A new procedure to optimally identifying the prediction equation of oil breakdown voltage with the barrier parameters’ effect is presented. The specified equation is built based on the results of experimental works to link the response with the barrier parameters as the inputs for hemisphere-hemisphere electrode gap configuration under AC voltage. The AC HV is applied using HV Transformer Type (PGK HB-100 kV AC) to the high voltage electrode in the presence of a barrier immersed in Diala B insulating oil. The problem is formulated as a nonlinear optimization problem to minimize the error between experimental and estimated breakdown voltages (VBD). Comprehensive comparative analyses are addressed using three recent innovative marine predators, grey wolf, and equilibrium optimization algorithms to reduce the error between the experimental and estimated breakdown voltages. In addition, the experimental results are expressed in two different models

The insulating oils are used for cooling and insulating purposes in the high voltage (HV) apparatus. The HV apparatus’s failure comes from defects in its insulating systems, such as oil in the transformer. Therefore, enhancing the dielectric strength of the insulating oil is the main task in research [

Several works discussed the effect of barrier parameters (barrier effect) on the dielectric strength of the insulating oil, such as gap space (d) between the electrode system, the barrier position in the gap space (a/d), the barrier diameter (D), the barrier thickness (th), the electrode configuration, and the oil temperature [

Gohneim et al. [_{r}), the hole radius in the barrier (hr), the barrier thickness (th), and inclined barrier angle (θ) based on the design of experiments (DOE). Jaroszewski et al. [

The researchers tried to build a prediction equation related to the barrier parameters and the VBD of the insulation oils [

In this work, a new prediction equation based on the experimental results was developed to explain the relationship between the response (VBD) and the barrier parameters (d, (a/d), and D) using the artificial intelligent optimization technique. The second-order quadratic form is suitable as in

Y is the response representing the VBD. X_{1}, X_{2}, and X_{3} are the input variables that affect the response referring to d, a/d, and D in the proposed work. The a_{0} is the constant term that gives the response value when the input variables are zero, and ε refers to the error. Finally, the a_{0}–a_{9} expresses the equations’ coefficients.

Continuous development in optimization algorithms encourages many researchers to develop these methods for their applications. Among these methods, there are three optimizers, with recent applications in the power system domain, called equilibrium optimizations [

The main contribution of the current work can be summarized as follows:

First, proposing three recent optimization techniques for finding the optimal parameters of the prediction equation of the VBD on the insulating oil in the presence of barriers.

Two applications, grey and real-based modules, are employed for proving the capability of the proposed optimizers.

The results are aggregated by applying the AC voltage on the hemisphere-hemisphere gap configuration in the presence of a barrier.

The prediction equation is built with only a few experiment runs then decreases the cost of the experiment.

The results indicated the optimization techniques, mainly marine predators, decreased the error between the measured and predicted VBD.

The solution robustness is proved

The high closeness between the measured and estimated models for testing data is observed.

The research gap related to the current work is how to use a few experimental observations to build a prediction equation to facilitate oil VBD computation based on the barrier parameters. Furthermore, how to reduce the errors between the actual and predicted oil VBD using the updated optimizers.

The schematic diagram of the experimental setup is illustrated in

The relationship between a/d% and VBD at different “d” was illustrated in

The effect of the barrier diameter (D) on VBD was illustrated in

Marine Predators Algorithm (MPA) is a modern nature-inspired optimizer simulated from different methods followed by the predators seeking to hunt the preys [

Firstly, the predators are in an exploration phase for observing the encircled region. They run at low speed while the preys have a much higher rate. This scenario is mathematically simulated as follows:

Z_{i} and Z* are the new and current positions of each prey (i), R refers to the vector of random uniformly distributed numbers within the range [0, 1]. The symbol _{m} is the vector of random normally distributed numbers that track the Brownian motion. E_{i} indicates the current position of the top predator (i) taken from the elite matrix. I and I_{max} are the current iteration and the maximum number of iterations, respectively.

In the second portion of the MPOA, the transition between exploration and exploitation processes is simulated where the relative speed between the predator and the prey is competitive. In this scenario, the prey is responsible for exploitation while the predator is responsible for exploration. This scenario is mathematically simulated as follows:

where R_{n} is a random distributed numbers vector that is based on Lévy motion. C_{It} indicates an adaptive dynamic coefficient that is iteratively varied to adapt to the predator’s locomotion. C_{It} is mathematically represented as:

In the last portion of the MPOA, the predators hunt the prey, so the predators run faster than their targets. This scenario simulates the exploitation mode in tracking their base. This scenario is mathematically formulated as follows:

where Pop_{S} is the population size.

where Z_{U} and Z_{L} are the vectors related to the upper and lower bounds of the control variables. F_{A} is the probability parameter that represents the effect of fish aggregation. U is the vector of random binary numbers.

After updating the prey’s positions, the Elite matrix and the fitness (Fit) of each prey are upgraded

where E* is the new Elite matrix of the updated positions related to the top predator. Z_{Bt} indicates the best position of the prey with the minimum fitness.

Grey Wolf Optimization (GWO) is an effective optimization technique developed in 2014 by Mirjalili et al. [

When the wolves attempt to kill a target, they work together on a successful hunting strategy described in a series of steps: finding, tracking, engirdling, and attacking. Next, GWO mentions the location of the wolves in addition to the site of the prey. It then changes the position of the wolf in line with the shifting positions of the prey. In GWO, Alpha (the best choice solution), beta, and delta have more excellent knowledge of the possible location of prey. So, the first three best solutions for the number of wolves (Nw) and the other pack members must change their positions in the light of the best three solutions. Such actions can be summarized as follows [

where, X_{α}, X_{β}, X_{Δ} represent the positions of the alpha, beta, and delta. D_{α}, D_{β}, and D_{Δ} are the difference between the alpha, beta, delta, and the actual solution (X). X_{new} is a new place for the dog. A and C are co-efficient vectors measured as follows:

where r is a random number. the co-efficient vector a is linearly decreased from 2 to 0 by the following equation:

where iter is the present iteration, and Max_{iter} is the cumulative number of iterations.

The equilibrium Optimization (EO) algorithm is a recent optimizer proposed, for the first time, by Farmarzi in 2020 in [

C_{initial} represents the particles’ initial concentrations; C_{min} and C_{max} are the defined limits of the decision optimization variables. The particles are then modified periodically regarding equilibrium solutions, which are selected as the best-of-the-art candidates. The EO upgrade function follows:

where the existing and new concentration vectors of the particle are X and X_{new}, respectively. X_{eq} is a random concentration vector to be drawn from the balance pool. μ is a random vector between 0 and 1; F is an exponential term specified as:

where G is the generation rate specified as follows:

b_{1} and b_{2} are constants (b_{1} = 2 and ba_{2} = 1); m1 is a random variable between 0 and 1; T and T_{max} are the current and cumulative iterations. r_{1} and r_{2} are random numbers between 0 and 1; GP is a given value called the generation likelihood (GP = 0.5). In each iteration, the considered objective shall be measured for the concentration of each particle to approximate its position. Moreover, each iteration of the balance pool is modified to include the best four particles so far.

Generally, the parameters prediction issue can be formulated as a nonlinear optimization problem that minimizes an objective function and subjects to different inequality constraints. The mathematical formulation of the parameters of

where the (*) refers to the optimal estimated condition.

An objective function is to find the best closeness between the estimated and experimental VBD to optimize these parameters. Therefore, this objective is simulated by minimizing the Squares Error Sum (SSE) between the experimental data and the simulation results. The proposed optimization problem is represented in

where F(u) is the objective function, u refers to the vector of predicted parameters. Nm is the number of recorded measurements. The minimum bounds of the control vector (u) and maximum limits u_{min} and u_{max}, respectively. The control vector (u) is represented as:

Thus, the inequality constraint in

The superscripts "max" and "min" indicate the maximum and minimum limits of each variable.

For handling the parameters prediction optimization problem, three optimization techniques, of GWO, EO, and MPA, are applied for the first time to define the breakdown voltage prediction equation concerning the barriers in the power transformer oil samples. Their development to deal with the parameters prediction problem is illustrated in

For GWO algorithm:

Initializing the control variables within the permissible boundaries u_{min} and u_{max} of

Extracting three grey wolves’ agents called Alfa, Beta, and Gamma agents.

Updating the movement vectors, A, C, and related

Updating position of each grey wolf according to the procedure

For the second optimization algorithm: EO

Initializing the control variables within ermissible boundaries u_{min} and u_{max} of

Specifying the equilibrium pool

Computing the generation rate and exponential factors using

Updating the equilibrium pool using

For the third optimization algorithm, MPA:

Initializing the control variables within the permissible boundaries u_{min} and u_{max} of

Estimating the adaptive coefficient of

Applying the memory savings is procced using

Evaluating the predicators' fitness and update the Elite matrix

Updating the marine’s predators position using

Checking the stopping criteria for all tested algorithms are reaching the maximum iteration.

In the first case, the proposed optimization methods are investigated to seek the optimal prediction equation parameters based on grey code developed in [

^{−8}) than EO and GWO of 0.0207 and 0.0551. The measured VBD and the BBD, EO, GWO, MPA are explained in

a_{i} |
Minimum | Maximum | BBD [ |
EO | GWO | MPA |
---|---|---|---|---|---|---|

a_{0} |
−20 | 40 | 30 | 30.00091 | 29.94738 | 29.9998 |

a_{1} |
−20 | 40 | 27.5 | 26.87355 | 26.8776 | 26.8750 |

a_{2} |
−20 | 40 | −0.25 | −0.8681 | −1.05392 | −0.87502 |

a_{3} |
−20 | 40 | 12.75 | 12.75086 | 12.7545 | 12.75 |

a_{4} |
−20 | 40 | −0.75 | −2.0068 | −2.22953 | −2.00006 |

a_{5} |
−20 | 40 | −3.75 | −3.74622 | −3.80697 | −3.74999 |

a_{6} |
−20 | 40 | −7.75 | −7.74384 | −7.77132 | −7.75002 |

a_{7} |
−20 | 40 | 9.375 | 8.754424 | 8.827906 | 8.7501 |

a_{8} |
−20 | 40 | 8.375 | 7.748127 | 7.739857 | 7.75008 |

a_{9} |
−20 | 40 | 13.87 | 14.49887 | 14.53897 | 14.5000 |

Indices | EO | GWO | MPA |
---|---|---|---|

1.21678 | 1.21910 | 1.21678 | |

1.22520 | 1.27610 | 1.21678 | |

1.283828 | 1.374087 | 1.21678 | |

0.020733 | 0.0551 | 4.61E−08 |

Experimental results (VBD) | BBD [ |
EO | GWO | MPA |
---|---|---|---|---|

17 | 19.75 | 18.49 | 18.46 | 18.5 |

16 | 20.75 | 20.77 | 20.81 | 20.75 |

81 | 76.25 | 76.25 | 76.68 | 76.25 |

72 | 74.25 | 70.50 | 70.11 | 70.5 |

15 | 9.25 | 9.88 | 9.88 | 9.88 |

44 | 42.25 | 42.88 | 43.00 | 42.88 |

70 | 71.75 | 71.12 | 71.24 | 71.13 |

84 | 89.75 | 89.13 | 89.14 | 89.13 |

29 | 32 | 32.62 | 32.75 | 32.63 |

74 | 73 | 73.61 | 73.81 | 73.63 |

46 | 47 | 46.37 | 46.19 | 46.38 |

60 | 57 | 56.39 | 56.16 | 56.38 |

31 | 30 | 30.00 | 29.95 | 30 |

30 | 30 | 30.00 | 29.95 | 30.00 |

29 | 30 | 30.00 | 29.95 | 30.00 |

OF | 1.30 | 1.22 | 1.22 | 1.22 |

Test results (VBD) | BBD [ |
EO | GWO | MPA |
---|---|---|---|---|

30 | 31.125 | 31.7489 | 31.7319 | 31.7499 |

77 | 73.375 | 74.0022 | 74.4964 | 74.0002 |

34 | 38.125 | 36.8809 | 36.6334 | 36.875 |

65 | 66.875 | 65.6288 | 65.6529 | 65.625 |

42 | 43.375 | 44.0208 | 44.1424 | 44.0002 |

67 | 56.625 | 57.2507 | 57.2409 | 57.2499 |

81 | 76.25 | 76.2518 | 76.6762 | 76.2502 |

72 | 74.25 | 70.502 | 70.1094 | 70.5 |

74 | 73 | 73.6107 | 73.806 | 73.625 |

44 | 42.25 | 42.8778 | 42.9982 | 42.875 |

46 | 47 | 46.3727 | 46.1892 | 46.375 |

88 | 89.375 | 86.2618 | 85.8246 | 86.25 |

60 | 57 | 56.3869 | 56.1556 | 56.375 |

31 | 30 | 30.0009 | 29.9474 | 29.9999 |

70 | 71.75 | 71.123 | 71.2444 | 71.125 |

17 | 19.75 | 18.4912 | 18.462 | 18.5 |

84 | 89.75 | 89.1324 | 89.1394 | 89.125 |

97 | 106.87 | 107.499 | 107.934 | 107.5002 |

29 | 32 | 32.6213 | 32.7544 | 32.625 |

42 | 43.375 | 44.0208 | 44.1424 | 44.0002 |

OF | 1.24524 | 1.20706 | 1.22244 | 1.207025 |

The MPA optimization technique is the best one to reduce the errors between the actual and estimated VBD. However, the average absolute error of MPA is 6.37%, but the absolute error between the actual and estimated VBD of each run is not high. The maximum errors of this technique are 11.75, 10.35, 15.02, 10.85, 11.53, and 11.58 % for run No. 2, 4, 7, 9, 10, and 11, respectively.

d (cm) | (a/d)% | D (cm) | VBD (kV) |
---|---|---|---|

1 | 50 | 10 | 44 |

1 | 100 | 10 | 42 |

2 | 25 | 10 | 74 |

2 | 100 | 5 | 46 |

2 | 100 | 10 | 60 |

2 | 50 | 7 | 31 |

2 | 50 | 5 | 30 |

2 | 100 | 7 | 34 |

2 | 50 | 10 | 67 |

2 | 75 | 5 | 41 |

3 | 25 | 7 | 81 |

3 | 100 | 7 | 72 |

3 | 50 | 5 | 70 |

3 | 25 | 5 | 77 |

3 | 50 | 7 | 65 |

3 | 100 | 10 | 88 |

3 | 25 | 10 | 97 |

3 | 50 | 10 | 84 |

3 | 100 | 5 | 71 |

3 | 75 | 5 | 75 |

a_{i} |
EO | GWO | MPA |
---|---|---|---|

a_{0} |
58.476202 | 68.2929815 | 68.6989009 |

a_{1} |
−3.3223119 | 1.49446585 | 5.45952649 |

a_{2} |
−0.3234591 | −1.1733735 | 0.83654452 |

a_{3} |
−17.115916 | −20.563137 | −22.460758 |

a_{4} |
−4.0325311 | 1.9992289 | −4.6400564 |

a_{5} |
−0.5825703 | −0.7982497 | −0.833439 |

a_{6} |
−3.2996987 | −0.4232859 | −3.308737 |

a_{7} |
9.1730157 | 7.4779745 | 7.5586654 |

a_{8} |
25.395654 | 0.1368574 | 25.634668 |

a_{9} |
1.6687208 | 1.8217036 | 2.0550708 |

Run | Exp. actual VBD | Estimated VBD | Absolute error % in estimated VBD | ||||
---|---|---|---|---|---|---|---|

EO | GWO | MPA | EO | GWO | MPA | ||

1 | 44 | 41.88 | 44.15 | 42.25 | 4.81 | 0.35 | 3.98 |

2 | 42 | 42.25 | 42.55 | 43.03 | 0.61 | 1.31 | 2.45 |

3 | 74 | 63.82 | 61.43 | 65.31 | 13.75 | 17.00 | 11.75 |

4 | 46 | 39.34 | 36.78 | 41.24 | 14.47 | 20.03 | 10.35 |

5 | 60 | 56.59 | 60.50 | 58.19 | 5.68 | 0.83 | 3.02 |

6 | 31 | 32.93 | 35.31 | 32.27 | 6.22 | 13.90 | 4.08 |

7 | 30 | 32.74 | 36.33 | 34.51 | 9.14 | 21.09 | 15.02 |

8 | 34 | 36.23 | 35.34 | 35.69 | 6.56 | 3.94 | 4.97 |

9 | 67 | 58.24 | 61.10 | 59.73 | 13.08 | 8.81 | 10.85 |

10 | 41 | 34.46 | 36.55 | 36.27 | 15.96 | 10.86 | 11.53 |

11 | 81 | 73.49 | 69.11 | 71.62 | 9.27 | 14.68 | 11.58 |

12 | 72 | 70.66 | 70.64 | 68.47 | 1.86 | 1.90 | 4.90 |

13 | 70 | 70.355 | 72.22 | 71.27 | 0.51 | 3.17 | 1.82 |

14 | 77 | 72.82 | 71.51 | 73.87 | 5.43 | 7.12 | 4.06 |

15 | 65 | 69.38 | 69.60 | 67.36 | 6.73 | 7.08 | 3.64 |

16 | 88 | 89.27 | 93.40 | 88.47 | 1.45 | 6.14 | 0.53 |

17 | 97 | 99.53 | 92.83 | 99.06 | 2.61 | 4.30 | 2.13 |

18 | 84 | 92.94 | 93.00 | 92.33 | 10.64 | 10.72 | 9.92 |

19 | 71 | 74.94 | 73.67 | 75.69 | 5.55 | 3.77 | 6.60 |

20 | 75 | 71.06 | 72.94 | 71.88 | 5.25 | 2.75 | 4.16 |

Average absolute error % | 6.98 | 7.99 | 6.37 |

EO | GWO | MPA | |
---|---|---|---|

Minimum | 1.03824998 | 1.16129312 | 0.99823917 |

Mean | 1.2525994 | 1.38249263 | 0.99824018 |

Maximum | 1.45761171 | 1.59560131 | 0.99824854 |

STD | 0.12817586 | 0.14788924 | 2.9381E−06 |

This paper has been developed new modeling of barrier effect on the breakdown voltage of transformer oil

The authors acknowledge the financial support received from Taif University Researchers Supporting Project (Number TURSP-2020/122), Taif University, Taif, Saudi Arabia.