Drytype aircore reactor is now widely applied in electrical power distribution systems, for which the optimization design is a crucial issue. In the optimization design problem of drytype aircore reactor, the objectives of minimizing the production cost and minimizing the operation cost are both important. In this paper, a multiobjective optimal model is established considering simultaneously the two objectives of minimizing the production cost and minimizing the operation cost. To solve the multiobjective optimization problem, a memetic evolutionary algorithm is proposed, which combines elitist nondominated sorting genetic algorithm version II (NSGAII) with a local search strategy based on the covariance matrix adaptation evolution strategy (CMAES). NSGAII can provide decision maker with flexible choices among the different tradeoff solutions, while the localsearch strategy, which is applied to nondominated individuals randomly selected from the current population in a given generation and quantity, can accelerate the convergence speed. Furthermore, another modification is that an external archive is set in the proposed algorithm for increasing the evolutionary efficiency. The proposed algorithm is tested on a drytype aircore reactor made of rectangular crosssection litzwire. Simulation results show that the proposed algorithm has high efficiency and it converges to a better Pareto front.
As an important apparatus applied to harmonic filtering, short circuit current limiting, and reactive power compensation, drytype aircore reactor plays a vital role in reducing failure and improving security of power system operation. In the past two decades, the design problem for reducing the production cost and the operation cost of drytype aircore reactor has received considerable attention. Most literatures about the design problem of drytype aircore reactors are mainly focused on the following aspects:
Actually, it is well known that the optimum design of drytype aircore reactor is a very complex problem as it requires the simultaneous minimization of two objective functions. The first objective deals with the minimization of the production cost by reducing aluminum wire weight. The second objective minimizes the operation cost by the minimization of power loss. And the voltage, height, package temperature rise and current density may be considered as constraint conditions. In theory, these two objectives are in conflict with each other. No single solution can be claimed as an optimum solution to multiple conflicting objectives, the resulting multiobjective optimization problem involves a number of tradeoff solutions. In order to provide a means to assess tradeoff between two conflicting objectives, one way is to formulate the optimum design of drytype aircore reactor as a multiobjective optimization problem.
This paper is concerned with the optimum design of drytype aircore reactor made of rectangular crosssection litzwire as a multiobjective optimization problem. Due to the nondifferentiality, nonlinearity, highconstraint, and highdiscretion of the optimum design problem, most traditional optimization or search approaches (e.g., Sequential Search, Tabu Search, etc.) not only often fall into local maxima and minima, but also are not easily applied to solve the optimization problem with multiobjectives. As one of the elegant approaches, the elitist nondominated sorting genetic algorithm II (NSGAII) [
In this study, the NSGAII algorithm with certain modifications is applied to settle the multiobjective problem. As an approach to improve the NSGAII algorithm, a local search strategy based on the Covariance Matrix Adaptation Evolution Strategy (CMAES) [
The paper is organized as follows. The optimum design of drytype aircore reactor is formulated as a multiobjective optimization problem with a number of equality constraints and inequality constraints in Section
The drytype aircore reactor, the diagram of which is shown in Figure
Diagram of drytype aircore reactor.
The optimal design of drytype aircore reactor is equivalent to finding a set of best decision vectors that minimizes the two competing objective functions, the production, and the operation cost, subject to a number of equality constraints and inequality constraints. Apparently, the multiobjective formulation is a choice way to treat the nonlinear and constrained optimal problem. Its mathematical model can be described as follows.
Since minimizing the aluminum wire weight is equivalent to minimizing the production cost, the first objective function is to minimize the aluminum wire weight, usually defined as
It is well known that the smaller power loss results in the less operation cost, so the second objective function can be represented by the following expression:
The objective functions are subjected to following some equality constraints and inequality constraints.
In order to design a high performance drytype aircore reactor, some additional equality constraints are proposed. There are two most common additional constraints which are layer resistance drop balance constraint and package temperature rise balance constraint. Layer resistance drop balance can assure that the aircore reactor has the minimum power loss; package temperature rise balance can assure the best heat sinking effect. In addition, in order to make the aircore reactor have more compact structure and reduce the manufacturing cost, the package height balance constraint should be considered too. The three additional equality constraints are expressed as follows.
Layer resistance drop balance constraint:
Package height balance constraint:
where
Package temperature rise balance constraint:
where
where
It is now evident from the above discussion that the optimal design problem of drytype aircore reactor is a nonlinear engineering optimization problem without analytical expression, that is, the objective functions cannot be expressed as the functions of the design variables in analytical form. However, the variable values in the objective functions can be obtained from the above four mutually independent design variables in the process of design.
NSGAII algorithm has been demonstrated as one of the most efficient and famous algorithms for multiobjective optimization. It uses the fast nondominated sorting to rank the population fronts and a parameter called crowding distance is calculated in the same front. Then, tournament selection is made between two individuals randomly selected from parent population. The individual with lower front number is selected if the two individuals come from different fronts. The individual with higher crowding distance is selected if the two individuals are from the same front. Then, both the crossover and the mutation operators are used to generate a new offspring population. Finally, the parent and offspring populations are combined together where a fast nondominated sorting and crowding distance assignment procedure is used to rank the combined population and only the best
NSGAII is proposed on basis of nondominated sorting genetic algorithm (NSGA) [
Memetic algorithms perform global exploration by evolutionary algorithms and local exploitation by a local search strategy, respectively. It has global search ability of evolutionary algorithm and local search ability of local search strategy simultaneously. It is reported that memetic algorithm converges to high quality solutions more efficiently than evolution algorithms [
Local search can enhance the search capability of evolutionary algorithms by carrying out local exploitation, and the global and local searches may be well balanced. The CMAES is probably one of the most powerful selfadaptation mechanisms for continuous search spaces. It uses a covariance matrix to construct the mutation distribution and adapts this covariance matrix from cumulative paths of successful mutations. Firstly, it samples a number of new candidate solutions from a multivariate normal distribution and then updates the distribution by means of two major strategies: step size update and covariance matrix adaptation. The process in five steps is given and explained as follows.
Initialize
At each generation
Update the covariance matrix
Update the global step size
The computation is repeated until the maximum number of generations is met.
It should be pointed out that the local search strategy based on CMAES cannot be directly used to handle multiple objectives. To apply the local search strategy based on CMAES to the constrained twoobjective optimization problem in this paper, the following overall objective
Although the local search strategy may improve the performance of the original NSGAII, it implies an additional cost, which may become prohibitive in the optimization design of drytype aircore reactor. When the quality of an individual is very poor, the application of local search strategy also seems to be waste of the computing time. Thus, the local search (i.e., CMAES) is applied only to nondominated individuals randomly selected from the current population obtained from elitism of a given generation. Actually, our experience has shown that in drytype aircore reactor the numbers of winding package and winding layer have a great influence upon the results. Therefore, in order to get good search results, the local search is applied only to the inner diameter and average side length.
Since the overall objective function is used in the CMAES method, the points obtained from the CMAES should be checked for nondominance before they are accepted in the current population. If the point obtained by the CMAES dominates any point in the current population, the dominated point must be replaced by the new point.
In each generation of the original NSGAII, a new offspring population is generated. The parent and offspring populations are combined together where a fast nondominated sorting procedure is used to sort the combined population and only the best
In order to increase the evolutionary efficiency especially at the early generation, an external archive of size
Let
Flowchart of NSGACMA algorithm.
Input the initial parameters of the 50 kVar drytype aircore reactor and the proposed algorithm; randomly generate an initial population; set
Calculate the values of the objective functions and constraints for each individual in the current population.
Apply nondominated sorting to classify the current population into different nondominated fronts and then calculate the crowding distance of each individual in the same front.
Perform tournament selection between two individuals randomly selected from the parent population and external archive.
Apply a crossover operator to each of the randomly selected
Combine parent and offspring population to generate a combined population and then apply fast nondominated sorting and crowding distance assignment to the combined population. Select
If
In the new generated parent population, individuals in the first nondominated front are stored in the external archive. When the archive exceeds the maximal size
If
To verify the effectiveness of the proposed algorithm, NSGAII and NSGACMA are used for the optimization design of a 50 kVar (i.e., 317.5 V, 157.5 A, and 6.42 mH) drytype aircore reactor made of rectangular crosssection litzwire, and comparisons are made between their results.
There are four mixed design variables in the optimization design of drytype aircore reactor. The number of winding package
The main parameters of a 50 kVar drytype aircore reactor are the extreme values of design variables and the constraint conditions, which are given in Table
Parameters for a 50 kVar drytype aircore reactor.
Parameters 










Values  2  7  400  1000  2  4  0.9  1.5  75 
The parameter settings of CMAES, which are discussed in [
Default parameter settings for CMAES.
Parameters 








Values 







In the proposed algorithm, the crossover probability is denoted as
Parameters for NSGACMA.
Parameters 

















Values  100  100  100  0.9  0.01  0.5  20  20  5  4  10  2  5  3  70  0.1 
In order to evaluate the performance of the proposed algorithm, the convergence metric [
The convergence metric measures the extent of convergence towards a reference set, and lower values of the metric represent good convergence ability. Let
To keep the convergence metric within
Let
To compare the proposed algorithm with NSGAII, the two algorithms run for 20 times, respectively, with 20 initial populations of random solutions (one for each run). For the optimization problem discussed in this paper, there is no known true Pareto front, so a reference set is used to calculate the performance metric. The reference set represented as
To demonstrate the effectiveness of the algorithm, the parameter function evaluation is used, and its budget is set to 10000. Since the function evaluation always consumes most of the time of the algorithm for the engineering optimization problems, and the overall number of function evaluations is high.
Figure
Curve of average convergence metric with 10000 function evaluations.
Figure
Curve of average set coverage value with 10000 function evaluations.
In order to intuitively compare the distribution of the solutions obtained by the two algorithms, Figures
Pareto fronts of NSGAII and NSGACMA at the 2500th function evaluation.
Pareto fronts of NSGAII and NSGACMA at the 10000th function evaluation.
It can be seen from Table
A set of Pareto optimal design results.
Serial number 







1  5  23  570.67  2.216  69.85  1682.38 
2  6  19  543.02  2.452  70.80  1665.53 
3  6  18  588.00  2.538  71.64  1636.41 
4  6  19  597.86  2.489  72.62  1609.73 
5  6  19  554.11  2.513  73.83  1574.07 
6  6  19  538.23  2.532  74.71  1546.59 
7  6  19  592.44  2.544  75.56  1532.94 
8  6  19  549.88  2.572  76.87  1492.68 
9  6  19  486.59  2.586  77.41  1472.08 
10  5  16  599.44  2.850  78.93  1435.91 
11  5  16  594.87  2.867  79.70  1417.04 
12  5  16  575.60  2.889  80.88  1393.91 
13  5  16  535.82  2.901  81.51  1382.52 
14  5  16  543.09  2.928  82.76  1351.21 
15  5  16  537.47  2.936  83.15  1343.45 
In this paper, the multiobjective optimization design of drytype aircore reactor made of rectangular crosssection litzwire is studied considering the minimal production cost as well as the minimal operation cost simultaneously. A modified nondominated sorting genetic algorithmII (called NSGACMA) is then proposed to solve this multiobjective optimal problem. Within the NSGACMA, the global exploration is done by NSGAII and the local exploitation by the local search strategy based on CMAES to improve the search ability and accelerate the convergence speed. In order to guarantee the search efficiency, the local search is applied to nondominated individuals in a given generation and quantity. In addition, an external archive is established for improving the evolutionary efficiency. From the simulation results, it is clearly seen that NSGACMA has smaller convergence metric value and more nondominated solutions which dominate the nondominated solutions obtained by NSGAII. This implies that NSGACMA has better search ability and it obtains better Pareto front than NSGAII. For its promising performance, the NSGACMA algorithm is certainly more suitable and effective than NSGAII for solving the multiobjective design problems of drytype aircore reactor. Also, it is one of the efficient potential candidates in solving other complicated multiobjective problems.
The authors declare that there is no conflict of interests regarding the publication of this paper.