370
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
Line-to-Line Short-Circuit-Based Finite-Element Performance and Parameter Predictions of Large Hydrogenerators René Wamkeue, Member, IEEE, Innocent Kamwa, Senior Member, IEEE, and Mama Chacha
Abstract—A two-dimensional (2-D), time-stepped finite-element (FE) method is used to model and successfully replicate saturated line-to-line and three-phase short-circuit test responses recorded on a 40-pole 13.75-MVA hydrogenerator at Hydro-Quebec’s Rapides-des-Quinze generating station. Three levels of line-to-line and sudden three-phase short-circuit tests (0.13, 0.25, and 0.48 p.u.) are simulated numerically using the FE-based model. While symmetrical faults are only used for parameter determination, the computed line-to-line waveforms are thoroughly compared to real data, with special attention given to field current responses. According to IEEE Std.-115-1995, the -axis dynamic reactances and time-constants are computed from three-phase short-circuit tests while the negative-sequence reactance is derived from the line-to-line short-circuit test resulting in a rated armature current. The obtained simulated tests responses and parameter values, from both symmetrical and asymmetrical faults, support the effectiveness of the proposed FE-based model in incorporating the saturation phenomenon, large number of poles, and detailed damper representation to achieve an accurate dynamic performance assessment together with negative-sequence reactance and dynamic constants prediction. Index Terms—Performance and parameter predictions, synchronous machines, transient electromagnetic analysis.
I. INTRODUCTION
A
N accurate modeling of synchronous machines requires both a realistic physical approach and an increasingly precise mathematical representation [1]–[4]. To better conciliate these two objectives, finite-element method has been widely used for accurate prediction of step response tests such as flux decay and three-phase sudden short-circuits from which parameters may be extracted [5]–[11]. These approaches however suffer some shortcomings: reduced-order models are often used (only one damper winding per axis), the studied machines have small number of poles (not more than four), since dealing with a large number of poles seriously complicates the circuit arrangements in the armatures slots and FE domain mesh.
Furthermore, in most of the previous papers, simulation results are not supported by real data. More recently, line-to-neutral and line-to-line short-circuits faults have been simulated using FE methods [12], [13]. Despite their theoretical originality, these papers failed to remove some of the above-mentioned limitations such as (1) unsaturated machine assumption and (2) lack of a comprehensive experimental validation based on real data. In addition, no guidance was provided on how to extract machine parameters from these line-to-line short circuit tests which are well documented in IEEE Std.-115-1995. This paper presents a two-dimensional (2-D) nonlinear, time-stepped finite-element simulation of a line-to-line short circuit on a large salient-pole synchronous generator with multiple rotor circuits. The work focuses on the complete synchronous machine model including a large number of poles, the damper windings, eddy current effects, and detailed field calculations for parts which have complex shapes and contain nonlinear magnetic materials. The end-winding resistors and inductors are also taken into account. The rotor movement is simulated using the air-band technique [14], [15]. Computations are performed for three different short-circuit test levels (0.13, 0.25, and 0.48 p.u.) of line-to-line short-circuit and sudden three-phase short-circuit. FE simulated and actual three-phase sudden short-circuit tests (not presented in the paper, since it is well documented [8]–[11], [16], [17]), are performed in order to compute -axis transient reactances and time-constants for per-unit rated armature current [18], [19]. According to the IEEE Std. 115 test procedure (8.9.5: Method 3) [18], the negative-sequence reactance for per-unit rated armature current is derived from FE simulated and measured line-to-line fault data, respectively, using the -axis per-unit rated current subtransient reactance previously computed from the sudden three-phase short-circuit. II. MATHEMATICAL FORMULATION
Manuscript received July 25, 2001; revised May 13, 2002. This work was supported in part by the National Science and Engineering Research Council of Canada and in part by the Quebec Research Aid Fund. R. Wamkeue is with Université du Québec en Abitibi-Témiscamingue (UQAT), Rouyn-Noranda, QC J9X 5E4, Canada (e-mail:
[email protected]). I. Kamwa is with Hydro-Québec/IREQ, Power System Analysis, Operation and Control, Varennes, QC J3X 1S1, Canada (e-mail:
[email protected]). M. Chacha is with Ryerson Polytechnic University, Toronto, ON M5B 2K3, Canada. Digital Object Identifier 10.1109/TEC.2003.815672
The 2-D transient magnetic equation describing the field within the machine cross-section is (1) where and are, respectively, the inverse of the magnetic peris meability and the electrical conductivity of the material; the magnitude of the gradient of the electric potential (voltage)
0885-8969/03$17.00 © 2003 IEEE
WAMKEUE et al.: FINITE-ELEMENT PERFORMANCE AND PARAMETER PREDICTIONS OF HYDROGENERATORS
Fig. 2. Line-to-line short-circuit negative-sequence impedance.
371
diagram
for
determination
of
Fig. 1. Finite-element domain, different voltage definitions, and different circuit connections of the synchronous machine.
applied to the finite-element region whose effective length is the magnetic vector potential, is (Fig. 1); where is the unit vector in the -direction. and are the Cartesian coordinates in the cross section of the machine and the -axis lies along the axis of the machine (Fig. 1). The total current in a solid conductor is obtained from the applied voltage and induced eddy currents as (2) Skin effects are neglected in conductors that are small in comparison with the skin depth defined by (3) where is the frequency of the source. The external circuit connections allow the operating conditions of the synchronous machine to be simulated with the real power supply connection. Fig. 1 illustrates the configuration of the circuit connection. The most basic component in the finite-element region is the bar, which is defined as a single conductive region of length in the -direction. Considering this figure, a number of bars may be connected together in series to form a coil. All bars in a given coil carry the same total current but, in opposite directions successively. The coil leads (modeled and reactance ) are brought with lumped resistance out of the finite-element region and connected to a voltage given by source (4) or and represents the polarity of bar while where is is the sum of the voltages across the bars comprising coil . This equation serves to couple the finite-element region, represented by , to the external circuits and sources, represented by
Fig. 3. External circuit connections for the synchronous-machine model.
, , and . Moreover, sets of coils may be connected in parallel to form a single circuit at the external terminals of the device. When this single circuit is connected to a source , and reactance , the corresponding with internal resistance circuit equation is (5) Salon [20], Lombard, and Meunier [21] give an extensive description of the circuits coupling. This coupling allows all kinds of conductors to be modeled as well as connections with electrical circuits and supplies, and three-dimensional (3-D) end effects. The new rotor position is determined from the angular velocity , by solving the mechanical equation (6) is the load torque, and where is the moment of inertia, is the friction coefficient. The electromagnetic torque developed is computed from the magnetic field solution. the generator Galerkin formulation is applied to the field and current equations. The governing equations are then discretized in the time domain using Crank-Nicholson scheme and linearized by the Newton-Raphson algorithm. A direct coupling is used between the mechanical, electrical, and magnetic equations, leading to a
372
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
TABLE I INITIAL CONDITIONS
TABLE II
d-AXIS REACTANCES AND TIME-CONSTANTS
(a)
(b)
Fig. 4. Open-circuit characteristic curve of the generator with the test points.
(c) Fig. 6. (a) Test #1: field current after the line-to-line short circuit. (b) Test #1: phase A current after the line-to-line short circuit. (c) Test #1: phase B current after the line-to-line short circuit.
Fig. 5.
Unloaded generator steady-state phase voltages (test #1).
linear symmetric matrix (7), which is solved using a direct block solver (7)
The matrix comprises a finite-element matrix, electrical matrices, a mechanical matrix, and matrices including coupling vector is composed of the unknowns for each terms. The and iteration : the change in magnetic time step on each node of the mesh for the magnetic unpotential , the change knowns, the change in voltage across each bar
WAMKEUE et al.: FINITE-ELEMENT PERFORMANCE AND PARAMETER PREDICTIONS OF HYDROGENERATORS
373
(a)
(b)
(c)
(d)
Fig. 7. (a) Test #2: field current after the line-to-line short circuit. (b) Test #2: phase A current after the line-to-line short circuit. (c) Test #2: phase B current after the line-to-line short circuit. (d) Test #2: damper current after the line-to-line short circuit.
in current in each coil , and the change in terminal voltage for the electrical unof each set of parallel-connected coils for the mechanical knowns. The change in the rotor position unknown. The vector is composed of the source terms at step and terms which depend on the value of the unknowns at and iteration , as well as at step . The latter terms step take into account the induced currents appearing in conducting parts. The advantage of this direct solving method is the reduced cost of iteration convergence wherever system (7) is of reasonable size. III. TEST SETUP The machine under test is a 40-pole, Y-connected three-phase turbine generator at Hydro-Quebec’s Rapide-des-Quinze generating station, rated at 13.75-MVA, 13.2-kV, 0.8-power factor, 60-Hz, and 180-r/min. The line-to-line short-circuit tests and three-phase sudden short-circuit tests are performed according to the standard procedure described in IEEE Standard 115 [18] and confirmed by IEC Standard [22]. The technical equipment used for data acquisition is fully explained in the test report [23]. As advised in Standard 115, the unloaded generator is driven at rated speed by some prime mover and the field voltage is maintained constant during the
whole test. Line voltage and short-circuit current are recorded from voltage and current transformers, respectively, both connected to a device which provides active power, all in conformity with IEEE Standard 115-1995 (Fig. 2). IV. NUMERICAL SIMULATIONS A. Finite-Element Model The insulating materials on the laminated core and the stator and rotor windings are not taken into account because of their small physical dimensions and minimal influence on the magnetic-flux distribution [20]. Advantage is taken of the magnetic and geometric symmetry to reduce the size of the calculation domain. The same geometric and winding configuration is found every set of 57 slots of the stator, corresponding to 10 poles. Thus, the FE model is reduced to ten poles (i.e., one quarter of the machine). The meshing resulted in a total number of 17 078 second-order triangular elements and 36 486 nodes (see A of the Appendix). Each phase comprises two identical circuits connected in parallel. Each branch or circuit constituting a series of 38 coils is arranged symmetrically in one half of the stator. A brief listing of the machine characteristics and design parameters is provided in Table IV. The armature-winding arrangement for one quarter of the machine (i.e., the current polarity
374
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
(a)
(b)
(c)
(d)
Fig. 8. (a) Test #3: field current after the line-to-line short circuit. (b) Test #3: phase A current after the line-to-line short circuit. (c) Test #3: phase B current after the line-to-line short circuit. (d) Test #3: damper current after the line-to-line short circuit. TABLE III NEGATIVE-SEQUENCE REACTANCE x
connected in series. A small square beside a component indicates the polarity: the current is considered to be positive when it flows toward the small square. In Fig. 3, and represent the end resistance and reactance of the corresponding circuit, respectively. C. Symmetrical and Asymmetrical Short-Circuit Tests
according to the -axis positive direction, for the three phases in the finite-element domain is given in Table V. B. Electric Circuits Fig. 3 shows the external circuit connection model of the simulated unloaded generator. At the top of the figure is a damper circuit; there is a total of ten (one per rotor pole in the model). Each damper circuit consists of three solid conductors short-circuited together at the ends. The excitation circuit is the one in the middle, while the stator three-phase circuits connected in a Y-configuration are drawn at the bottom of the figure. Each stator and rotor circuit comprises two coils corresponding to the go and return parts of a macro-coil in the finite-element domain. This macro-coil is physically equivalent to a set of several coils
1) Initial Conditions: Before starting the fault simulation, it is necessary to reach the unloaded steady-state condition numerically. The armature open-circuit is modeled by keeping switches S1, S2, and S3 in the off position, or equivalently, when switches are closed, by adding large resistance value in each phase [9], [11]. 2) Symmetrical and asymmetrical short-circuit tests: After the unloaded steady-state has been obtained, the symmetrical short-circuit test is simulated by closing switches S1, S2, and S3 (or a zero resistance value in each phase) and the asymmetrical test (line-to-line short-circuit) by switching on S1 and S2 (or a zero resistance value in phases a and b), S3 being maintained in the off position (or adding a large resistance value in the phase c).
WAMKEUE et al.: FINITE-ELEMENT PERFORMANCE AND PARAMETER PREDICTIONS OF HYDROGENERATORS
375
Fig. 10. Line-to-line negative-sequence reactance variation with respect to negative-sequence current; extrapolation to rated current value. TABLE IV GENERATOR DESIGN AND ELECTRICAL PARAMETERS
=
0 0 005 s + 0 005 s
Fig. 9. (a) Test #2: flux distributions at t tsc : (tsc: short-circuit time). (b) Test #2: flux distributions at t tsc : , corresponding to the first peak in the field current after the short circuit (tsc: short-circuit time).
=
Once the FE model was generated, performing each test (steady state and fault simulation, up to eight cycles after fault occurrence) required three to four days computation time on a PC 350-MHz, 64-MB RAM. V. NUMERICAL AND EXPERIMENTAL RESULTS A. Performance Prediction Table I gives the initial condition values before tests. Each value of operating field current is specified in Fig. 4. The FE numerical steady state phase voltages are illustrated in Fig. 5. The above induced phase voltages show sinusoidal behavior. This can be explained by the fact that the unloaded generator is not heavily saturated at this voltage level (about 0.13 p.u. for the first test). But, even if the machine were saturated, these voltages will remain sinusoidal due to the filtering properties of the windings that reduce some harmonics as shown in [9]. Figs. 6(a)–8(a) show FE prediction and measurement field current excursions for the three levels of line-to-line short-circuit. Following the short circuit, high peaks are observed in the short-circuit transient with continuous damping. Phase A armature currents for the three levels of test are shown in Figs. 6(b)–8(b) while those in phase B are shown in Figs. 6(c)–8(c). In each test, symmetrical behavior is observed for the currents in both phases. This result is in agreement with the classical line-to-line short-circuit oscillograms since phase C is kept open during the whole test so that the neutral connection does not influence armature
currents. The FE simulated waveforms of currents and experimental results are very close. The small discrepancy observed is due to the shift in time between computed and experimental quantities. This could be explained by the difficult matching of the initial rotor angle used in the simulated test with that pre-existing in the actual records.The maximum deviation in the amplitude of the armature currents is less than 15% for all three tests performed. Another source of discrepancy is the fact that several machine parameters used for the finite-element analysis, such as the winding-end inductance and characteristics of the damper circuits, are design data (a priori given values) instead
376
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
Fig. 11.
One quarter of the machine with the boundary conditions. The finite-element model comprises ten poles.
TABLE V STATOR WINDING ARRANGEMENT IN THE FINITE-ELEMENT REGIONS. NUMBERS REFER TO THE SLOTS, LETTERS TO DIFFERENT PHASES, POSITIVE-ORIENTED CURRENT, AND IS FOR NEGATIVE ORIENTATION OF THE CURRENT
0
of measured or actual parameters. The discrepancy between experiments and computations of the field current diminishes with the increasing field-voltage input and vanishes at very high excitation voltage values [see Fig. 8(a) for example]. Figs. 7(d) and 8(d) show the current variation in some damper circuits after application of the short circuit. Damper circuits withstand an extremely high instantaneous current during transients and act to attenuate abrupt variations of the flux. Fig. 9(a) and (b) for test #2, show flux distributions in the machine at different times, 5 ms before and after the short circuit, respectively.
same test conditions. Finally, the corrected value from (8) as described in [18] as follows:
+ IS FOR
is obtained
(8) (9) (10) (11)
B. Computation of the Negative-Sequence Reactance Table II shows per-unit dynamic -axis parameters computed from standard three-phase sudden short-circuit oscillograms for rated short-circuit current [18], [19]. The rated current parameters have been estimated graphically using curve extrapolation technique. Fig. 10 illustrates the variation of the line-to-line negative-sequence reactance against the negative-sequence current. Extrapolating the trend-line of Fig. 10 to the nominal cur) leads to the per-unit negative-serent value (i.e., . Following IEEE quence reactance rated current value Std. 115-83 (8.9.5: method 3), the definite negative-sequence using reactance is obtained from the correction made on computed for the per-unit direct-axis subtransient reactance
and are root-mean-square fundamentals of where and , respectively (Fig. 2). The values of obtained using both tests data and FE simulation results are presented in Table III together with the manufacturer’s. The small difference between test and FE negative-sequence reactance (less than 0.3%) attests the accuracy of the obtained results. VI. CONCLUSIONS A time-stepping FE-based model has been used for detailed analysis and prediction of saturated line-to-line short-circuit tests of a 13.5-MVA, 40-pole utility grade hydrogenerator. The negative-sequence reactance of the machine has been computed
WAMKEUE et al.: FINITE-ELEMENT PERFORMANCE AND PARAMETER PREDICTIONS OF HYDROGENERATORS
from asymmetrical transient faults responses while its -axis dynamic constants were obtained from standard symmetrical short-circuit tests. Using commercially available software, the work built and successfully tested a complete FE-based framework for performance and parameter prediction of large hydrogenerators equipped with a large number of poles, based on both balanced and unbalanced faults. Saturation is taken into account at each time step of the FE model in order to account for the very large faults currents. In addition, ten poles representing a quarter of the 228-slots machine are fully simulated, which brought the computation burden to a maximum of four days of cpu time for a single ten-cycle simulation performed on a small 350-MHZ/64-MB personal computer. Comparisons of FE numerical and experimental test responses in addition to the resulting machine parameter values confirmed the great potential and usefulness of the proposed methodology. The small gap observed between computed and experimental field current waveforms could be ascribed to the fact that the damper circuits characteristics and other physical data used to tailor the FE model are a priori design information from the manufacturer, which not necessarily match very accurately the individual machine data. Better improvement of results could be obtained by first estimating through adequate identification process the end turn reactance for the stator windings, electrical characteristics of the damper circuits, and actual initial rotor angle of generator as well as a better fitting geometry. However, this preliminarily processing step needs by itself an entire paper. APPENDIX Table IV provides a brief description of the generator while Fig. 11 shows an approximate structure of the studied generator. The Dirichlet boundary conditions are used on the outer limit of the stator part and the inner rotor edge. Since an even number of poles is implied in the symmetry consideration, cyclic conditions and . Nodes beare needed on the symmetry lines longing to two types of boundary conditions at the same time are treated with the one which is easier to apply. The moving-air-gap concept is used to allow rotor motion at aconstantangular velocity of 180 r/min during the finite-element analysis. ACKNOWLEDGMENT They would also like to thank engineer Manon Lessard-Bélanger of Hydro-Quebec for the machine design parameters and her kind collaboration. REFERENCES [1] P. Kundur, Power System Stability and Control. New York: McGrawHill, 1994. [2] S. A. Tahan and I. Kamwa, “A two-factor saturation model for synchronous machines with multiple rotor circuits,” IEEE Trans. Energy Conversion, vol. 10, pp. 609–616, Dec. 1995. [3] 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., vol. 26, no. 1, pp. 93–108, 1998. [4] , “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.
377
[5] R. P. Schulz, R. G. Farmer, C. J. Goering, S. M. Bennet, D. A. Selin, and D. K. Sharma, “Benefit assessment of finite-element based generator saturation model,” IEEE Trans. Power Syst., vol. 2, pp. 1027–1033, Nov. 1987. [6] S. R. Chaudhry, S. Ahmed-Zaid, and N. A. Demerdash, “Coupled finiteelement/state-space modeling of turbogenerators in the ABC frame of reference, Parts I & II,” IEEE Trans. Energy Conversion, vol. 10, pp. 56–73, Mar. 1995. [7] H. C. Karmaker and K. R. Weeber, “Time-stepping finite-element analysis of large synchronous motors with AC adjustable-speed drives for acoustic noise control studies,” in Proc. IEEE Int. Elect. Mach. Drives Conf., pp. WC2-5.1–WC2-5.3. [8] S. I. Nabeta, A. Foggia, J.-L. Coulomb, and G. Reyne, “A time-stepped finite-element simulation of a symmetrical short-circuit in a synchronous machine,” IEEE Trans. Magn., vol. 30, pp. 3683–3686, Sept. 1994. , “A nonlinear time-stepped finite-element simulation of a symmet[9] rical short-circuit in a synchronous machine,” IEEE Trans. Magn., vol. 31, pp. 2040–2043, May 1995. [10] K. Weeber, “Determination of dynamic parameters of large hydro-generators by finite-element simulation of three-phase sudden short-circuit tests,” in Proc. IEEE Int. Elect. Mach. Drives Conf., pp. MC1-13.1–MC1-13.3. [11] T. H. Pham, P. Wendling, S. J. Salon, H. Tsai, and A. Windhorn, “Load short circuit transient analysis of a generator using Flux2D with mechanical motion and electric circuit connections,” Intell. Motion Syst., pp. 44–56, Sept. 1996. [12] S. I. Nabeta, A. Foggia, J.-L. Coulomb, and G. Reyne, “Finite element simulation of unbalanced faults in a synchronous machine,” IEEE Trans. Magn., vol. 32, pp. 1561–1564, May 1996. [13] T. Renyuan, H. Yan, L. Zhanhong, Y. Shiyou, and M. Lijie, “Computation of transient electromagnetic torque in a turbogenerator under the cases of different sudden short circuits,” IEEE Trans. Magn., vol. 26, pp. 1042–1045, Mar. 1990. [14] E. Vassent, G. Meunier, A. Foggia, and G. Reyne, “Simulation of induction machine operation using a step by step finite-element method coupled with circuit and mechanical equations,” IEEE Trans. Magn., vol. 27, pp. 5232–5234, Mar. 1991. [15] I. A. Tsukerman, A. Konrad, J. D. Lavers, K. Weeber, and H. Karmaker, “Finite element analysis of static and time-dependent fields and forces in a synchronous motor,” in Proc. Int. Conf. Elect. Mach., vol. 2, Paris, France, 1994, pp. 27–32. [16] P. J. Turner, “Finite-element simulation of turbine-generator terminal faults and application to machine parameter prediction,” IEEE Trans. Energy Conversion, vol. EC-2, pp. 122–131, Mar. 1987. [17] J. P. Sturgess, M. Zhu, and D. C. Macdonald, “Finite-element simulation of a generator on load during and after a three-phase fault,” IEEE Trans. Energy Conversion, vol. 7, pp. 787–793, Dec. 1992. [18] Test Procedures for Synchronous Machines, IEEE Std. 115 (1995). [19] G. Shackshaft, “New approach to the determination of synchronous-machine parameters from tests,” Proc. Inst. Elect. Eng., vol. 121, no. 11, pp. 1385–1392, Nov. 1974. [20] S. J. Salon, Finite-Element Analysis of Electrical Machines. Norwell, MA: Kluwer, 1995. [21] P. Lombard and G. Meunier, “A general method for electric and magnetic combined problems for 2D, axisymmetric and transient systems,” IEEE Trans. Magn., vol. 29, pp. 1737–1740, Mar. 1993. [22] M. Pilote, “Centrale Rapide-des-Quinze—Essais de Réception des Alternateurs A1 à A4 Convertis de 25 Hz à 60 Hz,” Hydro-Québec, Service Essais et Expertises Techniques, Montréal, QC, Canada, EMC-93 015, 1993.
René Wamkeue (S’95–M’98) received the B.Eng. degree in electrical engineering from the University of Douala, Cameroon, in 1990. He received the Ph.D. degree in electrical engineering from École Polytechnique de Montréal, Montréal, QC, Canada, in 1998. Currently, he is Professor of Electrical Engineering at the Université du Québec en Abitibi-Témiscamingue, Rouyn-Noranda, QC, Canada, where he has been since 1998. From 1991 to 1992, he was a Professor of Electrical Engineering at the University of Douala. 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 reviewer committee of IEEE PES and Technical Committee of IASTED on ’Modeling and Simulation’.
378
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 18, NO. 3, SEPTEMBER 2003
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, in 1984 and 1988, respectively. Currently, he is a Senior Researcher in the Power System Analysis, Operation, and Control Department at Hydro-Québec Research Institute (IREQ), where he has been since 1988. He is also an Associate Professor of Electrical Engineering at Laval University. Dr. Kamwa is a member of the IEEE Power Engineering and Control System societies.
Mama Chacha received the B.Eng. degree in power systems and heat transfer and the Ph.D. degree in mechanical engineering from the University of Provence, Marseille, France, in 1991 and 1995, respectively. After graduation, he occupied contractual research positions at CNRS, France; CNR, Italy; and, the University of Québec, Rouyn-Noranda, QC, Canada. In 2000, he became a Research Scientist with Ryerson Polytechnic University, Toronto, ON, Canada. His research interests include finite-element modeling of electrical machines and computational flow dynamics/heat and mass transfer.
