WO2021237848A1 - 一种永磁电机多目标优化的参数化等效磁网络建模方法 - Google Patents

一种永磁电机多目标优化的参数化等效磁网络建模方法 Download PDF

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WO2021237848A1
WO2021237848A1 PCT/CN2020/097904 CN2020097904W WO2021237848A1 WO 2021237848 A1 WO2021237848 A1 WO 2021237848A1 CN 2020097904 W CN2020097904 W CN 2020097904W WO 2021237848 A1 WO2021237848 A1 WO 2021237848A1
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permanent magnet
magnetic
grid
stator
air gap
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PCT/CN2020/097904
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English (en)
French (fr)
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赵文祥
曹东辉
吉敬华
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江苏大学
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Priority to US17/414,964 priority Critical patent/US11373023B2/en
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K41/00—Propulsion systems in which a rigid body is moved along a path due to dynamo-electric interaction between the body and a magnetic field travelling along the path
    • H02K41/02—Linear motors; Sectional motors
    • H02K41/03—Synchronous motors; Motors moving step by step; Reluctance motors
    • H02K41/031—Synchronous motors; Motors moving step by step; Reluctance motors of the permanent magnet type
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/06—Multi-objective optimisation, e.g. Pareto optimisation using simulated annealing [SA], ant colony algorithms or genetic algorithms [GA]
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K15/00—Methods or apparatus specially adapted for manufacturing, assembling, maintaining or repairing of dynamo-electric machines
    • H02K15/02—Methods or apparatus specially adapted for manufacturing, assembling, maintaining or repairing of dynamo-electric machines of stator or rotor bodies
    • H02K15/03—Methods or apparatus specially adapted for manufacturing, assembling, maintaining or repairing of dynamo-electric machines of stator or rotor bodies having permanent magnets
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02K—DYNAMO-ELECTRIC MACHINES
    • H02K2213/00—Specific aspects, not otherwise provided for and not covered by codes H02K2201/00 - H02K2211/00
    • H02K2213/03—Machines characterised by numerical values, ranges, mathematical expressions or similar information

Definitions

  • the invention relates to a parametric equivalent magnetic network modeling method for multi-objective optimization of a permanent magnet motor, which belongs to the field of electromagnetic field calculation.
  • the Equivalent Magnetic Network (EMN) model has the advantages of fast analysis speed and low calculation cost.
  • the accuracy of the EMN model based on the grid method can be comparable to that of the finite element analysis.
  • the EMN model has not yet been innovatively applied.
  • the calculation principle is simple, the accuracy is high, and the simulation time is short, which brings higher design efficiency.
  • the electromagnetic parameters such as the motor torque are quickly solved under multi-variable, and the multi-objective optimization algorithm is used to realize the high-efficiency performance optimization design of the surface-mounted permanent magnet motor.
  • the purpose of the present invention is to provide a parametric equivalent magnetic network modeling method for multi-objective optimization of permanent magnet motors, which mainly includes dynamic grid permeance dissection in the disordered magnetic field line area in the motor, magnetic circuit permeance in the regular magnetic field line area, etc. Effectiveness and the establishment of the magnetic network matrix equations.
  • the parameterized equivalent magnetic network model is used to complete the sensitivity analysis of multiple structural variables and establish the response surface model, and the multi-objective optimization algorithm is used to determine the optimal parameter combination to realize the rapid multi-objective optimization of the permanent magnet motor.
  • the technical solution adopted by the present invention is: a parametric equivalent magnetic network modeling method for multi-objective optimization of surface-mount permanent magnet motors, which includes the following steps:
  • Step 1 Divide the disordered and regular areas of magnetic field lines in the motor
  • Step 2 Construct a dynamic mesh model of the disordered area of magnetic field lines in the motor
  • Step 3 Build a magnetic circuit model in the area of the magnetic field lines in the motor
  • Step 4 Connect the dynamic grid model and the magnetic circuit model to establish a parametric equivalent magnetic network model of the motor;
  • Step 5 Simultaneous nonlinear matrix solving equations, solving the magnetic potential of each node, and further obtaining the torque characteristics of the motor;
  • Step 6 select optimization variables and determine the optimization goal, and use the parameterized equivalent magnetic network model to complete the parameter sensitivity analysis;
  • Step 7 select high-sensitivity variables, and establish response surface models of average torque and torque ripple respectively;
  • Step 8 Substitute the response surface model into the multi-objective optimization algorithm to obtain the Pareto front and determine the optimal parameter combination.
  • the surface mount permanent magnet motor is a 48-slot/44-pole three-phase motor, which is composed of a stator, an air gap, a rotor permanent magnet, and a rotating shaft;
  • the stator includes a stator yoke, stator teeth, and stator poles.
  • Shoe, stator slot and armature winding, the material of the stator core is silicon steel sheet B20AT1500, and the armature winding adopts the centralized winding method of fractional slots;
  • the air gap is between the stator and the rotor, and the thickness of the air gap is 1mm;
  • the rotor is without
  • the iron core cylindrical structure is composed of only permanent magnets and rotating shafts.
  • the surface of the cylindrical rotating shaft is slotted and surface-mounted permanent magnets made of N42UH.
  • the surface-mounted permanent magnets are approximately rectangular in cross section and are uniformly arrayed with Halbach permanent magnets.
  • the main permanent magnets in the array are embedded in the slots on the surface of the rotating shaft to facilitate positioning and installation of auxiliary permanent magnets, reducing the impact of manufacturing tolerances on assembly;
  • the motor shaft is made of non-magnetic material stainless steel 304, which is solid Cylindrical shape with slots for positioning on the surface and coaxial connection with the rotor.
  • the finite element software is used to obtain the magnetic field line distribution cloud map in the motor.
  • the disordered area of the magnetic field lines in the motor is mainly concentrated in the stator pole shoes and air gap, and there are more spatial harmonics in this area; the magnetic field line regular area is mainly concentrated in the stator.
  • the yoke and stator teeth have a regular trend of internal magnetic field lines, and the magnetic flux leakage can be ignored.
  • the rotor shaft adopts non-magnetic materials, so the magnetic field line distribution can be viewed equivalently. In order to show the distribution of magnetic field lines in the air, no modeling is performed.
  • the specific process of dynamic grid modeling for the disordered area of magnetic field lines in the motor is: according to the pole shoe size and air gap size of the motor, a cross-shaped permeance grid with variable length and width is adopted Carry out meshing, taking into account the effect of magnetic leakage, meshing the pole shoe and the air gap between the pole shoe at the same time in the circumferential direction, and the distribution of the two grids is strictly limited to the respective regional boundaries, and the size setting is strictly in accordance with the respective regional specifications
  • the pole shoe structure changes only to affect the grid shape of the corresponding area; for the air gap part modeling, the number and width of the upper pole shoe grid must be kept consistent in the circumferential direction, and the grid height is solely based on the air gap Length judgment.
  • the width of the pole shoe grid changes, the grid in the air gap needs to be updated accordingly to ensure that the magnetic permeability connection relationship between the upper and lower grids is constant.
  • the specific process of modeling the magnetic circuit in the area of the magnetic field line of the motor is: for the stator teeth and the yoke, the general magnetic circuit model is used to perform the flux equivalent treatment, and a single tooth is equivalent It is a single permeance, the stator yoke is equivalent in segments by the number of teeth to establish a connection permeance between the teeth; for the permanent magnet of the rotor, each permanent magnet block is separately equivalent according to the direction of Halbach magnetization. A radial permeance is established in the main permanent magnet, and a tangential permeance is established in the auxiliary permanent magnet. According to the law of the flow direction of the Halbach magnetic field lines, the permanent magnet permeance is connected at the nodes in turn.
  • the bottom node of the stator tooth permeance is sequentially connected to the upper node of the stator pole shoe grid, and the upper node of the air gap grid between the stator pole shoes is vacant; the bottom grid node of the air gap grid is based on The corresponding relationship is connected with the main permanent magnet flux in the Halbach permanent magnet array of the rotor, and the judgment is based on whether the main permanent magnet’s mapping area on the air gap grid is greater than half, and the bottom flux of the air gap grid above the auxiliary permanent magnet is not empty. Connection: To update the position of the motor rotor, the rotation angle needs to be re-judged, and the connection relationship between the air gap grid and the main permanent magnet permeance of the rotor is reset.
  • the core nonlinear parameter BH curve is introduced, the permeability value is obtained through interpolation during iterative calculation, and the permeability is updated by the super relaxation iterative algorithm, and its new value is weighted iteratively with the last calculated value.
  • step 6 select 8 parameter variables of stator yoke thickness, stator tooth width, stator tooth length, pole shoe height, slot width, main permanent magnet length, auxiliary permanent magnet width and pole arc coefficient, and change the magnetic
  • the permeance value of the circuit and the shape of the permeance grid are adjusted.
  • the parameterized equivalent magnetic network model is used to analyze the variable sensitivity of the average torque and torque ripple, and four high-sensitivity variables are selected.
  • the Box-Behnken design design test method is used to sample the selected 4 high-sensitivity variables 29 times.
  • the average torque and rotation of the 29 parameter combination methods are designed respectively.
  • To calculate the torque ripple generate a response surface model of average torque and torque ripple in Design Expert software. The closer the multivariate correlation coefficient R 2 is to 1, and the smaller the p value, the higher the accuracy of the model; further based on R 2 and The p-value adjusts the number of variables in the response surface model accordingly to improve the accuracy of model fitting.
  • the response surface model of average torque and torque ripple is used to import the multi-objective differential evolution algorithm (MODE-RMO) based on the sort mutation operator, and the branch allocation solution is divided through mutation, crossover, and selection. External files, and use the crowded distance to analyze the average distance between the two solutions to ensure that the best individual solution will survive to the next iteration. Finally, the non-dominated solution set Pareto front surface is obtained, and the optimal non-dominated solution is determined using the optimization objective function constraint.
  • MODE-RMO multi-objective differential evolution algorithm
  • the surface-mounted permanent magnet motor is comprehensively modeled using mesh division and magnetic circuit permeance model, while taking into account the high precision of the finite element method and the low time consumption of the magnetic circuit method, which helps Improve the calculation accuracy and efficiency of the equivalent magnetic network.
  • a cross-shaped permeance grid with adjustable length and width is used for partial division.
  • the shape of the grid in the area is equivalently changed, which avoids repeated modeling when the parameters are changed. , Improve the versatility and dynamics of the model.
  • the grid of the air gap area and the grid fixing node of the pole shoe area when the pole shoe structure affects the pole shoe grid, the air gap grid is adjusted with equal width to ensure the stability of the magnetic permeability connection relationship sex;
  • connection relationship between the grid of the air gap area and the permanent magnet of the rotor is judged by the mapping area, which avoids the error of adjusting the air gap permeance according to the mapping area, and helps to improve the accuracy of model calculation.
  • the equivalent magnetic network model with adjustable parameters is used for multi-objective optimization design, the simulation time is short, the design efficiency is high, and the electromagnetic parameters such as the motor torque can be solved quickly under multi-variable, which is beneficial to high-efficiency realization
  • the structure optimization design of the surface mount permanent magnet motor is beneficial to high-efficiency realization
  • Figure 1 is a 2D structure diagram of the motor used in the present invention.
  • Figure 2 is an analysis diagram of magnetic lines of force of the motor used in the present invention.
  • Figure 3 is a structural diagram of the magnetizing direction of the permanent magnet of the motor
  • FIG. 4 is a structural diagram of the magnetic network model of the dynamic mesh division of the present invention.
  • FIG. 5 is a partial enlarged structure diagram of the magnetic network model of the dynamic mesh division of the present invention.
  • Figure 6 is a schematic diagram of the dynamic grid changes when the parameters of the present invention change; (a) the initial grid structure; (b) the parameterized structure adjustment;
  • Figure 7 is a schematic diagram of the structural parameters, sensitivity analysis, response surface and Pareto front surface of the multi-objective optimization process; (a) schematic diagram of the structural parameters; (b) histogram of sensitivity analysis; (c) the result of the response surface model (tooth width and Notch width); (d) Pareto front surface;
  • Figure 8 is a schematic diagram of the comparison of the magnetic network, finite element and experimental results of the motor used in the present invention before and after optimization.
  • Fig. 9 is a flowchart of the modeling method and optimization process of the present invention.
  • Figure 1 is the topological structure diagram of the motor, and 1 is the stator yoke, 2 -1 is the stator teeth; 2-2 is the stator teeth pole shoes, 2-3 is the air gap between the stator teeth pole shoes, 3 is the winding, 4 is the air gap between the stator and rotor, 5-1 is the main part of the Halbach array Permanent magnets, 5-2 is the auxiliary permanent magnet in the Halbach array, 6 is the non-magnetic rotating shaft; the embodiment of the present invention is a 48-slot/44-pole three-phase motor, consisting of four permanent magnets: stator, air gap, rotor, and rotating shaft Part of the composition; the stator contains the stator yoke, stator teeth, stator teeth pole shoes, stator slots and armature windings.
  • the material of the stator core is silicon steel sheet B20AT1500, and the armature winding adopts fractional slot concentrated winding method;
  • the gap is between the stator and the rotor, and the thickness of the air gap is 1mm;
  • the rotor is an iron-core cylindrical structure consisting only of permanent magnets and a rotating shaft.
  • the surface of the cylindrical rotating shaft is slotted and surfaced with a permanent material of N42UH.
  • the surface-mounted permanent magnets are approximately rectangular in cross-section, and are uniformly arranged on the surface of the rotating shaft with a Halbach permanent magnet array.
  • the main permanent magnets in the array are embedded in the slots on the surface of the rotating shaft to facilitate positioning and installation of auxiliary permanent magnets, reducing manufacturing tolerances.
  • the influence of assembly The motor shaft is made of non-magnetic material stainless steel 304, which is a solid cylindrical shape, with slots for positioning on the surface, and is coaxially connected with the rotor.
  • Step 1 Divide the disordered and regular areas of the magnetic field lines in the motor.
  • Fig. 2 is an analysis diagram of magnetic lines of force of a motor used in an embodiment of the present invention. It can be seen from the distribution of magnetic field lines that the disordered area of magnetic field lines in the motor is mainly concentrated in the air gap between the stator tooth pole shoes and the stator and rotor. This area is the key area for electromagnetic energy conversion.
  • the motor magnetic flux is distorted by the uneven air permeability. Therefore, the trend of magnetic field lines is complicated and irregular, and there are a lot of magnetic flux leakage between teeth and space harmonics.
  • the area of magnetic field lines in the motor is mainly concentrated in the stator teeth, stator yoke and rotor permanent magnets. This area is the path through which the magnetic flux guides the air gap.
  • the shape is regular, there is basically no magnetic flux leakage, the magnetic field lines tend to be regular, and the magnetic field lines are all in the same direction.
  • Step 2 Construct a dynamic mesh model of the disordered area of magnetic field lines in the motor.
  • a dynamically adjustable cross-shaped permeance grid is used for division, and the influence of magnetic flux leakage is considered.
  • the air gap between the shoes is meshed, and the distribution of the two grids is strictly limited to their respective regional boundaries.
  • the size setting is strictly in accordance with the respective regional specifications.
  • Fig. 4 is a structural diagram of a magnetic network model of the dynamic mesh division of the present invention
  • Fig. 5 is a partial enlarged structural diagram of the magnetic network model of the dynamic mesh division of the present invention; according to the actual size of the modeling area, while considering the magnetic leakage between teeth And the magnetic field line loop, including pole shoe height, pole shoe length, notch width, air gap outer diameter, air gap height, etc., and choose different meshing schemes in different areas accordingly.
  • the dynamically adjustable cross-shaped permeance grid is the key to modeling.
  • the calculation formulas of the tangential permeance G t and the radial permeance G h are:
  • L a is the axial length
  • w cell is a grid width
  • h cell is a grid height
  • ⁇ 0 is the permeability of vacuum
  • ⁇ r is the relative magnetic permeability of ferromagnetic material.
  • the area where the magnetic field of the stator part is complex is the air gap 2-3 between the pole piece 2-2 of the stator tooth and the pole piece 2-3.
  • the grid height is equal to h 1
  • the grid width is equally divided by the width of the modeling area.
  • the air gap between the pole shoes is a grid with 1 row and 2 columns of equal width
  • the pole shoe is a grid with 1 row and 9 columns of equal width
  • the 9 upper nodes of the grid in the pole shoes are sequentially connected to the lower nodes of the stator tooth 2-1 ,
  • the two upper nodes of the air gap grid between the pole shoes are left unconnected.
  • the air gap 4 is an important place for permanent magnet motor energy conversion, and it is also the most complex area of the magnetic field. Through reasonable design, the air gap 4 is divided into two layers.
  • the number of air gap grids is the same as the number of grids in the lower layer of the stator.
  • the total number is 48 slots/44 poles, which is the least common multiple of 528, which ensures the symmetry of the grid. Equality.
  • the air gap grid and the lower stator grid not only maintain the same number, but also have the same grid width, which ensures that the upper grid of the air gap and the lower grid of the stator are connected correspondingly.
  • the lower grid of the air gap is connected to the permanent magnets of the rotor.
  • the permanent magnet array of Halbach array is used on the rotor, which is composed of main permanent magnet 5-1 and auxiliary permanent magnet 5-2.
  • the main permanent magnet 5-1 is magnetized radially, and the auxiliary permanent magnet 5-2 sprints tangentially to enhance the unilateral air gap magnetic field.
  • the auxiliary permanent magnet 5-2 is not connected to the lower node of the air gap 4 grid, and only the main permanent magnet 5-1 that generates tangential magnetization Connect all the air gap 4 grids within the width of the main permanent magnet 5-1.
  • the grid in the air gap needs to be updated accordingly to ensure that the magnetic permeability connection relationship between the upper and lower grids is constant.
  • Step 3 construct the magnetic circuit model of the regular area of magnetic field lines in the motor.
  • the magnetic field lines in the stator yoke 1 and the stator teeth 2-1 have a regular pattern, basically no magnetic flux leakage, and the magnetic field lines are all in the same direction.
  • the general magnetic circuit permeance model is modeled according to its structural parameters, and the permeance calculation formula is:
  • ⁇ is the magnetic permeability of the iron core
  • S is the cross-sectional area in the magnetizing direction
  • l is the magnetizing length
  • l t is the tooth length
  • ⁇ 0 is the vacuum permeability
  • ⁇ r is the relative permeability of the ferromagnetic material.
  • w is the width of the structure
  • L a is the axial length.
  • the permeance of the magnetic circuit is equivalent.
  • the number of teeth is equivalent to the segmental equivalent, and the connection permeance between the teeth is established.
  • the calculation formula is the same as the calculation formula of the permeance of the tooth; the nodes at both ends of the permeance of the yoke They are respectively connected with the upper end nodes of the correspondingly connected tooth magnetic guides.
  • each permanent magnet block is equivalent to the permeance, the radial permeance is established in the main permanent magnet, and the tangential permeance is established in the auxiliary permanent magnet.
  • the calculation formula is the same In the sub-permeability calculation formula, the permanent magnet is approximately regarded as the permeability of the air, and the relative permeability is set to 1. At the same time, considering the permeance path, according to the law of Halbach magnetic field line flow, the lower end node of the radial permeance of the main permanent magnet is connected with the tangential permeance node of the auxiliary permanent magnet to establish a permeance network interconnected between permanent magnets.
  • Step 4 Connect the dynamic grid model and the magnetic circuit model to establish a parametric equivalent magnetic network model of the motor;
  • the bottom node of the stator tooth flux guide is connected to the upper node of the stator tooth pole shoe grid in turn, the upper node of the air gap grid between the stator teeth pole shoes is empty; the bottom grid node of the air gap grid corresponds to the rotor according to the corresponding relationship
  • the permeance of the main permanent magnet in the Halbach permanent magnet array is connected, and the judgment is based on whether the mapping area of the main permanent magnet on the air gap grid is greater than half, and the permeance at the bottom of the air gap grid above the auxiliary permanent magnet is empty and not connected; the motor rotor To update the position, it is necessary to re-judge the angle of rotation, and reset the connection relationship between the air gap grid and the permeance of the main permanent magnet of the rotor.
  • the permeance model of the stator tooth 2-1 is connected with the corresponding 9 grids in the first row of the tooth under the pole shoe 2-2, and the relationship is fixed.
  • the tooth under the pole shoe The grid of 2-2 is fixedly connected to the grid of air gap 4.
  • the connection relationship within the grid is fixed, and the connection relationship between the grid and the stator pole shoe 2-2 grid is also fixed. Therefore, when the rotor rotates in the circumferential direction of the Z axis, there is only the air gap and the rotor.
  • the permeance connection relationship changes with the rotation angle.
  • determining the connection method of the rotor Halbach permanent magnet array and the air gap grid is the key to establishing the rotating magnetic network model.
  • the nodes on the radial permeance of the main permanent magnet 5-1 are respectively connected to the corresponding permeance in the mapping area, and the judgment is based on whether the mapping area of the main permanent magnet on the air gap grid is greater than half
  • the air gap 4 grid node in the coverage area connected to the main permanent magnet 5-1 will change.
  • the connection relationship between the rotor main permanent magnet's permeance and the air gap grid needs to It is constantly updated during the rotation of the motor.
  • Figure 6 is a schematic diagram of the dynamic grid change when the parameters of the present invention change.
  • the equivalent magnetic network model of the motor is established by combining the grid and the magnetic circuit model, its parameterization function can dynamically adjust the grid size and magnetic circuit permeability Value realization.
  • it can mainly realize the dynamic adjustment of notch width, pole shoe height, air gap length and other variables.
  • the height of the pole shoe increases, the height of the entire row of grids located in the pole shoe layer increases to match the new pole shoe height.
  • the width of the notch becomes larger, which reduces the length of the pole shoe, so The width of the grid in the notch is elongated, and the width of the grid in the pole shoe is shortened.
  • the principle of keeping the grid and the boundary of the modeling area must be ensured, that is, the grid that coincides with the boundary of the modeling area must ensure that the boundary cannot be shifted, and at the same time in the area
  • the height and width of all grids are equally divided, which keeps the equality of the grid model and avoids the drawbacks of repeated modeling during variable adjustment.
  • the grid width in the air gap needs to be adjusted accordingly to ensure that the grid widths of the upper and lower regions are the same, so that the grid permeance is fixedly connected.
  • the parameterized magnetic circuit permeance adjustment it can mainly realize the dynamic adjustment of the stator yoke thickness, stator tooth width, stator tooth length, permanent magnet height, pole arc coefficient and other variables.
  • the trend of magnetic field lines in this area is parameterized directly by adjusting the cross-sectional area or length in the permeance calculation formula. It only changes the permeance value of the magnetic circuit between some nodes, and does not affect the judgment of the node connection relationship.
  • Step 5 Simultaneous nonlinear matrix solving equations, solving the magnetic potential of each node, and further obtaining the torque characteristics of the motor.
  • the complete model of the parameterized equivalent magnetic network of the motor has a total of 4456 nodes. According to the nodal flux law in the magnetic field, the permeance matrix G, the magnetomotive force matrix F, and the flux matrix ⁇ are connected to establish the permeance solution matrix equation. Calculated as follows:
  • the magnetomotive force matrix F in the formula includes the magnetomotive force generated by the permanent magnet and the magnetomotive force of the winding.
  • the magnetomotive force of the winding is set to 0 when there is no load, and its calculation formula is:
  • N is the number of winding turns
  • i is the current injected into the winding
  • B i,j (F(i)-F(j)) ⁇ G(i,j)/S i,j , where B i, j is the magnetic flux density between nodes i and j, F(i) is the magnetomotive force of node i, F(j) is the magnetomotive force of node j, and G(i,j) is between node i and j S i,j is the cross-sectional area between nodes i and j; the BH curve parameters of the ferromagnetic material B20AT1500 used in the motor are introduced, and linear interpolation is used for the BH curve:
  • H is the magnetic field strength
  • H n is the magnetic field strength of the nth iteration
  • H n+1 is the magnetic field strength of the n+1th iteration
  • B is the magnetic flux density
  • B n is the magnetic flux density of the nth iteration
  • ⁇ new iteration for the new magnetic flux density obtained.
  • F (k+1) i is the k+1 iteration magnetomotive force of node i
  • F (k) i is the k iteration magnetomotive force of node i
  • w is the weight coefficient
  • G(i,j) is the node mutual conductance
  • F (k+1) j is the iterative magnetomotive force of node j at the k+1th time
  • F (k) j is the k-th node j
  • ⁇ (i) is the magnetic flux at node i.
  • B is the calculated magnetic induction intensity
  • F(i) is the magnetomotive force of node i
  • F(j) is the magnetomotive force of node j
  • G(i,j) is the permeance between nodes i and j
  • S is the magnetizing direction Cross-sectional area.
  • the electromagnetic characteristics of the motor flux ⁇ , no-load back EMF, load torque T can be calculated, and the calculation formulas are as follows:
  • S is the cross-sectional area in the magnetizing direction
  • N is the number of winding turns
  • B is the magnetic flux density
  • T out is the output torque
  • T ave is the average torque
  • T cog is the cogging torque
  • m is the number of phases
  • I is the input current amplitude
  • ⁇ i is the internal power angle
  • W is the magnetic field energy
  • ⁇ is the position angle.
  • Step 6 select optimization variables and determine the optimization goal, and use the parameterized equivalent magnetic network model to complete the parameter sensitivity analysis.
  • Figure 7(a) is a schematic diagram of the structural parameters of the multi-objective optimization process. Multi-objective optimization of the motor is carried out, and the following 8 parameter variables are selected: stator yoke thickness, stator tooth width, stator tooth length, pole shoe height, slot width, main permanent magnet length, auxiliary permanent magnet width and pole arc coefficient .
  • the pole arc coefficient ⁇ is the ratio of the main permanent magnet radian ⁇ PM1 to a permanent magnet combined radian, ⁇ PM2 is the auxiliary permanent magnet radian, and its calculation formula is:
  • sensitivity analysis is performed on them first, and the high sensitivity parameters that have a higher influence on the average torque and torque ripple are screened out. Based on the output torque of the parameterized magnetic network, the sensitivity analysis is carried out one variable at a time, and the sensitivity index is further generated to indicate the degree of influence of the parameter on the performance.
  • the sensitivity index H(x i ) calculation formula can be expressed as:
  • Ey/x i represents when x i is a constant? Is the average of y, and V(Ey/ xi ) and V(y) are the variances of Ey/xi and y, respectively.
  • the positive and negative indicators of its sensitivity indicate that the design parameters can improve or inhibit performance accordingly.
  • Figure 7(b) is a comparison diagram of single sensitivity and comprehensive sensitivity analysis of selected variables.
  • the sensitivity analysis of the average torque and torque ripple are carried out in their respective value ranges.
  • a comprehensive sensitivity analysis is used to show the impact of a single individual's sensitivity on the overall torque performance.
  • the calculation formula can be expressed as:
  • G (x i ) is the comprehensive sensitivity factor
  • H out (x i ) is the average torque sensitivity index
  • H ri (x i ) is the torque ripple sensitivity index
  • the parameterized equivalent magnetic network model is used to analyze the variable sensitivity of the average torque and torque ripple, and select the high-sensitivity variables.
  • Step 7 select high-sensitivity variables, and establish response surface models of average torque and torque ripple respectively.
  • w t is the stator tooth width
  • B s0 is the stator slot opening
  • w PM2 is the width of the auxiliary permanent magnet
  • ⁇ is the pole arc coefficient
  • Fig. 7(c) is the interaction diagram between the tooth width and the notch width in the obtained response surface model. It can be seen from the figure that the average torque and torque ripple are nonlinear relationships between variables, and the combination of their values is difficult to determine directly. Therefore, the use of optimized algorithms for compromise design is of great significance.
  • Step 8 Substitute the response surface model into the multi-objective optimization algorithm to obtain the Pareto front and determine the optimal parameter combination.
  • the multi-objective differential evolution algorithm based on the sorting mutation operator is introduced, and the MODE-RMO is adopted in the present invention.
  • the key to this model lies in the differential evolution of mutation operators based on sorting.
  • MODE-RMO combines fast non-dominated sorting and crowding distance, and speeds up the convergence speed.
  • three different individuals x g r1 , x g r2 and x g r3 are randomly selected from the target population.
  • the mutation operator m i g+1 can be expressed as:
  • F is the mutation range interval, which is the mutation operator based on sorting. It is often taken between [0,1], and r 1 , r 2 , and r 3 are mutually exclusive integers different from i.
  • test vector v ij g+1 It can be expressed as:
  • rand is the random value between [0,1]
  • CR is the cross constant between [0,1]
  • x ij g is the selected individual.
  • the last step is to use a greedy operator to select a better individual.
  • the selection process can be expressed as:
  • v i g+1 and x i g are two competing individuals, and x i g+1 is the selected individual.
  • the population is further sorted by fast non-dominated sorting and crowded distance sorting. First, the members with a dominance number of zero will be put into a separate list Q, and these members belong to the non-dominated solution set. Second, analyze the average distance between two solutions using the crowding distance, and calculate the absolute normalized difference of two adjacent solutions to predict the population density. Finally, according to the population ranking, the best individuals in the population will survive to the next generation. Repeat the above process until it converges and produces the best value of the Pareto front,
  • Figure 7(d) is a schematic diagram of the generated Pareto front. It can be seen that the parameterized magnetic network matches well with the Pareto front obtained by the finite element method. The main reason for the deviation is that the average torque obtained by EMN is smaller than the result of finite element analysis.
  • the particle position includes the information of the understanding set, and the important area is divided into high-quality particles and low-mass particles. Based on the Pareto front, the predicted optimal average torque is 266.1Nm, and the torque ripple is 1.04nm. Compared with the initial design with an average torque of 252.6Nm and torque ripple of 1.87Nm, the torque performance is A very good improvement.
  • Figure 8(a) is a schematic diagram of the comparison of no-load back EMF before and after optimization. It can be seen that the magnitude of the back EMF of the initial design is about 170V, and the magnitude of the back EMF of the optimized design is increased by about 15V, and the back EMF obtained by the finite element analysis and EMN is basically the same in the initial design and the optimized design. The difference in the peak value of the waveform is caused by human error in the calculation of the equivalent magnetic circuit and the calculation of the permeability of the air gap connection.
  • Figure 8(b) is a schematic diagram of comparison with the experimental no-load back EMF. The EMN, FEA and the real-time measured no-load line potential are compared. Although the error in the phase potential is slightly increased in the online potential calculation, it can still be very good. Good matching, higher calculation accuracy.
  • Figure 8(c) is a schematic diagram of the comparison of cogging torque before and after optimization, which verifies the reduction of cogging torque.
  • the optimized torque ripple peak-to-peak value is reduced from 1.9N.m to 1.3N.m, which shows the effectiveness of torque ripple suppression.
  • Figure 8(d) is a schematic diagram of torque comparison before and after optimization. The results show that the average torque has increased from the initial 252.6Nm to the optimized 268.3Nm, the average torque has increased by 15.7Nm, and the torque ripple has been reduced from 1.87Nm to 1.21Nm.
  • the rotation speed is within the allowable error range.
  • the moment performance is basically well predicted. This difference mainly lies in the finite order of the response surface function.
  • the parametric equivalent magnetic network modeling method for multi-objective optimization of a permanent magnet motor of the present invention includes dividing the disordered and regular regions of magnetic field lines in the motor, and adopting a dynamic mesh section with parameterized characteristics for the disordered regions.
  • the conventional magnetic circuit and permeance equivalent model is used, and the dynamic grid model and the magnetic circuit model are connected to establish the parameterized equivalent magnetic network model of the motor; the simultaneous nonlinear matrix solves the equation and solves the magnetic potential of each node , To further obtain the torque characteristics of the motor; by changing the magnetic circuit permeance value and adjusting the permeance grid shape, using the parameterized equivalent magnetic network model to perform variable sensitivity analysis on the average torque and torque ripple, and select high-sensitivity variables, The response surface models of average torque and torque ripple are established respectively, and the response surface model is substituted into the multi-objective optimization algorithm to obtain the Pareto frontier, determine the optimal parameter combination, and compare and verify with the finite element analysis and experimental results.
  • the present invention performs parametric equivalent magnetic network modeling suitable for multi-objective optimization for a surface-mounted permanent magnet motor for the first time, and the provided scheme can provide reference research for modeling and optimization of this type of permanent magnet motor.

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Abstract

一种永磁电机多目标优化的参数化等效磁网络建模方法，首先划分电机内磁力线无序区域与规律区域，构建无序区域的动态网格模型，构建有序区域的磁路模型；再连接动态网格模型与磁路模型，建立电机的参数化等效磁网络模型，联立非线性矩阵求解方程，求解各节点磁位，获得电机的转矩特性；然后利用参数化等效磁网络模型对平均转矩和转矩脉动进行变量灵敏度分析，选择高灵敏度变量，分别建立平均转矩和转矩脉动的响应面模型，将响应面模型代入多目标优化算法，获得帕累托前沿，确定最优参数组合。该方法首次针对表贴式永磁电机进行适用于多目标优化的参数化等效磁网络建模，为该类型永磁电机建模及优化提供参考研究。

Description

一种永磁电机多目标优化的参数化等效磁网络建模方法 技术领域
本发明涉及一种永磁电机多目标优化的参数化等效磁网络建模方法，属于电磁场计算领域。
背景技术
表贴式永磁电机具有优良的动态性能，十分适用于伺服系统中DC/AC电机。其性能优劣的关键在于优化手段。传统的单目标优化一次针对一个变量优化，该方法操作简单，具有一定的效果，但无法避免多个参数间的影响冲突。多目标优化的方法常用来改善参数间的相互作用，其是由Pareto最优性集即最优解集来求解的。多目标优化可以平衡所选参数变量，同时优化各种性能参数，避免多个因素间的影响。然而，由于采用了有限元分析(Finite Element Analysis，FEA)，多目标优化存在时间消耗大、操作繁琐等缺点。因此，提出了响应面法，通过应用代理模型来减少过多的有限元分析。尽管如此，由于采样点的多维性，全局优化仍然存在仿真耗时过大的缺点。
等效磁网络(Equivalent Magnetic Network，EMN)模型作为有限元分析的替代方法，具有分析速度快、计算成本低等优点。通过考虑空间谐波含量的影响，基于网格法的EMN模型精度可以与有限元分析相媲美。然而在电机结构优化方面，EMN模型尚未得到创新性的应用。通过对表贴式永磁电机进行参数化等效磁网络建模，不需要依靠重复建模操作，其计算原理简单，精度高，且仿真耗时短带来较高的设计效率，即可在多变量下迅速求解出电机转矩等电磁参数，利用多目标优化算法，实现表贴式永磁电机的高效率性能优化设计。
发明内容
本发明的目的在于提供一种永磁电机多目标优化的参数化等效磁网络建模方法，主要包括电机内无序磁力线区域动态网格磁导剖分、规律磁力线区域的磁路磁导等效以及磁网络矩阵等式的建立求解。利用参数化等效磁网络模型，完成多个结构变量的灵敏度分析和建立响应面模型，利用多目标优化算法确定最优参数组合，实现永磁电机的快速多目标优化。
为实现上述目的，本发明采用的技术方案是：一种表贴式永磁电机多目标优化的参数 化等效磁网络建模方法，包括以下步骤：
步骤1，划分电机内磁力线无序与规律区域；
步骤2，构建电机中磁力线无序区域的动态网格模型；
步骤3，构建电机中磁力线规律区域的磁路模型；
步骤4，连接动态网格模型与磁路模型，建立电机的参数化等效磁网络模型；
步骤5，联立非线性矩阵求解方程，求解各节点磁位，进一步获得电机的转矩特性；
步骤6，选择优化变量并确定优化目标，使用参数化等效磁网络模型完成参数灵敏度分析；
步骤7，选择高灵敏度变量，分别建立平均转矩和转矩脉动的响应面模型；
步骤8，将响应面模型代入多目标优化算法，获得帕累托前沿，确定最优参数组合。
进一步，所述表贴式永磁电机为48槽/44极的三相电机，由定子、气隙、转子永磁体和转轴四个部分组成；定子中包含定子轭部、定子齿部、定子极靴、定子槽与电枢绕组，定子铁芯的材料为硅钢片B20AT1500，电枢绕组采用分数槽集中式绕制方式；气隙介于定子和转子之间，气隙厚度为1mm；转子为无铁芯圆筒形结构，只由永磁体与转轴构成，在圆筒状转轴表面开槽并表贴了材料为N42UH的永磁体，表贴永磁体截面近似呈矩形，并以Halbach永磁阵列均匀安置在转轴表面，阵列中主永磁体卡嵌在转轴表面开槽中，以便定位安装辅助永磁体，减小制造工差对装配的影响；电机转轴由不导磁材料不锈钢304制成，为实心圆柱状，其表面有起定位作用的开槽，并与转子同轴连接。
进一步，所述步骤1中，采用有限元软件得到电机内磁力线分布云图，电机内磁力线无序区域主要集中在定子极靴和气隙中，该区域空间谐波较多；磁力线规律区域主要集中在定子轭部和定子齿部，由于转子永磁体磁性能强，且处于Halbach阵列下，其内部磁力线走势规律，漏磁可忽略；转子转轴由于采用了不导磁材料，因此其磁力线分布可以等效视为为空气中磁力线分布，不进行建模。
进一步，所述步骤2中，对于电机中磁力线无序区域动态网格建模的具体过程为：根据电机的极靴尺寸与气隙尺寸，采用一种长宽可变的十字型磁导网格进行剖分，同时考虑漏磁影响，在周向上同时对极靴和极靴间气隙进行网格剖分，且两处网格分布严格限于各自的区域边界，尺寸设置严格按照各自的区域规格，数量固定后极靴结构变化时只影响其相应区域的网格形状；对于气隙部分建模需在周向上和上层极靴网格保持数量和宽度的一致，其网格高度单独依据气隙长度判断。当极靴网格宽度变化时，气隙内网格需随之进行宽度的更新，以保证上下层网格间的磁导连接关系恒定。
进一步，所述步骤3中，对于电机中磁力线规律区域磁路建模的具体过程为：对于定子齿部和轭部，采用通用的磁路模型进行磁导等效处理，单个的齿部等效为单个磁导，定子轭部以齿的数量进行分段等效，建立成齿间的连接磁导；对于转子永磁体，根据Halbach充磁方向，将每一个永磁体块进行分别等效，在主永磁体内建立径向磁导，在辅助永磁体内建立切向磁导，按照Halbach磁力线流向规律，依次将永磁体磁导在节点处相连。
进一步，所述步骤4中，定子齿部磁导底部节点与定子极靴网格的上方节点依次相连，定子极靴间气隙网格的上方节点空置；气隙网格的底层网格节点根据对应关系与转子Halbach永磁阵列中的主永磁体磁导连接，判断依据为主永磁体在气隙网格上的映射面积是否大于一半，辅助永磁体上方的气隙网格底部磁导空置不连接；电机转子位置更新需重新判断转过角度，重置气隙网格与转子主永磁体磁导的连接关系。
进一步，所述步骤5中，建立磁导G、磁动势F和磁通Ф的矩阵等式G·F＝Ф，根据节点磁位差计算两点间的磁感应强度，公式为：B i,j＝(F(i)-F(j))·G(i,j)/S i,j。引入铁芯非线性参数B-H曲线，在迭代计算时通过插值获得磁导率值，并采用超松弛迭代算法进行磁导率更新，其新值与上一次计算值加权迭代。当两次迭代计算之间齿部磁密差值ΔB≤0.5％时，可认为迭代已收敛，更新转子位置，计算下一个转子位置角度；完成一个电角度周期计算后，通过齿部磁密计算出电机磁链Ф、反电势E等电磁参数；带载情况下在电枢绕组中加入正弦电流，产生正弦交变的齿部磁动势，进一步可以计算电机的输出转矩。
进一步，所述步骤6中，选择定子轭厚、定子齿宽、定子齿长、极靴高度、槽口宽度、主永磁体长度、辅助永磁体宽度和极弧系数8个参数变量，通过改变磁路磁导值和调整磁导网格形状，利用参数化等效磁网络模型对平均转矩和转矩脉动进行变量灵敏度分析，并选出其中4个高灵敏度的变量。
进一步，所述步骤7中，使用Box-Behnken design设计试验方法，对选出4个高灵敏度变量进行29次采样，在磁网络模型中分别对设计的29种参数组合方法进行平均转矩和转矩脉动的计算，在Design Expert软件中生成平均转矩和转矩脉动的响应面模型，多元相关系数R 2越接近于1，p值越小，则模型的精度越高；进一步根据R 2和p值相应调整响应面模型中的变量个数，以提高模型拟合精度。
进一步，所述步骤8中，利用平均转矩和转矩脉动的响应面模型，导入基于排序变异算子的多目标差分进化算法(MODE-RMO)，通过变异、交叉和选择，划分分支配解外部档案，并利用拥挤距离分析两个解之间的平均距离，保证最优秀的个体解将存活到下一迭代。最终获得非支配解集合帕累托前沿面，使用优化目标函数约束确定最优非支配解。
本发明具有以下有益效果：
1、本发明中，分别使用网格剖分和磁路磁导模型对表贴式永磁电机进行综合建模，同时兼顾有限元法的高精度和磁路法的低耗时，有助于提高等效磁网络计算准确度与效率。
2、本发明中，采用一种长宽可调的十字型磁导网格进行局部剖分，建模区域结构变化时等效改变其区域内网格形状，避免了参数变化时的重复建模，提高模型通用性和动态性。
3、本发明中，气隙区域的网格与极靴区域的网格固定节点，当极靴结构影响极靴网格，气隙网格与之等宽调节，保证了磁导连接关系的稳定性；
4、本发明中，气隙区域的网格与转子永磁体连接关系通过映射面积判断，避免了根据映射面积调整气隙磁导的误差，有助于提高模型计算准确度。
5、本发明中，利用参数可调的等效磁网络模型进行多目标优化设计，仿真耗时短，设计效率较高，在多变量下迅速求解出电机转矩等电磁参数，有利于高效实现表贴式永磁电机的结构优化设计。
附图说明
图1是本发明所用电机的2D结构图；
图2是本发明所用电机的磁力线分析图；
图3是电机永磁体充磁方向的结构图；
图4是本发明动态网格剖分的磁网络模型结构图；
图5是本发明动态网格剖分的磁网络模型局部放大结构图；
图6是本发明参数变化时动态网格变化的示意图；(a)初始网格结构；(b)参数化结构调整；
图7是多目标优化过程结构参数、灵敏度分析、响应曲面和帕累托前沿面示意图；(a)结构参数示意图；(b)灵敏度分析柱状图；(c)响应面模型结果图(齿宽与槽口宽度)；(d)帕累托前沿面；
图8是本发明所用电机优化前后磁网络、有限元和实验结果对比示意图。(a)优化前后空载反电势对比；(b)与实验空载反电势对比；(c)优化前后齿槽转矩对比；(d)优化前后转矩对比；(e)实验测量转矩；
图9为本发明的建模方法和优化过程的流程图。
具体实施方式
下面将结合本发明实施例中的附图，对本发明实施例中的技术方案进行清楚、完整地描述。
为了能够更加简单明了地说明本发明的有益效果，下面结合一个具体的表贴式永磁电 机来进行详细的描述：图1为该电机的拓补结构图，图中1为定子轭部，2-1为定子齿部；2-2为定子齿部极靴，2-3为定子齿部极靴间气隙，3为绕组，4为定转子间气隙，5-1为Halbach阵列中主永磁体，5-2为Halbach阵列中辅助永磁体，6为不导磁转轴；本发明实施例为为48槽/44极的三相电机，由定子、气隙、转子永磁体和转轴四个部分组成；定子中包含定子轭部、定子齿部、定子齿部极靴、定子槽与电枢绕组，定子铁芯的材料为硅钢片B20AT1500，电枢绕组采用分数槽集中式绕制方式；气隙介于定子和转子之间，气隙厚度为1mm；转子为无铁芯圆筒形结构，只由永磁体与转轴构成，在圆筒状转轴表面开槽并表贴了材料为N42UH的永磁体，表贴永磁体截面近似呈矩形，并以Halbach永磁阵列均匀安置在转轴表面，阵列中主永磁体卡嵌在转轴表面开槽中，以便定位安装辅助永磁体，减小制造工差对装配的影响；电机转轴由不导磁材料不锈钢304制成，为实心圆柱状，其表面有起定位作用的开槽，并与转子同轴连接。
如图9所示的流程图，分为以下步骤实现：
步骤1，划分电机内磁力线无序与规律区域。
图2为本发明实施例所用电机的磁力线分析图。在磁力线分布中可以看出，电机内磁力线无序区域主要集中在定子齿部极靴和定转子间气隙，该区域是电磁能量转换的关键区域，电机磁通被不均匀空气磁导扭曲，因此磁力线走势复杂无规律，存在大量的齿间漏磁与空间谐波；电机内磁力线规律区域主要集中在定子齿部、定子轭部和转子永磁体，该区域是磁通导向气隙的路径，形状规则，基本无漏磁，磁力线走势规律，且磁力线均处在同一个方向。
步骤2，构建电机中磁力线无序区域的动态网格模型。
根据电机的定子齿部极靴尺寸与气隙尺寸，采用一种动态可调的十字型磁导网格进行剖分，同时考虑漏磁影响，在圆周方向上同时对定子齿部极靴和极靴间气隙进行网格剖分，且两处网格分布严格限于各自的区域边界，尺寸设置严格按照各自的区域规格，网格数量固定后定子极靴结构变化时只影响其相应区域的网格形状；对于气隙部分建模需在圆周方向上和上层定子齿部极靴网格保持数量和宽度的一致，其网格高度单独依据气隙长度判断，当定子齿部极靴网格宽度变化时，气隙内网格需随之进行宽度的更新，以保证上下层网格间的磁导连接关系恒定。
图4是本发明动态网格剖分的磁网络模型结构图，图5是本发明动态网格剖分的磁网络模型局部放大结构图；按照建模区域的实际尺寸，同时考虑齿间漏磁和磁力线回路，包括极靴高度、极靴长度、槽口宽度、气隙外径、气隙高度等，并相应地在不同区域选择不 同的网格剖分方案。其中，动态可调十字型磁导网格是建模的关键，其切向磁导G t与径向磁导G h的计算公式为：
Figure PCTCN2020097904-appb-000001
式中，L a是轴向长度，w cell是网格宽度，h cell是网格高度，μ 0是真空磁导率，μ r是铁磁材料相对磁导率。同时考虑漏磁影响，在圆周方向上同时对定子齿部极靴和极靴间气隙进行网格剖分，且两处网格分布严格限于各自的区域边界，尺寸设置严格按照各自的区域规格，网格数量固定后定子极靴结构变化时只影响其相应区域的网格形状。
定子部分磁场复杂的区域为定子齿部极靴2-2与极靴间的气隙2-3，其剖分网格高度相等均为h 1，其网格宽度由建模区域宽度均等分，极靴间气隙中为1行2列等宽网格，极靴中为1行9列等宽网格，且极靴中网格的9个上节点与定子齿2-1下节点依次相连，极靴间气隙网格的2个上节点空置不连。
气隙4是永磁电机能量转换的重要场所，也是磁场最为复杂的区域。通过合理设计，对气隙4进行2层网格剖分，气隙网格数量与定子下层网格剖分数量相同，其总数为48槽/44极的最小公倍数528，保证了网格的对称均等性。气隙网格与定子下层网格不仅数量保持一致，其网格宽度也完全保持一致，保证了气隙上层网格与定子下层网格对应相连。另一方面，气隙下层网格与转子永磁体相连。转子上使用Halbach阵列的永磁体阵列，由主永磁体5-1与辅助永磁体5-2组成。主永磁体5-1径向充磁，辅助永磁体5-2切向冲刺，起到了单边气隙磁场增强的作用。对于Halbach永磁阵列，由于实际中辅助永磁体仅产生切向磁力线，因此辅助永磁体5-2不与气隙4网格下层节点相连，仅由产生切向充磁的主永磁体5-1连接所有主永磁体5-1宽度范围内的气隙4网格。当定子齿部极靴网格宽度变化时，气隙内网格需随之进行宽度的更新，以保证上下层网格间的磁导连接关系恒定。
步骤3，构建电机中磁力线规律区域的磁路模型。
对于所述电机的定子部分，定子轭部1和定子齿部2-1中的磁力线走势规律，基本无漏磁，且磁力线均处在同一个方向。对于定子齿部，根据其结构参数进行通用磁路磁导模型的建模，其磁导计算公式为：
Figure PCTCN2020097904-appb-000002
式中，μ为铁芯磁导率，S为充磁方向截面积，l为充磁长度，l t为齿长，μ 0是真空磁导 率，μ r是铁磁材料相对磁导率，w是结构宽度，L a是轴向长度。
对于定子轭部的磁路磁导等效，以齿的数量进行分段等效，建立成齿间的连接磁导，其计算公式同齿部磁导计算公式；轭部磁导的两端节点分别与对应连接的齿部磁导上端节点相连接。对于转子永磁体，根据Halbach充磁方向，将每一个永磁体块分别进行磁导等效，在主永磁体内建立径向磁导，在辅助永磁体内建立切向磁导，其计算公式同定子磁导计算公式，永磁体近似视为空气磁导率，相对磁导率设为1。同时考虑到磁导路径，按照Halbach磁力线流向规律，在主永磁体径向磁导下端节点，与辅助永磁体切向磁导节点进行相连，建立成永磁体间互联的磁导网络。
步骤4，连接动态网格模型与磁路模型，建立电机的参数化等效磁网络模型；
定子齿部磁导底部节点与定子齿部极靴网格的上方节点依次相连，定子齿部极靴间气隙网格的上方节点空置；气隙网格的底层网格节点根据对应关系与转子Halbach永磁阵列中的主永磁体磁导连接，判断依据为主永磁体在气隙网格上的映射面积是否大于一半，辅助永磁体上方的气隙网格底部磁导空置不连接；电机转子位置更新需重新判断转过角度，重置气隙网格与转子主永磁体磁导的连接关系。
由于定子极靴区域使用了动态网格剖分，因此定子齿部2-1磁导模型与对应的齿下极靴2-2第一行9个网格相连且关系固定，同时齿下极靴2-2的网格与气隙4的网格固定相连。对于气隙4中的网格，其网格内连接关系固定，其与定子极靴2-2网格的连接关系也固定，因此转子沿Z轴周向旋转时，仅有气隙与转子间的磁导连接关系随旋转角度的不同发生变化，因此确定转子Halbach永磁体阵列与气隙网格的连接方法，是建立旋转磁网络模型的关键。当转子旋转到特定角度时，主永磁体5-1的径向磁导上节点分别与映射面积内的对应磁导相连，判断依据为主永磁体在气隙网格上的映射面积是否大于一半；在其范围内大约有6至7个气隙网格与之相连，辅助永磁体范围内大约也有6至7个气隙网格，其保持空置。每当转子永磁体旋转一个角度步长，与主永磁体5-1连接的覆盖范围内的气隙4网格节点将发生变化，转子主永磁体磁导与气隙网格的连接关系需要在电机转动过程中不断更新。
图6是本发明参数变化时动态网格变化的示意图，当电机的等效磁网络模型通过网格和磁路模型结合建立后，其参数化功能可以通过动态调整网格大小与磁路磁导值实现。对于参数化网格调整，其主要可以实现槽口宽度、极靴高度、气隙长度等变量动态调整。如图6所示，当极靴高度增加时，位于极靴层的整行网格的高度均增加以匹配新的极靴高度，同时由于槽口宽度变大，缩减了极靴的长度，因此槽口内的网格宽度拉长，而极靴内的网 格宽度缩短。当定子网格随参数化要求动态调整时，需要保证网格与建模区域边界保持统一的原则，即与建模区域重合边界的网格需保证该条边界不可发生位置偏移，同时区域内所有网格的高度与宽度为均等分处理，保持了网格模型均等性的同时，避免了变量调整时重复建模的弊端。另外，当定子部分网格在宽度方面发生变化时，需要相应地调整气隙中的网格宽度，保证上下两个区域网格宽度一致，实现网格磁导固定相连。对于参数化磁路磁导调整，其主要可以实现定子轭厚、定子齿宽、定子齿长、永磁体高度、极弧系数等变量动态调整。该区域磁力线走势规律，其参数化通过调整磁导计算公式中的截面积或长度直接实现，仅改变部分节点间磁路磁导值，并不影响节点连接关系判断。
步骤5，联立非线性矩阵求解方程，求解各节点磁位，进一步获得电机的转矩特性。
电机的参数化等效磁网络完整模型共有4456个节点，根据磁场中的节点磁通定律，联列磁导矩阵G、磁动势矩阵F、和磁通矩阵Ф，建立磁导求解矩阵方程，计算公式如下：
G·F＝Φ           (3)
式中的磁动势矩阵F包括了永磁体产生的磁动势与绕组磁动势。绕组磁动势空载时设为0，负载时其计算公式为：
F＝Ni             (4)
式中，N为绕组匝数，i为绕组注入电流。
将式(3)扩写成矩阵形式，可以得到：
Figure PCTCN2020097904-appb-000003
且
Figure PCTCN2020097904-appb-000004
F＝[F(1) … F(4456)] T
Φ＝[Φ(1) … Φ(4456)] T
根据节点磁位差计算两点间的磁感应强度，公式为：B i,j＝(F(i)-F(j))·G(i,j)/S i,j，其中，B i,j为节点i、j之间的磁通密度，F(i)为节点i的磁动势，F(j)为节点j的磁动势，G(i,j)为节点i、j之间的磁导，S i,j为节点i、j之间的横截面积；引入所述电机所采用的铁磁材料B20AT1500的B-H曲线参数，同时对B-H曲线采用线性插值：
Figure PCTCN2020097904-appb-000005
其中，H为磁场强度，H n为第n次迭代磁场强度，H n+1为第n+1次迭代磁场强度，B为磁通密度，B n为第n次迭代磁通密度，B n+1为第n+1次迭代磁通密度，μ new为迭代得出的新磁通密度。
进一步进行超松弛迭代算法求解矩阵方程，所使用的牛顿迭代法公式为：
Figure PCTCN2020097904-appb-000006
其中，F (k+1) i为节点i第k+1次的迭代磁动势，F (k) i为节点i第k次的迭代磁动势，w为权重系数，G(i,i)为节点自导，G(i,j)为节点互导，F (k+1) j为节点j第k+1次的迭代磁动势，F (k) j为节点j第k次的迭代磁动势，Ф(i)为节点i磁通。
选择每次计算出的48个齿部磁密中与上次计算值误差最大的项作为收敛判断标准，当两次迭代计算之间齿部磁密差值ΔB≤0.5％时，可认为迭代已收敛，更新转子位置，计算下一个转子位置角度；当一个电角度周期计算完成后，得到等效磁网络中各节点的磁位，进一步计算出节点间磁导上的磁通密度，其计算公式为：
Figure PCTCN2020097904-appb-000007
其中，B为计算磁感应强度，F(i)为节点i磁动势，F(j)为节点j磁通势，G(i,j)为节点i、j间磁导，S为充磁方向截面积。
进一步，根据电机齿部磁通密度，可以计算得到电机磁链Ф、空载反电势E、负载转矩T等电磁特性，其计算公式分别如下：
Figure PCTCN2020097904-appb-000008
Figure PCTCN2020097904-appb-000009
Figure PCTCN2020097904-appb-000010
其中，S为充磁方向截面积，N为绕组匝数，B为磁通密度，T out为输出转矩，T ave为平均转矩，T cog为齿槽转矩，m为相数，p为极数，I为输入电流幅值，θ i为内功率角，W 为磁场能量，α为位置角度。
步骤6，选择优化变量并确定优化目标，使用参数化等效磁网络模型完成参数灵敏度分析。
图7(a)是多目标优化过程结构参数示意图。对所述电机进行多目标优化，选取了如下8个参数变量：定子轭厚、定子齿宽、定子齿长、极靴高度、槽口宽度、主永磁体长度、辅助永磁体宽度和极弧系数。其极弧系数α为主永磁体弧度θ PM1占一个永磁体组合弧度的比例，θ PM2为辅助永磁体弧度，其计算公式为：
Figure PCTCN2020097904-appb-000011
对于所选取的8个参数变量，首先对其进行灵敏度分析，筛选出对平均转矩和转矩脉动影响性较高的高灵敏度参数。基于参数化磁网络的输出转矩，采用一次一个变量的方法进行灵敏度分析，进一步生成敏感度指标，表示参数对性能的影响程度，其灵敏度指标H(x i)计算公式可以表示为：
Figure PCTCN2020097904-appb-000012
式中，Ey/x i代表当x i是常数时？为y的平均值，V(Ey/x i)和V(y)分别是Ey/x i和y的方差。其敏感度的正、负指标表明设计参数可以相应地提高或抑制性能。
图7(b)是选取变量的单灵敏度和综合灵敏度分析对比图。对于所选取的8个参数变量，在各自的取值范围内分别对平均转矩和转矩脉动进行灵敏度分析。同时，基于对单一输出的影响，利用综合敏感性分析以表明单个个体的灵敏度对总体转矩性能的影响，其计算公式可以表示为：
G(x i)＝λ 1|H out(x i)|+λ 2|H ri(x i)|       (13)
式中，G(x i)为综合灵敏度因数，H out(x i)为平均转矩灵敏度指标，H ri(x i)为转矩脉动灵敏度指标，λ 1和λ 2分别是代表重要程度的权重系数，并且满足λ 1+λ 2＝1。在本发明中，由于同时对平均转矩和转矩脉动进行优化且重要性相等，取λ 1＝λ 2＝0.5。
通过改变磁路磁导值和调整磁导网格形状，利用参数化等效磁网络模型对平均转矩和转矩脉动进行变量灵敏度分析，并选出其中高灵敏度的变量。
步骤7，选择高灵敏度变量，分别建立平均转矩和转矩脉动的响应面模型。
由图7(b)中可以看出，对于综合灵敏度系数而言，齿宽、槽口宽度、辅助永磁体宽 度和极弧系数具有着较高的综合灵敏度，因此将这四组参数归并为高灵敏度系数，并进行响应面的建模。而剩下的定子轭厚、定子齿长、定子极靴高度和主永磁体长度则具有相对较低的的灵敏度系数，对其采用单参数优化方法。进一步利用高灵敏度系数，通过BBD(Box-Behnken Design)采样建立平均转矩和转矩脉动的响应面方程。与CCD相比，BBD采样具有点数少的优点，其关键是用多项式函数代替优化模型的替代模型。然后利用Design Expert软件建立响应面方程。基于BBD中的-1、0和1三个层次，EMN仿真只需要29个采样点。根据输入输出之间的响应关系，初步建立了响应曲面的数学函数表达式。然而初始响应面模型是原始的，需要调整以获得更好的性能。根据相关统计理论，当多元相关系数R 2接近1时，拟合度较好。两个初始模型的R 2值均在0.92以上，说明该模型具有良好的适应性和适用性。此外，p为假设检验结果参数，p值小于0.05的项被认为是提高准确度的重要项，其余的项则根据最小拟合不足进行调整。最终调整后获得的平均转矩T avg和转矩脉动T rip响应面模型如下所示：
Figure PCTCN2020097904-appb-000013
Figure PCTCN2020097904-appb-000014
其中，w t为定子齿宽，B s0为定子槽开口，w PM2为辅助永磁体宽度，α为极弧系数。
图7(c)是获得的响应曲面模型中齿宽与槽口宽度的相互作用图。由图中可以看出，平均转矩和转矩脉动是变量之间的非线性关系，其值的组合很难直接确定。因此，采用优化算法进行折衷设计具有重要意义。
步骤8，将响应面模型代入多目标优化算法，获得帕累托前沿，确定最优参数组合。
利用平均转矩和转矩脉动的响应面模型，导入基于排序变异算子的多目标差分进化算法，本发明中采用的是MODE-RMO。该模型的关键在于基于排序的变异算子的差分进化。MODE-RMO结合了快速的非支配排序和拥挤距离，并且加快了收敛速度。在突变过程中中，从目标群体中随机选择三个不同的个体x g r1、x g r2和x g r3，突变算子m i g+1可以表达为：
Figure PCTCN2020097904-appb-000015
式中，F是变异范围区间，也就是基于排序的变异算子，常在[0,1]间取值，r 1，r 2，r 3分别是不同于i的互斥整数。
进一步，对m ij g+1？执行交叉以处理，并且试验向量v ij g+1？可以表示为：
Figure PCTCN2020097904-appb-000016
式中，rand是[0,1]间随机值，CR是[0,1]间交叉常数,x ij g为被选择个体。
最后一步是用贪婪算子选择更好的个体，其选择过程可以表示为：
Figure PCTCN2020097904-appb-000017
式中v i g+1和x i g分别是两个竞争个体，x i g+1为被选择个体。
在算法中，种群P g＝{x g 1，x g 2，…，x g Np}与x g i＝{x gi 1，x gi 2，…，x g iD}(i＝1，…，Np)在搜索空间从EMN变量范围内初始化。进一步对种群进行快速非支配排序和拥挤距离排序。首先，支配数为零的成员将被放入一个单独的列表Q中，这些成员属于非支配解集。其次，利用拥挤距离分析两个解之间的平均距离，并计算两个相邻解的绝对归一化差分，预测种群密度。最后，根据种群排序，种群中最优秀的个体将存活到下一代。重复上述过程，直到收敛，并产生帕累托前沿的最佳值，
图7(d)是所产生的帕累托前沿面示意图。可以看出，参数化磁网络与有限元所得到的帕累托前沿匹配良好。其偏移的原因主要是是EMN得到的平均转矩小于有限元分析结果。粒子位置包括了解集的信息，而重要区划分高质量粒子和低质量粒子。基于帕累托前沿面，预测的最优平均转矩为266.1N.m，转矩脉动为1.04n.m，与平均转矩为252.6N.m，转矩脉动为1.87N.m的初始设计相比，其转矩性能得到了很好的改善。
图8(a)是优化前后空载反电势对比示意图。可以看出，初始设计的反电动势幅值约为170V，优化设计的反电动势幅值提高了约为15V，而且有限元分析和EMN得到的反电动势在初始设计和优化设计中基本一致。在波形峰值的差异是等效磁路计算和气隙连接磁导率计算中的人为误差造成的。图8(b)是与实验空载反电势对比示意图，将EMN、FEA和实现测得的空载线电势进行对比，虽然其相电势中的误差在线电势计算中略有增大，但仍能很好地匹配，计算精度较高。
图8(c)是优化前后齿槽转矩对比示意图，其验证了齿槽转矩的降低。优化后的转矩脉动峰峰值从1.9N.m减小到1.3N.m，表明了转矩脉动抑制的有效性。进一步，图8(d)是优化前后转矩对比示意图。结果表明，平均转矩从初始的252.6N.m提高到优化的268.3N.m，平均转矩提高了15.7N.m，转矩脉动从1.87N.m降低到1.21N.m，与预测结果相比，在允许误差范围内，转矩性能基本得到了较好的预测。这种差异主要在于响应面函数 的有限阶次。另一方面，在初始设计和优化设计中，EMN计算结果在波形和周期上与有限元分析结果吻合较好。然而，EMN得到的平均转矩比有FEA略低，约为0.65N.m，这是由于网格粗糙和有限元精细细分造成的反电势幅值差异。图8(e)是实验测量转矩示意图，显示了额定负载下的测量扭矩，平均测量扭矩Torque约为253N.m，而EMN的平均扭矩约为268.3N.m。这种差异主要是由机械损耗和杂散损耗引起的，在EMN建模中忽略了这些因素。实验中的转矩脉动虽然受到传感器噪声的影响，但仍然表现出平滑的输出转矩，证明了优化过程的有效性。
综上，本发明的一种永磁电机多目标优化的参数化等效磁网络建模方法，包括划分电机内磁力线无序与规律区域，对无序区域采用具有参数化特性的动态网格剖分，对于有序区域采用常规磁路磁导等效模型，连接动态网格模型与磁路模型，建立电机的参数化等效磁网络模型；联立非线性矩阵求解方程，求解各节点磁位，进一步获得电机的转矩特性；通过改变磁路磁导值和调整磁导网格形状，利用参数化等效磁网络模型对平均转矩和转矩脉动进行变量灵敏度分析，选择高灵敏度变量，分别建立平均转矩和转矩脉动的响应面模型，将响应面模型代入多目标优化算法，获得帕累托前沿，确定最优参数组合，并分别与有限元分析与实验结果进行对比验证。本发明首次针对表贴式永磁电机进行适用于多目标优化的参数化等效磁网络建模，所提供的方案可以为该类型永磁电机建模及优化提供参考研究。
虽然本发明已以较佳实施例公开如上，但实施例并不是用来限定本发明的。在不脱离本发明之精神和范围内，所做的任何等效变化或润饰，均属于本申请所附权利要求所限定的保护范围。

Claims (10)

  1. 一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，包括以下步骤：
    步骤1，划分永磁电机内磁力线无序区域与规律区域；
    步骤2，构建永磁电机中磁力线无序区域的动态网格模型；
    步骤3，构建永磁电机中磁力线规律区域的磁路模型；
    步骤4，连接动态网格模型与磁路模型，建立永磁电机的参数化等效磁网络模型；
    步骤5，联立非线性矩阵求解方程，求解各节点磁位，进一步获得永磁电机的转矩特性；
    步骤6，选择优化变量并确定优化目标，使用参数化等效磁网络模型完成参数灵敏度分析；
    步骤7，选择高灵敏度变量，分别建立平均转矩和转矩脉动的响应面模型；
    步骤8，将响应面模型代入多目标优化算法，获得帕累托前沿，确定最优参数组合。
  2. 根据权利要求1所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述永磁电机为48槽/44极的三相表贴式永磁电机，包括定子、气隙、转子；定子中包含定子轭部、定子齿部、定子齿部极靴、定子齿部之间的定子槽与定子槽内的电枢绕组，定子铁芯的材料为硅钢片，电枢绕组采用分数槽集中式绕制方式；气隙介于定子和转子之间；转子为无铁芯圆筒形结构，由转子永磁体与转轴构成，在圆筒状转轴表面开槽并表贴了永磁体，表贴永磁体截面呈矩形，并以Halbach永磁阵列均匀安置在转轴表面，阵列中主永磁体卡嵌在转轴表面开槽中，以便定位安装辅助永磁体；电机转轴由不导磁材料制成，为实心圆柱状，其表面有起定位作用的开槽，并与转子永磁体同轴连接。
  3. 根据权利要求2所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤1的具体过程为：
    采用有限元软件得到永磁电机内磁力线分布云图，电机内磁力线无序区域主要集中在定子齿部极靴和气隙中，该区域空间谐波较多；磁力线规律区域主要集中在定子轭部、定子齿部和转子永磁体；转子转轴由于采用了不导磁材料，因此其磁力线分布可以等效视为空气中磁力线分布，不进行建模。
  4. 根据权利要求2所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤2的具体过程为：
    根据电机的定子齿部极靴尺寸与气隙尺寸，采用一种动态可调的十字型磁导网格进行剖分，同时考虑漏磁影响，在圆周方向上同时对定子齿部极靴和极靴间气隙进行网格剖分，且两处网格分布严格限于各自的区域边界，尺寸设置严格按照各自的区域规格，网格数量固定后定子极靴结构变化时只影响其相应区域的网格形状；对于气隙部分建模需在圆周方向上和上层定子齿部极靴网格保持数量和宽度的一致，其网格高度单独依据气隙长度判断，当定子齿部极靴网格宽度变化时，气隙内网格需随之进行宽度的更新，以保证上下层网格间的磁导连接关系恒定。
  5. 根据权利要求2所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤3的具体过程为：
    对于定子齿部和定子轭部，采用通用的磁路模型进行磁导等效处理，单个的定子齿部等效为单个磁导，定子轭部以齿的数量进行分段等效，建立成齿间的连接磁导；对于转子永磁体，根据Halbach充磁方向，将每一个永磁体块进行分别等效，在主永磁体内建立径向磁导，在辅助永磁体内建立切向磁导，按照Halbach磁力线流向规律，依次将永磁体磁导在节点处相连。
  6. 根据权利要求2所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤4的具体过程为：
    定子齿部磁导底部节点与定子齿部极靴网格的上方节点依次相连，定子齿部极靴间气隙网格的上方节点空置；气隙网格的底层网格节点根据对应关系与转子Halbach永磁阵列中的主永磁体磁导连接，判断依据为主永磁体在气隙网格上的映射面积是否大于一半，辅助永磁体上方的气隙网格底部磁导空置不连接；电机转子位置更新需重新判断转过角度，重置气隙网格与转子主永磁体磁导的连接关系。
  7. 根据权利要求1所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤5的具体过程为：
    建立磁导G、磁动势F和磁通Ф的矩阵等式G·F＝Ф，根据节点磁位差计算两点间的磁感应强度，公式为：B i,j＝(F(i)-F(j))·G(i,j)/S i,j，其中，B i,j为节点i、j之间的磁通密度，F(i)为节点i的磁动势，F(j)为节点j的磁动势，G(i,j)为节点i、j之间的磁导，S i,j为节点i、j之间的横截面积；引入铁芯非线性参数B-H曲线，在迭代计算时通过插值获得磁导率值，并采用超松弛迭代算法进行磁导率更新，其新值与上一次计算值加权迭代；当两次迭代计算之间齿部磁密差值ΔB≤0.5％时，可认为迭代已收敛，更新转子位置，计算下一个转子位置角度；完成一个电角度周期计算后，通过齿部磁密计算出电机磁链Ф、反电势E等电磁 参数；带载情况下在电枢绕组中加入正弦电流，产生正弦交变的齿部磁动势，进一步可以计算电机的输出转矩。
  8. 根据权利要求1所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤6的具体过程为：
    选择定子轭厚、定子齿宽、定子齿长、定子齿部极靴高度、槽口宽度、主永磁体长度、辅助永磁体宽度和极弧系数8个参数变量，通过改变磁路磁导值和调整磁导网格形状，利用参数化等效磁网络模型对平均转矩和转矩脉动进行变量灵敏度分析，并选出其中高灵敏度的变量。
  9. 根据权利要求1所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤7的具体过程为：
    使用Box-Behnken Design设计试验方法，对选出高灵敏度变量进行多次采样，在磁网络模型中分别对设计的多种参数组合方法进行平均转矩和转矩脉动的计算，在Design Expert软件中生成平均转矩和转矩脉动的响应面模型，R 2为多元相关系数，p为假设检验结果判断参数，R 2越接近于1，p值越小，则模型的精度越高；进一步根据R 2和p值相应调整响应面模型中的变量个数，以提高模型拟合精度。
  10. 根据权利要求1所述的一种永磁电机多目标优化的参数化等效磁网络建模方法，其特征在于，所述步骤8的具体过程为：
    利用平均转矩和转矩脉动的响应面模型，导入基于排序变异算子的多目标差分进化算法，通过变异、交叉和选择，划分非支配解外部档案，并利用拥挤距离分析两个解之间的平均距离，保证最优秀的个体解将存活到下一迭代，最终获得非支配解集合帕累托前沿面，使用优化目标函数约束确定最优非支配解。
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