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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. Digital Object Identifier 10.1109/ACCESS.2017.Doi Number
Moth-Flame algorithm for accurate simulation of a non-uniform electric field in the presence of dielectric barrier M. Talaat1,2, Abdulaziz S. Alsayyari3, M. A. Farahat1 and Taghreed Said4 1
Electrical Power & Machines Department, Faculty of Engineering, Zagazig University, Egypt Electrical Engineering Department, College of Engineering, Shaqra University, Dawadmi, Ar Riyadh, Saudi Arabia Computer Engineering Department, College of Engineering, Shaqra University, Dawadmi, Ar Riyadh, Saudi Arabia 4 Electrical Engineering and Computer, Higher Technological Institute, 10th of Ramadan City, Egypt. 2 3
Corresponding author: M. Talaat (e-mail:
[email protected];
[email protected] ).
ABSTRACT In this paper a proposed Moth-Flame Optimization (MFO) technique has been investigated for obtaining an accurate simulation of the non-uniform electric field represented by needle-to-plane gap configuration. The needle electrode connected to the high voltage (HV) terminal while the earthed terminal connected to the plane electrode. In addition to the non-uniformity of the field, a transverse dielectric barrier has been presented and investigated along the gap with different thickness and location. The MFO works to optimize the error given by a numerical equation published before for calculating this field problem in the presence of a transverse barrier. This numerical equation was based on a correction coefficient called (𝛽𝛽), which is depended on three values; relative permittivity, barrier location and barrier thickness. The MFO is working to minimize the error given by 𝛽𝛽 using two new optimization factors in the 𝛽𝛽 equation. To ensure the accurate validation of MFO with minimum error for field problem simulation, various Artificial Intelligence (AI) optimization techniques have been compared with the MFO obtained results. The comparative study shows that MFO is more effective, especially at 30 % of the gap length from the HV electrode which represents the region of highly non-uniform field along the gap configuration. The numerical results of the field simulation that held by different types of AI techniques are compared with those obtained from accurate simulation results using Finite Element Method (FEM). The value of the error between the numerical and simulation results shows that MFO is the most effective optimization techniques that can be used in the numerical equation to obtain the best value of the correction factor. With MFO a good agreement has been reached between the proposed numerical equation and the accurate simulation values of the electric field problem. INDEX TERMS Artificial Intelligence, Moth-Flame Optimization (MFO), Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Dielectric Barrier, Electric Field Simulation, Finite Element Method.
I. INTRODUCTION
Recently we succeeded in finding an equation to simulate the non-uniform electric field in the presence of a transverse barrier [1]. However, there was an error between the calculated field from the proposed equation and the simulated one using finite element method (FEM). Therefore, a correction factor called (𝛽𝛽) was proposed by [1] for correcting this equation. This factor was based on three variables i.e., relative permittivity, barrier location and barrier thickness. Although 𝛽𝛽 was assumed, there was an error in the calculated value of the field especially near the high voltage electrode. So, the needing of artificial
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intelligence (AI) has been raised to optimize the error given by 𝛽𝛽. The using of AI applications in the electrical power system, especially in the fault diagnosis have been raised recently [2]. In addition, AI optimization techniques are widely used in solving non-uniform field problems especially at high voltage (HV) applications [3]. One of the most non-uniform field problems in the HV is the using of the transverse layer of a Dielectric Barrier (DB) in the field gap configuration [1, 4-5]. In recent years the air gap field strength is increased by inserting DB at optimum location between the HV electrodes. This enhance the air gap field characteristics 1
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without abolishing demand on environmental effect and preserving cost efficiency. The challenge now for the researchers in the using of the DB in the gap configuration is to select the suitable position, dielectric type and thickness of the DB. The optimum selection prevents the arc discharge and increases the value of the field that required for breakdown conditions [4-10]. Several experimental investigations were introduced by other for determining the optimum characteristics of DB. Also, the effect of DB on the electric field distribution in case of high impulse, DC and AC voltage [11-13]. In addition, different mathematical models were presented to describe the field distribution in the case of the existence of DB in the field gap [4, 5, and 14]. A new mathematical equation for simulating the non-uniform field in the case of using DB along the gap configuration was given by [1]. But the maximum error from this equation reaches 2.74 percent, especially at the 30 percent of the gap length from the HV electrode. Despite the use of a coefficient to reduce error, this coefficient called correction factor 𝛽𝛽 [1]. The error was estimated by comparing the proposed numerical equations with accurate simulation of the field problems using Laplace’s or Poisson’s equations. Different simulation techniques utilizing for this target such as, charge simulation methods (CSM) [3, 5, 10, 15], and FEM [1, 4, 1617]. Different AI optimization techniques are used for the high voltage field problems. The advantage of AI is to obtain accurate field computation with minimum error. The most popular optimization technique in the HV application is the genetic algorithm (GA) [3, 10, 18]. But, the new techniques such as; MFO did not use as a widely technique in the HV applications. GA is the first stochastic optimization algorithm, was introduced by Jon Holland in 1960 [19] and proposed to mitigate the drawbacks of the traditional mathematical algorithms. GA was developed based on Darwinian Theory and the natural process of reproduction behavior. The MFO algorithm is an evolutionary based algorithm developed by Seydali-Mirjilil in 2015 [20]. The idea of this algorithm depends on modeling the spiral flying of moths around moon light. MFO algorithm is a global optimization algorithm to determine the fitness variation with no need to use mathematical procedures [20-24]. Particle swarm optimization (PSO) is a population technique which introduced by motivating the social movement of bird flocks or fish school searching for food by Kennedy and Eberhart in 1995 [25]. In PSO algorithm, Swarm (particle) flies in a wide search space. During flight, each particle follows its track in the search domain associated with the best position VOLUME XX, 2017
(P𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ) visited so far from its previous memory. Also, track the nearest particle to the food (G𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ), the best particle among all the particles. Finally, is the step size for a particle that attempts to move in next turn determined by flight velocities [3]. This paper presents a new approach for simulating the nonuniform field in the presence of DB. The new approach using MFO technique to achieve the accurate simulation of the field problem. MFO technique utilizes to minimize the error obtained from the correction factor 𝛽𝛽 that presented by [1] for solving this field problem. To assess the accuracy of MFO it has compared with different AI optimization techniques that have been used for solving the same problem. The obtained results of the different types of AI have been compared with those obtained from FEM simulation for the same problem. The error has been investigated along the field gap configuration, especially at 30 percent of the gap length. The estimated error between the numerical and simulation proved that, MFO is the most effective techniques compared with other AI such as GA and PSO that can be used for minimizing the error of the correction factor. II. NUMERICAL EQUATION OF THE FIELD PROBLEM A. FIELD CALCULATION PROBLEM
The numerical field equation that can be used to solve the field problem in the case of the presence of the DB was given by [1]. The field problem for the rod-to-plane gap configuration can be described in Fig. 1. z-axis
εr1 εr2
Air
V=1 V Air DB V=0 V
HV electrode
R
En
(0,0)
Earthed electrode
z1
z2
z
r-axis
FIGURE 1. Field problem considers the presence of DB at rod-to-plane gap configuration [1].
So, the electric field, along the gap, 𝐸𝐸𝑔𝑔 , in the presence of DB can be calculated by the equation given by [1] as; 1 𝜀𝜀𝑟𝑟2 𝜎𝜎1 𝑧𝑧1 𝑧𝑧1 𝐸𝐸𝑔𝑔 = 𝛽𝛽 �𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚. − 𝐸𝐸𝑛𝑛 � − 1� �1 − � �1 − �� (1) 2 2 2 𝜀𝜀𝑟𝑟1 𝜎𝜎2 𝑧𝑧 �𝑅𝑅 + 𝑧𝑧1
where, 𝛽𝛽 represents the correction factor. This value of 𝛽𝛽 was given by [1] as, 9
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𝛽𝛽 = �1 +
𝑧𝑧2 � (𝜀𝜀𝑟𝑟2 + 2) × 𝑧𝑧1
(2)
where, 𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚. is the maximum value of the electric field without the DB, 𝐸𝐸n is the normal field component at the interference between air and DB, 𝜀𝜀𝑟𝑟1 is the relative permittivity of air, 𝜀𝜀𝑟𝑟2 is the relative permittivity of the DB, 𝜎𝜎1 is the electrical conductivity of air, and 𝜎𝜎2 is the electrical conductivity of the DB, 𝑧𝑧1 is the gap between the HV electrode and the DB, 𝑅𝑅 represents the radius of the DB, 𝑧𝑧 is the total gap length from rod to plate, and 𝑧𝑧2 represents the thickness of the DB, see Fig. 1. The value of the maximum percentage error of this equation when compared with an accurate simulation model was 2.74 % . This value of error obtained at the non-uniform field region (30 % of the gap length). Although this error can be accepted in the highly non-uniform field, but it can be minimized using AI optimization techniques. The main target of this research is to indicate the best optimization technique that can obtain the minimum percentage error for accurate numerical equation. B. PROPOSED OPTIMIZATION TECHNIQUE
To achieve accurate simulation of field problem, the equation of the correction factor rearranges in terms of two factors 𝑓𝑓1 and 𝑓𝑓2 . The first factor 𝑓𝑓1 is used to control the gap spacing with respect to the DB position and the second factor 𝑓𝑓2 is used to control the DB thickness. 𝑓𝑓2 𝑧𝑧2 𝛽𝛽 = �1 + � (3) (𝜀𝜀𝑟𝑟2 + 2) × 𝑓𝑓1 𝑧𝑧1
The inequality constraints of this problem are the factors lower and upper bounds: 0 ≤ 𝑓𝑓1 ≤ 1 & 0 ≤ 𝑓𝑓2 ≤ 1 The problem now is to determine the optimal values of the two factors for minimum error between 𝐸𝐸𝑔𝑔 , and the simulated one, 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 The simulated value of the electric field, 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 can be obtained in the concept of FEM using COMSOL Multiphysics software. 𝐸𝐸𝑔𝑔 − 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 % 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = � � × 100 (4) 𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠 After accuracy the numerical equation can be used for different thickness, position and the material types of the DB without building the shape and boundary conditions in each case such as in the simulation model. The problem is now customized to obtain the value of the two factors that satisfy minimum percentage error.
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III. DIFFERENT OPTIMIZATION TECHNIQUES A. MOTH FLAME OPTIMIZATION
In this algorithm the candidate solutions are considered as moths, each moth [𝑓𝑓1 , 𝑓𝑓2 ] defined by its position in the search space. The best positions visited so fare over the course of iteration are called by flames; flames are the artificial lights in MFO algorithm. Moths move toward the flames to determine the objective value (% error according to Eqn. (4)) depending on their positions [20-24]. The MFO algorithm, see Fig. 2, is processed as follows: 1- Moths are initially generated randomly to spread out in the feasible search space. 2- Evaluate and sort fitness of all population 3- Flames equal to sorted population. 4- While iteration < 𝑚𝑚𝑚𝑚𝑚𝑚.iteration • The flame number can be determined according to the following equation: 𝑁𝑁 − 1 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑁𝑁𝑁𝑁. = 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟(𝑁𝑁 − 𝑙𝑙 ∗ ) (5) 𝑇𝑇 𝑙𝑙, is the current iteration, 𝑁𝑁, is the maximum flames number, 𝑇𝑇, is the maximum number of iterations. • The distance, 𝐷𝐷𝑖𝑖 , between the 𝑖𝑖 𝑡𝑡ℎ moth (𝑀𝑀𝑖𝑖 ) with respect to its corresponding 𝑗𝑗𝑡𝑡ℎ flame (𝐹𝐹𝑗𝑗 ) can be obtained from, 𝐷𝐷𝑖𝑖 = �𝐹𝐹𝑗𝑗 − 𝑀𝑀𝑗𝑗 � (6) • Update the algorithm constant 𝑎𝑎 and 𝑡𝑡 −1 𝑎𝑎 = −1 + 𝑙𝑙 × � � 𝑇𝑇
(7)
𝑡𝑡 = (𝑎𝑎 − 1) × 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 + 1
(8)
𝑆𝑆�𝑀𝑀𝑖𝑖 , 𝐹𝐹𝑗𝑗 � = 𝐷𝐷𝑖𝑖 . 𝑒𝑒 𝑏𝑏𝑏𝑏 . cos 2𝜋𝜋𝜋𝜋 + 𝐹𝐹𝑗𝑗
(9)
where, 𝑡𝑡 is a random number between [−1, 1]. • Update the position of moth with its corresponding flame (𝐹𝐹𝑗𝑗 ) according to spiral function (𝑆𝑆) which simulates the transverse orientation of the moths around the moon with the following equation,
where, 𝑏𝑏 is the spiral shape constant • Update and sort the fitness for all search agents • Update the flames • Iteration =iteration+1 End while 5- Return the global best position. Changing the sequence of the flames emphasis the exploration process for avoiding stagnation in a local minimum in contrast degrading the flames number to only one flame at the end of generation brings the MFO 9
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exploitation. Balancing between the exploration and exploitation to find a rough approximation of the global optimum and then improve its accuracy. The random parameter (𝑡𝑡) increases the convergence rate over the course of generations.
Start
Initialize the Moth’s population (M)
Determine the error for each Moth
Sort and assign flame
Update flame No. using Eqn. (5)
Determine the distance between flames and Moths using Eqn. (6)
Update the algorithm constant a and t
Update Moths position in Eqn. (6)
No Iter=Iter+1
l≥T Yes
Global best Moth
End FIGURE 2. Moth-flame optimization algorithm flowchart.
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B. PARTICLE SWARM OPTIMIZATION
In PSO algorithm the candidate solutions are known as particles, each particle [𝑓𝑓1 , 𝑓𝑓2 ] defined by its position and flying velocity in the search domain. The best position obtained for each particle is represented as 𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 and the best position between all particles is called by 𝐺𝐺𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 . Particles move around 𝐺𝐺𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 to determine the objective value (% error according to Eqn. (4)) depending on their positions and velocities. The process of the PSO technique, see Fig. 3, is arranged as follows: 1- At initial, particles are generated at random positions 𝑋𝑋𝑖𝑖 and flight velocities 𝑉𝑉𝑖𝑖 (is the step size for a particle that attempts to move in next turn) in the search domain. 2- The fitness value is evaluated for each particle in the current iteration. 3- Update the local and global best positions (𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 and 𝐺𝐺𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 ). 4- Update the inertia weight (𝑤𝑤) 𝑤𝑤 = 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 −
𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 × 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚
(10)
where, 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 are the final and initial weights` 5- The position and speed of each particle are updated according to the following equations. 𝑉𝑉𝑖𝑖𝑘𝑘+1 = 𝑤𝑤𝑉𝑉𝑖𝑖𝑘𝑘 + 𝑐𝑐1 𝑟𝑟1 �(𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 )𝑘𝑘𝑖𝑖 − 𝑋𝑋𝑖𝑖𝑘𝑘 � + 𝑐𝑐2 𝑟𝑟2 �(𝐺𝐺𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 )𝑘𝑘𝑖𝑖 − 𝑋𝑋𝑖𝑖𝑘𝑘 �
(11)
𝑋𝑋𝑖𝑖𝑘𝑘+1 = 𝑋𝑋𝑖𝑖𝑘𝑘 + 𝑉𝑉𝑖𝑖𝑘𝑘+1
(12)
𝑉𝑉𝑖𝑖𝑘𝑘
where, is the velocity of individual 𝑖𝑖 at iteration 𝑘𝑘, 𝑐𝑐1 , 𝑐𝑐2 are the acceleration coefficients, 𝑟𝑟1 and 𝑟𝑟2 are the random numbers given between 0 and 1, 𝑋𝑋𝑖𝑖𝑘𝑘 is the position of individual 𝑖𝑖 at iteration 𝑘𝑘, (𝑃𝑃𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 )𝑘𝑘𝑖𝑖 is the best position of individual 𝑖𝑖 at iteration 𝑘𝑘, and (𝐺𝐺𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 )𝑘𝑘𝑖𝑖 is the best position of the group until iteration 𝑘𝑘 6- Check the stopping criteria.
In this algorithm, a large value of the inertia weight at the beginning allows particles to move around the search space and emphasizes exploration and then gradually decreased in inertia rate force particles move toward best points instead of moving towards the population best prematurely. C. GENETIC ALGORITHM
The stochastic components of this algorithm are the selection, re-production, and mutation which are the reasons behind the successes of GA and avoiding local minimum stagnation due to the ability of selection and reproduction of best chromosomes (individuals) [3]. 9
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The flow chart of GA is shown in Fig. 4. According to this flow chart the GA is arranged as follows: 1- GA is initiated with a random number of chromosomes. 2- Each chromosome’s fitness is evaluated. 3- The best chromosomes are selected 4- Generate new offspring chromosomes by exchanging information’s between selected best chromosomes via crossover instead of the worst ones. 5- Mutate a randomly selected percentage of chromosomes insure the exploration. 6- Obtain new populations. 7- Check the stopping criteria
are obtained. The input parameters of the optimization algorithms are given in Table I. Start
Generate a population of chromosomes
Evaluate the error for each individual of Eqn. (4)
Start
Select the best chromosomes (parent) for reproduction
Generate initial swarm with random position and velocity
Perform crossover and mutation to produce offspring’s
Evaluate the error for each particle of Eqn. (4)
Assign and incorporate new generation
Record personal best fitness (error) of all individual
Iter. ≥ Max.Iter
Update the global best particle
Yes
Iter. = Iter. +1
No Display best factors corresponding to minimum error
Iter. ≥ Max.Iter
Yes
End
Update particle
FIGURE 4. The flowchart of genetic algorithm.
No
End
Update particle velocity in Eqn. (11)
Update particle position in Eqn. (12) FIGURE 3. The particle swarm optimization method flowchart.
TABLE I OPTIMIZATION ALGORITHMS PARAMETER GA PSO MFO Parameters Parameters Parameters Population Size:50
Swarm Size:50
Moth Size:50
No. of Generation: 100 Crossover function: Scattered Mutation function: Gaussian
No. of Generation: 100
No. of Generation: 100
C1, C2: 2.0, 2.0
b:1
𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 , 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 : 0.4, 0.9
a: decreased linearly from -1to-2
IV. SIMULATION MODEL
A. FINITE ELEMENT METHODS
The first step for any optimization technique used in this paper is to call the simulation model for obtaining the field simulation. Then, estimate the fitness value of each optimization, using to minimize the error between numerical and simulation model. At each stage the objective function should be satisfied and the values of the two factors 𝑓𝑓1 and 𝑓𝑓2
where, ∇ is the normal operator, D, is the electric flux density (𝐶𝐶 ⁄𝑚𝑚2 ) and the value of the electric flux density can be
VOLUME XX, 2017
For an accurate simulation of the distribution of the electric field in the presence of DB a FEM is used with the concept of Poisson’s equation considering the charge accumulated at the DB [30, 31], as follows; (𝐶𝐶 ⁄𝑚𝑚3 ) ∇ ∙ D = 𝜌𝜌𝑣𝑣 (13)
9
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V= 1 Volt
V= zero Volt
(14)
𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑎𝑎 + 𝑎𝑎 � (15) 𝜕𝜕𝜕𝜕 𝑟𝑟 𝜕𝜕𝜕𝜕 𝑧𝑧 where, V is the applied voltage in Volt, (the value of the applied voltage at high the voltage electrode and earthed electrode are used as boundary condition in the FEM), see Fig. 5. 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 −∇ ∙ 𝜀𝜀 � 𝑎𝑎𝑟𝑟 + 𝑎𝑎 � = 𝜌𝜌𝑣𝑣 (𝐶𝐶 ⁄𝑚𝑚3 ) (16) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑧𝑧 𝐸𝐸 = − �
𝜕𝜕2 𝑉𝑉 𝜕𝜕 2 𝑉𝑉 𝜌𝜌𝑣𝑣 + �=− (17) 𝜕𝜕𝜕𝜕 2 𝜕𝜕𝜕𝜕 2 𝜀𝜀 The value of the volume charge density at the beginning of the simulation is given as zero charge, which satisfied the Laplace’s equation. Then, at each iteration the value of this charge increased with time according to the increasing of the accumulated charges on DB surface. �
𝜌𝜌𝑣𝑣 = 𝑛𝑛𝑛𝑛 (18) where, 𝑛𝑛 is the total number of the charges at the DB domain and 𝑞𝑞 is the induced charge from high voltage electrode at the DB which given by [1] as follows; 𝜎𝜎
𝜕𝜕𝑉𝑉𝐷𝐷𝐷𝐷 𝜕𝜕𝜕𝜕 = 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕
𝑡𝑡
𝑞𝑞 = � 𝜎𝜎 0
𝜕𝜕𝑉𝑉𝐷𝐷𝐷𝐷 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕
(19)
(20)
where, 𝜎𝜎 is the DB conductivity and 𝑉𝑉𝐷𝐷𝐷𝐷 is the simulated value of the voltage at the DB surface at each iteration. So, Eqn. (17) can be used to obtain the value of the potential at any point in the field problem domain, then the value of the VOLUME XX, 2017
From [1], the high voltage rod can be represented by cylindrical, with hemi-spherical head of radius 10 𝑚𝑚𝑚𝑚 and a cylindrical length of 0.25 𝑚𝑚 suspended in air and cylindrical earthed with a length of 10 𝑚𝑚𝑚𝑚 and 150 𝑚𝑚𝑚𝑚 radius, see Fig.1 and Fig. 5. The initial boundary condition was set to be (𝑉𝑉 = 1𝑉𝑉) at the high voltage electrode and (𝑉𝑉 = 0𝑉𝑉) at the earthed electrode and the voltage is supposed to be uniformly graded to the other boundary condition surfaces. A Polycarbonate material was used as a DB in the rod to plane gap at different position along the gap. This DB was 150 𝑚𝑚𝑚𝑚 in radius and with (1, 3 and 5 𝑚𝑚𝑚𝑚) thickness. RESULTS AND DISCUSSIONS
A. FITNESS FUNCTION APPROACH
FIGURE 5. The boundary condition in the FEM model.
where,
B. FIELD PROBLEM CONFIGURATION
V.
r-axis (mm)
∇ ∙ 𝜀𝜀E = 𝜌𝜌𝑣𝑣 (𝐶𝐶 ⁄𝑚𝑚3 )
electric field distribution can be simulated.
Figures from 6 to 18 show the convergence rate obtained from the application of GA, PSO and MFO to the error function. Also, these figures investigate the robustness and effectiveness of MFO that provide the best solution in minimum iteration number compared to GA and PSO. Figures 6 to 9 show the convergence graphs of GA, PSO and MFO algorithms to the optimal solution with successive generations for a barrier thickness 1mm and at (5, 10, 20 and 30 %) of the air gap from HV rod respectively. Figures 10 to 13 show the convergence graphs of GA, PSO and MFO algorithms to the optimal solution with successive generations for a barrier thickness 3𝑚𝑚𝑚𝑚 and at (5, 10, 20 and 30 %) of the air gap from HV rod respectively. Figures 14 to 18 show the convergence graphs of GA, PSO and MFO algorithms to the optimal solution with successive generations for a barrier thickness 5𝑚𝑚𝑚𝑚 and at (5, 10, 20, 30 and 40 %) of the air gap from HV rod respectively. In most cases the MFO algorithm convergence rate tends to be accelerated as iteration increases. By avoiding stagnation in local by searching toward the best solution obtained so far. 0.025 GA PSO MFO
0.02
Fitness Function (% error)
z-axis (mm)
given as, (D = 𝜀𝜀E) where 𝜀𝜀 is the permittivity of the medium (𝐹𝐹 ⁄𝑚𝑚) and E is the electric field (𝑉𝑉 ⁄𝑚𝑚), and 𝜌𝜌𝑣𝑣 is the volume charge density.
0.015
0.01
0.005
0 10
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Iteration Number
FIGURE 6. GA, PSO & MFO convergence with barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟓𝟓𝒎𝒎𝒎𝒎 from HV electrode. 9
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0.04 0.4
GA
0.035
GA
PSO
PSO
0.35
MFO
MFO
Fitness Function (% error)
0.03
Fitness Function (% error)
0.3
0.25
0.2
0.15
0.025 0.02 0.015 0.01 0.005 0
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Iteration Number
FIGURE 7. GA, PSO & MFO convergence with barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟏𝟏𝟏𝟏𝒎𝒎𝒎𝒎 from HV electrode. 0.752
FIGURE 11. GA, PSO & MFO convergence with barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟏𝟏𝟏𝟏𝒎𝒎𝒎𝒎 from HV electrode. 0.1
GA
GA
0.75
PSO
PSO MFO
MFO
0.08
Fitness Function (% error)
Fitness Function (% error)
0.748 0.746 0.744 0.742 0.74
0.06
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0.738
0
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Iteration Number
FIGURE 8. GA, PSO & MFO convergence with barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟐𝟐𝟐𝟐𝒎𝒎𝒎𝒎 from HV electrode. 0.96
FIGURE 12. GA, PSO & MFO convergence with barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟐𝟐𝟐𝟐𝒎𝒎𝒎𝒎 from HV electrode. 0.05
GA
GA PSO
PSO
MFO
MFO
0.04
Fitness Function (% error)
Fitness Function (% error)
0.955
0.95
0.945
0.03
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0
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FIGURE 9. GA, PSO & MFO convergence with barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟑𝟑𝟑𝟑𝒎𝒎𝒎𝒎 from HV electrode. 0.5
FIGURE 13. GA, PSO & MFO convergence with barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟑𝟑𝟑𝟑𝒎𝒎𝒎𝒎 from HV electrode. 1
GA
GA PSO
PSO
MFO
MFO
0.8
Fitness Function (% error)
0.4
Fitness Function (% error)
100
Iteration Number
Iteration Number
0.3
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0
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FIGURE 10. GA, PSO & MFO convergence with barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟓𝟓𝒎𝒎𝒎𝒎 from HV electrode.
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10
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FIGURE 14. GA, PSO & MFO convergence with barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟓𝟓𝒎𝒎𝒎𝒎 from HV electrode.
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B. OPTIMIZATION RESULTS
0.12 GA PSO
Fitness Function (% error)
0.1
MFO
0.08
0.06
0.04
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FIGURE 15. GA, PSO & MFO convergence with barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟏𝟏𝟏𝟏𝒎𝒎𝒎𝒎 from HV electrode. 0.2
GA PSO MFO
Fitness Function (% error)
0.15
C. SIMULATION RESULTS
0.1
0.05
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FIGURE 16. GA, PSO & MFO convergence with barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟐𝟐𝟐𝟐𝒎𝒎𝒎𝒎 from HV electrode. 0.035
GA PSO
0.03
MFO
Fitness Function (% error)
0.025
0.02
0.015
0.01
0.005
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FIGURE 17. GA, PSO & MFO convergence with barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟑𝟑𝟑𝟑𝒎𝒎𝒎𝒎 from HV electrode. 0.07
GA PSO
0.06
MFO
0.05
Fitness Function (% error)
To achieve accuracy the maximum values of error obtained by MFO compared with that obtained by PSO and GA see Table II. From the results in Table II the MFO provides the minimum values for error and error reaches to zero values in almost cases, so that the MFO algorithm is the most effective algorithm among the previous optimization techniques. The numerical results in Table II show that The MFO algorithm has the ability to find an initial point in the search domain and improve it to converge to the global minimum, so it is still the best choice for obtaining the optimal factors (𝑓𝑓1 and 𝑓𝑓2 ) corresponding to the minimum values of error for different barrier thickness and position in the gap.
0.04
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FIGURE 18. GA, PSO & MFO convergence with barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟒𝟒𝟒𝟒𝒎𝒎𝒎𝒎 from HV electrode.
VOLUME XX, 2017
Figures 19 to 22 show the contour distribution of electric field in the air gap of rod to plane arrangement with a polycarbonate barrier with 1mm thickness at (5, 10, 20, and 30 %) of the air gap from HV rod respectively. It is clearly shown that the presence of dielectric barrier in the rod-plane gap change the electric field distribution in the gap. Also, the maximum electric field is increased on rod tip and for a small distance around it by moving the dielectric barrier toward HV rod. Figures 23 to 26 show the contour distribution of electric field in the air gap of rod to plane arrangement with a polycarbonate barrier with 3𝑚𝑚𝑚𝑚 thickness at (5, 10, 20, and 30 %) of the air gap from HV rod respectively. Besides, the changing of electric field distribution and increasing of maximum field values on rod tip and around it by positioning barrier near to the HV rod. The maximum field values are increased also by increasing barrier thickness. Figures 27 to 31 show the contour distribution of electric field in the air gap of rod to plane arrangement with a polycarbonate barrier with 5𝑚𝑚𝑚𝑚 thickness at (5, 10, 20, 30 and 40 %) of the air gap from HV rod respectively. Figures from 19 to 31 investigate the contour distribution lines of the electric field simulation along the air gap of needle-to-plane configuration with a polycarbonate barrier. It is clearly shown that the presence of dielectric barrier in the rod-plane gap affect the electric field distribution along the gap. Also, the maximum electric field is increased on the HV electrode tip. In addition to non-uniformity of the electric field distribution between DB and plane gap configuration. The maximum value of electric field increased by increasing the barrier thickness and moving DB toward HV electrode.
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TABLE II Barrier position (mm) 5 10 20 30 40 50 60 70 80 90 Barrier position (mm) 5 10 20 30 40 50 60 70 80 90 Barrier position (mm) 5 10 20 30 40 50 60 70 80 90
Without optimization -0.24 -2.14 -1.74 -0.94 -0.79 -0.98 -0.89 -0.31 -1.00 -1.17
f1 0.9610 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Without optimization 2.74 -0.81 -1.74 0.16 -0.09 -0.51 -0.56 -0.86 -0.49 -0.42
f1 0.6967 0.9521 0.9029 0.2339 1.0000 1.0000 0.4336 0.8591 1.0000 1.0000
Without optimization 1.79 -0.33 -2.24 1.86 0.40 -0.09 0.66 -0.05 -0.16 0.63
f1 0.8585 0.8635 0.9604 0.6146 0.4604 1.0000 0.9714 0.8567 1.0000 0.9133
. CONVERGENCE OF GA, MFO AND PSO Barrier thickness=1mm GA PSO f2 % error f1 f2 0.9024 8.08E-07 1.0000 0.9389 0.0000 1.34E-01 0.2296 0.0000 0.0000 7.36E-01 0.3487 0.0000 0.0000 9.41E-01 0.3260 0.0000 0.0000 7.86E-01 0.3951 0.0000 0.0000 9.83E-01 0.9459 0.0000 0.0000 8.88E-01 0.7533 0.0000 0.0000 3.09E-01 0.0255 0.0000 0.0000 1.00E+00 1.0000 0.0000 0.0000 1.17E+00 0.0801 0.0000 Barrier thickness=3mm GA PSO f2 % error f1 f2 0.8799 6.06E-06 0.7915 1.0000 0.8169 2.36E-07 0.4636 0.3982 0.3733 2.16E-06 1.0000 0.4133 0.0190 1.51E-07 0.5734 0.0474 0.0000 9.03E-02 0.5744 0.0000 0.0000 5.08E-01 0.1667 0.0000 0.0000 5.59E-01 0.0894 0.0000 0.0000 8.61E-01 0.4633 0.0000 0.0000 4.93E-01 0.8303 0.0000 0.0000 4.22E-01 0.2153 0.0000 Barrier thickness=5mm GA PSO f2 % error f1 f2 0.9521 2.17E-05 0.7344 0.8160 0.8319 1.46E-06 0.8999 0.8664 0.5192 1.19E-06 0.7877 0.4257 0.3500 8.28E-07 1.0000 0.5695 0.0743 1.10E-08 0.9665 0.1575 0.0000 8.90E-02 1.0000 0.0000 0.3853 6.79E-08 1.0000 0.3967 0.0000 4.82E-02 0.5927 0.0000 0.0000 1.61E-01 0.5845 0.0000 0.5229 2.06E-07 1.0000 0.5724
FIGURE 19. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟓𝟓𝒎𝒎𝒎𝒎 from HV electrode.
VOLUME XX, 2017
% error 1.55E-04 1.34E-01 7.36E-01 9.41E-01 7.86E-01 9.83E-01 8.88E-01 3.09E-01 1.00E+00 1.17E+00
f1 1.0000 1.0000 1.0000 1.0000 0.9982 1.0000 0.4209 1.0000 0.5080 1.0000
MFO f2 0.9390 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
% error 0.00E+00 1.34E-01 7.36E-01 9.41E-01 7.86E-01 9.83E-01 8.88E-01 3.09E-01 1.00E+00 1.17E+00
% error 3.67E-03 5.82E-03 5.04E-04 3.09E-03 9.03E-02 5.08E-01 5.59E-01 8.61E-01 4.93E-01 4.22E-01
f1 0.7918 0.7567 1.0000 1.0000 1.0000 0.4488 0.3946 1.0000 0.0407 1.0000
MFO f2 1.0000 0.6492 0.4134 0.0812 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
% error 1.50E-14 3.26E-12 0.00E+00 1.77E-14 9.03E-02 5.08E-01 5.59E-01 8.61E-01 4.93E-01 4.22E-01
% error 3.51E-02 5.34E-03 9.88E-04 5.74E-05 3.85E-03 8.90E-02 6.19E-06 4.82E-02 1.61E-01 9.03E-05
f1 0.9017 1.0000 1.0000 1.0000 1.0000 0.5385 1.0000 0.4256 0.9747 0.9887
MFO f2 1.0000 0.9634 0.5406 0.5695 0.1614 0.0000 0.3967 0.0000 0.0000 0.5660
% error 0.00E+00 0.00E+00 1.72E-14 0.00E+00 0.00E+00 8.90E-02 0.00E+00 4.82E-02 1.61E-01 0.00E+00
FIGURE 20. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟏𝟏𝟏𝟏𝒎𝒎𝒎𝒎 from HV electrode.
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FIGURE 21. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟐𝟐𝟐𝟐𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 25. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟐𝟐𝟐𝟐𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 22. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟏𝟏𝒎𝒎𝒎𝒎 at position 𝟑𝟑𝟑𝟑𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 26. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟑𝟑𝟑𝟑𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 23. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟓𝟓𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 27. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟓𝟓𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 24. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟑𝟑𝒎𝒎𝒎𝒎 at position 𝟏𝟏𝟏𝟏𝒎𝒎𝒎𝒎 from HV electrode.
VOLUME XX, 2017
FIGURE 28. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟏𝟏𝟏𝟏𝒎𝒎𝒎𝒎 from HV electrode.
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All the previous figures approved that: the small barrier thickness at the appropriate position (at 20~30% of the air gap) reduced the maximum values of the electric field at the same time uniform the field distribution in barrier to plane gap which increase the breakdown voltage. CONCLUSIONS
FIGURE 29. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟐𝟐𝟐𝟐𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 30. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟑𝟑𝟑𝟑𝒎𝒎𝒎𝒎 from HV electrode.
FIGURE 31. The field strength distribution in a rod-plate gap with a barrier thickness equal to 𝟓𝟓𝒎𝒎𝒎𝒎 at position 𝟒𝟒𝟒𝟒𝒎𝒎𝒎𝒎 from HV electrode.
The maximum values of the simulated electric field at the needle tip varies with the changing in barrier thickness and position as given in Table III. TABLE III MAXIMUM SIMULATED ELECTRIC FIELD IN AT NEEDLE TIP Barrier thickness 1mm 2mm 3mm Barrier position (mm) Maximum field in (V/m) 5 84.79 94.53 100.56 10 80.72 85.13 88.86 20 79.66 81.28 82.48 30 79.31 80.20 81.60 40 79.34 79.90 80.30 50 79.14 79.52 79.85 60 79.18 79.44 80.41 70 79.62 79.18 79.82 80 79.05 79.45 79.72 90 78.91 79.50 80.34
VOLUME XX, 2017
In this paper, a metaheuristic optimization algorithm inspired from spiral moths flying around the moonlight has been investigated for solving a field problem. The non-linear field problem is more complicated in case of non-uniform electric field in addition to the presence of transverse DB. The nonuniform field arises from using needle-to-plane configuration. In addition to the non-uniformity of the field causing by the existing of the transverse layer of the DB across the gap. The new metaheuristic optimization algorithm called MFO proved its efficiency to obtain the optimum solution for the field simulation with minimum error. After accurate investigation by using MFO the following points can be illustrated: • With MFO a good agreement has been reached between the proposed numerical equation after minimizing its error using MFO, and the accurate simulation of the electric field distribution in the presence of DB. • MFO algorithm proved its effectiveness in solving non-uniform electric field problems, especially that are more complicated, such as using different barrier thickness and positions along the field gap. • Minimizing error using MFO in case of the thin barrier which extend to be limited by barrier position in the first 30% of the air gap for which the maximum percent error reaches to 0.74% and vice versa for the thick barrier the error is optimized using MFO for all barrier positions and reached zero% in most of the first 40% of the air gap. • Stagnation of PSO in local minimum is highly obvious from the results, but for MFO and GA reached easily with global minimum. • A comprehensive comparative study was conducted approved that MFO is the most efficient approach for minimizing error in the present field problem. REFERENCES [1] M. Talaat, M.A. Farahat and T. Said, “Numerical investigation of the optimal characteristics of a transverse layer of dielectric barrier in a nonuniform electric field,” Journal of Physics and Chemistry of Solids, Vol. 121, pp. 27-35, 2018. [2] M. Talaat, M.H. Gobran and M. Wasfi, “A hybrid model of an artificial neural network with thermodynamic model for system diagnosis of electrical power plant gas turbine,” Engineering Applications of Artificial Intelligence, Vol.68, pp. 222–235, 2018. [3] D. Rabah, C., Abdelghani, H., Abdelchafik, “Efficiency of some optimisation approaches with the charge simulation method for calculating
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2018.2889155, IEEE Access
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using Moth-Flame Optimizer”, Neural Computing and Applications, Vol. 30, No. 6, pp. 1889–1904, 2018. [24] A. H. Gandomi, A. R. Kashani, “Construction Cost Minimization of Shallow Foundation Using Recent Swarm Intelligence Techniques”, IEEE Transactions on Industrial Informatics, Vol. 14, No. 3, pp. 1099 – 1106, 2018. [25] J. Kennedy and R. Eberhart, "Particle Swarm Optimization", Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ, pp. 1942-1948, 1995 M. Talaat (Associate Professor) was born in El-Sharkia, Egypt in 1979. He received the B.Sc., M.Sc. and Ph.D. degrees from the Faculty of Engineering, Zagazig University, Egypt in 2000, 2005 and 2009 respectively, he is Associate Professor at the Electrical Power and Machines Department, Faculty of Engineering, Zagazig University, Egypt, and now he is Associate Professor at the Electrical Engineering Department, Shaqra University, Dawadmi, Ar Riyadh, Saudi Arabia. He is Editor Board Member for International Journal of Electromagnetics and Applications (USA) and Journal of Electrical Engineering (Romania). He has many published papers in IEEE Transaction, Electric Power System Research, Renewable Energy, and Journal of Electrostatics. The research interest covers different types of High voltage application, Electric fields, Plasma science, Renewable energy, Mathematical calculation, and Computer programs using simulation models. Abdulaziz S. Alsayyari is an assistant professor at the Computer Engineering Department at Shaqra University, Saudi Arabia, where he has been serving as the dean of the College of Engineering since February 2016. Previously, He had served as the dean of the College of Computing from October 2014 to February 2016. His research interest covers multiple areas in wireless networks including wireless sensor networks, internet of things, and cellular networks. Dr. Alsayyari graduated with a B.S., M.S., and Ph.D. degree all in computer engineering from Florida Institute of Technology, USA, in the year 2005, 2007, and 2013, respectively. M. A. Farahat (Professor) was born in ElSharkia, Egypt in 1959. He received the B.Sc. from the Faculty of Engineering, Al-Azahar University, Egypt in 1983, and M.Sc. from the Faculty of Engineering, Menofia University, Egypt in 1991 and Ph. D. degrees from the Faculty of Engineering, Zagazig University, Egypt with Scholarship Channel System Hannover University, Germany in 1996. Now he is now the Head of the Electrical Power & Machines Department, Faculty of Engineering, Zagazig University. The research interest covers different types of renewable energy, load forecasting and Distribution Systems. Taghreed Said (Ph. D. Electrical Engineering) was born in Sharkia, Egypt. She received the B.Sc. and M.Sc. degrees from the Faculty of Engineering, Zagazig University, Egypt in 2008 and 2014 respectively, and now she is Assistant Lecturer, Electrical and Computer department, Higher Technological Institute 10th of Ramadan City. The research interest covers different types of high voltage insulation, Computer programs using simulation models and Artificial Intelligence.
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2169-3536 (c) 2018 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
