AUTOMATIC INPUT/OUTPUT MODELING OF A SQUIRREL-CAGE INDUCTION MOTOR DRIVE SYSTEM USING NEURAL NETWORK J. F. Martins+*, A. J. Pires+, J. A. Dente* + - Escola Superior de Tecnologia, Instituto Politécnico de Setúbal; Portugal. * - Centro de Automática da Universidade Técnica de Lisboa, Instituto Superior Técnico; Portugal.
Abstract: This paper presents an alternative input/output modelling of a squirrel-cage induction motor drive system. Usually these systems are modelled using state variables like rotor currents or linking fluxes, some of them extremely difficult to obtain. A neural network based learning through examples algorithm is used to represent the motor dynamics, using only input/output information. This is very important when implementing modern control techniques which require a precise modelling of the drive. The learning algorithm performance is verified through experimental results. Keywords: Variable Speed Electrical Drive, Squirrel-Cage Induction Machine, Input/Output Modelling, Learning Through Examples Algorithm, Artificial Neural Network
INTRODUCTION
The benefits of using squirrel-cage induction machines – high robustness and low maintenance – make it widely used through various industrial processes. Variable speed drives (Figure 1) equipped with this type of machine play a key role in modern industrial processes, with growing economical and performing demands. Elaborated control methods [1,2] achieve good dynamic performances in variable speed drive applications. An essential condition for this achievement is the accurate knowledge of state variables dynamics and parameters. However the measurement of the induction machine linking fluxes or rotor currents is a difficult task. Constructive changes should be made on the machine, which compromises the machine robustness and constructive simplicity. The use of rotor variables observers or extremely
sophisticated mathematical models which represent the internal dynamics of the ac-machine depends on the precise knowledge of the machine parameters and dynamics. The use of learning through examples algorithms can be a powerful tool for automatic modelling variable speed drives [3,4]. A complex functional relationship representative of the output variable dynamic behaviour (1) can be obtained, using information about all inputs – u – and all state variables – x.
y = gx 0 ( x , u)
(1)
In this paper an input/output modelling learning through examples algorithm is proposed to avoid the measurement of the linking fluxes or rotor currents, and still obtain an accurate representation of the electrical drive dynamic behaviour.
Figure 1: Schematic diagram of the ac-drive system
AC-DRIVE SYSTEM DESCRIPTION
drive mathematical model with the consequent profitable use of learning through examples algorithms.
The electrical drive system considered consists in a squirrel-cage induction machine supplied by a power inverter, presented in figure 1.
For operation and security purposes an internal control loop for the ac-machine stator currents is usually considered. Using sliding mode control [5], which is well adapted to the power inverter structure, it is possible to assume that the stator currents are controlled [6]. In this conditions the mathematical model can be simplified into (3). However fundamental problems such as the good measurement of parameters and linking fluxes are still present.
The current regulated PWM power inverter uses GTO thyristors as switches. It operates in one of eight conduction modes to produce one of six non zero voltage vectors. Table I presents the voltage supplied to the stator machine. UK denotes the voltage vector, Si denotes the command signal for each inverter arm, Ui denotes the voltage applied to each phase of the load, Uαβ denotes the voltage applied to the load, in a αβ reference frame, and E denotes the dc rectifier average voltage TABLE I POWER INVERTER SWITCH STATES AND CORRESPONDING OUTPUT VOLTAGES Uk U0 U1 U2 U3
Sa 0 0 0 0
Sb 0 0 1 1
Sc 0 1 0 1
Ua 0 -E/3 -E/3 -2E/3
U4 U5 U6 U7
1 1 1 1
0 0 1 1
0 1 0 1
2E/3 E/3 E/3 0
Ub 0 -E/3 2E/3 E/3
Uc 0 2E/3 -E/3 E/3
Uα 0 -E/√6 -E/√6 √(2/3)E -E/3 -E/3 √(2/3)E -2E/3 E/3 E/√6 E/3 -2E/3 E/√6 0 0 0
Uβ 0 -E/√2 E/√2 0 0 -E/√2 E/√2 0
Assuming ideal behaviour for the three-phase power source and electronic switches, and neglecting the magnetic saturation of the machine iron, the system can be represented by (2), where ωR is the speed of the rotating dq frame considered. di ds 1 1−σ 1−σ 1− σ 1 i + ω R i qs + u = − + ψ + ωψ qr + στ r ds στ r M dr σM σL s ds στ s dt di qs 1 1−σ 1−σ 1−σ 1 i − u = −ω R i ds − + ωψ dr + ψ + στ r qs σM στ r M qr σL s qs στ s dt dψ dr 1 M i ds = − ψ dr + ω R − ω ψ qr + τr τr dt dψ qr 1 M i = − ω R − ω ψ dr − ψ qr + τr τ r qs dt 1 dω M B = − i dsψ qr + i qsψ dr − ω − Text JLr J J dt 2 1 1 1 1 cosθ R c1 + − cosθ R + senθ R c 2 + − cosθ R senθR c 3 e u ds = 6 2 6 2 3 1 1 2 1 1 senθR c1 + cosθ R + senθR c 2 + − cosθR + senθ R c 3 e u qs = − 3 2 6 2 6 2 1 i = i ds cosθ R − i qs senθR c1 + i ds − cosθ R + 3senθ R + i qs 3 cosθR + senθR c 2 + 3 6 + 1 i − cosθ + 3senθ + i − 3 cosθ + senθ c ds qs R R R R 3 6 i = i − i dLe 1 C = i dt C C di L 1 = − (u + e) L dt u = max u a , u b , uc − min u a , u b , uc
(
(
)
)
(
=−
)
( (
( (
)
}
{
(
)
))
(
1 τr
(
(
)
ψ dr + ω r − ω ψ qr +
)
= − ω r − ω ψ dr −
1 τr
The number of equations in the electrical drive system model (2) and their non-linear nature compromise the use of learning algorithms to extract its functional relationship (1). Even with the simplifications assumed it is desirable to conjugate these algorithms with other methodologies such, as internal control loops. This procedure allows the simplification of the electrical
(3)
i qsref
To avoid the use of inaccessible information, such as linking fluxes and rotor currents and parameters, the automatic modelling algorithm should be able to model the system using only simple obtainable information. It should be able to establish the evolution of the induction machine rotor speed from the knowledge of reference stator currents and rotor speed previous evolution. Using fixed sampling intervals it is possible to express the dynamic behaviour of the system as a discrete model (4). xk denotes the state variables – linking fluxes and rotor speed –, uk the inputs – reference rotor currents –, and yk the system output – induction machine rotor speed.
( )
}
τr
i dsref
INPUT/OUTPUT MODELLING
(
(2)
τr M
ψ qr +
x k +1 = f x k , u k y k = h x k
))
M
M 1 B − i ds ψ qr + i qs ψ dr − ω − Text ref ref JL r J J
)
(
{
dψ dr dt dψ qr dt dω = dt
)
,
x0
(4)
Assuming that the non-linear system (4) is observable, and that for any set of values of u(.) exists an unique solution of the functional relationship (1), the output of the non-linear drive system can be determinated exclusively from the previous input and output variables, as in (5) [7].
(
ω k+1 = g ω k , ω k-1 ,ω k-2 , iqsrefk , iqsrefk-1 , iqsrefk-2 , idsrefk , idsrefk-1 , idsrefk-2
)
(5)
It is difficult to assure the existence of this highly non-linear relationship and verify the relative significance of its arguments. To verify this a linearization, around a steady state operating point, is performed, obtaining a functional relationship (6),
where the coefficients α, β, γ depend on the operating point considered.
ω δ k +1 = ∑ (α i ω δ k − i + βi i qsrefδ k − i + γ i i dsrefδ k − i ) (6) 2
i=0
The linearized model presents a limited accuracy. However its analysis allow us to introduce further simplifications in the learning algorithm. Each term in (6) has a relative influence over ωδk+1, that depends on the operating point and sampling interval considered. As an example figure 2 presents the αβγ-coefficients dependence on the sampling interval. The continuos line refers to the k instant, the dashed line to the k-1 instant, and the dotted line to the k-2 instant.
LEARNING ALGORITHM
There are several methodologies that can be used to extract the complex unknown input/output functional relationship of the electrical drive system. This paper considers error-backpropagation, which is a well adapted training algorithm for neural network based learning algorithms [8].
1 .0 0 .6 0 .2 - 0 .2 0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
- 0 .6
Figure 3: Artificial Neural Network
- 1 .0
S a m p lin g I n te r v a l [s e c ]
A one hidden layer neural network, with sigmoid transfer function, for the hidden layer, and linear transfer function, for the output layer, is used. The weights and biases are updated using the generalized delta rule with adaptive learning rate and momentum.
Figure 2.1: Evolution of the α coefficients 1 .0 0 .6 0 .2 - 0 .2 0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
EXPERIMENTAL RESULTS
- 0 .6 - 1 .0
S a m p lin g I n te r v a l [s e c ]
To verify the performance of the proposed methodology the experimental rotor speed – ω – and the speed generated by the learning algorithm – ω* – are compared, as shown in figure 4.
Figure 2.2: Evolution of the β coefficients 1 .0 0 .6 0 .2 - 0 .2 0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
- 0 .6 - 1 .0
S a m p lin g I n te r v a l [s e c ]
Figure 2.3: Evolution of the γ coefficients
These results state the influence of the previous speed values; the growing influence of the reference stator currents, in the k instant; and these arguments influence, in the k-1 and k-2 instant, for sampling intervals close to the linking fluxes time constant (0,13 sec.). For sufficient small time intervals these two last influences can be neglected and the number of arguments in the functional relationship (5) can be decreased, as presented in (7).
(
ω k +1 = g ω k , iqref k , idref k
)
(7)
Figure 4: Block diagram used to compare the model response with the experimental system
The speed error (8) is evaluated over a test data set, distinct from the training data set used to extract the functional relationship. These two data sets are presented in figure 5, where the test data is represented in bold lines.
eω = ω − ω ∗
(8)
The similarity of the results allow us to simplify the neural network reducing its inputs, and representing the drive speed dynamics considering less information. This can be confirmed by the error parameters presented in Table II, where there is only a slight improvement for the functional relationship (11).
w [pu]
0.2 0.1 0
-0.1
TABLE II ERROR PARAMETERS FOR (7) AND (11) - ∆T = 0,002 SEC.
-0.2
σeωω 0,41 0,31
0.5
(7) (11)
0.5 0
0
iq [pu] -0.5 -0.5
id [pu]
Figure 5: Training and test data set domain
To evaluate the learning algorithm performance the standard deviation error (9) and the band error (10) are evaluated for each N-sampled test.
(
1 N ∑ eω − eω N − 1 i =1 i
σeω =
)
2
(9)
∆eω ω 3,40 2,97
There is a practical advantages in extending the sampling interval. However when it reaches values where the linking fluxes dynamic is important a more complex neural network is required. As an example figure 7 shows the results considering a larger sampling interval – ∆T=0,032 sec. Three distinct neural networks are considered, regarding three distinct functional relationships with different amount of information considered – (7), (11) and (5). 6
∆eω = max(eω ) − min(eω )
5 4
Speed Error [%]
(10)
Sampling Intervals
3 2 1 0 -1 0
1
2
3
4
5
6
-2 -3 -4 -5 -6
T im e [ s e c ]
In figure 6 we present the results, considering the same test data set with a sampling interval ∆T=0,002 sec., for two simplified functional relationships – (7) and (11).
(
Speed Error [%]
6 5 4 3 2 1 0 -1 0 -2 -3 -4 -5 -6
) (11)
6 5 4
Speed Error [%]
ω k +1 = g ω k , ω k -1 , iqref k , iqref k -1 , idref k , idref k -1
Figure 7.1: Speed-error evolution for functional relationship (7)
1 0 -1 0
1
2
3
4
5
6
-2 -4 -5 -6
1
2
3
4
5
6
7
8
9
10
T im e [ s e c ]
Figure 7.2: Speed-error evolution for functional relationship (11)
Figure 6.1: Speed-error evolution for functional relationship (7)
6 5 4
Speed Error [%]
Speed Error [%]
2
-3
T im e [s e c ]
6 5 4 3 2 1 0 -1 0 -2 -3 -4 -5 -6
3
3 2 1 0 -1 0
1
2
3
4
5
6
-2 -3 -4 -5
1
2
3
4
5
6
7
8
9
10
-6
T im e [ s e c ]
T im e [s e c ]
Figure 6.2: Speed-error evolution for functional relationship (11)
Figure 7.3: Speed-error evolution for functional relationship (5)
The results and respective error parameters, presented in Table III, state the learning algorithm improvement when a larger amount of information is considered.
TABLE III ERROR PARAMETERS FOR (7), (11) AND (5) - ∆T = 0,032 SEC.
∆eω ω 6,60 4,63 4,47
0,25 0,20 0,15 0,10
Speed [pu]
σeωω 1,55 1,29 1,05
(7) (11) (5)
considering information from this new working domain areas (13 to 17 sec).
0,05
ω
0,00 -0,05 0
2
4
6
8
10
12
14
16
12
14
16
ω∗
-0,10
Generalisation and Continuos Learning
-0,15 -0,20 -0,25
Time [sec] 20 15
Speed Error [%]
The previous results state that the learning algorithm present good performances even for points not contained in the training data set, presenting generalisation capabilities. This capabilities will decrease if the training data set shows a poor spanning of the working domain, as presented in figure 8.
10 5 0 -5 0
2
4
6
8
10
-1 0 -1 5 -2 0
T im e [ s e c ]
Figure 10: Speed and speed-error evolution with continuos learning
0.2
w
0.1 0
APPLICATIONS
-0.1 -0.2 0.5
Control strategies as model based predictive control [9] benefit from models that present simple structure and remain precise through time.
0.5 0
0
iqs ref -0.5 -0.5
ids ref
Figure 8: Training (with a poor spanning of the working domain) and test data set domain - ∆T=0,002 sec.
Speed Error [%]
Figure 9 presents the results for the previous test data set considering the training algorithm previously presented in figure 8. 6 5 4 3 2 1 0 -1 0 -2 -3 -4 -5 -6
1
2
3
4
5
6
7
8
9
The model obtained from the previous learning algorithm is intended to be used, as a predictor, in a predictive control structure, as shown in figure 11. Since rotor speed evolution is generated by the model itself, its prediction error, through time, depends on its own error.
10
T im e [s e c ]
Figure 9: Speed-error evolution for training and test data sets presented in figure 8 - ∆T=0,002 sec.
These results show the local characteristic of generalisation, for these systems. The speed error increases for areas not covered by the training data set. To avoid this drawback a permanent information of the working domain should be kept as a training data set. figure 10 show the results for different phases of the learning process. From 0 to 4 sec. the model is obtained from a theoretical training data set. The use of experimental information considerably improves the speed error (4 to 8,5 sec.). However when the speed drive goes into working areas not cover by the training data set the speed error increases (8,5 to 13 sec.). The error only improves with a subsequent learning
Figure 11: Block diagram used to compare the predictor response with the experimental system
As an example, figure 12 shows the results, where the experimental (continuos line) and model (dashed line) results are compared in two distinct situations. In the first two seconds of the test, the relationship obtained from the learning algorithm are used as a model, and in the last second as a predictor.
0,25 0,20
Speed [pu]
0,15 0,10 0,05 0,00 -0 ,0 5 0
ω
1
2
3
ω∗
-0 ,1 0 -0 ,1 5 -0 ,2 0 -0 ,2 5
The results presented in this paper validate the proposed methodology. Presents good generalisation capabilities and can be used as a model or as a predictor. However to achieve good results as a predictor the amount of information given to the learning algorithm should be increased.
T im e [s e c ]
Acknowledgements
Figure 12.1: Speed-error evolution for functional relationship (7)
This work is supported PBIC/C/TPR/2368/95.
by
JNICT,
contract
0,25 0,20
Speed [pu]
0,15
References
0,10 0,05 0,00 -0 ,0 5 0
ω
1
2
3
ω∗
[1]
-0 ,1 0 -0 ,1 5 -0 ,2 0 -0 ,2 5
T im e [s e c ]
[2]
Figure 12.2: Speed-error evolution for functional relationship (11) [3] 0,25 0,20
Speed [pu]
0,15 0,10 0,05 0,00 -0 ,0 5 0
ω
1
2
3
ω∗
[4]
-0 ,1 0 -0 ,1 5 -0 ,2 0
[5]
-0 ,2 5
T im e [s e c ]
Figure 12.3: Speed-error evolution for functional relationship (5)
In spite of the good results as a model the learning algorithm presents poor results as a predictor when simplified functional relationship is used. Since the precision of the predictor depends on the model precision it is necessary to decrease this error. It is essential to enlarge the amount of information given to the model increasing the number of arguments of the functional relationship representative of the drive behaviour. However the increased number of arguments, implies a neural network with more inputs and subsequent learning effort.
[6]
[7]
[8]
[9]
Blaschke, F.; “The Principle of Field Orientation Applied to the New Transvector Closed Loop Control System for Rotating Field Machines”; Siemens Rev., May 1972, vol. 39, pp. 217-220. Pires, A. J.; Dente, J. A.; “A New Methodology for the Induction Machine Control”, Proceedings of MELECON’94, Antalya, Turkey, 1994. pp, 781-784. Maia, J. H.; Branco, P. J.; Dente, J. A.; “Automatic Modeling of Electrical Drives”, Proceedings of Modern Electrical Drives in NATO Advanced Study Institute, Antalya, Turkey, 1994, pp. 73-78. Hofmann, W.; Liang, Q.; “Neural Network-Based Parameter Adaptation for Field-Oriented AC-Drives”, Proceedings of EPE’95, 1995, pp.391-396. Utkin, V. I.; “Discontinuous Control Systems: State of Art in Theory and Applications”, Proceedings of 10º IFAC, Munich, RFA, 1987, pp. 75-94. Kazmierkowski, M. P.; Sulkowski, W.; “A Novel Vector Control Scheme for Transistor PWM Inverter-Fed Induction Motor Drive”, Transactions on Industrial Electronics, February 1991, Vol. 38, No. 1, pp. 41-48. Martins, J. F.; “Automatic Input/Output Modelling of a Squirrel-Cage Induction Motor Drive System”, Master Thesis, IST, UTL, Lisbon, Portugal, December 1996. Dagli, C. H.; 1994; “Artificial Neural Networks for Intelligent Manufacturing”, Chapman & Hall, London, 1994. Clarke, D. W.; 1994; “Advances in Model-Based Predictive control”, in ‘Advances in Model-Based Predictive Control’, Oxford University Press, Oxford, 1994, pp. 3-21.
Adresses of the authors CONCLUSIONS
This paper proposes a learning through examples algorithm to extract a functional relationship, that models the behaviour of the squirrel-cage induction motor drive system, from experimental input/output data. This is an important feature since there is no need of considering all state variables, such as the ones difficult to obtain.
J. F. Martins, A. J. Pires Escola Superior de Tecnologia, Instituto Politécnico de Setúbal. Rua do Vale de Chaves, Estefanilha 2910 Setúbal, Portugal. Phone: 351-65-761621. fax: 351-65-721869 E-Mail:
[email protected]
J. A. Dente Centro de Automática da Universidade Técnica de Lisboa, Instituto Superior Técnico,; SMEEP. Av. Rovisco Pais 1, 1096 Lisboa Codex, Portugal. Phone: 351-1-8417435 fax: 351-1-8417167 E-Mail:
[email protected]
