Review of Sensorless Vector Control of Induction Motor Based on … of... · induction motor drive;...
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International Journal of Computer Architecture and Mobility (ISSN 2319-9229) Volume 1-Issue 6, April 2013
Available Online at: www.ijcam.com
Review of Sensorless Vector Control of Induction Motor
Based on Comparisons of Model Reference Adaptive System
and Kalman Filter Speed Estimation Techniques
Payal Thakur1, Rakesh Singh Lodhi
2
Indore,India
1 [email protected] 2 [email protected]
Abstract - In recent year, sensorless vector control of induction
motor has been most popular research topic. Continuing
research has to concentrate on elimination of the problem of
parameter variation of induction motor drive. This technical
review will be of assistance in reaching a dynamic modeling of
induction motor and sensorless vector control of induction
motor using efficient speed estimation approach : Model
Reference Adaptive System and Kalman Filter and its
application in area of electrical vehicles (EV’s) propelled by an
induction motor drive; compare the results of both methods
Keywords - Vector Control, Sensorless Control, Model
Reference Adaptive System (MRAS), Kalman Filter (KF),
Speed Estimation, Induction Motor Drives.
I.INTRODUCTION
Induction motors have been widely used in industry because of
the advantage of simple construction, ruggedness, reliability, low
cost and minimum maintaince.[1]
Mathematical model of induction motor is non linear and
presents strong coupling between the input, output and
intervariable, such as torque, speed or flux. So, its control is very
complex. Many schemes have been proposed for control of
induction motor drive, among which the field oriented control [2, 3]
or vector control has been accepted as most effective methods. In
vector control, the knowledge of the rotor speed is necessary, this
necessity requires additional speed sensor which add the cost and
complexity of the drive system.
Over the past few year, ongoing research has concentrated on the
elimination of speed sensor at the machine shaft without
deteriorating the dynamic performance of the drive control system
[4]. In order to achieve good performance of sensorless vector
control, different speed estimation scheme have been proposed and
a variety of speed estimator’s exists now a day’s [5], such as Direct
Calculation, Model Reference Adaptive System, Extended
Luenberger Observer, Kalman filter etc.
Out of various methods, Model Reference Adaptive System,
and Kalman filter based sensorless speed estimation has been used.
In MRAS, speed is estimated using difference between the
reference model output and adaptive model output. The biggest
problem of MRAS approach is the integration of pure voltage
signals. This problem is solved by modify the pure integration in
voltage model to the low pass filter. The Kalman filter has a good
dynamic behavior and it can work even in standstill position. The
filter implementation required model of AC motor must be
calculated in real time which is very complex problem .The
Extended Kalam Filter is a full order stochastic observer for the
estimation of a non linear dynamic system in real time by using
signals that are corrupted by noise.
This paper is organized as follows: In section 2, Dynamic
modeling of induction motor is reviewed. The principle of vector
control and sensorless speed estimation technique: MRAS and KF
are proposed in section 3, 4 and 5.Vechile model is given in section
6. Simulation results are shown in section 7. Finally, concluding
remark is provided in section 8.
II. DYNAMIC MODELING OF INDUCTION MOTOR
Mathematical model of three phase induction motor referring to
rotating reference frame (d-q) can be expressed as follows. [6]
𝑝𝑖𝑑𝑠𝑝𝑖𝑞𝑠
= −𝐴1 𝜔𝑠−𝜔𝑠 −𝐴1
𝑖𝑠𝑑𝑖𝑠𝑞
+
𝐿𝑚
𝜎𝐿𝑠𝐿𝑟𝑇𝑟𝐴2𝜔𝑟
−𝐴2𝜔𝑟𝐿𝑚
𝜎𝐿𝑠𝐿𝑟𝑇𝑟
−𝛹𝑟𝑑−𝛹𝑟𝑞
+ 𝐴3 00 𝐴3
𝑉𝑠𝑑𝑉𝑠𝑞
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𝑝Ψ𝑑𝑟𝑝Ψ𝑞𝑟
=
𝐿𝑚
𝑇𝑟0
0𝐿𝑚
𝑇𝑟
𝑖𝑠𝑑𝑖𝑠𝑞
+
−1
𝑇𝑟(𝜔𝑠−𝜔𝑟)
−(𝜔𝑠−𝜔𝑟)−1
𝑇𝑟
−𝛹𝑟𝑑−𝛹𝑟𝑞
(1)
Where 𝑝 = 𝑑
𝑑𝑡 and
𝑇𝑒𝑚 − 𝑇𝑙 = 𝑗𝑑𝜔𝑟
𝑑𝑡+ 𝐷𝜔𝑟 2
Where 𝑇𝑒𝑚 =3
2 𝑝
𝐿𝑚
𝑇𝑟 𝛹𝑟𝑑 𝑖𝑠𝑞 − 𝛹𝑟𝑞 𝑖𝑠𝑑 and 3
𝐴1 = 𝑅𝑠
𝜎𝐿𝑠+
1−𝜎
𝜎𝑇𝑟 ; 𝐴2 =
𝐿𝑚
𝜎𝐿𝑠𝐿𝑟 ; 𝐴3 =
1
𝜎𝐿𝑠
𝜎 = 1 − 𝐿𝑚
2
𝐿𝑠𝐿𝑟 ; 𝑇𝑟 =
𝐿𝑟
𝑅𝑟 ; 𝜔𝑔 = 𝜔𝑠 - 𝜔𝑟
𝜔𝑔 = Slip frequency;
𝜔𝑠 = Electrical synchronous stator speed; 𝜔𝑟 = Electrical rotor speed; 𝜎 = Linkage coefficient; 𝑇𝑟 = Rotor time constant; 𝑇𝑒𝑚 ,𝑇𝑙 = Electromagnetic torque and mechanical load or
disturbance torque;
J, D = Moment of inertia and viscous coefficient of motor;
𝐿𝑠 , 𝐿𝑚 , 𝐿𝑟 = Stator inductance, mutual inductance and rotor inductance.
III. PRINCIPLE OF FIELD ORIENTED CONTROL OR
VECTOR CONTROL
In 1972, Blaschke has introduced the principle of Field Oriented
Control to realize D.C motor characteristics in an induction motor.
In D.C motor, mmf produced by the armature current is
perpendicular to the field flux produced by stator. Being
orthogonal, there is no net interaction on one another is produced by
these two fluxes. DC motor flux can be controlled by adjusting the
field current and by adjusting armature current torque can be
controlled independently of flux [7].
In AC machine, the interaction between stator and rotor fields
whose orientation is not held at 90 degrees. So, AC machine is not
simple. For DC machine like performance, the field and armature
fields is orthogonal oriented and holding fixed in an AC machine by
orienting the stator current with respect to rotor flux so as to attain
independently controlled torque and flux. Such a scheme is called
vector control. [2]
Fig.1 : Basic block diagram of vector control
A. Basic Theory
Fig.2: Block diagram of Sensorless Induction Motor control
Sensorless vector control of induction motor block diagram
shown in fig 2, Sensorless control of induction motor means the
control of speed of induction motor without speed encoder. A speed
encoder in a drive is undesirable because it add cost and reliability
problem, besides the need for mounting arrangement and shaft
extension. The inverter is used for controlling the motor by
providing switching pulses. The speed and flux estimators are used
to estimate the speed and flux respectively. PI controller is used for
controlling such signals and compared with reference values.
IV. THE MODEL REFRENCE ADAPTIVE SYSTEM
In 1987, Tamai [8] has proposed speed estimation technique
based on Model Reference Adaptive System (MRAS).Two year
later, Schauder [9] presented an more effective and less complex
alternative MRAS scheme. Adaptive control has emerging potential
solution for implementing high performance control system. The
MRAS approach having two model one is reference model which
does not involve the estimated quantity (𝜔𝑟 ) and another is adaptive
model which involve the quantity to be estimated. The output of
adaptive model is compare with the output of reference model
difference is produced which is used to drive the adaptive
mechanism whose output is the quantity to be estimated (rotor
speed) .A number of MRAS based speed sensorless scheme have
been described in the literature for field oriented induction motor
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drive[10-11],[12-13]. The block diagram of MRAS is shown in fig
3.
A. Reference Model
The stator voltage in d-q equivalent circuit is given by
𝑉𝑑𝑠 = 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑠 𝑑𝑖𝑑𝑠𝑑𝑡
+𝑑Ψ𝑑𝑚𝑑𝑡
4
Where Ψ𝑑𝑚 = 𝐿𝑚
𝐿𝑟 𝛹𝑑𝑟 − 𝐿𝑟 𝑖𝑑𝑠
Put Ψ𝑑𝑚 in (4) gives
𝑉𝑑𝑠 = 𝐿𝑚
𝐿𝑟 𝑑Ψ𝑑𝑟
𝑑𝑡+ 𝑅𝑠 + 𝜎𝑠𝐿𝑠 𝑖𝑑𝑠 or
𝑑Ψ𝑑𝑟
𝑑𝑡=
𝐿𝑟
𝐿𝑚𝑉𝑑𝑠 −
𝐿𝑟
𝐿𝑚 𝑅𝑠 + 𝜎𝑠𝐿𝑠 𝑖𝑑𝑠 5
Similarly,
𝑑Ψ𝑞𝑟
𝑑𝑡=
𝐿𝑟
𝐿𝑚𝑉𝑞𝑠 −
𝐿𝑟
𝐿𝑚 𝑅𝑠 + 𝜎𝑠𝐿𝑠 𝑖𝑞𝑠 6
The equation (5) and (6) together give stationary frame equations
for reference model.
B. Adaptive Model
The rotor in d-q equivalent circuit equation in d-q equivalent is
given by
𝑑Ψ𝑑𝑟
𝑑𝑡+ 𝑅𝑟 𝑖𝑑𝑟 + 𝜔𝑟 Ψ𝑞𝑟 = 0
𝑑Ψ𝑞𝑟
𝑑𝑡+ 𝑅𝑟 𝑖𝑞𝑟 − 𝜔𝑟 Ψ𝑑𝑟 = 0 7
Adding 𝐿𝑚 𝑅𝑟
𝐿𝑟 𝑖𝑑𝑠 and
𝐿𝑚 𝑅𝑟
𝐿𝑟 𝑖𝑞𝑠 on both side of equation (7)
and substitutes
Ψ𝑑𝑟 = 𝐿𝑚 𝑖𝑑𝑠 + 𝐿𝑟 𝑖𝑑𝑟
Ψ𝑞𝑟 = 𝐿𝑚 𝑖𝑞𝑠 + 𝐿𝑟 𝑖𝑞𝑟
We have
𝑑Ψ𝑑𝑟
𝑑𝑡=
𝐿𝑚
𝑇𝑟𝑖𝑑𝑠 − 𝜔𝑟 Ψ𝑞𝑟 −
1
𝑇𝑟 Ψ𝑑𝑟 8
𝑑Ψ𝑞𝑟
𝑑𝑡=
𝐿𝑚
𝑇𝑟𝑖𝑞𝑠 − 𝜔𝑟 Ψ𝑑𝑟 −
1
𝑇𝑟 Ψ𝑞𝑟 9
The above equation (8) and (9) gives the rotor flux values as a
function of stator current and rotor speed. In above fig.3 adaptation
algorithms is used to tune the speed so that error 𝜉 = 0.we can
drive the following speed estimation relation using Popov criteria
for hyper stability for a asymptotically stable system[14].
𝜔𝑟 = 𝜉 𝐾𝑝 +𝐾𝐼𝑠
Where 𝜉 = Ψ𝑑𝑟^ Ψ𝑞𝑟 − Ψ𝑞𝑟
^ Ψ𝑑𝑟
The estimated speed from MRAS control is feedback to the speed
controller where it is compared with reference speed to get the
commanded output.
Fig.3: Block diagram of MRAS speed estimation.
V. THE KALMAN FILTER
The Kalman Filter is a deterministic type linear observer,
derived to meet a optimality stochastic condition. The kalman filter
has two forms: basic and extended. The extended kalman filter is
basically a full order stochastic observer can be used for non linear
system. The Kalman filter allows obtaining no measured state
variables with usage measured state variable and as well noise and
measurements statistics. [16]
The block diagram of extended kalman filter for speed
estimation shown in fig.4, where machine model is indicated at the
top. The extended kalman filter algorithms use the full machine
dynamic model, where speed is considered a parameter as well a
state. The augmented machine model [14] can be given by
𝑑𝑋
𝑑𝑡= 𝐴𝑋 + 𝐵𝑉𝑠 (10)
Y = CX (11)
Where,
A =
−
𝐿𝑚2 𝑅𝑟+ 𝐿𝑟
2 𝑅𝑠
𝜎𝐿𝑠𝐿𝑟2 0
𝐿𝑚𝑅𝑟
𝜎𝐿𝑠𝐿𝑟2
𝐿𝑚𝜔𝑟
𝜎𝐿𝑠𝐿𝑟0
0 − 𝐿𝑚
2 𝑅𝑟+ 𝐿𝑟2 𝑅𝑠
𝜎𝐿𝑠𝐿𝑟2
−𝐿𝑚𝜔𝑟
𝜎𝐿𝑠𝐿𝑟
𝐿𝑚𝑅𝑟
𝜎𝐿𝑠𝐿𝑟2 0
𝐿𝑚𝑅𝑟
𝐿𝑟
00
0𝐿𝑚𝑅𝑟
𝐿𝑟
0
−𝑅𝑟
𝐿𝑟 −𝜔𝑟 0
𝜔𝑟 −𝑅𝑟
𝐿𝑟 0
0 0 0
:
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B =
1
𝜎𝐿𝑠0
01
𝜎𝐿𝑠
000
000
: C = 1 0 0 0 00 1 0 0 0
X = 𝑖𝑑𝑠 𝑖𝑞𝑠 Ψ𝑑𝑟 Ψ𝑞𝑟 𝜔𝑟 𝑇
𝑉𝑠 = 𝑉𝑑𝑠 𝑉𝑞𝑠 𝑇 Is an input vector.
Fig.4 : The block diagram of extended kalman filter for speed estimation
equation (10) is of 5th order and speed 𝜔𝑟 is a state.If variation in
speed is negligible then 𝑑ω𝑟
𝑑𝑡= 0. With 𝜔𝑟 as a constant parameter,
the machine model used in extended kalman filter is linear. For
digital implementation of an extended kalman filter, the model
[17]must be discretized in following form:
X (K +1) = f ( X(K), U(K), K ) + W(K)
Y(K) = h (X(K), K) + V(K)
Where
Y(K) : is a vector containing the d-q component of
stator current space vector.
U(K) : is a vector of excitation signals.
X(K) : is a vector containing the state.
W(K) and V(K) : are the process and the measurement noise
vectors at time K.
E { W(K) } = 0, E{ W(K) W(j)T } = Q 𝛿kj ,Q ≥ 0
E { V(K) } = 0, E{ V(K) V(j)T } = R 𝛿kj , R ≥ 0
Where Q and R are respectively the process and measurement
covariance matrices.
The extended kalman filter equation is [6]
K(k) = F(k) P(k) HT [ H P(k) HT + R]-1
𝑋 = (k+1) = f (𝑋 (k) ,U(k)) + K(k) [Y(k) - H𝑋 (k)]
P(k+1) = F(k) P(k) FT(k) + Q – K(k) [H P(k) HT + R] KT(k)
Where 𝑋 (k) is state estimate;
P(k) is estimated error covariance matrix;
K(k) is Kalman gain matrix.
F(k) and H is given by
F(k) = 𝜕
𝜕𝑋 { f (X(k), U(k), K) } X ̂(k) ,U(k)
H = 𝜕
𝜕𝑋 { h (X(k), K) } R ̂(k) ,U(k)
VI.VECHILE MODEL
The vehicle model is based on mechanics and aerodynamics
principle [18-19].The total tractive effort is given by
Fte = Frr + Fad + Fhc + Fla + Fwa
Where Frr : is the rolling resistance force;
Fad : is the aerodynamic drag;
Fhc : is the hill climbing force;
Fla : is the force required to give linear acceleration;
Fwa : is the force required to give angular acceleration to the
rotating motor.
The power required to drive a vehicle at a speed v is given by:
Pte = v Fte = v (Frr + Fad + Fhc + Fla + Fwa )
VII.CONCLUSION
In this reveiw paper, sensorless vector control of induction
motor uses two different approaches: MRAS and KF have been
proposed. Sensorless control gives the benefits of vector control
without using any shaft encoder. This paper also elaborates the
dynamic model of induction motor and principle of vector control.
The comparative study between MRAS and KF response based on
speed reversal and step change in load conclude that MRAS is
better than extended KF response. But, extended KF shows a stable
behavior after a certain time has passed for settling. Torque
disturbances are reduced in extended KF as compare to MRAS.The
application of MRAS and KF approach in area of EV’s is also
explained in this review paper .The advantage of automotive speed
sensorless drive are increased reliability, lower cost, reduced size of
drive system and elimination of sensor cables. At low speed, speed
estimation methods responsible for poor drive performance due to
parameter variation. Such problem is over come by KF and MRAS.
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BUT MRAS shows better results than KF approach in the field of
EV’s propelled by an induction motor drive.
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