calculation of voltage profiles along transmission lines

7/27/2019 Calculation of Voltage Profiles Along Transmission Lines

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CALCULATION OF VOLTAGE PROFILES ALONG TRA NSMISSION LINESL. Marti (M)Ontario HydroCanada

ABSTRACTA method for calculating transient voltages and currents at alarge number of intermediate points along a transmission lineis presented. In the family of EMTP-type programs, these"profile" calculations are normally made by explicitlyconnecting a number of short line sections to createintermediate nodes where voltages and currents can bemonitored. This approach is time-consuming, computationallyintensive and can be inaccurate if the number of intermediatenodes is large. The model presented here allows the accurateevaluation of voltages and currents at an arbitrary number ofequally-spaced points along the line. The frequencydependence of the line parameters is taken into account. It iscomputationally fast and easy to use.

1. INTRODUCTIONSwitching surge voltage transients are an important factor ininsulation coordination studies of overhead transmission lines.In such studies, the overvoltages are normally obtained at thesending and receiving ends of a line. In most instances,overvoltages are most severe at the ends of the line, due tothe combined effects of incident and reflected waves. Thereare situations, however, where overvoltages at intermediatepoints along the line are higher than those at the ends of theline. In such cases, it becomes necessary to calculate a"profile" of voltages as a function of distance, to determinethe maximum overvoltages, as well as their location along theline. Such a case arises if metal oxide surge arresters are atthe ends of the line, instead of pre-insertion resistors in orderto limit switching surges in extra high voltage lines [ I ] , [2],[7], [ l 11. During line energization or re-energization, a metaloxide arrester effectively limits the overvoltage at theprotected end of the line below a fixed value. This moves thelocation of the maximum overvoltage away from the end ofthe line, which makes i t necessary to calculate the voltageprofile along the line.Profile calculations are also needed to find the sheathovervoltages in high voltage cables between solidly groundedpoints, or between points grounded through metal oxide

0-7803-3522-8196 $5.000 1996 IEEE

H.W. Dommel (F)The University of British ColumbiaCanada

arresters. Another application is the calculation of voltagesinduced from adjacent energized lines along intermediatepoints along grounded lines which have been taken out ofservice for maintenance. Profile calculations have also beenused to produce travelling wave movies for educationalpurposes [3,4,5]. hese movies show transient phenomena asa function of time and space.The traditional way for obtaining voltage profiles in programssuch as the EMTP is to calculate the voltages at intermediatenodes, which are created by connecting a number of shorterline segments together [ I l l . This is time consuming,especially when frequency dependent line models are used.If the line is broken down into a large number of sections, theaccumulation of errors degrades the accuracy of thesimulation.Another disadvantage of the traditional method for calculatingprofiles is the fact that the step size of a transient simulationmust be smaller than the travel time of the shortest distributedparameter line. If the time step has to be reduced toaccommodate the shorter line segments, CPU time, as well asstorage requirements for past history terms for all the lineswould increase.The method for profile calculation presented here addressesthe major disadvantages of the traditional method. It iscomputationally faster, more accurate, and it does not tie thestep size of the entire simulation to a value that depends onthe spatial resolution of the profile. Since it is based on theJMARTI [ 6 ] ine model, it takes the frequency dependence ofthe line parameters into account.This method was first implemented in 1982 as the FrequencyDependent Profile Model "FDPROFILE" in the University ofBritish Columbia version of the EMTP. In thisimplementation, FDPROFILE calculations were imbedded inthe time step loop of the EMTP. Because of the recentinterest in profile calculations, the original implementation forbalanced lines has been extended to handle untransposed ines.A "stand-alone" version of the FDPROFILE program, whichis independent of the particular version of the EMTP used, isunder development.

2. CALCULATION OF VOLTAGES AND CURRENTSAT INTERMEDIATE POINTSThe FDPROFILE model needs the voltages and currents atthe endpoints of the line at each time step of the simulationas input parameters. These voltages and currents at the

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terminals of the line must be calculated with a program suchas the EMTP with a host line model. Because theFDPROFILE calculations take the frequency dependence ofthe line parameters into account, the host line model shouldalso take frequency dependence into account, in order toproduce consistent results. The preferred host model is theJMARTI line model of the EMTP, since it is based on similarmathematical representations as the FDPROFILE model.In the calculation of the voltages and currents at N equally-spaced points along the line, it is assumed that the modaltransformation matrix [Q], which diagonalizes the matrixproduct [Yphue]Zphase],s real and constant. Phase voltagesand currents are converted into modal voltages and currentswith the linear transformation defined by

With this transformation, an n-phase line is then analyzed asa set of n single-phase lines in the modal domain. After thesolution in the modal domain has been obtained, phasevoltages and currents are found from the inverse relationshipsof equations (1) and ( 2 ) . The assumption that [Q] is constantis only strictly correct when the line is balanced or perfectlytransposed; otherwise, it is an approximation. The accuracyof this approximation depends on the geometry of the tower,the number of circuitsltowers on the same right-of-way, andthe type of surges [9]. All overhead transmission line modelspresently implemented in the EMTP assume that [Q] isconstant and real. There is an EMTP model which takes intoaccount the frequency dependence of [Q] in directly buriedcables [IO], but this model has not yet been extended tooverhead transmission lines, though work on this extension isin progress.Consider then, the single-phase line or a single mode of amultiphase line of length d shown in Figure 1. Let xi(j=1,2, ..,N) be the distance from the sending end, where NVoltages and currents are to be calculated at equally spacedpoints. Therefore,

xj = j -Ax ( 3 )d

N + lAX =-v vi r t T f

(4)

...........................I

I I I I cXx2 ... xj ... IX i

In the frequency domain, the voltages and currents at a pointx are given by

V, + ZcZ, = F, = A F , _ , , (5)

V, - ZcZ, = B, = A B , + , , (6)

where Z, is the characteristic impedance and A is thepropagation function for a line segment of length Ax . F, andB, are defined as the forward and backward travellingfunctions. Addition and subtraction of (5) and (6) produces(9 )V, = F, + B,

2 Z c I x = F, - B, (10)Let F,, and B, be the forward and backward travellingfunctions evaluated at the sending and receiving ends of theline; that is, F,, = F,,, and B, = Bxd. f the solution of theline at the endpoints is known, F, and B, are either abyproduct of the solution process of the host model, or can becalculated from the voltages and currents using equations ( 5 )and (6); that is,

(1 1), + ZcId = FdVo - ZcIo = Bo (12)

F, and B, can then be calculated from F,)and B, withFA, = A F ,

B j A x = A B ( f + l ) A x (j = N-1, N - 2 , ..., 1)from which the intermediate voltages and currents can bereadily obtained using equations (9) and (10).

3. TIME DOMAIN CALCULATIONS

Fig. I . Single-phase transmission ine.264


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The propagation function for a line segment Ax isapproximated with rational functions, of the form

where the poles and zeroes of (15) are real and lie in the righthand side of the complex plane, and 7 is the travel time ofthe fastest frequency component which propagates along theline. Therefore, the convolutions in (14) can be expressednumerically asm

&A.&t) = C k $ A x , k ( t - A t ) + d k f u - l ) A x , k ( t - t ) (16)k = l

where ck, d,, and ek are constants which depend on the typeof integration rule used. Intermediate voltages at time t arethen obtained directly from

To calculate intermediate currents, it is also necessary toapproximate he characteristic impedance as rational functions

where the poles and zeroes of (19) are real and lie in the righthand side of the complex plane. From equation (lo), theintermediate currents in the time domain can be calculatedfrom

e,@) = zeqi,(t) + h,( t-At)

where zq is constant and h,(t-At) depends on past historyvalues of e,(t) and i,(t) [6].In the solution of equations (16), (17), and (22), an "internal"step size At is used. This step size is chosen so that the traveltime T of a line section of length Ax becomes an integermultiple of At, regardless of the time step At' used by the hostmodel. The use of an internal step size for profilecalculations reduces storage and CPU requirements. In the

EMTP, the step size of the entire transient simulation must besmaller than the travel time of the shortest distributedparameter line in the system. When short line segments withtravel times 7 smaller than At' are used to calculateintermediate voltages, the step size must be decreased by afactor of TIAt', and execution time increases accordingly.Storage requirements for the past history terms of alldistributed parameter lines in the system will also increase bythe same amount. With the FDPROFILE model, the globaltime step of the entire simulation is unaffected by the numberof intermediate voltages, and the storage requirements of otherlines in the system are unchanged.To evaluate equations (16) and (17) at time t, past historyvalues of f, and b, must be known at t-7-At. If dAt is notan integer number, f,(t-z-At) and b,(t-z-At) have to becalculated by interpolation of recorded values at t-nAt and t-(n+l)At, where (n+l)At > T > nAt. If the magnitude of At iscomparable to the magnitude of 2, interpolation errorsbecome significant. In the case of multiphase line models inthe EMTP, it is not possible to choose At' such that dAt' isan integer for all modes and for all lines. In the FDPROFILEmodel, however, 7IAt is always an integer for all modes.This reduces interpolation errors in the evaluation of modalvoltages considerably . On the other hand, a different At forthe solution of each mode results in voltages evaluated atslightly different points in time, and linear interpolation isneeded to calculate phase voltages with equations (1) and (2).Since phase voltages and currents are output quantities notused in any additional FDPROFILE calculations, the amountof error due to these interpolations is small and does notaccumulate from time step to time step.The FDPROFILE model is relatively insensitive to the qualityof the rational functions approximation of the propagationfunction A. Since the solution of the line at the end points isprovided at each time step by the host line model, profilecalculations are always local to that time step. In otherwords, errors in FDPROFILE calculations do not propagatefrom time step to time step.When the propagation function of a line segment of length Axis used in FDPROFILE calculations or when N+l linesegments are modelled explicitly, the following assumptionsare implicit

While errors in the approximation of A(o) re normally verysmall, for large N, the effective length and the attenuation ofthe line can change significantly. For example, for thedouble-circuit line used in the examples shown in this paper,the zero sequence value of z for a 50 km section is 0.287 ms,while T for the 500 km line is 3.17 ms. This means that ten50 km sections do not represent exactly a 500 km line, butrather a line that is 10% shorter for zero sequencecalculations. Figure 2 shows the errors in the approximation26 5


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i5 . 0 0 N-point profile - 3 N + 4N + l line sections 4 N + 4-~=

i5 . 0 0T r ~ T r l T r l .'1"12 ' A ' ' T 2 Id " i T 2 d " i 6 2 ' d i o 2 4 i o

Frequency (Hz)

Fig. 2: Relative error fo r the approximation ofA(o);sol id:ten 5 km sections: dashed: one 50 km section.of the magnitude of A(o) for a 50 km segment (dashed trace)and the effective errors of ten 50 km segments (solid line).Even though the approximation of A(w) is remarkably good,magnitude errors accumulate exponentially and errors in theevaluation of T accumulate linearly with N. While theFDPROFILE model is also affected by the accuracy of theapproximation of A(w), errors do not accumulate from timestep to time step. Also, since the calculation of F, and B,start at opposite ends of the line, there is an additionalaveraging effect; that is, errors are distributed more or lessuniformly through all intermediate points.

3. IMPLEMENTATION IN THE EMTPIn the implementation of the FDPROFILE model in the UBCversion of the EMTP, the intermediate voltages and currentsare calculated inside the time step loop of the EMTP. Sinceat any given time step f,(t) and bd(t) are intermediate variablesreadily available from the JMARTI line model, equation (18)can be used directly. A quantitative estimate of the relativecomputational speed of the FDPROFILE routines can beobtained from the number of numerical convolutions evaluatedper time step. Assuming that the order of the approximationsof A(o) and Z,(o) is roughly the same and that onlyintermediate voltages are requested, then the performance ratiois

N-point projile - 2 N + 4N+1 line sections 4 N + 4--r =

In the explicit connection of N+l sections there is oneconvolution with a(t) and one convolution with y,(t) for bothsending and receiving ends, for a total of 4 convolutions persection. In FDPROFILE calculations for voltages, 4convolutions account for the solution at the endpoints, andthere are 2 convolutions with a(t) for each intermediatevoltage. If the intermediate currents are also requested, thenthere is the additional overhead of one convolution with z,(t)per sectionFor N=10, the performance ratio for a 10-point profile is 0.54if only voltages are requested, and 0.77 if voltages and

currents are requested (assuming that the simulation time stepis smaller than the travel time of the shortest line segment).Actual execution speed is much higher than what these ratiosindicate. Depending on the amount of output requested, andsystem configuration, a simulation with the FDPROFILEmodel can be 2 to 8 times faster than a simulation withexplicitly segmented lines.

4. IMPLEMENTA TION AS A POST-PROCESSORThe main advantages of implementing the FDPROFILE modelinside the EMTP time step loop are computational speed andsimplicity of the interface. Since the characteristic impedanceis the same for the FDPROFILE model and the host model,it does not have to be approximated again. The onlyadditional information required, other than the number ofintermediate points, is the approximation of A(o) for a linesegment of length Ax. On the other hand, profile calculationsrequire extra memory for storage of past history terms, andmanagement of the large amount of output quantitiesproduced. In versions of the EMTP where memory is limited,it makes sense to move the profile calculations to a post-processing program. A post-processing program also has theadvantage that it can be used in more than one version of theEMTP.A post-processing version of the FDPROFILE model is underdevelopment. In order to be compatible with most versionsof the EMTP and, possibly with host models other thanJMARTI, sending and receiving end voltages and currents ateach time step are required. To initialize past history termsin (16), (17) and (22), steady-state conditions at the endpointsprior to time zero must also be known. From the voltagesand currents at the endpoints, f,(t) and bd(t) are calculatedfrom equations (9), (10) and (21)

f J t ) = vd(t ) + ze q d@ ) + hd( t -At ) (27)b,(t) = v,(t) - z,i ,(t) - h,( t -At)

Both types of implementation are equally accurate as long asvoltages and currents are known with full precision. Theimplementation as a post-processor will be somewhat slowerthan its EMTP-based counterpart because of the additionalconvolutions for calculating f,,(t) and bd(t). However, theadvantages in terms of compatibility justify the small loss inperformance.

5. SIMULATION RESULTSTo illustrate the differences between the FDPROFILE modeland conventional profile calculations, the energization of adouble-circuit line has been simulated using the EMTP-basedimplementation. Figure 3 shows the one-line diagram of the

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2 . ooT-m

I Y& &- --Fig. 3: Test circuit for line energization with arresters

test system. One three-phase circuit is energized, while theother circuit is grounded at both ends. The energized circuitis terminated with surge arresters to limit voltages to 1.50p.u.The line is 500km long, and voltages are calculated at 50 kmintervals. Figure 4 shows the voltages measured 250km fromthe source when ten 50 km line segments are connected toobtain the intermediate voltages (solid trace). The referencesolution has been obtained by connecting only two 250 kmsections (dashed trace). Figure 5 shows the results obtainedwith a nine-point FDPROFILE model. It can be seen fromthis simulation that the FDPROFILE is virtually identical tothe reference simulation solution. Figure 6 shows the profileof maximum overvoltages for both the energized and thegrounded circuit. It can be seen from these results that even

Time (ms)

2.00, I1.

1.

m O .

2mul

- 0 .-1.-1.- 2 . 0 0 : . , . . , I , . I , . . , , , , , , , , , , 1

40.0 42.0 44.0 46.0 48.0 50.Time (ms)

Fig 4: Midpoint voltages, energization with arrester.Solid: IO-section model; dashed: two 250 kmsections.

1 . 0 03

0.50:0 -

- 0 . 5 0 --1.00-

/ I- Z . O O l , 1 . I , I . I , , , , , I , , , , , , , ,\, 10 2 . 0 4 . 0 6.0 8 . 0 l o . (Time (ms)

2.00,

-1.5c-j \ /v- 2 . 0 0 , 1 1 . I I . . I I I , , , , , , , , , , , , ,

4 0 . 0 4 2 . 0 4 4 . 0 4 6 . 0 4 8 . 0 5 . (Time (ms)

Fig 5: Midpoint voltages, energization with arrester.Solid: FDPROFILE model; dashed: two 250 kmsections.2 . 5 0 h

0 100.0 200.0 300.0 400.0 500.0Distance (Ian)

Fig 6: Profile of maximum overvoltagesin a 10-segment case, the FDPROFILE model produces moreaccurate results that the traditional approach. Errors in thetraditional approach become more significant at higherfrequencies (see Figure 2) and when the number ofintermediate points increases. For this particular example, theFDPROFILE simulation was only 20% faster than thetraditional approach. However, gains in computational speedalso increase with the number of intermediate points and thecomplexity of the rest of the system.

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6. CONCLUSIONSThis paper describes the calculation of voltage and currentprofiles on transmission lines when the solution at theendpoints is known. The calculation of the intermediatevoltages and currents is done in the modal domain, under theassumption that the modal transformation matrix [Q] is realand constant. The frequency dependence of the lineparameters is taken into account.The FDPROFILE model is computationally fast and easy touse. It offers an accurate alternative to the time consumingand computationally slow process of subdividing a line intosmaller segments. A stand-alone post-processing versionwhich is compatible with most versions of the EMTP is underdevelopment.

REFERENCES[I] A.C. Legate, J.H Brunke, J.J. Ray, E.J. Yasuda, "Eliminationof Closing Resistors of EHV Circuit Breakers". IEEETransactions on Power D elivery, vol. 3, no. 1, January 1989,pp. 223-231.[2] J.H. Brunke, "Application of Metal Oxide Arresters for theControl of Line Switching Transients". InsulationCoordination Seminar, Canadian Electric AssociationMeeting. Toronto, Ontario, May 1990.131 L.F. Woodruff, "Transmission Line Transients in MotionMovies". Transactions AIEE , vol. 57, July 1938, pp. 391-400.[4] L. Marti., "Voltage and Current Profiles and Low-orderApproximation of Frequency-dependent Transmission LineParameters". Department of Electrical Engineering, The

University of British Columbia. MASc. Thesis, April 1982,pp. 22-27.[5] S.J. Salon, "An Interactive Package for Electric PowerEngineering Education". IEEE Transactions on PowerApparatus and Systems, January 1982, pp. 147-157.[6] J.R. Marti, "Accurate Modelling of Frequency-DependentTransmission Lines in Electromagnetic TransientSimulations". IEEE Transactions PAS-101,January 1982, pp.147-155.[7] R.S. Tallam, T.G. Lundquist, S.R. Atmuri, D.A. Selim,"Design Studies for the Mead-Phoenix 500 kV acTransmission Project", IEEE Transactions Power System

Delivery,,Vol. 10, No. 4, pp.1862-1874, October 1995.[8] A. Semlyen and A. Dabuleanu "Fast and Accurate SwitchingTransient Calculations on Transmission Lines with GroundRetum Using Recursive Convolutions". IEEE Transactions,PAS-94, MarchlApril 1975, pp. 561-571.[9] H.W. Dommel, J.R. Marti, L. Marti, V. Brandwajn,"Approximate transformation matrices for unbalancedtransmission lines". Ninth Power Systems ComputationConference, Cascais, Portugal August-September 1987, pp.416-424.

[101 L. Marti, "Simulation of Transients in UndergroundCables with Frequency-Dependent ModalTransformation Matrices". IEEE Transactions onPower Delivery, vol. 3 , no. 3, July 1988, pp. 1099-1110.[I l l J.R. Ribeiro, M.E. McCallam, "An Application of MetalOxide Surge Arresters in the Elimination of Need forClosing Resistors in EHV Breakers", IEEE Transactions onPower Delivery, Vol. 4, No. 1, January 1989.

BIOGRAPHIESLuis Marti (M'79) received an undergraduate degree in ElectricalEngineering from the Central University of Venezuela in 1979,MASc and PhD degrees in Electrical Engineering in 1983 and1987, respectively, from The University of British Columbia. Hedid post-doctoral work in cable modelling in 1987-1988, andjoined Ontario Hydro in 1989, where he is currently working inthe Technical Support Department - Grid System Strategies &Plans Division.Hermann W. Dommel (F '79) was born in Germany in 1933. Hereceived the Dipl. Ing and Dr. Ing. degrees in electricalengineering from the Technical University Munich, Germany in1959 and 1962, respectively. From 1959 to 1966 he was withthe Technical University Munich , and from 1966 to 1973 withBonneville Power Administration ,Portland, Oregon. Since July1973 he has been with the University of British Columbia inVancouver, Canada. Dr. Dommel is a Fellow of IEEE and aregistered professional engineer in British Columbia, Canada

APPENDIXThe tower data for the transmission line an d simulationdata used in the simulation described in Figure 3 is asfollows:Voltage class 230 kV, double circuit.Number of conductoribundle = 3Separ atio n of conductors in bundle = 18"dc resista nce of main ACSR conductors = 0.05215 a m iDiameter = 1.602" (thicknesddiameter = 0.3636)dc resistance of (solid) ground w ires = 2.65 WmiDiameter of ground wires 1.602".Earth resistivity 100 SZm

cond I x(ft) I y(ft )1 I -18.25 I 120.

The source impedance used in Figure 3 is 10 SZ. Th earrester in Figure 3 is modelled as a nonlinear resistancewith a simple two-slope characteristic (-,.005 Q) , andVsat = 1.5.268

calculation of voltage profiles along transmission lines

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