1. Introduction
Flywheel energy storage (FES) is a physical energy storage method that converts electrical energy into the high-speed rotating kinetic energy of the flywheel, which has the advantages of high power density, fast response time, long service life and simple maintenance, and has broad application prospects in distributed power sources, hybrid vehicles, aerospace and other fields [
1,
2,
3]. However, there are still many technical difficulties that limit its application in engineering practice, among which the problems of suspension support and high-speed operation are particularly prominent. Traditional FES is mostly supported by magnetic suspension bearings to reduce system operating losses, but the structure has a large axial length, low critical speed, complex structure, and high maintenance costs [
4]. On the basis of retaining the excellent characteristics of the magnetic bearing and the reluctance machine, the bearingless switch reluctance machine further improves the high-speed performance and operating efficiency through the active control of its suspension force. It is introduced into the FES to form a bearingless flywheel machine, which can reduce system loss and volume, improve suspension performance, critical speed and power density.
In the 1990s, the Japanese scholars Chiba A and Takemoto M first carried out related research and proposed a typical 12/8 dual-winding structure [
5,
6]. The machine adopts a double winding structure, and its air gap magnetic field is generated by the combined action of the current-carrying main windings and the suspension windings, which makes the suspension force and electromagnetic torque of this machine exhibit a strong electromagnetic coupling characteristic. Besides this, there is a dead zone in the suspension force and its control is difficult, which has become the main bottleneck restricting its entry into engineering applications. Recently, some scholars have tried to weaken the electromagnetic coupling from the perspective of the machine structure to achieve the high-speed and stable operation of the machine and have proposed various structures such as dual stators [
7,
8], hybrid rotors [
9,
10], hybrid stators [
11,
12], and permanent magnet bias [
13,
14]. The proposal of these new structures effectively weakened the electromagnetic coupling problem. However, when used for an FES system, there are still many shortcomings. For example, the dual stator structure does not easily integrate the machine and flywheel, and the windings of the internal and external stator structure experience serious temperature rise and heat dissipation, which leads to low efficiency when the machine is running at high speed. The hybrid rotor structure causes the axial length of the machine to still be long, and the critical speed of the flywheel is limited. The hybrid stator and the permanent magnet bias structure adopt the outer stator structure, the machine is not easy to integrate with the flywheel and the permanent magnet is on the outer stator side, which consumes more permanent magnetic materials and is not cost-effective. In [
15], the authors proposed an axial split-phase topology, the structure of which uses its two-phase suspension windings distributed axially to achieve four radial degrees of freedom suspension of the rotor, greatly reducing the axial length of the mechanical or electromagnetic bearings at both ends and increasing the critical speed. At the same time, the inner stator structure is adopted, and the machine rotor and the flywheel are integrated, which reduces the size of the whole machine and further increases the stiffness of the flywheel rotor. However, the bias magnetic flux from the permanent magnet of this machine needs to frequently pass through the insulation layer between the silicon steel sheets, the axial magnetic path loss is large, and the utilization rate of the permanent magnet is low. Besides this, its stator adopts a rectangular tooth, which makes it difficult to wind more windings in a limited radial space, and the output power density of the machine is unsatisfactory. This paper proposes a novel axially split-phase bearingless flywheel machine with magnetic sleeve and pole-shoe tooth. By adopting a pole-shoe tooth, the slot space of the machine torque and suspension poles is effectively increased for more windings, as with the torque and suspension output. At the same time, a magnetically permeable sleeve is added to increase the utilization of permanent magnets and further enhance the suspension performance.
Previous studies have shown that the tooth profile parameters of machines have a significant effect on the output performance of machines, and optimizing the stator and rotor structure in a limited radial space to improve the radial space utilization and output performance has attracted the attention of many scholars [
16,
17,
18]. However, due to the nonlinear coupling of the internal magnetic field, it is difficult to apply traditional analytical formulas directly, so scholars have tried different optimization methods, such as the extreme learning machine (ELM) [
19], least squares support vector machine [
20], response surface methodology (RSM) [
21] and chaos harmony search [
22] to achieve the optimal selection of parameters, all of which have achieved certain optimization effects, but there are still insufficient practical applications. For example, the ELM needs to directly use the finite element model multiple times to obtain the relationship between the machine performance and key structural parameters, and the calculation efficiency is low. The support vector machine has low modeling accuracy under small amounts of sample data, and the traditional response surface methodology has multiple sets of optimal solutions and verification is tedious and subjective, while the selection of parameters in chaos harmony search, such as the harmony memory considering rate and pitch adjusting rate, lacks a theoretical basis; thus, the parameter values are subjectively blind.
In summary, this paper proposes a novel machine structure and tooth profile parameter optimization method aiming at solving the shortcomings of the existing machine structure and parameter optimization method. Based on the basic structure and working principle of the proposed machine, the three-dimensional finite element analysis (FEA) model of the machine is established, and the Box–Behnken design method for finite element simulation is utilized; then, the RSM and comprehensive objective optimization function are adopted to construct the average torque and suspension force of the machine. Furthermore, the differential evolution (DE) algorithm is introduced to find the optimal solution of this objective function; that is, the optimal tooth profile parameter combination. Finally, the machines before and after the optimization are compared and analyzed to verify the tooth profile optimization method, and a prototype is designed for further experimental exploration.
3. Tooth Profile Parameter Optimization Principle and Process
3.1. Response Surface Methodology
The essence of the RSM is to use the least square method to approximately establish a polynomial with a clear expression to express the implicit functional function between each response and the design variables, thereby facilitating the use of other excellent optimization algorithms to solve practical optimization problems [
26], assuming the functional relationship between the optimization target
and the selected design variables
is as follows:
where
represents the selected
i-th basis function, and
is the error. Assuming that the
m design variables are used for experimental design and
n experiments are performed, the response under
n corresponding parameter combinations can be obtained, which is recorded as:
where
, in which
,
is the value of the
j-th basis function in the
i-th experiment.
The matrix form can be expressed as
which can be abbreviated as
where
,
,
.
The estimated value
of
can be calculated using the least square method:
The fitted model of the final response surface function can be expressed as
where
is the
i-th row corresponding element of the column vector
.
Studies have shown that when the independent variable changes in a relatively small interval, a low-degree polynomial fitting can obtain a better approximate mathematical model [
17]. In engineering practice, if the response surface function is a polynomial of first degree, then Equation (6) is
where
m is the number of selected design variables. Similarly, the widely used second-order response surface function can be written as
3.2. Differential Evolution Algorithm
The DE algorithm is an emerging evolutionary computational algorithm. It performs a random heuristic search by simulating the “survival of the fittest” competition strategy in natural biological populations to approach or reach the global optimal solution of the optimization problem, which has the advantages of fewer undetermined coefficients, good robustness and global search, and has gained widespread attention in the field of parameter optimization [
27,
28,
29]. The flow chart of the DE algorithm is shown in
Figure 5, and the specific steps are as follows:
(1) Population initialization: Randomly generate
individuals to form the initial population, determine the maximum evolution number
of the population, and let the
i-th individual in the population be
where
,
is the number of individual independent variables, and
,
and
are the upper and lower bounds of the
j-th independent variable of the individual, respectively.
(2) Calculate the fitness values of the population: Bring in and calculate the corresponding fitness values of all individuals in the population and determine whether the current generation number g has reached the maximum evolutionary number : if it has not reached, continue with the subsequent operations; otherwise, this means that the parameter combination obtained is the best, and the optimizing process is ended.
(3) Mutation: Randomly select three different individuals
from the population, multiply the difference between any two individuals by the scaling factor, and add the third individual to obtain the new mutant individual:
where
is the new individual obtained by mutation,
F is the scaling factor,
represents the current generation, and
.
(4) Crossover: Crossover between the new individual
and the previous generation population with a certain probability, and generate a new individual
to increase the diversity of the population:
where
CR is the crossover probability and
is a random integer between [1, d].
(5) Selection: Compare the fitness of the newborn and the previous generation, and select the individual with the best fitness to enter the next iteration:
where
is the selected fitness function.
The above operations are repeated continuously until the maximum evolutionary number and the optimal parameter combination of the selected fitness function can be obtained.
3.3. DE Algorithm to Obtain Optimal Parameters of Response Surface Function
RSM combined with experimental design and mathematical modeling methods can fit the functional relationship between design variables and responses under a small number of experiment iterations and obtain the optimal output response and design variable combination [
30]. However, practice shows that the optimal parameters automatically obtained by the traditional RSM using Design Expert analysis software finally provide multiple sets of design schemes. As the artificial selection of the optimal combination depends on experience and requires comparison and verification one by one, the operation is still tedious. Therefore, based on the approximate response surface function obtained by RSM, this paper obtains the optimal parameter combination by establishing the corresponding multi-objective optimization function and introducing the DE algorithm with a global search capability to avoid blind selection and improve the accuracy of optimal parameter combination, effectively simplifying the experimental verification process.
The detailed process of obtaining the optimal parameter combination of the multi-objective response surface model based on the DE algorithm is described below, and the optimization process is shown in
Figure 6.
(1) Determine the optimization goals;
(2) Select optimization parameter variables and their level factors to establish a response surface experiment schedule;
(3) The finite element simulation obtains the response under each experimental combination and analyzes and establishes the appropriate fitness function;
(4) Initialize the DE algorithm parameters, such as the maximum evolution number , the number of individuals in the population , the cross probability CR, the scaling factor F, the upper and lower bounds of the population individuals , and so on;
(5) The DE algorithm is adopted to calculate the fitness values of various groups based on the fitness function built, and global optimization is performed on the parameters to be optimized;
(6) Determine whether the current algorithm has met the end condition: if it is satisfied, proceed to the next operation; if not, then skip to step (4) and perform the operation again;
(7) Obtain the global optimal parameter combination, verify it by comparison experiments, and determine whether the optimization goal is met: if not, go to step 2) to re-optimize; otherwise, the obtained parameter combination is the selected optimal value.
6. Discussion
Combined with the results of the research in this paper, the authors aim to explore the following research topics in the future.
(1) The actual operation verification of the prototype: At present, further experimental verification is being carried out in an orderly manner. After the commissioning is completed, the actual operating performance of the prototype will be further analyzed and verified.
(2) Improvement of the used RSM: An accurate objective function model is the key to structural parameter optimization. Although the mathematical model between the response and the structural parameters established in this paper has met the corresponding accuracy requirements, it can be seen from
Figure 9 that it can still be further improved. Therefore, we are going to make some improvements to the RSM used in this paper. We intend to use methods such as a dual response surface to establish a higher-precision response surface function and compare the accuracy and effectiveness of the two modeling methods.
(3) Multi-objective optimization of machine performance based on other structural parameters: This article only optimizes the machine stator tooth profile parameters based on the torque and suspension performance. However, during actual operation, other performance parameters of the machine, such as iron loss, torque ripple, and suspension force ripple, will also affect the machine’s stable operation. In order to further improve the high-speed and stable running performance of the machine, the next step is to further analyze and optimize the other corresponding machine structure parameters.