Issue
Eur. Phys. J. Appl. Phys.
Volume 90, Number 2, May 2020
International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering (ISEF 2019)
Article Number 20904
Number of page(s) 6
Section Physics of Energy Transfer, Conversion and Storage
DOI https://doi.org/10.1051/epjap/2020200052
Published online 01 July 2020

© EDP Sciences, 2020

1 Introduction

Wireless Power Transfer Systems (WPTSs) could be used for the onboard battery charge in the electric vehicles instead of the battery chargers that use plugs and cables to transfer energy to the vehicle [13]. The WPTSs are based on two coupled coils that transfer power from the transmitting coil to a receiving coil by means of electromagnetic induction [47]. Figure 1 shows a simplified schema of the First Harmonic Equivalent (FHE) circuit of a WPTS [8]. In general, the transmitting coil is casted under the road surface and is supplied by a high frequency voltage. The receiving coil is placed on the vehicle bottom. The best coupling between transmitting and receiving coils is reached when the coils are aligned. It can be noted that the performance of the WPTSs are related to the static power converters and also to the coupled coils. The aim of the paper is the use of optimization algorithms to design the coils of the WPTS system in order to obtain the prescribed mutual inductance and at the same time minimizing the copper volume. In the design problem a pair of coils endowed with series resonant capacitors and sized to charge the battery of a mini-car [911] was considered. The optimization algorithm used is the NSGA-II Non-dominated Sorting Genetic Algorithm [1214]. This method is well known and has shown good results in many fields of application [1521].

The optimization results are a set of non-dominated individuals; each of them represents a different feasible coil geometry that could enhance the performance of the available WPTS prototype. Simulations check the performance of each solution in terms of power transfer efficiency and transferred power.

thumbnail Fig. 1

Electrical FHE circuit of the wireless charger system.

2 The forward problem

Figure 2 shows the geometry of the coupling coils [9]. The geometry is a 2D axisymmetric. The forward problem is solved usingfinite-element analysis. The mutual inductance between the coils is computed solving a magnetic time-harmonic problem. In the proposed geometry two aligned pancake inductors are considered, each of characterized by a number of turns N (N = 12, 15 or 20), the internal diameter Di, and the turn-step Ts. The two ferrite cores are placed at a distance Gp from the pancake-inductor winding andhave diameter Df. The transmitting coil, generally positioned on the road plane, is supplied at a frequency f of 85 kHz according to the Society of Automotive Engineers (SAE) specification, whereas the receiving coil, generally is positioned under the car chassis. The distance between the two coils is set to 14 cm to comply with the ground clearance of the mini-car. The analysis problem is performed using Finite Element Analysis based on the model in Figure 1 and solving a time-harmonic problem (Flux 2D, manufactured by Altair1).

thumbnail Fig. 2

Geometry of the coupled coils (with design variables).

3 The inverse problem

The inverse problem searches for the coil pair geometry that fulfills the desired mutual inductance M minimizing the copper wire length lCu of the winding, which is proportional to the sum of all the radius of the turns. The mutual inductance M is a function of the design variables according to: (1)

The suitable values of M have been previously assessed in [8] by searching for a tradeoff between the power transfer efficiency and the voltage solicitation of the power supply. Indeed, when the series compensation is used in both the transmitting and the receiving side of the WPTS, as shown in Figure 1, the voltage induced across the receiving coil is equal to the voltage applied to the equivalent load RL while the supply voltage Vs must balance the voltage induced across the transmitting coil. Thus, given the required load voltage VL and load current IL, if M is high, a low current injected in the transmitting coil is enough to induce the voltage VL across the receiving coil. A sensible fraction of the total losses in the WPTS arises from the Joule effect in the coils and in the switches of the power converters; for this reason, decreasing the current in the transmitting coil by increasing M leads to an increase of the power transfer efficiency. On the other hand, for high values of M, the given load current IL flowing in the receiving coils induces a higher voltage across the transmitting one and requires the supply system to generate a higher voltage. The WPTS considered in [8] operates in nominal condition with M = 30 μH and theoretically its supply system could generate enough voltage up to M = 62 μH, so that this must be considered as the upper limit for M. In this paper the analysis has been performed on a wide interval of M, ranging from its maximum down to less than half of the nominal value, fixing the lower limit to M = 10 μH. The non-suitable solutions were discarded using an exterior penalty technique. In particular, the following objective functions were minimized: (2)

with M0 = 62 μH as our particular choice and (3)

where NS is the number of turns of the coil and the mutual inductance M(x) is computed according to (4) by the FEM model coupled to the circuit, supplying the transmitting coil with a current having amplitude It and measuring the open circuit voltage Voc,r(x) induced across the receiving coil. (4)

Considering a different number of turns in the inductor windings (12, 15 or 20), the design variables are: the diameter of the ferrite plates, Df [150, 550] mm, the internal diameter of the coils, Di [100, 200] mm, the turn step, Ts [4.6, 10] mm, axial distance between coiland plate, Gp [1, 5] mm.

The optimization problem reads: identify the coil geometry such that the length of copper-made conductor is minimized, fulfilling the prescribed value of mutual inductance and for a finite set N = 12, 15, 20 of the turn number in the coils. The improved solutions are represented by the Pareto front trading off mutual inductance and conductor length obtained using NSGA-II algorithm.

4 Results

The NSGA-II is a classical genetic algorithm that uses the natural selection in order to find the Pareto front of sub-optimal solutions. It uses 20 individuals, 50 generations in order to evaluate the improved Pareto front. The optimized solutions considering coils with 12, 15, and 20 are shown in Figure 3 using the green diamonds for the 12 turns and the yellow and black circles for the 20 and 15 turns, respectively. For the sake of a comparison, the value relevant to an existing laboratory prototype made of 15 turns is shown by a red cross. Analysis of the figure reveals that the Pareto front relevant to the 15 turns extends to a larger region of the M-coller wire length map and nearly covers the regions relevant to the individuals with lower and higher number of turns. It can be concluded that the coils made of 15 turns offer a larger margin in the device design so that only these are considered in the following discussion. The geometrical parameters of the 15 turns individuals laying on the Pareto front of Figure 3 are reported in Table 1; it reports also the external diameter De of the winding that represents the maximum size of the inductor. In all the cases the maximum diameter of the winding is smaller than the one of the ferrite cores, Df. Finally, for each value of the mutual-inductance, Table 1 shows the corresponding value of the efficiency, η, and supply voltage, Vs, both evaluated using the circuit model in Figure 1 considering a power transferred to the battery, PB = 560 W, an equivalent battery resistance RL = 6.2 Ω as the load resistance, the supply angular frequency ω = 2πf, RR and RS the parasitic resistances of the coil windings (RR = RS = 0.5 Ω). (5) (6)

From the point of view of the decision making, a criterion for selecting a single optimal solution out of the Pareto front could well be an efficiency higher than the one of the prototype or a supply voltage lower than the one of the prototype.

Specifically, the solution highlighted by the bold font in Table 1, close to the prototype (row 7 in Tab. 1, bold), could be considered as the separation point between the Pareto front region on the right, which contains thesolutions that lead to a higher efficiency at the expenses of longer windings and higher supply voltage, and the region of the left, where minimization of the winding length and supply voltage prevails on the efficiency maximization.

thumbnail Fig. 3

Comparison of the Pareto points of the FEM problem obtained considering N =12, 15 and 20 turns.

Table 1

Design variables [mm] and objective function values for 20 individuals on the pareto front considering inductors with 15 turns. In bold experimental prototype values; in italic point A and B in Figure 3.

5 Sensitivity analysis

The sensitivity of the coils wire length and mutual inductance to variation of the geometrical parameters of the coils themselves has been investigated by selecting some individuals from the Pareto front, forcing small variation on their design variables and finally checking the ensuing variation of the objective functions. Table 2 reports the results coming from variations equal to +1% and −1% on the actual values of the design variables, X0, of three individuals. The first individual is similar to the prototype while the two other individuals are highlighted by the italic fonts in Table 1; the one described in the first row corresponds to the extremity Aof the Pareto front in Figure 3 while the point relevant to the individual listed in the 14th row, denoted with B, lays in the middle between the prototypal point and the point A. The value f1, i.e. the mutual inductance related objective function, is considered in the sensitivity evaluation. In particular, the difference between the mutual inductance related to the design variables set at X0X and the one with design variables set to X0 − ΔX was computed. The analysis of Table 2 shows that the mutual inductance of individual A is the most sensitive to the variation of design variables, indeed the variation of M with design variable values changing from X0 −ΔX to X0X is 2.86 μH. For the prototype-like individual, which lays about in the middle of the Pareto front, the difference is at its minimum, and it reaches about 1.75 μH. This result confirms that the prototype represents a good tradeoff for the realization of the coils because it has a low sensitivity to parameter variation, does not require a big quantity of copper to be built and still maintains a large enough mutual inductance to assure a good power transfer efficiency. Finally, the individual B shows a sensitivity to design variables variation in the middle between the upper extreme A in Pareto front and the prototypal point.

Table 2

Sensitivity to design variable variations.

6 Simulation results

The soundness of the design solutions proposed by the optimization algorithm have been tested by simulation using an on-purpose model developed in the Matlab-Simulink environment to reproduce the behavior of the circuit of Figure 1. For each set of design variables reported in Table 1, the relevant M and the self-inductances LT = LR have been inserted in model, the capacitors CT and CR have been sized according to (7) to resonate at the supply frequency with the relevant coil inductance, and RS, RT and RL have been set to the values reported in the previous Section. (7)

In a first setof tests the supply voltage has been considered sinusoidal, with the amplitude taken from Table 1 and computed by (5). The simulation time has been set to 250 supply periods in order to reach the system steady state and the efficiency and the power delivered to the load have been assesses by considering the last 5 supply periods. The power transfer efficiency is reported as a function of the mutual inductance in Figure 4. The red line is used for the data coming from the simulation while the blue circles represent the values taken from Table 1 and computed by (6); the red star is relevant to the prototypal coil pair, characterized by M = 32 μH and η = 0.914. The figure highlights a nearly perfect correspondence between the computed values and the obtained results, thus confirming the validity of expressions (5) and (6).

The power delivered to the load is plotted in Figure 5. It results lower than the expected 560 W, but this is an obvious result because the supply voltage obtained from (5) does not consider the non-ideal efficiency of the system; in any case, the transferred power approached the reference one highlighted by the blue dashed line, as M increases because this leads to an increase of the power transfer efficiency. Also in this case, the red star represent the power delivered by the prototypal coils, equal to about 545 W.

A second series of test have been performed considering a more realistic condition in which the transmitting coil is supplied by a high frequency inverter with a DC side voltage VDC = 365 V and controlled with the phase shift techniques. The later one generates at the inverter output the quasi-square wave voltage having the waveform reported in Figure 6 and whose parameter α is adjusted according to (8) in order to obtain a first harmonic component with amplitude equal to the desired supply voltage Vs. (8)

In this case, despite the harmonic content of the supply voltage, the resonance condition of the series capacitors forces the waveform of the current in the transmitting coils to be nearly sinusoidal, as shown in Figure 7, while that in the receiving coil is practically not affected by the supply voltage harmonic contents. Both the efficiency and the transferred power obtained using the phase shift technique are very near to those plotted in Figures 4 and 5 thus confirming that the results obtained using the proposed optimization method, which is based on the FHE of the WPTS circuitry, can be applied to a real-world protoype. The efficiency obtained using the quasi-square supply voltage in most of the considered cases is smaller than the efficiency obtained with the sinusoidal supply. As a matter of fact, the first should be always lower than the second because the currents in the receiving coil is nearly perfectly sinusoidal so that the induced voltage across the transmitting coil is sinusoidal as well; then, the transferred power depends only on the first harmonic component of the current in the transmitting coil while all the other current harmonics components, originated by the quasi-square supply voltage cause additional losses in the coil resistances. This effect is not fully confirmed by the simulations results probably because of numeric issues in computing the transferred power.

thumbnail Fig. 4

Computed (circles) and simulated (red solid line) power transfer efficiency vs. mutual inductance. Star is the experimental point.

thumbnail Fig. 5

Actual transferred power (red solid line) and reference power (blue dashed line) vs. mutual inductance. Star is the experimental point.

thumbnail Fig. 6

Voltage waveform at the inverter output obtained with the phase shift technique.

thumbnail Fig. 7

Supply voltage and current obtained with the phase shift.

7 Conclusion

The paper presented a design method for the coil of a WPTS that optimizes the power transfer efficiency while minimizing the copper length. The validity of this approach has been confirmed by comparison with an existing prototype and by simulation.

Author contribution statement

All the authors were involved in the preparation of the manuscript. All the authors have read and approved the final manuscript.

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Cite this article as: Manuele Bertoluzzo, Michele Forzan, Paolo Di Barba, Maria Evelina Mognaschi, Elisabetta Sieni, Pareto optimal solutions of a wireless power transfer system, Eur. Phys. J. Appl. Phys. 90, 20904 (2020)

All Tables

Table 1

Design variables [mm] and objective function values for 20 individuals on the pareto front considering inductors with 15 turns. In bold experimental prototype values; in italic point A and B in Figure 3.

Table 2

Sensitivity to design variable variations.

All Figures

thumbnail Fig. 1

Electrical FHE circuit of the wireless charger system.

In the text
thumbnail Fig. 2

Geometry of the coupled coils (with design variables).

In the text
thumbnail Fig. 3

Comparison of the Pareto points of the FEM problem obtained considering N =12, 15 and 20 turns.

In the text
thumbnail Fig. 4

Computed (circles) and simulated (red solid line) power transfer efficiency vs. mutual inductance. Star is the experimental point.

In the text
thumbnail Fig. 5

Actual transferred power (red solid line) and reference power (blue dashed line) vs. mutual inductance. Star is the experimental point.

In the text
thumbnail Fig. 6

Voltage waveform at the inverter output obtained with the phase shift technique.

In the text
thumbnail Fig. 7

Supply voltage and current obtained with the phase shift.

In the text

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