Method for estimating backflashover rates on HV transmission lines based on EMTP-ATP and curve of limiting parameters

Electrical Power and Energy Systems 78 (2016) 127–137

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Method for estimating backflashover rates on HV transmission lines based on EMTP-ATP and curve of limiting parameters P. Sarajcev ⇑, J. Vasilj, D. Jakus University of Split, FESB, Department of Power Engineering, R. Boskovica 32, HR-21000 Split, Croatia

a r t i c l e

i n f o

Article history: Received 1 April 2015 Received in revised form 19 November 2015 Accepted 25 November 2015

Keywords: Backflashover Transmission line EMTP-ATP Bivariate Log-normal distribution Insulator string flashover Electrogeometric model

a b s t r a c t This paper presents a general method for estimating the backflashover rate (BFOR) on high voltage (HV) transmission lines. The method employs a state-of-the-art model of HV transmission line (TL) for backflashover (BFO) analysis, assembled within the EMTP-ATP software package and subsequently used—with EMTP running in batch mode—to construct a curve of limiting parameters (CLP). The CLP brings into relationship incident lightning currents with critical currents for the BFO occurrence and, furthermore, provides means for estimating the BFO probability. Namely, the probability of BFO occurrence at a particular TL tower is obtained by computing the volume under the surface of the probability density function of the bivariate log-normal statistical distribution of lightning current parameters, bounded by the CLP in the appropriate coordinate space. This probability in combination with the estimated number of direct lightning strikes to TL towers—obtained from the application of the electrogeometric model—provides the associated BFOR, assuming it is further normalised on the basis of 100 km-years. The usage of the proposed method is demonstrated on a typical HV transmission line. A sensitivity study for estimating the BFOR is provided, along with results comparison between different methods. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction High voltage (HV) transmission lines are exposed to lightning strikes, where only direct lightning strikes (to shield wire(s), phase conductors and tower tops) are of engineering concern. Nearby lightning strikes have no influence on the HV transmission line (TL) performance. Direct lightning strikes to phase conductors, where shield wire(s) is(are) present on the tower, are accompanied by lightning currents with limited amplitudes, due to the shielding effect of the wire(s); nonetheless, these strikes can and do provoke a flashover of the TL insulation (i.e. insulator string flashover). The rate at which this is to be expected, per 100 km-years of transmission line, is often termed the shielding failure flashover rate (SFFOR). Direct lightning strikes to shield wire(s) and tower tops can also provoke a flashover of the TL insulation, where the strikes to the tower tops are more prominent in producing insulator flashovers (and statistically speaking more probable) then the strikes to mid-spans. The rate at which this is to be expected, per 100 kmyears of transmission line, is often termed the backflashover rate (BFOR). The flashovers away from insulator strings, both in SFFOR and BFOR analysis, are regarded as being far less probable than the

⇑ Corresponding author. Tel.: +385 (21)305806; fax: +385 (21)305776. E-mail address: [email protected] (P. Sarajcev). http://dx.doi.org/10.1016/j.ijepes.2015.11.088 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

insulator string flashovers, viz. only flashovers at the TL insulator strings are considered possible in the numerical treatment of the phenomenon. Although it is the intention of this paper to analyse solely the BFOR of HV transmission lines, it ought to be kept in mind that the TL performance is gauged in regard to both BFOR and SFFOR. Moreover, in terms of the actual TL lightning performance, it is close to impossible to separate flashovers due to shielding failures from the backflashovers. Direct lightning strike to the transmission line tower, or to the shield wire, initiates a rather complicated travelling-wave process in the system of lightning channel, shield wires, TL tower, and tower footing impedance. Accompanying current and voltage transient states, at the various points of the TL tower, are established through the complex propagation pattern of current and voltage travelling waves, including reflection and transmission of those waves on various points of travelling paths wave-impedance discontinuities. This process results with a transient voltage being applied on the TL insulator strings with the possibility of their flashover, i.e. when the transient potential of the tower crossarm exceeds the critical flashover voltage of the insulator strings (itself being a non-linear function of the applied voltage), biased to some extent by the power-frequency phase voltage, it causes a flashover. Since the flashover is initiated from the tower arm, rather than the phase conductor, it is traditionally termed the

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backflashover. Analysis of this complex travelling-wave process and the associated backflashover (BFO) occurrence has been tackled by means of both analytical and numerical techniques. The analytical methods are extensively described by the IEEE WGs [1,2] and CIGRE WGs [3], with additional details provided in references cited therein. A comparison between these recommendations has been provided in Ref. [4]. Further extensive exposition of analytical methods, with many exceptional details, is provided in Ref. [5]. When it comes to analytical methods, particularly notable is the so-called CIGRE method, which is exposed with considerable detail in [5, Ch. 10]. Nowadays, it is far-more common to treat the BFO occurrence on HV transmission lines in terms of the numerical simulations, carried-out by means of the Electromagnetic Transients Programs (EMTP), [6–9]. With the numerical approach to the transient analysis of TL lightning surges, detailed, and often quite sophisticate, models of TL components are needed, some of which exhibit non-linear behaviour, frequency-dependence, non-TEM mode of propagation, etc. The IEEE WGs and CIGRE WGs offer extensive guidelines when it comes to representing transmission line elements for numerically simulating fast-front (i.e. high-frequency) transients [9–11]. Interested reader is at this point advised to consult the extensive treatment of modelling guidelines for TL lightning-surge numerical simulations provided in [12, Ch. 2] and references cited therein. Further important simulation details, concerning in-particular the backflashover analysis on HV transmission lines, can be found in Refs. [13–18]. When it comes to the numerical analysis of the BFO probability of occurrence and the BFOR per 100 km-years, different approaches utilising the Monte Carlo method have been proposed, e.g., [13,19–24]. The main downside of the application of the Monte Carlo method stems from the long CPU times associated with carrying-out simulations for obtaining the BFO probability. This is particularly true if one needs to examine multitude of different scenarios, which is often the case in practice. As many as 40,000 simulation runs, for a single scenario, have been reported in literature, e.g. see Ref. [22], necessitating several hours of computing time on modern PC architectures. This paper presents a different and novel method for estimating the BFO probability and BFOR per 100 km-years of HV transmission lines. The proposed method employs a state-of-the-art model of HV transmission line for backflashover analysis, assembled within the EMTP-ATP software package and subsequently used—with EMTP running in batch mode—to construct a so-called curve of limiting parameters (CLP) [25–28]. The authors believe that this particular approach to the CLP construction and subsequent application has not been proposed thus-far, although the CLP has been used before (and derived in analytical form). The CLP brings into relationship incident lightning currents with critical currents for the BFO occurrence and, furthermore, provides means for estimating the BFO probability. Namely, the probability of the BFO occurrence at a particular TL tower (featuring certain geometry and soil resistivity) is obtained by computing the volume under the surface of the probability density function of the bivariate log-normal statistical distribution of lightning current parameters, bounded by the CLP in the coordinate space defined by lightning current amplitude and wave-front duration. The BFO probability, in combination with the estimated number of direct lightning strikes to TL towers—obtained from the application of the electrogeometric model—provides the associated BFOR, assuming it is further normalised on the basis of 100 km-years. It will be demonstrated that the here proposed CLP method provides computational results with significantly fewer number of simulation runs than the Monte Carlo based methods, resulting in considerably lower CPU times. The method of estimating BFOR on HV transmission lines, proposed in this paper, aims to take into the account following aspects

of the phenomenon: TL route keraunic level(s); statistical depiction of lightning-current parameters (including statistical correlation between the parameters); electrogeometric model (EGM) of the lightning attachment process (assuming only vertical strokes); frequency-dependence of TL parameters with electromagnetic coupling between conductors; tower geometry and surge impedance; tower footing impulse impedance (with soil ionisation if present); lightning-surge reflections from adjacent towers; nonlinear behaviour of the insulator strings flashover characteristic; TL span length; statistical distribution of lightning strokes along the TL span; power frequency voltage. The proposed method, through numerical simulations, provides insight into the BFOR behaviour, which is due to many influential factors. Furthermore, considering the established influence of the tower grounding impulse impedance and insulator strings flashover characteristic on the BFOR, these aspects will be investigated within the sensitivity analysis provided in the paper. Finally, results obtained from the proposed method will be compared against results obtained from both the Monte Carlo approach and the CIGRE method. The paper is organised in the following manner. In Section ‘‘Tr ansmission line modelling for backflashover analysis”, a brief outline of the TL model for the BFOR analysis is provided, which is suitable for implementation in the EMTP-ATP software package. S ection ‘‘Statistical parameters of lightning currents” provides the necessary statistical treatment of the lightning current parameters. In Section ‘‘Number of direct lightning strikes to TL shield wire(s)”, an estimation of the number of direct lightning strikes to transmission line is provided, by means of implementing the traditional EGM of lightning attachment. Section ‘‘Curve of limiting parameters” provides a crucial information on the algorithm for constructing the curve of limiting parameters, from the EMTP simulation runs, and its application for obtaining the BFO probability and BFOR per 100 km-years. A test case of the HV transmission line, along with the sensitivity analysis, is provided in Section ‘‘Test case transmission line and sensitivity analysis”. A comparison of results obtained from the proposed method with those obtained from the Monte Carlo approach and from the application of the CIGRE method is presented in Section ‘‘Comparison of results and discussion”, which is followed with a conclusion in Section ‘‘Conclusion”. Transmission line modelling for backflashover analysis The EMTP model of the HV transmission line for lightning surge transient simulation in general, and backflashover analysis in particular, has been thoroughly studied and widely published, e.g. see Refs. [8–11,29–31]. A brief outline of the EMTP-ATP model, as employed for the purpose of this paper, will be presented in this Section. The model consists of several components [9]: (i) TL phase conductors and shield wire(s), including spans, line terminations and power frequency voltage, (ii) TL tower, (iii) tower grounding impedance, (iv) insulator string (i.e. archorn) flashover characteristic, (v) lightning current and lightning-channel impedance. Phase conductors, shield wire(s), spans, line termination, powerfrequency voltage High voltage transmission line phase conductors and shield wire(s) are modelled as distributed-parameters, untransposed, frequency-dependent, multiphase transmission line, by means of employing the so-called LCC component of the EMTP-ATP which utilises the JMarti TL model [6,7,32]. Phase conductors and shield wire(s) positions on the tower (from the most-representative tower within the TL route) are used, along with their maximum allowed sags, cross-sectional dimensions, DC resistances, ground resistivity of the ground return path, etc. The electromagnetic

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coupling between phase conductors and shield wire(s) are accounted for by this model, but the effects of corona are neglected. Five spans of the transmission line, at each side of the tower being struck by lightning, are modelled in this way, using eight decades with ten points per decade in the JMarti frequency-dependent model, with the modal transformation matrix computed at the dominant frequency [6,7]. After the fifth span, at each side of the struck tower, additional length of the transmission line is modelled in the same way (in order to eliminate reflections), and the line model is then terminated by an ideal, grounded, power-frequency, three-phase voltage source (with shield wire(s) grounded). Power-frequency voltage can be taken in thirty-degree angles, can be drawn randomly from the uniform distribution of phase-angle degrees, or fixed in accordance with the recommendation provided in [10], which is followed here. Transmission line towers The steel-lattice towers of HV transmission lines are usually represented as a single conductor, distributed-parameter, frequency-independent, transmission lines (i.e. Clark component), [6]. The towers of EHV and UHV transmission lines, on the other hand, are represented using more complicated so-called multistorey tower models, which are not considered adequate for HV transmission lines; e.g. see Ref. [15]. The single value of the tower surge impedance is computed from the analytical expressions, based on the theoretical background provided in Ref. [33], which depend on the tower configuration and can be found in Ref. [10]. For the conical tower configuration, following expression is often utilised:


 Z t ¼ 60  ln ðh=r eq Þ  1

ð1Þ

where h is the tower height and r eq is the equivalent radius obtained from splitting the tower at the waist (h ¼ h1 þ h2 ) and using r eq ¼ ðr 1 h2 þ r 2 h þ r 3 h1 Þ=2h, with r 1 ; r 2 and r3 standing for the radius of the tower top, waist and base, respectively. Several alternative expressions are also available, see [12, Ch. 2]. The velocity of the surge propagation along the steel-lattice tower is assumed to be equal to the speed of light in free space (although some authors assume somewhat lower value). Additionally, tower arms could be modelled as transmission line stubs (distributed parameter, single conductor) with constant value of surge impedance; some authors use inductances instead. However, influence of the tower arms could be neglected altogether. Each of the towers, eleven of them in total, is modelled in this way. Tower grounding impulse impedance The model of the lightning-struck TL tower grounding impedance is important factor influencing the subsequent formation of the overvoltage on its tower top (and its arms), due to subsequent reflections of travelling waves formed by the lightning strike, where reflections from the tower base will arrive much sooner at the tower top then reflections from adjacent towers. Hence, the influence of the (apparent) TL tower footing (i.e. grounding) surge impedance on the tower top transient voltage is determined by its response time and current dependence. In cases where the tower grounding (system) cannot be regarded as being concentrated (e.g. there are counterpoises installed covering distances greater than cca. 30 m), frequency-dependent tower grounding impedance model is needed. Its implementation can be somewhat complicated, e.g. see Ref. [17]. On the other hand, within some 30 m of the tower base (i.e. concentrated tower grounding system), the tower grounding impedance exhibits only current dependence, which can be modelled in accordance with

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guidelines provided in Ref. [10] and implemented in EMTP-ATP software package by means of the MODELS language [34]. The following equation is utilised for that purpose [10]:

R0 Ri ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ It =Ig

ð2Þ

where R0 is the tower grounding resistance at low-frequency and low-current magnitudes, It is the lightning current through the tower footing impedance, and Ig is the lightning current level which determines the soil ionisation inception process. This current is determined from the expression Ig ¼ qE0 =ð2pR20 Þ, where E0 is the soil ionisation electric field gradient and q is the apparent soil resistivity. It is evident that the inception of the soil ionisation depends on the value of the so-called soil ionisation electric field gradient and the low frequency value of the tower grounding resistance (which in-turn depends on the soil resistivity and type of the grounding system employed). The value of the soil ionisation electric field gradient usually ranges between 300 kV/m and 1000 kV/m and is taken at 400 kV/m for the purpose of this paper. Grounding impedances of the adjacent towers are modelled with a simple resistance, the value of which is equal to the tower grounding resistance at low-frequency and low-current magnitudes. Namely, the influence of the models of these impedances on the struck tower’s top overvoltage formation is negligible.

Insulator strings flashover characteristic The insulator strings (i.e. archorn) flashover characteristic is a non-linear function of the applied impulse voltage and it exhibits complicated behaviour in nature, which is rather difficult to fully reproduce. Interested reader is advised to consult Refs. [14,31,35,36] for more information. Hence, it is usually modelled within the EMTP-ATP software package by means of the voltagecontrolled switches. The flashover characteristic itself is programmed using the MODELS language [34]. Sometimes, a parallel capacitor is added to the controlled switch in order to simulate the coupling effects between conductors and tower structure [10]. There are several different models of the insulator flashover, having various degrees of sophistication, where more sophisticate models include the equal-area method and the leader progression model, e.g. [9,14,35]. There are several different variants of the leader progression model, but the particular one used in this paper is based on the solution of the following differential equation:

  d‘ uðtÞ ¼ k  uðtÞ   El0 dt dg  ‘‘ ðtÞ

ð3Þ

where dg is the insulator strings (i.e. archorn) length, ‘‘ ðtÞ is the leader length, uðtÞ is the actual voltage (absolute value) on the insulator strings, and k; El0 are constants which are found to be dependent on the type of the insulator and here assume following values [3]: k ¼ 1  106 (m2 v2 s1), El0 ¼ 670 (kV/m). It ought to be emphasised that the usage of different values for these parameters, as well as usage of different leader progression models, will result with insulation flashovers at different time and/or voltage instants, leading to different TL backflashover performance. The differential equation is solved during the EMTP simulation, by means of the MODELS language, for the length of the leader at each simulation time-step. If this length attains or exceeds the gap length (i.e. the length of the insulator strings), the associated TACS switch is closed, signifying the occurrence of the insulator strings flashover.

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Lightning current and lightning-channel surge impedance

Table 1 Lightning current statistical parameters.

A direct lightning strike in the EMTP-ATP software package is modelled as an ideal current source in parallel with the resistance, the value of which represents the lightning-channel surge impedance. This value is hereafter assumed at 400 Xm. The current source can be of different types: ramp-slope type, CIGRE type, double-exponential type, Heidler type, to name the few mostprominent ones [32], where Heidler type will be utilised for the purpose of this paper. Using a different type of current source will result with different overvoltages being formed on the TL tower top. This ideal current source is connected to the top of the TL tower being struck by lightning. Statistical parameters of lightning currents Lightning current is depicted with an amplitude, wave-front duration and wave-tail duration. In HV transmission line studies, predominantly negative downward lightning strikes are of engineering interest; hence, only these will be presented hereafter, [37,38]. It is well-known that lightning-current parameters each individually follow a log-normal distribution, in which case the probability density function (PDF) of the statistical variable can be given by the following expression [38]:

  ðln xln x Þ2 exp  2r2 l ln x f ðxÞ ¼ pffiffiffiffiffiffiffi 2p  x  rln x

ð4Þ

where xl represents the median value and rln x represents the associated standard deviation of the ln x, which holds for any of the lightning-current parameters. The associated cumulative distribution function (CDF) is depicted with the following expression [38]:

1 Fðx P x0 Þ ¼ pffiffiffiffi 

Z

p

1

2

eu du ¼

u0

1  erfcðu0 Þ 2

ð5Þ

pffiffiffi with u0 ¼ ðln x0  ln xl Þ=ð 2 rln x Þ and where erfc represents the complementary error function from the general statistics. However, a statistically significant correlation between the lightning-current amplitudes and wave-front durations has been found [38]. This necessitates usage of the joint (i.e. bivariate), as well as conditional, probability distributions in their treatment. The bivariate log-normal probability density function, in case of amplitudes and wave-front durations, can be described by the following relationship [38]:

h i 1 f 2 þf 3 exp  f2ð1 q2c Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ðI; t f Þ ¼ 2p  I  t f  rln I  rln tf  1  q2c

Lightning

Original set Alternative set

Amplitude

Front time

Correlation

Il

rln I

tf l

rln tf

qc

31.1 34.0

0.484 0.740

3.83 2.0

0.550 0.494

0.47 (0.47)

Ref.

[38] [13]

Table 1 features parameters of the statistical distribution of lightning currents (of negative downward lightning), which will be utilised hereafter [13,37,38]. Two different data sets are introduced in order to account for the fact that there are considerable differences between lightning data statistics provided by different researchers (e.g. CIGRE proposes piecewise log-normal distribution), and in order to demonstrate its influence on the backflashover occurrence rates. It needs to be emphasised that these parameters of the log-normal distribution (median value, standard deviation, and statistical correlation) are not, unfortunately, uniquely defined. Significant differences exist between different geographical locations, which are influenced by climate and other parameters (e.g. seasonal changes in thunderstorm activity, terrain topography and elevation, etc.). Furthermore, differences arise between statistical data provided by different researchers due to differences in measuring equipment and other factors (e.g. sample size, differences in positions of sensors on measuring towers, different trigger thresholds, etc.), which subsequently influence the parameters derived for the log-normal distributions. The lightning detection networks, now in operation for some time, will bring improvement in this regard. Further differences arise from the definition of the lightning-current wave-shape itself, in terms of the wave-front duration t 10=90 ; t 30=90 ; tm (minimum equivalent front), and from the different levels of statistical correlation observed between these parameters and lightning-current amplitudes, [38]. Finally, difficulties arise from applying different currentsource types, within the EMTP-ATP software package, for simulating lightning currents with these statistical parameters. As an example, Fig. 1 graphically depicts the bivariate probability density function (in terms of a heat map) of the log-normal distribution of lightning-current parameters, utilising the data from Table 1. Duration of the lightning-current wave tail—considering its negligible importance in TL backflashover analysis—is in all cases fixed at 77.5 (ls), which is the median value, or at the mean (average) value obtained from  x ¼ xl expðrln x =2Þ, where xl and rln x

ð6Þ

with

f1 ¼

 2 ln I  ln Il

f 2 ¼ 2qc 

f3 ¼

ð7Þ

rln I

ln I  ln Il ln t f  ln tf l 

rln I

ln tf  ln t f l

rln tf

rln tf

ð8Þ

!2 ð9Þ

where Il ; rln I and t f l ; rln tf represent median value and standard deviation of lightning current amplitudes and wave-front durations, respectively, and qc is the coefficient of correlation between them.

Fig. 1. Bivariate log-normal probability density functions of lightning-current amplitude and wave-front duration obtained for the ‘‘original set” (left) and the ‘‘alternative set” (right) of lightning data.

P. Sarajcev et al. / Electrical Power and Energy Systems 78 (2016) 127–137

stand for the median value and standard deviation, respectively, of the associated log-normal distribution. Number of direct lightning strikes to TL shield wire(s) The problem of estimation of the number of direct lightning strikes to transmission lines has been traditionally tackled by means of the electrogeometric model of lightning attachment. Interested reader is advised to consult, e.g., [5,12,20,39] for the in-depth treatment of this subject. Only a brief outline of the method, as far as it is needed for the purpose of this paper, will be presented hereafter. According to the EGM of lightning attachment to transmission lines, and considering only vertical lightning strikes, the number of lightning strikes to shield wire(s) depends on their exposure areas, which are determined in terms of the lightning striking distance and tower geometry. According to the theory presented in Ref. [5, Ch. 7], following expression for estimating the number of direct lightning strikes to shield wire(s) can be obtained:

Z Ngw ¼ 2LN g 

Im

Z Dg ðIÞf ðIÞ dI þ 2LN g 

0

1 Im

D0g ðIÞf ðIÞ dI þ LNg Sg

ð10Þ

where N g ¼ 0:04T 1:25 in km2 year1 is the annual average ground d flash density (T d is the long-term average annual number of thunderstorm days); f ðIÞ is the probability density function of the lightning current amplitudes distribution; L is the transmission line length; Sg is the distance between shield wires (in case of a single shield wire Sg ¼ 0 m); Im is the maximum shielding failure current; Dg ðIÞ and D0g ðIÞ are exposure distances for the shield wire(s) as a function of lightning current amplitudes. The maximum shielding failure current and exposure distances for the shield wire(s) depend on the EGM that is being applied to the TL geometry (there are several possible EGM models to choose from). Strictly speaking, expression (10) is valid only for transmission lines with horizontal conductor configurations, but it could be easily adopted for the TLs with vertical conductor configurations. If the TL route traverses through terrains with different keraunic levels, the route is then split into sections having different keraunic levels and Eq. (10) is solved for each of the sections separately. The EGM brings into functional relationship lightning current amplitude with its striking distance to phase conductors, shield wire(s) and to ground surface, and is of the form r ¼ A  Ib , where A and b are model parameters and I is the lightning-current amplitude. Table 2 lists model parameters for several different EGMs [5, Ch. 6]. The maximum shielding failure current, which features in (10), can be determined from the following expression:

Im ¼

 1=bg r gm Ag

ð11Þ

with [12, Ch. 6]:

r gm ¼

ðh þ yÞ=2 1  ðr c =rg Þ  sin tan1



ð12Þ

a hy

Table 2 Electrogeometric model parameters. r g (m)

r c (m)

EGM

Ag

bg

Ac

bc

Wagner Armstrong & Whitehead Brown & Whitehead Love Mousa

14.2 6.0 6.4 10.0 8.0

0.42 0.80 0.75 0.65 0.65

14.2 6.7 7.1 10.0 8.0

0.42 0.80 0.75 0.65 0.65

131

where r c ; r g depict striking distances to phase conductor(s) and ground surface, respectively; h, y are heights of the shield wire(s) and phase conductor(s) at the tower, respectively; a is the length of the tower arm(s) carrying the phase conductor(s). Tower dimensions are provided in metres. Parameters Ag and bg in (11) are taken from Table 2 with respect to EGM being applied. Exposure distances for the shield wire(s) can be determined from the following expressions [5, Ch. 7]:

 Dg ¼ r c  cos tan1

D0g

 a b hy

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 r 2c  ðrg  hÞ rg P h ¼ :r rg < h c

with 1

b ¼ sin

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ ðh  yÞ 2r c

ð13Þ

ð14Þ

ð15Þ

Lightning strikes are not uniformly distributed along the TL span length, which has been shown in Ref. [40]. If one considers only lightning strikes to TL tower tops (and their near-vicinity), which is often the case with the BFOR analysis, then the expression (10) needs to be corrected with the appropriate coefficient which takes into the account the actual statistical distribution of lightning strikes along the TL span length. A value of 0.6 can be assumed for this coefficient; e.g. see [5, Ch. 10] for more details. Curve of limiting parameters The curve of limiting parameters brings into relationship incident lightning currents with the critical currents for backflashover occurrence and can be constructed in the coordinate space of lightning-current amplitudes and wave-front durations. It is here derived directly from the EMTP simulation runs and is subsequently applied for estimating the BFO probability. The computational procedure for obtaining the CLP can be decomposed into three separate stages: pre-processing, numerical simulation of the backflashover occurrence, and post-processing, with first and third stages implemented in a purposefully developed computer program. In the first stage, a preparation of the input data is carried out, as will be explained in a moment. The second stage is implemented by means of the EMTP running in batch mode, with interventions on its input and output files carried-out between simulation runs with the developed computer program. This stage produces a curve of limiting parameters. In the third stage, computation of the BFO probability and BFOR per 100 km-years is performed, using the data provided by the numerical simulations (i.e. the CLP) and additional computations of the number of lightning strikes to transmission line. The algorithm for constructing the curve of limiting parameters is graphically depicted in Fig. 2. The outer loop runs across lightning current wave-front durations and for each front time the inner loop uses a type of bisection search method to find the minimum value of the lightning-current amplitude (i.e. critical current) for which a flashover is still possible (in accordance with the EMTP model)—which establishes a single point on the curve of limiting parameters. Any lightning current amplitude above this ‘‘critical” value (for that particular wavefront duration) is certain to produce a backflashover, while any amplitude below this threshold level (due to the determinism of the EMTP computational framework) cannot produce a backflashover. The complete run of the outer loop yields a curve of limiting parameters—defined point-by-point in the coordinate space of wave-front durations and amplitudes (the same coordinate

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and 350 kA, because this range generally contains above 99.99% of all values, in accordance with the appropriate log-normal distribution. The inner loop usually needs around 7–8 runs for achieving the prescribed tolerance, which yields around 1500 runs in-total, for constructing the curve of limiting parameters (with a TL model described in Section ‘‘Transmission line modelling for backflashover analysis”). This is the most time-consuming part of the computation and takes about 10 min of CPU time on modern PC architectures. It ought to be emphasised that Monte Carlo methods have been reported needing up to 40,000 runs, compared with only 1500 runs with the proposed method. The savings of the CPU time with proposed method are enormous, compared with Monte Carlo approach, particularly if one wants to examine multitude of different scenarios, which is often the case in practice. Once the curve of limiting parameters is constructed, the volume under the bivariate PDF of the log-normal distribution of lightning-current parameters—bounded by this curve—is numerically computed (by using region subdivision and double integration based on the globally adaptive Clenshaw-Curtis quadrature), yielding the BFO probability for a particular TL geometry and tower footing impedance. Using this probability, the BFOR can be estimated as follows:

BFOR ¼ 0:6  Ngw  P B

ð16Þ

with P B , as already stated, obtained from

ZZ

PB ¼

f ðI; t f Þ dI dt

ð17Þ

X

where X defines a half-open region in the coordinate space of lightning-current amplitudes and wave-front durations ‘‘above” the appropriate curve of limiting parameters. Test case transmission line and sensitivity analysis Fig. 2. Algorithm for constructing the curve of limiting parameters.

space where the surface of the bivariate PDF of the log-normal distribution of lightning current parameters exists); see Section ‘‘Stat istical parameters of lightning currents”. The shaded block in Fig. 2 embodies three distinctive actions, performed successively within the double loop. First, an ‘‘atp” model file, initially created with the ATPDraw [32], is manipulated in order to structure an associated ‘‘dat” file—featuring the appropriate lightning wave-front duration and amplitude values. Second, EMTP is invoked (through a batch file) and executed using a dedicated OS command. Third, newlycreated ‘‘lis” file is examined for the onset of flashover (considering the fact that the position of the TACS switch of the insulator flashover model has been broadcast). These steps are automated as part of the developed computer program, which also checks for any EMTP errors, etc. Lightning current wave-front times are chosen starting at 0.1 ls and ending at 32 ls, because this range contains above 99.99% of all values, in accordance with the appropriate log-normal distribution. From 0.1 ls to 10 ls the range is divided in 0.1 ls increments, from 10 ls to 20 ls the range is divided in 0.2 ls increments, and above that value the range is divided in 0.3 ls increments. This gives in-total 190 individual values of lightning wave-front times for the outer loop (a finer subdivision could be used with a penalty of longer execution times). This unequal subdivision of the wavefront times is designed for high accuracy with small number of individual values (i.e. short execution time), by considering the form of the CLP and the shape of the bivariate PDF of lightning currents. At the same time, the inner loop is bounded between 5 kA

Presented method for estimating the BFOR on HV transmission lines will be demonstrated on a typical single-circuit 110 kV TL, featuring vertical conductor configuration and steel-lattice towers. Tower geometry is typical for wind pressures between 750– 1500 N/m2, with individual spans of 350 m, typical for 750 N/m2 wind pressure and 65 N/m2 of maximum allowed conductors tensile strength. Average ground flash density is taken at 1 km2 year1 for the entire TL route. Tower height equals 25 m, with distance from the top to the highest arm of 3 m, distance between tower arms of 2 m; top console length of 2.5 m, middle console length of 3 m and bottom console length of 3.5 m. Phase conductor DC resistance 0.114 X/km with 10.95 mm diameter, shield wire 0.304 X/km with 8 mm diameter. Insulator string length equals 0.9 m. Fig. 3 presents curves of limiting parameters, obtained for several different values of TL tower footing impedances. They have been further superimposed on the bivariate PDF of the log-normal distribution of lightning-current parameters in Fig. 4, assuming ‘‘original set” of lightning data. Tower footing impedances used in all of the figures presented in this paper are low-current and low-frequency values, obtained from the tower grounding system configuration and soil resistivity. It is clear from these figures that the minimal values of lightning-current amplitudes (i.e. critical currents) that can still provoke a backflashover increase as the wave-front duration is increased. This is expected. In fact, for very long wave-front times the associated amplitudes attain the value of 350 kA (or more), meaning that the flashover is extremely improbable, regardless of the tower footing impedance. It could also be deduced from these figures that the BFO probability increases with the increase of the tower footing impedance, which is again expected [5].

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In addition, Fig. 5 presents the BFO probability and BFOR per 100 km-years obtained from Brown and Whitehead EGM and the ‘‘original set” of lightning data. Analysis could be further extended by considering different possible EGMs (from Table 2). Hence, Fig. 6 depicts BFOR per 100 km-years, obtained from different EGMs, using the original set of lightning data.

It is important to emphasise that the provided results accounted for the statistical correlation between lightningcurrent parameters. The statistical correlation, as it turns out, is very important in establishing the BFO probability. This is quite evident from Fig. 7, which depicts the BFO probability and BFOR per 100 km-years obtained with three different levels of statistical correlation, for the original set of lightning data and the EGM according to Brown and Whitehead. It is clear that by neglecting the statistical correlation, one significantly increases the BFO probability and, hence, increases the expected BFOR per 100 km-years. This is important finding and ought to be noted, considering that some authors still neglect the correlation and treat all lightning current parameters as independent statistical variables. Analysis provided thus-far can be repeated using the ‘‘alternative set” of lightning-current parameters (from Table 1). Hence, Fig. 8 provides the BFOR per 100 km-years, obtained for several different EGMs and the alternative set of lightning data. The strong influence of lightning-current parameters on the expected BFOR is self-evident from this figure, corroborating findings from Ref. [24]. This is expected since the parameters associated with this ‘‘alternative set” produce a skewing to the ‘‘left” of the associated PDF of the bivariate log-normal distribution (observe Fig. 1), providing larger volume under this function once it is bounded by the CLP (which stays the same). Furthermore, in order to provide a sensitivity analysis, one can make account of the several different possible treatments of various TL model components—most notable of which are the models of insulator flashover characteristic and tower grounding transient impedance. For that purpose, several different model combinations are assumed, along with lightning parameters from both data sets, as provided in Table 3. Simple switch model depicts here a TACS switch which is closed (signifying flashover) when the voltage across the insulator string exceeds a value of 605 d, with d being the length of the insulator strings. The IEEE model, as already stated in Section ‘‘Tower grounding impulse impedance”, assumes concentrated tower grounding system and current dependence due to soil ionisation. Simple (tower grounding) resistance, on the other hand, assumes that the tower footing surge impedance is equal to its low-current and low-frequency value and does not change during simulation. For the purpose of illustration, Fig. 9 displays the curves of limiting parameters obtained for the ‘‘Model C” of the transmission line at hand (to be compared with Figs. 3 and 4). Additionally, Fig. 10 presents the expected BFO probability, obtained with different model combinations from Table 3, revealing important

Fig. 5. BFO probability from CLPs and BFOR per 100 km-years using the Brown and Whitehead EGM, with original set of lightning data.

Fig. 6. BFOR per 100 km-years from different EGMs, using the original set of lightning data.

Fig. 3. Curves of limiting parameters for several different values of TL tower footing impedances.

Fig. 4. Superposition of the bivariate PDF of log-normal distribution with curves of limiting parameters.

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Fig. 7. BFO probability and BFOR per 100 km-years obtained with three different levels of statistical correlation between lightning-current amplitude and wavefront duration.

Fig. 9. Curves of limiting parameters obtained for ‘‘Model C” of transmission line.

Fig. 10. BFO probability for different combination of TL model components and lightning-current parameters. Fig. 8. BFOR per 100 km-years using alternative set of lightning data and several different EGMs.

Comparison of results and discussion Table 3 Several different combinations of TL model components and lightning parameters. Model Model Model Model Model Model

A B C D E

Flashover char.

Tower grounding

Lightning data

Leader progression Simple switch Leader progression Simple switch Leader progression

IEEE model Simple resistance Simple resistance IEEE model IEEE model

Original set Original set Original set Original set Alternative set

differences. A particularly notable influence emanates from the statistical parameters of lightning currents, with statistical correlation featuring prominently. Further differences are to be expected with these models, in terms of the BFOR per 100 km-years, due to differences between EGMs proposed by different authors. These findings are in general agreement with those reported elsewhere, e.g. [23,24,36]. It is noteworthy to mention that several additional parameters influence the BFOR estimation on TLs, not accounted-for here, such as: positive lightning strikes, subsequent strikes within the lightning flash, inclination of the lightning incidence path, upward answering leaders initiated by the descending step leader in the lightning attachment process, terrain topology (i.e. hilltop effects), etc. These all have tendency of increasing the expected backflashover rate.

In order to further confirm the validity of the proposed CLP method, some of the heretofore presented results will be compared with those obtained from the approach based on the Monte Carlo method and from the CIGRE method. Furthermore, in order to remove the influence of the EGM from the comparison, it will be confined (without loss of generality) to the estimation of the BFO probability. The Monte Carlo method which will be utilised for the purpose of comparison is presented in Ref. [24], according to which the BFOR can be estimated using the following relationship:

BFOR ¼ 0:6  Ngw  ½p  ðpu ; po Þ

ð18Þ

with p ¼ k=N, where N is the number of simulation runs and k is the obtained number of flashovers. Assuming Binomial distribution for this probability, a two-sided confidence interval for the probability can be determined from the following asymptotic relationship [24]:

pu po



0

k2 @k þ q  kq ¼ 2 N þ k2q 1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 kðN  kÞ kq A þ N 4

ð19Þ

where pu ; po are lower and upper confidence limits, respectively, and kq is the quantile of the standardised normal distribution of order q ¼ ð1 þ Þ=2, with  being the desired confidence coefficient.

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Additionally, a brief description of the CIGRE method, in accordance with [3,5], is provided in the Appendix A of this paper, for completeness. Interested reader ought to consult e.g. [5, Ch. 10] for the in-depth treatment of the CIGRE method, IEEE method, and their simplifications and comparisons. Consequently, Fig. 11 provides the BFO probability obtained from independently applying the CLP method and the Monte Carlo method to the TL at hand. Lightning-current parameters from the ‘‘original set” were employed in both methods. Furthermore, Monte Carlo method has been applied with the equal-area and the leader-progression models of the insulator string flashover. The shaded areas of the figure represent confidence limits on the BFO probability determined for the 95 % confidence level. In addition, Fig. 12 presents the BFO probability obtained from independently applying the CLP method and the CIGRE method to the TL at hand. Following input data were used in the CIGRE method (in addition to those already mentioned): Z g ¼ 360 X, CFO = 700 kV, Z T ¼ 170 X; K PF ¼ 0:4; C ¼ 0:35; E0 ¼ 400 kV/m, with two different values of the important model parameter q=R0 being examined in applying the CIGRE method [5, Ch. 10]. A rather good agreement between these three different methods can be observed from Figs. 11 and 12, for all tower footing resistances considered, further confirming the validity of the proposed approach. Moreover, the CIGRE method provides results which are in a fairly good agreement with those obtained from more sophisticate (numerical) methods, which has been confirmed by other authors and, apparently, is valid as long as the tower footing impedance can be assumed to be concentrated in nature. If, on the other hand, the tower grounding system cannot be considered concentrated, then the CIGRE method breaks down, as confirmed in [23]. The computation of the BFOR, as can be seen, is composed of (i) estimating the BFO probability and (ii) estimating the number of direct lightning strikes to transmission line, which makes the here proposed method completely general, in a sense that it can be combined with different methods of estimating the number of TL direct lightning strikes. In fact, apart from the EGMs, there are other possible models serving the same purpose, principally based on the lightning leader progression and inception analysis, e.g. [41], or dynamic simulation of lightning leader movements, e.g. [42]; see also [39] for more information on this subject. Inclusion of these different models will, inevitably, provide different estimated BFORs, notwithstanding the BFO probability.

Fig. 11. BFO probability obtained using Monte Carlo method and here proposed CLP method.

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Fig. 12. BFO probability obtained using CIGRE method and here proposed CLP method.

Conclusion This paper presented a general method for the analysis of the backflashover occurrence rates on HV transmission lines. As far as the authors are informed, this particular approach to the CLP construction and subsequent application has not been published thus-far, although the CLP has been used before in different circumstances (and mainly derived in analytical form). Majority of the influential factors affecting the BFO probability and BFOR per 100 km-years on HV transmission lines have been accounted for. This method can be applied equally-well on a specific portion of the TL, on the particular transmission line as a whole (i.e. accounting for different keraunic levels), or as a means of estimating the generic BFOR per 100 km-years. The proposed method yields computational results much faster then the Monte Carlo methods, which makes it superior choice for the large number of different simulation runs, testing for different scenarios and TL model component influences. The proposed method (as described heretofore) necessitates around 1500 simulation runs, unlike Monte Carlo methods which are reported needing as many as 40,000 simulation runs. A single simulation run of the EMTP transmission line model takes exactly the same amount of CPU time regardless of the underlying method, meaning that the CLP method proposed here provides answers in a fraction of time needed for the Monte Carlo solution. Sensitivity analysis revealed several factors influencing the expected BFO probability and their relative importance. On top of this list are the lightning-current parameters incident to transmission line, with the statistical correlation between parameters featuring prominently. Neglecting correlation increases the expected BFO probability. Also important is the model of the insulator string flashover, along with the type of EGM being used. Tower surge impedance is of lesser importance for HV lines (due to their restricted height), although the BFOR increases with the increase of the tower height. It should be mentioned that the type of lightning current source used in simulations have influence on the BFO probability; however, Heidler source is generally recommended in EMTP lightning surge transient analysis. Particular importance is to be paid to the tower grounding system modelling, especially if it cannot be regarded as being concentrated (which has been invariably assumed throughout this paper). The presented method could find its application in determining the BFO probability on the first several TL towers emanating from the HV substation, which is very important in designing the station overvoltage protection and selecting the parameters of metal-oxide surge arresters (i.e. in station lightning insulation coordination studies). In other words, presented method could be

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between the nominal system voltage and the critical flashover voltage (depends also on the tower geometry); V LN is a peak value of the power frequency phase-to-earth voltage; C is a coupling factor between phase conductor(s) and shield wire (under corona conditions); K TA ; K TT are coefficients taking into account reflections of travelling waves from the tower itself; K SP is a coefficient accounting for the reflections from adjacent towers. Coefficients K TA and K TT can be determined from the expression

K TX ¼ Re þ aT Z T 

TX tf

ðA:3Þ

by introducing appropriate T X value (T A or T T ), while coefficient K SP is obtained from the Bewley’s lattice diagram (assuming linearlyrising front) as follows:

K SP ¼ 1  aR ð1  aT Þ 1 X ðaR aT Þk ½1  2ðk þ 1Þss H½t  2ðk þ 1Þss  

ðA:4Þ

k¼0

with ss ¼ T s =tf and where: H is a Heaviside function; aR ; aT are coefficients of reflection of voltage travelling waves; T s ¼ ‘s =c is a surge travel time along a single span (i.e. span length divided by the speed of light in free space); T A ¼ h=c; T T ¼ y=c are tower surge travel times (obtained from the tower geometry); Re is an equivalent resistance formed by the surge impedance of the grounding wire(s) with the impulse impedance of the tower footing; Z T is a tower surge impedance; t f is a representative lightning-current wave-front duration. Coefficients aR and aT are provided by

Fig. A.1. Algorithm for obtaining the BFO probability using the CIGRE method.

instrumental in optimising station overvoltage protection (i.e. lightning risk mitigation), aiming for specific mean time between surges (MTBS) levels. Also, the presented method could find its application in determining the BFO probability for the several TL towers which are found to be exposed to frequent direct lightning strikes (e.g., part of the TL crossing a mountain ridge or traversing through a terrain associated with a high keraunic level), which is important for the transmission line arrester applications. Appendix A. The CIGRE method The CIGRE method for estimating the BFO probability and the BFOR per 100 km-years on HV transmission lines is an analytical method. It is very easy to implement and is computationally extremely fast. According to [3,5], the BFOR is found from the following expression:

BFOR ¼ 0:6  Ngw  FðI P Ic Þ

CFONS  K PF  V LN ðK TA  C  K TT ÞK SP

Zg Z g þ 2Ri

ðA:5Þ

aT ¼

Z T  Ri Z T þ Ri

ðA:6Þ

while the equivalent resistance Re is computed from

Re ¼

ðA:2Þ

where CFONS is a non-standard critical flashover voltage of the insulator strings; K PF is a power frequency factor dependent on the ratio

Ri Z g Z g þ 2Ri

ðA:7Þ

where Z g stands for the surge impedance of the shield wire (under corona conditions) and Ri is the tower footing impulse impedance, determined from the following expression:

R0 Ri ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ IR =Ig

ðA:8Þ

with IR ¼ ðRe =Ri ÞIc and Ig ¼ qE0 =ð2pR20 Þ, where R0 is a tower footing low-frequency and low-current impedance, E0 is a soil ionisation electric field gradient and qis the apparent soil resistivity. Furthermore, the non-standard CFO can be determined from the following expression (based on regression analysis) [5]:

CFONS ¼ CFO

ðA:1Þ

where FðI P Ic Þ stands for the BFO probability and can be obtained from the cumulative distribution function (CDF) of the lightningcurrent amplitudes log-normal distribution; see Section ‘‘Statistical parameters of lightning currents”. The critical current Ic , featuring prominently in (A.1), is determined from the following analytical expression [5, Ch. 10]:

Ic ¼

aR ¼

      2:82 DV DV V PF 1þ 1  0:2 1 þ 0:977 þ V IF V IF CFO s       10 DV DV t f exp   1  0:09 1 þ V IF 13 s V IF

ðA:9Þ

with

DV aT Z T ðT A  C  T T Þ ¼ V IF Re ð1  CÞt f

ðA:10Þ

and s ¼ ðZ g =Ri ÞT s , where CFO stands for the standard critical flashover voltage of the insulator strings (CFO gradient ranges from 489 kV/m at 16 ls to 822 kV/m at 2 ls of wave-front time); this expression is valid for front-times ranging from 0.5 ls to 5 ls. The above presented set of equations needs to be solved iteratively for the current Ic . An algorithm for achieving that is graphically presented in Fig. A.1. Initial value of the lightning-current

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