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PIERS Proceedings, Xi’an, China, March 22–26, 2010
502
The Application of the Equal Area Law in Ferroresonance for
Distribution Power System
Zheng-Wang Du1 , Heng-Xu Ha2 , Lei Zhai3 , Hai-Quan Zhou2 ,
Song-Bo Gou1 , and Chong-Shan Zhong1
1
Shenli Oil Field Power Company, Dong Ying, China
Dept. of Electric & Electronics Engineering, Shandong University of Technology, Zibo, China
3
School of Electrical Engineering & Automation, Hebei University of Technology, Tianjin, China
2
Abstract— Ferro-resonance frequently occur and become a serious threat to the safe operation
in the power system, in order to effectively suppress the complex phenomena of the system, this
paper introduces the time-varying phasor, while, establishes the time-varying phasor mathematical model corresponded it. Based on this, the Equal Area Rule is applied to the ferro-resonance
circuit analysis as a innovative way, that makes the analysis of ferro-resonance increased from
qualitative analysis to quantitative analysis, while, defines the energy function which gives the calculation method of the system stability margin. This paper verify the actual of the time-varying
Equal Area Rule applied in the ferro-resonance analysis through simulating the simple resonant
circuit with AT-P, that provide the theoretical basis to effectively prevent the ferro-magnetic
resonance.
1. INTRODUCTION
In the power system, there are many capacitive and inductive components, such as transformers,
generators, arc-suppression coil, reactor, circuit wire inductor, line wires and alternate with of
capacitance, compensation electrical appliances, the stray capacitance of the high-voltage equipment, etc. When operating the system or failure occur, the capacitors and the inductors formed
oscillation circuit components which may have resonance phenomenon. Resonance will lead to the
system or component over-voltage and thus endanger the equipment or insulation, and to produce
over-current caused equipment overheating and even burning [1].
In distribution network, it is a very common phenomenon that nonlinear ferro-resonance caused
by the saturation of voltage transformer core. So, over-voltage and over-current caused by ferroresonance often lead to burning of the voltage transformer [2, 3]. According to statistics, in 2007,
the transformer burning occurred 16 times in ShengLi Oilfield, which lead to 36 transformers are
burned down, it involved 9 substations and more than 50 lines cut off, and caused huge economic
losses.
At present, there are further research based on the ferro-resonance at home and abroad which
give a number of criteria and the corresponding analytical methods, such as Peterson’s theorem [4],
graphical method [5–7], analytical method [8, 9], digital simulation analysis method [10–13], wavelet
identification method [14], nonlinear dynamic analysis method [15–20], among which [4] through
a large number of tests given the conclusions that the ratio range between the zero-sequence capacitance and the inductive reactance in rated line voltageto determine the resonant type; the
literature [5–7] is simple and easy, which analyze the fundamental ferro-resonance through the
qualitative graph and roughly give the condition which the resonance occur; literature [8, 9] obtain
the system critical conditions of the fundamental ferro-resonance through complex mathematical
transform, filter out other frequencies; literature [10–13] analyze the fundamental resonance by
using the simulation software to simulate digitally; literature [14] introduce the wavelets to identify the type of resonance; literature [15–20] use the nonlinear dynamics to analyze the nonlinear
circuits, which give the method of the resonance or not and the resonant type by the phase-plane
characteristics. But the above methods all have limitations, some are not universal and suitable
only for the fundamental resonance research, and some are too imprecise and qualitative, and some
are too cumbersome to analyze the complex systems, and some give the criterion beside giving how
to eliminate the resonance, and so on.
In this paper, it introduces the time-varying phasor to analyze the system parameters timely,
while, establishes the time-varying phasor mathematical model corresponded it. Based on this, the
Equal Area Rule is applied to the ferro-resonance circuit analysis as a innovative way, that makes
the analysis of ferro-resonance increased from qualitative analysis to quantitative analysis, while,
defines the energy function which gives the calculation method of the system stability margin. This
Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010
503
paper verify the actual of the time-varying Equal Area Rule applied in the ferro-resonance analysis
through simulating the simple resonant circuit with ATP, that provide the theoretical basis to
effectively prevent the ferro-resonance.
2.
BASIC PRICIPLE
2.1. Time-varying Phasor
Sinusoidal voltage or current generated, from the rotating DC magnetic field, and the projection of
the real axis of the rotating phasor is defined sine wave. Steady-state phasor model was not fixed,
because the observer have the synchronization with the rotation. At the same time, assuming that
the phasor rotation speed is still the power angular velocity, when the system is disturbed, but the
phasor magnitude and the initial phase will change over time, then, the changed phasor shall be
time-varying, as shown in the next diagram:
Figure 1: Comparion of the steady phasor and the time-varying phasor.
In the two diagram, the left is the trajectory map which produce by the phasor rotate in space,
and the right is the projection on the real axis. Meanwhile, the above shows the steady-state
phasor, which rotating trajectory is a round, the below shows the time-varying phasor, which
rotating trajectory is irregular.
Assuming the sinusoidal voltage as:
h
i
h
i
u(t) = Um cos(ω 0 t + ϕ0 ) = Re Um (t)ejϕ(t) ejω0 t = Re U̇ (t)ejω0 t
which is shown as Figure 2.
Then, for the arbitrary variable x(t), we can express it as follow:
h
i
x(t) = Re Ẋm (t)ejω0 t
the number of its derivation is:
"
#
dx (t)
dẊm (t) jω0 t
= Re
e
+ jω0 Ẋm (t)ejω0 t
dt
dt
(1)
2.2. The Time-varying Phasor Mathematical Model
A simple series circuit, for example, comparing the steady phasor and buliding the time-varying
phasor mathematical model as in Figure 3.
R(t) is the total circuit resistance, L is the equivalent nonlinear inductance of the voltage
transformer, e(t) is the line voltage, UC (t), UL (t) is, respectively, the equivalent capacitance voltage
and the equivalent nonlinear inductance voltage.
PIERS Proceedings, Xi’an, China, March 22–26, 2010
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Figure 2: The time-varying phasor.
Figure 3: Series ferroresonance circuit model.
According to Figure 2, we establish the series resonance time-domain (steady-state phasor)
Mathematical Model:

dϕ


= UL


dt


 dU
I
C
= g(ϕ)
dt
C




e(t)
=
Ri

L + UL + UC


iL = g(ϕ)
When the introduction of the time-varying phasor concept, the mathematical expression of the
system parameters has changed, which is no longer a simple scalar, but the amplitude and phase
over time and change.
Required from the formula (1), we have:
h
i


ϕ(t) = Re Φ̇(t)ejωt


"
#
(2)
dϕ(t)
dΦ̇(t) jωt

jωt

=
Re
e
+
jω
Φ̇(t)e
=
U̇
−
jω
ϕ̇(t)
L

dt
dt
Other variables will be derivated such as (2), not go into details here.
By (2) combined with Figure 3, we can establish the time-varying model as follows:

dϕ̇(t)


= Ė − U̇C − jω0 Φ̇ − RI˙

 dt

dU̇C
I˙
=
− jω0 U̇C



dt
C


I˙ = G(Φ̇)
When the system is running at the equilibrium point, that is,

dΦ̇(t)



= Ė − U̇C − jω0 Φ̇ − RI˙ = Ė − ∆U̇ (Φ̇) = 0
dt
˙


 dU̇C = I − jω0 U̇C = 0
dt
C
By (3) and (4), we can obtain:
∆U̇ (Φ̇) = jω0 Φ̇ − j
G(Φ̇)
= U̇L (Φ̇) − U̇C (Φ̇)
ω0 C
(3)
(4)
(5)
2.3. The Balance Point
Changing the form of Equation (5), we can get the following equation:


∆U = |UC − UL |
∆U̇ = U̇C − U̇L








G(Φ̇)
G(Φ̇)
⇒ UC =
U̇C =


ω0 C
jω0 C






U
=
jω
U̇L = jω0 Φ̇
0 Φ̇
L
(6)
Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010
505
Through the Equation (6), in the system the relationship between parameters and flux linkage
can be showed in the picture 4.
In picture, the x-coordinate represents the flux linkage, the y-coordinate represents the voltage,
the blue dashed line represents the voltage across the nonlinear inductance, the green dashed
line represents the voltage across the capacitor, the red curve represents ∆U , the green solid line
represents the system voltage.
From the picture, we conclude that the A, B and C points are three balance points of the
system, two points among them are stable balance point (A and C), another is unstable balance
point (B), it is impossible for the system to operate at the unstable balance point. of the stable
balance points, one is normal operating point (A), and the other is stable resonance point (C).
The transient resonance is caused by the system oscillation at the stable balance point, the
oscillation frequency depends on the operating point and the capacitance, the oscillation amplitude
is determined by the disturbance quantity, the amplitude and frequency of the flux linkage directly
influence the over-voltage size.
Whether the stable resonance happens, network parameters and operation parameters play a
decisive role, most important parameters are C and Φ. If C can be adjusted according to the Φ
at the normal operation point, to make the ∆U curve monotone, then the system will lack the
resonance condition. From the qualitative analysis, we can draw the conclusion that the smaller
and bigger C can make the curve monotone, which is also consistent with the Peterson Theorem.
2.4. The Equal Area Rule
Section 2.3 shows the relationship between system resonance and parameters through the static
analysis of the system, and also put forward two balance points, one is normal working point,
another is stable resonance point. But the qualitative analysis is not enough, only quantitative
resonance condition or resonance criterion can put forward methods to solve the resonance problem,
to reduce economic losses. So the equal area rule play an essential role in system dynamic analysis.
In following picture, the x-coordinate represents flux linkage (P.U), the y-coordinate represents
voltage (P.U), the blue solid curve represents initial voltage E0 , the green solid curve represents
post-disturbance voltage E. A, A0 and C 0 respectively represent the normal working point at the
initial voltage, the normal working point and resonance point at the post-disturbance voltage point.
In order to better solve resonance problem, we introduce the Eaqual Area Rule, defining the
black area to acceleration area (S2 ) and red area to deceleration area which is stability margin (S1 ).
If the acceleration area is larger than the deceleration area, the system can enter the resonance
zone, and the system will operate at the C point, which result the happening of stable resonance;
on the contrary, the system will operate at the A0 point. At the same time, because I˙ = Ġ(Φ) is
time variant, the Equal Area Rule is time variant, that is to say, if the acceleration area is bigger
than the deceleration area, then the system satisfies the resonance condition, and if the time of
disturbance is appropriate, the resonance will happen, and the system will operate at the resonance
point. So if the time of disturbance is appropriate and the system satisfies the following condition:
When S2 > S1 , the system can enter resonance condition, and operates at resonance balance
point C 0 ;
When S2 > S1 , the system can not enter resonance condition, and operates at the normal
working point;
When S2 = S1 , the system is at the critical resonance state.
If we define the energy function S (calculating area) is as follows:
Zb
S=
|∆U − E|dϕ
(7)
a
where a, b are the x-coordinates of balance points (A, A0 ) for acceleration area, and balance points
(A0 , B 0 ) for deceleration area, ∆U is the Equation (6).
2.5. Experiment Example and Simulation Verdication
We establish a simple series resonant modethe fault type is the voltage rise, as shown below:
U1 is the initiative voltage, and U2 is the risen voltage at last, in the simulation we let U2 =
3000 V, the a switch’s working time is −1 ∼ ts and the b switch’s working time is t ∼ 10 s, and
R = 6000 Ω, C = 0.011 Uf, the excitation characteristic curve of nonlinear inductance is as follows
(calculated under p.u):
PIERS Proceedings, Xi’an, China, March 22–26, 2010
506
B'
Figure 4: ∆U and the flux.
Figure 5: Analysis of the equal area.
From the Equation (8), three balance points can be got:
Where E is the voltage of source.
Through the MATLAB program, we can get A0 = 1.049 and B 0 = 1.434. According to the
Equation (7), we can get S1 = 0.0212, the critical voltage which satisfies S2 = S1 is U1 = 2321.3 V.
According the theory of Equal Area Rule, at the appropriate time, when U1 < 2321.3 V, that is
S2 > S1 ,
∆U = |UC − UL | = E
(8)
the system can enter resonance condition; when 2321.3 V < U1 < 3000 V, that is S2 < S1 , the
system can return to the normal working point A0 .
1)
2)
3)
4)
when
when
when
when
t = 0.223 s, U1 = 1500 V, other conditions don’t change, the current is as Figure 8;
t = 0.222 s, U1 = 2000 V, other conditions don’t change, the current as Figure 9;
t = 0.01 s, U1 = 2000 V, other conditions don’t change, the current is as Figure 10;
t = 0.012 s, U1 = 2500 V, other conditions don’t change, the current is as Figure 11.
Contrast to Figure 8, Figure 9 and Figure 11, the Equal Area Rule theory is validated; contrast
to Figure 9 and Figure 10, we can validate that the fault time influence the happening of the
resonance. Through the four pictures, we find that the time variant Equal Area Rule is applicable
in resonance problem, which provide an important theoretical basis for avoiding resonance.
Figure 6: Model of the equal area rule.
Figure 7: The fitting excitation curve.
Progress In Electromagnetics Research Symposium Proceedings, Xi’an, China, March 22–26, 2010
507
Figure 8: The current waveform at U1 = 1500 V.
Figure 9: The current waveform at U1 = 2000 V.
Figure 10: The current waveform at U1 = 2000 V.
Figure 11: The current waveform at U1 = 2500 V.
3. CONCLUSION
The paper first introduce the new conception of time-variant vector, and establish time-variant
vector mathematical model of a simple series resonance model, on the basis of it, we put forward
the conclusion of three balance points (one is normal working point, one is unstable working point,
another is frequency resonance point).
Secondly, on the basis of static analysis, we define the energy function, and make dynamic
analysis to the system. We put forward the Equal Area Rule on the Basis of time-variant vector,
and we calculate the stability margin S1 through MATLAB and ATP:
When the acceleration area is larger than S1 , we find the industrial frequency resonance point
through simulation.
When the acceleration area is smaller than S1 , the system operates at the normal working point.
In summation, we validate the correctness and practicality of the Equal Area Rule in the analysis of resonance problem, which provides theoretical basis for the future research of restraining
resonance.
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