EVALUATION OF THE THEORETICAL ERROR FOR THE LIGHTNING PEAK CURRENT AS MEASURED BY LIGHTNING LOCATION SYSTEMS

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1 X nternational Symosium on Lightning Protection 6 th -30 th Novemer 007 Foz do guaçu, razil EVALUATON OF THE THEORETCAL ERROR FOR THE LGHTNNG PEAK CURRENT AS MEASURE Y LGHTNNG LOCATON SYSTEMS Cinzia Rovelli Marina ernardi CES CES Cinzia.Rovelli@cesi.it Marina.ernardi@cesi.it c/o CES Centro Elettrotecnico Serimentale taliano, Via Ruattino 54, 0134 Milan -T Astract - The lightning eak current is one of the arameters that a LLS can measure. This numer lays a fundamental role in the evaluation of rotection systems of structures and ower lines. The most common LLS technology evaluates the lightning eak current alying a simle formula that correlates the current and the field emitted from the lightning. n this work we firstly examine the semi-emirical formulas to otain the eak current value as a function of the emitted field, as was determined y Orville [1] and done []. We estimate a theoretical error of the eak current calculation, using the error roagation method and with the assumtion that all the involved variales are indeendent ones. 1. NTROUCTON A lot of LLS are installed in the world, most of which ased on MPACT (Vaisala) technology. All these instruments give an estimation of the lightning eak current for each event. All these measures ring with them an uncertainty, oth due to emirical factors and to theoretical aroximations. The emirical errors are very different from case to case and from time to time, eing linked to sensor and lightning ositions, outages, transmission and other exerimental situations. On the other hand, an indication of the theoretical error involved in the method should e calculated. Also in the theoretical errors nonetheless some rolems arise, for examle eing these linked to the assumtions of the TL model and to the return stroke velocity values. n a LLS the eak current is derived y the values of the field signal strength detected y the sensors, alying the formula [3]: eak [ka] = K RS [LLP-unit] ( where RS is the signal strength detected y each sensor contriuting to the stroke calculation, normalized as if the stroke occurred at km: RS = SS ( ), () given SS as the measured signal for each sensor and as the distance etween the sensor and the stroke. K is a conversion factor, defined as 0.3 in the sensor manual and derived assuming a Transmission Line TL model [4, 5] aroximation, with a return stroke velocity of 1/3 the seed of light. The signal strength SS detected y each stroke is usually exressed as a magnetic signal, or more recisely a calirated magnetic field or a normalized as an equivalent magnetic field, deending on the sensor model. The same relation ( can e actually exressed as well using the electric field, and calculating the relevant K.. PEAK CURRENT TO FEL SGNAL RELATON The relation etween the electric and/or magnetic eak field emitted y a lightning stroke and the lightning stroke eak current, as reorted in (, was originally formulated y Orville and done [1, ]. n their studies ( is exressed as: [ka] = k M [LLP-unit] (3) - M is the mean of the normalized magnetic signal strengths M, exressed in LLP-units (an aritrary unit used y LLS equiment), - k is the emirical coefficient. - M is the magnetic signal detected y each sensor articiating in the solution, normalized to a distance of km. Each M signal detected y a sensor should e roortional to the magnetic flux density, roduced at the sensor site y the relevant return stroke.

2 For stroke-sensor distances larger than 10 km, only the radiation comonent of the electric field is suosed to e involved, thus, following the TL model: µ V 0 rs (, t / c) = ( t) + (4) eak, πc - c is the seed of light - µ 0 is the free sace ermeaility - V rs is the return stroke roagation seed, assumed constant - the distance etween the stroke and the sensor assuming signal roagation without losses, that is infinite ground conductivity. Formula (4) deends on the distance arameter as -1 and the signal strength can e normalized as: M = M (/), (5) like in ()..1 k coefficient The k value was exerimentally otained y comarison etween triggered lightning events and F sensor detections. n 1991 Orville examined 18 triggered lightning events. Each event was contemorarily detected y F sensors (magnetic direction finding antennas) of the NLN (National Lightning etection Network for continental USA). He otained the following relation from a est-fit linear interolation of the results: = M (6) The correlation coefficient resulted in r = n 1993, done analyzed the Orville dataset and added some more exerimental results increasing the numer of triggered lightning events, for a total of 57 strokes. done found a different correlation law etween the eak current and the mean value of the normalized magnetic signal strengths, otaining the following equation: = M (7) where the relevant correlation coefficient was r = and the standard deviation was σ = 4.6 ka. oth the regression equations (6) and (7) have a nonhysical intercet, showing a non-zero eak current for a zero magnetic signal. Orville forced a zero intercet recalculating the regression line etween M and the eak current values larger than.3 ka. The new equation was: = 0. M (8) done referred to aroximate the equation (7), without recalculation, in: = (9) M. The signal attenuation coefficient oth Orville and done considered also the roagation of the normalized signal (M ) over a non-erfectly conducting ground, introducing a value in the exression for the signal normalization that ecame: ( / ) M M = (10) The data analysis gave = for Orville and = for done. With = the equation (7) ecomes: = M (1 n 1996 GA-Gloal Atmosherics nc. (now art of Vaisala) used this equation to determine the eak current algorithm for LLS sensors; forcing the intercet to zero and recalculating the linear regression [6, 7] on done dataset, they otained: = ( M 3. THEORETCAL ERROR To evaluate the theoretical error for the stroke eak current calculation, we use the method of the roagation of errors. f we have a measure z(x,y), where x and y are two indeendent variales, the δz error is: δz δx δy + (13) z x y We consider the eak current exressed as a function of the magnetic flux density at the time (t+c/), reminding formula (4): µ v + (14) πc 0 rs (, t c ) = ( t),

3 We use this formula to evaluate the eak current error and the constant factors are gathered as follows: h 1 = µ 0 v πc rs (15) Moreover the magnetic flux density has to e normalized at km, considering the attenuation factor : = (16) Exressing (16) as a function of, it assumes the following exression:, = h (17) ( ) where =1.13, suosing that the roagation soil is homogeneous and uniform. From the error roagation method, the current error will e given y: δ h δ + δ + δ (18) For each of the variales involved we have to aly again the error roagation calculation, to get each of the comonents: - the magnetic flux density error (δ), - the distance error (δ), - the attenuation coefficient error (δ). 3.1 The δ error For the magnetic density flux error δ the worst case is verified when the magnetic signal reaches the sensor antennas with an azimuthal angle of 45 degrees; in this case the signal have to e aout 40% higher to trigger the sensor, as exlained in [8]; this ercentage can e considered also as the angle-error sread for the eak current detection. f we suose that the stroke electromagnetic wave reaches the two sensor antennas with an azimuthal angle α, the magnetic flux density can e exressed in function of its azimuthal comonents North-South ( ) and West- East ( WE ): The error calculated with the error roagation method, is: ( δ) = ( δ ) + ( δα) α that, exliciting the artial derivatives, ecomes: ( δ) = ( δ ) + ( δα) (0) 1 cos α ( 4 senα senα where δ 0.4 and δα=1. The total magnetic density flux error, disregarding minor contriutions, results as δ 0.4; we can assume the worst-case scenario δ = 0.4. f we consider that each stroke is detected y a numer of sensors varying from as a minimum to 8 as a reasonale maximum, the error on should thus e evaluated making a statistic comination of each sensor s error on. We can consider that they are indeendent events, eing each detection indeendent from the others. The total error can e exressed y the following formula: δ = 0.4 / () tot N sensor where N sensor can vary from to 8, thus giving: 0.14 < tot < 0.3 as a worst case scenario. 3. The δ error is the distance etween the single sensor and the stroke imact oint. The comination of the two stroke localization methods (Time of Arrival and irection Finding) used in the LLS technology considered allows to otain four measures ( azimuthal angles and times of arrival) for each stroke imact determination (see figure ; with these four measures the network otains: - stroke latitude, - stroke longitude, - stroke time. = = WE (19) sen α cos α

4 ( h) = k log o, (6) then the error on can e exressed as: δ δ = δ = k (7) where k = = (8) ln10 Fig. 1 Scheme of the calculation method used for determining the lightning imact oint From figure 1 it can e noted that, if 1 and are two detecting sensors: 1 = v t 1 and = v t (3) - t 1 and t are resectively the different signal time of arrivals at the sensors, - v is the electromagnetic signal seed; we suose the ground conductivity as infinite and v c. Usually the random error on t is considered as δt = 1.5µs. From δ = vδ t (4) we calculate the error on, getting δ = 450 m. 500 m. Considering tyical distances for of some hundreds of km, we get δ The δ error The attenuation factor was determined emirically y Orville s exerimental analysis, as already seen, ut using the Cooray formula [9] we can write: ( 6 γ ) = k0 log (5) with: γ σ = where - is the distance etween the single sensor and the stroke imact - σ is the ground conductivity (the ground is assumed homogeneous and uniform) f in (5) k 0 = and h = it results: σ f it is suosed that the distance is of some hundreds of km, like the usual situation in a national network with tyical aseline of 00 km, it is: δ 10 4 (9) This is a very small quantity comared to the other comonents and can e neglected in the error roagation for δ. 3.4 The δ error Finally, after the evaluation of each error comonent, we can derive the δ. n the exression (17) we calculate the artial derivatives for and, otaining: + 1 = k = k ( + (30) h where k =. (3 The exression (18) can e re-arranged in: δ δ k ( + h h δ k ( + [( + δ ] + 1 (3) δ h = and after some mathematical assages, this ecomes: δ δ [( + δ+ 0.3 ] = ( (33) from which the ercentage error on current is:

5 δ δ 1 3 ( + ) + 0. (34) eing δ 10 3, we can say the ercentage error on the stroke eak current results: δ 0,1 0,3 (35) [8] W. Schulz and G. iendorfer, "etection Efficiency and Site Errors of Lightning Location Systems, nt. Lightning etection Conference, Tucson, Novemer 1996 [9] V. Cooray and H. Perez, Proagation effects on the first return stroke radiation fields: homogeneous aths and mixed two section aths, Proceedings of the nd nt. Conf. Lightning Protection (LCP), R1a-06, udaest, Setemer 1994 where this considers the worst situation for the error on, with the azimuthal angle of incidence at 45 degrees. oth the inferior and suerior limits can imrove if the incidence of the signal is at a minor angle. 4. CONCLUSO We considered the semi-emirical formulas used in the most common technology of LLS for the calculation of the lightning stroke eak current. On the current calculation different kind of error comonents have to e considered: the theoretical errors due to the formula used, the systematic errors rought in y the oeration quality of the network (thresholds, outages, site errors, waveform criteria), the Gaussian exerimental errors introduced y noise, tilt of channels and so on. We considered only the theoretical errors, suosed as Gaussian, and alied the error roagation to evaluate the whole theoretical error on the current calculation. A maximum value for the theoretical ercentage error results around 30%, in the worst-case scenario. The next ste of this study should e to comare our result to exerimental tests on eak current detection, otained y different grous. 5. REFERENCES [1] R.E. Orville, Caliration of a magnetic direction finding network using measured triggered lightning return stroke eak currents, J. Geohys. Res., 96, , 1991 [] V.P. done et al., A re-examination of the eak current caliration of the national lightning detection network, J. Geohys. Res., 98. No 10, , 1993 [3] G. iendorfer et al., Results of a erformance analysis of the Austrian lightning location network ALS, Proceedings of the nd nt. Conf. Lightning Protection (LCP), R1-01, udaest, Setemer 1994 [4] M. Uman and.k.mclain, Magnetic field of lightning return stroke, J. Geohys. Res., 74, ,1969 [5] M. Uman et al., The electromagnetic radiation from a finite antenna, Am.J. Phys., 43, 33 38, 1975 [6] K. L. Cummins et al., NLN eak current estimates ,Gloal Atmosherics technical note, 1996 [7] K. L. Cummins et al., A comined TOA/MF technology ugrade of the US National Lightning etection Network, J. Geohys. Res., 103. No 8, , 1998