Multiobjective Optimal Design of Wireless Power Transfer Devices using a Genetic Algorithm and Accurate Analytical Formulae A. Desmoort, Student Member, IEEE, Z. De Gr` eve, Member, IEEE, and O. Deblecker, Member, IEEE University of Mons, Electrical Power Engineering Unit, Bd Dolez 31, 7000 Mons, Belgium {alexis.desmoort,zacharie.degreve,olivier.deblecker}@umons.ac.be Abstract—In this work, the Non-dominated Sorting Genetic Algorithm (NSGA) is employed for the multiobjective optimal design of Resonant Inductive Power Transfer (RIPT) devices. A thorough review of the literature has been performed in order to propose accurate analytical formulae for computing the lumped parameters of the system equivalent circuit. A particular attention is paid on the representation of the skin and proximity effects, and on the consideration of any relative position between the coils. The tool permits to observe design trends by comparing optimal individuals in the Pareto front, and is illustrated on an electric vehicle battery charging application. In that case, a design able to transfer 3 kW at 60 kHz, with 93.53 % efficiency through 25 cm air with a fixed radial space footprint of 25 cm, and capable to support a lateral misalignment of 10 cm, was obtained. Index Terms—Design Optimization, Genetic Algorithms, In- ductive Power Transfer I. I NTRODUCTION Designing a Resonant Inductive Power Transfer (RIPT) device is not a trivial task, especially when high power (≥ 1 kW) needs to be transferred (e.g. for applications such as electric vehicles batteries charging). Indeed, it requires to achieve multiple and often conflicting objectives, such as to maximize the active power transmitted to the load and to maximize the efficiency of the transfer. The minimization of the working frequency is also a capital objective when high power is needed, since the technological constraints inherent to the power electronics supplies increase with the product of the power and the frequency. Moreover, the relations between the design parameters (e.g. the geometry of the coils) and the output quantities such as power and efficiency are not linear. Several deterministic design procedures are proposed in the literature. However, they often require the action of the designer in order to guide the optimization, and relate to an assisted trial-and-error methodology [1]. Evolutionnary algorithms looking automatically for the optimal design, such as Genetic Algorithms (GAs), are investigated in [2] and [3]. Nevertheless, the first approach relies on mono-objective optimization and on approximate analytical formulae, and concerns low power/high frequency (up to several MHz) applications. Higher power systems are considered in the second work, but cumbersome numerical models are employed instead of analytical formulae. The design space related to the coil pattern is moreover very restricted. In this work, the multiobjective Non-dominated Sorting Genetic Algorithm II (NSGA-II) [4] is employed for the design of WPT systems. Thanks to the use of the Pareto’s theory, NSGA-II is able to determine the best tradeoffs between two or more contradictory objectives. Thus, besides being able to design a specific setup, the proposed tool provides information about design trends by observing the optimal relation between the objective values of the individuals of the final population. The design procedure is thus more dynamic, making the strength of the proposed tool, much more interesting in com- parison with deterministic tools. The WPT device is studied using a full analytical model based on the system equivalent circuit. Coils are modeled by lumped parameters computed with analytical formulae. Those formulae are derived from a thorough literature review and guarantee accuracy (by the inclusion of skin and proximity effects in the coil equivalent resistance) as well as more generality (by computing the magnetic coupling between coils in arbitrary relative position). Section II describes the WPT system analytical model em- bedded in the GA procedure. A brief review of the analytical formulae used for the evaluation of the coils equivalent pa- rameters is also presented. In section III, the optimization tool is described. For illustrational purpose, some results involving an electric vehicle application are presented and discussed in section IV. Finally, the section V draws the conclusions and perspectives of this paper. II. MODELING OF THE WPT SYSTEM The GA needs a model of the transfer device to link the de- sign variables (geometry of the coils and characteristics of the source/load) to the objectives (output power and efficiency). A. Equivalent circuit In this work, the system is replaced by an equivalent circuit where the coils are modeled by lumped parameters (parasitic resistance R i , self-inductance L i , i = 1 for primary and i = 2 for secondary, and mutual inductance M 12 ). The source is replaced by its Thevenin equivalent circuit (V S and R S ) and the load is represented by a resistance R L . The compensation capacitances C i (i =1, 2) are computed using the resonant frequency f 0 value (shared by both primary and secondary) and the estimated self-inductance L i of each coil (C i = L −1 i (2πf 0 ) −2 ). The different positions of the compensation capacitances yield four possible topologies : series at both primary and secondary (SS), series at primary and parallel at secondary (SP), parallel at primary and series at secondary (PS) and parallel at both primary and secondary (PP). As an example, Fig. 1 shows the equivalent circuit for the SS topology. For each case, the currents and voltages in the corresponding circuit are solved by the use of the Kirchhoff’s