CN106529085A - Mathematical Model of Motor and Optimization Method of Permanent Magnet Synchronous Linear Motor - Google Patents
Mathematical Model of Motor and Optimization Method of Permanent Magnet Synchronous Linear Motor Download PDFInfo
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Abstract
The invention relates to a mathematical model of a motor and an optimization method of a permanent magnet synchronous linear motor, wherein the optimization method comprises the following steps: performing cross operation optimization on the width of the magnetic pole of the motor by combining a PSO algorithm and a genetic algorithm; the optimization algorithm and the optimization method overcome the defects that the motor in the prior art has a complex structure, a large volume and high cost, the motor generates large oscillation during operation, and the motor has low operation rotating speed, poor reliability and the like, are favorable for enabling the no-load counter electromotive force waveform of the motor to be close to sine, reducing the thrust fluctuation of the linear motor and optimizing the performance of the motor.
Description
Technical Field
The invention relates to a method for optimizing a linear motor based on Particle Swarm Optimization (PSO) combined with genetic algorithm cross operation, which optimizes the performance of the linear motor by applying the algorithm to enable the no-load counter electromotive force of the motor to be close to a sine waveform so as to reduce the thrust fluctuation of the motor.
Background
In recent years, linear motors have been rapidly developed and widely used in various fields, and have gradually replaced common servo drive systems in some high-speed machining fields. The linear motor directly generates electromagnetic thrust through electric energy, has the advantages of simple structure, small size, large thrust, low loss, high dynamic response speed and the like, and the performance of the linear motor is greatly influenced by the fluctuation of the thrust, so that the motor can generate overlarge vibration in the operation process, and the servo operation performance of the motor is deteriorated. In order to improve the driving performance, reducing the thrust fluctuation of the linear motor is a part of the linear motor which is in urgent need of improvement.
The invention mainly researches the influence of the no-load counter electromotive force waveform of the linear motor on the thrust fluctuation of the linear motor.
The PSO algorithm is derived from the research on the bird predation behavior and is a general heuristic search technology. And searching the optimal solution particles in the population through iterative updating of the speed positions of the particles in the population.
Disclosure of Invention
The invention aims to provide a mathematical model of a motor, so as to facilitate structural optimization of the motor.
In order to solve the above technical problem, the present invention provides a mathematical model of an electric machine, which is represented by an equivalent magnetization spatial distribution function m (x) of the electric machine, that is, the mathematical model
Wherein,
in the formula: b isrResidual magnetization of permanent magnet, mu0Is air permeability, τmIs the width of the permanent magnet, tau is the polar distance of the permanent magnet, mnIs an intermediate variable, and n is the number of magnetic poles of the motor.
In another aspect, the present invention further provides an optimization algorithm, including:
and carrying out cross operation optimization on the particles by combining a PSO algorithm and a genetic algorithm.
Further, the method for optimizing the cross operation of the particle by combining the PSO algorithm and the genetic algorithm comprises the following steps:
the particle velocity is updated as follows:
v＝w×v+c1×rand()×(pBest-present)+c2×rand()×(gBest-present)；
the particle position is updated as follows: present + v;
expression for inertial weight:
in the formula: v is the current velocity of the particle; the present is the current position of the particle;
rand () is a random constant between 0 and 1;
pBest and gBest are an individual extreme value and a global extreme value of a certain particle respectively;
c1、c2acceleration weights respectively corresponding to the individual extreme value and the global extreme value of a certain particle;
w is the inertial weight; the present is the current position of the particle;
wmax、wminrespectively representing the maximum value and the minimum value of the inertia weight;
K'maxis the maximum number of iteration steps; k' is the current iteration step number;
and (5) searching a global extreme value in the population through multiple times of optimization.
In a third aspect, the invention further provides an optimization method of the permanent magnet synchronous linear motor, so as to overcome the defect that motor oscillation is brought by thrust fluctuation in the prior art.
The optimization method of the permanent magnet synchronous linear motor comprises the following steps: and performing cross operation optimization on the width of the magnetic pole of the motor by combining a PSO algorithm and a genetic algorithm.
Further, the method for optimizing the cross operation of the magnetic pole width of the motor by combining the PSO algorithm and the genetic algorithm comprises the following steps:
step S1, initializing setting;
step S2, establishing an expression of the change of the air gap flux density along with the position angle theta;
step S3, the magnetic pole width is optimized by the crossover operation.
Further, the method for initializing the setting in step S1 includes:
the permanent magnet synchronous linear motor is suitable for adopting an asymmetric magnetic pole structure, and the width of the magnetic pole is theta1With adjacent poles of width theta2、θ3、……、θnN is the number of magnetic poles of the motor, and theta is selected1、θ2For reference variable, letIs the ratio of the width of a pole of the motor to the width of its adjacent pole, and k is a positive number.
Further, the method for establishing the expression of the variation of the air gap flux density with the position angle θ in the step S2 includes:
performing Fourier decomposition on the air gap flux density waveform, wherein the expression of the air gap flux density along with the change of a position angle theta is as follows:
in the formula (11), anThe amplitude of the nth harmonic is expressed as follows:
in the formula (12), B1、B2Respectively, the air gap flux density amplitude under different polarity magnetic poles, n is a positive integer
θ1、θ2The relationship between (A) and (B) is as shown in formula (10), i.e. theta1＝kθ2(13)。
Further, the step S3 optimizes the magnetic pole width by a crossing operation, i.e. the step S3 is performed by a crossing operation
Taking the width of a magnetic pole of the motor as an optimization target;
taking the k value as a speed variable in a particle swarm algorithm, taking a magnetic pole shoe as a dependent variable, searching an individual extreme value pBest of each independent variable particle through self learning, and obtaining a global extreme value gBest through learning cross comparison of other particles in the particle swarm, wherein the global extreme value gBest is the optimal value of the magnetic pole shoe; the proper k value is selected to be the optimal ratio of the width of the magnetic pole of the motor to the width of the adjacent magnetic pole, so that the pole shoe of the magnetic pole obtains the optimal performance, namely the optimal magnetism.
The optimization algorithm and the optimization method have the beneficial effects that the defects that the motor in the prior art is complex in structure, large in size and high in cost, the motor generates large oscillation during operation, the motor has the defects of low operation rotating speed, poor reliability and the like, the no-load counter electromotive force waveform of the motor is close to sine, the thrust fluctuation of the linear motor is reduced, and the performance of the motor is optimized are overcome.
Drawings
The invention is further illustrated with reference to the following figures and examples.
FIG. 1 is a flow chart of the PSO algorithm of the present invention;
fig. 2 is a flowchart of an optimization method of the permanent magnet synchronous linear motor of the present invention.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings. These drawings are simplified schematic views illustrating only the basic structure of the present invention in a schematic manner, and thus show only the constitution related to the present invention.
The mathematical model of the permanent magnet synchronous linear motor is a high-order, nonlinear and strongly coupled multivariable system. The excitation effect of the permanent magnet is analyzed by an equivalent magnetization method, and the equivalent magnetization spatial distribution function M (x) can be represented by Fourier series:
wherein,
in the formula: b isrResidual magnetization of the permanent magnet; mu.s0Air permeability; tau ismIs the width of the permanent magnet; τ is the pole pitch of the permanent magnet, mnIs an intermediate variable.
Establishing a Poisson equation set of an air gap area and a permanent magnet area according to a Maxwell equation set:
in the formula A1、A2The vector magnetic potential of the air gap region and the permanent magnet region, respectively.
The air gap region and the permanent magnet region satisfy the following boundary conditions:
substituting formula (4) for formula (3) yields the air gap flux density:
wherein,
in the formula:An1、Bn1,Tnis an intermediate variable; h ismIs the thickness of the permanent magnet; g is the air gap length.
The PSO algorithm is an intelligent optimization algorithm for simulating bird foraging, whether the optimization variables are optimal solutions or not is judged by iterating the optimization variables into an objective function, the optimization algorithm based on an iteration mode has more obvious advantages in the aspect of finding the optimal solutions for continuous variables, therefore, the variation range of each optimization variable is set, in each iteration process, the particles in a particle swarm determine the motion direction of the next step through self learning and learning of the optimal particles in the swarm, namely, the particles update themselves by tracking two extreme values, one is the optimal solution which can be found by the particles, namely, an individual extreme value pBest, and the other is the optimal solution in a particle swarm, namely, a global extreme value gBest. The specific steps of the algorithm are shown in fig. 1.
Arranged in m-dimensional space, there are n particles, each having coordinates Xi＝(xi1，xi2,…,xim) And has a fitness (usually directly considering the objective function as a fitness of the particles) related to the optimization objective function f (x), while each particle has a respective velocity Vi＝(vi1,vi2,…,vim). For the ith particle, the historical best position it has experienced is noted as Pi＝(pi1,pi2,…,pim) Note that all particles pass through the best position as Pg＝(pg1,pg2,…,pgm)。
The PSO algorithm is described below in conjunction with the crossover operation in the genetic algorithm. And setting pBest and gBest as the historical optimum and the current global optimum of a certain particle respectively, and enabling elements at a certain random corresponding position in the two vectors to be crossed so as to enhance the learning of the historical optimum of the particle to the global optimum, wherein the crossed historical optimum and the current global optimum become pBest and gBest respectively. pBest and bBest simultaneously determine the speed update of the particles, the historical optimum of each particle is not changed, the current global optimum of the whole particle swarm is not changed, and the intersection process only influences the speed update of the particles.
Particle velocity update:
v＝w×v+c1×rand()×(pBest-present)+c2×rand()×(gBest-present) (7)
particle position updating: present ═ present + v (8)
In formulas (7) and (8): v is the particle velocity, present is the current position of the particle, rand () is a random constant between 0 and 1, c1、c2The acceleration weights respectively corresponding to the individual extreme value and the global extreme value of a certain particle are 2 for a normal number; w is the inertia weight, the introduction of the inertia weight enables the global search capability and the local search capability of the PSO algorithm to be obviously improved, the PSO algorithm cannot fall into the local optimal solution, and the present is the current position of the particle.
Expression for inertial weight:
in formula (9): w is amax、wminRespectively representing the maximum and minimum values of the inertial weight, K'maxIs the maximum number of iteration steps; k' is the current iteration step number.
As shown in fig. 2, the above embodiment further provides an optimization method for a permanent magnet synchronous linear motor, so as to overcome the defect that motor oscillation is caused by thrust fluctuation in the prior art.
The optimization method of the permanent magnet synchronous linear motor comprises the following steps: the magnetic pole width of the motor is optimized through the cross operation of a PSO algorithm and a genetic algorithm, and the method comprises the following steps:
step S1, initializing setting;
the invention adopts an asymmetric magnetic pole structure, and the width of the magnetic pole is theta1With adjacent poles of width theta2、θ3、……、θn(n is the number of magnetic poles of the motor), selecting theta1、θ2For reference variable, let(k is a positive number) (10);
step S2, establishing an expression of the variation of the air gap flux density along with the position angle theta, namely
Performing Fourier decomposition on the air gap flux density waveform, wherein the expression of the air gap flux density along with the change of a position angle theta is as follows:
in the formula (11), anThe amplitude of the nth harmonic is expressed as follows:
in the formula (12), B1、B2The magnetic flux density amplitude of the air gap under the magnetic poles with different polarities is shown, and n is a positive integer
θ1、θ2The relationship between (A) and (B) is as described in equation (10): theta1＝kθ2(13)
In the formula (13), k is a positive number, and the magnetic pole width variable is closely related to the value of k.
Step S3, the magnetic pole width is optimized by cross operation
The k value is used as a speed variable in the particle swarm optimization, the pole shoe of the magnetic pole is used as a dependent variable, each independent variable particle searches an individual extreme value pBest of the independent variable particle through self learning, and a global extreme value gBest is obtained through learning cross comparison of other particles in the particle swarm, so that the extreme value gBest is the optimal value of the pole shoe of the magnetic pole.
And selecting a proper k value to obtain the optimal ratio of the width of the magnetic pole of the motor to the width of the adjacent magnetic pole, and determining the optimal performance of the pole shoe of the magnetic pole so as to optimize the magnetism.
The k value determines the width of the magnetic pole of the linear motor, optimizes the structure of the motor, reduces the thrust fluctuation of the motor and enhances the stability of the motor in operation.
In light of the foregoing description of the preferred embodiment of the present invention, many modifications and variations will be apparent to those skilled in the art without departing from the spirit and scope of the invention. The technical scope of the present invention is not limited to the content of the specification, and must be determined according to the scope of the claims.
Claims (8)
1. Mathematical model of an electrical machine, characterized in that it is represented by the equivalent magnetization spatial distribution function m (x) of the electrical machine, i.e. it is a mathematical model
Wherein,
in the formula: b isrResidual magnetization of permanent magnet, mu0Is air permeability, τmIs the width of the permanent magnet, tau is the polar distance of the permanent magnet, mnIs an intermediate variable, and n is the number of magnetic poles of the motor.
2. An optimization algorithm, comprising:
and carrying out cross operation optimization on the particles by combining a PSO algorithm and a genetic algorithm.
3. The optimization algorithm of claim 2,
the method for optimizing the cross operation of the particle by combining the PSO algorithm and the genetic algorithm comprises the following steps:
the particle velocity is updated as follows:
v＝w×v+c1×rand()×(pBest-present)+c2×rand()×(gBest-present)；
the particle position is updated as follows: present + v;
expression for inertial weight:
in the formula: v is the current velocity of the particle; the present is the current position of the particle;
rand () is a random constant between 0 and 1;
pBest and gBest are an individual extreme value and a global extreme value of a certain particle respectively;
c1、c2acceleration weights respectively corresponding to the individual extreme value and the global extreme value of a certain particle;
w is the inertial weight; the present is the current position of the particle;
wmax、wminrespectively representing the maximum value and the minimum value of the inertia weight;
K'maxis the maximum number of iteration steps; k' is the current iteration step number;
and (5) searching a global extreme value in the population through multiple times of optimization.
4. A method for optimizing a permanent magnet synchronous linear motor is characterized in that,
and performing cross operation optimization on the width of the magnetic pole of the motor by combining a PSO algorithm and a genetic algorithm.
5. The optimization method according to claim 4,
the method for optimizing the cross operation of the width of the magnetic pole of the motor by combining the PSO algorithm and the genetic algorithm comprises the following steps:
step S1, initializing setting;
step S2, establishing an expression of the change of the air gap flux density along with the position angle theta;
step S3, the magnetic pole width is optimized by the crossover operation.
6. The optimization method according to claim 5,
the method for initializing the setting in step S1 includes:
the permanent magnet synchronous linear motor is suitable for adopting an asymmetric magnetic pole structure, and the width of the magnetic pole is theta1With adjacent poles of width theta2、θ3、……、θnN is the number of magnetic poles of the motor, and theta is selected1、θ2For reference variable, letIs the ratio of the width of a pole of the motor to the width of its adjacent pole, and k is a positive number.
7. The optimization method according to claim 6,
the method for establishing the expression of the variation of the air gap flux density with the position angle theta in the step S2 comprises the following steps:
performing Fourier decomposition on the air gap flux density waveform, wherein the expression of the air gap flux density along with the change of a position angle theta is as follows:
in the formula (11), anThe amplitude of the nth harmonic is expressed as follows:
in the formula (12), B1、B2Air gap flux density amplitudes under different polarity magnetic poles are respectively;
θ1、θ2the relationship between (A) and (B) is as shown in formula (10), i.e. theta1＝kθ2(13)。
8. The optimization method according to claim 7,
in the step S3, the magnetic pole width is optimized by the crossover operation, i.e. the magnetic pole width is optimized
Taking the width of a magnetic pole of the motor as an optimization target;
taking the k value as a speed variable in a particle swarm algorithm, taking a magnetic pole shoe as a dependent variable, searching an individual extreme value pBest of each independent variable particle through self learning, and obtaining a global extreme value gBest through learning cross comparison of other particles in the particle swarm, wherein the global extreme value gBest is the optimal value of the magnetic pole shoe;
the proper k value is selected to be the optimal ratio of the width of the magnetic pole of the motor to the width of the adjacent magnetic pole, so that the pole shoe of the magnetic pole obtains the optimal performance, namely the optimal magnetism.
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| CN108021039A (en) * | 2017-12-23 | 2018-05-11 | 西安交通大学 | A kind of Linear motor feeding system electromechanics integrated modelling approach |
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| 李全武 等: "改善永磁同步电机反电势波形的磁极宽度优化方法", 《万方数据库期刊库》 * |
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| CN108021039A (en) * | 2017-12-23 | 2018-05-11 | 西安交通大学 | A kind of Linear motor feeding system electromechanics integrated modelling approach |
| CN108021039B (en) * | 2017-12-23 | 2020-03-17 | 西安交通大学 | Electromechanical integration modeling method for linear motor feeding system |
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Inventor after: Zhang Fuchun Inventor after: Yang Yifei Inventor after: Cheng Limin Inventor after: Gu Lipeng Inventor after: Wang Lina Inventor before: Zhang Fuchun Inventor before: Yang Yifei Inventor before: Cheng Li Inventor before: Gu Lipeng Inventor before: Wang Lina |
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Application publication date: 20170322 |