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Yüksek gerilim hatlarında yürüyen dalgalara korona etkilerinin incelenmesi

Analysis of corona effects for travelling waves on transmission lines

  1. Tez No: 14412
  2. Yazar: MEHMET ÖZALTINOK
  3. Danışmanlar: PROF.DR. MUZAFFER ÖZKAYA
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Aşırı gerilimler, Enerji iletim hatları, Korona, Yüksek gerilim hatları, Yürüyen dalgalar, Overvoltage, Energy transmission lines, Corona, High voltage lines, Travelling waves
  7. Yıl: 1991
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: Belirtilmemiş.

Özet

ÖZET Yüksek gerilim enerji iletim hatlarında genellikle yıldırım gibi dış aşırı gerilimlerin etkisi altında meyda na gelen yürüyen dalgalar, hat boyunca yayılması sırasında çeşitli noktalardaki yansıma ve kırılmaları yanında, koro na etkisiyle zayıflama ve bozulmaya da maruz kalırlar. Bir hatta uygulanan gerilim hattın korona başlangıç geri liminden yüksek olması durumunda hat üzerinde korona adı verilen elektriksel kısmi boşalmalar olur. Bu olay hattın eşdeğer çapının büyümesine dolayısıyla hat kapasitesinin artmasına ve yürüyen dalga hızının azalmasına sebep olur. Korona, aynı zamanda, yürüyen dalganın genliğine ve şekli ne bağlı olan enerjisinin azalmasına yol açar. Bu çalışmada, koronanın yürüyen dalgaların zayıf lama ve bozulmasına etkisi incelenmiş ve bu konuda veri len modeller yardımıyla bu etkinin aşırı gerilim çalışma larında ve yalıtım koordinasyonunda nasıl hesaba katıla cağı gösterilmiştir. Bu çalışmada, tekil ve demet iletkenli bir fazlı hatlarda, hattın yük-gerilim karakteristiği parça parça lineer olarak gözönüne alınmış, dalga denklemlerinin çözü mü sonlu farklar yöntemiyle yapılmıştır. Denklemler bilgi sayar yardımıyla çözülüp dalga şekillerinin çizdirilmesiy- le, standart yıldırım darbe gerilimi ve dik cepheli darbe gerilimi şeklindeki yürüyen dalganın korona etkisiyle ve uzaklıkla zayıflaması, cephesinin yatıklaşması ve şekli nin bozulması gösterilmiştir. - iv -

Özet (Çeviri)

SUMMARY ANALYSIS OF CORONA EFFECTS FOR TRAVELLING WAVES ON TRANS MISSION LINES Overvoltages and transient phenomena have been stu died since the advent of transmission and distribution networks. Because of their effects on outages and equip ment failures, transients have been analyzed for their cause and behavior on power systems. Complete books and many technical articles have been devoted to this subject. Causes of surges include lightning, system faults, and breaker reclosings or switching operations. Effects of surges include equipment failure and damage, lighting ar rester breakovers, and circuit breaker openings. The waveform that is impressed upon a piece of equ ipment is usually different from the initial waveform measured at the point of injection into the system. Atte nuation, distortion and reflections modify the waveform as it travels from the point of origin to the point of interest. In the insulation co-ordination design of substa tion, the basic insulation level (BIL) of electrical equ ipment (such as transformers and cables) and characteris tics of protective devices (such as v-i characteristics of lightning arresters) must be optimized with regard to reliability and economics. Overvoltage level predictions require accurate representation of wave propagation thro ugh the incoming overhead transmission line, taking into account corona attenuation and distortion. In this study, corona effects for travelling waves on transmission lines are analyzed and described. Some corona models are considered about this phenomena for com puter studies. Multiple terminations and changes in surge impedan ce, usually at a substation, cause reflections and refrac tions of the waveform. The starting point for waveform analysis is the linear solution to the telegraph equation including loss terms: v(z,t) - V e“azcos(wt-3z) (1) = Re{V e3ü)V(a+3p)z} max - v -Various methods have been used to attain the solu tion to the travelling wave equation. L.V.Bewley [l] pro posed the use of Duhamel ' s integral method to breakdown the input waveform into a finite combination of step func tions. A solution to the telegraph equation is found for a step function input, and superposition is applied to each step to simulate a given input waveform. Magnusson alternatively solves the telegraph equation for a sinuso idal wave of a given frequency. The input waveform is de composed into a Fourier series, each element of the seri es solved by Magnusson' s method, and superposition appli ed to recombine the components for a given time and dis placement. The latter method is more conducive when incor porating frequency dependent effects, such as skin effect. Magnusson' s method provides a solution for a sing le frequency. The other physical parameters associated with a given frequency are defined in closed form. These parameters include surge impedance, surge velocity, and attenuation and phase functions, which are all functions of frequency. Corona effects on the waveform have been studies 2] _ since the early 1900' s, when voltages high enough to ionize air could be produced in a laboratory. Corona oc curs when the maximum dielectric strength of air is excee ded by an electric field. The air is ionized, causing small pathlength breakdowns and generating an audible crackling sound accompanied by a bluish glow of light. The ionization causes radio interference or electromagne tic radiation, and creation of ozone, O3. F.W.Peek was the foremost pioneer in this field and published various articles [3], along with a book on the”subject. In 1931, E.W.Boehne proposed a corona sheath model where capacitan ce increased with voltage above the corona threshold V^h* Although described qualitatively only in a discussion to an A.I.E.E. transaction paper, this was the basis for la ter analytic discussion. O.Brune and J.R.Eaton in 1931, L.V.Bewley [l], in 1933, and H.H.Skilling and P. de K. Dykes [6] in 1937, analyzed and described corona effects for transients. Their material on the subject was descriptive and experimental, but did not include a rigorous mathematic model. Recently, an excellent current source model for corona [7, 8] has been developed for different purposes. In 1956, Wagner and Lloyed presented additional experimental data. This paper also presented an incomple te analytical model. Therein the capacitance was shown to change with voltage, but an explicit function was not given. Energy lost per unit length of line was also defi ned. Although for alternating voltages and surges the same phenomena of air ionization occurs, different - vi -characterizations evolve because of the unipolarity of the surge. For unipolar waves, negative surges have dif ferent corona effects than positive surges. Wagner and Lloyd showed via charge-voltage hysteresis oscillograms that ac corona incurs much greater energy losses than dc corona. Most researchers [2 J are cautious about applying ac corona results to impulse corona. Differences do oc cur, but a unifying theory has not been derived. Transient voltage waveforms have very high rates of rise and thus their frequency domain equivalents have significant components at higher frequencies. Therefore skin effect, usually defined as a frequency dependent pa rameter, does have an effect on the distortion and attenu ation of a travelling wave. Because skin effect has been studied and is well-defined analytically, it is important to quantify this effect in an overall model. In order to eliminate the assumption of a perfect ly conducting ground plane, J.R.Carson first developed the fundamental model for earth to be a homogeneous mass below the surface with a resistivity greater than zero, and a relative permeability of one. A later article in 1931 by W.H.Wise re-derived Carson's equations, but kept permeability as a parameter. The resultant equation thus allowed variations in both resistivity and permeability. Various authors have modelled the earth as stratified layers of homogeneous material with constant physical properties. When there is only a single layer, Carson's equations again appear. Wedepohl and Efthymiadis [9-ll] have been involved with grounding effects and have deve loped the most general equations for a single layer earth. Hedman [12] has used the methods of images to incorporate imperfect grounding into his model. Frequency appears as a parameter in these ground plane studies, so that it is again impartant to compare their effect with that of vol tage magnitude dependent corona. It is possible to determine the supplementary pro pagation attenuation and deformation to which the wave is subject, by calculation alone. Practically speaking, a charge- voltage cycle is measured at several voltage le vels. When the voltage applied is lower than the conduc tor critical voltage, the charge is purely capacitive and a linear function of voltage such that: q“aA = C.u (2) ^geom geom Above critical voltage u the measured charge may be written as follows (Figure 1) : * - qgeom + Q-SP = Cgeom'u + <?SP (3) in which qgpf the supplementary charge due to corona, is a function of applied voltage u and also, very probably, - vn -of its derivative du/dt. Two possible interpretations may be imagined of propagation distortion along a line based on this type of cycle. Let us consider (Figure 2) an overvoltage wave cha racterised by the equation u^(t) at point on the abscissa line and by u~ (t) on abscissa 4A in the direction of pro pagation. Figure 1. General aspect of cycle of charge q(u) obtained from a laboratory sample of con ductor. First interpretation: Propagation delay-On the basis of the two propagation equations applied to a line element d£: du _ T di dx ^ dt (4. a) di dx du dt (4.b) In the case with which we are concerned, we may write: di = d2 dx dt (5) in which q, by definition, is none other than the charge q(u) which appears in the plotting of the cycle. We may then say: dq = dcj du dt du ”dt (6) In this equation, dq/du has the dimensions of a capacitance which we may call dynamic capacitance C^. This capacitance is a function of u and given by the ins tantaneous slope of the cycle (Fig.l); it is always grea ter than the geometric capacitance, CqeQI_ of the conduc tor. The second equation of the system (4.b), in the - Vlll -presence of corona thus becomes ; di = _, du dx <3 - dt (7) it. ûi (H) Time Figure 2. Interpretation of propagation distortion from delay of propagation of wave front. The solution of the new system depicts a propagation speed :.L.C (8) It may be seen that v is a function of C^ and thus of u. Distortion suffered by the voltage wave may then be interpreted, assuming that each section of the wave bet ween u and u+Au moves at a different speed, lower than the speed of light c and corresponds to instantaneous dynamic capacitance C *. Delay At, characteristics of the distortion, is expressed by: At = Ax(i - i) (9) Second interpretation: power dissipation-Power dissipated by space charges during a voltage cycle may obviously be written: w = § u dq cycle ^ (10) and may consequently be calculated from the surface of the cycle. On a line, the phenomenon is more difficult to analyse. We assume, in this case, that each elementary vertical wave section (Fig. 3) is always propagated at the speed of light in spite of power dissipation. The capaci tance which determines the speed of propagation must be cgeom under these conditions and it will then be assumed - ix -that the surge impedance of the line is given in every case by t -/ (11) geom Energy carried by a voltage wave is therefore given by: W = | / u2 (t) dt o (12) Assuming these hypotheses, the energy dissipated along a section A£ is plainly equal to: oo W1-W2 = | / [u2 (t) - u2 (t)] dt (13) Provided that the section is sufficiently short, the fol lowing may then be written : 00 I / [u2(t) - u2 (t)]dt = W dx (14) In fact, the energy“w”of a cycle depends on amp litude and form of the voltage wave u(t), so that the cycle to be considered in the expression above, in order to calculate total distortion over a propagation length L, evolves constantly, as the wave u(t) is modified. The practical calculation must therefore be separated into elements Mi which are sufficiently short for wave modifi cation to remain low in order to be able to use the same cycle over the whole length of this element (Fig. 3) Voltaoe U(t) Time Figure 3. Interpretation of propagation distortion from energy dissipated. - x -Two types of propagation equations have been pro posed and they are as follows: a) Those based upon 3q/3v in the charge voltage characteristics (q-v curve) of line sections. b) Those based upon the corona loss expressed by an area enclosed by the q-v curve. In this study the author analyzes results of mea surements he made previously using method (a) above, and demon st rates the effectiveness of the propagations equati ons and their solution. Figure 4 gives an example of q-v curves of line sections for impulse voltage. The q-v curve shows complex variation at a region very close to the peak voltage value as it is strongly affected by the voltage waveform due to the involved corona discharge ge neration mechanism. The q-v curves are simplified by rela tionships (1), (2) and (3) in Fig. 4 for this study. To determine voltage and current at a location in the line and at a given time, the initial conditions assumed as v (x,o) = o (15) i (x,o) = o and the boundary conditions v(o,t) = e(t) (16) v(£,t) = o where, e(t): Applied voltage at beginning of line. £: Length of line. The terminating end of the line is grounded. The propagation equations are solved by the finite difference method. Figure 5 gives a T-type equivalent lum ped constant circuit which is divided into infinitesimal distances of Ax for the calculations of Ea. (4) under 3q/3v=C. 'o(t) n(t) j2(t) jVı(t) û(t).-rw^f/ırrk - tp-t-hît*- rinrw-'W>-i r Ax, Ax, Ax_. Ax t Ax, Ax CAx ui(t) CAxT Mt) CAxTMO Figure 5. T-type equivalent lumped constant circuit for 3q/3v=C. - XISeveral examples of calculations are shown in Figs. 6-8. 2 ü* - 3d - Dİ - Jİ - Figure 6, Positive 1600 kV surge on 1.265 mm dia. Wire travel distance a) 0 b) 352.5 m c) 705 m d) 1057.5 m e) 1410 m f) 1762.5 m. S t (ne) Figure 7. Positive 1540 kV surge on 1.265 mm dia. Wire travel distance a) 0 b) 352.5 m c) 705 m d) 1057.5 m e) 1410 m f) 1762.5 m. Ficmre 8, tens) Positive 1540 kV surge on 1265 mm dia. Wi re travel distance a) 0 b)1762.5m c) 3525m, d)5287.5m, e)7050m, f)8812.5 m. - xn -

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