^{1}

^{*}

^{2}

In this paper, the main objective is to identify the parameters of motors, which includes a brushless direct current (BLDC) motor and an induction motor. The motor systems are dynamically formulated by the mechanical and electrical equations. The real-coded genetic algorithm (RGA) is adopted to identify all parameters of motors, and the standard genetic algorithm (SRGA) and various adaptive genetic algorithm (ARGAs) are compared in the rotational angular speeds and fitness values, which are the inverse of square differences of angular speeds. From numerical simulations and experimental results, it is found that the SRGA and ARGA are feasible, the ARGA can effectively solve the problems with slow convergent speed and premature phenomenon, and is more accurate in identifying system’s parameters than the SRGA. From the comparisons of the ARGAs in identifying parameters of motors, the best ARGA method is obtained and could be applied to any other mechatronic systems.

The mechanical commutator of the brushless direct current (BLDC) motor [

Many literatures [

Recently, due to rapid improvements in power devices and microelectronics, the field-oriented control and feedback linearization techniques have increased induction motor drives for high-performance applications possible [

Genetic algorithm (GA) is a searching process based on natural selection, and now is used as a tool for searching the large, poorly understood spaces that arise in many application areas of science and engineering [

In this paper, the real-coded encoding scheme of fixed length to randomly generate the initial population is used by means of roulette wheel. This standard genetic features by use of only three basic genetic operators: selection operator, crossover operator and mutation operator, simplifies the process of genetic evolution, and easy to be understood. However, the fitness value may occur slow convergent speed and premature phenomenon in a traditional real-coded genetic algorithms (RGA) [

This paper is organized as follows. Firstly, the BLDC and induction motors’ equations are established. Secondly, the algorithms in the SRGA and ARGAs are presented and discussed. Thirdly, comparisons between the SRGA and ARGA in numerical simulations and experimental results are discussed and it is concluded that the ARGAs are better than SRGA.

The RGA [

It is important that crossover probability and mutation probability are set for genetic algorithms, the improper settings will cause falling into local optimum algorithms in search and the premature convergence. Therefore, an efficient method for a fast setting is essential. For this point, a mechanism to adjust the crossover probability and mutation probability according to the algorithmic performance is considered [

In Equations (4) and (5), the crossover probability will be reduced to preserve excellent chromosomes; on the contrary it will be added to evolutionary excellent chromosomes. And then, the mutation probability will be reduced to preserve excellent chromosomes; on the contrary it will increase the diversity of the population and avoid to falling into local optimum [

A. Encoding: The parameters of the BLDC motor and induction motor are composed by real-coded values.

B. Initialization: A collection of individuals is referred to as a population. A population size of 100 is used to generate final segmentation boundaries.

C. Fitness function: The fitness function is adopted as follows.

where n is the total number of sampling point,

D. Selection: In this stage, the expected time of an individual being selected for recombination is proportional to its fitness value relative to the rest of the population. This operation is to achieve a mating pool with the fittest individuals selected according to a probabilistic rule that allows these individuals to be mated into new populations. The selection is carried out by using the roulette wheel method.

E. Crossover and Mutation: The crossover is the breeding of two parents to produce a single child, who has features from both parents and thus may be better or worse than either parent according to the objective function. The primary purpose of mutation is to introduce variation and help bring back some essential genetic traits, and also to avoid the premature convergence of entire feasible space caused by some super chromosomes [

To reduce the premature convergence and improve convergence rate of the SRGA, the adaptive probabilities of crossover and mutation are presented in the ARGA. The probabilities of crossover

where

From Equations (2) and (3), it is known that the adaptive

The GAs have been extensively used in different domains as a type of robust optimization method. However, the GA to demonstrate a more serious question is a premature convergence problem, less capable local optimization, the late slow convergence and can’t guarantee convergence to the global optimal solution and so on. In recent years, many researches [

where,

bility;

As a result, the adaptive

An important problem in usage of the RGA is premature convergence, and the searching process may trap in a local optimum before the global optimum is found. This section employs an ARGA which adjusts mutation probability dynamically based on average square deviation (ASD) of population fitness value, which shows the population diversity to solve the premature problem. From compared analysis, it is shown the proposed ARGA efficiently avoid the premature problem [

The selection operation reduces the diversity of population, the crossover operation does not decrease the diversity of population, and the mutation operation can advance the diversity. Mainly, all these issues produce two effects, the lack of diversity in the population and a disproportionate exploitation or exploration relationship, cause the premature problem [

A. Selection operator: The selection is carried out using the roulette wheel method in this paper.

B. Crossover operator: The crossover operation does not decrease the diversity of population, and the crossover probability is fixed.

C. Mutation operator: This section employs an adaptive method to adjust mutation probability dynamically based on the ASD value. When the ASD decreasing, mutation probability will be increased to advance the population diversity. The relationship between mutation probability and ASD is given as follows.

where

where

In this section, a new GA with two species is proposed for the ARGA. The dual-specie GA composes of two sub-populations that constitute of same size individuals. The sub-populations have different characteristics, such as crossover probability and mutation operator. In one sub-population, the parents with higher similarity are cross with higher probability, and mutate with general mutation operator. In the other sub-population, the parents with smaller similarity are cross with higher probability and mutate with big mutation probability. Therefore, the new algorithm can obtain good exploitation and exploration ability [

Multi-population GA [

For the dual-population GA, one has the following operators:

A. Selection operator: The proposed algorithm establishes two separate sub-populations by random initialization, and then carries on evolution inside single sub-population and migration between two sub-populations. Here, the roulette selection is employed.

B. Crossover operator: The crossover probability is correlated with parents’ similarity. The similarity between two individuals is defined as:

where

The crossover1 emphasized local search ability, and crossover probability

C. Mutation operator: Mutation1 is the normal mutation which proceeds with constant probability. Mutation 2 needs to have ability to robustly explore the solution space and to escape from local peak. The probability

of mutation 2 is adaptive mutation probability, and the probability value

where

The different four ARGAs will be applied to the BLDC and induction motors. At first, it is needed to show the governing equations of the motor system, and find out what are parameters to be identified.

The BLDC motor [

A commonly used second-order linear model for a BLDC motor [

where

The field-oriented induction motor drive can be applied for high-performance industrial applications, and the controllers implemented in induction motor drives are generally based on the system mathematical model. The parameter identification in a rotation rotor is very useful in monitoring and testing a high-power induction motor drive, and then its performance depends heavily on the motor parameters [

The complete electrical and mechanical models [

where

state variables and the electrical voltages

In the numerical simulations, the input voltage is defined as follows:

where

time. In this paper,

In order to investigate the ARGAs, compare them and find the best one for system identification of the electrical fan, the parameters

From the electrical input voltages and rotation output speed, by using the ARGAs and SRGA the identified parameters and fitness values can be obtained as shown in

From comparisons in

In an induction motor, any vector in a rotating coordinate can be described as follows:

According to Euler’s formula, the three-phase part can be rewritten as:

Therefore, the relation formula can be obtained as follows:

Parameters | Assigned | Feasible Domains | Identified Values / Error Percentages | ||||
---|---|---|---|---|---|---|---|

SRGA | ARGA (M1) | ARGA (M2) | ARGA (M3) | ARGA (M4) | |||

2 | 0 - 4 | 2.78/39.90% | 1.92/4.20% | 1.49/25.37% | 1.53/23.41% | 1.77/11.59% | |

4 | 0 - 8 | 3.64/8.91% | 4.11/2.85% | 3.94/1.41% | 3.59/10.32% | 4.80/19.88% | |

1 | 0 - 2 | 0.80/20.50% | 0.68/31.60% | 0.52/47.90% | 0.69/31.50% | 0.39/60.90% | |

1 | 0 - 2 | 1.18/18.00% | 1.00/0.00% | 1.02/1.90% | 1.23/23.40% | 0.81/19.90% | |

0.15 | 0 - 0.3 | 0.14/8.80% | 0.15/1.05% | 0.16/3.79% | 0.15/2.15% | 0.16/4.15% | |

1 | 0 - 2 | 0.80/20.00% | 1.00/0.00% | 1.10/10.00% | 1.00/0.00% | 0.90/10.00% | |

Fitness value | 1.658 | 2.540 | 2.314 | 2.249 | 2.391 | ||

Error % of Rotation Speed | 0.51% | 0.20% | 0.29% | 0.31% | 0.30% | ||

Convergence Generation | 994 | 626 | 812 | 790 | 864 |

The stator transformation formula between the three-phase coordinate and d-q axis is shown as follows:

If there is a voltage amplitude

The input electrical voltages

The input voltage is with alternative current (AC) and is shown as Equation (24). If the fixed frequency is defined as

where

In order to investigate the ARGAs for system identification of the electrical fan, the parameters

From the electrical input voltages and rotational output speed, the identified parameters and fitness values by using the ARGAs and SRGA can also be obtained and shown in

Assigned | Feasible Domains | Identified Values/Error Percentages | |||||
---|---|---|---|---|---|---|---|

SRGA | ARGA (M1) | ARGA (M2) | ARGA (M3) | ARGA (M4) | |||

83 | 82 - 84 | 82.41/0.71% | 83.30/0.36% | 83.61/0.74% | 82.74/0.31% | 83.64/0.77% | |

53 | 52 - 54 | 52.30/1.32% | 52.00/1.88% | 53.87/1.54% | 52.17/1.56% | 53.99/1.86% | |

86 | 70 - 90 | 72.82/15.33% | 86.51/0.59% | 84.81/1.85% | 86.65/0.76% | 85.49/0.59% | |

86 | 70 - 90 | 75.00/12.82% | 85.28/0.83% | 84.85/1.34% | 87.28/1.49% | 84.81/1.38% | |

82 | 70 - 90 | 79.71/2.79% | 83.11/1.35% | 81.76/0.29% | 79.62/2.90% | 80.81/1.45% | |

33 | 20 - 40 | 33.96/2.91% | 33.83/2.52% | 34.18/3.57% | 32.14/2.58% | 33.25/0.77% | |

55 | 40 - 60 | 56.40/2.55% | 55.16/0.28% | 54.47/0.96% | 55.50/0.91% | 53.95/1.91% | |

Fitness value | . | . | 392.37 | 1178.70 | 1049.55 | 1098.13 | 1085.16 |

Error % of rotation speed | . | . | 0.1463% | 0.0248% | 0.0758% | 0.0956% | 0.0204% |

Convergence Generation | . | . | 368 | 130 | 317 | 291 | 271 |

fitness values of the ARGAs and SRGA are compared in

From comparisons in

In experiments, the electrical input voltage and rotation output speed of a real motor are to be measured, and the ARGAs are to be employed to obtain system’s parameters.

In experimental results, three different input voltages are given to the BLDC motor for system’s identification. From numerical simulations, the method1 of ARGA, which not only accurately search for parameters, but also has small generation number inquick convergence, is the best method to identify parameters, and is also applied to the experimental system.

In order to compare experimental results, three different input voltages are employed by hand. The input voltages are ascendant before 1.5 sec and the stable after 1.5 sec, and shown in

curves of the experimental and identified speeds are compared. In ^{th} generation.

The feasible domains and identified parameters by method1 of ARGA are shown in

The electrical input voltages and rotation output speeds of a real induction motor are measured, and the ARGAs are employed to identify system’s parameters in experiments. The experimental setup is shown in

Parameters | Feasible domains | Identified values for a BLDC motor | ||
---|---|---|---|---|

10.6V | 11.5V | 12V | ||

1 - 2 | 1.12 | 1.06 | 1.13 | |

4 - 5 | 4.06 | 4.03 | 4.65 | |

1 - 10 | 1.83 | 2.77 | 2.79 | |

1 - 10 | 1.67 | 1.07 | 1.58 | |

2.5 - 5.5 | 4.07 | 4.61 | 4.09 | |

0 - 2 | 0.00 | 1.14 | 0.50 | |

Fitness Values | . | 1.26 | 1.67 | 1.18 |

The input voltages are given to the induction motor for system’s identification in experimental results. From numerical simulations, the method 1 of ARGA, which not only accurately search for parameters but also has small generation number in quick convergence, is the best method to identify parameters, and is also applied to the experimental identification.

In order to obtain experimental results,

seen the rotation speeds of the induction motor obtained from LabVIEW are low before 10 sec, and the induction is unstable during this interval. Therefore, the identification is performed after 10 sec when the system is stable.

centages are very small, and the errors of the ARGA are smaller than the SRGA. The fitness value with respect to total number of generations is shown in ^{th} generation. It is seen that the ARGA has faster convergence near the 50^{th} generation and higher fitness value.

In order to validate the identified parameters by the SRGA and ARGA in Equation (16), the input voltage as an exponential function in experiments is taken as follows:

where

voltages into (16), the rotation speeds are obtained. The rotation speeds for these three voltages are shown in Figures 8(b)-(d), respectively.

The electric currents

The feasible domains and the identified parameters between the SRGA and method 1 of ARGA are compared in

Feasible domains | Identified values for an induction motor | ||
---|---|---|---|

SRGA | ARGA | ||

0.0 - 1.5 | 0.507 | 0.489 | |

0.0 - 1.5 | 0.689 | 0.720 | |

0.0 - 0.5 | 0.186 | 0.202 | |

0.0 - 0.5 | 0.302 | 0.276 | |

0.0 - 0.5 | 0.218 | 0.215 | |

0.0 - 0.5 | 0.061 | 0.085 | |

0.0 - 0.5 | 0.021 | 0.020 | |

Fitness value | . | 0.0088 | 0.0097 |

Convergence generation | . | 191 | 53 |

is bigger than the SRGA, and it means the ARGA parameters are more accurate and correct. For the convergence in generations, the ARGA converges at the 53thgeneration, and is faster than the SRGA.

This paper attempts to improve crossover and mutation operators in the traditional genetic algorithm by the adaptive technique. Effectiveness of the algorithm in identifying system’s parameters is verified by the BLDC motor and induction motor. It is found that the SRGA and ARGA methods are feasible to system identification. The results show that the ARGA is found to have the faster convergence and the larger fitness value than the SRGA. In numerical simulations, the ARGAs with the identified parameters’ errors percentages are less than 5% with respect to the assigned parameters. In this paper, method 1 of ARGA is found to be the best one to identify parameters of the BLDC and induction motors, and some experimental results are also compared.

The financial support from Ministry of Science and Technology of the Republic of China (MOST103-2221- E-327-009-MY3) is gratefully acknowledged.

Rong-FongFung,Chun-HungLin, (2015) Adaptive Real-Coded Genetic Algorithm for Identifying Motor Systems. Modern Mechanical Engineering,05,69-86. doi: 10.4236/mme.2015.53007