Abstract—In this work, the performance of three differenttransmission line models included in the professional transientprogram ATP for a statistical study of switching overvoltages (SOV), also known as Monte Carlo study, is analyzed. The models under consideration are Bergeron, Semlyen and Marti models, which are compared to a frequency domain methodbased on the numerical Laplace transform and the superpositionprinciple.

Index Terms—Electromagnetic transients, numerical Laplacetransform, statistical studies, switching transients.

I. INTRODUCTION

RANSIENT overvoltages in power systems are mainlyrelated to switch maneuvers, fault occurrence and

clearance and lightning phenomena. For voltage levels higherthan 230kV, switching and fault disturbances, commonlynamed internal disturbances, can be more important thanlightning surges. This is simply due to the fact that internalovervoltages are directly related to the nominal voltage of thesystem.

Given their severity, overvoltages resulting from theenergization of long transmission lines are of particularconcern, and a deep knowledge of these disturbances isfundamental for insulation design and selection of protectivedevices.

Early line models for electromagnetic transients were basedon the lossless case, for which the line equations are just aparticular case of the Wave Equation. D’Alembert solution tothis equation in terms of traveling waves is well known and isthe basis for the Bergeron model, which was originallydeveloped for analyzing hydraulic systems in 1949 and waslater adapted to transmission lines [1]. Since the late 1960’s and followed by the creation of the most important digitalsimulation program for electromagnetic transients, theElectromagnetic Transients Program (EMTP) [2], several transmission lines models have been proposed in the literature [3-11], and some of them have found their use in generalsimulation programs such as the Alernative TransientsProgram (ATP) and the Electromagnetic Transients Programfor Direct Current (EMTDC).

This work was supported by the National Polytechnic Institute of México under project 20070211.

P. Gómez, is with the Grad. Program in Electrical Eng., SEPI-ESIME-Zacatenco, National Polytechnic Institute, Mexico City, MEXICO (e-mail:pgomezz@ipn.mx).

In practice, the electrical parameters of transmission linesare frequency dependent, due to skin effect in conductors and in ground plane. Consequently, as an electromagnetic wavepropagates along a transmission line, its shape suffers a gradual distortion since the different frequency components of the wave travel at different speeds and with differentattenuations. Therefore, one of the most important aspects forthe accurate modeling of transmission lines has been the inclusion of these frequency dependent effects for timedomain transient studies.

In this article, the performance of three transmission linemodels included in ATP is analyzed, namely the basic constant parameter model from Bergeron’s theory [1], and thefrequency dependent models by Semlyen [6] and Marti [7].These models are used to perform a statistical or Monte Carlo study of switching overvoltages in a 400kV transmission line.Validation of results is obtained by comparison with a frequency domain method [12-14], which is based on thenumerical Laplace transform and the superposition principle.

II. MONTE CARLO STUDY

Accurate computation of switching overvoltages on transmission systems is fundamental for insulation design and selection of protective devices.

In general, the poles of a 3-phase circuit breaker do notclose at the same time, but in a sequential manner. Besides,the insulation design of equipment is not based on consideringclosing times for which the most severe overvoltages areobtained (times of source peak values for each phase), sincethese values are of low probability, resulting in overestimatedovervoltages which would be economically unpractical.

A more efficient way of computing switching transients ontransmission systems for insulation coordination is byobtaining the probability distribution of switchingovervoltages. For this purpose, the Monte Carlo method isusually applied, which consists of performing severalswitching simulations with random closing times for each phase [15-17].

In a more general sense, a Monte Carlo simulation is anumerical procedure applied to a problem which involvesrandom variables. For obtaining the probability distribution of switching overvoltages, the process consists of three mainsteps:

1. Generation of random closing times2. Computation and recording of overvoltages3. Statistical analysis of results for a number of events

Validation of ATP Transmission Line Models for a Monte Carlo Study of Switching Transients

Pablo Gómez, Member, IEEE

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978-1-4244-1726-1/07/$25.00 c© 2007 IEEE

Typically, one hundred ore more switching events areconsidered, keeping in mind that the accuracy of results is dependent of the number of events.

A. Determination of closing times

Each pole of a circuit breaker can be represented by twocontacts, as presented in Fig. 1 [15-17]:

An auxiliary contact which determines the instant ofthe incident sinusoidal wave in which the closingevent starts. Closing of this contact is considered to occur according to a uniform probability distribution with a typical range of 0 to 0.5/f or 0 to 1/f, being fthe nominal frequency of the system. This time is usually considered as the same for the 3 phases, andis represented by Taux in Fig. 1. A main contact which establishes the actual closingtime. Closing of this contact follows a normal(Gaussian) distribution. This time is usuallyconsidered as different for the 3 phases, and isrepresented by TAr, TBr and TCr in Fig. 1.

A time delay between closing of the auxiliary and maincontacts can also be taken into account; this is represented as

A, B and C in Fig. 1. The exact closing time for each phaseor pole can be defined as

rCCauxC

rBBauxB

rAAauxA

TTT

TTT

TTT

, (1)

Besides, the standard deviation of the main contacts is givenby

6

mpsM (2)

where mps is the maximum pole span.

B. Probability Distribution Functions

Maximum overvoltages registered after N sequentialclosing events are used to obtain the probabilistic distributionof overvoltages using two types of graphics [18]:

Probability distribution functions (histograms)Cumulative distribution functions.

The probability of a random variable X to take a certainvalue x is defined as the probability distribution function andcan be written as

)()( xXpXf (3)

Any probability distribution function must comply with thefollowing:

xxXp ,0)( (4a)

N

x

xXp1

1)( (4b)

This is, the probability of occurrence of any value x must begreater or equal to 0 and all the probabilities corresponding to different values of x must add 1.

A 3 M

0.5/f

Auxiliarycontact

Main

contact A

t

TAr

Taux

TBr

B 3 M

TCr

C 3 M

Main

contact B

Main

contact C

A 3 M

0.5/f

Auxiliarycontact

Main

contact A

t

TAr

Taux

TBr

B 3 M

TCr

C 3 M

Main

contact B

Main

contact C

Fig. 1. Probability distribution of closing times

On the other hand, the probability of a random variable Xto take values smaller or equal to x is defined as thecumulative distribution function of X and is denoted by

)()( xXpXF (5)

These two types of functions are very important for computing the risk of failure of a transmission system for insulation coordination purposes.

III. FREQUENCY DOMAIN METHOD FOR SWITCHING

For the most accurate modeling of transmission lines a frequency domain method can be used, which considers thefrequency dependence of the electrical parameters in a simplerand more straightforward manner than time domain methods.However, it is known that frequency domain methods canonly be applied to linear and time-invariant systems, and thatthe analytical inversion of Fourier or Laplace transforms is only possible for simple systems. These two problems can still be addressed using the superposition principle and a numericalfrequency-time transformation, namely the Numerical Laplacetransform [12-14]. This method will serve as base solution tovalidate ATP line models in the Monte Carlo study.

A. Superposition principle for closing

Assuming a switch as initially open, the potentialdifference between its terminals can be represented by a voltage source Vsw. Closing can be simulated by a seriesconnection of another source Vsw2 with equal magnitude thanVsw but opposite polarity, as shown in Fig. 2. The voltagesource required to close the switch at time tc 0 is given by

)()(2 cswsw ttutvLV (6)

where vsw(t) is the time domain waveform of Vsw and Lindicates the Laplace transform.

2007 39th North American Power Symposium (NAPS 2007) 125

+ Vsw - + Vsw2 -+ Vsw -

open closed

j k j k

inicial condition for t > tc

+ Vsw - + Vsw2 -+ Vsw -

open closed

j k j k

inicial condition for t > tc

Fig. 2. Superposition principle for switch closing

If nodal analysis is applied, instead of the ideal voltagesource Vsw2 a Norton equivalent with current sourceIsw2=Vsw2/Rsw can be used, being Rsw a resistance needed to perform the source transformation, which must be smallenough to approximate the ideal source or it can take someparticular value to represent a contact resistance.Alternatively, the modified nodal analysis (MNA) can be usedto allow the direct insertion of ideal voltage sources [19].

The complete voltage response is obtained by adding thesystem response before switch closure to that resulting fromapplying the current source Isw2. Therefore, the completesolution corresponding to a closure between nodes j and k of the system can be expressed as follows

)1()1()0( IYVV bus (7)

where V(0) is the nodal voltages vector before switching, )1(busY

is the admittance matrix modified by the inclusion of Rsw andI(1) is an injected current vector containing only the elementscorresponding to source Isw2 connected between nodes j and k.

B. Numerical Laplace Transform Algorithm

Considering a finite integration range, the direct and inverse Laplace transforms can be written as:

Ttjct eetfjcF

0)()( (8a)

0)(Re)( tj

ct

ejcFe

tf (8b)

where f(t) is a real and causal function and F(s) is its image inthe Laplace domain; is the angular frequency, T is theobservation time and is maximum frequency of thespectrum. Term ( ) is a weighting function, also known as window function, used to attenuate the Gibbs errors produced by the truncation of the frequency range. Also, since timedomain function f(t) obtained by numerical evaluation of (8b) will necessarily be distorted by aliasing, the Laplace stability constant c can be used to attenuate the associated errors by “smoothing” the frequency response. A value of c=2 ,obtained empirically by Wilcox [20], is used in this work.Finally, a numerical form of equations (8) that allows usingthe Fast Fourier Transform (FFT) is obtained [21]:

1,,2,1,2

exp1

0

NmN

mnjDfF

N

nnnm (9a)

1,,2,1,2

expRe1

0

NnN

mnjFCf

N

mmmnn (9b)

where

)12( mjcFFm (10a)

)( tnffn (10b)

N

njtcntDn exp (10c)

N

njtcnCn exp

2 (10d)

)12( mm (10e)

TN

Tt , (10f),(10g)

TN

Tt , (10h),(10i)

being the spectrum integration step and t the timediscretization step.

IV. MONTE CARLO SWITCHING STUDY ON A 400KV LINE

In this section, Bergeron, Semlyen and Marti line modelsincluded in the ATP simulation program are used to perform a Monte Carlo study of switching overvoltages on atransmission line of 230km, with the tower configurationpresented in Fig. 3, which is a typical configuration for 400kVin Mexico. The frequency domain method described inSection III is used as base solution. The circuit implementedin ATP’s graphical interface known as ATPDraw is shown inFig. 4.

One hundred switching events are simulated, fixing thetime step as equal for the different time domain models and the frequency domain method, and considering an observationtime of 35ms with 1024 time samples for each event. A range of 0 to 1/f for the auxiliary contacts and equal time delays of5ms are considered for the 3 phases. Also, main contacts havea standard deviation of 0.833ms.

Fig. 3. 400kV tower configuration

126 2007 39th North American Power Symposium (NAPS 2007)

Fig. 4. Implementation in ATPDraw

0 0.005 0.01 0.015 0.02 0.025 0.03-3

-2

-1

0

1

2

3

time (s)

volta

ge (

p.u.

)

FD

Bergeron

(a)

0 0.005 0.01 0.015 0.02 0.025 0.03-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time (s)

volta

ge (

p.u.

)

FD

Semlyen

(b)

0 0.005 0.01 0.015 0.02 0.025 0.03-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time (s)

volta

ge (

p.u.

)

FD

Marti

(c)

Fig. 5. Transient overvoltage at phase A of the receving end of the line:(a) Bergeron, (b) Semlyen, (c) Marti

To get a better insight of the performance of the linemodels for the Monte Carlo study, the overvoltage obtained atphase A of the line’s open end for the 30th switching event is presented in Figs. 4, 5 and 6 for Bergeron, Semlyen and Martimodels, respectively, comparing in all cases the results withthose of the base solution in the frequency domain (FD). Random closing times for this event are obtained as 6.058,5.708 and 4.589 ms for phases A, B, and C, respectively.

Although Bergeron model is able to reproduce the naturalfrequency of the line, this is not enough to get a good approximation of the maximum overvoltage, and thereforethis value is overestimated. On the other hand, the frequencydependent line models (Semlyen and Marti) are able toreproduce not only the natural frequency of the transmissionline but also the distortion and attenuation due to skin effect,and therefore the results are closer to those of the base solution. However, while differences between Semlyen modeland FD method are still readily seen, Marti model has a very close agreement to the base solution. In Table 1, the maximumovervoltages and times at which these values occur are listed,comparing the results from the base solution and the timedomain line models. Marti model results are by far better thanthose of the other time domain models under consideration.

An analysis of the results for the complete Monte Carlostudy with the different models can be obtained by comparingthe cumulative probability curves of switching overvoltagesfrom each model, as shown in Fig. 6. A summary of the study is also included in Table 2.

Table 1. Results from event 30.

ModelMaximum

overvoltage(p.u.)

Time (s)

FD 2.0931 0.01599

Bergeron 2.4631 0.01945

Semlyen 2.2873 0.01599

Marti 2.0906 0.01599

1 1.5 2 2.5 3 3.50

20

40

60

80

100

voltage (pu)

cum

ulat

ive

prob

abili

ty (

%)

BergeronSemlyenMartiFD

Fig. 6. Cumulative distribution of switching overvoltages

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Table 2. Summary of the Monte Carlo Study

Overvoltage (p.u.)

Model Maximum Minimum MeanStandarddeviation

FD* 2.708673 1.530819 2.213836 0.198515

Bergeron 3.307891 1.827554 2.655602 0.296618

Semlyen 2.867659 1.682979 2.287310 0.200750

Marti 2.670574 1.542537 2.206774 0.198382 * Base solution

V. DISCUSSION

In the Monte Carlo study, only transmission lines modelscorresponding to the ATP version of EMTP were included.However, another versions of EMTP, such as EMTDC and EMTP-RV include more recent line models that considerfewer approximations than Marti model and could thereforebe more accurate; for example, the Phase Domain Model [11] and the Wide Band Model [9]. Validation of these modelsusing as base solution the frequency domain method for thesame type of Monte Carlo study would be of interest.

Marti model had by far the best performance, which couldbe seen from a particular event (Fig. 5) or from the wholeMonte Carlo study (Table 2). The difference of this modelwith respect to the base solution was minimal but still a smalloverestimation of overvoltages from this model can be noticedin Fig. 6.

Marti model considers that the modal transformation matrixis constant for all the frequency range of the study, which is ingeneral not true, but still a good enough approximation forbalanced or nearly balanced transmission lines. However, ifthe line is untransposed and also an unsymmetricalconfiguration for the tower is considered, the performance of this model is questionable. Validation of the model for thistype of configurations would also be of interest.

VI. CONCLUSIONS

In this work, the performance of three different transmission line models for a Monte Carlo study of switchingovervoltages was analyzed, using as base solution a frequencydomain method based on the numerical Laplace transform and the superposition principle.

Marti model had very close agreement to the frequencydomain method, since both the natural frequency of the lineand distortion of the waveforms due to skin effect could be well reproduced, in such a way that the maximumovervoltages for each switching event could be accurately computed and recorded. However, as stated in the previous section, this does not necessarily mean that Marti model willhave correct results for any Monte Carlo study. This is ingeneral highly dependent on the complexity of the towerconfiguration.

Previous work by this author has shown the importance of the frequency domain analysis as a complementary tool for

assessing time domain results and new models, ore even as a standalone method to use alternatively to time domainmethods when the accuracy of results is mandatory [22-25]. This could be even more important for a practical studyintended for insulation coordination, such as the Monte Carlostudy of switching overvoltages presented in this work.

VII. REFERENCES

[1] L. Bergeron, Water Hammer in Hydraulics and Waves Surge in Electricity, John Wiley, NY, 1961.

[2] H. W. Dommel, “Digital Computer Solution of ElectromagneticTransients in Single and Multiphase Networks”, IEEE Trans. PowerApparatus and Systems, vol. PAS-88, pp. 388-399, April 1969.

[3] A. Budner, “Introduction of Frequency Dependent Line Parameters into an Electromagnetic Transients Program”, IEEE Trans. Power Apparatusand Systems, vol. PAS-89, pp. 88-97, January 1970.

[4] J. K. Snelson, “Propagation of Travelling Waves on Transmission Lines-Frequency Dependent Parameters”, IEEE Trans. Power Apparatus andSystems, vol. PAS-91, pp. 85-91, January 1972.

[5] W. S. Meyer and H. W. Dommel, “Numerical Modeling of Frequency-Dependent Transmission-Line Parameters in an ElectromagneticTransients Program”, IEEE Trans. Power Apparatus and Systems, vol.PAS-93, pp. 1401-1409, September 1974.

[6] A. Semlyen, A. Dabuleanu, “Fast and Accurate Switching TransientCalculations on Transmission Lines with Ground Return Using Recursive Convolutions”, IEEE Trans. Power Apparatus and Systems,vol. PAS-94, no. 2, March/April 1975.

[7] J. R. Martí, “Accurate Modelling of Frequency-Dependent TransmissionLines in Electromagnetic Transient Simulations”, IEEE Trans. PowerApparatus and Systems, vol. PAS-101, no. 1, pp. 147-157, January 1982.

[8] F. J. Marcano and J. Martí, “Idempotent Line Model: Case Studies”,Proc. of the International Conference on Power Systems Transients, Seattle, Washington, June 1997.

[9] H. V. Nguyen, H. W. Dommel, J. R. Marti, “Direct Phase DomainModelling of Frequency Dependent Overhead Transmission Lines”, IEEE Trans. Power Delivery, vol. 12, no. 3. pp. 1335-1342, July 1997.

[10] B. Gustavsen and A. Semlyen, “Combined Phase Domain and ModalDomain Calculation of Transmission Line Transients Based on VectorFitting”, IEEE Trans. Power Delivery, vol. 13, no. 2, pp. 596-604, April1998.

[11] A. Morched, B. Gustavsen and M. Tartibi, “A Universal Model forAccurate Calculation of Electromagnetic Transients on Overhead Linesand Underground Cables”, IEEE Trans. Power Delivery, vol. 14, no. 3,pp. 1032-1037, July 1999.

[12] N. Nagaoaka and A. Ametani, “A Development of a GeneralizedFrequency–Domain Transient Program – FTP”, IEEE Trans. PowerDelivery, vol. 3, no. 4, pp. 1996-2004, October 1988.

[13] P. Moreno, R. de la Rosa and J. L. Naredo, “Frequency DomainComputation of Transmission Line Closing Transients”, IEEE Trans.Power Delivery, vol. 6, pp. 275-281, January 1991.

[14] P. Moreno, P. Gómez, J. L. Naredo, J. L. Guardado, “Frequency DomainTransient Analysis of Electrical Networks Including Non-linear Conditions”, ELSEVIER Int. Journal of Electrical Power & EnergySystems, vol. 27, pp. 139-146, 2005.

[15] A. M. Gole, D. W. Durbak et al., “Task Force Report: Modeling Guidelines for Switching Transients”, IEEE PES Switching TransientsTask Force 15.08.09.03.

[16] H. W. Dommel, Electromagnetic Transients Program Reference Manual(EMTP Theory Book), Prepared for Bonneville Power Administration,Portland, USA, 1986.

[17] J. A. Martínez, R. Natarajan, E. Camm, “Comparison of Statistical Switching Results Using Gaussian, Uniform and Systematic Switching Approaches”, IEEE Power Engineering Society Summer Meeting, vol. 2,pp. 884-889, Seattle, Washington, July 2000.

[18] H. Torres, Metodología en Pruebas de Laboratorio para Aplicación a la Coordinación de Aislamiento Estadístico, Publicaciones de la Facultad de Ingeniería de la Universidad Nacional de Colombia, March 1988 (inspanish).

[19] C. W. Ho, A. E. Ruehli and P. A. Brennan, "The modified nodalapproach to network analysis," IEEE Trans. Circuit and Systems, vol.CAS-22, no. 6, pp. 504-509, Jun. 1975.

128 2007 39th North American Power Symposium (NAPS 2007)

[20] J. Wilcox, “Numerical Laplace Transformation and Inversion,” Int. J. Elect. Enging. Educ., Vol 15, pp. 247-265, 1978

[21] A. Ametani, "The Application of the Fast Fourier Transform to Electrical Transient Phenomena," Int. J. Elect. Eng. Educ. Vol. 10 (4),pp. 277-286, 1972.

[22] P. Gómez, P. Moreno and J. L. Naredo, “Modeling Non-LinearConditions in Transmission Network Transients Using the NumericalLaplace Transform”, Proc. of the 34th North American PowerSymposium 2002 (NAPS’02), Tempe, Arizona, USA, August 2002.

[23] P. Gómez, P. Moreno, J. L. Naredo and M. Dávila, “Modeling of Non-Uniform Transmission Lines in the Frequency Domain”, Proc of the 35th

North American Power Symposium 2003 (NAPS’03), Rolla, Missouri,USA, October 2003.

[24] P. Gómez, P. Moreno, J. L. Naredo, “Frequency Domain TransientAnalysis of Non-Uniform Lines with Incident Field Excitation”, IEEETrans. Power Delivery, vol. 20, no. 3, pp. 2273-2280, July 2005.

[25] P. Gómez, P. Arellano, R. O. Mota, “Frequency Domain Transient Analysis Applied to Transmission System Restoration Studies”, Proc. of the 7th Internacional Conference of Power Systems Transients (IPST’07), Lyon, France, July 2007.

VIII. BIOGRAPHY

Pablo Gómez (IEEE Member) received the B.Eng. degree in mechanical and electrical engineering fromUniversidad Autonoma de Coahuila, Mexico, in1999 . He received the M.Sc. and Ph.D. degrees inelectrical engineering from CINVESTAV, Guadalajara, Mexico in 2002 and 2005, respectively.Currently, he is a professor with the Electrical Engineering Department of SEPI-ESIME Zacatenco, National Polytechnic Institute, Mexico City, Mexico. His research interests are in the modeling and

simulation of electromagnetic transient phenomena.

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