ESTIMATION OF LIGHTNING ELECTROSTATIC PARAMETERS ...

2006

19th International Lightning Detection Conference 24-25 April • Tucson, Arizona, USA 1st International Lightning Meteorology Conference 26-27 April • Tucson, Arizona, USA

ESTIMATION OF LIGHTNING ELECTROSTATIC PARAMETERS BASED ON ATMOSPHERIC ELECTRIC FIELD MEASUREMENTS AND GENETIC ALGORITHMS Daniel Aranguren, Andrés Delgadillo, Mauricio Vargas, Ernesto Pérez and Horacio Torres Research Program of Acquisition and Analysis of Signals, National University of Colombia, Bogotá D.C., Colombia.

1. INTRODUCTION Colombia, a country located in tropical region, presents one of the greatest lightning activities of the world. Torres (1996). For this reason, many studies focused lightning stroke parameters and lightning protection have been developed; however, only few measurements of atmospheric electric field have been carried out. In order to characterize the spatial and temporal thunderclouds behavior and contrasting lightning parameters in Bogotá, an electric field measurement network was designed covering around 60% of the city area (about 196 km2) by using five electric field mill stations (Figure 1).

Atmospheric electric field variations are produced due to the electric charge generation and motion into the thunderclouds. During the thundercloud and lightning discharges formation, changes of magnitude and polarity of the atmospheric electric field take place. On the basis of cloud electric field measurements, it is possible to know the total thunderstorm duration, the different storm stages characteristics and some lightning parameters such as transferred charge and lightning location. McGorman (1998) Since 1920 many researchers have used the analysis of lightning electric field changes to identify the cloud electric structure and charge lightning characteristics. Wilson (1920) Nowadays the study of the atmospheric electric field is mainly based on the large ground-based networks of electric field mills which can measure the vertical electric field and the sudden field changes that are produced by lightning discharges. One of the most important field mill networks in the world is located on the Kennedy Space Center (KSC) and the Cape Canaveral Air Force Station (CCAFS) in Florida – USA. It is conformed by 31 electric field mill stations, and an algorithm based on the least squares optimization procedure is used to analyze the electric field variation measures. Koshak and Krider (1989)

Figure 1. Field mill network of Bogotá

The aim of this work is to present a methodology for estimating lightning parameters using genetic algorithms, applied to an electric field mill network located in Bogotá, Colombia. By utilizing genetic algorithms tools, the estimation errors are reduced considerably and it is possible to take into account more complex leader models spending less computation time.

The amount of field mills and their location were determined based upon the following criteria: a) typically, the thunderstorm electric field is detected at a distance not higher than 20 km. b) the number of field mills depends on the number of unknown parameters of the lightning model used. c) cloud-to-cloud and cloud-to-ground lightning discharges can be discriminated by means of a large network. Krider (1989). Table 1 shows the separation distances between field mills stations.

2. ELECTRIC FIELD MILL NETWORK OF BOGOTÁ

1

By considering the SC model, the total electric field change on ground produced by a totally extended leader (L) is given by (1).

TABLE 1. Distances between field mill

sites Stations U. Nacional – Fontibón

Distance [m]

∆E = −

7020

Fontibón – Suba

9810

Suba – Usaquen

8600

Usaquen - B. Unidos

7040

B. Unidos - U. Nacional

5980

U. Nacional – Suba

11530

Fontibón – Usaquen

17050

λq 2πε

 1 1 + H T2  −  D H T2 + D 2

(

)

1/ 2

  

(1)

Where D is the horizontal distance between the leader and the measurement point, which is given by: D = ( x − xi ) 2 + ( y − y i ) 2

The field mills used to measure the lightning electric field variations were designed and constructed by the National University of Colombia (Aranguren 2005). Each mill detects fields between -20 kV/m and 20 kV/m at a rate of 500 samples per second. The measurements are digitized with an accuracy of 40 V/m and a band-pass filter was used in order to eliminate the field components at 60 Hz and over 1 kHz. 3. LIGHTNING LEADER MODELS Figure.2. Source-charge leader model (Unidirectional propagation). Adapted from MacGorman (1998)

Lightning occurrence produces a change in the magnitude and distribution of the charge regions into the thundercloud. Lightning leader and return stroke models are required for the development and operation of the lightning location and parameters estimation algorithms. In the case of field mill networks, it is necessary to include models that consider the electric charge along the leader channel as it propagates toward ground and the transferred charge to ground during the return stroke stage. In this work, the point-charge and bidirectional leader models were used for the stepped leader and return stroke process.

If the process of stepped leader and return stroke (L+RS) are taken into account, the final electric charge change is reduced to a point charge variation ∆Q= λ q H T, because the return stroke neutralizes the leader electric charge. The total electric field change produced by the L+RS process is given by (2) B

∆E = −

3.1. Source-Charge Model

B

B

2∆QH T 4πε ( H T2 + D 2 ) 3 / 2

(2)

Expressions (1) and (2) have four unknown parameters which are found by the location algorithms based on the simultaneous electric field measurements.

The Source-Charge Model or Point-Charge Model (SC), proposed by Scholand (1938), considers the unidirectional and unipolar leader propagation (MacGorman, 1998). Figure 2 illustrates the SC model, where H T is the height of the inception point and λ q is the constant charge distribution along the leader channel. Because of the simplicity of this model, it is the most widely used in the world for cloud-to-ground discharges. B

B

3.2. Bidirectional Leader Model

B

B

The Bidirectional Leader Model (BD), proposed by Kasemir (1950), considers a bidirectional and bipolar leader propagation (MacGorman, 1998). This model assumes two leader channels propagating from an inception

2

point located at H T (Figure 3.) with opposite directions and polarities. In this case the charge distribution λq varies linearly as a function of the leader channel length. B

B

Where ∆Q = KH T

2

4. GENETIC ALGORITHM

By assuming the same velocity for the positive and negative leaders, the total electric field change on ground produced by the leader process (L) for a fully extended leader channel is given by (3):

∆E = +

K 2πε

 HT H  + T  2 2 1/ 2 H D D ( 4 + )  T

 K  D ln 2 2 1/ 2  2πε  (2H T + (4H T + D ) ) 

Where

Genetic algorithms (GA) are a search techniques based on the process of natural evolution where the more adapted individuals have greater probabilities of surviving and to transfer its genetic material to the following generations. The fundamental idea of GA is to find an acceptable solution of a problem by improving sets of individuals whose function of evaluation corresponds to a problem solution. This optimization is made by means of selective processes and of genetic information exchange.

(3)

λq = − KH T

The algorithm works with the information stored in genes. Every individual are constituted by genes chains denominated genome that represents a point in the search space of the problem. Thus, the objective of the genetic algorithm is to find the individual that contains the best possible characteristics for solving the problem (Fig. 4). (Martinez, 1999)

Figure 3. Bidirectional leader model. Adapted from MacGorman (1998)

For the leader and return stroke processes (L+RS), the electric charge of the downward channel is neutralized whereas the charge of the upward one is increased. The final electric field change on ground due to this process can be calculated by the expression (4). McGorman (1998). Figure 4. Flow Chart of a Genetic Algorithm 4.1. Genetic Codification

2H + (4HT2 + D2 )1 / 2  ∆Q  − 2HT + ln T ∆E =  1/ 2 2 2 2 4πεHT  4HT + D D  (4)

(

)

The genetic codification indicates the form to represent the variables of the problem,

3

optimized within the genome. Each individual Cx of the genetic algorithm population is made by a genes array whose size corresponds to the size of the problem.

one represents a model parameter, for example, in the SC model the individual was codified by 4 real genes, which represent the location (x,y), height H, and charge variation ∆Q. Eq. (5).

4.2. Selection Process

C x = {x, y, H , ∆Q}

This process chooses individuals from the population which will give origin to the future generations. Generally, the selection depends on the evaluation function value of each individual. This process has two stages: probability assignment and individual selection.

(5)

The objective function (OF) was defined as error between the electric field measured in the mills and the electric field simulated using the parameters found with the GA. In this way the goal of the GA is to minimize this error, finding the best parameter estimation. Eq. (6).

4.3. Cross Process

5

OF = ∑ Emesi − Esimi (x, y, H , ∆Q ) (6)

A mixing the genetic information of two individuals is made with the aim of generating new individuals. This process is executed in several stages: pair assignation, reproduction strategy and cross.

i =1

The methodology validation was carried out by the next procedure and criteria: Cloud-to-ground lightning discharges were simulated in steps of 200 m over the covering area of the field mills network. For each model, the values of electric field that would appear in the mills were found.

4.4. Mutation Process The individual genetic characteristics are modified in order to increase the exploration probability in the space search and diminishing the risk of stagnation of the algorithm in a local optimal. Each gene has a mutation probability that determines the alteration of the same gene to a new gene.

In this stage they were included both, the signal digitization rate (40 V/m) and the measurement error of the field mills, where ±10% was considered.

4.5. Adaptation Process

The GA was used to find the parameters of every lightning discharge simulated and using both SC and BD model.

The genetic algorithms use adaptation operators to fit their basic parameters with the aim to optimize the process search.

The process was made in two stages, the first execution began with random values of parameters, and the obtained results were used as the initial solution of a second execution of the GA

5. LIGHTNING LOCATION METHOD BASED ON GENETIC ALGORITHMS In order to estimate the parameters of lightning discharges on the basis of the lightning leader models SC and BD, a software was developed using C++ language and UNGenético 2.01 (Delgadillo, 2004). This allows implementing diverse kinds of optimization problems using Genetic Algorithms easily.

Finally the errors in the parameters estimation were found for every model and over the network covering region.

The Genetic Algorithm individuals were codified by means of real genes, where each

6. RESULTS AND ANALYSIS

1

UNGenético 2,0 is a C++ library to implement and to develop Genetic Algorithms, it is of free distribution under license GNU, and it was developed in the National University of Colombia in the 2004.

First of all, a least square optimization procedure similar which was presented by Koshak and Krider (1989) was used to find the

4

parameter of lightning discharges into the field mill network of Bogotá using the point-charge model Q. In this case a small area with location errors less than 500 m in the centre of network and some regions with an error in the parameter ∆Q less than 5% was found, however, a large zone presented much higher errors, more than 8 km in the location and 100 % in the charge variation ∆Q. Those results are shown by Aranguren (2005).

Bidirectional leader (BD) models, considering the stepped leader process (L) and the total leader and return stroke process (L+RS). Next, these results will be shown. Tables 2, 3 and 4 have the errors for the process L+RS when are used the models SC and BD. In this cases were considered the digitalization accuracy and measurement error of the field mills. Section 5 explains these assumptions.

Afterwards, the results by using the location method based on genetic algorithms proposed in this work are shown. In order to compare the results, maps of the network errors and graphics of the accumulate error distribution were made.

In the case of the location error (Table 2), the data are shown in three groups: percentage of the total map area with location errors less than 100 m, percentage of area with location errors less than 500 m and percentage of area with location errors less than 1000 m.

In figure 5 it is shown the distribution of the location error, in meters, when the SC model was used, considering the total leader and return stroke process L+RS. Figure 5a shows the map of the geographical distribution of the location error and the figure 5b present a histogram (blue blocks) and an accumulated location error distribution (red line). In figure 5b, the number of data is the number of lightning discharges simulated into the network and the % of data is the number of lightning discharges expressed in percentage.

In general, when it is used the SC model there are obtained larger areas with smaller errors than when it is used the BD model. Nevertheless, in both cases are obtained percentages of area between 30% and 40% with location errors less than 100 m. and more of 80% of the area has errors less than 1000m. The last are very good results taking account the first results obtained by Aranguren (2005). Besides, the errors in the parameters H and ∆Q were analyzed in three groups: percentages of area with errors less than 5 %, 10 % y 20 % (see tables 3 and 4).

As we can see in the map (Figure 5a) the largest part of the area into de network has location errors less than 300 m. In the histogram (Figure 5b) is able to see that 40% of the total area shown in the map, including the regions out the network, has location errors less than 83 m, also in the accumulate distribution is able to see that 70 % of the total area has location errors less than 498 m.

Tables 5, 6 and 7 show the errors for the L process when the models Q and BD were used. In these cases are not included the digitalization accuracy and measurement error of the field mills. The errors using the SC and BD models are similar and very low.

In the same way it was analyzed the errors in the parameters H and ∆Q which are shown in figures 6 and 7. 54% of the map area has errors in the charge variation ∆Q less than 2%. In addition, 35% of the area has errors of the inception height H less than 44 m for a lightning started at 3 km.

Between 30 % and 40 % of the total area has location errors less than 100 m in both models and around 80 % has location errors less than 1000 m; in addition, more than 80 % of the total area has errors in the parameter H less than 20 % and finally between 60 % and 70 % of the map area present errors in the parameter ∆Q not higher than 10%.

Finally, this analysis was used to compare the different errors in the network when are using the Source-charge (SC) and

5

Usaquen

Suba

Location Error Distribution QLRS0

3500

5000

99%

4500

89%

4000

79%

3500

69%

3000

60%

2500

50%

2000

40%

1500

30%

1000

20%

500

10%

3000

8000

[m]

2000 6000 1500 4000

B. Unidos

number of data

2500

Location error [m]

10000

1000

Fontibón 2000

% of data

12000

500 0

U. Nacional -6000

-4000

-2000

0 [m]

2000

4000

6000

0

0

83

166

249

(a)

332

415 498 581 Location error [m]

830

747

664

0%

(b)

Figure 5. Location error by using the SC model for the total leader and return stroke process. (a) Location error map. (b) Location error distribution. Parameter H, error distribution, Q-LRS0

Suba

Usaquen

12000

5000

99%

4500

89%

4000

79%

3500

69%

3000

60%

2500

50%

2000

40%

1500

30%

1000

20%

500

10%

30 4000

B. Unidos 20

Fontibón 2000

% of data

40 6000

Number of data

50

8000

Error parameter H [m]

60

10000

[m]

70

10 0

U. Nacional -6000

-4000

-2000

0

2000

4000

6000

0

0

44

88

132

[m]

(a)

176 220 264 308 Error Parameter H [m]

352

396

440

0%

(b)

Figure 6. Error in the parameter H by using the SC model for the total leader and return stroke process. (a) Map. (b) Location error distribution (HT = 3 km). ,

Parameter dQ, error distribution, Q-LRS0

8000

[m]

50 6000 40 B. Unidos 4000

30

Fontibón

Number of data

60

Error Parameter dQ [% ]

70

10000

5000

99%

4500

89%

4000

79%

3500

69%

3000

60%

2500

50%

2000

40%

1500

30%

1000

20%

500

10%

80

20

2000

10 0

U. Nacional -6000

-4000

-2000

0 [m]

2000

4000

6000

0

0

2

4

6

8 10 12 14 Error Parameter dQ [%]

20

16

18

0%

(a) (b) Figure 7. Error in the parameter ∆Q by using the SC model for the total leader and return stroke process. (a) Map. (b) Location error distribution.

6

% of data

Usaquen

Suba

12000

TABLE 2. Total area percentage with location errors less than 100, 500 and 1000 m (SC and BD models considering the L+RS process) Location error: < 100 m < 500 m < 1000 m Model / Lightning stage

SC/L+RS BD/L+RS

44.3 % 31.9 %

72.2 % 68.9 %

87.9 % 84.1 %

TABLE 3. Total area percentage with errors in the parameter H less than 5%, 10% and 20% (SC and BD models considering the L+RS process) Error parameter H: