Constrained Minimization for Parameter Estimation of Induction Motors ...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 52, NO. 5, OCTOBER 2005

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Constrained Minimization for Parameter Estimation of Induction Motors in Saturated and Unsaturated Conditions Maurizio Cirrincione, Member, IEEE, Marcello Pucci, Member, IEEE, Giansalvo Cirrincione, Member, IEEE, and Gérard-André Capolino, Fellow, IEEE

Quadrature component of the stator current in the rotor-flux oriented reference frame. Space vector of the rotor magnetising currents in the rotor-flux oriented reference frame. Space vector of the rotor currents expressed in the general reference frame. Space vector of the rotor currents in the stator reference frame. Space vector of the stator flux linkages in the general reference frame. Space vector of the stator flux linkages in the stator reference frame. Space vector of the rotor flux linkages in the general reference frame. Space vector of the rotor flux linkages in the stator reference frame. Stator inductance. Rotor inductance. Total static magnetising inductance. Total dynamic magnetising inductance. Rotor leakage inductance. Resistance of a stator phase winding. Resistance of a rotor phase winding. ; transient time constant of the machine. Stator time constant. Modified rotor time constant. ; total leakage factor. Stator leakage factor. Rotor leakage factor. Rotor inertia. Mechanical damping factor. Number of pole pairs. Angular rotor speed of the general reference frame. Angular rotor speed (in mechanical angles per second). Angular rotor speed (in electrical angles per second). ; inverse of the rotor time constant . . Electromagnetic torque. Load torque.

Abstract—This paper presents the analytical solution of the application of the constrained least-squares (LS) minimization to the online parameter estimation of induction machines. This constrained minimization is derived from the classical linear dynamical model of the induction machine, and therefore it is able to estimated the steady-state value of the electrical parameters of the induction motor under different magnetization levels. The methodology has been verified in simulation with a dynamical model which takes into account iron path saturation effects. After a description of the experimental setup and its signal processing systems, the methodology is verified experimentally under saturated and unsaturated working conditions, and the results are discussed and compared to those obtained with a classical unconstrained ordinary LS technique. Index Terms—Constrained minimization, identification, induction motor drives, least-squares (LS), parameter estimation.

NOMENCLATURE

,

,

Space vector of the stator voltages in the general reference frame. Space vector of the stator voltages in the stator reference frame. Space vector of the stator voltages in the rotor-flux oriented reference frame. Direct and quadrature components of the stator voltages in the stator reference frame. Space vector of the rotor voltages in the general reference frame. Space vector of the rotor voltages in the stator reference frame. Space vector of the stator currents in the general reference frame. Space vector of the stator currents in the stator reference frame. Space vector of the stator currents in the rotor-flux oriented reference frame. Direct and quadrature components of the stator currents in the stator reference frame.

Manuscript received January 15, 2003; revised April 15, 2005. Abstract published on the Internet July 15, 2005. M. Cirrincione and M. Pucci are with the ISSIA-CNR Section of Palermo, Institute on Intelligent Systems for the Automation, 90128 Palermo, Italy (e-mail: [email protected]; [email protected]). G. Cirrincione and G.-A. Capolino are with the Department of Electrical Engineering, University of Picardie-Jules Verne, 80039 Amiens, France (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TIE.2005.855657

I. INTRODUCTION

I

T IS well known that present high-performance induction motor drives, e.g., field-oriented control (FOC) and directtorque control (DTC), suffer from the uncertainty of the knowl-

0278-0046/$20.00 © 2005 IEEE

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edge of electrical parameters, which results in a decrease of control performances. Thus, it is necessary not only to know these parameters before the operation of the drive to fine tune the controllers (self-commissioning), but also to track their variation during normal conditions (online tracking). A great deal of work has attacked the first problem: in [1]–[3], automated procedures are proposed that can be simply implemented on a microprocessor and are capable of retrieving the motor parameters by separate purposely designed methods for each of them, without tracking them, however. In this respect, [4] not only proposes a self-commissioning scheme but proposes also a method for the online identification of the rotor time constant in an FOC drive by computing the active and reactive power of the motor. Other methods instead rely on the flux estimation to retrieve the motor parameters, like the extended Luenberger observer (ELO) [5], extended Kalman filter (EKF) [6], [7], and model reference adaptive systems (MRAS) [8]–[11]. As recalled in [12] about the complexity of the algorithms, the drawback of EKF lies in the difficulty of properly selecting the noise covariance matrices and the initial values of the algorithm, which can easily lead to instability. As for MRAS systems, the main problem lies in the choice of those parameters of the adaptation law which guarantee global stability. As for ELO, the main drawback lies in the correct choice of the gain matrix. With regard to the computational time required by these algorithms, EKF can be cumbersome in its implementation on a digital signal processor (DSP) (even a matrix inversion is required), and, moreover, its performance decreases with the increase of the sampling time of the control system, while ELO and MRAS systems are less demanding in terms of computational time. In any case, in all of the above algorithms, the parameter estimation is performed at the same time as the flux estimation, which means that they are more useful when flux estimation is also needed (flux feedback-controlled drives) and that they must update the parameter estimation with the same time rate as that of the flux estimation. There are other methodologies, however, which permit the parameter estimation to be performed independently from the flux estimation and can therefore be suitably employed both in grid-supplied machines and in converter-fed drives, working at updating times higher than the sampling time required by the control system (electrical parameter variations are slow compared to the variations of other electrical variables of the machine). The parameter estimation methodology presented here belongs to this framework. For online estimation of slowly varying parameters, as in the case of induction motors, the method of least-squares (LS) is certainly one of the most suitable both for its simplicity and its low computational burden. In [4], a LS method was presented for the self-commissioning phase. In [13]–[19], an LS approach was developed for the first time to estimate all of the electrical parameters on the basis of the stator currents, voltages, and rotational speed. These methods have always presented some difficulties, because auxiliary variables and constraints are to be introduced for retrieving the physical parameters, and, unfortunately, not all of them are easy to detect. The identification problem with LS algorithms is usually solved by means of an unconstrained minimization of the 2-norm of the error with a simple gradient descent algorithm [11] to [19]. This approach

fails in computing one of the parameters ( , see Section III for more details) because of the particular structure of the data problem) and because of matrix which is ill-conditioned ( the flatness of the LS error surface along one direction (see [20, Fig. 5]). Moreover, numerical scaling does not help in solving this problem. Only in [18] has a more complete method been developed to overcome this problem; however, no proof of converproblem even causes the paramgence is given. In [21], the eter estimation procedure to be split into two different phases, the first of which is focused essentially on the estimation of the parameter. In [22], a follow-up of the method of [18] is presented to test its validity under different working conditions, requiring, however, the a priori knowledge of the stator resis-problem. In [23], a total LS method tance, thus avoiding the has been used for the estimation of the electrical parameters of the motor, but it gives accurate results only for one parameter in accordance with the operating conditions. After describing the parameter estimation trough LS constrained minimization, the paper gives an analytical solution to this problem and verifies this method experimentally. II. EQUATION OF THE INDUCTION MACHINE FOR THE APPLICATION OF LS The employment of any LS technique for real-time identification of induction machines requires the mathematical model of the machine itself to be rearranged. It is well known that the induction motor model can be described by the following stator and rotor space-vector voltage equations in a general reference frame, which rotates at a given speed [10]

(1) In the above equations, the different electrical parameters could be constant or time- and space-varying depending on preliminary assumptions. By eliminating the space vector of the rotor currents, which are not measurable, and expressing the direct and quadrature components in the stator reference frame, the following matrix is obtained under the assumption ) and by defining the of slowly varying speed ( following parameters, called -parameters:

(2)

CIRRINCIONE et al.: CONSTRAINED MINIMIZATION FOR PARAMETER ESTIMATION OF INDUCTION MOTORS

and

(3) Between the

- parameters, the following relationship exists: (4)

From the -parameters, not all of the five electrical param, , , and ) can be retrieved as no rotor eters ( , measurements are available. In fact, the -parameters deter, mine only four independent electrical parameters, that is, , , and , in the following way:

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critical, which is confirmed This makes the estimation of problem). Two paths can be then followed to in [13]–[19] ( overcome this difficulty. The former employs an unconstrained minimization which takes into account the constraint indirectly: it uses in fact the relationships (5) without using the computed . This is the method followed by [18], but no proof regarding the convergence of the algorithm has been shown. Also, in [19], an LS constrained minimization has been suggested, but too many constraints are present due to the high order of the differential equations derived from (3). This makes the method unsuitable for real-world applications. This paper instead presents a constrained analytical minimization which also overcomes the problem. This method is fully explained in the following. III. CONSTRAINED MINIMIZATION: ANALYTICAL SOLUTION In the upper equation of (6), the vector is computed by an OLS algorithm, which means that the following function error should be minimized:

(5a) (7a) (5b)

where

The subscripts and in (3) refer, respectively, to the component along the axis (direct axis) and the axis (quadrature axis ) of the stationary reference frame fixed to the stator. Matrix (3) together with the constraint (4) can be written in the following form:

is the autocorrelation matrix (5 5), while is the mutual correlation vector ( ). The presence of the lower equation of (6) implies that a constrained LS minimization should be performed, which can be achieved by following two paths. The first reduces the equations into a canonical form, which permits a geometrical insight into the problem, and then uses a Lagrangian optimization technique [24]. The second is explained here and is simpler to deduce, however, it does not convey the same geometrical meaning. the total cost funcLet be the Lagrangian multiplier and tion to be minimized, defined as

(6)

(7b)

(5c) (5d)

The gradient of this cost function is given by is the constraint function (4), where is an 5 where is the vector matrix (data matrix), 1 vector (observation of the five unknowns, and is the vector). The first equation in (6) can be solved for the -parameters both in steady-state and transient in real time by using an ordinary least-squares (OLS) method, because the main cause of errors, i.e., the modeling error due to the assumption of slowly varying speed and the second derivatives are present in the observation vector. The drawback, as pointed out in [13]–[19], is that a suitable signal processing system must be designed which employs voltage and current sensors, analog low-pass filters, digital low-pass filters, and differentiator filters. By inspection of (3), it is easy to recognize that the second column of the matrix has lower values than those of the other columns. This means two things. 1) A gradient descent method is unsuitable as it is too slow to converge. 2) If a constrained minimization is used, it is intuitive that the difference between the true value, that is, the solution satisfying the constraints, and the 2-norm solution, that is, the unconstrained minimum, can be quite apart from each other along the direction of minimum gradient ( ).

(8)

If the following matrix

is introduced:

(9)

then (8) can be written as (10) which implies

or (11)

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If the value given in (11) is substituted into (8), a scalar equation in the unknown is obtained, which can be easily solved for with a nonlinear numerical method, i.e., the Newton–Raphson method, the bisection method, or even with the old “regula falsi,” a variation of the secant method [25, p. 338]. The vector is eventually obtained with (11). As for the error estimate, the same considerations made in [18] with respect to the unconstrained minimization can be made. If the unconstrained minimization is used, [18, (42)] can be applied, i.e.,

ical behavior of the induction motor in saturated condition, are [10]

(12)

(13) where (parametric error index) indicates the amount by which the th component of could vary without causing , i.e., more than a doubling of the minimum residual error the residual error computed with the solution of the unconindicates the th (diagonal) strained least square method; element of the matrix inside the brackets. From (11), a similar formula can be obtained for the constrained case, i.e.,

(14) with (15) (16) (17)

where (parametric error index) indicates the amount by which the th component of could vary without causing more than a doubling of the residual error (parametric error index), defined as the value obtained by when is obtained and by (11), i.e., the minimum residual error with a constraint. From the definition of the matrix , it follows that

For the symbols, see the Nomenclature. The effect of the magnetic saturation has been taken into account by look-up tables (LUTs) (with linear interpolation) which give the nonlinear re, , , , and lationship between the electrical parameters and the rotor magnetizing current space vector , as shown in Fig. 1. V. ESTIMATION OF THE MAGNETIZATION CURVE

and, since , then it results that . This means that, with the constrained minimization, the parametric error index diminishes. A large parametric error index indicates less accuracy in the results [18]. Thus, with constrained minimization, the accuracy of the results is more than in the case of the unconstrained minimization. In any case, it should be remarked that, with this OLS approach, the errors are assumed to be confined mainly in the observation vector and because the assumption of the slowly varying speed affects the oservation vector. This assumption is in a way acceptable because there exist second derivatives in the observation vector. If the uncertainty in the data matrix is also to be accounted for, then a constrained total least squares (TLS) [20] technique should be used, as, e.g., in the case of current and voltages affected by noise. IV. MATHEMATICAL DYNAMICAL MODEL OF THE INDUCTION MOTOR TAKING INTO ACCOUNT THE MAGNETIC SATURATION This section shows the mathematical dynamical model of the induction motor which takes into account the magnetic saturation of the iron paths. This model has been used to simulate the dynamical behavior of a real induction motor employed as the machine under test. It is assumed in this model that the effects of the leakage flux saturation are neglected. The well-known space-vector equations, expressed in the rotor-flux oriented reference frame, which describe the dynam-

A direct consequence of the retrieval of the four electrical parameters is the possibility to estimate the magnetizing curve. For this purpose some tests should be made at different magnitudes of the supply voltage and at the same supply frequency in order to make the machine work under different magnetizing excitations. At each supply voltage after a speed transient from zero to steady-state speed (start-up test) the electrical parameters rotor time constant , stator inductance , stator resisand global leakage factor are estimated by means of tance the constrained algorithm. At the same time the rotor magneis computed by means of the well known flux tizing current model based on the rotor equations of the induction motor in the rotor flux reference frame [10], [11]. It should be remarked that, because the start-up tests are made at no-load, the steady-state does not suffer from any inaccurate value of the estimated parameter needed by the flux estimator. In fact, guess in the the ratio between the true and the estimated rotor-flux linkage is given by the following expression [26]:

(18) is the slip where the “ ” stands for estimated variables and frequency, depending on the load torque. From (18) and from Fig. 2, where this equation is plotted for different values of , it


Fig. 1.

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Block diagram of the mathematical model of the induction machine including iron path saturation effects.

However, it should be noticed that only the global leakage factor can be estimated and not the stator or rotor leakage factors individually. Consequently, the estimation of the static magnetizing inductance has been made under the usual assumption [27], [28]. Under this assumption, and are that computed from by [27], [28] (19a) (19b)

Fig. 2. Sensitivity of the flux model versus slip parameterized in T .

Therefore, under each magnetic excitation, the total magneis computed from the estimated values of tizing inductance and as (20)

is apparent that the magnetizing current estimation is indepenused in the flux model at no-load dent from the parameter (zero slip frequency), as in the case under test. and , is estiAfterwards, from computed values of mated for each operating point as explained below. Finally, at the end of the tests at different voltage levels, the magnetization curve is estimated.

Finally, the magnetization curve of the machine, which gives the nonlinear relationship between the rotor magnetizing current and the rotor magnetic flux , is computed as (21)

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Fig. 4. Frequency response of the low-pass analog Bessel filter.

Fig. 3.

Block diagram of the signal processing system.

VI. SIGNAL PROCESSING SYSTEM The parameter estimation algorithm employs the signals of the stator voltages and currents, their derivatives (up to the second order for the current and the first order for the voltage), and the rotor speed. Since the motor can be supplied by both the electric grid and a VSI inverter, filters for stator voltage and current signals are needed. The presence of filters, however, causes time delays of the processed signals which, therefore, at each time instant, should be synchronized with one another in order to satisfy the dynamic equation of the induction machine. Fig. 3 shows the complete scheme used to process all of the signals used by the identification algorithm. In the signal processing system, the following filters have been designed to process the voltage and current signals correctly: • Four analog low-pass anti-aliasing Bessel filters of fourth block in order with linear phase characteristic ( Fig. 3) and cut-off frequency of 800 Hz which filter the stator voltage and current signals from the voltage and current sensors in the drive. • four digital finite-impulse-response (FIR) low-pass filters block in Fig. 3) and cut-off frequency of 20th order ( of 200 Hz reducing high-order harmonics and the noise

of the stator voltage and current signals which can be amplified by the differentiator filters which follow. Digital low-pass FIR filters have been chosen to perform this last task, as FIR filters can be easily designed with an exactly linear phase characteristic, i.e., with exactly constant group delay. • Six digital FIR differentiator filters of tenth order ( block in Fig. 3) for obtaining the derivatives of the stator voltages (up to the first order) and currents (up to the second order) of the drive. FIR filters have been chosen again for their characteristic of having an exactly linear phase diagram. As all of the signals processed by the estimation algorithm must be synchronized, whenever a stator voltage or current signal is processed by the differentiator filter, the other signals, which are not differentiated and are used in the identification algorithm, must be delayed in time with the group delay of the differentiator filter. In the experimental application, all digital low-pass and differentiator filters have been implemented on a DSP, along with the identification algorithm, while an electronic board with six low-pass fourth-order Bessel filters has been designed and built. In particular, Fig. 4 shows the frequency response of the Bessel filter board which has been built together with the frequency response of the ideal filter. Figs. 5 and 6 show the frequency response of the digital low-pass FIR filter and of the differentiator FIR filter which have been implemented on the DSP. It should be remarked that, even if differentiator filters tend to amplify high-frequency content of noise, the low-pass FIR filter cuts off any harmonic component of both voltages and current signals above 200 Hz. Moreover, because of the linearity of the phase characteristics of these filters and of the antialiasing analog Bessel ones, the input signals to the identification algorithm are not distorted. VII. DESCRIPTION OF THE TEST BENCH The LS methodologies for the estimation of the electrical parameters of an induction motor have been tested experimentally.


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TABLE I PARAMETERS OF THE INDUCTION MOTOR

Fig. 5.

Frequency response of the low-pass FIR filter.

permits the instantaneous value of the dc link voltage to be taken into account for the modulation. In particular, the employed current sensors present an accuracy of 0.65% with a linearity error less than 0.15% and with a 1-dB bandwidth of 200 kHz, while the voltage sensors present an accuracy of 0.9% with a linearity error less than 0.2% and with a response time of 400 s. Moreover, the six analog signals have been acquired with six 16-b analog–digital converters (ADCs), multiplexed in groups of three, with 1 s sampling time for each channel; therefore, the associated quantization error, where is the number of bits of the evaluated as ADC, is 7.6 . On the basis of the above, the quantization error is negligible in comparison with the transducer errors; in particular, the global error associated with the acquisition of the current signals is 0.800 76% and the corresponding one associated with the voltage signals is 1.100 76%. VIII. SIMULATION RESULTS

Fig. 6. Frequency response of the differentiator FIR filter.

A test bench has been built for this purpose, which consists of [29] the following: • A three-phase induction motor with rated values shown in Table I. • An electronic power converter (three-phase diode rectifier and VSI composed of three IGBT modules without any control system) of rated power 7.5 kVA. • An electronic card with voltage sensors (model LEM LV 25-P) and current sensors (model LEM LA 55-P) for monitoring the instantaneous values of the stator phase voltages and currents. • A voltage sensor (Model LEM CV3-1000) for monitoring the instantaneous value of the dc link voltage. • An electronic card with analog fourth-order low-pass Bessel filters and cut-off frequency of 800 Hz. • An incremental encoder (model RS 256-499, 2500 pulses per round). • A dSPACE card (model DS1103) with a floating-point DSP. The VSI is driven by an asynchronous space vector modulakHz) implemented by tion technique (switching frequency software on the dSPACE card, while the dc link voltage sensor

The proposed methodology has been tested in simulation by using the above dynamical model where the nonlinear relation, , , , and ship between the electrical parameters and the rotor magnetizing current have been obtained experimentally [27] on the real induction motor (see Section VII). At first, a comparison has been made between the estimation of the electrical and magnetic parameters obtained with the proposed algorithm (constrained minimization) and a classical ordinary recurrent LS algorithm (RLS), like the one already implemented by the authors in [13] and [17]. In particular, various tests have been performed for verifying the capability of the parameter estimation algorithm to work correctlyunderbothslowandfastspeedtransientsandunderdifferent steady-state magnetization levels. Different steady-state magnetizationlevelsinfactcorrespondtodifferentvaluesofthemagnetic parameters of the machine. Since the equations employed for the identification model come from the linear mathematical model of the induction machine which do not take into consideration the effects of magnetic saturation of the iron path, the constrained minimization algorithm can compute only the steady-state values of the parameters themselves, that is, the values corresponding to the steady-state magnetization of the machine, as highlighted in [13]. Thismeans,therefore,thatnocorrectestimationof theparameters is possible during the speed transient. For this purpose, a set of start-up tests has been done under different voltage levels at a frequency of 50 Hz to make the machine work under different steady-state magnetization excitaand . Correspondingly, tions, i.e., with different values of

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TABLE II ESTIMATION ERRORS OBTAINED WITH THE RLS

TABLE III ESTIMATION ERRORS OBTAINED WITH THE CONSTRAINED Fig. 7. Rotor speed and stator current at start-ups under a 155-V, 50-Hz supply (simulation).

Fig. 8. Estimated and real electrical parameters of the machine under a 155-V, 50-Hz supply (simulation).

both slow and fast speed transients have been created by feeding, respectively, with low-voltage/frequency and high-voltage/frequency ratios. At the end of each test, the four electrical parameters , , , and have been retrieved with both methods. current waveFig. 7 shows the rotor speed and the stator forms, during a start-up test under a 155-V and 50-Hz supply. Fig. 8 shows the corresponding curves of the estimated electrical parameters in comparison with the real ones, obtained with the constrained minimization. It shows that, as obviously expected, the magnetic parameters of the machine ( , , and ) vary during the speed transient of the machine from zero to steady-state speed because of the magnetization of the machine. Tables II and III show the relative error of the estimations obtained with the RLS method, and the constrained minimization algorithm referred to the true values of the parameters given by the LUTs of the model described in Section IV. Fig. 9 summarizes the results for each voltage level in the form of histograms and clearly shows that, except for low voltage levels, the results obtained with the constrained minimization method are superior to those obtained with the classical unconstrained LS method. The variation of the estimation error according to the load torque has also been investigated. Fig. 10 shows the percent es-

Fig. 9. Percent estimation errors on the electrical parameters (simulation).

timation error of every -parameter under different load conditions ranging from no load to rated load. It can be observed that and have a low percent error which is almost inboth and are heavily dependent from load conditions while load-dependent and have higher percent errors. An explanation can be given by considering that the columns of the data matrix and have a magnitude which is much corresponding to and . Moreover, the higher than those corresponding to are higher than those in as the corresponding errors in column of the data matrix of this last parameter is dependent on the load: the higher the load, the higher the stator current, the higher the value of the corresponding column in the data maparameter than the one trix, and the lower the error of the of the parameter.


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Fig. 10. Percent estimation errors on the k -parameters according to the variation of the load (simulation). Fig. 13. Estimated and no-load and locked-rotor tests electrical parameters of the machine under a 55-V, 50-Hz supply (experiment).

Fig. 11. Rotor speed and stator current during start-up tests under, respectively, 55-V and 105-V, 50-Hz supply (experiment). Fig. 14. Estimated and no-load and locked-rotor tests electrical parameters of the machine under a 105-V, 50-Hz supply (experiment).

Fig. 12. Rotor speed and stator current during start-up tests under, respectively, 155- and 220-V, 50-Hz supply (experiment).

IX. EXPERIMENTAL RESULTS The presented methodology has been verified experimentally on the test setup described in Section VII. Table I shows the nameplate data and the electrical parameters of the employed induction machine, obtained with the no-load and locked rotor

tests. As explained in Section V, a series of start-up tests have been made under different voltage levels at the frequency of 50 Hz with no-load, to make the machine work under different steady-state magnetization excitation, i.e., with different values and (the same kind of tests as in Section V). Figs. 11 of and 12 show the rotor speed and the stator current waveforms, during four start-up tests, respectively, under a 55-, 105-, 155-, 220-V and 50-Hz supply. Figs. 13–16 show the corresponding curves of the estimated electrical parameters in comparison with those obtained with the usual no-load and locked-rotor tests. It should be noted that, depending on the steady-state magnetization level of the machine, the estimated electrical parameters, after convergence, can be either closer or not to the values of the parameters obtained with the no-load and locked-rotor tests. For example, the steady-state estimation obtained under 220-V, 50-Hz supply is closer to the of obtained with the no-load test than the estimation of at the same supply conditions to the obtained with the locked-rotor test. This is easily explained by the fact that, in computing with the locked-rotor test, the supply voltage is reduced to ensure rated current, with resulting unsaturated working the no-load test is conditions, while for the estimation of

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Fig. 15. Estimated and no-load and locked-rotor tests electrical parameters of the machine under a 155-V, 50-Hz supply (experiment).

Fig. 16. Estimated and no-load and locked-rotor tests electrical parameters of the machine under a 220-V, 50-Hz supply (experiment).

Fig. 18. Estimated T versus i interpolating curve (experiment).

: experimental results and corresponding

Fig. 19. Estimated , and versus i : experimental results and corresponding interpolating curves (experiment).

Fig. 17. Estimated L and L versus i : experimental results and corresponding interpolating curves (experiment).

Fig. 20. Estimated magnetization curve: experimental corresponding interpolating curve (experiment).

employed, which is carried out under a condition similar to this experiment, which is also at no-load [4]. Several more tests have been made under different voltage levels at the frequency of 50 Hz by employing (16)–(18), so that the curves in Figs. 17–20 , have been obtained, which represent the variation of ,

, , , , and , respectively, as a function of . Each set of experimental data of each electrical parameter has been , then interpolated with a polynomial curve. In particular, , and and the magnetization curves have been interpolated with a third-order polynomial while the , , and

results

and


curves have been interpolated with a fifth-order polynomial, as shown in figures. X. CONCLUSION This paper presents an analytical method for the retrieval of the four electrical parameters of an induction motor by a constrained minimization. The proposed methodology has been derived from the classical linear dynamical model of the induction machine which permits the four electrical parameters to be computed in the steady state. This means that, although the parameters can vary their values according to the temperature and the magnetization conditions, the method is able to compute their values correctly in the steady state. To assess the method, first a series of tests have been made in simulation to retrieve the electrical parameters of an induction motor simulated with its nonlinear model which takes the saturation effects into account. Then, the experimental verification of this methodology has been performed on a real low-power induction machine: a series of tests has been made under different voltage/frequency ratios to have different steady-state magnetization conditions and correspondingly slow and fast speed transients. Moreover, the steady-state values of the electrical parameters of the machine as a function of the rotor magnetizing current and the corresponding magnetization curve have been computed. In the end, a comparison has been made with the results obtained with the classical minimization technique. It can be concluded that the employment of the constrained minimization algorithm permits the -parameters to be better estimated than in the case of the use of an unconstrained classical LS method. The proposed methodology in general offers the following advantages: • the possibility to work with the motor supplied either by the sinusoidal voltage waveform from the electric grid or by the voltage waveform generated by a converter; • no need for an a priori knowledge of the electrical parameters or all of the nameplate data of the machine (except for the rated voltage and frequency). • the simultaneous estimation of the four electrical parameters; Work is in progress regarding the use of the TLS method to increase the robustness of the estimated parameters to the unavoidable errors present both in the observation vector and in the data matrix. REFERENCES [1] W. Leonhard and G. Heinemann, “Self-tuning field oriented control of an induction motor drive,” in Proc. Int. Power Electronics Conf., Tokyo, Japan, 1990, pp. 465–472. [2] A. M. Khambadkone and J. Holtz, “Vector-controlled induction motor drive with self-commissioning scheme,” IEEE Trans. Ind. Electron., vol. 38, no. 5, pp. 322–327, Oct. 1991. [3] T. Kudor, K. Ishihara, and H. Naitoh, “Self-commissioning for vector controlled induction motors,” in Conf. Rec. IEEE-IAS Annu. Meeting , Oct. 1993, pp. 528–535. [4] M. Sumner and G. M. Asher, “Autocommissioning for voltage-referenced voltage-fed vector-controlled induction motor drives,” Proc. Inst. Elect. Eng., vol. 140, no. 3, May 1993. [5] H. Kubota, K. Matsuse, and T. Nakano, “DSP-based speed adaptive flux observer of induction motor,” IEEE Trans. Ind. Appl., vol. 29, no. 2, pp. 344–348, Mar./Apr. 1993.

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[6] A. Dell’Aquila, F. Cupertino, L. Salvatore, and S. Stasi, “Kalman filter estimators applied to robust control of induction motor drives,” in Proc. IEEE IECON’98, vol. 2, Aug. 31–Sep. 4, 1998, pp. 2257–2262. [7] S. Stasi, L. Salvatore, and F. Cupertino, “Comparison between adaptive flux observer- and extended Kalman filter-based algorithms for field oriented control of induction motor drives,” in Proc. Eur. Conf. Power Electronics and Applications, 1999. [8] D. J. Atkinson, J. W. Finch, and P. P. Acarnley, “Estimation of rotor resistance in induction motors,” Proc. Inst. Elect. Eng.—Electr. Power Appl., vol. 143, no. 1, pp. 87–94, Jan. 1996. [9] D. J. Atkinson, P. P. Acarnley, and J. W. Finch, “Observers for induction motor state and parameter estimation,” IEEE Trans. Ind. Appl., vol. 27, no. 6, pp. 1119–1127, Nov./Dec. 1991. [10] P. Vas, Vector Control of AC Drives, A Space-Vector Approach. Cambridge, U.K.: Cambridge Univ. Press, 1994. , Parameter Estimation and Condition Monitoring. Cambridge, [11] U.K.: Cambridge Univ. Press, 1996. [12] C. Ilas, A. Bettini, L. Ferraris, G. Griva, and F. Profumo, “Comparison of different schemes without shaft sensor for field oriented control drives,” in Proc. IEEE IECON’94, 1994, pp. 1579–1588. [13] M. Pucci and M. Cirrincione, “Estimation of the electrical parameters of an induction motor in saturated and unsaturated conditions by use of the least-squares method,” in Proc. IEEE ACEMP’01, Kusadasi, Turkey, Jun. 2001, pp. 288–294. , “A direct-torque control of an AC drive based on a recursive-least[14] squares (RLS) method,” in Proc. IEEE SDEMPED’01, Grado, Italy, Sep. 2001, pp. 651–657. [15] L. A. de S. Ribeiro, C. B. Jacobina, and A. M. N. Lima, “The influence of the slip and the speed in the parameter estimation of induction machines,” in Proc. IEEE PESC’97, vol. 2, Jun. 1997, pp. 1068–1074. [16] M. Velez-Reyes, K. Minami, and G. C. Verghese, “Recursive speed and parameter estimation for induction machines,” in Conf. Rec. IEEE-IAS Annu. Meeting, vol. 1, Oct. 1–5, 1989, pp. 607–611. [17] M. Cirrincione and M. Pucci, “Experimental verification of a technique for the real-time identification of induction motors based on the recursive least-squares,” in Proc. IEEE AMC’02, Maribor, Slovenia, Jul. 2002, pp. 326–334. [18] J. Stephan and M. Bodson, “Real-time estimation of the parameters and fluxes of induction motors,” IEEE Trans. Ind. Appl., vol. 30, no. 3, pp. 746–759, May/Jun. 1994. [19] C. Moons and B. De Moor, “Parameter identification of induction motor drives,” Automatica, vol. 31, no. 8, pp. 1137–1147, Aug. 1995. [20] M. Cirrincione, M. Pucci, G. Cirrincione, and G. A. Capolino, “A new experimental application of least-squares techniques for the estimation of the induction motor parameters,” IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 1247–1256, Sep./Oct. 2003. [21] L. A. Ribeiro and C. B. Jacobina, “Real-time estimation of the electrical parameters of an induction machine using sinusoidal PWM voltage waveforms,” IEEE Trans. Ind. Appl., vol. 36, no. 3, pp. 743–754, May/Jun. 2000. [22] C. Cecati and N. Rotondale, “On-line identification of electrical parameters of the induction motor using RLS estimation,” in Proc. IEEE IECON’98, vol. 4, Aug.-Sep. 1998, pp. 2263–2268. [23] J. L. Zamora and A. Garcìa-Cerrada, “Online estimation of the stator parameters in an induction motor using only voltage and current measurements,” IEEE Trans. Ind. Appl., vol. 36, pp. 805–816, May/Jun. 2000. [24] M. Cirrincione, M. Pucci, G. Cirrincione, and G. Capolino, “Constrained least-squares method for the estimation of the electrical parameters of an induction motor,” COMPEL, Int. J. Comput. Math. Elect. Electron. Eng. (Special Issue: Selected Papers From the International Conference on Electrical Machines (ICEM)), vol. 22, no. 4, pp. 1089–1101, 2002. [25] A. Ralston and P. Rabinowits, A First Course in Numerical Analysis, 2nd ed. New York: McGraw-Hill, 1978. [26] P. L. Jansen and R. D. Lorenz, “A physically insightful approach to the design and accuracy assessment of flux observers for field oriented induction machine drives,” IEEE Trans. Ind. Appl., vol. 30, no. 1, pp. 101–110, Jan./Feb. 1994. [27] H. Klaassen, “Selbsteinstellende, Feldorientierte Regelung einer Asynchronmaschine und Geberlose Drehzahlregelung,”, TU Braunschweig, 1999. [28] G. R. Slemon, “Modeling of induction machines for electric drives,” IEEE Trans. Ind. Appl., vol. 25, pp. 1126–1131, Nov./Dec. 1989. [29] M. Cirrincione, M. Pucci, and G. Vitale, “A test-bench for experiments on control, identification and EMC of high performance adjustable speed drives with induction motors,” in Proc. SPEEDAM , Ravello, Italy, Jun. 2002, pp. B1:13–B1:20.

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Maurizio Cirrincione (M’03) received the Laurea degree engineering from the Politecnico of Turin, Turin, Italy, in 1991, and the Ph.D. degree from the University of Palermo, Palermo, Italy, in1996, both in electrical engineering. Since 1996, he has been a Researcher with the Section of Palermo of ISSIA-CNR (Institute on Intelligent Systems for the Automation), Palermo, Italy. His current research interests are neural networks for modeling and control, system identification, intelligent control, electrical machines and drives. Dr. Cirrincione was the recipient of the “E.R.Caianiello” Prize for the best Italian Ph.D. dissertation on neural networks in 1997.

Marcello Pucci (M’03) received the Laurea and Ph.D. degrees from the University of Palermo, Palermo, Italy, in 1997 and 2002, respectively, both in electrical engineering. In 2000, he was a visiting student with the Institut of Automatic Control of the Technical University of Braunschweig, Braunschweig, Germany, where he was involved with the field of control of ac machines. Since 2001, he has been a Researcher with the Section of Palermo of ISSIA-CNR (Institute on Intelligent Systems for the Automation), Palermo, Italy. His current research interests are electrical machines, control, diagnosis and identification techniques of electrical drives, intelligent control, and power converters. Dr. Pucci is a member of the Editorial Board of the Journal of Electrical Systems.

Giansalvo Cirrincione (M’04) received the Laurea degree in electrical engineering from the Politecnico of Turin, Turin, Italy, in 1991, and the Ph.D degree from the Laboratoire d’Informatique et Signaux (LIS) de l’Institut National Polytechnique de Grenoble (INPG), Grenoble, France, in 1998. He spent a year in a postdoctoral position with Leuven University, Leuven, Belgium, in 1999, and, since 2000, he has been an Assistant Professor with the Department of Electrical Engineering, University of Picardie-Jules Verne, Amiens, France. His current research interests are neural networks, data analysis, computer vision, brain models, and system identification.

Gérard-André Capolino (A’77–M’82–SM’89– F’02) received the B.Sc. degree in electrical engineering from Ecole Supérieure d’Ingénieurs de Marseille, Marseille, France, in 1974, the M.Sc. degree from Ecole Supérieure d’Electricité, Paris, France, in 1975, the Ph.D. degree from University Aix-Marseille I, Marseille, France, in 1978, and the D.Sc. degree from Institut National Polytechnique de Grenoble, Grenoble, France, in 1987. In 1978, he joined the University of Yaoundé, Cameroon, as an Associate Professor and Head of the Department of Electrical Engineering. From 1981 to 1994, he was an Associate Professor with the University of Dijon, Dijon, France, and the Mediterranean Institute of Technology, Marseille, where he was founder and Director of the Modeling and Control Systems Laboratory. From 1983 to 1985, he was a Visiting Professor with the University of Tunis, Tunisia. From 1987 to 1989, he was the Scientific Advisor of the Technicatome SA company, Aix-en-Provence, France. In 1994, he joined the University of Picardie-Jules Verne, Amiens, France, as a Full Professor, Head of the Department of Electrical Engineering (1995–1998), and Director of the Energy Conversion and Intelligent Systems Laboratory (1996–2000). He is now Director of the Graduate School in Electrical Engineering at the University of Picardie-Jules Verne. In 1995, he was a Fellow European Union Distinguished Professor of Electrical Engineering with the Polytechnic University of Catalunya, Barcelona, Spain. Since 1999, he has been the Director of the Open European Laboratory on Electrical Machines (OELEM), a network of excellence in between 50 partners from the European Union. He has published more than 250 papers in scientific journals and conference proceedings since 1975. He has been the advisor of 13 Ph.D. and numerous M.Sc. students. In 1990, he founded the European Community Group for teaching electromagnetic transients, and he coauthored the book Simulation & CAD for Electrical Machines, Power Electronics and Drives (Brussels, Belgium: ERASMUS Program Edition, 1991). His research interests are electrical machines, electrical drives power electronics, and control systems related to power electrical engineering. Prof. Capolino is the chairman of the France Chapter of the IEEE Power Electronics, Industrial Electronics, and Industry Applications Societies and the vice-chairman of the IEEE France Section. He is also a member of the AdCom of the IEEE Industrial Electronics Society. He is the cofounder of the IEEE International Symposium for Diagnostics of Electrical Machines Power Electronics and Drives (IEEE-SDEMPED) that was held for the first time in 1997. He is a member of steering committees for several highly reputable international conferences. Since November 1999, he has been an Associate Editor for the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS.