Flux observer for induction machine control. Part II - Semantic Scholar

Eur. Phys. J. AP 14, 25–43 (2001)

THE EUROPEAN PHYSICAL JOURNAL APPLIED PHYSICS c EDP Sciences 2001

Flux observer for induction machine control. Part II – Robust synthesis and experimental implementation E. Delmottea , B. Semail, B. Robyns, and J.P. Hautier ´ ´ ´ Laboratoire d’Electrotechnique et d’Electronique de Puissance (L2EP) de Lille, Ecole Centrale de Lille, BP 48, 59651 Villeneuve d’Ascq Cedex, France Received: 26 April 2000 / Revised and Accepted: 24 November 2000 Abstract. This paper proposes an approach to determine the gain of closed loop flux observer for an induction machine with respect to robustness and stability criteria. In the companion paper, a sensitivity analysis of flux observer has pointed out the influence of the sampling rate and the parameters mismatch. From this study, an accurate observer can be deduced. The analytical results are checked experimentally, in steady state and during transients. For that aim, an experimental rig has been installed. This paper deals also with a new description of the induction machine drive thanks to Causal Informational Graph. This description is useful for the simulation and for the implementation of the experimental rig. PACS. 84.50.+d Electric motors – 84.60.Bk Performance characteristics of energy conversion systems; figure of merit – 84.70.+p High-current and high-voltage technology: power systems; power transmission lines and cables (including superconducting cables) – 89.20.+a Industrial and technological research and development

Nomenclature (φd , φq )R σ =1− ωR ωS Ω f

MC2 LCR LCS

Introduction (Wb)

Two-phase-equivalent rotor magnetic fluxes Leakage coefficient

(Id , Iq )S

(A)

J LCR LCS MC

(J s) (H) (H) (H)

p RR RS Te

(Ω) (Ω) (s)

Speed of rotor phase Speed of stator phase Mechanical speed Coefficient of viscous friction Two-phase-equivalent stator currents Inertia moment Cyclic rotor inductance Cyclic stator inductance Mutual cyclic inductance between stator and rotor Pole pairs number Rotor resistance Stator resistance Sampling time

(s)

Equivalent stator current

Teq =

1 1 1−σ + σTS σTR

LCR TR = RR (Vd , Vq )S a

(rad/s) (rad/s) (rad/s) (N m s)

time constant (s)

Rotor time constant

(V)

Two-phase-equivalent stator voltage

e-mail: [email protected]

In the previous paper [1], it was proved that both bad knowledge of the parameters and sampling effects lead to errors in flux estimation. Moreover, this analysis allows observer gain values capable of minimising parameters and sampling rate influences to be distinguished. This is what is proposed in this paper. In the first part, the flux reconstitution errors are studied as a function of observer gain for the two sampling methods in case of parameter inaccuracy or not. An optimal gain value is deduced. As the flux modulus and orientation errors are computed assuming a steady state condition, the observer transient behaviour has to be verified. The simulated results will next be compared to experimental trials in order to prove the validity of the closed-loop observer gain computation.

1 Observer error analysis in steady state In this part we search the observer gain K which reduces the parameter value uncertainties and the sampling rate influence [6,8]. The analysis follows several steps: (1) With a null gain, the most sensitive operating point at steady state is determined, i.e. the pair of speed and torque for an observed flux which give the greater modulus or orientation errors. Thus the results of the previous study are used for each discrepancy source;

26

The European Physical Journal Applied Physics

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Fig. 1. Modulus and orientation error (nominal flux, Te = 0.8 ms).

(2) For these sensitive points, the gain K which minimises the modulus and orientation errors is deduced from the steady state analysis; the observer stability is verified, then the result of the synthesis is carried out; (3) The gain is tested for all operating points at steady state and the stability is verified; (4) For one or more sensitive points, speed and flux transients are simulated in order to test the observer dynamic behaviour; (5) Finally, experimental and simulated results are compared. This method is illustrated with some examples.

1.1 Sample rate influence The consequences of the sampling method on flux estimation have already been studied in the previous paper [1]. With the assumption that the parameters of the model are accurate, the error analysis has brought to light the following results [6]. In the case of the full order sampling method, the modulus and orientation errors are large at high speed because of the limited development and the orientation error depends on the torque. The most sensitive point is found at high speed (here equal to 2000 rpm) and at no load. In the case of the reduced order sampling method, the modulus error is insignificant (lower than 0.6%), the orientation error increases with the speed and does not depend

on the torque. The most sensitive point, which is studied, is at high speed: 2000 rpm for a torque equal to 20 N m. The gain K that minimises the modulus and orientation error is now sought for these sensitive points. The results are presented in Figure 1. The gain matrix is still defined as: " # k1 k2 sign(Ω) σo MCo K= · (1) 1 − σo −k2 sign(Ω) k1 The errors are computed for different values of the pair (k1 , k2 ). In the case of the full order sampling method, the modulus and orientation errors are reduced nearly to zero for the following gain: " # 0 −0.1sign(Ω) σo MCo K= · (2) 1 − σo 0.1sign(Ω) 0 In the case of the reduced order sampling method, the orientation error decreases whatever the value of K different to zero. Therefore another criterion is introduced in order to choose a gain K: the observer eigenvalues are imposed to be close to null in order to increase the response time of the observer: " # −1 −3.5sign(Ω) σo MCo K= · (3) 1 − σo 3.5sign(Ω) −1 This study was carried out in the steady state for two particular operating points. Now it must be verified that

E. Delmotte et al.: Flux observer for induction machine control


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the observer behaviour is satisfactory over a large range of operating points. The results are given in Figure 2. In the case of the full order sampling method, the modulus error is decreased and the orientation error is lower than 2◦ . In the case of the reduced order sampling method, the orientation error is reduced but the modulus error is increased. Nevertheless this error remains insignificant (lower than 2%). Thought to these results, the observer decreases the modulus and orientation errors due to the sample rate and the sampling method. Naturally the observer stability was verified by computing the eigenvalues the modulus of the eigenvalues varies from 0 to 0.998 according to the speed. 1.2 Influence of rotor resistance This study concerns the influence of parameters uncertainties in flux reconstitution [9]. With a sampling rate equal to 0.8 ms, the modulus and orientation errors are analysed for the two sampling methods. The parameters are assumed to be the same for the observer and the machine, except the rotor resistance. As previously in the first paper, an error is artificially introduced in the rotor resistance machine for the simulations and the calculations as follows: rRr = 1 −

RR = −0.33 . RRo

(4)

The effects of both sampling rate and rotor resistance variation are studied.

The previous analysis has highlighted the discrepancies in flux reconstitution due to these effects (Fig. 7 of the previous paper). In the case of the full order sampling method, the influence of the sampling rate is greatest at high speed while at low speed the rotor resistance variation is prominent. The modulus and orientation errors depend on the torque. Two sensitive points therefore appear: at high speed (here equal to 2000 rpm) and without load; at standstill and at high load (here the torque is equal to 20 N m). In the case of the reduced order sampling method, the rotor resistance variation is prominent: its effects add to the sampling rate effects. The most sensitive point is at high torque and at high speed (here respectively equal to 2000 rpm and 20 N m). As before the gain K, which minimises the modulus and orientation error for the sensitive points, is sought (Fig. 3). In the case of the full order sampling method, two gains are obtained for the two sensitive points:

σo MCo At 2000 rpm : K = 1 − σo at 0 rpm : K =

σo MCo 1 − σo

0

−0.075

0.075

0

−1 0 0 −1

(5)

·

(6)

28









Speed varying gain therefore appears to be the right solution. For example the following law is possible: " # k1 k2 sign(Ω) σo MCo K= (7) 1 − σo −k2 sign(Ω) k1  |Ω|   k = − exp −  1 Ω1 with |Ω|    k2 = 0.075 1 − exp − Ω2 ( and

Ω1 = 200 tr/mn Ω2 = 500 tr/mn.

In the case of the reduced order sampling method, from Figure 3, the appropriate gain K seems to be: σo MCo −1 0 K= = Kt . (8) 1 − σo 0 −1 The two gains are now tested for all the operating points. The results of modulus and orientation errors for these gains are given in Figure 4. In the case of the full order sampling method, the modulus error is decreased in comparison to the computation with a null gain; the orientation error is lower than 6◦ . In the case of the reduced order sampling method, the orientation and modulus errors are reduced. So in conclusion, the observer decreases the modulus and orientation errors due to the sample rate and the rotor

resistance variation. Once again, the observer stability was verified. A similar approach may be used to analyse observer robustness together with other electrical machine parameters. For example a variation of the mutual cyclic inductance simulates a change of the magnetic state. It is possible to take into account the saturation with inductances that depend on the magnetizing current [10,12]. In the case of the observer of the rotor flux, the influence of the stator resistance is neglected with regard to the rotor resistance [6]. These studies enable choice of an appropriate observer gain, able to cope with parameter uncertainty. However, the analysis was carried out for the steady state assumption. It is now also necessary to verify the gain choice validity for transient state by simulation and experimental trials. We now propose a new way to describe the experimental and simulated system in order to precisely define the experimental and simulated conditions.

2 Description of the experimental and simulated system 2.1 Electrotechnical system functional analysis In order to control the induction machine speed, the system is built along two axes: the power axis and the control axis [2] (Fig. 5). The power axis contains the power source (a three phase rectifier), the static converter (a three phase inverter



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Fig. 4. Modulus and orientation errors with a rotor resistance error equal to 33% (nominal flux, Te = 0.8 ms).

based on IGBT transistors) and the load (a three kilowatt and four poles induction machine loaded by a direct current machine). The control axis is described by its different global functions: the Commutation Control Block (CCB) generates the gate signals to the semiconductors and assures their protection according to their specifications and to the connections rules; the Near Control Automaton (NCA) performs the logic states from the output of the Process Control Automaton (PCA), for example pulse width modulation; the PCA executes the algorithm which controls the power transfer from the source to the load; finally the System Mode Control Automaton (SMCA) oversees the process. All these block works with information about the system, which has been brought by the observation bus.

2.2 Process modelling methodology The process, naturally heterogeneous, is modelled by using a formalism, which is founded on the previous functional decomposition [3]. The power axis is naturally divided between a control part and a processing part (Fig. 6 presents this as applied to the studied system). This concept is in fact induced by the multiplicity of solutions, which leads to the creation of connections between various elements of a circuit making use of different semiconductor switch associations. The control part results from the choice of the converter structure. In all cases, it is modelled by means of a status Petri Net which explicitly shows how connections are linked together in the processing part. In the case under study the switches are considered totally controllable: the internal electrical variable does not influence the connections if the switches are considered as perfect. So the control part is reduced to three objects: the applications

30

The European Physical Journal Applied Physics Control Axis

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Fig. 5. Electrotechnical system functional analysis.

Fig. 6. Knowledge modelling of power axis.

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which transform the connection matrix F to the voltage MV or current MI conversion matrix and the relation between the gate voltages and the connection matrix F which is a constant gain because the switches are assumed to be perfect. The connection matrix F contains the connection functions [2,5]: • fsc = 1 when the switch is closed; • fsc = 0 when the switch is opened. The index s indicates the switch number in the commutation cell index c. These functions thus allow to write the equations between the switch electrical variables and the electrical sources, which are connected to this switch. The structure is characterised by conversion matrices whose elements, designated conversion functions, are quantities that control the transfer of energy. The processing part is divided into two blocks: a continuous block representing state equations (load and source equations) and a discontinuous block representing the connections effectively made by the power switches. Figure 6 shows these parts for the system under study. The switched quantities are the induction machine stator currents (Ias and Ibs ) and the rectifier voltage E. The modulated quantities are two line voltages (uac and ubc ) and the rectifier current Ie . The output variables are on one hand the mechanical speed Ω, the stator currents (Ias and Ibs ) and the rotor flux ϕdr of the induction machine, and on the other hand the inverter voltage E. External quantities are represented by the resistant torque cr and the current Id which is delivered by the rectified network. The scheme is transformed into a Causal Informational Graph (CIG) [2]: its greatest advantages are that all the relations between the variables are represented, and that the inversion of the graph gives the different functions of the command axis. This graph also represents all the relations between outputs and inputs by displaying the causal properties. The outputs and inputs are defined. The relations of the Causal Informational Graph are now written.

2.3 Knowledge modelling This part deals with the equations of the three components of the power axis: the static converter, the source and the load.

2.3.1 Static converter The converter makes the connections between the source and the load. Therefore the connection functions theory is used. In the case under study, the following relation is obtained (Fig. 7):   iaS ie   = F  ibS  (9) −ie icS with

F=

f11 f12 f13

f21 f22 f23

F is called the connection matrix. The stator currents sum is null. So we have: iaS ie = MI (10) ibS with

RFmi :

MI = (f11 − f13

f12 − f13 )

MI is called the current conversion matrix. As the converter does not consume power, it was proven that [4]: RX : with

RFMv :

uac ubc

= MV E

MV =

f11 − f13 f12 − f13

MV is called the voltage conversion matrix. All these equations are discrete.

(11)

32


2.3.2 Source The current source iD symbolises the inductive nature of network, which is not modified by the rectifier (Fig. 8). The modelling equations are:    1    0 −R C  vC1  d vC1   1 1   =    dt vC2 1  vC2   0 −    R 2 C2    RSource : 1   C 0    1  (i − ie )  +   1  D    0   C2    E = vC1 + vC2 . (12) 2.3.3 Load The load is made up of the induction machine and the direct current machine. In a two- phase reference frame characterised by θS and θR (Fig. 1 of the previous paper), the induction motor equation can be written (Appendix A): d φd dθS π φd Id Vd = RS + + P Vq S Iq S dt φq S dt 2 φq S

This transform conserves power. Therefore it is possible to write the torque expressions as a function of rotor flux: Cem = p

with

φd Id Id = LCS + MC φq S Iq S Iq R

Cem = p

θS = θR + pθ .

The Park transforms are defined by: uac Vd RPark : = T22 P(θS ) ubc Vq S 1 0 −1 with T22 = 0 1 −1 T32 r √ 2 1/2 3/2 √ = 0 3 S 3 0 RPark :

with

Ia Id 0 = T22 P(θS ) Ib S Iq S 100 T022 = 0 1 0 T32 r 2 1 √0 = · 3 −1/2 3/2

MC φdR IqS . LCR

(18)

This choice involves simplification of the induction machines equations: LCS dIdS = VdS − RS + (1 − σ) RR IdS R1 : σLCS dt LCR (19) LCS φdR dθS − (1 − σ) − σLCS IqS MC T R dt

R3 :

σLCS

dIqS LCS = VqS − RS + (1 − σ) RR IqS dt LCR (20) LCS dθS pΩφdR + σLCS IdS − (1 − σ) MC dt

R2 :

TR

dφdR = MC IdS − φdR dt

(21)

where TR = LCR /RR is the rotor time constant

φd Id Id = LCR + MC φq R Iq R Iq S and

(17)

With the application of the vector control principle, the above equations are reduced in the rotor flux frame φqR = 0:

d φd dθR π φd 0 Id = RR + + P Iq R dt φq R dt 2 φq R 0 R (13)

MC (φdR IqS − φqR IdS ) . LCR

MC φdR IqS LCR

R4 :

Cem = p

R5 :

dθR MC IqS = · dt TR φdR

(22)

(23)

(14)

The direct current machine, which is used as generator mostly, establishes a torque that depends linearly on speed. Therefore the following mechanical equation is con(15) sidered where Cr represents a perturbation torque (dry friction) and Ω is the rotor speed: dΩ = Cem − f Ω − Cr dt J : inertia moment F : viscous friction coefficient. R6 : J

(24)

All these relations (source, static converter and load) are (16) now included in a global informational scheme. 2.3.4 Causal Informational Graphs of the power axis This first scheme (Fig. 6) is translated into a Causal Informational Graph (Fig. 9).


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The relations R1 , R2 , R3 , R4 , R5 and R6 were described above in the load modelling. The relation Rsource sums up the source modelling equations (Eq. (12)). RPark 0 and RPark symbolise the special Park transforms defined in the induction machine modelling (Appendix A respectively Eqs. (A4) and (A5)). The relations RFMv and RFmi are the definitions of, respectively, voltage (Eq. (11)) and current (Eq. (10)) conversion matrix. The relation RX is a multiplier. Relations with a double arrow represent rigid and reversible relations, i.e. independent of time, and single arrow relations represent causal and irreversible relations, i.e. time dependent. It may be noted that the gate voltages control two distinct branches: the load branch and the source branch. In fact, in the case of controlling the load (respectively source) outputs (speed for example), the voltage (respectively current) conversion matrix is required. For vector control, the load outputs, i.e. the voltage conversion matrix, are required. This knowledge modelling is well adapted for the simulation, but the presence of both discrete and continuous variable makes control design difficult. A solution is to introduce some average variables known as average

functions [3]:

RFg :

   f11g (k) =      f12g (k) =        f13g (k) =

1 R (k+1)Te f11 (τ )dτ Te k Te 1 R (k+1)Te f12 (τ )dτ Te k Te 1 R (k+1)Te f13 (τ )dτ Te k Te

RFgMvg : RFgMig :

MV g = MIg = (f11g − f13g

f11g − f13g

(25)

f12g − f13g

(26)

f12g − f13g ) .

(27)

Introducing these functions in the Causal Informational Graph model of Figure 9, Figure 10 is obtained. The relation RFg defines a demodulator, which allows the change from discrete functions to continuous functions. The relation RFgMvg and RFgMig respectively correspond to RFMv and RFmi . They transform the generator connection matrix into a generator conversion matrix.

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2.4 Control definition The control axis organisation can now be deduced by reversing the causal informational graph of the load branch. The global strategy of vector control requires the rotor flux to be held constant in order to obtain a better dynamic response for the speed. So two variables are imposed by the command: the reference rotor flux and the reference speed. The application of reversibility rules on the causal graph leads to Figure 11. The subscript symbolises a reference variable. Only six relations are directly reversed because they −1 −1 −1 0 are rigid relations: RPark , R4−1 , RPark , RX , RFgMvg , −1 and RGate . As the other load relations are causal, four new relations must be introduced: R20 , R10 , R60 and R30 (Appendix B). It is respectively the flux corrector, the Id current corrector, the speed corrector and the Iq current corrector, which require measurement of the regulated variable. Therefore four sensor appear: the flux sensor RSφ , two current sensors grouped in the relation RSI and the speed sensor RSΩ . In order to reconstitute the Id and Iq currents from the three-phase currents, it is necessary to build an angle θS sensor: RSθ . The index m shows that a variable is either measured or estimated. Finally the 0 modulator RFg is used in order to generate the reference connections matrix. The modulator is based on a pulse width modulation [4]. The matrix, which is represented by the relation RFgMvg , is not a regular matrix. Therefore it is not mathematically invertible but the relation is reversible. So the −1 relation RFgMvg has many expressions [5] but only one

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property: the generator functions must have their values between 0 and 1 in order to keep the system linear. One possibility is:   1 −1/2 1 2  (28) Fgref = +  −1/2 1  MVgref . 2 3 −1/2 −1/2 The flux corrector and speed corrector are so-called integral proportional correctors and the current correctors are so-called proportional integral correctors. The corrector coefficients as well as the sampling rate are given in Appendix B. 2.5 Hardware implementation The purpose of the following step is to make concretise all these relations, i.e. to choose the hardware in experimenting on the control and to compare this with the simulation, which is performed from the same scheme [6]. From the graph of Figure 11, it is necessary to know the actual rotor flux value. As it can not be measured, the previously developed observer will be useful. Because of the calculations complexity, a Digital Signal Processor (DSP) is necessary and specifically the DSP96002 by Motorola is chosen. It is implanted on a card inserted in a personal computer. The DSP capacities allow not only the observer (relations RSφ and RSθ ) but also the algorithmic command (relations R0 1 , R0 2 , R0 3 , R4−1 , −1 0 R60 , RPark and RPark ), the dynamic linearisation (relation


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