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Impacts of Neutral Earth Resistance on the Stability Domain of Ferroresonance Modes in Power Transformers including MOV Surge Arrester HAMID RADMANESH Electrical Engineering Department, Aeronautical University of Science& Technology, Shahid Shamshiri Street, Karaj Old Road, Tehran, IRAN Tel: (+98-21)64032128, Fax: (+98-21)33311672 [email protected] Mehrdad Rostami Electrical Engineering Department,Shahed University, End of Khalij-e-Fars High way, Infront of Imam Khumaini holly shrine, Tehran-1417953836 , IRAN Tel: (+98-21)51212020, Fax: (+98-21)51212021 [email protected] Abstract – this work studies the effect of neutral earth resistance on the controlling ferroresonance oscillation in the power transformer including MOV surge arrester. A simple case of ferroresonance circuit in a three phase transformer is used to show this phenomenon and the three-phase transformer core structures including nonlinear core losses are discussed. The effect of MOV surge arrester and neutral earth resistance on the onset of chaotic ferroresonance and controlling chaotic transient in a power transformer including nonlinear core losses has been studied. It is expected that these resistances generally cause into ferroresonance control. Simulation has been done on a power transformer rated 50 MVA, 635.1 kV with one open phase. The magnetization characteristic of the transformer is modelled by a single-value two-term polynomial with q=7, 11. The core losses are modelled by third order in terms of voltage. The simulation results reveal that connecting the MOV arrester and neutral resistance to the transformer, exhibits a great impact on ferroresonance over voltages. Keywords: MOV Arrester, Neutral Earth Resistance, Control of Chaos, Bifurcation, Ferroresonance, Power Transformers, Nonlinear core losses effect I. Introduction In simple terms, ferroresonance is a series of “resonance” involving nonlinear inductance and capacitances. It typically involves the saturable magnetizing inductance of a transformer and a capacitive distribution cable or transmission line connected to the transformer. The word ferroresonance first appeared in the literature in 1920 [1], although papers on resonance in transformers appeared as early as 1907 [2]. Practical interest was generated in the 1930’s when it was shown that use of series capacitors for voltage regulation caused ferroresonance in distribution systems [3], resulting in damaging over voltages. The first analytical work was done by Rudenberg in the 1940’s [4]. More precise and detailed work was done later by Hayashi in the 1950’s [5]. Subsequent research has been divided into two main areas: improving the transformer models and studying ferroresonance at the system level. Sensitivity studies on power transformer ferroresonance of a 400 kV double circuit is given in [6], Novel analytical solution to fundamental ferroresonance in [7] investigated a major problem with the traditional excitation characteristic (TEC) of nonlinear inductors, in that the TEC contains harmonic voltages and/or currents, and has been used the way as if it were made up of pure fundamental voltage and current. Stability domain calculations of period-1 ferroresonance in a nonlinear resonant circuit have been investigated in [8]. Application of wavelet transform and MLP neural network for ferroresonance identification was done in [9]; in this paper an efficient method for detection of Ferroresonance in distribution transformer based on wavelet transform is presented. Using this method ferroresonance can be discriminate from other transients such as capacitor switching, load switching, transformer switching. Impacts of transformer core hysteresis formation on stability domain of ferroresonance modes were done in [10]. In this paper, impacts of various formations of hysteresis on the stability domain of ferroresonance modes of a voltage transformer (VT) have been studied. Based on four different hysteretic and two single-valued polynomial models, ferroresonance behaviors of the VT are studied. 2-D finite-element electromagnetic analysis of an autotransformer experiencing ferroresonance was given in [11]. Experimental and simulation analysis of ferroresonance in single-phase transformers considering magnetic hysteresis effects is investigated in [12]. An accurate hysteresis model for ferroresonance analysis of a transformer showed an accurate transformer core model, using the Preisach theory, to represent the core magnetization characteristic [13]. A new modeling of MATLAB transformer for accurate simulation of ferroresonance shows a new modeling of transformers in Simulink/MATLAB enabling to simulate slow transients more accurate than the existing models used in the software [14]. Effect of circuit breaker shunt resistance on chaotic ferroresonance in voltage transformer was shown in [15], in this work it has also been shown C.B shunt resistance successfully can cause ferroresonance drop out and can control it. Then controlling ferroresonance has been investigated in [16], it is shown controlling ferroresonance in voltage transformer including nonlinear core losses by considering circuit breaker shunt resistance effect, and clearly shows the effect of core losses nonlinearity on the system behavior and margin of occurring ferroresonance. In [17], electromagnetic voltage transformer has been studied in the case of nonlinear core losses by applying metal oxide surge arrester in parallel with it and Simulations have shown that MOV successfully can cause ferroresonance drop out. Effect of neutral earth resistance on the controlling ferroresonance oscillations in power transformer has been studied and it has been shown that system has been greatly affected by neutral resistance [18]. This paper studies the effect of MOV arrester and neutral earth resistance on the global behavior of a ferroresonance circuit with nonlinear core losses and controlling these oscillations. II. SYSTEM MODELING INCLUDING MOV ARRESTER WITHOUT NEUTRAL EARTH RESISTANCE The system studied here consists of a source feeding an unloaded power transformer with one supply conductors being interrupted (Fig.1) [19]. The transformer is then energized through the capacitive coupling with the other phases. Fig.2 shows the circuits that feed the disconnected coil trough the capacitive coupling [19]. 1 2 Cm v1 v2 v3 Cg 3 Figure.1. model of ferroresonance circuit including line capacitance 2 Cm v1 v2 1 Cm Cm Cg Cg Cg Figure.2. Circuit that feeds disconnected coil In order to derive the mathematical description of the above circuit, Thevenin’s theorem will be used to obtain an equivalent circuit. The equivalent capacitance can be found by decreasing the two remaining voltage source of phase 1 and 2. In doing so, both corresponding ground capacitance and transformer windings are shorted and can be omitted as well. Therefore, the remaining equivalent circuit will consist of two mutual and one ground potential. Thus, the equivalent capacitances can be grouped as [19] C C g 2Cm (1) And the equivalent source voltage can be derived as E Cm Cg 2Cm (2) Under loaded operating conditions, the flux induced in the primary transformer windings are immediately compensated by the current flow in the secondary windings so that the core losses in the transformer can be neglected. However, in the case of an unloaded (or very lightly loaded) power transformer, current can develop in the secondary side and this causes the flux to emanate from the transformer leg and flow through the iron core. This in turn increases the transformer core losses and therefore it can no longer be neglected. Fig.3 shows the final reduced equivalent circuit with the inclusion of R, which represents the transformer core losses [19]. C L R E MOV the same porcelain housing due to required protecting voltage. Size of each disc is related to its power dissipation capacity. The nonlinear V-I characteristic of each column of the surge arrester is modeled by combination of the exponential functions of the form I V Ki I Vref ref (4) Where: V represents resistive voltage drop, I represents arrester current and K is constant and is nonlinearity constant. This V-I characteristic is graphically represented as follows (on a linear scale and on a logarithmic scale). 5 2 Figure3. Equivalent circuit of system including MOV surge arrester hysteretic and eddy current characteristics [vm iRm ] , respectively. The iron core saturation characteristic due to the explanation above is given by: x 10 MOV V-I characteristic 1.5 1 MOV voltage Fig.3. shows the equivalent circuit of system which was described above. The magnetization branch is modeled by a nonlinear inductance in parallel with a nonlinear arrester accordingly and represents the nonlinear saturation characteristic [ iLm ] and nonlinear 1/ i 0.5 0 -0.5 -1 -1.5 -2 -5000 0 5000 10000 MOV current iLm a bq Figure 5. V-I characteristic of MOV surge arrester (3) Exponent q depends on the degree of saturation. It was found that for adequate representation of the saturation characteristics of a power transformer the exponent q may acquire values 7 and 11. Fig.4 shows simulation of these iron core characteristic for q=7, 11. The Surge Arrester block is modeled as a current source driven by the voltage appearing across its terminals. Therefore, it cannot be connected in series with an inductor or another current source. As the Surge Arrester block is highly nonlinear, a robust integrator algorithm must be used to simulate the circuit. MATLAB ode23t with default parameters usually gives the best simulation speed. For continuous simulation, in order to avoid an algebraic loop, the voltage applied to the nonlinear resistance is filtered by a first-order filter with a time constant of 0.01 microseconds. This fast time constant does not significantly affect the result accuracy [17]. In this paper, the core loss model adopted is described by a third order power series which coefficients are fitted to match the hysteresis and eddy current nonlinear characteristics given in [17]: (i) (ii) 1.7 (iii) (iV) (V) , pu 1.6 1.5 1.4 1.3 1.2 1.1 i 3.14 0.417 9 2 (ii) i 10 (i) (iii) i 0.28 0.7211 102 (iV) Transformer (V) i 11 10 2 0.2 0.4 0.6 0.8 1 1.2 i, pu Figure. 4. Nonlinear characteristics of transformer core with different values of q [16] III. Metal Oxide Surge Arrester Model Surge Arrester is highly nonlinear resistor used to protect power equipment against over voltages. MOV can be arranged by cascading several metal oxide discs inside iRm h0 h1 Vm h2Vm h3Vm 2 (5) Per unit value of 3 (iRm ) is given in (6) iRm 0.000001 0.0047Vm 0.0073Vm 0.0039Vm 2 (6) The differential equation for the circuit in fig. 3 before connection of MOV arrester can be presented as follow: 3 vl d dt (13) h0 u 1 C u dE 2 dt (7) ic C dvc , vc E vl dt (8) ic il iRm (14) y (t ) CX (t ) (9) dE dv C C l dt dt 2 3 h0 h1vl h2vl h3vl a bq iMOV y (t ) v(t ) 0 (10) dvl dt dE 1 1 v 2 3 h0 h1vl h2vl h3vl a bq l dt C C k det A 0 x1 (t ) 1 x2 (t ) (15) (16) 1 det a bx1 (t ) q 1 (h1 h2 x2 (t ) h3 x2 (t )2 0 C C (17) (11) Where E is the peak value of the voltage source as shown in fig.3 and is the flux linkage of the transformer. Typical values for system parameters without MOV arrester due to the equation (3) are as given below: Table 1: Typical value of q and its coefficient q 11 7 Coefficient (a) 0.0028 3.14 Coefficient (b) 0.0072 0.41 (h1 h2 (1.41) h3 (1.41) 2 ) p C q a C 2 p q 0 1, 2 (18) p p 2 4q 2 (19) Bode Diagram 60 Magnitude (dB) 377 rad/sec Cpu 0.7955 Vbase 635.1 kv , I base 78.72 A Phase (deg) d (0) 0.0; vl (0) 2 dt and d / dt 0 -4 as state 10 -3 10 -2 10 -1 10 0 10 Frequency (rad/sec) Fig.5. Bode diagram of the given parameters IV. Simulation Results 1 dE h0 h1 x 2 (t ) h2 x 2 (t ) 2 h 3 x 2 (t ) 3 ax1 (t ) bx1 (t ) q C dt 0 x1 (t ) x (t ) a qb x1 (t ) q 1 2 C 90 -90 x1 (t ) x (t ) x (t ) 2 1 v d / dt x2 (t ) v x 2 (t ) 20 0 180 Initial conditions: The state-space formulation variables is as given below: 40 (12) x1 (t ) 1u (h1 h2 x2 (t ) h3 x2 (t ) 2 ) x2 (t ) 1 C 1 Ferroresonance in three phase systems can involve large power transformers, distribution transformers, or instrument transformers. The general requirements for ferroresonance are an applied source voltage, a saturable magnetizing inductance of a transformer, a capacitance, and little damping. The capacitance can be in the form of capacitance of underground cables or long transmission lines, capacitor banks, coupling capacitances between double circuit lines or in a temporarily-ungrounded system, and voltage grading capacitors in HV circuit breakers [20]. In this section of simulation, system has Bifurcation Diagram with q=7 including Nonlinear core losses and MOV effect 2.5 Voltage of Transformer(perunit) been considered with MOV arrester and Time-domain simulations were performed using the MATLAB programs. Figs.6 and 7 show the phase plan diagram of system states with MOV arrester for E=3 p.u. Figs.8 and 9 show the corresponding bifurcation diagrams of the system states with MOV arrester and including nonlinear core losses for q=7, 11 which depicts chaotic behavior. Phase Plan Diagram with q=7 including Nonlinear core losses and MOV effect 2 1.5 1 0.5 2.5 2 0 0 1 2 3 Input voltage(perunit) 4 5 6 Figure8. Bifurcation diagram with q=7, without MOV arrester 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 Flux Linkage of Transformer 1 1.5 2 Figure6. Phase diagram of system with q=7, with MOV arrester In Fig.6 Input voltage has been plotted against the flux of the power transformer. When the magnitude of the input voltage is 3p.u, trajectory of the system has chaotic behavior and amplitude of the flux and ferroresonance overvoltage reaches up to 3p.u. by increasing in the degree of core nonlinearity, q, the behavior of the system has being more chaotic as shown in fig.7. In this case q=11 and oscillation in the voltage of the transformer goes up. Fig.7 shows the Phase plan diagram that has been simulated by q=11, it shows that chaotic resonance is highly dependent on the transformer modeling. Phase Plan Diagram with q=11 including Nonlinear core losses and MOV effect Fig.8 shows the abnormal phenomena in the power transformer with the core model nonlinearity q=7, it clearly shows when the q degree changes from 7 to 11; ferroresonance begins in 2p.u. by comparing fig.9 with fig.8 margin of the beginning ferroresonance has been decreased. Fig.9 shows the bifurcation diagram of the over voltage with q=11, it shows that with q=11, system behavior reaches up to 1.6 p.u. Bifurcation Diagram with q=11 including Nonlinear core losses and MOV effect 1.8 1.6 Voltage of Transformer(perunit) Voltage of Transformer 1.5 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 Input voltage(perunit) 4 5 6 Figure9. Bifurcation diagram with q=11, without MOV arrester 2.5 2 Table (2) shows different values of E, considered for analyzing the circuit in the absence of MOV arrester. Voltage of Transformer 1.5 1 0.5 Table 2: Simulation results without neutral earth resistance q/E 1 2 3 4 5 6 Chaotic Chaotic 7 P1 P1 P3 P3 0 -0.5 -1 11 P1 *Pi: Period i -1.5 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 Flux Linkage of Transformer 1 1.5 P1 P1 chaotic P3 P3 2 Figure7. Phase diagram of system with q=11, with MOV arrester Another tool that can show manner of the system in vast variation of parameters is bifurcation diagram. In fig.8, voltage of the system has been increased up to 6p.u and ferroresonance has been begun in 3p.u, it is shown that if input voltage goes up due to the abnormal operation or switching action, ferroresonance occurs, because of the nonlinearity in core losses, ferroresonance began in the big value of the input voltage. V. System Modeling With MOV Arrester And Neutral Earth Resistance In the case of simulation, time domain simulations were performed using fourth order Runge-Kutta method and validated against MATLAB SIMULINK. In this case, the system which was considered for simulation is shown in fig.10. Phase Plan Diagram of for q=7, Including NonLinear Core Losses and Neutral Resistance C 3 Rcore Lcore vl MOV Voltage of Transformer 2 1 0 -1 -2 E -3 -2 Rneutral vRn -1.5 -1 -0.5 0 0.5 Flux Linkage of Transformer 1 1.5 2 Figure11. Phase diagram of system with q=7, with MOV arrester and neutral earth resistance Phase Plan Diagram of for q=11, Including NonLinear Core Losses and Neutral Resistance 3 Figure.10. Equivalent circuit of system with MOV arrester and neutral earth resistance Typical values for various system parameters has been considered for simulation were kept the same by the case 1, while neutral earth resistance has been added to the system and its value is given below: The differential equation for the circuit in fig.10 can be presented as follows: dvl dE 1 v 2 3 h0 h1vl h2 vl h3vl a bq l dt dt C k dv dv d d 2 dvl Rn .(h1 l 2h2 vl l 3h3vl a qbq1 dt dt dt dt dt 1 1 dv l k dt ) (20) Where λ is the flux linkage and V is the voltage of transformer. Voltage of Transformer 2 1 0 -1 -2 -3 -1.5 -0.5 0 0.5 Flux Linkage of Transformer 1 1.5 Figure.12. Phase diagram of system with q=11, including MOV arrester and neutral earth resistance The use of geometric graphical methods like phase plane projections and bifurcation can be applied to obtain a better understanding of ferroresonance. Ferroresonance in the above three phase nonlinear core distribution transformer can be periodic or non periodic. Some of the periodic modes of ferroresonance may contain sub harmonics, but still have strong power frequency components, but take longer than one fundamental cycle to repeat. This occurs more typically for very large values of q. The “higher energy modes” of ferroresonance involving relatively large capacitances and little damping can produce a non-periodic voltage on the open phase(s). These voltage waveforms can be quite similar to those of Doffing’s equation. VI. SIMULATION RESULTS Bifurcation Diagram with q=7 Including NonLinear Core Losses and Neutral Resistance 1.8 1.6 1.4 Voltage of Transformer The nonlinear behavior of ferroresonance falls into two main categories. In the first, the response is a distorted periodic waveform, containing the fundamental and higher-order odd harmonics of the fundamental frequency. The second type is characterized by a nonperiodic, or chaotic, response. In both cases the response’s power spectrum and simulation results contain fundamental and odd harmonic frequency components. In the chaotic response, however, there are also distributed frequency harmonics and sub harmonics. Figs.11and 12 shows the corresponding phase diagram for the system in fig.10. Also figs.13 and 14 show the bifurcation diagrams for corresponding system including MOV arrester and neutral earth resistance. It is shown that chaotic region mitigates by applying neutral earth resistance. -1 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 Input Voltage(perunit) 4 5 6 Figure13. Bifurcation diagram with q=7, with MOV arrester and neutral earth resistance In figs. 13 and 14, system behavior is shown by the proper bifurcation diagram. In fig13, the degree of transformer core is q=7, in this figure, one jump has occurred in the trajectory of the system and voltage of the transformer has period 3 oscillation. In fig.14, q=11 and by considering MOV arrester in the system, there is no nonlinear phenomena and voltage of the transformer has a period 3 oscillation. By increasing the core nonlinearity, behavior remains in periodic oscillation and MOV arrester can cause ferroresonance drop out. Bifurcation Diagram with q=9 Including NonLinear Core Losses and Neutral Resistance Voltage of Transformer 1.5 makes over voltage on power equipment and is a dangerous phenomenon. MOV surge arrester is a nonlinear resistance and can cause ferroresonance dropout for some range of system parameters, but MOV only can limit these over voltages and cannot remove it successfully. By connecting neutral earth resistance to the neutral point of the power transformer is shown that system has been greatly affected by this resistance. The presence of the neutral earth resistance results in clamping the ferroresonance non-conventional over voltages in the power system. This resistance can remove the ferroresonance oscillation completely and system shows less sensitivity to the initial conditions. 1 IX. REFERENCES [1] 0.5 0 0 1 2 3 Input Voltage(perunit) 4 5 6 Figure14. Bifurcation diagram with q=11, with MOV arrester Table (3) includes the set of cases which are considered for analyzing the circuit including MOV arrester. For cases including MOV arrester, it can be seen that ferroresonance drop out will be occurred. Table 3: Simulation results (with MOV arrester) q/E 1 2 3 4 5 6 7 P2 P1 P1 P1 P2 P2 9 P1 P1 P2 P3 P3 P3 11 P1 P1 P3 P3 P3 P3 *Pi: Period i VII. Comparative discussions In this paper power transformer has been studied with nonlinear core losses model and finally effect of MOV surge arrester and neutral resistance on the controlling of nonlinear phenomena and chaos has been investigated. In the real system, transformer core has many kind of losses and is strictly nonlinear, the model has been used in this work was chosen from [19] , therefore other mathematic and real models for studying the transformer behavior when one unwanted phenomena such as switching action and lightning has been occurred. it has been shown the nonlinear core losses can cause system works better and margin of occurring ferroresonance has been decreased. By considering neutral resistance, ferroresonance over voltages has been ignored and even if unwanted phenomena appear, power transformer can works in the safe operation region and there is no dangerous condition in the power system. VIII. CONCLUSIONS This paper investigated the ferroresonance oscillation in the power transformer. Ferroresonance in power system B.A. Mork, D.L. Stuehm, Application of nonlinear dynamics and chaos to ferroresonance in distribution systems, IEEE Transactions on Power Delivery 9 (1994) 1009_/1017 [2] J. 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