Download OPTIMAL EFFICIENCY BASED PREDICTIVE CONTROL OF

OPTIMAL EFFICIENCY BASED PREDICTIVE CONTROL OF
INDUCTION MACHINE INCLUDING MAGNETIC SATURATION
AND SKIN EFFECT
K. Barra*, T.Bouktir* and K.Benmahammed**
* Department of Electrical Engineering,
University of Larbi Ben M’hidi,
Oum El Bouaghi, 04000
Algeria.
** Department of Electronics,
University of Setif, 19000
Algeria.
Abstract- In this paper a minimum-loss control algorithm MLCA is applied to improve high efficiency of an
induction motor. The efficiency optimisation is done by adjusting the rotor flux level to an optimal value as a
function of the operating conditions (such as torque and speed). Magnetic saturation and skin effect are taken
into account and their influence is discussed. The application of the generalized predictive control GPC
algorithm is based on the input-output linearised model. Simulation studies show the performance of the
proposed algorithm.
Key-Words : minimum loss control, magnetic saturation, skin effect, Generalized predictive control, Direct
field oriented control method, induction motor.
In addition to its advantages, such as speed capability,
robustness, cheapness and ease of maintenance, when
used with a field oriented control scheme, the induction
machine is the most widely used electrical machines.
The usually used direct field oriented control methods
keeps rotor flux at constant rated value and ignores the
machine losses. Consequently, the efficiency of the motor
is poor especially when the load torque is low or zero.
The main losses, about 90 % of the total losses are
copper, core losses and stray-load losses [1-4]. A good
machine design dictates that operation around rated
operating condition corresponds to high efficiency
operating point.
Whenever it is possible to obtain the required torque with
less than the rated flux, it is possible to reduce the flux
decreasing the magnetising current . A reduction in the
magnetic flux results in a reduction in the core losses and
a decrease in the magnetising current results in a decrease
in the stator and rotor copper losses, due to the smaller
magnetising current needed [2],[3]. One way to reduce
the losses is to adjust the flux level as a function of the
operating conditions such as torque and speed.
2. Induction motor model
It is well known that the linear model of a squirrel-cage
induction motor neglect the equivalent core losses
resistance (generally connected in parallel with the statorrotor mutual inductance), with that model, all parameters
are considered constant and the core losses are not taken
into account.
In order to consider such losses, the proposed model in
[2] is used, where the equivalent core losses resistance Rfs
is connected in series with the stator-rotor mutual
inductance.
The core loss resistance is a function of frequency and
flux level, but the change of Rfs value depends on the
variation of frequency larger than that of rotor flux [1].
Therfore Rfs is approximately a function of frequency
only.
The stator equivalent core loss resistance is determined
from the classical experimental no-load test data as
shown in figure1. by :
R fs = a. fs + b. fs 2
(1)
a and b are the coefficients of hysteresis and eddy current
losses respectively.
fs is the stator flux frequency.
At low speed, Rfs can be neglected comparing to stator
resistance but at high speed, the stator flux frequency is
almost the same as the rotor speed frequency (the slip
frequency is nearly zero and the rotor core losses can be
neglected).
14
Stator c urrent v alues
12
Equivalent c ore los ses res is tanc e [Oh m]
1. Introduction
fit ted curve
0.25 A
0.5 A
1A
10
8
6
4
2
0
0
5
10
15
20
25
30
frequency [Hz]
35
40
45
50
Fig. 1. Stator core losses equivalent resistance versus
stator current frequency.
The induction motor model used under field oriented
control FOC can be expressed in the synchronously
rotating d-q reference frame as follows:
σ r .R fs 

di
.i ds + σ .L s . ds − σ .L s .w s .i qs +
V ds =  R s +

dt
1+ σ r 

R fs
dφ dr
(1 − σ )(. 1 + σ s )
.φ dr
+
dt
(1 + σ r ).L m
di
σ r .R fs 

.i qs + σ .L s . qs + σ .L s .w s .i ds +
V qs =  R s +

dt
1+ σ r 

(2)
(1 − σ )(. 1 + σ s ).ws .φ dr
dφ
Tr . dr + φ dr = L m .i ds
dt
w s = w + w sl
L m .i qs
w sl =
Tr .φ dr
Tem = p.(1 − σ )(
. 1 + σ s )φ dr .i qs
Assuming that none of the motor parameters has any
dependence on the rotor flux, under the constant load
torque, we can obtain the rotor flux providing the
minimum loss by setting the derivative of the equation
(8) to zero
∂Ploss
(9)
=0
∂φ dr
The relationship that minimises the criterion Ploss as
function of torque and speed can be obtained by:
∗ 1/ 2
(φ dr * ) S = k opt . ( Tem )
(10)
with :
1/ 2
kopt
where
σs =
ls
lr
1
, σr =
, σ = 1[(1 + σ s ).(1 + σ r )]
Lm
Lm
1/ 2

σ r .R fs  

Rr

R
+
+
 
 s

Lm
(
1 + σ r )2 (1 + σ r )  

=
 
Rs + R fs
.1+σs )
 p.(1 − σ )(
 


 



(11)
400
- proposed
350
. conventional
(
P js = R s . i ds 2 + i qs 2
)
(3)
Power losses [W]
300
2.1 Induction machine losses model
Stator copper losses :
Rotor copper losses:
)
Stator core losses:
(
Rr
(1 + σ r )
)
2
2
 φ




.  dr − i ds  + i qs 2 
  Lm




R fs 

φ
Pfs =
.σ r . ids 2 + iqs 2 + dr .ids 
1+σr 
Lm

0
L2m
)
1
1.5
Fig. 2. Loss map
(5)
1.5
1
Rotor speed
0.5
75 rpm
750 rpm
1500 rpm
0
0
0.5
1
1.5
Torque [p.u]
Fig. 3. Appropriate rotor flux versus torque
0.9
1500 rpm
0.8
0.7
0.6
750 rpm
Efficiency
Pin = Vds .ids + Vqs .iqs
Pout = Tem .Ω
+ R fs
0.5
Torque
where
s
75 rpm
750 rpm
0
Ploss = Pin − Pout
(R
150
50
(4)
2.2 Losses minimisation
In conventional direct field-oriented control method ids* is
the reference flux current and iqs* is the reference torque
current and the rotor flux level is maintained constant at
its rated value for every value of the load torque.
Consequently, the core losses are ignored and the
efficiency of the motor is poor especially when the load
torque is low.
In steady state, assuming that the machine control ensures
no static error, then we can write:
dφ dr
=0
dt
(6)
Tem
i qs =
p.(1 − σ ).(1 + σ s ).φ dr
Total losses is the input power minus the output power
Ploss =
1500 rpm
200
100
Appropriate rotor flux [p.u]
(
P jr = R r . i dr 2 + i qr 2 =
250
(7)
σ r .R fs 

Rr
+

 Rs +
2
(
1 + σ r )  2 −2
(
)
σ
+
1

r
.φdr2 + 
.Tem .φdr
2
. 1 + σ s )]

 [ p.(1 − σ )(




(8)
- proposed
. conventional
0.5
0.4
75 rpm
0.3
0.2
0.1
0
0
0.5
1
Torque [p.u]
Fig. 4. Efficiency map
1.5
6
5
Rotor rés is tanc e [Ohm ]
The total losses for the proposed method is lower than
that for the conventional method, which is constant rated
rotor flux control, in the region of low torque as it is
shown in figure 2.
The appropriate rotor flux for maximum operating
efficiency given by (10) is plotted in figure 3.
Figure 4. shows an efficiency map. It is clear that the
proposed method is superior to the conventional method
over a wide range of torques.
4
3
2
Fitted curve
measured points
1
3. Magnetic saturation and skin effect
Rotor flux variation given with (2) , is valid only under
the assumption of linear magnetic circuit. Such an
assumption in real and exact cases is not admissible
because inductances are nonlinear functions of currents. It
is therfore necessary to modify the equation of the rotor
flux in (2), so that it accounts for the main flux saturation.
The synchrounous speed test gives the stator total
inductance ( Ls=Lm+ls ) by measuring the reactive power.
Figure 5. shows the stator total inductance as function of
stator current .
0
0
5
10
15
20
25
30
Rotor current frequency [Hz]
35
40
45
50
Fig. 6. Rotor resistance versus rotor current
frequency
Figure 7. shows the relationship given by (8) for
various load torques. For the points noted by asterisk, the
minimum loss point correspond to operating point that
satisfy the relation (10) in which magnetic saturation is
ignored. The points noted by circle correspond to the case
considering magnetic saturation.
1100
1000
0.6
Fitted curve
Measured points
0.5
. ignoring saturation
Power losses [W]
800
0.4
0.3
Stator induc tanc e [H ]
- considering saturation
900
0.2
700
600
torque 1.5 p.u
500
400
1 p.u
300
0.5 p.u
200
0.1
100
0
0
0
0.5
1
1.5
2
2.5
Stator c urrent [ A]
3
3.5
4
0
0.2
0.4
0.6
0.8
1
Rotor flux [p.u]
1.2
1.4
1.6
1.8
Fig. 7. Total losses versus rotor flux
Fig. 5. Stator inductance as function of
stator current
An acceptable approximation is that, two inductances
are defined: an unsaturated inductance Ls0 which applies
for stator current lower than Is0, and a saturated one for
higher Is current.
L
I s < I s0 
L s =  s 0
(12)
L
f
I
I
I s > I s 0 
(
)
−
−
s
s0
 s0
The rotor resistance and the leakage inductances are
determined by locked-rotor tests with stator frequencies
from 10 to 50 Hz, and the determined constants are
extrapolated down to a few hertz to take into account the
skin effect in the rotor.
On figure 6, one can see that the rotor resistance changes
considerably due to skin effect [2].
4. Generalized predictive controller
Generalized predictive control (GPC) introduced by
CLARKE ( 1987-1988 ) can be summarized through the
three following points.
If we want to make the plant output coincide in the future
with a setpoint or a known trajectory, it is necessary:
First to predict the plant output over a
•
defined horizon.
then to calculate the future control
•
values which will minimize the errors
between the predicted outputs and the
setpoints values.
Finally to apply only the first optimal
•
control value, and repeat all this at the
next sampling period, with a receding
horizon.
4.1 Model and cost function
The Controlled AutoRegressive Intergrated Moving
Average model (CARIMA) is commonly used in GPC, as
it is applicable to many single-input single output plants
with dead-time equal to one :[6]
A(q −1 ) y (t ) = q −1 B(q −1 )u (t ) +
C ( q −1 )
∆( q −1 )
ξ (t )
(13)
~
U = [∆u (t ) , ... , ∆u (t + N u − 1)]T
where u (t ) , y (t ) are the input and output of the plant at
the sample instance t,
ξ (t ) is an uncorrelated random sequence and the use of
−1
∧
∧
∧

Y =  y (t + N1 ) ,... , y (t + N 2 )


−1
the operator ∆(q ) = 1 − q ensures an integral control
law.
A(q −1 ) , B(q −1 ), C (q −1 ) are polynomials in the backward
g N1 −1 ....
0 
 g N1


....
0 
g N1 +1 g N1



M
....
....
0
G=

....
....
g1 
g N u


....
....
M 
M

g N
 2 g N 2 −1 .... g N 2 − N u +1 
A(q −1 ) = 1 + a1 q −1 + ... + a na q − na
B(q −1 ) = b0 + b1 q −1 + ... + bnb q − nb
The criterion which is minimized is a weighted sum of
squares of predicted future errors and increments of
control values:
2
(14)
so that:
[
−1
.G T .(W − F) .
[
the first line of the matrix G T .G + λ.I N u
(21)
]
−1
.G T has to
4.3 Choice of different parameters
the backward shift operator q-1 and obtained by solving
the following Diophantine equations :
∆ (q −1 ). A(q −1 ).E j (q −1 ) + q − j .F j (q −1 ) = 1
G j (q −1 ) + q − j .H j (q −1 ) = B (q −1 ).E j (q −1 )
[
[
[
(16)
]
]
]
degree E j (q −1 ) = degree G j (q −1 ) = j − 1
degree F j (q −1 ) = degree A(q −1 )
degree H j (q −1 ) = degree B ( q −1 ) − 1
The first term of equation (15) is called ‘free response’,
as it represents the plant predicted output ω(t+j), when
there is no future control action. The second term is called
‘forced response’ , as it represents the output prediction
due to the future control actions u (t + j − 1) , j ≥ 1 .
If we define the following vector formed with the
polynomial solutions of equations (16) :
F = [ f (t + N 1 ) , ... , f (t + N 2 )]T
and, if we denote :
]
∂J
~ =0,
∂U
(15)
for N1 ≤ j ≤ N 2 , where F j , G j , H j are polynomials in
]
]
]
Uˆ opt = GT .G + λ .I N u
(20)
computed off line and only the first control value of the
sequence is applied on the system.
forced response
[
[
[
(19)
and the optimal control law comes from
G j (q ).∆u (t + j − 1)
144424443
with :
∧
~
~
~ ~
J = (G.U + F - W)T .(G.U + F - W) + λ .U T .U
∧
free response
The output prediction has the following form :
The criterion J can be rewritten in a matrix form
The previous CARIMA model is now used to elaborate
the predicted outputs under the form:[5],[6]
−1
(18)
Y = G.Uˆ + F
with the assumption : ∆u (t + j ) ≡ 0 for j ≥ N u
where : N1, N2 are the costing horisons
Nu is the control horison
λ is the control weighting factor
w(t + j ) is the future setpoint.
4.2 Minimization of the criterion
y (t + j ) = F j (q −1 ). y (t ) + H j (q −1 ).∆u (t − 1) +
14444442444444
3
(17)
W = [w(t + N1 ) , ... , w(t + N 2 )]T
shift operator q-1 and in most cases C (q −1 ) = 1 .
∧
Nu
N2 

J = ∑ . w(t + j ) − y (t + j ) + λ ∑ ∆u 2 (t + j − 1)
j = N1 
j
= N1

T
There is in fact no particular rule that enables an
optimal choice of N1,N2,Nu and λ . Morever it is possible
to note the four following points : [5],[8],[9]
* It is better to choose N1 so that at least one element of
the first row of G is nonzero; that implies that N1 should
be greather than the maximum expected dead-time of the
process.7
* N2 should be chosen in order to satisfy N2.T equal to the
time response of the process (T sampling period).
* Very often Nu is chosen so that Nu<<N2 and we
previously stressed the fact Nu =1 is very interesting.
* λ is often hard to determine a priori. If the matrix Gt.G
is itself invertible, even λ=0 gives a solution. But in most
cases, it seems better to choose λ from the empiric
relation:
( )
λ = tr G t .G
(22)
4.4 Identification of the system
Before the design of the control law, an identification of
the system is necessary to define the CARIMA model.
The identification algorithm can be executed off-line at
the beginning of the test, or when an important change of
the system is detected.
The basic idea is to generate a pseudo random binary
sequences (P.R.B.S) to calculate the correlation functions.
The model that we are looking for has the following form
:
q −1 B ( q −1 )
A(q −1 )
200
(23)
where na and nb are chosen a priori and can be changed
if the results are not satisfying, then a recursive least
squares method provides the unknown parameters
[− a1 ,.....,−a na , b0 , b1 ,....., bnb ] .
180
160
140
motor speed [rd/s]
G (q −1 ) =
120
100
80
60
5 Simulation results
40
To prove the interest of the MLCA, a GPC control law
has been applied both to the speed loop and rotor flux
loop and the tuning that has been chosen is: (sampling
period =5ms)
Speed loop:
20
0
0
5
10
15
time [s]
Fig. 8. Motor speed and speed
reference
[N11, N12 , Nu1, λ1 ] = [1 , 48 , 5 , 1165.4]
flux loop:
15
[N 21, N 22 , Nu 2 , λ2 ] = [1 , 37 , 1 , 1.364]
In this paper, an IM model in direct field oriented frame
including core losses, magnetic saturation and skin effect
have been presented. Based on this model, a loss
minimisation algorithm was proposed and some
simulated results have been presented, showing the
validity of the proposed algorithm.
The method is based on the choice of an optimal value of
the rotor flux level depending on the operating conditions
(torque, speed) and taking into account the inductances
and the rotor resistance variations. Steady state losses
minimisation method is an important way to minimise
losses when the motor torque is low or zero or when the
motor operates at low speeds. The efficiency of the steady
state optimised MLCA (varying flux) trajectories for the
whole cycle is better than the optimal constant flux
usually used in FOC.
torque [N. m]
load torque
5
0
0
5
10
15
time [s]
Fig.9. Motor torque and load torque
3.5
3.4
3.3
Rotor res is tanc e [Ohm ]
6 Conclusions
10
3.2
3.1
3
2.9
2.8
2.7
2.6
0
5
10
15
time [s]
Fig. 10. Rotor resistance
0.65
0.6
0.55
0.5
Stator to tal induc tanc e [H]
The proposed MLCA is applied to the direct control
method of induction motor drive where rotor flux
reference is used from (10). The chosen example of motor
speed reference and load torque trajectories are given by
figure 8. and figure 9. On these figures, we can see that
motor speed and torque follow their references without
steady state errors nor overshoot.
Figure 10. and figure 11. show both the
rotor
resistance and stator total inductance variations. We can
see the variations of rotor resistance due to skin effect
and stator inductance due to saturation effect as given by
figure 5. and figure 6.
On figure 12., It is clear that the optimal rotor flux
follows the torque and speed variations as given by
equation (10). At t=0, the initial rotor flux is not zero in
order to deliever high torque value when the machine
starts.
motor torque
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0
5
10
time [s)
Fig. 11. Stator total inductance
15
1.4
[4]. K.Matsuze, T. Yoshizumi and S.Katsuta, High
reponse flux control of direct field oriented induction
motor with high efficiency taking core loss into account,
IAS Annual meetings, pp.410-417,1997.
rotor flux norm
rotor flux reference
1.2
Rotor flux [Wb]
1
0.8
[5]. D.Dumur
and P.Boucher, Predictive control
Application in the machine-tool field,Advances in ModelBased Predictive control, Oxford Science Publications,
Oxford University Press, 1994, pp.471-482.
0.6
0.4
0.2
0
0
5
10
15
time [s]
Fig. 12. Rotor flux norm
[7]. Mademlis C. and N. Margaris, Loss minimization in
vector–controlled interior permanent synchronous motor
drives, IEEE Tran. on Ind. Electronics, Vol. 49, N°.6,
December 2002.
speed signal c ontrol
3
speed loop
2
(incréments de commande)
1
0
-1
0
5
10
15
time [s]
0.6
rotor flux loop
flux s ignal control
[6]. Bitmead. R.R., M. Gevers and V. Wertz, Adaptive
Optimal Control: The thinking Man’s GPC, Prentice Hall
International, 1990.
0.4
0.2
[8]. D.Dumur, P.Boucher and J. Roder, Design of an open
architecture structure for implementation of Predictive
controllers for motor drives, Proc. Of the 1998 IEEE, Int.
Conf. On Cont. Applications, pp.1307-1311, 1998.
0
-0.2
0
500
1000
1500
time [s]
2000
2500
3000
Fig. 13. Signal control
Motor rated data: 1.1KW, 1500rpm, 220/380V, 3.4A, 7
N.m, Rs=8Ω, Rr=3.1Ω, Ls=Lr=0.47H, Lm=0.443H,
J=0.06kg. m 2 , f=0.0042kg. m 2 , φr 0 =1.14Wb.
References
[1] S.Taniguchi,T.Yoshizumi and K.Matsuze, A method
of speed sensorless control of direct-field-oriented
induction motor operating at high efficiency with core
loss consideration, Proc.of the IEEE/IECON’ 2001.
[2]. E.Mendes, A.Baba and A.Razek, Losses
minimization of a field oriented controlled induction
machine, Proc. of the ‘Electrical machines and drives‘
conf.,pp.310-314.
[3]. R.D.Lorenz and S.M.Yang, Efficiency optimized flux
trajectories for closed-cycle operation of field orientation
induction machine drives, IEEE Trans.on Ind. App.,1992,
vol.28, n°3, pp.574-580.
[9] D.W. Clarke, C. Mohtadi and P.S. Tuffs, Generalized
predictive control – Part I. The basic algorithm, Part II.
Extensions and interpretations, Automatica, pp.23-2:137160,1987.