Unbalanced transients-based maximum likelihood ... - IEEE Xplore

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 1, MARCH 2003

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Unbalanced Transients-Based Maximum Likelihood Identification of Induction Machine Parameters René Wamkeue, Member, IEEE, Innocent Kamwa, Senior Member, IEEE, and Mama Chacha

Abstract—The paper describes an effective formulation of a maximum-likelihood identification algorithm for linear estimation of the equivalent-circuit parameters of cage-type (single cage and double cage) or deep-bar induction motors with measurement and process noises. A complete generalized model for symmetrical and asymmetrical test analysis of induction machines is developed for this purpose. The paper outlines the theory and reasoning behind the proposed statistical-based treatment of online data derived from generalized least-squares estimator and a Kalman filter. The method is successfully applied to online double-line independent finite-element (FE) short-circuit-simulated records of a deep-bar-type induction motor. Index Terms—Finite-element (FE) simulation, identification, induction machines, modeling.

I. INTRODUCTION

I

N THE PAST few years, the performance, prediction, and control of electrical machines have become topics of great interest. It is essential to have some knowledge of the machine characteristics, at least from previous applications, in order to formulate a model and compute its parameters. If a high degree of parameter accuracy is required, it may be imperative to turn to accurate model structures and more sophisticated and effective estimation methods, particularly in a time-varying noisy environment. The machine structures frequently used in the literature are fixed-order models generally developed for single squirrel cage or double-cage motors and generators [1], [2]. By the fact, they are not flexible. For example, a single squirrel cage induction machine model cannot be accurately used to analyze a deep-bar rotor induction motor. Since they do not include the neutral of the machine, they are unable to efficiently predict its asymmetrical transient performances in the generator mode. In fact, more recently, authors have shown that unsymmetrical transients such as line-to-line short circuits, due to the sinusoidal characteristic of their input signals, which cover a large spectral density, are more convenient than symmetrical step response tests for exciting machine modes [3], [4]. These transients, therefore, appear to be suitable candidates for parameter identification of alternating-current (ac) machines. In the field of electrical machine identification, it has been common practice to accept a model provided if it Manuscript received October 11, 2000; revised May 13, 2002. This work was supported by the National Science Engineering Research Council of Canada. R. Wamkeue is with the Université du Québec en Abitibi-Témiscamingue, Rouyn-Noranda, QC, J9X 5E4, Canada (e-mail: [email protected]). I. Kamwa is with the Hydro-Québec/IREQ, Power System Analysis, Operation and Control, Varennes, QC, J3X 1S1, Canada (e-mail: [email protected]). M. Chacha is with the Ryerson Polytechnic University, Toronto, ON, M5B 2K3, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TEC.2002.808383

Fig. 1. Generalized Park’s equivalent circuits of Wye-Grounded-connected induction machine (a) d-axis; (b) q -axis; (c) zero-sequence axis.

leads to a fitting that “looks good.” Yet this is a very simplistic definition of the precision that a model should offer. To be rigorous, yet without requiring unrealistic hypotheses or prior knowledge of the process and measurement noise sequences, a good statistical estimation program should, ultimately, ensure white innovation sequences, perfect proof of convergence to the optimal value of the machine parameter vector [3], [4]. In this paper, an accurate formulation of a maximum-likelihood identification method is used for time-domain estimation of induction-machine parameters. The main objectives are • to develop a voltage-controlled state space model of an induction machine in generator or motor mode, including adjustable rotor circuits and an equivalent Park neutral connection for Wye-connected generators, well adapted for both symmetrical and asymmetrical transient-performance predictions; • to present a suitable maximum-likelihood identification procedure for induction-machine modeling;

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 1, MARCH 2003

• to provide a technique for building simulated noisecorrupted data for identification practice problems from an independent flux two-dimensional (2-D) finite-element simulation; • to show the effectiveness of the proposed algorithm in the model derivation of a deep-bar rotor-type induction motor from line-to-line short-circuit tests.

and coupling-axis matrices, respectively. and are is for the the -order identity matrix. The coefficient is for the generator behavior of the machine motor and . All submatrices of (Fig. 1). The order of the system (2) is (1)–(5) are defined in the Appendix. Equation (2) can be organized in the space state form with the machine currents as state variables as follows: (6)

II. GENERALIZED MODELS OF INDUCTION MACHINES AND PARAMETERIZATION

with

(7)

A. Electrical Equations—State Space Model The generalized dq0 Park equivalent circuits of a three-phase asynchronous machine can by summarized by Fig. 1 where arbitrary number of rotor circuits has been added in order to adapt the model for different types of machines. The classical single-cage and double-cage types of induction machines are and , obtained by setting respectively. The deep-bar rotor induction machine can be or [3], [5]. A number of rotor modeled with windings suitable for accounting for skin effects could also be chosen to enhance the accuracy of the model. It is established in [5] that the equation of an accurate induction-machine model, including the neutral connection for both symmetrical and asymmetrical test analysis, can be described by the following (1)–(5): (1)

or, equivalently, with flux linkages as state variables

(8) or

(9) B. Deterministic/Stochastic State-Space Model and Parameterization It can be proven that in a noisy environment, when state and measurement errors are taken into account, the general form of a deterministic/stochastic discrete linear state-space model of the machine is given by (10)–(11) shown at the bottom of the page where (12)

(2) (3)

(13) (14)

(4)

(5) is the nominal frequency in rad/s and also the base where is the armafrequency for per-unit transformation purpose; ture current frequency and the rotation speed of the mmf of the is the slip of the machine, positive for machine; motors and negative for self-excited induction generators; and is the rotor speed in per unit; and and are the derivative

(15) (16) (17) (18) vector in (10) is a function not only The state variable but also of the of the system-machine parameter vector noise parameter vector . Equations (17)–(18) define various

(10) (11)

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covariance matrices, where is the process noise covariance constraints relating matrix. The equations in (11) are , , and a priori which may ensure system stability denotes the th and accelerate algorithm convergence. eigenvalue of . The initial-condition variables for simulation purposes are computed from the steady-state analysis by setting or (9) equal to zero for (6) equal to zero for for a given slip , or equivalently (19)

III. NUMERICAL MODELING OF UNBALANCED FAULTS The armature of the connected wye-grounded induction machine is represented in Fig. 2. Appropriate selection of the switches allows the desired test to be performed on the induction generator or motor. The switch defines a grounded ) and open ( ), an unmachine when closed ( ) [3]. Prior to fault application, the grounded machine ( ) machine input voltages are symmetrical ( , , and their expressions for any instant are , where and

Fig. 2. Armature of a connected wye-grounded induction machine.

TABLE I THREE-PHASE FAULTS AND TEST PRINCIPLE

(20) is the per-unit terminal voltage. For the ungrounded where or delta-connected generator, the zero-sequence voltage equals zero at any instant, as given by the following equation: (21) The principles of the tests and their corresponding input voltages in the abc reference frame are summarized in Table I. In the dq0 reference frame, the control voltage vector is derived from the abc corresponding input voltages for a given test using Park’s matrix transformation [8]. IV. MAXIMUM-LIKELIHOOD IDENTIFICATION More recently, an accurate approach of the maximum-likelihood estimator (MLE) has been successfully used for synchronous-machine parameter identification [4]. The likelihood function remains an unbiased, consistent estimator for stochastic system modeling. The maximum-likelihood estimate of the parameter vector is the value of which maximizes the is equal to the actual measurements joint probability that in hands. This is equivalent to minimizing the negative logarithm of (22)

(22) (23) is the innovation sequence and is the corwhere is the predicted noisy responding covariance matrix.

observations. A serious difficulty in using the maximum-likelihood identification method is how to estimate the unknown . Inspired by the generalized least-squares covariance of the machine method, the maximum likelihood estimate can be computed using the following parameter vector three-step procedure [4]: and minimize (22) with respect to ; 1) set from the resid2) compute uals of the previous step; 3) form the cost function (22) and solve the minimization problem. Steps 2 and 3 are repeated until convergence is attained. are updated at step 2; thus, At each iteration, and (22) behaves like the weighted least-squares estimator, which will be reweighted during future iterations. This is why the procedure is also called the iteratively reweighted least-squares for two distinct iterations estimator, since and . The predicted state and output variables are computed using the discrete form (24) obtained by combining the discrete

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 1, MARCH 2003

Fig. 3. Finite-element domain, different voltage definitions, and circuit connections of the induction machine.

Fig. 5. Phase a armature current after line-to-line short-circuit test of deep-bar induction motor.

Since is unknown, an adequate value of is chosen to initialize the Kalman filter. Further estimates of , together with the system parameter vector of (14), are obtained by the estimation procedure. Finally, the minimization problem is defined by (29) subject to r

(29)

Equation (29) is a nonlinear optimization problem, which can be solved, starting with an initial guess of the parameter vector , by means of Newton-type iterative algorithms using finite-difference computation such as proposed in [9]. V. FLUX2D TIME-STEP-BASED SIMULATED DATA

Fig. 4. Influence of type of induction generator on phase b current for line-to-line short circuit.

deterministic state-space model of the system with the Kalman prediction-correction formulation

A. Noise-Corrupted Simulated Data for the Identification Process One of the major problems of identification is the availability of actual system data. If a set of true parameters for a given system machine is known, then simulated data can be used to test the the performance and robustness of the estimation algorithm, as shown in this section. Naturally, the obtained results remain valuable in the real world with actual data for a physical machine. The discrete state linear model of process noise can be is the process state noise vector and defined by (30); where is the process noise output variable vector

(24) where denotes the predicted variable values and is the is the limiting steady-state number of lines of matrix . Kalman gain matrix defined by (25) and , a solution of the Riccatti matrix (26)

(30)

(31) (25) (26) (27)

(10). The measurement noise vector is given by It can be rewritten as shown in (31), where is a positive coefficient allowing an adequate signal/noise ratio to be selected.

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If is the exact parameter vector for a given machine, the noise-corrupted data are modeled by determin.output

process

measurement

(32) Therefore, the innovation sequence vector can also be defined by the first part of (33) (33) is the reference covariance matrix of innovation where sequences. The latter (33) shows that the innovation vector is a superposition of process noise and measurement noise. For the vector is computed present work, the deterministic output using finite-element modeling of 25% line-to-line short-circuit test of the machine in operating condition. Noise-corrupted simulated data are then obtained by adding to the deterministic a given amount of residuals with an apoutput . propriate signal/noise ratio and a fixed covariance matrix The estimation algorithm converges to the optimal machine parameter vector for when corresponding residuals become white at the end of the procedure. B. Machine Design—Finite-Element Simulated Data for a Double-Line Short-Circuit Test In order to provide the output to serve as data for the estimation procedure, precise experiments are primordial; otherwise, a complete and realistic approach of the induction machine based on a rigorous mathematical description should be adopted. The design (geometric and electric) parameters of the machine given in Table IV were computed following the procedure defined in [10]. To conduct the analysis, the finite-element model of the induction machine was developed using Flux2D [11]–[13]. The 2-D case has the following usual form in the static frame of reference (stator) and the moving frame of reference (rotor): (34) where and are, respectively, the inverse of the magnetic peris meability and the electrical conductivity of the material; the electric scalar potential (voltage) applied to the finite-element region; and is the effective length of the machine. is the magnetic potential. The total current in a conductor is given by the following current-voltage relationship: (35) and are the Cartesian coordinates in the crosswhere section of the machine. The -axis lies along the axis of the machine. Also, skin effects are neglected in conductors that are small in comparison with the skin depth. Galerkin’s weighted-residuals method is applied to the field and current equations to yield the finite-element matrix equations. The external circuit connections allow diverse operating conditions

Fig. 6. Phase b armature current after a line-to-line short-circuit test of a deep-bar induction motor.

of the induction machine to be simulated with the real powersupply connection. Fig. 3 illustrates a general configuration of the circuit connection. A number of bars (bar 1, bar 2, …, bar ) or stator conductors are connected together in series to form coils [12]. All bars in a given coil carry the same but in opposite directions successively. The total current coil leads are brought out of the finite-element region and given by connected to a voltage source (36) where is 1 or 1 and represents the polarity of bar while is the sum of the voltages across the bars comprising coil . Equation (36) serves to couple the finite-element region, represented by , to the external circuits and sources, represented , , and . A number of deep bars are shorted toby gether by two end rings to form the rotor electric circuit. In each deep bar, the current is given by the current voltage (35) [9]. The motion of the rotor is governed by the following mechanical equation: (37) is the electromagnetic where is the moment of inertia, is the load torque, is the the friction coef(EM) torque, is the angular acceleration, is the ficient, is the rotation angle. The governing angular velocity, and equations are discretized in the time domain using an implicit method and linearized by the Newton—Raphson algorithm. A direct coupling is used between the mechanical, electrical, and magnetic equations, leading to a complete matrix system with crossed terms given by (38) matrix comprises a finite-element matrix, electrical maThe trices, a mechanical matrix, and matrices including coupling vector is composed of the unknowns for each terms. The

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 1, MARCH 2003

Fig. 7. Phase c armature current after a line-to-line short-circuit test of a deep-bar induction motor.

Fig. 8. Evolution of residuals based on phase a current.

time step and iteration. The rotor movement is simulated using the moving-air-gap technique [12], [13]. VI. ESTIMATION RESULTS The studied motor was a deep-bar rotor-type induction machine. Flux2D and electric characteristics are given in Table IV of the Appendix. The electric parameters of the machine used as ) have been obtained from the nominal reference values ( data of the machine and the analytical algorithms proposed in [10]. The noise-corrupted simulated data have been obtained from (32), where the deterministic output vector or noncorwere computed from Flux2D method rupted data vector ). The measurement noise vector using parameters ( and the process noise were computed from (31) and (30), respectively. As previously mentioned in Section II, the deep-bar rotor induction machine has been modeled with in the generalized Park reference frame (Fig. 1). This affirmation is clearly justified by the illustration example given in Fig. 4 for a 1875-hp, 2.402-kV, 60-Hz, three-phase induction machine obtained from reference [5], showing that line-to-line short-circuit test oscillograms for a deep-bar-rotor machine and a double-cage-type rotor are close. Starting with an initial parameter vector linearly perturbed ), the proposed algorithm should lead to the true ( ) and at the parameter vector, otherwise ( , the order of the end of the estimation process. Since system (10) is 7. Initial conditions for identification algorithm , , and . The are initial steady-state conditions for motor mode of the machine and . were computed from (19) with the slip in (11) with , defines constraints on parameters. (39) Table III shows the covariance matrices of current residuals in the dq0 reference frame at different steps of estimation process.

Fig. 9. Evolution of residuals based on phase b current.

At the step 1 of the three-step procedure given in Section IV, this first iteration of the identification algorithm behaves like the classical least-squares estimator (column 2 of Table III) and provides the initial estimated (Figs. 5—7) with initial residuals (innovations) shown in Figs. 8 and 9. It is observed from previous figures that initial estimated are very far from optimal results. These results attest that least-squares algorithm cannot provide efficient estimated parameters without prior knowledge of the process and measurement noise sequences as previously mentioned in the introduction. At the step 2, (column 3 of Table III), where is computed from the step 1. Step 3 provides the final estimated (column 4 of Table III) as given in Figs. 5—7 with corresponding residuals (Figs. 8 and 9). The results of Tables II and III demonstrate the effectiveness of the proposed estimation method while Figs. 5—7 show the accuracy of the process. The evolution of the estimation procedure is shown in Figs. 8 and 9 where it can be seen that the innovation sequences are near white. Otherwise, the spectra density of final residuals contains

WAMKEUE et al.: UNBALANCED TRANSIENTS-BASED MAXIMUM IDENTIFICATION OF PARAMETERS

TABLE II ESTIMATED PARAMETERS OF THE EQUIVALENT CIRCUIT

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TABLE IV FLUX2D AND ELECTRIC PARAMETERS OF THE MOTOR

TABLE III PARAMETERS OF COVARIANCE MATRICES OF RESIDUALS

only high-frequency zero mean and normal distributed signals. Consequently, the process yields optimal machine parameter values. It can be also observed that the computed electric parameter from Levi’s formula [10] and estimated parameters are very close (Table II). Furthermore, the reference covariance matrix of residuals added to Flux2D simulated data is obtained at the end of estimation process as illustrated by the last two columns of Table III and columns two and four of Table II. The great advantage of designing a machine in the proposed work is that we are sure the computed parameters are true parameters of the machine. This insurance is not guaranteed for a real machine. Therefore, the effectiveness of the estimation algorithm is easily proved by the fact that it computes previous true parameters of the machine from a given initial guess. In fact, it may be observed from these tables that thus , where is the estimated parameter vector (column four of Table II). To be rigorous, the statistical inference on the parameters should be evaluated, since the procedure yields optimal results. In view of the fact that the machine studied is ungrounded through a high neutral-impedance value, ). the zero-sequence and neutral currents are zero ( Although induction motors are usually ungrounded, the neutral impedance was added to the machine in order to extend the proposed machine modeling to studies of self-excited induction ) of the generators. The unknown impedance ( ) of the electrical model (Fig. 1) of the masecond rotor ( chine was estimated through the identification process from an assumed initial value (Table II).

VII. CONCLUSION An efficient and concise method for estimating the induction machine parameters from an asymmetrical short-circuit test is proposed. A generalized induction-machine model (for selfexcited induction generator and motor mode) and an unbalanced fault analysis are developed for this purpose. By combining the maximum-likelihood estimator and a Kalman filter, the method ultimately produces an optimal parameter vector and white innovation sequences. The paper has attempted to built a framework for identification experiments when actual data are available or not. Finally, the identification algorithm is successfully tested in an accurate modeling of a deep-bar-loaded induction motor based on line-to-line short-circuit Flux2D finite-element simulated records. Since the level of the short circuit is low (25%), the present analysis does not include the machine’s iron saturation or the rotor motion that strongly

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influences short-circuit currents occurring under rated operating conditions but, in the near future, there are plans to study their impact on current prediction and estimated machine parameters. APPENDIX (a.1) (a.2) (a.3) (a.4) (a.5)

[5] R. Wamkeue, N. E. E. Elkadri, I. Kamwa, and M. Chacha, “Unbalanced transient-based finite-element modeling of large generators,” Elect. Power Syst. Res., vol. 56, pp. 205–210, Feb. 2000. [6] A. Dell’ Aquila, F. S. Lovecchio, L. Salvatore, and S. Stani, “Induction motor parameter estimation via EKF,” in Proc. EPE Firenze, 1991, pp. 3-543–3-549. [7] M. Ferfra, “Contribution to modeling and identification of induction,” Ph.D., Faculty of Engineering and Sciences, Univ. Laval, Quebec City, QC, Canada, 1993. [8] R. Palma, “Transient Analysis of Induction Machines Using Finite Elements,” Ph.D., Rensselaer Polytechnic Inst., Troy, NY, 1989. [9] A. A. F. Seber and C. J. Wild, Nonlinear Regression, New York: Wiley, 1988. [10] P. Kundur, Power System Stability and Control. New York: McGrawHill, 1994. [11] A. Grace, Optimization Toolbox for Use With Matlab. Natick, MA: The MathWorks, Inc., 1992. [12] E. Levi, Polyphase Motors: A Direct Approach to Their Design. New York: Wiley, 1984. [13] Flux2D, Electromagnetic Software Solutions: Magsoft Corp., 1998.

(a.6) (a.7)

(a.8) (a.9)

.. .

.. .

(a.10)

REFERENCES [1] P. T. Lagonotte, H. Al Miah, and M. Poloujadoff, “Modeling and identification of parameters of saturated induction machine operating under motor and generator conditions,” Elect. Mach. Power Syst. J., pp. 107–121, Feb. 1999. [2] R. Wamkeue, I. Kamwa, and X. Dai-Do, “Numerical modeling and simulation of unsymmetrical transients on synchronous machines with neutral included,” Elect. Mach. Power Syst. J., vol. 1, pp. 93–108, Jan. 1998. [3] , “Line-to-line short-circuit test based maximum likelihood estimation of stability model of large generators,” IEEE Trans. Energy Conversion, vol. 14, pp. 167–174, June 1999. [4] R. Wamkeue and I. Kamwa, “Generalized modeling and unbalanced fault simulation of saturated self-excited induction generators,” Elect. Power Syst. Res., vol. 61, pp. 11–21, 2002.

René Wamkeue (S’95—M’98) received the B.Eng. degree in electrical engineering from the University of Douala, Cameroon, in 1990 and the Ph.D. degree in electrical engineering from École Polytechnique de Montréal, Montreal, QC, Canada, in 1998. Currently, he is Professor of electrical engineering at Université du Québec en Abitibi-Témiscamingue, Rouyn–Noranda, QC, Canada, where he has been since 1998. His research interests include control, power electronics, modeling, and identification of electric machines and power system cogeneration by induction generators. Dr. Wamkeue is a member of Technical Committee of IASTED on Modeling and Simulation.

Innocent Kamwa (S’83–M’88–SM’98) received the B.Eng. and Ph.D. degrees in electrical engineering from Laval University, Quebec City, QC, Canada, in 1984 and 1988, respectively. Currently, he is an Associate Professor of Electrical Engineering at Laval University. He has been with the Hydro-Québec Research Institute, IREQ, since 1988. His current interests include system identification, synchronous-machine advancement, as well as control and real-time monitoring of electric power systems. Dr. Kamwa is a member of the Synchronous Machine and Stability Controls Subcommittees of the IEEE PES. He was elected a member of the New York Academy of Sciences in 1992.

Mama Chacha received the B.Eng. degree in power systems and heat transfer and the Ph.D. degree in mechanical engineering from University of Provence, Marseille, France, in 1991 and 1995, respectively. In 2000, he joined Ryerson Polytechnic University, Canada, as Research Scientist. He has been with CNRS, France; CNR, Italy; and University of Québec, Rouyn-Noranda, QC, Canada. His research interests include finite-element modeling of electrical machines and computational flow dynamics/heat and mass transfer.