energies Article
Study and Optimal Design of a Direct-Driven Stator Coreless Axial Flux Permanent Magnet Synchronous Generator with Improved Dynamic Performance Wenqiang Wang , Weijun Wang * , Hongju Mi, Longbo Mao, Guoping Zhang, Hua Liu and Yadong Wen Department of Electrical Engineering, Army Logistics University of PLA, Chongqing 401331, China;
[email protected] (W.W.);
[email protected] (H.M.);
[email protected] (L.M.);
[email protected] (G.Z.);
[email protected] (H.L.);
[email protected] (Y.W.) * Correspondence:
[email protected]; Tel.: +86-023-8673-6189 Received: 22 October 2018; Accepted: 12 November 2018; Published: 15 November 2018
Abstract: In this paper, the study and optimization design of stator coreless axial flux permanent magnet synchronous generators is presented for direct driven variable speed renewable energy generation system applications while considering the requirement of reliability and dynamic performance with unstable input conditions. The dynamic analytical model is developed based on the investigation of the axial flux permanent magnet synchronous generator (AFPMSG) structure and basic electromagnetic equations to find out the relationship between generator parameters and dynamic performance. Simulation via the MATLAB/Simulink platform is carried out to obtain the sensitivity of dynamic performance to generator parameters. An integrated optimization model that takes the key parameters as variables is proposed, aiming to improve the mechanical dynamic performance of AFPMSG. For accurate design, the design procedure is modified by combining the nonlinear iterative genetic algorithm (GA) to perform the calculation. A 3_kW AFPMSG is optimally designed to minimize the output voltage overshooting—the index of dynamic performance for direct driven variable speed generation application. Finally, a three-dimensional (3D) finite element model of the generator is established by Maxwell ANSOFT, and the simulation results confirm the validity of the dynamic performance analysis and optimal design procedure. Keywords: direct driven variable speed; stator coreless axial flux permanent magnet synchronous generator; mechanical dynamic performance; optimization design
1. Introduction With the rapid development of society and technology, renewable energy resources have emerged as the most viable and sustainable response to the increasing energy demand, limited fossil fuel supplies as well as political pressure to reduce carbon emissions. Among the renewable energy technologies, electric power that is converted via mechanical energy is the most popular, in particular, those involving kinetic and potential energy forms such as wind energy and wave energy. Compared to the traditional gearbox connecting the energy harvesting device to the generators, the recently developed direct driven coupling method has the advantages of simple structure, high efficiency, and reliability [1,2]. However, the inherent randomness and fluctuation of renewable energies results in the output of direct driven coupling with low speed and instability, which requires advanced system control and suitable generators. Therefore, specification design and optimization of the generator have always been highlighted in the research of renewable energy exploitation and utilization, as the generator plays a key role in the direct driven energy conversion.
Energies 2018, 11, 3162; doi:10.3390/en11113162
www.mdpi.com/journal/energies
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The axial flux permanent magnet synchronous generator (AFPMSG) has been widely used in electric vehicles, energy storage systems, and renewable energy systems due to its benefits of high power density, big diameter axial ratio, and convenient multipolar structure layout, as well as other features [3,4]. Depending on the stator and rotor structure, the AFPMSG can be single-sided or double-sided, core or coreless, with surface-mounted or embedded permanent magnets, single stage or multi stage or other topological structures [5]. Due to the high torque density, simple construction, excellent electromagnetic property and other advantages, AFPMSG with two outer rotors and an inner coreless stator is increasingly being researched and widely used. Moreover, when considering the coreless stator, the structural mass of the generator is decreased; meanwhile, the efficiency and operation stability are increased resulting from no stator core loss and cogging torque [6,7]. Because of the high generator diameter to the axial length ratio and two outer rotors that consist of PM and rotor yoke, the proposed generator has bigger rotational inertia which can also be taken as energy storage. The special structure introduced above makes the AFPMSG better regarding its dynamic and anti-disturbance performance, and it is suitable for low speed and high torque direct driven variable speed generation application. With the increasing application of AFPMSG in direct driven renewable energy power systems, studies on their analysis and optimization have drawn great attention recently. Analytical analysis, the equivalent magnetic circuit approach and finite element simulation are the main AFPMSG research directions. An AFPMSG design based on the dynamic generator and the power system model has been presented as well as the system control for a variable speed wind power system in [8]. In [9], the optimization design of high-speed coreless stator AFPMSG was performed by a hybrid method combining analytical and finite element analysis. Design and optimization, taking a different stator, rotor, permanent magnet (PM), and other structure parameters as variables, have been studied in [10–13]. Considering the wind speed and wind turbine characteristic, the AFPMSG structure parameters were optimized to minimize the cost while maximizing the annual power output with the particle swarm optimization algorithm and three-dimensional (3D) model simulation in [14]. Wedge-shaped air gap was proposed to deal with the issue of coil space factor at the inner diameter of AFPMSG to increase electromagnetic loading in [15]. In the literature, studies on AFPMSG mainly focus on the design and optimization of cost, efficiency, structure and dimension parameters as well as generator controls, while a minority of them look into the dynamic performance of AFPMSG. For the unstable output condition of direct driven renewable power generation systems, this paper concentrates on the study and improvement of the dynamic performance of AFPMSG by its dynamic analytical model and structure parameter optimization, while keeping the overall dimension and output performance of the original prototype unchanged. Recently, computer-aided techniques boosted by advanced optimization algorithms and computer technology have been applied in the electromagnetic analysis and design. References [16,17] presented the development of computer-aided electromagnetic field computation and multi-objective optimization in detail. A novel procedure combining the multi-objective into a single objective was proposed to optimize the electromagnetic field design in [18]. In [19], the dimensions of a coreless AFPMSG were optimized, including the magnet pole, the number of winding turns and air gap distance by implementing a hybrid pattern search-genetic algorithm. Magnetic field calculation and the optimal design of electrical machine require searching for solutions to a multidimensional nonlinear problem, resulting in the wide use of the heuristic algorithm to improve the calculation efficiency and accuracy, such as genetic algorithm (GA), particle swarm optimization (PSO), and simulated annealing method [20–22]. GA is a random search method that imitates natural selection and genetic mechanisms, and it has been widely used as an optimization algorithm. GA searches for the best solution by simulating natural evolution. GA was adopted to optimize the design of AFPMSG with the aims of reducing the cost, increasing the power density and decreasing the torque ripple, as in [23,24]. Global optimization of AFPMSG is calculated by the sequence-genetic algorithm and fuzzy genetic algorithm to achieve higher accuracy and efficiency, as in [25–27].
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In this paper, a complete dynamic analytical model is developed from the AFPMSG dynamic model and swing equation, through which the key structure parameters are analyzed. Aiming at improving the dynamic performance of AFPMSG to achieve reliability in direct driven renewable energy generation system applications, an optimal design model considering the key parameters as variables is constructed and calculation is carried out by a modified design procedure that combines the nonlinear iterative genetic algorithm. In the following, the complete dynamic analytical model that is based on the structure and electromagnetic relationship investigation of AFPMSG is given in Section 2. Key parameters of AFPMSG are analyzed for dynamic performance based on the analytical model in Section 3. Section 4 introduces the integrated optimization model and the modified design procedure. In Section 5, the validity of the optimal design is simulated and compared through the dynamic performance. Finally, the conclusions are drawn in Section 6. 2. AFPMSG and Dynamic Analytical Model The dynamic performance analysis and optimal design of stator coreless AFPMSG requires an overall study on the structure and basic electromagnetic relationship, as well as the dynamic analytical model. Hence, it is important to introduce the stator coreless AFPMSG and dynamic analytical model in detail. 2.1. AFPMSG The AFPMSG proposed in this paper is an axial flux permanent magnetic machine topology with two outer rotors and an inner coreless stator. Sectorial PMs that are mounted on the surface of two rotor yokes make up the rotor while the stator armature is formed by epoxy-resin-casted non-overlapping concentrated winding. The configuration and simulation model schematic, as well as the specifications referring to the real AFPMSG, are given in Figure 1 and Table 1. Figure 2 illustrates the rotor and stator winding in detail. When compared to other topologies, the stator coreless AFPMSG can make full use of the low-grade energy for its low detent torque and stable operation characteristic in the low speed condition. Due to the high moment of inertia contributed by the structure, the generator can also be used as energy storage to absorb the input disturbance energy, which increases system’s anti-disturbance capacity and reliability. Electromagnetic relation is the basis of motor analysis and research. The basic electromagnetic relations of AFPMSG are the steady state equations and main sizing equations [3], as shown below. Equations (1), (2), and (4) indicate the relation between the EMF (electromotive force), electromagnetic torque, electromagnetic power and structural parameters. Equation (3) represents the influence of electrical parameters established by structural parameters on the operation state. From the electromagnetic relation, the transformation principle inside the generator can be known. It can be seen that the steady state equations and main sizing equations represent the operation characteristic under different conditions and the relationship between the rated design power and structure dimensions, respectively. Also, it is obvious that the structure and dimensions have great influence on the output performance of the generator: Ef =
√
2π pN1 k w1 φ f ns = KE ns
m T = √1 pN1 k w1 φ f Ia = KT Ia 2 →
→
→
→
(1) (2)
→
E f = U + Ia R1 + j Iad Xsd + j Iaq Xsq
(3)
√ Pe =
2π 3 3 ns k w A av αi Bδ Dout (1 + k d )(1 − k2d )η cos ϕ 32
(4)
where Xad and Xaq are armature reaction reactance, R1 , X1 , Ef and Ia are the stator winding impedance, leakage reactance, induced voltage and effective phase current. U is the terminal voltage. ns , kw , Aav ,
analytical model. Hence, it is important to introduce the stator coreless AFPMSG and dynamic analytical model in detail. 2.1. AFPMSG The AFPMSG Energies 2018, 11, 3162
proposed in this paper is an axial flux permanent magnetic machine topology 4 of 22 with two outer rotors and an inner coreless stator. Sectorial PMs that are mounted on the surface of two rotor yokes make up the rotor while the stator armature is formed by epoxy-resin-casted nonα1 , Bδ , Dout concentrated and kd are thewinding. rotational windingand factor, averagemodel electrical load, magnet width overlapping Thespeed, configuration simulation schematic, as well as the to pole pitch ratio, flux density of the air gap, outer diameter, and radio of outer diameter to inner specifications referring to the real AFPMSG, are given in Figure 1 and Table 1. Figure 2 illustrates the diameter of the winding generator, rotor and stator inrespectively. detail.
(a)
(b)
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Figure 1. Axial flux permanent magnet synchronous generator (AFPMSG) configuration without the stator core and simulation model. (a) Assemble model; (b) Ironless stator structure; (c) simulation model. Rotor yoke Table 1. Design data of the proposed axial flux permanent synchronous generator (AFPMSG).
Parameters Rated output power Pout Rated RMS voltage at rated 300 rpm: U out Coreless stator
Number of phase: m Frequency: f Number of pole pairs: p
Value 3 kW
Parameters Outer diameter Dout
0.418 m
220 V
Out-in diameter ratio:
1.8
3
Magnet thickness: hM
15 mm
50 Hz
Winding turns per phase: N Air gap length: l g
434
(c)
10
Value
1 mm
Pole-arc coefficient:magnet 0.8 generator Rotor yoke thickness: 10 mm the Figure 1. Axial flux permanent synchronous (AFPMSG) configuration without stator core and simulation model. (a) Assemble model; (b) Ironless stator structure; (c) simulation model. Rotor yoke Permanent magnets Stator coil
Ferromagnetic materials
(a)
(b)
Figure 2. Structure of the ironless AFPMSG. (a) Rotor and permanent magnet (PM) structure; Figure 2. Structure of the ironless AFPMSG. (a) Rotor and permanent magnet (PM) structure; (b) (b) Ironless stator structure. Ironless stator structure.
When compared to other topologies, the stator coreless AFPMSG can make full use of the lowgrade energy for its low detent torque and stable operation characteristic in the low speed condition. Due to the high moment of inertia contributed by the structure, the generator can also be used as energy storage to absorb the input disturbance energy, which increases system’s anti-disturbance capacity and reliability. Electromagnetic relation is the basis of motor analysis and research. The basic electromagnetic relations of AFPMSG are the steady state equations and main sizing equations [3], as shown below. Equations (1), (2), and (4) indicate the relation between the EMF (electromotive
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Table 1. Design data of the proposed axial flux permanent synchronous generator (AFPMSG). Parameters
Value
Parameters
Value
Rated output power Pout Rated RMS voltage at rated 300 rpm: Uout Number of phase: m Frequency: f Number of pole pairs: p Pole-arc coefficient: α
3 kW 220 V 3 50 Hz 10 0.8
Outer diameter Dout Out-in diameter ratio: γ Magnet thickness: h M Winding turns per phase: N Air gap length: l g Rotor yoke thickness: ∆
0.418 m 1.8 15 mm 434 1 mm 10 mm
2.2. Dynamic Analytical Model The direct driven variable speed generation system under consideration consists of stator coreless AFPMSG that is directly coupled to the energy harvest device without any gear box via a simple shaft. Therefore, to find out the dynamic response under the disturbance input and the key parameters that affect the performance, the dynamic analytical model including the swing equation and dynamic model of stator coreless AFPMSG is presented below. 2.2.1. Dynamic Model of AFPMSG The dynamic model of AFPMSG is derived from the dual-reaction principle of permanent magnet synchronous machine based on d-q rotating reference frame [28,29]. According to the needs of practical application, when considering the conditions and assumptions, such as no loss, torque pulsation, unsaturated magnetic circuit and ignoring the complex magnetic field effects, the second order reduced model is adopted for dynamic study [30–32] and the dynamic equations of output voltage are given by: u d = −r · i d +
dϕd − we ϕq dt
(5)
u q = −r · i q +
dϕq + we ϕd dt
(6)
Considering the influence of stator current, the flux linkage equations are given by: ϕd = − Ld · id + ϕ PM
ϕq = − Lq · iq
(7)
where r is the stator resistance, we is the angular electrical speed, ud , id , uq and iq are the d and q axes stator voltage and current, respectively, φd , Ld , φq , and Lq are the inductances and flux linkages along Ke d and q axes, and φPM is the permanent magnet flux linkage given by ϕ PM = 2π p , where Ke is the induced EMF constant of AFPMSG. Dynamic voltage equations can be derived from the above four equations. Figure 3 shows the equivalent circuit diagram of AFPMSG. did + we Lq iq dt
(8)
diq − we Ld id + we ϕ PM dt
(9)
u d = −r · i d − L d u q = −r · i q − L q
The equivalent circuit of AFPMSG can not only be used in steady state analysis but it is also suitable for the transient process. The electromagnetic power generated by AFPMSG can be obtained as below: 3 Pe = (ed · id + eq · iq ) (10) 2 where ud , uq are the d and q axes stator voltage and current, respectively, ed = we Lq iq and eq = −we Ld id + we ϕ PM . The transient power and torque developed is given by:
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3 we [ ϕ PM iq − ( Ld − Lq )id iq ] 2 3 Pe = p[ ϕ PM iq − ( Ld − Lq )id iq ] Te = wm 2 Pe =
(11) (12)
where wm is the mechanical speed, p is the number of pole pairs. In (12), the first part is torque component of PM, which is linearly related to stator current, the second part is the magnetic resistance torque component. For surface mounted AFPMSG, there is Ld = Lq . Therefore, the torque is linear to iq and Te can be modified: 3 Te = pϕ PM iq (13) 2 Based on the modeling equations above, the MATLAB/Simulink model of AFPMSG is shown in Figure 4. The model in Figure 4 contains all of the main parameters that are needed for dynamic performance. From the model, it is obvious that the dynamic performance can be simulated with different input Energies 2018, 11, conditions and it can be improved by choosing the proper generator parameters. 6 of 22
Energies 2018, 11,
(a)
(b)
7 of 22
Figure3.3.Equivalent Equivalentcircuit. circuit.(a) (a)ininqqaxis; axis;(b) (b)ininddaxis. axis. Figure
The equivalent circuit of AFPMSG can not only be used in steady state analysis but it is also r suitable for the transient process. The electromagnetic power generated by AFPMSG can be obtained as below:
ud
1 s
1L 3 Pe (ed id eq iq ) d 2
Lq
id
(10)
where ud , uq pare the d and q axes stator voltage and current, respectively, ed we Lqiq and
wm eq we Ld id we PM . The transient power and torque developed is given by: 1.5* p
Pe
PM
Ld
Te
Te
3 we [ PM iq ( Ld Lq )id iq ] 2
Pe 3 p[ PM iq ( Ld Lq )id iq ] wm 2 1 Lq
where wm is the mechanical speed, p is the number of pole pairs. In
(11) (12)
1 s (12),
the first part i is torque q
uq of PM, which is linearly related to stator current, the second part is the magnetic component resistance torque component. For surface mounted AFPMSG, there is Ld = Lq. Therefore, the torque is linear to iq and Te can be modified: r
3 2
Te ofpthe PMproposed iq Figure 4. Dynamic model AFPMSG.
(13)
Based on the modeling equations above, the of MATLAB/Simulink model of AFPMSG is shown in Figure 4. Dynamic model the proposed AFPMSG. Figure 4. The model in Figure 4 contains all of the main parameters that are needed for dynamic 2.2.2. AFPMSGFrom Swingthe Equation performance. model, it is obvious that the dynamic performance can be simulated with different input conditions and itiscan improved by choosing theequation proper generator parameters. the The swing equation, which alsobecalled the rotational inertia [33,34], representing connection relationship between the energy harvesting device and AFPMSG, is essential for the dynamic performance analysis. The swing equation characterizes the rotor balance relation of the generator between electromagnetic torque and mechanical torque. In a direct driven generation
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2.2.2. AFPMSG Swing Equation The swing equation, which is also called the rotational inertia equation [33,34], representing the connection relationship between the energy harvesting device and AFPMSG, is essential for the dynamic performance analysis. The swing equation characterizes the rotor balance relation of the generator between electromagnetic torque and mechanical torque. In a direct driven generation system, the generator is connected to the energy harvesting device through a shaft without any gear box and therefore a simple lumped mass model has been adopted and the differential equation is defined by: J
dwm = Tm − Te − Kwm dt
(14)
where J is the moment of inertia of rotor mass, Tm and Te are developed by the energy harvesting device and generator, and K is the friction coefficient. The moment of inertia mainly comes from permanent and rotor yoke, and the equation is given by: J
= Jsha f t + JFe + JPM = Jsha f t +
m PM = 2α p
2 + D2 Dout in 8
(15)
· (m Fe + m PM )
2 − D2 ) 2 − D2 ) π ( Dout π ( Dout in in h M m Fe = 2 ∆ 4 4
(16)
where J is the moment of inertia developed by the shaft, m Fe and m PM are the mass of rotor yoke 8 of 22 and PM, and h M , ∆ are the thickness of PM and rotor yoke. The MATLAB/Simulation model of the AFPMSG swing equation is shown in Figure 5.
sha f t11, Energies 2018,
wm
Tm
1 s
1J
Te
p Theta
1 s
K
we
wt
Figure 5. Simulink model of swing function.
2.2.3. Dynamic Analytical Model of AFPMSG Figure 5. Simulink model of swing function. The dynamic analytical model of AFPMSG combines the dynamic model of APMSG and swing 2.2.3. Dynamic Analytical Model6.ofStructure AFPMSGparameters, electrical parameters, torque and movement equation, as illustrated in Figure analysis beenanalytical consideredmodel in theof analytical model. The dynamic response ofof AFPMSG theswing direct Thehave dynamic AFPMSG combines the dynamic model APMSGtoand driven variable speed condition can be obtained by adjusting the input condition and the generator equation, as illustrated in Figure 6. Structure parameters, electrical parameters, torque and movement parameters. Three are takenintothe analyze the dynamic AFPMSG.ofFirst, the dynamic analysis have beensteps considered analytical model. performance The dynamicofresponse AFPMSG to the response of the generator under different input conditionsbyis adjusting found by changing input torque or direct driven variable speed condition can be obtained the input the condition and the speed condition according to the typical input disturbance. Second, the control variable method is generator parameters. Three steps are taken to analyze the dynamic performance of AFPMSG. First, applied to change one parameter, whileunder the others are kept unchanged the sensitivity of the the the dynamic response of the generator different input conditionstoisfind found by changing main parameters of AFPMSG to the performance. The last step is to adjust the generator parameters input torque or speed condition according to the typical input disturbance. Second, the control on the basis of optimization improve dynamic performance. variable method is applied totochange onethe parameter, while the others are kept unchanged to find the
sensitivity of the main parameters of AFPMSG to the performance. The last step is to adjust the generator parameters on the basis of optimization to improve the dynamic performance.
AFPMSG
generator parameters. Three steps are taken to analyze the dynamic performance of AFPMSG. First, the dynamic response of the generator under different input conditions is found by changing the input torque or speed condition according to the typical input disturbance. Second, the control variable method is applied to change one parameter, while the others are kept unchanged to find the sensitivity of 3162 the main parameters of AFPMSG to the performance. The last step is to adjust Energies 2018, 11, 8 ofthe 22 generator parameters on the basis of optimization to improve the dynamic performance.
AFPMSG
Figure 6. Dynamic characteristic analysis block diagram. Figure 6. Dynamic characteristic analysis block diagram.
3. Dynamic Performance and Key Factor Analysis 3. Dynamic Performance and Key Factor Analysis 3.1. Dynamic Performance Analysis
3.1. Dynamic Performance Analysis system without any speed adjustment gear, the energy including In the direct driven generation the fluctuation disturbance by the any generator, will directly reflect in theincluding output In the directand driven generationreceived system without speed adjustment gear, the energy performance, which depends on the structure parameters of generator. In this paper, dynamic the fluctuation and disturbance received by the generator, will directly reflect in the output performance researched startingparameters process andofresponse of the voltage when performance,analysis which is depends on via the the structure generator. In output this paper, dynamic there is disturbance. to Table 1, the moment of inertia of statorof coreless AFPMSG is 0.957 performance analysisAccording is researched via the starting process and response the output voltage when 2 , the friction coefficient is 0.09, stator resistant is 3.3 Ω, and induction is L = L = 0.008 H. Those kgm q d there is disturbance. According to Table 1, the moment of inertia of stator coreless AFPMSG is 0.957 parameter data arecoefficient obtained from theory of Maxwell kgm2, the friction is 0.09, statorcalculation resistant isand 3.3 the and induction is Ld = LqANSOFT. = 0.008 H. Those ,simulation Figure data 7 shows the output voltage of theand starting up with 100 Nm rated input torque parameter are obtained from theorycurves calculation the simulation of Maxwell ANSOFT. (a) and voltage response to the disturbance with 5 Nm step torque input (b). It can be observed that the Figure 7 shows the output voltage curves of the starting up with 100 Nm rated input torque (a) generator needs to taketosome time ∆t (about s) tostep reach the steady or settlethat down and voltage response the disturbance with1.5 5 Nm torque input state (b). Itoperation can be observed the from the disturbance. the voltage response toreach the disturbance inputoperation is damped oscillation, generator needs to takeBesides, some time 1.5 s) to the steady state or settle down t (about peak value of which greatly influences the power quality and reliability operation. The output voltage from the disturbance. Besides, the voltage response to the disturbance input is damped oscillation, fluctuation bygreatly unstable input needs to be considered in a directThe driven generation peak valuecaused of which influences the power quality andparticularly reliability operation. output voltage system with no speed adjustment gear. Therefore, according to the curves and the principle of the control system [35,36], the regulating time ∆t and maximum voltage overshoot Mp representing the dynamic stable time and the disturbance deviation of the system, respectively, is presented to illustrate the dynamic performance of AFPMSG. The maximum voltage overshoot Mp is defined by: Mp =
|U − Us |max Us
(17)
where U is the transient output voltage, US is the rated steady state voltage value and |U − Us |max is the maximum difference between the transient output voltage and the rated steady state voltage. According to the dynamic analytical model, the rotational inertia J, EMF factor Ke , number of pole pairs N, stator winding resistance r, inductance is Ld , Lq and damping coefficient K are the main model variables that affect the dynamic performance of AFPMSG. The damping coefficient is obtained from the experience formula and it is not a designable parameter of the generator, which is neglected during the optimization. The variables of the AFPMSG analytical model are structure parameters or developed by them, while one structure parameter might affect more than one model variable, as in the relation presented in Section 2. Therefore, it is essential to find out the impact of structure parameters on the dynamic performance to optimize the overall structure of AFPMSG for improving the output dynamic performance, which is the main target of this paper.
neglected during the optimization. The variables of the AFPMSG analytical model are structure parameters or developed by them, while one structure parameter might affect more than one model variable, as in the relation presented in Section 2. Therefore, it is essential to find out the impact of structure parameters on the dynamic performance to optimize the overall structure of AFPMSG for Energies 2018, 11,the 3162 9 of 22 improving output dynamic performance, which is the main target of this paper.
U(A) U(B) U(C)
Tm
400 108
Output Voltage(V)
Input Torque(N·m)
200 104 100 96
0
-200
92
0.0
0.3
0.6
0.9
1.2
1.5
Time (s)
1.8
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Time (s)
Output Electromagnetic Torque(N·m)
Speed response(r/min)
0.2
n
300
Te
100
200
100
0 0.0
-400 0.0
0.3
0.6
0.9
1.2
1.5
50
0
1.8
0.0
0.3
Time (s)
0.6
0.9
1.2
1.5
1.8
Time (s)
Energies 2018, 11,
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(a) Tm 6
U(A) U(B) U(C)
8 6 4
Output Voltage(V)
Input Torque(N·m)
4
2
2 0 -2 -4 -6
0
-8 -10 1
10
2
3
6
4
2
0 1
2
Output Electromagnetic Torque(N·m)
n
8
Speed response(m/min)
1 3
Time (s)
2
3
Time (s) Te
2
1
0
3
1
Time (s)
2
3
Time (s)
(b) Figure7.7.Output Outputperformance performanceofofstarting startingand andinput inputdisturbance. disturbance.(a) (a)Start Startwith withconstant constanttorque torqueinput; input; Figure (b) (b)dynamic dynamicperformance performancewith withstep steptorque torqueinput. input.
3.2. 3.2.Key KeyParameter ParameterAnalysis Analysis The Thegenerator generatorstructure structureparameters parameterscan canbe bedivided dividedinto intotwo twotypes typesaccording accordingtotothe thedynamic dynamic analytical model and structure schematic as shown in Figures 1 and 2. The first type are structure analytical model and structure schematic as shown in Figures 1 and 2. The first type are structure size size parameters, which mainly include the outer diameter of the generator, ratio of outer diameter parameters, which mainly include the outer diameter of the generator, ratio of outer diameter to inner todiameter, inner diameter, thickness of PM and rotorand yoke, the of length of air gap. Thetype other typenumber is the thickness of PM and rotor yoke, theand length air gap. The other is the
parameter of the structure components such as the number of pole pairs and winding turns. Some generator structure parameters are determined by the rated design parameters of AFPMSG or they are limited by the overall structure size, which are not considered in the parameter analysis. Main structure parameters are analyzed by simulation based on the structure and electromagnetic relationship of the AFPMSG in the following.
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number parameter of the structure components such as the number of pole pairs and winding turns. Some generator structure parameters are determined by the rated design parameters of AFPMSG or they are limited by the overall structure size, which are not considered in the parameter analysis. Main structure parameters are analyzed by simulation based on the structure and electromagnetic relationship of the AFPMSG in the following. 3.2.1. Ratio of Main Dimensions
8
6
6
4
4
2
2 1.6
1.8
2.0
2.2
Diameter ratio
1.6
2.4 Ke J
1.6
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8 1.4
1.6
1.8
2.0
2.2
5.5
t
Mp
5.0
5.5 5.0
4.5
4.5
4.0
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5 1.4
2.4
Maximum overshoot Mp(%)
2
10
8
1.4
Rotary inertia J(kq*m )
12
Regulating time t(s)
Ld&Lq r
10
Stator resistance r()
12
EMF constant Ke
Induction Lq&Ld(mH)
The ratio of the main dimensions is defined as the ratio of the shaft length to the outer diameter [37], while means the ratio of the outer diameter to the inner diameter in AFPMSG. The ratio is the vital geometry parameter of motor design, which not only decides the diameter of the generator but also affects the variables of the dynamic analytical model, including the rotational inertia, EMF factor, resistance and inductance. The influence of the ratio on the variables and dynamic performance is simulated via parameterization, as shown in Figure 8. The value of all dynamic analytical model variables increase as the ratio increases. The change of EMF factor that is caused by the ratio is greater than other variables, resulting in the output dynamic performance. The maximum output voltage overshoot increases while the regulating time decreases with the effect of changing variable value caused2018, by the Energies 11, adjustment of the main dimension ratio. 11 of 22
1.6
1.8
2.0
2.2
2.4
Diameter ratio
Diameter ratio
(a)
(b)
Figure Dynamic characteristic characteristicresponse. response. Figure8.8.Influence Influenceofofdifferent differentdiameter diameterratio: ratio: (a) (a) Model Model variables; (b) Dynamic
3.2.2. Thickness Thickness of of the the Permanent Permanent Magnet Magnet 3.2.2.
12
14
16
18
Length of magnets hM(mm)
10
12
14
16
Length of magnets hM(mm)
(a)
20 J Ke
18
20
1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8
6.0
6.0 t Mp
5.5
5.5
5.0
5.0
4.5
4.5
4.0
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
Maximum overshoot Mp(%)
2
Rotary inertia J(kq*m )
10 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8
10 9 8 7 6 5 4 3 2 1
Regulation time t(s)
Ld&Lq r
Stator resistance r()
10 9 8 7 6 5 4 3 2 1
EMF constant Ke
Induction Lq&Ld(mH)
The permanent is is thethe source of the field and is also magnetic The permanentmagnet magnet(PM) (PM) source of magnetic the magnetic fieldit and it aispart alsoofathe part of the circuit. The electrical time constant as well as factor inductance is affected due todue the magnetic circuit. The electrical time constant as the wellEMF as the EMFand factor and inductance is affected magnetic resistance being closed to air. The EMF factor and rotational inertia increases while the to the magnetic resistance being closed to air. The EMF factor and rotational inertia increases while induction decreases when the PM becomes thicker. The dynamic performance changes because of the induction decreases when the PM becomes thicker. The dynamic performance changes because thethe increasing thickness ofofthe decreasing slightly slightly of increasing thickness thePM PMwith withthe themaximum maximumoutput outputvoltage voltage overshoot overshoot decreasing and the regulating time reducing. When considering the cost and saturation of the magnetic circuit, and the regulating time reducing. When considering the cost and saturation of the magnetic circuit, the thickness of the PM needs to be set properly to achieve the desirable output design and dynamic the thickness of the PM needs to be set properly to achieve the desirable output design and dynamic performance. Figure Figure 99 illustrates illustrates the the influence influence on on the the generator generator caused performance. caused by by the the thickness thickness of of PM. PM.
1.0 10
12
14
16
18
20
Length of magnets hM(mm)
(b)
Figure 9. Influence of different PM thickness: (a) Model variables; (b) Dynamic characteristic response.
12
1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8
14
16
18
Length of magnets hM(mm)
10
12
14
16
20 J Ke
18
1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8
6.0
6.0 t
5.5
Mp
5.5
5.0
5.0
4.5
4.5
4.0
4.0
3.5
3.5
3.0
3.0
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2.5
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1.5
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Maximum overshoot Mp(%)
2
Rotary inertia J(kq*m )
10
10 9 8 7 6 5 4 3 2 1
Regulation time t(s)
Ld&Lq r
Stator resistance r()
10 9 8 7 6 5 4 3 2 1
EMF constant Ke
Induction Lq&Ld(mH)
to the magnetic resistance being closed to air. The EMF factor and rotational inertia increases while the induction decreases when the PM becomes thicker. The dynamic performance changes because of the increasing thickness of the PM with the maximum output voltage overshoot decreasing slightly and the regulating time reducing. When considering the cost and saturation of the magnetic circuit, the thickness the PM needs to be set properly to achieve the desirable output design and dynamic Energies 2018, 11,of 3162 11 of 22 performance. Figure 9 illustrates the influence on the generator caused by the thickness of PM.
1.0 10
20
12
14
16
18
20
Length of magnets hM(mm)
Length of magnets hM(mm)
(a)
(b)
Figure characteristic response. response. Figure9.9.Influence Influenceofofdifferent differentPM PMthickness: thickness: (a) (a)Model Model variables; variables; (b) Dynamic characteristic
3.2.3. Thickness Thickness of of Rotor Rotor Yoke Yoke 3.2.3.
1.2
1.1
1.1
1.0
1.0
0.9
0.9
0.8
0.8 8
10
12
14
16
14
r 10 Ld&Lq 9 8 7 6 5 4 3 2 16
Thickness of rotor yoke (mm)
10 9 8 7 6 5 4 3 2 8
10
12
4.5
t
4.5
Mp
4.0
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0 8
Thickness of rotor yoke (mm)
10
12
14
Maximum overshoot Mp(%)
1.3
Regulation time t(s)
1.2
EMF constant Ke
J Ke
1.3
Stator resistance r()
Induction Lq&Ld(mH)
2
Rotary inertia J(kq*m )
The rotor rotor yoke yoke is is aa component component of of the the magnetic magnetic circuit circuit as as well well as as the the supporting supporting structure structure of of the the The PM. Therefore Therefore when ofof thethe magnetic circuit andand no PM. when deciding decidingthe thevalue valueofofPM PMthickness, thickness,the thesaturation saturation magnetic circuit deformation of the structure need to be taken into account. With electrical and structural demands no deformation of the structure need to be taken into account. With electrical and structural demands satisfied, an anincrement incrementinin the thickness of the rotor magnetic causes no impact thefactor, EMF satisfied, the thickness of the rotor magnetic yokeyoke causes no impact on theon EMF factor, synchronous inductance or stator resistance, but it increases the rotational inertia effectively, synchronous inductance or stator resistance, but it increases the rotational inertia effectively, which which lowers the maximum voltage overshoot and increases the regulation time, as indicated lowers the maximum outputoutput voltage overshoot and increases the regulation time, as indicated in in Figure 10. Energies 2018, 11, 12 of 22 Figure 10.
16
Thickness of rotor yoke (mm)
(a)
(b)
Figure response. Figure10. 10.Influence Influenceof of rotor rotor yoke yoke thickness: thickness: (a) (a) Model Model variables; variables; (b) Dynamic characteristic response.
3.2.4. Length Length of of the the Air Air Gap Gap 3.2.4.
1.2
0.9
0.9
0.6
0.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Air gap hg(mm)
10
4.0
Ld&Lq r
4.5 10
8
8
6
6
4
4
2
2
0.0
0.5
1.0
1.5
2.0
2.5
Air gap h (mm)
3.0
3.5
4.0
4.5
t
5
Mp
5
4
4
3
3
2
2
1
1
0.0
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1.0
1.5
2.0
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Air gap hg(mm)
3.0
3.5
4.0
4.5
Maximum overshoot Mp(%)
1.5
Regulation time t(s)
J Ke
1.2
EMF constant Ke
1.5
Stator resistance r()
Induction Lq&Ld(mH)
2
Rotary inertia J(kq*m )
The length airair gapgap is ais key influencing the EMF the inductance of AFPMSG. The lengthofofthe the a factor key factor influencing thefactor EMFand factor and the inductance of Generally, the air gap length of PM motors is larger than that of the rotary motors and the value is AFPMSG. Generally, the air gap length of PM motors is larger than that of the rotary motors and the usually over 1 mm. The EMF factor and inductance decreases as the air gap increases while the former value is usually over 1 mm. The EMF factor and inductance decreases as the air gap increases while dropped significantly, resulting in the regulating time increasing and the maximum voltage the formermore dropped more significantly, resulting in the regulating time increasing and the maximum overshoot decreasing, as shown in Figure 11. voltage overshoot decreasing, as shown in Figure 11.
1.2
0.9
0.9
0.6
0.6
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Air gap hg(mm)
10
4.0
4.5
Ld&Lq r
10
8
8
6
6
4
4
2
2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
t
5
Mp
5
4
4
3
3
2
2
1
1
0.0
4.5
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Maximum overshoot Mp(%)
1.2
Regulation time t(s)
1.5
J Ke
EMF constant Ke
1.5
Stator resistance r()
Induction Lq&Ld(mH)
2
Rotary inertia J(kq*m )
The length of the air gap is a key factor influencing the EMF factor and the inductance of AFPMSG. Generally, the air gap length of PM motors is larger than that of the rotary motors and the value is usually over 1 mm. The EMF factor and inductance decreases as the air gap increases while the former dropped more significantly, resulting in the regulating time increasing and the maximum Energies 2018, 11, 3162 12 of 22 voltage overshoot decreasing, as shown in Figure 11.
4.5
Air gap hg(mm)
Air gap hg(mm)
(a)
(b)
Figure11. 11.Influence Influence different air length: gap length: (a) Model variables; (b) Dynamic characteristic Figure of of different air gap (a) Model variables; (b) Dynamic characteristic response. response.
3.2.5. Turns of Coil
3.2.5.The Turns of Coil stator of the proposed stator coreless AFPMSG is post winding coils that are cast in epoxyThe resin or other materials. coil turns are greatly dependent stator of thenonmagnetic proposed stator corelessThe AFPMSG is post winding coils that on aredesign cast in needs, epoxy winding configuration and stator space. However, the turns are also related to the EMF factor, resin or other nonmagnetic materials. The coil turns are greatly dependent on design needs, winding stator resistance andstator inductance, which affect output electrical and The turns configuration and space. However, thethe turns are also related todynamic the EMFperformance. factor, stator resistance demonstrated a positive proportional with EMF factor, with theThe results and inductance, which affect the outputrelation electrical andthe dynamic performance. turnsdemonstrating demonstrated that the larger the number of turns, the higher the EMF factor will be with the increasing of resistance a positive proportional relation with the EMF factor, with the results demonstrating that the larger and induction at the same time. The maximum voltage overshoot increases and the regulating time the number of turns, the higher the EMF factor will be with the increasing of resistance and induction Energies 2018, 11, 13 22 decreases fortime. an increasing number of coil turns. The dynamic response is shown in Figure 12. forofan at the same The maximum voltage overshoot increases and the regulating time decreases
1.2
1.0
1.0
0.8 375
400
425
450
475
0.8 500
Lq&Ld r
Turns per coil N
12
10
10
8
8
6
6
4
4
2
2 375
400
425
450
475
500
6
6
t Mp
5
5
4
4
3
3
2
2
1
1
0 360
380
400
420
440
460
480
Maximum overshoot Mp(%)
1.2
Regulation time t(s)
1.4
EMF constant Ke
Ke J
Stator resistance r()
12
2
Rotary inertia J(kq*m )
1.4
Induction Lq&Ld(mH)
increasing number of coil turns. The dynamic response is shown in Figure 12.
0 500
Turns per coil N
Turns per coil N
(a)
(b)
Figure coil: (a) (a) Model Modelvariables; variables;(b) (b)Dynamic Dynamiccharacteristic characteristic response. Figure12. 12.Influence Influenceof ofdifferent different turns per coil: response.
4. 4. Integrated Integrated Optimization Optimization Design Design 4.1. Optimization Model 4.1. Optimization Model As discussed previously, the dynamic response of AFPMSG varies with different structure As discussed previously, the dynamic response of AFPMSG varies with different structure parameters, and the two dynamic performance indicators that regulate time and maximum output parameters, and the two dynamic performance indicators that regulate time and maximum output voltage overshoot are affected. In terms of the cross impact of different structure parameters, voltage overshoot are affected. In terms of the cross impact of different structure parameters, optimization once at a time or simple superposition optimization is not enough to achieve the optimal optimization once at a time or simple superposition optimization is not enough to achieve the optimal target. When optimizing the dynamic performance of AFPMSG, the integrated optimization is essential target. When optimizing the dynamic performance of AFPMSG, the integrated optimization is when taking all of the structure parameters into account. The mathematic equation connecting the essential when taking all of the structure parameters into account. The mathematic equation dynamic performance indicator and the structure parameters of AFPMSG is given by: connecting the dynamic performance indicator and the structure parameters of AFPMSG is given by: GG ( x(1x,1x, 2x2· · · xxnn)) = M pp + kk∆t t
where x1 , x2
(18) (18)
xn are variables of the structure parameters, M p and t are the two dynamic
performance indicators of overshoot and regulation time respectively, and k is the weighting coefficient depending on the optimization target. In a direct driven generation system, the dynamic performance is especially concerned because the unstable output voltage amplitude has a direct
optimization once at a time or simple superposition optimization is not enough to achieve the optimal target. When optimizing the dynamic performance of AFPMSG, the integrated optimization is essential when taking all of the structure parameters into account. The mathematic equation connecting the dynamic performance indicator and the structure parameters of AFPMSG is given by: Energies 2018, 11, 3162
G( x1, x2
13 of 22
xn ) M p k t
(18)
are the the two two dynamic where xx11,,xx22 · · · xxnn are variables variables of of the the structure structure parameters, parameters, MMp p and and ∆tt are performance indicators of overshoot and regulation time respectively, and k is the weighting coefficient performance indicators of overshoot and regulation time respectively, and k is the weighting depending on the optimization target. In a direct generation the dynamic coefficient depending on the optimization target.driven In a direct drivensystem, generation system, performance the dynamic is especially concerned because the unstable output voltage amplitude has a direct impact the performance is especially concerned because the unstable output voltage amplitude has a on direct safety and reliability operation of the generator with the disturbance conditions. Thus, to reduce the impact on the safety and reliability operation of the generator with the disturbance conditions. Thus, amplitude of the dynamicofoutput voltage, an integrated aiming model at the maximum to reduce the amplitude the dynamic output voltage, optimization an integratedmodel optimization aiming at voltage overshoot is proposed with taking the structure parameters as variables. the maximum voltage overshoot is proposed with taking the structure parameters as variables. Two are taken taken to to determine determinethe theobjective objectivefunction. function.The Thefirst first develop variables Two steps steps are is is toto develop thethe variables of of the dynamic analytical model from the generator structure parameters via the calculation of the the dynamic analytical model from the generator structure parameters via the calculation of the electromagnetic equation. Then thethe object function is constructed by the electromagnetic relationship relationshipand andthe thebasic basicsize size equation. Then object function is constructed by variables through the dynamic analytical model and equation of maximum output voltage overshoot. the variables through the dynamic analytical model and equation of maximum output voltage The processThe of determining the object function is illustrated Figure 13.in Figure 13. overshoot. process of determining the object function isin illustrated Basic size and electromagnetic relation
Optimization parameters
Dynamic model and maximum overshoot function
Model variables
Objective function
Figure 13. Objective function deriving block diagram.
The accurate definition of the constraints in terms of optimization variables is a key step in the optimal design procedure of stator coreless AFPMSG. The optimization in the paper is to improve the dynamic performance on the condition of keeping the output power and the overall size of the generator unchanged. Hence constraints containing the overall size of the generator, the demand of non-saturation of magnetic circuit, and electrical output performance are considered in the optimization. The mathematic optimization model of maximum output voltage overshoot is presented as follows: minMP : M p = G (γ, h M , ∆, l g , N ) s.t. Pout ≥ P0 (19) η ≥ η0 B ≤ B yoke myoke size( Dout ; l ) where γ, h M , ∆, l g , N are the ratio of the outer diameter to inner diameter, thickness of PM, thickness of rotor yoke, length of air gap and turns of coils. Pout , η and P0 , η0 are the power and efficiency of the generator before and after optimization, Byoke is the flux density of the rotor yoke, which should be smaller than the saturated maximum flux of the rotor yoke Bmyoke , and the overall size of the generator mainly refers to the diameter and axial length. In addition to the limit of output power and efficiency, the rated design parameters are considered as the limitation, such as the rated output voltage and frequency. 4.2. Optimization Design The dynamic performance optimization design of the stator coreless AFPMSG is the nonlinear multi parameters optimization according to the previous model and analysis, where the algorithm adopted is important to the optimization results and efficiency. Direct search and random search are the two main nonlinear optimization algorithms [38]. Nowadays heuristic optimization algorithms including the genetic algorithm (GA) and Simulated Annealing (SA), have been widely developed and
Energies 2018, 11, 3162
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used in the optimization of electrical machines to overcome the limitations of initial conditions, local optimization, the difficulty of considering multi parameters as well as other deficiencies. The GA is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). GA are commonly used to generate high-quality solutions to optimization and search problems by relying on bio-inspired operators such as mutation, crossover and selection [39]. GA has been applied to the optimization design due to the advantages of simple process, global searching ability, and high expansibility, as in [18,19]. In this paper, the improved GA is used to perform the optimization, as discussed in [40]. In terms of the calculation of an integrated optimization model for dynamic performance, the modified design procedure combing the GA is proposed. GA was adopted in the paper to perform the calculation as a part of design procedure for the best optimization result. Thus simple and utility algorithm is preference. To avoid the problems of slow convergence and “precocity”, adjustable crossover probability and mutation probability are used in the GA process. Figure 14 illustrates the basic structure of the modified design procedure. A preliminary design is obtained through the sizing equation and basic electromagnetic relation on the condition of rated design parameters. The main structure parameters are optimized by the GA applied to search for the best solution from Energies 2018, 11, the variables of the generator for the object function. 15 of 22 Start
Determination of initial design parameters
Initial design results of AFPMSG
Initial population by random generation New generation
Evaluation the population fitness
Crossover & mutation with adjustable probabiility No
GA stopping criteria
Select elitist
Yes Opitmial design results of AFPMSG
Simulation and Validation
End
Figure Figure 14. 14. Flowchart Flowchart of of the the genetic genetic algorithm algorithm (GA) (GA) optimization optimization design design procedure. procedure.
The optimization optimization design design is is carried carried out out by by MATLAB MATLAB GA tools that are based on the mathematic mathematic optimization model. The The structure structure parameters before and after the the optimization optimization for for maximum maximum output output voltage overshoot overshoot target target are given in Table 2. When When comparing comparing the the data data of of Table Table 2, the the structure structure parameters of the generator are optimized with minor adjustments to the overall diameter and axial length. ofof PM areare decreased after optimization, which willwill reduce the length. The Theturns turnsofofcoils coilsand andthickness thickness PM decreased after optimization, which reduce EMF factor and and loss loss of the as well as the density of the and and output power. On the EMF factor of generator the generator as well as flux the flux density ofair thegap air gap output power. the the decreasing of theofdiameter ratio and of the of airthe gapair can notcan only make the On contrary, the contrary, the decreasing the diameter ratiolength and length gap not onlyup make influence mentioned above but also reduce the dynamic output voltage overshoot. Besides, although the mass of generator increases, the increasing of the rotor yoke thickness without changing the whole size greatly increases the rotational inertia and lowers the output voltage overshoot because the increment of mass concentrates on the rotor.
Energies 2018, 11, 3162
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up the influence mentioned above but also reduce the dynamic output voltage overshoot. Besides, although the mass of generator increases, the increasing of the rotor yoke thickness without changing the whole size greatly increases the rotational inertia and lowers the output voltage overshoot because the increment of mass concentrates on the rotor. Table 2. Design variables of the AFPMSG before and after optimization. Design Variable
Initial Value
Optimal Value
Outer diameter: Dout Out-in diameter ratio: γ Magnet thickness: h M Winding turns per phase: N Air gap length: l g Rotor yoke thickness: ∆ Axial length: l
418 mm 1.8 15 mm 434 1 mm 10 mm 62.1 mm
420 mm 1.75 13 mm 420 0.6 mm 15 mm 67.5 mm
5. 3D Finite Element Analysis 3D finite element analysis is performed to evaluate the validity of the dynamic analytical model and the optimization design procedure presented in the previous sections. The 3D model of the stator coreless AFPMSG is developed and simulated via the electromagnetic field simulation software Maxwell ANSOFT (Maxwell v16, ANSYS, Inc., Pittsburgh, PA, USA). The finite element models of the proposed AFPMSG are based on Table 2. Figure 15a illustrates the magnetic field distribution for the whole AFPMSG and the effective plane of the air gap. Asymmetrical distribution of flux density in the rotor yoke is caused by the armature reaction when considering the influence of stator current. As mentioned in Section 4, the optimization design of generator dynamic performance with the consideration of rotational inertia will increase the thickness of rotor yoke, which results in lower values of the maximum flux density being presented in the rotor yoke. Along with the structure characteristic of the stator coreless, it is obvious that the magnetic circuit is not saturated. The flux density of air gap increases with the optimization of air gap length and the thickness of PM, which guarantees the demand of electrical output. The flux density distributions of the air gap along the radial and axial direction are given in Figure 15b. The optimal three phase voltage as well as their harmonic contents are obtained and shown in Figure 16a. It is obvious that the optimized generator has a sinusoidal output voltage with the value meeting the design requirements and the harmonic contents can be negligible. Figure 16b compares the output voltage and power before and after optimization. The output voltage and power of optimization are bigger than that before optimization, indicating that the improved structure parameters enhance the electrical performance. The comparison also confirms the validity of structure parameter optimization for electrical performance. To compare the dynamic performance, the output voltage response to the same continuous disturbance torque input is simulated and analyzed before and after the optimization of the AFPMSG. The output voltages before and after the optimization are shown in Figure 17a, while Figure 17b illustrates the envelop curve of the maximum output voltage. The output voltage response of the optimized generator is slower and smaller than the preliminary one. It is observed that the dynamic response of output voltage after optimization became more stable, which exerts a better inhibition of voltage leap in the case of heavy disturbance. The comparison of maximum output voltage overshoot with different input torques is indicated in Figure 17c. The markings in Figure 17b,c indicate that the output voltage becomes more stable and the maximum output voltage overshoot is reduced, making the system more safe and reliable after optimization. Moreover, the inhibition of voltage leap for the optimized generator is more apparent when the input is heavily disturbed. It is concluded are obtained that the dynamic performance of the optimized AFPMSG is better and the optimized AFPMSG can operate more stably and reliable with unstably input conditions.
generator dynamic performance with the consideration of rotational inertia will increase the thickness of rotor yoke, which results in lower values of the maximum flux density being presented in the rotor yoke. Along with the structure characteristic of the stator coreless, it is obvious that the magnetic circuit is not saturated. The flux density of air gap increases with the optimization of air gap length and flux Energies 2018, 11, 3162the thickness of PM, which guarantees the demand of electrical output. The 16 of 22 density distributions of the air gap along the radial and axial direction are given in Figure 15b.
(a)
initial value optimal value
1000
Initial value Optimal value
800
750
Mag B of air gap Bg(mT)
Mag B of air gap Bg(mT)
900
600 450 300 150
600
400
200
0 0
20
40
60
80
100
120
0
Radial distance(mm)
10
20
30
40
50
60
Circumstance distance(mm)
(b) Figure 15. 15. Distribution Mag B before andand after optimization: (a) Magnetic flux Figure Distributionand andcomparison comparisonofof Mag B before after optimization: (a) Magnetic density distribution in the APFMSG andand on on thethe middle plane The flux density distribution in the APFMSG middle planeofofthe theeffective effectiveair air gap; gap; (b) The circumferentialdistribution distributioncurve curveat ataverage averageradius radiusand and radial radial distribution distributioncurve curve of of Mag Mag BB before before and and circumferential after optimization. optimization. after
The the preliminary and optimal designare are obtained given in Table 3, where The comparisons optimal threebetween phase voltage as well asdesign their harmonic contents and shown in the items toItbeiscompared arethe mainly the output electrical performance, consumption PM, Figure 16a. obvious that optimized generator has aoutput sinusoidal output voltage with theofvalue and the dynamic is the obvious that the optimized meeting the designperformance. requirements It and harmonic contents can begenerator negligible.complies with design requirements with slight improvement in terms of economy and efficiency. Besides, the output peak voltage is reduced under the same disturbance condition, which improves the safety and reliability of the generator. Additional simulation is carried out on the output voltage versus rotor speed and input torque for a constant resistive load, as shown in Figure 18. The result shows that the voltage versus rotor speed and input torque appear to be a linear relationship respectively, even in the low speed range, making it suitable for direct driven power generation such as direct couple wave power generation and low speed wind turbines.
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Energies 2018, 11,
17 of 22 Output voltage UA,UB,UC
FFT analysis
optimaldmodel_load
375.00
Curve Info
optimaldmodel_load
375.00
rms
Curve Info mag(InducedVoltage(WindingA)) Setup1 : Transient
UA 238.3232 Setup1 : Transient UB 237.1617 Setup1 : Transient UC 239.1178 Setup1 : Transient
mag(InducedVoltage(WindingA)) [V]
250.00
250.00
Voltage(V)
125.00
0.00
125.00
-125.00
-250.00
-375.00
0.00 0.00
5.00
10.00
15.00 Time [ms]
20.00
25.00
30.00
0.00
100.00
200.00
300.00
400.00
500.00
Freq [Hz]
(a)
Initial value Optimial value
300
3000
200
2500
100 0 -100 -200
2000 1500 1000 500
-300
0
-400
-500 0
5
10
15
20
25
Initial value Optimal value
3500
Output power Pout(W)
Output voltage U(V)
400
30
0
Time (s)
5
10
15
20
25
30
Time (s)
(b) Figure 16. 16. Distribution Distribution and and comparison comparison of of output output performance: performance: (a) (a) Output Output voltage voltage after after optimization optimization Figure and Fast Fourier Output voltage andand power comparison before and after and FourierTransform Transform(FFT) (FFT)analysis; analysis;(b)(b) Output voltage power comparison before and optimization. after optimization.
Figure 16b compares the output voltage and power before and after optimization. The output Table 3. Output and dynamic performances of the AFPMSG before and after optimization. voltage and power of optimization are bigger than that before optimization, indicating that the improved structure parameters enhance the electrical performance. Initial The comparison also confirms Parameters Value Optimal Value the validity of structure parameterRated optimization for electrical performance. output power: Pout 2893 W 3078 W To compare the dynamic output voltage to the same 237 continuous Ratedperformance, RMS voltage atthe rated 300 rpm: Uout response 220 V V Output performance Total loss:analyzed Ploss 325.8the W optimization 298.6 W disturbance torque input is simulated and before and after of the Maximum efficiency: η 0.9 0.912 AFPMSG. The output voltages before and after the optimization are shown3 in Figure 17a, 3while PM material volume: VPM 0.0024 m 0.002 m Figure 17b illustrates the envelop curve of the maximum output voltage. The output voltage response time:than ∆t the preliminary 1.5 s It is observed 1.9 that s of Dynamic the optimized generator is slowerRegulating and smaller one. the performance Maximum voltage overshoot: M p 4.45% 3.37% dynamic response of output voltage after optimization became more stable, which exerts a better inhibition of voltage leap in the case of heavy disturbance. The comparison of maximum output voltage overshoot with different input torques is indicated in Figure 17c. The markings in Figure 17b,c indicate that the output voltage becomes more stable and the maximum output voltage overshoot is reduced, making the system more safe and reliable after optimization. Moreover, the inhibition of voltage leap for the optimized generator is more apparent when the input is heavily disturbed. It is concluded are obtained that the dynamic performance of the optimized AFPMSG is better and the optimized AFPMSG can operate more stably and reliable with unstably input conditions.
18 of 22 18 of 22
Input Torque
110
400
100
300
95
200
90 0.0
2.5
5.0
7.5
450
Output Voltage(V)
Partial output voltage
105
12.5
10.0
15.0
Time(s)
17.5
Output Voltage(V)
Input Torque(Nm)
Energies 2018, 11, 3162 Energies 2018, 11,
20.0
Output Voltage
300 150 0 -150
100 0 -100 -200 -300
-300 -450 0.0
2.5
5.0
7.5
12.5
10.0
15.0
17.5
-400 4.4
20.0
4.8
4.6
5.0
5.2
5.4
5.6
Time(s)
Time(s)
(a) Upper envelope_initial deisign Lower envelope_initial design Upper envelope_optimal deisign Upper envelope_optimal deisign
Envelope of output voltage(V)
450 300 150 0 -150 -300 -450 0
5
10
15
20
time(s)
(b) 140
Initial design Optimal design
Maximum overshoot Mp(%)
120 100 80 60 40 20 0 0
20
40
60
80
100
120
140
160
Input disturbance torque T(Nm)
(c) Figure Dynamic performance analysis: Output voltage with variable torque input optimal Figure 17.17. Dynamic performance analysis: (a)(a) Output voltage with variable torque input forfor optimal design and partial detail; (b) Envelop comparison of the maximum output voltage before and after design and partial detail; (b) Envelop comparison of the maximum output voltage before and after optimization; (c) Maximum overshoot with different input disturbance torque. optimization; (c) Maximum overshoot with different input disturbance torque.
The comparisons between the preliminary design and optimal design are given in Table 3, where the items to be compared are mainly the output electrical output performance, consumption of PM, and the dynamic performance. It is obvious that the optimized generator complies with design requirements with slight improvement in terms of economy and efficiency. Besides, the output peak voltage is reduced under the same disturbance condition, which improves the safety and reliability of the generator.
Additional simulation is carried out on the output voltage versus rotor speed and input torque for a constant resistive load, as shown in Figure 18. The result shows that the voltage versus rotor speed and input torque appear to be a linear relationship respectively, even in the low speed range, Energies 2018, 11, 3162 for direct driven power generation such as direct couple wave power generation 19 of 22 making it suitable and low speed wind turbines. InducedVoltage vs n
InducedVoltage vs T
optimaldmodel_load(variable_n)
600.00
optimaldmodel_load(variable_T)
600.00
Curve Info
Curve Info
InducedVoltage(WindingB) Setup1 : Transient n='100rpm'
InducedVoltage(WindingA) Setup1 : Transient T='40New tonMeter'
InducedVoltage(WindingB) Setup1 : Transient n='150rpm'
InducedVoltage(WindingA) Setup1 : Transient T='60New tonMeter'
InducedVoltage(WindingB) Setup1 : Transient n='200rpm'
400.00
InducedVoltage(WindingA) Setup1 : Transient T='80New tonMeter'
400.00
InducedVoltage(WindingB) Setup1 : Transient n='250rpm'
InducedVoltage(WindingA) Setup1 : Transient T='100New tonMeter' InducedVoltage(WindingA) Setup1 : Transient T='120New tonMeter'
InducedVoltage(WindingB) Setup1 : Transient n='350rpm'
200.00
InducedVoltage(WindingA) [V]
InducedVoltage(WindingB) [V]
InducedVoltage(WindingB) Setup1 : Transient n='300rpm'
InducedVoltage(WindingB) Setup1 : Transient n='400rpm' InducedVoltage(WindingB) Setup1 : Transient n='450rpm' InducedVoltage(WindingB) Setup1 : Transient n='500rpm'
0.00
-200.00
InducedVoltage(WindingA) Setup1 : Transient T='140New tonMeter'
200.00
InducedVoltage(WindingA) Setup1 : Transient T='160New tonMeter' InducedVoltage(WindingA) Setup1 : Transient T='180New tonMeter'
0.00
-200.00
-400.00
-400.00
-600.00
-600.00 0.00
10.00
20.00
30.00 Time [ms]
40.00
50.00
60.00
0.00
10.00
20.00
(a)
30.00 Time [ms]
40.00
50.00
60.00
(b) 500
Torque vs E Speed vs E
InducedVoltage E(V)
400
300
200
100
0 0
100
200
300
400
500
Speed(r/min)&Torque(Nm)
(c) Figure Figure 18. 18. Induced InducedVoltage Voltage curves curves with with varied varied speed speed & & torque. torque. (a) (a) simulation simulationcurves curves of of induced induced voltage voltagewith withvaried variedspeed speedinput; input;(b) (b)simulation simulationcurves curvesof ofinduced inducedvoltage voltagewith withvaried variedtorque torqueinput; input; (c) (c)comparison comparisonon onthe theinduced inducedvoltage voltagebetween between varied varied speed speed and and torque torque input. input.
Conclusions 6.6.Conclusions In this this paper, performance study and an optimization design design of statorofcoreless In paper,a adynamic dynamic performance study and an optimization stator AFPMSG coreless was presented for direct driven variable speed power generation application. The dynamic AFPMSG was presented for direct driven variable speed power generation application. The analytical dynamic model was developed based on the structure a dynamic of AFPMSG swing analytical model was developed based on the analysis structureand analysis and amodel dynamic model of and AFPMSG equation. Moreover, dynamic performance indexes were considered through the simulation of a dynamic analytical model under starting up and disturbance input conditions. Key structure parameters were analyzed based on the dynamic analytical model and an integrated optimization model with multiple structure parameter variables was developed for the optimization design of the generator dynamic performance. Considering the demand of dynamic performance for direct driven generators, based on a nonlinear iterative genetic algorithm, a modified optimization design procedure was proposed. A 3 kW 300 r/min stator coreless AFPMSG was optimally designed through the design procedure. Theoretical calculation and simulation analysis were performed on the electrical and dynamic performance of the generator before and after optimization, the results of which showed that the dynamic performance was improved after optimization without reducing the electrical output. Moreover, the 3D finite element simulation outcomes and comparison also demonstrated the accuracy of the theoretical analysis and model in the paper.
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Author Contributions: W.W. (Wenqiang Wang) and Y.W. designed the main parts of the research, including objective function establishment and optimal methodology designing. W.W. (Weijun Wang) was responsible for guidance, a number of key suggestions and manuscript editing. L.M. and H.M. mainly contributed to the writing of the paper and were also responsible for project administration. H.L. and G.Z. made contributions to simulations in Ansoft and gave the final approval of the version to be published. Funding: This research received no external funding. Acknowledgments: The authors would like to acknowledge the technological and financial support from “The National Science and Technology Support Program (supported by the Ministry of Science and Technology of P.R.C. No. 2014BAC01B05)”, “Key projects of PLA (BY113C001)”, “Army Funding for military graduate students (2016JY486”. The authors are grateful for the comments and suggestions provided by the anonymous reviewers and editor helped to improve the quality of the paper Conflicts of Interest: The authors declare no conflict of interest.
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