Improvements and New Features in...

UGM 2016 – BOULDER, COLORADO

Copyright © 2016 SES ltd. All rights reserved. -1

1 ADVANCEMENTS IN THE TREATMENT OF NONLINEAR

DEVICES IN THE FREQUENCY DOMAIN

Stéphane Franiatte(1,2), Octavio Ramos(1), Marc-André Joyal(1), Simon Fortin(1),

Alain April(2), Jean-Marc Lina(2), and Farid P. Dawalibi(1)

(1) Safe Engineering Services & Technologies ltd.

Email: [email protected], Web Site: www.sestech.com

(2) École de Technologie Supérieure

Web Site:www.etsmtl.ca

Abstract

Frequency Domain (FD) methods are a valuable complement to Time Domain (TD) methods for the analysis of

electromagnetic transients in power systems. However, it is still difficult to include non-linear elements in the FD.

This article presents some improvements on a technique developed several years ago for analyzing systems

including non-linear elements in the FD. The article focuses on the modeling of surge arresters on distribution lines

for the analysis of insulation failure due to lightning. This work includes improvements to the Simulated Annealing

metaheuristic (SA) technique presented last year (solutions are found using realistic MOV arresters parameters)

and an example is given where results obtained by the proposed method are compared to those obtained with the

widely accepted Time Domain tool ATP-EMTP.

1 Introduction

When studying the lightning performance of surge arresters using time domain (TD) solvers such

as ATP-EMTP, an equivalent electric circuit including the surge arresters is built to represent the

transmission line system under study. A lightning strike on the line is simulated and the circuit is

solved entirely in the time domain. Hence, the frequency dependence of the elements in the circuit

has to be obtained by other methods or even neglected in some cases. Approximations are usually

required in computing the equivalent impedances in the circuit.

Fortin et al. [1] proposed an alternative approach, studying the lightning performance of surge

arresters directly in the frequency domain (FD) and solving the complete network in the frequency

domain. Once the response (voltage and current) of the network including surge arresters is

obtained, the voltage and current through the surge arresters in the time domain are obtained

using the inverse Fourier transform.

The importance of developing Frequency Domain methods for the analysis of Electromagnetic

Transients resides in the fact that most power system elements are frequency dependent and these

can be modeled in a straightforward manner within FD methods. Since the basic principles of FD

methods are different from those of the Time Domain, FD methods are very useful in verifying

TD methodologies.

This frequency-based approach requires solving of a large number of coupled non-linear

equations. Last year, an automated method for finding solutions of the nonlinear system of

equations through stochastic global optimization was proposed [2]. The original problem as

stated in [1] (nonlinear equations system solving) was transformed into a global optimization one

by synthesizing objective functions whose global minima, if they exist, are also solutions to the

original system. The global optimization task was carried out by the stochastic method known as

Simulated Annealing (SA). This article describes progress in using this method. Tests of the

mailto:[email protected]

http://www.sestech.com/

PART III: SES CONTRIBUTIONS

Page 1-2 Copyright © 2016 SES ltd. All rights reserved.

approach for realistic values of alpha are carried out, and the results are compared with those

obtained using a well-known Time Domain solver named ATP-EMTP.

2 Mathematical Model

2.1 Time Domain Compensation Method

The Time Domain compensation method is used in time domain solvers such as ATP-EMTP to

solve networks containing nonlinear elements. According to the Compensation Theorem, any

network element can be replaced by a current source whose current is equal to the current flowing

through the replaced element. This theorem is applied in [2] alongside an LU decomposition of

the network’s nodal admittance matrix to simulate the effects of nonlinear elements, without the

need of re-computing the admittance matrix. To take into account the effects of the nonlinear

element, the following equation system is solved first, for each time step:

𝑣𝑘𝑚 = 𝑣𝑘𝑚(0)

+ 𝑧𝑇𝑖𝑘𝑚 (1)

𝑣𝑘𝑚 = 𝑓(𝑖𝑘𝑚 ,𝑑𝑖𝑘𝑚

𝑑𝑡, 𝑡)

(2)

where 𝑣𝑘𝑚(0)

indicates the voltage drop between nodes 𝑘 and 𝑚 without the nonlinear element while

𝑧𝑇 is the Thevenin equivalent impedance (Figure 1).

Note: In this article the following conventions are followed in formulas; lowercase letters are used

to represent time domain variables, uppercase letters are used to represent frequency domain

variables, boldface letters are used to represent vectors or matrices and normal letters are used to

represent scalar values.

Equations (1) and (2) are solved simultaneously for 𝑖𝑘𝑚, the current through the nonlinear branch.

The final voltage solution is found by superposing the voltage drop obtained when applying the

non-linear current source 𝑖𝑘𝑚:

𝒗 = 𝒗(0) + 𝒛 ∙ 𝑖𝑘𝑚 (3)

As shown in [3], the compensation method can also be used with 𝑀 parallel nonlinear branches:

𝒗 = 𝒗(0) + ∑ 𝒛(𝑗) ∙ 𝑖(𝑗)

𝑀

𝑗=1

(4)

Figure 1 Compensation method: left) linear network without the nonlinear element; right) the equivalent

circuit.


Copyright © 2016 SES ltd. All rights reserved. Page 1-3

2.2 Frequency Domain Compensation Method

To fully take into account frequency dependence of the elements of the network, a new Frequency

Domain Compensation Method was proposed in [1]. The model for the network model is stated

directly in the frequency domain, and can be seen as the frequency counterpart to Eq. (4):

𝑽𝑖(𝜔) = 𝑽𝑖

(0)(𝜔) + ∑ 𝒁(𝑖,𝑗)(𝜔) ∙ 𝑰(𝑗)(𝜔)

𝑀

𝑗=1

(5)

For a network containing 𝑀 nonlinear elements, the voltage across the 𝑖𝑡ℎ nonlinear element is

obtained from this equation. 𝒁 and 𝑽𝑖(0)

are frequency dependent constants. The current 𝑰(𝑗) is

obtained from the model H for the nonlinear element by applying a Fourier transform:

𝒊(𝑡) = 𝐻(𝒗(𝑡), 𝑡) ↔ 𝒗(𝑡) = 𝐻−1(𝒊(𝑡), 𝑡), 𝒊, 𝒗 ∈ ℝ𝑁 (6)

A self-consistency equation is then derived by combining the linear Eq. (5) with the nonlinear

element’s equation (6):

𝒗𝑖(𝑡) = ℱ−1 [𝑽𝑖(0)

(𝜔) + ∑ 𝒁(𝑖,𝑗)(𝜔) ∙ ℱ[𝐻(𝑗)(𝒗𝑗(𝑡), 𝑡)]

𝑀

𝑗=1

],

𝒗 ∈ ℝ𝑁

(7)

Eq. (7) represents a self-consistency condition on the voltage across the nonlinear elements, and

can be solved iteratively. A suitable parameterization of 𝒗𝑖 as a function of time must first be

obtained, then the parameters can be adjusted to satisfy equation (7); this can be stated as a root-

finding problem:

𝑓𝑖(𝒗) = 𝒗∗,𝑖 − ℱ−1 [𝑽∗,𝑖

(0)(𝜔) + ∑ 𝒁(𝑖,𝑗)(𝜔) ∙ ℱ[𝐻(𝑗)(𝒗∗,𝑗)]

𝑀

𝑗=1

] = 0,

𝒗 ∈ ℝ𝑁×𝑀

(8)

Considering the set of 𝑀 nonlinear elements in the network for which (8) needs to be solved, a

system of nonlinear equations is constituted:

𝐹(𝒗) = [

𝑓1(𝒗)

𝑓2(𝒗)⋮

𝑓𝑀(𝒗)

] = 0, v∈ ℝ𝑁×𝑀 (9)

The nonlinear equations system solving task is transformed into an optimization problem by

synthesizing objective functions whose global minima, if they exist, are also solutions to the

original system. The Simulated Annealing metaheuristic method [4] is used to solve this system.



3 Application to Surge Arrester Lightning Performance

Analysis

3.1 System Description

The case presented here is the same as that presented in [1]. Figure 2 shows the simulation circuit.

The circuit under test is composed of 7 poles with an earth impedance of 24 ohms at power

frequency. The phase conductors are 4/0 ACSR with a radius of 7.15 mm. The neutral conductor

is a 1/0 ACSR conductor with a 5.05 mm radius. Surge arresters are located at poles #2 to #6.

Poles #1 and # 7 are terminated with the line surge impedance value of 555 ohm. As was done in

[1], we study the case in which lightning strikes Phase C at mid-span between Poles #4 and #5. It

is considered that phases C and A are more susceptible to lightning strikes due to their positions.

To reduce computation time, we have modeled only Phase C in the study, since the transient

behavior is dominated by this phase during the lightning strike.

Figure 2 Seven distribution poles modeled.

The nonlinear elements are represented as gapless metal oxide varistors (MOV). The MOV

nonlinear V-I relationship, corresponding to Eq. (6), is expressed by the following equation:

𝒊(𝑡) = 𝐻(𝒗(𝑡), 𝑡) = 𝑖𝑠 (

𝒗(𝑡)

𝑣𝑠)

𝛼

, 𝒊, 𝒗 ∈ ℝ𝑁 (10)

In Eq. (10), , sometimes referred to as squareness, can be different for different arresters. Typical

values for lie between 2 and 75. The constant value 𝑣𝑠 represents the voltage threshold of the

device: the arrester begins to conduct significantly when the voltage across its terminals reaches

or approaches this value. The value 𝑖𝑠 is the amount of current flowing through the device at the

voltage threshold .Figure 3 displays the V-I characteristic of a typical MOV arrester for several 𝛼

values, with 𝑖𝑠 = 700 𝐴 and 𝑣𝑠 = 15 000 V.



Figure 3 Typical MOV arrester V-I characteristic for different 𝛼 values.

The lightning surge is modeled as an ideal current source characterized by a 1.2/20s wave; two

lightning models were studied, the double-exponential, Eq. (11), and the more widely used Heidler

model [5], equation (12):

𝑖(𝑡) = 𝑖𝑚(𝑒−𝑎𝑡 − 𝑒−𝑏𝑡) (11)

𝑖(𝑡) = 𝐴 ∙ 𝐺(𝑡, 𝜏1𝐴, 𝜏2

𝐴) + 𝐵 ∙ 𝐺(𝑡, 𝜏1𝐵 , 𝜏2

𝐵)

𝐺(𝑡, 𝜏1, 𝜏2) =1

𝛾∙

(𝑡𝜏1⁄ )2

1 + (𝑡𝜏1

⁄ )2∙ 𝑒

−𝑡𝜏2⁄

𝛾 = 𝑒−√2

𝜏1𝜏2

⁄

(12)

Figure 4 Heidler lightning surge.



3.2 Frequency Domain Solver: HIFREQ and Simulated Annealing

The HIFREQ and FFTSES engineering modules of the CDEGS software package are used to

perform the computations. The HIFREQ module is used to compute the currents and voltages

throughout the conductor network in the frequency domain, while the FFTSES module is used to

convert the results into the time domain. The Simulated Annealing approach to solve nonlinear

Eq. (9) is programmed in Matlab.

Since last year [2], improvements have been made in the convergence capabilities of the

Simulated Annealing implementation. Solutions can be found for realistic 𝛼 values up to 45.

Figure 5 and 6 below display results for 𝛼 = 25 and 𝛼 = 45 , respectively. These results were

obtained using an input signal based on the Heidler model of Eq. 12, with 𝐴 = 25 kA.

5-A

Voltage across the

arresters.

It can be seen here that

voltages are properly

clamped around the value

of 15 kV.

5-B

Current through the

arresters.

Figure 5 Voltage and Current solutions, 𝛼 = 25.



6-A

Voltage across the

arresters.

6-B

Current through the

arresters.

Figure 6 Voltage and Current solutions, α = 45.

3.3 Time Domain Solver

In order to validate the proposed methodology, the widely accepted Time Domain solver ATP-

EMTP is used to reproduce the experiment. The ATP circuit model is presented in Figure 7. MOV

surge arresters are modeled as nonlinear resistances given that it was impossible to utilize the

available MOV models in ATP with the given nonlinear characteristics and time step.

Transmission lines were modeled using the frequency dependent Marti model with a frequency

range of 0.01 Hz up to 100 MHz with a selected modal conversion matrix at 10 MHz. The

simulation time step was selected to be 0.1 ns. Notice that the smallest travel time in the system

is of 0.145 s.



Figure 7. ATP-EMTP simulation circuit.

Results obtained with this approach are compared with the ones obtained with the Frequency

Domain method in the next section.

4 Comparison between time and frequency domain

approaches

Figure 8 shows the results with both methodologies when the lighting surge is of a double

exponential form. The Frequency Domain results are plotted in black and the Time Domain

results are plotted in colors. All signals present a similar rise time and a similar oscillation

frequency of 1.7 MHz. This value corresponds to a resonance that is seen when reflections occur

at the arresters. However, the Time Domain results present less attenuation for current in the

arresters at poles 4 and 5 resulting in a difference of up to 3 kA.

Figure 8 Results for a double-exponential lightning surge, 25 kA input signal. Results obtained with the

Frequency Domain method are in black and results obtained with the Time Domain method are in color.

Figure 9 presents the current for the arrester at pole 5 which is similar to the current for the

arrester at pole 4. In this case, the results from the Frequency Domain method present more

attenuation than those obtained with the Time Domain method. Figure 10 presents a close up on

Figure 9 where it can be seen that the current difference between the two methodologies is around

V

II

HH

LCC

0.087 km

LCC

0.087 km

LCC

0.087 km

LCC

0.087 km

LCC

0.087 km

LCC

0.043 km

LCC

0.043 km

R(i)

I

R(i)

I

R(i)

I

R(i)

I

R(i)

I



3 kA. On the other hand, the rise time and the oscillation frequency of the two signals, (Time

Domain and Frequency Domain), are very similar. This resonance frequency of about 1.7 MHz

corresponds to a half wavelength separation between the arresters, causing reflected waves to be

sustained during the period of the surge.

Figure 11 to Figure 16 present comparisons for arresters at poles 2, 3 and 6. In general these figures

show that the Time Domain results for these arresters behave in the opposite fashion to those

obtained with the Frequency Domain method regarding the magnitude of oscillations, i.e., the

Time Domain signals present oscillations of smaller magnitude in the first microseconds and

oscillations of larger magnitude in the middle part of the time window, whereas the Frequency

Domain results present larger magnitudes at the beginning of the signal and attenuate as time

goes by.

Due to the fundamental differences between the Time Domain and Frequency Domain methods,

multiple causes could explain the discrepancies stated above. One of them is the ability of the

proposed Frequency Domain method to take the frequency dependence of the network into

account, which is something more difficult for Time Domain methods. For instance, the

impedance of the poles in the ATP software was set to a constant value of 24 Ω given that Time

Domain methods do not have dedicated models capable of representing grounding structures for

a wide range of frequencies. On the other hand, since the actual poles were modeled with HIFREQ

in the proposed method, their frequency dependence was fully taken into account. As an

illustrative example, Figures 17 and 18 show the resistance and the reactance of one pole with

respect to the frequency. It is seen that for very low frequencies, the pole impedance is about the

same as that used in ATP, but for higher frequencies, the differences are more pronounced. As a

matter of fact, an important frequency to look at is the system’s resonance frequency of 1.7 MHz.

At that frequency, the amplitudes of the oscillations are highly dependent on the reflection

coefficient at the arresters, which is determined from the pole impedance assuming that the

arresters impedance is negligible when the lightning strike occurs. By looking more closely at

these results, we find that at this resonance frequency, the pole resistance is 84.4 Ω whereas its

reactance is 261j Ω. Similar observations could be done with all components included in the

network, showing the potential lack of accuracy of Time Domain methods with respect to

Frequency Domain methods.



Figure 9. Current trough arrester 5.

Figure 10. Close up on Figure 9.



Figure 17 Resistance of the pole with respect to the frequency, computed with HIFREQ.

Figure 18 Reactance of the pole with respect to the frequency, computed with HIFREQ.



5 Conclusion

In this article a technique aiming to solve nonlinear elements in the Frequency Domain first

introduced in [1] has been improved. A difficulty encountered by Fortin et al. in their work to solve

networks containing nonlinear devices in the frequency domain was related to the method used

to solve the self-consistency equation (7). The optimization algorithm used to solve the system

often failed to converge for high values of the nonlinearity constant 𝛼, and required the user to

frequently update some parameters to eventually find a solution, even for lesser values of 𝛼. A

new approach based on the Simulated Annealing metaheuristic technique was presented in [2],

and improved since then: as demonstrated in this article, solutions were obtained for 𝛼 values as

high as 45 and no human intervention during the convergence process was required. The

proposed Frequency Domain method was implemented and compared to a widely accepted Time

Domain method based on travelling waves, i.e. ATP-EMTP. Comparisons between results

obtained with both approaches show that the results obtained with the proposed method match

the general behavior of those obtained with the Time Domain tool. The method presented here is

still a work in progress and it is expected to be refined in the years to come. This work represents

a first step towards the accurate solution of nonlinear elements in the Frequency Domain, a task

that has proven to be challenging.

6 References

[1] S. Fortin, W. Ruan, F. P. Dawalibi, and J. Ma, "Optimum and Economical Deployment Method of Surge Arresters on Distribution Lines for Insulation Failure due to Lightning - An Electromagnetic Field Computation Analysis," 2002.

[2] S. Franiatte, S. Fortin, A. April, J.-M. Lina, M.-A. Joyal, and F. P. Dawalibi, "Treatment of Nonlinear Devices in the Frequency Domain," presented at the SES Users Group Conference, San Diego, California, USA, 2015.

[3] H. W. Dommel, "Nonlinear and Time-Varying Elements in Digital Simulation of Electromagnetic Transients," Power Apparatus and Systems, IEEE Transactions on, vol. PAS-90, pp. 2561-2567, 1971.

[4] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by Simulated Annealing," Science, vol. 220, pp. 671-680, 1983.

[5] F. Heidler and J. Cvetic, "A Class of Analytical Functions to Study the Lightning Effects Associated With the Current Front," European Transactions on Electrical Power, vol. 12, pp. 141-150, 2002.

Improvements and New Features in...

Documents

Transcript of Improvements and New Features in...