Accurate Computation of Transient Profiles Along Multiconductor Transmission Systems by Means of the...

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 29, NO. 5, OCTOBER 2014 2385

Accurate Computation of Transient Profiles AlongMulticonductor Transmission Systems by Means of

the Numerical Laplace TransformRodrigo Nuricumbo-Guillén, Pablo Gómez, Senior Member, IEEE, Fermín P. Espino-Cortés, Member, IEEE, and

Felipe A. Uribe, Member, IEEE

Abstract—In this paper, transient internal profiles along multi-conductor transmission lines are obtained from a model describedin the - domain (spatial frequency–temporal frequency). In suchdomain, the telegrapher equations of the line are defined in a di-rect algebraic form. Applying boundary conditions, these equa-tions can be easily solved for and . Then, the in-verse numerical Laplace transform is applied successively (twice)to obtain and , that is, the voltage and current pro-files along the line. Besides, since the line’s electrical parametersare defined in the frequency domain, the frequency dependenceof such parameters is included in a straightforward manner. Testcases corresponding to an overhead line, a transformerwinding, anunderground power cable, and a transmission network are evalu-ated in order to demonstrate the accuracy and versatility of theproposed method. The results are compared with those obtainedwith Alternative Transients Program/Electromagnetic TransientsProgram and PSCAD/EMTDC at specific points along the trans-mission systems under analysis, given that the space discretizationis cumbersome and prone to error accumulation in these time-do-main simulation programs.

Index Terms—Numerical Laplace transform (NLT), transientinternal overvoltages, transmission-line modeling.

I. INTRODUCTION

T HE TELEGRAPHER equations define the propagation ofvoltage and current waves along transmission lines.When

defined in the - (space-time) domain for an -conductor case,they consist of equations which comprise first-order partialderivatives. If the frequency dependence of the line parametersis taken into account, convolution terms appear [1]. However, ifthe Laplace transform is applied, the problem is reduced to first-order total derivatives (without convolutions) in the - (space-temporal frequency) domain. Considering boundary conditions,

Manuscript received September 01, 2013; revised January 31, 2014; acceptedMarch 20, 2014. This work was supported by Secretaría de Investigación y Pos-grado, Instituto Politécnico Nacional (SIP-IPN) and CONACyT. Date of pub-lication April 14, 2014; date of current version September 19, 2014. Paper no.TPWRD-01012-2013.R. Nuricumbo-Guillén, P. Gómez, and F. P. Espino-Cortés are with the Elec-

trical Engineering Department, SEPI-ESIME Zacatenco, Instituto PolitécnicoNacional (IPN), Mexico City 07738, Mexico (e-mail: [email protected]).F. A. Uribe is with the Department of Mechanical and Electrical Engineering,

Graduate Program on Electrical Engineering, CUCEI, Universidad de Guadala-jara, Guadalajara 44430, Mexico.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRD.2014.2313526

the line can be represented bymeans of a two-port model, so thatvoltages and currents at the line terminals can be computed inthe frequency domain.In this paper, the Laplace transform is applied twice to the te-

legrapher equations in the - domain to obtain algebraic equa-tions in the - domain (spatial frequency–temporal frequency).After considering boundary conditions, these equations can beeasily solved to obtain voltages and currents in such domain.The solution of voltages and currents profiles in the - domainis obtained by means of the successive application of the inversenumerical Laplace transform (NLT) [2], including the frequencydependence of the line parameters. This algorithm has been ex-tensively used to obtain the transient time-domain response ofpower system devices (see, for instance, [8]–[14]). It has alsobeen extensively used as a very accurate validation tool for newmodels andmethods [2], [4], [8]–[10], [14]. However, this paperreports its first application in a successive manner for the com-putation of transient internal overvoltages and currents alongpower lines and networks, transformers windings, and under-ground power cables. Recent work has successfully introduceda similar approach applied to electronic transmission systemsconsidering constant electrical parameters (frequency indepen-dent [3]. In contrast, frequency dependence of the electrical pa-rameters is accounted for in this paper. It is well known thatincluding this dependence can be very important to achieve ac-curate results for power system transient analysis.Techniques based on Electromagnetic Transients Program

(EMTP)-type programs to compute transient profiles have alsobeen proposed [7], [15], [16]. However, inclusion of frequencydependence of line parameters in time-domain models is farmore complicated than in frequency domain ones and canproduce numerical errors in highly unbalanced systems.The contributions of this paper can be summarized as follows.a) It is demonstrated that the inverse NLT algorithm (as re-vised in [2] and [4] for power system transient analysis)can be successfully applied not only for temporal fre-quency to time transformations, but also for spatial fre-quency (wave number) to space transformations.

b) A method (alternative to time-domain methods) to com-pute transient voltage and current profiles along powertransmission elements and networks is developed, withthe inherent advantage of a direct inclusion of frequencydependence. Its application to transformer windings isalso presented.

This paper is the basis of ongoing developments, includingextension to transmission lines excited by indirect electromag-

0885-8977 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

2386 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 29, NO. 5, OCTOBER 2014

netic (EM) fields, as well as the application of the method toother power components where the computation of internal volt-ages or current profiles can provide important information fordesign, protection purposes, and insulation coordination.

II. SOLUTION OF THE TELEGRAPHER EQUATIONSIN THE - DOMAIN

The telegrapher equations of a uniform multiconductor lineare defined in the - domain in matrix form as follows:

(1)

where and are the vectors of voltages and cur-rents along the propagation axis ; and and are the matricesof series impedances and shunt admittances per unit length, re-spectively. Applying the Laplace transform to (1) with respectto coordinate , it follows that:

(2)

where and are the voltages and currents at 0,and is an identitymatrix (size for a line of conductors).Solving (2) for

(3)

The voltages and currents at can be obtained from theline’s nodal matrix in the -domain and modal decomposition

(4)

where

(5a)

(5b)

(5c)

(5d)

No line-balancing conditions are assumed in order to derive(4) and (5). Vectors and are the current sourcesconnected at the line’s ends. If the line is excited at by asource represented by its Norton equivalent (see Fig. 1), then

(6a)

(6b)

Substituting (6) in (4) and solving for , it yields

(7)

where

(8)

Fig. 1. Circuit representation for the calculation of .

For the calculation of , one has to consider that the currentinjected by the source is divided in two parts, as seenin Fig. 1: one circulating through (obtained as ),and the other one corresponding to . From (6a) and (7), itfollows that:

(9)

Substituting (7) and (9) in (3), the solution for the voltagesand currents in the - domain is obtained

(10)

By means of a successive application of the inverse algorithmof the NLT, (10) can be transformed into the - domain to ob-tain and , that is, the voltage and current profilesalong the multiconductor line. It can be noticed that since andare defined in the frequency domain, (10) can incorporate fre-

quency-dependent parameters in a straightforward manner.Equation (10) is only applicable to the case of single lines ex-

cited at the sending node. The extension of the method to trans-mission networks (multiple lines with multiple sources) will betreated in Section V.

III. INVERSE NLT

A. Transformation From to

For a real function, the inverse Laplace transform is expressedas

(11)

In (11), complex temporal frequency is defined as, where is a positive real constant and is the angular

frequency. Also, the integral is truncated taking as the max-imum frequency. Considering a maximum observation timeand including a window function to reduce truncation er-rors (Gibbs oscillations), the discrete form of (11) is

for (12)

NURICUMBO-GUILLÉN et al.: ACCURATE COMPUTATION OF TRANSIENT PROFILES 2387

where

forfor

(13a)

(13b)

(13c)

(13d)

(13e)

is a factor with a value between 1 and 1 , andis the number of discrete samples. The term between bracketsin (12) corresponds to the inverse fast Fourier transform (IFFT)algorithm. The reader is referred to [2] and [4] for a detaileddescription of the inverse NLT.

B. Transformation From to

The procedure to obtain the inverse NLT with respect to thecoordinate is completely analogous to the procedure for thetransformation from to . The analytical expression for thecorresponding inverse Laplace transform is

(14)

In (14), complex spatial frequency is defined as ,where is a positive real constant and is the angular spatialfrequency. The integral is truncated taking as the maximumspatial frequency. The discrete form of (14) is

for (15)

where

forfor

(16a)

(16b)

(16c)

(16d)

(16e)

In (16), is the line’s length, is the number of discretesamples, and takes the same values defined before.

C. Successive Partial Numerical Inversion

Defining the partial inverse Laplace transforms as

(17)

Fig. 2. Zigzag connection of a multiconductor line model for a three-turnstransformer winding model.

and

(18)

and since the Laplace transform is a linear operator, the fol-lowing property can be applied:

(19)Therefore, the voltage and current profiles and

along the line can be obtained from the partial numerical in-version of and by applying (12) and (15) in anyorder.

IV. APPLICATION TO TRANSFORMERS WINDINGS

For the simulation of fast front transients, a transformerwinding can be modeled as a multiconductor line, where eachconductor represents a section of the winding (turn or groupof turns) [18]. In order to establish the winding continuitybetween sections, a zigzag-type connection is applied, in whicha high admittance connects the end of a conductor withthe beginning of the next one, as seen in Fig. 2.Using (10), the voltage and current solutions in the - do-

main are found. Finally, by applying (12) and (15), the voltageand current profiles and along each turn of thewinding are obtained. An example of voltage profiles along theturns of a winding is provided in Section VI-B.

V. APPLICATION TO TRANSMISSION NETWORKS

The method described in Section II can be easily extendedand applied to the analysis of transients in transmission net-works. For this purpose, the voltages and currents at each nodeof the network have to be computed first.The voltages at each node of the network can be obtained

from the network’s nodal matrix in the frequency domain

......

(20)


Fig. 3. Calculation of from the line’s equivalent circuit.

where is the number of lines, is the network’s admit-tance matrix, and is the current source connected at node. Once the voltages have been computed, the currents at eachnode of the line can be obtained from the line’s -equivalentcircuit. From Fig. 3, it follows that:

(21)

and are the voltages and currents at the sendingnode of the th line, is the voltage at the receiving node,and and are computed from (5a) and (5b) for the th line.After that, it is possible to obtain the voltages and currents ofeach line in the - domain

... . . ....

. . ....

(22)

... . . ....

. . ....

(23)

where

(24)

and are the series impedance and shunt admittance ma-trices of the th line, respectively.

Fig. 4. 3D plot of voltage profile along phase A of the line.

Fig. 5. 3D plot of voltage profile along phase C of the line.

Finally, by applying (12) and (15) to each line, the voltageand current profiles in the time domain are obtained. An ex-ample corresponding to a four-node network is provided inSection VI-D.

VI. TEST CASES

Four cases are considered in this section: 1) a three-phaseaerial transmission line; 2) a transformer winding; 3) an under-ground cable arrangement; and 4) a transmission network. Forsimplicity, the number of samples used for the calculation forboth time and distance is 1024 ( ). However, this is not a re-quirement of the methods; distance and time sampling can bedifferent, as long as the Courant–Friedrichs–Lewy condition isfulfilled [6]. A Hanning window and a value ofwere used in all cases when applying the inverse NLTs (12),(15).Due to space limitations, the results at only some representa-

tive measuring nodes are shown in the presented cases.Mean relative differences obtained for each case (corre-

sponding to the comparisons shown in Figs. 6–9, 12–13,17–20, and 24–25) are listed in Tables III–VI.

A. Multiconductor Aerial Transmission Line

A three-phase aerial line is considered. The conductors radiusis 0.0254 m, their height above ground is 16 m, the horizontalseparation between adjacent conductors is 5 m, and the totallength of the line is 1000 m; observation time of the transientresponse is 50 s. Phase A is excited by means of a doubleexponential voltage waveform (1 p.u. 1.2 s/50 s), while theother conductors are left open at both sending and remote ends.


Fig. 6. Comparison between inverse NLT (continuous lines) and ATP/EMTP(dotted lines) for phase B. The line is divided into 32 segments in ATP/EMTP.

Fig. 7. Comparison between inverse NLT (continuous lines) and ATP/EMTP(dotted lines) for phase C. The line is divided into 32 segments in ATP/EMTP.

Fig. 8. Comparison between inverse NLT (continuous lines) and ATP/EMTP(dotted lines) for phase B. The line is divided into four segments in ATP/EMTP.

Fig. 9. Comparison between inverse NLT (continuous lines) and ATP/EMTP(dotted lines) for phase C. The line is divided into four segments in ATP/EMTP.

Fig. 10. Voltage profile along the first turn of the winding.

Fig. 11. Voltage profile along the sixth turn of the winding.

Fig. 12. Comparison between the proposed method (continuous lines) and afrequency-domain two-port model (dotted lines) at the beginning of the turns.

The frequency dependence of the series impedance of the line(due to skin effect in ground and conductors) is taken into ac-count by means of the concept of complex penetration depth [5].The ground resistivity is 100 m, and the conductor resistivityis 3.21 m.The voltage and current distributions along the line are ob-

tained by successively applying the inverse NLT to (10). Figs. 4and 5 show the voltage profiles along phases A and C of the line.For comparison purposes, a simulation is performed in ATP/

EMTP using the frequency-dependent line model developed byMarti [17] and dividing the line in 32 segments to obtain mea-surements at internal points along the line. Figs. 6 and 7 showthe comparison between the results from the inverse NLT andATP/EMTP at phases B and C of the line. For illustrative pur-poses, only four measurements are provided (corresponding toplots every 256 samples in the -coordinate for the results fromthe inverse NLT and every eight segments for the results fromATP/EMTP). However, there are some noticeable differences


Fig. 13. Comparison between the proposed method (continuous lines) and afrequency-domain two-port model (dotted lines) at the middle point of the turns.

Fig. 14. 3D plot of voltage profile along the core of phase A.

Fig. 15. 3D plot of voltage profile along the shield of phase B.

Fig. 16. 3D plot of current profile along the shield of phase C.

between the results obtained from the inverse NLT and the ATP/EMTP, which become more evident as time increases. Further

Fig. 17. Comparison between inverse NLT (continuous lines) and PSCAD/EMTDC (dotted lines) for the cables’ conductors at the middle point of the ca-bles (transient voltage).

Fig. 18. Comparison between inverse NLT (continuous lines) and PSCAD/EMTDC (dotted lines) for the cables’ shield at the middle point of the cables(transient voltage).

Fig. 19. Comparison between inverse NLT (continuous lines) and PSCAD/EMTDC (dotted lines) for the cables’ conductors at the middle point of the ca-bles (transient current).

simulations were performed in ATP/EMTP by increasing thetime sampling, but the results did not show significant variation.A different approach consisted of reducing the number of line

segments in ATP/EMTP. Divided into eight segments, the re-sults from ATP/EMTP approached those from the inverse NLT,and with four segments, the results are very close, as shown inFigs. 8 and 9. The Courant–Friedrichs–Lewy condition was sat-isfied in all cases.The issues presented by the ATP/EMTPwhen discretizing the

line in several segments (modeled using the Marti setup [17])are attributed to error accumulation, which reduces the accuracyof the results [15]. Similar problems were reported in [7] when


Fig. 20. Comparison between inverse NLT (continuous lines) and PSCAD/EMTDC (dotted lines) for the cables’ shield at the middle point of the cables(transient current).

Fig. 21. Transmission network used for the test case D.

Fig. 22. 3D plot of voltage profile along phase A of line 2.

Fig. 23. 3D plot of voltage profile along phase C of line 2.

computing internal overvoltages using the EMTP-RV by cas-cading line segments represented by means of the universal linemodel (ULM). This suggests that an EMTP-type program can be

Fig. 24. Comparison between inverse NLT (continuous lines) and ATP/EMTP(dotted lines) for line 2.

Fig. 25. Comparison between inverse NLT (continuous lines) and ATP/EMTP(dotted lines) for line 3.

unreliable in some cases for computing transient voltage profilesalong transmission lines. The method presented in this paperdoes not present this problem and, in consequence, it can ac-commodate large discretization sampling along the line. There-fore, in the following cases, low discretization is considered forthe time-domain programs used for comparison purposes.

B. Transformer Winding

A two-disc, 26-turns transformer (13 turns each disc) is con-sidered for this case. Each turn is 1.5636 m long and the obser-vation time is 0.2 s. Electrical parameters of this arrangementare listed in [17]. An ideal dc source is connected at the begin-ning of the first turn, and the end of the last turn is grounded.The voltage and current profiles along each turn are obtained

as described in Section IV. Figs. 10 and 11 show the voltageprofile along turns 1 and 6 of the winding.For comparison purposes, a simulation is performed using

a two-port model in the frequency domain. In this simulation,the multiconductor line representing the winding is divided intofour segments connected in series to obtain voltages at interiorpoints of the turns. Figs. 12 and 13 show a comparison betweenthe results obtained with the successive application of the in-verse NLT and the results from the analysis in the frequencydomain. This comparison is performed at the beginning and atthe middle point of the second, sixth, and last turn. It can be seenfrom these figures that there is a good level of agreement in allcases.Comparing the results from Fig. 12 with those from Fig. 13,

some differences in the transient behavior at the beginning andat the middle of each turn can be noticed (particularly at turn 6


TABLE ILINES’ LENGTH AND CONDUCTORS’ RADIUS

TABLE IIPOSITION OF THE CONDUCTORS

in this case). It is expected for these kinds of differences to bemore significant for large transformers. Thus, this type of de-tailed analysis along each turn can provide additional informa-tion for insulation design of windings (if the voltage distributionchanges along the turn, the dielectric stress will also change). Itcan also be useful to find exact locations for large interwindingovervoltages [18] or for dealing with windings consisting of alarge amount of turns. In the latter case, a group of turns (a diskfor example) can be treated as a single segment, and the methodpresented in this paper can be applied to approximate the tran-sient behavior inside each segment.

C. Three-Phase Cable Arrangement

A three-phase underground cable arrangement is consideredfor this case. Each cable consists of a copper core, EP insula-tion, lead sheath, and PVC jacket. The cables system is located0.762m below ground, the distance between phases is 0.1524m,and its length is 16093.44 m. Additional geometrical and elec-trical data of this arrangement can be found in [2]. Frequencydependence is considered for the computation of the electricalparameters, as described in [21]. The core of phase A is excitedby means of an ideal unit step on its sending end, while its re-ceiving end is left open. All of the remaining cores and shields(on both sending and receiving ends) are connected to ground.Figs. 14 and 15 show the voltage profiles along the core of phaseA and shield of phase B, respectively. Fig. 16 shows the currentprofile along the shield of phase C. Figs. 17 – 20 show compar-isons between the results obtained by applying the inverse NLTand a simulation performed using the PSCAD/EMTDC. Theseplots correspond to the middle point of the cable.

D. Transmission Network

In order to demonstrate the extension of the method to net-works consisting of several transmission lines, the three-phasetransmission network shown in Fig. 21 is considered. An idealthree-phase sinusoidal source is connected at node 1. There areinductive loads of 0.01 and 0.05 H connected at each phase ofnodes 2 and 4, respectively. The ground resistivity is 100 mand the conductor resistivity is 3.21 m. The observa-tion time is 15 ms. The length and radius of the conductors of

TABLE IIIMEAN RELATIVE DIFFERENCE OF TRANSIENT VOLTAGE FOR CASE A

TABLE IVMEAN RELATIVE DIFFERENCE OF TRANSIENT VOLTAGE FOR CASE B

TABLE VMEAN RELATIVE DIFFERENCE OF TRANSIENT VOLTAGE

AND CURRENT FOR CASE C

TABLE VIMEAN RELATIVE DIFFERENCE OF TRANSIENT VOLTAGE FOR CASE D

each line are listed in Table I; the position of the conductors ofthe lines is shown in Table II. Frequency dependence is consid-ered when computing the electrical parameters of each line.Transient profiles along each line are obtained as described in

Section V. Figs. 22 and 23 show the voltage profile along phasesA and C of line 2. Figs. 24 and 25 show the comparison betweenthe results from the inverse NLT and ATP/EMTP at the middlepoint of lines 2 and 3. Similar to all previous test cases, a highlevel of agreement can be observed.

VII. CONCLUSION

The inverse NLT algorithm has been applied in this paper in apartial and successive manner for the solution of a 2D problem:the propagation along a transmission system.Conversely to the - domain, in which the telegrapher equa-

tions of an -phase transmission line are defined as a set ofPDEs, or the - domain, in which these equations are defined asa set of ODEs, in the - domain, the system is purely algebraicand the solution for voltages and currents is easily obtained byusing the boundary conditions of the line. Moreover, it can ac-commodate the frequency dependence of electrical parameters


in a straightforward manner, which is not the case with time-do-main models.Comparisons with ATP/EMTP and PSCAD/EMTDC results

have shown the effectiveness and accuracy of this approach forthe computation of transient profiles along multiconductor linesand cables. It has also been shown that the method is suitable forthe analysis of internal overvoltages in a transformer winding,providing very detailed profiles of transient voltage distribution.In addition, it has been shown that the method can be easily ex-tended to compute transient profiles along each line of a trans-mission network.Frequency-domain methods have proven in the past to be a

useful alternative to time-domain methods when high accuracyof the solution is required, as well as for the validation and as-sessment of new models, methods, or numerical techniques fortransient calculation.Finally, the method presented in this paper can be particularly

suitable for the accurate computation of transient internal over-voltages along transmission lines, windings, cables, or other dis-tributed parameter components for insulation coordination ordesign purposes.

REFERENCES

[1] R. Radulet, A. Timotin, A. Tugulea, and A. Nica, “The transient re-sponse of electric lines based on the equations with transient line-pa-rameters,” Rev. Roumaine Sci. Technol.-Elect.-chn. Energ., vol. 23, no.1, pp. 3–19, 1978.

[2] P. Gómez and F. A. Uribe, “The numerical Laplace transform:An accu-rate tool for analyzing electromagnetic transients on power system de-vices,” Int. J. Elect. Power Energy Syst., vol. 31, no. 2–3, pp. 116–123,Feb./Mar. 2009.

[3] L. Brancik, “Simulation of multiconductor transmission line circuitscombining 1D and 2D Laplace transformations,” in Proc. 10th Int.Conf. Electron., Circuits Syst., Sharjah, United Arab Emirates, Dec.2003, pp. 774–777.

[4] P. Moreno and A. Ramírez, “Implementation of the numerical Laplacetransform: A review,” IEEE Trans. Power Del., vol. 23, no. 4, pp.2599–2609, Oct. 2008.

[5] C. Gary, “Approche complète de la propagation multifilaire en hautefrequence par utilization des matrices complexe,” EDF Bull. Dir.Études Rech., no. 3/4, pp. 5–20, 1976.

[6] J. D. Hoffman, Numerical Methods for Engineers and Scientists.New York, USA: McGraw-Hill, 1992.

[7] B. Gustavsen and J. Mahseredjian, “Simulation of internal overvolt-ages on transmission lines by an extended method of characteristicsapproach,” IEEE Trans. Power Del., vol. 22, no. 3, pp. 1736–1742,Jul. 2007.

[8] F. A. Uribe, J. L. Naredo, P. Moreno, and L. Guardado, “Electromag-netic transients in underground transmission systems through the nu-merical Laplace transform,” Int. J. Elect. Power Energy Syst., vol. 24,no. 3, pp. 215–221, Mar. 2002.

[9] P. Moreno, P. Gómez, J. L. Naredo, and L. Guardado, “Frequency do-main transient analysis of electrical networks including non-linear con-ditions,” Int. J. Elect. Power Energy Syst., vol. 27, no. 2, pp. 139–146,Feb. 2005.

[10] P. Gómez, P. Moreno, and J. L. Naredo, “Frequency domain tran-sient analysis of non-uniform lines with incident field excitation,” IEEETrans. Power Del., vol. 20, no. 3, pp. 2273–2280, Jul. 2005.

[11] X.M. Lopez-Fernandez and C. Alvarez-Mariño, “Computationmethodfor transients in power transformers with lossy windings,” IEEE Trans.Magn., vol. 45, no. 3, pp. 1863–1866, Mar. 2009.

[12] P. Gómez, F. de León, and I. A. Hernández, “Impulse response analysisof toroidal core distribution transformers for dielectric design,” IEEETrans. Power Del., vol. 26, no. 2, pp. 1231–1238, Apr. 2011.

[13] B. Gustavsen, A. P. Brede, and J. O. Tande, “Multivariate analysisof transformer resonant overvoltages in power stations,” IEEE Trans.Power Del., vol. 26, no. 4, pp. 2563–2572, Oct. 2011.

[14] P. Gómez and J. C. Escamilla, “Frequency domainmodeling of nonuni-form multiconductor lines excited by indirect lightning,” Int. J. Elect.Power Energy Syst., vol. 45, no. 1, pp. 420–426, Feb. 2013.

[15] L. Marti and H. W. Dommel, “Calculation of voltage profiles alongtransmission lines,” IEEE Trans. Power Del., vol. 12, no. 2, pp.993–998, Apr. 1997.

[16] C. Y. Evrenosoglu and A. Abur, “Time series modeling of voltage pro-files along transmission lines,” IEEE Trans. Power Syst., vol. 22, no.1, pp. 172–178, Feb. 2007.

[17] F. J. Quiñónez, P. Moreno, A. Chávez, and J. L. Naredo, “Analysis offast transient overvoltages in transformers windings using the methodof characteristics,” in Proc. 33rd North Amer. Power Symp., CollegeStation, TX, USA, Oct. 15–16, 2001, pp. 521–526.

[18] F. de León, P. Gómez, J. A. Martinez-Velasco, and M. Rioual, “Trans-formers,” in Power System Transients: Parameter Determination, J. A.Martinez-Velasco, Ed. Boca Raton, FL, USA: CRC, 2009, ch. 4, pp.177–250.

[19] J. Martí, “Accurate modelling of frequency-dependent transmissionlines in electromagnetic transient simulations,” IEEE Trans. PowerApp. Syst., vol. PAS-101, no. 1, pp. 147–157, Jan. 1982.

[20] M. M. Kane and S. V. Kulkarni, “MTL-based analysis to distinguishhigh-frequency behavior of interleaved windings in power trans-formers,” IEEE Trans. Power Del., vol. 28, no. 4, pp. 2291–2299, Oct.2013.

[21] B. Gustavsen, T. Noda, J. L. Naredo, F. A. Uribe, and J. A. Martinez-Velasco, “Insulated cables,” in Power System Transients: ParameterDetermination, J. A. Martinez-Velasco, Ed. Boca Raton, FL, USA:CRC, 2009, ch. 3, pp. 137–175.

Rodrigo Nuricumbo-Guillén received the B.Sc. degree in electrical engi-neering from Instituto Tecnológico de Tuxtla Gutiérrez, Tuxtla Gutiérrez,Mexico, in 2009 and the M.Sc. degree in electrical engineering fromSEPI-ESIME Zacatenco, National Polytechnic Institute, Mexico City, Mexico,in 2014.From 2009 to 2011, he was with Instituto de Investigaciones Eléctricas, Cuer-

navaca, Mexico. His research interests are in electromagnetic transient analysisof transmission lines.

Pablo Gómez (S’01–M’07–SM’14) was born in Zapopan, México, in 1978. Hereceived the B.Sc. degree in mechanical and electrical engineering from Uni-versidad Autonoma de Coahuila, Monclova, Mexico, in 1999, and the M.Sc.and Ph.D. degrees in electrical engineering from CINVESTAV, Guadalajara,Mexico, in 2002 and 2005, respectively.Since 2005, he has been a Full-Time Professor with the Electrical Engineering

Department, SEPI-ESIME Zacatenco, National Polytechnic Institute, MexicoCity, Mexico. From 2008 to 2010, he was on postdoctoral leave at the Poly-technic Institute of New York University, Brooklyn, NY, USA. His researchinterests are in modeling and simulation for electromagnetic transient analysisand electromagnetic compatibility.

Fermín P. Espino-Cortés (S’00–M’07) received the B.S. degree in electricalengineering from the Autonomous Metropolitan University, Mexico City,Mexico, in 1995, the M.S. degree in electrical engineering from SEPI-ESIMEZacatenco, National Polytechnic Institute, Mexico City, in 1998, and the Ph.D.degree in electrical engineering from the University of Waterloo, Waterloo,ON, Canada, in 2006.Since 2007, he has been a Full Time Professor with the Electrical Engineering

Department, SEPI-ESIME-Zac, National Polytechnic Institute, Mexico City,Mexico. His research interests are in insulating materials, modeling of high-voltage equipment, and industrial applications of high-voltage engineering.

Felipe A. Uribe (M’96) was born in Guadalajara, Mexico, in 1970. He re-ceived the B.Sc. andM.Sc. degrees in electrical engineering from the Universityof Guadalajara, Guadalajara, Mexico, in 1994 and 1998, respectively, and thePh.D. degree in electrical engineering from the Center for Research and Ad-vanced Studies of Mexico (CINVESTAV), Campus Guadalajara, in 2002.During 2001, he was a Visiting Researcher at the University of British Co-

lumbia, Vancouver, BC, Canada. Currently, he is a Professor at the Universityof Guadalajara. His primary interests are in electromagnetic transient analysisand electromagnetic simulation of propagating waves in biological tissues.

Accurate Computation of Transient Profiles Along Multiconductor Transmission Systems by Means of the...

Documents

Transcript of Accurate Computation of Transient Profiles Along Multiconductor Transmission Systems by Means of the...