Design optimization of double-acting hybrid magnetic thrust bearings with control integration using multi-objective evolutionary algorithms
Introduction
A magnetic bearing is a system that is supposed to levitate a rotor by using magnetic force with or without active control. As compared to conventional bearings, magnetic bearings have unique characteristics of absence of wear and lubrication ensuring higher reliability of the system with longer life and clean room environment [21]. Magnetic bearings where the magnetic flux is generated solely by electromagnets are called active magnetic bearings (AMB). If the magnetic flux is generated solely by permanent magnets they are called permanent magnetic bearings (PMB). If the magnetic flux is generated by the combination of electro- and permanent magnets, they are called hybrid magnetic bearings (HMB).
Still the state of the art magnetic bearing technology has no standardized methods on optimum designs. This requires one’s expertise in their design. Allaire et al. [1] presented the design of a prototype of thrust magnetic bearing for the high load-to-weight ratio. A design method for magnetic devices with the topology and the material optimization was described by Dyck and Lowther [10]. Zeisberger et al. [24] studied the optimization of levitation forces for an ideal super-conductive magnetic bearing.
The classical design of magnetic bearings depends on the ideal magnetic circuit theory, which deviates drastically from experiments [6], [14], [21]. Groom and Bloodgood [11] proposed a model by adding the loss and leakage factors to ideal models with and without bias permanent magnets. Subsequently, Bloodgood et al. [3] applied the theory for the optimal design of a thrust magnetic bearing with bias permanent magnets.
Schroder et al. [19] studied the multi-objective optimization of a nonlinear controller for a magnetic bearing. Afterward, Schroder et al. [20] studied on online genetic tuning of magnetic bearing using multi-objective genetic algorithms (MOGAs). Sequential Quadratic Programming was used by Bloodgood et al. [3] for optimization of the hybrid magnetic thrust bearing (HMTB) to minimize the power consumption as the objective. The optimization of radial magnetic bearings using finite element techniques and differential evolution algorithms was studied by Stumberger et al. [22]. The optimal design of radial active magnetic bearings integrated with their control using SOGA was studied by Chang and Chung [4]. The integration of control was done on constraints but not in objectives. Hu et al. [12] investigated the design of magnetic bearings with additional constraints of small air gaps. Chen and Chang [5] attempted a single objective genetic algorithm (SOGA) problem for the optimization of PID controller for magnetic bearings. The optimization of the load-to-weight ratio and the powerloss of a HMTB by using SOGA was studied by Rao and Tiwari [15]. Rao and Tiwari [16] presented a design methodology for single acting AMBs by using MOGAs, however, the permanent magnets and controller were not considered. Moreover, only two objectives were considered for the optimization.
In the present paper, multi-objective optimization of double-acting hybrid magnetic thrust bearings (DAHMTBs) integrating both the actuator and the controller as a single system has been investigated using real coded genetic algorithms. An actuator or a controller could be independently optimized. But when the actuator is optimized independent of the controller, the chosen actuator might not be a feasible one when constraints of the controller are considered. This requires redesign of the actuator. Similarly, when the controller is optimized independent of the actuator, the chosen controller might not be a feasible one when constraints of the actuator (such as the current density in the coil and the magnetic flux density in the stator iron) are taken into consideration. Moreover, the performance tradeoffs of the actuator and the controller, when they are optimized independently might not be the optimum when considered as a unified system. Hence, it is worthwhile to integrate the design of the actuator and the controller as a unified system and the same is investigated in the present paper. Three objectives namely the powerloss, the weight of bearing and the load capacity have been considered for the actuator. Whereas, two objectives namely the input performance index and the dynamic performance index have been considered for the controller. Constraints considered for the actuator are the maximum allowed current density in the coil, maximum allowed flux density in the stator iron, maximum space available, and maximum powerloss allowed. The stiffness and eigen values of characteristic equation have been considered as constraints for the controller.
Section snippets
Macro geometry of the actuator
The macro-geometry of a double-acting hybrid magnetic thrust bearing (DAHMTB) is shown in Fig. 1. The different design parameters are shown in Fig. 2, and are described in Nomenclatures. The radius of the shaft rs, the clearance between the shaft and the inner radius of the magnetic bearing lc, the inner radius of the bearing ri, the outer radius of the bearing ro, the inner radius of the coil rci, the outer radius of the coil rco, the height of the coil hc, the thickness of the coil tc, the
Fundamental relations
Fundamental relations related to the single-acting and double-acting HMTBs are provided in this section. These relations can be reduced for AMBs, by ignoring parameters related to permanent magnets to zero. From the magnetic circuit theory, according to Amperes’ circuital law [11], the magnetic flux density can be expressed aswhere n is the number of turns, ib is the bias current, ic is the control current, Br is the remnant flux density of the
Multi-objective problem formulations
The multi-objective optimization problem consists of objective functions to be optimized by design variables while satisfying certain constraints in the process of optimization. A model of multi-objective optimization problem is provided in Table 1.
Expressions for objective functions and constraints have been presented for the optimum design of double-acting hybrid magnetic thrust bearings in the following subsections.
Numerical results
For the present optimization problem five objectives with fourteen design variables and fourteen constraints have been presented in Section 4. Due to the complexity involved in the optimization problem, obtaining a closed form formula for the solution is very difficult; and one has to opt for a numerical optimization. Population (stochastic) based methods such as genetic algorithms have proven to land on the global optimum within the provided search space with higher probability in contrast to
Conclusions
In the present work an optimal design of double-acting hybrid magnetic thrust bearing has been carried out. Double-acting actuators and controller are optimized as a unified system. Two different geometries have been assumed for the individual actuators of the double-acting bearing. For the present case the uni-mode control has been assumed. Real coded genetic algorithm has been implemented to carry out the constrained multi-objective optimization of the magnetic bearing. Results have been
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