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 ...
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
