# On Minimisation of Earthing System Touch Voltages

^{*}

## Abstract

**:**

## 1. Introduction

_{1}…d

_{n}are appropriate ES dimensions and g

_{1}…g

_{n}, h

_{1}…h

_{n}are the spatial dimension constraints. This is because the designing engineer should mainly focus on decreasing the overall EPR to lower the corresponding touch voltages as much as possible. Decreasing the EPR is most effectively done through decreasing the resistance to earth of the ES design. The described procedure of (2) and (3) in all situations would mean lengthening the ES horizontal dimensions up to the outer boundary of the horizontal constraints and burying the ES near the burial depth lower constraint. This procedure might be also expected, regardless of most often used horizontally stratified soil model if the simplified formula of (1) is used.

## 2. Ansys Maxwell Optimisers

#### 2.1. Quasi Newton (QN)

#### 2.2. Pattern Search (PS)

#### 2.3. Sequential Non-Linear Programming (SNLP)

#### 2.4. Sequential Mixed Integer Non-Linear Programming (SMINLP)

#### 2.5. Genetic Algorithm (GA)

#### 2.6. Matlab Optimiser (MO)

## 3. Earthing System Optimised Model

_{1}was buried at depth h

_{1}and the outer ring with diameter D

_{2}was buried at depth h

_{2}. The earthing system was made of FeZn strip with cross section 30 × 4 mm

^{2}. The earthing system was modelled in Ansoft Maxwell [16,17] with horizontally stratified soil model with two layers—the surface one with resistivity ρ

_{1}and thickness H and the bottom layer with resistivity ρ

_{2}. The soil was modelled by semi-sphere with radius of 200 m and the analysis percent error was set to 1%. For the illustrative purpose, the earthing system model in Figure 1a,c is depicted with an improper soil scale. The earthing system was excited by a DC current [17] of 30 A (red arrow in bottom right picture of Figure 1c) injected at the centre of the ES, where the ES conductor is brought up above the ground and it is considered as accessible for touch.

## 4. Earthing System Sensitivity Analysis Results

_{1}= 1, 2, and 3 m, and outer ring always greater than inner ring D

_{2}= 4 and 5 m. In the case of burial depths of both rings h

_{1}and h

_{2}, seven burial depths were used for each ring as 0.3–1.5 m with step of 0.2 m. I.e., for one set combination of ES rings diameters D

_{1}and D

_{2}all 49 combinations of ES design burial depths were modelled by Ansys Maxwell and the results were used in the sensitivity analysis. By this way, about 2000 ES design variations were analysed in this analysis. For each of these ES design variations, the potential profile on the earth surface along x direction (Figure 1b) was read and the TV distribution have been determined in the vicinity of the ES. Although the potential profile was read with step of 0.25 m, for the purpose of optimisation evaluation throughout this paper four main performance values have been determined—TV

_{1m}, TV

_{2m}and TV

_{3m}as TV 1, 2, and 3 m apart from the centre of the earthing system, respectively (i.e., potential difference between EPR and point 1, 2 and 3 m apart). Additionally, the total EPR was read. For set diameter D

_{2}(used here to represent the dimensional constraint), the optimum burial depth was manually searched whilst changing other three dimensions D

_{1}, h

_{1}, and h

_{2}. Incorporating the optimisation of inner ring diameter D

_{1}whilst manually searching for optimum design is quite challenging as the number of analysed variations increase rapidly. The results of the sensitivity analysis are introduced in Figure 2, Figure 3, Figure 4 and Figure 5 and Table 2, Table 3, Table 4 and Table 5.

_{2}= 4 m, three TV

_{1m}surfaces are depicted for different inner ring diameter D

_{1}and each surface is consisting of results of all 49 burial depth variations. In this compact form, the optimum burial depth of both rings can be immediately read. Within the dimensional constraints of h

_{1}, h

_{2}(being in range of 0.3–1.5 m) and D

_{2}(being fixed as 4 m), the ‘global’ minimum of the solution surface can be expected for D

_{1}= 2 m, h

_{1}= 0.3 m, and h

_{2}= 1.5 m. In case of D

_{1}= 3 m, a remarkable saddle point (h

_{1}= 0.7 m, h

_{2}= 0.6 m) is formed by a significant change in slope direction, where two areas with local (h

_{1}= 1.5, h

_{2}= 0.5 m) and global (h

_{1}= 0.3, h

_{2}= 1.5 m) minima are formed. Although this saddle point might not be as much remarkable for all simulated inner ring diameters, it is still present among all the surfaces.

_{2m}and TV

_{3m}. The results of these analysis are summarized in tabular form in Table 2, Table 3, Table 4 and Table 5. In this compact tabular form, only the remarkable points of the sensitivity analysis surfaces are listed for different soil models and for different ES dimensions whilst preserving the clarity of the results. The burial depths of ES designs were selected for two extreme cases and for all three analysed TV

_{xm}objective functions:

- Min TV where the burial depth of the design is selected for lowest TV
_{xm}. - Max TV where the burial depth of the design is selected for the highest TV
_{xm}.

_{1}, D

_{2}, and selected soil model are listed and they can be compared for different objective functions TV

_{xm}, EPR objectives etc.

_{1}from 2 m to 3 m and with set constraint of outer ring D

_{2}= 4 m. Even though the best possible design assessed based on EPR would be with both rings in the bottom layer h

_{1}= h

_{2}= 1.5 m and D

_{1}= 2 m, based on TV

_{1m}the best design would be a different one with only the outer ring in the bottom layer and the inner ring in the surface layer h

_{1}= 0.3 m, h

_{2}= 1.5 m and D

_{1}= 2 m. Additionally, if TV

_{2m}and TV

_{3m}objectives would be assessed a completely different designs are optimal. These findings might not be as much intuitive. However, with the increasing number of possible assessed option, the situation might get even more tricky.

_{1m}and TV

_{2m}and for constraint of outer ring diameter D

_{2}= 4 m. As it might be expected by some experienced engineers, the better designs are with an outer ring burial depth greater than the thickness of the surface layer. The overall best design, as already stated in the previous paragraph, is with D

_{1}= 2 m, h

_{1}= 0.3 m, and h

_{2}= 1.5 m. However, this might not be true if other constraints are applied, e.g., dimensional constraint of both rings maximum burial depth of only 0.9 m and both TV

_{1m}and TV

_{2m}should be assessed. In the region of constraint burial depth of only 0.9 m, there is an evident change of pattern in optimal burial depths (between (a) and (b) of Figure 3) where the optimum design depending on the objective function is either for inner ring close to the surface with h

_{1}= 0.3 m and h

_{2}close to bottom layer, or the inner ring close to bottom layer and outer ring close to the surface, respectively. Additionally, design with increased inner ring diameter of 3 m gives better results than 2 m as in the case of TV

_{1m}objective. From both figures it is evident that if both TV

_{1m}and TV

_{2m}objectives need to be evaluated, the contradiction of the solution will need some sort of more complicated addressment, e.g., weighting of risk, etc.

_{1m}, burying the inner ring only to a shallow depth of 0.3 m led to decreasing the risk. The rest of the designs are either comparable, or even worse. Burying both rings to the bottom layer is worse by slightly more than 40% than the optimal shallow inner ring placement (h

_{1}= 0.3 and h

_{2}= 1.5 m). The TV surface for inner ring diameter of 1 m for HoL has yielded in almost all cases to higher TVs and is thus excluded here.

_{1m}, TV

_{2m}, and TV

_{3m}can be obtained. From Figure 5a, it is obvious that the optimum design in case of objective TV

_{1m}is with deep outer ring h

_{2}= 1.5 m and with shallow inner ring h

_{1}= 0.3 m. However, this chosen design is by no means also optimal when also assessed to TV

_{2m}.

## 5. Ansys Maxwell Earthing System Optimisation Results

_{1}, h

_{1}, and h

_{2}as D

_{1i}, h

_{1i}and h

_{2i}) and through the optimisation the ‘optimised’ dimensions have been obtained and are denoted by the subscript ‘o’ (i.e., D

_{1o}, h

_{1o}, h

_{2o}). The outer ring diameter D

_{2}was set as a dimensional constraint equal to 4 m in most of the simulations. The burial depth constraint was set as in region 0.3–1.5 m for both rings and the maximum inner ring diameter constraint was set as D

_{1max}= 0.9xD

_{2}, thus so the inner ring is always smaller than the outer ring. The minimization of TV

_{1m}was set as the optimisation objective. The optimisation was carried out by five different optimisers—QN, SNLP, SMINLP, PS, GA. After the optimisation process some other parameters have been also recorded for possible comparison and benchmarking of different optimisers. I.e., TV

_{2m}, TV

_{3m}, the total EPR of optimised ES design, total time needed for finding the optimal design t, the number of performed simulations in total ItN (optimisation iterations), and a number of iterations (out from total of ItN iterations) where the optimal design was found ItO. As an optimisation analysis stopping criterion was also set the maximum number of optimisation iteration that was in most cases 300, or in some cases only 100. The optimisation results, together with the performed stopping option, are listed in Table 6 (Parts 1 and 2). The stopping option is in the last column ‘Status’ as either one of the following three options:

- S—for solved, stopped by finding minimum by Ansys Maxwell optimizer, the obtained result is expected to be the global minimum.
- M—for stopped by maximum number of iterations, not necessary global minimum found, the best suiting result was selected.
- F—for simulation failed. Again, the best suiting result was selected, however the optimisation ended prematurely. The failure might had happened due to more different problems. Either Ansys Maxwell adaptive mesher failed to build the mesh of finite elements, or the optimizer failed in finding the dimensions of the next design or the optimizer generally failed without description.

_{2}= 4 m (Figure 2) should be 0.3 and 1.5 m for h

_{1}and h

_{2}, respectively, and inner ring D

_{1}= 2 m (as was found in sensitivity analysis Table 2, Table 3 minimum TV

_{1m}). For simplicity and comparability, first two optimisation analysis had been performed just with only burial depths h

_{1}and h

_{2}being optimised (lines 26, 27 of Table 6). This is also due to the fact that the simulated designs in sensitivity analysis were simulated with quite great step of inner ring diameter of 1 m, as the possible number of designs rapidly increase when a lower step would had been used. These two optimisation analysis were performed with different initial conditions. In both cases, Ansys Maxwell found ES design close to the expected optimal solution. In the second case, the outer ring depth was only 1.4 m where the optimiser evaluated visiting the depth of 1.5 m as unnecessary. This might be caused by the level of solution noise, i.e., with the set analysis percent error of 1% (Section 3) the changes in solution data of 5–15 V might be expectable due to randomness of creation of finite element mesh and also due to not sufficiently fine mesh elements. Better results might be expected for percent error of about 0.3% [17], however this will cause about three times or even greater increase in solution time.

_{1m}surfaces from Figure 2 it can be expected that the optimal design should has inner ring diameter in 2–3 m region, and the burial depths as h

_{1}= 0.3 m and h

_{2}= 1.5 m. This expectation had been met in most of the results for SNLP optimiser. However, in case of the other optimisers, kind of invalid and different results had been found. This might have happened in some cases due to stucking in actual local minima, stucking in local minima caused by the solution noise, failing due to Ansys Maxwell Adaptive Mesher, etc.

_{1}~ 2 m, h

_{1}~ 0.3 m, h

_{2}~ 1.5 m) only about three times out of total 1134 optimisation iterations. The optimal design was already found after about the first half of all iterations in the 664th iteration, however the optimiser did not evaluate it as the global optimum and was further searching for better solution.

_{1}= 3.5 m). However, in case of this local minimum the TV

_{1m}value is about 15–25 V higher that the expected global minimum in the region of D

_{1}= 2–2.5 m. Thus, from results of lines 24 and 28 in (and also 2 QN and 5 PS) Table 6, it can be seen that, in the case of unfavourable initial conditions, the global minimum is not always found and the optimiser might be unable to overcome this problem on its own.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**TV1m solution surfaces for ES with diameters D

_{1}= 1–3 m, D

_{2}= 4 m, ρ

_{1}= 500 Ωm, ρ

_{2}= 100 Ωm, H = 2 m.

**Figure 3.**Touch voltage (TV) surfaces, 500 Ωm/100 Ωm/1 m, D

_{2}= 4 m, (

**a**) TV

_{1m}surface; (

**b**) TV

_{2m}surface.

**Figure 5.**TV profiles along x direction for HoL model ρ

_{1}= 500 Ωm, ρ

_{2}= 100 Ωm, D

_{1}= 2 m, D

_{2}= 4 m, h

_{1}= 0.3 m; (

**a**) surface layer thickness H = 2 m; and, (

**b**) Surface layer thickness H = 1 m.

Soil Model No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

HoL | HoL | HoL | Uniform | LoH | LoH | LoH | |

ρ_{1} (Ωm) | 500 | 1000 | 500 | 100 | 100 | 100 | 100 |

ρ_{2} (Ωm) | 100 | 100 | 100 | 100 | 500 | 1000 | 500 |

H (m) | 2 | 2 | 1 | ∞ | 2 | 2 | 1 |

**Table 2.**Sensitivity analysis results for ρ

_{1}= 500 Ωm, ρ

_{2}= 100 Ωm, H = 2 m, D

_{2}= 4 m, D

_{1}= 2 m.

Objective | Min TV Option | Max TV Option | ||||||
---|---|---|---|---|---|---|---|---|

h_{1} | h_{2} | TV | EPR | h_{1} | h_{2} | TV | EPR | |

(m) | (m) | (V) | (V) | (m) | (m) | (V) | (V) | |

TV_{1m} | 0.3 | Any | 160 | 870 * | 1.5 | 1.5 | 315 | 785 |

TV_{2m} | 1.5 | 0.3 | 235 | 935 | 0.3 | 1.3 | 460 | 935 |

TV_{3m} | 1.5 | 1.5 | 545 | 785 | 0.3 | 0.3 | 805 | 1220 |

**Table 3.**Sensitivity analysis results for ρ

_{1}= 500 Ωm, ρ

_{2}= 100 Ωm, H = 2 m, D

_{2}= 4 m, D

_{1}= 3 m.

Objective | Min TV Option | Max TV Option | ||||||
---|---|---|---|---|---|---|---|---|

h_{1} | h_{2} | TV | EPR | h_{1} | h_{2} | TV | EPR | |

(m) | (m) | (V) | (V) | (m) | (m) | (V) | (V) | |

TV_{1m} | 0.3 | 1.5 | 170 | 820 | 1.5 | 1.5 | 290 | 750 |

TV_{2m} | 1.5 | 0.3 | 185 | 815 | 1.5 | 1.5 | 390 | 750 |

TV_{3m} | 1.5 | 0.5 | 490 | 805 | 0.3 | 0.3 | 780 | 1190 |

**Table 4.**Sensitivity analysis results for ρ

_{1}= 100 Ωm, ρ

_{2}= 500 Ωm, H = 2 m, D

_{2}= 4 m, D

_{1}= 2 m.

Objective | Min TV option | Max TV option | ||||||
---|---|---|---|---|---|---|---|---|

h_{1} | h_{2} | TV | EPR | h_{1} | h_{2} | TV | EPR | |

(m) | (m) | (V) | (V) | (m) | (m) | (V) | (V) | |

TV_{1m} | 0.3 | Any | 25 | 510 * | 1.5 | 1.5 | 75 | 515 |

TV_{2m} | 1.5 | 0.3 | 45 | 515 | 1.5 | 1.5 | 120 | 515 |

TV_{3m} | 0.9 | 0.5 | 155 | 515 | 0.3 | 0.3 | 185 | 550 |

**Table 5.**Sensitivity analysis results for ρ

_{1}= 500 Ωm, ρ

_{2}= 100 Ωm, H = 2 m, D

_{2}= 5 m, D

_{1}= 2 m.

Objective | Min TV Option | Max TV Option | ||||||
---|---|---|---|---|---|---|---|---|

h_{1} | h_{2} | TV | EPR | h_{1} | h_{2} | TV | EPR | |

(m) | (m) | (V) | (V) | (m) | (m) | (V) | (V) | |

TV_{1m} | 0.3 | 1.5 | 135 | 720 | 1.5 | 1.5 | 255 | 655 |

TV_{2m} | 1.5 | 0.3 | 245 | 795 | 0.3 | 1.5 | 345 | 720 |

TV_{3m} | 1.5 | 0.3 | 375 | 795 | 0.3 | 0.3 | 480 | 980 |

No. | Solver | ρ_{1} | ρ_{2} | H | D_{2} | D_{1i} | h_{1i} | h_{2i} | D_{1o} | h_{1o} | h_{2o} | TV_{1m} | TV_{2m} | TV_{3m} | EPR | t | ItO | ItN | Status |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

- | - | Ωm | Ωm | m | m | m | m | m | m | m | m | V | V | V | V | m | - | - | - |

1 | QN | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2.45 | 0.33 | 0.65 | 167 | 376 | 703 | 1092 | 69 | 24 | 31 | S |

2 | QN | 500 | 100 | 2 | 4 | 3.4 | 0.9 | 0.3 | 3.57 | 1.5 | 0.3 | 179 | 157 | 435 | 753 | 120 | 42 | 47 | S |

3 | QN | 500 | 100 | 2 | 4 | 3.6 | 1.5 | 1.5 | 3.40 | 0.63 | 1.5 | 199 | 296 | 489 | 785 | 89 | 30 | 31 | F |

4 | PS | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2.38 | 0.37 | 0.7 | 170 | 386 | 694 | 1076 | 61 | 22 | 27 | S |

5 | PS | 500 | 100 | 2 | 4 | 3 | 1.5 | 0.3 | 3.53 | 1.5 | 0.36 | 180 | 178 | 436 | 753 | 60 | 14 | 21 | S |

6 | PS | 500 | 100 | 2 | 4 | 3.6 | 1 | 1.5 | 2.43 | 0.31 | 1.5 | 153 | 394 | 559 | 849 | 112 | 31 | 34 | S |

7 | GA | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2.34 | 0.52 | 0.96 | 207 | 418 | 658 | 1006 | 38 | 13 | 16 | F |

8 | GA | 500 | 100 | 2 | 4 | 2 | 0.3 | 0.6 | 2.34 | 0.52 | 0.96 | 207 | 418 | 658 | 1006 | 40 | 13 | 16 | F |

9 | GA | 500 | 100 | 2 | 4 | 2.2 | 0.3 | 0.6 | 2.34 | 0.52 | 0.96 | 207 | 418 | 658 | 1006 | 39 | 13 | 16 | F |

10 | GA | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2.36 | 0.35 | 1.46 | 164 | 422 | 613 | 748 | 3714 | 663 | 1134 | F |

11 | SMINLP | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2.4 | 0.3 | 0.8 | 164 | 408 | 693 | 1064 | 57 | 22 | 23 | S |

12 | SMINLP | 500 | 100 | 2 | 4 | 1 | 0.3 | 0.3 | 2.4 | 0.3 | 0.8 | 165 | 407 | 691 | 1063 | 64 | 27 | 27 | S |

13 | SMINLP | 500 | 100 | 2 | 4 | 1 | 1.5 | 1.5 | 2.4 | 0.3 | 0.8 | 165 | 407 | 691 | 1063 | 64 | 27 | 27 | S |

14 | SMINLP | 500 | 100 | 2 | 4 | 3.6 | 0.3 | 0.3 | 2.4 | 0.3 | 0.3 | 165 | 407 | 691 | 1063 | 64 | 27 | 27 | S |

15 | SMINLP | 500 | 100 | 2 | 4 | 3.6 | 1.5 | 1.5 | 2.4 | 0.3 | 0.8 | 165 | 407 | 691 | 1063 | 64 | 27 | 27 | S |

No. | Solver | ρ_{1} | ρ_{2} | H | D_{2} | D_{1i} | h_{1i} | h_{2i} | D_{1o} | h_{1o} | h_{2o} | TV_{1m} | TV_{2m} | TV_{3m} | EPR | t | ItO | ItN | Status |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

- | - | Ωm | Ωm | m | m | m | m | m | m | m | m | V | V | V | V | m | - | - | - |

16 | SNLP | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2.08 | 0.3 | 1.5 | 160 | 440 | 599 | 872 | 270 | 70 | 100 | M |

17 | SNLP | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2.29 | 0.31 | 1.5 | 147 | 412 | 580 | 858 | 845 | 240 | 300 | M |

18 | SNLP | 500 | 100 | 2 | 4 | 1 | 0.3 | 0.3 | 2.36 | 0.3 | 1.5 | 152 | 404 | 575 | 853 | 433 | 126 | 152 | S |

19 | SNLP | 500 | 100 | 2 | 4 | 1 | 0.3 | 0.3 | 2.36 | 0.3 | 1.5 | 152 | 404 | 575 | 853 | 434 | 126 | 152 | F |

20 | SNLP | 500 | 100 | 2 | 4 | 1 | 0.4 | 0.4 | 2.32 | 0.3 | 1.5 | 147 | 412 | 581 | 858 | 836 | 251 | 300 | M |

21 | SNLP | 500 | 100 | 2 | 4 | 1 | 1.5 | 1.5 | 2.26 | 0.31 | 1.49 | 147 | 415 | 584 | 861 | 616 | 118 | 207 | S |

22 | SNLP | 500 | 100 | 2 | 4 | 1 | 1.5 | 1.5 | 2.26 | 0.31 | 1.49 | 147 | 415 | 584 | 861 | 616 | 118 | 207 | S |

23 | SNLP | 500 | 100 | 2 | 4 | 3.6 | 0.3 | 0.3 | 2.38 | 0.3 | 1.48 | 148 | 403 | 577 | 857 | 930 | 66 | 299 | S |

24 | SNLP | 500 | 100 | 2 | 4 | 3.6 | 1.5 | 1.5 | 3.6 | 1.5 | 0.33 | 177 | 162 | 430 | 748 | 543 | 74 | 188 | F |

25 | SNLP | 500 | 100 | 2 | 4 | 3.6 | 0.4 | 0.4 | 2.41 | 0.31 | 1.49 | 150 | 397 | 571 | 581 | 858 | 87 | 300 | M |

26 | SNLP | 500 | 100 | 2 | 4 | 2 | 0.5 | 0.7 | 2 | 0.31 | 1.5 | 164 | 448 | 604 | 876 | 191 | 16 | 66 | S |

27 | SNLP | 500 | 100 | 2 | 4 | 2 | 1.4 | 1.4 | 2 | 0.30 | 1.40 | 172 | 457 | 625 | 907 | 89 | 12 | 28 | F |

28 | SNLP | 500 | 100 | 2 | 4 | 3.6 | 1.4 | 1.4 | 3.6 | 1.49 | 0.32 | 177 | 163 | 431 | 749 | 473 | 138 | 180 | F |

29 | SNLP | 500 | 100 | 2 | 5 | 2 | 0.5 | 0.7 | 2.37 | 0.34 | 1.46 | 125 | 310 | 435 | 718 | 331 | 19 | 99 | F |

30 | SNLP | 500 | 100 | 2 | 5 | 2 | 0.6 | 0.8 | 2.39 | 0.31 | 1.5 | 122 | 311 | 435 | 711 | 391 | 83 | 118 | F |

31 | SNLP | 500 | 100 | 1 | 4 | 2 | 0.5 | 0.7 | 2.15 | 0.3 | 1.43 | 37 | 118 | 179 | 349 | 734 | 153 | 300 | M |

32 | SNLP | 100 | 100 | NA | 4 | 2 | 0.5 | 0.7 | 3.59 | 1.29 | 0.3 | 29 | 31 | 105 | 265 | 141 | 74 | 89 | S |

33 | SNLP | 100 | 500 | 2 | 4 | 2 | 0.5 | 0.7 | 2.6 | 0.37 | 1.45 | 24 | 86 | 152 | 518 | 81 | 62 | 78 | F |

_{1}not optimised, increasedD

_{2}= 5 m, different soil model—red text. * Optimisation ended in local minimum—light red, * Global optimum reached—dark green. * Only burial depths optimised—light green, * Iteration limit set to 100 only—light orange. * Uniform and LoH soil models—light blue.

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**MDPI and ACS Style**

Vycital, V.; Ptacek, M.; Topolanek, D.; Toman, P.
On Minimisation of Earthing System Touch Voltages. *Energies* **2019**, *12*, 3838.
https://doi.org/10.3390/en12203838

**AMA Style**

Vycital V, Ptacek M, Topolanek D, Toman P.
On Minimisation of Earthing System Touch Voltages. *Energies*. 2019; 12(20):3838.
https://doi.org/10.3390/en12203838

**Chicago/Turabian Style**

Vycital, Vaclav, Michal Ptacek, David Topolanek, and Petr Toman.
2019. "On Minimisation of Earthing System Touch Voltages" *Energies* 12, no. 20: 3838.
https://doi.org/10.3390/en12203838