Optimization of a Transverse Flux Linear Generator for the Harnessing of Wave s Energy

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1 Optimization of a Transverse Flux Linear Generator for the Harnessing of Wave s Energy Tiago Barroca IST UTL Lisbon, Portugal tiagobarroca@ist.utl.pt 1 Abstract Energy from ocean waves is a renewable energy source which, until the present, still remains one of the most under-exploited. Even though several projects have been developed over the past years, no specific technology has stood out as the most reliable, contributing to what is called a nontechnological stabilization. Specifically, for this work, the harnessing of ocean wave s energy is embodied in the analysis and optimization of a transverse flux linear generator. This paper aims to analyze and optimize, through computational tools, the single-phase topology of the linear generator. To obtain a compact yet economically competitive generator, optimizations were carried out on the weight and cost. Studies were also conducted regarding the notorious importance of the magnetic dispersion, from which it was possible to gather constraints to be implemented in the optimization programs. This work seeks to contribute to a better understanding of the linear generator through the implementation of optimization methodologies applied to finite element models. Keywords wave energy, linear generator, transverse flux, magnetic dispersion, finite element analysis, optimization, weigth, cost. I. INTRODUCTION Daily, huge amounts of energy are extracted, converted, distributed and consumed. Even though 81% of the worldwide consumed energy origins from fossil fuel [1], scientists from all around the world are already studying the possibility that the oil exploration has reached its peak [2] and how the emissions of carbon dioxide is causing climate changes. Accordingly, it is important to develop new forms of energy which are sustainable and do not affect the environment dramatically. Energy from ocean waves is a renewable energy source which, until the present, still remains one of the most under-exploited. Among the numerous models and prototypes of converters for this type of energy, there is the Archimedes Wave Swing (AWS) [3] (shown in Figure 1). The AWS consists of an upper part (the oater) of the underwater buoy moves up and down in the wave while the lower part (the basement or pontoon) stays in position. The periodic changing of pressure in a wave initiates the movement of the upper part (Figure 1(a)). The oater is pushed down under a wave top (Figure 1(b)) and moves up under a wave trough (Figure 1(c)). To be able to do this, the interior of the system is pressurized with air that serves as an air spring. The air spring, together with the mass of the moving part, is resonant with the frequency of the wave [3]. The mechanical power required to damp the free oscillations is converted to electrical power by means of a linear electrical generator. Figure 1 Operating principle of an AWS device This paper aims to analyze and optimize, through computational tools, the single-phase topology of the linear generator. For the finite element simulation the FEMM program was chosen and for the optimization purposes was used the commercial program MATLAB as well as the optimization functions available on it. However, the dimensioning should also fulfill the specifications described in Table 1. Table 1 Design specifications of the linear generator Parameter Designation Value v Linear speed p Number of pole pairs PN Nominal Power J Current Density II. OPERATING PRINCIPLE OF A SINGLE-PHASE TRANSVERSE FLUX LINEAR GENERATOR In Figure 2 it is shown a single pole of the transverse flux machine (TFM) where its working principle can be understood. This machine consists of a fixed part (stator) and a moving part (translator). The machine s stator has two iron pieces, one U shaped and another I shaped and between them are two neodymium magnets that will change their polarity periodically, as the translator moves. Therefore, the magnetic flux in the iron core will vary and it will generate an electromotive force on a turn winding [6]. Figure 2 Pole morphology and magnetic flux density plot

2 2 III. GEOMETRICAL, ELECTRICAL AND MAGNETIC CHARACTERISTICS OF THE SINGLE-PHASE MFT This section will provide an analysis about the geometrical, electrical and magnetic characteristics of the linear generator. Knowing that stands for the electrical frequency of the generator, the mean value of the electromotive force,, can be defined as: (3) Due to the low electrical frequency observed in the generators for this type of energy harnessing, only the Joule losses will be considered for the calculation of the efficiency. Considering the total length of the linear generator and as the copper resistivity, the Joule losses can be defined as follows: (4) Thus, considering efficiency shall be: the total generated power, the (5) Figure 3 Two pole pairs of the single-phase MFT Figure 3 shows a schematic of two pole pairs of the generator where is possible to observe the parameters related to the physical dimensions as well as the symbology associated. Therefore, the geometry of the generator can be completely described within thirteen parameters: four from Table 1 and nine from Table 2. Table 2 Main parameters of the single-phase MFT geometry Parameter Comments Units D Total width of the generator H U shaped iron piece c U shaped iron piece d U shaped iron piece / Magnet u I shaped iron piece e Distance between poles g Air gap h Magnet N Winding turns IV. NUMERICAL STUDY OF THE MAGNETIC FIELD The magnetic dispersion and its influence on the magnetic are crucial aspects that determine the dimensioning of the MFT. Although considering that the magnetic in the stator is uniform (due to the high magnetic permeability of the ferromagnetic material), the same cannot be applied to the air gaps. Thus, simplified magnetic circuits that try to recreate real situations will now be used in order to determine the behavior of the magnetic flux dispersion. Three critical situations can be considered: the analysis of the influence of the distance between poles; the analysis of the influence of the distance between the rows of magnets of the translator and the analysis of the air gap dimension. A. Analysis of the influence of the distance between poles Since the translator moves and the magnets will change their polarity periodically, if the magnets of the same row are very close to each other, the magnetic lines will close between themselves and the magnetic in the stator would be less intense. It is therefore necessary to quantify that distance in relation of the behavior of the magnetic. Figure 4 shows the simplified circuit used for this simulation. Defining as the fill factor, the current intensity given by the generator is: (1) Considering a magnetic circuit without leakage and an uniform magnetic flux along the circuit, once the crosssectional area of the U shaped iron piece and of the air gap are the same, so will be their respective magnetic s and. Considering as the magnetic remanence and as the relative magnetic permeability of the magnet, the following equation is given: (2) Figure 4 Simplified circuit used in the analysis of the distance between poles Thus, fending the upper circuit, calculating the magnetic in the lower one and varying the parameter, allows to understand the behavior that the magnetic adopts. It is

3 3 now possible to observe in Figure 5 the magnetic in relation to the ratio of the parameters and. It was considered a very low dimension for the parameter in order to minimize the effects of the air gap dimension. This fact is in accordance to equation (2), where is possible to understand that the magnetic is inversely proportional to the air gap dimension. Figure 6 - Simplified circuit used in the analysis of the distance between the rows of the translator Varying the parameters and and calculating the magnetic in the U shaped iron piece allows to know the variation of the magnetic. The Figure 7 was made based on the obtained results and it represents the magnetic in relation to the ratio of the parameters and. Figure 5 Adequate polynomial of the variation of the magnetic In Figure 5 can be seen five curves of the magnetic, one for each value of the parameter simulated. However, it is now important to discover the points where each curve stabilizes, because that is the point where the distance between poles is no longer relevant. With the fitting software Ezyfit [10] it is possible to find an adequate polynomial to represent those points. Once the ratio between the parameters should be in the adequate polynomial or in the region to the right of it, assuming that: it is now possible to define a mathematical relation between the magnetic and, as follows: (6) Figure 7 Adequate polynomial of the variation of the magnetic Through the observation of Figure 7 is possible to, once again, distinguish five curves of the magnetic, one for each value of the simulated parameter. Similarly to what was preciously carried out, it is now of interest to verify where the curves stabilize and create a mathematical relation to ensure that the ration between the parameters is on the adequate polynomial or in the region to its right. Assuming that: B. Analysis of the influence of the distance between the rows of magnets of the translator Since the distance between poles is now analyzed, it is of interest to understand how the magnetic varies in relation to the distance between the two rows of the magnets. For this simulation it is used a simplified model that seeks to resemble to a pole of the single-phase MFT and it is in Figure 6. Is it also important to refer that the parameter adopted a very low dimension for the same reason that was mentioned in the previous simulation. the magnetic is related to C. Analysis of the air gap dimension as follows: The air gap is the region of the space between the magnets and the stator and it allows the movement of the translator. Once the magnetic reluctance of the air is very high, the dimensions of the air gap will be responsible for the dispersion of the magnetic lines. Moreover, the smaller the air gap, the stronger the magnetic in the iron core. First of all, in this section, it will be made a joint analysis of the relation between the air gap dimension (parameter ) and the magnet parameters and. To finalize, it will be carried out an (7)

4 4 analogous analysis but now focused on the parameter of the magnet and the air gap dimension. The first simplified magnetic circuit used in these simulations is identical to the one expressed in Figure 6. However, it is now the parameter the one who varies along with the parameter. Now, once is intended to study the influence of the dimension of the air gaps, the parameter is a fixed value and it is large enough to minimize the effects of the distance between the rows of the translator, emphasizing the effects of the air gap dimension. Through the obtained results of the simulation, it is now possible to present Figure 8. Figure 9 - Simplified circuit used in the analysis of the air gap dimension Based on the results that were obtained from the simulation, it is now possible to present Figure 10. Figure 8 - Adequate polynomial of the variation of the magnetic This simulation states what was theoretically expected according to equation (2), once the magnetic is inversely proportional to the parameter and directly proportional to the parameter. To gather this information in a mathematical equation, considering that: Figure 10 Adequate polynomial of the variation of the magnetic In Figure 10 is possible to distinguish several curves of the magnetic, one for each value of the simulated parameter. Therefore, to find the points where the magnetic stabilizes and where the minimal influence of the air gap dimension is guaranteed, considering that: the magnetic is related to as follows: (8) the ratio of the parameters shall be on the adequate polynomial or in the region to the right of it, as the follow equation demonstrates: It is now of interest to compare the equation (2) that occurs on a system where the magnetic dispersion is not considered, with the equation (8). It should also be noticed that the two equations mentioned above were purposely arranged in the same morphology for a better analysis and comparison. Therefore, once the magnetic remanence of the magnets is equal to, it can be considered that the magnetic remanence is the numerator of equation (8). So, the biggest difference relies in the relative magnetic permeability of the magnet, as it appears to increase from (original value) to, thereby decreasing the magnetic in the iron core when the magnetic dispersion is considered. For a further analysis of the dimension of the air gap, it is now necessary to quantify the magnetic dispersion during the variation of the parameters and. Unlike the circuit in Figure 6 that is referent to a front view, the simplified circuit used for this simulation tries to show the generator s pole from a side view and it is explicit in Figure 9. V. BOUNDS OF THE PARAMETERS It is now necessary do adopt an upper and lower bounds within the parameters of Table 2 may vary. Those bounds are expressed in Table 3. Table 3 Bounds for the parameters of the linear generator Parameter Comments Range / Magnet / Magnet u Distance between poles Air gap Magnet Winding turns (9)

5 5 VI. OPTIMIZATION CONSTRAINTS After the study realized in section IV it is now possible to gather more constraints about the magnetic properties of the linear generator. The inequality and equality constraints are expressed in Table 4 and Table 5, respectively. Although it is not a constraint, equation (8) is used for calculation purposes of the magnetic in the U shaped iron piece. Table 4 Inequality constraints for the optimization Objective Comments Inequality Parameter s Geometrical aspects Parameter t Geometrical aspects Avoid magnetic saturation U shaped iron piece I shaped iron piece Avoid Air gap dimension magnetic dispersion Distance between poles Distance between the rows of magnets Table 5 - Equality constraints for the optimization Objective Comments Equality Generated power Voltage Generated power Current intensity VII. OBJECTIVE FUNCTIONS This paper goal is to minimize the total active weight and the total cost of the single-phase MFT. The total active weight is important for the generator to be practical, easy to transport and easy to install. On the other hand, the cost is a keyword for any project nowadays. First of all, a description of the functions to optimize must be made. Adopting as the total weight function, it can be described as follows: (10) where, and are the total volume of each used material and, and are the respective material density. Regarding the total cost function, it includes more than just the cost of the necessary materials, i.e., it also includes direct and indirect costs. Direct costs, in addition to the material prices, also represent the cost of the material treatment. On the other hand, indirect costs are related to the power losses, to the maintenance of the generator and even with the availability of the materials in the market [7]. As the generator in question is linear, it requires a less frequent maintenance compared to rotary generators, thus, the costs associated with maintenance will not be counted. Moreover, the costs related to the availability of the materials can also be neglected since they all are commercially available and in large quantity. Assuming that the materials cost and their treatment can be expressed by a specific cost per weight unit, the direct costs,, can be defined as follows [7]: (11) weight of each used material. Proceeding to the indirect costs, the only cost accounted for will be the one related to the power losses. So, the indirect costs,, can be defined as [4]: (12) where is the lifetime, in years, of the generator, stands for the annual energy dissipation and is the specific costs related to the losses. One last cost that is relevant to approach is the cost of the structure that will support the stator and all the forces that it is subjected [5]. Thus, considering that the structure cost,, can be related to a specific cost of per length unit, where represents that specific cost and is the total length of the generator, the cost of the structure can be defined as follows: (13) From the equations (11),(12) and (13) the total cost function can be defined as: (14) Due to the fluctuation of the material costs in the market, especially of the copper and the electricity price, along with the difficulty of finding updated specific costs, is hard to find indisputable values. So, the best method to accept the uncertainty of the costs is not to find a value but a range of values for each variable. Thus, it is possible to analyze the sensitivity of each parameter of the design of the generator in relation to the cost variation. However, the total cost function is designed to be used in an optimization process and should not be seen as the actual cost or constructing cost of the generator. Nevertheless, the adopted values in Table 6 were acquired with the intent to be as close to reality as possible [4][5][8]. Table 6 Specific costs and parameters of the total cost function Parameter Comments Lower cost Higher cost Direct cost Direct cost Direct cost Indirect cost Inirect cost Structure cost In Table 7 all the cases that represent all the possible combinations of costs are expressed. Table 7 Combinations of the materials cost Case where, and are the specific costs of the material and, and stands for the total

6 6 VIII. OPTIMIZATION OF THE TOTAL WEIGHT FUNCTION The chosen optimization function is fmincon [9] from MATLAB. The results of the optimization for the specifications expressed in Table 1 are presented in Table 8. Table 8 Optimized results for the total weight function Variable Comments Result / Magnet / Magnet Distance between poles Air gap Magnet Winding turns Spatial period Total length Available area/electr. circ. Efective area/electr. circ. linkage Electromotive force Current intensity Electrical frequency Joule losses Efficiency Objective function min max In Figure 11 it is shown the generator with the obtained optimized dimensions from Table 8. Figure 11 Weight optimized single-phase MFT for a nominal power of 500 kw In order to understand which parameters are the most important and the ones which have more influence for the generator s design, the value of the produced voltage is now fixed to and the generated power suffers a variation from to. However, this study will be made to two different minimum values of the air gap dimension: and. The results are shown in Figure 12. Figure 12 Variation of the nominal power and sensibility of the parameters of the total weight optimization It is recommended to build generators for higher nominal power because as it increases the efficiency raises its levels. Another relevant aspect is that as the nominal power increases, the generator s length remains almost unchanged and its width and height grows (parameters and, respectively). IX. OPTIMIZATION OF THE TOTAL COST FUNCTION The results of the optimization for the specifications expressed in Table 1 are presented in Table 9. Table 9 Optimized results for the total cost function Variable Comments Result / Magnet / Magnet Distance between poles Air gap Magnet Winding turns Spatial period Total length Available area/electr. circ. Efective area/electr. circ. linkage Electromotive force Current intensity Electrical frequency Joule losses Efficiency Objective function max min max

7 7 From the examination of Table 9 it is possible to conclude that all the parameters related to iron usage have increased their dimension, unlike the parameters and that are related to the copper in the generator. This was expected, once the specific cost of the iron is the lowest one. It should be noted that from Table 9 was obtained considering the lowest cost of the materials, i.e., case 1 from Table 7. In Figure 13 it is shown the generator with the optimized results from Table 9. Figure 13 - Cost optimized single-phase MFT for a nominal power of 500 kw Once more, it will be carried out an analysis on the sensitivity of the parameters for a range of values of the nominal power. Unlike before, it will only be adopted one value for the air gap dimension, which is, and it will be considered all the cases from Table 7. Figure 14 - Variation of the nominal power and sensibility of the parameters of the total cost optimization of the generator, even those which are directly related to the dimensions of the copper or the magnets. Thus, it can be concluded that the specific cost is the variable that all the parameters are more sensitive to. X. CONCLUSION In Table 1 it is possible to find the design specifications for the linear generator. Despite the optimization program requires some base conditions, the constraint and imposition of values to the parameters removes degrees of freedom to the optimization. Moreover, when a fixed value of is imposed, the possibility of the optimization program to make variations on this parameter, for efficiency purposes, is withdrawn. To reinforce this idea, knowing that the Joule losses are related to the generator s temperature variation, as a future work and to have more realistic results, an analysis relating the variation of and the associated thermal effects should be done. In what concerns the optimizations, it can be concluded that the variable with more influence for the efficiency of the generator, focusing on equation (4), for the total weight function is the total length (parameter ). Regarding the total cost function, that parameter is the available area for the electric circuit (parameters and ). In this paper the single-phase topology of the linear generator was analyzed. Based on the obtained results, it is possible to state that this generator is a reliable option and deserves further consideration. REFERENCES [1] International Energy Agency World Energy Outlook 2011 Factsheet. Technical report, [2] HIRSCH, R.; BEDZEK, R.; WENDLING, R. Peaking of world oil production: Impacts mitigation and risk management.energy Bulletin, Março [3] CRUZ, J.M. B. P.; SARMENTO, A. J. N. A. Energia das ondas: Introdução aos Aspectos Tecnológicos, Económicos e Ambientais, [4] POLINDER, H.; MECROW, B. C.; JACK A. G.; DICKINSON P. G.; MUELLER M. A. Conventional and TFPM Linear Generators for Direct-Drive Wave Energy Conversion, Junho [5] WOLFBRANDT, A. Automated Design of a Linear Generator for Wave Energy Converters A Simplified Model, Julho [6] BEIRÃO, G. F. Prótotipo de um Gerador Linear para Aproveitamento de Energia das Ondas num Sistema AWS. Dissertação (Mestrado), Maio [7] GRAUERS, A. Design of Direct-driven Permanent-magnet Generators for Wind Turbines. Doctoral Degree Project, Outubro [8] McDONALD, A. S.; CROZIER. R.; CARAHER, S.; MUELLER, M. A.; CHICK, J. P. Integrated design of direct-drive linear generators for wave energy converters. Sustainable Power Generation and Supply SUPERGEN 09, [9] MEIRELES, J. F. B. Análise Dinâmica de Estruturas por Modelos de Elementos Finitos Identificados Experimentalmente. Dissertação (Doutoramento), [10] available in As it can be seen in Figure 14, the consideration of the cases from Table 7 does not cause a significant alteration in the dimensioning of the parameters. In the efficiency figure, the recommendation of a designed generator for a high nominal power is supported. However, the same figure shows two sets of results which converge to two distinct values. The difference between them focuses on the fact that one group refers to the cases relative to the minimal cost of the iron (higher efficiency) and the other group to the maximum cost of the iron (lower efficiency). This characteristic is reflected in all the parameters