Robust Optimization for Misalignment Tolerance in Inductive Power Transfer Coupler Design

Coupler parameters, such as transmitter inductance (LTx), receiver inductance (LRx), and the coupling coefficient (k) or mutual inductance (M), vary with the misalignments. Their variations must be quantified and incorporated into the design and optimization stage as they impact performance indices such as efficiency and leakage magnetic field. Most of the solutions proposed in the literature compromise the performance of the coupler in search of a misalignment-tolerant design. Therefore, this article presents a method to incorporate the misalignments into the design and optimization process of an inductive power transfer (IPT) coupler by utilizing robust optimization, where misalignments are considered as noises or disturbances or uncertainties to a perfectly aligned coupler. The optimized designs resulting from this method will not compromise the performance of the coupler in search of a misalignment-tolerant design. Effectiveness of the proposed robust optimization method is experimentally verified with a 3.7 kW hardware prototype.


I. INTRODUCTION
Inductive power transfer (IPT) technology has matured over the last few decades to find its applications in charging the batteries of mobile phones, laptops, medical implants, and electric vehicles.The recent developments in standards (STD) such as SAE J2954/1 [1], IEC 61980-1:2015 [2], and ISO 19363:2020 [3] are prime examples of recognizing this technology among the industrial partners.An IPT system consists of input-side and output-side power electronics converters to control and transfer the power through the coupler, depending on the load dynamics [4], [5].The coupler should operate efficiently while minimizing the leakage magnetic fields to ensure living beings' safety [5], [6], [7], [8], [9].In addition, the literature emphasizes improving the power density [9], [10] while reducing the overall cost and weight of these coils [8], [10], [13].
The EVs are expected to operate in nonideal environments where coils' misalignments are likely to occur, as shown in Fig. 1 for static wireless charging applications.The receiver coil will be misaligned longitudinally, vertically, laterally, and angularly compared to the transmitter.These misalignments impact the parameters such as transmitter inductance (L Tx ), receiver inductance (L Rx ), and the coupling coefficient (k), or mutual inductance (M), which affects the IPT system's operating performance.
Several approaches are proposed in the literature to address misalignments or improve an IPT system's misalignment tolerance [14].The first approach is through optimized circuit structures with optimal closed-loop controllers that maintain Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
stable output with misalignments [14], [15], [16].Phase shift modulation of the inverter is proposed to achieve stable output with coupling and load variation [17].DC-DC converters are used on the transmitter and receiver sides to control the input voltage of the IPT system or the equivalent output load to regulate power [18].Alternatively, active rectifiers are proposed to control the output to achieve high efficiency [19], [20].
The second approach is to design higher order compensation topologies to realize an optimal functional relation between output variables and mutual inductances [21], [22], [23], [24], [25], [26], [27], [28].A misalignment tolerant S-SP compensation topology with constant current or voltage characteristics is proposed in [24].The parameters of the compensation topologies are also optimized to improve the misalignment tolerance [27].Moreover, hybrid compensation topologies are proposed to achieve stable output current with load and coupling coefficient variations [22], [23], [28].However, hybrid solutions are complex and inherit higher losses due to a higher number of passive components.
The third approach is to design coil structures with improved misalignment tolerance.For example, various coil structures such as circular, double DD, DDQ, bipolar, and tripolar are proposed for EV applications, and their misalignment tolerance (MT) in the three-dimensional (3-D) space varies due to different winding arrangements and geometries [5].Couplers such as DDQ, bipolar, and tripolar have enhanced flux linkage during misalignments [5].The fourth approach improves the misalignment tolerance of existing couplers through optimization techniques, where optimal design variables corresponding to a misalignment tolerant design are realized [13], [35], [36], [37], [38], [39].This article's focus and contributions are toward the fourth approach.
When considering the fourth approach, the literature presents numerous multiobjective optimization strategies for couplers.Table 1 summarizes the coupler configuration, optimized objectives, and multiobjective optimization methodology of notable contributions from the literature.The proposed solutions consider multiple objectives for optimization.The common objectives considered are core loss, system cost, weight, power transfer (PT), power transfer efficiency (ɳ), k, leakage field (B L ), efficiency between perfectly aligned and misaligned coils (δɳ M ), gravimetric power density (γ ), area power density (α), and misalignment tolerances.The objectives vary in these proposed solutions due to the requirements of the optimization problem considered.
Most of these solutions do not incorporate misalignments into the optimization or design process except for the last seven designs shown in Table 1.Table 2 summarizes the type of misalignment considered, the method used to incorporate misalignment, and their advantages and disadvantages.Lateral (L), vertical (V), horizontal (H), and rotational/angular (R) misalignments of coils are considered in these studies to derive the misalignment tolerant design.There are several techniques followed to incorporate misalignments into the design flow.In [8], misalignments are considered as an objective, along with efficiency and coil costs.Mohamed et al. [13] consider the average coupling coefficient (k) during all misalignment scenarios.In [35], the lateral misalignment is regarded as a variable during the optimization in determining the optimal design variables.In [36], misalignments in six degrees of freedom are categorized into subregions, and the probability of alignment in each region is used as an index to prioritize misalignment and incorporate it into the objective function: figure of merit (FOM).In [37], the efficiency difference at misaligned and perfectly aligned operations is considered as an objective to minimize, while separate Pareto fronts are derived for perfectly and misaligned scenarios in [39].The magnetic core size is optimized to maximize coupling with misalignments using the 3-D topology optimization technique [40].
The pros and cons of the proposed solutions are summarized in Table 2.One of the significant disadvantages of the proposed solutions is the performance at the aligned position is compromised in the search for a misalignmenttolerant design.Therefore, the coupler will operate below its maximum performance point as it accommodates misalignments.The root cause for it is the approach taken to define misalignment.The misalignments are considered to be an integral part of the operation.This approach will result in a compromised performance at an aligned position.However, another approach is to consider misalignments as noise or disturbance to a perfectly aligned scenario.In this approach, the performance at the aligned position is not compromised while enhancing the misalignments performances.This is the approach proposed in this article.The proposed method incorporates misalignments into the design flow without compromising the performance at the aligned position, and the best misalignment tolerant coupler is selected for prototyping.

TABLE 2. Misalignment Tolerant Optimization Strategies
The contribution of the proposed methodology compared to the literature can be summarized as follows.
1) The misalignments are considered noises or disturbances or uncertainties to a perfectly aligned coupler.2) Introduces a method to quantify the impact of misalignment scenarios for a given objective or parameters, such as coil inductances.3) Introduces a method to remove the designer bias in selecting a design from Pareto-front by incorporating misalignment as a means of decision-making criteria.4) Introduces a method to incorporate misalignment into the optimization process without compromising the performances at the aligned position.The design with minimal dispersion in optimization objectives due to misalignments is identified.In addition, the values of L Tx , L Rx , and k of a fabricated coupler vary from the design/optimized values due to the manufacturing tolerances of the component used for coil materials and fabrication tolerances of the coupler [6], [7].Therefore, tolerances and uncertainties of design variables are also incorporated into the design process to present a robust optimization solution.

II. IPT SYSTEM DESIGN AND MISALIGNMENTS
This section briefly explains the relationship between the power electronics system design and coupler parameters.Later, it introduces the various misalignments and their impact on the operations of the IPT system.

A. IPT SYSTEM DESIGN
The circuit diagram of the IPT system considered in this article is shown in Fig. 2, where series-series (SS) compensation  is utilized.The subscripts Tx and Rx stand for the transmitter and receiver sides, and L, R, and C are the inductances, coil resistances, and compensation capacitances of the IPT system.The R L is the load resistance of the system.The  basic operating principle of an SS-compensated IPT system is derived here to understand the relationship between the power electronics system parameters and the coupler's parameters.The resonant capacitors to maximize the power transfer can be determined with C i = 1/(ω 2 0 L i ), where i= Tx, Rx.L and ω 0 are the corresponding coil's inductance and the angular resonant frequency of the IPT system.The IPT system is modeled based on the fundamental frequency approximation since the first non-zero harmonic is 30 dBA lower than the fundamental operating frequency [6].
The load of the SS-compensated IPT system can be approximated to an equivalent resistance (R e ) given by the following: where P is the power rating of the IPT system, V Rxdc is the voltage at the terminal of the load resistance R L or the voltage across the battery, as shown in Fig. 2. The load factor for an SS-compensated IPT system is defined as follows: The optimum load factor (l, f (opt) ) corresponding to maximum transmission efficiency for an SS-compensated IPT system is given by the following [6]: where Q Tx and Q Rx are the quality factors of transmitter and receiver coils.The quality factor of a coil is given by the following [5]: where i= Tx, Rx.E L is the peak energy stored in the coil, P L is the average power loss in winding, magnetic materials, and passive shields, and f is the operating frequency.Often, Q i can be approximated as shown by the right-hand side of (4), assuming that core losses and shielding losses are minimal compared to winding losses.R i is the equivalent series ac resistance of the transmitter or receiver coil with inductance L i .
The IPT systems for EVs operate at a high-frequency range of 81.38-90.00kHz to comply with SAE J2954 [1].Therefore, coils have a larger unloaded quality factor.For large quality factors, the l f(opt) simplifies to k.Therefore, the efficiency can be maximized if the load factor is as follows: R e ω 0 L Rx ≈ k. (5) The maximum theoretical power transmission efficiency (η m ) at the matched optimal load is given by the following [6]: The "figure of merit (FOM)" is given by: FOM = k 2 Q T x Q Rx .Larger values of FOM result in higher transmission efficiencies.Since FOM is related to k and the quality factors Q Tx and Q Rx , it is beneficial to increase these parameters to achieve higher η m .Moreover, the nominal coupling coefficient should be less than the critical coupling coefficient (k c = R e ω 0 L Rx ) to avoid bifurcation.Fig. 3 is used to explain the bifurcation phenomenon.The x-axis of all the subfigures in Fig. 3 is the normalized frequency derived by dividing the frequency with resonance or operating frequency.In this example, the resonance frequency is 85 kHz.Therefore, "1" on the x-axis corresponds to the resonance frequency.Fig. 3(a) shows the variation of the input impedance magnitude (Z in ) of an SS-compensated IPT system with frequency for coupling coefficients in distinct ranges of an SS-compensated IPT system.Fig. 3(b) and (c) show the corresponding variations in phase angle (Arg(Z in )) of input impedance and voltage transfer ratio (VTR).k 0 is the nominal coupling coefficient of a design, while k c is the critical coupling coefficient.The bifurcation occurs if k > k c , and there are two minima in Z in , three zero phase angle frequencies, and two peaks in the VTR.The system will operate in the zero-current switching region instead of the zero-voltage switching region, increasing losses.However, the bifurcation is avoided by ensuring k < k c .There will be a single zero phase angle, VTR, and a minimum in Z in , as shown in Fig. 3. ZVS can be realized slightly above the resonant frequency as the phase angle is always positive above the resonant frequency.Furthermore, as the design's k 0 moves away from k c where k 0 < k c , the higher frequency harmonics of the transmitter current will experience a larger input impedance.Therefore, it also reduces the total harmonics distortion of the current.
Therefore, L Rx is designed to satisfy the inequality given by ( 7) to reduce the harmonic distortion of Tx-current due to the nonlinearity of the inverter and to avoid the frequency bifurcation phenomena: The K pmin and K pmax correspond to factor K p 's minimum and maximum values.The factor K p is selected to be in the range of 0.65-0.85as it has minimal impact on the maximum transmission efficiency given by ( 6) and is relatively constant within this range.
The transmitter coil's inductance is determined based on the desired voltage transfer ratio for a given R e and radian operating frequency.For an optimally matched load condition, the Tx-coil inductance is given by the following: The voltage gains at the fundamental angular frequency are given by: G v (ω 0 ) = V Rxdc /V T xdc .The desired G v (ω 0 ) at optimal efficiency operation should marginally satisfy the coupler's L Tx and L Rx as follows: This article considers these design rules or constraints during the finite element analysis (FEA) based optimization process.

B. MISALIGNMENT
In stationary wireless charging systems, the Tx is fixed on the floor while the Rx is on the EV.However, the position of the Rx changes depending on the dynamics of the EV and creates different misalignment scenarios.Each scenario has a different degree of freedom (DOF) in terms of the movements.These scenarios can be categorized into the following: 1) horizontal misalignment; 2) vertical misalignment; 3) angular misalignment.Fig. 1 shows some of the misalignment scenarios of an EV.Horizontal misalignment is due to inaccurate vehicle parking in the charging zone.It occurs laterally (y-axis) and longitudinally (x-axis).The lateral misalignment is shown in Fig. 1(b).The vertical and the angular misalignment occur due to the weight distributions in a vehicle or misalignments in tires due to wear and tear, as shown in Fig. 1(c) and (d).The coupler's air gap is often fixed to abide by the ground clearance ranges defined under the J2954 std [1].Therefore, vertical misalignment (z-axis) is any initial fixed air gap deviation.The vertical misalignment has a single DOF, as it can move only in the z-direction.The angular displacement has three DOFs (roll (θ x ), pitch (θ y ), and yaw (θ z ) along the x-, y-, and z-axes, respectively), which depend on the weight distribution of the vehicle.Therefore, misalignment (M a ) has six DOFs for an EV M a = x, y, z, θ x , θ y , θ z (10) The acceptable limits of these misalignments are defined in the SAE J2954/1 STD and are listed in Table 3 [1].The misalignments will impact the coupler's self-inductances and the coupling coefficient or mutual inductance.The impact on the power transfer can be understood by rearranging (1) using ( 9) The dc voltages V Rxdc and V Txdc are adjusted accordingly to achieve the optimum efficiency operating condition under misalignment, as discussed in [41].

III. ROBUST OPTIMIZATION OF COUPLER
Fig. 4 shows the critical steps of this article's proposed robust optimization approach.The details of the steps shown in Fig. 4 are as follows:

A. DESIGN SPECIFICATION
The robust optimization is applied to a circular coupler of a 3.7 kW IPT system, which belongs to the WPT 1 power class according to the SAE J2954 [1].However, the proposed design methodology shown in Fig. 4 can be applied to other coupler structures at different power classes.The power electronics parameters are listed in Table 4.The SS-compensated IPT system discussed in Section II is considered in this article.

B. PROBLEM FORMULATION
Once the design specifications are identified, the multiobjective optimization problem is formulated.The problem formulation consists of the following steps.

1) DESIGN OBJECTIVES IDENTIFICATION
Objectives such as power transmission efficiency (η m ) given by ( 6) and leakage magnetic fields (B L ) depend on parameters such as k or M, L Tx , and L Rx .However, misalignments alter these parameters and hence will impact the objectives.Therefore, this paper considers maximizing η m and minimizing B L as the objectives for optimization.
where d 1 , d 2 , . . ...d k d are design variables of the coils as elaborated in Section III-B3.The ANSYS Maxwell 3-D is used to simulate the couplers and derive the transmission efficiency of coils.The coupler losses consist of winding losses, losses in the magnetic material, and passive shielding losses.MnZn Ferrite (HP400 from Mahindra) and aluminum are the magnetic and passive shield materials used in this optimization.Litz wire with 800 strands of AWG 37 is used for the coil winding.Loss characteristics of MnZn ferrite material used are incorporated into ANSYS Maxwell to determine core losses.The ac resistance considers the skin effect and the internal proximity effect.Winding losses are determined using this ac resistance and current through the coils at the resonant frequency.At the same time, eddy current losses are also determined using the ANSYS Maxwell software's eddy current solver.

2) DESIGN CONSTRAINTS IDENTIFICATION
The coupler design constraints are extracted from the SAE J2954 STD, industrial practices, and power electronics system design.The constraints resulting from the design of the SS-compensated IPT system are considered during the optimization to restrict parameters such as k, L Rx , and L Tx in the FEA domain.The L Rx must satisfy the inequality given by ( 7) While the L Tx must satisfy the inequality given (8) The coupling coefficient is limited to less than the critical coupling coefficient to avoid bifurcation The air gap is limited to 150 mm A nominal operating frequency of 85 kHz is used within the 81.38-90.00kHz [1] The maximum coil area is restricted to 0.30 m 2 by considering the coil dimensions required to transfer power over an air gap of 150 mm [1] The flux density of the ferrite bars is restricted to a maximum of 200 mT to avoid saturation Table 5 summarizes these constraints.

3) DESIGN VARIABLES DEFINITION
The circular coil design variables are shown in Fig. 5.The design variables vector is given by the following: where r w and r s are the radius of the conductor and the shield, r i and r f are the starting radius of the coil and the ferrite bars, r c , h wf , and h fs are the separations between the two adjacent conductors, the conductor, and the ferrite bar and the shield and the ferrite bar.l f , w f , and h f are the ferrite bars' length, width, and thickness.t s is the thickness of the shield, while N is the number of turns.The design variables' ranges are listed in Table 6.The design variables with subscript T in Table 6 correspond to the

4) DESIGN VARIABLE'S TOLERANCES
These design variables are associated with manufacturing and fabrication tolerances.Manufacturing tolerances are variations in ferrite, litz wire, and shielding material dimensions due to imperfections in the manufacturing process.Fabrication tolerances are mainly due to difficulties in accurately realizing design variables such as space between the turns.The coefficient of variation (COV) of design variables is defined to incorporate the tolerances of design variables into the design process.COV is the ratio between the design variables' standard deviation (SD) and mean value with a normal distribution.COV of this study is provided in Table 6.The process capability index is used to determine the SD of the manufacturing process The upper and lower tolerance limits are U SL and L SL .C Pk for an established manufacturing process is around 1.33.
Once the problem is formulated, as discussed above, the next optimization step is the sensitivity analysis (SA), as shown in Fig. 4.

C. DESIGN VARIABLE SENSITIVITY ANALYSIS
The SA identifies important design variables related to the optimization objectives and constraints.The design of the experiment approach is used to evaluate the reliability of the numerical model.The Latin hypercube sampling (LHS) is used to scan the multidimensional input variables to create the meta-model of optimal prognosis (MOP) [42].This model can represent the original problem, and model accuracy can be verified by evaluating the coefficient of prognosis [43].A Latin hypercube is an m-dimensional object representing m different analyzed design variables.
Design variables are listed in Table 6, and 100 FEA designs were solved to create the MOP.Furthermore, the MOP can be used to analyze various design configurations without additional simulations.This feature of the MOP is useful for optimization problems solved through FEA due to their high  and k ) derived from the MOP.It is the relative indication of a variable's impact on objectives and constraints.For example, the receiver side r i has a 15.5% impact on efficiency, while r c has 13.5% on efficiency.
Therefore, when looking into the percentage contributions, the design variables r i , r w , and r c of the Tx-coil and the Rx-coil have the highest impact on the efficiency, leakage magnetic field, coil inductances, and coupling coefficient.The design variables such as h wf , h wfT , h fs , h fsT , t s , and t sT have the least contribution toward the objectives and constraints, as shown in Table 7, with low percentage contributions.Therefore, these design variables are kept at their nominal values, as they contribute little improvements (as identified by percentage contribution in Table 7) at the expense of higher computational cost.However, in this article, h wf , h wfT , h fs , and h fsT are the only design variables kept constant at their initial nominal values, as the computational cost was not a major limitation due to high-performance computing availability.
The variations of objectives, constraints, and parameters due to design variables can also be derived from sensitivity analysis, as shown in Fig. 6.Fig. 6(a) shows the variation of efficiency with design variables r c and r i of the Rx coil.Fig. 6(b) shows the efficiency variation for the Tx coil.Fig. 6(c) shows the coupling coefficient variation with the Tx coil's r c and r i .The results are interpolated to visualize the pattern depicted by each response.The next step of the design flows is multiobjective optimization, as shown in Fig. 4.

D. MULTIOBJECTIVE OPTIMIZATION
The multiobjective optimization (MOO) is performed for objectives ( 12) and ( 13) for design variables defined in (21).
The Pareto-front of size m derived using each n iteration of the optimization algorithm is given by the following: This study uses the non-dominated sorting genetic algorithm II (NSGA-II) to derive the Pareto-front.NSGA II is preferred as it has fast and efficient convergence and can search a wide range even if the starting solution is unfeasible.Fig. 7 shows the multiobjective optimization's Pareto-front (red line) with Pareto optimal designs labeled as 1, 2, 3, 4, 5, and 6.Black dots not on the Pareto front represent non-Pareto optimal designs that did not violate any constraints.Red dots represent the designs that violate the constraints identified in Section III-B2.These designs are ignored when deriving the Pareto-front.

E. MISALIGNMENTS
This section explains how misalignments are incorporated into the design flow.

1) DEFINITION OF MISALIGNMENTS
Initially, misalignments considered in the optimization problem are identified.This article assumes the circular coupler's angular misalignment has a single DOF, where θ x = θ y = θ .The horizontal misalignment in the y-direction is considered as it has the maximum offset limit compared to the x-direction according to the SAE J2954 standard [1].Fig. 8 shows the 2-D side views of the circular receiver coil's angular and horizontal misalignments considered in this article.In addition, the combinations of angular and horizontal misalignments are also considered.

2) MISALIGNMENT SENSITIVITY
Misalignment sensitivity analysis is performed on an optimized design on the Pareto-front shown in Fig. 7 to identify the impact of misalignments on k, L Tx , and L Rx .The method followed here is similar to the design variables sensitivity analysis explained in Sections III-C.The results are listed in Table 8 in terms of the percentage contribution of each misalignment on parameters k, L Tx , and L R .The horizontal misalignment significantly impacts the k, L Tx , and L Rx compared to angular misalignment, as seen in Table 8.
The variations of k, L Tx , and L Rx due to misalignments can also be derived from misalignment sensitivity analysis, as shown in Fig. 9. Fig. 9(a) shows the coupling coefficient's variation with angular and horizontal misalignment, and it reduces significantly with the horizontal misalignments (Delta x).The impact of angular misalignment is minimal.Fig. 9(b) and (c) show the variation of the L Tx and L Rx with misalignments, where both show similar behavior, and the impact of horizontal misalignment is dominant compared to angular misalignment.The variations shown in Fig. 9 complement the results listed in Table 8.Therefore, the misalignment sensitivity analysis provides an overall idea of the coupler's responses with respect to misalignments and will be used to define the misalignment tolerance evaluation criteria in the next section.3) MISALIGNMENT TOLERANT CRITERIA Fig. 10 shows an example of a Pareto-front for two objectives.Design 1 and design 2 correspond to two Pareto-optimal designs on Pareto-front derived from MOO.These designs are similar to designs 1-6 shown in Fig. 7. Design 1's optimal objective 1 and objective 2 values are O 1 and O 2 .The optimal design 1 on the Pareto-front will deviate from their optimal operating point (O 1 , O 2 ) due to misalignments to a new operating point as indicated by m 1 ,m 2 ,m 3 …., m n on Fig. 10.m n is the nth Pareto-optimal design.A similar deviation will occur for Pareto-optimal design 2 as shown by their new operating points x 1 ,x 2 ,x 3 ….,xn .However, this dispersion due to misalignment will vary from one optimal design to another, as shown in Fig. 10 for design 1 and design 2. Design 1 has a high dispersion compared to design 2 .
In this article, power transmission efficiency and leakage field are functions of the parameters k, L Tx , and L Rx , as seen in (6).Therefore, minimizing dispersion on these parameters will reduce dispersion in power transmission efficiency and leakage field objectives.Thus, the SD in the k, L Tx , and L Rx is considered as the indices to measure a design's robustness against misalignments or misalignment tolerance.The variation in L Tx and L Rx due to misalignments is minimal compared to k, as shown in Fig. 9, and misalignment sensitivity analysis results in Table 8.Therefore, the SD of k is prioritized compared to L Tx and L Rx .The SD for each parameter can be derived using the following: where σ P is the SD of the parameter.y i is each value of the dataset, while μ and N are the mean and size of the dataset.The limits of the L Rx given by design ( 14) can also be violated due to both design variables' tolerances and misalignments.These influences can be defined as follows: The probability of violating the limits defined by ( 14) due to misalignments and design variable tolerances is calculated to ensure the selected design's conformity with the requirements of the power electronics design.This can be expressed as follows: P T (F ) is the target probability and P(F ) is the probability of failure.Therefore, the design with minimum SD in coil parameters in the order of k, L Rx , and L Tx within the limits of L Rx given by ( 14) is considered the best misalignment tolerant design for implementation.

4) MISALIGNMENT METHODOLOGY
The proposed method to incorporate misalignments is implemented as follows: The misalignment tolerance of each optimized designs 1 to 6 on the Pareto-front shown in Fig. 7 is evaluated with 100 FEA designs sampled with advanced LHS [44], [45], [46], [47].In this method, each optimized design on the Pareto-front will be subjected to 100 different misalignment combinations.The LHS distribution can accurately predict the outputs of the possible designs based on a relatively small sample of the FEA simulations.Table 9 lists the SD of each optimized design on the Pareto-front due to misalignment scenarios.Design 3 exhibits the minimum SD in k and L Rx with misalignments and tolerances of design variables.
Furthermore, the probability distribution of the L Rx of design 3 with misalignments is shown in Fig. 11.The conformity of the design with respect to the upper and lower limit of L Rx  given by ( 14) is calculated in terms of sigma levels.Sigma level measures how a design varies from the perfect/optimized design, based on the number of defects or violations per million designs.The short-term reliability is evaluated in this article, where a target of 4.5 sigma (4.5 σ ) is considered for the upper and lower limit of L Rx given by ( 14) [44], [45].4.5 sigma corresponds to a probability of P(F ) = 3.40 × 10 −6 or 3.4 violations per million in the short term due to the variation of the design variables' misalignments and tolerances.This is also the target probability defined in (26).The calculated upper bound is 5.02 σ , while the lower bound is 4.62 σ for design 3, as shown in Fig. 11.The upper bound and lower bound are shown in red color.Therefore, it is safe to assume that the robust criterion of 4.5 σ is achieved by design 3 with misalignments.
Designs 1, 2, 4, 5, and 6 are Pareto optimal.However, these designs have a higher SD in k and L Rx than design 3 as shown in Table 9.The minimum SD in k, L Rx , and L Tx are prioritized criteria in selecting the best misalignment tolerant design, as discussed in Section III-E-3.Therefore, designs 1, 2, 4, 5, and 6 will not be the best misalignment tolerant design.Design 1 has the highest SD in k.Therefore, design 1 is compared with the misalignment tolerant design 3 to demonstrate the efficacy of the proposed method in Section IV. 945 FEA simulations were required to derive the misalignment tolerant design using the proposed methodology.
In a scenario where all the designs do not conform to the robust or misalignment tolerance criteria, the safety factors or constraints must be modified, as shown in Fig. 4. Multiobjective optimization is performed again to obtain the new Pareto front.As discussed above, the same design steps are followed to select the misalignment tolerant design.A misalignmenttolerant design can often be realized with a second iteration.However, this optimization problem did not require a second iteration.

IV. EXPERIMENTAL VERIFICATION
Designs 1 and 3 are selected for implementation to validate the efficacy of the proposed methodology in a 3.7 kW hardware prototype.The design variables of the coils are listed in Table 6.The coils are manufactured using a litz wire with 800 strands of AWG 37. MnZn Ferrite (HP400 from Mahindra) with a permeability of 2200 is used as the magnetic material for the core.Aluminum (Al) is used for the passive shield.Moreover, the air gap of both designs is 150 mm, with an area of less than 0.3 m 2 .

A. EXPERIMENTAL SETUP DETAILS
Fig. 12(a) shows the experimental setup, which includes the input-side inverter, output-side rectifier, load resistor, resonant capacitors (C Tx and C Rx ), and the coupler.Fig. 12(b) shows the measurement instruments, semiconductor devices, and controller board used in the experimental setup.The power electronics system parameters corresponding to the experimental setup with design 3 are provided in Table 4. Similar values were used for the experimental setup with design 1. 1 nF of EPCOS metalized polypropylene film capacitors are used to realize the resonant capacitances (C Tx and C Rx ) values of each design given in Table 10.The input dc-link voltage at rated power 3.7 kW is 349 V.A three-phase source is rectified to obtain the dc voltage.CoolMOS TM IPW60R031CFD7 power MOSFETS are used for the inverter.Two MKPC.4B 1.5 μF snubber capacitors are used to protect the inverter switches from switching stresses.The operating frequency of the IPT system is 85 kHz and complies with the operating frequency range defined in SAE J2954 STD [1].Two modules of IXYS DSEP2x31-06A, fast recovery diodes, are used in the rectifier.The output DC voltage across the rated load (R L = 24.3 ) is 300.5 V. ELP/DCM9715 electronic load, and a custom-manufactured variable load with a rated power of 7.2 kW are used to model the load or output resistor (R L ).TMS320F28335 DSP controller is utilized to implement the control and generate switching signals for the inverter.
Yokogawa PX8000 precision power scope with Agilent N2782B AC/DC probes and Tektronix differential p5200 voltage probes were used to measure the efficiency.IPT systems are usually low power factor systems, and PX8000 supports measurements of such systems at high frequencies.The deskew function of the PX8000 was used to compensate for the differences between current waveform and voltage waveform due to measurement sensors and input characteristics of a small power factor IPT system.Agilent Technologies 3000 DSO-X 3014A captured the voltage and current waveforms of the IPT system.The couplers' leakage field is measured using a Narda ELT 400 magnetic field meter.

B. ALIGNED PERFORMANCE EVALUATION
The simulated and experimental results are listed in Table 10.The coupling coefficients of designs 1 and 3 are 0.227 and 0.229, at nominally aligned positions.These values are less than the critical coupling coefficients shown in Table 10.Therefore, frequency bifurcation is avoided.This is achieved by incorporating the inequality ( 16) into the optimization process.The phase angle and magnitude of the input impedance of the IPT system with the fabricated design 3 are shown in Fig. 13.The system has a single zero phase angle, as it avoids the bifurcation phenomenon.
Table 10 also lists the parameters and objectives of the fabricated coils corresponding to designs 1 and 3.The L Tx , L Rx , ac resistance of the transmitter ( R Tx ), ac resistance of the receiver ( R Rx ), M 0, and k 0 coil parameters are measured at aligned positions using an LCR meter.The coil parameters vary slightly compared to the optimized/simulated parameters due to manufacturing and fabrication tolerances of the design variables, as shown by parameter difference ( ) in Table 10.Moreover, the ac resistances of receiver and transmitter coils are lower due to the use of litz wire.The quality factors of the transmitter and receiver are also provided in Table 10.Both coils have a high-quality factor at 85 kHz.High-quality factors are required to achieve higher efficiency, according to (6).The slight variations in simulated and experimental quality factors are due to inaccuracies in estimating the ac resistance of the coil and variations in inductances from the simulated values during coil fabrication.
The objectives ɳ and B L correspond to the efficiency and the leakage magnetic field measured at 650 mm from the center of the coupler at the aligned position.The transmission efficiency at the aligned position of design 3 is ∼96.90%, and design 1 is ∼96.95%.These values are almost equal to the optimized values on the Pareto-front, as shown in Fig. 7.
The measured leakage field of coupler design 3 is 7.78 and 7.40 μT for design 1.These values are about 0.68 μT (design 3) and 0.598 μT(design 1) higher than the Paretooptimal designs, as shown in Fig. 7.The leakage fields of both designs are well below the upper limit of 27 μT defined in the SAE J2954 STD or ICNIRP guidelines [1], [2].The input and output side current and voltage waveforms (i Tx (t), i Rx (t), v Tx (t), and v Rx (t)) of the IPT system with coupler design 3 or misalignment tolerant design are shown in Fig. 14.Efficiency and leakage field results verify that the design operates as intended in the aligned position.
Additionally, the transmission efficiency of both designs at aligned positions is evaluated by varying the load resistance (R L ) from 2 to 25 .Fig. 15 shows the variation of transmission efficiency with load resistance.Designs 1 and 3 reach  maximum efficiency at 20 .This maximum efficiency is 96.92% for design 3 at 20 , while 96.96% for design 1.The efficiencies at rated load (R L = 24.3 ) are 96.95% and 96.90% for designs 1 and 3.The efficiency variation between 20 and the rated load is less than 0.05% for both designs.High-frequency profiles over wide load variation attest to the quality of the optimization solutions derived in this work.

C. MISALIGNMENT PERFORMANCE EVALUATION
The horizontal, angular, and combinations of angular and horizontal misalignments are evaluated using the hardware prototype.Twenty-one misalignment measurements were recorded during the experiments, corresponding to different misalignments.Fig. 16 shows a snapshot of six out of twenty-one misalignment scenarios evaluated.The subfigure descriptions of Fig. 16 provide the horizontal and rotational misalignment position corresponding to each figure.The images corresponding to the rest are not provided in the interest of space.The variations in the parameters k, L Rx , and L Tx with the fabricated receiver coil's angular (Angle) and horizontal (Delta X) misalignments are shown in Fig. 17    The coupling coefficient reduction ( k) due to misalignment is worse in the case of coupler design 1 compared to design 3, as seen in Table 10, Fig. 17  that design 1 had a higher deviation than design 3 and justifies the selection of design 3 as the misalignment-tolerant design.
The impact of angular misalignment on L Rx and L Tx is minimal compared to the horizontal misalignment, and this is true for both designs, as seen in Fig. 17 10.However, the L Rx of design 3 lies within the range defined by (14).Therefore, its safety margins defined in Section III-E-3 and 4 are not violated, as confirmed during the design stage, as shown in Fig. 11 for L Rx .
Fig. 18 shows the variation of L Tx and L Rx with air gaps.The air gap is the distance between the transmitter's top surface and the receiver's lowest contact point.This lowest contact point of the receiver varies with angular misalignment, as shown in Fig. 16.Therefore, the air gap variation happens due to angular misalignments.The angular misalignments are measured at 0 0 , ±2 0 , ±4 0 , and ±6 0 intervals.The air gaps at these angular misalignment positions are approximately 150, 139, 129, and 119 mm for both designs, as shown in Fig. 18.Therefore, the maximum and minimum air gaps between the coils are 150 and 119 mm for designs 1 and 3.The maximum inductance variation in transmitter and receiver inductances given above is observed within this air gap range.
The maximum deviation of the transmitter side's resonant frequency due to inductance deviation is 0.63 kHz for design 1 and 0.48 kHz for design 3. The maximum variation of the receiver side's resonant frequency due to inductance deviation is 0.64 kHz for both designs 1 and 3.The resonant frequency variation is significantly smaller because of the smaller variation range in inductances due to misalignments.The change in self-inductances due to misalignments has minimal impact on the quality factors of the coil.According to (6), the efficiency will depend on the coils' coupling coefficient and quality factors.Both coupling and quality factors change simultaneously with misalignments.However, the coupling variation is significant (-34.5%)compared to the quality factor (-1.5%) for design 1.This observation is valid for design 3, as well.This behavior is also observed in Fig. 9 during the design stage.Therefore, coupling variation significantly impacts efficiency more than the quality factors or self-inductance variations when considering misalignments.The measured efficiency with misalignments of the receiver is shown in Fig. 19(a) and (b).The coupler design 3 has 94.52%, corresponding to a 100 mm horizontal offset and 0 0 angular offset.This is the coupler's lowest efficiency for any misalignment considered in this paper.However, coupler design 1 has 93.62% as the lowest efficiency, corresponding to the worst-case misalignment position of design 3. Therefore, the maximum efficiency deviation due to misalignments is 2.38% for design 3 and 3.33% for design 1. Design 3 has approximately 1% less deviation with misalignments.This confirms that design 3 has a better misalignment tolerance than design 1 and validates the efficacy of the proposed method.The reduction in efficiency in design 1 compared to design 3 is mainly related to the coupling coefficient in design 1 at the worst misalignment position is lower than that of design 3.
The leakage field is measured 650 mm from the center of the coupler using a Narda ELT 400 magnetic field meter.The leakage field plots corresponding to misalignments are shown in Fig. 19(c) and (d) for coil designs 3 and 1.The leakage magnetic field variation is minimal with angular misalignments, as seen in Fig. 19(c) and (d).However, horizontal misalignment has a higher impact on the leakage field.This was correctly identified during the design stage by misalignment sensitivity analysis, where Table 8 showed that horizontal misalignment had a high impact compared to angular misalignments.The coupler's leakage field increases to a maximum of 13.18 μT at (0 0 , 100 mm) in design 3 and 15.28 μT at (0 0 , 100 mm) in design 1.These results correspond to the highest leakage field reported for misalignments considered in this article.Both couplers' leakage field is well below the 27 μT limit defined in SAE J2954 STD and ICNIRP guideline, even with misalignments.However, design 3 has a lower leakage field compared to design 1.

D. DISCUSSION
The comparison between design 3 and design 1 shows that design 3 has higher efficiency and the lowest leakage magnetic field at the misaligned position.Moreover, the performances at the aligned position align with the Pareto-front results.Therefore, it validates that design 3 has a higher misalignment tolerance than design 1.
In literature, selecting a design for implementation from a conventional Pareto front of viable solutions lacks rational decision-making.More often, the objectives' values vary slightly, and it is hard to distinguish between the designs.In this optimization problem, the variation in efficiencies and leakage magnetic fields of designs 1 and 3 is 0.05% and 0.08 μT at the aligned position, as seen in Table 10 and Fig. 7.The decision to select a design for prototyping will be based on the designer's choice of this marginal difference, as both designs are Pareto-optimal.This article introduces misalignment tolerance as a decision-making factor in selecting a design from Pareto-front.It avoids user bias in choosing a Pareto-optimal design while incorporating misalignments in the design process.Design 3 had superior efficiency and the lowest leakage field at the worst misaligned position compared to design 1.Moreover, the efficiency difference is 0.9%, and the leakage field difference is 2.1 μT between design 1 and 3 at the worst-case misaligned positions.These differences are significant compared to differences at aligned positions and validate the significance of the proposed methodology.
This work demonstrated the efficacy of the proposed methodology with a circular coupler.However, it can be applied to other existing coupler structures because most coupler structures have similar design variables to circular couplers and experience similar misalignment scenarios discussed in Section II-B.The typical design variables of any coupler are the number of turns, the separation between the turns, the copper wire radius, dimensions of magnetic material and passive shield, the separation between winding and magnetic material, the separation between the magnetic material and passive shield, starting point of the windings, and starting point of the magnetic material if pieces are used.The proposed robust optimization method will optimize these design variables to derive the Pareto-front.Then, it will evaluate the misalignment tolerance of each design on the Pareto-front against the misalignment tolerance criteria defined in Section III-E-3 to determine the best misalignment-tolerant design.Therefore, the proposed method can be easily incorporated to realize a misalignment tolerant coupler configuration without compromising its performance at the aligned position.
As a future work, it would be interesting to perform a comparative study with different coupler configurations (DD, DDQ, Bipolar pad, and Tripolar pad) using this method to determine the improvement in misalignment tolerance of each coupler compared to conventional methods.Section I discusses different approaches to improving misalignment tolerance.Therefore, it would be interesting to develop a hybrid design methodology incorporating all these approaches to realize a superior misalignment-tolerant IPT system.

V. CONCLUSION
The coupler optimization is critical for improving the overall performance of an inductive power transfer system at aligned and misaligned positions.Therefore, this article introduced a method that aids in identifying the misalignment tolerant coupler design for hardware prototyping by considering misalignments as uncertainties to a perfectly aligned coupler.The proposed methodology also eliminates the designer bias in selecting a design for implementation from a Pareto-front.The misalignment criteria for determining the best design are defined based on the understanding gained through misalignment sensitivity analysis, where the SD of k, L Tx , and L Rx due to misalignments and the probability of violating the design limitations on L Rx were considered.The experimental results demonstrated that the best misalignment-tolerant design outperforms other designs at worst-case misalignments while not compromising its Pareto-optimal performance at the aligned position.

FIGURE 3 .
FIGURE 3. IPT system characteristics.(a) Variation of magnitude of input impedance with normalized frequency.(b) Variation of phase angle of input impedance with normalized frequency.(c) Variation of voltage transfer ratio (VTR) with normalized frequency.

FIGURE 5 .
FIGURE 5. Design variables of the circular coil.

TABLE 4 .
IPT System ParametersANSYS Maxwell software is used to model the IPT coupler and determine parameters such as k, L Tx , and L Rx and the objectives' values (efficiency and leakage field).The coils are analyzed using a 3-D FEA model.

FIGURE 6 .FIGURE 7 .
FIGURE 6. Impact of design variables on the responses.(a) Efficiency with r c and r i .(b) Efficiency with r cT and r iT .(c) Coupling coefficient with r cT and r iT .

FIGURE 13 .
FIGURE 13.Magnitude of input impedance(|Z in |) and zero phase angle of input impedance (Arg(Z in )) of the optimized design 3.

FIGURE 14 .
FIGURE 14.Current and voltage waveforms of the IPT system.

FIGURE 15 .
FIGURE 15.Variation of transmission efficiency with load resistance. .

Fig. 17
(a)-(c) show the variations for coupler design 1. Fig.17 (d)-(f) show variations for design 3. The responses evaluated at selected misalignment positions are interpolated to depict the general pattern of parameter variation due to misalignments in each figure.
(b), (c), (e), and (f).The maximum variation of transmitter-side (L Tx ) and receiver-side (L Rx ) due to misalignment are 2.61 and 1.94 μH for design 3, while 3.34 and 2.04 μH for design 1 from their experimental nominal values given in Table

FIGURE 17 .
FIGURE 17. Misaligned experimental results.(a) Variation of k with angular and horizontal misalignment of design 1.(b) Variation of L Tx with angular and horizontal misalignment of design 1. (c) Variation of L Rx with angular and horizontal misalignment of design 1. (d).Variation of k with angular and horizontal misalignment of design 3. (e) Variation of L Tx with angular and horizontal misalignment of design 3. (f) Variation of L Rx with angular and horizontal misalignment of design 3.

FIGURE 18 .
FIGURE 18. Variation of inductance with air gaps.(a) Receiver side inductance variation with air gaps.(b) Transmitter-side inductance variation with air gaps.

FIGURE 19 .
FIGURE 19.Misaligned experimental results.(a) Variation of efficiency with misalignments of design 1.(b) Variation of efficiency with misalignments of design 3. (c) Variation of leakage field with misalignments of design 1.(d) Variation of leakage field with misalignments of design 3.

TABLE 6 . Design Variables of the Coupler transmitter
coil values, while the others correspond to receiver coil values.

TABLE 7 . Coefficient of Prognosis Matrix for Design Variables computational
cost.Table 7 lists each variable's contribution percentage towards the objectives (ɳ m and B s ) and constraints (L Tx , L Rx,