Implementation of the Numerical Laplace Transform: A Review

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008 2599

Implementation of the NumericalLaplace Transform: A Review

Task Force on Frequency Domain Methods for EMT Studies, Working Group on Modeling and Analysis of System TransientsUsing Digital Simulation, General Systems Subcommittee, IEEE Power Engineering Society

Pablo Moreno and Abner Ramirez

AbstractIn this paper a detailed description, an analysis, andan assessment of a frequency-domain technique highly applicableto power systems transient analysis (i.e., the numerical Laplacetransform) are presented. The errors due to truncation and sam-pling when converting a frequency-domain signal to the time do-main are analyzed. Additionally, the use of odd and regular sam-pling is discussed. Two major goals of the paper are the revival ofthe mentioned technique and its friendly description for power sys-tems engineers. As an application, a transmission-line transient ispresented.

Index TermsFourier transform, frequency-domain (FD) anal-ysis, Laplace transform, transient analysis.

I. INTRODUCTION

F REQUENCY dependence is an intrinsic characteristic ofelectrical/electronic devices. Its effects are clearly visiblein the distortion of voltage/current waveforms.

A natural choice for mathematically handling frequency de-pendent elements is the frequency domain (FD). However, onanalyzing transients one needs to go back to time domain (TD)where a more comprehensive way of interpreting results ex-ists. In general, the mathematical FD expressions resulting frompractical studies are difficult, if not impossible, to be solved byanalytical means. This difficulty is, however, overcome by nu-merical methods.

Initially, the Inverse Fourier transform was applied to con-vert an FD signal into the TD. The direct integration brought upcertain errors caused by discretization and truncation of the FDsignal. Later on, the modified Fourier transform (MFT) and win-dowing techniques were introduced in order to decrease theseerrors. Finally, slight modifications to the latter produced thenumerical Laplace transform (NLT) in its actual form.

This paper intends first to revive the interest in the NLTthrough its detailed description and analysis.

Manuscript received October 14, 2007; revised December 17, 2007. First pub-lished May 7, 2008; current version published September 24, 2008. Paper no.TPWRD-00600-2007.

P. Moreno and A. Ramirez are with CINVESTAV, Guadalajara 44550,Mexico (e-mail: [email protected]).

Task Force Members: J. L. Naredo (Task Force Chairperson), A. Ametani,S. Carneiro, M. Davila, V. Dinavahi, J. A. de la O, F. de Leon, P. Gomez, J.L. Guardado, B. Gustavsen, J. A. Gutierrez-Robles, J. R. Marti, J. A. Martinez-Velasco, F. Moreira, P. Moreno, A. Morched, N. Nagaoka, W. A. Neves, T.Noda, V. H. Ortiz-Muro, A. I. Ramirez, M. Rioual, A. C. Siqueira de Lima, F.A. Uribe, K. Strunz, N. Watson, D. J. Wilcox.

Digital Object Identifier 10.1109/TPWRD.2008.923404

As a second objective, this work describes in a comprehen-sive manner the numerical issues of the NLT in such a way thata power system engineer (without being a specialist in digitalsignal processing) can understand the basic theory and developa FD program for handling transients. Since some of the refer-ences mentioned in this paper are old and/or out of reach formost of power engineers, we repeat here many of the theory al-ready contained in those papers.

Although the NLT technique is applied in this work to thepower systems transients area, it is deemed here that its appli-cability can be expanded to other Engineering areas.

Finally, a code section is included (Appendices A and B) thatallows to apply the NLT in Matlab.

II. HISTORYOne of the earliest numerical methods to invert Laplace

transform was developed in the 1960s by Bellman, Kalaba, andLockett [1]. From 1964 to 1973, Day, Mullineux, Battisson,and Reed approached the problem of analyzing power systemtransients using Fourier transforms and reported their resultsin [2][5]. These researchers realized that, due to truncation ofthe FD signal, the corresponding time functions were affectedby Gibbs phenomenon [2]. Additionally, since FD functionshave to be sampled, the corresponding TD waveforms wereaffected by aliasing. To alleviate Gibbs errors they introducedthe use of window functions [4], and to decrease aliasing errorsthey proposed the use of an artificial exponential damping, i.e.,shifting the integration path away from the imaginary axis [3].These authors named their technique the MFT.

Wedepohl and Mohamed, in 1969 and 1970, adopted the MFTand applied it to the calculation of transients on multiconductorpower lines [6]. These authors further extended the techniqueto applications including switching manoeuvres [7]. In 1973,Wedepohl and Wilcox applied the MFT to the analysis of un-derground cable systems [8].

A major problem with the MFT was that it required very longcomputational times. In 1973 Ametani introduced the use of thefast Fourier transform (FFT) algorithm and the MFT became amuch more attractive transient analysis alternative [9]. In 1979,Ametani proposed a numerical Fourier Transform with expo-nential sampling for handling electrical transients where a verywide range of frequency or time is required [10].

In 1978, Wilcox formulated the MFT methods in terms of theLaplace Transform theory and introduced the term numericalLaplace transform [11].

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2600 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 23, NO. 4, OCTOBER 2008

An application of the NLT to the solution of boundary valueproblems using a similar approach to that due to Wilcox waspresented in 1979 [26].

An FD transient program that includes switching manoeuvresand nonlinear lumped elements (using piecewise techniques)was reported by Nagaoka and Ametani in 1988 [12].

An alternate approach was published in 1998 to the numericalinversion of the Laplace transformation [27]. In this techniquethe exponential function in the integrand is approximated bya series expansion. The method presents truncation errors andthe errors associated with the approximation of the exponentialfunction. It is also reported the existence of difficulties whendealing with some singularities in the excitation functions (i.e.,the Laplace transform of a square function, and singularitiespresent in the response of distributed constant circuits).

Recently, a group of researchers have been successfully ap-plying the NLT to analyze electromagnetic transients [13][16].

III. MATHEMATICAL BASIS

Let be a transient waveform and be its FD image.These functions are related by the Laplace transform [25]

(1a)

and the inverse Laplace transform

(1b)

where the complex frequency is given by . corre-sponds to the angular frequency variable and is a positive realconstant. Alternatively, (1a) and (1b) can be expressed as

(1c)

and

(1d)

From (1c), one can observe that the Laplace transformis equivalent to the Fourier transform of the damped signal

. This exponential damping makes it possible to dealwith dc components when using the Fourier transform insteadof the Laplace transform. In fact, this is the approach of themodified Fourier transform introduced by Day, Mullineux,Battisson, and Reed [2][5].

With being a real and causal function, the inverseLaplace transform can be written as

(2)

IV. ERRORS INCURRED IN THE NLT

Numerical evaluation of the inverse Laplace Transform intro-duces two types of errors: Gibbs oscillations due to truncationof the integration range and aliasing due to discretizationof the continuous variable . Truncation errors are reducedby introducing a window function . Aliasing errors aredecreased by smoothing the frequency response of the systemby an appropriate choice of the convergence (damping) factor

[11].Although we are dealing with the NLT, in this Section, the

illustration of the errors incurred while converting from the FDto the TD is presented using the Fourier transform as a basis.

A. Truncation Error

Consider the rectangular pulse [see Fig. 1(a)]

(3a)

The Fourier image of is given by [see Fig. 1(a)]

(3b)

where the period of oscillation is equal to .Suppose that a data window with maximum frequencyis defined, we would like to obtain back from the trun-

cated integral

(4)

where

(5a)

and the TD image of is

(5b)

with a period of oscillation . Fig. 1(b) depictsand a partial view of .

From (4), it is observed that the transform of the FD productis equivalent to perform the TD convolution [illustrated inFig. 1(c)]

(6)

Additionally, from (6), we notice that the original functionhas been degraded to due to the truncation of the infi-

nite frequency range, as shown in (4). Fig. 1 shows graphically

MORENO AND RAMIREZ: IMPLEMENTATION OF THE NLT: A REVIEW 2601

Fig. 1. Illustration of truncation phenomenon. (a) Original function. (b) Datawindow. (c) Truncation effect.

the truncation process described by (3)(6). The ringing phe-nomenon appearing in [see Fig. 1(c)] is known as Gibbsoscillations and has a period of oscillation that is equal to[17]. The value of also corresponds to the minimum rise timeof ; that is, due to frequency truncation, the resulting func-tion not only presents Gibbs oscillations, it also has a finite risetime.

One way to attenuate truncation errors is to introduce aweighting factor that averages over one period of theGibbs oscillations, i.e.,

(7a)

where is known as the window function in TD. For instance,consider that corresponds to the rectangular time windowfunction shown in Fig. 2(a). Then, the substitution of (4) into(7a), and using , gives

(7b)

Interchanging the integration order in (7b) and performing theinner integral yields

(7c)

where is the Lanczos window given in the FD as [seeFig. 2(a)]

(7d)

Notice that for numerical implementation, only the centrallobe of Lanczos window is taken. The effect of truncating andintroducing the window function is represented in the FD andin the TD by

(8a)

(8b)

respectively, and is illustrated by Fig. 2. Comparison of thegraphs in Figs. (1c) and (2b) shows the attenuation of Gibbsoscillations in the latter due to the use of the window function.

B. Discretization Error

In the numerical computation of the inverse Fourier trans-form, what it has actually done is transform an FD sampledsignal (instead of a continuous one) into the TD as follows:

(9)

where corresponds to a train of regularly spaced Diracimpulses [17]

(10)


Fig. 2. Effect of windowing. (a) Lanczos window. (b) Windowed function .

In the TD, (9) becomes

(11)

where is the TD image of given by

(12)

Substitution of (12) into (11) gives

(13)As an illustration of the discretization error, the functions de-

scribed by (9)(13) are shown in Fig. 3 for a specific triangularsignal. Fig. 3(a) shows the TD triangular function and its FDcounterpart. The sampled FD signal is depicted in Fig. 3(b) andits corresponding TD counterpart in Fig. 3(c).

As it can be noted from (13), the resulting TD function ob-tained after discretization of its FD image consists of a periodicfunction with period . According to the samplingtheorem, overlapping will exist if [17].

To decrease the aliasing phenomenon, the NLT takes aim intothe fact that the inversion involves the function attenuated by a

Fig. 3. Illustration of discretization phenomenon. (a) TD original waveform and its FD image . (b) Sampled version of and (c) TD wave-form of the sampled version.

damping coefficient . Several formulae have been proposed forcalculating this damping coefficient; some of these have beenobtained in a heuristic way. We will only mention those pro-posed by Wilcox [11] and by Wedepohl [18], respectively

(14a)(14b)


Fig. 4. Illustration of basic definitions: (a) TD and (b) FD.

Wilcoxs criterion often permits an accuracy of about ,whereas with Wedepohls criterion, the numerical error levelcan be brought up to . This is shown in Section VI. Inthe authors experience, the criteria presented provide good re-sults; however, it is worthy to note that up to now, there is nosound basis for choosing a suitable value for . On the one hand,choosing a small value will not displace far enough from theimaginary axis the integration path and, therefore, waveformtails will not be damped enough. On the other hand, if is toolarge, while the frequency response may then be very smooth,the truncated integration contour might not enclose the signif-icant poles and the resulting response would be unacceptablydistorted.

V. NUMERICAL FORMULATION

A. Odd Sampling

The numerical evaluation of (2) requires the consideration ofsampling intervals and , as shown in Fig. 4. The numer-ical formulation that will be described uses an odd discretizationin the FD with spacing of , and normal time steps of ;therefore, the following definitions are made:

for (15a)

and

for(15b)

where for time and frequency, we use equally spaced sam-ples. Additionally, since the numerical integration of (2) re-quires finite integration limits, let us define as the maximumfrequency and as the observation time. The observation timecorresponds to the waveform repetition period which in this caseis given by

(15c)

Thus, we can establish the following relations [23]:

(16a)(16b)

Equation (16b) is defined by the odd sampling procedureand (16a) represents the minimum useful time step. This valuecomes from the fact that truncation of the spectrum produces arise time of

(16c)

and the ratio gives samples.By using the sampling scheme defined by (15) (and including

the window function), (2) can be approximated numerically as

(17)Substitution of definitions (15) and (16) into (17) yields

(18a)

where

(18b)

The term inside the square brackets in (18a) permits us toutilize the fast Fourier transform (FFT) algorithm [17] whichmakes the calculation of in a very efficient manner whenis equal to an integer power of two.

B. Choice of , , andIn this respect, one has several options [11]. For instance,

once the total observation time in the transient analysis ischosen, is automatically defined by (16b); according to thebandwidth of (or the type of transient under analysis), onecan choose the maximum frequency , and will be givenby (16b); is then calculated by (16a). Note that can becalculated by either (14a) or (14b).

C. Regular SamplingAlternatively, (2) can be approximated numerically by using

regular sampling (i.e., equally spaced samples with spacing


equal to ). In this case, the relation corresponding to (17) isgiven by

(19)where is defined as in (16a), and (16b) becomes

(20a)

and (and ) has been discretized as follows:

(20b)Note that in (20b), the samples at the left hand side of the

imaginary axis have been flipped to the right and conjugated. In(20), the selection of , , and should be modified accord-ingly.

D. Direct TransformFor completeness of this paper, the numerical formulation of

the direct Laplace transform is provided in this subsection.In order to obtain a useful formula for the frequency spec-

trum, the discretized TD function can be converted to a contin-uous analog time function. Since the behavior between samplesis unknown, it suffices to assume that time point defines atime rectangle of magnitude from to (sampleand hold operation). Hence, for the case of FD odd sampling,(1c) can be expressed as follows:

(21a)

Solving the integral in (21a) by applying the rectangular ruleof integration gives the following formula that allows using theFFT algorithm:

(21b)

Similarly, for the case of regular sampling

(21c)

where , , and , are given by (15a), (15b), and (20b),respectively.

Since the analog function formed from is a train of rectan-gular pulses, a different formula for the frequency spectrum canbe obtained by summing the Laplace transforms of all pulses. Inthe authors experience, (21b) and (21c) provide good results.

Fig. 5. Error from choosing the damping factor corresponding to expressions:(a) (14a) and (b) (14b).

VI. ASSESSMENT OF THE NLT

A. Damping Factor and Number of SamplesFor the sake of illustration, we have taken the shifted unit step

(22a)

whose Laplace image is

(22b)

The NLT is evaluated by first transforming into the TDand then comparing the result with the exact function given by(22a). In the graphs of Fig. 5, the delay has been assumedto be equal to 0.1 s and the decimal logarithm of theerror is presented for each of the two choices for selecting[i.e., (14a) and (14b)]. The number of samples in the NLT hasbeen taken as 512, 1024, and 2048. The spikes seen in the plotsof Fig. 5 correspond to the error at the discontinuity of the unitstep, which is caused by the different rise times between theexact formula in (22a) and the numerical solution.


TABLE IDAMPING FACTOR

Fig. 6. Error from using different window functions.

Table I presents the damping coefficient corresponding tothe results shown in Fig. 5. Notice that from criterion (14a)remains constant regardless the number of samples. Note alsothat using (14b) the damping factor increases as increases.

B. Window Functions

The analysis of the NLT is now directed to the differencesgiven by the use of different window functions. As a basis, weuse the same function given by (22), choose from (14b), anduse 1024 samples. In Fig. 6, the errors obtained by using thewindow functions by Hanning, Lanczos, and Blackman are pre-sented. Additionally, the error without window function is alsoshown. From Fig. 6, we notice that for engineering purposes,the three windows give excellent results.

C. Odd Versus Regular Sampling

The comparison between the odd and regular sampling [ex-pressions (18) and (19)] is shown in Fig. 7 for the specific caseof using Hanning window function with and ap-plying (14b) in the numerical inversion of (22b). From Fig. 7,one can notice that the error of the numerical inversion usingregular sampling is bigger than the one from using odd sam-pling. Besides, Gibbs phenomenon is less attenuated with theformer, as seen in Fig. 7. An explanation for the statement aboveis that given the same observation time and number of samples,while the actual samples spacing in both methods is the samethe maximum frequency for the odd sampling is twice the max-imum frequency for the regular sampling.

The Matlab v.7 code that has been used for obtaining the re-sults in Figs. 5 and 6 and Table I is presented in Appendix A.In that code, the user can easily modify the number of samples,the choice of , and the type of window function.

Fig. 7. Odd sampling versus even sampling errors.

Fig. 8. Switching simulation. (a) Closing. (b) Opening.

VII. SWITCHING MANEUVERS SIMULATION

A. FD Switch Representation

Switching operations produce changes in network topologythat turn the network into a time-varying system, precluding atfirst sight, the use of FD methods. However, for a linear networkby using the superposition principle, the NLT can still be applied[12], [19].

On the one hand, an opened switch can be represented by avoltage source that is equal to the potential difference be-tween its terminals. Switch closing is accomplished by the seriesconnection of a voltage source with an equal magnitudebut opposite polarity to that of so that

[see Fig. 8(a)]. Therefore, the voltage source required to closethe switch at is given by

(23)

where is the TD waveform of the voltage between theswitch terminals for the whole observation time with the switchopened and indicates the Laplace transform.

On the other hand, a closed switch is represented by a cur-rent source that is equal to the current flowing across it.Switch opening is performed by connecting in parallel toa current source of equal magnitude but opposite polarity,so that [see Fig. 8(b)]. Opening the switch


Fig. 9. (a) Ideal switch and (b) equivalent circuit.

when the current reaches its first zero value after a specifiedopening time is accomplished by injecting the current source

(24)

where is the current zero-crossing time and is the TDwaveform of the current flowing through the closed switch forthe whole observation time.

It should be mentioned that when using a nodal formulation,ideal voltage sources cannot be used to simulate switch closures.Hence, the injection of voltage must be accomplishedthrough a Norton equivalent current source

(25)

where is a resistance with a very small value (or, alterna-tively, equal to the switch resistance).

Regarding the switch initial condition, a conductance of zerois assigned for an initial state of open and a value of for aninitial state of closed. In order to implement switching simula-tions, a topological change in the network nodal matrix must beaccomplished when a maneuver occurs. This topological changeis performed by introducing (closure) or extracting (opening)

from the network matrix before solving for the state due tothe injected current.

From the discussion before, a practical switch model suitablefor simulating closures and openings can be as the one shownin Fig. 9. The injected current is given by

closureopening (27a)

and the Norton conductance is given by

closureopening. (27b)

When is positive, is introduced in the admittancematrix, representing the connection between the switch nodesand when is negative, is subtracted, leaving the switchnodes disconnected.

After a switching operation, the complete system responseis obtained by adding the response existing before switching to

Fig. 10. Scheme of closing condition.

Fig. 11. Transmission system model.

that resulting from injecting the current source that performsthe switch maneuver. As can be seen in (23) and (24), switchingsimulation requires going back and forth from the FD to the TD,therefore to speed the development of an inhouse tool by thereader. In Appendix B, a Matlab function for the direct Laplacetransform is provided.

B. Numerical ExampleConsider the three-phase transmission line represented in

Fig. 10. Using a two-port admittance model for the transmis-sion line (see Fig. 11) and the model shown in Fig. 9 for thecircuit breaker, the network nodal equation can be expressed asfollows:

(28)where and are the two-portadmittance submatrices for the transmission line with charac-teristic admittance and propagation matrix [6], [23], [28],[29]. is the vector of Norton equivalent currents, isa diagonal conductance matrix that represents the three phaseswitch condition, is a diagonal matrix whose elements are

, and is the load admittance ( for an openended line). When , (28) represents the system statebefore switching. For an initially opened three-phase switch

. In (28), all submatrices are of order 3 3; there-fore, the system matrix is of order 9 9.

As an application example, consider the energization of a203 km long transmission line. The line is equipped with twoBluejay ACSR subconductors per bundle separated by 450 mm.The average height and horizontal separation of the bundle cen-ters are 40 m and 8.8 m, respectively. The earth resistivity is


Fig. 12. Voltage waveform calculated at the receiving end of phase B.

taken equal to 100 m. The three poles of the circuit breakerare closed sequentially. The closing times are 3.0, 5.0, and 8.0ms for phases A, B (center), and C, respectively. The switch re-sistance is set to 0.1 , and the receiving endof the transmission line is left open. The number of samples is

.

Consider the first maneuver (closing of phase A). In order toperform the operation, is set to zero, and the injected currentis determined as follows:

(29)

where is the unit step function. The topological change inthe admittance matrix is done by adding the switch conductancematrix

(30)

After solving (28) for the node voltages due to the injec-tion of , the complete voltage response is obtained by addingthe system response due to initial conditions (previous systemstate). Closing the rest of the circuit-breaker poles is performedin a likewise manner.

Fig. 12 shows the resulting voltage waveform of phase A atthe receiving end after closing the three phases using the methodpresented in this work and using the PSCAD program. The cputime for this simulation with 1024 samples was 0.8125 s in aPentium IV with a 3 GHz processor.

VIII. DISCUSSION AND CONCLUSIONThe basic theory of the NLT has been described in detail. We

have analyzed the main errors incurred in the numerical imple-mentation of the NLT and discussed the way to diminish thoseerrors.

The NLT is deemed here as a powerful tool for the analysis ofelectromagnetic transients. Its evolution has permitted to refineits numerical characteristics through the use of window func-tions, the choice of an integration path ( parameter), and theuse of the FFT.

As shown in Section VI-B, one can choose any of the threewindow functions used here (Hanning, Lanczos, and Blackman)achieving an accuracy far beyond the one required by engi-neering applications (in the order of ). Nevertheless, theNLT permits the implementation of any of the window func-tions from the literature.

Concerning the choice of parameter , in the authorsexperience, (14b) gives the best results. Apparently (seeTable I), increasing produces better results; however, oneshould be careful because using a larger produces anamplification of Gibbs oscillations due to the term in(2). In fact, the NLT technique suffers from numerical errorsin the waveform tails. This forces discarding the last 5 to10% of the TD samples.

The NLT technique presented in this work uses odd samplingand the utilization of the FFT speeds up the solution dramatically.In the cases of electromagnetic transients, we can say that basedon (14b), 512 or 1024 samples are reasonable numbers for agood definition of the TD response. Some algorithms usinga nonuniform sampling have been proposed [10], [20][22].The problem with jagged behavior of the FD responsescaused by poles close to the integration axis was addressedin [22], where an adaptive sampling scheme was proposed.Aliasing was here avoided by integrating analytically betweensamples and using linear variation, similarly as in [12]. Theuse of splines was proposed in [24]. Although it is claimedthat such procedures could save computing time, their usecan become very cumbersome when switching occurs. Infact, the most general sampling is the uniform one; it doesnot need to know beforehand the behavior of the transient(i.e., the fast dynamics appearing at the beginning followedby the slow dynamics).

Despite the NLT being fundamentally oriented to linear prob-lems, it is shown here that it can still be applied to nonlinearevents such as sequential switching [7], [8], [19]. Moreoverthe NLT is capable of handling nonlinear elements in a piece-wise manner [12], [14]. In addition to simulation of generaltransients, the inverse transform is deemed here to be an in-valuable tool for validation of frequency-dependent transmis-sion-line models.

The authors believe that this paper will help power engineersdevelop further insight into the field of electromagnetic tran-sients. In addition, the presented NLT technique is an invaluablecomplement for the more familiar TD analysis.

APPENDIX A

The Matlab v.7 code used for the assessment of the NLT inSection VI is provided:


clear all;

% Laplace data;

; ; error ; ;;

% choose a criterion to determine c;

% ;

;

; ; ;

;;

% function in the s-domain;

; ;

% choose a window function;

% ; %Hanning;

% ;%Lanczos;

%; %Blackman;

% function in the time domain through NLT;

; %comment this line to make it withoutwindow function;

;

;

% function in the time domain analytical;

;;

%plotting;

;

figure, plot(t(1:Np),real(ftd(1:Np)),t(1:Np),ft(1:Np));figure, plot(t(1:Np),log(abs(real(ftd(1:Np))-ft(1:Np))));

APPENDIX B

Matlab v.7 code for the direct Laplace transform with FD oddsampling:

function

%;

% ftd: values of the waveform for 0;

% dt, ;

% c: damping factor;

% dt: time step;

%;

;

;

;

;

;

.

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Implementation of the Numerical Laplace Transform: A Review

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