Implementation of hybrid pattern search genetic algorithm into optimizing axial-flux permanent magnet coreless generator (AFPMG)

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1 DOI 1.17/s ORIGINAL PAPER Implementation of hybrid pattern search genetic algorithm into optimizing axial-flux permanent magnet coreless generator (AFPMG) C. Long. Lok 1 B. Vengadaesvaran 1 S. Ramesh 1 Received: 23 April 216 / Accepted: 22 September 216 Springer-Verlag Berlin Heidelberg 216 Abstract Complex real-world problems can be solved by heuristic optimization efficiently. Improved hybrid optimization method using pattern search (PS) algorithm and genetic algorithm (GA) in MATLAB is presented in this paper, and the optimization is based on the popular multi-objective sizing equation. This hybrid model utilizes concepts from GA and invents new-generation chromosomes not only through mutation and crossover operation but also by mechanisms of PS. In the design procedure, hybrid optimization model with some predefined constraints for the objective function has been taken into consideration which includes the physical limitations and performance characteristics. The dimensions of the machine are optimized with multiple adjustments to the number of magnet pole, the number of winding turns and air-gap distance to gain the highest power density within desired dimensional constraints. Moreover, the electromagnetic field and electromagnetic characteristics of the chosen generator are subject to finite element analysis. A finalized low-power AFPM generator is fabricated, examined and testified to produce desired output. It has been observed that the experiment result agreed with the simulation result. Keywords AC generator AFPM generator (AFPMG) Direct search algorithm Electric machine Hybrid GA PS Power generator Sizing equation B B. Vengadaesvaran venga@um.edu.my 1 Department of Physics, Faculty of Science, University Malaya, 563 Kuala Lumpur, Malaysia Nomenclature AFPM EMF FEA GA MATLAB PS RFPM RPM 1 Introduction Axial-flux permanent magnet Electromotive force Finite element analysis Genetic algorithm MATrix LABoratory Pattern search Radial-flux permanent magnet Rotation per minute In the topic of permanent-magnet electric machine construction, recent research works have shown that the usage of neodymium iron boron (NdFeB) permanent magnet has drastically increased over the last decades mainly due to cheaper cost and the awareness of non-pollution energy alternative, particularly the wind energy generator. AFPM machines can be single-sided or double-sided, core or coreless, consist of surface-mounted or interior PM and are single- or multi-staged configuration [1]. Although there are many categories of permanent magnet generator available, only the axial-flux permanent magnet generator is discussed here. Among those types of AFPM machine, double-sided AFPM electric machine is most widely researched and used in the industry. The structure of double-sided AFPM machines has the highest torque-to-volume ratio, especially if the machine is designed with high number of pole pairs [2]. This type of generator has a simple and sturdy structure together with its excellent electro-thermal properties [3]. The main advantage in this study is the elimination of iron core, and thus, the reluctance in the magnetic equivalent circuit has been critically minimized. With no stator iron core, there are

2 eddy currents induced in the stator conductors due to rotating magnetic field flux from the permanent magnets [4]. According to Wang et al., their study has shown that about 18 % of the losses are attributed to the eddy current losses in axial-flux machine [5]. Some papers have shown that only very little cogging torque exists within the air gap of an iron-less stator [6,7]. According to [8], AFPM coreless machine design could eliminate the cogging torque completely. Nevertheless, parasitic torque ripples still exist due to limitation in machine design. In this study, only simple results of the eddy current and cogging torque between an iron core stator and coreless stator are compared. Therefore, detailed work dealing with cogging torque reduction and eddy current elimination will be studied in the future. Over the years, various optimization techniques and theories were developed by engineers and scientists to improve the electric machine structure with the aims to achieve desired characteristics. Huang et al., sizing equation was derived to compute the power density equation for RFPM machines - in this paper, effective method to compare the topologies of various electric machines was presented too. Other papers also derived the sizing equation for AFPM electric machines, but no study on electric machine optimization was developed [9]. AFPM electric machine can be optimized with dimensional adjustment via geometrical parameters, output characteristic, analytical methods or soft computing methods. For instance, analysis of coreless axial-flux permanent magnet generator was done by Virtic et al. using magnetic equivalent circuit, and the proposed method is lack of dimensional modification [1]. Chung et al. presented 3D electromagnetic finite element analysis and basic structure adjustment, but only four parameters (output power density, machine s efficiency, maximum current density and maximum rms phase voltage) were considered as optimization variables, and the optimization method used was the Conjugate Gradient which is not suitable to solve a non-linear multi-objective function [11]. Mirzaeian et al. presented a multi-objective optimization model that utilizes the Genetic-Fuzzy Algorithm (GFA) for optimal design of a Switched Reluctance Motor (SRM) with high efficiency and low torque ripple as objective functions [12]. In [13], the authors introduce soft computing based on genetic algorithm method to optimize multi-objective slotted TORUS AFPM machine parameters; this method shows the tradeoffs between machine output power density and size. It is a method that minimizes the AFPM TORUS machine using various parameter considerations that include multiple dimensional variables in a non-linear fitness functions. In the aforementioned research subject, the design procedures were not considered using hybridization of two different optimization algorithms for multi-objective function coreless AFPM machine search. Moreover, problems exist in the Evolutionary Algorithm (EA) using genetic algorithm such as slow convergence speed, easily fall into the partial optimum [14]. As a result, the design parameter obtained by utilizing genetic algorithm may not be that accurate. In this paper, analytical and impact study of the hybrid optimization method into a coreless AFPM generator (AFPMG) is presented; the paper highlights the hybrid optimization used, the design of such generator, and the reason why such options are selected to achieve the desired objectives, through simulation and experiment. 2 Machine parameter analysis 2.1 Power analysis using sizing equations A good electric machine design incorporates substantially good design knowledge and machine design theory. The sizing equation is one of the basic machine design knowledge acquired into this design. The sizing equation shows that the electric machine output power is interdependent on the design dimensions. According to Huang et al., sizing equation derivatives [9], if the stator leakage resistance and inductance are neglected, the general output power for any electric machine is calculated using: ( ) T m P o = η e(t) i(t)dt = mηk p E pk I pk (1) T where EMF e(t) and E pk are air gap phase EMF and the peak value. The current i(t) and I pk are the phase current and the peak phase current value, m is the number of machine phases, η is the machine efficiency and T is the period of one cycle of EMF. f i (t) = i(t)/i pk and f e (t) = e(t)/e pk are the expressions for current waveform and normalized EMF. Meanwhile in a typical sinusoidal waveform generator, the value for electrical power power-waveform factor, K p is.5 and the current-waveform factor, K i is 2[15]. The peak EMF of electric machine is defined as: E pk = d dt = K e N ph B g f P (1 λ2 )D 2 o (2) where Λ is the air-gap flux linkage in each phase, K e is the EMF factor, N ph is the winding turns per phase, f is the machine frequency, B g is the flux density in air gap, P is the machine pole number in pair, and the winding factor, K w. The diameter ratio, λ is equal to D i /D o, where D o is the outer diameter of surface and D i is the inner diameter of surface. The peak current is defined as: I pk = Aπ K i (1 + λ) 2 D 2m 1 N ph (3) where m 1 and A are the number of phases in each stator and electrical loading value. The derived sizing equation in terms of D 2 L e is presented as:

3 1 m π 1 + K m 1 P o = 2 K e K i K p K L ηb g A f P ( 1 + λ (1 λ 2 ) 2 ) D 2 L e (4) where L e is the effective length in axial direction for AFPM machine, K = is ratio of electrical loading on rotor and stator in a no-armature winding machine topology, K L is the co-efficient considering the factor of temperature rise, losses and design specifications [15]. The total power density of AFPM machine is defined as: P den = P o π 4 D 2 tot L tot where D tot and L tot are the machine total outer diameter and total effective stack length. The outer diameter of surface, D o can be expressed as: D o = 3 P o πm 4m 1 K e K p K i AB g η f P ( 1 λ 2 ) (1 + λ) In simplicity, the variables of the AFPM coreless generator are carefully chosen to maximise the power density which is subject to the following parameters: P den D o P o E pk 15 V I pk 1A η 85 % f subject to.5 Bg mm g 6. mm.4 λ.75 P N ph Le 1 mm A 2.2 Cogging torque analysis Cogging torque can be a problem for iron-stator core in AFPM machine; in addition to undesired noises and vibrations, the inherent cogging torque in machine can cause problems during start-up. Nevertheless, little cogging torque is present in a coreless AFPM generator design. The cogging torque is derived as [16]: = 1 dr 2 2 dθ T cog (8) (5) (6) (7) where is the air-gap flux, θ is the angular position in electrical degree and R is the air-gap reluctance. To reduce cogging torque in electric machine, magnet skew can be used in most iron-stator core design [17]. Alternatively, fringing in magnetic flux path may also result in torque reduction. 2.3 Losses and efficiency Losses in bearing of an AFPM machine vary with the bearing friction and machine structural design. Intuitively, compared with conventional rotating machines, bearing loss of a largediameter AFPM machine may be proportionate to the rotor mass. The bearing loss known as mechanical losses can be derived as: P b = 3k bg t θ 1π where θ is the angular speed of the machine rotor, k b is a factor that lies between 1 and 3 [18], G t is the rotor mass in metric kg. For single-rotor double stator configuration, the bearing axial load can be reduced by aligning the rotor precisely between two stators. Alternating field in conductors in an AFPM stator induces eddy current. When an AFPM machine is operating in relatively high-frequency alternating field, these induced eddy currents may lead to serious additional losses in the machine and decrease the machine efficiency [5]. Thus, the winding eddy current losses can be derived as: P eddy_cu = (2π B g fd wire 1 3 ) 2 32ρ cu V cu (1) where B g is the air-gap flux density, D wire is the diameter of coil wire, f is the frequency of rotor, ρ cu is the conductor resistivity, and V cu is the volume of the copper winding [19]. A paper by Atallah at el. shows that eddy current losses may also appear in the rotor disc under the permanent magnets, since the fundamental air-gap field usually rotates in synchronism with the rotor and the time harmonics in the current waveform and space harmonics in the winding distribution are generally small. Therefore, these eddy current loss components are usually neglected [2]. In short, the theoretical calculation of efficiency is to assess the losses in every aspect of the machine. Efficiency of an electric machine is expressed as: n = (9) P o P o + P mech + P cu + P eddy_cu + P core (11) where P o, P mech, P cu and P eddy_cu, P core are the output power, rotational loss, copper loss, eddy current loss due to static high frequency and high-frequency magnetic field and core loss =, respectively [21].

4 3 Optimization algorithms 3.1 Genetic algorithm (GA) GA is a meta-heuristic search that is inspired by the process of natural evolution which generates solutions to optimization problems. It utilizes processes such as mutation, selection, crossover and inheritance [22]. Advantages of GA neither depend on the starting point of the function searches nor require any information of the fitness function/constraint function. Often, GA searches for global optima rather than local optima of a fitness function. Since GA does evolve candidate parameters in a population, each candidate is coded as a long binary string of and 1 called the chromosome representation. The chromosome is decoded and evaluated of its fitness performance using a performance function in the optimization tool. For example, in optimizing an AFPM generator, the chromosomes of the optimized model are N ph, B g, A, g, P out, E pk, I pk,λ,η,d, L e (refer Table 1). After an evaluation is complete, a random generator randomly chooses multiple pairs of higher-quality chromosome to perform subsequent GA evaluation called the crossover and mutation. Upon multiple crossover and mutation evaluation, the weaker chromosomes previously selected are replaced by the stronger chromosomes in the current generation until the set stopping criteria are reached [23]. With some improvements by the genetic operators, the real-coded GA obtained a better fitness solution than the binary-coded GA for continuous problems [24]. 3.2 Pattern search optimization tool (PS) Traditional optimization method uses information about gradient of a derivative to search for an optimal point by looking for best-fitted value among a set of points in a region. As opposed to that, pattern search solves a given optimization problem without the need to acquire information gradient of the objective function [25]. Pattern search tool is also known as Direct Search Algorithm that searches a pool of points around the current points [26]. This method can solve problem for any non differentiable or even continuous objective function, and it works by computing a number of points that get comparatively close to the optimal point. In this process, the tool searches a pool of points called a mesh around the current points (the points computed in the previous step). Next, this algorithm formulates a mesh by adding the current points to a scalar multiple of a fixed set of vectors called pattern [27]. Upon continuous iteration, the algorithm is terminated if a point in the mesh is shown to have improved the objective/fitness function value of the current point; the newly selected point replaces the current point to be in the next step in the algorithm [28,29]. 3.3 Hybrid GA PS algorithm Crossover and mutation are the core of GA; it has better robustness and adaptability into any objective functions interdependent on any constraint function. Besides, the problem involved does not need to be in continuous or differentiable form. As known, GA is best at solving complicated functions Fig. 1 a Flowchart of the GA PS optimization. b Rosenbrock s wave function. c Comparison using Rosenbrock s test function using PS, GA and Hybrid GA PS

5 Fig. 1 continued and it does not easily fall into local optimum of a fitness function. However, the algorithm occasionally does not converge to the actual depth of the global minimum [14]. So, GA and PS algorithm are combined to overcome the above problem. In the hybrid model, GA in MATLAB initializes the initial population based on preset generated population and all the candidates are to be vectorized in m n matrix, usually in the form of multiple-column single-row matrix as shown in Sect. Crossover Operator. Then, the first fitness value based on minimized fitness function variables is calculated and the best fitness value (Elitist) is selected. Followed by crossover, genes from each parent are intervened and the similar number of new child/offspring generation is created. Finally, a small mutation probability ratio is imposed on randomly selected chromosomes. Upon reaching the set stopping criteria, the best value of variables with the best fitness function value is delivered to the next pattern search algorithm to perform search computation. In the pattern search algorithm, all previously returned values are added to the mesh size in a pattern vector. The algorithm calculates the objective function at mesh points until it spots one for which the value is smaller than the previous smallest value [29]. With a large population, the hybrid GA PS searches the solution space more thoroughly, and thereby confirms that the function converges to the global minimum. However, this

6 Fig. 2 a Field analysis of three-dimensional auto-mesh generation. b 1/4 of meshed model of the AFPM generator. c CAD design and FEA flux density of the 1/4 model. d Coil windings with iron-stator core (center) Fig. 3 a Eight poles single-rotor double-stator assembly. b Alignment of twelve air-core stator coils on a stator plate. c Hardware setup of experiment test bench algorithm runs more slowly with a larger population size subject to preset generation. Although there are multiple tools to perform hybridization, this GA PS hybrid model is chosen as it could provide more consistent and more accurate best fitness value, and it ensures the function converges to the actual depth of the local optimum on the PS mesh pattern. Another reason is because the algorithm is easily accessible in the MathWorks development environment or programmable in other programming language. Figure 1c shows the preliminary comparison between GA, PS and hybrid GA PS using the famous Rosenbrock s test function [3]. It is shown that the best possible solution obtained is by the hybrid GA PS algorithm, the best fitness value is which is relatively close to the actual global optimum value of zero of the Rosenbrock s function as shown in Fig. 1b. To implement this hybrid algorithm into optimizing the AFPM machine, a MATLAB GUI optimization tool had been developed that follows the GA PS optimization process flow as shown in

7 Table 1 Design data of proposed AFPM generator Symbol Quantity Value P out Rated output power 12W V rms Rated RMS voltage at rated 14/ 2 1rpm 2 P Number of poles 8 m Number of phases 3 f Frequency 65 Hz N ph Winding turns per phase 195/3 η Efficiency 95 % D Outer diameter 235 mm D i Inner diameter 132 mm L pm Magnet thickness 6 mm γ p Pole pitch 45 L cs Stator-yoke thickness 2 mm L cr Rotor-yoke thickness 6 mm Q Number of stator or slots 12 B g Air-gap flux density.75 T g Air-gap length 4. mm P den Optimized power density W/m 3 (a) Best Fitness Value, f(x) (b) Max Constraint GA Hybrid GA-PS PS Popula on GA Hybrid GA-PS PS Popula on Fig. 1a. Results are discussed in the results and discussions section. 4 Design requirements and simulation 4.1 Generator modelling using finite element analysis Despite the hybrid GA PS algorithm has facilitated the process to acquire the maximum power density of the proposed generator, the computed parameters obtained in MATLAB GUI in the optimization tool needed to be further analyzed to validate the practicality. To achieve that, three-dimensional FEA is incorporated to analyze the magnetic flux saturation, to evaluate power density and validate induced back-emf during static and dynamic simulation. The FEA software used is the ANSYS 3D Maxwell. Figure 2a shows the auto-mesh results of the proposed AFPM generator. The top stator part has been hidden to access the mesh pattern of the machine s rotor. Using symmetrical characteristic of this AFPM generator, a quarter of the design has been extracted relative to the vertical axis in axial direction as seen in Fig. 2b, and this has drastically reduced the model analysis time. Figure 2c shows the flux density obtained in FEA transient analysis, the rotor has been captured at 5 rotated out of the phase current coils, it can be seen that around the circumference of each circular surface-mounted permanent magnet, it indicates the strongest magnetic flux density compared to other part of (c) Execu on Time (s) GA Hybrid GA-PS PS Popula on Fig. 4 a Fitness value versus population. b Maximum constraint versus population. c Execution time versus population the rotor. Figure 3a, b shows the proposed AFPM generator that has been fabricated based on the dimensions optimized by hybrid GA PS algorithm. Table 1 shows the list of finalised dimensions and specifications for the proposed generator. These parameters are chosen in regards to the proposed hybrid GA PS algorithm and results from FEA simulation. 5 Results and discussions 5.1 Optimization results Among the results obtained using PS, GA and hybrid GA PS in MATLAB, a more consistent and better accurate fitness value is shown by hybrid GA PS model. As seen in Fig. 4a, the optimized machine s maximum power density is computed based on the objective function in Eqs. (5)

8 (a) (b) Name X Y m m m1 EMF m2 12 coils for best adjusted model Curve Inf o 15. InducedVoltage(Winding3) Setup1 : Transient InducedVoltage(Winding1) Setup1 : Transient InducedVoltage(Winding2) Setup1 : Transient Y2 [V] Cogging Torque (Nm) With Iron Core Without Iron Core Name Delta(X) Delta(Y) Slope(Y) InvSlope(Y) d( m1,m2) N/A (c) Voltage (V) Time [ms] Time (ms) Predicted -2. Experiment (d) Voltage (V) Mechanical Angle (deg) Predicted Experiment Speed (rpm) Fig. 5 a Induced back-emf in FEA. b Cogging torque between the iron-stator core and air-core stator of AFPM generator. c Measured back-emf waveform of the AFPM prototype. d No load back-emf versus speed and (6) subject to several constraints in Eq. (7). The maximum power density computed using hybrid GA PS is at W/m 3, and the value remains constant throughout the population from 1 to 1; GA-based calculated power density fluctuates between and W/m 3 within the same population range; this clearly shows that GA does not converge to the actual global optima, thus giving less accurate best-fitness value at low population size. The worst result of fitness value is obtained in PS; the power density fluctuates between and W/m 3. In Fig. 4b, the computed maximum constraint value for hybrid GA PS is regardless of the number of population size. However, PS and GA show a drastic decline of maximum constraint value as the population size increases. For comparison of execution time in Fig. 4c, result shows that the hybrid GA PS has a convergence rate that is relatively slower compared to PS and GA. It can be seen that in order for the hybrid GA PS to converge to the actual depth of the local optimum, the time taken for computation increases two to three fold. As the population size increases, the convergence rate for hybrid GA PS increases. On the other hand, PS shows a better convergence rate at average execution time of 31.8 seconds within 1 population. In conclusion, the overall performance of hybrid GA PS model shows a better outcome in optimization method for this research particularly at lower population size. Unlike GA and PS, population size has only minor effect in the hybrid GA PS on local optimum search. For instance, the hybrid GA PS has obtained a very consistent fitness value even though when the population size is only as low as Generator performance Results were obtained from the simulation of a one-fourth designed structure shown in Fig. 2b; simulation of this particular structure requires around 25 min to complete. During simulation, the auto-mesh generates six-node tetrahedral elements onto target structure. The meshing process is essential to evaluate the magnetic flux density for all components of the generator such as the flux density in air gap and induced back-emf. An experiment is designed to testify the perfor-

9 Fig. 6 a No-load losses versus speed. b Output power versus speed. c Torque versus speed (a) No-load Losses (W) (b) Speed (rpm) 1 Ohm Load eddy Predicted eddy Measured bearing Predicted bearing Measured Output Power (W) Ohm Load 2 Ohm Load 1 Ohm Load (Predicted) (c) Speed (rpm) 1 Ohm Load Torque (Nm) ohm Load 2 Ohm Load speed (rpm) 1 Ohm Load (Predicted) mance of the designed coreless AFPM prototype generator as shown in Fig.3c. The experiments include the no-load test, load test, output power, speed torque acquisition, speed temperature acquisition and efficiency. Figure 5a depicts the generator three phase back-emf sinusoidal waveform measured at 1 rpm. At 65Hz, the measured EMF peak value is 14V which is lower than the peak EMF value by 6 % as predicted from design calculations as shown in Fig. 5c. Figure 5d shows a plot of output voltage versus speed (rpm), the plot shows the terminal voltage is proportional to the rotational speed (rpm) and the generator generates higher terminal voltage as rotational speed increases. Throughout the rotational speed from to 1 rpm, the maximum percentage voltage regulation is about 7 % between experiment and prediction. To compare the no-load cogging torque, a coreless-stator core design in Fig. 2b and an iron-stator core design in Fig. 2d are compared in 3D FEA. Figure 5b shows the comparison of no-load cogging torque in iron-core and ironless stator based on the proposed AFPM generator. The peak value of cogging torque is ±.252 Nm for the ironless (coreless) AFPM generator, while that for the iron core configuration is ±1.328 Nm. The cogging torque is computed from the noload air-gap flux density distribution as in Eq. (8) through simulation. Since this coreless generator is mainly fabricated from non-ferromagnetic material such as laminated wood and plastic, the armature reaction in ironless stators is insignificant due to very small cogging torque in the generator. By comparison, it is clear that the cogging torque in a coreless stator AFPM generator is significantly lower by

10 about 5 times than the AFPM generator with iron-stator core. Hence, lower cogging torque increases the electric machine efficiency. To determine the no-load losses, no-load tests were carried out on the generator. The major losses that occur are the windings eddy current losses and bearing losses. Figure 6a shows the no-load losses versus speed plot between experiment and prediction based on Eqs. (9) and (1). Results are clear that the major component of the no-load losses in this generator is by the shaft bearing. For instance, there is about 83 % bearing losses out of the total no-load losses at 1 rpm. In the same figure, eddy current losses in stator windings are less significant compared to bearing losses even though the rotational speed reaches at 1 rpm. This result has shown that good less-friction shaft bearing used in rotating machine can improve performance of the generator. In load tests, it was carried out using variable resistive loads. Throughout the test, the load resistance is set at 2, 6 and 1, while the generator speed is kept constant to measure the output characteristics. Figure 6b shows the output power comparison for the three load resistance values. From the result, it is clear that the load resistance is proportional to the rotational speed. For instance, for the load resistance of 1 at 1 rpm, the measured output power is about 5 % less than the predicted output power due to losses in the bearing friction, and eddy current losses. A network of dummy resistive loads is connected to the output of the generator to simulate load condition to measure the speed torque characteristics. Figure 6c shows the speedtorque characteristics comparison between simulation and experiment results at different resistive load. For instance, at 1 rpm, the measured torque at load 1 is lower than the predicted torque value by.6 Nm. This is because in experiment, there is resistive loss which causes the temperature of stator coils to raise and, hence, further increases the resistance in the windings that results in lower current flow. Therefore, a larger diameter conductor wire can be used to increase the conductivity of current. Also, fan cooling can be another conventional solution to this problem. 5.3 Efficiency Figures below show the graph of efficiency and temperature rise versus rotational speed. Figure 7a shows that machine efficiency versus speed with load resistance. It can be observed that the machine efficiency is proportional to the rotational speed as well as load resistance. The lowest efficiency obtained is 83 % at 2 resistance, while the highest recorded efficiency is 94 % at 1 rpm with a resistance of 1. The measured efficiency has a maximum discrepancy of about 6 % compared to the predicted efficiency at 1 load. The reason is that when the generator is operating at low speed, losses in the bearing friction dominates the total (a) Efficiency (%) (b) Temp Rise (oc) Ohm Load 6 Ohm Load 2 Ohm Load 1 Ohm Load (Predicted) Speed (rpm) 1 Ohm Load 6 Ohm Load 2 Ohm Load Speed (rpm) Fig. 7 a Efficiency versus speed. b Temperature versus speed losses. At high speed, major losses are attributed to the eddy current losses in the windings. The stator winding s temperature is obtained with an infrared thermometer. In Fig. 7b, the graph shows that the temperature rises linearly with rotational speed; the highest temperature at 2 load is at around 61 C which is about twice higher than the ambient temperature of 31 Cintest lab. The temperature is inversely proportional to load resistance due to losses in the form of heat in stator windings; this result corresponds to the output power characteristics of this generator. Overall, the predicted efficiency is 95 % and the measured efficiency is 94 % at 1 rpm, both of which are in good agreement. 6 Conclusion An optimized low-speed AFPM generator using hybrid GA PS algorithms was built, tested, and its performance was verified using comprehensive computer software and advanced laboratory equipments. Hybrid GA PS algorithmbased sizing equation was used to compute the most accurate highest power density over volume for a 12 W eight-pole AFPM generator. Several types of stator coil windings and design features were simulated and compared to get the best optimized model. Based on the simulation and experimental results obtained, the output characteristics of the proposed generator are in good agreement. In general, this method of

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