Search for Network Parameters Preventing Ferroresonance Occurrence

Search for Network Parameters Preventing Ferroresonance Occurrence Jozef Wisniewski Technical University of Lodz Institute of Electrical Power Engineering Lodz, Poland jozef.wisniewski@p.lodz.pl Edward Anderson* ), Janusz Karolak** ) Institute of Power Engineering Department of Network Research and Analyses Warsaw, Poland * ) edward.anderson@ien.com.pl ** ) janusz.karolak@ien.com.pl Abstract Application of the continuation and bifurcation method for studying the influence of network parameter values (e.g. source voltage, network capacitance or damping resistance loading the broken delta of voltage transformers (VT) tertiary windings) on network stability is described in the article. It is possible to estimate in this way sensitivity of the network to ferroresonance occurrence after a fault such as breaker switching or ground short-circuit. The advantages and limitations of the method are given. The examples of the method used for 6 kv auxiliary substation in the power plant and for kv networks are presented. The ferroresonance may be a result of the interaction between the VT inductance and the network capacitance or grading capacitance mounted in the breaker. The results of field measurements and calculations using the XPPAUT continuation package and simulations using the EMTP program are shown. Keyword: ferroresonance, continuation method, bifurcation I. INTRODUCTION Susceptibility of electrical network to ferroresonance occurrence can be treated as one of the parameters describing quality of power supply. Appearance and prolonged duration of ferroresonance may cause damage to the substation gear such as a voltage transformer (VT), power transformer or cable insulation. The ferroresonance phenomenon in the power network has been known and described for many years [,2,3]. However, it is difficult to investigate this phenomenon because of its significant sensitivity to even small changes in network parameters. Also, the form and parameters of the network equivalent scheme of devices like VT, power lines and breakers exert an effect on the phenomenon character during computer simulations. It is important to know the network parameters that prevent ferroresonance occurrence. Looking for the range of network parameters in relation to ferroresonance appearance by means of the simulation method with gradually changing network parameters [4], is a longterm process and does not guarantee finding a proper solution. There are many methods for ferroresonance investigation. The field measurements method gives the most credible results [5,6] but because of costs and organizing problems its application is limited. Application of the continuation and bifurcation method for finding the range of the chosen network parameters characterizing its specific state of work has been shown. The obtained results were verified by comparing them to calculations made with EMTP program and to field measurements. The continuation method consists in analyzing stability of the solution of the ordinary difference equations (ODE) set describing the examined net for a given value of network parameters. This set of equations is linearized in the vicinity of the current parameter value. The analysis of Floquet multipliers resulting from the state equation matrix provides the information on net stability and possibility of ferroresonance appearance. It is not necessary to solve the set of equations in time domain. Stability of the system is examined at a continuously varying chosen network parameter like supply voltage,

equivalent capacitance of network or dumping resistance loaded broken triangle of VTs. This method allows to find the ranges of network parameters values, where the ferroresonance phenomenon may appear and be characterized as periodic with source frequency, subharmonic frequency, quasiperiodic or chaotic shape. The results of calculations are presented on bifurcation diagrams, where bifurcation points indicate values of the parameter studied, in which the character of phenomenon is changing. The advantages and limits of the method are described. The cases of using the method for medium voltage and high voltage network, where the ferroresonance is an effect of the interaction between nonlinear inductivity of VT and network capacitance or breaker grading capacitance, respectively, are presented. For calculations of dynamic system performance while changing the parameter value with the continuation method, the specialist programs, continuation package like AUTO, CONTENT or CANDYS can be used. Calculations presented in the article were performed using the XPPAUT program [7,8,9]. It works in the UNIX system or X-Windows server. The program consists of two basic units. Module XPP (X- Windows Phase Plane) includes the BVP tool (Boundary Value Problem Solver). It allows to calculate the starting point for continuation method. The starting point must be defined by steady state work parameters. Usually, it is searched for in the range of safe values of the parameter studied (e.g. at small value of supply voltage). Data for this module should be prepared as autonomous set of first order differential equations (ODE). Further presented nonautonomous ODEs describing investigated network should be transformed in this way to fulfil this condition. AUTO module is the main calculating tool of continuation package. It enables to calculate and draw the bifurcation diagram. At the same time, it enables to observe the position of Floquet multipliers (in the left bottom corner of the screen). The screenshot of the AUTO module interface is shown in Fig.. On investigation of the properties of periodic oscillations in the nonlinear system, the values q i of Floquet multipliers are significant. This permits to find out how the periodic solution of nonlinear ODEs at small changes in initial conditions or in the studied parameter is altering. The condition of asymptotic stability of periodic solution is that each i-th Floquet multiplier q <. i The bifurcation can appear in one of three characteristic manners in case of this unfulfilled condition (Fig. 2): Im{q} c) Re{q} Figure 2. Possibilities of bifurcation appearance (observation of the Floquet multipliers) Floquet multiplier reaches the value of (fold bifurcation, LP - limit point or turning point of a branch), Floquet multiplier reaches the value of - (PD - period doubling bifurcation), c) The pair of complex-conjugate values q i exceeds the unit circle, then the periodic solution changes into quasiperiodic (HB - Hopf bifurcation). LP, PD, HD - are the symbols which the AUTO module places on bifurcation diagram. II. INVESTIGATION ON POSSIBILITIES OF FERRORESONANCE OCCURRENCE IN MEDIUM VOLTAGE NETWORK Conditions of ferroresonance occurrence in 6 kv auxiliary substation of the power plant have been investigated. This network works with an isolated neutral point. Its equivalent scheme is presented in Fig. 3 and equivalent scheme of VT is shown in Fig. 4. Figure. Screenshot of AUTO module interface of XPPAUTO package

U U N U C C E VT Figure 3. Three-phase equivalent scheme of medium voltage network for ferroresonance investigation (fragment) R s VT ϑ u R resistance R broken triangle of VTs) or capacitance C E (at fixed amplitude of supply voltage and unloaded VTs). In Fig. 5 bifurcation diagrams indicating ranges of stable and unstable solutions for fixed equivalent network capacitance C E (2 nf 2 µf) and for varying supply voltage U m at unloaded broken triangle of VTs are presented. Flux linkage, Ψ (Wb turn) 5 4 3 2 C=2 nf 2 nf 2 nf stable unstable 2 nf 3 6 9 2 5 i(ψ) R m Figure 4. Equivalent scheme of VT The scheme of the investigated network can be described by equations (): dψk 3 U = [ U N + U k Rs ( i( Ψk ) + dt K2 k =,2,3 + du N i 3 U N K = dt 3 CE + K Rs where: K K 3 R m R = + 2 ϑu 2 3 u o ( s o ; R N s 2 ; Rm K = + R K 3 s = ϑ R + K R ) ; U = U sin( ω t + φ ) - k m k i )] source phase voltage (k - phase number), i = i( Ψ) + i( Ψ2) + i( Ψ3) - sum of VTs magnetizing n currents, i( Ψ ) = a Ψ + a n Ψ - VT magnetizing current in relation to flux linkage (a = 7,8-4, a n = 7,3-5, n = 9), C E - equivalent capacitance of network (phase to ground), R - resistance loaded broken triangle of VTs, R s = 2 Ω, R m = 9 Ω - parameters of VT equivalent scheme. The investigation of network susceptibility to ferroresonance occurrence was performed by varying the following parameters: amplitude of supply voltage U m (at fixed equivalent network capacitance C E and unloaded or loaded by () Amplitude of supply voltage, U m (V) Figure 5. Bifurcation diagram - flux linkage Ψ vs. amplitude of supply voltage U m (at fixed equivalent network capacitance C E) The calculations show that for e.g. network capacitance C E = 2 nf (the substation works alone, the cables are unplugged), at nominal supply voltage, the network is susceptible to ferroresonance occurrence. If the network capacitance rises, this susceptibility decreases, but only at capacitance approximately C E = 2 µf and more, the network is resistant to ferroresonance. For the amplitude of the supply voltage below 3 V, the ferroresonance does not appear. In Fig. 6 bifurcation diagrams for varying network capacitance C E for fixed values of the supply voltage are shown. These diagrams point that the steady state of work is impossible at nominal and higher supply voltage and network capacitance below nf. For supply voltage which can appear at fault states (U m = 3 7 kv), the range of capacitance C E = nf nf is extremely dangerous. Flux linkage, Ψ (Wb turn) 25 2 5 stable unstable 5.. Network capacitance, C E (nf) U m=7 V 6 V 5 V 4 V 3 V Figure 6. Bifurcation diagram - flux linkage vs. equivalent network capacitance C E (at fixed amplitude of supply voltage U m)

Fig 7a presents the results of ferroresonance measurements in 6 kv substation [5]. The substation works with cables unplugged. Ferroresonance was induced by a short time arc ground fault. During repetition of this fault, the ferroresonance did not always appear. It might be caused by the randomize moments of initiation and cutting off this fault or its stochastic process. Fig. 7b presents the results of computer simulation of the same case. The calculation was performed using the EMTP program. 3U III. INVESTIGATION ON POSSIBILITIES OF FERRORESONANCE OCCURRENCE IN HIGH VOLTAGE NETWORK In high voltage network series ferroresonance occurring after switching off the breaker is possible. It is caused by an interaction between breaker grading capacitance and nonlinear inductivity of VT magnetizing branch [4,]. The equivalent scheme of phenomenon simulation is shown in Fig. 8, where: - breaker, C - grading capacitance, C E - phase to ground capacitance, VT - inductive voltage transformer. The aim of simulation is to check in which circumstances and in which range of network parameters the ferroresonance can appear, as well as how it is possible to prevent it. C VT U L i Rs U L2 U(t) C E i 3 Ψ(i) i 2 i Rp U L3 I z 3U U L U L2 U L3 I z 25 [V] 25-25 -25...2.3.4 [s].5 (f ile A_przek.pl4; x-v ar t) v :3U 5. 7.5. -7.5-5....2.3.4 [s].5 (f ile A_przek.pl4; x-v ar t) v :SA 5. 7.5. -7.5-5....2.3.4 [s].5 (f ile A_przek.pl4; x-v ar t) v :SB 5. 7.5. -7.5-5....2.3.4 [s].5 (f ile A_przek.pl4; x-v ar t) v:sc 2 [A] - -2...2.3.4 [s].5 (f ile A_przek.pl4; x-v ar t) c:z - Figure 7. Ferroresonance in 6 kv substation: field measurements [5], computer simulation (3U - residual voltage in broken triangle of VTs, U L,L2,L3 - phase voltage on substation busbars, I z - ground fault current) The results of computer simulation and field measurements are similar. They are in agreement with the continuation method. Figure 8. One phase equivalent scheme of network for ferroresonance calculation at the breaker switching off The scheme can be described by equations (2): dψ dt du dt µ = U µ n i2( Ψ ) = / K [ U µ [ K2 + K3 ( a + n an Ψ )] C + ω U m cos( ω t)] CE Rs K = ( + ) ( + ) ; K C R p = C R p 2 ; CE K 3 = Rs ( + ) (2) C where: Ψ -flux linkage, U ( t) = U m cos( ω t) - source voltage, R s =3,3 kω; R p = 9 Ω; U m = kv * 2 / 3. Magnetizing characteristic of VT can be described by the n equation: i 2 ( t) = a Ψ + a n Ψ, where: a = 3,7-6, a n =,25-6, n = 5. Fig. 9 presents a bifurcation diagram showing VT flux linkage vs. grading capacitance C. The lower line indicates values of flux linkage during the steady state. The upper line shows parameters of ferroresonance which can appear at the moment of breaker switching off.

Flux linkage, Ψ (Wb turn) 2 chaos 8 6 Hz 5 Hz 6 4 stable unstable 2 Grading capacitance, C (pf) and C E = 7 pf. The transients before ferroresonance occurrence are a result of ignitions between poles of the breaker. uz iz up ip Figure 9. Bifurcation diagram - flux linkage vs. grading capacitance C (at fixed network capacitance C E =7 pf) Numerous computer simulations for studying possibilities of ferroresonance occurrence at a different network parameters combination were performed. The measurements and simulations give the similar results. Further results of simulations can be considered reliable. Fig. presents the results of many simulations using the EMTP program. It shows the ferroresonance parameters which occur after switching off the breaker. The grading capacitance is C = pf, equivalent VT capacitance is C p = 39 pf, [] and the network capacitance varies in the range of C E = pf. µf. The lower line in the diagram means values of flux linkage in steady states. The points which lie higher represent the ferroresonance parameters after switching off the breaker. ic uz iz up 9 45-45 -9..5..5.2.25.3.35 [s].4 (f ile juk23_f az.pl4; x-v ar t) v:a 6 [ma] 3-3 -6..5..5.2.25.3.35 [s].4 (f ile juk23_f az.pl4; x-v ar t) c:a -B 9 45-45 Peak value, Umax (kv) 8 6 4 2 8 6 4 2 steady state transient ferroresonance (chaotic) stable ferroresonance (3T) Network capacitance, C E (pf) Figure. Ferroresonance parameters at breaker switching off (fixed network parameters: C = pf, C E = pf. µf, R p = 9 Ω, C p =39 pf) Fig. a shows the results of field measurements and Fig. b shows the results of computer simulations for ferroresonance which appears after switching off the breaker. Network parameters are: U m = kv * 2 / 3, C = 39 pf ip ic -9..5..5.2.25.3.35 [s].4 (f ile juk23_f az.pl4; x-v ar t) v:d 4 [ma] 2-2 -4..5..5.2.25.3.35 [s].4 (f ile juk23_f az.pl4; x-v ar t) c:c -D 4 [ma] 2-2 -4..5..5.2.25.3.35 [s].4 (f ile juk23_f az.pl4; x-v ar t) c:d - Figure. Ferroresonance after switching off the breaker: field measurements, computer simulation (fixed parameters: C = 39 pf and C E = 7 pf). Descriptions: uz - supply voltage, iz - source current, up - voltage ip - VT current, ic - current in the VT capacitance. In the range of network capacitance C E =2 nf after the breaker switching off (Fig. ), the permanent ferroresonance with period 3T appears. It is shown in Fig. 2a. The case of transient chaotic ferroresonance is presented in Fig. 2b. For the illustration of ferroresonance oscillation period, the Poincare diagrams are used. They are created by transient sampling performed once per basic period. The number of fixed values informs us about subharmonic number. The lack of fixed values confirms the lack of steady state or chaotic ferroresonance.

(f ile juk23_3f az.pl4; x-v ar t) m:ue 9 45-45 -9..4.8.2.6 [s] 2. (f ile juk23_3f az.pl4; x-v ar t) v:da 9 * 3 6 3-3 -6..4.8.2.6 [s] 2..5 [A].75. -.75 -.5..4.8.2.6 [s] 2. (f ile juk23_3f az.pl4; x-v ar t) c:da -EA 2 - -2..4.8.2.6 [s] 2. (f ile juk23_3f az.pl4; x-v ar t) v:da 6 * 3 8-8 -6..4.8.2.6 [s] 2. (f ile juk23_3f az.pl4; x-v ar t) m:ue.2 [A].. -. -.2..4.8.2.6 [s] 2. (f ile juk23_3f az.pl4; x-v ar t) c:da -EA - Poincare diagram VT current, - Poincare diagram VT current, Figure 2. Transients at switching off the breaker, C = pf: C E =398 pf- permanent ferroresonance with period 3T, C E =25 pf - transient chaotic ferroresonance IV. CONCLUSIONS The results received by using the continuation method do not allow an unambiguous determination of network parameters at which the ferroresonance does not appear. The situation seems to be safe basing on the bifurcation diagram, however due to a fault, e.g. a transient ground short circuit, ferroresonance may occur. The method allows rather to state in which range of investigated parameters (network parameters or resistance in broken triangle of VTs) the stable work of network will certainly not appear or the danger of ferroresonance occurrence will be significant. On the basis of the authors experience obtained through a large number of simulations with the EMTP program, confirmed by the field tests, it can be stated that the ranges of parameters visible on bifurcation diagrams as unstable (fat line) should be extended in both directions of parameters. In those extended ranges of network parameters, the risk of ferroresonance is high, especially after the appearance of fault such as ground short circuit. The limited accuracy of the used network device models influences the obtained results. Its improvement, e.g. taking under consideration the full loop of hysterezis instead of magnetizing curve, will certainly provide better quality of results. If the work of the examined network in the range of device parameters guaranteeing the resistance to ferroresonance is not possible, the additional means should be used. These may be: grounding the neutral point of the network by properly selected resistor, installation of varistor overvoltage limiters or special damping ferroresonance devices in the broken triangle of VTs. Susceptibility of network to ferroresonance occurrence should be treated as one of supply quality ratios. REFERENCES [] P. Ferraci, Ferroresonance, Cahier technique, No. 9, Groupe Schneider, 998. [2] J. Horak, A Review of Ferroresonance, 57th Annual Conference for Protective Relay Engineers, Texas A&M University, 24. [3] M. Iravani, Modeling and Analysis Guidelines for Slow Transients - Part III: The Study of Ferroresonance, IEEE Transactions on Power Delivery, No, 2. [4] M. Escudero, I. Dudurych, M. Redfern, Characterization of Ferroresonant Modes in HV Substation with CB Grading Capacitors, International Conf. on Power Systems Transients (IPST), Montreal, Canada, June, 25. [5] E. Anderson, J. Karolak, M. Kumanowski, Z. Piątek, Investigation of internal overvoltages in 6 kv network of the Kozienice power plant (in Polish), EBA/6/E/26, Warszawa, 26. [6] E. Anderson, J. Karolak, Z. Piątek, J. Wiśniewski, The choice of the manner of neutral point work in 6 kv network in the aspect of ferroresonance phenomena (in Polish), Wiadomości Elektrotechniczne, 4, 26. [7] B. Ermentrout, Simulating, analyzing, and animating dynamical systems. A guide to XPPAUT for researchers and students, SIAM, Philadelphia, 22, (http://www.pitt.edu/~phase/). [8] F. Wornle, D. Harrison, C. Zhou, Analysis of a Ferroresonant Circuit Using Bifurcation Theory and Continuation Techniques, IEEE Transactions on Power Delivery, No, 25. [9] J. Wiśniewski, E. Anderson, J. Karolak, The use of continuation and bifurcation method for calculation of ferroresonance appearance conditions (in Polish), XIII Międzynarodowa Konferencja Naukowa: Aktualne Problemy w Elektroenergetyce, Jurata, 6, 27. [] E. Stawowy, Ferroresonance in 22 kv voltage transformers (in Polish), Przegląd Elektrotechniczny - Konferencje, Nr, 25. [] Results of voltage - current characteristic of the combined voltage transformer JUK23a measurements (in Polish), (unpublished). Instytut Energetyki, 24.