TRANSMISSION LINE MODELING FOR REAL-TIME SIMULATIONS
TRANSMISSION LINE MODELING FORREAL-TIME SIMULATIONSMaria Isabel Silva Lafaia SimõesInstituto Superior TécnicoAv. Rovisco Pais, 1049-001 Lisboa, PortugalEmail: maria.simoes@ist.utl.ptAbstract—Real-time simulation of power systems transientsis an important tool when there is a need to include physicalelements in the system under study, rather than theirmathematical models. Real-time is hard to achieve in digitalsimulations, where accuracy runs oppositely to processingspeed. It is therefore necessary to combine parallel processingwith efficient numerical techniques for model computation.Transmission lines allow parallel processing in power systemsstudies, by dividing large networks into smaller independentsubnetworks. The need to account for frequency dependentparameters, poses a challenge on the definition of an adequateline model. The goal of this work is to establish adequatenumerical techniques for approximating the propagationparameters for transmission line modeling, allowing real-timesimulations.with examples in real-time modeling literature, namely [8],[9]. The WB Line application computed by the EMTP-RV 2.3is also considered and taken as a reference of accuracy.Keywords: Real-time simulations, transmission line modeling,frequency dependent line model, RT WB Line, optimized modaldelay computation, optimized modal poles assignment.1) CP Line – constant parameters line model: this linemodel is based on the work by Dommel [2]. It is based onmodal analysis and each of the n line modes is characterizedin terms of the corresponding characteristic admittance Ȳc andpropagation function H̄, which are computed through a realconstant transformation matrix. The model approximates theline as an ideal lossless line (R 0, Geachp 0). Therefore, jωτmodal function simply becomesȲ C/LandH̄ e,c where C, L and τ d LC are the capacitance, inductanceand propagation delay of the corresponding mode. This modelcan include the effect of losses by inserting constant lumpedresistances in discrete points of the line.2) FD Line – frequency dependent line model: the FDLine is based on the work by J. Marti[3]. This model isbased on the characteristic impedance Z̄c and propagationfunction H̄ of each mode. These functions are computed fromthe line parameters in phase domain, using a real constanttransformation matrix. For each line mode, the characteristicimpedance and propagation function are approximated throughAsymptotic Fitting[3] in the s-domain, as:I. I NTRODUCTIONReal-time simulation of power systems transients is an important tool when there is a need to include physical elementsin the system under study, rather than their mathematicalmodels. However, the real-time is hard to achieve in digitalsimulations, where accuracy runs oppositely to processingspeed. It is therefore necessary to combine parallel processingwith efficient numerical techniques for model computation.Transmission lines play an important role for allowing parallel processing in power systems studies, by dividing largenetworks into smaller independent subnetworks.The accurate representation of a transmission line requiresthe use of its frequency dependent parameters. This poses achallenge on the definition of an adequate line model. Thegoal of this work is to establish adequate numerical techniquesfor approximating the propagation parameters for transmissionline modeling, allowing real-time simulations.The study of existing line models provides the basis for thedevelopment of the RT WB Line, which is a reformulationof the EMTP-RV model WB Line (based on the UniversalModel [1]), in-line with the real-time simulation target. Toensure additional accuracy with reduced fitting resources, twooptimizations are suggested, concerning the computation ofthe modal delays and the assignment of the modal poles.The validation of the developed model consists of tests infrequency and time domain conditions and uses an applicationof the RT WB Line which order of approximations is in-lineII. EMTP-RV AND TRANSMISSION LINE MODELINGThe EMTP-RV 2.3 is a specialized software for the simulation of electromagnetic, electromechanical and control systemstransients in multiphase power systems. The transmission linemodels available in the EMTP-RV provide a summary of theevolution of line modeling.A. EMTP-RV models reviewH̄(s) NzXkxs pxx 1!NhXkye sτmins pyy 1Z̄c (s) k0 (1)(2)where Nz and Nh are the number of poles used to approximatethe corresponding modal functions and τmin is the minimumpropagation delay of the mode.
3) WB Line – wide-band line model: this model is basedon the Universal Line Model, introduced in a work by Gustavsen et al.[1]. The WB Line describes the line in phasedomain through the frequency dependent matrices Ȳc andH̄. The characteristic admittance matrix Ȳc is fitted columnby-column through Vector Fitting[4]. The elements of thepropagation matrix H̄ are all approximated with the samepoles and delays, defined by the modes:NyXkxjω pxx 1!NnkXXcmkije jωτkH̄ij (ω) jω pmkm 1Ȳcij (ω) k0 (3)Fig. 2: Open-end frequency response according to theEMTP-RV line models. Voltage at the receiving end of phase1 (CP Line – thin, FD Line – dotted, WB Line – bold).(4)k 1As regards the short-circuit scan, all line models generateinaccurate values for low frequencies. However, the approximations are more accurate for higher frequencies, except forthe CP Line which is generally very inaccurate. An exceptionis observed near the 1 kHz, for which all models agree.This is the frequency used by EMTP-RV to compute thetransformation matrix of all models and the parameters of theCP Line. Concerning the open-end condition, the FD Lineand the WB Line provide very accurate results for all therange observed – 0.1 Hz to 1 MHz. The CP Line is also veryaccurate, except for frequencies beyond the 2 kHz.2) Line energization and single-phase short-circuit: Thistest concerns two common transient conditions: line energization and single phase short-circuit. Figure 11 illustrates thecircuit used, where the line described in appendix is connectedthrough ideal switches to a three-phase ideal source of 1Vpeak voltage and 50 Hz. The three phases are connected tothe source simultaneously at t 20 milliseconds. After thetransient of line energization, a short-circuit occurs at t 180milliseconds in phase 3 of the line. The voltage at the receivingend of phase 1 is observed. The results of this test are plottedin figures 3 and 4, which represent only the CP Line and theWB Line approximations. This is done for simplicity, giventhe FD Line generates practically the same results as the WBLine.where: n is the number of line modes,Ny is the number of poles to fit the j th column of Ȳc ,Nk is the number of poles defined by the k th modeτk is the minimum delay defined by the k th mode.The poles of Ȳc are generally real, whereas those of H̄may be real or complex. The modal poles and delays thatapproximate H̄ are obtained by applying Vector Fitting[4]to each modal propagation function. The residues cmkij arecomputed from a set of samples of H̄, by solving a linearleast squares problem.B. EMTP-RV line models testingThis section presents the major conclusions of a set of teststo compare the line models provided by the EMTP-RV 2.3.The line represented by the models throughout these tests isdescribed in appendix.1) Frequency scans: The first test consists of frequencyscans in short-circuit and open-end conditions, plotted infigures 1 and 2.Fig. 3: Response to line energization at t 20 ms accordingto the CP Line – thin, and to the WB Line – bold, in termsof the voltage at the receiving end of phase 1.Fig. 1: Short-circuit frequency response according to theEMTP-RV line models. Current at the sending end of phase1 (CP Line – thin, FD Line – dotted, WB Line – bold).2
Fig. 4: Response to a short-circuit at the receiving end ofphase 3 at t 180 ms according to CP Line – thin, and tothe WB Line – bold, in terms of the voltage at the receivingend of phase 1.Fig. 6: Current at the sending end of phase 3 induced byenergization of phase 1, during the first second of thetransient according to the EMTP-RV line models (CP Line –thin, FD Line – dashed, WB Line – bold).The time responses according to the FD Line and the WBLine show very good agreement for both transient conditions.The CP Line, however, generates inaccurate results, speciallyin approximating the energization transient, by clearly neglecting the distortion phenomena. The attenuation is anotherinaccurate aspect in the time response of the CP Line.C. Notes on the EMTP-RV models performanceThe tests have showed that the FD Line and the WB Lineare adequate models to simulate typical transient conditions,such as line energization and simple short-circuit or open-endconditions. On the other hand, the CP Line provides inaccurateresults for all the tests, and should therefore be avoided intransient studies. Though the FD Line and the WB Line providevery similar results in many conditions, there are some criticaltests in which only the WB Line succeeds in providing resultsphysically acceptable, as the test of induced currents shows.Furthermore, the WB Line uses a total of 11 21 32 polesto fit Ȳc and H̄ compared to the 51 69 120 poles used bythe FD Line to fit Z̄c and H̄. Therefore, the WB Line presentsa great improvement in efficiency, by obtaining better resultswith less resources. The efficiency of the WB Line is reinforcedby the fact of being a phase-domain model – there is no needto convert from phase to modal quantities, and vice-versa, ateach simulation step.3) Current induced by phase coupling: This is a test inwhich all models show significantly different results. Considerfigure 14, where the transmission line is short-circuited atall terminals except at the sending end of phase 1, which isconnected to a DC voltage source of 1V, at time t 1 ms,by an ideal switch. The current induced at the sending end ofphase 3, ik3 (t), is observed. The results are plotted in figures5 and 6. The FD Line and the WB Line agree only for thevery initial period of the transient. An important aspect in thistime response is that the induced current must decline to zeroin steady-state, due to extinction of the coupling phenomena.The FD Line and, particularly, the CP Line are very inaccuratein approximating this behavior. Therefore, the WB Line is theonly model of the test providing results physically acceptable.III. W IDE - BAND MODEL FOR REAL - TIME SIMULATIONSThe developed line model RT WB Line is a reformulation ofthe EMTP-RV model WB Line (based on the universal model[1]), in-line with the real-time simulation target. Therefore, itis a phase domain model based on the rational approximationsof Ȳc and H̄ using the scheme described through equations(3) and (4). The following text presents two optimizationprocedures, concerning the computation of the modal delaysand the assignment of the modal poles, which are introducedin the developed model to ensure additional accuracy withreduced fitting resources.A. Optimal modal delay identificationConsider the propagation function of the k th line mode:Fig. 5: Current at the sending end of phase 3 induced byenergization of phase 1, during the first 20 miliseconds ofthe transient according to the EMTP-RV line models (CPLine – thin, FD Line – dashed, WB Line – bold).H̄ k (ω) e (αk (ω)d jωτk (ω))(5)where τk (ω) is the propagation delay of the k th mode. Eachfunction H̄ k (ω) may be approximated by a rational function3
0TABLE I: Effect of optimized assignement of modal poleson the average error of approximation of H̄, according todifferent order applications of the RT WB Lineand a constant time delay factor e jωτk :k0kH̄ (ω) H̄ (ω)e jωτk0 NkXcnjω pnn 1!0e jωτk(6)Total poles69121518where Nk is the number of poles used to fit the k th mode.The extraction of a constant propagation delay allows a moreaccurate fitting for the same number of poles.The constant delay τk0 may be computed so that H̄ 0k (ω)becomes approximately a minimum phase shift function. According to [10], this is done through: 1 π d ln H̄ k (ω1 ) 0(7)τk τk (ω) ω1 ωω 2 d ln ω1Distributionof poles[2 3 1][4 4 1][5 0 7][5 1 9][8 1 9]Improvement on theapproximation error 28% 6% 55% 43% 15%As table I shows, the optimized assignment of modal poleshas a positive impact on the approximating propagation matrix,with a reduction of at least 6 % in relation to the errorobtained with equal distribution of modal poles, for most ofthe orders tested for the RT WB Line. A deeper look to tableI shows that, for a low number of approximating poles, theoptimized distribution tends to assign more poles to modes1 and 2, whereas for a higher approximating order, modes 1and 3 are preferred. Nevertheless, only in exceptional casesone mode is neglected, being assigned zero fitting poles, asshowed in table I for a total of 12 poles. Therefore, thereis not an explicit tendency of optimal pole distribution thatallows to define a single strategy that works both for low andhigh orders of approximation. As a consequence, in order toachieve the optimal approximation of the propagation matrixH̄, the developed procedure tries all the possible assignementsof modal poles. This is not the most efficient process, but itcertainly reaches the most accurate result, based on the averageerror of the fitting matrix H̄.computed for a frequency ω such that H̄ k (ω) 0.1.However, according to [10], the computed delay may notcorrespond to the most accurate fitting of H̄ k (ω). This motivates the construction of a routine to compute the modaldelays leading to the most accurate approximation as definedin (4). Tests involving different lines have showed that a goodestimation for τk0 can be found in the interval from 0.9 τk0 to1.1 τk0 , where τk0 is given by (7). Tests have further shownthat generally it is enough to search with an iteration of 1%of the base modal delay.Though the accuracy in the approximated modes is notdirectly related to the phase domain results [10], tests haveshowed that, for certain orders of the approximating modalfunctions, the use of optimized modal delays leads to improvedaccuracy both in the modal and phase domain, when comparedto the approximating functions obtained by simply using thelossless modal delays. Nevertheless, it must be noted thatthe improvements introduced by this process vary with theparticular line and with the order of the model.IV. RT WB L INE MODEL VALIDATIONThe validation of the developed model consists of frequency and time domain simulations in the EMTP-RV 2.3environment, using an application of the RT WB Line, whichperformance is compared to that of the WB Line, computedby the EMTP-RV and taken as a reference of accuracy. Theapplication of the WB Line uses 21 11 32 poles to fitH̄ and Ȳc , whereas the RT WB Line application uses only9 9 18 poles, respectively. The total number of poles usedby this application is in-line with the examples in real-timeline modeling literature, namely [8], [9].B. Optimal modal poles assignmentThe developed model approximates the propagation matrixusing the poles and delays defined by the modes, as expressedin (4). In order to respect the pre-defined order of the model,the sum of the poles assigned to each mode, Nk , must beequalP to the maximum number of poles allowed for H̄, thatis,Nk Nmax . Practical tests have showed that assigningan equal number of poles to each mode generally does not leadto the most accurate fitting of H̄. Therefore, it is advantageousto optimize the number of poles assigned to each modalpropagation function. This is done by trying all the possibledistributions of the number of poles among the modes, withthe requirement that the total number of poles must respectthe pre-defined order of the model. The following numericalexample illustrates and justifies this procedure.1) Numerical example: Consider the line described in appendix. Table I shows the decrease in the average approximation error thanks to optimizing the number of poles assignedto each mode, for several model orders.A. Frequency response – short-circuited and open-ended lineThis test evaluates the accuracy of the line models by comparing their frequency responses with the expected behavior,according to analytical expressions computed with the exactline functions, which are rewritten, respectively, as: 1 Īk short Ȳc I H̄2I H̄2 V̄source(8) 1V̄m open I H̄2H̄ 2V̄source(9)where Vsource is the phasor representing the 1V–RMS, threephase symmetrical source feeding the line. Given the symmetry of the problem, it is sufficient to analyze one phase of theline (phase 1 was chosen).4
Figure 7 plots the relative error of the approximating shortcircuit responses according to the WB Line and RT WB Lineapplications. The approximating responses, specially in whatconcerns the developed model, are inaccurate for the rangeof frequencies up to 50 Hz. However, for higher frequencies,both models provide accurate approximations, except whenapproximating the various peaks in the expected frequencyresponse. Figure 8 shows a zoom of the relative errors ofthe approximating responses from 700 Hz to 10 kHz, whichincludes the switching transients frequencies. For this range,the RT WB Line is more accurate than the WB Line, despiteusing lower order approximations.700 Hz to 10 kHz. As verified in the short-circuit scan, theRT WB Line is more accurate than the WB Line for this rangeof frequencies.Fig. 9: Relative error of the open-end frequency response(100 Hz - 1 MHz) – WB Line (bold) and RT WB Line(thin).Voltage at the receiving end of phase 1.Fig. 7: Relative error of the short-circuit frequency response(0.1 Hz - 1 MHz) – WB Line (bold) and RT WB Line (thin).Current at the sending end of phase 1.Fig. 10: Detailed relative error of the open-end frequencyresponse (700 Hz - 10 kHz) – WB Line (bold) and RT WBLine (thin). Voltage at the receiving end of phase 1.B. Line energization and single phase short-circuitThis test evaluates how the models represent the behavior ofthe line under typical transient conditions. Figure 11 illustratesthe circuit used, where the line is connected by ideal switchesto a three phase symmetrical source of 1 V peak voltage and50 Hz. The line energization occurs at t 20 ms, with theclosure of the switches connecting the line to the source. Afterreaching steady-state, a new transient is originated, at t 180ms, by closing the receiving end switch (short-circuit on phase3). The voltage at the receiving end of phase 1 is observed. Theresults of this test are plotted in figures 12, which concernsthe energization transient, and 13 regarding the short-circuitof phase 3.Both figures show very good agreement between the twomodels performance. A difference is perceived only in theenergization condition, where the approximating line responseaccording to the RT WB Line denotes a slight weaker attenuation of the voltage at the receiving end of phase 1.Fig. 8: Detailed relative error of the short-circuit frequencyresponse (700 Hz - 10 kHz) – WB Line (bold) and RT WBLine (thin). Current at the sending end of phase 1.Figure 9 plots the relative errors of the approximatingopen-end responses, and it shows that both the WB Lineand the RT WB Line applications are very accurate for lowfrequencies. However, the errors tend to be higher for the approximating peaks in the open-end response. Figure 10 showsa zoom into the relative errors of the open-end responses, from5
as plotted in figure 16. The induced current in steady-stateis another important aspect, which approximation is moreaccurately computed using the WB Line.Fig. 11: Circuit used to study the response to lineenergization followed by single-phase short-circuit, accordingto the WB Line and to the RT WB Line applications.Fig. 14: Circuit used to study the current induced in phase 3by phase coupling, according to the WB Line and to theRT WB Line appli
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