WO2021044325A1 - Method for sensorless estimating rotor position and rotor angular speed of a synchronous reluctance machine - Google Patents

Method for sensorless estimating rotor position and rotor angular speed of a synchronous reluctance machine Download PDF

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Publication number
WO2021044325A1
WO2021044325A1 PCT/IB2020/058192 IB2020058192W WO2021044325A1 WO 2021044325 A1 WO2021044325 A1 WO 2021044325A1 IB 2020058192 W IB2020058192 W IB 2020058192W WO 2021044325 A1 WO2021044325 A1 WO 2021044325A1
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Prior art keywords
flux linkage
observed
stator
projection vector
error signal
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PCT/IB2020/058192
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French (fr)
Inventor
Anantaram VARATHARAJAN
Gianmario PELLEGRINO
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Politecnico Di Torino
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    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P6/00—Arrangements for controlling synchronous motors or other dynamo-electric motors using electronic commutation dependent on the rotor position; Electronic commutators therefor
    • H02P6/14—Electronic commutators
    • H02P6/16—Circuit arrangements for detecting position
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H02P21/141—Flux estimation
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H02P21/18—Estimation of position or speed
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/24—Vector control not involving the use of rotor position or rotor speed sensors
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/24—Vector control not involving the use of rotor position or rotor speed sensors
    • H02P21/26—Rotor flux based control
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P6/00—Arrangements for controlling synchronous motors or other dynamo-electric motors using electronic commutation dependent on the rotor position; Electronic commutators therefor
    • H02P6/14—Electronic commutators
    • H02P6/16—Circuit arrangements for detecting position
    • H02P6/18—Circuit arrangements for detecting position without separate position detecting elements

Definitions

  • the present invention generally relates to sensorless control of a synchronous reluctance machine.
  • the present invention relates to a method for estimating rotor position and rotor angular speed of a synchronous reluctance machine.
  • sensorless control consists of a closed loop controlling a synchronous reluctance machine without using a rotor position encoder for synchronizing the control reference frame to the rotor phase angle or for evaluating the angular speed of the rotor itself.
  • the rotor angular position and the rotor angular speed are estimated by the digital controller by manipulating electrical quantities and machine parameters via a real time software.
  • Said real-time software is normally called “observer” and includes a flux observer and an angular position observer.
  • stator resistance As is known, the methods for estimating the flux and the angular position of a rotor depend on the stator resistance.
  • the direct measurement of the stator resistance is possible when the machine is not running (cold machine); however, the stator resistance value while the machine is in use (warmed-up machine) differs considerably from the result obtained on a cold machine.
  • stator resistance varies with the machine temperature during operation, which results in imprecise estimate of the machine internal voltage and ultimately errors in flux and angular position estimates.
  • the angular position observer is strongly coupled with the flux observer.
  • the susceptibility of flux observer to internal voltage estimate errors drives the angular position observer to instability.
  • the Applicant has perceived that the known sensorless machine drives are prone to stator resistance variation or require to be augmented with additional online resistance adaptation.
  • the Applicant has tackled the problem of providing a method for estimating the rotor position and/or the rotor angular speed of a synchronous reluctance machine which overcomes the aforesaid drawbacks.
  • the present invention relates to a method for sensorless estimation of a position error signal for a synchronous reluctance machine, said synchronous reluctance machine comprising a stator and a rotor, said method comprising: receiving an observed flux linkage of said stator; receiving a current model based flux linkage estimate of said stator; selecting a first projection vector wherein said first projection vector is expressed as: wherein:
  • G is a gain matrix
  • • / is an identity matrix; so that a rotor angular position error, due to a stator resistance error, calculated, at maximum torque per ampere condition, by means of said first projection vector is within a band of one electrical degree; calculating a position error signal as a function of said observed flux linkage said current model based flux linkage estimate and said first projection vector.
  • the present invention relates to method for sensorless estimation of at least one of observed rotor angular position and observed rotor angular speed of a synchronous reluctance machine comprising: calculating said position error signal according to embodiments of the present invention; calculating at least one of observed rotor angular position and observed rotor angular speed as a function of said position error signal.
  • the present invention relates to a control system for a synchronous reluctance machine, said synchronous reluctance machine comprising a rotor and a stator, said control system comprising: - a hybrid flux observer configured for: calculating an observed flux linkage of said stator based on estimated values of electric quantities associated with said synchronous reluctance machine; calculating a current model based flux linkage estimate of said stator based on detected values of electric quantities; an error evaluation block configured for: receiving said observed flux linkage and said current model based flux linkage estimate from said hybrid flux observer; selecting a first projection vector wherein said first projection vector is expressed as: wherein:
  • G is a gain matrix
  • FIG. 1 shows a block diagram illustrating a control system of a synchronous reluctance machine according to the present invention
  • FIG. 2 shows a block diagram illustrating a hybrid flux observer
  • FIG. 3 shows a block diagram of an error evaluation block according to the present invention.
  • estimate quantities and “observed quantity” indicate a quantity calculated by means of a numerical model; estimated and/o observed quantities are represented by the superscript
  • FIG 1 shows a block diagram of a control system 100 controlling a synchronous reluctance machine 5 (also indicated as “SyR machine” in the following), as example, the SyR machine could be a SyR motor 5.
  • SyR machine also indicated as “SyR machine” in the following
  • the SyR machine could be a SyR motor 5.
  • the synchronous reluctance machine 5 comprises a rotor and a stator.
  • the control system 100 is a sensorless control system.
  • control system 100 comprises a proportional integral speed controller 1 , a Maximum Torque Per Ampere (MTPA) controller 2, a current vector control 3, a voltage source inverter 4.
  • MTPA Maximum Torque Per Ampere
  • a target rotor angular speed is set to drive the rotor at an actual angular speed w,.
  • the MTPA controller 2 implementing a maximum torque per ampere control strategy, selects a target current vector (in an estimated rotor reference frame that satisfies the maximum torque per ampere condition at the target rotor angular speed
  • MTPA controller is known in the art, as described in:
  • the current control 3 receives the target current vector from the MTPA controller 2 and generates a target voltage vector
  • the target voltage vector is converted to the stator reference frame by a first conversion and transformation block 3a, generating an estimated voltage vector (the control voltage is also used as voltage estimate feeding the flux observer).
  • the first conversion and transformation block 3a is an inverse Park transformation.
  • the estimated voltage vector is sent, as an input after a further conversion and transformation block 3b, to the voltage source inverter 4 which generates a three-phase voltage machine supply v abc .
  • the further conversion and transformation block 3b is an inverse Clarke transformation.
  • the three-phase voltage machine supply is provided in input to the SyR machine 5 to drive the same.
  • the rotor and/or stator quantities are estimated and used by the control system 100 to drive the SyR machine 5.
  • a mathematical model of the SyR machine 5 is given.
  • the SyR machine mathematical model is expressed in the reference frame fixed to the rotor dq (i.e. a rotating reference frame).
  • the rotating reference frame in use is the estimated rotating reference frame
  • the voltage equation of the SyR machine 5 is expressed as: wherein s is the Laplace variable, R s is the stator resistance at a rated machine temperature, (T superscript stands for “transposed” vector or matrix) is the stator flux linkage vector, is the stator voltage vector, is the stator current vector, is the estimated rotor angular speed and J is the orthogonal rotational matrix
  • the stator flux linkage vector and its time-derivative can be expressed in terms of an incremental inductance matrix l, an apparent inductance matrix L and a position error wherein q is an actual rotor angular position (i.e. the correct rotor angular position), is an estimated rotor angular position, as shown in the equations [2a] and [2b]: wherein the components of the inductance matrices are: where l d , l q respectively represent the incremental inductance along direct d and quadrature q axis, while l dq represent the cross-saturation term.
  • the estimated electromagnetic torque is given by: wherein N is the number of pole pairs, is the stator flux linkage vector and is the stator current vector.
  • the SyR machine 5 is driven in a maximum torque per ampere condition (also indicated as "MTPA condition” in the following).
  • the analytical expression of the MTPA condition can be derived from equation [4], differentiating with respect to a current angle g, wherein the current angle g is obtaining the following expression:
  • the control system 100 comprises a second conversion and transformation block 4a.
  • the second conversion and transformation block 4a is a Clarke's transformation block configured to convert a three-phase measured current i abc , acquired by means of current transducers on a machine supply line, into the stator reference frame ab, obtaining an actual stator current
  • a third conversion and transformation block 4b converts the three-phase measured current into the estimated rotor reference frame to serve as a feedback for close-looped current vector control.
  • the third conversion and transformation block 4b is a Park transformation.
  • the control system 100 comprises an observer block 6.
  • the observer block 6 preferably, receives in input the estimated rotor angular position the estimated voltage vector and the actual stator current respectively provided by the first conversion and transformation block 3a and the second conversion and transformation block 4a - and calculates a current model based stator flux linkage estimate and an observed stator flux linkage
  • the observer block 6 is a Hybrid Flux Observer (HFO) 6.
  • the HFO 6 is configured for: calculating an observed stator flux linkage of said stator based on estimated values of electric quantities, as example, as a function of the estimated voltage vector associated with said synchronous reluctance machine 5; calculating a current model based stator flux linkage estimate of said stator based on detected values of said electric quantities, as example, as a function of the actual stator current
  • the HFO 6 in the stator reference frame is defined by: wherein wherein is a 2x2 gain matrix, is the observed stator flux linkage in the stator reference frame, is the time-derivative of the observed stator flux linkage in the stator reference frame, is the estimated voltage vector in the stator reference frame, is the actual stator current and is the estimated inductance matrix.
  • A is the flux maps lookup table of the SyR machine 5.
  • the HFO 6 is defined by:
  • the HFO 6 calculates - as a function of the actual stator current the estimated voltage vector and the estimated rotor angular position - the current model based stator flux linkage estimate and the observed stator flux linkage by means of equations [7], [8], [9] and [10]
  • the error evaluation block 11 calculates a position error signal
  • the position error signal is defined as the projection of the difference of the observed stator flux linkage and the current model based stator flux linkage estimate
  • the position error signal is defined as: as described in i. and further developed in ii. i. M. Hinkkanen, S. E. Saarakkala, H. A. A. Awan, E. Molsa, and T. Tuovinen, Observers for Sensorless Synchronous Machine Drives: Framework for Design and Analysis,” IEEE Transactions on Industry Applications, p. 1 , 2018; and ii. Varatharajan and G. Pellegrino, “Sensorless Control of Synchronous Reluctance Machine Drives: Improved Modeling and Analysis beyond Active
  • the first projection vector is expressed as: wherein w is the rotor angular speed, is an auxiliary flux linkage vector.
  • the first projection vector in steady-state condition is expressed as: wherein is the first projection vector in steady-state operating condition, w 0 is the rotor angular speed in such steady state operating condition, is an auxiliary flux linkage vector in such steady state operating condition
  • an observed rotor angular speed and/or an observed rotor angular position are calculated as a function of the position error signal
  • the observed rotor angular speed and the observed rotor angular position are calculated by means of a position and angular speed estimation block 12.
  • the observed rotor angular speed and the observed rotor angular position are calculated as: wherein is the observed rotor angular speed, are the proportional and integral gains respectively and is the integral output of the phase locked loop proportional-integral regulator, and is the rotor angular speed error
  • the stator flux error defined as: wherein is the stator flux error, is the actual stator flux vector and is the observed stator flux vector
  • the flux error dynamic can be represented as: wherein and considering invariance of the incremental inductance, the following expression is obtained:
  • J is the orthogonal rotational matrix
  • l is the incremental inductance matrix
  • the steady state position error can be expressed as:
  • Equation [23] and equation [6] show that, as long as the MTPA condition is respected, the rotor position error calculated by means of the first projection vector is basically not affected by the stator resistance error
  • the position error owing to the stator resistance error calculated by means of said first projection vector is basically zero at MTPA condition.
  • the measured position error calculated by means of said first projection vector is within a band of one electrical degree at MTPA condition.
  • the aforesaid measured position error is due to, as example, white noise and/or inaccuracies in flux-maps.
  • said band of one electrical degree at MTPA condition is proportional to: the accuracy of magnetic model of machine under test; and to deviations from MTPA trajectory.
  • a fundamental component of a voltage error due to inverter dead time is in phase with the actual stator current
  • the voltage error due to inverter dead time is reflected in the stator resistance error
  • the position error calculated by means of said first projection vector is independent from the voltage error due to the inverter dead time.
  • the observed rotor angular position and/or the observed rotor angular speed is/are used to drive the SyR machine 5.
  • a conventional phase lock loop with a proportional integral controller is employed to drive the position error signal to zero.
  • stator resistance adaptation is provided.
  • the error evaluation block 11 calculates a resistance error signal
  • the resistance error signal is defined as: wherein is a second projection vector.
  • control system comprises a resistance calculation block 13.
  • the resistance calculation block 13 calculates an observed stator resistance as a function of said resistance error signal
  • the estimated stator resistance is calculated as: wherein k r is a constant.
  • the resistance error signal is used to implement a software stator resistance adaptation of the estimated stator resistance.
  • the software stator resistance adaptation compensates for non-compensated voltage error estimate caused by the inverter dead-time. As example, inaccurate dead-time compensation results in constant voltage error that is independent of the operating point, currents or speed. As example, considering an experiment wherein a dead-time error of 1 ms result in a constant voltage error of about 6 V. If the amplitude of the present operating point is 1A, the software stator resistance adaptation increases the estimated resistance by 6 W to compensate the 6V voltage error. Conversely, at a higher load, if the operating current is 2A, the estimated resistance will be 3 W higher than the real resistance.
  • the position error signal and the resistance error signal are estimated independently.
  • the second projection vector is a vector orthogonal to the first projection vector Even more preferably, the second projection vector is a vector orthogonal to the first projection vector and is scaled proportionally with operating point to have a constant loop gain.
  • a cumulative error vector e is defined as wherein is a projection vector matrix.
  • the present invention provides important advantages.
  • the present invention provides the following advantages:
  • the use of the first projection vector in the control system 100 makes the control inherently stable at low speed; - the position error calculated by means of said first projection vector is independent from the voltage error due to inverter dead time, so precise MTPA torque control is possible, resulting in maximized efficiency;
  • the second projection vector compensates resistance variations when the machine is deliberately controlled out of MTPA conditions; - if the second projection vector is used, inverter identification for inverter voltage error compensation can be omitted: this simplifies the electric drive preliminary commissioning stage.

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  • Engineering & Computer Science (AREA)
  • Power Engineering (AREA)
  • Control Of Ac Motors In General (AREA)
  • Synchronous Machinery (AREA)
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Abstract

It is disclosed a method for sensorless estimation of a position error signal (ϵθ) for a synchronous reluctance machine (5), said synchronous reluctance machine (5) comprising a stator and a rotor, said method comprising: receiving an observed flux linkage formula (i) of said stator; receiving a current model based flux linkage estimate formula (ii) of said stator; selecting a first projection vector formula (iii) wherein said first projection vector formula (iii) is expressed as formula (I), wherein: ω is the rotor angular speed; formula (iv) and formula (v) are components of an auxiliary flux linkage vector formula (vi) of said stator, wherein said auxiliary flux linkage vector formula (vi) is an orthogonal rotational matrix; G is a gain matrix; and I is an identity matrix; so that a rotor angular position error formula (vii), due to a stator resistance error formula (viii), calculated, at MTPA condition, by means of said first projection vector formula (iii) is within a band of one electrical degree; calculating a position error signal (ϵθ) as a function of said observed flux linkage (formula i) said current model based flux linkage estimate (formula ii) and said first projection vector formula (iii).

Description

METHOD FOR SENSORLESS ESTIMATING ROTOR POSITION AND ROTOR ANGULAR SPEED OF A SYNCHRONOUS RELUCTANCE MACHINE
Technical field The present invention generally relates to sensorless control of a synchronous reluctance machine. In particular, the present invention relates to a method for estimating rotor position and rotor angular speed of a synchronous reluctance machine.
Background art As is known, sensorless control consists of a closed loop controlling a synchronous reluctance machine without using a rotor position encoder for synchronizing the control reference frame to the rotor phase angle or for evaluating the angular speed of the rotor itself.
The rotor angular position and the rotor angular speed are estimated by the digital controller by manipulating electrical quantities and machine parameters via a real time software. Said real-time software is normally called “observer” and includes a flux observer and an angular position observer.
As is known, the methods for estimating the flux and the angular position of a rotor depend on the stator resistance. The direct measurement of the stator resistance is possible when the machine is not running (cold machine); however, the stator resistance value while the machine is in use (warmed-up machine) differs considerably from the result obtained on a cold machine.
In particular, the stator resistance varies with the machine temperature during operation, which results in imprecise estimate of the machine internal voltage and ultimately errors in flux and angular position estimates.
In a sensorless machine control drive, the angular position observer is strongly coupled with the flux observer. In particular, the susceptibility of flux observer to internal voltage estimate errors drives the angular position observer to instability.
To circumvent this problem, several methods have been explored: • a stator resistance observer for salient synchronous machines is proposed in M. Hinkkanen, T. Tuovinen, L. Harnefors, and J. Luomi, “A Combined Position and Stator-Resistance Observer for Salient PMSM Drives: Design and Stability Analysis,” IEEE Transactions on Power Electronics, vol. 27, no. 2, pp. 601-609, 2012;
• a sliding mode observer is developed in D. Liang, J. Li, and R. Qu, “Sensorless Control of Permanent Magnet Synchronous Machine Based on Second-Order Sliding-Mode Observer With Online Resistance Estimation,” IEEE Transactions on Industry Applications, vol. 53, no. 4, pp. 3672-3682, 2017.; and
• a recursive least square approach is resorted to identify the resistance online in Y. Inoue, Y. Kawaguchi, S. Morimoto, and M. Sanada, “Performance Improvement of Sensorless IPMSM Drives in a Low-Speed Region Using Online Parameter Identification,” IEEE Transactions on Industry Applications, vol. 47, no. 2, pp. 798-804, 2011.
Summary of the invention
The Applicant has perceived that the known sensorless machine drives are prone to stator resistance variation or require to be augmented with additional online resistance adaptation.
In particular, the Applicant has perceived that the known flux and angular position observers present some drawbacks arising from internal voltage estimate errors due to stator resistance variation or to non-ideal compensation of the inverter voltage error. As said above, several on-line adaption techniques have been proposed, on top of the flux and angular position estimates scheme, requiring extra computation and extra calibration effort.
In view of the above, the Applicant has tackled the problem of providing a method for estimating the rotor position and/or the rotor angular speed of a synchronous reluctance machine which overcomes the aforesaid drawbacks.
According to a first aspect, the present invention relates to a method for sensorless estimation of a position error signal for a synchronous reluctance machine, said synchronous reluctance machine comprising a stator and a rotor, said method comprising: receiving an observed flux linkage of said stator; receiving a current model based flux linkage estimate of said stator; selecting a first projection vector wherein said first projection vector is expressed as:
Figure imgf000005_0001
wherein:
• w is the rotor angular speed;
• are components of an auxiliary flux linkage vector of said
Figure imgf000005_0002
Figure imgf000005_0003
stator, wherein said auxiliary flux linkage vector
Figure imgf000005_0004
• J is an orthogonal rotational matrix;
• G is a gain matrix; and
• / is an identity matrix; so that a rotor angular position error, due to a stator resistance error, calculated, at maximum torque per ampere condition, by means of said first projection vector is within a band of one electrical degree; calculating a position error signal as a function of said observed flux linkage said current model based flux linkage estimate and said first projection vector.
According to a second aspect, the present invention relates to method for sensorless estimation of at least one of observed rotor angular position and observed rotor angular speed of a synchronous reluctance machine comprising: calculating said position error signal according to embodiments of the present invention; calculating at least one of observed rotor angular position and observed rotor angular speed as a function of said position error signal.
According to a third aspect, the present invention relates to a control system for a synchronous reluctance machine, said synchronous reluctance machine comprising a rotor and a stator, said control system comprising: - a hybrid flux observer configured for: calculating an observed flux linkage of said stator based on estimated values of electric quantities associated with said synchronous reluctance machine; calculating a current model based flux linkage estimate of said stator based on detected values of electric quantities; an error evaluation block configured for: receiving said observed flux linkage and said current model based flux linkage estimate from said hybrid flux observer; selecting a first projection vector wherein said first projection vector is expressed as:
Figure imgf000006_0001
wherein:
• w is the rotor angular speed;
• are components of an auxiliary flux linkage vector
Figure imgf000006_0003
Figure imgf000006_0002
• J is an orthogonal rotational matrix;
• G is a gain matrix; and
• / is an identity matrix; so that a rotor angular position error, due to a stator resistance error, calculated, at maximum torque per ampere condition, by means of said first projection vector is within a band of one electrical degree; calculating a position error signal as a function of said observed flux linkage, said current model based flux linkage estimate and said first projection vector; a position and angular speed estimation block, configured to calculate an observed rotor angular position and/or an observed rotor angular speed as a function of said position error signal; wherein said control system is configured to drive said synchronous reluctance machine based on said observed rotor angular position and/or said observed rotor angular speed. Brief description of the drawings
The present invention will become clearer from the following detailed description, given by way of example and not of limitation, to be read with reference to the accompanying drawings, wherein:
- Figure 1 shows a block diagram illustrating a control system of a synchronous reluctance machine according to the present invention;
- Figure 2 shows a block diagram illustrating a hybrid flux observer;
- Figure 3 shows a block diagram of an error evaluation block according to the present invention.
Detailed description of preferred embodiments of the invention
In the following, vectors in a rotating reference frame, associated with the rotor, are denoted by subscript "dq". Vectors in a stationary reference frame, associated with the stator, are denoted by subscript Finally, the subscript " abc " indicates
Figure imgf000007_0003
machine phase quantities.
In the following, the expressions "estimated quantity" and "observed quantity" indicate a quantity calculated by means of a numerical model; estimated and/o observed quantities are represented by the superscript
Figure 1 shows a block diagram of a control system 100 controlling a synchronous reluctance machine 5 (also indicated as "SyR machine" in the following), as example, the SyR machine could be a SyR motor 5.
The synchronous reluctance machine 5 comprises a rotor and a stator. Preferably, the control system 100 is a sensorless control system.
Preferably, the control system 100 comprises a proportional integral speed controller 1 , a Maximum Torque Per Ampere (MTPA) controller 2, a current vector control 3, a voltage source inverter 4.
As shown in Figure 1 , a target rotor angular speed is set to drive the rotor at
Figure imgf000007_0002
an actual angular speed w,.
Preferably, the MTPA controller 2, implementing a maximum torque per ampere control strategy, selects a target current vector (in an estimated rotor reference
Figure imgf000007_0001
frame that satisfies the maximum torque per ampere condition at the target rotor angular speed
Figure imgf000008_0007
It has to be noted that the MTPA controller is known in the art, as described in:
L. Xu, X. Xu, T. A. Lipo and D. W. Novotny, "Vector control of a synchronous reluctance machine including saturation and iron loss," in IEEE Transactions on Industry Applications, vol. 27, no. 5, pp. 977-985, Sept.-Oct. 1991.
S. Morimoto, M. Sanada and Y. Takeda, "Wide-speed operation of interior permanent magnet synchronous machines with high-performance current regulator," in IEEE Transactions on Industry Applications, vol. 30, no. 4, pp. 920-926, July-Aug. 1994.
The current control 3 receives the target current vector from the MTPA
Figure imgf000008_0001
controller 2 and generates a target voltage vector The target voltage vector
Figure imgf000008_0003
Figure imgf000008_0002
is converted to the stator reference frame by a first conversion and transformation block 3a, generating an estimated voltage vector (the control voltage is
Figure imgf000008_0004
also used as voltage estimate feeding the flux observer).
Preferably, the first conversion and transformation block 3a is an inverse Park transformation.
The estimated voltage vector is sent, as an input after a further conversion
Figure imgf000008_0005
and transformation block 3b, to the voltage source inverter 4 which generates a three-phase voltage machine supply vabc.
Preferably, the further conversion and transformation block 3b is an inverse Clarke transformation.
Clark and Park transformations are known in the art and will not be described in the following. The three-phase voltage machine supply is provided in input to the SyR
Figure imgf000008_0006
machine 5 to drive the same.
During operation, the rotor and/or stator quantities, such as rotor angular position and/or rotor angular speed, are estimated and used by the control system 100 to drive the SyR machine 5. In the following, a mathematical model of the SyR machine 5 is given. The SyR machine mathematical model is expressed in the reference frame fixed to the rotor dq (i.e. a rotating reference frame). In absence of a rotor position transducer (sensorless control), the rotating reference frame in use is the estimated rotating reference frame
Figure imgf000009_0011
As example, the voltage equation of the SyR machine 5 is expressed as:
Figure imgf000009_0001
wherein s is the Laplace variable, Rs is the stator resistance at a rated machine temperature, (T superscript stands for “transposed” vector or matrix)
Figure imgf000009_0006
is the stator flux linkage vector,
Figure imgf000009_0003
is the stator voltage vector,
Figure imgf000009_0002
is the stator current vector,
Figure imgf000009_0007
is the estimated rotor angular speed and J is the orthogonal rotational matrix
Figure imgf000009_0004
The stator flux linkage vector and its time-derivative can be expressed
Figure imgf000009_0008
Figure imgf000009_0009
in terms of an incremental inductance matrix l, an apparent inductance matrix L and a position error
Figure imgf000009_0012
wherein q is an actual rotor angular position (i.e. the correct rotor angular position),
Figure imgf000009_0013
is an estimated rotor angular position, as shown in the equations [2a] and [2b]:
Figure imgf000009_0010
wherein the components of the inductance matrices are:
Figure imgf000009_0005
where ld, lq respectively represent the incremental inductance along direct d and quadrature q axis, while ldq represent the cross-saturation term. The estimated electromagnetic torque is given by:
Figure imgf000010_0001
wherein N is the number of pole pairs, is the stator flux linkage vector and is
Figure imgf000010_0002
Figure imgf000010_0003
the stator current vector.
Preferably, the SyR machine 5 is driven in a maximum torque per ampere condition (also indicated as "MTPA condition" in the following).
The analytical expression of the MTPA condition can be derived from equation [4], differentiating with respect to a current angle g, wherein the current angle g is obtaining the following expression:
Figure imgf000010_0004
Figure imgf000010_0005
Incorporating the inductances defined in the expression [5]
Figure imgf000010_0006
simplifies to:
Figure imgf000010_0007
Preferably, the control system 100 comprises a second conversion and transformation block 4a. As example, the second conversion and transformation block 4a is a Clarke's transformation block configured to convert a three-phase measured current iabc, acquired by means of current transducers on a machine supply line, into the stator reference frame ab, obtaining an actual stator current
Figure imgf000010_0008
Preferably, as shown in Figure 1 , a third conversion and transformation block 4b converts the three-phase measured current
Figure imgf000010_0009
into the estimated rotor reference frame to serve as a feedback for close-looped current vector control. Preferably, the third conversion and transformation block 4b is a Park transformation. According to embodiments of the present invention, as shown in Figure 1 , the control system 100 comprises an observer block 6.
The observer block 6, preferably, receives in input the estimated rotor angular position the estimated voltage vector and the actual stator current
Figure imgf000011_0013
respectively provided by the first conversion and transformation block 3a and the second conversion and transformation block 4a - and calculates a current model based stator flux linkage estimate and an observed stator flux linkage
Figure imgf000011_0012
Figure imgf000011_0014
Preferably, the observer block 6 is a Hybrid Flux Observer (HFO) 6.
In particular, the HFO 6 is configured for: calculating an observed stator flux linkage of said stator based on
Figure imgf000011_0004
estimated values of electric quantities, as example, as a function of the estimated voltage vector associated with said synchronous reluctance
Figure imgf000011_0006
machine 5; calculating a current model based stator flux linkage estimate of said
Figure imgf000011_0005
stator based on detected values of said electric quantities, as example, as a function of the actual stator current
Figure imgf000011_0011
Preferably, the HFO 6 in the stator reference frame is defined by:
Figure imgf000011_0001
wherein
Figure imgf000011_0003
wherein is a 2x2 gain matrix, is the observed stator flux linkage in the stator
Figure imgf000011_0007
reference frame, is the time-derivative of the observed stator flux linkage in the
Figure imgf000011_0008
stator reference frame, is the estimated voltage vector in the stator reference
Figure imgf000011_0009
frame, is the actual stator current and
Figure imgf000011_0010
is the estimated inductance matrix. With reference to equation [8], A is the flux maps lookup table of the SyR machine 5.
Preferably, in the estimated rotor reference frame, the HFO 6 is defined by:
Figure imgf000011_0002
Figure imgf000012_0001
It has to be noted that, for the equivalence holds.
Figure imgf000012_0015
Figure imgf000012_0002
As shown in Figure 2, the HFO 6 calculates - as a function of the actual stator current the estimated voltage vector and the estimated rotor angular
Figure imgf000012_0003
position - the current model based stator flux linkage estimate and the
Figure imgf000012_0005
observed stator flux linkage by means of equations [7], [8], [9] and [10]
Figure imgf000012_0004
As shown in Figure 3, the current model based stator flux linkage estimate
Figure imgf000012_0006
and the observed stator flux linkage are sent in input to an error evaluation block
Figure imgf000012_0007
11.
The error evaluation block 11 calculates a position error signal
Figure imgf000012_0009
Preferably, the position error signal
Figure imgf000012_0008
is defined as the projection of the difference of the observed stator flux linkage and the current model based stator
Figure imgf000012_0010
flux linkage estimate
Figure imgf000012_0011
As example, the position error signal is defined as:
Figure imgf000012_0014
Figure imgf000012_0012
as described in i. and further developed in ii. i. M. Hinkkanen, S. E. Saarakkala, H. A. A. Awan, E. Molsa, and T. Tuovinen, Observers for Sensorless Synchronous Machine Drives: Framework for Design and Analysis,” IEEE Transactions on Industry Applications, p. 1 , 2018; and ii. Varatharajan and G. Pellegrino, “Sensorless Control of Synchronous Reluctance Machine Drives: Improved Modeling and Analysis beyond Active
Flux,” in Electric Machines and Drives Conference (IEMDC), IEEE International, 2019.
Wherein is a first projection vector.
According to embodiments of the present invention, the first projection vector
Figure imgf000012_0013
is expressed as:
Figure imgf000013_0001
wherein w is the rotor angular speed, is an auxiliary flux linkage
Figure imgf000013_0002
vector.
In the following, the flux error and position error dynamics are expressed with reference to steady-state operation conditions. Steady-state operating point quantities are signified by the subscript 0.
The first projection vector in steady-state condition is expressed as:
Figure imgf000013_0003
Figure imgf000013_0004
wherein is the first projection vector in steady-state operating condition, w0 is the rotor angular speed in such steady state operating condition,
Figure imgf000013_0005
is an auxiliary flux linkage vector in such steady state operating condition As shown in Figure 1 and Figure 3, an observed rotor angular speed
Figure imgf000013_0014
and/or an observed rotor angular position
Figure imgf000013_0012
are calculated as a function of the position error signal Preferably, the observed rotor angular speed and the observed
Figure imgf000013_0015
Figure imgf000013_0013
rotor angular position
Figure imgf000013_0011
are calculated by means of a position and angular speed estimation block 12. As example, the observed rotor angular speed
Figure imgf000013_0010
and the observed rotor angular position are calculated as:
Figure imgf000013_0006
wherein is the observed rotor angular speed, are the proportional and
Figure imgf000013_0017
integral gains respectively and is the integral output of the phase locked loop
Figure imgf000013_0016
proportional-integral regulator, and is the rotor angular speed error
Figure imgf000013_0009
Figure imgf000013_0018
Considering equation [13] and the stator flux error defined as:
Figure imgf000013_0007
wherein is the stator flux error, is the actual stator flux vector and is the
Figure imgf000013_0008
observed stator flux vector, the flux error dynamic can be represented as:
Figure imgf000014_0006
Figure imgf000014_0001
wherein
Figure imgf000014_0002
and considering invariance of the incremental inductance, the following expression is obtained:
Figure imgf000014_0003
By means of the above expression, an auxiliary flux linkage vector is defined
Figure imgf000014_0007
as:
Figure imgf000014_0004
Wherein J is the orthogonal rotational matrix,
Figure imgf000014_0008
is the estimated inductance matrix, l is the incremental inductance matrix, and is the stator current vector in steady
Figure imgf000014_0009
state operating condition. Therefore, the flux error dynamic simplifies to:
Figure imgf000014_0010
Figure imgf000014_0005
hence, the position error signal is defined as:
Figure imgf000014_0011
Figure imgf000014_0012
If the system is stable, the condition holds at steady state.
Figure imgf000014_0013
Manipulating the equations [20] and [21], the steady-state position error owing to
Figure imgf000014_0015
a stator resistance error is expressed as follows:
Figure imgf000014_0014
Figure imgf000015_0001
Substituting the expression of the proposed projection vector [13] (i.e., the first projection vector at steady-state condition) in equation [22], the steady state
Figure imgf000015_0015
position error can be expressed as:
Figure imgf000015_0016
Figure imgf000015_0002
Equation [23] and equation [6] show that, as long as the MTPA condition is respected, the rotor position error calculated by means of the first projection
Figure imgf000015_0003
vector is basically not affected by the stator resistance error
Figure imgf000015_0005
According to the mathematical model, the position error
Figure imgf000015_0006
owing to the stator resistance error calculated by means of said first projection vector is
Figure imgf000015_0004
Figure imgf000015_0007
basically zero at MTPA condition. In other words, the measured position error
Figure imgf000015_0009
calculated by means of said first projection vector is within a band of one electrical degree at MTPA condition.
Figure imgf000015_0008
In particular, the aforesaid measured position error
Figure imgf000015_0010
is due to, as example, white noise and/or inaccuracies in flux-maps.
It has to be noted that, said band of one electrical degree at MTPA condition is proportional to: the accuracy of magnetic model of machine under test; and to deviations from MTPA trajectory.
Moreover, advantageously, a fundamental component of a voltage error due to inverter dead time is in phase with the actual stator current
Figure imgf000015_0011
In particular, the voltage error due to inverter dead time is reflected in the stator resistance error In other words, the position error
Figure imgf000015_0014
calculated by means of said
Figure imgf000015_0012
first projection vector is independent from the voltage error due to the inverter
Figure imgf000015_0013
dead time.
As shown in Figure 1 , the observed rotor angular position
Figure imgf000016_0008
and/or the observed rotor angular speed
Figure imgf000016_0007
is/are used to drive the SyR machine 5.
As example, a conventional phase lock loop with a proportional integral controller is employed to drive the position error signal to zero.
Figure imgf000016_0014
The control of a Synchronous Reluctance Machine by means of observed quantities is known in the art, and it will not be discussed in the following.
It has to be noted that the MTPA condition, imposed by the MTPA controller 2, deviates from the maximum torque per ampere values for small loads due to the imposition of a fundamental excitation current
Figure imgf000016_0003
Another case of deviation from the MTPA condition is the so called "Flux weakening", but at high speed, the stator resistance error effects are negligible. Advantageously, If the SyR machine 5 is
Figure imgf000016_0006
not at MTPA condition, a steady-state position error calculated as described
Figure imgf000016_0004
above in equation [23] (i.e. , as a function of the first projection vector occurs
Figure imgf000016_0005
although the control remains stable.
Preferably, when the MTPA condition is not respected, a stator resistance adaptation is provided.
Preferably, the error evaluation block 11 calculates a resistance error signal
Figure imgf000016_0010
Preferably, the resistance error signal is defined as:
Figure imgf000016_0009
Figure imgf000016_0001
wherein is a second projection vector.
Preferably, the control system comprises a resistance calculation block 13. The resistance calculation block 13 calculates an observed stator resistance as a
Figure imgf000016_0012
function of said resistance error signal
Figure imgf000016_0011
As example, the estimated stator resistance is calculated as:
Figure imgf000016_0013
Figure imgf000016_0002
wherein kr is a constant.
Preferably, the resistance error signal is used to implement a software stator resistance adaptation of the estimated stator resistance. The software stator resistance adaptation compensates for non-compensated voltage error estimate caused by the inverter dead-time. As example, inaccurate dead-time compensation results in constant voltage error that is independent of the operating point, currents or speed. As example, considering an experiment wherein a dead-time error of 1 ms result in a constant voltage error of about 6 V. If the amplitude of the present operating point is 1A, the software stator resistance adaptation increases the estimated resistance by 6 W to compensate the 6V voltage error. Conversely, at a higher load, if the operating current is 2A, the estimated resistance will be 3 W higher than the real resistance.
Preferably, the position error signal and the resistance error signal are
Figure imgf000017_0005
Figure imgf000017_0002
estimated independently.
Preferably, the second projection vector
Figure imgf000017_0004
is a vector orthogonal to the first projection vector Even more preferably, the second projection vector is a
Figure imgf000017_0006
Figure imgf000017_0003
vector orthogonal to the first projection vector and is scaled proportionally with
Figure imgf000017_0007
operating point to have a constant loop gain.
As example, a cumulative error vector e is defined as
Figure imgf000017_0001
wherein is a projection vector matrix.
Figure imgf000017_0008
As described, the present invention provides important advantages. In particular, the present invention provides the following advantages:
- no need for online resistance adaptation at MTPA condition: this simplifies the control algorithm calibration and execution;
- the use of the first projection vector in the control system 100 makes the control inherently stable at low speed; - the position error calculated by means of said first projection vector is
Figure imgf000017_0010
Figure imgf000017_0009
independent from the voltage error due to inverter dead time, so precise MTPA torque control is possible, resulting in maximized efficiency;
- the second projection vector compensates resistance variations when the
Figure imgf000017_0011
machine is deliberately controlled out of MTPA conditions; - if the second projection vector is used, inverter identification for inverter voltage error compensation can be omitted: this simplifies the electric drive preliminary commissioning stage.

Claims

1. A method for sensorless estimation of a position error signal for a
Figure imgf000019_0001
synchronous reluctance machine (5), said synchronous reluctance machine (5) comprising a stator and a rotor, said method comprising: receiving an observed flux linkage of said stator;
Figure imgf000019_0002
receiving a current model based flux linkage estimate of said stator;
Figure imgf000019_0003
selecting a first projection vector wherein said first projection vector
Figure imgf000019_0004
is expressed as:
Figure imgf000019_0019
wherein:
• w is the rotor angular speed;
• are components of an auxiliary flux linkage vector of said
Figure imgf000019_0005
Figure imgf000019_0006
stator, wherein said auxiliary flux linkage vector
Figure imgf000019_0007
• J is an orthogonal rotational matrix;
• G is a gain matrix; and
• / is an identity matrix; so that a rotor angular position error due to a stator resistance error
Figure imgf000019_0008
Figure imgf000019_0009
calculated, at a maximum torque per ampere condition, by means of said first projection vector is within a band of one electrical degree;
Figure imgf000019_0010
calculating a position error signal as a function of said observed flux
Figure imgf000019_0013
linkage said current model based flux linkage estimate and said
Figure imgf000019_0011
Figure imgf000019_0014
first projection vector
Figure imgf000019_0012
2. The method according to claim 1 wherein said position error signal is
Figure imgf000019_0015
calculated as the projection of a difference between said observed flux linkage and said current model based flux linkage estimate on said first
Figure imgf000019_0017
Figure imgf000019_0016
projection vector
Figure imgf000019_0018
3. The method according to any of the previous claims wherein said method further comprises: selecting a second projection vector said second projection vector being orthogonal to said first projection vector
Figure imgf000020_0001
calculating a resistance error signal as a function of said observed flux
Figure imgf000020_0003
linkage said current model based flux linkage estimate and said
Figure imgf000020_0004
Figure imgf000020_0002
second projection vector
Figure imgf000020_0005
4. The method according to claim 3 wherein said resistance error signal is
Figure imgf000020_0006
calculated as the projection of a difference between said observed flux linkage and said current model based flux linkage estimate on said second
Figure imgf000020_0008
Figure imgf000020_0007
projection vector
Figure imgf000020_0009
5. A method for sensorless estimation of at least one of observed rotor angular position and observed rotor angular speed
Figure imgf000020_0010
of a synchronous reluctance
Figure imgf000020_0012
machine (5) comprising: calculating said position error signal according to anyone of claim 1 or
Figure imgf000020_0011
2; calculating at least one of observed rotor angular position and observed
Figure imgf000020_0014
rotor angular speed
Figure imgf000020_0013
as a function of said position error signal
Figure imgf000020_0015
6. The method according to claim 5 wherein said method further comprises: calculating said resistance error signal
Figure imgf000020_0017
according to claim 3 or 4; calculating an observed stator resistance as a function of said
Figure imgf000020_0016
resistance error signal
Figure imgf000020_0018
7. A control system (100) for a synchronous reluctance machine (5), said synchronous reluctance machine (5) comprising a rotor and a stator, said control system (100) comprising:
- a hybrid flux observer (5) configured for: calculating an observed flux linkage
Figure imgf000020_0019
of said stator based on estimated values of electric quantities associated with said synchronous reluctance machine (5); calculating a current model based flux linkage estimate of said
Figure imgf000020_0020
stator based on detected values of electric quantities; an error evaluation block (11) configured for: receiving said observed flux linkage and said current model based
Figure imgf000021_0002
flux linkage estimate from said hybrid flux observer (5);
Figure imgf000021_0003
selecting a first projection vector wherein said first projection vector
Figure imgf000021_0004
is expressed as:
Figure imgf000021_0005
Figure imgf000021_0001
wherein:
• w is the rotor angular speed;
• are components of an auxiliary flux linkage vector
Figure imgf000021_0006
Figure imgf000021_0007
• J is an orthogonal rotational matrix;
• G is a gain matrix; and
• / is an identity matrix; so that a rotor angular position error
Figure imgf000021_0008
due to a stator resistance error calculated, at a maximum torque per ampere condition, by means
Figure imgf000021_0010
of said first projection vector is within a band of one electrical
Figure imgf000021_0009
degree; calculating a position error signal
Figure imgf000021_0012
as a function of said observed flux linkage said current model based flux linkage estimate and
Figure imgf000021_0013
Figure imgf000021_0011
said first projection vector
Figure imgf000021_0014
and a position and angular speed estimation block (12), configured to calculate an observed rotor angular position
Figure imgf000021_0015
and/or an observed rotor angular speed
Figure imgf000021_0019
as a function of said position error signal
Figure imgf000021_0016
wherein said control system (100) is configured to drive said synchronous reluctance machine (5) based on said observed rotor angular position
Figure imgf000021_0017
and/or said observed rotor angular speed
Figure imgf000021_0018
8. The control system (100) according to claim 7 wherein said control system (100) is configured to drive said synchronous reluctance machine (5) at a maximum torque per ampere condition.
9. The control system (100) according to claim 7 or 8 wherein said error evaluation block (11) is configured for: selecting a second projection vector said second projection vector
Figure imgf000022_0001
being orthogonal to said first projection vector
Figure imgf000022_0002
- calculating a resistance error signal as a function of said observed flux
Figure imgf000022_0008
linkage said current model based flux linkage estimate and said
Figure imgf000022_0005
Figure imgf000022_0003
second projection vector
Figure imgf000022_0004
said control system (100) comprising a resistance calculation block (13) configured to calculates an observed stator resistance as a function of said
Figure imgf000022_0006
resistance error signal
Figure imgf000022_0009
10. The control system (100) according to claim 9 wherein said control system (100) comprises a software stator resistance adaptation block based on said observed stator resistance
Figure imgf000022_0007
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