The present invention relates to a method for determining an estimate
for the stator flux of an asynchronous machine when the stator current, stator
voltage, supply frequency, stator inductance, stator resistance or an estimate
therefor, and short-circuit inductance of the machine are known. A stator resistance
estimate for the machine can also be determined by the method.
In frequency converter-based control of an asynchronous machine,
the object is often to make the torque generated by the machine to behave in a
desired way when the current and voltage supplied to the machine are known. In
that situation, one attempts to influence the electric torque, which in terms of
the stator flux and stator current is:
Tm= k(Ψs ×
Tm = electric torque,
k = constant coefficient,
Ψs = stator flux, and
is = stator current.
Controlled torque regulation therefore requires that besides the
current is, the stator flux or a commensurate variable
(such as the rotor flux or air gap flux) of the machine is known. This will not
present any problem with operation at high frequencies, in which situation integration
of the voltage supplied to the machine is known to give a good estimate for the
Ψs= ∫usdt = us / (jωs),
us = stator voltage, and
ωs = supply frequency.
Ψs is easy to calculate from equation
2 when the supply voltage and its frequency are known.
It can also be seen from this equation that when ωs
diminishes, below a specific nominal frequency the voltage must be reduced in order
for the flux not to increase too much and the machine not to become saturated.
Yet equation 2 is not practicable with low frequencies, since in
reality the voltage to which the windings of the machine are subjected deviates
from the supply voltage to the extent of the voltage loss developed in the winding
resistances. Thus the relative proportion of the loss component in the voltage
increases when us has to be reduced as ωs
diminishes. With low frequencies the loss component should thus be taken into
account, i.e., the flux estimate should be calculated from the equation:
Ψs = ∫(us-Rsis)dt,
where Rs = stator resistance (see GB-A-2 239 320).
The accuracy of the flux estimate calculated by means of this equation
is, however, strongly dependent on the accuracy of the Rs estimate
employed and on the operating frequency, such that the error in the steady state-of
the flux estimate increases in direct proportion to the error in the
Rs estimate and in inverse proportion to the frequency. On the
other hand, the Rs estimate must always be distinctly smaller
than the actual stator resistance to enable stable control by the integrating
method according to equation 3. Therefore, with the mere integrating method one
can in practice hardly attain frequencies below 10 Hz without a significant steady
state error in the flux estimate.
This problem related to the integrating method can be solved with
the use of either direct or indirect vector control. In the first case, the stator
flux is measured directly with a measuring element incorporated in the machine,
whereas in the latter method it is calculated indirectly on the basis of the stator
current and speed information obtained from a tachometer disposed on the shaft
of the machine. In both cases, the torque of the machine can also be controlled
at zero frequency, but bot methods require an extra measuring element which is
relatively costly and diminishes reliability.
The above problems can be avoided without any need for extra measuring
elements incorporated in the machine by the method of the present invention according
to claim 1.
In the following the invention will be set forth in greater detail
with reference to the accompanying drawings, in which
Figure 1 shows an example of a stator current vector as a function of time,
and the dependence of the difference variable ε on the stator current and
Figure 2 shows an example of function f as a function of the supply frequency,
Figures 3a and 3b show examples of angle &thetas; as a function of the supply
frequency when the torque is a) positive and b) negative, and
Figure 4 shows a method of the invention for calculating the stator flux of
an asynchronous machine.
For deduction of the expression for the reference current, let us
first look at certain known basic equations for the steady state in an asynchronous
machine in stator coordinates:
Ψr = rotor flux,
ir = rotor current,
ωr = slip frequency,
Rr = rotor resistance,
Ls = stator inductance,
Lr = rotor inductance, and
Lm = main inductance.
Employing equations 5 and 6, the rotor flux and rotor current can
be expressed by means of the stator flux and stator current:
Ψr= Lr / (Lm)
(Ψs-σLsis) ir= 1 / (Lm) (Ψs-Lsis),
σ = 1 - L2mLsLr
= dispersion coefficient, and
= short-circuit inductance.
It follows from equation 4 that
In other words, the rotor current in steady state is perpendicular to the rotor
flux, and thus the notation is:
where "&peseta;" = scalar product.
By inserting equations 7 and 8 in equation 10 we have
An incorrect stator flux estimate will not normally satisfy equation
11, and thus the magnitude of the error in the flux estimate may be denoted by
difference variable ε, which is determined as follows:
where Ψse is the stator flux estimate.
As a next step, the electric torque Te is determined
in such a way that
Te = Ψse × is
where isq is the perpendicular component of the stator current
relative to the stator flux estimate.
Now, the scalar product of the flux and current in equation 13 may
be written as
Ψse&peseta;is=Ψseisd=Ψse sqrt(i2s-i2sq)= sqrt(Ψ2sei2s-T2e),
where isd is the component of the stator current having the direction
of the stator flux estimate.
Inserting equation 15 in equation 13 gives the following dependence
between the flux and torque estimates and the square of the stator current:
Ψ2se - (Ls + σLs) sqrt(Ψ2sei2s-T2e)
The aim is to correct the stator flux estimate such that ε is
zeroed in equation 16. In that situation, the absolute value of the stator current
approaches the reference value iref which satisfies the equation:
Ψ2se - (Ls + σLs) sqrt(Ψ2sei2ref-T2e)
where iref represents the current the value of which the absolute
value of the stator current vector should have in the steady state if the machine
had a stator flux of Ψse and a torque of Te.
Thus the square of the reference current obtained from equation 17
as a function of the flux and torque estimates is:
However, calculating the reference current from the statement of
equation 18 is rather cumbersome and also unnecessary, as it can be shown that
In other words, the difference variable ε calculated in equation
12 is positive if the amplitude of the stator current is lower than the reference
current, and vice versa. This dependence has been illustrated in Figure 1. Thus,
using the difference variable it is possible to correct the flux estimate such
that the stator current will be equal in amplitude to the reference current.
In the present invention the correction of the flux estimate is performed
indirectly in such a way that first a correction term proportional to ε is
subtracted from the voltage estimate, wherefrom the flux estimate is subsequently
calculated by integration, i.e. (cf. equation 3):
Ψse = ∫(us-Rseis-εwuc)dt,
correction term for voltage estimate
amplification coefficient (>0) for correction of voltage estimate, and
direction vector for correction of voltage estimate.
Coefficient wu has bearing on how close to the
reference current the measured current is set. The higher the value of
wu, the closer the current will be to the reference and the smaller
ε will also be, in other words, wu is comparable to the
P factor in a conventional controller. It should preferably be selected to be as
high as possible in order for the noise in ε not to have too much influence
on the flux estimate.
The direction vector c is selected so as to form a
predetermined angle &thetas; relative to the flux estimate:
c = ej&thetas; Ψse
In order for the control based on the present method to be stable,
the direction &thetas; of correction of the voltage estimate should be selected
and f(ωs) = odd function as shown in Figure 2. This
receives the value zero when the absolute value of the frequency exceeds a predetermined
threshold frequency ωL. It is piecewise monotonic decreasing
in the range -ωL...ωL, receiving
its minimum and maximum values -&thetas;L and &thetas;L
at zero frequency. ωL and &thetas;L are
machine-dependent to some extent, so that ωL is 10%...20%
from the nominal frequency and &thetas;L is 50...80°.
Thus the direction of correction of the voltage estimate is dependent
on the frequency and torque existing in the machine as shown in Figures 3a and
3b. When the torque is positive, which situation is illustrated in Figure 3a,
with positive frequencies the machine serves as a motor, and in that case the voltage
estimate is only corrected in the direction of the flux estimate (&thetas; = 0).
On the generator side above the threshold frequency -ωL
said angle is turned as a function of the frequency in the negative direction,
so that the angle -&thetas;L
is achieved with zero frequency. Respectively
with a negative torque, which situation is illustrated in Figure 3b, the machine
serves as a motor when the frequency is negative, and in that case &thetas; = 0.
With a positive frequency one operates on the generator side, in which case the
angle is reduced as a function of the frequency starting from the value &thetas;L,
so that above the threshold frequency ωL, &thetas; = 0.
In the calculation of the estimate Rse for the
stator resistance employed in equation 20, one makes use of the finding that a
lower estimate than the actual stator resistance will cause an error in the flux
calculated by the integrating method (equation 3), which will result in too low
a stator current in a no-load situation and on the motor side, and too high a stator
current on the generator side. Respectively, a higher Rse than
actual causes a reverse error in the stator current. By adding to the integrating
method a term correcting the stator voltage estimate (equation 20), the effect
of Rse on the stator current can be considerably diminished,
but also in that case it has a small effect of a similar direction on the current
and thereby also on the difference variable ε, so that on the motor side:
and on the generator side:
Therefore, it is possible to adjust Rse by means
of the difference variable ε and equations 24 and 25 to equal the actual
stator resistance. Thus in the present invention Rse is calculated
Rse = ∫(wrε)dt,
and wR is a positive constant.
The estimate for the stator resistance is thus obtained by integrating
the difference variable ε weighted by coefficient wr (equation
26). In accordance with equation 27, wr is selected in a no-load
situation and on the motor side (q ≥ O) to equal the constant wR
on the generator side (q < 0) to equal the constant -wR, in
consequence of which Rse increases on the motor side and diminishes
on the generator side with a positive ε value. The coefficient wR
determines how fast Rse follows variations in the actual stator
resistance which are mainly due to variations in the temperature of the stator
of the machine dependent on load variations. In practice, wR
should preferably be selected to be rather small, since the actual Rs
can only change very slowly.
With correction of Rse, one achieves setting of
the current vector in steady state at its reference value (ε = 0).
The greater wR, the faster the setting is; yet too high
wR will cause instability. wR is comparable
to the I factor in a conventional controller.
The method of the invention is illustrated as a flow chart in Figure
4. The input variables are the measured stator current is
and stator voltage us of the asynchronous machine 1. Furthermore,
the stator inductance Ls, short-circuit inductance σLs
and supply frequency ws are presumed to be known. The method
gives as an output variable an estimate Ψse for the
stator flux of the machine, in addition to which an estimate Rse
for the stator resistance is also calculated in the method.
Calculation of the stator flux estimate employs equation 20, according
to which first in block 3 the product of the estimates of the stator current and
stator resistance calculated in block 2 is subtracted from the stator voltage
us. Block 4 subtracts the correction term εwuc
from the voltage estimate us-Rseis
obtained as an output from block 3, and the resultant difference is further integrated
in block 5 to obtain a stator flux estimate Ψse.
The stator resistance estimate Rse is calculated
on the basis of equation 26 by integrating in block 12 the product of the difference
variable ε and a weighting factor wr, which has been calculated
in block 11. The weighting factor wr is given by the selector
of block 15, whose output receives the value wR if q ≥ 0,
or the value -wR if q < 0 (equation 27).
To determine the correction term εwuc
for the voltage estimate, angle &thetas; is first formed in block 18, the selector
of which gives as an output either zero if q ≥ O, or a function f(ωs)
of the supply frequency ωs
calculated in block 17 (Figure
2) if q < 0, in accordance with equation 22. From angle &thetas; a unit vector
ej&thetas; is formed in block 19; the unit vector
is multiplied in block 20 by the stator flux estimate obtained from block 5 as
feedback to give a direction vector c for the voltage estimate (equation
21). The resultant direction vector is multiplied in block 21 by the difference
variable ε weighted by factor wu obtained from block 16,
which gives as the output from block 21 the correction term for said voltage estimate.
The difference variable ε is determined by means of a scalar
product in accordance with equation 12. To obtain the first factor of the scalar
product, the stator current is is first multiplied by
the stator inductance Ls in block 6 and the product thus obtained
is subtracted in block 8 from the stator flux estimate Ψse
obtained as feedback from block 5. Respectively, the other factor in said scalar
product is obtained by multiplying the stator current is
by the short circuit inductance σLs in block 7 and subtracting
the product thus obtained in block 9 from the stator flux estimate Ψse
obtained from block 5. Finally, in block 10 a scalar product is calculated from
the outputs of blocks 8 and 9 to give the difference variable ε.
The variable q is determined on the basis of equation 23 by first
calculating in block 13 a cross product of the current is
and the stator flux estimate Ψse obtained as feedback
from block 5, i.e. a torque estimate Te (equation 14) which is
subsequently multiplied in block 14 at supply frequency ωs
to give the variable q.
In practice, the calculation method illustrated in Figure 4 can be
realized either as an analog system or as a time-discrete system based on sampling.
In an analog system the stator flux estimate produced has a direct feedback effect
on the inputs of blocks 20, 8, 9 and 13. In a time-discrete system the input of
said blocks is in practice constituted by a previous value for the stator flux
estimate. However, the selected mode of operation has no effect on the actual method
and its practicability, and both modes of operation are encompassed by the scope
defined in the appended claims.
Verfahren zur Bestimmung einer Schätzung für den Statorfluß einer Asynchronmaschine,
wenn Speisefrequenz ωs, Statorinduktivität Ls,
Statorwiderstand Rs oder eine Schätzung Rse
dafür und Kurzschlußinduktivität σLs bekannt sind, in dem
der Statorstrom is und die Statorspannung
us gemessen werden,
eine Differenzspannung durch Subtrahieren des Produkts aus dem Statorwiderstand
oder seiner Schätzung und dem Statorstrom von der Statorspannung berechnet wird
die Differenzspannung über die Zeit integriert wird, was eine Statorflußschätzung
dadurch gekennzeichnet, daß vor der Integration ein auf
der zurückgeführten Statorflußschätzung ψse basierender
und zum Produkt aus der Differenzvariablen ε und dem dafür gegebenen Richtungsvektor
c proportionaler Korrekturterm von der Differenzspannung subtrahiert
wird, wobei die Differenzvariable aus der Gleichung
bestimmt wird und der Richtungsvektor c aus der zurückgeführten Statorflußschätzung
ψse gebildet wird, indem diese auf den Winkel &thetas;,
der von der Betriebsweise der Maschine abhängt, gedreht wird.
Verfahren nach Anspruch 1,
dadurch gekennzeichnet, daß der Winkel &thetas; gleich
0 ist, wenn die Maschine als Motor arbeitet und der Winkel &thetas; eine Funktion
der Speisefrequenz ωs ist, wenn die Maschine als Generator
Verfahren nach Anspruch 1 oder 2, bei dem im Verfahren auch eine Schätzung
Rse für den Statorwiderstand zur Berechnung der Differenzspannung
dadurch gekennzeichnet, daß die Statorwiderstandsschätzung
Rse für den durch Integrieren des Produkts aus der Differenzvariablen
ε und einem von der Betriebsweise der Maschine abhängigen Faktor
wr über die Zeit bestimmt wird.
Verfahren nach Anspruch 3,
dadurch gekennzeichnet, daß der Faktor wr
gleich WR ist, wenn die Maschine als Motor arbeitet, und der
Faktor wr gleich -wR ist, wenn die Maschine
als Generator arbeitet, wobei wR eine positive Konstante ist.
A method for determining an estimate for the stator flux of an asynchronous
machine when the supply frequency ωs, stator inductance
Ls, stator resistance Rs
or an estimate
Rse therefor, and short-circuit inductance σLs
are known, in which method
the stator current is and stator voltage
us are measured,
a difference voltage is calculated by subtracting the product of the stator
resistance or its estimate and the stator current from the stator voltage and
integrating said difference voltage in relation to time to give a stator flux
estimate Ψse, characterized
in that prior to
the integration a correction term based on the feedback stator flux estimate
Ψse and being proportional to the product of the difference
variable ε and the direction vector c given therefor is subtracted
from said difference voltage, said difference variable being determined from the
and the direction vector c being formed from the feedback stator
flux estimate Ψse by turning it for the angle &thetas;
which is dependent on the operational mode of the machine.
A method as claimed in claim 1, characterized
in that when the machine
serves as a motor the angle &thetas; is 0, and when the machine serves as a generator
the angle &thetas; is a function of the supply frequency ωs.
A method as claimed in claim 1 or claim 2 where an estimate Rse
for the stator resistance for calculating said difference voltage is also determined
in the method, characterized in that the stator resistance estimate
Rse is determined by integrating the product of said difference
variable ε and a factor wr dependent on the operational
mode of the machine in relation to time.
A method as claimed in claim 3, characterized
in that when the machine
serves as a motor the factor wr is wR, and
when the machine serves as a generator the factor wr is -wR
where wR is a positive constant.
Procédé pour déterminer une estimation du flux statorique d'une machine asynchrone
lorsqu'on connaît la fréquence d'alimentation ωs, l'inductance
Ls du stator, la résistance Rs du stator ou une estimation
Rse de celle-ci, et l'inductance de court-circuit σLs,
procédé dans lequel :
on mesure le courant statorique is et la tension statorique
on calcule une tension de différence en soustrayant le produit de la résistance
statorique, ou l'estimation de celle-ci, et du courant statorique de la tension
on intègre ladite tension de différence par rapport au temps pour obtenir une
estimation du flux statorique Ψse,
caractérisé en ce que, avant l'intégration, un facteur de correction basé sur l'estimation
du flux statorique Ψse
en retour et proportionnel au produit
de la variable de différence ε et du vecteur de direction c obtenu
pour celle-ci est soustrait de ladite tension de différence, ladite variable de
différence étant déterminée d'après l'équation
(ψse - Lsis)&peseta;(ψse
- σLsis) = ε
et le vecteur de direction c étant obtenu à partir de l'estimation du flux statorique
ψse en retour en le faisant tourner de l'angle &thetas; fonction
du mode de fonctionnement de la machine.
Procédé selon la revendication 1,
caractérisé en ce que, lorsque la machine sert de moteur, l'angle &thetas; = 0,
et lorsque la machine sert de générateur, l'angle &thetas; est fonction de la fréquence
Procédé selon la revendication 1 ou 2, dans lequel on détermine aussi une estimation
Rse de la résistance statorique pour calculer ladite tension de différence,
caractérisé en ce que l'estimation de la résistance statorique Rse est
déterminée en intégrant le produit de ladite variable de différence ε et d'un
fonction du mode de fonctionnement de la machine par rapport
Procédé selon la revendication 3,
caractérisé en ce que,lorsque la machine sert de moteur, le facteur wr
est wR, et lorsque la machine sert de générateur, le facteur wr
est -wR, wR étant une constante positive.