Dual three-phase permanent magnet synchronous motor harmonic suppression method based on virtual back electromotive force identification

Abstract

The invention discloses a dual three-phase permanent magnet synchronous motor harmonic suppression method based on virtual back electromotive force identification, and the method comprises the following steps: S1, building a driving system model of a DTP-PMSM; s2, constructing a harmonic model of the DTP-PMSM, analytically decomposing the nonlinear characteristic and the non-sinusoidal counter electromotive force of the inverter into dq and dqz subspaces, introducing the concept of virtual harmonic counter electromotive force, and unifying the nonlinearity of the inverter, the space harmonic counter electromotive force and cross coupling terms related to harmonic current into a single expression form; and S3, developing an identification principle based on current mapping, including amplitude and phase estimation, gain design and delay compensation, so as to estimate the amplitude and phase deviation of the virtual harmonic back electromotive force on line to realize harmonic suppression and avoid a complex algorithm and a large amount of parameter adjustment. According to the method, the operation effect is good in a wide rotating speed range, and meanwhile complex algorithms and a large amount of parameter adjustment are avoided.

Description

Technical Field

[0001] This invention relates to the field of dual three-phase permanent magnet synchronous motor technology, specifically to a control method for dual three-phase permanent magnet synchronous motors based on virtual harmonic back electromotive force identification and compensation. Background Technology

[0002] Multiphase motors have gained significant attention in electric vehicles, marine propulsion, and aerospace applications due to their advantages such as low torque ripple, high reliability, and strong fault tolerance. Among them, DTP-PMSMs, also known as asymmetrical six-phase motors, are widely considered a practical and efficient configuration. DTP-PMSMs consist of two sets of three-phase stator windings with a 30° electrical angle offset between them and mutually isolated neutral points. They are driven by a six-phase inverter (VSI). Figure 1 As shown in the diagram, this structure achieves a good balance between simplicity and performance.

[0003] To achieve efficient operation, the stator current of a DTP-PMSM should ideally be six balanced sinusoidal waveforms forming two independent three-phase groups. However, in actual operation, achieving the ideal sinusoidal air-gap flux is difficult due to factors such as slot effect, winding distribution, magnet geometry, and manufacturing tolerances. Furthermore, the nonlinear characteristics of the PWM inverter, such as dead time, voltage drop of switching devices, diode recovery time, and digital or PWM delay, introduce voltage harmonics characterized by 6k±1. After vector space decomposition, the distribution of these harmonics is non-uniform: the fundamental and 12th-order components combine in the dq subspace, while the 5th and 7th-order components appear in the harmonic xy subspace, where they primarily increase copper losses and reduce efficiency. Compared to traditional three-phase motors, the DTP-PMSM has a smaller effective impedance in the harmonic subspace, thus making it more likely for harmonic components to generate larger harmonic currents. In particular, the 12th harmonic mapped to the dq subspace is attenuated less at low electrical frequencies due to the smaller motor impedance, resulting in significant torque pulsation that cannot be ignored.

[0004] Harmonic currents can be mitigated by optimizing the electromagnetic structure of the motor. For example, the magnetomotive force of harmonics can be reduced through skewed slot design, rotor tilting, or pole configuration design. However, these measures cannot completely eliminate harmonic currents. Fortunately, harmonic current suppression has been extensively studied, and widely adopted methods include the Multiple Synchronous Rotating Coordinate Transformation (MSRFT) method and the Resonant Control (RC) method.

[0005] Multiple Synchronous Rotating Coordinate Transformation (MSRFT) has been widely used in DTP-PMSM for harmonic suppression because it enables selective tuning of specific frequency components by converting them into DC quantities in their corresponding synchronous coordinate systems. This framework provides a simple and straightforward method for achieving zero steady-state error for specific harmonics using PI controllers. However, this significantly increases the number of PI controllers and low-pass filters used, complicating the tuning and implementation process. Furthermore, the low-pass filters used for harmonic extraction introduce a large phase delay, which reduces dynamic response and can potentially affect system stability. Their effectiveness is further reduced when the frequency spacing between the fundamental and harmonic components is small, leading to insufficient harmonic suppression. Additionally, digital and PWM delays in higher-order coordinate systems cause additional phase hysteresis, reducing stability margin unless explicit compensation is introduced. Although several improvements have been proposed, such as closed-loop detection schemes to improve extraction accuracy in high-speed applications and deadbeat predictive controllers (which combine harmonic suppression with online parameter identification), the overall framework remains complex, delay-sensitive, and difficult to implement in practical applications.

[0006] In contrast, the RC method requires no additional filters; multiple resonant regulators can simultaneously suppress specific harmonic currents without introducing steady-state errors. A paper titled "Current control methods for an asymmetrical six-phase induction motor drive" proposes a general model for a six-phase induction motor and applies a resonant controller to suppress the 5th and 7th harmonic currents. However, the RC method requires careful tuning of the controller parameters to maintain effective harmonic suppression across varying operating speeds, making its practical application challenging. Furthermore, the performance of the resonant controller is highly sensitive to the discretization method; shifting the resonant poles at the resonant frequency leads to a decrease in harmonic suppression capability at high speeds. A paper titled "Effects of discretization methods on the performance of resonant controllers" evaluates several discretization methods, including zero-order hold, bilinear transformation, forward Euler method, and zero-pole matching; significant steady-state errors may occur at high frequencies due to resonant pole displacement. To address this issue, a paper titled "Digital current control of an asymmetrical dual three-phase flux-switching permanent magnet machine" directly designed and optimized a resonant controller in the z-domain to improve stability margin. However, this approach still presents a trade-off between accurately tracking the target value and the system's anti-interference capability.

[0007] Furthermore, Active Disturbance Rejection Control (ADRC) has been extensively studied in harmonic suppression, with recent research focusing on improving observer design to enhance disturbance estimation and compensation capabilities. These developments can be broadly categorized into two types. The first involves optimizing the observer structure, with the Cascaded ESO (CESO) architecture being widely used. By cascading multiple ESO units, CESO improves disturbance estimation performance, but the expanded observation range inevitably reduces steady-state accuracy. The second type incorporates auxiliary components into the observer loop. A typical example is the Generalized Integral ESO (GI-ESO), which extends LESO by embedding multiple resonant controllers of different frequencies into the estimation loop. This method can extract and suppress periodic disturbances under specific harmonics. Despite its significant effect on harmonic suppression, GI-ESO still suffers from the same limitation as quasi-resonant structures: the sensitivity of resonant poles to discretization, which inevitably leads to a decrease in suppression performance in the high-frequency region. Other control-based strategies, such as vector resonant control, repetitive control, and adaptive linear neuron controllers, have also been used in harmonic suppression research. However, these methods are generally more complex and require detailed parameter tuning. Summary of the Invention

[0008] To address the aforementioned problems, this invention proposes a method for identifying and compensating for virtual harmonic back electromotive force. This method performs well over a wide speed range while avoiding complex algorithms and extensive parameter adjustments.

[0009] The specific plan is as follows:

[0010] A control method for a dual three-phase permanent magnet synchronous motor based on virtual harmonic back EMF identification and compensation includes the following steps:

[0011] S1. Establish the drive system model of DTP-PMSM;

[0012] S2. Construct a harmonic model of DTP-PMSM, analytically decompose the nonlinear characteristics of the inverter and the non-sinusoidal back EMF into the dq and dqz subspaces, and introduce the concept of virtual harmonic back EMF to unify the inverter nonlinearity, spatial harmonic back EMF and cross-coupling terms related to harmonic current into a single representation.

[0013] S3. Develop an identification principle based on current mapping, including amplitude and phase estimation, gain design, and delay compensation, to estimate the amplitude and phase deviation of the virtual harmonic back electromotive force online to achieve harmonic suppression, while avoiding complex algorithms and a large number of parameter adjustments.

[0014] Furthermore, in step S1, establishing the drive system model of DTP-PMSM includes:

[0015] The six-phase voltage-source inverter (VSI) used in the DTP-PMSM driver generates 64 different switching states, which determine the instantaneous values ​​of the stator phase voltages; these voltages can be simply represented as a six-dimensional vector:

[0016]

[0017] Among them, u s It is the phase voltage;

[0018] To facilitate decoupling control and harmonic analysis, the stator voltage vector u is first converted using VSD. s Projected onto two orthogonal subspaces —αβ and —xy;

[0019] For decoupling control and harmonic analysis, the stator voltage vector is decomposed into two subspaces, αβ and xy, using VSD:

[0020]

[0021] Among them, T VSD It is the VSD transformation matrix;

[0022] Defined in the form of a decomposition matrix:

[0023]

[0024] To simplify harmonic analysis, the xy subspace is further transformed into a frame that focuses the sixth component at a fixed spectral location; the resulting dq-dqz frame voltage vector is:

[0025]

[0026] Among them, T dqdqz ( ) is the transformation matrix from the stationary coordinate system to the rotating coordinate system. It refers to the position of the motor rotor, and

[0027]

[0028] Assuming the existence of an isolated neutral point, the zero-sequence component will be naturally suppressed. Under common assumptions such as sinusoidal winding distribution, symmetrical winding parameters, and neglect of core losses, the voltage equation of the DTP-PMSM in the dq-dqz coordinate system is expressed as:

[0029]

[0030] Where u dqThese are the voltages in the dq coordinate system, u and u. dqz Let i be the voltage in the dqz coordinate system. dq Let i be the current in the dq coordinate system. dqz Let ω be the current in the dqz coordinate system. e L is the electric angular velocity. d and L q These are the equivalent inductances in the dq coordinate system, L and L. dz and L qz R represents the equivalent inductance in the dqz coordinate system, respectively. s ψ is the stator winding resistance. f This refers to the magnetic flux linkage of a permanent magnet.

[0031] Furthermore, in step S2, two factors are considered: inverter nonlinearity, which is caused by dead time and voltage drops of devices and diodes; and non-sinusoidal back electromotive force generated by space flux harmonics.

[0032] (A) Harmonic modeling of back electromotive force

[0033] For DTP-PMSM, due to the non-sinusoidal nature of the flux distribution, the back electromotive force in the phase domain contains multiple harmonic components. To characterize the spatial harmonic content of the permanent magnet flux linkage, in addition to the fundamental wave, the expression for the phase domain back electromotive force is extended to include the 5th, 7th, 11th, and 13th harmonics.

[0034]

[0035] Where, ψ n Let θ represent the magnetomotive force amplitude of the nth harmonic, and θ represent the magnetomotive force amplitude of the nth harmonic. n That is the corresponding phase offset.

[0036] By applying these transformation matrices The back electromotive force component in the subspace is obtained as:

[0037]

[0038] As can be seen from equation (6), in the dq subspace, the 11th and 13th harmonics are mapped to a unified 12th component, which directly affects the generation of torque; in contrast, in the dqz subspace, the 5th and 7th harmonics are mapped to a unified 6th component, which leads to a reduction in motor efficiency.

[0039] (B) VSI nonlinear modeling

[0040] The phase output voltage error is related to the direction of the phase current, and is expressed as:

[0041]

[0042] Where i s Represents phase current, A p These are coefficients related to device characteristics;

[0043] To obtain a more accurate representation, the dead time duration T was considered. dead On-delay T on and shutdown delay T off In one switching cycle T s The average voltage error is given by the following formula:

[0044]

[0045] Where V dc This is the DC bus voltage, while V ce and V d These represent the voltage drop across the conducting device; V ce and V d These represent the saturation voltage of the active switch and the forward voltage of the anti-parallel diode, respectively.

[0046] The distortion in the inverter output itself contains multiple harmonics, and its Fourier series expansion is as follows:

[0047]

[0048] By applying the transformation matrix The distortion voltage component in the stationary dq-dqz coordinate system can be obtained as follows:

[0049]

[0050] and

[0051]

[0052] in, and Let be the distortion voltage component in the αβ coordinate system. and The distortion voltage component in the xy coordinate system;

[0053] For the nth harmonic, the motor response depends largely on the frequency-dependent impedance value in each decoupling subspace; the impedance vector is expressed as:

[0054]

[0055] in This represents the equivalent inductance value in the dq-subspace and dqz-subspace, and the corresponding values ​​are:

[0056]

[0057] As the frequency increases, the impedance increases, and the attenuation of higher harmonics also increases. Conversely, at low frequencies, the impedance remains relatively small, which reduces the motor's natural filtering capability for lower harmonics. This phenomenon explains why, at low speeds, the 12th perturbation in the dq subspace has a significant impact on torque fluctuations, while the 6th perturbation in the dqz subspace exacerbates the decrease in efficiency.

[0058] Considering the distortion and non-sinusoidal back EMF caused by the inverter, the complete voltage equation of the motor is finally expressed as:

[0059]

[0060] (C) Equivalent harmonic circuit modeling

[0061] To facilitate clear harmonic analysis, each variable is decomposed into its basic components and harmonic components.

[0062]

[0063] in, , , These represent the fundamental components distributed in the subspaces dq and dqz, respectively. , , These represent the corresponding harmonic components in the same subspace;

[0064] Substituting equation (15) into the voltage equation, the equivalent harmonic circuit model can be expressed as:

[0065]

[0066] The subscripts “12” and “6” represent the effective harmonic orders in the dq and dqz subspaces, respectively.

[0067] To achieve a unified expression, virtual harmonic back EMF is defined as including nonlinear inverter interference, spatial harmonic back EMF, and circuit mutual coupling terms related to harmonic currents; their definitions are as follows:

[0068]

[0069] in and Let represent the 12th virtual harmonic back electromotive force in the dq subspace, and respectively. and These represent the 6th virtual harmonic back electromotive force in the dqz subspace;

[0070] Substituting (17) into (16), we obtain the simplified harmonic voltage equation for DTP-PMSM as follows:

[0071]

[0072] The above equations show that, under harmonic excitation, the DTP-PMSM can be modeled as an equivalent RL circuit driven by the virtual harmonic BEMF.

[0073] Furthermore, in step S3, based on the harmonic circuit model of DTP-PMSM, the influence of amplitude and phase errors on the system under feedforward compensation conditions is analyzed, providing a theoretical basis for the online identification algorithm of harmonic back electromotive force.

[0074] (A) Virtual harmonic back EMF estimation and current mapping

[0075] Based on the harmonic circuit model of DTP-PMSM, the true virtual back electromotive force is represented as:

[0076]

[0077] in and These represent the equivalent amplitude and phase vectors of the virtual back electromotive force, respectively, including d-, q-, dz-, and qz- components; for the dq subspace, n=12, and for the dqz subspace, n=6;

[0078] If the inverter generates a harmonic compensation voltage, it is represented as:

[0079]

[0080] in, and These represent the estimated amplitude vector and phase vector, respectively;

[0081] The controller measures real-time current. Harmonic currents are extracted using RC subtraction:

[0082]

[0083] in This represents the reference current, which remains constant under steady-state conditions.

[0084] Therefore, winding voltage Represented as:

[0085]

[0086] Therefore, when When the harmonic current is eliminated, the harmonic current will be eliminated; however, due to the impedance angle, the harmonic current in each subspace will be eliminated. It will be relative to the applied voltage Lag; these impedance angles are expressed as:

[0087]

[0088] Among them, all values All are within the range of (0°, 90°);

[0089] This mapping shows that the estimated harmonic voltage With respect to actual harmonic back electromotive force The amplitude or phase deviation between them will be directly reflected in the harmonic current. Therefore, harmonic current serves as a practically observable variable for identifying and quantifying such deviations.

[0090] Average harmonic current It is by... The result is obtained by summing up the data from each half-cycle and then dividing by the number of samples:

[0091]

[0092] in For the accumulated current, The number of current samples;

[0093] Furthermore, the half-cycle average current is updated at each zero-point crossover.

[0094]

[0095] Ensure its It can always accurately reflect the average current value within the most recent half-cycle;

[0096] (B) Amplitude and phase identification principle:

[0097] and The characteristic mapping between them and the harmonic currents provide clear and robust characterizations of amplitude and phase mismatch; using these characterizations, and its half-cycle average It is considered an observable to drive the online estimation scheme developed below;

[0098] 1) Phase identification:

[0099] When the average harmonic current is detected This indicates that the estimated harmonic voltage and actual harmonic back electromotive force There is a 90° phase deviation;

[0100] when Leading When the estimated harmonic voltage vector passes through the zero point of the positive half-axis, the instantaneous harmonic current satisfies the condition; conversely, when... When the negative half-axis crosses the zero point, Similarly, when behind At that time, the above relationship is reversed; therefore, and The relative phase relationship between them is directly from and It is inferred from the polarity;

[0101] Based on this relationship, the phase update rule is expressed as follows:

[0102]

[0103] in It is the phase update gain vector. This refers to the electrical angular velocity used to ensure that the correction direction is consistent with the motor rotation direction;

[0104] 2) Amplitude identification:

[0105] Since the virtual harmonic back electromotive force maintains a fixed amplitude and phase under steady-state conditions, the estimated amplitude must be adaptively adjusted to ensure that the phase angle between the harmonic voltage and current is close to 90°, thereby triggering phase identification.

[0106] The reference magnitude vector will be updated using the following control rules:

[0107]

[0108] in This represents the amplitude gain vector, where all elements are positive constants; therefore, the amplitude will change according to... polarity and The symbols are iteratively modified;

[0109] 3) Gain design and delay compensation:

[0110] Since the frequency of amplitude identification is much higher than that of phase identification, because the former is updated twice in each harmonic voltage cycle, a larger gain is required at low speeds to ensure fast convergence, while a smaller gain is required at high speeds to ensure stability. In addition, the gain design must take into account both the dq subspace and the dqz subspace, because their harmonic frequencies are different.

[0111] Amplitude gain vector Represented as

[0112]

[0113] Where k p It is a positive constant;

[0114] Amplitude and phase compensation terms were introduced as follows:

[0115]

[0116] The following calculations are based on the given compensation benefits and phase terms:

[0117]

[0118] in These represent the amplitude compensation gain, while This represents the phase compensation value.

[0119] The beneficial effects of this invention are:

[0120] 1) This invention analytically decomposes the nonlinear characteristics and non-sinusoidal back electromotive force of the inverter into dq and dqz subspaces. Analysis shows that the 12th harmonic in the dq subspace has a significant impact on torque ripple, especially at low speeds, while the 6th harmonic in the dqz subspace mainly leads to increased copper losses and reduced efficiency.

[0121] 2) A concept of virtual harmonic back EMF is constructed, which is jointly represented by inverter distortion, spatial harmonic back EMF, and dq / dqz circuit cross-coupling terms related to harmonic currents. This concise representation simplifies the handling of multi-source interference and provides a basis for online compensation.

[0122] 3) A current-mapping-based identification method is proposed, which determines the amplitude and phase deviation between the virtual harmonic back EMF and the observed harmonic current response. An iterative update mechanism is derived, including coordinated gain design and delay compensation, to ensure stable and accurate convergence over a wide speed range. Attached Figure Description

[0123] Figure 1 This is a diagram of a dual three-phase motor drive scheme.

[0124] Figure 2 The experimental back electromotive force spectrum of DTP-PMSM at 500 r / min is shown in (a) phase back electromotive force waveform and (b) harmonic spectrum.

[0125] Figure 3 The diagram shows the nonlinear characteristics of VSIs, (a) the equivalent circuit of the phase bridge arm; and (b) the gate drive signal and phase voltage output.

[0126] Figure 4 The circuit model diagrams are for the sixth harmonic: (a) the d-axis circuit model of the sixth harmonic; and (b) the q-axis circuit model of the sixth harmonic.

[0127] Figure 5 The diagram shows the characteristics of harmonic currents under different amplitudes and phase deviations, (a) Advanced (b) Lag (c) (d) .

[0128] Figure 6 This is a diagram illustrating the amplitude identification process of the proposed method.

[0129] Figure 7 It is the overall control block diagram and the flowchart for identifying the virtual back electromotive force.

[0130] Figure 8 The figures show the robustness of the proposed controller under varying parameters. (a) Nominal parameter; (b) 500% of nominal parameter; (c) 20% of nominal parameter.

[0131] Figure 9 The following are the steady-state performance diagrams of the proposed method at low speeds: (a) phase current; (b) amplitude, phase, and current of the principal plane; (c) amplitude, phase, and current of the harmonic plane.

[0132] Figure 10 The following are steady-state diagrams of the proposed method at high speed: (a) phase current; (b) amplitude, phase and current of the principal plane; (c) amplitude, phase and current of the harmonic plane.

[0133] Figure 11 The figures show a comparison of the steady-state performance of the RC method and the proposed method at different speeds: (a) RC method at 500 r / min; (b) proposed method at 500 r / min; (c) RC method at 800 r / min; (d) proposed method at 800 r / min.

[0134] Figure 12 Here are the anti-interference performance diagrams of the proposed method, (a) phase current i αβ (b) Amplitude, phase, and current i in the principal plane αβ (c) Amplitude, phase, and current i in the harmonic plane xy .

[0135] Figure 13 Here is a dynamic tracking performance graph of the proposed method; (a) Phase current i αβ (b) Amplitude, phase, and current i in the principal plane αβ (c) Amplitude, phase, and current i in the harmonic plane xy . Detailed Implementation

[0136] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the present invention.

[0137] This invention provides a control method for a dual three-phase permanent magnet synchronous motor based on virtual harmonic back electromotive force identification and compensation, comprising the following steps:

[0138] S1. Establish the drive system model of DTP-PMSM;

[0139] S2. Construct a harmonic model of DTP-PMSM, analytically decompose the nonlinear characteristics of the inverter and the non-sinusoidal back EMF into the dq and dqz subspaces, and introduce the concept of virtual harmonic back EMF to unify the inverter nonlinearity, spatial harmonic back EMF and cross-coupling terms related to harmonic current into a single representation.

[0140] S3. Develop an identification principle based on current mapping, including amplitude and phase estimation, gain design, and delay compensation, to estimate the amplitude and phase deviation of virtual harmonic back electromotive force online to achieve harmonic suppression, while avoiding complex algorithms and a large number of parameter adjustments.

[0141] S4. Experimental results confirm the effectiveness of the proposed method over a wide speed range, demonstrating that it achieves significant harmonic suppression, reduces torque ripple, and improves efficiency in the DTP-PMSM drive system.

[0142] In step S1, establishing the drive system model of DTP-PMSM includes:

[0143] like Figure 1 As shown, the six-phase voltage-source inverter (VSI) used in the DTP-PMSM driver generates 64 different switching states, which determine the instantaneous values ​​of the stator phase voltages; these voltages can be simply represented as a six-dimensional vector:

[0144]

[0145] To facilitate decoupling control and harmonic analysis, the stator voltage vector u is first converted using VSD. s Projected onto two orthogonal subspaces —αβ and —xy;

[0146] For decoupling control and harmonic analysis, the stator voltage vector is decomposed into two subspaces, αβ and xy, using VSD:

[0147]

[0148] Defined in the form of a decomposition matrix:

[0149]

[0150] To simplify harmonic analysis, the xy subspace is further transformed into a frame that focuses the sixth component at a fixed spectral location; the resulting dq-dqz frame voltage vector is:

[0151]

[0152] as well as

[0153]

[0154] Assuming the existence of an isolated neutral point, the zero-sequence component will be naturally suppressed. Under common assumptions such as sinusoidal winding distribution, symmetrical winding parameters, and neglect of core losses, the voltage equation of the DTP-PMSM in the dq-dqz coordinate system is expressed as:

[0155]

[0156] In step S2, two factors are considered: inverter nonlinearity, which is caused by dead time and voltage drops of devices and diodes; and non-sinusoidal back electromotive force generated by space flux harmonics; these models form the basis for subsequent compensation and control design.

[0157] (A) Harmonic modeling of back electromotive force

[0158] For DTP-PMSM, due to the non-sinusoidal nature of the flux distribution, the back electromotive force in the phase domain contains multiple harmonic components; the measured phase back electromotive force waveform of the DTP-PMSM is as follows: Figure 2 As shown in (a), the corresponding harmonic spectrum is as follows: Figure 2 As shown in (b), it can be seen that, in addition to the lower harmonics of the 3rd, 5th, and 7th, the 11th and 13th harmonics are also clearly visible, with amplitudes of 0.86% and 0.99% of the fundamental frequency, respectively. Although their magnitudes are relatively small compared to the lower harmonics, the 11th and 13th harmonics are particularly important because they map into the dq subspace and thus interact directly with the torque output.

[0159] To characterize the spatial harmonic content of the permanent magnet flux linkage, in addition to the fundamental frequency, the expression for the phase domain back electromotive force is extended to include the 5th, 7th, 11th, and 13th harmonics:

[0160]

[0161] By applying these transformation matrices The back electromotive force component in the subspace is obtained as:

[0162]

[0163] As can be seen from equation (6), in the dq subspace, the 11th and 13th harmonics are mapped to a unified 12th component, which directly affects the generation of torque; in contrast, in the dqz subspace, the 5th and 7th harmonics are mapped to a unified 6th component, which leads to a reduction in motor efficiency.

[0164] (B) VSI nonlinear modeling

[0165] The nonlinear effects of the inverter are explained by analyzing a single-phase branch, such as... Figure 3 As shown in (a), a brief shielding time is inserted during the switching process to prevent conduction short circuits caused by the limited switching speed of semiconductor devices; at the same time, coupled with the voltage drop of the switch and diode, this shielding interval introduces an average output voltage error, the polarity of which depends on the direction of the phase current. Figure 2 The equivalent circuit in (b) shows that when both switches are closed, either the upper diode or the lower diode is conducting, resulting in a voltage offset. The phase output voltage error is related to the direction of the phase current, and is expressed as:

[0166]

[0167] To obtain a more accurate representation, the dead time duration T was considered. dead On-delay T on and shutdown delay T off In one switching cycle T s The average voltage error is given by the following formula:

[0168]

[0169] The distortion in the inverter output itself contains multiple harmonics, and its Fourier series expansion is as follows:

[0170]

[0171] By applying the transformation matrix The distortion voltage component in the stationary dq-dqz coordinate system can be obtained as follows:

[0172]

[0173] and

[0174]

[0175] For the nth harmonic, the motor response depends largely on the frequency-dependent impedance value in each decoupling subspace; the impedance vector is expressed as:

[0176]

[0177] The corresponding values ​​are:

[0178]

[0179] As the frequency increases, the impedance increases, and the attenuation of higher harmonics also increases. Conversely, at low frequencies, the impedance remains relatively small, which reduces the motor's natural filtering capability for lower harmonics. This phenomenon explains why, at low speeds, the 12th perturbation in the dq subspace has a significant impact on torque fluctuations, while the 6th perturbation in the dqz subspace exacerbates the decrease in efficiency.

[0180] Considering the distortion and non-sinusoidal back EMF caused by the inverter, the complete voltage equation of the motor is finally expressed as:

[0181]

[0182] However, in practical applications, variations in inverter parameters pose challenges to accurate modeling, and the harmonic components of the back electromotive force cannot be precisely determined. Furthermore, because the harmonic plane impedance of the DTP-PMSM is smaller than that of a traditional three-phase motor, the negative impact of harmonics on the entire drive system is more severe.

[0183] (C) Equivalent harmonic circuit modeling

[0184] To facilitate clear harmonic analysis, each variable is decomposed into its basic components and harmonic components.

[0185]

[0186] Substituting equation (15) into the voltage equation, the equivalent harmonic circuit model can be expressed as:

[0187]

[0188] To achieve a unified expression, virtual harmonic back EMF is defined as including nonlinear inverter interference, spatial harmonic back EMF, and circuit mutual coupling terms related to harmonic currents; their definitions are as follows:

[0189]

[0190] Substituting (17) into (16), we obtain the simplified harmonic voltage equation for DTP-PMSM as follows:

[0191]

[0192] The above equations show that, under harmonic excitation, the DTP-PMSM can be modeled as an equivalent RL circuit driven by the virtual harmonic BEMF. Figure 4 The harmonic circuit model for the 6th component in the dqz subspace is shown, with the equivalent circuit for the 12th component in the dq subspace also shown.

[0193] In step S3, based on the harmonic model of DTP-PMSM, the influence of amplitude and phase errors on the system under the condition of feedforward compensation (online identification of harmonic back EMF) is analyzed, providing a theoretical basis for the online identification algorithm of harmonic back EMF.

[0194] (A) Virtual harmonic back EMF estimation and current mapping

[0195] Based on the harmonic circuit model of DTP-PMSM, the true virtual back electromotive force is represented as:

[0196]

[0197] If the inverter generates a harmonic compensation voltage, it is represented as:

[0198]

[0199] The controller measures real-time current. Harmonic currents are extracted using RC subtraction:

[0200]

[0201] Current It remains constant under steady state;

[0202] therefore, Figure 4 winding voltage Represented as:

[0203]

[0204] Therefore, when When the harmonic current is eliminated, the harmonic current will be eliminated; however, due to the impedance angle, the harmonic current in each subspace will be eliminated. It will be relative to the applied voltage Lag; these impedance angles are expressed as:

[0205]

[0206] This mapping shows that the estimated harmonic voltage With respect to actual harmonic back electromotive force The amplitude or phase deviation between them will be directly reflected in the harmonic current. Therefore, harmonic current serves as a practically observable variable for identifying and quantifying such biases; to illustrate this more clearly, Figure 5 It demonstrates the effects under different amplitudes and phase deviations. The following are relevant observations regarding its characteristics:

[0207] 1) When both amplitude and phase deviations exist, harmonic currents will precede voltage by more than 90°, such as... Figure 5 As shown in (a) and (b), the average current over each half-cycle is approximately zero, regardless of whether the harmonic current leads or lags.

[0208] 2) In Figure 5 In (c) and (d), when there is only amplitude deviation and no phase error, the average current within half a cycle is... The sign depends on and The relative size. Specifically, in In the positive half-cycle, if the estimated amplitude Greater than the actual amplitude The average harmonic current satisfies Conversely, if Smaller, then .

[0209] These phenomena highlight The effectiveness of the virtual harmonic back electromotive force identification index. Average harmonic current. It is by... The result is obtained by summing up the data from each half-cycle and then dividing by the number of samples:

[0210]

[0211] Furthermore, the half-cycle average current is updated at each zero-point crossover.

[0212]

[0213] This ensures its It can always accurately reflect the average current value within the most recent half-cycle;

[0214] (B) Amplitude and phase identification principle:

[0215] and The characteristic mapping between them and the harmonic currents provide clear and robust characterizations of amplitude and phase mismatch; using these characterizations, and its half-cycle average It is considered an observable to drive the online estimation scheme developed below;

[0216] 1) Phase identification:

[0217] When the average harmonic current is detected This indicates that the estimated harmonic voltage and actual harmonic back electromotive force There is a 90° phase deviation;

[0218] like Figure 5 As shown in (b), when Leading When the estimated harmonic voltage vector passes through the zero point of the positive half-axis, the instantaneous harmonic current satisfies the condition; conversely, when... When the negative half-axis crosses the zero point, Similarly, when behind At that time, the above relationship is reversed; therefore, and The relative phase relationship between them is directly from and It is inferred from the polarity;

[0219] Based on this relationship, the phase update rule is expressed as follows:

[0220]

[0221] It should be noted that, due to measurement noise in actual systems, it is difficult to determine precisely. Therefore, a small threshold is typically applied in the implementation (e.g., 0.1 in this study), and phase correction is only triggered when the deviation exceeds this limit.

[0222] 2) Amplitude identification:

[0223] Since the virtual harmonic back electromotive force maintains a fixed amplitude and phase under steady-state conditions, the estimated amplitude must be adaptively adjusted to ensure that the phase angle between the harmonic voltage and current is close to 90°, thereby triggering phase identification.

[0224] Once the phase deviation is eliminated, it can be based on Figure 5 The current reflection relationship summarized in (c) and (d) is used to determine the amplitude error. Therefore, the reference amplitude vector will be updated using the following control law:

[0225]

[0226] 3) Gain design and delay compensation:

[0227] Since amplitude identification occurs at a much higher frequency than phase identification, as the former updates twice per harmonic voltage cycle, the amplitude loop requires careful gain design. Specifically, a larger gain is needed at low speeds to ensure fast convergence, while a smaller gain is needed at high speeds to ensure stability. Furthermore, the gain design must consider both the dq and dqz subspaces, as their harmonic frequencies differ.

[0228] This indicates the magnitude gain vector The choice of gain is crucial because it should be precisely matched to the impedance characteristics of the motor. Therefore, the gain can be expressed as...

[0229]

[0230] Equation (28) ensures that the proposed method can achieve an appropriate balance between convergence speed and stability under various operating conditions.

[0231] For the phase identification loop, gain The design requirements are not stringent because the amplitude loop updates more frequently and plays a dominant role in the overall dynamics. In practical applications, Only the monotonic correction of the phase needs to be guaranteed, while the amplitude loop dominates the overall convergence speed and robustness.

[0232] Considering the discrete implementation of the algorithm, delay compensation must also be incorporated. Since the frequency of the harmonic signal is much higher than the fundamental frequency, the effects of digital delay and PWM delay become significant. To mitigate these effects, amplitude and phase compensation terms are introduced as follows:

[0233]

[0234] The following calculations are based on the given compensation benefits and phase terms:

[0235]

[0236] like Figure 6 As shown, this mechanism ensures the estimated amplitude Through iterative updates, it gradually converges to the actual harmonic back electromotive force. The overall control block diagram of the proposed identification scheme is as follows: Figure 7 As shown.

[0237] In step S4, the performance of the proposed method was evaluated using a DPMSM drive system. This test platform included an oscilloscope, two inverters, a real-time control system (RTU-BOX206), a dual three-phase motor, and a load motor driven by a load cabinet. The system operated at a sampling frequency of 10kHz, a DC bus voltage of 310V, and a dead time of 3μs. For the experiment, the controller parameter was set to K. a,s =[10010044] T and K p,s =[0.050.050.050.05] T .

[0238] 1) Robustness assessment of controller parameters

[0239] The robustness of the controller parameters under conditions of 1000 r / min and 6 Nm is evaluated. The proportional gain vector of the virtual harmonic back electromotive force is also considered. It is designed to be inversely proportional to the impedance of each subspace, which is determined by the stator inductance and resistance. In this way, dynamic response and steady-state performance can be coordinated.

[0240] like Figure 8As shown in (a), under nominal parameters, the proposed algorithm can accurately identify the amplitude and phase of the virtual harmonic back electromotive force in the fundamental and harmonic subspaces. Therefore, the phase current THD is reduced from 28.14% to 5.12%, while maintaining excellent transient and steady-state performance. Figure 8 In (b), when the stator inductance and resistance are increased to 500% of their nominal values, the gain K... pa The inductance was reduced to approximately 20% of its original value. As a result, the recognition response slowed, but steady-state accuracy was maintained, and the THD decreased to 5.01%. In contrast, when the stator inductance and resistance were each reduced to 20% of their nominal values, the gain vector... This is almost a fivefold increase, resulting in a very rapid dynamic response, but also increased current fluctuations, such as... Figure 8 As shown in (c), this demonstrates strong robustness and maintains effectiveness over a wide range of parameter deviations.

[0241] 2) Steady-state performance evaluation

[0242] The steady-state performance of the proposed method was verified at low speed (200 r / min) and rated speed (2000 r / min), with a load torque of 6 Nm. Figure 9 As shown, at 200 r / min, the d-axis current exhibits a significant 12th-order pulsation, resulting in a THD of 1.5% for the αβ current. Simultaneously, the 6th-order current amplitude in the dqz subspace reaches 1.03 A, leading to a harmonic current i xy The ripple is 2.78 A, ultimately resulting in a phase current THD of 20.23%. After activating the proposed compensation method, the amplitude and phase of the virtual harmonic back EMF are rapidly and accurately estimated. Therefore, the ripple of the d-axis current is significantly reduced, the THD of the αβ current is reduced to 0.89%, and the 6th component in the dqz subspace is reduced to 0.089 A. This reduction will... xy The pulsation was reduced to 1.16%, and the overall phase current THD was increased to 6.47%, indicating that the motor was operating efficiently.

[0243] exist Figure 10 Similar conclusions can be drawn at the rated speed of 2000 r / min. Although the harmonic back electromotive force is stronger at high speeds, where i xy The pulsation reached 4.19A, but the proposed method still effectively suppressed the harmonic current, reducing the phase current THD from 40.93% to 6.11%.

[0244] Figure 11The steady-state performance of the proposed method was further compared with that of the PR method under a constant load torque (4 Nm). At a speed of 500 r / min, the superiority of the proposed method is evident: the phase current THD is reduced to 8.05%, while that of the PR method is 9.47%, and no activation-induced oscillations were observed. When the speed is further increased to 800 r / min, the PR controller fails and generates persistent oscillations in the fundamental frequency subspace. This is attributed to the strong sensitivity of the resonant controller to discretization, which significantly reduces its harmonic suppression capability at higher speeds, making it ineffective in reducing the 12th harmonic in the dq subspace.

[0245] Figure 12 and Figure 13 The dynamic and disturbance rejection performance of the proposed method was further evaluated. Figure 12 In the experiment, the motor operates at 600 r / min with an initial load torque of 4 Nm, which then increases to 6 Nm in a step. Under this disturbance, the estimated virtual harmonic back EMF responds rapidly and adjusts to the ideal value, thereby maintaining the total harmonic distortion rate of the current at 1.26% and reducing the harmonic subspace current i xy The pulsation is limited to 1.13 A, and the overall phase current total harmonic distortion rate is kept at 6.44%. Figure 13 The dynamic response of the system was evaluated when the rotational speed changed abruptly from 1000 r / min to 1600 r / min. During this abrupt change, the virtual harmonic back electromotive force converged rapidly to its steady-state value, stabilizing the total harmonic distortion rate of the αβ axis current at 2.55%, and reducing the i xy The ripple is limited to 0.98 A, and the total harmonic distortion of the phase current is maintained at 6.44%.

[0246] This invention proposes a method for identifying and compensating harmonics based on virtual harmonic back EMF to suppress harmonics in DTP-PMSM drives. By establishing a harmonic modeling framework, it is revealed that factors such as non-sinusoidal back EMF and inverter nonlinearity introduce 12th and 6th harmonic disturbances in the dq and dqz subspaces, respectively, which significantly affect torque ripple and system efficiency. To address these issues, the concept of virtual harmonic back EMF is introduced, unifying inverter distortion, spatial harmonics, and cross-coupling terms within a concise expression framework. Subsequently, an online identification strategy based on current mapping is developed for real-time estimation of amplitude and phase deviations, and the convergence and robustness of the algorithm are ensured through gain design and delay compensation. Experimental results show that the proposed method can effectively suppress harmonic currents, reduce torque ripple, and improve system efficiency over a wide speed range, while avoiding complex algorithms and cumbersome parameter tuning. Overall, this research provides a simple, robust, and real-time harmonic suppression solution for DTP-PMSM drive systems.

[0247] The technical means disclosed in this invention are not limited to those disclosed in the above embodiments, but also include technical solutions composed of any combination of the above technical features. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of this invention, and these improvements and modifications are also considered within the scope of protection of this invention.

Claims

1. A harmonic suppression method for a dual three-phase permanent magnet synchronous motor based on virtual back electromotive force identification, characterized in that: Includes the following steps, S1. Establish the drive system model of DTP-PMSM; S2. Construct a harmonic model of DTP-PMSM, analytically decompose the nonlinear characteristics of the inverter and the non-sinusoidal back EMF into the dq and dqz subspaces, and introduce the concept of virtual harmonic back EMF to unify the inverter nonlinearity, spatial harmonic back EMF and cross-coupling terms related to harmonic current into a single representation. S3. Develop an identification principle based on current mapping, including amplitude and phase estimation, gain design, and delay compensation, to estimate the amplitude and phase deviation of the virtual harmonic back electromotive force online to achieve harmonic suppression, while avoiding complex algorithms and a large number of parameter adjustments.

2. The control method for a dual three-phase permanent magnet synchronous motor based on virtual harmonic back EMF identification and compensation according to claim 1, characterized in that, In step S1, establishing the drive system model of DTP-PMSM includes: The six-phase voltage-source inverter (VSI) used in the DTP-PMSM driver generates 64 different switching states, which determine the instantaneous values ​​of the stator phase voltages; these voltages can be simply represented as a six-dimensional vector: ； Among them, u s It is the phase voltage; To facilitate decoupled control and harmonic analysis, the stator voltage vector u is first decomposed using voltage space decomposition (VSD). s Projected onto two orthogonal subspaces —αβ and —xy; For decoupling control and harmonic analysis, the stator voltage vector is decomposed into two subspaces, αβ and xy, using VSD: ； Among them, T VSD It is the VSD transformation matrix; Defined in the form of a decomposition matrix: ； To simplify harmonic analysis, the xy subspace is further transformed into a frame that focuses the sixth component at a fixed spectral location; the resulting dq-dqz frame voltage vector is: ； Among them, T dqdqz ( ) is the transformation matrix from the stationary coordinate system to the rotating coordinate system. It refers to the position of the motor rotor, and ； Assuming the existence of an isolated neutral point, the zero-sequence component will be naturally suppressed. Under common assumptions such as sinusoidal winding distribution, symmetrical winding parameters, and neglect of core losses, the voltage equation of the DTP-PMSM in the dq-dqz coordinate system is expressed as: ； Where u dq These are the voltages in the dq coordinate system, u and u. dqz Let i be the voltage in the dqz coordinate system. dq Let i be the current in the dq coordinate system. dqz Let ω be the current in the dqz coordinate system. e L is the electric angular velocity. d and L q These are the equivalent inductances in the dq coordinate system, L and L, respectively. dz and L qz R represents the equivalent inductance in the dqz coordinate system, respectively. s ψ is the stator winding resistance. f This refers to the magnetic flux linkage of a permanent magnet.

3. The control method for a dual three-phase permanent magnet synchronous motor based on virtual harmonic back EMF identification and compensation according to claim 1, characterized in that, In step S2, two factors are considered: inverter nonlinearity, which is caused by dead time and voltage drops of devices and diodes; and non-sinusoidal back electromotive force generated by space flux harmonics. (A) Harmonic modeling of back electromotive force For DTP-PMSM, due to the non-sinusoidal nature of the flux distribution, the back electromotive force in the phase domain contains multiple harmonic components. To characterize the spatial harmonic content of the permanent magnet flux linkage, in addition to the fundamental wave, the expression for the phase domain back electromotive force is extended to include the 5th, 7th, 11th, and 13th harmonics. ； Where, ψ n Let θ represent the magnetomotive force amplitude of the nth harmonic, and θ represent the magnetomotive force amplitude of the nth harmonic. n This is the corresponding phase offset; By applying these transformation matrices The back electromotive force component in the subspace is obtained as: ； As can be seen from equation (6), in the dq subspace, the 11th and 13th harmonics are mapped to a unified 12th component, which directly affects the generation of torque; in contrast, in the dqz subspace, the 5th and 7th harmonics are mapped to a unified 6th component, which leads to a reduction in motor efficiency.

4. (B) VSI nonlinear modeling The phase output voltage error is related to the direction of the phase current, and is expressed as: ； Where i s Represents phase current, A p These are coefficients related to device characteristics; To obtain a more accurate representation, the dead time duration T was considered. dead On-delay T on and shutdown delay T off In one switching cycle T s The average voltage error is given by the following formula: ； Where V dc This is the DC bus voltage, while V ce and V d These represent the voltage drop across the conducting device; V ce and V d These represent the saturation voltage of the active switch and the forward voltage of the anti-parallel diode, respectively. The distortion in the inverter output itself contains multiple harmonics, and its Fourier series expansion is as follows: ； By applying the transformation matrix The distortion voltage component in the stationary dq-dqz coordinate system can be obtained as follows: ； and ； in, and Let be the distortion voltage component in the αβ coordinate system. and The distortion voltage component in the xy coordinate system; For the nth harmonic, the motor response depends largely on the frequency-dependent impedance values ​​in each decoupling subspace; the impedance vector is expressed as: ； in This represents the equivalent inductance value in the dq-subspace and dqz-subspace, and the corresponding values ​​are: ； As the frequency increases, the impedance increases, and the attenuation of higher harmonics also increases. Conversely, at low frequencies, the impedance remains relatively small, which reduces the motor's natural filtering capability for lower harmonics. This phenomenon explains why, at low speeds, the 12th perturbation in the dq subspace has a significant impact on torque fluctuations, while the 6th perturbation in the dqz subspace exacerbates the decrease in efficiency. Considering the distortion and non-sinusoidal back EMF caused by the inverter, the complete voltage equation of the motor is finally expressed as: ； (C) Equivalent harmonic circuit modeling To facilitate clear harmonic analysis, each variable is decomposed into its basic components and harmonic components. ； in, , , These represent the fundamental components distributed in the subspaces dq and dqz, respectively. , , These represent the corresponding harmonic components in the same subspace; Substituting equation (15) into the voltage equation, the equivalent harmonic circuit model can be expressed as: ； The subscripts "12" and "6" represent the effective harmonic orders in the dq and dqz subspaces, respectively. To achieve a unified expression, virtual harmonic back EMF is defined as including nonlinear inverter interference, spatial harmonic back EMF, and circuit mutual coupling terms related to harmonic currents; their definitions are as follows: ； in and Let represent the 12th virtual harmonic back electromotive force in the dq subspace, and respectively. and These represent the 6th virtual harmonic back electromotive force in the dqz subspace; Substituting (17) into (16), we obtain the simplified harmonic voltage equation for DTP-PMSM as follows: ； The above equations show that, under harmonic excitation, the DTP-PMSM can be modeled as an equivalent RL circuit driven by the virtual harmonic BEMF.

5. The control method for a dual three-phase permanent magnet synchronous motor based on virtual harmonic back EMF identification and compensation according to claim 1, characterized in that, In step S3, based on the harmonic circuit model of DTP-PMSM, the influence of amplitude and phase errors on the system under feedforward compensation is analyzed, providing a theoretical basis for the online identification algorithm of harmonic back electromotive force. (A) Virtual harmonic back EMF estimation and current mapping Based on the harmonic circuit model of DTP-PMSM, the true virtual back electromotive force is represented as: ； in and These represent the equivalent amplitude and phase vectors of the virtual back electromotive force, respectively, including d-, q-, dz-, and qz- components; for the dq subspace, n=12, and for the dqz subspace, n=6; If the inverter generates a harmonic compensation voltage, it is represented as: ； in, and These represent the estimated amplitude vector and phase vector, respectively; The controller measures real-time current. Harmonic currents are extracted using RC subtraction: ； in This represents the reference current, which remains constant under steady-state conditions. Therefore, winding voltage Represented as: ； Therefore, when When the harmonic current is eliminated, the harmonic current will be eliminated; however, due to the impedance angle, the harmonic current in each subspace will be eliminated. It will be relative to the applied voltage Lag; these impedance angles are expressed as: ； Among them, all values All are within the range of (0°, 90°); This mapping shows that the estimated harmonic voltage With respect to actual harmonic back electromotive force The amplitude or phase deviation between them will be directly reflected in the harmonic current. Therefore, harmonic current serves as a practically observable variable for identifying and quantifying such deviations. Average harmonic current It is by... The result is obtained by summing up the data from each half-cycle and then dividing by the number of samples: ； in For the accumulated current, The number of current samples; Furthermore, the half-cycle average current is updated at each zero-point crossover. ； Ensure its It can always accurately reflect the average current value within the most recent half-cycle; (B) Amplitude and phase identification principle: and The characteristic mapping between them and the harmonic currents provide clear and robust characterizations of amplitude and phase mismatch; using these characterizations, and its half-cycle average It is considered an observable to drive the online estimation scheme developed below; 1) Phase identification: When the average harmonic current is detected This indicates that the estimated harmonic voltage and actual harmonic back electromotive force There is a 90° phase deviation; when Leading When the estimated harmonic voltage vector passes through the zero point of the positive half-axis, the instantaneous harmonic current satisfies the condition; conversely, when... When the negative half-axis crosses the zero point, Similarly, when behind At that time, the above relationship is reversed; therefore, and The relative phase relationship between them is directly from and It is inferred from the polarity; Based on this relationship, the phase update rule is expressed as follows: ； in It is the phase update gain vector. This refers to the electrical angular velocity used to ensure that the correction direction is consistent with the motor rotation direction; 2) Amplitude identification: Since the virtual harmonic back electromotive force maintains a fixed amplitude and phase under steady-state conditions, the estimated amplitude must be adaptively adjusted to ensure that the phase angle between the harmonic voltage and current is close to 90°, thereby triggering phase identification. The reference magnitude vector will be updated using the following control rules: ； in This represents the amplitude gain vector, where all elements are positive constants; therefore, the amplitude will change according to... polarity and Iterative modification of the symbols; 3) Gain design and delay compensation: Since the frequency of amplitude identification is much higher than that of phase identification, because the former is updated twice in each harmonic voltage cycle, a larger gain is required at low speeds to ensure fast convergence, while a smaller gain is required at high speeds to ensure stability. In addition, the gain design must take into account both the dq subspace and the dqz subspace, because their harmonic frequencies are different. Amplitude gain vector Represented as ； Where k p It is a positive constant; Amplitude and phase compensation terms were introduced as follows: ； The following calculations are based on the given compensation benefits and phase terms: ； in These represent the amplitude compensation gain, while This represents the phase compensation value.

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