Robust Optimization of a Wireless Power Transfer Coupler Against Manufacturing and Fabrication Tolerances

The design variables are optimized to improve the overall performance of wireless power transfer (WPT) coils or couplers. However, failure to consider their uncertainties and tolerances will lead to variations in the performance targets of a WPT system and mass-scale production of couplers cannot afford performance variation from one coupler to another. Therefore, this article proposes a stochastic-based optimization method to incorporate the uncertainties and tolerances of design variables into the optimization process to realize a robust WPT coupler. The proposed robust optimization methodology is compared with the existing deterministic optimization methodology to achieve reduced transmitter and receiver side current harmonics, maximum efficiency, and minimum leakage magnetic field. The results demonstrate that the design resulting from the robust optimization has 1.59% and 1.66% fewer harmonics distortion in the receiver and transmitter side currents compared to the deterministic optimization, along with coil-to-coil efficiency of 96.5% and leakage field of 7.4 μT measured 650 mm from the center of the coupler. Moreover, the fabricated coupler from robust optimization has a superior efficiency profile under load variations.


I. INTRODUCTION
Inductive power transfer (IPT) technology is key to fully autonomous electric vehicles (EV), where conventional conductive chargers are replaced with IPT chargers.The coupler or the coils is the heart of an IPT charger.Coils with different geometrical shapes and winding arrangements are proposed to improve the objectives such as coupling coefficient, mutual inductance, quality factor, power density, and efficiency while reducing the cost, weight, and leakage magnetic field [1].Moreover, the coil design must consider constraints such as the coil's area, air gap, frequency of operation, coupler material properties, and converter ratings [1].Most limits on coil design objectives and constraints are defined in the SAE J2954 EV standard [2].WPT system optimization problems usually take two approaches.One approach is to optimize the coupler or coil design variables, while the other is to optimize the WPT system parameters, such as coil inductances and compensation capacitors.However, this article focuses on the optimization of the design variables of the coupler.
Multi-objective optimization techniques are introduced to solve these coil optimization problems with multiple objectives and constraints, as shown in Table 1.Table 1 also summarizes the power ratings, finite element analysis (FEA) models, and optimization algorithms used for circular couplers.They search for the best design variables that meet the objectives and comply with the constraints of the design.Both analytical and numerical methods are proposed to optimize the IPT couplers [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].
The IPT coupler optimization proposed in the literature is carried out for deterministic model properties, as shown in Table 1, where uncertainties and tolerances of the design variables are ignored [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].Therefore, solutions are not robust.Circular coils  1 as it is the frequently used coupler, and observations are valid for other coupler structures such as DD, DDQ, BBP, and tripolar [8], [11], [12].The design variable uncertainties arise from the coils' manufacturing and fabrication tolerances (MFT).The conductor's diameter, ferrite's dimensions, and passive shield's dimensions vary by a few millimeters from the designed optimized values due to inaccuracies in these components' manufacturing process.The space between two conductors in a winding, starting radius of a coil, ferrite placement, etc., also varies from their optimized values during the coil fabrication process.These variations contribute to the MFT of the coils.These tolerances are found in every industrial manufacturing process.
The failure to consider uncertainties of design variables during the optimization process will push the optimized design into extreme boundaries of the optimization space and has the risk of violating design objectives and constraints when implemented, as the exact values of the optimized design variables may not be precisely realizable during the production process due to abovementioned uncertainties and tolerances.In a series-series compensated IPT system, the coupling coefficient is restricted during optimization to be less than the critical coupling coefficient to avoid bifurcation [3], [9].However, there is always the risk of violating such constraints for designs closer to the boundary of critical coupling.An increase in total harmonic distortion in the implemented WPT system is reported in [9] due to violations of constraints in the implemented system compared to simulations.Moreover, efficiency deviations in the implemented design by a large margin are reported in [7] compared to the optimized/simulated design.Such variations question the validity of solutions resulting from conventional deterministic multi-objective optimization (DMOO).These deviations are also evident from Table 1, as the fabricated coupler's objectives and parameters vary significantly (maximum of 13%) compared to the simulation results from the optimization process [3], [4], [5], [6], [7], [8], [9].Deviations in objectives and constraints may result in poorly performing IPT chargers [3], [7], [9].
The cause of these variations is unlikely to occur due to the tolerances of measurement instruments such as LCR meters and power analyzers, as they have a worst-case tolerance of ±0.5%.One might avoid these performance deviations by altering the design variables in a laboratory environment.However, such activities question the validity of the DMOO process and design methodology, and one cannot follow a trial-and-error approach in mass-scale production in a commercial environment.Mass-scale production of WPT couplers requires a formalized design approach that can quantify these uncertainties and tolerances at the design stage to ensure that the fabricated coupler is operating as intended during the design stage.
Therefore, this article proposes a method to include the design variable uncertainties in the optimization process to realize a robust multi-objective optimization (RMOO) process.The optimization problem is formulated to achieve a WPT system with high efficiency, a lower leakage field, and lower total harmonics distortions (THD) in currents.Reducing the THD of an IPT system is beneficial as it impacts electromagnetic interference (EMI) issues.The designer must ensure all constraints related to THD are met to achieve lower THD.The proposed optimization method will ensure that all design objectives and constraints are met by the manufactured IPT system, even with the tolerances and uncertainties of design variables, to guarantee performance targets set during the design stage.Moreover, the effectiveness of the proposed methodology is compared with conventional DMOO.
The major contribution of this article compared to the existing literature is summarized as follows: 1) Introduces a method to quantify the impact of design variables on objectives and constraints using the design of experiment (DOE) technique.2) Introduces a method to quantify tolerances of design variables of a WPT coupler.This is achieved by defining each design variable's coefficient of variation by considering the manufacturing process's standard deviation determined via the process capability index.3) Introduces safety margin to quantify failures or violations of design targets due to uncertainties and tolerances of design variables.4) Introduces a method to incorporate reliability into the coil manufacturing process in terms of the number of defects per million designs in the short term.

II. IPT SYSTEM DESIGN
Initially, the impact of the power electronics system and compensation topology design requirements on the coil parameters, such as transmitter (L Tx ) and receiver side (L Rx ) inductances, mutual inductance (M), and coupling coefficient (k), is identified.These parameters usually depend on the operating frequency, rated load, voltage gain, etc., of the IPT system.This is critical as these requirements constrain coil parameters to achieve efficient operation of the IPT system.The proposed robust coil optimization was carried out once the impact of these requirements on the design parameters was identified.Because L Tx , L Rx , M, and k change with coils' design variables and their associated manufacturing and fabrication tolerances.In this article, the relationship is demonstrated for SS-compensation topology.However, similar relationships can be derived for other compensation topologies proposed in the literature [1].

A. IMPACT OF SS-COMPENSATION ON COUPLER DESIGN
The SS compensated IPT system shown in Fig. 1 is preferred for EV applications as the capacitances of Tx (C Tx ) and Rx coils (C Rx ) are independent of the coupling coefficient (k) and load variation, high efficiency at low k, etc. [14].

sin (n
where n is the odd harmonic number and t on is the ON time of the applied voltage pulse.Fundamental frequency (n=1) can be used to model the loosely coupled inductive power transfer system since the fundamental component is significantly larger than the harmonic components.The rectifier's output is usually connected across a battery bank, which can be approximated to a resistive load (R L = V 2 L /P) with the power rating (P) equal to the rated power of the IPT system.V L is the rated voltage across the load [3].The ac equivalent resistance looking into the rectifier (R ac ) is given by: Fig. 1(c) shows the independent decoupled voltage source model of the circuit shown in Fig. 1(a).The terms (-jωMI Rx ) and (-jωMI Tx ) are reflected impedances of the input and output side.The operating frequency (f 0 ) is given by: (3) The coupling coefficient (k) of the coils is given by: The resonant inverter and rectifier of the IPT system have non-linear characteristics.As a result, it can cause harmonic distortion in the Tx-side and Rx-side currents.These distortions can interfere with the external electronic systems of the EV or radio signal receivers [9].This article considers a class D rectifier, which is frequently used for IPT applications [1], [3], [8].The harmonics of the I Rx due to the nonlinearity of the class D rectifier can be minimized by satisfying the following condition [15]: where Q e is the external quality factor Q e = (ω 0 L Rx / R ac )) Similarly, to reduce the harmonic distortion of the transmitter (Tx) current due to the non-linearity in the resonant inverter and the frequency bifurcation phenomena in the input impedance, the coupling coefficient for IPT systems that use D, E, and DE resonant converters should be limited to [9]: where k 0 is the coupling coefficient at the nominal position of the coils, k c is the critical coupling coefficient, and k s is the split coupling coefficient [9].Equations ( 5) and ( 6) contain Q e , which is a function of L Rx [9].Therefore, the range of L Rx to meet the design criteria stipulated by ( 5) and ( 6) simultaneously can be found by: The maximum value of k 0 that satisfies the equality given by ( 7) is given by [9]: Therefore, any design with a coupling coefficient higher than the k om will increase harmonics, and the leakage magnetic field measured at a particular point from the coil [9].Even though (8) restricts the k of the coil and its size, it is still important to maximize the coupling coefficient and quality factors of the coils to maximize the power transmission efficiency [3], [9].The quality factor (Q c ) of a coil is given by: L is the coil's inductance, while R c is the parasitic AC resistance of the coil.Equation (9) assumes that the core and passive shielding losses are smaller than the copper losses [3].The voltage transfer ratio (V TR (ω 0 )= V Rx1 / V Tx1 ) between the input voltage (V Rx1 ) and output voltage (V Tx1 ) is critical for the safe operation of the IPT system, as it provides a guideline for the ratings of the semiconductor devices of the PE system [9]: The approximation in (10) considers R Tx <<R e , R Rx <<R e , and k=k 0 =k s [9].Since L Rx is determined by (7), the size of the L Tx depends on the selected V TR (ω 0 ).
The requirement of low harmonic currents in the Tx and Rx side of the IPT systems can only be fulfilled if ( 5)-( 8) are met by the coupler design, while (10) ensures the required voltage gain of the IPT system.Therefore, the coupler parameters such as k, L Tx , and L Rx are constrained by the inequalities (5)-( 8) and (10).However, the uncertainties in design variables will impact the k, L Tx, and L Rx .Therefore, some of these constraints defined in ( 5)-( 8) and ( 10) might be violated if the impact of uncertainties of design variables are ignored.These violations result in performance deviations compared to targeted performance levels.This impact is shown in Sections IV and V of this article by comparing the proposed robust and conventional deterministic optimization methods' designs.

B. LOAD DEPENDENCE CHARACTERISTICS
The load characteristics are derived at resonance.Therefore, the inductive and capacitive reactance at the primary and secondary are considered to cancel off each other.The reflected input impedance can be derived from Fig. 1(b) and (c) as: Z Rx is the secondary equivalent impedance given by R Rx + R ac .Moreover, the primary equivalent impedance (Z T x ) will be equal to R T x .Using Kirchhoff's law, the transmitter current can be derived as: The output power of a SS-compensation system is given by: Therefore, it is evident that the output power increases with the increase in load.The efficiency (η e ) of the IPT system can be derived as: where P in is the input power given by (V s I Tx ).Equations ( 14) and ( 15) will be used to analyze the load independence characteristics of the designs resulting from the conventional DMOO and proposed RMOO.

A. DESIGN SPECIFICATIONS
Initially, the impact of the power electronics system and compensation topology design requirements on the coil parameters are identified.These requirements will vary for different applications and compensation topologies.Section II derived the relationship for SS-compensation topology, and similar relationships can be derived for other compensation topologies proposed in the literature.

B. OPTIMIZATION PROBLEM FORMULATION
The next step of the proposed optimization process is identifying the design variables, design variables' tolerances, objectives, and constraints to formulate the optimization problem.They will vary depending on the designer's requirements.Design variables will depend on the coupler structures.The common coupler structures for EV are DD, DDQ, bipolar, tripolar, etc [1].Most of the design variables are common to these couplers with minor variations.These common design variables are the separation between the turns, conductor radius, ferrite dimensions, separation between winding and ferrite, the separation between ferrite and passive shield, starting point of the windings, starting point of the ferrites, number of turns, and dimensions of the passive shield [1].Moreover, these design variables have their associated fabrication and manufacturing tolerances except for the number of turns.
The common optimization objectives are efficiency, leakage magnetic field, area power density, volumetric power density, cost, weight, coils' quality factor, and coupling coefficient [1].The designer can select which ones to optimize depending on their requirements.Constraints are usually determined based on the power electronics and compensation topology design requirements as demonstrated in Section II and industrial standards such as the SAE J2954 EV standard [2].The design example shown in Section IV will explain how these objectives, design variables, and constraints are used to formulate the optimization problem.

C. SENSITIVITY ANALYSIS
The sensitivity analysis provides insight into the behavior of independent design variables on the performance of a given objective and constraint functions.If a particular design variable does not impact these functions, it can be excluded from the optimization process to simplify the problem and reduce the computational cost [17].
The next step is performing the optimization.This article provides steps of the conventional DMOO and proposed RMOO to demonstrate how manufacturing and fabrication tolerances or uncertainties can be incorporated into the design process.

D. DETERMINISTIC MULTI-OBJECTIVE OPTIMIZATION STEPS
The conventional DMOO will based on the following steps to determine the Pareto-front (PF) of the design [25].
r Minimize the functions: r Provided that: r For design variables where f n (d ) is the optimization functions of the design with the constraint inequality functions  d k d ) optimization variables.The PF of size y derived using each q iteration of the optimization algorithm is given by [16]: Fig. 2 provides a summarized flowchart of a conventional DMOO proposed for wireless power transfer couplers [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].The optimization problem is formulated by considering the design specification, design variables, objectives, and constraints.Then, the DMOO is performed using a suitable optimization algorithm to derive the PF.Finally, a Pareto-optimal design can be selected for implementation.It is important to note that sensitivity analysis was hardly considered in conventional DMOO and can be easily performed after the problem formulation.

E. ROBUST MULTI-OBJECTIVE OPTIMIZATION STEPS
The DMOO given by ( 16)- (20) does not incorporate stochastic restrictions (SR).Therefore, SRs are introduced in RMOO to ensure the predefined failure probabilities are not violated, and target variances are met.The design reliability can be expressed as [25], [26]: P(F) is the probability of failure, while P T (F) is the target probability.Equation (22) ensures that: will not exceed the predefined P T (F), under the consideration of random influences: where f X (x) is the joint probability density function of random variables and n = 12,...,k g limit state functions g n (x)≤ 0. The following modifications ( 22)-( 24) ensure that the design variables d j in (19) satisfy the constraint given by ( 22) to predefined probabilities.Therefore, the design variables d j becomes: are the means of the k r random influences X.The objectives and constraints defined by ( 16)-( 18) become nondeterministic functions due to random influences.Therefore, the objective ( 16) is improved with the requirement to minimize the variances σ 2 X i [26].
It is important to evaluate the reliability of the final design before implementation.The definition of the probability of failure is based on histogram fitting with a probability density function.The literature recommends a reliability analysis for sigma levels greater than two [27].The frequently used reliability analysis methods are the First-order reliability method, global approximation with support points from robustness samples, and Adaptive Response Surface Method (ARSM).The method used will vary depending on the nature of the problem solved.[27].
The proposed optimization method is summarized, as shown in Fig. 3, so it can be easily adapted to any wireless power transfer optimization problem.Initially, the IPT system design specifications are identified, and the optimization problem is formulated where the design variables, objectives, and constraints are determined for the selected coil structure.The constraints are determined based on specifications of the power electronics system, compensation requirements, industry standards, and the characteristics of the coil structures.The sensitivity analysis identifies the impact of design variables on the objectives and constraints of the coil design.The number of design variables considered for the optimization process can be reduced depending on the outcome of the sensitivity analysis.The RMOO optimization performed using an optimization algorithm determines the Pareto-optimal solutions, and the safety level is calculated using robust evaluation.If a design violates the robust criteria, the constraints are modified to meet the target safety levels, and the optimization is repeated with the modified constraints.The accuracy of the final robust design is verified through reliability analysis, and if the design fails the reliability analysis, the algorithm will revert to the modification of constraints.Section IV will demonstrate the application of the proposed RMOO methodology to realize a robust coupler.

IV. DESIGN EXAMPLE
This article utilizes SS-compensated IPT system with a circular coupler to demonstrate the efficacy of the proposed robust optimization methodology.However, it can be applied to other coupler structures and compensation topologies proposed in literature by identifying their relevant design variables, objectives, and constraints [1].

A. DESIGN SPECIFICATIONS OF THE WPT SYSTEM
Initially, the power electronics system parameters of the SScompensated IPT system are identified depending on the application.The parameters related to this work are listed in Table 2.The optimization is carried out for a 3.7 kW IPT system.The maximum input and output DC voltages are restricted to 350V.The AC equivalent resistance (R ac ) is derived using (2).The operating frequency of the IPT system is 85 kHz, while I Tx(peak) and I Rx(peak) are the AC peak values of transmitter and receiver side currents.The impact of selecting an SS-compensation topology is discussed under constraints Section IV-B4.The coupler optimization is carried out for a 3.7 kW IPT system operating at 85 kHz.

B. PROBLEM FORMULATION
This section identifies the design variables, objectives, and constraints to formulate the optimization problem discussed in this article.

1) DESIGN VARIABLE IDENTIFICATION
A circular coupler is characterized by the following variables, as shown by Fig. 4: radius of the conductor (r w ), starting radius of the coil (r i ), separation between two adjacent conductors (r c ), length (l f ) of ferrite (FE), thickness (h f ) of FE, width (w f ) of FE, separation between the conductor and FE (h wf ), separation between the FE and shield (h fs ), thickness(t s ) and radius(r s ) of the shield, FE starting (r f ) and number of turns (N).The design variables matrix is given by: 2) DESIGN VARIABLE TOLERANCES Traditional wireless power transfer optimization problems do not consider the tolerances of design variables [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].Therefore, this article introduces the coefficient of variation (COV) to quantify the tolerances of design variables.It is given by: μ and σ are the mean and standard deviation of the design variables that have a normal distribution.The standard deviation of the design variables' manufacturing process can be estimated using the process capability index.It is given by: where U SL and SL are the upper and the lower tolerance limits.C pK for an existing manufacturing process is around 1.33 [18].Therefore, the design variables tolerances are determined using ( 29) and (30) for a standard manufacturing process and from the manufacturer datasheets.They are listed under Table 3 in terms of the coefficient of variance, and a statistically normal distribution is assumed.Note that the number of turns does not have tolerances.

3) OBJECTIVES IDENTIFICATION
The proposed RMOO aims to maximize coil-coil to efficiency (ɳ cc,max ) [19] and minimize leakage magnetic field (B s ) evaluated at 650 mm from the center of the coil, with the assumption that the Rx coil is fixed at the center of a 1.7 m wide small vehicle.Moreover, ɳ cc,max > 95% and B s <27 μT constraints are also considered in this design example.The considerations for achieving this will be discussed in Section IV-E and F.

4) CONSTRAINTS IDENTIFICATION
The coil design needs to consider the constraints imposed by the power electronics system and compensation topology design, STDs, and other limitations identified in the literature.The selected compensated network and its design methodology impose constraints on inductances and coupling coefficient, as identified in Section II.Equation ( 7) imposes a limit on the Rx-side inductance to reduce the current harmonics due to the non-linearity of the inverter and rectifier.Therefore, L Rx ≥ 186 μH according to (7).A voltage transfer ratio of √ 2 will result in identical coils and it also implies that L T x ≥ 186 μH according to (10).The coupling coefficient of the design should be approximately less than 0.191, according to (8), to avoid the frequency bifurcation phenomena in the input impedance, reduce current harmonics.150 mm air gap (h) is considered in this article.The coil area (A C ) is limited to 0.30 m 2 by considering the flux height of a circular coil is 1/4 of its diameter [2].The operating frequency (f) is 85 kHz as the TIR J2954 STD limits the operating frequency to 81.38 -90.00 kHz band to avoid interferences with other electronics of an EV.Flux density of ferrite bars (B Fe ) limited to 200 mT as this application falls under WPT 1 power class according to TIR J2954 STD [2].Table 4 summarizes these constraints.

C. SENSITIVITY ANALYSIS
The coils are designed using the ANSYS Maxwell finite element analysis software, and the RMOO is performed with the tools available in the optiSLang software [28].The lower and upper bounds of a circular coil's design variables considered for optimization are listed in Table 3.
Initially, a sensitivity analysis is performed.The influence of the design variables on the objectives and the constraints are evaluated using 100 design samples with the Advanced Latin Hypercube (ALH) sampling using the design of experiment (DOE) technique [20].The Meta-model of optimal prognosis approach quantifies the importance or the sensitivities of the design variables [20].The results are illustrated in Table 5 as a coefficient of prognosis matrix.Table 5 shows the impact of each design variable on objectives and constraints in terms of percentages.The design variables r i , r c , and r w have the highest impact on all the objectives and the coil's constraints.The coil radius and radius of the shield (r s ) are the main parameters that impact the magnetic field measured at a point in the surrounding environment according to Biot Savart laws.In this article, coil radius is a function of r i , r c , and r w .The self-inductances of planar coils are a function of r c , r w , r i , as shown in [21].A generalized analytical model to calculate the mutual inductance of two planar coils is derived from Neumann's integral of constant current-carrying filament with a Taylor expansion approach [22].According to the derived analytical equations, the radius of the coil mainly impacts the mutual inductance.Therefore, mutual inductance is also a function r c , r w , r i .The coupling coefficient (k) value depends on self and mutual inductances.However, both self and mutual inductances are a function of r c , r w , r i .Therefore, k is also a function of r c , r w , r i .Another analytical method to determine the coupling coefficient-based magnetic flux is given in [23].The magnetic flux is a function of r f .Therefore, the k is also a function of r f .The transmission efficiency is a function of the coils' self-inductances, k, and losses, as shown in [3], [9].The losses of the coils consist of winding losses, core losses, and shielding losses [1].The winding losses are a function of the design variables r w, while shielding losses are a function of r s as it is a variable in calculating the resistance of a shield according to [24].The core losses depend on the volume of the ferrite, and the volume is a function of r f .Therefore, transmission efficiency is a function of r c , r w , r i, r s and r f .The proposed sensitivity analysis identified these relationships, as shown in Table 5.
The design variables h fs , h wf , and t s have a negligible impact on the design objectives and constraints and can be neglected during optimization.The rest of the design variables have a significant impact, as shown by higher percentage contributions in Table 5.Therefore, r c , r w , r i , r s , and r f , are considered in the optimization process, whilst the other design variables are kept constant at their initial design values.Sensitivity analysis is a relative measure of impact for the formulated optimization problem and may not be true for different optimization scenarios and coil structures.
r Minimize or minimize the functions: where f ncc,max (d ) and f Bs (d ) are the coil-to-coil efficiency and leakage field optimization functions.
r Provided that:  r For design variables (40) is similar to the design variable matrix defined by (28).
d L and d U are the upper and lower search bounds of the design variables defined in Table 3.The sensitivity analysis results were used to determine the upper and lower limits.The Pareto-front is derived using the ( 21) for the design variables defined by ( 28) for a circular coupler.Fig. 5 shows the results of the Pareto-optimization of both DMOO and proposed RMOO.The design points of DMOO and proposed RMOO are superimposed in Fig. 5.These data points were generated using a non-dominated sorting genetic algorithm II (NSGA II) optimization algorithm in conjunction with ANSYS Maxwell simulations.
The Pareto-front PF1 shown in Fig. 5 corresponds to the results from the DMOO.Design points 1, 2 and 3 are on the PF1, while design points 7 and 8 are non-pareto optimal designs corresponding to DMOO.These two designs did not violate the constraints identified in Table 4.But they are not optimal.Other gray points represent the designs that violated these constraints.Note that the gray points shown in Fig. 5 are for both DMOO and RMOO results.All these gray design points are not critical as they violated the design constraints.Therefore, these designs are not suitable for implementation.The results corresponding RMOO are discussed in Section IV-F.

E. ROBUSTNESS ANALYSIS
This article introduces a safety margin to quantify the failures or violations of the target due to the uncertainties of design variables discussed in Section IV-B2.A safety level is introduced in terms of Sigma.Sigma is a statistical measure of variability in a manufacturing process.Usually, a 4.5 sigma (4.5 σ ) level is acceptable for the manufacturing process, as the area of a normal distribution beyond 4.5 sigma from the mean will result in a probability of or 3.4 failures/violations per million designs due to the uncertainties of design variables  [20].Therefore, 4.5 σ is considered for efficiency, leakage flux, coupling coefficient, and Tx and Rx side inductances.
The robustness of each designs on the PF1 of the DMOO shown in Fig. 5 is evaluated using 100 design samples with ALH sampling in optiSLang [28].The probability distribution of the objectives of design 3 is plotted with their safety margins, as shown in Fig. 6(a) for efficiency and in Fig. 6(c) for leakage magnetic field.The lower safety limit of efficiency is defined as 95%, while the upper limit of leakage magnetic field is 27 μT as defined under the SAE J2954 STD to evaluate the robustness of the design [2].Both limits are more than 10σ safe when compared to the target safety level 4.5 σ .Therefore, the efficiency and leakage field objectives are robust.
However, as shown in Fig. 7(a), the lower bound of Rx inductance L Rx is 186 μ according to design (7), which is 3.36 sigma safe.The target limit is 4.5 σ for L Rx as well.This means the design does not meet the target for L Rx .
The upper bound for coupling coefficient(k) is 0.191, according to (8), which is 0.90σ safe as shown in Fig. 7(c).Yet again coupling coefficient also does not meet the target safety level of 4.5 σ .This violation is visible in Fig. 7(c) due to tolerances, as several designs are above 0.191.Therefore, the optimized design 3 from DMOO is not robust in terms of the constraints L Rx and k.Therefore, the implemented IPT system may violate inequalities ( 5)-( 8) and ( 10) due to uncertainties of design variables.This is often the shortcoming of deterministic optimization.Table 6 summarizes the results of the other two Pareto optimal designs (1 and 2) from DMOO in meeting the 4.5 sigma robustness level.The results are demonstrated for the efficiency, leakage magnetic field, L Rx , and k within brackets, while Yes (Y) and No (N) demonstrate whether the design meets the robust criteria.Designs 1 and 2 also do not meet the robust criteria.Detailed evaluations of these designs are not provided in the interest of space.

F. PROPOSED ROBUST OPTIMIZATION EXAMPLE
As discussed in Section III-E, DMOO given by ( 31)-(41) did not incorporate stochastic restrictions (SR).Therefore, SRs are introduced in robust optimization according to (22) to ensure the predefined probabilities of failure are not violated and target variances are met.In this work, the target TABLE 6. Evaluation of Robustness Criteria of Designs From DMOO and RMOO probability P T (F) defined in ( 22) is 3.40 x10 −6 or 3.4 failures/violations per million designs due to the uncertainties of design variables.Equation (22) ensures that designs will not exceed the predefined P T (F), under the consideration of random influences defined by (24).These random influences are the tolerances of design variables determined using ( 29) and ( 30) and shown in Table 3 in terms of COV.
These considerations ensure that the objectives and constraints satisfy the predefined probabilities defined in (22) under the random influences of design variables.The objectives and constraints defined by ( 31)-(39) become nondeterministic functions due to random influences.Therefore, the objectives and constraints are improved with the requirement to minimize the variances σ 2 X i defined by (27).The objectives become: The constraints become: The Pareto-optimization is performed using NSGA II algorithm in conjunction with ANSYS Maxwell.The PF2 shown in Fig. 5 is the PF corresponding to pareto-optimal designs resulting from robust optimization process.Design points 4, 5, and 6 are on the PF2, which are Pareto-optimal.Design point 9 is a non-pareto optimal design corresponding to RMOO.This design did not violate the constraints identified in Table 4.However, it is not Pareto-optimal.Other gray points represent the designs that violated these constraints, as discussed in DMOO.Note that the results of DMOO and RMOO are superimposed in Fig. 5 for a better visual comparison.
The robust analysis discussed in Section III-E is performed for design 6 shown on the PF2 of the proposed RMOO, as shown in Fig. 5.The efficiency and leakage field are more than 10σ safe, as shown in Fig. 6(b) and (d).More importantly, the safety margins corresponding to L Rx and k are 5.11 σ and 5.01 σ , as shown in Fig. 7(b) and (d), which is greater than targeted 4.5 σ safety margin.Therefore, this design meets the robust design criteria.Table 6 summarizes the results of the other two Pareto optimal designs (4 and 5) from proposed RMOO in meeting the 4.5 sigma robustness level.The results are demonstrated for the efficiency, leakage magnetic field, L Rx , and k within brackets, while Yes (Y) and No (N) demonstrate whether the design meets the robust criteria.Therefore, designs 4 and 5 also meet the robust criteria.Detailed evaluations of these designs are not provided in the interest of space.

G. RELIABILITY ANALYSIS OF THE OPTIMIZED DESIGN
When comparing the reliability analysis methods listed in Section III-E, ARSM uses fewer solver evaluations, and extrapolation is not used in the approximation [27].Therefore, the probability of failure is estimated accurately by using ARSM.The calculated failure of probabilities for constraints considered in design 6 is as such: L Rx (P(F)=1.21* 10 -7 < P T (F)=3.4* 10 -6 ) and k (P(F)=1.11* 10 -7 < P T (F)=3.4* 10 -6 ).Both probabilities of failures are below the target P T (F) =3.4 * 10 -6 .These calculations indicate that the RMOO design has fulfilled the target probability of failure and is suitable for mass-scale production.

V. EXPERIMENTAL VERIFICATION
This section will demonstrate the details of the experimental procedure and experimental results of both DMOO and proposed RMOO methods.

A. EXPERIMENTAL SETUP DETAILS
The system parameters of the IPT system are listed in Table 2.The winding, core, and passive shield are fabricated using Litz wire, ferrite, and aluminium.The Litz wire is made up of 800 strands of AWG 37. Ferrite with a permeability of 2200 was used.Fig. 8 shows the hardware setup used in this study.Fig. 8(a) shows the schematic diagram of the hardware setup with semiconductor devices, measuring devices, and controllers used, while Fig. 8(b) shows the actual hardware setup of the IPT system.The inverter is made up of four Infineon IPW60R031CFD7 high-voltage power MOSFETs.Texas Instrument's TMS320F28335 digital signal processor was used to implement the control related to inverter operation.The rectifier comprises two modules of IXYS DSEP2x31-06A fast recovery diodes.Two MKPC.4B 1.5 μF snubber capacitors are connected across the DC bus of the inverter to protect the switches from high-frequency switching stresses.C Tx and C Rx are realized using combinations of 1nF of EPCOS Metallized polypropylene film capacitors (700 VAC).A resistor bank was used as the load (R L ) to dissipate the power.Yokogawa PX8000 precision power scope was used to measure the efficiencies of the system with Agilent N2782B AC/DC probes and Tektronix differential p5200 voltage probes.Additionally, Agilent Technologies InfiniiVision 3000 DSO-X 3014A was used to capture the voltage and current waveforms.The leakage magnetic field is measured from the edge of the coupler in the lateral direction from the Tx using the Narda ELT 400 field meter.In the interest of space, the figures corresponding to RMOO are provided.However, numerical values corresponding to both DMOO and RMOO are also provided.

B. CONFORMITY AND VIOLATIONS OF CONSTRAINTS
Design 3 from DMOO and design 6 from proposed RMOO were implemented.Table 7 the values of L Rx , L Tx , k, and coil quality factors of transmitter (Q Tx ) and receiver (Q Rx ) coils of both designs.The coils have high-quality factors at 85 kHz, which is critical for achieving high efficiency.The discrepancies in quality factors of simulated and implemented designs are mainly due to inductance variation and inaccuracies associated with estimating the parasitic resistances of the coils.Table 8 lists the design criteria ( 5)- (8), and (10) satisfied or violated by the simulated and implemented designs from DMOO and RMOO.The simulated here refers to the designs from the optimization process.The simulated designs from the DMOO and RMOO method satisfied all the criteria, as these designs were selected from the PFs shown in Fig. 5.
However, the fabricated coupler resulting from DMOO marginally violates the inequality (7) for L Rx , while the inequality ( 8) is violated by a margin of +2.61%.It also violates the design criterion (5) due to violating the design inequality (7).The fabricated coupler from deterministic optimization conforms with the inequality ( 6) and (10).This observation is predicted during the design stage, as shown in Fig. 7(a) and (c).As discussed in Section IV-E, it failed to achieve the robust criteria on constraints (7) and (8).The fabricated coupler resulting from the proposed RMOO does not violate any inequality and conforms to the robust criteria, as shown in Fig. 7(b) and (d).
The major contributing factor for violating the inequality (8) in deterministic optimization is the design variable r c , as it is challenging to achieve uniform r c between the coil turns.As shown in Table 5, a small variation will significantly impact k as r c is a design variable with the highest impact on k.Moreover, the design variable r i of the robust optimization design is slightly varied within their assumed tolerance range from the optimized value given in Table 3.This was deliberately done to check the effectiveness of the proposed RMOO.A slight variation in design variable r i will significantly impact L Rx , L Tx , and k as it has the highest impact on these parameters, as shown in Table 5.

C. EFFICIENCY AND LEAKAGE MAGNETIC FIELD ANALYSIS
The efficiency and leakage magnetic field performances of the IPT systems with the fabricated coupler from both DMOO and proposed RMOO are listed in Table 7. Power losses are 133.2W for the implemented design from the conventional DMOO and 129.5 W for the proposed RMOO.Therefore, the robust coupler design has a lower loss component than the conventional design from DMOO.
The efficiency is measured using Yokogawa PX8000 precision power scope (20MHz bandwidth) with Agilent N2782B AC/DC probes (DC to 50 MHz, maximum current 30 Arms) and Tektronix differential p5200 voltage probes (50 MHz, ±1300 V).The transmit/receive part of the wireless power transfer system has a low power factor.The PX8000 supports the measurement of low-power-factor systems operating at very high frequencies.Moreover, its deskew function was used to compensate for and eliminate the differences between voltage and current introduced by the sensor and input characteristics in low-power-factor systems.The maximum coil-to-coil efficiency of 96.40% is reported for DMOO design, while 96.50% for RMOO design at the rated power.There is a 1 % variation compared to the final optimized solutions, but both are within the safety margin or 4.5σ boundary.More importantly, they satisfy the initial design criterion of efficiency above 95%.Fig. 9 shows the measured Tx and Rxside current and voltage waveforms.v Tx and v Rx correspond to the transmitter and receiver side voltages, while i Tx and i Rx correspond to transmitter and receiver side currents.The frequency of these waveforms corresponds to the IPT system's resonance frequency.
The leakage magnetic field measured at 650 mm from the center of the coupler is 7.5 μT for the DMOO design and 7.4 μT for the RMOO design.Both designs are well within the safety margin and are less than 27 μT.Furthermore, the simulated and measured leakage magnetic fields are shown in Fig. 10 for RMOO design, and the maximum variation is 1.2 μT at any given point.In this design example, both objectives are satisfied by the designs derived from DMOO and RMOO.
In both designs, k< k c , which corresponds to inequality (6) to avoid bifurcation.Therefore, both designs have a singlephase angle in input impedance, and the frequency bifurcation   is avoided.This is also evident from the high dc-dc efficiency listed in Table 7.
The efficiency of fabricated couplers from DMOO and the proposed RMOO method is also measured by displacing the receiver coil in each design along the horizontal axis at 20 mm intervals up to 140 mm.Fig. 11 shows the results of the efficiency variation with horizontal misalignments for DMOO and RMOO.Efficiency drops with an increase in misalignment for both solutions, which is expected for a wireless power transfer system due to a reduction in mutual inductance or coupling.The RMOO design has superior performance, up to 60 mm horizontal misalignment compared to DMOO.However, the efficiency is around 93.85% for RMOO, while 94.6% for DMOO at 100 mm.The 100 mm is critical, as it is the maximum admissible horizontal offset defined in the SAE J2954 standard for EV [2].

D. TOTAL HARMONICS DISTORTION ANALYSIS
The THD of the I Tx and I Rx currents of DMOO and RMOO designs are evaluated to understand the impact of constraints defined by ( 5), ( 6), (7), and (8).Table 7 lists the THDs of both designs.The THD(I Rx ) of the fabricated design from DMOO is 1.81% higher than the optimized design.This is due to the fabricated coupler's marginal violation of constraint (5).A 0.42% increase in the THD is observed for RMOO even though it is not violating (5).However, when comparing the THD(I Rx ) of the implemented system from RMOO and DMOO.The THD(I Rx ) of RMOO is 1.59% lower than the DMOO.
The THD(I Tx ) of the fabricated design from DMOO is 0.99% higher than the optimized design.This is due to the fabricated coupler's violation of constraint (6).A negligible increment is observed with the design resulting from RMOO.However, when comparing the THD(I Tx ) of the implemented system from RMOO and DMOO.The THD(I Tx ) of RMOO is 1.66% lower than the DMOO.This observation can also be validated by plotting the input impedance of the IPT system, as shown in Fig. 12.The impedance difference at the 3rd harmonic is enlarged for better visualization.The harmonics of RMOO experience a larger input impedance than DMOO, hence the lower THD.

E. LOAD DEPENDENCE ANALYSIS
As shown in Table 2, the IPT system is designed for a 3.7 kW rated power, corresponding to a rated load resistance (R L ) of 33.11 .Therefore, the load resistance is varied from 2 to 34 to understand the impact of variation in load resistance on output power.Fig. 13 shows the results of the variation of the output power of design 6 from RMOO with load resistance.The theoretical behavior is also plotted using (14).A maximum of ±15W discrepancy is observed between the theoretical and implemented design of RMOO.A similar behavior is also observed for DMOO design.It is observed  that for both DMOO and RMOO, the output power increases with load resistance as output power is a function of load resistance, according to (14).
The load resistance is also varied from 2 to 34 to understand the impact of variation in load resistance on efficiency.Fig. 14 shows the results of the variation of efficiency with load resistance.The theoretical behavior is also plotted using (15) for the simulated parameters.It is observed that for both DMOO and RMOO, the efficiency increases at higher load conditions.The efficiency peaks around 21 for RMOO and 23 for DMOO.The efficiency difference is minimal between the efficiency at peak load resistance (33.11 ) and the resistances (21 and 23 ) where efficiency peaks.However, the implemented RMOO has a superior efficiency profile compared to the implemented DMOO.
It is evident from the experimental results that the design from DMOO had higher total harmonics distortions due to violations of constraints related to the total harmonics distortions.However, the RMOO methodology ensured that these constraints were not violated, even at the expense of these tolerances.Hence, lower total harmonics distortions.Moreover, design 6 from RMOO has a superior efficiency profile with load variations than design 3 from DMOO.Therefore, the proposed robust optimization is beneficial in the massscale production of couplers, where one can incorporate these tolerances into the design process to realize a robust system.Additionally, the proposed methodology can be extended to incorporate tolerances of the compensation capacitors, resonant frequency, driving frequency, and other related tolerances of the power electronics components.

VI. CONCLUSION
The impact of uncertainties of design variables is presented in this article.The fabricated designs resulting from deterministic and robust optimization meet the efficiency and leakage field design objectives.However, the fabricated design from deterministic optimization violates THD criteria considered during optimization.This increased the THD of receiver current by 1.59% and transmitter current by 1.66% compared to robust optimization.The performance deviations are because of not incorporating uncertainties of design variables.The proposed robust optimization method demonstrated that the fabricated coupler met all the design objectives and constraints as expected.Therefore, mass-scale production of couplers in a commercial environment will benefit from the proposed methodology as the proposed design methodology incorporates the uncertainties of design variables and will produce consistently performing IPT systems.It also improves the reliability of the overall system.

Fig. 1
(a) shows the circuit model where S 1 , S 2 , S 3 , and S 4 are semiconductor switches while D 1 , D 2 , D 3 , and D 4 are fast recovery diodes.V DC is the DC bus voltage supplied to the inverter.R Tx and R Rx are parasitic resistance of transmitter, and receiver coils with inductances L Tx and L Rx, respectively.R L is the equivalent load resistance of the battery.Fig. 1(b) shows the simplified fundamental harmonic mutual inductive model of Fig. 1(a).M is the mutual inductance between the transmitter and receiver coils, and I Tx and I Rx are the currents in the transmitter and receiver side.The relationship between the input DC voltage (V DC ) and equivalent AC input voltage (V s ) or output voltage of the inverter can be defined as: Vs = 4V dc π n=1,3,5,..

FIGURE 3 .
FIGURE 3. Proposed robust multi-objective optimization (RMOO) flowchart for a wireless power transfer coupler.

FIGURE 4 .
FIGURE 4. Two-dimensional side view of a circular coil structure illustrating its design variables.

FIGURE 5 .
FIGURE 5. Pareto optimization results for DMOO method and RMOO method.

FIGURE 6 .
FIGURE 6. Probability density functions of designs: (a) Efficiency of DMOO design 3, (b) Efficiency of RMOO design 6, (c) Leakage magnetic field of DMOO design 3 with safety limit at 27 µT, (d) Leakage magnetic field of RMOO design 6 with safety limit at 27 µT.

FIGURE 8 .
FIGURE 8.A 3.7 kW Inductive power transfer system: (a) Schematic diagram of the hardware setup, (b) Hardware prototype.

FIGURE 9 .
FIGURE 9. Transmitter-side and receiver-side voltage and current waveforms of the proposed robust inductive power transfer coupler (design 6).

FIGURE 10 .
FIGURE 10.Simulated and measured leakage field from the proposed robust coupler (design 6).

FIGURE 11 .
FIGURE 11.Variation of efficiency with horizontal misalignments of design 3 from DMOO and design 6 from RMOO.

FIGURE 13 .
FIGURE 13.Variation of output power with load resistance of the design 6 from RMOO.

FIGURE 14 .
FIGURE 14. Variation of efficiency with load resistance of design 3 from DMOO and design 6 from RMOO.

TABLE 1 . Proposed Multi-Objective Optimization Solutions For Circular Coil Structures are
compared in Table