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The effect of corona on wave propagation on transmission lines Naredo V., José Luis A. 1992

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THE EFFECT OF CORONA ON WAVE PROPAGATION ON TRANSMISSION LINESbyJOSE LUIS A. NAREDO V.B. Eng., Universidad Anahuac, Mexico, 1984M. A. Sc., The University of British Columbia, 1987A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinDEPARTMENT OF ELECTRICAL ENGINEERINGTHE FACULTY OF GRADUATE STUDIESAUGUST 1992We accept this thesis as conformingto—th required standardTHE UNIVERSITY OF BRITISH COLUMBIA18 June 1992JOSE LUIS A. NAREDOV., 1992Signature(s) removed to protect privacyIn presenting this thesis in partial fulfilment of the requirements for anadvanced degree at The University of British Columbia, I agree that theLibrary shall make it freely available for reference and study. I furtheragree that permission for extensive copying of this thesis for scholarlypurposes may be granted by the Head of my Department or by his or herrepresentatives. It is understood that copying or publication of this thesisfor financial gain shall not be allowed without my written permission.AUGUST 1992The University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1W5Date: 6 August 1992Signature(s) removed to protect privacyABSTRACTFast transients on power transmission systems, such as the ones producedby lightning and faults, are usually modelled by the Telegrapher’s Equationswhich, because of the corona effect, are nonlinear. Although it has beenlong recognized that the method of characteristics of partial differentialequations (PDE’s) theory is the most adequate to tackle this problem, itsprevious applications have been very limited. A very general technique forthe simulation of transients on lines with corona, based on the method ofcharacteristics, is thus proposed in this thesis. This technique consists ofrepresenting the transmission lines by a system of first order quasilinearpartial differential equations (PDEs) and of solving them on a characteristicsystem of coordinates by applying interpolation techniques.A method of analysis and simulation is first developed by applying thetechnique of characteristics with interpolations to the 2x2 system ofquasilinear PDE’s representing a monophasic line with static corona. Thismethod is further implemented on a computer. The numerical examplesprovided show that this method overcomes the problem of numericaloscillations which is often found at the tails of waves simulated by meansof conventional methods based on constant discretization schemes. Anotherimportant feature of the developed method is that it requires substantiallyfewer discretization points than the conventional ones.The developed method is then extended to the time domain analysis ofmulticonductor lines both, the linear ones and the quasilinear ones withIIstatic corona. Most conventional methods for the analysis of multiconductorlines in the time domain are based, either directly or indirectly, on modaltransformations from frequency domain analysis. One problem with thisapproach is, however, that these transformations usually introduce complexquantities which lack physical meaning in the time domain. The extensiondeveloped here maintains the analysis in the domain of the real numbers.In the case of transmission lines with corona, an additional problem ofconventional modal transformations is that they presuppose linearity. Theextension developed here avoids this shortcoming.IIITABLE OF CONTENTSABSTRACTTABLE OF CONTENTS ivLIST OF FIGURES viACKNOWLEDGMENTS x1. INTRODUCTION 11.1. Problem Statement 11.2. Background 21.3. Scope and Aims of the Research 72. TRANSMISSION LINE MODELS FOR TRANSIENT STUDIES 102.1. Underlying Theory of Transmission Line Modelling 102.2. Frequency Domain Transient Analysis of Lines 142.3. Transient Analysis of Lines in Time Domain Through theMethod of Bergeron 192.4. Preliminary Considerations for the Modelling of TransmissionLines with Corona 252.5. Remarks 323. CORONA MODELS FOR LINE TRANSIENT STUDIES 333.1. Preamble 333.2. Overview of the Phenomenon of Corona 343.3. Corona Models 383.4. Final Considerations and Recommendations 444. QUASILINEAR MODELS FOR MONOPHASIC LINES WITH CORONA 464.1. Preamble 464.2. Systems of Equations for Monophasic Transmission Lines 474.3. Monophasic Linear Line Model Based on Characteristics 504.4. Quasilinear Model for a Line with Static Corona 574.5. Performance of the Proposed Quasilinear Method 634.6. Observations and Remarks 705. LINEAR AND QUASILINEAR ANALYSIS OF MULTICONDUCTORTRANSMISSION LINES CORONA 755.1. Multiconductor Linear Lines 755.2. Numerical Solution of the Linear Multiconductor Line Equations825.3. Multiconductor Quasilinear Transmission Lines 875.4. Proposed Numerical Solution of the QuasilinearMulticonductor Line Equations 905.5. Remarks 936. CONCLUSIONS 956.1. Preamble 95iv6.2. Summary of Results.966.3. Future Research Recommendations 996.4. Concluding Remarks 1017. REFERENCES 103VLIST OF FIGURESFigure 2.1.— Frequency domain transient simulation.Figure 2.2.— a) Lossless line section. b) Equivalent circuit for a lossless linesection. c) Lossy line model.Figure 2.3.— Time domain transient simulation with a frequency independentline model.Figure 2.4.— Corona voltage/charge relation.Figure 2.5.— Double exponential impulse and characteristics for atransmission line with corona.Figure 3.1.— Piecewise linear approximation of the Q—v curve by using oneand two straight line segments from v to vc maxFigure 4.1.— Transmission line initial—boundary problem, a) Scheme of aterminated transmission line, b) x—t plane with the line problem boundaries.Figure 42.— Extension of the Solution from points D and E to F.Figure 4.3.— Extension of the Solution to Boundary Points, a) From a knownpoint E to a point S on the source boundary. b) From a known point D tovia point L on the load boundary.Figure 4.4.— a) and b), two possible meshes of characteristics for lineartransient simulation.Figure 4.5.— Simulation of a linear double ramp traveling wave.Figure 4.6.— a) Regular grid for nonlinear line calculation. b) Finding theinterpolation points.Figure 4.7.— Application of the method of characteristics with interpolations.a) Transmission line diagram. b) Distortion of a linear double ramptravelling along the line depicted in fig 4.7a with corona. c) Partial view ofthe map of characteristics.Figure 4.8.— Simulation of a double ramp on a line terminated in its linearsurge impedance.Figure 4.9.— Transient wave at 3 km distance from the source for both,open ended and semi—infinite line. The double of the semi—infinite lineresponse is included as reference.Figure 4.10.— Simulations involving reflections, a) Open ended line. b) Shortcircuited line end.viiFigure 4.11.— Open ended line transient simulation with 270 line segments.Figure 4.12.— Open ended line transient simuLation with 18 line segments.Figure 4.13.— Open ended line transient simulation with 12 line segments.Figure 4.14.— Map of characteristics in which some of the curves belongingto the same family seem to merge.Figure 4.15.— Comparison between measurements and a computer simulation.a) Measurements performed by Wagner, Gross and Lloyd and reported inref. [1 1]. b) Simulation of Wagner, Gross and Lloyd’s experiment.Figure 4.16.— a) Corona capacitance obtained experimentally by Köster andWeck and reported in ref. [42]. b) 0—v curve form for a continuouslyvarying corona capacitance. c) Rounded form of the 0—v characteristic. Theslope between P1 and P2 is negative.Figure 5.1.— Characteristics of a multiconductor linear line passing through apoint in the x—t plane.Figure 5.2.— Positive and negative slope characteristics intersecting the linet=T.Figure 5.3.— Characteristics of a multiconductor quasilinear line passingviiithrough a point in the x—t plane.ixACKNOWLEDGMENTS.To the memory of my father Antonio Naredo G. (1919—1991) for whom thecompletion of this thesis was at least as important as it is for me. Tomy mother Caritina Villagrn whose love, encouragement and support havebeen vital for the pursuit of my lifelong goals; the Ph. D. degree, amongthem. To my wife Jacqueline for her love, her endless patience and,particularly, for being my companion in the good as well as in thedifficult times. To my sister Lupita for her love, her encouragement andher readiness to help. To my brothers Antonio and lgnacio, whose faith inme has been always a source of strength. To Harold Steiman for whomfriendship is unconditional and who put his house, his knowledge of theEnglish language, his time and even his savings at my disposal so I couldfinish this thesis. To Dr. Avrum C. Soudack for his wise advise and forhelping me stretch my intellectual horizons. To Dr. Hermann Dommel forproviding me with practical insights and for bringing me down to earth inrepeated occasions during my Ph. D. program. To Dr. Martin Wedepohl whopointed me in the right directions and who got me going in graduatestudies. To Dr. Jose Mard, Dr. Sandoval Carneiro and Dr. Nelson Santiagowhose technical advice has been invaluable. To the Instituto delnvestigaciones EIctricas de Mxico for granting me a leave of absence toconduct graduate studies. Finally, to The National Council of Science andTechnology of Mxico (CONACYT), to The Bank of Mxico and to TheNational Science and Engineering Council of Canada whose economicsupport I gratefully acknowledge.x1. INTRODUCTION.1.1) Problem Statement.Modern insulation design of transmission lines and of substations isincreasingly becoming more dependent on digital simulations. Due to progress inresearch and development over the past number of years, the simulationmethodology for linear lines has achieved a satisfactory level [27,28,53,81,95].Often in practice, however, one has to deal with the corona effect whichintroduces a distributed nonlinearity in the transmission lines. Although theproblem of wave propagation on lines with corona has been studied for a longtime, progress has been slow and there is still much to do. A thorough accountof the most relevant work in lines with corona, since its beginnings, is given ina survey paper by Carneiro [72].The minimization of corona by overdesigning transmission systems wouldbe extremely expensive and, furthermore, it may not be desirable. Corona tends toattenuate the parts of a waveform that are above the corona threshold level at amuch higher rate than the ones below it. In order to bring corona into designconsiderations, however, the techniques for the analysis of its nonlinearcharacteristics would have to be brought to a similar level of development astheir linear line modelling counterparts. There are, at present, several difficulties.Firstly, the phenomenon of corona in itself is highly complicated and is still avery active field of research. Secondly, reliable results from field and laboratoryare very difficult to obtain and, therefore, scarce. Thirdly, a nonlinear line theoryis expected to be far more complicated than the linear one. The work reported inthis thesis focuses on the third issue.It was felt that a method for analyzing wave propagation on nonlineartransmission lines, which was based on as few assumptions as possible, wouldbe a valuable tool for advancing the subject. Firstly, such a method wouldprovide an insight into the required features of the corona models. This methodwould serve to test hypotheses, to be an aid in the design of experiments andin the interpretation of their results and to help to establish, through sensitivityanalysis, which parameters of the corona models are relevant for transientstudies. Finally, a design methodology would be established.1.2) BackgroundInitial studies of corona go as far back as 1911. Between this year and1929, Peek published the results of his pioneering research. Some of his resultsare still in effect, as for instance his well known corona inception law [5].Progress since then has been slow. Some other important results are thedetermination of the hysteretic characteristic of corona by Ryan and Henline in1924 [1] and the initial studies of wave distortion due to corona by Skilling andDykes in 1937 [8]. In the 50’s, Wagner, Lloyd and Gross conducted an extensive—2—research program on corona and its effects on wave propagation [10,1 1]. Theymade several tests on a short transmission line. Their results, published in 1954[10), are often taken as benchmarks for new simulation methods. Wagner andLloyd also did experiments aimed at the characterization of the phenomenon ofcorona [11]. It was perhaps due to the limitations of their measuring equipmentthat they concluded wrongly that corona was an essentially static phenomenon. Inthis context, a phenomenon or its representation is said to be static when thespatial charge is a function of the instantaneous voltage only and not of its rateof change with respect to time “avIat”. If, on the other hand, the spatial chargealso depends on av/at, the phenomenon or its representation is said to bedynamic. Recent experiments show that corona is dynamic [42]; however, in spiteof the modern equipment and techniques, the effect of aviat doesn’t seem to befully determined yet.Wagner and Lloyd also proposed a finite difference technique to evaluatetravelling wave distortion due to corona [11]. It seems that this was the firstapplication of a digital computer to this type of problem. A further digitaltechnique was proposed later, in 1965, by Stafford, Evans and Hingorani [23].These authors suggested that further developments should make use of thecharacteristic curves of the transmission line partial differential equations (PDE’s).These curves, known as characteristics, have as a property that, along them, theline PDE’s turn into ordinary differential equations (ODE’s). In 1970 Zielinskypresented a graphical method based on characteristics [30]. Its applicability was,however, very limited.In the 70’s and until the middle 80’s, the emphasis of power transientresearch was directed to linear line models. Two complementary techniquesemerged. One based on frequency domain methods [28,95] and, the other, on time—3—domain methods [27,37,53]. Several successful developments led to thedevelopment of the Electromagnetic Transients Program (or EMTP) which hasbecome one of the most used programs of its kind [27]. Its availability, alongwith the fact that the time domain is far better suited for dealing withnonlinearities, has encouraged the development of EMTP compatible techniques tosimulate lines with corona. Among them are the one proposed by K. C. Lee [59],the one by Semlyen and Wei—Gang [70,76], the one by Hamadami—Zadeh [68] and,more recently, the one by Carneiro, MartT and Dommel [91,94].Further progress in the topic requires a deep understanding of the coronaphenomenon. Some knowledge has been provided by the experiments conducted atThe Hydro—Qubec Institute of Research (IREQ) [42] and at Electricite de France(E DF) [43]. However, many more experiments are needed. As they are verydifficult and expensive to conduct, it may be convenient to coordinate them withthe development of mathematical models of the corona effect which are basedon the actual physical processes. An example of such a mathematical/physicalmodel is the one proposed recently by Abdel.-Salam and Stanek [71). Thesimulation of transients, on the other hand, necessitates computationally lessintensive corona models. Simpler models that aim to preserve the featuresrelevant to transient wave propagation have been already proposed. Among themare the static corona models by Gary, Dragan and Cristescu [85], the dynamicmodel based on shells of spatial charge by Harrington and Afghahi [55], therefined model of shells by Semlyen and Wei—Gang [70] and the dynamic modelbased on the time delay of charge formation by Li, Malik and Zhao [87]. Thesemodels should still be subjected to further tests that would establish their rangeof validity. The tests, however, would require a very reliable and flexible methodfor calculating nonlinear transients as well as a set of well documented lineexperiments.-4-In addition to the line experiments conducted by Wagner, Lloyd and Gross,there are only few more that are well documented. Among them are the onesreported by Gary, Dragan and Cristescu [43], the ones by Ouyang and Kendall [35]and the ones by Inoue [44,65). More experiments are needed and it is desirablethat some of them involve waves with multiple peaks and/or reflections [48].Concerning the methods for calculating nonlinear transients, in addition tothe ones already mentioned before, few more have been proposed. These methodscan be classified into the following three groups: 1) wavefront delay methods, 2)lumped element approximation methods and 3) finite difference methods. Thewavefront delay methods are essentially extensions of early graphical techniques.Since they are not meant to be very accurate [85), they will not be given furtherconsideration here. As for the lumped element methods, it is usually possible toestablish their equivalence with a finite difference method. The research reportedhere is thus focused on finite differences. Variational methods, such as the finiteelement method, are often considered better alternatives to finite differences;however, their theory and techniques for handling the nonlinear hyperbolic PDE’sarising from transmission lines with corona are still in an early stage ofdevelopment [73]. It is important, nevertheless, to keep track of their progress.Other techniques that should be considered in future studies are the analytic onesthat are currently being developed [90]. An example is the hodographtransformation which converts a homogeneous (lossless) nonlinear PDE into alinear PDE by exchanging the roles between dependent and independent variables[41). A recent extension by Fusco and Manganaro [86] permits the application ofthe hodograph transformation to nonhomogeneous PDE’s and, consequently, to theanalysis of lossy lines with corona.—5—Lumped element methods are very popular, perhaps because it is relativelyeasy to derive models of lines with corona from the well known ir—line models.In addition, charge/voltage relationships describing corona are often given ascircuits involving capacitors, voltage sources, diodes and sometimes resistors. Thetraditional ir—line models are often modified by adding lossless line sections asmodelling elements [83]. Additional elements may be inserted to account for thefrequency dependence of the line parameters [68,70]. Most of the EMTPcompatible methods, including the abovementioned ones, belong to the lumpedelement category.Concerning the finite difference methods, few more alternatives have beenproposed in addition to the ones by Wagner and Lloyd and by Stafford, Evansand Hingorani. In 1978 and, later, in 1985 Inoue published his experimental resultsthat have been mentioned earlier. Along with them, he proposed a finitedifference method [44,65] in which the lines are considered lossless. A furtherlimitation of lnoue’s work is that both the experiment and the simulations seemto deal only with nondecaying wavetails. In 1981 Kudyan and Shih presented afinite difference method, also for lossless lines [50). In their paper, theyintroduced some stability and convergence considerations. More refined finitedifference schemes have been introduced by Gary, Dragan and Cristescu [56), byM. T. Correia de Barros [64] and by Li, Malik and Zhao [87]. In the first method,a discrete convolution is incorporated into the discretization scheme to accountfor frequency dependence. It seems that the authors didn’t use recursiveconvolution [37,53] and it is thus possible that the computational performance oftheir method could be improved. In the second method, the line PDE’s are turnedinto a differential—difference system. The resulting differential equations areintegrated through Gear’s method. In the third method, a hybrid discretizationscheme based on forward and backward differences is applied, producing a—6--tridiagonal system of equations. The method is combined with the model ofcorona proposed also by the same authors [80].Extensive research on numerical methods for the simulation of lines withcorona was conducted at The University of British Columbia between 1987 and1989 [72,78,91,94]. Several important results have come from this project. First, itwas established that the corona shunt conductance losses have negligible effecton the propagation of transient waves [72). Second, that the frequencydependence effects are negligible for line distances shorter than 10 km. A similarconclusion was reached by Gary [56]. Third, that the space disretization, either inthe lumped or in the finite difference methods, causes artificial reflections whosemagnitudes, under certain circumstances, become considerable [78,91]. In additionto this last problem, Janischewskyji and Gela [48) have pointed out that most ofthe existing methods do not satisfactorily reproduce the slow decay of travellingwavetails.1.3) Scope and Aims of the ResearchIt seems from the previous two sections that a new method of simulatingpropagation on lines with corona, which complements the already existing ones, isrequired. Such a method should enable the representation of corona as adistributed phenomenon. In addition, a desirable feature is that its extension tomulticonductor lines does not require linear modal analysis or any other techniquebased on the linear superposition principle. In a preliminary study by this author,it became clear that the transmission line PDE’s are quasilinear when corona isrepresented by a static model. As a recall, a PDE is said to be quasilinear whenthe highest order derivatives of its dependent variables (that is, the ones thatdetermine the order of the equation) occur only to the first degree [15,33,45]. Itbecame clear also that the method of characteristics is well suited forrepresenting distributed corona. This method is, in fact, regarded by many authorsas the best one for handling quasilinear hyperbolic PDE’s [41,45,79]. Apart fromZielinski’s work, the characteristics haven’t been adopted as a general tool for theanalysis and simulation of lines with corona. The aim of the research reportedhere is, therefore, to develop a methodology for the analysis and simulation oftransmission lines with corona that is based on the method of characteristics.A further consideration made in the preliminary study was that thequasilinear line equations could develop shocks [33]. In this case, the method ofcharacteristics would provide the most convenient way for detecting them [14],analyzing them [15] and handling them [13]. Apparently, the possibility of a shockon a line with corona hasn’t been considered before. The closest hint of it, foundby the author, is on a recent CIGRE report by Gary, Dragan and Cristescu [93]where the authors take note of the ambiguity that arises when applying thewavefront delay method to the crest of travelling waves.In order to apply characteristics to the numerical analysis of transmissionlines with corona, a decision has to be made as to whether the nonlinear lineequations should be handled as a system of first order PDE’s [14] or as anequation of higher order [45]. Since there is a general theory for quasilinearhyperbolic systems [15,33,39], the former approach is adopted here. It has theadvantage that its extension to multiconductor lines is straightforward and that itdoesn’t require the use of the linear superposition principle. A further advantageof the adopted approach stems from the fact that nonlinear PDE’s can always beturned into quasilinear systems of PDE’s [15,39]. The methods developed in thethesis can thus be extended to analyze lines with dynamic corona in a rigorousmanner. Because of time limitations, this very important issue was not pursuedfurther here. It is proposed, instead, as a future project.—8—From all the above, the goals of the research reported in this thesis arestated as follows:1. To propose, develop and implement a method to analyze the propagation ofmonopolar impulses on lossy monophasic lines with static corona.2. To obtain, from the numerical applications of the implemented method, newideas concerning the features that are required from the adopted coronamodels.3. To propose and develop an extension of the method of characteristics formulticonductor lines with corona.Finally, in passing, an alternative to the conventional methods for handlingfrequency independent models of multiconductor linear lines in the time domain isalso proposed.—9-.2. TRANSMISSION LINE MODELS FOR TRANSIENT STUDIES.2.1) Underlying Theory of Transmission Line Modelling.Consider a transmission line formed by the ground plane and a wirerunning horizontally at a height h. If one assumes that the wire and the groundare perfect conductors, that the air between them is a perfect dielectric and thatnone of the wavelengths involved are smaller than 4h, then the electric andmagnetic fields are transversal (TEM) to the direction of the wire [26,63).Maxwell’s first equation (Faraday’s Law) yields [26]:av2= Lg ( .la)where v is the voltage of the wire relative to the ground, i is the current carriedby the wire and Lg is the line inductance per unit length. Because of itsdependence on the sectional line geometry, this inductance is called geometric. Asecond relationship, companion of (, is obtained by applying Maxwell’s Second—10—equation (Ampere’s Law) to the ideal line [26]:ai av= Cg ( Cg is the line capacitance per unit length which is also known asgeometric because of its dependence on the sectional geometry of the line. Infact, Cg and Lg are inversely related as follows:L9Cg = ( and e0 are, respectively, the magnetic permeability and the electricpermitivity of the air. These two constants are practically identical to the onesof the vacuum and either side of ( is equal to the inverse of the speed oflight squared.Equations ( and ( describe in fact an ideal (lossless) transmissionline in which waves travel without distortion. Actual lines, however, consist ofimperfect (lossy) conductors and dielectrics. In consequence, propagating wavesusually suffer distortion. The standard practice to account for these losses is tomodify ( and ( as follows [26]:av ai—-b-— = L.-- + R.i (2.2a)and—-w— = C.T + G.v (2.2b)where L is the line inductance, C is the line capacitance, R is the seriesresistance that accounts for the imperfect conductor losses and G •is the shuntconductance that accounts for the dielectric losses. All these parameters are inper unit length. Equations (2.2a) and (2.2b) are known as the Telegrapher’sequations. In a strict sense, the L and C parameters differ somewhat from theirgeometric counterparts Lg and C9, because they must account now for the—11—penetration of the fields (magnetic and electric, respectively) into the imperfectconductors. Nevertheless, in the case of aerial lines, the value of C is veryclosely approximated to the one of Cg for a large range of frequencies and ofground resistivities which encompasses most cases of practical interest [51]. Inaddition to this, the shunt conductance of these lines is negligible [67]. Thefollowing equation can thus be used instead of (2.2b):ai av= Cg (2.2c)A more rigorous analysis of lossy transmission, lines via Maxwell’sequations, is only possible after making certain simplifying assumptions concerningthe distribution of the electric and magnetic fields. A number of theseassumptions, known as quasi—transversal electromagnetic or quasi—TEM, lead to thefollowing general expressions [2,4,6,7]:dV—a--- = (jcL + R).I (2.3a)and-= (jcoC + G).V (2.3b)where V and I are the Fourier transforms of the actual voltage and currentwaveforms, respectively. Note that (2.3a) and (2,3b) are ordinary differentialequations (ODE’s) in the frequency domain. The quasi—TEM assumptions arejustified for a wide range of frequencies and ground resistivities which comprisesmost power transient analysis and power line carrier applications [38,40].Despite the similarities between (2.3a) and (2.2a), these two equations arenot equivalent. Since in the former equation the R and L parameters are functionsof the frequency, its inverse Fourier transform would involve a convolution. Forexample, Radulet, Timotin and Tugulea have proposed the following expression asa time domain equivalent of (2.3a) [29]:—12—av t= Lg +--f r(t — .(2.4)where r(t) is defined as the unit step resistance of the line. As for (2.3b), it isclear that its inverse Fourier transform yields (2.2c).Equations (2.3a) and (2.3b) provide the basis for a frequency domaintechnique to calculate transmission line transients [28]. As for the time domaintechniques, most of them are based on (2.2a) and (2.2c). The models derived fromthese equations are said to be frequency independent. A major reason for using(2.2a), instead of a convolution expression like (2.4), is that frequency independentmodels are simple and adequate for many practical situations [95]. In addition,frequency dependence features can be incorporated later on to the models bymeans of recursive convolution techniques [37,53]. An important and apparentlystill unresolved issue, concerning the development of frequency independent linemodels, is the systematic determination of the values of the R and L parametersthat better represent a transmission line for a given simulation situation. Anotherapparently unresolved issue is the establishment of criteria for determining whichtransient analysis cases require frequency dependent modelling.The analysis leading to equations (2.2a) and (2.2c) is based on theassumption that transmission lines respond linearly. This is not the case, however,if a line is affected by corona for which equation (2.2c) has to be modified[56,65]. In many practical studies, this modification results in the representation ofthe line capacitance by a function of the line voltage [44]. Due to the fact thatthis function is multivalued, it is considered here that corona cannot berepresented adequately in the frequency domain. The approach adopted in thisthesis is thus based on the time domain. Nevertheless, a frequency domainmethod is also employed here as an aid for the selection of parameters for—13—frequency independent line models. Section 2.2 thus provides a broad descriptionof this method. Section 2.3 provides a summary of current time domain methodsfor the analysis of transients on transmission lines. Section 2.4 providespreliminary considerations for the analysis of transients on lines with corona.Section 2.5, finally, provides the remarks of this chapter.2.2) Frequency Domain Transient Analysis of Lines.It is customary to write (2.3a) and (2.3b) in the following form:—= Z.I (2.5a)= Y.V (2.5b)where Z is the line series impedance and Y is the line shunt admittance, both inper unit length. For a transmission line consisting of a thin wire above asemi—infinite ground plane, this impedance is made up of three terms:Z = jLJLg + Zcond + Zgnd (2.5c)The first one is the geometric impedance due to the magnetic flux in the air, thesecond one accounts for the effects of this flux inside the wire and the thirdone for the effects of this flux inside the ground. Details for the calculation ofthese terms are provided in references [20) and [74]. As for the admittance, ithas been atready mentioned that the shunt capacitance per unit length of anaerial line is practically identical to the geometric line capacitance; therefore:Y = JøCg (2.5d)Equations (2.5a) and (2.5b) are further manipulated to produce the twofollowing second order ODE’s:—14-d2V= ZY.V.(2.6a)andd2 I——--= YZ.I (2.6b)The solution of (2.6a) is readily obtained as follows:V(x) = C1 exp (—-yx) + C2 exp (‘yx) (2.7a)where C1 and C2 are integration constants and(2.7b)is the propagation constant of the line. The real part of this constantcorresponds to the line attenuation and its imaginary part to the line phase delay,both are in per unit length. Whereas the term of (2.7a) with the positiveexponential factor represents a voltage wave travelling forwards along the line,the other one with the negative exponential factor represents a voltage wavetravelling backwards [24]. The respective amplitudes of these two travelling waves,C1 and C2 are determined from the line boundary conditions; that is, from theconnections at both ends of the line section. As for the solution of (2.6b), thefollowing expression is obtained by applying (2.7a) to (2.5a):1(x) = Y [ C1 exp (—7x) + C2 exp (7x)] (2.7c)where= (2.7d)or= (2.7e)is the characteristic admittance of the line.Consider now a homogeneous line section of length -e. Let V and bethe voltage and current at the side of the line considered the input side; that is,—15—at x = 0. Let V0,. and 10ut be the voltage and current at the output side of theline; that is, at x = . By applying expressions (2.7a) and (2.7c) to this linesection, the following relationship between the aforesaid input and output variablescan be obtained [18):l A B= (2.8a)‘out B A V0where A and B are defined as follows:A= c coth(7e) (2.8b)andB=cosech(’y) (2.8c)Expression (2.8a) provides a convenient frequency domain model for ahomogeneous line section [54,95]. Although this model presupposes a puresinusoidal excitation, its extension to transient analysis is possible through theuse of the superposition principle of linear systems theory. In the methodadopted here, first the excitation waveform is decomposed into its harmoniccomponents, then the response of the line section to each of these componentsis obtained through expression (2.8a) and finally the total response is obtained bysuperposing the individual harmonic responses [28]. The processes of harmonicdecomposition and of superposition are conveniently performed by the FastFourier Transform Algorithm (FFT). Since these processes involve the discretizationof the frequency range, the undesirable effects of Gibbs oscillations and aliasingare inevitable. Nevertheless, these effects are reduced by applying the complexfrequency concept which Consists of replacing the imaginary frequency jc by acomplex variable whose real part is a damping factor and its imaginary part isthe frequency itself [46].—16—Figure 2.1.— Frequency domain transient simulation.An attractive feature of the above described frequency domain method isthat the numeric error bound can be controlled. In fact, the number of harmoniccomponents, the truncation frequency (that is, the highest harmonic frequencyconsidered) and the damping factor determine this error bound [46,54]. Anapplication example is provided next. A transient is simulated on a semi—infiniteline consisting of an aluminum conductor of radius r = 2.54 cm at a heighth = 18.9 m above a ground plane whose resistivity is 100 m. Figure 2.1 showsthe applied excitation impulse which is a 1.0 p. u. [2/20j.ts] double linear ramp;that is, a ramp with a maximum amplitude of one unit, which reaches thismaximum value in 2 js and which decays to 50 % of this maximum in 20 gs.Figure 2.1 also shows the impulse after it has travelled 500, 1000, 1500, 2000 and3000 m along the line. The comparison of these different waveforms shows thesmoothing effect on the corner of the impulse as it propagates. Fig. 2.1simulation has been made with 256 samples spread over an observation time of33 js and with a damping factor that has been chosen for a maximum numericerror of 0.1 %. Note from this figure that only the first 30 js of simulationzIi=0=500m4 /X15OOm141/_____/ / , i = 3000 m20.0Time— ILS-.17—have been displayed. Since the aliasing error tends to increase in the last portionof the time scale, it is a common practice to display only the first 90 % portionof the time scale [46].The extension of the previously described simulation method tomulticonductor lines is straightforward provided the superposition property holds.This extension is outlined as follows for an homogeneous line section. Considerthe following multiconductor line equations in the frequency domain [54,67]:= Z•I (2.9a)and= Y•V (2.9b)dxwhere V is the vector of conductor voltages, I is the vector of conductorcurrents, Z is the matrix of series impedances per unit length of the line and Yis the matrix of shunt admittances of the line. The off—diagonal elements ofthese two matrices account for the mutual impedances and admittances betweenconductors, respectively. Note that vector and matrix quantities are denoted bybold letters. Let T be the matrix that diagonalizes the ZY matrix product asfollows:T1ZY.Tv A (2.10)where A is the diagonal matrix of eigenvalues of ZY. Let F be the diagonalmatrix whose nonzero elements are the square roots of the diagonal elements ofA with positive or zero real part. The general solution of (2.9a) and (2.9b) can beexpressed as follows in terms of F [54]:V(x) = C1 exp (—jix) + C2 exp (&x) (2.lla)1(x) = ‘‘cC1 exp (—Jix) + YC2 exp (ijix) (2.llb)—18—where= Tv.F.Tv1 (2.11c)and= z-1.r (2.lld)Note that (2.1 la) and (2.1 ib) are matrix generalizations of (2.7a) and (2.7c). Notefurther, from the comparison between (2.7c) and (2.llb), that Y is a matrix ofcharacteristic admittances.Consider now a multiconductor line section of length-e for which V,, andl-, represent the vectors of the voltages and currents at the input side of theline section and and 1out represent the vectors of voltages and currents atthe output side. The application of expressions (2.1 la) and (2.1 ib) to this linesection yields [54):AB Vm= (2.12a)1out B A V0whereA=‘f coth(ix) (2.12b)andB = Yc cosech(4cx) (2.12c)Expression (2.12a) provides the desired generalization of model (2.8a) formulticonductor transmission lines.2.3) Transient Analysis of Lines in Time Domain Through the Method of Bergerori.Several methods have been proposed for the analysis of transients ontransmission lines in the time domain. Among them, the method of Bergeron isperhaps the most widely used one [27]. In addition, some of the most important—19—line models of the Electromagnetic Transients Program (EMTP) are based on thismethod [67]. For these reasons, this section focuses on the method of Bergeron.Consider first a transmission line in which the resistive losses areneglected. After some elementary manipulations, equations (2.2a) and (2.2b)become:32v= LC (2.13a)anda2i a2i= LC (2.13b)Their general solution can be stated as follows [27]:v(x,t) = F1(x — Ct) + F2(x + Ct) (2.14a)andi(X,t) = (lIZc)Fi(X — ct)— (l/Zc)F2x + Ct) (2.14b)where c is the wave velocityc = ii1/t, (2.14c)Z is the characteristic impedance of the linez, = (2.14d)and F1 and F2 are functions determined by the initial and boundary conditions ofthe line. Expressions (2.14a) and (2.14b) can be further combined yielding:v + Z.i = 2F1(x — Ct) (2.15a)v—Z.i = 2F(x + ct) (2.15b)Note that (2.15a) represents a constant wave travelling forwards; that is, in thedirection defined as positive. Similarly, (2.15b) represents a constant wave—20—kwax.,wkRax waxT rLOSSLESS LINE mmFigure 22.- a) Lossless line section. b) Equivalentcircuit for a lossless line section. c) Lossy linekzcEh,st,ka)b)mzcEhj,mWSSLESS LINEzw.c)model.travelling backwards; or, in the negative direction of the line. The left hand sideterms of (2.15a) and (2.15b) are the Riemann invariants of equations (2.13a) and(2.13b) [60]. In the context of power line transient analysis, these two termssometimes are called characteristics; although in this thesis this term is reservedfor those curves of the x—t plane along which a PDE becomes an ODE. Thesolution of equations (2.13a) and (2.13b), at any point along the lossless line, canthus be expressed as a superposition of these invariants.Consider now the lossless line section between nodes k and m, of lengthx which is illustrated in fig. 2.2a. According to expressions (2.15a) and (2.lSb),the voltage and current at node k are related to their values at node m asfollows:vk(t) — Zcik(t) = Vm(t — r) — Zcim(t — r) (2.16)where r is the travel time of the line section:r = ax/c.Since the right hand side of (2.16) is constituted of past values, it is referred toas the history term and is denoted by Ehjstk [27]:Ehjst,k = vm(t — r)— Zcim(t — r) (2.17a)Relation (2.16) becomes thus:Vk(t) — Zcik(t) Eh/st,k (2.17b)In an analogous manner, the following expression is established for node m;Vm(t) — Zcim(t) = Ehist,m (2.17c)In this case, the history term Ehjst,m is given as follows in terms of the pastvalues of voltage and current at node k:—22—Eh/st,m Vk(t —r)— ZCik(t—r)..(217d)Expressions (2.17b) and (2.17c) constitute the basic lossless line model of theEMTP [27]. The circuit equivalent to these equations is shown in fig. 2.2b.Figure 2.2b model can be applied to the simulation of a frequencyindependent lossy line by considering that the resistive losses are lumped at fewpoints along its length [27]. Figure 2.2c illustrates this method as implemented inthe EMTP [67]. Note from the figure that the losses are lumped at three pointsas follows: one fourth at one end, one half at the middle and one fourth at theother end of the line. The two lossless line sections between these points arerepresented by fig. 2.2b model. Numerical experiments have shown that fig. 2.2.cline representation is accurate inasmuch as R.Ax << Z [67]. In order to meetthis condition, a long transmission line can be subdivided into several linesections and each one can be represented by fig. 2.2c model. As an example,consider a 50 km transmission line whose parameters are: L 1.25 mH/km,C = 9.664 nF/km and R = 11.74 a/km. Consider that this line is terminated in a360 load which corresponds to the characteristic impedance when the resistivelosses are neglected. A 1.0 p. u. [1O/ double ramp impulse is applied atx = 0 kms. Figure 2.3 shows this impulse as well as its waveshape after it hastravelled 5.17, 10.34, 15.52 and 20.69 kms. One important aspect of frequencyindependent line models, which is apparent in this figure, is that they do notreproduce the gradual smoothing of a waveform corner as it travels. Figure 2.3simulations are made with EMTP ©MicroTran by dividing the transmission line into87 lossy line sections. Essentially the same results are obtained through a finitedifference technique presented in chapter 4.There is one development that permits us to account for losses as well asfor frequency dependence effects by means of essentially the same model as forlossless lines depicted in fig. 22b [53]. In this development, Z as well as thehistory terms expressions (2.17a) and (2.17d) are modified. While in the losslessline case the Z element Consists of a pure resistance, in the frequencydependent case it is a network that approximates the characteristic impedance ofthe line within a large range of frequencies. As for the history terms, in thefrequency dependent case they involve a convolution relationship. Fortunately, thisconvolution can be performed recursively [37,53].p.u.1,00.8o.q0.00 ao ç o 80 fOO 1Z0 40 1O(A.- Sec.Figure 2.3.— Time domain transient simulation witha frequency independent line model.The abovementioned time domain models presuppose monophasic lines.Their extension to multiconductor lines is customarily performed by applying amodal transformation borrowed from the frequency domain techniques described inthe previous section. One problem here is, however, that these transformations arenormally complex and introduce imaginary quantities which lack physical meaningin time domain analysis, This problem is usually approached in two ways. One—24—way is to force the transformation to be real, either by assuming that the line isperfectly balanced or by neglecting the series resistive losses in their calculation.The other way is to calculate the complex transformation matrix at a suitablefrequency and to discard its imaginary part [67].Another problem related to the application of modal transformations in thedevelopment of multiconductor line models in the time domain is that thesetransformations are functions of frequency. In many practical cases, theirapproximation by a constant matrix provides acceptable results. There are cases,however, in which the frequency dependence effects of the modal transformationmatrices have to be considered [81,95). A time domain model that takes intoaccount these effects has been devQloped for underground cables [81).2.4) Preliminary Considerations for the Modelling of Transmission Lines with Corona.When the voltage of a line conductor reaches a value for which theelectric field in the neighborhood is higher than the dielectric stength of the air,many of the air molecules break and an ion layer is formed around thisconductor. This is the corona effect whose description is often made by adiagram like the one shown in fig. 2.4, which is known as the Q—v curve. In thisfigure, segment AB corresponds to the region where Q and v maintain a linearrelation. The slope of this segment is the geometric capacitance of the line. Thepoint B where the Q—v relation ceases to be linear is called the corona inceptionpoint and v, the value of the voltage there, is called the corona inceptionvoltage. As the conductor’s voltage increases above v, the ions that accumulatearound determine the shape of the segment BC. It is assumed for this segmentthat the voltage is increased monotonically from v to its maximum value Vmax.Segment CD corresponds to the region where the voltage decreases monotonicallyfrom the maximum value Vmax towards zero. The slope of this segment is—25-.almost constant and it is usually in practice considered equal to the geometricline capacitance.Figure 2.4.— Corona voltage/charge relation.Assume that the corona generated spatial charges move in the transversaldirections to the line conductor only. These movements do not affect the first ofthe Telegrapher’s equations; that is, (2.2a). The effects on (2.2c), the second ofthe Telegrapher’s equation, are considered next. Let Q represent the line chargeper unit length. From the law of conservation of charge [65):aci218ax — atAssume now that the corona effect can be considered static, as in the previouslymentioned Q—v curve. The charge depends thus on the voltage only:ci = F(v) (2.19)On applying the chain rule [16), (2.18) becomes:a Cvc Vmax V—26—ai Q V220By introducing the following definition:C (2.21)— avequation (2.20) takes the same form as (2.2c); the only difference is that thecapacitance C is now a function of the line voltage. Note from the Q—v curvethat Q is a double valued function. Its derivative with respect to the voltageshould thus be evaluated by considering one branch of this curve at a time.Equations (2.2a) and (2.20) represent a monophasic line with static corona.These two equations can be combined into a single second order equation byeliminating the current i as follows. On differentiating (2.2a) with respect to x:a2v____ai=+ R (2.22)On differentiating (2.20) with respect to t:a2i a av2 aci a2vaxat = (Cc)(r) + (2.23)Now, the terms containing i are eliminated from (2.22) by applying (2.20) and(2.23):= R(Cc) + L(fCc)()2 + L(Cc)4 (2.24)In practice, this single second order POE is equivalent to the two equations (2.2a)and (2.20) [45). Note in this equation the term involving the derivative of thecapacitance with respect to the voltage and the square of the derivative of thevoltage with respect to time. Due to the fact that the second order derivativesof v occur in (2.24) to the first degree only, this equation is quasilinear. Forease of manipulation, (2.24) is expressed as follows:a2v a2v= a-b-p- + b (2.25a)—27—where a and b are two variables defined as follows:a = LCc (2.25b)andb = R(Cc) + L(fCc)()2 (2.25c)A set of ODE’s are derived next from (2.25a).Let the following definitions be introduced:p=-;a2v=a2vaxata2vWTTIn terms of these definitions, equation (2.25a) becomes:r— a•w— b = 0 (2.26)The total differentials of p and q are:dp r•dx + s•dt (2.27)anddq s.dx + w•dt (2.28)By applying (2.27) and (2.28) into (2.26) r and w are eliminated yielding:—— + as-— b = 0 (2.29)On multiplying (2.29) by (dt/dx):dt2 dp dt dq dt—s[(-—) (2.30)By imposing the following condition on (2.29):—28-.(dt)2— a = 0 .(2.31)the term containing s is eliminated:— a.dq— b.dt = 0 (2.32)Condition (2.31) has two roots, each of which determines an equation hereaftercalled a characteristic equation. When the roots are real, each of the characteristicequations determines a family of curves in the x—t plane. Along any of thesecurves, which are known as characteristic curves or simply as characteristics, PDE(2.25a) is equivalent to the pair of ODE’s constituted by (2.32) and by thefollowing one [45]:dv = p.dx + q.dt (2.33)This last equation is the total differential of the voltage v in terms of p and q.If in addition to being real, the roots of (2.31) are different, equation(2.25a) is said to be hyperbolic. These two roots would imply that there arealways two different characteristics, one from each family, passing through eachpoint of the x—t plane [45]. In consequence, the characteristics constitute asystem of coordinates in which the ODE’s (2.32) and (2.33) are equivalent to PDE(2.25a). It follows from (2.25b) and (2.31) that (2.25a) is hyperbolic as long as thecorona capacitance defined as aQ/av is positive. As in most studies of wavepropagation on lines with corona, this assumption is adopted here. Nevertheless,the possibility of aQ/av taking negative values is reviewed briefly in chapter 4.From a physical point of view, the characteristics represent trajectories ofthe x—t plane along which wave disturbances propagate. The slope of acharacteristic corresponds to the inverse of the local speed at which a wavedisturbance propagates. The existence of two characteristic speeds at every point(x,t), with equal magnitude and opposite sign, agrees with the fact that wave—29—propagation occurs in two directions. Consider now a double exponential impulsebeing injected at the beginning of a transmission line by means of a purevoltage source. This impulse is depicted in fig. 2.5 along with the correspondingpositive slope characteristics. Note that the points A, B, C and D marked on thisimpulse are related to the ones on the Q—v curve of fig. 2.4. Since the linecapacitance for segment AB is constant, the characteristics between points A andB’ are parallel. As the waveform voltage increases from point B to C, thecapacitance increases, the propagation velocity decreases and the characteristicsdiverge. As soon as the voltage waveform starts decreasing, right after point C,the propagation velocity increases suddenly and the characteristics in theneighborhood of point C’ in fig. 2.5 tend to converge.tFigure 2.5.— Double exponential impulse andcharacteristics for a transhiission line with corona.A situation that can arise in nonlinear wave modelling is the crossing overof characteristics belonging to the same family. This situation, hereafter called ashock, is illustrated in fig. 2.5. Its occurrence implies that the model beingemployed doesn’t provide a unique response any longer. Even though there areV CDx—30--physical phenomena that admit multivalued solutions, a shock is usually regardedas a breakdown of the used model which calls for further refinement [33]. Asfor the modelling of transmission lines with corona, it doesn’t seem to be clearyet whether or not shock conditions can arise. The finalization of this issuewould require a very specialized analysis beyond the scope of this thesis. As analternative, the approach adopted here is to implement a shock detectionmechanism in the developed simulation programs. It is clear, from the aboveanalysis, that a simulation program based on characteristics would facilitate thisimplementation. In addition, in the event of a shock occurrence the characteristicsprovide a means to deal with it [13].The quasilinear form of the above line equations is a direct consequenceof the assumption of a static representation of corona. Because of this form, thetheory of characteristics can be applied directly to transform these equations intoan equivalent system of ODE’s. Laboratory experiments indicate, however, thatcorona is a dynamic phenomenon. This means that, in addition to the voltage, itdepends on aviat and even on higher derivatives of v with respect to t [56]:Q = F ( v , ) (2.34a)andav a2vQ = F(v,---,-ã----,...). (2.34b)If any of these two representations are adopted instead of (2.19), the resultingline equations are nonquasilinear. Since these equations can always be convertedinto a system of first order quasilinear PDE’s [15], the theory of characteristicscan still be applied to the analysis of transients on lines with dynamic corona.—31—2.5 Remarks.An overview of current techniques for the analysis of transients on lineshas been presented in this chapter as a background for the development of amethod of analysis for transmission lines with corona. It has been determinedhere that, for this development, the time domain techniques are better suited thanthe frequency domain ones. Nevertheless, a method based on frequency domaintechniques has been considered convenient as a tool for the construction offrequency independent time domain line models.The analysis of a monophasic line under the assumption that corona canbe modelled as a static phenomenon has produced a second order quasilinearPDE. The further application of the theory of characteristics of PDE’s hasconverted this equation into an equivalent system of ODE’s. From the numericalpoint of view, it is more convenient to deal with this equivalent system ofODE’s than with the second order PDE directly [45]. In addition, the use ofcharacteristics facilitates the detection and the handling of shocks [13].Although the preliminary analysis of lines with corona presented here isbased on their description by a second order PDE, this can be done also byhandling the two Telegrapher’s equations as a system of two first order PDE’s[79]. An attractive feature of this alternate approach is the possibility of itsextension to a system of equations with a larger number of first order PDE’s.Such an extension is needed for the analysis and simulation of multiconductorlines as well of lines with dynamic corona.—32—3. CORONA MODELS FOR LINE TRANSIENT STUDIES.3.1) Preamble.The objectives of the research reported in this thesis are to propose, todevelop and to implement a model of lines with corona by applying the methodof characteristics of partial differential equations theory. The application of sucha model requires a representation of the corona effect. To this end, a coronamodel is chosen from among the many that have been proposed in thespecialized literature, It should be mentioned that the study of corona as well asits modelling still require further research.This chapter discusses some aspects of the physics and modelling ofcorona as a background for the developments presented in the following chapters.A very general description of the physics of the phenomenon is thus provided insection 3.2. Section 3,3 follows with an overview of some of the mostcommonly adopted models for the analysis of transients on transmission lines,—33—Finally, section 3.4 provides the considerations for the selection of a coronamodel for this work, as well as suggestions for future research aimed atimproving the representation of corona on transmission line transient studies.3.2) Overview of the Phenomenon of Corona.Corona consists essentially of the ionization of an electrically stressedregion in the air or in any other gaseous dielectric that surrounds a conductor ata high voltage. It is generally accepted that this ionization is initiated by thefree electrons that are being continuously produced, mostly, by natural radiation.These electrons are accelerated by the presence of an electric field and, as theytravel, they undergo multiple collisions against the air molecules. Although themajority of these collisions are elastic, the few inelastic collisions are the onesthat cause the ionization.In an elastic collision the electron transfers some of its kinetic energy tothe molecule. An inelastic collision, on the other hand, may result in the moleculebeing broken into a positive ion and an electron; this depends on the kineticenergy accumulated by the electron. If the molecule is not broken, part of theelectron’s kinetic energy is absorbed by one of the molecule’s atoms, which isnow said to be in a high energy state or to be excited. The electron, after this,may continue its travel or become attached to the molecule thus creating anegative ion. Excited atoms will eventually release the extra energy in the formof photons which may reach, later on, the conductor with the negative polarity(that is, the cathode) and extract electrons through the photoelectric effect [31].The generation of free charges through electron—molecule collisions isconsidered to be the principal mechanism of ionization. For this reason, it isreferred to as the primary ionization process [31,34]. The other mechanisms, which—34—are grouped as secondary processes, belong to one of the following twocategories: 1.— cathodic electron emissions and 2.— ion molecule ionizingcollisions. An example of the first category is the previously mentioned electronextraction through photoelectric effect. Other causes of cathodic electron emissionare positive ion bombardment, Schottky effect and Malter effect [31].The quantitative study of the ionization phenomena is performed throughthe following coefficients: the first ionization coefficient of Townsend, thegeneralized second ionization coefficient of Townsend and the attachmentcoefficient. They are denoted, respectively, as a, y and r. The first one isrelated to the primary ionization process and is defined as the average numberof ionizing collisions per unit distance of electron travel. The second one isrelated to all the secondary processes which, despite their basic differences, canbe described by a single generalized coefficient [31]. The third coefficient, theone of attachment, is defined as the inverse of the mean distance travelled byan electron before becoming attached. The three coefficients are functions of theelectric field intensity E and of the pressure p. They are in fact often expressedas functions of E/p [34,92]. A derived coefficient is the one of effectiveionization which is denoted by a and is obtained by subtracting from a.Consider now a region in which the electric field is uniform. If primaryionization and electron attachment were the only processes to be considered, thespatial distribution of electrons would be given by the following expression[89,92]:n = n0 exp(ax0) (3.1)where n0 is the number of initial electrons at x = 0 and n is the number ofelectrons at the distance x = x0. One can notice from (3.1) that if a is greaterthan zero, the electron density increases with x. From the definition of a, it35:follows that for a selfsustained ionization the primary ionization coefficient mustbe greater than the attachment coefficient. When secondary processes areintroduced into the analysis leading to (3.1), the following expression is obtainedinstead [31]:aexp(ãx0)—no (32)a — -y[exp(ax —1]One important aspect of uniform fields is that a selfsustained ionizationprocess leads, almost invariably, to a total breakdown [31]. As a matter of fact,the following breakdown criterion is obtained by equating the denominator of (3.2)to zero:[exp(ãx0)—1] = aSince the corona effect consists of a series of partial breakdowns, rather than atotal one, its analysis requires the consideration of a nonuniform electric field. Inthis case, expressions (3.1) and (3.2) become [92]:xo—n = no exp(f0 a(x)dx (3.3)and1 +n = no (3.4)1 — fxo(x[fxoa(z)dz]dxPractical application of (3.3) and (3.4) to the analysis of corona would stillrequire the following: 1.the introduction of additional spatial dimensions, 2.— thedetermination of the functional dependence of a, i and y on the electric fieldand 3.— the determination of the spatial distribution of the electric field.Concerning the first point, since the present work focusses on transmission lines,the analysis can be confined to two dimensions. Any pair of coordinates thatdetermines the plane transversal to the line is adequate. Concerning the secondpoint, there are well established empirical expressions for a and . However, very—36—little is known about ‘y [31,92]. As for the third point it can be said that, evenwithout corona, electric field calculations are difficult [47]. Further complicationsarise with corona because of the presence of spatial charges.Other important aspects of corona, which highlight its complexity, becomeapparent in macroscopic observations in the laboratory [34]. They show, forinstance, that there are substantial differences between the corona near the anodeand the one near the cathode. The former is called positive corona and the latteris called negative corona. Another important difference is the one between thefast occurring phenomenon and the slow occurring one. The first is in the orderof microseconds or less, while the second is above this order. Fast positivecorona, for instance, produces streamerlike avalanches, while fast negative coronaproduces featherlike ones [31,34]. These basic differences can be attributed to thedirection of electron travel. In the first case, the electrons are being attractedtowards the electrode and, in the second one, they are being repelled from it. Asfor the slow occurring corona, this is a more complicated phenomenon whosegeneral description is provided in the following two paragraphs.It is often possible to distinguish three stages in positive slow corona[31,34]. The first one starts as soon as the electric field surpasses the coronainception value; that is, the value at which the ionization coefficient becomesbigger than the attachment coefficient. This stage is characterized by theappearance of onset streamers that resemble the ones from fast positive corona.As the field is increased, the second stage commences. It is characterized by thedisappearance of the streamers and, in their place, the appearance of a glowregion in which the electric field is quasiuniform. Further increases in the fieldintensity brings the phenomenon into the third stage which is characterized by theappearance of long breakdown streamers and, if the field becomes strong enough,—37—a total breakdown can happen.For slow negative corona, three stages can also be detected [34]. The firstone is marked by the appearance of Trichel pulses [31] whose frequency variesbetween 2 kHz and a few MHz and depends directly on the field intensity. Asthis intensity increases, the second stage may or may not appear. It ischaracterized by a glow which can be inhibited by the presence of electronegativegases; that is, of gases to which electrons can attach easily. Oxygen is one ofthese. The third stage, finally, is characterized by the appearance of negativestreamers.3.3) Corona Models.It is clear from the previous section that corona is a very complicatedphenomenon and that its modelling, on the basis of the actual physicalmechanisms, would require a considerable effort. The models proposed so far, forpractical applications to transmission lines, are therefore based on simplifyingassumptions. The three most important ones are: 1.— The Deutsch assumption, 2.—The Kaptsov assumption and 3.-. The Peek empirical formula for the coronainception value of the electric field. The Deutsch assumption states that thespatial charges have an effect on the magnitude of the electric field but not onits direction. The Kaptsov assumption states that the electric field intensityremains at the inception value on the surface of the conductor for as long asthe ionization is ongoing. The Peek empirical formula, which provides the coronainception value of the field for cylindrical conductors, can be stated as follows[80]:= 3x106m5(1 + 3//)where m is a factor that accounts for the irregularities in the conductor’s surfaceand for the polarity, 5 is the relative air density and r0 is the conductor radius—38—in centimeters. Peek’s formula is used to calculate the corona inception voltagefor cylindrical conductors. It is also applied to bundled conductors by taking theequivalent radius of the bundle as r0.Several models of corona have been proposed for line transientsimulations. They range from the very simple to the very sophisticated [72]. Muchresearch must still be done, however, to determine the features of thephenomenon that are relevant to transmission line transient analysis, as well asfor establishing the degree of sophistication required from its models. Asimplistic model, on the one hand, may not allow us to obtain dependable resultsfrom the transient simulations. A sophisticated model, on the other, could imposean insurmountable computational burden on the line simulations. The ideal model,thus, should provide the essential aspects of the phenomenon and, at the sametime, be computationally efficient. In addition, another desirable feature is that themodel can be derived entirely from line specifications.aV1 Vmax VFigure 3.1,— Piecewise linear approximation of theQ—v curve by using one and two straight linesegments from V to Vmax,Perhaps the simplest model of corona is the one obtained by adding aconstant value to the line capacitance when the line voltage is increasing andbeyond the corona inception voltage v and, as soon as the voltage startsdecreasing or when it is lower than v, the added value is zero. Since this modelis equivalent to the Q—v curve approximation shown in fig. 3.1, it is known aspiecewise linear. Note from the figure that this approximation can also includemore than one segment between the inception voltage point v and the point ofmaximum voltage Vmax.A further improvement in the corona representation is obtained byapproximating the Q—v curve segment between v and Vmax by a generalizedparabola which is, in its most general form, given by the following expression:Q = k + k2(1 +where k1, k2 and k3 are parameters chosen to match measured Q—v curves andk3 is greater than 1.0. This model is known as parabolic. As an example,consider the model proposed by Inohue [44,65) which uses two parabolicsegments between v and vmax. This model unfortunately relies very strongly onmeasurements, Another parabolic model is the following one proposed by Gary,Cristescu and Dragan [85,93):CgV vvc, aviat>OQ = Cgvc()k3 v<v, av/at>0 (3.5a)CgV av/at 0where Cg is the geometric line capacitance. This model is proposed along withthe following empirical formulas for k3 which cover all cases of practicalinterest. For positive polarity on a single conductor:—40—k3 = O.22r0 + 1.2.(3.5b)where r0 is the conductor radius in centimeters as before. For positive polarityon a bundled conductor:k3 = 1.52— O.lSIogn (3.5c)where n is the number of subconductors in the bundle. For negative polarity on asingle conductor:k3 = OO7r0 + 1.12 (3.5d)For negative polarity on a bundled conductor:k3 = 1.28— O.OBIogn (3.5e)Note that this model introduces a discontinuous jump in the total line capacitancegiven by the slope of the Q—v curve aQ/av.The previously described model has been derived by applying a statisticalcurve fitting process to the vast amount of data that has been collected throughseveral years of experimental research conducted at the Electricite de Franceresearch facilities [85]. Another approach in the development of corona models isto make use of spatial charge considerations. Harrington and Afghahi, for instance,have proposed a model in which the spatial charges are assumed to beconcentrated on several hypothetical concentric shells [55]. These authors reported,however, that their algorithmic implementation of the multiple shell model wasprone to numeric oscillations. They thus used a single shell for their linesimulations [57,58]. A modification to the multiple shell model has been proposedby Semlyen and Huang [70]. Unfortunately, it requires either field or laboratorymeasurements for the determination of some of its parameters. Two additionalmodels have been proposed more recently, one by Al—Tai et. a!. [83] and theother by Li, Malik and Zhao [87]. Both of them are based on spatial chargeconsiderations and can be derived entirely from line information.—41—Often, the proposed corona models provide the spatial charge as a functionof the local voltage only. These models are known as static and are equivalentto Q—v curve representations. Since the early days of research on corona it hasbeen proposed, however, that the derivative of the local voltage with respect totime av/at has also an important influence in the evolution of the phenomenonand, consequently, in the propagation of transient waves on transmission lines, Ithas even been suggested that higher order derivatives of the voltage with respectto time could also be important [56]. Fairly recent experiments [42,77] indicatethat, at least, the effect of avIat should be considered, The models that accountfor it are known as dynamic.One of the most important features of corona, that are influenced byaviat, is the inception voltage. In reality, corona is a probabilistic phenomenonwhose initiation requires the presence of at least one free electron inside theregion where the electric field has reached its inception value, This causes a timedelay between the point at which the voltage (or the field) reaches the inceptionvalue and the point at which the ionization process starts, The concept of meantime delay can thus be established by statistical means, This concept provides anexplanation to the fact that measured inception voltages tend to be bigger forfast rising impulses than for slow rising ones [87]. Another feature of coronathat is affected by the rate of growth of the voltage is the formation of spatialcharge, This effect can be attributed to the mobility of the ions which can beneglected only in the fast occurring phenomena.A dynamic model has been proposed by M. M. Suliciu and I. Suliciu onthe basis of the similarities between corona and viscoplastic phenomena [52]. Onedrawback of this model, as of many others, is the requirement of experimentalinformation. The required information must include, in this case, different q—v—42-curves obtained for different rates of rise of voltage in order to convey thedynamic features of the phenomenon. The model parameters have to be obtained,in addition, through a fairly complicated identification process whose applicationrequires a great deal of expertise [68]. A more convenient approach seems to bethe one proposed by Li, Malik and Zhao. In their corona model, the dynamicfeatures from both the inception voltage and the formation of spatial charge areaccounted for by introducing a time delay parameter [87].An ambitious model, in which the three assumptions mentioned at thebegining of the section are removed, has been proposed by Abdel—Salam and hiscoworkers [71,84]. This model simulates the principal mechanisms of corona.Spatial charges are represented by longitudinal lines of charge. Many of theselines, in the order of the hundreds, are assumed to be distributed around theconductor throughout the entire ionization space. The conductors are alsorepresented by lines of charge which are obtained through the well known chargesimulation method [62]. In order to keep the computations within manageablelimits, the model has been implemented for cylindrical symmetry only. Even inthis case the amount of computation is considerable; consequently, its applicationto line transient simulation is not yet deemed feasible. It is suggested here,nevertheless, that this model is invaluable as an aid for laboratory experiments aswell as for providing benchmarks for the more simple corona models.There are, finally, two important aspects in the modeling of corona fortransient wave propagation that have received scant consideration. They are thecorona effect on multiconductor lines and the behavior of corona during excitationimpulses having more than one peak above the corona threshold. Concerning thefirst aspect, it seems that only recently has a first approximation been proposed[85]. It consists of, first, assuming that the spatial charge is axially symmetrically-43-.distributed around the coronating conductor, then, to form the matrix of potentialcoefficients of Maxwell and, finally, to invert this matrix to obtain the self andmutual capacitances. An advantage of using the potential coefficients matrix, asan intermediate step in the calculation of the line capacitances, is that theassumed distribution of charge only affects the self potential coefficientcorresponding to the coronating conductor. As for the second aspect, Köster andWeck have quoted experimental evidence [49] which leads to the following Q—vrepresentation: after a maximum peak, the effective capacitance takes the linegeometric value for as long as successive peaks remain below vmax; that is, thevoltage reached by the maximum peak. If this peak is surpassed, the Q—v curvecontinues along its parabolic segment from the point at which v = Vmax.3.4) Final Considerations and Recommendations.An overview of the physics of corona and of the techniques to model itseffects on transmission lines has been presented in the previous two sections. Itis apparent that the two issues are highly specialized. It is also apparent thatfurther progress is required for the successful application of corona models topractical analysis and design. Concerning the research work reported here, a modelof corona is needed in the implementation of the proposed transmission linemodel. This model has been selected from among the ones described in section3.3. In order to keep the transmission line equations strictly quasilinear, theselection has been restricted to the static models. The parabolic one proposed byGary, Cristescu and Dragan is the most convenient one for the purposes of thethesis. Firstly, because it has been obtained from experimental data, secondly,because its implementation from some given line information is straightforwardand, thirdly, because the supplementary formulas (3.5b) to (3.5e) cover all casesof practical interest.—44—Furture research work on the corona effect and its modelling should put aspecial emphasis on experimental aspects. In addition, in the opinion of thisauthor, a preliminary stage consisting of computational simulations is highlydesirable. A study of sensitivity on the line models, for instance, could helpdetermine the relevance of the different features of corona to propagationphenomena. It could help, in addition, determine the effect of other factors thatare unrelated to corona on the measured waves, as for instance, the effect ofconductors’ sagging.In addition to the field experiments and to their computer simulation, morestudies should be conducted on scaled down transmission lines, as the onesreported in references [75] and [97]. One advantage of scaled down lines is thatone can eliminate from them many of the factors that corrupt the coronaexperiments on actual transmission lines. Wave propagation measurements alsoprovide valuable information about the corona effect itself. Reference [97]provides q—v curves derived from measured travelling waves. A collection ofthese curves, obtained at different points along the line, would show the effectof the shape of the voltage impulse on the corona characteristic. This effect isvery closely related to the dynamic features of the phenomenon of corona.Finally, it is recommended that further laboratory experiments of the coronaeffect be performed in coordination with studies on an advanced model like theone proposed by Abdel—Salam et. al. [84].—45—4. QUASILINEAR MODELS FOR MONOPHASIC LINES WITH CORONA.4.1) Preamble.As in most studies of lines with corona, in this chapter the Telegrapher’sEquations are adopted as a basis of mathematical line models. However, insteadof combining them into a single second order PDE as in chapter 2, they will betreated as a 2x2 system of first order quasilinear equations. This approach hasthe advantage that the techniques developed can be extended to PDE systems ofhigher dimensions. These extensions are needed to deal with multiconductor linesand with nonlinearities that do not belong to the quasilinear category [15).The Telegrapher’s Equations do not take into account the frequencydependence effects caused by the Skin Effect of the line conductors and of theground plane. Whereas this is acceptable for most studies of lightning and arcingimpulses, where the line sections involved are usually less than 10 km long [94),more general situations may require that frequency dependence be accounted for.—46—This may be done through numeric convolutions [29,56] which, in order todecrease the computational burden of the models, should be performed recursively[37,53].4.2) Systems of Equations for Monophasic Transmission Lines.Equations 2.2a and 2.2b can be put in the form of a 2x2 system of PDE5:+ AfU + BU = o, (4.1)where:= [ 1 (42a)0 1/CA= i/L 0 (4.2b)andB= [ : : I (4.2c)By assuming a static model of corona, where the capacitance “C” is a functionof the voltage “v” only, equation 4.1 becomes quasilinear. The two eigenvalues ofA are:= +i/vi (4.3a)and= —ii/t (4.3b)Since X1 and X2 are distinct and real, (4.1) is a strictly hyperbolic system [60].—47—Associated with the elgenvalues are the following matrices of right and lefteigenvectors, respectively:1 11/?E7i:-V7[ (.)and1 Vt71 (4.5)A and X2 are equal to the characteristic directions obtained in chapter 2 for thesecond order PDE. They thus correspond to the wavefront velocities. Since thesecond row elements of Er have the dimensions of an admittance and the onesin the second column of E have those of an impedance, the following twodefinitions are introduced:= V7E (4.6a)z = VE7 (4.6b)Notice that L and C are evaluated at the frequency considered representative ofthe study in which equation 4.1 is to be applied. Z, differs, thus, from the surgeimpedance as defined in [22], for instance. As it also differs from thecharacteristic impedance, as defined elsewhere, it will be called here “the waveimpedance”. Similarly, Y will be called “the wave admittance”.Equation (4.1) is turned next into an equivalent 2x2 system of ordinarydifferential equations (ODE’s). First, let (4,1), be multiphed by the first row ofE :—48—av ai av ai+ Z-- + Xi + + X1Ri = 0.Then, let the terms in the same dependent variables be grouped:( + X1) + Zw( + X1) + X1R1 = 0.Finally, by restricting this equation to the curves of the x—t plane defined bydx= X1, (4.7)the following ODE is obtained:dv di .dx+ Z-- + Ri--- = 0. (4.8)A second ODE:dv di .dx—Z/ + R = 0, (4.9)is obtained by multiplying (4.1) by the second row of E and by restricting itto the curves in the x—t plane defined by the following expression:dx= X2 (4.10)The two families of curves defined by (4.7) and (4.10) are the characteristiccurves or simply, characteristics of (4.1). As the strict hyperbolicity of (4.1)guarantees that two characteristics pass through each point of the x—t plane,these curves can be taken as a new coordinate system where (4.8) and (4.9) forma 2x2 system equivalent to (4.1) [15]. The main advantage of this newformulation lies in the availability of powerful techniques to deal with systemsof ODE’s.In the lossless case—49—B=O.(4,11)and equation 4.1 is said to be homogeneous. Its solution leads to the Riemanninvariants:v + Zi = k1andv—Zi = k2,where k1 and k2 are two constants. These are the invariants encountered insection 2.3. A numerical method of approximating the solution of the lossy lineequations that removes assumption 4.11 is presented in the next section. Themethod is extended later, in section 4.4, to the quasilinear case for the simulationof transmission lines with corona.4.3) Monophasic Line Linear Model Based on Characteristics.Consider the terminated transmission line depicted in figure 4,la and thecorresponding x—t plane of coordinates of fig., in which the boundaries ofthe wave propagation problem have been represented, It is assumed here that thesolution has been determined at the initial boundary line (I. e., at t = 0 in from the initial conditions of the transmission line. This initial solution canbe extended to neighboring points of the initial boundary through the techniquesdescribed as follows and, furthermore, to any point of the region delimited bythe additional boundaries at x = 0 and at x = I of fig. by theirconsecutive application. In addition to the usual assumptions of time and distanceinvariance of the line parameters, the one of linearity is made in this section, Itthus follows that the characteristics are straight lines, A further implication oflinearity is that the characteristics can be obtained beforehand, since theintegration of (4.7) and (4.10) is independent from that of (4.8) and (4.9).—50—TRANSMISSION LINEFigure 4.1.— Transmission line initial—boundaryproblem, a) Scheme of a terminated transmissionline. b) x—t plane with the line problemboundaries.II IiFigure 4,2.— Extension of the solution from pointsD and E to F.a)SOURCEtLOADSOURCESIDEBOUNDARYLOADSIDEBOUNDARY(xO)INITIAL VAWE BOUNDARY LINEb)xAt/2Ix—51—Let D and E be two points of the x—t plane where the solution is known.They are represented in fig. 4.2, along with their characteristics whoseintersections define two additional points F and F’. Only F however, in thedirection of the t—coordinate increase, is of interest for wave propagationanalysis. Let (X , T) and (X+x , T) be the respective coordinates of D and E.Thus, from fig. 4.2, the coordinates for F are:(X+Ax/2 ,where:= X (4.12)Wave behavior between points D and F is described by equation 4.8, which canbe approximated as follows:(\TfVd) + Zw(I f’ d) + --(I f +1d) = 0, (4.13)where Vd and ‘d are the values of voltage and current at point D, and Vf andare the corresponding ones at point F. Note that, in the third term of (4.8),the current is approximated by the average of its values at the end points D andF. In the same way, (4.9) is approximated between E and F by the followingexpression:TfVe)— Zw(IfIe) Ie+3f) 0, (4.14)where Ve and ‘e are the respective values of voltage and current at point E.Equations 4.13 and 4.14 can be rearranged, respectively, in the following form:Vf + (Z + )If Vd + (Zw )Id (4.15)and—52—Vf— (Zw + Ve — (Z— )Ie .(4.16)Further manipulation of (4.15) and (4.16) yields the following formulas:Vf (Vd+Ve)/2 + (IdIe)Z2 (4.17)If (VdVe)/2Z1 + (Id+Ie)Z2/2Z1, (4.18)whereRxZ, = Z +—— (4.19)andRxZ2 Z—i--- (420)Note that expressions 4.15 and 4.16 lead to the EMTP lossy line modeldepicted in figure 2.2c which was obtained by intuitive and empirical means[27,67]. Expressions 4.15 and 4.16 provide, thus, an additional justification for it.Expressions 4.17 and 4.18, on the other hand, provide Vf and Ic explicitly interms of known values. They are therefore adopted here for algorithmiccomputations. Even though the points D and E of figure 4.2 have been selectedwith the same t—coordinate, the solution extension can be performed with anypair of solved points, provided they are not along the same characteristic.Extensions are also possth)e for points on any of the two additional boundarylines at x = 0 and at x = I. Each of these points, however, has to be at theintersection of the boundary with a characteristic that passes through a solvedpoint. The characteristic provides one equation and the other one, required forsolving the two unknowns of voltage and current, is derived from the boundaryconditions. The technique is illustrated next through two elementary examples.—53—Consider a point S at the intersection of the boundary at x 0 and acharacteristic passing through the solved point E as in fig. 4.3a. Let thecoordinates of E be (x/2 , T). The ones for S are thus (0, T+t/2) and (4.9)is approximated as follows:s1e+IeZ2)/Z,where Z1 and Z2 are as in (4.19) and (4.20). If the boundary condition is that ofan ideal voltage source whose waveform is a function “f” of time only, thesecond equation is:V, = f(T+t/2)and point S is entirely determined.Consider now the point L of fig. 4.3b at the intersection of the boundaryline at x = I and a characteristic passing through a solved point D whosecoordinates are (l—Ax/2 , T). The approximation of (4.8) yields:V1 + Z1 = Vd + Z2Id. (4.21)If the boundary condition is that of a pure resistive load Ri, V1 and Ii arerelated as in the following expression:V1 = Rj•1jwhich, along with (4.21), determines point L solution.In order to apply the above extension techniques to transient simulation ofa line, like the one of fig., a mesh of characteristics has to be constructedin the x—t plane. The construction is as follows: first, the transmission line isdivided into equal length sections by a number of points. For each of them, there—54--tFigure 4.3.— Extension of the solution to boundarypoints, a) From a known point E to a point S onthe source boundary. b) From a known point D toa point L on the load boundary.Figure 4.4,— a) and b), two possible meshes ofcharacteristics for linear transient simulation.E— ——x(a) (b)IINITIAL VALUE POINTS(a) (b)—55—is another corresponding point, hereafter called the initial value point, on theboundary line of initial values of the x—t plane. This is illustrated in figure 4.4a.Next, the characteristics of each initial value point are plotted in the direction ofthe t—coordinate increase and, as soon as one of them intersects one of theadditional boundaries (either the one at t = 0 or the one at t = I) the plot iscontinued along the second characteristic of the intersection point. Finally, theplot is continued until it spans the observation time required for the simulation.Figures 4.4a and 4.4b show two possible mesh constructions. The one shown infig. 4.4b was adopted here, since it is equivalent to the one required fornonlinear transient simulations.Figure 4.5 shows the simulation of a 10/90 jis double ramp impulsetraveling along a 50 km long transmission line which is terminated in a 360 2load. The line parameters are L = 1.25 mH/km, C = 9.664 nE/km andR = 11,74 2/km. The line was divided for the simulation into 174 sections. Theplots correspond to the travelling wave obtained at 0, 5.17, 10.34 15.52 and20.69 kms distance from the source. The same results were obtained with EMTP©MicroTran (see fig. 2.3). Two of the main features of frequency independentmodels of lossy lines are apparent in fig. 4.5. The first is that the corner of thedouble ramp is preserved. The second is that the width of the pulse broadens asit travels along the line.Concerning the simulation of linear transients by the method ofcharacteristics, there are three issues that should be addressed; namely,convergence, accuracy and number of line subdivisions. The three of them areinter—related. The convergence of discrete approximations of linear hyperbolicPDEs is guaranteed by the Courant—Friederichs—HHbert (CFL) condition [3] which,—56—for equation 4.1, can be written as follows:ií/t (4.22)Note that expression 4.12 is in fact the equal sign option of (4.22). It thusfollows that the method of characteristics fulfills implicitly the CFL condition.Regarding the accuracy, it has been shown [27] that the fig. 2.3 model is anaccurate representation of a lossy line inasmuch as:RAx << Z. (4.23)It has been shown through numerical experiments [67] that when the value of Zis ten times larger than that of Rx, the accuracy is acceptable for engineeringapplications. Furthermore, it is always possible to meet condition 4.23 byincreasing the number of line subdivisions. In this respect, it can also be saidthat (4.23) determines a lower bound for the number of line subdivisions. Anotherfactor to consider is the specification of a maximum sampling interval t, sinceit determines through (4.12) the maximum length x of the line sections and,therefore, the minimum number of subdivisions that are required.4.4) Quasilinear Model for a Line with Static Corona.When the line capacitance is a function of voltage, the eigenvalues of A in(4.1) are also functions of the voltage and the characteristics become curved linesin the x—t plane. In consequence, the integration of (4.7) and (4.10) cannot bedone independently from that of (4.8) and (4.9). The extension of solutions toneighboring points can be done, nevertheless, through iterations. Two techniqueshave been proposed to do this [14]. In one of them, for each pair of solvedpoints that are not on the same characteristic, each iteration yields a betterapproximation to the coordinates and the solution of the crossing point of the—57—Figure 4.6.— a) Regular grid for nonlinear linecalculation. b) Finding the interpolation points.In00±>d80.0 Z5.0 50.0 5.O 100.0 i2..o 150.0Time jiec.Figure 4.5.— Simulation of a linear double ramptraveling‘ISD EAF-t—1-TJrx(a)Fx+4’e+ 1XxeC+(b)—58--characteristics in the t—coordinate direction. Since the characteristics are curved, itis clear that this technique produces an irregularly distributed grid of solvedpoints. In the other technique [9], a regular grid of points is chosen beforehand.The solution points are then forced to coincide with the nodes of the regulargrid by means of interpolations. The first technique, hereafter called “purecharacteristics”, is a complement to analytic studies. The latter one, hereaftercalled “characteristics with interpolation”, is more suitable for engineeringapplications and, for this reason, it is adopted in this section.Consider a regular grid as shown in fig. 4.6a. Usually in practice, the timestep At is determined by the minimum time resolution needed for an adequatedescription of the important features, such as rise time, of the particularwaveform excitation under study. From this value, the distance step Ax is fixed insuch a way that the ratio Ax/At is at least equal to the largest possible Aeigenvalue or wave speed. Suppose now that the solution is known at the nodepoints on the line t T of fig. 4.6a. Let D, E and F be three of these points,whose respective values of voltage and current Vd, ‘d. Ve, 1e Vf and I,c are tobe used in determining the point G solution; that is, Vg and Let thecharacteristics be approximated, between lines t = 0, t = At, t = 2At, etc., bystraight segments. Let Xd, Xe, Xf and Xg represent the positive eigenvalues of Aat points D, E, F and G. The following steps are then applied:1. Assume Xe initially as the eigenvalue of A at G. Let Axd”e be the distancefrom point D”, where the G—characteristic with positive slope intersects theline t = T, to point E and Axef” the distance from point E to point F”,where the G—characteristic with negative slope intersects t = T. Their valuesare calculated as follows (see fig. 4.6b):Axd”e XgAT—59—tXef” = Xd”e2. Obtain first order estimates of current and voltage at points D” and F”through the following linear interpolation expressions:Vd” = Vdkl + Ve(1k1)Idkl + I0(lk1)Vf” = Vfk1 + Ve(1k1)If” Ifk1 + Ie(lki)where k1 = Xde/Ax3. Obtain a first order estimate of the voltage and current of point G, fromthe following expressions:VgVd” + Zwd,,(Ig d”) + (I g + Id”)RXd”e/2 0VgVf + Z1,,(Ig1 f”) + (I g + I f”)RXef”/2 = 0where Zwd. and Zwff. are the wave impedances evaluated at D” and F”.4. Obtain from V9, the eigenvalue X9 of A at point G. Obtain two newdistances.xd”e and xef” from the following expressions:txef” = T(Ag +5. Obtain second order estimates of Vd”, ‘d” Vf” and Ii” through thefollowing quadratic interpolation expressions [32]:Vd” Ve - k1 (VfVd) + k2 (“d +Vf 2Ve)I— kl(IfId) + k2(Id+If 21e)Vf” = Ve + k3(Vf-Vd) +4(Vd+Vf2Ve)—60—1e + k3(IfId) + k4(Id+If 21e)where k1 = AXd”e/ k2 = (k1)2/ k3 = ef”/x and k4 = (k3)2/6. Obtain a second order estimate of Vg and 1 from the expressions:VgVd” + + (Ig+Io”)RXd”e/2 0VgVf + (IgIf”)(Zwf+Zwg)/2+ (Ig+I f”)RXef”/2 = 07. Repeat steps 4 to 6, until the desired convergence is attained. Theconvergence criterion used here is that the relative difference between thenew value of V9 and the one from the previous iteration is equal or lessthan 10-6.The extension of the solution to boundary points is similar to thepreviously described procedure. In much the same way as in the linear algorithm,a general boundary condition is described by a difference equation which isincorporated into the iterative process along with the approximation of thecorresponding propagation equation. Very complex boundary conditions can behandled by combining the above described procedure code with a general timedomain transient simulation package, such as the EMTP.As an example, the iteration method is applied to the transmission linedepicted in figure 4.7a, whose linear parameters are C0 = 9.664 nF/km,L = 1.25 mH/km and R = 11.743 2/km. The model used for corona is the staticone proposed by Gary, et. a?. in [85] and given by equations 3.10 and 3.11. For aconductor radius r = 2.54 cm, expression 3.11 yields a value of = 1.7588.Figure 4.7b shows the distortion of a 2/20 j.ts double ramp impulse with a1.0 p. u. initial amplitude as it propagates. The corona inception voltage v0 hasbeen taken at 0.3 p. u. The plotted waveforms correspond to x = 0 (the source0InCD0.2Et0oa.0)C4-I’%dCU.‘qoto.9H1o1€)UU.4-U(J.(II0.JNI-.’-Q0LI)00(.40m000C)IU’Uo-CN(0oU)U)o°-c0c4i00side), x = 571 m, x = 1142.9 m, x = 1714.3 m and x = 2285.7 m. The line hasbeen made long enough, x = 9 km, so that the reflections from the load sidedo not play any role in the simulation during the observation time t 30jis.Figure 4.7c shows a portion of the map of characteristics corresponding to fig.4.7b. The same simulation is made for a shorter line length (I = 3 km) and aload impedance equal to the surge impedance:Z8 == 360.0 2The resulting waveforms are shown in figure 4.8 where one can observe that thisload termination produces some reflection.One of the important differences between linear and nonlinear lines isillustrated by fig. 4.9, where the same parameters as for the simulation of fig.4.7b have been assumed. The line is made long enough to exclude the reflectionsfrom the load end during the simulation time and the travelling waveform atx = 3 km is obtained and plotted in the figure. If the line was open at themeasuring point, instead, one would expect from experiences with linear lines thatthe previous waveform would double. Figure 4.9 presents both, the simulationresult and the double of the original waveform. One can notice there that theformer wave is smaller than the latter one.4.5) Performance of the Proposed Quasilinear Method.A problem with many of the methods of discretization of the lineequations is that the chosen step lengths, X and T, provoke numericosciflations [23] which could result in a large ripple at the tail of the simulatedpropagating wave. Such methods, therefore, are not adequate for studyingphenomena that involve reflections. With the method proposed here the numeric—63--Figure 4.8.— Simulation of a double ramp on a lineterminated in its linear surge impedance.Figure 4.9.— Transient wave at 3 km distance fromthe source for both, open—ended and semi—infiniteline. The double of the semi—infinite line responseis included as reference.Time— ILSO.O 2&OTime ——64-oscillations are minimized. It can be, in fact, observed that figs. 4.7b, 4.8 and 4.9are practically free of them. Two additional simulations are made for the line offig. 4.7a with a line length of I = 3 km. The first one with the load side openended and the second one with a shortcircuited load. These terminations arechosen in order to produce large reflections as a test for the proposed method.Figure 4.lOa shows the results of the first simulation. One can notice there thereflected wave with the same polarity as the incident wave which is characteristicof open ended lines. Figure 4.lOb shows, on the other hand, the reflected wavewith polarity opposite to the one of the incident wave due to the shortcircuitedtermination.In order to show the effect of the discretization step length on theaccuracy of the method of characteristics with interpolation, fig. 4.lOa simulationis repeated with different numbers N of line sections ranging from N = 12 toN = 270. Figures 4.11, 4.12 and 4.13 show the respective results for 270, 18 and12 line sections. These plots should be compared among themselves and with fig.4.lOa plot which was obtained with 105 line sections. Figures 4.lOa and 4.11 arepractically identical. They both predict almost the same magnitudes for theovervoltage peaks and the wavefront delays. The finer resolution of the latterfigure is apparent only after a detailed examination; see for example the secondpeak of the waveform obtained at the distance x = 571 m from the source side.The comparison of figs. 4.12 and 4.13 with fig. 4.lOa shows that even thecoarsest discretizations yield acceptable results. This is remarkable, especially forfig. 4.13 simulation, where the time step iT = 0.8689 jis implies that the initialwavefront is described by two samples only. The corresponding distance intervalof X = 250 m contrasts favorably with the more common figure of= 10 m that is normally required by other methods in similar simulationconditions [651.—65—U)C04-t‘Ib-C.6w>t—z00>b.6ö4-I..C-C00zEai1wdvCa)u-rnC0EIa(0(0owiwaoncroau-u-nd—C04-.E04-.Ca)0C(aI-4-4)CVa)-DoC.-’a)cEa)0)a000a)I.Cri.4.0NC’1a)NCoU)a)C04-LuzE04-.Ca)U)C(UI.-4-Li)a4)0a)0CU)Ca)’ISII;cr0ococaoaejoThe abovementioned features of the proposed method can be attributed inpart to its iterative nature. The number of iterations that are required in thenonlinear part of the wavefront is usually three and a maximum of four has beendetected. The first implementation showed convergence difficulties at the pointswhere the corona capacitance model had a discontinuous jump; that is, at thepoint at which the wavefront voltage reaches the corona inception voltage and atthe wavecrest where av,at changes sign. The largest differences found at thesepoints were below 1 %. All the convergence difficulties at the crest and most ofthem at the corona inception point disappeared when the jumps were replaced byfast but continuous transitions. They were implemented by performing linear orquadratic interpolations inside a distance interval AX. This approach has the addedadvantage that finer discretizations automatically provide better approximations ofdiscontinuities. As for the physcal possibility of discontinuous jumps of thecapacitance, this is an issue that requires further consideration. More about it ismentioned in the next section.The possibility of a shock condition developing at the crest of a wave ona line with corona has been a concern throughout the research reported here.Nevertheless, such a condition has not been detected so far. The map ofcharacteristics of fig. 4.7c, for instance, shows that some of these curves becomecloser as the time increases. The effect is more evident in fig. 4.14 which is acontinuation of fig. 4.7c. Although it seems there that some characteristics aremerging, a detailed review of the numerical data shows that they only approacheach other asymptotically.Figure 4.15a shows the results of one of the field tests performed byWagner, Gross and Lloyd [11]. To establish a comparison between the actualexperiment and a simulation, a 1.3/6.2 jis double exponential impulse with an—68—0‘UEU)Ca)U)CLU4-Ia)•1..a)•0.a)Ca)0.0)‘a)I-—iiEOW.C>o=I..a)E(fla)U)4-I.a)‘-‘U...4..ULUI_4-Ia)0)og0.a)(0.0a)I.>oa)U).C4-II.0..0)o;ss?1—auiijamplitude V = 1560 kV is considered. This excitation is injected at the pointx = 0 of a line whose parameters are assumed to be the following;L 1.73 RH/km, C 7.8 pF/km and R = 11.35 2/km. The parameterscorrespond to a conductor radius r 2.54 cm, a conductor mean height abovethe ground h 18.9 m, an earth resistivity p 30 m and a frequencyf = 192.3 kHz, The frequency is chosen as the inverse of four times the time topeak of the excitation impulse. The excitation impulse is plotted in fig. 4.15b,along with the simulated travelling waves at x = 655.6 m, x = 131125 m andx = 2185.4 m. These waves compare well with the experimental curves, It can beobserved that the model reproduces the transfer of energy from the wavefront tothe wavetail. There are noticeable differences between the two sets of curves atthe nonlinear portion of the wavefront. They are attributed here mostly to thefollowing factors:1. In the simulation corona is represented by a static model instead of by adynamic model,2. In the real line the conductor height varies between towers from 15.24 m to26.2 m. In addition to the line impedance variations, the corona inceptionvoltage is affected too.3. The real line is a polyphasic system consisting of two horizontal threephase lines in parallel and with their respective ground wires, whereas theline model is monophasic.4. The earth resistivity is an estimate only.4.6) Observations and remarks,It is shown in this chapter that the method of characteristics of the partialdifferential equations theory is well suited for the analysis and simulation oftransients on transmission lines with corona, The implementation presented here—70—400TIME- usa)Figure 4.15.— Comparison between measurementsand a computer simulation, a) Measurementsperformed by Wagner, Gross and Lloyd andreported in ref. [1 1]. b) Simulation of Wagner,Gross and Lloyd’s experiment.76007200800Ui 2 3 4 6Time —b)—71—solves some of the problems found with other methods; namely: numericaloscillations, very fine discretization requirements and not predicting well theevolution of travelling wave tails. The advantages, however, come at the price ofan increase in the number of computations, If the average number of iterationsrequired is three, one could say that the proposed method takes four times thenumber of calculations required by a non—iterative one (one time for the firstestimate plus one for each iteration). This is by considering that both methodshave the same discretization level, The computational increase, nevertheless, iscompensated for by the ability of the method to perform well with much coarserdiscretizations. In addition to the aforesaid features of the proposed method, itshould be said that its major advantage perhaps is the possibility of extensionsin a rigorous manner for the handling of dynamic corona and of multiconductorlines.The examples of the chapter suggest at least two points to consider in theelaboration of further models of the corona effect. The first one is concernedwith the discontinuous jumps of the line capacitance implied by several coronamodels, The second one is concerned with travelling waves with multiple crestsabove the corona inception voltage.From a physical point of view, a discontinuous jump of the capacitanceinvolves an instantaneous transfer of energy. At the corona inception voltage, itmay be possible that the actual capacitance makes a sharp but continuoustransition. See for instance fig. 4.16a which was obtained experimentally by Kösterand Weck [49]. As for the wavecrest, a Q—v curve like the one illustrated in fig.4.16b would be required to keep the transition continuous. It is interesting tonote that such a characteristic has been reported in some experimental studies[12,97]. A rounded corner, on the other hand, like the one depicted in fig, 4.16c—72-.would introduce a negative value of 3q/8v in addition to the jump. This negativevalue would make the line equations elliptic which implies that, if they still havephysical meaning, they do not describe wave propagation anymore.The analysis of propagation of waves with more than one peak above thecorona inception voltage requires the corona model to be able to represent thephysical phenomena ongoing between peaks. In this chapter, the simulations thatinvolved reflections were made by assuming the geometric capacitance as thecorona capacitance at the intermediate voltages below a prior maximum peakvalue. This assumption is suggested by experimental observations made by Kösterand Weck [49].From all the above, it is apparent that further progress in the subjectrequires more experimentation aimed at the determination of the features of thecorona effect that are relevant to wave propagation, especially to the dynamicones. Meanwhile, the author considers that the method proposed here is avaluable tool for the analysis of data from experimental lines.—73—4pF/m32a)10Figure 4.16.— a) Corona capacitance obtainedexperimentally by Köster and Weck and reportedin ref. [42]. b) Q—v curve form for a continuouslyvarying corona capacitance. c) Rounded form ofthe q—v characteristic. The slope between P1 and2 is negative.500 1000U0curve 1 : measured corona capacity of the conductor1500 kV 2000curve 2 corona capacity from equation (3)cib) c)vc Vmax V-74-5. LINEAR AND QUASILINEAR ANALYSIS OFMULTICONDUCTOR TRANSMISSION LINES.5.1) Multiconductor Linear Lines.The following equations describe a linear multiconductor transmission lineunder the assumption of frequency independent parameters [17,18]:= L.-- + RI ( C- (Sib)where L, R and C are, respectively, the matrix of series inductances, the matrixof series resistances and the matrix of shunt capacitances, all in per unit length,and V and I are the vectors (or column matrices) of voltages and currents. If theline has n conductors, the order of L, R and C is nxn.Let ( and ( be written in the following form:—75—aV 0 C1aV 0 0 VãT + + =0 .(52)L1 0 1 0 L1R Iwhich constitutes a system of 2n first order partial differential equations (PDE’s)with x and t as independent variables. By definition, (5.2) is hyperbolic if thematrix0 C1A = (5.3)L1 0is algebraically similar to a diagonal matrix whose nonzero elements are real [60].In order to show that this is indeed the case, the inverse of (5.3)0 L= (5.4)Cois analyzed as follows. It will also be shown that the eigenvalues andeigenvectors of (5.3) and (5.4) are related to the ones of the LC matrix product.Let T be the matrix that diagonalizes the LC product in the followingform [69]T1 LC T = (5.5)According to the conventional modal theory of transmission lines [19], Tvcorresponds to the matrix whose columns are the voltage modes of atransmission line without resistive losses. The nonzero elements of the diagonalmatrix A are the inverses of the squared speeds of the corresponding modes[18]. The physics thus requires the diagonal elements of A to be real andpositive. Transmission line matrices L and C are real and symmetric. Since it isassumed here that they are positive definite, the LC eigenvalue problemLCV=XV—76--is equivalent to the generalized eigenvalue oneC V = AL1Vwhich is known to have real eigenvalues [25). In practice, as the diagonalelements of LC are positive and much bigger than the off—diagonal ones, theGershgorin theorem [82] can be used to show on a case by case basis that theeigenvalues are positive.Another important matrix is T, the one of column current modes, whichdiagonalizes the CL product; that is:T CL T1 = A (5.6)Tv and T1 are related as follows [19]:TZ, = T1 (5.7)Two additional matrices from conventional modal theory are still required. Theyare denoted here by L’ and C’ and arise as a direct consequence of thediagonalization of LC product. From (5.5),T,1 L T1) C Tv = A (5.8)By defining:L’ = T,1LT (5.9a)andC’ = T 1 CTv (5.9b)(5.8) becomes:L’C’ = A (5.9c)It can be shown that L’ and C’ are diagonal matrices [19,54).—77—Let E L and Er be the two 2nx2n matrices such that:EL Er = 1 (5.10)andF1 0ELAEr = (5.11)0 F2where 1 denotes the unit matrix and F1 and F2 are two diagonal nxn matrices.E L is called the (left hand side) matrix of row eigenvectors, while Er is calledthe (right hand side) matrix of column eigenvectors. Let it be assumed that Erhas the following form:TM TvMEr = a (5.12)T1N .-T1Nwhere M and N are proportionality matrices to be determined later. According to(5.10), EL must have the following form:MT,’ N-1TEL = a (5.13)MT—NT11with a = (1 /i/2).On applying (5.12) and (5.13) to (5.11):p11 p12ELA1rp21 p22whereP11 1/2(MTLTN + NTjCTvM) = F1p12 = (—MT,LTN + N Tj’CTM) = 02l =1/2QV[’TLTN- N’T1CTvM) = 0—78—=—N1TjCTM) = I’2By further applying (5.9a) and (5.9b) in the above expressions:p1 l/2-1L’N + N.1C’M) = F1 (5.14a)= 1/2(_M1L’N + N1C’M) = 0 (5.14b)l’2l ‘/2Qvi1L’N — N1C’M) = 0 (5.14c)p22 l/2(_M1L’N — W1C’M) 0 (5.14d)it is seen then, that if Er and EL are of the forms given by (5.12) and (5.13):r =-r2 (5.15)andM1L’N = N1C’M (5.16)By further choosing M = 1, (5.16) is satisfied by the following diagonal form ofmatrix NN = /L’ 1C’ (5.17)and (5.14a) becomes:1 = F1 (5.18)It is straightforward to check that:(F1)2 = (F2) = A (5.19)which also agrees with (5.9c). Expression (5.19) shows that the eigenvalues ofA 1 are real, and this implies that the ones of A are real too. Due to the factthat N has the dimensions of an admittance and that N and N 1 will playimportant roles later on, the following definitions are introduced:Z VL’C’- 1 (5.20)and—79—= /L’ 1c’.(5.21)In addition to providing support to the assumption that (5.2) is hyperbolic,the above results provide a convenient way to obtain the elgenvalues andeigenvectors of the system. First, because the diagonalizing process is applied tothe nxn matrix LC instead of to the 2nx2n matrix A 1 and, second, because thenew eigenvalues/eigenvectors are related to the more familiar ones from theconventional modal theory [54]. Summarizing:T TvEr= (1/V2) (5.22a)T1Y TjYwT .1I WIE L = (1/V2)T 1(5.22b)andr1 o r1 o= (5.22c)0 F2 0 —F1whereF1 = (5.22d)orF = (5.22e)Expression 5.2 can be transformed to a more convenient form byleft—multiplying it by EL’ yielding:( + ZW(Hm+Ffm) + FT RI = 0 (5.23a)and(Fm_rFm) -- 1’T,Rl = 0.(523b)where:Vm = T1V, (524)— T1l 525m— iandF = (I’ 1 ) 1 = 1C’ 1 (5.26)Note that Vm and ‘m correspond to the vectors of voltage and current in themodal domain from conventional modal theory with the difference that thepresent analysis is based on the real matrix LC rather than on the complex ZYone. Note also that the j—th diagonal element of r, is the velocity of thej—th mode.Expressions (5.23a) and (5.23b) define 2n pairs of equations in the followingform:.Ymj +7J.mJ) + ZJ(.mj+7mJ)+Rjk 1k) = 0 (5.27a)and( )— ZWJ(4mJ_7J.mJ)— 7i(k/ikIk) = 0 (5.27b)where the single sub—index j denotes the j—th element in the case of the columnmatrices Vm and tm and the j—th diagonal element in the case of the diagonalmatrices F and Z. R]k is the element on j—th row and k—th column of matrixR’, which is defined as follows:R’ = TREquations (5.27a) and (5.27b)become:—81—dVmj + ZwjdImj + (ZRjkIk)dX = 0 .(5.28a)anddVmj — ZwjdImj + (ZRjkIk)dX = 0 (5.28b)along their respective characteristics, namely:dx= (5.28c)and=—y, (528d)Expressions (5.28a), (5.28b), (5.28c) and (5.28d) should be compared with theirmonophasic counterparts; i. e., (4.8), (4.9), (4.7) and (4.10). It can be said, in theform of a summary, that the linear transformation defined by E L turns themulticonductor line equations 5.2 into a system of 2n first order ODE’s.5.2) Numerical Solution of the Linear Multiconductor Line Equations.Consider the two lines parallel to the x—axis shown in figure 5.1. Considerfurther that the solution of the multiconductor line equation is known along theline t = T and that it is to be extended to the point G on the linet = T + AT. According to the results of the previous sections, there are 2ncharacteristics passing through G which provide an equal number of equations inthe form of (5.28a) or (5.28b). These equations are sufficient to determine the 2nunknowns at point G in terms of the data at the points where the characteristicsintersect the line t = T. In case of two or more equal eigenvalues, whichimplies that their characteristics become one, the associated equations will stillbe different inasmuch as the eigenvectors are linearly independent. Consider nowthat the solution along t = T has been produced numerically and, therefore, isdetermined only at a finite number of points. It is clear that these points will—82—tAplane.Figure 5.1.— Characteristics of a multiconductorlinear line passing through a point in the x—tAtFigure 5.2.— Positive and negative slopecharacteristics intersecting the line t— 1.—83--not necessarily coincide with the characteristic intersection points and, furthermore,it may not be possible to coordinate the different characteristic meshes in orderto produce a useful distribution of solution points in the region of the x—t planedelimited by the initial—boundary condition lines. A solution to this problem isproposed next. It consists of the use of interpolations in much the same way aswas done in chapter 4 to generate a regular grid of characteristics.Let denote the j—th positive eigenvalue of A with j = 1 n. Letdenote its negative value which is also an eigenvalue of A. Suppose that thepositive eigenvalues are ordered as follows:0 <7, 7fl..1Suppose also that the solution points are equally spaced along a segment of theline t = T and that AX is the spacing between two successive points. Figure 5.1depicts three of these points denoted by D, E and F. Let AT be determined asfollows:AT = AX/7 (5.29)Note that such AT is the smallest one possible. Let denote the characteristicwith slope 7+j and F..j the one with slope ‘y.j. Note also from fig. 5.1 that thepoint G on t T + AT is chosen in such a way that the F.4. i characteristicpasses through D and that the F. 1 one passes through F. Equations 5.28a and5.28b can be approximated along .. and F..,respectively, as follows:V,1 + Zw(IiI1)+ = 0 (5.30a)andF F AX FViVmi + Zw(IiImi) + Z Rk(Ik+Imk) = 0 (530b)k=1where V, 1. 1, 1 and 1 represent the values of the first component ofthe modal voltage vector at the points D, E, F and G, respectively, 1. 1,and I, 1 the values of the first component of the modal current vector atpoints D, E and F and Rjk is the element on the j—th row and k—th column ofthe matrix R”, which is defined as follows:R” = (5.31)Consider now any other two characteristics and f’.1, where j = 2, ..., n.Along them (5.28a) and (5.28b) become:V,1 + Zw(IjI,j) + 2kl;k(rnk+rn) = 0 (5.32a)and— Zw(I,JIj)— k,k)I1k+tt) = 0 (5.32b)where D1 and F1 are the intersection points of the and P.1 characteristicswith the line t = T. It can be seen in fig 5.2 that D1 and F1 are at a distancefrom point E given by:= ‘yJAT (5.33)The evaluation of the dependent variables at points D1 and F1 can be donethrough interpolation. From the quadratic interpolation formulas of section 4.4,Fjand Vmj become:Dj D E FVmj QijVmj + 2jVfl7J + a3jVmj (5.34a)andV, = a3jVj + a21V,1 + a11V, (5.34b)where:DaijVmj = r1 + ‘p /2= 1 — (rj)2 (5.34d)a31v1 = (rj)2/2— rj (5.34e)and—85—r1 = AXi/x=.(5.34f)The same formulas apply for the other dependent variables ...,Fj Fj Fj .‘mi, ‘m2 ‘mn• Equations (5.30a) and (5.32a) with the interpolation formulasincorporated into them can be put into the following more convenient form:+ (2w +R”)l = a1 [ V +(ZwR”)l ]2[ V+(Zw-R”)I )a[ V+w-R”)I ] (5.35a)where:= diag (X,AX2 (5.35b)a1 = diag (1,a 2 ,a1) (5.35c)a2 = diag (O,a22 a) (5.35d)a3 = diag (O,a32 ,a3fl) (5.35e)In the same manner, equations (5.30b) and (5.32b) with the interpolation formulaslead to the following expression:V- (Z+R”)I = a3 [ V-(ZR”)I IEa2[ VmwR )1ma 1 [ ] (5.35f)Expressions (5.35a) and (5.35f) provide the solutions of voltages andcurrents of point G in terms of the known values of these variables at points D,E and F. Note that the addition of these two expressions yields an explicitsolution for the voltage at G, while their subtraction yields an expression for theunknown currents at G without the unknown voltages. An additional feature of(5.35a) and (5.35f) is that the matrices involved in them are constant; inconsequence, their numerical implementation is computationally very efficient.—86—5.3) Multiconductor Quasilinear Transmission Lines.The relatively simple expressions obtained in the previous two sections formulticonductor linear lines were possible because the elgenvectors of the systemwere constant. In the quasilinear case they are variable and, furthermore, theycannot be determined beforehand as they depend on the solution. Consider nowthat the elements of matrix C of (5.2) depend on the components of the voltagevector V. For convenience, (5.2) wiW be represented as follows:+ A.-! + BU = 0 (5.36a)whereVU = , (5.36b)0A (5.36c)L1 0and00B = (5.36d)0 L1RNote that (4.1) is a special case of (5.36) in which A and B are 2x2 matrices. LetEr, EL and F be the right eigenvectors matrix, the left eigenvectors matrix andthe diagonal matrix of eigenvalues of A, respectively. They have the forms givenby expressions (5.22a), (5.22b) and (5.22c); however, the sub—matrices T, T, Z,and Y depend now on some of the components of U. Let Urn be the vector ofdependent variables in modal domain; that isU = ErUm (5.37)The substitution of (5.37) into (5.36a) yields [60];—87-.Erm +AErm + (r+Ar)u + BU = 0By applying (5.11):Er(frn+1jm) + (.r+A+B)U = aand by multiplying by E}1m + rm + (E}1 i.r+rE-1 r+E1B)J = 0 (5.38)Expression 5.38 represents 2n equations of the form:+71.mJ + E Lj(’ +yJr +B)U = 0 (5.39)where E is the j—th row of E L Along the curvedx= -7] (5.40a)equation (5.38) becomes:dU 2nmj + k=ljkmk = 0 (5.40b)where bjk is the element on the j—th row and k—th column of the matrix B’,which is defined as follows:B’ = L+FEL’+ELBEr (5.40c)Note that a consequence of having variable transformation matrices, Er and E L isthat the partial derivatives of Er with respect to x and t are present in each ofthe 2n equations of the form (5.39). An alternate approach to the solution of(5.2) is presented as follows [15]. Let (5.36a) be left—multiplied by EL:+ ELA.- + ELBU = 0—88—On applying (5.11), the following expression is obtainedE L + 1’E L + E LBU = 0 (5.41)which contains 2n equations of the formE Lj(7j) + E L1BU = 0 (5.42a)They are sufficient for determining the 2n unknown dependent variables. Theparenthesized part of (5.42a) can be interpreted as the differentiation of U in thedirection given by:= —‘y,, (5.42b)where (542b) defines a curve in the x—t plane which is hereafter denoted by I’(with j = , 2, ..., 2n). Along this curve (5.42a) becomes:+ ELJBU = 0(5.42c)An interpretation of expression (5.41) is illustrated in figure 5.3. Let G be apoint on the line t T + T; the 2n characteristics passing through it arecurved now. They intersect the line t = T at the points denoted by D1, D2D2. The 2n equations of the form (5.42c) relate the solution at these points tothe one at point G.In the linear case the first n characteristics were mirror images of theadditional ones. In the quasilinear case, however, this will not always be true. Thelack of symmetry, along with the variability of the eigenvalues and eigenvectorsof A, makes the nonlinear case much more complicated than the linear one thatwas discussed in sections 5.1 and 5.2. One consequence of this is that a moreexplicit form of (5.41), which is obtained by applying the definitions of EL, I’, U—89--and B given by (5.22b), (5.22c), (5.36b) and (5.36d), is cumbersome. For this reasonfurther development is based on (5.41).tt= TxFigure 5.3.— Characteristics of a multiconductorquasilinear line passing through a point in the x—tplane.5.4) Proposed Numerical Solution of the Quasiliriear Multiconductor Line Equations.From the two alternate ways for dealing with the quasilinear multiconductorline equations presented in the previous section, the second one which led to(5.41) seems to be the most convenient one for numerical implementation. Let theeigenvalues of A be denoted now by ‘y (with j = 1, 2 2n) and let thefollowing order be assumed among them:7fl+1 7fl-2 .... 72fl < 0 < 7 7fl.j •... ‘LiThe integration of (5.42c) along a segment of between D1 and G can beperformed by appealing to the mean value theorem, yielding:=T +T—90—E LjQ - uDJ) + T(E LjPU = 0 .(5.43)where the tUde denotes the mean value of the corresponding variable. Anumerical approximation of (5.43) is obtained by replacing each mean value by itstwo end point average:+ E’?X1..P - + + E)B(U’ + uDJ) = 0 (5.44)Further algebraic manipulation leads to the following expression:( + + lTBG = (Ej +- 1BJ) (5.45)Suppose now that the intersection points D1, D2,..., D2n are always between thesolution points D and F, as indicated in fig 5.3. uD1 in expression (5.45) can thusbe obtained as follows by means of quadratic interpolation:uDj = ai1LP + a21UE + a3ji! (5.46a)where= (rj)2/2— rj (5.46b)a2JV,J = 1 — (rj)2 (5.46c)a3JVJ = rj + (rJ)2/2 (5.46d)and= j(AT/Ax) (5.46e)Similarly, is determined as follows:Dj D E FELI = a1JELJ + a2IELJ + a3JELJ (5.47)By applying (5.46a) and (5.47), (5.45) becomes:C D E F T CLI+a1jELI+a2IELJ+a3JELJ)(1+.—B =a 1 jE+a 11E+a21E1+a31E)(14.IB)iJ’ +—91—G D E F T Ea21(E U -Fal 1E U +a21EU +a31ELj)(1B)U +a31 +a23 ’)(1-4.IB)L( (5.48)The 2n equations defined by (5.48) can be grouped into the following matrixexpression:E’B’LP = aiEBUD + a2EBUE + aaEBUC (5.49a)wherea = diag (a1 1 ,a 1 2 ,a1 ,,) (5.49b)a2 = diag (a2 i ,a2 2 ,a2fl) (5.49c)a3 = diag (a3 ,3 2 a.3fl) (5.49d)E’ = E + a1E + a2E + a3E’ (5.49e)B’ = I + ‘B (5.49f)B” = 1— #‘B (5.49g)Expression 5.49a can be used to determine by means of an iterative process,such as the one proposed next.1. Initialize by assumingC EEL ELand2. Assume that F, the diagonal matrix of mean eigenvalues, is equal to andobtain the initial matrices of interpolation coefficients a1, a2 and a3 byapplying (5.46b), (5.46c), (5.46d) and (5.46e).3. Obtain an initial estimate of by applying (5.49a).4. Evaluate matrix A at point G by using the most recent estimate of—92—G G5. Obtain new EL and F from the new matrix A.6. Obtain the diagonal elements for a new matrix of mean eigenvalues F asfollows:= + a1 [/3 + 2 + a3 = 1,2 ,...,2n)7. Obtain new matrices of interpolation coefficients a1, a2 and a3 byapplying the new 3j’5 to expressions (5.46b), (5.46c), (5.46d) and (5.46e).8. Obtain a new by applying (5.49a).9. If the new and its previously estimated value are equal within apreselected tolerance, convergence has been attained; otherwise, take the newU as the most recent estimate of and repeat the process from step 4onwards.5.5 Remarks.The extension of the method of characteristics to multiconductor linesproposed in this section provides an attractive alternative, even in the linear case.One problem of the more traditional approaches for handling these lines in thetime domain stems from the fact that they based on the eigenvalue/eigenvectoranalysis of the ZY matrix product. This introduces imaginary terms that lackphysical meaning in time domain. By using the modes of the LC product, instead,the analysis is confined here to the real domain.As for the analysis of transients on multiconductor lines with corona, mostmethods so far proposed resort to constant modal transformation matrices aswell as to the linear superposition principle. The removal of these twoassumptions is made possible in the method proposed here at the expense of anincrease in the number of computations. It is believed here, nevertheless, that itsnumerical implementation will prove valuable as a principal method in practical—93—analysis as well as in providing benchmarks for simpler techniques.-94--6.- CONCLUSIONS.6.1.— Preamble.The application of the method of characteristics of PDE theory to theanalysis of lines with corona had been already suggested by some researchers[23,30,52). As the actual progress towards this end previously has been verylimited [30), the research work reported in this thesis was devoted to this issue.As a result, a new methodology for the analysis and simulation of transients ontransmission lines with corona, based on the aforesaid theory, has been proposedand developed.First, a new method for the analysis and simulation of transients onmonophasic lines with and without corona has been proposed, developed andimplemented on a computer. The effectiveness of this method has been furtherestablished through numerica tests on its computer implementation. Includedamong these tests is the simulation of a field experiment. Also, a new method—95—for the analysis and simulation of multiconductor linear lines has been proposedand developed as an extension of the method for monophasic lines. Finally, anew method for the analysis and simulation of multiconductor linear lines withcorona has been proposed and developed, also as an extension of the aforesaidmonophasic method. The following section provides a summary of the mainfeatures of these three new methods as well as of the main results of thisthesis. Section 6.3 provides suggestions for future research in the area oftransient simulations on transmission lines. Section 6.4, finally, provides theconcluding remarks for this thesis.6.2.— Summary of Results.The method of characteristics of PDE’s provides in principle a better wayfor dealing with transients on nonlinear lines, such as the ones affected bycorona, than more conventional methods based on constant discretization steps[23,30]. The author considers that the results presented in this thesis establish thepracticality of this method. The constant discretization steps of conventionalmethods introduce numerical errors in the form of artificial reflections which areoften observed as large oscillatory waves superimposed on the tail of thesimulated travelling waves [78]. In the method of characteristics, the sizes of thediscretization steps are adjusted in a way which minimizes these artificialreflections. The step size variations, however, result in the irregular distribution ofthe solution points throughout the x—t plane of coordinates. Nevertheless, thisundesirable effect is overcome by combining the method of characteristics withan interpolation process [14].The approach adopted here consists of expressing the line equations as asystem of first order PDE’s and applying the method of characteristics withinterpolations. A major advantage of this approach is that it performs equally—96—well with the multiconductor line case as with the monophasic one, in the lattercase, the resulting method is closely related to Bergeron’s method on which theEMTP is based [67]. It has been shown here, in fact, that the application of themethod of characteristics with interpolations to a monophasic Iossy linear lineresults in a well known line model of the EMTP (see fig. 2.1). [67].The absence of errors caused by reflections is readily apparent in all thesimulated transients on monophasic lines with corona that are presented in thisthesis. This feature represents an advantage over more conventional methodsbased on fixed discretizations wherein numerical errors are bound to occur. Mostof the simulations presented are done for a double linear ramp excitation,because its sharp corner provides a more stringent test than the rounded peaksfrom other more commonly used waveforms. The absence of artificial reflectionspermitted the simulation of travelling waves with multiple peaks above the coronathreshold. As it is observed in the examples provided in chapter 4, these multiplepeaks can arise from wave reflections in the line.The computer implementation of the method proposed for monophasic lineswith corona performs very well with coarse discretization schemes, This featurehas been tested by simulating a monophasic line case with different discretizationlevels. The widest step length that was used allowed only one point to representthe wavefront, It is remarkable that, even with such a low discretization level, theimplemented method delivers meaningful results. To produce equivalent results,other more conventional methods require step sizes that are five to ten timessmaller.In addition to the previously described simulations, a field experimentconducted by Wagner and Evans [10] has been reproduced by means of theaforesaid computational method. The simulation results are in satisfactory—97--agreement with the experimental ones, particularly with respect to the evolutionof the wavetail as it propagates. Most other methods are not able to reproduceaccurately the tails of waves propagating on a line with corona [48].A new method for the simulation of multiconductor linear lines, based onthe technique of characteristics with interpolations, has been developed in thisthesis. Although the discovery of this method was a byproduct of the analysis ofmulticonductor lines with corona, it is an important result in its own right. Manyconventional methods for analyzing multiconductor linear lines in the time domainmake use of frequency domain modal transformations [67]. The problem is,however, that these transformations introduce complex quantities which lackphysical meaning in the time domain. The new method developed here confinesthe analysis to the real domain.A new simulation method has also been developed in this thesis formulticonductor lines with corona, It consists of applying the method ofcharacteristics with interpolations to the quasilinear system of PDE’s which isobtained when representing transmission lines under the assumption that corona isa static phenomenon. Most other methods are based on conventional modalanalysis which poses two particular problems. First, complex quantities areintroduced; and second, the line is presumed to respond linearly. This new methodavoids these two shortcomings. Another important feature, is that the method canbe extended, in a rigorous manner, to multiconductor lines with dynamic corona.The abovementioned new methods proposed here are based on aneigenvalue/eigenvector analysis. Although there are some basic differences betweenthis analysis as applied here and the conventional line modal analysis [19], it hasbeen shown in this thesis that the two of them are related, This relationshipprovides a computationally efficient way to derive the new eigenvector—98—transformations from the conventional modal ones.6.3.— Future Research Recommendations.A number of future research topics became apparent throughout thedevelopment of this project. These topics came out partly from existing needs inthe field of transient analysis in general, from needs in the subject of transientanalysis of lines with corona and from new possibilities opened by thetechniques developed in this thesis. The author proposes that these topics bedealt with by means of research projects described as follows.The issues related to general line simulations, which require furtherdevelopment, can be grouped in one project. The aim of the main project wouldbe the further development of the methods based on characteristics withinterpolations for their application to practical linear line analysis. The first stageof this project would consider the development of a systematic way for selectingfrequency independent parameters of lines and for determining the cases thatrequire frequency dependent modelling. The second stage of this project couldconsist of the numerical implementation of the multiconductor linear line methodthat is developed in this thesis and, perhaps, its incorporation into a transientsimulation package like the EMTP. A third stage of this project could be theincorporation of frequency dependence modelling capabilities into the method ofcharacteristics with interpolations. It is possible, however, that this last stagedeserves to be considered another project by itself.Concerning the issues related to the analysis of lines with corona, it isproposed here that one project be devoted to the practical implementation of themethod developed in this thesis fo multiconductor !ines with static corona. Thefirst. stage of this project would consist of the algorithmic implementation of this—99—method and of the refinement of this implementation through numerical tests andcomparisons with suitable field tests. The second stage of this project wouldconsist of the incorporation of this resulting algorithm into a host transientsimulation program, possibly the EMTP.A second project proposed here for transmission lines with corona wouldbe concerned with the implementation of frequency dependent and of dynamiccorona features on the quasilinear methods proposed in this thesis. The firststage of this project would be the incorporation of frequency dependent featuresinto these two methods, the monophasic and the multiconductor one. The secondstage would consist of the development of a simulation method for monophasiclines with dynamic corona and its implementation in a computer program. Thethird stage would consist of applying this computer program to sensitivity studiesaimed at determining the need for dynamic corona modelling for practical analysisof transmission lines. It would be desirable that this third stage includedcomparisons with field tests; however, these comparisons are subject to theavailability of suitable field test results. The results of this third stage wouldindicate whether the next two stages are required or not. The fourth stage wouldconsist of the extension of the method developed in the second stage tomulticonductor lines. The fifth stage, finally, would consist of the implementationof frequency dependent features into the methods of simulation of lines withdynamic corona.The need for results from field experiments on actual lines, in addition tothe ones that are available from the specialized publications [10,44,43], has beenstressed in several sections of this thesis. Hence, this author considers that aresearch project consisting of an extensive program of field experiments isessential for the advancement of the techniques for the analysis of transients on—100—lines with corona. Afthough the required experiments are very difficult andexpensive to perform, it is proposed here that the use of computer simulationscan help to increase their effectiveness and, consequently, to decrease the costof a field experimental project. It is therefore proposed here that the methods ofanalysis developed in this thesis be used for the design of field experiments ontransmission lines with corona, for the interpretation of their results and for thedetermination of the effects that other factors unrelated to corona have on theseresults.It is highly desirable that another research project consisting of laboratoryexperiments on corona be conducted in coordination with the abovementioned fieldexperiments. This author proposes that, in much the same manner as suggestedabove for the field tests, laboratory experiments on short conductor sections andon electrodes could be made more effective through the application of anadvanced corona simulation model; e. g., the one proposed by Abdel—Salam [84].This author proposes also that a laboratory experimental project include transientpropagation experiments on scaled down transmission lines [97].6.4.— Concluding Remarks.A new methodology for the analysis of lines with corona has beenproposed in this thesis. The application here of this methodology in thedevelopment of specific methods has demonstrated its usefulness in the analysisof power transmission lines with or without corona. Thiz methodology overcomessome important technical problems of conventional methodologies. 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