A TVAR Parametric Model Applying for Detecting Anti-electric-Corona Discharge Zhe Chen, Hongyu Wang, and Tianshuang Qiu School of Electronic and Information, Dalian university of technology, Dalian, P.R. China 116023
[email protected] http://dsp.dlut.edu.cn
Abstract. It is well known that the time-varying parametric model based on wavelet neural network has excellent performance on modeling a signal. The spark discharge signal is a typical nonstationary one. The spark discharge has two types in a static dust catcher. one is normal discharge, another is antielectric-corona discharge. A few simulations indicate that the performance of the TVAR model on modeling a discharge signal is fineness, especially, on distinguishing between normal discharge and anti-electric-corona discharge.
1 Introduction The time-varying autoregressive (TVAR) parametric model is representative and used widely [1]. Assume x(n) is a nonstationary random signal with zero-mean. Grenier proved that the TVAR parametric model does not always exist [2], In this paper, how to prove the model exists or not is not touched upon. it is supposed that the model exist is true, and its analytical expression is shown in (1). x(n) + a (n,1) x(n − 1) + " + a (n, p ) x(n − p ) = e(n) .
(1)
Accordingly, its time-varying transform function which is denoted by H(n, z) can be written as
1
H (n, z ) =
p
1 − ∑ a ( n, i ) z
−i
.
(2)
i =1
Where, a(n, i), i = 0, 1, … , p, are the time-varying parameters of the model. e(n) is 2 variance for exciting the model. the white noise with zero mean and For some special random signals, Kalman filter algorithm, polynomial-algebraic algorithm etc. can be used to solve the parameters of model [1], [3]. But, decomposing a(n, i) to linear combination of base function is a general method [1]. The base function can be chose freely. In general, They are at least linear independent, selfcontained, may as well orthogonal to reduce the number of parameters (such as Walsh
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F. Yin, J. Wang, and C. Guo (Eds.): ISNN 2004, LNCS 3174, pp. 607–612, 2004. © Springer-Verlag Berlin Heidelberg 2004
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function). Assume that base function is fk(n), k = 0, 1, … and si(k) are the weighted coefficients, then a(n, i) can be written as ∞
(3)
a (n, i ) = ∑ si (k ) f k (n) . k =0
Put (3) in (2), H (n, z ) =
1 p
∞
1 − ∑ ∑ s i ( k ) f k ( n) z − i
(4)
.
i =1 k = 0
By this way, the problem with time-varying coefficients can be converted to the one with constant coefficients (all of si(k) are constant). Certainly, other costs have to be suffered. Because of the number of unknown coefficients increasing, the chief cost is the scale of problem. When the time-varying problem is considered, the cost will be worth paying out. To get these coefficients(viz. si(k)), Recursive Least Square (RLS) algorithm or polynomial-algebraic method are available [1], [3]. For all practical purposes, limited precision a(n, i) may be also accepted. Therefore, superior limit of k need not be up to , Assume it is up to m-1. The “m” is named “base degree”[1]. The wavelet neural network (WNN) [4] for function approximation can be write as
∞
m −1
f (t ) = ∑ wk ϕ ( k =0
t − gk )+d . hk
(5)
Where, “m” is the number of wavelet function (viz. base degree); “gk” is the kth shift factor; “hk“ is the kth scale factor; “wk” is the kth weight coefficient; “ (t)” is a kind of mother wavelet. “d” is a constant for approximating to non-zero mean function. In expressions (1), substitute a(n, i) by the WNN shown in expressions (5), and change wavelet to a discrete form. The TVAR based on WNN can be obtained as
ψ
p m −1
x(n) + ∑ [ ∑ wik ϕ ( i =1 k =0
n − g ik ) + d i ] x ( n − i ) = e( n ) . hik
(6)
After the wavelet neural network is introduced into TVAR parametric model, How to estimate the parameters of model is converted to how to train the neural network. There are several training algorithms available to solve the model parameters in (6). Zhang used the random gradient algorithm in his WNN [4]. Szu used conjugation gradient algorithm to solve parameters [5]. Other algorithms, such as orthogonal Least Square algorithm are available [6]. To reduce computation cost, a new LMS algorithm based on input signal whitened is presented [7]. Because only part data is used in an iterative training, its computation cost will be very low.
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2 The TVAR Parametric Model Based on WNN Applying for Detecting Anti-Electric-Corona Discharge The electric sketch map of static dust catcher is shown in Fig.1(a). A high voltage charge source builts a static field between two pole board. When a dust particle passes the room between two pole board, ionization will happen. The ionization results in the dust particle is decomposed into a electron and a dust particle with positive charge. Any electriferous particle will be forced to move in electric field as shown in Fig.1(a). When a dust particle reach negative pole board, the electric neutralization makes it discharge its charge. When a dust particle passes through the median of two pole board, The minimum voltage to insure any dust will not escape from the dust catcher is U min =
md 2 v 2 2
qL
(7)
.
Where, d is the distance of two pole board. U is the voltage of two pole board. L is the length of pole board. m is the mass of a dust particle. q is the charge amount of a dust particle. d
L
+ + +
v
_ _ _
+
U
-
high voltage charge source
_ _ _
+ + +
dust flow (a)
(b)
Fig. 1. Electric sketch map of static dust catcher
For a kind of dust, m and q are usually considered as a constant. When a dust catcher is made of, d and L are almost not changed. To process more dust particles, v is needed to increase. So U is also needed to increase. By limit of air discharge intensity, U does not increase unboundedly. Because the property of dust particles changes momently, discharge between two pole board may occur(shown in Fig.1(b)). The power of high voltage source is usually small. When a discharge occurs, the voltage between two board will be very low. A spark discharge will not transition an arc discharge. To avoid the discharge is very difficult, but, the times need to be controlled. If not, the efficiency of dust catcher will be very low. After discharge, dust particles reveal neutral electricity character. But they still stay up in virtue of adhere force. Obviously, the discharge of dust particles on surface have to pass inner dust particles. While the resistance of dust layer is very high, it will results in discharging difficultly. In fact, dust layer on negative pole board comes into being a layer with positive charge, which is shown in Fig.2(a), and it is zoomed in
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Fig.2(b). In Fig.2, the electric field largens the adhere force of dust particles, and dust is wiped off hardly by vibration. On the other hand, dust layer close with negative board, the discharge happens easily. (anyone can easily find the baby blue glimmer on negative board surface) This kind of discharge is called anti-electric-corona discharge. The anti-electric-corona discharge is usually expected, because it makes the electric field intensity descend, and results in dust can not be wiped off effectively.
_ _ _
-
+
+ + +
_ + _ + _ +
high voltage charge source
_ _ _
+ + +
dust layer
dust layer (a)
(b)
Fig. 2. Sketch map of anti-electric-corona discharge U2
U1
I
R _ _ _
+ + +
+
high voltage charge source
Fig. 3. Discharge signal collection sketch map
When a static dust catcher is working, its work state is needed monitor, especially, if anti-electric-corona discharge has occurred. If it is, all of steps must be adopted to wiped off dust on pole board. As we know, the mechanism of discharge is that there is a plasm channel between two electriferous object. Because plasm is very unsteady, the electric character of plasm is also nonstationary. How to monitor dust catcher also becomes very difficult. In this paper, The TVAR parametric model based on WNN is adopted for discharge signal modeling. By the TVAR parametric model, signal character can be extracted easily. In Fig.3, The signal collection sketch map is shown. In Fig.3, U1 denotes the voltage between two board. U2 denotes the current between two board by ohm law (I = U2/R). They are collected by digital oscillograph. Usually, a broad way attenuation network is needed to avoid to damage the input device of oscillograph.
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3 Simulation and Conclusion For normal discharge and anti-electric-corona discharge, four sets data including voltage and current signal are collected. The total number of examples is twenty. The TVAR parametric model based on WNN is used to model the collection data, and the training algorithm in [7] is used to solve the parameters of model. In the TVAR model, p equals 2, base degree equal 3. The initial value of WNN parametrics are according to reference [5]. Because digital oscillograph is working in trigger-mode, The are a lot of noise data before-and-after valid data. To improve modeling effect, shorten the training time, All of original data are needed to preprocess properly. The preprocessing process includes seven step. They are (1)high pass filter with linear phase to wipe off DC component; (2) wipe off unuseful noise data by short time windowed energy; (3)obtain main component period (Tmain) by classical power spectrum density; (4) extract valid data including about ten Tmain; (5) high pass filter again with linear phase to wipe off DC component; (6)find out absolute maximum; (7) normalize all data by the absolute maximum. 1.2
1.2 1.0
0.8
TVAR origin
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -500
0
mormalize amplitude
normalize amplitude
1.0
0.8
TVAR origin
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -500
500 1000 1500 2000 2500 3000 3500 4000
0
sample
sample
(a) Normal voltage
(b) Anti-electric-corona voltage
1.2 0.8
TVAR origin
0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -200
0
200 400 600 800 1000 1200 1400 1600
sample
(c) Normal current
mormalize amplitude
mormalize amplitude
1.0 0.6
500 1000 1500 2000 2500 3000 3500 4000
1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 -200
TVAR origin
0
200 400 600 800 1000 1200 1400 1600
sample
(d) Anti-electric-corona current
Fig. 4. A few modeling results (TVAR denotes the output of model)
A few results of modeling are shown in Fig.4. It can be found that the avail is acceptable. According to same procedure, every data example has its model. Based on these models, the equivalent attenuation coefficient of model can be calculated [8],
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and they are listed in table 1. The equivalent attenuation coefficients of normal discharge voltage fasten on 0.0007, and the ones of anti-electric-corona discharge voltage fasten on 0.0015. the equivalent attenuation coefficients of normal discharge current fasten on 0.0036, and the ones of anti-electric-corona discharge current fasten on 0.0033. They can be easily distinguish from each other, so, they can be parametric for monitoring dust catcher directly or input for other pattern sorter. The wavelet transform has excellent character on time-frequency field, and it is also called microscope on math. The neural network has powerful learning ability, and it is fit for nonlinear problem. The WNN combine. When the WNN is introduced in the TVAR model, their virtue is exerted, and very fit for model of nonstationary signal. Table 1. Equivalence attenuation coefficients by algorithm in this paper discharge state(voltage)
normal
anti-electric-corona
coefficients 7.263e-004 7.120e-004 7.584e-004 7.193e-004 7.088e-004 1.623e-003 1.587e-003 1.597e-003 1.548e-003 1.633e-003
discharge state(current)
Normal
anti-electric-corona
coefficients 3.649e-003 3.646e-003 3.827e-003 3.800e-003 3.538e-003 2.946e-003 3.515e-003 3.346e-003 3.437e-003 3.257e-003
Acknowledgements. The research is supported by: National Natural Science Foundations of China (No. 60172072, No. 60372081).
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