search for network parameters preventing ferroresonance
TRANSCRIPT
Search for Network Parameters Preventing
Ferroresonance Occurrence
Jozef Wisniewski Technical University of Lodz
Institute of Electrical Power Engineering Lodz, Poland
Edward Anderson*), Janusz Karolak**) Institute of Power Engineering
Department of Network Research and Analyses Warsaw, Poland
*) [email protected] **) [email protected]
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 110 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 [1,2,3]. 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 long-term 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. 1.
Figure 1. Screenshot of AUTO module interface of XPPAUTO package
On investigation of the properties of periodic oscillations in the nonlinear system, the values qi 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 1<iq .
The bifurcation can appear in one of three characteristic manners in case of this unfulfilled condition (Fig. 2):
Imq
Req a) b)
c)
1
1
Figure 2. Possibilities of bifurcation appearance (observation of the Floquet multipliers)
a) Floquet multiplier reaches the value of 1 (fold bifurcation, LP - limit point or turning point of a branch),
b) Floquet multiplier reaches the value of -1 (PD - period doubling bifurcation),
c) The pair of complex-conjugate values qi 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.
UN
U1 UC1 R0
VT
CE
Figure 3. Three-phase equivalent scheme of medium voltage network for ferroresonance investigation (fragment)
Rs
VT ϑu
i(ψ)
Rm
Figure 4. Equivalent scheme of VT
The scheme of the investigated network can be described by equations (1):
⋅+⋅⋅+
⋅⋅
−=
=
⋅−⋅+⋅−+⋅=
∑
∑
s
N
E
N
sNkskN
k
RKKUi
CdtdU
3 2, 1,kK
iRUiRUU
Kdtd
1
1
32
13
31
)]3
)(([1 ΨΨ
(1)
where: oum RR
K⋅
+= 2131
ϑ;
m
sRRK +=12 ;
)1( 12
3 sou RKRK ⋅+⋅⋅=ϑ ; )sin( kmk tUU φω +⋅= -
source phase voltage (k - phase number),
∑ ++= )()()( 321 ΨΨΨ iiii - sum of VTs magnetizing
currents, nnaai ΨΨΨ ⋅+⋅= 1)( - VT magnetizing current in
relation to flux linkage (a1 = 7,08⋅10-4, an = 7,03⋅10-15, n = 9), CE - equivalent capacitance of network (phase to ground), R0 - resistance loaded broken triangle of VTs, Rs = 2000 Ω, Rm = 109 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 Um (at fixed equivalent network capacitance CE and unloaded or loaded by
resistance R0 broken triangle of VTs) or capacitance CE (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 CE (2 nF÷2 µF) and for varying supply voltage Um at unloaded broken triangle of VTs are presented.
0
10
20
30
40
50
0 3000 6000 9000 12000 15000
Amplitude of supply voltage, Um (V)
2000 nF
200 nF 20 nF
C=2 nF
stableunstable
Flux
link
age,
Ψ (W
b⋅tu
rn)
Figure 5. Bifurcation diagram - flux linkage Ψ1 vs. amplitude of supply voltage Um (at fixed equivalent network capacitance CE)
The calculations show that for e.g. network capacitance CE = 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 CE = 2 µF and more, the network is resistant to ferroresonance. For the amplitude of the supply voltage below 3000 V, the ferroresonance does not appear.
In Fig. 6 bifurcation diagrams for varying network capacitance CE 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 1 nF. For supply voltage which can appear at fault states (Um = 3÷7 kV), the range of capacitance CE = 1 nF÷10 nF is extremely dangerous.
5
10
15
20
25
Network capacitance, CE (nF)
Um=7000 V
6000 V
5000 V
4000 V
3000 V
stableunstable
0.01 0.1 1 10 100 1000
Flux
link
age,
Ψ (W
b⋅tu
rn)
Figure 6. Bifurcation diagram - flux linkage vs. equivalent network capacitance CE (at fixed amplitude of supply voltage Um)
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.
3U0
UL1
UL2
UL3
Iz
a)
(f ile A_przek.pl4; x-v ar t) v :3U0 0.0 0.1 0.2 0.3 0.4 0.5[s]
-250
-125
0
125
250[V]
(f ile A_przek.pl4; x-v ar t) v :SA 0.0 0.1 0.2 0.3 0.4 0.5[s]
-15.0
-7.5
0.0
7.5
15.0[kV]
(f ile A_przek.pl4; x-v ar t) v :SB 0.0 0.1 0.2 0.3 0.4 0.5[s]
-15.0
-7.5
0.0
7.5
15.0[kV]
(f ile A_przek.pl4; x-v ar t) v :SC 0.0 0.1 0.2 0.3 0.4 0.5[s]
-15.0
-7.5
0.0
7.5
15.0[kV]
(f ile A_przek.pl4; x-v ar t) c:Z - 0.0 0.1 0.2 0.3 0.4 0.5[s]
-2000
-1000
0
1000
2000[A]
b)
3U0
UL1
UL2
UL3
Iz
Figure 7. Ferroresonance in 6 kV substation: a) field measurements [5], b)
computer simulation (3U0 - residual voltage in broken triangle of VTs, UL1,L2,L3 - phase voltage on substation busbars, Iz - ground fault current)
The results of computer simulation and field measurements are similar. They are in agreement with the continuation method.
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 [4,10].
The equivalent scheme of phenomenon simulation is shown in Fig. 8, where: Q - breaker, CQ - grading capacitance, CE - 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.
U(t)
Rs
Rp
CQ
Ψ(i)
i
i2 i1
i3
CE
VT
Q
Figure 8. One phase equivalent scheme of network for ferroresonance calculation at the breaker switching off
The scheme can be described by equations (2):
⋅⋅⋅+
−⋅⋅+⋅+⋅−⋅=
=
−
)]cos(
)()]([[/1 211321
tU
CianaKKUK
dtdU
Udt
d
m
Q
nn
ωω
ΨΨ
Ψ
µµ
µ
)1()1(1 +⋅+=p
s
Q
ERR
CCK ;
pQ RCK
⋅=
12 ;
)1(3 +⋅=Q
Es C
CRK (2)
where: Ψ -flux linkage, )cos()( tUtU m ⋅⋅= ω - source
voltage, Rs=30,3 kΩ; Rp=1⋅109 Ω; Um = 3/2*110 kV . Magnetizing characteristic of VT can be described by the equation: n
naati ΨΨ ⋅+⋅= 12 )( , where: a1 = 3,17⋅10-6, an = 1,025⋅10-16, n = 5.
Fig. 9 presents a bifurcation diagram showing VT flux linkage vs. grading capacitance CQ. 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.
0
200
400
600
800
1000
1200
1 10 100 1000 10000 100000
Grading capacitance, CQ (pF)
50 Hz 16 Hz
chaos
stableunstable
Flux
link
age,
Ψ (W
b⋅tu
rn)
Figure 9. Bifurcation diagram - flux linkage vs. grading capacitance CQ (at fixed network capacitance CE=700 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. 10 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 CQ = 1000 pF, equivalent VT capacitance is Cp = 1390 pF, [11] and the network capacitance varies in the range of CE = 1 pF ÷ 0.1 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.
0
20
40
60
80
100
120
140
160
180
1 10 100 1000 10000 100000
steady state transient ferroresonance (chaotic) stable ferroresonance (3T0)
Peak
val
ue, U
max
(kV
)
Network capacitance, CE (pF)
Figure 10. Ferroresonance parameters at breaker switching off (fixed network parameters: CQ=1000 pF, CE=1 pF ÷ 0.1 µF, Rp=109 Ω, Cp=1390 pF)
Fig. 11a shows the results of field measurements and Fig. 11b shows the results of computer simulations for ferroresonance which appears after switching off the breaker. Network parameters are: Um= 3/2*110 kV , CQ = 390 pF
and CE = 700 pF. The transients before ferroresonance occurrence are a result of ignitions between poles of the breaker.
uz
iz
up
ip
ic
a)
(f ile juk123_1f az.pl4; x-v ar t) v :A 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40[s]
-90
-45
0
45
90[kV]
(f ile juk123_1f az.pl4; x-v ar t) c:A -B 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40[s]
-60
-30
0
30
60[mA]
(f ile juk123_1f az.pl4; x-v ar t) v :D 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40[s]
-90
-45
0
45
90[kV]
(f ile juk123_1f az.pl4; x-v ar t) c:C -D 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40[s]
-40
-20
0
20
40[mA]
(f ile juk123_1f az.pl4; x-v ar t) c:D - 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40[s]
-40
-20
0
20
40[mA]
uz
iz
up
ip
ic
b)
Figure 11. Ferroresonance after switching off the breaker: a) field measurements, b) computer simulation (fixed parameters: CQ = 390 pF and
CE = 700 pF). Descriptions: uz - supply voltage, iz - source current, up - voltage on VT, ip - VT current, ic - current in the VT capacitance.
In the range of network capacitance CE=2÷10 nF after the breaker switching off (Fig. 10), the permanent ferroresonance with period 3T0 appears. It is shown in Fig. 12a. The case of transient chaotic ferroresonance is presented in Fig. 12b. 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 juk123_3f az.pl4; x-v ar t) m:UE 0.0 0.4 0.8 1.2 1.6 2.0[s]
-60
-30
0
30
60
90*103
(f ile juk123_3f az.pl4; x-v ar t) v :DA 0.0 0.4 0.8 1.2 1.6 2.0[s]
-90
-45
0
45
90[kV]
(f ile juk123_3f az.pl4; x-v ar t) c:DA -EA 0.0 0.4 0.8 1.2 1.6 2.0[s]
-0.150
-0.075
0.000
0.075
0.150[A]
Voltage on VT, phase A
Voltage on VT, phase A - Poincare diagram
VT current, phase A
a)
(f ile juk123_3f az.pl4; x-v ar t) v :DA 0.0 0.4 0.8 1.2 1.6 2.0[s]
-200
-100
0
100
200[kV]
(f ile juk123_3f az.pl4; x-v ar t) m:UE 0.0 0.4 0.8 1.2 1.6 2.0[s]
-160
-80
0
80
160*103
(f ile juk123_3f az.pl4; x-v ar t) c:DA -EA 0.0 0.4 0.8 1.2 1.6 2.0[s]
-0.2
-0.1
0.0
0.1
0.2[A]
Voltage on VT, phase A
Voltage on VT, phase A - Poincare diagram
VT current, phase A
b)
Figure 12. Transients at switching off the breaker, CQ=1000 pF: a) CE=3980 pF- permanent ferroresonance with period 3T0, b) CE=251 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 [1] P. Ferraci, Ferroresonance, Cahier technique, No. 190, Groupe
Schneider, 1998. [2] J. Horak, “A Review of Ferroresonance”, 57th Annual Conference for
Protective Relay Engineers, Texas A&M University, 2004. [3] M. Iravani, “Modeling and Analysis Guidelines for Slow Transients -
Part III: The Study of Ferroresonance”, IEEE Transactions on Power Delivery, No 1, 2000.
[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, 2005.
[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/06/E/2006, Warszawa, 2006.
[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, 2006.
[7] B. Ermentrout, Simulating, analyzing, and animating dynamical systems. A guide to XPPAUT for researchers and students, SIAM, Philadelphia, 2002, (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 1, 2005.
[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, 06, 2007.
[10] E. Stawowy, “Ferroresonance in 220 kV voltage transformers” (in Polish), Przegląd Elektrotechniczny - Konferencje, Nr 1, 2005.
[11] “Results of voltage - current characteristic of the combined voltage transformer JUK123a measurements” (in Polish), (unpublished). Instytut Energetyki, 2004.