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Investigations into the Influence of Matrix Dimensions and Number of Iterations on the Percolation Phenomenon for Direct Current

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  • Pawel Zukowski

    (Department of Economy, Vincent Pol University in Lublin, 2, Choiny Str., 20-816 Lublin, Poland)

  • Paweł Okal

    (Department of Electrical Devices and High Voltage Technology, Lublin University of Technology, 38D, Nadbystrzycka Str., 20-618 Lublin, Poland)

  • Konrad Kierczynski

    (Department of Electrical Devices and High Voltage Technology, Lublin University of Technology, 38D, Nadbystrzycka Str., 20-618 Lublin, Poland)

  • Przemyslaw Rogalski

    (Department of Electrical Devices and High Voltage Technology, Lublin University of Technology, 38D, Nadbystrzycka Str., 20-618 Lublin, Poland)

  • Sebastian Borucki

    (Department of Electric Power Engineering and Renewable Energy, Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology, 76, Proszkowska Str., 45-758 Opole, Poland)

  • Michał Kunicki

    (Department of Electric Power Engineering and Renewable Energy, Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology, 76, Proszkowska Str., 45-758 Opole, Poland)

  • Tomasz N. Koltunowicz

    (Department of Electrical Devices and High Voltage Technology, Lublin University of Technology, 38D, Nadbystrzycka Str., 20-618 Lublin, Poland)

Abstract

The paper presents studies of the site percolation phenomenon for square matrixes with dimensions L = 55, 101 and 151 using the Monte Carlo computer simulation method. The number of iterations for each matrix was 5 × 10 6 . An in-depth analysis of the test results using the metrological approach consisting of determining the uncertainty of estimating the results of iterations with statistical methods was performed. It was established that the statistical distribution of the percolation threshold value is a normal distribution. The coefficients of determination for the simulation results in approximations of the percolation threshold using the normal distribution for the number of iterations 5 × 10 6 are 0.9984, 0.9990 and 0.9993 for matrixes with dimensions 55, 101 and 151, respectively. The average value of the percolation threshold for relatively small numbers of iterations varies in a small range. For large numbers of iterations, this value stabilises and practically does not depend on the dimensions of the matrix. The value of the standard deviation of the percolation threshold for small numbers of iterations also fluctuates to a small extent. For a large number of iterations, the standard deviation values reach a steady state. Along with the increase in the dimensions of the matrix, there is a clear decrease in the value of the standard deviation. Its value is about 0.0243, about 0.01 and about 0.012 for matrixes with dimensions 55, 101 and 151 for the number of iterations 5 × 10 6 . The mean values of the percolation threshold and the uncertainty of its determination are (0.5927046 ± 1.1 × 10 −5 ), (0.5927072 ± 7.13 × 10 −6 ) and (0.5927135 ± 5.33 × 10 −6 ), respectively. It was found that the application of the metrological approach to the analysis of the percolation phenomenon simulation results allowed for the development of a new method of optimizing the determination and reducing the uncertainty of the percolation threshold estimation. It consists in selecting the dimensions of the matrix and the number of iterations in order to obtain the assumed uncertainty in determining the percolation threshold. Such a procedure can be used to simulate the percolation phenomenon and to estimate the value of the percolation threshold and its uncertainty in matrices with other matrix shapes than square ones.

Suggested Citation

  • Pawel Zukowski & Paweł Okal & Konrad Kierczynski & Przemyslaw Rogalski & Sebastian Borucki & Michał Kunicki & Tomasz N. Koltunowicz, 2023. "Investigations into the Influence of Matrix Dimensions and Number of Iterations on the Percolation Phenomenon for Direct Current," Energies, MDPI, vol. 16(20), pages 1-19, October.
    • Handle: RePEc:gam:jeners:v:16:y:2023:i:20:p:7128-:d:1261957
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    References listed on IDEAS

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    1. Wilkinson, Leland & Friendly, Michael, 2009. "The History of the Cluster Heat Map," The American Statistician, American Statistical Association, vol. 63(2), pages 179-184.
    2. Li, Daqing & Zhang, Qiong & Zio, Enrico & Havlin, Shlomo & Kang, Rui, 2015. "Network reliability analysis based on percolation theory," Reliability Engineering and System Safety, Elsevier, vol. 142(C), pages 556-562.
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    Cited by:

    1. Pawel Zukowski & Pawel Okal & Konrad Kierczynski & Przemyslaw Rogalski & Vitalii Bondariev, 2023. "Analysis of Uneven Distribution of Nodes Creating a Percolation Channel in Matrices with Translational Symmetry for Direct Current," Energies, MDPI, vol. 16(22), pages 1-14, November.
    2. Pawel Zukowski & Pawel Okal & Konrad Kierczynski & Przemyslaw Rogalski & Vitalii Bondariev & Alexander D. Pogrebnjak, 2023. "Monte Carlo Simulation of Percolation Phenomena for Direct Current in Large Square Matrices," Energies, MDPI, vol. 16(24), pages 1-14, December.

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