O2-4 Lovric D Full

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LEAST SQUARES ESTIMATION OF DOUBLE-

EXPONENTIAL FUNCTION PARAMETERS

Double-exponential function parameters can be estimat-

ed in four cases by solving a set of m nonlinear equations

using the Marquardt method [3], [5], depending on the

available input requirements (Fig. 2):

Case 1: m = 2, E1 = R 1 and E2 = R 2

Case 2: m = 3, E1 = R 1, E2 = R 2 and E3 = R 3

Case 3: m = 3, E1 = R 1, E2 = R 2 and E3 = R 4

Case 4: m = 4, E1 = R 1, E2 = R 2, E3 = R 3 and E4 = R 4

In Fig. 2, a flowchart is presented that describes the es-

timation of the double-exponential function parameters $, %

and & introduced in (1). In each rth iteration, parameters

tmax, $ and t1 have to be computed, where the auxiliary pa-

rameters tmax and t1 are computed by solving the corre-

sponding nonlinear equation. The abbreviation MSE in

Fig. 2 stands for Marquardt method for a Single nonlinear

Equation. It is used for estimation of values of t1 or tmax.

The parameter $ is computed from the linear equation

given in Fig. 2. Then the parameters % and & are computed

by the Marquardt method from the corresponding set of

nonlinear equations in the rth iteration.

The nonlinear equation for computing the parameter tmax

can be obtained from the following requirement:

0dt

di

maxtt

=

=

(10)

The following normalized nonlinear equation for com-

putation of tmax can be obtained from (10):

max1k r

max1k r

tr tr

max

k eeF

!!"#!"$!

"#+"$!= (11)

When estimating tmax, the initial value is taken to be

tmax = 0.9'T1.

The nonlinear equation of computing the parameter t1

can be obtained from the following requirement:

10 ttfor I1.0i =!= (12)

Fig.2 – Least squares estimation of double-exponential function parameters.



Using (2), the following normalized nonlinear equation

for computation of t1 can be obtained:

1ee1.0

1F 1

1k r 1

1k r tt

1

k !"#$%

&' !(

)(=

!! (*!(+! (13)

When estimating t1, the initial value is taken to bet1 = 0.1'T1.

In Fig. 2, the matrix [D] is a diagonal matrix whose di-

agonal elements are identical to the diagonal elements of

matrix [A] defined by the following equation:

[ ] [ ] [ ]JJA T

!= (14)

where the Jacobian matrix [J] can be computed by:

[ ]

( ) ( )

( ) ( )!!!!!!

!!

"

#

$$$$$$

$$

%

&

'(

')(

)(

')(

'(

')(

)(

')(

=

*

**

*

**

*

**

*

**

1r

1r 1r m

1r

1r 1r m

1r

1r 1r 1

1r

1r 1r 1

,E,E

,E,E

J !! (15)

Vector {B} can be computed using following expression:

{ }

( )

( )!!

"

!!

#

$

!!

%

!!

&

'

()

()

=

**

**

1r 1r m

1r 1r 1

,E

,E

B ! (16)

Partial derivatives of nonlinear equations (6-9) whichare required for computation of matrix [J] in (15) can be

computed analytically using the following expressions:

!"

"=

#$

$

!"

"%=

&$

$ "#%"&%

9.0

etR ;

9.0

etR 22 t21

t21 (17)

!"

"=

#$

$

!"

"%=

&$

$ "#%"&%

5.0

etR ;

5.0

etR hh th2

th2 (18)

20

03

20

03

Q

IR ;

Q

IR

!"#"=

!$

$

%"#"&=

%$

$ (19)

( )

( ) !

"

#$%

&

'()

'+*(

+(=

',

,

!"

#$%

&

*()

'+*(

+(=

*,

,

222

0

2

04

222

0

204

2

12

W

IR

2

12

W

IR

(20)

NUMERICAL EXAMPLES

The first numerical example features Case 4, i.e. the

simultaneous solving of four nonlinear equations. The input

data taken from IEC 62305-1 Ed. 2 [4] represent the maxi-

mum values of lightning current quantities of the first posi-

tive impulse for Lightning Protection Level III-IV:

I0 = 100 kA, T1/T2 = 10/350 µs, Q0 = 50 C and

W0 = 2.5 MJ/(. Double-exponential function parameters

are estimated using a computer program that implements

the method described in the previous section. The following

parameters have been obtained: $ = 0.9511, % = 2121.76 s-1

and & = 245303.6 s-1

. Double-exponential function with

these parameters is depicted in Fig. 3.

Fig.3 – Double-exponential function approximation of the first posi-

tive impulse.

Other often used lightning current waveshapes some-

times used for designing low-voltage power lines withinstructures are the T1/T2 = 0.2/5 µs waveshape, the

T1/T2 = 4/16 µs waveshape and T1/T2 = 1.2/50 µs

waveshape [1, 6]. Since these waveshapes are only defined

by T1 and T2 values, only two nonlinear equations are sim-

ultaneously solved. Results of the estimation for these three

waveshapes are presented in Table I. Corresponding dou-

ble-exponential functions approximating these waveshapes

are depicted in Fig. 4. All functions have a current peak

value of I0 = 1 A for plotting clarity purposes. Peak values

can be changed accordingly since the parameters in Table I

are independent of the peak current value.

Table I

Double-exponential function parameters for fast-decaying light-ning current waveshapes.

Waveshapes

T1/T2

Double-exponential function

parameters

! " (s-1

) # (s-1

)

0.2/5 µs 0.93269 152921.46 11887358.7

4/16 µs 0.27475 117598.38 252722.5

1.2/50 µs 0.95847 14732.18 2080312.7

Fig.4 – Double-exponential function approximation of fast-decayinglightning current waveshapes.

However, in the case of communication lines which

have a different exposure to lightning than power lines,

different waveshapes are used for designing lightning pro-

tection system. These waveshapes are characterized by a

relatively sharp rise followed by a very slow current decay:



the T1/T2 = 10/700 µs waveshape and the

T1/T2 = 10/1000 µs waveshape [6]. Again, these

waveshapes are only defined by T1 and T2 values, so only

two nonlinear equations are simultaneously solved.

Results of the estimation for these two waveshapes are

presented in Table II. Double-exponential functions approx-

imating these waveshapes are depicted in Fig. 5.Table II

Double-exponential function parameters for slow-decaying light-ning current waveshapes.

Waveshapes

T1/T2

Double-exponential function

parameters

! " (s-1

) # (s-1

)

10/700 µs 0.97423 1028.39 257923.7

10/1000 µs 0.98135 712.41 262026.6

Fig.5 – Double-exponential function approximation of slow-

decaying lightning current waveshapes.

In addition to standardized lightning current, the pre-

sented least squares method can easily be used to approxi-

mate the waveshapes of various recorded impulse stroke

currents. The recorded waveshape taken from [7] is depict-

ed on Fig. 6, along with the double-exponential function

approximation. The input data of the recorded waveshape

current taken from [7] is: I0 = 5 A, T1 = 9.3 µs and T2 = 90

µs. The resulting double-exponential function parameters

are: $ = 0.80917, % = 10054.37 s-1

and & = 197765.3 s-1

.

Fig.6 – Recorded impulse current and its double-exponential func-

tion approximation.

In the following example a typical negative first stroke

current waveshape is considered. The recorded waveshape

taken from [8] is depicted on Fig. 7, along with the double-

exponential function approximation. One can observe from

this figure the inadequacy of the double-exponential func-

tion approximation. The recorded lightning current is char-

acterized by a slow rise at the very beginning followed by a

much steeper rise. On the other hand, the double-

exponential function has the steepest rise in the time t = 0

and can not approximate the recorded waveshape accurate-

ly. Much better results can be obtained using the Heidler

function [3, 9] or Javor function [10]. The input data of the

recorded waveshape current taken from [8] is: I0 = -1 A, T1

= 8 µs and T2 = 100 µs. The resulting double-exponential

function parameters are: $ = 0.86481, % = 8421.53 s-1

and& = 265585.9 s

-1.

Fig.7 – Recorded lightning current and its double-exponential func-

tion approximation.

CONCLUSION

In this paper, a robust and effective algorithm for the

least squares estimation of double-exponential function

parameters is presented. Using this algorithm various

standardized and recorded lightning current waveshapes

can be approximated by the double-exponential function.

This algorithm can be easily modified to estimate the pa-

rameters of an arbitrary lightning current function.

REFERENCES[1] V. Cooray: “ Lightning protection”, V. Cooray, 2007, London,

pp. 67-72.

[2] C. E. R. Bruce, R. H. Golde: “The lightning discharge”, J. Inst.

Elect. Eng, Vol. 88, Part 2, 1941, pp. 487-520.

[3] S. Vujevi#, D. Lovri#, I. Juri#-Grgi#: “Least squares estimation

of Heidler function parameters”, European Transactions on

Electrical Power, Vol. 21, 2011, pp. 329-344.

[4] IEC 62305-1 Ed. 2, Protection against lightning – Part 1: Gen-

eral principles, 2010.

[5] D.W. Marquardt: “An algorithm for least-squares estimation of

nonlinear parameters”, Journal of the Society for Industrial

and Applied Mathematics, Vol. 11, No. 2, 1963, pp. 431-441.

[6] M.A. Uman: “The art and science of lightning protection”,

Cambridge University Press, 2008, New York.

[7] Z. Stojkovi# et al.: “Sensitivity analysis of experimentally

determined grounding grid impulse characteristics”, IEEE

Transactions on Power Delivery, Vol. 13, No. 4, 1998, pp.

1136-1142.

[8] K. Berger, R. B. Anderson, H. Kroninger: “Parameters of

lightning flashes”, ELECTRA, No. 41, 1975, pp. 23-38.

[9] D. Lovri#, D. Vujevi#, T. Modri#: “On the estimation of

Heidler function parameters for reproduction of various

standardized and recorded lightning current waveshapes”, In-

ternational Transactions on Electrical Energy Systems, Vol.

23, 2013, pp. 290-300.

[10] V. Javor: “New function for representing IEC 62305 Standardand other typical lightning stroke currents”, Journal of Light-

ning Research, Vol. 4, 2012, pp. 50-59.

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