least squares heidler

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2011; 21:329–344Published online 28 April 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.445

Least squares estimation of Heidler function parameters

*C

NayE-

Co

Slavko Vujevic*, Dino Lovricy and Ivica Juric-Grgic

University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture,

Croatia

SUMMARY

The aim of the proposed paper is to present an effective numerical algorithm for the computation of Heidlerfunction parameters. The basic six channel-base current quantities can be prescribed: current peak value,front duration, time to half value, current steepness factor, charge transfer at the striking point and specificenergy. The approximation of the unknown three lightning current parameters for Heidler function isachieved using the least squares method. For the purpose of better convergence, the Marquardt least squaresmethod has been applied. The proposed algorithm can be successfully applied for lightning currentmodelling in power engineering as well as in research on electromagnetic compatibility. Copyright #

2010 John Wiley & Sons, Ltd.

key words: lightning current; Heidler function parameters; least squares method; Marquardt method

1. INTRODUCTION

In recent years, a number of experiments have been conducted with the purpose of directly measuring

and recording the current waveform produced by a lightning strike. The most important data were

accumulated and analysed by Berger, who recorded on top of a telecommunications tower situated on

the mountain San Salvatore in Switzerland [1,2]. Based on this data, a new analytical function has been

proposed by Heidler in Reference [3]. A variant of this new analytical function, for a current steepness

factor n¼ 10, has been incorporated in IEC 62305–1 as the power function recommended for lightning

current modelling in both lightning research and engineering applications [4]. This new function

permits a good separation of the characteristic lightning current quantities.

The computation of all Heidler function parameters simultaneously requires solving a system of

nonlinear equations. In Reference [5], the authors claim that the system of nonlinear equations often

has no real solution, and they instead use a set of approximated functions that represent the current rise

time and current decay time. In this way, they avoid solving the nonlinear integral equations. The

authors then use a graphical algorithm in combination with these approximations to obtain Heidler

function parameters.

In this paper, Heidler function parameters are computed by simultaneously solving a system of two,

three or four nonlinear equations. The numerical algorithm developed is a robust and reliable tool,

which can be used to compute Heidler function parameters for any combination of the input

requirements. Furthermore, with minor modifications, this universal numerical algorithm can be used

to compute the parameters of any other lightning current approximation function, for example the often

orrespondence to: Slavko Vujevic, University of Split, Faculty of Electrical Engineering, Mechanical Engineering and

val Architecture, Croatia

mail: [email protected]

pyright # 2010 John Wiley & Sons, Ltd.

330 S. VUJEVIC, D. LOVRIC AND I. JURIC-GRGIC

used double exponential approximation. In relation to the prescribed set of lightning current quantities,

four different cases are possible.

2. HEIDLER REPRESENTATION OF THE LIGHTNING CURRENT

According to Reference [4], the lightning current approximation of the first stroke can be represented

by the plot shown in Figure 1, where I0 is the current peak value, t0 is the virtual starting time, t1 is the

time to 10% of peak value, t2 is the time to 90% of peak value, th is the total time to half value of the

peak value, tmax is the time to the peak value, T1 is the front duration and T2 is the (virtual) time to half

value.

Figure 1. Lightning current approximation by Heidler function.

The current function most frequently used in the lightning research is Heidler function, which can be

expressed by the following equations [3–5]:

i tð Þ ¼ I0

h� xðtÞ � yðtÞ ¼ I0

h� x � y (1)

x ¼ x tð Þ ¼tt1

� �n

1 þ tt1

� �n (2)

y ¼ y tð Þ ¼ e� t

t2 (3)

where h is the correction coefficient of the current peak value, x(t) is the rise function, y(t) is the decay

function, t1 is the time constant determining current rise-time, t2 is the time constant determining

current decay-time, and n is the current steepness factor.

Copyright # 2010 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2011; 21:329–344

DOI: 10.1002/etep

HEIDLER FUNCTION PARAMETERS 331

3. SET OF NONLINEAR EQUATIONS

According to Figure 1, the two basic requirements for the estimation of lightning current parameters h,

t1 and t2 can be written as:

i ¼ 0:9 � I0 for t ¼ t2 (4)

i ¼ 0:5 � I0 for t ¼ th (5)

Two additional requirements are deduced from the value of the charge transfer at the striking point

Q0 and the specific energy W0:

Z10

i � dt ¼ Q0 (6)

Z10

i2 � dt ¼ W0 (7)

The specific energy W0 represents the energy dissipated by the lightning current in a unit resistance

[4].

From Equations (1) and (4)–(7), the following four normalized nonlinear equations are obtained:

R1 ¼ 1

0:9 � h � x t2ð Þ � y t2ð Þ � 1 ¼ 0 (8)

R2 ¼ 1

0:5 � h � x thð Þ � y thð Þ � 1 ¼ 0 (9)

R3 ¼ I0

Q0 � h�Z10

xðtÞ � yðtÞ � dt � 1 ¼ 0 (10)

R4 ¼ I20

W0 � h2�Z10

x2ðtÞ � y2ðtÞ � dt � 1 ¼ 0 (11)

Integrals in nonlinear Equations (10) and (11) are solved using a combination of the Simpson’s rule

for interval t2 [0, tg] and analytical integration for t> tg:

R3 ¼ I0

Q0 � h�Ztg0

xðtÞ � yðtÞ � dt

|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Simpson0s rule

þ I0 � t2

Q0 � h� e�

tgt2 � 1 (12)

R4 ¼ I20

W0 � h2�Ztg0

x2ðtÞ � y2ðtÞ � dt

|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Simpson0s rule

þ I20 � t2

W0 � h2 � 2� e�

2�tgt2 � 1 (13)

where:

tg ¼ t1 �ffiffiffiffiffiffiffi104n

p(14)


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is chosen in such a way that, for t> tg, the rise function x(t)� 1 and Heidler function (1) can be quite

accurately approximated by:

i tð Þ � I0

h� yðtÞ ¼ I0

h� e�

tt2 (15)

According to Equations (2) and (14), the rise function x(tg)¼ 0.9999 and Heidler function (1), for

t> tg, is approximated by Equation (15) with a percentage error less than 0.01%.

4. LEAST SQUARES ESTIMATION OF t1 AND t2

In Figure 2, a flow chart is presented that describes the estimation of the lightning current parameters h,

t1 and t2 introduced by Equations (1)–(3). Parameters t1 and t2 are computed by the Marquardt least

squares method from the corresponding system of nonlinear equations, while the parameter h is

computed from a linear equation at the beginning of the iterative procedure as well as at the end of each

iteration.

The current steepness factor n, current peak value I0, front duration T1 and time to half value T2 are

prescribed lightning current quantities for all cases. However, in some cases the charge transfer at the

striking point Q0 and/or the specific energy W0 can be additional input data. The estimation of Heidler

function parameters t1 and t2 is based on the Marquardt least squares method (also known as

Levenberg–Marquardt method) [6,7].

In each rth iteration, the parameters tmax, h and t1 have to be computed, where the auxiliary

parameters tmax and t1 are computed by solving the corresponding nonlinear equations, while the

coefficient h can be computed using Equation (B.2) derived in Appendix B:

h ¼ rh ¼tmaxrt1

� �n

1 þ tmaxrt1

� �n � e�tmax

rt2 (16)

The lightning current parameters t1 and t2 can be estimated in four cases depending on the available

input requirements. The requirements described by Equations (4) and (5) are known in all cases, while

the value of the charge transfer at the striking point Q0 and the specific energy W0 described by

Equations (6) and (7) are known in some cases. This gives a total of four possible combinations of input

requirements. In all cases, the unknown parameters t1 and t2 can be estimated by solving the set of m

nonlinear equations using the least squares method, where one needs to minimize the objective

function representing the sum of squares of several nonlinear functions depending on the input set of

requirements:

S ¼Xmi¼1

E2i ¼ minimum (17)

where:

Case 1: m¼ 2; E1¼R1 and E2¼R2

Case 2: m¼ 3; E1¼R1, E2¼R2 and E3¼R3

Case 3: m¼ 3; E1¼R1, E2¼R2 and E3¼R4

Case 4: m¼ 4; E1¼R1, E2¼R2, E3¼R3 and E4¼R4

The system of nonlinear Equations (8), (9), (12) and (13) is solved simultaneously using the

Marquardt least squares method:

Aþ l � D½ � � Dt1

Dt2

� �¼ Aþ 3 � D½ � � Dt1

Dt2

� �¼ B (18)

where l is an adjustable positive parameter which is used to control the iteration.


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Figure 2. Estimation of the lightning current parameters.


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Observing the left hand side, as l!1, matrix A becomes insignificant with respect to the dominant

elements of the matrix l�D. Consequently, the solution of the current iteration is influenced only by the

elements of the matrix l�D, and this reduces the Marquardt least squares method to the steepest

descents method. On the other hand, if the parameter l! 0, then the solution of the current iteration is

influenced only by the elements of the matrix A and this reduces the Marquardt least squares method to

the Gauss–Newton or generalized least squares method [7]. The steepest descent method is known to

be convergent but slow, while the Gauss–Newton method is rapid but the convergence is less reliable.

The main advantage of the Marquardt least squares method [6] is the possibility of choosing the

parameter l as to follow the Gauss–Newton method as much as possible to ensure rapid convergence,

while at the same time retaining certain characteristics of the steepest descent method to prevent

divergence. After a number of numerical experiments, the parameter l was chosen to have the value 3,

which enables rapid and reliable convergence in all cases.

The matrix D is a diagonal matrix whose diagonal elements are identical to the diagonal elements of

matrix A [8,9], which is defined by the following equation:

A ¼ JT � J (19)

where the Jacobian matrix J can be computed approximately:

J ¼

DE1

Dt1

DE1

Dt2

..

. ...

DEm

Dt1

DEm

Dt2

2664

3775 (20)

Approximations of partial derivatives are used, and they are computed as follows:

DEi

Dt1

¼ Ei t1 þ 1; t2ð Þ � Ei t1; t2ð Þ; i ¼ 1; 2; :::;m (21)

DEi

Dt2

¼ Ei t1; t2 þ 1ð Þ � Ei t1; t2ð Þ; i ¼ 1; 2; :::;m (22)

where m is the index number of Equations (8), (9), (12) and (13) as stated before.

According to the Marquardt least squares method, the matrix on the right hand side in the Equation

(18) is:

B ¼ �JT �

E1

E2

..

.

Em

8>><>>:

9>>=>>; (23)

The iterations will continue until the conditions for the convergence of parameters t1 and t2 are

achieved. In all cases the condition for the convergence of the values of t1 and t2 is:

S ¼Xmi¼1

E2i < 10�14 (24)

where m¼ 2 for the first case, m¼ 3 for the second and third cases and m¼ 4 for the fourth case. When

this condition is achieved the iteration process is over, and the final values of parameters t1, t2 and h are

acquired.

5. COMPUTATION OF PARAMETER tMAX

As presented in Figure 3, the input data for computation of the unknown auxiliary parameter tmax are

the current steepness factor n and the time constants obtained in the rth iteration rt1 and rt2.


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Figure 3. Computation of tmax.


In each rth and kth iteration, the parameter tmax can be computed using nonlinear Equation (C.7)

derived in Appendix C:

Fmax ¼ktmax

rt1

� �nþ1

þktmax

rt1

� �� n � rt2

rt1

¼ 0 (25)

The nonlinear Equation (25) is solved using a special case of the Marquardt least squares method for

a single nonlinear equation. Analogously, as in Equation (18), it can be stated:

J2max � 1 þ lð Þ � Dtmax ¼ B (26)

where:

B ¼ �Jmax � Fmax (27)


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In each rth and kth iteration, the Jacobian Jmax can be computed using Equation (C.9) derived in

Appendix C:

Jmax ¼ @Fmax

@ðktmaxÞ¼ 1

rt1

� 1 þ nþ 1ð Þ �ktmax

rt1

� �n (28)

where Fmax is described by Equation (25).

Then the growth of the parameter Dtmax can be computed by:

Dtmax ¼ � Fmax

Jmax � 1 þ lð Þ ¼ � Fmax

4 � Jmax

(29)

where l ¼ 3.

The value of tmax for the kth iteration is computed from the following expression:

ktmax ¼ k�1tmax þ Dtmax (30)

As stated in Equation (30), the value of parameter tmax will grow in every iteration by the value

of Dtmax. The iterations will continue until the condition for the convergence of the parameter tmax is

achieved:

S ¼ F2max < 10�14 (31)

6. COMPUTATION OF PARAMETER t1

The input data for computation of the unknown auxiliary parameter t1 are the current steepness factor n,

coefficient h and the time constants obtained in the rth iteration rt1 and rt2. The procedure for

estimating the parameter t1 is presented in Figure 4.

In each rth and kth iteration, the parameter t1 can be computed using nonlinear Equation (D.3)

derived in Appendix D:

F1 ¼ 0:1 � hþkt1rt1

� �n

� 0:1 � h� e�

k t1rt2

� �¼ 0 (32)

The nonlinear Equation (32) can be solved in the same way as described for the

parameter tmax. Again, in the first iteration (k¼ 0), the first step is to assume the value of the

variable t1. Analogously, as in Equations (26)–(29), the growth of the iteration for the parameter t1 can

be found from:

Dt1 ¼ � F1

J1 � 1 þ lð Þ ¼ � F1

4 � J1

(33)

where l ¼ 3.

In each rth and kth iteration, the Jacobian J1 can be computed using Equation (D.5) derived in

Appendix D:

J1 ¼ @F1

@ðkt1Þ¼ n

rt1

�kt1rt1

� �n�1

� 0:1 � h� e�

k t1rt2

� �þ 1

rt2

�kt1rt1

� �n

�e�k t1rt2 (34)

where F1 is described by Equation (32).

The value of t1 for the kth iteration is then computed from the following expression:

kt1 ¼ k�1t1 þ Dt1 (35)

As presented in Equation (35), the value of parameter t1 will grow in each iteration by the

value of Dt1. The iterations will continue until the condition for the convergence of parameter t1 is

achieved:

S ¼ F21 < 10�14 (36)


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Figure 4. Computation of t1.


7. COMPUTING THE MAXIMUM OF THE CURRENT STEEPNESS

The maximum of the current steepness didt

� �max

can be computed from the requirement:

d2i

dt2

� �t¼ts

¼ 0 (37)

where ts is the time when the current derivative attains its maximum. The flow chart of the procedure

for computing the maximum of the current steepness is presented in Figure 5. In the first step, the

unknown value ts is computed using the Marquardt least squares method, while, in the second step, the

maximum of the current steepness can be computed using a simple linear equation.


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Figure 5. Computation of the maximum of the current steepness.


In each kth iteration of the Marquardt least squares method, according to Equations (1)–(3) and

Equation (37), the following nonlinear equation Fs and its Jacobian Js can be obtained:

Fs ¼ t22 � f 00 � 2 � t2 � f 0 þ f ¼ 0 (38)

Js ¼@Fs

@ðktsÞ¼ t2

2 � f 000 � 2 � t2 � f 00 þ f 0 (39)

where the function f and its derivatives are given by:

f ¼ktst1

� �n

1 þ ktst1

� �n (40)

f 0 ¼ n

t1

�ktst1

� �n�1

1 þ ktst1

� �nh i2(41)


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k� �n�2 n� 1 � nþ 1ð Þ � kts

� �n

f 00 ¼ n

t21

� ts

t1

� t1

1 þ ktst1

� �nh i3(42)

f 000 ¼ n

t31

�kts

t1

� �n�3

�n� 2ð Þ � n� 1ð Þ � 4 � n2 � 1ð Þ � kts

t1

� �n

� nþ 1ð Þ � nþ 2ð Þ � ktst1

� �2�n

1 þ ktst1

� �nh i4(43)

The value of parameter ts in each kth iteration can be computed using the following relations:

kts ¼ k�1ts þ Dts (44)

Dts ¼ � Fs

Js � 1 þ lð Þ ¼ � Fs

4 � Js(45)

where l ¼ 3. The iterations will continue until the condition for the convergence of the parameter ts is

achieved:S ¼ F2

s < 10�14 (46)

Using the computed ts, the maximum of the current steepness is given by:

di

dt

� �t¼ts

¼ di

dt

� �max

¼ I0

h� f 0 � f

t2

� �� e�

tst2 (47)

In Heidler grapho-analytical algorithm [5], the maximum of the current steepness is computed using

the approximation y(t) � 1, where y(t) is decay function defined by Equation (3). However, in this

paper, decay function approximation is not used.

8. NUMERICAL EXAMPLES

The following three test cases are shown to illustrate the application of the proposed procedure.

8.1. Example 1

In the following example, the unknown parameters of Heidler function are computed by

simultaneously solving the nonlinear equations E1, E2 , E3 and E4 (Case 4). The input data are

taken from Reference [4], and they represent the maximum values of lightning current quantities for

Table I. Computed parameters of Heidler function for 10/350ms for I0¼ 200 kA, Q0¼ 100 C andW0¼ 10 MJ/V.

n t1 (ms) t2 (ms) h didt

� �max

ðkA=msÞ

3 5.824 463.498 0.936 30.3494 7.731 468.095 0.940 28.7235 9.630 470.705 0.941 28.0266 11.516 472.391 0.940 27.6827 13.403 473.546 0.938 27.4798 15.286 474.391 0.935 27.3569 17.168 475.036 0.933 27.275

10 19.048 475.544 0.930 27.22211 20.928 475.954 0.927 27.18512 22.806 476.291 0.924 27.15913 24.684 476.574 0.921 27.14114 26.561 476.815 0.917 27.12715 28.438 477.021 0.914 27.118


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Figure 6. Heidler representation of the first short stroke 10/350ms for I0¼ 200 kA, Q0¼ 100 C andW0¼ 10 MJ/V.


the first short stroke for Lightning Protection Level I: I0¼ 200 kA, T1/T2¼ 10/350ms, Q0¼ 100 C and

W0¼ 10 MJ/V. Parameters t1, t2, h and (di/dt)max are computed using a computer program written in

MATLAB that implements the Marquardt least squares method described previously. The results of the

computations are shown in Table I for the current steepness factor n 2 3; 4; :::; 15f g, while the Heidler

lightning function for n¼ 5, 10, 15 is shown in Figure 6.

8.2. Example 2

In this example, the unknown parameters of Heidler function are computed by simultaneously solving

the nonlinear equations E1 and E2 (Case 1). The input data are taken from Reference [4], and they

represent the maximum values of lightning current quantities for the subsequent short stroke for

Lightning Protection Level I: I0¼ 50 kA, T1/T2¼ 0.25/100ms. Parameters t1, t2, h and (di/dt)max are

computed using the Marquardt least squares method, as before, and are presented in Table II for the

current steepness factor n 2 3; 4; :::; 15f g, while the Heidler lightning function for n¼ 5, 10, 15 is

shown in Figure 7.

Table II. Computed parameters of Heidler function of the subsequent stroke 0.25/100ms for I0¼ 50 kA.

n t1 (ms) t2 (ms) h didt

� �max

ðkA=msÞ

3 0.129 142.442 0.991 328.6164 0.177 142.777 0.992 303.3595 0.224 142.955 0.993 292.3366 0.270 143.064 0.993 286.5297 0.317 143.137 0.993 283.0948 0.363 143.189 0.993 280.8959 0.409 143.228 0.993 279.404

10 0.455 143.258 0.993 278.34711 0.501 143.282 0.993 277.57212 0.547 143.302 0.992 276.98713 0.593 143.318 0.992 276.53514 0.639 143.332 0.992 276.17915 0.685 143.343 0.992 275.894


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Figure 7. Heidler representation of the subsequent stroke 0.25/100ms for I0¼ 50 kA.

Table III. Comparison of the computed Heidler function parameters for typical subsequent return stroke.

h t1 (ms) t2 (ms)

Marquardt least squares method 0.837 0.265 3.607Heidler grapho-analytical algorithm [5] 0.84 0.27 3.5Percentage difference (%) �0.358 �1.887 2.966


8.3. Example 3

As a final example, in Table III, a comparison is made between Heidler function parameters computed

by the Heidler grapho-analytical algorithm [5] and those computed by the Marquardt least squares

method for typical subsequent return stroke with input data: I0¼ 13 kA, n¼ 5 and Q0¼ 50 mC.

In this case, the results computed by the proposed numerical algorithm and the Heidler grapho-

analytical algorithm [5] are relatively close. Although the relatively close results in this case, Heidler

grapho-analytical algorithm does not enable the separation of the input parameters, while the proposed

numerical algorithm can find the best estimation of lightning current parameters depending on the used

set of input parameters.

9. CONCLUSION

In this paper, the lightning current parameters of Heidler function are computed using the Marquardt

least squares method to solve several different systems of nonlinear equations. Four cases for obtaining

the lighting current parameters are presented for the chosen set of input parameters. In the first case,

two nonlinear equations require simultaneous solving, in the second and third cases, there are three

nonlinear equations, and in the fourth case, there are four nonlinear equations to solve simultaneously.

In all cases, four basic input parameters are prescribed, while an additional two input parameters—

charge transfer at the striking point and the specific energy—may be prescribed. The iterative process

of computing the lightning current parameters is illustrated in detail in four flow charts, which describe


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the various segments of the solution process and thus provide general guidance for implementing these

algorithms in a computer program.

The proposed numerical algorithm represents an advancement in relation to the Heidler grapho-

analytical algorithm. The proposed algorithm is much more robust and suitable for software

implementation than the Heidler grapho-analytical algorithm. Besides these advantages, the proposed

numerical algorithm can take into account the specific energy of the lightning strike as an additional

input parameter. This parameter is important in determining the mechanical effects of the lightning

stroke, more specifically the thermal effects. Since each of the lightning current parameters tends to

dominate each failure mechanism, introduction of the specific energy as the input parameter is an

advancement in analysing the mechanical effects of lightning strike. Furthermore, disregarding the fact

that this numerical algorithm improves the lightning current parameter computation, it also enables the

separation of the input parameters, while the Heidler grapho-analytical algorithm does not allow for the

separation of the input parameters.

Unlike the Heidler grapho-analytical algorithm, the proposed numerical algorithm can be easily

modified to accommodate the need of computing the parameters of any other lightning current function

approximation used in electric power engineering as well as in research on electromagnetic

compatibility.

REFERENCES

1. Berger K, Anderson RB, Kroninger H. Parameters of lightning flashes. Electra 1975; 41:23–37.

2. Anderson RB, Erikson AJ. Lightning parameters for engineering application. Electra 1980; 69:65–101.

3. Heidler F, Cvetic J. A class of analytical functions to study the lightning effects associated with the current front.

European Transaction on Electric Power (ETEP) 2002; 12(2):141–150.

4. IEC 62305-1: Protection Against Lightning - Part 1: General Principles, 2006.

5. Heidler F, Cvetic J, Stanic BV. Calculation of lightning current parameters. IEEE Transactions on Power Delivery

1999; 14(2):399–404.

6. Marquardt DW. An algorithm for least-squares estimation of nonlinear parameters. Journal of the Society for

Industrial and Applied Mathematics 1963; 11(2):431–441.

7. Lill SA. Optimization in Action. Academic Press: London, 1976.

8. Vujevic S, Kurtovic M. Direct and iterative automatic interpretation of resistivity sounding data. International Journal

for Engineering Modelling 1992; 5(3–4):65–118.

9. Vujevic S. A combined analysis of earthing grids in karstic soil. PhD Thesis, University of Split, Faculty of Electrical

Engineering, Mechanical Engineering and Naval Architecture: Split 1994. (in Croatian).

APPENDIX A: NOMENCLATURE

(di/dt)max m

Copyright # 2

aximum of the current steepness

F1 s
tarting nonlinear equation for computing the parameter t1 Fmax s tarting nonlinear equation for computing the parameter tmax
Fs s
tarting nonlinear equation for computing the parameter ts I0 c urrent peak value
n c
urrent steepness factor
Q0 c
harge transfer at the striking point
T1 f
ront duration
T2 (
virtual) time to half value
t0 v
irtual starting time
t1 t
ime to 10% of peak value
t2 t
ime to 90% of peak value
tg t
ime parameter used for approximation of Heidler function
th t
otal time to half value of the peak value
tmax t
ime to the peak value
ts t
ime when the current derivative attains its maximum
W0 s
pecific energy
010 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2011; 21:329–344

DOI: 10.1002/etep


x(t) r

Copyright # 2

ise function

y(t) d
ecay function
h c
orrection coefficient of the current peak value
l a
rbitrarily chosen constant with a positive value
t1 t
ime constant determining current rise-time
t2 t
ime constant determining current decay-time
APPENDIX B

The current i(t) attains its peak value I0 at the time tmax. After substituting I¼ I0 and t¼ tmax in

Equations (1)–(3), the following equation is obtained:

I0 ¼I0h�

tmax

t1

� �n

1þ tmaxt1


t2 (B.1)

After dividing both sides of Equation (B.1) by I0/h, the equation for computing correction

coefficient h is obtained:

h ¼tmaxt1

� �n

1þ tmaxt1


t2 (B.2)

APPENDIX C

The time derivative of Heidler function described by Equations (1)–(3) equals zero for t¼ tmax when

the lightning current attains its peak value:

di

dt

t¼tmax

¼ I0h� dx

dt� yþ x � dy

dt

� � t¼tmax

¼ 0 (C.1)

where the time derivative of the rise function x described by Equation (2) is:

dx

dt¼

nt1� t

t1

� �n�1� 1þ t

t1

� �nh i� n

t1� t

t1

� �n�1� t

t1

� �n

1þ tt1

� �nh i2 ¼nt1� t

t1

� �n�1

1þ tt1

� �nh i2 (C.2)

while the time derivative of the decay function y described by Equation (3) is:

dy

dt¼ � 1

t2� e�

tt2 (C.3)

After substituting Equations (2), (3), (C.2) and (C.3) into Equation (C.1), the following equation for

t¼ tmax can be obtained:

nt1� tmax

t1

� �n�1�e�

tmaxt2 � 1þ tmax

t1

� �nh i� tmax

t1

� �n� 1t2� e�

tmaxt2

1þ tmax

t1

� �nh i2 ¼ 0 (C.4)

The numerator of Equation (C.4), which is equal to zero, can be written in the next form:

e�tmax

t2 � n

t1� tmax

t1

� �n�1

� tmax

t1

� �n

� 1t2

� tmax

t1

� �2�n� 1t2

" #¼ 0 (C.5)

010 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2011; 21:329–344

DOI: 10.1002/etep


From Equation (C.5) follows equation:

tmax

t1

� �2�nþ tmax

t1

� �n

� n � t2t1

� tmax

t1

� �n�1

¼ 0 (C.6)

which can be transformed into the next nonlinear equation for computing tmax:

Fmax ¼tmax

t1

� �nþ1

þ tmax

t1

� �� n � t2

t1¼ 0 (C.7)

The Jacobian Jmax of Fmax is defined by:

Jmax ¼@Fmax

@tmax(C.8)

Substitution of Equation (C.7) into Equation (C.8) and partial differentiation with respect to tmax

yields:

Jmax ¼@Fmax

@tmax¼ nþ 1

t1� tmax

t1

� �n

þ 1

t1¼ 1

t1� 1þ nþ 1ð Þ � tmax

t1

� �n (C.9)

APPENDIX D

The current i(t) attains 10% of the peak current value I0 at the time t1. After substituting I¼ 0.1�I0 and

t¼ t1 in Equations (1)–(3), the following equation is obtained:

0:1 � I0 ¼I0h�

t1t1

� �n

1þ t1t1

� �n � e�t1

t2 (D.1)

After dividing both sides of Equation (D.1) by I0/h, follows the next equation:

0:1 � h ¼t1t1

� �n�e�

t1t2

1þ t1t1

� �n (D.2)

Rearranging the previous expression the next nonlinear equation for computing t1 is obtained:

F1 ¼ 0:1 � hþ t1t1

� �n

� 0:1 � h� e�t1

t2

� �¼ 0 (D.3)

The Jacobian J1 of F1 is defined by:

J1 ¼@F1@t1

(D.4)

Substitution of Equation (D.3) into Equation (D.4) and partial differentiation with respect to t1 yields:

J1 ¼@F1@t1

¼ n

t1� t1

t1

� �n�1

� 0:1 � h� e�t1

t2

� �þ 1

t2� t1

t1

� �n

�e�t1t2 (D.5)


DOI: 10.1002/etep

least squares heidler

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