The Equilibrium Optimizer (EO), Grey Wolf Optimizer (GWO), and Whale Optimizer (WO) algorithms are being recently developed for engineering optimization problems. In this paper, the EO, GWO, and WO algorithms are applied individually for a brushless direct current (BLDC) design optimization problem. The EO algorithm is inspired by the models utilized to find the system’s dynamic state and equilibrium state. The GWO and WO algorithms are inspired by the hunting behavior of the wolf and the whale, respectively. The primary purpose of any optimization technique is to find the optimal configuration by maximizing motor efficiency and/or minimizing the total mass. Therefore, two objective functions are being used to achieve these objectives. The first refers to a design with high power output and efficiency. The second is a constraint imposed by the reality that the motor is built into the wheel of the vehicle and, therefore, a lightweight is needed. The EO, GWO, and WOA algorithms are then utilized to optimize the BLDC motor’s design variables to minimize the motor’s total mass or maximize the motor efficiency by simultaneously satisfying the six inequality constraints. The simulation is carried out using MATLAB simulation software, and the simulation results prove the dominance of the proposed algorithms. This paper also suggests an efficient method from the proposed three methods for the BLDC motor design optimization problem.

A DC motor is an electrical machine that transforms direct current electrical energy into mechanical energy. The popular forms depend on the magnetic field forces that produce. The BLDC motors have been implemented for practical applications due to their features, such as more efficient, less noisy operation, and more output power. An integrated inverter or switching power supply is used to power the BLDC motors with a DC electric source, which generates AC power to power the motor. In real-time applications, such as computers, automotive applications, and electronics, they have been used successfully [

The Lagrange multipliers and the Lagrangian function could be used in an improved manner to optimization problems that have equality constraints. It allows you to find the maximum or minimum of a multi-objective function whenever the input parameters are constrained. The reality that Lagrange multiplier approaches are not always complex enables numerical optimization difficult. This can be dealt with by measuring the gradient magnitude since the magnitude zeros must be local minima. The downside of this method is however that it falsely expands the problem’s dimension [

Recently, evolutionary optimization techniques have been used extensively in real-world engineering optimization problems, including mechanical engineering, electrical engineering, thermal engineering, etc. [

In this paper, the optimized values of the motor’s design variables, which satisfies the design settings, can be achieved by a recent single-objective algorithm and applied to the empirical model of BLDC motors. There are two main objectives for the design optimization problem of the BLDC motor: minimization of total mass and maximization of efficiency. To deal with these multi-objective problems, multi-objective algorithms are required; however, a single-objective algorithm is used due to its computation complexity and implementation complexity. But the objective functions are handled separately to discover the optimal variables of the BLDC motor.

The remainder of the paper is organized as follows. Section 2 presents the problem formulation of the BLDC motor design optimization problem. Section 3 discusses the basic versions of EO, GWO, and WO and their application in the BLDC motor optimization problem. The simulation results are discussed in Section 4, and finally, Section 5 concludes the paper.

The BLDC motors are beneficial to conventional DC motors because they are highly powerful and need low maintenance because they do not have any brushes to carry the current. They are also more flexible, primarily due to their torque and speed capabilities. Built on the compact kit of BLDC motors, they are used in various computer and automotive applications [

The MATLAB model is required that can be selected from [_{e}_{s}_{d}_{cs}_{a}

Variables | Definitions | Boundaries |
---|---|---|

_{e} |
Airgap magnetic induction | (0.5–0.76) T |

_{s} |
Stator bore diameter | (0.15–0.33) m |

_{cs} |
Stator back iron induction | (0.6–1.6) T |

_{d} |
Teeth magnetic induction | (0.9–1.8) T |

Conductor’s current density | (2–5)e06 A/m^{2} |

Variables | Definitions | Boundaries |
---|---|---|

External diameter | ||

Internal diameter | ||

Total mass | ||

_{a} |
Temperature | |

Maximum magnetizing current | ||

Determinant utilized in slot height calculation |

Two objective functions for the BLDC motor design problem is given as follows.

Therefore, the objective of the function _{1} is to optimize the motor’s efficiency, which is the same as minimizing the power losses, and the objective of the function _{2} is to minimize the total mass of the motor.

A brief description of implemented meta-heuristic algorithms, such as GWO, WOA, and EO, is discussed in this section of the paper. These algorithms were chosen because they were not previously applied to the BLDC motor design optimization problem.

The GWO was introduced in 2014 and employs the hunting behavior and leadership skill of grey wolves [

where the current iteration is denoted as _{1} and _{2}.

The hunting activity of grey wolves is simulated mathematically because the alpha, beta, and delta know the prey’s possible better position. The first top three results generated so far are therefore saved, and the other search agents (including omegas) are forced to update their positions as per the location of the best search agents. In this regard, the following formulas are proposed.

By attacking the prey when it stops moving, the grey wolves end the chase., as described above. The value of

The WO algorithm was introduced in 2016 has become one of the intelligent meta-heuristic algorithms and is inspired by the hunting mechanism of a humpback whale [

where the current iteration is defined by

where

The idea behind the single-objective Equilibrium Optimizer was introduced in 2020 [

During initialization, EO uses a particle group, in which each particle describes the vector of concentration that includes the solution to the problem. The initial concentration vector is arbitrarily developed using the following formula in the search space.

where,

The update of the concentration allows EO to balance exploration and exploitation fairly.

where

_{2} is a constant to control the capacity to exploit. Another variable

The rate of generation is referred to as

where,

where the random numbers are denoted as _{1} and _{2} and varies between 0 and 1. The vector

The value of

The implementation procedure of the BLDC motor parameter design process for all selected algorithms is discussed as follows.

The initialization is the first step in optimization problems.

For position _{i}

_{i1}, _{i2}, _{i3}, _{i4}, and _{i5} are within the range as specified in

Initialize other parameters for all selected algorithms and adjust the parameters to get the required performance.

Two objective functions, such as _{1} and _{2}, are handled separately in this paper.

Evaluate the search agent’s position and look for the best position, and the position is updated using

The iteration count is updated by

If It is less than

The selected algorithms, such as GWO, WO, and EO, are tested for all the benchmark test functions and few real-world problems. However, none of these algorithms are tested for BLDC wheel motor design problems. Therefore, all the selected algorithms are applied directly to solve this design problem and optimize the design variables by minimizing the motor’s total mass or maximizing the motor’s efficiency. All the algorithms run 10 times for both the objective functions and the control parameters of all algorithms are listed in

S. No. | Parameters | Values |
---|---|---|

1 | Number of search agents | 50 |

2 | Maximum number of iterations | 500 |

3 | Convergence parameter (GWO and WO) | Linear decrease from 2 to 0 |

4 | Constants _{1} and _{2} (EO) |
2 and 1, respectively |

5 | Generation probability (EO) | 0.5 |

All the selected algorithms proved their capability in handling real-world engineering problems. The design of the BLDC motor mentioned in Section 2 is added to test the effectiveness of all algorithms further. For two objective problems, all algorithms also run 10 times, as stated earlier. Two different case studies, such as minimizing the mass and maximizing efficiency, are studied, and simulation is carried out.

Run No. | _{d} |
_{e} |
_{cs} |
_{s} |
^{2}) |
Efficiency |
---|---|---|---|---|---|---|

1 | 1.8000 | 0.6481 | 1.0547 | 0.2036 | 95.3035 | |

2 | 1.8000 | 0.6504 | 1.0954 | 0.2037 | 95.3094 | |

3 | 1.8000 | 0.6473 | 1.0541 | 0.2034 | 95.3024 | |

4 | 1.7912 | 0.6474 | 1.1027 | 0.2042 | 95.3089 | |

5 | 1.8000 | 0.6473 | 1.0539 | 0.2032 | 95.3082 | |

6 | 1.8000 | 0.6468 | 1.0708 | 0.2037 | 95.3098 | |

7 | 1.7977 | 0.6474 | 1.0082 | 0.2035 | 95.3100 | |

8 | ||||||

9 | 1.8000 | 0.6528 | 0.8461 | 0.2012 | 95.2973 | |

10 | 1.8000 | 0.6527 | 1.1371 | 0.2040 | 95.3035 |

^{*}Bold letters indicate the best results

Run No. | _{d} |
_{e} |
_{cs} |
_{s} |
^{2}) |
Efficiency |
---|---|---|---|---|---|---|

1 | 1.3194 | 0.7399 | 1.1645 | 0.2007 | 94.3151 | |

2 | 1.6835 | 0.7255 | 0.8953 | 0.1977 | 95.1365 | |

3 | 1.8000 | 0.7600 | 0.9523 | 0.1937 | 95.1536 | |

4 | 1.6209 | 0.7122 | 0.8176 | 0.1980 | 95.0899 | |

5 | 1.4660 | 0.7600 | 0.7353 | 0.1937 | 94.4536 | |

6 | 1.6936 | 0.7515 | 0.8201 | 0.1943 | 95.0556 | |

7 | 1.3983 | 0.7232 | 0.9098 | 0.1986 | 94.6522 | |

8 | 1.8000 | 0.7600 | 0.7212 | 0.1927 | 94.9887 | |

9 | ||||||

10 | 1.7756 | 0.7600 | 0.9973 | 0.1945 | 95.1348 |

^{*}Bold letters indicate the best results

Run No. | _{d} |
_{e} |
_{cs} |
_{s} |
^{2}) |
Efficiency |
---|---|---|---|---|---|---|

1 | 1.8000 | 0.6479 | 0.9148 | 0.2017 | 95.3172 | |

2 | 1.8000 | 0.6481 | 0.9001 | 0.2014 | 95.3176 | |

3 | 1.8000 | 0.6480 | 0.9045 | 0.2014 | 95.3175 | |

4 | ||||||

5 | 1.8000 | 0.6478 | 0.9287 | 0.2019 | 95.3170 | |

6 | 1.8000 | 0.6481 | 0.8975 | 0.2013 | 95.3177 | |

7 | 1.8000 | 0.6481 | 0.9089 | 0.2014 | 95.3176 | |

8 | 1.8000 | 0.6481 | 0.9173 | 0.2013 | 95.3174 | |

9 | 1.8000 | 0.6481 | 0.9109 | 0.2013 | 95.3176 | |

10 | 1.8000 | 0.6482 | 0.9022 | 0.2012 | 95.3177 |

^{*}Bold letters indicate the best results

From

Run No. | _{d} |
_{e} |
_{cs} |
_{s} |
^{2}) |
Total Mass (Kg) |
---|---|---|---|---|---|---|

1 | ||||||

2 | 1.7990 | 0.6545 | 1.5952 | 0.1886 | 10.5836 | |

3 | 1.8000 | 0.6524 | 1.5666 | 0.1886 | 10.5955 | |

4 | 1.8000 | 0.6564 | 1.6000 | 0.1855 | 10.5997 | |

5 | 1.8000 | 0.6525 | 1.5892 | 0.1893 | 10.5820 | |

6 | 1.8000 | 0.6522 | 1.6000 | 0.1902 | 10.5776 | |

7 | 1.7960 | 0.6533 | 1.5572 | 0.1879 | 10.6111 | |

8 | 1.8000 | 0.6509 | 1.5964 | 0.1929 | 10.5946 | |

9 | 1.8000 | 0.6544 | 1.6000 | 0.1885 | 10.5823 | |

10 | 1.7949 | 0.6492 | 1.5946 | 0.1942 | 10.6133 |

^{*}Bold letters indicate the best results

Run No. | _{d} |
_{e} |
_{cs} |
_{s} |
^{2}) |
Total Mass (Kg) |
---|---|---|---|---|---|---|

1 | 1.8000 | 0.6654 | 1.3681 | 0.1911 | 10.8286 | |

2 | 1.7999 | 0.6512 | 1.5999 | 0.1986 | 10.7011 | |

3 | ||||||

4 | 1.7791 | 0.6635 | 1.2312 | 0.1981 | 11.1998 | |

5 | 1.7540 | 0.7313 | 1.4037 | 0.1839 | 11.2242 | |

6 | 1.7495 | 0.7024 | 1.4589 | 0.1956 | 11.2207 | |

7 | 1.7971 | 0.6455 | 1.5075 | 0.2089 | 11.0934 | |

8 | 1.7894 | 0.7377 | 1.5958 | 0.1791 | 11.0671 | |

9 | 1.6677 | 0.6791 | 1.5999 | 0.1972 | 11.0646 | |

10 | 1.6706 | 0.6935 | 1.6000 | 0.1900 | 11.0214 |

^{*}Bold letters indicate the best results

Run No. | _{d} |
_{e} |
_{cs} |
_{s} |
^{2}) |
Total Mass (Kg) |
---|---|---|---|---|---|---|

1 | 1.8000 | 0.6499 | 1.6000 | 0.1919 | 10.5764 | |

2 | 1.7998 | 0.6616 | 1.5965 | 0.1893 | 10.6140 | |

3 | 1.8000 | 0.6511 | 1.6000 | 0.1902 | 10.5699 | |

4 | 1.8000 | 0.6437 | 1.6000 | 0.2008 | 10.7074 | |

5 | 1.8000 | 0.6511 | 1.6000 | 0.1901 | 10.5693 | |

6 | 1.7996 | 0.6526 | 1.6000 | 0.1886 | 10.5721 | |

7 | 1.8000 | 0.6497 | 1.5999 | 0.1922 | 10.5781 | |

8 | ||||||

9 | 1.8000 | 0.6506 | 1.6000 | 0.1908 | 10.5712 | |

10 | 1.8000 | 0.6520 | 1.6000 | 0.1889 | 10.5695 |

^{*}Bold letters indicate the best results

From

The performance of the selected algorithms, such as GWO, WO, and EO, are compared with other metaheuristic algorithms, such as PSO, ACO, and BA. The optimized design variables for both objective functions of the BLDC motor by all algorithms are listed in

Method | PSO | ACO | BA | GWO | WO | EO |
---|---|---|---|---|---|---|

_{s} |
202.1 | 201.2 | 202.2 | 202.4 | 193.9 | |

_{e} |
0.6476 | 0.6481 | 0.6535 | 0.6491 | 0.7598 | |

^{2}) |
2.0417 | 2.0437 | 2.0514 | 2.0022 | 2.3614 | |

_{d} |
1.8 | 1.8 | 1.8 | 1.7983 | 1.8 | |

_{cs} |
0.9298 | 0.8959 | 0.9792 | 1.0331 | 1.0924 | |

95.315 | 95.316 | 95.311 | 95.314 | 95.164 | ||

_{tot} |
15 | 15 | 14.95 | 14.98 | 14.75 | |

Evaluations | 1600 | 1200 | 1590 | 500 | 500 |

^{*}Bold letters indicate the best results

Method | PSO | ACO | BA | GWO | WO | EO |
---|---|---|---|---|---|---|

_{s} |
186.01 | 187.07 | 191.92 | 190.14 | 185.91 | |

_{e} |
0.6992 | 0.6636 | 0.6580 | 0.6514 | 0.6652 | |

^{2}) |
3.2406 | 2.5054 | 3.9728 | 3.837 | 3.673 | |

_{d} |
1.7731 | 1.7466 | 1.7710 | 1.7983 | 1.7982 | |

_{cs} |
1.5334 | 1.6000 | 1.5943 | 1.5970 | 1.5984 | |

94.221 | 94.775 | 94.494 | 93.807 | 93.145 | ||

_{tot} |
11.609 | 12.082 | 10.585 | 10.5770 | 10.6286 | |

Evaluations | 1600 | 1200 | 1590 | 500 | 500 |

^{*}Bold letters indicate the best results

It is observed from

Objective Function | Statistics | PSO | ACO | BA | GWO | WO | EO |
---|---|---|---|---|---|---|---|

1 | Min | −0.9531 | −0.9531 | −0.9531 | −0.953 | −0.9513 | −0.9532 |

Max | −0.9214 | −0.9245 | −0.9298 | −0.9379 | −0.9358 | −0.9355 | |

Mean | −0.9488 | −0.9358 | −0.9414 | −0.9518 | −0.9495 | −0.9527 | |

STD | 0.00754 | 0.00987 | 0.00654 | 0.00209 | 0.00416 | 0.0021 | |

2 | Min | 11.61 | 12.08 | 10.59 | 10.61 | 11.02 | 10.57 |

Max | 14.58 | 14.15 | 14.73 | 13.07 | 13.53 | 14.08 | |

Mean | 10.97 | 11.04 | 10.88 | 10.88 | 11.14 | 10.69 | |

STD | 0.4478 | 0.5878 | 0.4698 | 0.464 | 0.4556 | 0.4137 |

This paper deals with the optimal design of a brushless DC wheel motor, and the main aim of the optimization technique is to maximize the efficiency of the machine or minimize its total mass. An analytical model discussed in the literature was considered to accomplish this purpose, and numerous optimization techniques, such as GWO, WO, and EO, were applied. Two different objective functions are formulated and utilized to optimize the design variable of the BLDC motor. The results obtained from the selected algorithms are compared with the other metaheuristic algorithms, such as PSO, ACO, and BA, and show that the EO algorithm can give optimized and best results for both the objective functions. Also, in terms of Min, Max, Mean, and STD, the EO algorithm can give the lowest values than the other algorithms. It can thus be inferred that the EO algorithm is an effective algorithm to be applied to the problem of BLDC wheel motor design.

For future research, the authors have planned to investigate both the objective functions simultaneously to get the optimal trade-off between the motor mass and motor efficiency by applying the multi-objective versions of GWO, WO, and EO. In addition, the multi-objective versions can also be investigated to handle different uncertainties during the optimization process.

We extend our thankfulness to GMR Institute of Technology, Rajam, Andhra Pradesh, India, for providing the facility and allowing us to validate the performance of the algorithms at the laboratory.