A CHOICE OF VARIABLES IN AUTOMATIC MODELLING OF AC-DRIVE SYSTEMS J. F. Martins+, A. J. Pires+, J. A. Dente* + - Escola Superior de Tecnologia, Instituto Politécnico de Setúbal; Portugal. * - CAUTL; Secção de Máquinas Eléctricas e Electrónica de Potência; Instituto Superior Técnico; Portugal.
Abstract: This paper analysis the variables needed to represent the internal dynamics of a squirrel-cage induction motor drive system. This is an important issue when automatic input/output modelling, based on learning through examples algorithms, is considered. These modelling techniques allow a precise knowledge of the drive dynamics, fundamental topic in modern control approaches, without using variables that are difficult to obtain. Experimental results are presented to support the analysis.
Keywords: Variable Set, Variable Speed Electrical Drive, Squirrel-Cage Induction Machine, Power Inverter, Automatic Input/Output Modelling, Learning Through Examples Algorithm
INTRODUCTION
y = gx 0 ( x , u)
(1)
Variable speed drives, equipped with squirrel-cage induction machines, play a key role in modern industrial applications. The benefits of using this type of machine – high reliability and low maintenance – make it widely used through various industrial processes, with growing performance demands.
This paper describes a methodology used in order to choose an adequate variable set, which should be used as argument for the functional relationship (1). This study is significant to take the better profit from learning through examples algorithms.
New technologies, such as new materials, power electronics, microelectronics, optimised or predictive control methods, present new demands for optimisation and performance in variable speed drive applications. When fast performance is required, evolving highly nonlinear systems, classical methods present a limited representation of the real system. The use of learning through examples algorithms can be a powerful tool for automatic modelling variable speed drives [1]. 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.
AC-DRIVE SYSTEM
The electrical drive system considered, presented in figure 1, consists in a squirrel-cage ac-machine supplied by a current regulated voltage sourced pulse width modulated (VS-PWM) power inverter. The power inverter is constituted by a full bridge diode rectifier, as a DC source, a LC-filter, and the IGBT power inverter itself. This power inverter generates the current that feeds the motor stator coils.
M 3~ c1 , c2 , c3
ia
Command Figure 1: Schematic diagram of the electrical drive system
ib
MODELLING VARIABLES - EXTENDED SET
Assuming ideal behaviour for the three-phase power source and electronic switches, the PWM power inverter can be modelled by (2). 2 1 1 1 cos θ R c1 + − cosθ R + senθ R c 2 + − cosθ R u ds = 6 6 3 2 2 1 1 1 u qs = − senθ R c1 + cosθ R + senθ R c 2 + − cosθ R + 3 2 2 6 2 i = i cos θ R − i qs senθ R c1 + 3 ds 1 i ds − cos θ R + 3senθ R + i qs 3 cos θ R + senθ R c 2 + + 6 1 i ds − cos θ R + 3senθ R + i qs − 3 cos θ R + senθ R c 3 + 6 i L = i − i C de 1 dt = C i C di 1 L = − (u + e) L dt u = max{u a , u b , u c } − min{u a , u b , u c }
(
senθ R c 3 e (2) senθ R c 3 e 6
1
2 1
)
( ( ( (
( (
) )
))
))
The induction motor is usually modelled using electromechanical power conversion theory. Some assumptions can be made in order to simplify the model. Assuming that the magnetic saturation of the iron can be neglected, the machine can be represented by equation (3). This model assumes a dq rotating frame with speed ωR and position θR. dids 1 1− σ 1− σ 1− σ 1 i + ωR iqs + = − + ψ + ωψ qr + u στ r M dr σM σL s ds στ s στ r ds dt diqs 1 1− σ 1−σ 1− σ 1 i − = −ωR ids − + ωψ dr + ψ + u στ r M qr σL s qs στ s στ r qs σM dt dψ dr 1 M = − ψ dr + ω R − ω ψ qr + ids τr τr dt dψ qr 1 M = − ω R − ω ψ dr − ψ qr + iqs τr τr dt M B 1 dω dt = JL − idsψ qr + iqsψ dr − J ω − J Text r
(
(
(3)
)
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. In this way a new variable set should be derived from (4), considering only input and output variables. Assuming that, as in most variable speed drives, the relevant output variable is the drive speed, one can consider the previous model (2,3) in a general form (5), where x denotes the state variables, u the input variables and y the output variable. Φ(.) and Ψ(.) are non-linear functions defined as Φ:ℜn×ℜp→ℜn and Ψ:ℜn→ℜ, where n denotes the number of state variables and p the number of input variables.
x (t ) = Φ(x(t ), u(t )) , x 0 y(t ) = Ψ(x(t ))
Using fixed sampling intervals, expression (5) can be rewritten as (6), where x1 denotes the accessible state variables, x2 the non-accessible ones, u the input variables and y the output variable. k denotes the present time instant and k+1 the posterior time instant. f1(.), f2(.) and h(.) are non-linear functions defined as f1:ℜl×ℜm×ℜp→ℜl, f2:ℜl×ℜm×ℜp→ℜm and Ψ:ℜl→ℜ, where l denotes the number of accessible state variables, m the number of non-accessible state variables and p the number of input variables.
( (
)
From (2) and (3), and under the previous assumptions, an extended state, input and output variable set can be considered (4).
c1 , c2 , c3 , e, i L , i , (4) ids , iqs ,ψ dr ,ψ qr ,ω
MODELLING VARIABLES - INPUT/OUTPUT SET
When using the previous model (2,3), or some modern elaborated control methods [2,3,4], an essential condition for achieve a good dynamic performance is the accurate knowledge of state variables dynamics and parameters. However the measurement of the squirrel-cage induction machine linking fluxes, or even rotor currents, is a difficult task. Constructive changes should be made on the machine, which compromises the machine
) )
x1 = f1 x1 , x 2 , u k k k k +1 x = f x , x , u 2 k +1 2 1k 2k k y = h x k 1k
)
(
(5)
( )
(6)
From (6) it is desirable to obtain a functional relationship, that represents the dynamics of the electrical drive system, evolving only accessible variables. In this way it is necessary to consider the previous m measurements of the output variable (7), where m denotes the number of difficult to obtain variables. y k − ( m −1) = h x 1k − ( m −1) = h f1 x 1k − m , x 2 k − m , u k − m y k − ( m − 2 ) = h x 1k − ( m − 2 ) = h f 1 x 1k − ( m −1) , x 2 k − ( m −1) , u k − ( m −1) = h f 1 x 1k − ( m −1) , f 2 x 1k − m , x 2 k − m , u k − m , u k − ( m −1) y = h x 1k k = h f 1 f 2 x 1k -( m-1) , f 2 x 1k − m , x 2 k − m , u k − m , u k − ( m −1) ,
( (
((
( )
( (
) ( ( ) (( (
(
))
)
)
)) ))
(7)
) , u ) k −1
(
x 2 k − m = p y k , y k −1 ,
, y
k −( m − 2 )
, y k −( m−1) , x1k −1 ,
Assuming that the system is observable, one can consider the existence of a function p(.), that allows us to determinate the measurement of the difficult to obtain variables in the instant k-m (8).Combining expressions (8) and (6) the desired functional relationship, where the output is only a function of the accessible variables, can be stated by (9).
((
(
y k +1 = h f1 x1k , f2 x1k −1 ,
, p(⋅),, u ), u )) (9) k −1
k
In this conditions a possible variable set is given by (10), where the measurements in the previous m instants must be considered. The problem of measuring non-accessible variables is overcomed, however fundamental problems, such as the large number of variables, good measurement of parameters and the non-linearity of the drive system, are still presented.
c1 , c2 , c3 , e, i L , i , i , i ,ω ds qs
(10) k ... k − m
MODELLING VARIABLES - SIMPLIFIED SET
, x
1k − ( m −1) , x1k − m , u k −1 ,
, u
k − ( m −1 )
, u k−m
)
(8)
Assuming the existence of the respective jacobians, the previous model can be linearized around an operating point. Using fixed sampling intervals it is possible to express the dynamic behaviour of the system as a discrete model (12), where xδ denotes the deviation of the state variables, uδ the deviation of the input variables and yδ the deviation of the output variables, all of them around the considered operating point.
x δ k +1 = Px δ k + Qu δ k y δ k = Cx δ k
, x δ0
(12)
From the one-dimensional output equation it is not possible to reconstruct the state of the system. If the system is observable it is possible, by considering the output equation in (12) for consecutive time instants - as explained in the previous section -, to express the system output without using the linking fluxes (13). Equation (13), rewritten in (14), states the output of the non-linear drive system exclusively from the previous input and output variables.
ω δ k +1 = ∑ (α i ω δ k − i + βi i qsrefδ k − i + γ i i dsrefδ k − i ) (14) 2
i=0
For operation and security purposes an internal control loop for the ac-machine stator currents is usually considered. Using on-line sliding mode current feedback control, which is well adapted to the power inverter structure and accurately selecting all available voltage vectors in order to minimise the current error [5], it is possible to assume that the stator currents are controlled. In this conditions the mathematical model (2,3) can be simplified into (11). 1 M dψ dr dt = − τ ψ dr + (ωR − ω )ψ qr + τ idsref r r dψ qr 1 M = −(ωR − ω )ψ dr − ψ qr + iqsref d t τ τr r dω M B 1 = − idsrefψ qr + iqsrefψ dr − ω − Text d t JL J J r
(
ω δ k +1
This linearized functional relationship presents a limited accuracy. However its analysis allow us to discuss further simplifications in the variable set to be used with the learning algorithm. Each term in (14) has a relative influence over ωdk+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 continuous line refers to the α-coefficients, the dashed line to the β-coefficients, and the dotted line to the γ-coefficients.
(11)
)
−1 ω 0 0 C δ k-2 0 0 idsrefδ idsrefδ k-1 idsrefδ k-2 idsrefδ k-1 k -2 CP CQ ω = CP P P − 0 0 + Q i + Q i δ − qsrefδ k-2 qsrefδ k-1 CP 2 ω k-1 CPQ iqsrefδ k -2 CQ iqsrefδ k-1 δk
(13)
4,0 1,0
0,2 -0,2 0,0
0,1
0,2
0,3
0,4
0,5
0,6
Amplitude [pu]
3,0 0,6
2,0 1,0 0,0 0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
-1,0
-0,6
-2,0
-1,0
Time [sec]
Sampling Interval [sec]
Figure 2.1: Evolution of the αβγ coefficients regarding the k measurement 1,0 0,6
Figure 3: ψqrδ impulsive response due to idsrefδ
Figure 4 shows the ωδ impulsive response due to iqsrefδ, where the extinction time is very large (44 sec.). So for any smaller sampling interval the drive behaviour is influenced by its mechanical impulse response. 160
0,2 0,1
0,2
0,3
0,4
0,5
0,6
-0,6 -1,0
Sampling Interval [sec]
Amplitude [pu]
140 -0,2 0,0
120 100 80 60 40 20
Figure 2.2: Evolution of the αβγ coefficients regarding the k1 measurement
0 0
10
20
30
40
50
Time [sec]
1,0
Figure 4: ωδ impulsive response due to iqsrefδ
0,6 0,2 -0,2 0,0
0,1
0,2
0,3
0,4
0,5
0,6
-0,6 -1,0
As stated before the importance of observability in the formulation of the identification of a linear system is a fundamental issue. However constructive procedures, similar to those available for linear systems, do not exist for non-linear systems.
Sampling Interval [sec]
Figure 2.3: Evolution of the αβγ coefficients regarding the k2 measurement
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 sampling intervals sufficient smaller than the linking fluxes time constant the drive dynamic response due to input variations is slow. So its output variation will be influenced essentially by its own mechanical inertia. For large sampling intervals, close to the mechanical time constant (8,7 sec.) the free response can be considered extinct and the output evolution will depend essentially on the input references and previous output. As an example figure 3 shows the ψqrδ impulsive response due to idsrefδ. This response can be assumed extinct for 0,65 sec., so any sampling interval larger than this value makes the drive response independent of its electrical impulse response.
Assuming that the non-linear system (5) 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 (15).
(
ω*k+1 = g ω k ,ω k-1 ,ω k-2 ,iqsref k ,iqsref k-1 ,iqsref k-2 ,idsref k ,idsref k-1 ,idsref k-2
) (15)
For sufficient small time intervals the influence of the previous time instants can be neglected and the number of arguments in the functional relationship (15) can be decreased, as presented in (16).
(
ω*k +1 = g ω k ,iqref k ,idref k
)
(16)
For sufficient small sampling intervals the variable set (18) can be considered, while for larger sampling intervals one must consider the variable set (17). In this case the amount of information needed to obtain a reliable functional relationship is higher.
k
k -1
}
, ω k - 2 , iqsref k , iqsref k -1 , iqsref k -2 , idsref k , idsref k -1 , idsref k - 2 (17)
{ω
k
, i qrefk , i drefk
}
(18)
Speed Error [%]
{ω , ω
EXPERIMENTAL RESULT
eω = ω − ω
∗
(19)
It can be seen, from Figure 5, that for a small sampling interval (∆T=2 msec.) the similarity of the results allow us to use the simplified variable set (18) to achieve a good representation of the drive speed dynamics. For example, with the use of variable set (17) the error band is only improved by 14%. This is a particular important issue, since it allows the simplification of the learning algorithm.
3
4
5
6
7
8
9
10
However when the sampling interval reaches values where the linking fluxes dynamic is important more information is required. From figure 6, where a sampling interval of 32 msec. is considered, it can be seen that the use of variable set (17) is required in order to obtain better results. The use of this variable set improves the error band by 42%. 6 5 4 3 2 1 0 -1 0
1
2
3
4
5
6
-2 -3 -4 -5 -6
Time [sec]
Figure 6.1: Speed-error evolution for functional relationship (16) with variable set (18) - ∆T=32 msec. 6 5 4
1
2
3
4
5
6
7
8
9
10
Speed Error [%]
Speed Error [%]
2
Figure 5.2: Speed-error evolution for functional relationship (15) with variable set (17) - ∆T=2 msec.
6 5 4 3 2 1 0 -1 0 -2
1
Time [sec]
Speed Error [%]
To experimentally verify the variables influence in the electrical drive system automatic input/output modelling an error-backpropagation trained neural network is used as learning through examples algorithm [10]. 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. 1 The performance of both functional relationships - (15) and (16) - are compared by evaluating the experimental rotor speed – ω – and the output speed generated by the neural-network based model – ω*. The speed error is given by (19).
6 5 4 3 2 1 0 -1 0 -2 -3 -4 -5 -6
3 2 1 0 -1 0
1
2
3
4
5
6
-2 -3
-3 -4 -5 -6
-4 -5 -6
Time [sec]
Figure 5.1: Speed-error evolution for functional relationship (16) with variable set (18) - ∆T=2 msec.
Time [sec]
Figure 6.2: Speed-error evolution for functional relationship (15) with variable set (17) - ∆T=32 msec.
References
CONCLUSION
[1]
This paper analyses the variable set that could be used with a learning through examples algorithm that extracts, from experimental data, a functional relationship which models the behaviour of the squirrel-cage induction motor drive system. The results presented in this paper validate the use of a simplified data set that considers only input/output variables. This is an important feature since there is no need to consider all state variables, such as the ones difficult to obtain. There are practical advantages in extending the sampling interval. However when it reaches values where the linking fluxes dynamic is important a more complex data set is required, increasing the amount of information needed by the learning algorithm.
[2]
[3]
[4]
[5]
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. 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.: “Variable Speed Induction Motor Drive using a New Mwthodology for Controller Synthesis - Implementation Aspects”. Proceedings of EPE-EDD’94, Lausanne, Switzerland, 1994, pp. 295-300. Tiitinen, P.; Pohjalainen, P.; Lalu, J.: “The Next Generation Motor Control Method: Direct Torque Control (DTC)”. EPE Journal, Vol. 5, nº 1, March 1995. Martins, J. F.; Pires A.; Silva, J. F.: “A Novel and Simple Current Controller for Three-Phase PWM Power Inverters”. Proceedings of PESC’97, St. Louis, USA, June, 1997, pp. 1027-1032.
List of Symbols Addresses of the Authors uabc .......... full bridge rectifier input voltage u.............. full bridge rectifier output voltage L ............. LC-filter inductance C............. LC-filter capacitance iL ............. LC-filter inductance current iC ............. LC-filter capacitor current e.............. LC-filter output voltage i .............. LC-filter output current c123 .......... power inverter command signals uqs, uds ..... electrical machine stator voltages (dq frame) ids, iqs ....... electrical machine stator currents (dq frame) ψdr, ψqr .... electrical machine rotor fluxes (dq frame) ω ............. electrical machine rotor speed Text .......... electrical machine load torque σ ............. electrical machine magnetic dispersion coefficient τs ............. electrical machine stator time constant τr ............. electrical machine rotor time constant Ls ............ electrical machine stator self-inductance coefficient Lr ............ electrical machine rotor self-inductance coefficient J .............. inertia coefficient B............. friction coefficient
Acknowledgements This work is supported PBIC/C/TPR/2368/95.
by
JNICT,
contract
J. F. Martins Escola Superior de Tecnologia Instituto Politécnico de Setúbal. Rua do Vale de Chaves, Estefanilha 2910 Setúbal, Portugal. Phone: 351-65-790000. Fax: 351-65-721869. E-Mail:
[email protected]
A. J. Pires Escola Superior de Tecnologia Instituto Politécnico de Setúbal. Rua do Vale de Chaves, Estefanilha 2910 Setúbal, Portugal. Phone: 351-65-790000. Fax: 351-65-721869. E-Mail:
[email protected]
J. A. Dente Instituto Superior Técnico; CAUTL; SMEEP. Av. Rovisco Pais 1, 1096 Lisboa Codex, Portugal. Phone: 351-1-8417435. Fax: 351-1-8417167. E-Mail:
[email protected] 