Academia.eduAcademia.edu

Figure 1 – uploaded by Yannis L Karnavas

See full PDFdownloadDownload figure
Figure 1. Representation of the design problem considered: a) inner rotor PMSM cross section, b) detailed geometry of slots, airgap and permanent magnets (one pole) and (c) typical PMSM drive system. The PMSM design procedure relies on the amplitude of the fundamental airgap flux density Bs and 2.1. Magnetic properties

Figure 1 Representation of the design problem considered: a) inner rotor PMSM cross section, b) detailed geometry of slots, airgap and permanent magnets (one pole) and (c) typical PMSM drive system. The PMSM design procedure relies on the amplitude of the fundamental airgap flux density Bs and 2.1. Magnetic properties

Related Figures (8)

Figure 2. Magnetic considerations of PMSM design: a) airgap flux density variation with and without considering magnetic leakage, b) approximation of magnetic leakage as a function of poles number
Figure 2. Magnetic considerations of PMSM design: a) airgap flux density variation with and without considering magnetic leakage, b) approximation of magnetic leakage as a function of poles number
Figure 3. Simplfied illustration of Grey Wolf Optimizer algorithm, a) flowchart, b) pseudo-code As it is claimed by the authors in [19], the main advantage of GWO algorithm over most of the well- known meta-heuristic algorithms is that the GWO algorithm operation requires no specific input parameters and, additionally, it is straightforward and free from computational complexity. Further, its advantages include ease of transformation of such concept to the programming language and ease of comprehensibility. But, as it was stated in the Introduction, there is no meta-heuristic optimization technique well suited for all optimization problems. For this reason, we examine here the GWO suitability for the optimization problem desribed previously. The GWO algorithm mimics the leadership hierarchy and hunting mechanism of gray wolves pack in nature. Four types of grey wolves such as alpha (a), beta (f), delta (6), and omega (w) are employed for simulating the pack leadership hierarchy. Alpha wolves represent the top-level i.e. the absolute leaders (which also give the best solution to the optimization problem). Beta wolves represent a second-level group in hierarchy which actually help alphas in decision making while at the same time command the other lower-level wolves. Omega wolfs refer to the lowest hirarchy level, thus the wolves belong to this group have to submit to all the other dominant wolves. Finally, if a wolf can not be characterized as a, # or w, he/she is called delta. Delta wolves have to submit to alphas and betas, but they dominate the omega ones. In a nature- context environment, the delta wolves may be scouts, elders, sentinels or caretakers. In addition to this social hierarchy, group hunting is the other basic characteristic of GWO, because it provides interesting social behaviour of grey wolves. There are three main steps of hunting, searching for prey, encircling prey, and attacking prey. The mathematical implementation of these steps to GWO (or modifications of them) is actual the mean of performing optimization in a complex problem. An illustration of the GWO algorithm in flowchart and pseudo-code form is depicted in Figure 3. The basic mathematical expressions used by GWO (prey's encircling behavior) is formulated by,
Figure 3. Simplfied illustration of Grey Wolf Optimizer algorithm, a) flowchart, b) pseudo-code As it is claimed by the authors in [19], the main advantage of GWO algorithm over most of the well- known meta-heuristic algorithms is that the GWO algorithm operation requires no specific input parameters and, additionally, it is straightforward and free from computational complexity. Further, its advantages include ease of transformation of such concept to the programming language and ease of comprehensibility. But, as it was stated in the Introduction, there is no meta-heuristic optimization technique well suited for all optimization problems. For this reason, we examine here the GWO suitability for the optimization problem desribed previously. The GWO algorithm mimics the leadership hierarchy and hunting mechanism of gray wolves pack in nature. Four types of grey wolves such as alpha (a), beta (f), delta (6), and omega (w) are employed for simulating the pack leadership hierarchy. Alpha wolves represent the top-level i.e. the absolute leaders (which also give the best solution to the optimization problem). Beta wolves represent a second-level group in hierarchy which actually help alphas in decision making while at the same time command the other lower-level wolves. Omega wolfs refer to the lowest hirarchy level, thus the wolves belong to this group have to submit to all the other dominant wolves. Finally, if a wolf can not be characterized as a, # or w, he/she is called delta. Delta wolves have to submit to alphas and betas, but they dominate the omega ones. In a nature- context environment, the delta wolves may be scouts, elders, sentinels or caretakers. In addition to this social hierarchy, group hunting is the other basic characteristic of GWO, because it provides interesting social behaviour of grey wolves. There are three main steps of hunting, searching for prey, encircling prey, and attacking prey. The mathematical implementation of these steps to GWO (or modifications of them) is actual the mean of performing optimization in a complex problem. An illustration of the GWO algorithm in flowchart and pseudo-code form is depicted in Figure 3. The basic mathematical expressions used by GWO (prey's encircling behavior) is formulated by,
t is any current iteration. Especially for vector Aj, it can take values in the gap [-2a, 2a], so as when |A,|<1 the wolves are forced to attack the prey. This is translated as the method's exploitation ability while searching for prey is the exloration ability. When |A,|>1 the wolves pack is enforced to diverge from the prey. In any iteration, the wolves’ position update is performed by, 3.2. Genetic Algorithms and Particle Swarm Optimization a For fair comparison purposes, two other meta-heuristic optimization techniques are also applied to the same problem, genetic algorithm (GA) and particle swarm optimization (PSO). These techniques have been proved robust and efficient enough to most engineering problems in the past. Moreover, since they are well-known, their principles of operation have been extensively presented in the literature and therefore no relative information is given here. The reader can refer to [25],[26] for further implementation details.
t is any current iteration. Especially for vector Aj, it can take values in the gap [-2a, 2a], so as when |A,|<1 the wolves are forced to attack the prey. This is translated as the method's exploitation ability while searching for prey is the exloration ability. When |A,|>1 the wolves pack is enforced to diverge from the prey. In any iteration, the wolves’ position update is performed by, 3.2. Genetic Algorithms and Particle Swarm Optimization a For fair comparison purposes, two other meta-heuristic optimization techniques are also applied to the same problem, genetic algorithm (GA) and particle swarm optimization (PSO). These techniques have been proved robust and efficient enough to most engineering problems in the past. Moreover, since they are well-known, their principles of operation have been extensively presented in the literature and therefore no relative information is given here. The reader can refer to [25],[26] for further implementation details.
Figure 4. Convergence behavior of applied optimization methods to PMSM design problem, a) GWO,b) GA, c) PSO The left part of Table 3 shows the computational costs of the adopted and applied methods. Here the first place goes to GWO method which provides solutions in the order of magnitude of 2-3 seconds fo: our problem. PSO needs almost 1.5 times more computational time, while GA is the slowest optimizatior method with a response time of almost 10 seconds.
Figure 4. Convergence behavior of applied optimization methods to PMSM design problem, a) GWO,b) GA, c) PSO The left part of Table 3 shows the computational costs of the adopted and applied methods. Here the first place goes to GWO method which provides solutions in the order of magnitude of 2-3 seconds fo: our problem. PSO needs almost 1.5 times more computational time, while GA is the slowest optimizatior method with a response time of almost 10 seconds.
Figure 5. PMSM design obtained (CF) by GWO application. a) flux density distribution, b) hea density plot
Figure 5. PMSM design obtained (CF) by GWO application. a) flux density distribution, b) hea density plot
Table 2. Main objective quantities results through GWO, GA and PSO optimization methods
Table 2. Main objective quantities results through GWO, GA and PSO optimization methods
ble 3. Computational cost of optimization methods used and corresponding PMSM design solutions’ sl current density
ble 3. Computational cost of optimization methods used and corresponding PMSM design solutions’ sl current density
In terms of converging, there are also interesting results. F igure 4 shows the behavior of GWO, GA and PSO convergence throughout the iterative process (CF2 case). It can be seen that GWO converges to the 12-variable optimal combination in about 500 iterations giving a approximately twice iterations while trying to explore/exloit the so cost function value of 0.53. GA needs ution search space and converges to 0.7. PSO seems to converge prematurely scoring 0.87. Our investigation continues with the thermal calculations involving slot current density for each one of the geometries. These 3. It can be seen that all designs develop a current density lower t are summarized in the right part of Table hat 800A/cm? and can be realized as air- cooled machines (as stated in §2.2). One of the proposed by GWO algorithm optimized design (optimized for efficiency i.e. CF2) was implemented in FEM and thermal analysis operation under nominal load are shown in Figure 5, where t software and visualization instants of its he relevant geometry, the flux density distribution as well as the heat density plots are depicted. It can be seen that the maximum magnetic flux density, in any part of the machine is lower than the constraint set, with a maximum of approximately 1.42T (with a 1.6T constraint), while the maximum temperature rize is about 39°C (without an outer enclosure/frame structure). In other words, these results from both FEM and thermal analyses are judged absolutely satisfactory.
In terms of converging, there are also interesting results. F igure 4 shows the behavior of GWO, GA and PSO convergence throughout the iterative process (CF2 case). It can be seen that GWO converges to the 12-variable optimal combination in about 500 iterations giving a approximately twice iterations while trying to explore/exloit the so cost function value of 0.53. GA needs ution search space and converges to 0.7. PSO seems to converge prematurely scoring 0.87. Our investigation continues with the thermal calculations involving slot current density for each one of the geometries. These 3. It can be seen that all designs develop a current density lower t are summarized in the right part of Table hat 800A/cm? and can be realized as air- cooled machines (as stated in §2.2). One of the proposed by GWO algorithm optimized design (optimized for efficiency i.e. CF2) was implemented in FEM and thermal analysis operation under nominal load are shown in Figure 5, where t software and visualization instants of its he relevant geometry, the flux density distribution as well as the heat density plots are depicted. It can be seen that the maximum magnetic flux density, in any part of the machine is lower than the constraint set, with a maximum of approximately 1.42T (with a 1.6T constraint), while the maximum temperature rize is about 39°C (without an outer enclosure/frame structure). In other words, these results from both FEM and thermal analyses are judged absolutely satisfactory.
Related topics:
Computer ScienceElectrical Computer Engineering
580 California St., Suite 400
San Francisco, CA, 94104
© 2025 Academia. All rights reserved