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1052 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 3, MARCH 2007 Variable Bias Current in Magnetic Bearings for Energy Optimization M. Necip Sahinkaya1 and Ahu E. Hartavi2 Department of Mechanical Engineering, University of Bath, Bath, U.K. Faculty of Mechanical Engineering Automotive Controls and Mechatronics Research Center, Istanbul Technical University, Istanbul, Turkey We introduce an adaptive variable bias current control scheme to minimize the energy consumption of magnetic bearings without altering the dynamic performance. We developed analytical expressions for the optimum bias current settings for both differential and unidirectional modes of operation as a function of the orbit size and the desired bearing stiffness. The orbit size is measured in situ by a recursive Fourier coefficient calculation. The analytical results are presented in normalized variables, and hence can be applied to any size and type of magnetic bearing. Both zero and nonzero static load cases are considered. We show analytically that, under variable bias current operation, the differential mode provides better energy efficiency and stiffness capacity than the unidirectional mode. We present the energy consumption data obtained experimentally on a flexible rotor magnetic bearing rig. The data show that, with the proposed variable bias current approach, a significant energy saving can be obtained without deterioration of dynamic performance. Index Terms—Adaptive control, bias current, energy optimization, magnetic bearings. I. INTRODUCTION PPLICATION areas of magnetic bearings are steadily expanding because of the significant advantages they offer compared with conventional bearings [1]. They provide contact-free and oil-free support of rotating machinery without the losses encountered due to friction in oil film bearings. They can operate in hostile environments, at very high temperatures, and at high speeds. They are also an ideal choice in applications where contamination is an issue such as turbo-molecular pumps and artificial hearts. The power loss in magnetic bearings is lower than for fluid film bearings, but there is potential for further significant reduction. Low power consumption is important in several application areas, such as high-speed energy storage flywheel systems, transport applications including future military vehicles where available power is concentrated to critical operational functions, remote and unmanned air sea and land vehicles where the energy is provided by a battery, and micro/nano applications. Ohmic loss is one of the major energy losses in magnetic bearings, and this is proportional to the square of the current flowing in the coils. Other losses such as the eddy current and hysteresis in the rotor and stator, and circuit loss in the electrical equipment (cable, electronic cabinet) are also functions of coil current. Therefore, minimization of the current has a significant effect on the overall energy efficiency of magnetic bearings. Furthermore, the heat generated by the magnetic bearings will be reduced with a decrease in coil current. The design of low-loss magnetic bearings can be achieved by modifying hardware and/or software. A hardware approach is A Digital Object Identifier 10.1109/TMAG.2006.888731 A color version of Fig. 9 is available online at http://ieeexplore.org. described in [2], where the geometry is optimized for energy efficiency in an advanced energy storage system for future military vehicles. The effect of various design parameters on hysteresis, eddy current, and windage losses is studied experimentally in [3], and the effect of magnetic pole arrangement on the core loss of a magnetic bearing at high speed and high flux density operation is reported in [4]. A new type of magnetic bearing is proposed in [5], which has smooth flux distribution to reduce rotating loss. The use of permanent magnets to carry the static load provides savings in energy consumption at the expense of design complexity and cost. A survey of early work is given in [6], where experimental results are presented. A software approach to reduce power loss involves modifying the bias current and the controllers. Normally a fixed electric current is supplied to each electromagnet, which is referred as the bias current. The control current is superimposed on the bias current. Therefore, the bias current flows in the coils even if no force is required, causing power loss and rotor heating. The purpose of this dc bias current is to linearize the force-current and force-displacement relationships, to extend the dynamic force capacity, and to improve the dynamic performance. There are three commonly used operation modes of power amplifiers [7]. The most popular are Class-A power amplifiers, where the dc bias current is set at half of the maximum allowable current. The control current is added to the bias current in one coil and subtracted from the opposing coil according to the force direction. This is highly inefficient in terms of the energy consumption, but it provides maximum range for the dynamic force with good linearity and dynamic performance. A gain scheduled adaptive control algorithm is suggested in [8] to minimize the speed-synchronous control current. Class-B power amplifiers are more energy efficient as the bias current is set to a much lower value. The control current is added to the bias current of one of the coils only according to the force direction. This is appropriate for low bearing stiff- 0018-9464/$25.00 © 2007 IEEE SAHINKAYA AND HARTAVI: VARIABLE BIAS CURRENT IN MAGNETIC BEARINGS FOR ENERGY OPTIMIZATION ness and low vibration applications because the force slew rate and the dynamic performance and controllability are poor. Performance degradation of Class-B power amplifiers is studied in [9], [10]. Large feedback gains are required to achieve an acceptable bearing stiffness, and hence the possibility of current saturation is high. The design of nonlinear controllers to handle current saturations and stability problems is not trivial [11]. In the case of zero-bias Class-C modes of operation, nonlinearities are severe and the control singularity has to be mitigated. Linear controllers do not function for this mode of operations, but several forms of nonlinear controllers are suggested in the literature. A switching control strategy is introduced in [12], and used with a nonlinear controller involving an integrator backstepping approach to minimize power losses, to avoid singularities, and to ensure stability. Various other nonlinear control designs for a single one-degree-of-freedom active magnetic bearing are discussed in [13] via numerical simulations. The application of H-infinity compensation based on a control current switching rule is studied experimentally in [14]. Apart from controller complexity, the poor dynamic performance and lack of robustness against changes in the operating conditions such as changes in unbalance distribution and external excitation, are the main drawbacks of that approach. In this paper, a variable bias current approach is exploited to achieve a required dynamic performance with minimum power consumption under varying operating conditions. An earlier study [15] applied to turbo-molecular pumps showed that minimum bias current does not always produce minimum power consumption under dynamic loading such as that caused by unbalance disturbances. Magnetic bearings already suffer from a limited force capacity, and low bias current operation decreases the dynamic load capacity significantly and render the bearings vulnerable to changes in operating conditions. II. DYNAMIC PERFORMANCE The forces in an active magnetic bearing are generated by applying control currents to the coils. Two opposing poles generate a bidirectional force along the axis of pole centers as shown in Fig. 1. Defining as the distance of the shaft center from the bearing center towards the coil 1, and as the radial clearance, the force generated in the -direction can be written as follows [1]: 1053 Fig. 1. Schematic view of the opposing poles and the axis system. TABLE I LIST OF NORMALIZATION CONSTANTS are shown in Table I, where is the maximum allowable current. The normalized magnetic force generated in the -direction can now be written as (3) The coil currents and are normally generated from a control current in either the differential or unidirectional modes. In both modes, a constant bias current is supplied to opposing coils even if no force generation is required. In the differential mode (defined as Class-A mode with variable bias current), the control current is added to and subtracted from the bias current to generate driving currents for both coils as follows: (4) (1) The bearing constant is a function of the permeability of free space , the pole area , number of windings , and the half angle between double pole faces (2) In order to generalize the analysis, normalized variables are used throughout the paper. The normalized variables are denoted by capitalized symbols and the normalization constants In the unidirectional mode (defined as Class-B mode with variable bias current), only one of the coils receives the control current depending on its sign if if (5) In the presence of a nonzero normalized static load , the bias currents on the opposing poles are different to provide a constant static force to counteract the static load. If is defined as the mean bias current, i.e., the bias currents on two opposing coils 1054 are set to and centrally positioned rotor IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 3, MARCH 2007 , (3) gives the following for a The dynamic characteristics can be examined by assuming a steady-state rotor synchronous response of the following form, where is the normalized vibration amplitude in the -direction, and is the constant rotational speed of the rotor: (6) This would give a difference between bias currents in opposing . poles of The magnetic force expression in (3) can be linearized for a centered rotor around a zero control current (7) The negative bearing stiffness and the current gain coeffiare functions of the bias current and can be obtained cient by linearizing (3) as follows for both differential and unidirectional mode of operations: (8) For the unidirectional mode, due to discontinuity in (5), the current coefficient is the average of current coefficients for positive and negative control currents. Because of their inherent negative stiffness properties, magnetic bearings are unstable and require a position feedback for stability (9) giving the following linearized bearing dynamics: (13) and must lie It is important to note that the coil currents between 0 and 1. This imposes restrictions on the bias current as a function of the vibration amplitude and the bearing stiffness . In the differential mode, bias current limits are specified by , and , and can be written as follows after substituting for from (12): (14) For the unidirectional mode the bias current limits are and , which can be expressed in terms of the orbit size and the desired bearing stiffness as follows after eliminating : (15) Limits on the bias current settings can be interpreted as limitations on the achievable bearing stiffness as a function of the maximum orbit size or vice versa. The maximum bearing stiff, ness in the differential mode can be achieved when i.e., a Class-A bearing. This gives the following expression for the maximum bearing stiffness as a function of the orbit size : (10) The proportional feedback gain produce positive bearing stiffness should be large enough to (11) Alternatively, the following equation can be used to calculate a position feedback gain in order to achieve a given bearing stiffness as a function of the bias current and the static load: (16) For a unidirectional mode of operation, an expression for the maximum bearing stiffness coefficient can be obtained from the upper limit expression in (15). Differentiating this with respect to gives the following optimum bias current setting as a function of the orbit size , and the corresponding maximum achievable value of the bearing stiffness: (17) (12) To associate physical meaning to the normalized parameters, consider a Class-A power amplifier (i.e., ) with zero static load. The normalized bearing characteristics are and The force capacity along one axis is and . achieved by setting and can be as a funcFig. 2 shows the maximum achievable values of tion of and the corresponding settings in order to achieve the limiting performance. It can be seen from this figure that the differential mode gives a better dynamic range for the mag, the netic bearing. For example, for an orbit size of maximum achievable bearing stiffness under differential mode with , but under the unidirectional mode, is with . Thus, the differential mode provides 2.24 times more stiffness than the unidirectional mode. SAHINKAYA AND HARTAVI: VARIABLE BIAS CURRENT IN MAGNETIC BEARINGS FOR ENERGY OPTIMIZATION Fig. 2. Maximum achievable values of K and corresponding optimized bias current settings as a function of orbit size A in both modes of operations. Alternatively, for a given stiffness value of say , the magnetic bearing will saturate when the orbit size reaches in differential mode, but at in unidirectional mode. 1055 Fig. 3. Average power dissipation as a function of bias current for different bearing stiffness and orbit sizes (solid: Differential, dashed: Unidirectional, dashdot: Class-A). (i.e., ), and the other for the positive control ) regions giving the following current (i.e., average power dissipation: III. POWER DISSIPATION A unified measure of ohmic losses (or in normalized form) is introduced in order to compare the energy efficiency of different configurations by integrating the instantaneous power dissipation of opposing coils along one axis over one synchronous cycle and then averaging as follows: (18) For a Class-A power amplifier and zero static force, the above for the maximum force (i.e., ) measure gives for a steady-state centered position of the case, and rotor (i.e., ). Under variable bias current operations, the above cost measure can be minimized with respect to bias current as a func. tion of the orbit size and the desired bearing stiffness For a differential mode, if the current values under proportional control are inserted into (18), this yields the following average power dissipation expression as a function of the vibration orbit, desired bearing stiffness and bias current: (20) Fig. 3 shows the power consumption as a function of for two and different values of bearing stiffness coefficients ( ), and two different values of orbit sizes ( and ). The saturation and negative coil current regions are not plotted. It is obvious from the figure that for the unidirectional mode, there is an optimum bias current setting, which varies with the orbit size and the bearing stiffness. The smaller bias current does not always provide better energy efficiency under dynamic loading. Also in cases where higher dynamic performance is required, this mode of operation is of limited use. IV. BIAS OPTIMIZATION An expression for the optimum bias current settings to minimize energy consumption can be obtained by differentiating and then equating it to zero. (19) and (20) with respect to For the differential mode this gives (19) (21) The same procedure can be followed for the unidirectional mode, where, due to discontinuity, the integration should be broken into two regions; one for the negative control current It can be shown however that this optimum value is always less than the minimum allowable bias current given by (14), hence it has no practical significance. This is consistant with findings 1056 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 3, MARCH 2007 Fig. 4. Percentage energy saving when using optimum variable bias current in differential mode compared with constant bias current of I = 0:5 (Class-A). K values increase in the direction of the arrows from 0.1 to 1 with an increment of 0.1, and then from 1.5 to 5 with an increment of 0.5. reported in [16]. Therefore, the optimum bias current setting for the differential mode is Fig. 5. Optimum bias current settings and the corresponding average power dissipation as a function of orbit size. (22) This gives the minimum average power dissipation of (23) Fig. 4 shows the percentage energy saving, as a function of the orbit size and the bearing stiffness, by using the optimum variable bias current in (22) compared with the conventional Class-A constant bias current mode of operation. As expected, the savings are significant at low vibration levels. The value at which the constant bearing stiffness curves intersect with the zero energy saving line (horizontal axis) indicates the limiting case where the optimum bias current approaches the 0.5 value. The control currents would saturate for orbit sizes greater than those at the crossing points. This is not an issue for ) settings. In many applications, the norsoft bearing (low malized retainer bearing clearance is set to below 0.6; hence, the use of a normalized bearing stiffness of 0.7 or lower would ensure that no saturation would occur for vibrations within the clearance of the retainer bearings. The “best” value for bearing stiffness is a tradeoff between transmitted forces and the rotor vibrations. For the unidirectional mode, the optimum setting of the bias current is (24) by . Therefore, all regions in the figure may not be usable dedue to current pending on the value of the bearing stiffness limitations as set out by (14) and (15). The results clearly show the advantages of using differential mode variable bias current controllers in terms of the energy consumption. As a numerical example, consider a radial bearing for a ver. If tical rotor with a desired bearing stiffness of (i.e., 10% of the air gap) at a the shaft orbit size is steady-state operating running speed, under Class-A mode with , a position feedback gain of is required from (12). This gives an average power dissipation of . However, under variable bias control, the minimum energy con, and , sumption can be achieved with giving the same desired bearing stiffness and hence the same orbit size but with a much lower power dissipation figure of . That is, an energy saving of approximately 92% that results in quieter operation and less heat generation. If the orbit size changes due to external disturbances or transient effects, the to stay on the optimum energy controller can adjust and consumption conditions without affecting the system dynamics. If the controller is adaptive and selects the bearing stiffness in accordance with the operating conditions, such as fuzzy logic can easily be achieved controllers [15], then this desired by using the proposed optimized bias current, and the corre. A discussion of control strategies is outside the sponding scope of this paper, but the proposed control of the bias current can be used with any open-loop, closed-loop, or nonlinear controllers with slight modification. V. STATIC LOADING This gives a minimum average power dissipation of (25) Fig. 5 shows the optimum bias current settings and the corresponding average power dissipation curves as a function of the orbit size for both modes of operation. This figure is normalized In this section, the effect of the static load on the bearing dynamic properties, bias current optimization and energy consumption is studied. The static load carrying capacity is provided by the difference in bias current in the opposing coils as discussed earlier. The bias current in coils opposite to the direc, and for the tion of the static load is set to SAHINKAYA AND HARTAVI: VARIABLE BIAS CURRENT IN MAGNETIC BEARINGS FOR ENERGY OPTIMIZATION Fig. 6. Bearing stiffness capacity as a function of the orbit size for different static load levels for both mode of operations. The static load increases in the direction of arrows from 0 to 0.5 with increments of 0.1. opposing coils. The bias current denotes the average (mean) bias current, and can be calculated from (6). As shown in (8), the static load increases the negative bearing stiffness by , and an increase in position feedback gain is required in order to achieve the desired bearing stiffness as given by (12). This also reduces the stiffness capacity of the magnetic bearing. Fig. 6 shows the reduction of bearing stiffness capacity with increasing static load for both differential and unidirectional modes of operations. The zero static load curves are the same as shown in Fig. 2. The unidirectional mode again gives inferior bearing stiffness capacity, and saturates at a lower orbit size compared with the differential mode. Obviously, the design of a magnetic bearing should take into account the static load as discussed in [17]. An average power dissipation expression can be obtained by in (4) for the differential mode, and coil current inserting expressions into (18). This gives the following term to be added to (19): 1057 Fig. 7. Optimum bias current settings under static loading and the corresponding average power dissipation as a function of the orbit size A for K = 0:5 (The static load increases in the direction of arrows starting from 0 with increments of 0.1). Fig. 8. Percentage energy saving of using the optimized bias current compared with Class-A mode for K = 0:5. The static load increases in the direction of arrows starting from 0 with increments of 0.1. TABLE II DATA (26) Differentiating with respect to gives the following optimum bias current setting, which includes the effect of static load: (27) This converges to (22) for . Fig. 7 shows the optimum bias current settings for a typical bearing stiffness of and the corresponding average power dissipation figures as a function of the orbit size . An increase in static load requires a higher bias current setting, and hence there is higher energy consumption. The percentage energy saving compared with the Class-A mode is shown in Fig. 8. An increase in static load reduces the percentage energy savings. As a numerical example consider a horizontal rotor supported by two radial magnetic bearings. For a desired bearing stiffness , and a static load at each axis of the magnetic bearof to the vertical) of , the bias currents ings (which are at the lower and upper coils should be 0.4 and 0.6 respectively ) for Class-A mode of operation. If the orbit (i.e., size is at the steady-state operating speed of the rotor, 1058 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 3, MARCH 2007 Fig. 9. A view of the experimental rotor bearing system used in the simulations. then the average power dissipation is . However, under variable bias current differential mode, the optimum bias current , giving the lower and upper coil bias currents of apis ) with proximately 0.09 and 0.45, respectively (i.e., an average power dissipation of . This corresponds to an energy saving of approximately 58% without changing the bearing stiffness, and hence the same orbit size. VI. EXPERIMENTAL IMPLEMENTATION The experimental magnetic/bearing system used to validate the theoretical analysis consists of a uniform flexible steel shaft of length 2 m and radius 0.025 m, with four 10-kg disks of radii 0.125 m. The total rotor mass is 100 kg and it is mounted horizontally on two radial magnetic bearings. The control current of the magnetic coils was supplied through eight amplifiers. The magnetic bearings have an air gap of 1.3 mm and each is protected by a retainer bearing having a 0.75 mm clearance. The magnetic bearing data are provided in Table II. Eight eddy-current displacement sensors are placed at four planes with a 45 angle with the vertical line. A schematic view of the experimental setup showing the rotor, magnetic bearings, retainer bearings and sensors is provided in Fig. 9. A dSPACE digital signal processor (DSP), together with Simulink and Realtime Workshop software, was used to implement the control algorithm. A rigid rotor assumptions were made because the experiments were conducted at 10 Hz, which is well below the first flexural critical speed of 30 Hz. The orbit size was measured by a recursive calculation of synchronous Fourier coefficients along the axis of the magnetic forces. This approach enables changes in orbit size to be estimated without excessive delay. A PID controller was used to control the rotor, and the proportional gain set to achieve a bearing stiffness N/m (i.e., ). Derivative and integral of controller gains were set to achieve effective bearing damping N s/m, and integral action of N/m/s. These of physical values were divided by before being supplied to the PID controller in order to reflect the effect of changing bias current. The optimum bias current control system to achieve a specified bearing stiffness is shown in Fig. 10. Fig. 10. Bias current control to minimize energy consumption for a specified bearing stiffness k (RFT: recursive Fourier transform). The average power dissipation per Ohm of coil resistance was calculated by integrating the square of the measured coil currents over one synchronous period, and dividing the results by the syn. An average over 10 periods was taken to chronous period minimize the effect of measurement noise. The unknown unbalance distribution along the rotor was the only external forcing to generate a steady-state orbit. The rotor weight was supported by the two magnetic bearings. The coils were placed with a 45 angle to the vertical, hence the static load at each axis was N . The synchronous vibration amplitudes at each axis is calculated by a recursive Fourier transform algorithm. The maximum of the amplitude values along two axes was used as the orbit size for each magnetic bearings. Under the Class-A mode of A), bias currents for the upper and lower coils operation ( were set to 4.3 A and 5.7 A, respectively, giving vibration amplitudes of 86.4 m and 100.1 m along the two axes for MB-2 ). The predicted average power dissi(i.e., the orbit size pation for MB-2 was 102.63 W/ , compared with the measured consumption of 102.44 W/ . When the system was switched to variable bias current mode, the optimum bias current for MB-2 A. The measured average power settled at 2.04 A with dissipation was 25.80 W/ , which corresponds to 74.8% energy savings. The top plot in Fig. 11 shows the predicted and measured average power dissipation when the bias current was manually decreased from 5 A to 2.2 A with an increment of 0.2 A. The middle plot shows the predicted and measured percentage energy savings compared with the Class-A mode of operation. The significant energy saving introduced by the variable bias current SAHINKAYA AND HARTAVI: VARIABLE BIAS CURRENT IN MAGNETIC BEARINGS FOR ENERGY OPTIMIZATION 1059 The optimum bias current expression as a function of the orbit size and the required bearing stiffness are obtained analytically for both differential and unidirectional mode of operations, and these formed the basis of the suggested adaptive control of the bias current. The effect of static loading on the energy consumption and the maximum achievable bearing stiffness is also studied. The suggested bias current control technique was applied to an experimental rotor horizontally supported by two radial magnetic bearings. Experimental results showed the validity of the analytical predictions. The measured energy savings agreed well with the analytical predictions. The proposed method was able to achieve 74.8% energy saving compared with the standard Class-A mode of operation. REFERENCES Fig. 11. Experimental and predicted results for MB-2 when the bias current is manually decreased. Top plot: average power dissipation, Middle plot: percentage energy saving compared with Class-A mode of operation, Bottom plot: i for static force generation. 1 method can clearly be seen. The bottom plot shows the predicted , which provides the static lift force. Integral and measured action ensures zero steady-state errors adjusting the bias currents at opposing coils. The results suggest that the theoretical values were slightly higher than the measured difference. This may be due to errors in the linearized model and also misalignment errors of opposing poles. As described above, the optimum bias current was set in accordance with (27), which aims to achieve zero minimum current at lower coils. However, due to measurement and modeling errors, and also due to inaccuracies in the recursive Fourier transform especially under transient conditions, the control current may drive the coil current temporarily to negative values. Although negative currents are prevented by a zero saturation block, this saturation may not be desirable. This can be prevented by adding a safety margin to (27). Because such transient saturation was experienced during the experiments, further tests were carried out with safety margins of 0.1, 0.2, and 0.3 A. The power savings were slightly reduced to 73.9%, 72.8%, and 71.5%, respectively, compared with the 74.8% saving with zero safety margin. Because of the small cost of introducing a safety margin, it may be advisable to implement it if zero saturation may be a potential problem for the power electronics or for the system dynamics. It was observed that a safety value of 0.2 A was sufficient to ensure that no saturation was occurred during the experiments. VII. CONCLUSION Magnetic bearings offer contact free support of rotating machinery with losses less than that of the oil-film bearings. However, there is still a scope to achieve significant energy savings through implementing adaptive control of the bias current. [1] G. Schweitzer, “Active magnetic bearings—Chances and limitations,” in Proc. IFTOMM Sixth Int. Conf. Rotor Dynamics, Sydney, Australia, 2002, vol. 1, pp. 1–14. [2] M. A. Pichot and M. D. Driga, “Loss reduction strategies in design of magnetic bearing actuators for vehicle applications,” IEEE Trans. Magn., vol. 41, no. 1, pp. 492–496, Jan. 2005. [3] P. E. 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Necip Sahinkaya received the B.Sc. degree from the Istanbul Technical University, Istanbul, Turkey, in 1974, and the M.Sc. and Ph.D. degrees from the University of Sussex, Sussex, U.K., in 1976 and 1979, respectively. Following periods as an academic at the universities of Sussex, Strathclyde and Bath, he joined an international shipping company as a general manager in late 1988. He rejoined the University of Bath in May 2000, and is now a Reader in the Department of Mechanical Engineering. His research interests include active control of magnetic bearings, control and dynamic analysis of multi-physics systems. He was the Chair of the Organizing Committee for the UKACC Control 2004 conference. IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 3, MARCH 2007 Ahu E. Hartavi received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the Istanbul Technical University, Istanbul, Turkey, in 1997, 2000, and 2005, respectively. She is currently working as a Post Doctoral Researcher in the Automotive Controls and Mechatronics Research Center (AUTOCOM) at the Istanbul Technical University. Her research interests include modeling, design, and control of active magnetic bearings, power optimization, electrical machines and drives, modeling and control of powertrain, hybrid electric vehicles, solar and fuel cells. She is the Workshops Chair of the 2007 IEEE Intelligent Vehicle Symposium.