Multi-objective optimization of a double-sided air-cored tubular generator based on a new integrated modeling method

Abstract

Double-sided air-cored tubular generators (DSTG) have higher power density than traditional air-cored generators, and are desirable for applications in renewable energy conversion systems. In order to achieve best power quality and maximum efficiency with minimum cost, multi-objective optimization of the DSTG is carried out. Aim to decrease the computational time and guarantee the accuracy of the multi-objective optimization of DSTGs, a new integrated modeling method is proposed and focused in this paper. The new modeling method integrates the analytical models and the machine learning models together. The experimental results prove that the new integrated model can provide higher accurate calculation results than analytical models and need fewer samples than machine learning models. The optimization time needed by the new model is 5 times shorter than that needed by the FE model.

Keywords

Double-sided tubular generator machine learning multi-objective optimization

1. Introduction

Compared with slotted tubular generators [1–3], air-cored tubular generators can achieve excellent force ripple values and a lower moving mass, which effectively enhance the dynamic stability and the generated power quality [4]. However, the relatively large equivalent air gap of air-cored configurations constrains the generator’s power density. A double-sided air-cored tubular generator (DSTG) adapts two layers of quasi-Halbach permanent magnet (PM) arrays to generate the magnetic field. It can provide higher power density than other air-cored tubular generators, and is suitable for applications in energy conversion systems of ambient vibrations, free-piston engines, and tidal wave energy conversion systems.

The basic performances of a single-phase DSTG were analyzed in [5] by using equivalent magnetic circuit model. In order to enhance the calculation accuracy, analytical models of the double-sided tubular motor were derived in [6]. Analytical models of the three phase double-sided generators with account of the armature reaction field were established in [7]. However, these analytical models ignored the effects of ends of cores, eddy currents and the flux leakage, which would cause calculated errors in the flux density and the performances.

In this paper, the differences between the analytical models [7] and the FE models are analyzed systematically. A new integrated modeling method is proposed and emphasized. Then, based on the fast but accurate new integrated models, a multi-objective optimization of the DSTG is carried out. An optimization design of the DSTG with best power quality, maximum efficiency and minimize cost is obtained.

2. Topology of double-sided tubular generators

The basic construction of the DSTG of one pole pair is shown in Fig. 1. The generator consists of two cylinder stators and one cylinder mover. Two layers of cylinder quasi-Halbach arrays are installed on the external stator and the internal stator separately. Two PM arrays can generate magnetic flux closed-loop themselves. This PM topology can provide higher radial magnetic field density than traditional single PM topology. The air-cored mover is placed between the external stator and the internal stator. The windings are twined around the hollow resin axis, which make it has lower mover mass, lower iron loss and lower force ripple than slotted tubular generators. When the mover moves straight reciprocatingly, the windings on the mover will cut the flux line to generate electromotive force.

Fig. 1.

Construction of the DSTG.

In the DSTG, the two PM arrays always have the same thickness h _m and same composition structure. In the quasi-Halbach PM arrays, the width of the radial-magnetized PM is τ_mr, the width of the axial-magnetized PM is τ_mz. τ_mr + τ_mz = τ, where τ is the pole pitch of the PM array. The thickness of the windings is h _c. The width of one winding is τ_c, τ_c ≈ τ∕3. When the external diameter of the external PMs R ₂ and the thickness of air-gap are determined, the main dimensions need to be optimized in the DSTG is the τ_mr, h _m and h _c. With the three dimensions, the internal diameter of the internal PMs R ₁ and the internal diameter of the windings R _c1 both can be derived.

3. New integrated modeling method

It is well known that FE models have high accuracy but are quite time-consuming in calculation. In order to save calculating time, analytical models are often used in optimization designs instead of FE models [8]. However, the hypothesis conditions of the analytical models cause the calculation errors. To improve the accuracy, machine learning (ML) algorithm is put into use in machine modeling recently with the development of artificial intelligent technology. The ML modeling method uses FE results as learning samples. To get the high accurate learning result, the more outputs, the more samples and the longer calculation time is needed.

Generally, at least three or four performances need to be calculated in the multi-objective optimization of the electrical machines. In order to guarantee the calculation accuracy and decrease the numbers of the leaning samples, a new integrated modeling method is proposed. The overall diagram of the proposed integrated modeling method is shown in Fig. 2.

Fig. 2.

Overall diagram of the integrated modelling method.

This modeling method is a method utilizing difference concept. Analytical models are used as knowledge based coarse models, FE models are used as fine models. The differences between the coarse models and the fine models are used as samples to train ML models. The analytical models and the ML models are integrated to generate the proposed model.

3.1. Analytical modeling of DSTG

Analytical models of the three phase double-sided generators derived in [7] are based on the magnetic vector potential method with the following assumptions: (1) the relative permeability of PMs 𝜇_r = 1; (2) the effects of eddy currents are neglected; and (3) the z-axis length of the PM array is infinite. When the position of the windings is z, the introduced voltage across one phase can be described as Eq. ((1)). The electromagnetic force of one phase can be described as Eq. ((2)). $\begin{eqnarray}\displaystyle & \displaystyle e=-\mathop{\sum }_{n=1}^{\infty }\frac{4{\pi}pK_{dn}N_{1}v}{h_{c}}\left(\int _{R_{c1}}^{R_{c1}+h_{c}}rK_{Irn}dr\sin m_{n}\left(z-\frac{{\tau}}{2}\right)+\int _{R_{c1}}^{R_{c1}+h_{c}}r\mathop{\sum }_{m=1}^{phase}K_{\mathit{IIrmn}}dr\right) & \displaystyle\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle & \displaystyle f=-\mathop{\sum }_{n=1}^{\infty }\frac{4{\pi}pK_{dn}N_{1}i}{h_{c}}\left(\int _{R_{c1}}^{R_{c1}+h_{c}}rK_{Irn}dr\sin m_{n}\left(z-\frac{{\tau}}{2}\right)+\int _{R_{c1}}^{R_{c1}+h_{c}}r\mathop{\sum }_{m=1}^{phase}K_{\mathit{IIrmn}}dr\right) & \displaystyle\end{eqnarray}$ (2) where K _Irn is the amplitude of the nth harmonics in B _Ir, B _Ir is r-component of the PMs-due flux density, K _Irmn is the amplitude of the nth harmonics in B _IIrmn, B _IIrm is r-component of the flux density generated by the m-phase windings, i is the current, v is the speed of the mover, p is the number of pole pairs, N ₁ is the turn number of the windings, K _dn is the distribution factor of the windings, phase is the number of phases, m _n = (2n −1)π∕τ.

Table 1

Main dimensions of the DSTG (Unit: mm)

R ₂	R _m2	R _c2	R _c1	R _m1	R ₁	τ
110	102	101	91	89	81	50

Fig. 3.

Variation of the parameters with width of radial magnetized PMs.

3.2. Differences between analytical models and FE models

In order to analyze the accuracy of the analytical models, FE models of the DSTG with two pole pairs are built and compared with the analytical models. The main dimensions of the DSTG are shown in Table 1. With the variation of the main dimensions of the DSTG, differences between the analytical models and the FE models are systematically analyzed. Figure 3, Fig. 4 and Fig. 5 illustrate the variations of four important performances and their differences between two models, where blue solid lines are the FE simulation results, dashed lines are the analytical results, and dotted lines are the differences in the figures. Power density is the ratio of output power to PM volume. Force ripple is the ratio of thrust peak-to-peak value to thrust force mean value. V _thd is total harmonic distortion of output voltage as Eq. (3), where V ₁ is the amplitude of the fundamental wave in the output voltage, V _n is the amplitude of the nth harmonic in the output voltage. $\begin{eqnarray}\displaystyle V_{thd}=100\sqrt{\mathop{\sum }_{n=2}^{\infty }}V_{n}^{2}/V_{1}. & & \displaystyle\end{eqnarray}$ (3)

The comparison results show that the maximum difference of the power density between FE results and analytical results is smaller than 3.5%, and the maximum difference of the efficiency is smaller than 0.4%. The two performances calculated by analytical models are accurate enough for the optimization.

Fig. 4.

Variation of the parameters with thickness of PMs.

Fig. 5.

Variation of the parameters with thickness of windings.

By contrast, the analytical results of force ripple and the V _thd show big difference with the FE results. The dotted line in Fig. 3(c), Fig. 4(c) and Fig. 5(c) show that the difference of the force ripple between two models changes from 10% to 170%, which reaches its maximum when τ_mr is 36 mm. The dotted line in Fig. 3(d), Fig. 4(d) and Fig. 5(d) show that the difference of V _thd between two models changes from −10% to 28%, which reaches its maximum when the thickness of PM is 4 mm. The differences are too big to be ignored. The accuracy of the force ripple model and the V _thd model need to be improved. However, there is no regular relationship between the differences and the dimensions can be found.

3.3. Machine learning modeling

In the new integrated modeling method, machine learning algorithm are used to build a compensate model to improve the calculation accuracy of the analytical models. As shown in Fig. 2, the differences between the FE results and the analytical results are used as sample sets to training the machine learning models.

Different from the mechanism modeling, machine learning is a method to devise data-driven prediction models based on data analytics. In the machine learning modeling method, the size and the structure of the dataset would influence the accuracy of the machine learning method. With plenty of the sample data and the appropriate algorithm, a machine-learning model can provide high accurate prediction results, which makes it suitable for identifying nonlinear models for complex systems, such as electromechanical systems. KNN algorithm is used in [9] to establish a two-output model of a PMSLM based on 243 groups of FEA results, and the model accuracy is about 98%. Extreme learning machine is used in [10] to establish a nine-output model for a similar PMSLM based on 625 groups of FEA results. The number of the sample data increases quickly with the increment of the output numbers and the complex degree of the systems.

For multi-objective optimization of the DSTG, five performances, such as force ripple, efficiency, copper loss, V _thd and output power, should be calculated by the models. In order to simple the machine learning model, decrease the number of the sample data and maintain the accuracy of the calculation results, only parts of the performances are chosen to be calculated by machine learning models in the proposed new modeling method. The output power, the loss and the efficiency calculated by the analytical models are accurate enough as shown in the former section. Only the force ripple and V _thd should be predicted by the machine learning model. With the variation of width of radial magnetized PMs, the thickness of PMs and the thickness of windings, 300 groups of difference data between the FE results and analytical results are prepared as samples, in which 240 groups of data are used as train samples, and 60 groups of data are used as cross-validation samples.

In order to make accurate predictions possible, three different machine learning algorithms have been attempted. Linear regression is firstly used to explore and map the relationships between the dimensions and the performances as one of the most well-known and well-understood algorithms in statistics and machine learning. However, the fitting results in Fig. 6 show that the score of the linear regression models decreases quickly with the increment of the numbers of the original data. The accuracy of the seventh degree polynomial regression with 200 groups of original data is only 0.86.

Fig. 6.

Learning curves of the linear regression algorithms.

The learning curves of the gradient boosting regressor, Ada boost regressor and decision tree regressor are shown in Fig. 7. The three machine learning models are all based on decision tree algorithm, but Ada boost regressor is obviously more suitable for the prediction of sample data, which accuracy is about 0.96 when the number of the data is 200.

Fig. 7.

Learning curves of the decision tree regression algorithms.

K-nearest neighbors (KNN) algorithm is a simple but an effective algorithm for prediction the data in this paper. In KNN, predictions are made for a new data point by searching through the entire training set for the K most similar instances (the neighbors) and summarizing the output variable for those K instances. The learning curves of the KNN algorithm in Fig. 8 show that the accuracy of the KNN prediction results increase with the increment of the number of the original data. With all the 240 original data, the final scores of three different algorithms are calculated and shown in Table 2. It can be found that the KNN algorithm with K = 3 has the highest accuracy. Thus, the proposed integrated models of the DSTG can be obtained by summarizing the KNN machine learning models and the analytical models together.

Fig. 8.

Learning curves of the KNN algorithms.

Table 2

Scores of the machine learning algorithms

Algorithm	Linear regression (n = 3)	Linear regression (n = 5)	Linear regression (n = 7)
Score	<75%	75.6%	86.4%
Algorithm	Gradient boosting regressor	Ada boost regressor	Decision tree regressor
Score	<75%	96.4%	87.3%
Algorithm	KNN (n = 3)	KNN (n = 4)	KNN (n = 5)
Score	96.6%	96.5%	95.8%

3.4. Prototype and experiment

In order to verify the accuracy of the new model, a prototype is manufactured as shown in Fig. 9 are built. The prototype is driven by a tubular motor. The position of the mover is measured by an optical grating, and the force acted on the mover is measured by a force sensor. The output signals of the optical gating and the force sensor are collected by the NI data acquisition card and connected with the computer. When the resistance load is 303Ω, the output of one phase is measured. The experimental results and the results of the models are shown in Table 3. The force result of the experiment is not compared because of the influence of large frictional force in the end part of the mover. The comparison of the results shows that the integrated model results have higher accuracy than the analytical results. The differences between the experimental results and the integrated model results are smaller than 6.8%.

Fig. 9.

The prototype of DSTG.

Table 3

The experimental results and the results of the models

Models	V ₁ (V)	V _thd	F _mean (N)	F _ripple
Experimental results	33.225	21.67%	/	/
FE results	34.19	20.6%	6.13	11.63%
Analytical results	35.65	18.6%	5.813	10.14%
Integrated results	35.65	21.5%	5.813	11.78%

4. Multi-objective optimization of DSTG

Based on the new integrated models of the DSTG, a multi-objective optimization of the DSTG is carried out. The overall diagram of the multi-objective optimization is presented in Fig. 10. Firstly, the possibility of the optimization is analyzed and the dimension constrains are defined according to the FE and analytical results. Then, the multi-objective function is built according to the optimization goal, and PSO optimization algorithm is used to obtain an optimal solution.

Fig. 10.

Overall diagram of the proposed multi-objective optimization.

4.1. Objective function

The external PM array and the internal PM array has the same magnetizing structure and the same thickness in the DSTG. With the outer dimension of the external PM remained constant, the main dimensions need to be optimized are the magnetizing composition of the Halbach PM array (τ_mr), the thickness of the PM array and the thickness of the windings.

The results shown in Fig. 3 present that the force ripple gets the minimum when τ_mr is 36 mm, and the V _thd gets the minimum when τ_mr is 28 mm. The best τ_mr must exist between 28 mm and 36 mm to satisfy the combination optimization for force ripple and the V _thd. The results shown in Fig. 4 and Fig. 5 illustrate that the increment of the PM thickness would decrease the power density and enhance the efficiency. The increment of the winding thickness would enhance the power density and decrease the efficiency. Therefore, there is a tradeoff between the efficiency and the power density. According to the conclusions, the dimension constraint set as Eq. (4) can ensure that all the possible optimal dimensions are included in the optimization.

Since the force ripple and the V _thd could not be minimized at the same time according to the FE results, the performance constraint is set as Eq. (5) to guarantee the stationarity and reliability of the DSTGs.

In this paper, the objective is to maximize the output power density and the efficiency while minimizing the force ripple and the total harmonics distortion in the output voltage. Aggregating all objective functions together, a composite single-objective function as Eq. (6) can be created, in which F _ripple is the force ripple, P _cu is the copper loss of the windings, 𝜂 is the efficiency, P is the output power and V _pm is the PM volume. $\begin{eqnarray}\displaystyle & \displaystyle \left\{\begin{array}{@{}l@{}}25∼\text{mm}\leq {\tau}_{mr}\leq 40∼\text{mm}\\ 4∼\text{mm}\leq h_{m}\leq 10∼\text{mm}\\ 5∼\text{mm}\leq h_{c}\leq 12∼\text{mm}\end{array}\right. & \displaystyle\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle & \displaystyle \left\{\begin{array}{@{}l@{}}F_{ripple}<10\%\\ V_{thd}<20\%\end{array}\right. & \displaystyle\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle & \displaystyle OF=\frac{F_{ripple}+V_{thd}}{{\eta}}+\frac{P_{cu}∼^{\ast }∼10^{4}}{P/V_{pm}}. & \displaystyle\end{eqnarray}$ (6)

4.2. PSO optimization

Particle swarm optimization (PSO) algorithm is applied to optimize the construction of the DSTG based on the integrated models [11]. 30 sets of particles are searching for the optimum in the PSO algorithm. The particles update their velocities v _i according to the best positions P _g at each generation. For the s +1 generation, the velocity of the ith particle is: $\begin{eqnarray}\displaystyle v_{id}(s+1)=0.7v_{id}(s)+1.4r_{1}(P_{id}(s)-x_{id}(s))+1.3r_{2}(P_{gd}(s)-x_{id}(s)). & & \displaystyle\end{eqnarray}$ (7)

For the s +1 generation, the positon of the ith particle is: $\begin{eqnarray}\displaystyle x_{id}(s+1)=x_{id}(s)+v_{id}(s+1) & & \displaystyle\end{eqnarray}$ (8) Where r ₁ and r ₂ are random numbers uniformly distributed in the interval [0, 1], P _i is the existing minimum value of the ith particle in the objective function, d is the dth dimensions of the particle. The three dimensions of the particles correspond to the main three dimensions of the DSTG, i.e. the width of the radial magnetizing PM τ_mr, the thickness of PMs, and the thickness of windings.

In the optimization, the PSO algorithm is completed after 47 iterations. The variation of the fitness is shown in Fig. 11. The corresponding positions of the particles, the variation of three dimensions, are shown in Fig. 12. It is clear that the PSO has an excellent convergence rate. The optimal dimensions are thus found to be τ_mr = 35 mm, thickness of PMs h _m = 10 mm, and thickness of windings h _c = 6 mm. Compared with the original dimensions, the optimized DSTG has higher efficiency and much lower force ripple. The details of the comparison are listed in Table 4.

Fig. 11.

Variation of the fitness.

Fig. 12.

Variation of the dimensions.

5. Conclusion

In this paper, a new integrated modeling method is proposed, which integrates the analytical models and the machine learning models together. It takes advantage of the time-saving of the analytical model, simples the prediction algorithm of the machine learning models and is as accurate as the FE models.

Table 4

Comparison of the performances

	𝜂	Force ripple	V _thd	P∕V _pm ∕P _cu
Original design	0.8385	13.18%	21.49%	10536
Optimized design	0.8747	3.19%	17.83%	11323

Based on the integrated model, a multi-objective optimization of the DSTG is carried out. The efficiency of the optimized design is 4% higher than that of the original one. The force ripple of the optimized design is 10% lower than that of the original one. The total harmonic distortion of the optimized design is 3.6% lower than that of the original one.

The proposed integrated model is quite timesaving especially in the optimization. In the multi-objective optimization, about 1400 conditions are totally calculated by the integrated model, because 30 particles are compared in each iteration. Each condition will cost about 600 seconds by the FE models, but only 5 seconds by the new model. The calculation time need by the optimization based on the new model is about 52 hours with account of the calculation time of the samples. However, the calculation time need by the optimization based on the FE model is about 233 hours. The calculation time of the optimization based on the new model is 78% shorter than that of the latter one.

Footnotes

Acknowledgements

This work was supported in part by the National Science Foundation of China under Grant 51677172, in part by the Zhejiang Provincial Natural Science Foundation of China under Grant LY19E070006, LY18E070006 and LY17E070002.

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