Arc Model Parameter Extraction Techniques using Nonlinear Least Squares Bienvenido Rodríguez-Medina 1, Lionel Orama-Exclusa2, and Miguel Vélez-Reyes1 (1)
Electrical and Computer Engineering, (2) General Engineering Department University of Puerto Rico, M ayagüez, Puerto Rico, 00681-9042, e-mail:
[email protected],
[email protected],
[email protected] Abstract – This paper presents nonlinear least square GaussNewton parameter estimation for the Modified Cassie-Mayr arc model used to describe the arc behavior in the postcurrent zero period. The proposed algorithm avoids numerical differentiation presented in many parameter extraction techniques discussed in the literature. Experimental results are shown to present the methodology. Keywords – arc models, circuit breaker simulations, parameter extraction, system identification, nonlinear least squares
I. INTRODUCTION The way in which the current (electric arc) is extinguished in circuit breakers and interrupters is complicated. It can’t be explained in a quantitative way, so the use of models is helpful to study, characterize, and understand the basics of circuit interruption. In the past, power systems transients caused by circuit breakers were not represented accurately. Some of the approximations used to represent circuit breakers include: Ideal switch models, voltage, current or time controlled switches, and programmed time dependent or nonlinear resistances. Those approximations were good for the study of the Transient Recovery Voltage (TRV), but inappropriate for other applications such as the dielectric failure and thermal breakdown [1]. The modeling of the nonlinear behavior of the electric arc was needed to overcome this problem. Some research and development efforts have been directed to the use of computer methods for the evaluation and design of circuit breakers, leading to the use and creation of arc models to describe the circuit breaker behavior and their interaction with the electrical network. The use of computer methods reduces the cost and space needed when realizing synthetic test circuits and prototyping. A. The Electric Arc The electric arc is not found only in nature, it is also present whenever two conductors are separated to interrupt a circuit current [2]. The arc is generated before the current extinguished, so the process of extinguishing a current is consider extinguishing the arc. The electric arc is a self-sustained discharge having low voltage drop and able to support great amplitudes of current [3]. For this reasons it was thought that the arc was an
obstacle to current interruption. Now days the art and science of circuit interruption is to control the arc to act as a variable resistor, in such a way that current interruption is obtained [4]. The technical basis of circuit breakers or current interruption lies in creating arc plasma (electric arc) of as high a conductivity possible, carrying a large current, cooling it effectively and converting it into insulating gas space in a very short time. During circuit breakers operation the electric arc behaves as a nonlinear resistance [5]. After the opening of the contacts current flows through of the arc formed from the plasma. This plasma has been created between the breaker’s contacts. It is after the current has cross it zero value that the transient recovery voltage (TRV) builds up between the contacts, although after current interruption (current-zero) a small current called the post-arc current continues to flow a few microseconds (µs) after the current zero. This phenomenon is possible since the insulation medium will still hot and some ionization will still be present, letting post-arc current flow through the hottest paths [6]. B. Arc model Incorporation to Circuit Breaker Simulations The process of incorporating the electric arc models in circuit breaker simulations involves the procedure describe below [7,8]: • Choosing the software tool: In order to study the operation of circuit breakers and their interaction with the power system, models of the system and the interrupter have to be implemented in computer software. • Choose an arc model: The arc models are described by differential equations, which relate the rate of change of the conductance with the arc current and voltage. • Test Oscillograms: Voltage and current oscillograms must be obtained from field or laboratory tests on circuit breakers. These oscillograms are descriptive of the behavior of the arc during the interruption process. • Arc parameters evaluation: Models have experimental parameters that improve the arc representation. These parameters have to be determined or estimated.
•
Numerical Simulations: After all the above steps are completed we can proceed to reproduce the entire process of arc interruption in the circuit breaker and study its behavior under different circuit conditions. Current Zero
Post Arc Current
Equation 1 describes the rate of change of the conductance of the arc with respect to time; this equation can be seen as an energy balance equation [9]. In the modified Cassie-Mayr model the arc time constant and the power dissipation are dependent of the conductance of the arc. This dependency was postulated by Avdovin et al., during the study of puffer type gas circuit breakers in [10]. The constants P0 and t 0, and the experimental constant parameters a and ß should be found from voltage and current oscillograms of a tested device, before using the model for simulations of different electrical system conditions.
III. PARAMETER DETERMINATION M ETHODS
TRV t=to
A. Amsinck Method Figure 1 Transient Recovery Voltage and Post-arc Current on a High Pressure Gas Circuit Breaker.
C. High Pressure Gas Arc Model A modified Cassie-Mayr equation was the model selected to represent high pressure gas circuit breakers during this work. This model is a combination of the models postulated by Cassie and Mayr. The modified Cassie-Mayr model represents the conductance nonlinear behavior of the arc and states the following, 1 dg 1 ui = − 1 (1) g dt t (g) P(g)
This method applies only for the cases where a thermal breakdown occurs during the circuit breaker test [5,6,11]. Amsinck assumes that P(g) and t (g) are equal at points of equal conductance. To determine the parameters (P0, t 0, a and ß), the voltage and current oscillograms are used to create a table of the conductance (g = i/v) from some µs before to some µs after current zero. Then the arc model equation is applied to the points marked with ta and tb in Figure 2, from which we could see that:
g 'A =
1 i A2 −gp τ ( g p ) P( g p )
(4)
g
t (g) = τ 0 g
β
P(g) = P0 gα
(2) (3) Breakdown
where, g
g is the conductance of the arc, in Siemens. u
A
B
p
is the voltage drop across the arc, in Volts.
t
i is the current through the arc, in Amperes.
ta
t is the time constant of the arc, in seconds. The time con-
tb
Figure 2- Representation of Amsinck method.
stant can be interpreted, as a measure of the arc will to survive. t is a representation of the inertia inherent in
g 'B =
the arc, since it can not be extinguished instantaneously.
1 i B2 −gp τ (g p ) P(g p )
(5)
P is the power dissipated from the arc to the surrounding
g 'A = dg
g 'B = dg
gas, in Watts.
where
a is the parameter that influences the conductance de-
i A = i(t A ) and iB = i(t B ) . If we solve for g p in equa-
pendency of P. ß is the parameter that influences the conductance dependency of t.
dt t =t A
and
dt t = t B
,
also
tions 4 and 5, then we can yield to the following result;
τ (g p ) =
1 i 2A − iB2 P(g p ) g 'A − g B'
(6)
If now we substitute (6) in (4) we get
P( g p ) =
g i −g i g p . g 'A − g ' 2 A B
' 2 B A ' B
(
)
(7)
in the same way we can substitute (7) in (6)
τ (g p ) =
g p . i B2 − i 2A
(
)
i g −i g
' A
2 A
' B
2 B
(8)
The process describe above just determine a pair of values for P(g) and t (g) as a function of the conductance, so the process has to be repeated for various values of the conductance to obtain curves for P(g) and t (g). After the curves are obtained we can use regression to P0, t 0, a and ß. B. Generalized Method This method is based on the use of two sets of test oscillograms from the same device and test circuit [6]. Taking a look to Figure 3 the conductance traces of test A and B can be observed for the same device (circuit breaker). If we use the arc model equations on points A and B, we would find that P(g) and t (g) at g p would be:
g ' i2 − g ' i2 P( g p ) = A B ' B ' A g p gA − gB
(
τ (g p ) =
(
g p i B2 − i A2
)
(9)
)
(10)
i g − i g 'A 2 A
' B
2 B
As in the previous method the process should be repeated for various points, so then a regression could be done to obtain P0, t 0, a and ß.
time to our arc model equations. Solving for the product v.i dg v.i = P0 dt τ ( g ) + 1 (11) g Then the equation could be write as
g' v.i = f and g
we could think of v.i as the equation for a line (y=mx+b), where v.i is the dependent variable and the independent ' variable is g
g
. Now we could think of P(g).t (g) as the
slope of a line where all the conductance points are equal, and the intersection with the x- and y-axis of the line is described by the following equations: • •
x-axis intersection → v.i = 0 ∴ 1
τ (g ) = −
y-axis intersection → − g
'
g
g' g
= 0 ∴ P ( g ) = v.i
If we have various oscillograms, we could obtain a set of points for P(g) and t(g). Then the process is repeated n times, where n is the test number. Finally, as in the previous methods we could use regression to obtain the model parameters. v.i
P(g1)
g P(g2)
Test B
Test A
P(g3)
1 1 1 τ (g 1 ) τ (g 2 ) τ ( g 3 ) A
B
gp
t
-g' / g
Figure 4- Schematic representation of Ruppe method.
D. Glinkowski-Takahashi Method ta
tb
t=0
Figure 3- Representation of the generalized method.
C. Ruppe Method In this method different oscillograms are needed, the method is based on the following belief [6]; If P(g) and t (g) are function of the conductance they should follow the same behavior for different electric arcs (under similar conditions). So, to use this method we should perform various tests to the same device under the same test circuit conditions. For the purpose of demonstration, lets look one more
The method proposed by Glinkowski and Takahashi in [7] only uses a single test oscillogram of the device for the parameter determination process. They didn’t establish a preference between test oscillograms containing a successful interruption or a breakdown. In this case we are dealing with a more general method than those described earlier. In order to optimize the parameters P0, t0, a and ß the arc model equation is rewritten as
gˆ ' (t ) =
dgˆ 1 1 = v.i .g 1−α − β − g 1− β dt τ 0 P0 τ0
(12)
gˆ ' (t ) = where
a number of input and output measurements N, which can be set as
dgˆ = A'v.i. g a' − B ' g b' dt
[
introduction of A’, B’, a’ and b’ the optimization problem could be seen as separable into two steps, linear and a nonlinear. Where A’ and B’ are the linear parameters while a’ and b’ are the nonlinear ones. In the first step, as in all the methodologies above conductance is calculated (g=i/v). Then the derivative of the conductance is calculated point by point from
g 'p (t ) =
g p (t ) − g p (t − ∆t ) ∆t
[
error = max g ' − gˆ '
]
y = [y (1), y (2 ),.....y ( N )]
(14)
T
N
where u(t) and y(t) are the input and output signal of the system, respectively. This data is then fitted to a model (the model could be a white, grey or black box model)
yˆ (t ,θ ) = g (t ,θ ) where
yˆ (t, θ ) is the prediction of y (t ) , and θ is the
(
)
VN θ , Z N of the error between model and system output;
g (t ) = ' p
g p (t − ∆ t ) − g p (t + ∆t ) 2∆t
(15)
Second, the model is rewritten as
gˆ ' (t ) =
dgˆ g v.i = − 1 dt τˆ (g ) Pˆ ( g )
(
θˆ = arg min VN θ , Z N
[
]
error = ∑ g (t ) − gˆ (t ) i =1
' p
2
(20)
(
)
VN θ , Z N =
1 N ∑ l (ε (t,θ )) N t =1
(21)
ε (t, θ ) = y (t ) − yˆ (t ,θ ) equation (21) being the cost function,
(22)
ε (t,θ ) the error and
l(.) is some appropriate norm (common choices are the
norm-1, norm-2 and the infinity norm). After obtaining the parameters the final step would be to validate the model. Figure 5 exemplifies a system identification task, the process is assumed as having a single output for simplicity.
(16)
estimated values of τˆ(g ) and Pˆ (g ). Then an unconstraint optimization is performed on the parameters to minimize the error between equations (15) and (16) using Simplex method [12]. The final distinction of the method is how the error function is defined (see equation 17). ' p
)
where
noise
where gˆ ' (t ) is the value calculated by the model for the
p
(19)
parameter vector which has to be estimated. The parameter estimation is frequently done by solving an optimization problem which requires the minimization of a cost function
E. Austurian Method This method was proposed by Giménez et al. in [8], and as in the Glinkowski-Takahashi method only uses a single test oscillogram for the parameter determination process. The logic of this methodology is similar to the one presented by Glinkowski and Takahashi. There are three main differences between Austurian and Glinkowski-Takahashi methods. First, the derivative of the conductance is calculated point by point not from equation (13) but from
(18)
u N = [u (1), u (2 ),.....u (N )]T
(13)
Then an unconstraint optimization is performed on the parameters to minimize the error between equations (12) and (13) using fmins, a MATLAB built-in function [28]. The error is define by
]
Z N = y N ,u N ,
gˆ ' (t ) is the value calculated by the model. With the
(17)
Process
+
+ y +
error
u
_ Model
^y
F. System Identification Applied to Arc Modeling During the course of this work a system identification based methodology is proposed. This methodology should get rid of all the restrictions and assumptions made by the methodologies presented on the previous section. On the system identification theory a problem can be stated as follows (more specific information about system identification problems could be found in [13,14,15]). Given
Figure 5- System identification methodology, a model is adapted in order to represent a system behavior.
The parameter determination methodology developed on this work was obtained from using the norm-2 on the cost function, which results on a least squares optimization
problem. In fact, since the model we used is nonlinear in the parameters the problem results on a nonlinear least squares optimization. The standard method to compute the parameters estimates in a nonlinear least squares problem is the GaussNewton method [12,13]. This is an iterative method of the form (i ) ˆ(i +1 ) ˆ( i ) (23)
θ
=θ
The most common approximation is the fourth order Runge-Kutta approximation [16]. Let suppose we have the following equation
dy = f (t , y(t )) dt the fourth order Runge-Kutta will lead to the following
k1 = hf (t , y(t ))
+ γ .∆θ
ˆ (i )
k h k 2 = hf t + , y (t ) + 1 2 2 k h k 3 = hf t + , y (t ) + 2 2 2
( i)
where θ is the estimate at the i-th iteration and ∆θ is the Gauss-Newton search direction computed by solving the linear least squares problem
( )
−1
∆θ (i ) = argmin r (i ) − J (i ).∆θ = J T J JT r ( i)
where r is the residual vector and J matrix at the i-th iteration are given by ˆ r (i ) = ε t ,θ ( i ) , J (i ) = ∂y(t ,θ )
(
)
∂θ
(i )
is the Jacobian
θ =θˆ( i)
In order to implement the Guass-Newton method, lets rearrange equation (1) as
dgˆ g w = f (t, g,θ ) = −1 dt t (g) P(g) w = v.i . Now we may think of
(26)
where obtaining the model parameters as in system identification problems, w being the measured input and g the measured output. It may be convenient to recall that on the model implementation in circuit breaker simulations the arc is represented as a nonlinear resistance ( R
= g −1 ).
Right now the only problem we have to solve in order to implement the Guass-Newton method is how to obtain the Jacobian matrix, since J = ∂gˆ (t ,θ ) in our case (equation ∂θ 26) describe the rate of change of the conductance
dgˆ = gˆ ' . To solve this problem we can differentiate dt
equation (26) by each parameter and get the following results
∂gˆ ' ∂f (t , g,θ ) ∂f (t , g,θ ) ∂g = + . ∂θi ∂θi ∂g ∂θi
(27)
where h is the time step. The step size should be chosen reflecting the desired accuracy, and the simplest solution is to keep reducing the step size until the solution does not change within the desired tolerance. The compactness of this fourth order Runge-Kutta formula provides a nice compromise between implementation and execution effort. H. Gauss-Newton Implementation in MALAB The MATLAB software was used for the creation of our parameter extraction tool. The use of MATLAB offer two advantages : first, the software operates in matrix form so the test oscillograms could be enter by the user as an input data matrix; second, MATLAB possess a large variety of toolboxes containing algorithms and functions which facilitate the creation of more sophisticated codes. A general diagram describing the concept of the parameter extraction tool is shown in Figure 6. The main program can be seen as receiving two inputs, the test oscillograms (voltage and current oscillograms) in matrix form and the parameters initialization. During the course of this work the mathematical model was fixed, but model selection can be included in future versions. The Guass-Newton optimization routine contains all the equations and processes explained on the previous section. When the program reaches this stage, it starts the process of fitting the data to the mathematical model. Then the program will start to iterate until satisfying one of the following:
Because of the presence of ∂ g and g on the right hand
•
side we can not simply integrate with respect to t to ob-
•
∂θ i
calculate a new value of ∂g . Fortunately there are meth-
∂θ i
ods for solving ordinary differential equations (ODEs) that are relative straightforward to implement.
the error is smaller than a specified tolerance,
g − gˆ ≤ ε 1
∂θ i
tain ∂g . We will need some kind of iterative procedure to
(29)
k 4 = hf (t + h, y(t ) + k3 ) 1 y (t + h ) = y (t ) + (k1 + 2k 2 + 2k 3 + k 4 ) 6
(24)
(25)
(28)
the stopping criterion is satisfied,
θ i+1 − θ i ≤ ε 2 •
the program reach the maximum number of iterations. After the program satisfied any of those statements, it returns the optimized parameters which may be used within the models for the simulation of the circuit breaker.
Data in Matrix Form
perturbations in the data points [18, 19]. Thus, GlinkowskiTakahashi methodology may be used in occasions where high quality data (noise free data) is available.
Parameters Initialization
Statement of the Mathematical Model
500 0
Gauss-Newton Nonlinear Least Squares Optimization Routine
Arc voltage (Volts)
error< tol
-500
yes
no Stopping Criterion
-1000 -1500 -2000 -2500 -3000 -3500
yes
-4000 0
1
2
no
3
4
5
6
7 -7
Time (seconds)
no
x 10
(a) Iter. > Max. Iter. 0.2
yes Optimized Parameters
IV. COMPARISON OF RESULTS Lets us illustrate the properties of the different parameter extraction approaches by an example. The arc conductance 1 in Figure 7c will be approximated by the modified Cassie-Mayr model (equation 1). The voltage and current traces which are translated to the input w are shown on Figure 7a-b. In Table 1 there is a summary of the results obtained with four of the parameter extraction techniques including the one proposed on this work. Owing to the use of MATLAB built-in functions during the implementation of the other methodologies, ours needs more computational time. It results obvious from observing the results (Table 1) and Figures 8-11 that the proposed system identification technique performs better than the other methods. However, we should note that our results were close to those obtained by the Glinkowski-Takahashi methodology. Clearly this might bring the question: Which methodology is better? The answer to this question depends on the specific data under consideration, but some general guidelines can be given. In terms of the parameters initialization the Glinkowski-Takahashi method seems superior to the one proposed on this work, since their methodology takes advantage of initializing only the nonlinear parameters. This type of parameter initialization eliminates the worrying of needing four good initial parameter guesses. In terms of performance the identification technique proposed on this work is favorable with respect to noisy measurements. In Glinkowski-Takahashi methodology the conductance rate of change is calculated by Euler differentiation having an error calculation of ∆t , leading to large errors from small 1
Measurements data provided by P.H. Schavemaker [17].
Arc current (Amperes)
im-
0.1
0.05
0
-0.05
-0.1
-0.15 0
1
2
3
4
5
6
Time (seconds)
7 x 10
-7
(b) x 10
-4
8
Arc conductance (Siemens)
Figure 6 General diagram of the parameter extraction tool plemented in MATLAB.
0.15
6
4
2
0 0
1
2
3
4
Time (seconds)
5
6
7 x 10
-7
(c) Figure 7- Training data for fitting the model parameters; (a) voltage between the circuit breaker contacts, (b) current across the breaker contacts, (c) arc conductance.
Table 1- Comparison of the results for our Gauss-Newton implementation and other parameter extraction methodologies encounter in the literature. Gauss-Newton
Glinkowski-
Method
Generalized
RMSE*100
0.00246
0.00271
0.00605
0.01625
Time
89.1560 sec.
17.5310 sec.
2.5940 sec.
5.1250 sec.
P0
8.4894e5
4.3588e4
4.3542e6
1.8426e7
t0
2.1618e-6
4.1663e-6
3.1349e-6
1.0000e-6
α
0.81644
0.53706
-3.73300
1.32610
β
0.38510
0.46639
0.32986
0.30000
-4
-4
x 10
8
Measured conductance Model Error
x 10
Measured conductance Model Error
6 Arc Conductance (Siemens)
8 Arc Conductance (Siemens)
Austurian Takahashi
6
4
4
2
0
2
-2
0 0
1
2
3 (seconds) 4 Time
5
6
7
-4 0
-7
x 10
Figure 8- Model fitting results for the training data applying a system identification procedure.
1
2
3 4 Time (seconds)
5
6
7 -7
x 10
Figure 10- Model fitting results for the training data using the Austurian method. -4
x 10
-4
x 10
Arc Conductance (Siemens)
Arc Conductance (Siemens)
Measured conductance Model Error
8
Measured Conductance Model Error
8
6
4
2
6
4
2
0
0 0
0
1
2
3
4
Time (seconds)
5
6
7 x 10
1
2
3 4 Time (seconds)
5
6
7 -7
x 10
-7
Figure 9- Model fitting results for the training data using Glinkowski-Takahashi method.
Figure 11- Model fitting results for the training data using the Generalized method.
V. M ODEL VALIDATION We tested the quality of the best two approximations by choosing another set of data, where the device was under same operation condition. The results are presented on Table 2 and Figure 12. From the results we confirmed that the model resulting from our nonlinear least squares Gauss-Newton parameter estimation is of a better quality. -4
Arc Conductance (Siemens)
8
x 10
Measured NLS G-N Glinkowski-Takahashi 6
4
2
0 0
1
2
3
4
5
6
7
8
Time (seconds)
9 x 10
-7
Figure 12- Model approximation results for the validation data.
Table 2- Comparison of validation results
RMSE
Proposed Method 2.3433e-5
Glinkowski-Takahashi 2.8922e-5
VI. CONCLUSIONS A description of parameter extraction methodologies used in the literature as well as a detail description of our methodology was presented. The methodology proposed seems to be superior to ones presented on the literature, and eliminates the need of the assumptions and simplifications made in the past to obtain the model parameters. Also, it reduced the need of filtering the measured data by avoiding numerical differentiation.
ACKNOWLEDGMENTS For using their measurement data, the authors would like to thank the former international consortium (KEMA High-Power Laboratory, Delft University of Technology, Siemens AG, RWE Energie and Laborelec cv) that worked on the project “Digital Testing of High-Voltage Circuit Breakers” in the period 1997 - 2001 and was sponsored by Directorate General XII of the European Com-
mission in the Standards, Measurements and Testing programme under contract no. SMT4-CT96-2121.
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