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6СПОСТЕР1ГАЧ ШВИДКОСТ1 ОБЕРТАННЯ РОТОРА В СИСТЕМ1 ВЕКТОРНОГО КЕРУВАННЯ
АСИНХРОННОЮ МАШИНОЮ
Клюев О.
доцент кафедри електротехнгки та електромехатки Дтпровський державний технгчний утверситет,
м. Кам 'янське, Украша ORCID: 0000-0003-4542-3317
SPEED OBSERVER OF THE ROTATION ROTOR IN THE SYSTEM OF VECTOR CONTROL
ASYNCHRONOUS MACHINE
Klyuyev O.
Dnipro State Technical University ORCID: 0000-0003-4542-3317 DOI: 10.5281/zenodo.7618463
АНОТАЦ1Я
У статп здшснений синтез спостерпача швидкосп обертання ротора асинхронно! машини в бездат-чиковш системi полеорieнтованого керування. Шляхом перетворення диференцшних рiвнянь асинхронно! машини визначено структуру функцл адаптацi!, яка налаштовуе модель спостертача таким чином, щоб вщхилення розрахункових значень струмiв статора вщ !х справжнiх значень прямували до нуля. Таким способом функщя адаптацп, яка використовуе векторний добуток векторiв потокозчеплення ротора та струму статора, забезпечуе асимптотичну стiйкiсть руху щентифжатора швидкостi. У запропонованому спостерiгачi швидкосп в процесi дослiджень на математичних моделях не виявленi автоколивання оцшки швидкостi на вiдмiну ввд вщомого спостерiгача, в якому використовуються ввдхилення потокозчеплень. Уникнути автоколивань вдаеться не пльки за рахунок пiдбору коефщенпв пiдсилення регулятора, але i завдяки тому, що в структурi електромагштних контурiв спостерiгача вiдсутнi iнтегратори, не охоплеш вiд'емним зворотним зв'язком.
ABSTRACT
Purpose. The article aims to determine the structure of the observer, which provides an asymptotically stable process identification of the speed rotation of the rotor an asynchronous machine. Methodology. The structure of the observer in asynchronous electric drives must contain such an adaptation function that ensures the asymptotic stability of the observer itself to the reference model, which is a real control object. In the article, the synthesis of the observer speed rotation of the rotor of an asynchronous machine for its use in a sensorless system field-oriented control of an electric drive is carried out. Equations balance of voltages and flux linkages in an asynchronous machine, formulas for transformation of coordinates and vector product vectors of rotor flux linkage and stator current are used as initial mathematical models. The proposed approach to the synthesis of the observer allows to determine the structure of the observer and the adaptation function, which adjusts the model of electromagnetic processes in such a way that the difference between the measured variables and their estimates goes to zero, that is, the adaptation function, which uses the vectors of the flux linkage of the rotor and the stator current, provides an asymptotic stability of the speed observer. Results. The structure and parameters of an asynchronous machine speed observer are determined, which can be used in frequency and vector control systems for asynchronous electric drives with a linear speed PID controller. In this case, linear and relay controllers can be used in the internal control loops of stator currents. Originality. For the synthesis of observers, modal control methods or the second Lyapunov theorem on the stability of movement are most often used. These methods were not used in this article. The structure of the observer is determined as a result transformation of the equations electromagnetic circuits of the asynchronous machine. The parameters of the PI controller as part of the observer are selected by the method of mathematical modeling. Practical value. The use of speed sensors in electric drives can reduce their reliability and increase cost. The use of observers makes it possible to recover state variables that are not available for direct measurement and to exclude additional sensors, which contributes to the improvement of operational and cost indicators of control systems.
Ключовi слова: асинхронна машина; релейно-векторне керування; спостертач швидкосп обертання ротора; векторний добуток векторiв; потокозчеплення ротора, струм статора, асимптотична стшшсть.
Keywords: asynchronous machine; relay-vector control; observer of the speed rotation of the rotor; vector product of vectors; rotor flux linkage, stator current, asymptotic stability.
PROBLEM STATEMENT. The use of physical sensors in electric drive control systems requires taking into account their additional dynamics in the model of the control object, which can cause problems with the synthesis of regulators due to an excessive increase in
the order of the control object model. The use of observers makes it possible to restore state variables that are not available for direct measurement and to exclude additional sensors, which contributes to the improve-
ment of operational and cost indicators of control systems. Identification of the flux linkage vector and rotor speed is used in direct torque control systems and in vector control systems of asynchronous electric drives. A significant part of the methods of sensorless vector control of an asynchronous electric drive are based on various mathematical models of current loops and flux linkages of an electric machine and combine the calculation of the estimation of the rotor rotation speed with the identification of the flux linkage vector or the motor current, information about the spatial position of which must be known for the implementation of a multi-channel separate control system in active and reactive power channels [1].
To identify the speed of rotation of the rotor, observers are used based on the calculation of the vector product of the flux coupling vectors [1,2]. However, the structure of such observers contains open integrators, the output of which accumulates an integration error. As a result, observers with a comparison of flux linkages are prone to self-oscillations near some value of the observed rotation speed, which leads to the failure of control systems closed through such observers [3].
The identifiers of the speed of rotation of the rotor in electric drives based on asynchronous machines (AM) with a short-circuited rotor, in which the control is carried out along the stator circle, have become widespread. Sensorless control systems have been created and are effectively used in alternating current electric drives of industrial mechanisms of general purpose [4].
The method of adaptive system with a reference model has been widely used for the structural synthesis of speed of rotation observers [5]. This method is based on Lyapunov's second theorem on stability of movement. The analytical form of the adaptation function is
determined, which ensures the asymptotic convergence of the adaptive model of the rotor to the reference model of the stator. To estimate the speed of rotation of the rotor, artificial neural networks are used, the algorithms of which are implemented on digital signal processors [6]. Some observers use the equation of motion of the electric drive to determine the speed of rotation of the rotor under the condition of constancy of the magnetic flux AM. Analysis of the stability of dynamic modes is performed by the method of small oscillations [7]. The use of observers with a sliding mode makes it possible to increase the resistance of the control system to the influence of interference on the measuring channels, as well as to reduce the sensitivity to changes in the parameters of the equivalent circuit of an electrical machine [8,9].
The paper aims to synthesize an asymptotically stable observer of the rotor rotation speed, in which the adaptation function is calculated as a determinant of the vector product of the rotor flux coupling vectors and stator current inconsistencies in order to minimize the possibility of the appearance of self-oscillating processes.
MATERIEL AND RESULTS. Electromagnetic processes occurring in an electric machine contain information about the necessary mechanical variables, which can be obtained by measuring stator currents and voltages, if the parameters of the AM equivalent circuit are known.
The equations of the stator and rotor circles of an asynchronous machine in the u, v axes, composed according to Kirchhoffs second law, have the following form [10]:
- - d¥ - - - d¥ U = isRs + ^+M^* ; ur = irRr + ^
+ !(&,
j J k ß J r r r 7 /\I
at at
The flux coupling equations are as follows
% = ISLS + IrLm; % = IsLm + IrLr. Let us write equation (1) in orthogonal axes tt, ß for AM with a short-circuited rotor
usa = p%sa + RJsa; ußß = p^ßß + RJSß.
0 = p%ra + oWßß + RrIra; 0 = p%ßß - a№„ + RrIßß,
where p = d/dt - differentiation operator.
Expressions of flux linkages due to currents (2) take the form
^ = TT + LI
sa s sa m ra
¥ = TT + TT
ra r ra m sa
¥ = TT + TT
L sß Ts1sß+ Tm1rß-
¥ = TT + TT
L rß TrTrß+ TmTsß .
The components of the rotor current vector are determined from relations (6)
¥ - T T
j _ ¥rp TmT sp
¥ - TT
j _ _ ra m sa
T
-rß
T
Substitute expressions (7) into formulas (5) and after the transformations, the following will be obtained
¥ = Tm ¥
sa ra
+ -
TT - T
T
T
¥ = Tm ¥
-sß
T
rß
+ -
TT - Il
T
T
sß •
(1) (2)
(3)
(4)
(5)
(6)
(7)
(8)
Let us get the AM equation in the form of a structure (Pr, Is) written in the axes a, P. First, we substitute
the values of the rotor currents (7) into expression (4) and after the transformations we obtain the differential equations of the rotor flux linkages
R
P^ra =- Y *ra- Wp + RrKrIsa ;
R
pvrß = - R + + RrKrIsß, (9)
Lr
where K = LlL .
r mir
Next, we differentiate expressions (8), after which we obtain
TT - T2
pWsa = KrpWra + s r m pIsa ;
Lr
TT - T2
p^Sß = Krp^rß + s ; m piSß. (10)
Lr
In equation (3), instead of the derivatives of the stator flux linkages, we substitute expressions (10), which leads to the following formulas
TT - T
u = R I + K pW + ^^-m pi ;
sa s sa rr ra j r sa ^
Lr
TT - T
Usß = Rsisß + KrpWrß + s r m pisß . (11)
Lr
In equation (11), instead of the derivatives of the projections flux coupling vector of the rotor, we substitute their expressions (9). After transformations, we come to the following equations for currents
nT = KrRr W (Rs + KrRr)Lr I + Lm _W + Lr u
p sa LL -T ra LL -T sa LL -T w rß LL -T sa
s r m s r m s r m s r m
piß = Wß - R + KRr)Lr Iß--L^oWm (12)
ß LL -L2 ß LL -L2 sß LL -L2 LL -L2 sß
s r m s r m s r m s r m
If combine equations (9) and (12), we obtain the following system of equations of an open observer
pi = KrRr W (Rs + KrRr)Lr j + Lm 1 + Lr u
pl sa T T _ r2 Lra j j _ j2 1 sa + j j _ j2 ™ rß + j j _ j2 Usa
LsLr Lm LsLr Lm LsLr Lm LsLr Lm
p = KrRr W (Rs + KRr)Lr 1 - Lm mW + Lr u
Psß~ LL - Li Wß LL - Li sß LL - Li ra + LL - Li sß
m
R„
- MJ + R K . -
ra r r sa rß'
pWra=--^ Wra+ RrKrIsa -VW,
L
r
R
P*rp =--J- RrKrIp + GWra . (13)
Lr
It is possible to consider the unknown values of flux coupling and the speed of rotation of the rotor at known currents. Then equation (9) will be written as follows
R R
P^ra =- J- Xa- + RrKrIsa ; = - R %+ 3*aa + ^„Ip . (14)
r r
However, the observed quantities can be considered currents, and the speed can be taken as known. Then
equation (14) will be written as follows
R R
P^ra =- J- ^ra- + RrKrIsa ; P% =- J~ %+ ^ra + R*j,p ■ (15)
Lr Lr
Subtract equation (14) from equations (15) and obtain
0 = -0% + RrKrIsa + 0% - RrKrIsa ;
0 = O^ra + RrKrIsp - 0*ra - R„K„ISP ■ (16)
We will write system (16) using deviations of the currents
- (k - l-)=Yrt(&-a); isp - ^ = JTK" -• (17)
It follows from formula (17) that the current differences are sinusoidal values, because the expressions include the projections of the rotor flux linkages 1ra and 1rp, which change sinusoidally.
We multiply each equation (17) by the corresponding projection of the rotor flux coupling and find the sum
W2 ¥2 / \ / \
+ (ip - I J-% (lsa -I J. (18)
RrKr RrKr
Then the speed determination error is equal to
Aa = 3-a = 4¥ra (I sp - Itp )- % (iaa - Iaa )J, (19)
R K R K
where X = i „ r—I^- = —= const because the modulus of the rotor flux coupling vector is a con-
VI/2 I VI/2 VI/2
T ra + 1 rp 1 r
stant value.
Thus, in expression (19), the difference between the real and the observed speeds was obtained, expressed using errors in the estimation of the stator currents. No matter how the variables in the right-hand side of (19) change, it is always equal through the factor X of the difference between the calculated speed and the real one Aw = 5-5. If you introduce this difference to the PI controller, you can identify the speed rotation of the
rotor 5, bringing the error of its observation closer to zero. Then we get the law of adaptation
( — ( —
Kp + ^[Wralp - IsP)-lSlsa - Isa )J. (20)
V p J
We supplement the law of adaptation (20) with the equations of flux linkages and currents (13) and obtain a system of observer equations
PLa = -ailLa + «13Wra + auw1 rp + b11Usa ;
PIsp = -a22hp - a23W1ra + «241 p + b22Up ;
PWra = «31 isa - «33 Wm - 51 rp ; (21)
PWrp = «42 hp + 51 ra - «441 rp
( -
fKp + ^K^X - Isp)-Vrp(lsa - /J.
v P P
In equations (21), the values of the coefficients are follows:
_ _ _ KrRr _ _ L
^^ -, ; - ^O/l --i ; ^^ - ^M -
11 22 L L - L2 ' 13 24 L L - L2 ' 23 14 L L - L2 '
s r m s r m s r m
«31 = a42 = KrRr ; a33 = a44 = Rr/Lr ; ¿11 = b22 = ^ ,2 . (22)
L„L — L„
s r m
The structural diagram of the observer, which corresponds to the system of equations (21), is shown in Fig. 1. The signals of the observer of the speed estimation 5, the projections of the vector flux coupling of the rotor
1ra, 1rp on the axis a , p, are used in the vector control system of the electric drive. A mathematical model
of an asynchronous electric drive with a vector control system and a synthesized observer of the speed of rotation and flux coupling of the rotor was created in the Matlab program. Transient processes are calculated for an electric motor with the following data: Pn = 5,5 kW ; Un = 380F ; N = 3 [11].
KP + — p p
Figure 1 - Structural diagram of the observer speed rotation of the rotor of an asynchronous short-circuited motor
In vector control systems with flux-coupling stabilization, as a rule, the motor is excited first, then a start-up is carried out to a set speed with a load, or a load torque is applied after acceleration, as shown in Fig. 2a). Fig. 2b) illustrates the identification process when switching from a high to a low rotation speed.
After excitation of the electric motor, speed regulation proceeds with a constant modulus of the rotor flux coupling vector. The properties of the synthesized observer allow, when the speed feedback signal is connected through it, to realize its qualitative adjustment at reduced rotor rotation frequencies (Fig. 3a, 3b). For the
reduced speed mode, stabilization at a given level, feeding and unloading of the load with subsequent braking to an even lower speed is presented. From the graphs, it is possible to determine the quality of speed observation when the speed feedback is closed in the vector control system based on the observer's signal (21). The observer works out the speed of the rotor qualitatively because, as can be seen from the transient processes,
the speed 5r and the speed estimate 5r practically
coincide throughout the transient regimes. The graphs
in Fig. 2 and 3 show slight deviations of the speed estimate from its actual value during transitions from dynamics to statics and vice versa.
Closing feedback through signal of the observer slightly increases the oscillation of the system. The speed controller has an integrating component, so there should be no static error in the rotor speed control under load. The static error is indeed zero when a speed sensor is used in the feedback loop, as evidenced by the transients in Fig. 4. However, when the feedback is closed through the speed observer, a static error of rotor speed regulation occurs in the presence of astatism according to the speed estimation signal from the observer output. The static error of the rotor speed control and the slight discrepancy between the speed and its estimate when the electric motor is loaded are due to the fact that the equation of motion of the electric drive was not taken into account in the observer's model. Taking into account the equation of motion requires accurate
determination of the moment of inertia of the rotating parts of the electric drive and identification of the moment of static load, which is a complex problem that does not have an exact solution.
Therefore, the observer is built using the equations of electromagnetic processes of an asynchronous machine. A slight error in the estimation of the speed when the asynchronous motor is loaded is due to the fact that the developed
observer has a slight phase shift between the real flux couplings of the asynchronous machine and their
estimates , when the load moment is applied to the shaft AM. Thus, the speed observer (21) conditions the existence of a small static error in the control system depending on the speed of the rotor despite the integrating component in the speed controller.
100 99 98 97 96 95 94 93 92 91 90
- rar, fir,r]
ps
\r
t,
34
1.5 . a )
33 J 32
®r fir,rps
/ l a*.
tt
if
y t, s [
2 2 5 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4
b)
Figure 2 - Graphs of the speed of rotation of the rotor of the electric motor (red line) and
the signal of the observer of this speed (green line): a) when starting the electric motor and changing the moment of resistance on its shaft; b) when braking to a reduced speed of
rotation
©„, ©„ ,rps
t,
14 p
13 -12 -11 -10 -9 -8 -7 -6 -5 -
a)
®r, ®r,rps
Y
v t,
Figure 3 - Graphs of the rotation speed of the electric motor rotor (red line) and the signal of the observer of this speed (green line): a) at reduced rotation speed; b) when braking the
electric drive
40
39
38
37
36
35
4
13
2
10
9
8
7
0
0.5
2
100 90 80 70 60 50 40 30 20 10 0 -10
®r,rps
0 0.5
a)
®r,rps
t, s
-LL -2L.
2 2.5 3 3.5
b)
t, s
2 2.5 3 3.5
Figure 4 - Graphs of the speed of rotation of the rotor of the electric motor when using not an observer, but a speed sensor: a) regulation at significant speeds; b) regulation on reduced speed
2
10
8
6
4
2
.5
0 0.5
.5
CONCLUSIONS. The research of the vector field-oriented control system of an asynchronous machine in dynamic modes with a synthesized speed observer confirms its asymptotic stability. The stability of the entire control circuit when closing the feedback on the speed of rotation of the rotor through the observer is proven, and high accuracy of speed identification is recorded. A speed observer with an adaptation function formed as a vector product of the rotor flux linkage vector and a vector of stator current inconsistencies and their estimates along orthogonal axes has a greater margin
of stability than an observer that uses only flux linkage projections. As a consequence of the specified property of the observer, the research of the proposed control system did not reveal self-oscillating processes in different operating modes.
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