 120 > "> 100 s al 80 60 40 20 0 0 Lui ti u i... .„H, 400 500 600 Frequency [HZ] Figure 10. Poincare map - subharmonic ferroresonance Because of the dominance of the third harmonic in the harmonic spectrum of the voltage signal, the Poincare map consists of three points [3], [6]. 2.3 Chaotic ferroresonance The chaotic ferroresonance occurs when the values of the system parameters are the ones shown in Table 3. Table 3: System parameter values - chaotic ferroresonance Figure 12. Spectral analysis of the transformer voltage waveform - chaotic ferroresonance From the results of the spectral analysis of the transformer voltage waveform we see that the voltage spectrum is not discrete, i.e. it is a continuous signal which shows on irregular and unpredictable behavior. The phase plane for the case of the chaotic ferroresonance is shown in Fig. 13. 800 600 400 200 0,85 □,90 □ ,95 1,00 -10 10 5 100 200 300 700 800 900 1000 0 2 4 The transformer voltage waveform of the chaotic ferroresonance is shown in Fig. 11. Phase plane (chaotic ferroresonance) 500 > CB 0 -20 -15 -10 -5 0 5 10 15 20 Current [A] Figure 13. Phase-plane analysis - chaotic ferroresonance The phase plane is represented by a trajectory that is never closed to itself. The Poincare map for the case of the chaotic ferroresonance is shown in Fig. 14. Poincare map (chaotic ferroresonance) 1000 - £