US20070194742A1 - Angular position and velocity estimation for synchronous machines based on extended rotor flux - Google Patents
Angular position and velocity estimation for synchronous machines based on extended rotor flux Download PDFInfo
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- 230000004907 flux Effects 0.000 title claims abstract description 68
- 230000001360 synchronised effect Effects 0.000 title description 6
- 238000000034 method Methods 0.000 claims abstract description 7
- 238000004364 calculation method Methods 0.000 claims description 4
- 238000012937 correction Methods 0.000 abstract description 12
- 238000010586 diagram Methods 0.000 description 5
- 230000010354 integration Effects 0.000 description 3
- 238000005259 measurement Methods 0.000 description 2
- 238000013461 design Methods 0.000 description 1
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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
- 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
- H02P21/16—Estimation of constants, e.g. the rotor time constant
-
- 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
- H02P21/06—Rotor flux based control 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
- H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
- Y10S—TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y10S388/00—Electricity: motor control systems
- Y10S388/907—Specific control circuit element or device
- Y10S388/911—Phase locked loop
Definitions
- An aircraft generator is usable as a motor for engine starting when powered by an inverter. To reduce cost and improve reliability, it is very desirable to eliminate the mechanical shaft sensor for the engine starter.
- a novel sensorless synchronous dynamoelectric machine control based on dynamoelectric machine flux estimation, designated Extended Flux Sensorless (EFS) position sensing, is disclosed in U.S. Ser. No. 10/917,018, filed 12 Aug. 2004, to Markunas et al. and hereby incorporated by reference. Markunas et al. defines extended rotor flux, which aligns with the rotor field flux axis.
- the invention comprises a method of correcting the determination of extended rotor flux using a lag function and a correction algorithm that closely approximates a pure integrator function to correct for lag function errors that can extend the EFS control down to dynamoelectric machine speeds corresponding to as low as 10 Hz electrical frequency.
- FIG. 1 is a block diagram of operations performed within a controller for controlling a dynamoelectric machine that calculates extended rotor flux according to the prior art.
- FIG. 2 is a block diagram of elements within a controller for controlling a dynamoelectric machine that estimates rotor angular position and velocity of the dynamoelectric machine using a phase lock loop (PLL) according to the prior art.
- PLL phase lock loop
- FIG. 3 is a phasor diagram that shows extended flux sensorless correction according to the invention.
- FIG. 6 is an estimated flux sensorless block diagram with correction algorithm according to the invention.
- phase locked loop described in Markunas et al. and shown in FIG. 2 . Note that use of the phase locked loop forces the estimated rotor electrical frequency ⁇ circumflex over ( ⁇ ) ⁇ r to be equal to the synchronous electrical frequency ⁇ e of the terminal electrical potential applied to the dynamoelectric machine.
- FIG. 3 shows equations (5) through (12) graphically in phasor format.
- Phasors 2 and 4 represent net electrical potential phasors, v ⁇ ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ and v ⁇ ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ , respectively.
- FIG. 3 arbitrarily aligns the ⁇ -axis net electrical potential phasor 2 with the y-axis. The alignment of the ⁇ -axis net electrical potential phasor 4 must then extend along the x-axis for a positive sequence 3-phase electrical potential waveform applied to the synchronous dynamoelectric machine terminals.
- Phasors 6 and 8 represent the estimated stator flux linkages ⁇ circumflex over ( ⁇ ) ⁇ ⁇ and ⁇ circumflex over ( ⁇ ) ⁇ ⁇ , respectively. Phasors 6 and 8 lag their respective net electrical potential phasors 2 and 4 by less than 90°. Phasors 10 and 12 represent estimated stator flux linkages determined using a pure integrator and do indeed lag their respective net electrical potential phasors 2 and 4 by exactly 90°.
- FIG. 4 shows a right triangle formed by phasors 6 10 and 14 from FIG. 3 .
- Phasor 10 represents the hypotenuse of the right triangle, and it is proportional in length to the magnitude of the ⁇ -axis stator flux linkage estimated using a pure integrator.
- Phasor 6 represent the adjacent side of the right triangle, and it is proportional in length to the magnitude of the ⁇ -axis stator flux linkage estimated using a lag approximation.
- the magnitude of phasor 10 is proportional to 1/ ⁇ circumflex over ( ⁇ ) ⁇ r and the magnitude of phasor 6 is proportional to 1/ ⁇ square root over ( ⁇ circumflex over ( ⁇ ) ⁇ r 2 + ⁇ i 2 ) ⁇
- the magnitude of phasor 14 is necessarily proportional to ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ square root over ( ⁇ circumflex over ( ⁇ ) ⁇ r 2 + ⁇ i 2 ) ⁇ .
- FIG. 5 shows a similar right triangle formed by phasors 6 , 10 and 14 and dividing the magnitude of each of the sides by ⁇ square root over ( ⁇ circumflex over ( ⁇ ) ⁇ r 2 + ⁇ i 2 ) ⁇ .
- the adjacent side of the right triangle represented by phasor 6 now has a length of unity and the hypotenuse of the triangle represented by phasor 10 is formed by vectorially adding ⁇ circumflex over ( ⁇ ) ⁇ ⁇ and ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ .
- An ⁇ circumflex over (R) ⁇ s function 20 representing estimated stator resistance multiplies ⁇ -axis current on a signal path 22 by ⁇ circumflex over (R) ⁇ s to produce a signal ⁇ circumflex over (R) ⁇ s ⁇ i a on a signal path 24 .
- a summer 26 subtracts ⁇ circumflex over (R) ⁇ i ⁇ on the signal path 24 from the ⁇ -axis potential v ⁇ on a signal path 28 to produce a signal v ⁇ ⁇ ( ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ ) on a signal path 30 .
- a 1 s + ⁇ i lag function 32 as described above multiplies v ⁇ ⁇ ( ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ ) on the signal path 30 by 1 s + ⁇ i to produce a signal 1 s + ⁇ i ⁇ ( v ⁇ + ( R ⁇ s ⁇ i ⁇ ) ) on a signal path 34 that represents estimated ⁇ -axis stator flux ⁇ circumflex over ( ⁇ ) ⁇ ⁇ .
- An ⁇ circumflex over (L) ⁇ q function 36 representing estimated q-axis inductance of the dynamoelectric machine multiplies the ⁇ -axis current i ⁇ on the signal path 20 by ⁇ circumflex over (L) ⁇ q to produce a signal i ⁇ ⁇ circumflex over (L) ⁇ q on a signal path 38 .
- Another summer 40 subtracts i ⁇ ⁇ circumflex over (L) ⁇ q on the signal path 38 from ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on the signal path 34 to produce an uncorrected estimated ⁇ -axis extended rotor flux signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext unc as represented by ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ⁇ ( ⁇ ⁇ ⁇ circumflex over (L) ⁇ q ) on a signal path 42 .
- Uncorrected estimated ⁇ -axis extended rotor flux ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext unc is determined in a similar way.
- An ⁇ circumflex over (R) ⁇ s function 44 representing estimated stator resistance multiplies ⁇ -axis current on a signal path 46 by ⁇ circumflex over (R) ⁇ s to produce a signal ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ on a signal path 48 .
- Another summer 50 subtracts ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ on the signal path 48 from ⁇ -axis potential v ⁇ on a signal path 52 to produce a signal v ⁇ ⁇ ( ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ ) on a signal path 54 .
- Another 1 s + ⁇ i lag function 56 multiplies v ⁇ ⁇ ( ⁇ circumflex over (R) ⁇ s ⁇ i ⁇ ) on the signal path 54 by 1 s + ⁇ i to produce a signal 1 s + ⁇ i ⁇ ( v ⁇ - ( R ⁇ s ⁇ i ⁇ ) ) on a signal path 58 that represents estimated ⁇ -axis stator flux ⁇ circumflex over ( ⁇ ) ⁇ ⁇ .
- Another ⁇ circumflex over (L) ⁇ q function 60 representing estimated q-axis inductance of the dynamoelectric machine multiplies the ⁇ axis current i ⁇ on the signal path 46 by ⁇ circumflex over (L) ⁇ q to produce a signal i ⁇ ⁇ circumflex over (L) ⁇ q on a signal path 62 .
- Another summer 64 subtracts i ⁇ ⁇ circumflex over (L) ⁇ q on the signal path 62 from ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on the signal path 58 to produce an uncorrected estimated ⁇ -axis extended rotor flux signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext unc as represented by ⁇ ⁇ ⁇ (i ⁇ ⁇ circumflex over (L) ⁇ q ) on a signal path 66 .
- These operations together comprise an uncorrected extended rotor flux estimation calculation system 68 according to the prior art as described in Markunas et al. and shown in FIG. 1 .
- an estimated extended rotor flux correction algorithm 70 corrects deviations of ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext unc and ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext unc as described above to generate corrected values of estimated extended rotor flux ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext for the ⁇ -axis and ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext for the ⁇ -axis that closely approximate values that would result from operations with a pure integrator 1 s .
- a digital PLL 72 as described in Markunas et al. and shown in FIG.
- a comparator 74 compares a signal representing the estimated rotor electrical frequency ⁇ circumflex over ( ⁇ ) ⁇ r on a signal line 76 with a signal representing a minimum frequency ⁇ o on a signal path 78 to avoid dividing by zero or overflowing the resulting ratio, ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r , in fixed point math processors.
- the comparator 74 selects the higher level of these two signals to produce a signal representing a selected estimated rotor electrical frequency ⁇ circumflex over ( ⁇ ) ⁇ r on a signal path 80 .
- a divider function 82 divides the lag function corner frequency ⁇ i on a signal path 84 by the selected estimated rotor electrical frequency ⁇ circumflex over ( ⁇ ) ⁇ r to produce a signal ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r on a signal path 86 .
- a multiplier 88 multiplies the signal ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r on the signal path 86 by the signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on the signal path 58 to produce a signal ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on a signal path 90 .
- a summer 92 adds the uncorrected estimated ⁇ -axis extended rotor flux signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext unc as represented by ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ⁇ i ⁇ ⁇ circumflex over (L) ⁇ q on the signal path 42 to the signal ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on the signal path 90 to produce a signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ⁇ i ⁇ ⁇ circumflex over (L) ⁇ q +( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ representing ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext on a signal path 94 .
- a multiplier 96 multiplies the signal ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r on the signal path 86 by the signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on the signal path 34 to produce a signal ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on a signal path 98 .
- a summer 100 subtracts the signal ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ on the signal path 98 from the uncorrected estimated ⁇ -axis extended rotor flux signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext unc as represented by ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ⁇ i ⁇ ⁇ circumflex over (L) ⁇ q on the signal path 66 to produce a signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ⁇ i ⁇ ⁇ circumflex over (L) ⁇ q ⁇ ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ representing ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext on a signal path 102 .
- the signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ⁇ i ⁇ ⁇ circumflex over (L) ⁇ q +( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ representing ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext on the signal path 94 and the signal ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ⁇ i ⁇ ⁇ circumflex over (L) ⁇ q ⁇ ( ⁇ i / ⁇ circumflex over ( ⁇ ) ⁇ r ) ⁇ circumflex over ( ⁇ ) ⁇ ⁇ representing ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext on a signal path 102 serve as inputs to the digital PLL 72 .
- a multiplier 104 multiplies the corrected estimated ⁇ -axis extended rotor flux ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext with a feedback signal on a signal path 106 from a sine function 108 to produce an ⁇ -axis multiplier output signal on a signal path 110 .
- a multiplier 112 multiplies the corrected estimated ⁇ -axis extended rotor flux ⁇ circumflex over ( ⁇ ) ⁇ ⁇ ext with a feedback signal on a signal path 114 from a cosine function 116 to produce a ⁇ -axis multiplier output signal on a signal path 118 .
- a summer 120 subtracts the ⁇ -axis multiplier output signal on the signal path 110 from the ⁇ -axis multiplier output signal on a signal path 118 to produce a difference signal on a signal path 122 .
- a proportional plus integral regulator (PI) function 124 multiplies the difference signal on the signal path 122 by the function K ⁇ + K i s to produce a PI output signal on a signal path 126 , wherein K p and K i are the proportional and integral gains of the PI function 124 , respectively.
- An integral function 128 multiplies the PI output signal on the signal path 126 by the function 1 s to produce an integration output signal on a signal path 130 .
- the integration output signal on the signal path 130 serves as the input signal for both the sine function 108 and the cosine function 116 to provide the PLL.
- the integration output signal on the signal path 130 therefore represents the estimated rotor electrical position ⁇ r .
- a low pass filter (LPF) function 132 multiplies the PI output signal on the signal path 126 by the function ⁇ c s + ⁇ c , where ⁇ c is the corner frequency of the LPF function 132 to produce the estimated rotor electrical angular velocity ⁇ circumflex over ( ⁇ ) ⁇ r on the signal path 76 .
- the LPF function 132 is desirable to better attain a smooth signal for the estimated rotor electrical frequency ⁇ circumflex over ( ⁇ ) ⁇ r .
- the corrected EFS control algorithm is only effective above some minimum dynamoelectric machine fundamental electrical frequency. Below this frequency, the correction algorithm described in equations (13) and (14) becomes inaccurate due to sensor inaccuracies, digital word length effects, and measurement and computation noise. Test results to date were able to demonstrate acceptable accuracy above an approximate 10 Hz electrical frequency. Below this threshold speed, determination of rotor position for inverter commutation requires the use of alternative means.
- Described above is a method of correcting the determination of extended rotor flux using a lag function and a correction algorithm that closely approximates a pure integrator function to correct for lag function errors.
- the described embodiment of the invention is only an illustrative implementation of the invention wherein changes and substitutions of the various parts and arrangement thereof are within the scope of the invention as set forth in the attached claims.
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Abstract
Description
- The invention relates to rotor angular position and velocity sensing systems for mechanical shaft sensorless control of dynamoelectric machines, and more particularly to an improved system for resolving the position of a rotor for a dynamoelectric machine using an estimate of extended rotor flux.
- As the aerospace industry moves into the more electric era, inverter controlled dynamoelectric machine drives become more common onboard aircraft. Next generation dynamoelectric machine controllers must meet many new system and design challenges including cost reduction and reliability improvement. Shaft sensorless dynamoelectric machine control holds great promise for meeting these challenges.
- An aircraft generator is usable as a motor for engine starting when powered by an inverter. To reduce cost and improve reliability, it is very desirable to eliminate the mechanical shaft sensor for the engine starter. A novel sensorless synchronous dynamoelectric machine control based on dynamoelectric machine flux estimation, designated Extended Flux Sensorless (EFS) position sensing, is disclosed in U.S. Ser. No. 10/917,018, filed 12 Aug. 2004, to Markunas et al. and hereby incorporated by reference. Markunas et al. defines extended rotor flux, which aligns with the rotor field flux axis. The dynamoelectric machine rotor position and speed are estimated from the extended rotor flux, which is derived from dynamoelectric machine terminal electrical potential and current measurements (
FIG. 1 ), expressed in the α-β two-axis stationary reference frame well known in the electric machine technical community. Ideally, a pure integrator is required to reconstruct the flux. However, in practice, a pure integrator suffers from direct current (DC) drifting, initial value holding, and even stability problems. Markunas et al. proposes to alleviate these problems with a lag approximation to an integrator. A digital phase lock loop (FIG. 2 ) determines the rotor position and speed from the extended rotor flux. The lag approximation to a pure integrator becomes better at high rotor speeds, asymptotically approaching the characteristics of the integrator at very high speeds. However, at low dynamoelectric machine speeds or equivalently low synchronous electrical frequency, ωe, the error due to the lag approximation can become unacceptably large. - The invention comprises a method of correcting the determination of extended rotor flux using a lag function and a correction algorithm that closely approximates a pure integrator function to correct for lag function errors that can extend the EFS control down to dynamoelectric machine speeds corresponding to as low as 10 Hz electrical frequency.
- For a system that derives an estimated rotor electrical position {circumflex over (θ)}r and electrical frequency {circumflex over (ω)}r for the rotor of a polyphase alternating current (AC) dynamoelectric machine with an extended rotor flux estimation calculation system that generates uncorrected estimated values of extended rotor flux {circumflex over (λ)}α ext unc and {circumflex over (λ)}β ext unc based on derived values of estimated stator flux {circumflex over (λ)}α and {circumflex over (λ)}β the α-β two-axis stationary reference frame using a lag function
that approximates a pure integrator function
wherein ωi represents a corner frequency of the lag function, and a digital phase lock loop (PLL) to determine values of estimated rotor electrical position {circumflex over (θ)}r and electrical frequency {circumflex over (ω)}r based upon the estimated values of extended rotor flux {circumflex over (λ)}α ext and {circumflex over (λ)}β ext, one possible embodiment of the invention comprises a method of correcting the estimated α-axis and β-axis values of estimated extended rotor flux to more closely approximate values generated with a pure integrator function
comprising the steps of: dividing the lag function
corner frequency ωi by the estimated rotary frequency {circumflex over (ω)}r to generate a signal ωi/{circumflex over (ω)}r; multiplying the signal ωi/{circumflex over (ω)}r by the derived values of estimated stator flux {circumflex over (λ)}α and {circumflex over (λ)}β to produce signals (ωi/{circumflex over (ω)}r){circumflex over (λ)}α and (ωi/{circumflex over (ω)}r){circumflex over (λ)}β; adding signal (ωi/{circumflex over (ω)}r){circumflex over (λ)}β to {circumflex over (λ)}α ext unc as represented by {circumflex over (λ)}α−(iα×{circumflex over (L)}q) to produce a signal {circumflex over (λ)}α−(iα×{circumflex over (L)}q)+(ωi/{circumflex over (ω)}r){circumflex over (λ)}β that represents a corrected value of estimated extended rotor flux {circumflex over (λ)}α ext; and subtracting signal (ωi/{circumflex over (ω)}r){circumflex over (λ)}α from {circumflex over (λ)}β ext unc as represented by {circumflex over (λ)}α−(iβ×{circumflex over (L)}q) to produce a signal {circumflex over (λ)}β−(iβ×{circumflex over (L)}q)−(ωi/{circumflex over (ω)}r){circumflex over (λ)}α representing {circumflex over (λ)}β ext, -
FIG. 1 is a block diagram of operations performed within a controller for controlling a dynamoelectric machine that calculates extended rotor flux according to the prior art. -
FIG. 2 is a block diagram of elements within a controller for controlling a dynamoelectric machine that estimates rotor angular position and velocity of the dynamoelectric machine using a phase lock loop (PLL) according to the prior art. -
FIG. 3 is a phasor diagram that shows extended flux sensorless correction according to the invention. -
FIG. 4 is an estimated flux linkage triangle that shows extended flux linkage according to the invention. -
FIG. 5 is an estimated flux linkage similar triangle according to the invention. -
FIG. 6 is an estimated flux sensorless block diagram with correction algorithm according to the invention. - For Extended Flux Sensorless (EFS) control of a synchronous dynamoelectric machine, the following differential equations estimate the stator flux linkages in the α-β two-axis stationary reference frame using a lag approximation to a pure integrator:
where, -
- λα−α-axis stator flux estimate: V-sec
- λβ=β-axis stator flux estimate; V-sec
- vα=α-axis stator potential; V
- vβ=β-axis stator potential; V
- iα=α-axis stator current; A
- iβ=β-axis stator current; A
- {circumflex over (R)}s=estimated stator resistance; Ohm
- ωi=lag function corner frequency; sec−1
- For alternating current (AC) steady state analysis, the transformation of these differential equations into Laplace transform notation is:
where -
- s=Laplace operator; sec−1
- The magnitudes of these two stator flux linkage estimates at an estimated electrical frequency Of {circumflex over (ω)}r are:
and the angles are: - One convenient means for deriving the estimated rotor electrical frequency {circumflex over (ω)}r is the phase locked loop described in Markunas et al. and shown in
FIG. 2 . Note that use of the phase locked loop forces the estimated rotor electrical frequency {circumflex over (ω)}r to be equal to the synchronous electrical frequency ωe of the terminal electrical potential applied to the dynamoelectric machine. - As compared to the results for a lag approximation, the amplitudes and angles for stator flux estimates determined using a pure integrator are:
-
FIG. 3 shows equations (5) through (12) graphically in phasor format.Phasors FIG. 3 arbitrarily aligns the α-axis netelectrical potential phasor 2 with the y-axis. The alignment of the β-axis netelectrical potential phasor 4 must then extend along the x-axis for a positive sequence 3-phase electrical potential waveform applied to the synchronous dynamoelectric machine terminals. In the α-β two-axis stationary reference frame, α-axis quantities lead β-axis quantities in time for positive sequence.Phasors electrical potential phasors Phasors electrical potential phasors Phasors Phasors phasors phasors -
FIG. 4 shows a right triangle formed byphasors 6 10 and 14 fromFIG. 3 . Phasor 10 represents the hypotenuse of the right triangle, and it is proportional in length to the magnitude of the α-axis stator flux linkage estimated using a pure integrator.Phasor 6 represent the adjacent side of the right triangle, and it is proportional in length to the magnitude of the α-axis stator flux linkage estimated using a lag approximation. Since the magnitude ofphasor 10 is proportional to 1/{circumflex over (ω)}r and the magnitude ofphasor 6 is proportional to 1/√{square root over ({circumflex over (ω)}r 2+ωi 2)}, the magnitude ofphasor 14 is necessarily proportional to (ωi/{circumflex over (ω)}r)∞{square root over ({circumflex over (ω)}r 2+ωi 2)}. -
FIG. 5 shows a similar right triangle formed byphasors phasor 6 now has a length of unity and the hypotenuse of the triangle represented byphasor 10 is formed by vectorially adding {circumflex over (λ)}α and (ωi/{circumflex over (ω)}r)×{circumflex over (λ)}β. The mathematical expression for the corrected α-axis extended flux linkage is:
{circumflex over (λ)}α corrected={circumflex over (λ)}α+(ωi/{circumflex over (ω)}r)×{circumflex over (λ)}β (13)
and similarly, the mathematical expression for the corrected β-axis extended flux linkage is:
{circumflex over (λ)}β corrected={circumflex over (λ)}β−(ωi/{circumflex over (ω)}r)×{circumflex over (λ)}α (14) - The extended flux linkages can be determined with these corrected estimates of stator flux linkages as shown in
FIG. 1 and passed to the phase-locked-loop shown inFIG. 2 .FIG. 6 shows an overall block diagram of an EFS system 18 according to the invention with the correction algorithm included. As described in Markunas et al., a special lag function
substitutes for the pure integrator
wherein ωi is a corner frequency of the lag function. An {circumflex over (R)}sfunction 20 representing estimated stator resistance multiplies α-axis current on asignal path 22 by {circumflex over (R)}s to produce a signal {circumflex over (R)}s×ia on asignal path 24. Asummer 26 subtracts {circumflex over (R)}×iα on thesignal path 24 from the α-axis potential vα on asignal path 28 to produce a signal vα−({circumflex over (R)}s×iα) on asignal path 30. A
lag function 32 as described above multiplies vα−({circumflex over (R)}s×iα) on thesignal path 30 by
to produce a signal
on asignal path 34 that represents estimated α-axis stator flux {circumflex over (λ)}α. An {circumflex over (L)}qfunction 36 representing estimated q-axis inductance of the dynamoelectric machine multiplies the α-axis current iα on thesignal path 20 by {circumflex over (L)}q to produce a signal iα×{circumflex over (L)}q on asignal path 38. Anothersummer 40 subtracts iα×{circumflex over (L)}q on thesignal path 38 from {circumflex over (λ)}α on thesignal path 34 to produce an uncorrected estimated α-axis extended rotor flux signal {circumflex over (λ)}α ext unc as represented by {circumflex over (λ)}α−(−α×{circumflex over (L)}q) on asignal path 42. - Uncorrected estimated β-axis extended rotor flux {circumflex over (λ)}β ext unc is determined in a similar way. An {circumflex over (R)}s
function 44 representing estimated stator resistance multiplies β-axis current on asignal path 46 by {circumflex over (R)}s to produce a signal {circumflex over (R)}s×iβ on asignal path 48. Anothersummer 50 subtracts {circumflex over (R)}s×iβ on thesignal path 48 from β-axis potential vβ on asignal path 52 to produce a signal vβ−({circumflex over (R)}s×iβ) on asignal path 54. - Another
lag function 56 multiplies vβ−({circumflex over (R)}s×iβ) on thesignal path 54 by
to produce a signal
on asignal path 58 that represents estimated β-axis stator flux {circumflex over (λ)}β. Another {circumflex over (L)}qfunction 60 representing estimated q-axis inductance of the dynamoelectric machine multiplies the βaxis current iα on thesignal path 46 by {circumflex over (L)}q to produce a signal iβ×{circumflex over (L)}q on asignal path 62. Anothersummer 64 subtracts iβ×{circumflex over (L)}q on thesignal path 62 from {circumflex over (λ)}βon thesignal path 58 to produce an uncorrected estimated β-axis extended rotor flux signal {circumflex over (λ)}β ext unc as represented by λβ−(iβ×{circumflex over (L)}q) on asignal path 66. These operations together comprise an uncorrected extended rotor fluxestimation calculation system 68 according to the prior art as described in Markunas et al. and shown inFIG. 1 . - According to the invention, an estimated extended rotor
flux correction algorithm 70 corrects deviations of {circumflex over (λ)}α ext unc and {circumflex over (λ)}β ext unc as described above to generate corrected values of estimated extended rotor flux {circumflex over (λ)}α ext for the α-axis and {circumflex over (λ)}β ext for the β-axis that closely approximate values that would result from operations with a pure integrator
Adigital PLL 72 as described in Markunas et al. and shown inFIG. 2 derives estimated rotor electrical frequency {circumflex over (ω)}r and estimated rotor electrical position {circumflex over (θ)}r of the dynamoelectric machine rotor from the corrected estimated extended rotor flux {circumflex over (λ)}α ext for the α-axis and {circumflex over (λ)}β ext for the β-axis that thecorrection algorithm 70 generates. - The description of the
correction algorithm 70 is as follows. Acomparator 74 compares a signal representing the estimated rotor electrical frequency {circumflex over (ω)}r on asignal line 76 with a signal representing a minimum frequency ωo on asignal path 78 to avoid dividing by zero or overflowing the resulting ratio, ωi/{circumflex over (ω)}r, in fixed point math processors. Thecomparator 74 selects the higher level of these two signals to produce a signal representing a selected estimated rotor electrical frequency {circumflex over (ω)}r on asignal path 80. Adivider function 82 divides the lag function corner frequency ωi on asignal path 84 by the selected estimated rotor electrical frequency {circumflex over (ω)}r to produce a signal ωi/{circumflex over (ω)}r on asignal path 86. - A
multiplier 88 multiplies the signal ωi/{circumflex over (ω)}r on thesignal path 86 by the signal {circumflex over (λ)}β on thesignal path 58 to produce a signal (ωi/{circumflex over (ω)}r){circumflex over (λ)}β on asignal path 90. Asummer 92 adds the uncorrected estimated α-axis extended rotor flux signal {circumflex over (λ)}α ext unc as represented by {circumflex over (λ)}α−iα×{circumflex over (L)}q on thesignal path 42 to the signal (ωi/{circumflex over (ω)}r){circumflex over (λ)}β on thesignal path 90 to produce a signal {circumflex over (λ)}α−iα×{circumflex over (L)}q+(ωi/{circumflex over (ω)}r){circumflex over (λ)}β representing {circumflex over (λ)}α ext on asignal path 94. - Similarly, a
multiplier 96 multiplies the signal ωi/{circumflex over (ω)}r on thesignal path 86 by the signal {circumflex over (λ)}α on thesignal path 34 to produce a signal (ωi/{circumflex over (ω)}r){circumflex over (λ)}α on asignal path 98. Asummer 100 subtracts the signal (ωi/{circumflex over (ω)}r){circumflex over (λ)}α on thesignal path 98 from the uncorrected estimated β-axis extended rotor flux signal {circumflex over (λ)}β ext unc as represented by {circumflex over (λ)}β−iβ×{circumflex over (L)}q on thesignal path 66 to produce a signal {circumflex over (λ)}β−iβ×{circumflex over (L)}q−(ωi/{circumflex over (ω)}r){circumflex over (λ)}α representing {circumflex over (λ)}β ext on a signal path 102. - The signal {circumflex over (λ)}α−iα×{circumflex over (L)}q+(ωi/{circumflex over (ω)}r){circumflex over (λ)}β representing {circumflex over (λ)}α ext on the
signal path 94 and the signal {circumflex over (λ)}β−iβ×{circumflex over (L)}q−(ωi/{circumflex over (ω)}r){circumflex over (λ)}α representing {circumflex over (λ)}β ext on a signal path 102 serve as inputs to thedigital PLL 72. Amultiplier 104 multiplies the corrected estimated α-axis extended rotor flux {circumflex over (λ)}α ext with a feedback signal on asignal path 106 from asine function 108 to produce an α-axis multiplier output signal on asignal path 110. Likewise, amultiplier 112 multiplies the corrected estimated β-axis extended rotor flux {circumflex over (λ)}β ext with a feedback signal on asignal path 114 from acosine function 116 to produce a β-axis multiplier output signal on asignal path 118. - A
summer 120 subtracts the α-axis multiplier output signal on thesignal path 110 from the β-axis multiplier output signal on asignal path 118 to produce a difference signal on asignal path 122. A proportional plus integral regulator (PI)function 124 multiplies the difference signal on thesignal path 122 by the function
to produce a PI output signal on asignal path 126, wherein Kp and Ki are the proportional and integral gains of thePI function 124, respectively. - An
integral function 128 multiplies the PI output signal on thesignal path 126 by the function
to produce an integration output signal on asignal path 130. The integration output signal on thesignal path 130 serves as the input signal for both thesine function 108 and thecosine function 116 to provide the PLL. The integration output signal on thesignal path 130 therefore represents the estimated rotor electrical position θr. - A low pass filter (LPF)
function 132 multiplies the PI output signal on thesignal path 126 by the function
where ωc is the corner frequency of theLPF function 132 to produce the estimated rotor electrical angular velocity {circumflex over (ω)}r on thesignal path 76. TheLPF function 132 is desirable to better attain a smooth signal for the estimated rotor electrical frequency {circumflex over (ω)}r. - The corrected EFS control algorithm is only effective above some minimum dynamoelectric machine fundamental electrical frequency. Below this frequency, the correction algorithm described in equations (13) and (14) becomes inaccurate due to sensor inaccuracies, digital word length effects, and measurement and computation noise. Test results to date were able to demonstrate acceptable accuracy above an approximate 10 Hz electrical frequency. Below this threshold speed, determination of rotor position for inverter commutation requires the use of alternative means.
- Described above is a method of correcting the determination of extended rotor flux using a lag function and a correction algorithm that closely approximates a pure integrator function to correct for lag function errors. The described embodiment of the invention is only an illustrative implementation of the invention wherein changes and substitutions of the various parts and arrangement thereof are within the scope of the invention as set forth in the attached claims.
Claims (6)
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US11/358,437 US7265507B1 (en) | 2006-02-20 | 2006-02-20 | Angular position and velocity estimation for synchronous machines based on extended rotor flux |
GB0703266A GB2435356B (en) | 2006-02-20 | 2007-02-20 | Improved angular position and velocity estimation for synchronous machines based on extended rotor flux |
FR0701200A FR2897729B1 (en) | 2006-02-20 | 2007-02-20 | ENHANCED ESTIMATION OF ANGULAR POSITION AND SPEED FOR SYNCHRONOUS MACHINES BASED ON EXTENDED ROTOR FLOW. |
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US11/358,437 US7265507B1 (en) | 2006-02-20 | 2006-02-20 | Angular position and velocity estimation for synchronous machines based on extended rotor flux |
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GB2435356B (en) | 2008-03-12 |
GB2435356A (en) | 2007-08-22 |
GB0703266D0 (en) | 2007-03-28 |
FR2897729A1 (en) | 2007-08-24 |
US7265507B1 (en) | 2007-09-04 |
FR2897729B1 (en) | 2017-09-15 |
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