# A Methodology to Reduce the Computational Effort in the Evaluation of the Lightning Performance of Distribution Networks

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## Abstract

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## 1. Introduction

## 2. Procedure for the Evaluation of the Lightning Performance of Distribution Networks

- Random generation of a large number of lightning events N. Each event S
_{n}(with n = 1,…,N) is characterized by the point of impact ${P}_{F,n}=\left({x}_{F,n},{y}_{F,n}\right)$ and the channel-base current peak I_{0,n}. According to [7,8], the current is assumed to follow a log-normal Probability Density Function (PDF), while, for the point of impact coordinates, a uniform PDF is considered within a striking area that contains the power system of interest and all possible lightning events that can cause critical flashovers [3,4]. - Application of an attachment model (e.g., the electrogeometric model (EGM) [1]) that determines whether the selected event is a direct or indirect stroke. The direct lightning events can be simulated with the injection of a current source represented using Heidler’s function [9] in parallel with a suitable resistance accounting for the lightning channel. Otherwise, in case of an indirect strike, one has to resort to the equations describing the field-to-line coupling problem.
- For each event S
_{n}, with n = 1,…,N, the obtained overvoltage ${V}_{{S}_{n}}^{\text{max}}$ is compared with the line CFO in order to define a Boolean variable:$${b}_{n}=\{\begin{array}{cc}1& \text{if}{V}_{{S}_{n}}^{\text{max}}\ge CFO\\ 0& \text{otherwise}\end{array}$$ - Computation of the probability of having a dangerous overvoltage as the following ratio:$${p}_{N}=\frac{{\displaystyle \sum _{n=1}^{N}{b}_{n}}}{N}$$

## 3. A Methodology to Improve the Procedure Performances

#### 3.1. Reduction of the Number of Monte Carlo Runs

- Set a number K > 2 of Monte Carlo simulations, carried out in parallel, in which the independent model variables are maintained the same.
- Set $N=1$.
- Build up a set of K random events S
_{N,j}, for any j = 1,…,K. - Determine the probability p
_{N,j}according to Equation (2). - Calculate the means for each j = 1,…, K:$${\overline{p}}_{Nj}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{p}_{i,j}}$$
- Calculate the means of the means:$${\overline{p}}_{N}=\frac{1}{K}{\displaystyle \sum _{j=1}^{K}{\overline{p}}_{Nj}}$$
- Calculate values of the MSPE of the mean (MSPE
^{MEAN}):$$MSP{E}_{N}^{MEAN}=\frac{1}{K-1}{\displaystyle \sum _{j=1}^{K}{\left({\overline{p}}_{Nj}-{\overline{p}}_{N}\right)}^{2}}$$ - Compute the confidence interval for the probability, $\overline{p}$, to a significance level 1−α as$${\overline{p}}_{N}-{\delta}_{N}\le \overline{p}\le {\overline{p}}_{N}+{\delta}_{N}$$$${\delta}_{N}=\frac{\sqrt{MSP{E}_{N}^{MEAN}}}{\sqrt{K}}{t}_{1-\alpha /2,K-1}$$
- Define the Boolean quantity:$${B}_{N}=\left\{2{\delta}_{N}<\delta \text{or}{E}_{N}\epsilon \right\}$$$${E}_{N}=\frac{2{\delta}_{N}}{{\overline{p}}_{N}}$$
_{N}= 1 if at least one of the two conditions is verified, otherwise it is equal to 0). - If B
_{N}= 0, increase N of a unit, i.e., N = N + 1 and repeat all steps starting from Step 3, otherwise stop the Monte–Carlo procedure.

_{N}= 1, as in [11] it is shown that $\underset{N\to \infty}{\text{lim}}MSP{E}_{N}^{MEAN}=0$, which means that $\underset{N\to \infty}{\text{lim}}{\delta}_{N}=0$. This guarantees that the stopping criterion introduced at Step 10 is well defined and effectively enables an exit out of the loop.

_{N}is posed to relate the amplitude of the interval δ

_{N}to the mean probability value ${\overline{p}}_{N}$, but is likely to become ineffective for very small values of ${\overline{p}}_{N}$. This way, the condition on the absolute value of such interval 2δ

_{N}prevents us from performing an unnecessarily high number of Monte–Carlo runs.

#### 3.2. Reduction of the Number of Calls to the Field-to-Line Coupling Code

_{i}, each of them placed at a height h

_{i}(I = 1,…,M) over a lossy ground. In the reference frame depicted in Figure 1, a reference system Oxyz is placed with the x axis parallel to the direction of the line conductors, so that it is possible to define the position y

_{i}of each conductor. For the sake of simplicity and without losing generality, we can suppose that each line starts from the point (0, y

_{i}, h

_{i}) and ends at (L, y

_{i}, h

_{i}). Let us then suppose that a lightning strike occurs in a point P

_{F}at the ground level with coordinates (x

_{F}, y

_{F},0).

_{F}of all the possible events that can cause dangerous overvoltages has to be chosen. According to the geometry represented in Figure 1, such an area is here indicated as

_{min}and y

_{max}can be based on the following considerations:

- y
_{min}can be chosen as the value under which, for any peak current, the power system will always experience a direct strike. Recalling the electrogeometric criterion described in [1], one can express the lateral distance as a function of the peak current. As can be expected, such function is monotonically increasing; thus, the value corresponding to the minimum current can be selected as y_{min}. In the following, y_{min}will be set as 15.00 m, which corresponds to about 2 kA (that has a probability smaller than 0.016% according to the current lognormal probability density function presented in Equation (1) in [1] by using the parameters of Table 1 in [1]); - y
_{max}can be chosen as the horizontal distance at which, no matter the peak current, the resulting overvoltage will always be smaller than the CFO. According to the Rusk–Darveniza formula [12], one can set$${y}_{\text{max}}=\frac{38.80}{\text{CFO}}\left(h+\frac{0.15}{\sqrt{{\sigma}_{g}}}\right)\text{max}\left\{{I}_{0}\right\}.$$

_{max}is assumed to be 1500.0 m, corresponding to the most critical situation characterized by a peak current of 100.0 kA (that has a probability smaller than 0.0038% according to the current lognormal probability density function presented in Equation (1) in [1] by using the parameters of Table 1 in [1])), a ground conductivity of 0.001 S/m, and a CFO of 50 kV.

_{0}and impact point P

_{F}in area A, calculates the resulting overvoltage, i.e., the quantity:

_{s}, T

_{e}] is the duration of the lightning electromagnetic fields and ${V}_{i}\left(x,t\right)$ is the voltage at point x of the conductor i at time t. In this work, this calculation is performed using the algorithm and the PSCAD interface presented in [13].

_{0r}, r = 1,…,R of peak currents, a set of SxQxR events is available, named S

_{sqr}, and it is possible to write the 3D matrix

_{3}performs the 3D linear interpolation of the elements of matrix

**V**when the impact point is P

_{F}and the channel-base current amplitude is I

_{0}.

## 4. Numerical Results

- in Test1 (see Figure 2a), the network is fed by a 132 kV voltage source and contains an HV/MV transformer, a MV/LV transformer (both protected by surge arresters), and two laterals respectively at 250 m and 750 m from the HV/MV transformer and consisting of two MV/LV identical transformers with two 300-m-long LV cables.
- in Test2 (see Figure 2b), the same network of Test1 is considered, but without laterals.
- in Test3 (see Figure 2c), the same network of Test1 is considered, but without the surge arresters.

- the number of parallel runs K is 5 (according to [10]);
- two significance levels, 90% and 95% (corresponding to $\alpha =0.10$ and $\alpha =0.05$, respectively), are analyzed;
- the threshold constant values are $\delta =0.015$ and $\epsilon =0.15$, which means that the mean probability value, for a selected CFO, will belong, at level $1-\alpha $, to the confidence interval with an amplitude smaller than 0.015 or, in any case, smaller than 15% of the computed mean probability ${\overline{p}}_{N}$.

- the mean value of the probability ${\overline{p}}_{N}$
- the boundaries of the confidence strip ${\overline{p}}_{N}-{\delta}_{N}$ and ${\overline{p}}_{N}+{\delta}_{N}$
- the absolute and relative errors $2{\delta}_{N}$ and ${E}_{N}$ (in
**bold**the leading stop condition)

- the evolution of ${E}_{N}$ as a function of the number of runs for $\alpha =0.10$ and $\alpha =0.05$ (Panels a and b);
- the evolution of ${\delta}_{N}$ as a function of the number of runs for $\alpha =0.10$ and $\alpha =0.05$ (Panels c and d) with an indication of the number of runs selected to construct Table 2;
- the evolution of the confidence strip and the mean probability ${\overline{p}}_{N}$ as a function of the number of runs for $\alpha =0.10$ and $\alpha =0.05$ (Panels e and f).

_{0}and γ

_{L}being two quarters of the circles centered respectively on (0, y

_{min}) and (L, y

_{min}) with a radius of d-y

_{min}. As mentioned in the definition of the domain A, here, D can be reduced to D

_{+}if a perfect symmetry of the MTL is exploited.

## 5. Conclusions

## Conflicts of Interest

## References

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**Figure 2.**Distribution network topology for the three considered tests. Panels (

**a**–

**c**) show the networks for Test1, Test2, and Test3, respectively.

**Figure 3.**Graphical evolution of ${E}_{N}$ (Panel (

**a**) and (

**b**)), ${\delta}_{N}$ (Panel (

**c**) and (

**d**)), confidence strip and ${\overline{p}}_{N}$ (Panel (

**e**) and (

**f**)) as a function of N for CFO = 150 kV.

**Figure 6.**Comparison of the proposed approach performed on the original domain (blue) and on the reduced one (red). Panels (

**a**–

**c**) show the results for Test1, Test2, and Test3, respectively.

Cond. 1 | Cond. 2 | Cond. 3 | |
---|---|---|---|

height from ground | 8.0 m | 8.0 m | 8.6 m |

distance from y axis | −1.2 m | 1.2 m | 0.0 m |

conductor diameter | 0.64 cm | 0.64 cm | 0.64 cm |

CFO (kV) | N | ${\overline{\mathit{p}}}_{\mathit{N}}$ | ${\overline{\mathit{p}}}_{\mathit{N}}-{\mathit{\delta}}_{\mathit{N}}$ | ${\overline{\mathit{p}}}_{\mathit{N}}+{\mathit{\delta}}_{\mathit{N}}$ | $2{\mathit{\delta}}_{\mathit{N}}$ | ${\mathit{E}}_{\mathit{N}}$ | |
---|---|---|---|---|---|---|---|

$\alpha =0.05$ | 50 | 1500 | 0.233 | 0.216 | 0.250 | 0.034 | 0.145 |

100 | 1100 | 0.087 | 0.080 | 0.094 | 0.014 | 0.158 | |

150 | 1100 | 0.055 | 0.047 | 0.062 | 0.014 | 0.268 | |

200 | 900 | 0.042 | 0.035 | 0.050 | 0.014 | 0.344 | |

250 | 400 | 0.034 | 0.027 | 0.041 | 0.014 | 0.423 | |

$\alpha =0.10$ | 50 | 1100 | 0.235 | 0.217 | 0.252 | 0.034 | 0.146 |

100 | 800 | 0.088 | 0.081 | 0.095 | 0.014 | 0.158 | |

150 | 900 | 0.055 | 0.048 | 0.062 | 0.014 | 0.250 | |

200 | 700 | 0.043 | 0.036 | 0.050 | 0.014 | 0.320 | |

250 | 400 | 0.034 | 0.029 | 0.040 | 0.011 | 0.325 |

CFO (kV) | N | ${\overline{\mathit{p}}}_{\mathit{N}}$ | ${\overline{\mathit{p}}}_{\mathit{N}}-{\mathit{\delta}}_{\mathit{N}}$ | ${\overline{\mathit{p}}}_{\mathit{N}}+{\mathit{\delta}}_{\mathit{N}}$ | $2{\mathit{\delta}}_{\mathit{N}}$ | ${\mathit{E}}_{\mathit{N}}$ | |
---|---|---|---|---|---|---|---|

$\alpha =0.05$ | 50 | 1400 | 0.270 | 0.250 | 0.291 | 0.040 | 0.149 |

100 | 700 | 0.099 | 0.092 | 0.107 | 0.014 | 0.145 | |

150 | 2600 | 0.056 | 0.049 | 0.064 | 0.014 | 0.264 | |

200 | 1500 | 0.040 | 0.032 | 0.047 | 0.014 | 0.371 | |

250 | 1400 | 0.035 | 0.027 | 0.042 | 0.014 | 0.422 | |

$\alpha =0.10$ | 50 | 1100 | 0.235 | 0.217 | 0.252 | 0.034 | 0.146 |

100 | 1000 | 0.275 | 0.255 | 0.294 | 0.039 | 0.142 | |

150 | 400 | 0.098 | 0.091 | 0.105 | 0.014 | 0.148 | |

200 | 1200 | 0.056 | 0.049 | 0.064 | 0.014 | 0.265 | |

250 | 800 | 0.040 | 0.033 | 0.047 | 0.014 | 0.362 |

CFO (kV) | N | ${\overline{\mathit{p}}}_{\mathit{N}}$ | ${\overline{\mathit{p}}}_{\mathit{N}}-{\mathit{\delta}}_{\mathit{N}}$ | ${\overline{\mathit{p}}}_{\mathit{N}}+{\mathit{\delta}}_{\mathit{N}}$ | $2{\mathit{\delta}}_{\mathit{N}}$ | ${\mathit{E}}_{\mathit{N}}$ | |
---|---|---|---|---|---|---|---|

$\alpha =0.05$ | 50 | 1000 | 0.430 | 0.399 | 0.461 | 0.062 | 0.145 |

100 | 1700 | 0.174 | 0.162 | 0.187 | 0.026 | 0.147 | |

150 | 2900 | 0.087 | 0.079 | 0.094 | 0.014 | 0.172 | |

200 | 3900 | 0.053 | 0.045 | 0.060 | 0.014 | 0.283 | |

250 | 2800 | 0.039 | 0.032 | 0.047 | 0.014 | 0.380 | |

$\alpha =0.10$ | 50 | 700 | 0.436 | 0.406 | 0.466 | 0.060 | 0.138 |

100 | 1000 | 0.177 | 0.164 | 0.190 | 0.026 | 0.147 | |

150 | 1100 | 0.089 | 0.082 | 0.096 | 0.014 | 0.163 | |

200 | 2000 | 0.052 | 0.045 | 0.060 | 0.014 | 0.286 | |

250 | 1300 | 0.040 | 0.033 | 0.048 | 0.014 | 0.374 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Bendato, I.; Brignone, M.; Delfino, F.; Procopio, R.; Rachidi, F.
A Methodology to Reduce the Computational Effort in the Evaluation of the Lightning Performance of Distribution Networks. *Atmosphere* **2016**, *7*, 147.
https://doi.org/10.3390/atmos7110147

**AMA Style**

Bendato I, Brignone M, Delfino F, Procopio R, Rachidi F.
A Methodology to Reduce the Computational Effort in the Evaluation of the Lightning Performance of Distribution Networks. *Atmosphere*. 2016; 7(11):147.
https://doi.org/10.3390/atmos7110147

**Chicago/Turabian Style**

Bendato, Ilaria, Massimo Brignone, Federico Delfino, Renato Procopio, and Farhad Rachidi.
2016. "A Methodology to Reduce the Computational Effort in the Evaluation of the Lightning Performance of Distribution Networks" *Atmosphere* 7, no. 11: 147.
https://doi.org/10.3390/atmos7110147