A Diversified Multiobjective Simulated Annealing and ... - IEEE Xplore

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 26, NO. 2, MARCH 2016

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A Diversified Multiobjective Simulated Annealing and Genetic Algorithm for Optimizing a Three-Phase HTS Transformer Shabnam V. Daneshmand and Hossein Heydari, Member, IEEE

Abstract—In this paper, a diversified multiobjective optimization of a transformer built from high-temperature superconducting (HTS) windings is presented. The main goal is an effective approach for an optimal HTS transformer design that involves the determination of selective transformer parameters when selected objectives are optimized. However, multiobjective optimization parameters are usually complex functions of the design variables and available only from an analysis of a finite-element model of the structure. As such, this requires the need for advanced numerical techniques for simulation and analysis of the HTS transformer by FLUX software. In addition, Python software is used along with two-dimensional FLUX for running the optimal design concepts based on simulated annealing and the genetic algorithm for the multiobjective optimization of the HTS transformer, which is the main motivation of this paper. Index Terms—Genetic algorithm (GA), high-temperature superconducting (HTS) transformer, multiobjective optimization, simulated annealing (SA).

I. I NTRODUCTION

H

IGH-TEMPERATURE superconducting (HTS) transformers are one of the most important equipment using superconducting technology. They have several superiorities over the conventional transformers, such as compactness, lightness, increased efficiency, and lower environmental impact [1]. Although, the fundamental structure of HTS transformers is similar to the conventional ones; there are some differences such as ac loss and critical current, which must be considered in the design of HTS transformers [2]. As such, many researchers have tried to develop advanced economical HTS transformers with higher quality and reliability considering the diagnosis to avoid the unexpected failures [2]–[4]. For this purpose, several parts, including core, window, high-voltage (HV) and low-voltage (LV) windings, etc., must be designed in a way that some constraints are satisfied. In general, the transformer design process is essentially a complicated procedure, but sevManuscript received August 17, 2014; revised January 17, 2015, July 24, 2015, and December 10, 2015; accepted January 11, 2016. Date of publication January 21, 2016; date of current version February 18, 2016. This paper was recommended by Associate Editor P. J. Masson. S. V. Daneshmand is with the R&D Department, MAPNA Group, Tehran 1918953651, Iran (e-mail: [email protected]). H. Heydari is with the Center of Excellence for Power System Automation and Operation, Electrical Engineering Department, Iran University of Science and Technology (IUST), Tehran 1684613114, Iran (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TASC.2016.2519420

eral optimization techniques are proposed to obtain the best design variables [5]–[7]. The prior art for optimization algorithms used for superconducting devices is generally based on stochastic methods [8]–[11]. Since an optimal solution for HTS transformer design is a tradeoff between conflicted objectives however, heuristic methodologies can be useful for the optimization. In this paper, a prioritized multiobjective simulated annealing (SA) algorithm as well as a multiobjective continuous genetic algorithm (GA) are used for an optimal design of an exemplary three-phase HTS transformer. In the electromagnetic design of the 315-kVA HTS transformer windings via fieldcircuit coupling simulation, its inter-related parameters (i.e., ac loss is proportional to critical current and leakage flux) in an optimal design, for instance, are considered. In addition, ac loss increases, when the length of the windings increases. However, prior to performing magnetostatic simulation to calculate different parameters of the transformer, with an algorithm based on the finite-element method (FEM), the influence of the inter-related parameters on each other has to be specified. Thereafter, an appropriate tool for running optimization concepts is introduced for performing consecutive simulations. Python software is used for running optimization algorithms based on SA and GA [12]. These two softwares are coupled to each other and work together. That is, the iteration of the optimization algorithm in Python is accomplished by its corresponding simulations of the transformer using FLUX-2D software for the consecutive iterations until optimal objective results are achieved [13]. This requires a CPU with rather long time scale to comply with the large number of iterations, meshing the model, solve the problem, and calculation of the necessary objectives. The proposed algorithm includes the merits of SA and GA simultaneously, has the ability to find the global optimum, and not trapping in local optimum points, which is a crucial feature in optimization problems. As such, the use of this approach for an optimal design of an HTS transformer, which is an intrinsically complicated nonlinear problem, is yet the main motivation for imitating this paper.

II. M ODELING OF THE HTS T RANSFORMER For a better understanding of the present approach and showing how it may be brought into effect, reference will now be made by way of an example to the design of a three-phase

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TABLE I D ESIGN PARAMETERS OF HTS T RANSFORMER

functions with nv variables subject to nc constraints must be optimized, which is defined as follows [15]: min {fi (xj )} for all i = 1, 2, . . . , nf and j = 1, 2, . . . , nv . (1) Subject to gk (xj ) ≤ 0,

k = 1, 2, . . . , nc

where g(x) is the constraint, and T X = x1 x2 · · · xnv ,

xmin ≤ x ≤ xmax

(2)

(3)

where X is the variable set, and T is the symbol of matrix transpose. It must be noted that this is different from T (symbol of temperature) in the SA algorithm. Vector X ∗ is a Paretooptimal solution if and only if there is no other vector X, which can dominate it. That is, fi (X ∗ ) ≤ fi (X) for all i = 1, 2, . . . , nf and fi (X ∗ ) < fi (X) for at least one i. The goal of the multiobjective problem is to find many different Paretooptimal solutions as possible. In Appendix I, SA and GA are completely described. IV. R ESULTS AND D ISCUSSION

Fig. 1. Magnetization characteristic of the core.

three-limb core-type 315-kVA HTS transformer. Accordingly, the advantages of the present approach will be described with reference to such a transformer. Analytical and simulation results were obtained using the transformer data given in Table I. The core is manufactured by British Steel Corporation under the trade name Unisil, coded 30M5 was sheared from a coil of grain-oriented 3% silicon–iron of thickness 0.30 mm. The magnetization characteristic of the core material is shown in Fig. 1. III. M ULTIOBJECTIVE O PTIMIZATION U SING SA AND GA The majority of engineering design problems contains several conflicting design interrelated objectives that need to be compromised. These are called multiobjective optimization problems. In most scenarios, the objective functions are in conflict, that is, reduction in one objective function leads to increase in another. Therefore, a single optimal solution that is optimum with respect to all objectives does not exist, but there is a set of optimal solutions, which is known as the Pareto set (or tradeoff curve). The diversity assessment of Pareto front approximations is an important issue in the stochastic multiobjective optimization community. It is not possible to reduce any of the objective functions without increasing at least one of the other objective functions, and it is the main property of the Pareto-optimal solution [14]. In a multiobjective optimization problem, nf ≥ 2 objective

For an optimal design of an HTS transformer, different variables, objective functions, and constraints could be considered. As the cost of HTS material is high, reduction in the length of HTS windings can be an effective method for cost reduction of an HTS transformer. Volt per turn is an important parameter in transformer design, which can determine the dimension of the transformer, turn number, and length of the windings. It is known that the size and weight of a transformer are directly relevant with volt per turn. It therefore seems that volt per turn can be an effective variable for the optimal design of an HTS transformer. The transformer parameters mentioned below are calculated using the FLUX-2D software. This paper deals with three different multiobjective optimization problems. In Scenario 1, only one variable: voltage per turn ET for multiobjective design is considered, two objective functions: minimum core size and length of the windings are considered, and one constraint: efficiency of higher than 98.5% for the HTS transformer is considered. This case can be written as f1 (x1 ) = S(ET) f2 (x1 ) = l(ET)

(4)

g1 (x1 ) = η(ET) ≥ 0.985

(5)

and the constraint is

which can be considered as Min {S, l}

subject to :

1 − 1.0152 ≤ 0. η

(6)

In Scenario 2, two variables are considered as voltage per turn and maximum magnetic flux density in core (Bm), three objective functions are considered as size of the core (m2 ), windings’ length, and ac loss of the windings, which must be

DANESHMAND AND HEYDARI: SA AND GA FOR OPTIMIZING A THREE-PHASE HTS TRANSFORMER

minimized. The core size is described by the total surface area (m2 ) of the side view of the three-phase core, looking through the two core windows of the HTS transformer. The constraint is also considered as efficiency of the HTS transformer, which must be higher than 98.5%. This scenario can be considered as

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TABLE II O PTIMIZATION PARAMETERS OF SA

f1 (x1 , x2 ) = S(ET, BM) f2 (x1 , x2 ) = l(ET, BM) f3 (x1 , x2 ) = p(ET, BM)

(7)

and the final problem can be considered as Min {S, l, p}

subject to :

1 − 1.0152 ≤ 0. η

(8)

In Scenario 3, the three objective functions of Scenario 2 plus mechanical force acting on the windings, i.e., f1 (x1 , x2 ) = S(ET, BM) f2 (x1 , x2 ) = l(ET, BM) f3 (x1 , x2 ) = p(ET, BM) f4 (x1 , x2 ) = F (ET, BM)

(9)

and the final problem can be considered as Min {S, l, p, F }

subject to :

1 − 1.0152 ≤ 0. η

Fig. 2. Size variation of the core with number of iterations by SA.

(10)

The variation range of variables is considered as 20 ≤ ET ≤ 100,

1.2 ≤ BM ≤ 1.6.

(11)

Hence, Scenarios 1, 2, and 3 are two-objective, three-objective, and four-objective optimization problems, respectively. In each iteration, the ac loss of the windings and iron loss of the core should be calculated to compute the efficiency of the HTS transformer. The ac loss is composed of parallel and perpendicular components of magnetic field [3]. The detailed description of ac loss calculation is given in Appendix II. Based on fundamental postulates of electromagnetics, the current-carrying windings will experience a force due to the interaction between the electric and magnetic fields [16], i.e., F = (J × B).h.dx.dy (12) where h is the height of the model, and the integration must be performed over the winding area. A. Scenario 1 The optimization parameters of SA are shown in Table II. Figs. 2–4 show the variation of variables and objectives with number of iterations in the SA algorithm for T1 = 2 and T2 = 1 (see Appendix I.A). It can be seen that the objectives concentrate on an optimal point after about 400 iterations during cooling of the annealing process. The size of the core changes

Fig. 3. Length variation of the windings with number of iterations by SA.

from a minimum value of 1.60 to a maximum value of 3.65 and reaches 2.8. The length of the windings varies from 950 to 2300 and reaches 1138. Volt per turn also changes between 20 and 96.5 and finally reaches 67.5. The optimization parameters of GA are shown in Table III. Figs. 5 and 6 show the objective function for the number of iterations in GA for weights w1 = w2 = 0.5 (see Appendix I.B). By comparing these figures with Figs. 2 and 3, it is found that convergence to the optimum value is faster in the SA algorithm. Furthermore, contrary to GA, the objectives’ values in the SA algorithm extensively vary. However, in GA, there are a wide range of values for objectives. Both of these algorithms choose a large range of values randomly preventing them to be trapped in a local optimum

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Fig. 4. Variation of volt per turn with number of iterations by SA. TABLE III O PTIMIZATION PARAMETERS OF GA

Fig. 6. Length variation of the windings with number of iterations by GA.

Fig. 7. Pareto curves with SA and GA for Scenario 1. TABLE IV F INAL R ESULTS FOR PARETO F RONT W ITH SA

Fig. 5. Size variation of the core with number of iterations by GA.

solution instead of a global optimum solution. By changing initial temperatures and weights as the objective functions in SA and GA, different optimization results can be calculated, and Pareto curves for these 2-D problems are plotted and shown in Fig. 7. It can be seen that Pareto results for both the SA and GA algorithms are nearly the same. This implies that both of the proposed methods are suitable for optimal solutions for Scenario 1. B. Scenario 2 For this scenario, the optimization parameters for SA- and GA-based algorithms are the same as Scenario 1.

Tables IV and V show 15 optimization process results with SA- and GA-based algorithms, respectively. The initial temperatures’ variations of the objective in the SA-based algorithm


TABLE V F INAL R ESULTS FOR PARETO F RONT W ITH GA

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TABLE VI F INAL R ESULTS OF F OUR O BJECTIVE O PTIMIZATION W ITH SA

TABLE VII F INAL R ESULTS OF F OUR O BJECTIVE O PTIMIZATION W ITH GA

Fig. 8. Pareto fronts with SA and GA for Scenario 2.

are between 1 and 20. Similar to Scenario 1, optimal results in Pareto fronts of SA- and GA-based algorithms shown in Fig. 8 are approximately identical. Thus, these two methods are appropriate for solving nonlinear problems. C. Scenario 3 For this scenario, the optimization parameters for SA and GA are the same as those for Scenario 1, and Tables VI and VII show the results of 15 optimization processes for the SA and GA based on the corresponding algorithms, respectively. Figs. 9 and 10 show Pareto of SA and GA for four objective optimization problems, respectively. The solid colorful circles in the figures denote the fourth axes (mechanical force). D. Winding Field Calculation Magnetic field in the windings is calculated with FLUX-2D to determine conductor critical currents and ac losses [17]. The problem is considered and solved in a steady-state condition, and the 2-D magnetostatic solver has been used for leakage flux

Fig. 9. Pareto of Scenario 3 for SA.

distribution analysis in the simulation. The Dirichlet boundary condition has been used in the simulation, wherein the magnetic vector potential on the boundary is zero (A = 0), and the

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Fig. 10. Pareto of Scenario 3 for GA.

magnetic field is tangent to the boundary. Moreover, the flux is also zero outside the boundary region (infinite box). Mesh type and mesh shape are considered as automatic and rectangular for all the regions. Small elements are chosen for meshing the HTS windings, because of high-leakage flux density in these regions. The E-J power law model is used to describe the nonlinear characteristic of the HTS windings [16]. One of the optimized solutions of Scenario 2 is ET = 50.7 and BM = 1.59 being selected from the Pareto front of SA. Fig. 11(a) and (b) shows the leakage flux distribution of HV winding on the left limb of the core of the HTS transformer with conventional values and optimized values, respectively. The selected ET and BM amounts of the HTS transformer are based on trial-and-error design, which is the conventional method for transformer design. It can be seen that with optimized values of ET and BM, the leakage flux reduced from the maximum value of 56.47 mT to 46.75 mT. This reduction concludes to lower ac loss and thereby higher efficiency in the transformer. E. Comparison of the Time of Convergence of SA and GA To compare the time of convergence of SA- and GA-based algorithms, MATLAB software, being compatible for the concepts of the both optimization algorithms, is used. As MATLAB software does not have the ability to work with FLUX software and SA and GA algorithms cannot be directly performed in MATLAB software without the magnetic field computation for different iterations, the results of different runs of FLUX software must be imported in MATLAB software programs. For this purpose, FLUX software was run for 441 trials (considering 21 different values for variables ET and BM); different parameters such as radial and axial components of leakage flux, ac loss, core loss, and efficiency are calculated. The more number of runs in FLUX software, the higher accuracy in MATLAB software is achieved. The “interp” function in MATLAB software is used to compute parameters that are not calculated in 441 runs. Hence, parameters between calculated data are interpolated and computed. Using this method causes decrease in convergence time because MATLAB software is

Fig. 11. Leakage flux distribution of the HV winding of the HTS transformer with (a) conventional values and (b) optimized values.

not coupled with FLUX software and parameters are directly imported in MATLAB software. However, using interpolation makes this method not very accurate, but MATLAB software can be used for comparing the time convergence of two optimization algorithms. The method is executed on a 3.41-GHz Quad core CPU PC with 8-GB memory. The time of convergence of SA and GA algorithms for Scenario 1 (two-objective problem) is calculated as 0.530 and 0.612 s, respectively, and the convergence time difference will be very large by using FLUX and Python softwares being coupled to each other. It shows that the SA algorithm is faster than GA. V. C ONCLUSION In this paper, multiobjective optimization methods using SA and GA for an optimal design of a three-phase HTS transformer have been proposed. The algorithm must be reliable and encountered global optimum results rather than local optima. Here, three simple scenarios were considered for optimization: Scenario 1: one-variable and one-constraint two-objective functions, Scenario 2: two-variable and one-constraint threeobjective functions, and Scenario 3: two-variable and oneconstraint four-objective functions.


It can be seen that both optimization methods based on prioritized SA and multiobjective continuous GA have a good performance in the optimal design of an HTS transformer. However, in SA, convergence of the objectives to their optimum value is faster than GA. However, the multiobjective optimization parameters (i.e., complex functions of the design variables) were available from an analysis of an FEM of the structure. Thus, 2-D FLUX software was coupled with Python for multiobjective optimization of an exemplary three-phase HTS transformer based on SA and GA concepts. The nondominated solutions lying on the Pareto front will provide a variety of options to engineers to improve one objective without jeopardizing optimality. More importantly, it depends on the designer’s suit to select the constant parameters as variables and the other design parameters of the HTS transformer as objectives. A PPENDIX I Python software is used for running optimization algorithms based on SA and GA. FLUX 2-D software is used for the modeling of the transformer, solving the nonlinear problem using the FEM, and calculating the components of magnetic field and everything about the HTS transformer. SA and GA algorithms, which are described completely in the following, are written using Python. For the calculation of different parameters of the transformer, such as ac loss and mechanical force, FLUX 2-D software is needed, and so, Python and FLUX softwares must be coupled to each other and work together. In each iteration of the optimization algorithms in Python, the design parameters of an HTS transformer are changed, and new values must be imported into FLUX software to calculate new values. A. SA SA is a method that is derived from the simulation of solids during cooling. That is, SA achieves its principles from its nature and resembles as liquid being frozen and crystallized or the metals cool. Boltzmann probability distribution expresses the probability of a particle being in a special energy level. Kirkpatrick et al. [18] introduced SA for finding global extreme functions. They introduced a search method that accepts all the inferior moves based on Boltzmann probability distribution. By accepting some solutions that are worse, the algorithm has the ability of climbing out of local minima and finding a global optimum. This way, the algorithm will not stick in local optima [19], and it is the most important point of SA. In addition, SA does not require any functional derivative information and is unaffected by discontinuities and nonlinearities [14]. When a temperature reduction occurs, fewer inferior moves are accepted. Previous works show that SA is an effective approach for the optimal design of various apparatus and systems [20]–[22]. In this paper, a prioritized multiobjective stochastic algorithm based on SA is considered. In this algorithm, all objective functions are simultaneously considered, but it approximates the unique optimal solution that satisfies the given priority.

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The algorithm starts from an initial random solution and initial temperatures corresponding to the different objectives. It means that for different objectives, different initial temperatures are considered to give them different priorities. One of the objectives (F1 ) has the most priority. It is considered that there are nc constraints, and satisfaction of them also has the top priority, stated as follows [19], [20]: F1 (X) = f1 (X) + b

nc

Max {0, gk (X)}

(1A)

k=1

where b is a penalty parameter expressing the importance of constraints. Hence, it depends on the designer to choose the penalty factor, and it is obvious that by changing b, the Pareto results can also change. In this paper, our main purpose is introducing a method for the optimal design of the HTS transformer, and we selected a value for b arbitrarily. Designers can select design parameters such as b to suit their needs. For an initial random solution, all the nf objectives and nc constraints are calculated. Then, by using the nonuniformed mutation operator, a new random solution is generated as xnew randj e−iter/max = xiter + xmax − xiter j j j j if

bin = 0 = xiter − xiter − xmin xnew randj e−iter/max j j j j

if

iter

bin = 1

iter

(2A)

where iter is the iteration number, randj is a random number between 0 and 1, max iter is the maximum number of iterations, and j = 1, 2, . . . , nv . bin is a random binary number that may be 0 or 1 in each iteration. For a new solution, all the objectives are calculated, and probabilities of a new solution are calculated as ΔF1 = F1 (Xnew ) − F1 (Xiter ) ΔFi = fi (Xnew ) − fi (Xiter ), i = 2, . . . , nf exp(−ΔFi /Ti ), ΔFi ≥ 0 pri = , i = 1, . . . , nf . 1, ΔFi ≤ 0 (3A) Then, nf random numbers between 0 and 1 are generated, and variables of the next step are considered as xiter+1 = xnew , if all randi < pri j j for i = 1, 2, . . . , nf

and j = 1, 2, . . . , nv

xiter+1 = xiter j , if some randi < pri j for i = 1, 2, . . . , nf

and j = 1, 2, . . . , nv . (4A)

It means that better results for the new solution are firmly acceptable; otherwise, it is conditionally accepted, which is close to 1 at the beginning of the process, but it is close to 0 as the algorithm progresses.

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Now, the temperatures of all objective functions are reduced, and the number of iterations is increased, respectively, as Ti = Ti rb , iter = iter + 1.

i = 1, . . . , nf ,

0 < rb < 1

(5A) (6A)

The algorithm terminates if iter max iter and Xmax iter is the optimal solution. Thus, one set of Pareto-optimal solutions is achieved, corresponding to hierarchies of the objectives in one run of the algorithm. In this paper, objective functions and constraints are normalized to be in the same range. Except F1 , for objectives, the normalization of ΔFi can be done as ΔFinorm = (fi (Xnew ) − fi (Xold ))/fi (Xnew ) for i = 2, 3, . . . , nf .

(7A)

However, for F1 with constraints, it is done as follows. Each constraint is divided into its initial values, and then, it is multiplied with the initial value of F1 to convert its normalized value into a quantity with the same dimension with F1 as

nc

f1 (Xinitial )

gk (X) . (8A) max 0,

F1 (X) = f1 (X)+b gk (Xinitial )

k=1

ΔF1 can be obtained as ΔF1 = Δf1 (X) + bΔG(X)

(9A)

where (10A) Δf1 (X) = f1 (Xnew ) − f1 (Xold )

n c

f1 (Xinitial )

gk (Xnew ) max 0,

ΔG(X) = gk (Xinitial )

k=1

nc

f1 (Xinitial )

gk (Xold ) − max 0,

(11A) gk (Xinitial )

k=1

and the normalized value of ΔF1 is calculated with [27] ΔF1norm = 1 −

b f1 (Xold ) + ΔG(X). f1 (Xnew ) f1 (Xnew )

(12A)

B. GA The GA is a search heuristic that mimics the process of natural evolution, and its concept was developed by John Holland in 1975 [23]. During a GA optimization, a set of random solutions is chosen and considered as a population. Each individual in the population is assigned a fitness value by the evaluation of the objective function. Different pairs of individuals are selected from the population and considered as parents. Fig. 12 shows a flowchart of continuous GA [24]. Referring to Fig. 12, it is shown that a random initial population must be generated, and its cost should be calculated. In multiobjective problems, one of the methods for the GA is a weighted cost function being used in this paper and expressed as cost =

N

wn fn

(13A)

n=1

where fn is the cost function n for 0 ≤ fn ≤ 1, and wn is the weighting factor for N n=1 wn = 1.

Fig. 12. Flowchart of continuous GA.

Normalized constraints are also added to one of the objectives with a penalty factor. After that, costs and their associated chromosomes are ranked from the lowest to the highest cost. Some of the chromosomes from the top of the list are selected as parents while the others die. There are different methods for parent mating and rank weighting, which is independent problem and finding the probability from the rank, i.e., n, of the chromosome and used in this study as Nkeep − n + 1 (14A) pn = N keep n n=1

where Nkeep is the number of the chromosomes being constant, and n is the rank of the chromosome in the sorted list. The sorted list is based on the cost of the chromosomes that are ranked from the lowest to the highest cost, and n is the rank of the chromosome in the sorted list. Many different approaches have been tried for crossing over and generating children in continuous GA. The following approach is used in this paper. A variable in the first pair of parents is randomly selected. Parents are considered as parent1 = [pm1 pm2 , . . . , pmα , . . . , pmNvar ] parent2 = [pd1 pd2 , . . . , pdα , . . . , pdNvar ] where m and d show mother and father (dad). The selected variables are combined as pnew1 = pmα − β[pmα − pdα ] pnew2 = pdα − β[pmα − pdα ]

(15A)

where β is a random number between 0 and 1. Children are produced as (3) child1 = [pm1 pm2 , . . . , pnew1 , . . . , pdNvar ] child2 = [pd1 pd2 , . . . , pnew2 , . . . , pmNvar ]

(16A)

where Nvar is the number of variables in the chromosome string.


To prevent the overly fast convergence and trapping into local minima instead of global minima, the GA must be forced to explore the other areas of the surface cost by randomly introducing changes or mutations in some of the variables. In each run with one set of weights, one point of the Pareto front is calculated. For various weights of objectives, different solutions are calculated, and the Pareto front is achieved. A PPENDIX II For calculation of ac loss, two components of magnetic field (parallel and perpendicular) should be computed. The parallel component is then calculated by [25]–[27] ⎧2f CAB 2 p 3 2 ⎪ i , bac < iac + 3b i ac ac ⎪ ac 3μ0 ⎪ ⎪ ⎪ 2f CABp2 ⎪ 3 2 ⎪ b iac< bac< 1 +3b i ⎪ ac ac , ac 3μ ⎪ 0 ⎨ 2 p = 2f CABp bac 3+i2ac −2 1−i3ac (W/m) 3μ0 ⎪ ⎪ ⎪ 2 3 ) ) (1−i (1−i ⎪ ac ac 2 2 ⎪ ⎪ ⎪+6iac (bac −iac ) −4iac(bac −iac )2 , (bac> 1) ⎪ ⎪ 2 ⎩2f CABp bac 3+i2ac −2 1−i3ac , (bac< 1) 3μ0 (17A) where f is frequency; bac = Bac /Bp is the normalized parallel magnetic field component; Bac is the parallel magnetic field amplitude; Bp = μ0 Jc d is full field penetration; iac = Iac /Ic is normalized current; Iac and Ic are the alternative current amplitude and the tape critical current, respectively; Jc is the critical current density of the tape; C is an effective area depending on the geometrical configuration of the tape; and A is the total cross section. The perpendicular component when iac ≤ tanh(bac⊥ ) is [26], [28] P⊥ =

f μ0 Ic2 π ⎧ ⎛ ⎞ ⎨ 2 (1 + p )(1 − p ) + a 0 0 ⎠ 0 × 2coth−1 ⎝ ⎩ 2 2 2 2 (1 + p0 ) − a0 (1 − p0 ) − a0 1 − (1+p0) (1+p0)2−a20 +(1−p0) (1−p0)2 −a20 4 1 + p0 1 − p0 × cosh−1 + cosh−1 a0 a0 1 2 2 2 2 + (1+p0 ) −a0 − (1−p0 ) − a0 2 −1 1+p0 −1 1−p0 × (1+p0 )cosh +(1−p0)cosh a0 a0 2 1 + (1 + p0 )2 − a20 − (1 − p0 )2 − a20 4 1 2 2 2 2 − (1 + p0 ) − a0 − (1 − p0 ) − a0 2 ⎫ ⎬ (1+p0 )2 −a20 + (1−p0 )2 −a20 × (W/m) ⎭ (18A)

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where a0 = a/w, p0 = p/w, w is half of the width of the tape, bac⊥ = Bac⊥ /Bf is the normalized perpendicular magnetic field component, and Bac⊥ and bf = μ0 Jc /π are the perpendicular ac magnetic field amplitude and characteristic magnetic field, respectively. a and p are the width of the nonpenetrated region and the position of the center of this region, respectively, and are given by 1 − i2ac (19A) a =w cosh(bac⊥ ) p = wiac tanh(bac⊥ ).

(20A)

When iac ≥ tanh(bac⊥ ), the perpendicular component can be obtained as P⊥ =

f μ0 Ic2 π ⎧ ⎛ ⎞ ⎨ 2 (1 + p )(1 − p ) + a 0 0 0 ⎠ × −2coth−1 ⎝ ⎩ 2 2 (1 + p0 ) − a0 (1 − p0 )2 − a20 1 2 2 2 2 − (1+p0 ) (1+p0) −a0 −(1−p0) (1−p0 ) −a0 4 1 + p0 1 − p0 × cosh−1 − cosh−1 a0 a0 1 + (1 + p0 )2 − a20 + (1 − p0 )2 − a20 2 −1 1+p0 −1 1−p0 × (1+p0 )cosh +(1−p0)cosh a0 a0 ⎫ 2⎬ 1 − (1+p0)2 −a20 + (1−p0)2 −a20 (W/m). ⎭ 4 (21A) R EFERENCES

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Shabnam V. Daneshmand received the B.Sc. degree in electronic engineering from Amirkabir University, Tehran, Iran, in 2001 and the M.Sc. degree in power engineering and the Ph.D. degree in HTS transformers from Iran University of Science and Technology, Tehran, in 2004 and 2013, respectively. She is currently with the R&D Department, MAPNA group. Her main research interests include transformers, applied superconductivity, and optimization in power systems.

Hossein Heydari (M’09) received the B.S. degree in electrical engineering and the M.Sc. degree in power electronics from Loughborough University, Loughborough, U.K., in 1985 and 1987, respectively, and the Ph.D. degree in transformer core losses from the University of Wales, Cardiff, U.K., in 1993. Following graduation, he joined Iran University of Science and Technology (IUST), Tehran, Iran, as an Academic Member (Lecturer) of the Electrical Power Group, and was also appointed as the Director of the High Voltage and Magnetic Materials Research Center. He is currently with the Center of Excellence for Power System Automation and Operation, IUST. His research interests include EMC considerations in power systems, magnetic gears, fault current limiters, and applied superconductivity in power systems.