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    • 专利摘要:
    Procedure and computer program product to accelerate the steady state calculation of a squirrel cage induction motor. The present invention relates to a computer program method and product for accelerating the steady state calculation of a squirrel-cage induction motor. The method includes making an estimate of the initial conditions of the squirrel-cage motor, which allows determining the calculation of the steady state in a shorter time. (Machine-translation by Google Translate, not legally binding)
    公开号:ES2687868A1
    申请号:ES201830228
    申请日:2018-03-08
    公开日:2018-10-29
    发明作者:Alfredo BERMÚDEZ DE CASTRO LÓPEZ-VARELA；María Dolores GÓMEZ PEDREIRA；Marta PIÑEIRO PEÓN；María Del Pilar SALGADO RODRÍGUEZ
    申请人:Consorcio Instituto Tecnoloxico De Matematica Industrial；Consorcio Inst Tecnoloxico De Matematica Industrial；Universidade de Santiago de Compostela；
    IPC主号:
    专利说明:

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    DESCRIPTION
    Procedure and product of a computer program to accelerate the calculation of the steady state of a squirrel cage induction motor
    TECHNICAL SECTOR OF THE INVENTION
    The present invention relates to a computer program method and product for accelerating the calculation of the steady state of a squirrel cage induction motor. More specifically, it refers to a computer program method and product for calculating the steady state of a squirrel cage motor in which the stator coils are fed by periodic currents.
    STATE OF THE TECHNIQUE
    Numerical simulation is a key tool in the design stage of electric machines since it avoids the construction of unnecessary prototypes and significantly reduces the costs and time required to obtain new designs. However, these advantages may be diminished if excessive calculation time is required. Particularly, in the numerical simulation of induction motors there is a growing demand for having competitive simulation techniques in terms of computational efficiency since the design of this type of device generally requires long simulations.
    Electromagnetic simulation of squirrel cage induction motors using the finite element method involves solving a system of partial derivative equations (obtained from Maxwell's equations) coupled with electrical circuit equations. In the state of the art, the most extended electromagnetic model describes the active part of the motor using an induced currents model defined in a cross section thereof, while the squirrel cage rings and the stator coil heads are represented. through circuit elements. As a result, a two-dimensional (2D) mathematical model is formulated in the time domain that requires spatial and temporal discretization for its numerical approximation. The unknowns of this model are the currents and potential drops in the squirrel cage circuit and the magnetic potential vector in the motor cross section. In this type of engine, the most important difficulty in solving the problem described is that the time required to reach the steady state is generally very high. Therefore, if the model does not start with adequate initial conditions, its resolution may involve a high computational cost. Thus, if the so-called method of
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    brute force, consisting of advancing in time to reach a stationary solution, it may take more than ten days to reach steady state, even with the performance of current computer equipment.
    It is evident that a significant reduction in the calculation time allows for a greater number of simulations in the stage prior to the design of the machine and to expand the range of its possible configurations. Therefore, in recent years the problem of reducing the computational cost to reach the steady state in the numerical simulation of induction motors has been addressed using different methodologies, among which the following stand out:
    • TPFEM (Time Periodic Finite Elements) methods, which consist of raising the problem discretized in time at an interval where the solution is periodic and solving all time steps simultaneously (Takahashi, Y., et al., Time- Domain Parallel Finite-Element Method for Fast Magnetic Field Analysis of Induction Motors, IEEE Trans. Magn. 49 (5) (2013) 2413-2416). While avoiding solving the problem at each time step until it reaches steady state, this technique leads to solving nonlinear systems with extremely large and non-symmetrical matrices. Therefore, effective parallelization techniques are necessary to address the size of the problem.
    • TP-EEC (Time Periodic - Explicit Error Correction) methods, which aim to accelerate the convergence of the transitional model based on correction techniques used in more general iterative methods and properties of the TPFEM method (Katagiri, H., et al., Improvement of Convergence Characteristics for Steady-State Analysis of Motors With Simplified Singularity Decomposition-Explicit Error Correction Method, IEEE Trans. Magn. 47 (6) (2011) 1786-1789).
    • Methods based on providing the transient model with an initial condition obtained by solving a nonlinear induced current model raised in the frequency domain. The harmonic approach of the problem to determine the initial condition is based on the hypothesis that the temporal variation of the magnetic potential vector can be described by a complex exponential function. Nonlinear magnetic effects are included in this approach by introducing the so-called effective magnetization curve, while rotor movement is considered by modifying the electrical conductivity of the bars (Stermecki, A., et al.,
    Numerical analysis of steady-state operation of three-phase induction machines by
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    an approximate frequency domain technique. Elektrotech Inf. Tech 128 (3) (2011) 81-85).
    A common difficulty with TPFEM and TP-EEC techniques is to establish the periodicity condition in induction motors so that the method has a low computational cost. This is because the potential magnetic vector varies with very different frequencies in rotor and stator, so if you intend to work with a common period in both parts of the motor, a very large time interval may be required. However, it is convenient to define the periodicity condition in a sufficiently short period so that the procedures described are advantageous compared to a brute force method. In the TPFEM methods there are several proposals in this regard based on the fact that the solution exhibits a spatio-temporal symmetry, while in the TP-EEC methods the convergence acceleration in the two domains is applied separately, or only in one of them.
    The above methods approach the problem based on the transient model of induced currents coupled with electrical circuits. This starting point leads, in general, to problems with high computational cost because the periodicity condition affects a spatial field, the magnetic potential vector, and the circuit elements. In addition, as previously mentioned, different frequencies appear in each part of the engine.
    DESCRIPTION OF THE INVENTION
    Given the problems presented by the calculation methods of the steady state of induction motors existing in the state of the art, the need arises for a method that solves at least some of the aforementioned problems. It is the object of the present invention to satisfy said need.
    The present invention relates to a process for accelerating the calculation of the steady state of a squirrel cage induction motor in the case where the motor is fed by periodic currents in the stator coils (100). The process of the present invention makes it possible to calculate initial currents for the bars (102) of the motor squirrel cage which, when used as initial data to solve the usual system of partial derivative equations that characterize the electromagnetic behavior of the motor, result in a considerable reduction of the transitory part of the solution. In this way, by requiring a smaller number of cycles to reach the steady state of the machine, the computational cost is reduced and therefore the time required to perform the simulations.
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    In a first aspect, the invention relates to a method of calculating an induction motor that addresses the problem of temporal periodicity by focusing this condition on the currents of the squirrel cage bars (102). In this way, it will be possible to work only with the rotor frequency and with a very small number of unknowns compared to the mentioned methods. In addition, the computational cost of the calculation method object of the present invention is independent of the size of the rotor period.
    The calculation of the appropriate initial currents to reduce the transient is made by solving a system of overdetermined equations that is obtained by following the steps detailed below.
    The procedure for the accelerated calculation of the steady state of a squirrel cage induction motor comprises the following steps:
    a) Determine the dimensions and structure of the induction motor to determine the geometry of its cross section, which is divided into rotor (102, 104 and 106) and stator, the rotor being the digest of all moving parts, and the rest being The stator The cross section geometry includes, at least, the stator coils, the squirrel cage bars, the ferromagnetic rotor and stator cores, the air gap and the motor shaft.
    b) Model the electrical circuit corresponding to the squirrel cage of the motor, for this the structure of the squirrel cage located in the rotor of the induction motor is analyzed to determine the topology of the associated electrical circuit. Said circuit, which is modeled by a graph, is formed by the conductors associated with the bars (202) of the cage and the resistances (200) associated with the rings located at the ends of the motor, which were not included in the geometry of the cross section. The electrical circuit parameters of the squirrel cage include the resistance of the electrical circuit elements associated with the squirrel cage rings and the length l of the squirrel cage bars in the direction of the machine's axis of rotation .
    c) Establish the physical properties of the materials that constitute the different parts of the motor to determine: the magnetic reluctivity v of the materials that make up the induction motor and the electrical conductivity of the bars of the bars (202) of the cage of chipmunk.
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    d) Determine the conditions under which the motor operates, which includes determining the periodic current through the stator coils and the rotational speed of the machine rotor.
    e) Calculate the initial current in the squirrel cage bars.
    f) Calculate the steady state of the induction motor by solving the classical model of induced currents, or "eddy currents", from the initial currents obtained in the previous stage.
    The calculation of the initial current in the bars of the squirrel cage comprises proposing a mathematical model, hereinafter referred to as a complete model, which allows to calculate the electromagnetic fields in the cross section of the motor. This model has as a mystery the vector magnetic potential in the cross-section of the engine and the currents and potential drops in the squirrel cage circuit, and consists of two coupled submodels: a distributed submodel obtained from Maxwell's equations in low frequency and a submodel of concentrated parameters associated with the electric circuit of the squirrel cage. Thus, the calculation of the initial current in the squirrel cage bars comprises:
    to. propose a mathematical model, complete model, for the calculation of electromagnetic fields in the cross section of the induction motor;
    b. transform the complete model into a simplified model through algebraic manipulations to obtain a system of ordinary algebraic-differential equations;
    C. transform the simplified model into an approximate reduced model;
    d. solve the approximate reduced model using a numerical method; Y
    and. obtain the initial current in the bars of the squirrel cage from the solution of the approximate reduced model.
    The approach of the complete mathematical model includes:
    to. propose a distributed submodel based on Maxwell's equations for currents induced in a transient magnetostatic regime, called a magnetostatic distributed submodel, characterized in that the currents in the stator coils and rotor bars are assumed to be uniformly distributed;
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    b. rewrite the distributed submodel by changing the conventional reference system for another mobile in solidarity with the rotor;
    C. propose a sub-model of concentrated parameters for the electric circuit of the squirrel cage of the rotor obtained from Kirchhoff's laws, including as initial data the currents in the bars; Y
    d. propose the complete model by coupling the distributed sub-model of the electrostatics and the sub-model of concentrated parameters.
    Next, the complete model is transformed into a simplified model by algebraic manipulations to obtain a system of ordinary algebraic-differential equations whose unknowns are the currents in the squirrel cage bars and a scalar value, called Lagrange multiplier. The algebraic manipulations to be carried out are the following: first, for each time t> 0, an operator Ft (w) with as many mb components as bars (102) is defined in the motor rotor, Ft = (Ft , i, .., Ftm6); Each component of this operator is defined as the product of the resistance per unit length in each bar by the integral in the cross-section of said product bar between the solution of the distributed sub-model of magnetostatics at time t with data w (independent of time) and the electrical conductivity of the bar. The temporary derivative of this operator Ft (w) is then used to write the complete problem in terms of the currents and potential drops in the squirrel cage circuit. Finally, the currents in the rings and the potential drops in the entire squirrel cage of the previous system are eliminated by introducing a new scalar mystery, the Lagrange multiplier.
    The next step involves obtaining the approximate reduced model by transforming the simplified model of the previous stage. For this, the electrical period in the squirrel cage bars, Tb, is calculated from the number of pairs of motor poles, the frequency of the current in the stator coils (100) and the rotor rotation speed , and the simplified model is integrated in time in the intervals [0, t] and [0,7 ^]. Next, the order of integration is exchanged in terms that do not involve the Ft (w) operator and the term that involves the Ft (w) operator applied to the currents in the squirrel cage bars across the entire interval is removed. temporary [0.7 ^]. Finally, the currents in the squirrel cage bars are approximated by the main harmonic of their respective Fourier series developments (at frequency fb = 1 / Tb) written in terms of an amplitude common to all bars, and different angles phase relative to time
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    initial. In addition, the phases relative to the initial time of the current in the squirrel cage bars are written in terms of one of them, considering a constant lag between bars that is a function of the engine's operating conditions, thus obtaining a non-linear system overdetermining numerical equations or approximate reduced model.
    Finally, in the last stage the approximate reduced model is solved, which is an overdetermined system of nonlinear equations. In a particular embodiment of the invention the reduced model is solved by the least squares method. In turn, the overdetermined system requires the resolution of the distributed magnetostatic submodel that can be addressed by various techniques; in a particular embodiment it is solved with the finite difference method, in another particular embodiment it is solved by the finite element method. Once a solution of the approximate reduced model has been found, the first harmonic of the current in the squirrel cage bars is used to calculate the approximation of the initial current in said bars. These currents are used as initial data to calculate the steady state of the induction motor using a classic model of induced currents, or "eddy currents".
    In another aspect the invention relates to a computer program product comprising program instructions to cause a computer system to perform the procedure to calculate the steady state of a squirrel cage induction motor as described. In another aspect the invention relates to a computer program product that is stored in storage media. In another aspect the invention relates to a computer program product that is carried by a carrier wave.
    BRIEF DESCRIPTION OF THE FIGURES
    The modalities detailed in the figures are illustrated by way of example and not by way of limitation:
    Figure 1 shows a concrete embodiment of the cross section of a squirrel cage induction machine, in which the names of the different parts referred to in the description of the specific embodiment of the invention have also been indicated.
    Figure 2 shows a concrete embodiment of the directed graph of the electrical circuit
    associated with the squirrel cage of the induction motor, in which a
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    numbering that would correspond to the nodes and edges, in addition to a choice of the meaning of the edges.
    DETAILED DESCRIPTION
    A detailed description of the invention is set forth below for the particular case of a possible embodiment with reference to the figures.
    First of all, an analysis of the geometry and physical properties of the specific induction motor of which the stationary state is to be simulated must be carried out, comprising the following stages:
    a) Study the dimensions and structure of the induction motor to determine the geometry of its cross section, which will correspond to the geometric configuration at the initial moment of the transient simulations, which in the case of this embodiment includes (see the Figure 1):
    • squirrel cage bars (102),
    • stator coils (100),
    • ferromagnetic rotor and stator cores (106, 110),
    • the machine shaft (104),
    • the air gap of the machine that separates the rotor and stator cores, which is composed of air (108).
    The following notation is used for the different parts:, i = 1, ..., mb, for the rotor bars, ü¿, í = mb + 1, ..., m, for the stator coils and üm + 1 for the part of the domain that is not conductive, that is, air and ferromagnetic cores. In particular, the part corresponding to the axis of the machine will be treated as air.
    The üm + 1 domain breaks down into two parts:
    • ^ m + i (corresponding to the rotor),
    • ^ m + i (corresponding to the stator).
    Therefore, the two-dimensional domain (2D) at the initial moment will be fl = U ™ i1ü¿. All defined cross sections are considered to be sets of the xy plane while the motor axis is assumed in the z direction perpendicular to the previous plane. The vectors ex, ey and ez denote the elements of an orthonormal base of R3 in the corresponding directions.
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    b) Study the structure of the squirrel cage located in the rotor of the induction motor to determine the topology of the associated electrical circuit which, in the particular embodiment presented, is formed by the conductors associated with the cage bars and the associated resistances to the rings located at the ends of the motor, which are not included in the cross section geometry. In a particular embodiment of the invention the graph corresponding to the squirrel cage of mb bars (202) consists of 3mb edges (206) and 2mb nodes (204).
    c) Study the physical properties of the materials that constitute the different parts of the engine to determine:
    • magnetic reluctivity v of the materials that make up the induction motor. In a particular embodiment of the invention, reluctivity of the following components is provided: ferromagnetic cores, stator coils and squirrel cage bars of the rotor. The reluctivity of ferromagnetic materials is, in general, a non-linear function.

    The electrical conductivity of the squirrel cage bars; from said conductivity, the resistance per unit length of each bar is defined in the direction of the axis of rotation of the machine to be denoted by a ¿and is defined as follows:
    fnOi dxdy
    , i = 1, ..., mb.
    (one)
    • The Rt resistors of the electrical circuit elements associated with the squirrel cage rings.
    • The length l of the squirrel cage bars in the direction of the axis of rotation of the machine.
    d) Determine the conditions in which the motor operates: current in the stator coils In, n = mb + 1, ..., m, based on the time and speed of rotation of the rotor denoted by nr. In currents are periodic functions in time and their frequency is denoted by fc.
    Secondly, a mathematical model (complete model) is determined that allows to calculate the electromagnetic fields in the cross section of the motor. This model consists of two main parts: a distributed submodel obtained from Maxwell's equations in low frequency (magnetostatic distributed submodel) and a concentrated parameter submodel associated with the electric circuit of the squirrel cage.
    These two submodels will be coupled through the currents and an integral of the magnetostatic distributed submodel solution in the rotor bars.
    In order to obtain the distributed magnetostatic submodel, Maxwell's equations are considered at low frequency (that is, neglecting the term of
    5 electric displacement in Ampére's law) in a cross section of the motor and
    expressed in terms of the vector magnetic potential, which in cartesian coordinates has the form A = (0,0, Az (x, y)) J. In addition, both in the stator coils and in the rotor bars, the current is considered uniformly distributed ("stranded conductors"), thus obtaining a model in which time acts as a parameter.
    10 concrete embodiment presented, the distributed submodel is reduced to:
     curl (v curl A)  y¿ (0 "| fl¡l ez in fl¿ (t), í = 1, ..., mb, (2)
     curl (v curl A)  h (t) lflil6z in fl¿, í = mb + 1, ..., m, (3)
     curl (v curl A)  = 0 in air and cores, (4)
     TO  = 0 over 3fl, (5)
    where y¿ (í), i = 1, ..., mb, are the currents in the squirrel cage bars at the moment í> 0, 7¿ (í), í = mb + 1, m, are the currents in the stator coils at that moment and | fl¿ | It denotes the area of each section that is invariant over time. The function v will take on each domain the value corresponding to the reluctivity of the material that forms it. Next, the distributed model is rewritten by changing the reference system to a mobile one in solidarity with the rotor, that is, fl ™ + x is fixed and fl ^ l moves so that at time í, the domain fl ^ + í it has moved to the position r ^ fl ^^) where rt is the rotation whose angular velocity is the opposite of that of the rotor. Note that the air gap zone can be included in any of these subdomains or can be divided into two parts (one of them belonging to the rotor and the other to the stator). In this way, the system of
     resulting equations is:
     curl (v curl A)  y¡ (0 | fl¡l ez in fl¿, í = 1, ..., mb,
     curl (v curl A)  / ¡(O | fl¿ | ez in rt (fl¿), í = mb + 1
     curl (v curl A)  = 0 in air and cores,
     TO  = 0 over 3fl,
    For the writing of the sub-model of concentrated parameters, the Kirchhoff laws and the constituent laws of the circuit elements must be taken into account. In the specific case of the embodiment presented, the sub-model of concentrated parameters is:
    ^ y (t) = 0 (6)
    Dy (t) + <AJv (t) = 0 (7)
    where y (t) e B mb and v (t) and E2 ™ 6 are the vectors whose components are the currents at the edges and the voltages at the nodes of the graph associated to the circuit, respectively, ^ is the incidence matrix ( of dimension 2mb x3mb) of the graph associated to the circuit and D is the diagonal operator given by:
    (£ j aAz (x, y, t) dxdy + yi (t) 'J i = 1, ..., mb,
    fí¿y¿ (í) i = mb + 1, .., 3mb.
    The complete model is obtained by coupling the distributed magnetostatic submodel (2) - (5) with the concentrated parameter submodel (6) - (7), and adding an initial condition 10 for the current in the bars:
    ® ¿(y¿ (0) =
    y¿ (0) = y °, í = i, ..., mb.
    Thus, the complete model is written as follows:
    (8)
     curl (vcurl A)  y¿ (0 m¡iez in ü¿ (í), í = 1, ..., mb, (9)
     curl (vcurl A)  | fl¡ | 6z enrt (fl¡), í = mb + 1, ..., m, (10)
     curl (vcurl A)  = 0 in air and cores, (11)
     TO  = 0 over 3ü, (12)
     <Ay (t)  = ^, (13)
     'Dy (t) + ¿lTm  = ^, (14)
     and (0)  = y “, í = 1, ..., mb. (fifteen)
    Next, the complete model is rewritten in a simplified way to obtain the reduced model, whose unknowns are the currents in the squirrel cage bars and a scalar unknown (Lagrange multiplier) associated with the potential drop in one of 15 bars. To do this, the following steps are performed:
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    a) For each t> 0, the nonlinear operator Ft: Rmb given by:
    Fti (w): = ai I oAz (x, y, t) dxdy, i = l, ..., mb,
    Jai
    where the field A (x, y, t) = (0,0, Az (x, y, t)) T is the only solution of the magnetostatic distributed submodel (equations (2) to (5)) at time t considering as current the current in the bars given by y = w.
    b) Thanks to the Ft operator, and defining the following diagonal matrices:
    (Lb) ij = lSiJ, (: Rb) iJ = laiSiJ, i, j = 1, ..., mb,
    ~) íj ~ ^ í + m¡, ^ íj, i, j ~ 1, ■■■, 3w.b - mb,
    you can write the whole problem in a simplified way
     £ b ^ ftFt (y + Jlby b (t) + (¿Lbym = 0,  (16)
     Rry r (t) + (^ r) Tv (t) = 0,  (17)
     Aby b (t) + JFyr (t) = 0,  (18)
     yb (0) = yb, °,  (19)
    where yb is the vector whose components are the currents in the edges of the graph corresponding to the squirrel cage bars, and r is the vector whose components are the currents in the remaining edges of the graph associated with the squirrel cage and <Ab y Jlr are the parts of the incidence matrix of the graph corresponding to the edges that represent the bars and the rest of the edges, respectively.
    c) The reduced model is then obtained by eliminating the unknowns yr and v from the previous system through a series of algebraic manipulations. In a particular embodiment, the unknowns are eliminated by expressing the potential drop vector of the form:
    v (t) = B ~ Aby b (t) + A (t) (0),
    where e: = (1,., 1) Tel ”6 and B is the matrix obtained from Ar ^ ~ 1 {Ar) T by placing very large numbers in two diagonal positions. In the present invention the term very large numbers refers to large numbers compared to the orders of magnitude present in the system. In the specific embodiment it is proposed to choose these positions as those corresponding to the nodes
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    of the ends of the same squirrel cage bar and put numbers of the order of 10so in these positions. Thus, the unknown scalar A (t) represents the potential drop along this bar; In addition, equation (6) is verified, which in this case is reduced to:
    cAbyb {t) • = 0.
    d) In this way the reduced model is obtained whose unknowns are the currents in the squirrel cage bars and a scalar unknown (Lagrange multiplier) associated with the potential drop in one of the bars. This model turns out to be the following:
    £ b ^ Ft (yb (í)) + + (ctfh) TB_1 (^ h)) and h (t) + A (t) (^ h) T = 0,
    (twenty)
    (twenty-one)
    yb (0) = yb ’°.
    (22)
    The next step is to obtain the approximate reduced system from the previous reduced model. To perform this step, the following steps are performed:
    a) Calculate the electrical period of the current in the squirrel cage bars from the number of pairs of poles p of the machine, the frequency of the current in the stator coils, fc, and the rotational speed of the rotor nr:
    60
    Y = __________
    60fc - p nr ‘
    b) Integrate in time equations (20) and (21) of the reduced model in the interval [0, í], with te [0, r¡,] taking into account the initial condition (22) to obtain the system:
    t (y b (0) - F0 (y-M)) + (xb + (cAi ’) V1 (^))
    + J A (s) ds (<Ab) T (0) = 0
    image 1
    F
    and b (s) ds
    (2. 3)
    [y b (s) ds • Jo
    ^ UNCLE) -
    0.
    (24)
    c) Integrate in time equations (23) and (24) in the interval [0,7 ^] to obtain the system:
    ¿B (f bFt (y t)) dt-F0 (and b ’°) Tb ^ + (JZb + (cAb) TS-1 (^ b)) f and b (s) ds ^ dt
    5
    +
    image2
    image3
    (25)
    f b (f and b (s) ds) say • (^ by (°) = 0. (26)
    d) Exchange the order of integration in the terms of (25) and (26) that do not involve the Ft operator, obtaining:
    £ b (f * Ft (y t)) dt - FQ (yb, °) Tb ^ + (&> + (cAb) TB- ¿lb)) j fb -s) and b (s) ds
    + (| rb- s) A (s) ds ^ (Ab) T (0) = 0,
    l 6 (rb-s); and b (s) ds ^ (^ b) T (e) = 0.
    e) Disregard the first term of equation (27) obtaining the system:
    r ^ b
    —LbF0 (yb’0) Tb + (Rb + (c / lb) TB ~ 1Ab) I (Tb -s) and b (s) ds
    Jo
    + (f rb - s) A (s) ds) (Aby (°) = 0,
    -s) and s) ds • (^ b) T (°) = 0.
    (27)
    (28)
    (29)
    (30)
    f) Approximate the currents in the squirrel cage bars by the main harmonic of their respective Fourier series developments (at frequency fb: = 1 / Tb), written in terms of a common amplitude Y and the phase angles £ > n, n = l, .., mb, relative to the initial time
    I -
    (^ b) T
    0
    and
    I (Ycos (2nfbt + pn)).
    - * n
    (31)
    g) Write the phases relative to the initial time pn, n = 1, ..., mb, in terms of one of them considered a constant lag
    2 np a = -----.
    mb
    If the bar chosen for this is the first,
    Pn = Pi + (n - 1) a, n = 1, ..., mb.
    (32)
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    10
    h) Substitute the components of the current vector in the squirrel cage bars of the system (29) - (30) with the approximate currents, obtaining an over-determined non-linear system of numerical equations (approximate reduced model),
    - £ bF0 {Yv.1) Tb-1¡LY {jlb + (cAb) TB ~ 1¿lb) Ü2 + ^ (Ab)
    2n
    u2 - (c ^) T ^ -
    b ^ T (0 _
    and.
    0.
    0, (33)
    (3. 4)
    where
    rTb
    / u: = I (
    Jo
    (Tb - s) A (s) ds,
    and U1, u2 are vectors whose n-th components, n = l, .., mb, are respectively
    ul, n:
    (Ab) T I (
    in
    cos + (n — 1) a)
    U2, n:
    (Ab) T I (
    without + (n - l) a).
    i) Eliminate the unknown ^ in terms of the amplitude of the main harmonic of the current in the squirrel cage bars and the phase of the main harmonic of the current in one of the bars relative to the initial moment:
    -i
    (Rb + (cab) T B-1 Ab ^ LbF0 (Yv.1) Tb - (cdb)
    T '(!
    and
    F - -
    CBb + (cAb) T B-1 ^ b) ~ 1 (^ b) T - (c ^ b) T ^
    Thus, the approximate reduced model obtained from (33) - (34) is reduced to:
    - £ bF0 (Y: uí) Tb -Y ^ (3lb + (cAb) TB ~ 1¿lb) Ü2 2 n
    H—
    to
    {3lb + (cAb) T B ~ x Jlb) £ 1 bF0 (YUí) Tb - (cAb) T ((
    in
    (¿Iby (0) = 0,
    where
    (35)
    a: = (! Rb + (cAb) T B ~ x Jlb) 1 (¿lb) T (£) - (c ^) 7 ((
    e / e¿
    To effect the resolution of the approximate reduced model, a least squares method has been chosen. In this way, if defined
    f (Y, p. = - £ bF0 (YHí) Tb -Y2 ^ (Xb + (c / lb) TB ~ 1Ab) u2
    l
    He has
    (3lb + (cAb) T B-1 Jlb) £ 1 bF0 (YU1) Tb - (cAb) T ((16
    in
    Go
    Y
    y ^ (Y, pi): = l f (y, pi) \ 22, look for Y and ^ such that
    0 (7, £ i) = mm {0 (Z, y): ymin <Z <Ymax, 0 <and <2tt}.
    In particular, the calculation of F0 (YÜ.1) involves the resolution of the distributed sub-model of electrostatics with current data in the bars given by Yií1. To do this, in this specific embodiment of the invention, this sub-model of magnetostatics is discretized by means of finite nodal element techniques, and iterative algorithms are used to handle the non-linearities of the laws constituting the ferromagnetic materials of the motor cores.
    Once the solution (Y, ^ 1) of the approximate reduced model is found, an initial current for the current in the squirrel cage bars is obtained by evaluating the 10 expressions (31) at the time t = 0.
    Finally, to calculate the steady state of the induction motor, a classical model of induced currents ("eddy currents") is solved with the initial conditions obtained following the previous procedure.
    权利要求:
    Claims (1)
    [1]
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    1- A method for the accelerated calculation of the steady state of a squirrel cage induction motor, characterized in that it comprises the following stages:
    a) determine the geometry of a cross section of the induction motor, which is divided into rotor (102, 104 and 106) and stator, the rotor comprising those moving parts and the stator the rest of the motor parts;
    b) model the electrical circuit corresponding to the motor squirrel cage;
    c) establish the physical properties of the materials that constitute the engine parts and the electrical circuit parameters of the squirrel cage;
    d) determining the operating conditions of the motor comprising determining the current through the stator coils and the rotation speed of the machine rotor;
    e) calculate the initial current in the squirrel cage bars; Y
    f) calculate the steady state of the induction motor by solving the classic induced currents model, or "eddy currents", from the initial currents obtained in the previous stage.
    2- The method according to claim 1, characterized in that the physical properties of the motor materials comprise the magnetic reluctivity v of the materials that make up the induction motor, the electrical conductivity of the squirrel cage bars.
    3- The method according to claim 1, characterized in that the parameters of the electrical circuit of the squirrel cage comprise the resistance of the elements of the electrical circuit associated with the squirrel cage rings and the length l of the cage bars of squirrel in the direction of the axis of rotation of the machine.
    4- The method according to claim 1, characterized in that the geometry of a cross-section includes at least the stator coils, the squirrel cage bars, the ferromagnetic rotor and stator cores, the air gap and the shaft the motor.
    5- The method according to claim 1, characterized in that the modeling of the electrical circuit is carried out by means of a graph, which includes both the bars (202)
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    as the resistors (200) corresponding to the outer rings of the squirrel cage, not included in the cross section geometry.
    6- The method according to claim 1, characterized in that the calculation of the initial currents in the squirrel cage bars comprises:
    a) propose a mathematical model, complete model, for the calculation of electromagnetic fields in the cross section of the induction motor;
    b) transform the complete model into a simplified model by algebraic manipulations to obtain a system of ordinary algebraic-differential equations;
    c) transform the simplified model into an approximate reduced model;
    d) solve the approximate reduced model using a numerical method; Y
    e) obtain the initial current in the squirrel cage bars from the solution of the approximate reduced model.
    7- The method according to claim 6, characterized in that posing the complete mathematical model comprises:
    a) propose a distributed submodel based on Maxwell's equations for currents induced in a transient magnetostatic regime, called a magnetostatic distributed submodel, characterized in that the currents in the stator coils and rotor bars are assumed to be uniformly distributed;
    b) rewrite the distributed submodel by changing the conventional reference system with another mobile in solidarity with the rotor;
    c) propose a sub-model of concentrated parameters for the electric circuit of the squirrel cage of the rotor obtained from Kirchhoff's laws, including as initial data the currents in the bars; Y
    d) propose the complete model by coupling the distributed sub-model of electrostatics and the sub-model of concentrated parameters.
    8- The method according to claims 6 and 7, characterized in that transforming the complete model into a simplified model by algebraic manipulations comprises:
    a) Define for each time t> 0 a real nonlinear operator Ft (w) with Ft =
    (F1¡t, ..., Ft¡mb), mb being the number of bars of the motor rotor; every
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    This operator component is defined as the product of the resistance per unit length in each bar by the integral in the cross-section of said product bar between the solution of the distributed sub-model of magnetostatics, with data w (independent of time), and the electrical conductivity of the bar;
    b) use the temporary derivative of the operator Ft (w) to write the complete problem based on the currents and potential drops in the squirrel cage circuit;
    c) eliminate the currents in the rings and the potential drops throughout the squirrel cage of the entire problem by introducing an unknown scalar, Lagrange multiplier.
    The method according to claims 6 to 8, characterized in that transforming the simplified model into an approximate reduced model comprises:
    a) calculate the electric frequency in the bars of the squirrel cage Tb from the number of pairs of poles of the machine, the frequency of the current in the stator coils and the speed of rotation of the rotor;
    b) integrate into the time of the reduced model in the intervals [0, t] and [0, Tb], where t is any time in the interval [0, Tb];
    c) exchange the order of integration of the integrals of the previous stage in terms that do not involve the operator Ft (w);
    d) eliminate the nonlinear term that involves the operator Ft (w) applied to the currents in the squirrel cage bars throughout the entire time interval of integration;
    e) approximate the currents in the squirrel cage bars by their respective Fourier series developments, at frequency fb = 1 / Tb, written in terms of a common amplitude and the phase angles relative to the initial time; Y
    f) write the phases relative to the initial time of the current in the bars of the squirrel cage in terms of one of them, considering a constant offset that is a function of the number of poles of the motor and the number of bars, obtaining a system not linear overdetermined numerical equations or approximate reduced model.
    10- Computer program product that includes program instructions for
    cause a computer system to perform the procedure to calculate the
    twenty
    steady state of a squirrel cage induction motor according to any one of claims 1 to 9.
    11- Computer program product, according to claim 10 which is stored
    5 in storage media.
    12- Computer program product according to claim 10, which is carried by a carrier wave.
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    同族专利:
    公开号 | 公开日
    ES2687868B2|2019-07-29|
    WO2019170937A1|2019-09-12|
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    ES201830228A|ES2687868B2|2018-03-08|2018-03-08|Procedure and product of a computer program to accelerate the calculation of the steady state of a squirrel cage induction motor|ES201830228A| ES2687868B2|2018-03-08|2018-03-08|Procedure and product of a computer program to accelerate the calculation of the steady state of a squirrel cage induction motor|
    PCT/ES2019/000022| WO2019170937A1|2018-03-08|2019-03-08|Method and computer program product for speeding up the calculation of the stationary state of a squirrel-cage induction motor|
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