Optimal Coils and Control Matching in Wireless Power Transfer Dynamic Battery Chargers for Electric Vehicles

Wireless Power Transfer (WPT) is a key enabling technology towards the transportation electrification, able to overcome some limits of the plug-in charging of the Electric Vehicle (EV). In this frame, a special attention is paid to the dynamic WPT systems, where the EV is recharged during motion. This paper presents a model and a resultant method for the system level design of a series-series WPT Dynamic Battery Charger (WPT-DBC) for EVs. The model allows analyzing the joined influence of coil pair characteristics and power converter stages controls on the global performances of WPT-DBCs. The design method provides general guidelines for system level optimization, matching the characteristics of the power and control elements of a series-series WPT-DBC, to achieve the desired efficiency, receiver power loss and battery charge.


I. INTRODUCTION
The development of Wireless Power Transfer Dynamic Battery Chargers (WPT-DBC) for Electric Vehicles (EVs) has approached a quite good technological maturity level [1] [2]. Today several solutions implementing effective coil geometries and magnetic couplers, convenient power conversion architectures and control strategies are available [3]. A recent overview and comparative analysis on existing power conversion architectures and control methods in WPT-DBC for EVs is presented in [4]. Such a review paper discusses in details the typical magnetic couplers, the converter stages for the primary and the secondary side and the control handles associated with them, the compensation networks and their characteristics. Fig. 1 shows the block diagram of a WPT charger, including transmitter (TX) and receiver (RX) coil pair and relevant power conversion stages. The coil pair characteristics, the compensation topology, the power conversion stages architecture and relevant controls, and the vehicle trajectory and speed determine the overall performances of a WPT charger. Normally, the development of a WPT charger starts from the coils design, based on given power ratings. Coil design constraints involve minimum efficiency requirements, safety issues relevant to the limitation of the leakage magnetic field, misalignment performance, cost, volume and weight. Some limitations on these issues are defined under standards like SAE J2954/1 [5] and IEC 61980-1:2015 [6]. Given the desired constraints to fulfill, the coil pair design process involves a series of steps, starting from practical choices (e.g., the minimum coupling to be ensured or the type of ferrite to be adopted), passing through the adoption of numerical and analytical modeling, to the final validation with the thermal modeling of the coil pair. In such a chain, the impact of compensation techniques cannot be neglected, and can be studied by analyzing the input impedance seen by the power supply [7]. Recent progress in coil design for EVs are discussed in [8], where a summary of couplers types, objectives, constraints, optimization algorithms and methodologies adopted for their design is provided, with a full discussion on the main coil structures properties (e.g., system complexity, misalignment tolerances, etc.). For example, in [9] unipolar and double D (DD) rectangular-shaped coils for wireless EV chargers are investigated. The coil pair sizes are designed to achieve the maximum coupling coefficient and efficiency. In [10] a new set of coil design formulas for high-efficiency and low harmonic currents with low magnetic field leakage for highpower WPT chargers is presented. Since the lower transfer efficiency is directly connected to low coupling coefficient, many efforts have been done in order to adapt the loosely coupled coil pair in WPT chargers. A bipolar DD coupler structure with a compensation-integrated feature is also proposed in [11], where an additional coil with smaller size is placed between the main TX coil and a ferrite layer for coupling enhancement. A novel array-type primary resonant coil design is proposed in [12] allowing to mitigate coil misalignments impact and improve the power transfer efficiency thanks to multiple independent TX coils, selectively powered based on their relative position with respect to the RX coil. Several optimization design processes have been proposed in the literature to support the optimal and innovative design of new coil pair structures, aiming to achieve maximization of the coil pair efficiency [13], power transfer capability [14], power density [15], weight and volume [16]. Enhanced evolutionary algorithms have been also proposed in the literature. An interesting advanced machine-learning algorithm has been applied to EV coil pair design to find a ferrite core structure with high magnetic coupling between the TX and RX coils [17]. Improved pad structures and a systematic design approach to achieve optimal primary DD pad have been also presented [18], based on a joint finite element method simulations and genetic algorithm. Nowadays it is possible to design power stages with very high efficiency values, thus ensuring convenient power converters performances. In WPT static chargers it is also easy to set power converters control so as to achieve optimal performances within a given range of misalignment between the coils. However, in WPT-DBC additional goals must be accomplished, such as maintaining the power flow control regardless misalignment and velocity of the vehicles, and implementing segmentation control to energize the TX pad only when the vehicle is crossing it [19]. The simplest way to achieve these goals is based on controlling the power transfer on the transmitter side only, with a receiver side realized as a simple passive diode rectifier [20]- [22]. This simplifies the control, but limits the load (battery) regulation capability.
Active receiver regulation makes the control more complex and may require additional sensors and communication between the primary and secondary sides [23] [24]. An effective WPT-DBC control is based on the impedance regulation on the receiver side, achieved by modulating the post-regulator duty-cycle so that the value of the equivalent load resistance at the rectifier output maximizes the power or efficiency rate. In [22], an optimal design against misalignment conditions in a Series-Series (SS) WPT-DBC is presented, providing optimal load matching thanks to a DC-DC converter, which guarantees the highest average transfer efficiency. In [25], a new charging technique, based on a sensorless identification of the actual presence of the vehicle over the receiver, realizes the load resistance matching to obtain the maximum power transfer in condition of nominal mutual coupling. All cited prior studies are based on the use of nominal mutual coupling inductance for the WPT-DBC design. However, in WPT-DBC the mutual coupling inductance depends on the relative instantaneous positions of the TX-RX coils. Therefore, the magnetic coupling profile along the vehicle trajectory must be considered to achieve optimal performances. This is the real core of the WPT-DBC modeling and design discussed in this paper. In particular, we aim to achieve the system-level optimization of a WPT-DBC based on the optimal matching of coil pair mutual coupling profile and power conversion control setup. First, the paper provides the complete analytical model of coil pair and control system referring to a SS-WPT-DBC. Then, a novel system-level design approach is proposed, to effectively match coils characteristics and power electronics control setup on energy and efficiency performances. The proposed design approach is valid for any WPT-DBC architecture and is scalable, as the analysis and design equations are derived by using dimensionless factors, parameters and metrics. The paper is organized as follows. In Section 2, the unified analytical model of a SS-WPT-DBC is presented. In Section 3, a method allowing the set-up of optimal inverter and postregulator controls and mutual inductance profile is proposed. In Section 4, a real SS-WPT-DBC prototype is considered to validate the model and design method presented in the paper. The results of PSIM simulations confirm the theoretical model predictions.

II. WPT-DBC ANALYSIS
The block schematic of a dynamic wireless power transfer system is shown in Fig. 1. The inverter stagein its multiple configurations, e.g. voltage fed single-phase or multi-phase [26][27], current fed push-pull [28], etc.normally uses duty-cycle and/or phase-shift control actions, to realize the transmitter regulation. The receiver side is typically made of a full-bridge diode rectifier [20], with a DC-DC converter (buck, boost or buck-boost) with duty-cycle control connected off its output, followed by a possible battery management system [29]. In this paper, the SS-WPT-DBC shown in Fig. 2 is considered as reference case study to illustrate the proposed system-level modeling and design approach for WPT-DBC systems. Nevertheless, with no loss of generality, the same modeling and design approach can be applied to different compensation topologies and conversion architectures. where Ī 1 and Ī 2 are the currents in the TX and in the RX, respectively,  o is the chosen resonance angular frequency, M is the mutual coupling, R ac = V 2 /I 2 = 8 R L / 2 is the equivalent FHA resistance at rectifier input, and R L = V rect /I rect is the equivalent load resistance at the boost converter input [25]. Solving (1) yields the nominal amplitude of the currents on the two side of the system: where R 12 = 1 2 RR and the following dimensionless factors have been introduced: (6) D i and D p being the PWM inverter and boost converter dutycycles. From (2), the inverter input current and the load output current are then determined as given in (7): Equations (7) highlight that the DC input and output currents I in and I dc , which determine the power rating and the efficiency of the WPT-BDC, depend on the dimensionless factors given in the square brackets. These factors are determined by the normalized squared mutual inductance m, the inverter and post-regulator conversion ratios n i and n p , and the normalized equivalent load resistance r ac seen at the input of the rectifier stage. The vehicle trajectory and speed determine the mutual inductance time evolution m (x t , y t ), where (x t , y t ) are the coordinates at the time t of the trajectory followed by the RX coil mounted on-board the vehicle, moving with respect to the TX coil fixed to the ground. The inverter and post-regulator converter duty cycles are determined by the relevant controls, which are normally set to regulate the TX current at a given reference value I 1,ref and the DC load at the diode rectifier output at a given reference value r ac,ref , respectively. Accordingly, (7.a) and where the coefficients υ in and υ dc are given in (9) VOLUME XX, 2017 9 The inverter and post regulator converter duty cycle must fulfill the constraints  with n imin and n imax given by (5) for D i = D imin and D i = D imax , and n pmin and n pmax given by (6) for D p = D pmax and D p = D pmin , respectively. During the vehicle transit over the transmitter coil, the value of m varies from a minimum value (usually close to 0) to a maximum value (named as peak value, m pk ). In theory, according to (10)(11), there can be up to nine different possible operating conditions for the inverter and post regulator converter stages, depending on the values of m imin , m imax , m pmin and m pmax . In reality, some of these conditions never occur. In fact, the post-regulator can operate at the maximum value of the duty-cycle (D pmax ≈ 1) with consequent m pmin ≈ 0, according to (13.a). This condition allows an easier detection of the vehicle (RX coil) while approaching the TX coil, as shown in [25]. Moreover, the saturation down to D pmin of the post-regulator duty cycle is quite unlikely, as m pmax is normally very high, according to (13.b). Therefore, it makes sense to consider the inverter duty cycle saturation only, with the post-regulator duty cycle D p valid for m pmin  m p  m pmax . Based on (7) and (10), we get: Additionally, based on (4) and (6) (17) This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  (18) The charge absorbed by the system and the charge delivered to the battery during the vehicle transition over the TX coils are given by (19): (19.b) where t = t 1 -t 0 is the time interval the RX coil takes to transit over the TX coil (at constant speed), covering the distance between the start and the end points of the trajectory, where m ≈ 0. Based on (14)(15), we have:  (21.c) being t 0 < t the duration of the time interval during which m imin m  m imax (no inverter duty cycle saturation). Equations (20) provide the basis to investigate two different kind of problems: #1 optimize the WPT-DBC performances, given the operating conditions and the mutual inductance; #2 optimize the mutual inductance, given the operating conditions and the required WPT-DBC performances. Next Section discusses these design optimization goals.

A. INVERTER AND POST-REGULATOR OPTIMIZATION
Let us start by investigating the problem #1, by searching the values of r ac,ref and I 1,ref that can optimize the charge and efficiency performances, given the operating conditions and the normalized squared mutual inductance. The first key point to be clarified is the impact of the inverter duty cycle saturation, which involves the presence of the integrals  m1 and  m2 in (20). The values m imin and m imax that always allow to prevent the inverter duty cycle saturation are given by (22) If it is possible to set m imin and m imax so that the inverter duty cycle never saturates, then from (20) we get: where m av =  m0 /t 0 is the average value of m over the vehicle trajectory. Accordingly, (24) gives the resulting energy efficiency: It can not be excluded that accepting the occurrence of the inverter duty cycle saturation may provide some advantage, in terms of increased charge or increased efficiency compared to the non-saturating operation. However, the assessment of this eventuality is not easy to be analytically done, due to the integrals  m1 and  m2 . For this analysis, numerical computations are required, as it will be shown in Section IV.

B. MUTUAL INDUCTANCE OPTIMIZATION
Let us now consider the problem #2, by defining a criterion to determine the optimal values of m pk and m av ensuring the desired efficiency and charge performances. In particular, if we assume non-saturating inverter duty cycle operation, then we can analyze the battery charge and the energy efficiency when r ac,ref = r ac,opt and I 1,ref = I 1,opt . From (23) and (24) (28) Having m pk = K m av with K > 1, (27) and (28) highlight that increasing m av reduces the charge and increases the efficiency, while increasing m pk reduces the charge. Therefore, some other elements are needed to identify the best combination of m pk and m av values. A desirable goal is to limit the winding losses, especially in the RX coil. In fact, as the RX coil is on board the vehicle, it must be as small as possible and as light as possible. The size and weight of the RX coil are determined by the winding, the casing and the heatsink. This last can be quite large and heavy, if the RX coil power dissipation is high. Let (29) The active power dissipated by the RX coil, instead, is not constant during the time interval t. Therefore, we must calculate the average value of the active power over this interval based on (2.b), thus obtaining (30): (30) Equations (28), (29) and (30) provide the bases for the coil pair design, oriented at identifying the combination of m pk and m av allowing to: a) keep the RX coil power losses under a given value; b) ensure an efficiency above a given value; c) provide a charge above a given value.
For example, based on goal a), let us suppose that the RX coil dissipation must be limited within a threshold value P R2max . Based on (30), we must fulfill the condition (31): (31) where  = P R2max /P 0 . To keep the RX coil power losses under a given threshold, as suggested by goal a), solving (31) with respect to m pk yields (32): av pk m m m m (32) If the efficiency target is a given value , as suggested by goal b), then from (28)   where v =L/t is the constant speed of the vehicle, covering the distance L between the start and the end points of the trajectory.

C. OVERALL SYSTEM OPTIMIZATION
Based on the equations discussed in the previous two subsections, given the input voltage V in , the output voltage V dc , the TX coil resistance R 1 , a procedure for optimal RX coil design can be formulated as follows: step#1. calculate the minimum average value m avmin using (34), based on the required minimum energy efficiency ; step#2. calculate the minimum peak value m pkmin using (32), based on the maximum RX coil power dissipation P R2max ; step#3. select m av and m pk just above m avmin and m pkmin ; step#4. evaluate the maximum vehicle speed v max using (35), based on the required minimum charge Q dcmin ; step#5. optimize the control based on (26); step#6. design the RX coil so that the mutual inductance M and resistance R 2 fulfill the reference values of m pk and m av . In step#3, m av and m pk must be as close as possible to m avmin and m pkmin to allow higher vehicle speed according to (35).

A. SYSTEM ANALYSIS
The WPT charger configuration and operating conditions adopted in this case study are referred to a real SS-WPT-DBC investigated in the frame of the European H2020-EMPIR MICEV project [30][31], characterized by the values of the parameters listed in Table I. The details of inverter and post regulator implementation are described [25]. The coil pair of this WPT-DBC is shown in Fig. 3. Under low-speed conditions and in case of no lateral displacement (y t = 0), it has been proved that the analytical behavioral model of the mutual inductance of this coil pair is given by (36), as discussed in detail in [32] where M is the mutual inductance in μH, x t is the longitudinal displacement between the center of the RX and TX coil at the time instant t, in meters, and the coefficients p 0 , … , p 6 are listed in Table II.  According to (36), Fig.4 (a) shows the plot of the normalized squared mutual inductance factor m while the vehicle is moving (vehicle speed = 4 km/h) along the trajectory corresponding to the longitudinal symmetry axis of the TX coil, represented by the black arrow in Fig. 4(b). From (36), we can evaluate the peak value of the normalized squared mutual inductance as given in (37) where M 0 is the value of the mutual inductance when the centers of TX and RX coils are perfectly aligned. From (37), based on coefficients shown in Table II (27), the charge delivered to the battery is 36.6 C.
Let us now analyze the impact of the control setup. The WPT-DBC model based on (10)(14)(15)(17)(20)(21)   The plots of Fig. 8 compare the inverter and post regulator duty cycles and the inverter input current and the post regulator output current for r ac,ref = r ac,opt and r ac,ref = 0.85 r ac,opt . These numerical results show that there can be an incremental charge benefit while accepting inverter duty-cycle saturation to D imax , with almost negligible efficiency degradation. Unfortunately, the optimal r ac,ref setting determining the inverter duty cycle saturation, which provides the maximum charge increase, is not analytically predictable.
The predictions of WPT-DBC operation based on the model previously implemented in MATLAB have been further validated by means of PSIM numerical simulations, based on the schematic shown in Fig. 9. The instant value of M is generated by a C-block that calculates the mutual inductance as a function of the instant position x of the RX coil center, according to (37), given by a generator emulating the constant speed transit of the vehicle. The plots in Fig. 10 show the results of PSIM simulations for r ac,ref = r ac,opt . The inverter duty cycle and current (blue curves) and the post regulator duty cycle and current (red curve) have been compared to the corresponding theoretical predictions (black dotted curves), obtained by implementing the model previously discussed in a MATLAB routine. The PSIM simulations results clearly confirm the MATLAB theoretical model.

B. SYSTEM DESIGN
The purpose of this second case study is to discuss the system-level performance optimization. The physical design of the RX coil given the desired values of m av and m pk is not the goal of this case study analysis. Nevertheless, it is worth to point out that a moderate change of the m av and m pk values can be practically achieved by accommodating different ferrite pads on the TX and RX coils without changing the coil pair windings.
Let us suppose that our target is to achieve  min = 87%, Q dcmin = 50 C and P R2max = 500 W. From (32), (34) and (35), we get m avmin = 206, m pkmin = 311, and v max = 0.7775 m/s ( 2.8 km/h), respectively. The RX coil must be modified to achieve the new values of m av  m avmin and m pk  m pkmin . The coefficients p 0 , … , p 6 listed in Table III correspond to an RX coil providing m av = 206 and m pk = 311. Fig. 11 shows the modified shape of m along the nominal trajectory of the RX coil over the TX coil. By using these values of m av and m pk , from (26) we get I 1,opt = 38.5 A and r ac,opt = 14.3. The plots of Fig. 12 provide the MATLAB results of the inverter and post-regulator duty cycle values, and of the input and output current for the given values of r ac,opt and I 1,opt . Finally, the plots of Fig. 13 show the input and output energy and charge, thus confirming that the minimum required charge (Q dc = 50.7 C>Q dcmin ) and the minimum required efficiency ( = 87%) have been achieved.

V. CONCLUSION
A system-level approach is required in order to match the control parameters with the coil pair characteristics, in order to fulfill the efficiency, power dissipation and charge requirements of a Wireless Power Transfer Dynamic Battery Charger (WPT-DBC) for electric vehicles. To this purpose, a novel dimensionless model and a design method have been presented in this paper, providing guidelines for optimal inverter and post regulator controls setup in a series-series WPT-DBC. The analysis and design equations presented in the paper are scalable, since they are derived for dimensionless setup factors, physical parameters and performance metrics, and based on scale factors depending on the input voltage and the output voltage. The architecture of power converters and relevant control of a real seriesseries WPT-DBC prototype, previously discussed in the literature, have been considered to validate the theoretical predictions provided by the method presented in the paper.