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Behavioral modeling of Wireless Power Transfer System coils
Journal Pre-proof Behavioral modeling of Wireless Power Transfer System coils Kateryna Stoyka, Gennaro Di Mambro, Nicola Femia, Antonio Maffucci, Salvatore Ventre, Fabio Villone
PII: DOI: Reference:
S0378-4754(20)30005-7 https://doi.org/10.1016/j.matcom.2020.01.004 MATCOM 4928
To appear in:
Mathematics and Computers in Simulation
Received date : 15 October 2019 Revised date : 20 December 2019 Accepted date : 13 January 2020 Please cite this article as: K. Stoyka, G. Di Mambro, N. Femia et al., Behavioral modeling of Wireless Power Transfer System coils, Mathematics and Computers in Simulation (2020), doi: https://doi.org/10.1016/j.matcom.2020.01.004. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2020 International Association for Mathematics and Computers in Simulation (IMACS). ⃝ Published by Elsevier B.V. All rights reserved.
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Behavioral Modeling of Wireless Power Transfer System Coils Kateryna Stoyka1, Gennaro Di Mambro2, Nicola Femia1, Antonio Maffucci2, Salvatore Ventre2, Fabio Villone3 1
DIEM, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy
2
DIEI, University of Cassino and Southern Lazio, Via G. Di Biasio 43, 03043 Cassino (FR), Italy
3
DIETI, University of Naples Federico II, Via Claudio 21, 80125 Napoli, Italy
Corresponding author: Kateryna Stoyka, [email protected] Highlights
Wireless power transfer system coils were modeled.
An analytical model was identified to describe the mutual inductance of the coils.
A multi-objective genetic programming algorithm was used to identify the model.
The model is able to easily analyze the effects of the coils misalignment on the mutual inductance.
Abstract
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This paper proposes a technique to derive behavioral models for describing the mutual inductance between the coupled coils used in Wireless Power Transfer Systems for the electrical recharging of vehicles. These models describe analytically the dependence of the mutual inductance with respect to geometrical parameters related to the coils misalignments, to take into account the real operating conditions of such recharging systems. A Multi-Objective Genetic Programming (MOGP) algorithm has been adopted to discover behavioral models offering optimal trade-off between accuracy and complexity. The behavioral models are identified from a set of data evaluated by using literature analytical models and are then validated by using another set of such data and also by comparing the results with full 3D Finite Element numerical simulations. Keywords
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Coupled coils; behavioral modeling; electrical mobility; genetic programming; inductive power transfer. 1. Introduction
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Wireless Power Transfer (WPT) is an emerging technology for battery recharging that nowadays is undergoing intense investigations, given its potential breakthrough impact on the electrical mobility [1][2]. Being such a technology based on the inductive coupling, the derivation of suitable models of the coupled coils is essential in view of a reliable design and an effective performance analysis of the overall WPT System (WPTS) [3], including the sensitivity analysis with respect to variations of main operating parameters and coils mutual coupling [4]. Accurate and efficient calculations of the coupled coils are in particular needed to develop reliable techniques for measuring the power transferred on board the vehicle during the inductive charging and evaluate the compliance to human exposure standards, which are the primary goals of the H2020EMPIR project “Metrology for Inductive Charging of Electric Vehicles” (MICEV) [5]-[7].
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The mutual inductance, M, is a key parameter influencing the performance of the WPTs, in terms of efficiency, power transfer, harmonic distortion and so on [2][3]. Therefore the design and the optimization of such systems require the mapping of M in the range of variability of the geometrical and physical parameters. In static recharging systems, for instance, M can be expressed as function of the misalignment of the electrical vehicle with respect to the nominal position, and as function of the ferromagnetic and conducting properties of the materials. In dynamic systems, M should be also mapped in the different points of the vehicle trajectory. The above requests may be easily addressed if an analytical solution can be found for M, expressed in terms of the above geometrical and physical parameters. Unfortunately, analytical models (e.g., based on the classical Biot-Savart law [8]) or semi-analytical ones (e.g., based on Bessel and Struve functions [9] or Heuman’s lambda function [10]) are only available when simple coupling pair structures are analyzed. Instead, when studying real world WPTS, in all the cases of practical interest the coil pair systems are characterized by the presence of complicated 3D geometries including metallic parts (e.g., the vehicle chassis) and ferrite beds for controlling the magnetic flux lines. As a consequence, analytical or semi-analytical expressions of M are not available, and its evaluation requires experimental characterizations and/or numerical full 3D solutions of a magneto-quasi-static (MQS) problem [11]. Finite element solutions of the MQS problem are the most common approach used nowadays for studying these coil systems. For example, this approach is adopted in [12] to analyze different values of coils lateral misalignment and tilt angle, in [13] to optimize the ferrite arrangements and determine the coil turn number for multi-turn spiral coils, and in [14], where a parametric performance evaluation of different resonant coil designs have been performed for different values of axial, longitudinal and
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lateral misalignments of the coils. In all these cases, the mapping is provided by look-up tables of points obtained via full numerical simulation, possibly refined by means of a numerical interpolation of the look-up table values. In this paper, we propose a way to combine the accuracy of the full numerical models to the features of the analytical solutions, by deriving suitable behavioral models, widely adopted in computational electromagnetics. Specifically we propose a technique to identify analytical behavioral models for the WPTS coils mutual inductance, as a function of the coil mismatch, expressed in terms of vertical and horizontal shifts and of the reciprocal rotation. The models are obtained by means of a Multi-Objective Genetic Programming (MOGP) algorithm [15], by imposing trade-off conditions on the metrics associated to all the possible solutions provided by such an algorithm. This evolutionary algorithm was previously used for discovering power loss behavioral models of IGBTs, power inductors and inverter power modules [16]-[18]. An alternative way could be the use of Artificial Neural Networks (ANNs) or Support Vector Machines (SVMs) [19], but both do not provide compact behavioral model expressions and have not been investigated in this study. The paper is organized as follows. In Section 2, a WPTS coils pair studied in MICEV project is briefly presented, and the problem is formulated. In Section 3, two alternative behavioral models are derived expressing the dependence of the mutual inductance on the axial and lateral misalignments of the two coils and on their reciprocal rotation angle. Finally, the results are discussed in Section 4. 2. WPTS coils and problem formulation
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The coils analyzed here are taken from one of the WPTSs studied in MICEV project [20]. The geometry is shown in Fig. 1, highlighting the surfaces associated to the transmitting (TX) and receiving (RX) coils. The model also includes an aluminum structure (which simulates the vehicle chassis) and some ferrite parts. Their main parameters are summarized in Tab. 1, while Tab. 2 lists the measured values of the inductances [20] along with the simulated ones, obtained by means of a full numerical solution of a 3D Magneto-Quasi-Static (MQS) solver (CARIDDI code, [21]). (b)
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x
x
z
z
0.5
1 (m)
y 0
0.5
1 (m)
Fig. 1 The analyzed WPTS coils, with (a) the transmitting coil and (b) the receiver coil highlighted in magenta.
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Tab. 1 Parameters for the analyzed WPTS coils
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y
Parameter
Value
Frequency
85 kHz
TX and RX coil: wire diameter
6.0 mm
TX and RX coil: number of turns
10
TX coil: inner dimensions
1.5m x 0.5m
RX coil: inner dimensions
0.3m x 0.5m
Vertical distance between coils, Δx
20 cm
Chassis conductivity
22.4 MS/m
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Tab. 2 Computed and measured self and mutual inductance values for the WPTS Inductance
Computed (µH)
Measured (µH)
Self inductance (TX)
278.6
281.4
Self inductance (RX)
118.3
119.8
Mutual inductance, M
17.6
18.2
As previously pointed out, the final goal of this paper is the study of the sensitivity of the mutual inductance M with respect to the reciprocal position between the coils. In particular, we consider different values of the vertical distance, Δx (nominal value Δx = 20cm, see Tab. 1), and a misalignment in the horizontal plane (y, z), described by a displacement (Δy, Δz) and a rotation α, as shown in Fig. 2. The nominal position is given by Δy = Δz = 0cm, α = 0°. As an example, Fig. 3 shows the behavior of the mutual inductance M as a function of Δx, where all the other parameters are set to the nominal values. The meshing requires about 50.000 elements, and the solution of the magneto-static problem for any single combination of parameter values takes about 1.5 hours on a parallel cluster of 25 processors. This makes clear the need of using behavioral models. (b)
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Fig. 2 WPTS coils in: (a) nominal position; (b) misaligned position. The reciprocal position of the coils is given by the coordinates {Δx, Δy, Δz} of the RX coil center point PRX with respect to the TX coil center point PTX (located in the origin), and a by rotation angle .
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Fig. 3 Mutual inductance for the WPTS coils as a function of the vertical distance, Δx, in nominal conditions for the other parameters. The inductance is normalized to its nominal value 17.6 µH (referred to Δx = 20cm).
3. The behavioral model 3.1 Data set generation
In order to build a reliable behavioral model, large data sets are needed for the identification and validation phases, covering the range of interest of the displacement variables Δx, Δz and α with adequate resolution. Let us note that no longitudinal misalignment (Δy) has been considered for the analysis, as only the static WPT process has been simulated (no
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movement of the vehicle along the charging lane). As pointed out, the use of 3D solvers like CARIDDI would require long computation times to generate the required data sets. Thus, since the goal of this paper is that of assessing a technique to derive behavioral models, hereafter, for the sake of simplicity, we will study a simplified system where only the coils are present, without chassis and ferrite beds. For a two coil system in air, the mutual inductance can be computed by the Neumann formula [22], applied to the equivalent filamentary conductors passing along the median lines of the two coils:
nn M 0 1 2 4
l1 l2
dl1 dl 2 Rij
(1)
where dl1 and dl2 are the filaments elements, Rij is the distance between them, μ0 is the vacuum permeability, and n1 and n2 are the turn numbers of the two coils. Equation (1) has been numerically solved and its numerical solutions have been successfully validated by comparing to those obtained for the simplified system by the above-mentioned full 3D code CARIDDI: Fig. 4 shows, for instance, the two solutions as functions of the rotation angle, = {0°,10°,20°,30°,40°}, with the other parameters set to the nominal values (Δx = 20cm and Δy = Δz = 0 cm). The results confirm the reliability of the use of the Neumann model for fast and accurate mutual inductance calculation of the WPTS coils. Hereafter, the data set to be used for training the Genetic Programming (GP) algorithm, as well as that to be used for the model validation, are therefore calculated by means of the Neumann formula (1) under a certain number of test conditions (presented in Section 4). 25
MCARIDDI
10 5 0
10
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0
MNeumann
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20
20
30
40
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[°] Fig. 4 Mutual inductance versus the rotation angle: numerical solution of (1) compared to the reference solution (CARIDDI code).
3.2 Model generation via Genetic Programming
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The GP is a special evolutionary algorithm in which the population is composed of models. During its evolution, the GP algorithm transforms the current population of models into a new population of models by applying classical genetic operations, such as selection, cross-over, mutation, etc. The GP models are generally represented by means of tree structures. To construct such trees, the GP algorithm considers a non-terminal set of functions (sum, multiplication, division, logarithm, arctangent, hyperbolic tangent, sine, exponential and power function) and a terminal set of constant coefficients and input variables. Complexity factors (cf) are usually assigned to all the elements included in these sets. The following values have been herein assumed for the elements of the terminal set: cf = 0.6 for the multiplication of input variables and cf = 1 for all the other operations between input variables and for constant coefficients. Instead, the following values have been considered for the non-terminal set: cf = 1 for sum and/or multiplication of elementary functions, cf = 1.5 for all the other functions. To quantify the global complexity of each GP model, a term Fcomplexity has been introduced and calculated as follows: if a function is the argument of another function, the complexity factors cf of the two functions are multiplied;
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if two functions are multiplied or summed, their complexity factors cf are summed and subsequently multiplied by the complexity factor of a sum or a product, respectively.
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In this paper, the GP goal is the discovery of behavioral models of the mutual inductance as a function of axial (Δx) and lateral (Δz) misalignments and of the rotation angle of the WPTS coils. In order to achieve a kind of separation of the effects of the three geometric variables Δx, Δz and , two variables are selected as the model inputs, whereas the remaining variable is used as the parameter influencing the values of the behavioral model coefficients. Starting from this assumption, two different approaches for the behavioral model Mbehav formulation have been investigated:
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Approach 1: Mbehav(Δx,Δz,p()) is given as a function of Δx and Δz and is parametrized with respect to , where p() is a vector of model coefficients dependent on ;
Approach 2: Mbehav(Δz,,p(Δx)) is given as a function of Δz and and is parametrized with respect to Δx, where p(Δx) is a vector of model coefficients dependent on Δx. Let us note that the scope of investigating the above two approaches consists in the discovery of a compact behavioral model formulation as a function of two geometric variables, whose coefficients are given as functions of the third variable, used as model parameter. To such a scope, the above two approaches have been tested in parallel, so as to discover which of them provides better behavioral models from the point of view of accuracy, complexity and smoothness of the resulting model coefficients. For a more detailed description of the behavioral model identification by the GP algorithm, let us refer to Approach 1, but the same considerations hold also for Approach 2 by interchanging variables Δx and in Mbehav formulation. Let us consider a set of m rotation angle values j, j = 1,…,m, used as the behavioral model parameter. For each j value, n couples of axial and lateral misalignments (Δxi, Δzi) have been analyzed, with i = 1,…,n. For each of the n×m test conditions, a data vector has been created, including the test values (Δxi, Δzi) and the corresponding mutual inductance value Mij=M(Δxi,Δzi,j), evaluated by means of the Neumann formula (1). Such n×m data vectors compose the training data set T of the GP algorithm. The GP algorithm has to identify the behavioral model Mbehav(Δx,Δz,p()), such that the value of the function Mbehav computed for each test condition of the training data set T is as close as possible to the corresponding training value Mij, i {1, …, n} and j {1, …, m}. The structure of the function Mbehav has to be the same for all the rotation angle values j, j = 1,…,m, while the coefficients p vary with . To determine the coefficients p for each j, a Non-Linear Least Squares (NLLS) method, based on Levenberg-Marquardt optimization [23], has been applied to the n experimental data vectors relevant to j. Thus, the values of coefficients p have been determined by minimizing the χ-squared error, evaluated between the training mutual inductance values Mij and the GP-predicted values Mbehav(Δxi,Δzi,p(j)), for i = 1,…,n:
M behav xi , zi , p( j ) M ij M ij i 1
2
n
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1n 2 j
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(2)
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Then, the interpolating functions p(j) have been determined, as will be discussed in the next Section. To estimate the behavioral model accuracy, the Root Mean Square Error (RMSE) over the whole training data set has been evaluated according to (3): m
1 RMSE m 2j
(3)
j 1
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Eventually, a validation data set V has been created, including test conditions of Δx, Δz and different from the training set conditions, to verify the models generalization capability. A MOGP algorithm has been adopted to discover models offering an optimal trade-off between accuracy (RMSE) and complexity (Fcomplexity). In this paper, an elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) has been used [24], which searches for the models jointly minimizing RMSE and Fcomplexity values. Multiple runs of the MOGP algorithm have been executed, to discover the most repeatable GP models. Specifically, each model generated by the GP has been scored on the basis of the following metrics: - Nrun – the number of runs during which the algorithm has discovered the model; - Ngen – the average number of generations during which the model survived in the population; - {μerr, σerr, errmax} – the mean value, standard deviation and maximum value of the model percent error over the training data set T; - errVmax – the maximum value of the model percent error over the validation data set V; - Nmon – the average number of intervals over which the model coefficients p change their monotonicity with respect to the selected model parameter ( for Approach 1, Δx for Approach 2). The best models are those ones maximizing the first two metrics and minimizing the last five metrics. In general, the use of simpler behavioral models with a lower value of the Nmon metric results in more regular coefficients p, which can be more easily approximated as functions of the selected model parameter.
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4. Results and discussion 4.1 Approach 1 The coils misalignment parameters chosen to assemble the training data set T of the GP algorithm are listed in the left-hand side of Tab. 3: all the combinations of such values have been simulated, resulting in 300 test conditions. The right-hand side of Tab. 3 lists the misalignment parameters describing the validation data set V adopted in Approach 1, consisting of 120 test conditions. It should be noted that the values of Δx and Δz, chosen for the validation data set, are different from the values adopted for the training data set. Tab. 3 Coils misalignment parameters adopted for training and validation data set in Approach 1 Parameter
Training Values
Validation Values
Δx
{15, 20, 25, 30, 35} cm
{18, 23, 28, 33} cm
Δz
{0, 15, 25, 28, 30, 35} cm
{12, 22, 32} cm
{0, 20, 40, 60, 80, 100, 120, 140, 160, 180}°
α
{0, 20, 40, 60, 80, 100, 120, 140, 160, 180}°
On the contrary, the values of have been maintained unchanged, so as to use the same model coefficient values p(), discovered during the models identification, also to validate the models.
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The MOGP algorithm has been executed over 100 runs, with a population of 500 models evolving over 300 generations for each run. Only the non-dominated Pareto-optimal solutions having the repeatability of at least 10 runs and the maximum percent error lower than 15% have been considered for further comparison. Such solutions are represented as blue markers in Fig. 5 and labelled with increasing solution numbers. The related model expressions are provided in Tab. 4, while the respective values for the metrics {Nrun, Ngen, μerr, σerr, errmax, errVmax , Nmon} are given in Tab. 5. Let us note that Fig. 5 contains several model solutions discovered by the GP algorithm during certain runs, which are dominated by some other solutions discovered in the other independent runs. Since such models still present high repeatability and good error metrics, they have also been considered for the final comparison. In particular, models #2 and #3 have been discovered by the GP algorithm under two different formulations, which correspond to the same equivalent model expression, thus increasing the overall model repeatability. In principle, model #6 dominates models #2 and #3 in terms of better RMSE and Fcomplexity values, but has lower values of Nrun and Ngen metrics. The other models have a lower repeatability, or higher Nmon values (i.e., more irregular coefficient trends), or both. Hence models #2 and #3, reformulated as given in (4), present a good trade-off among all the metrics and has been therefore selected as an optimal behavioral model of the mutual inductance for Approach 1:
M behav p 0 z 1.5 exp p1x p 2 exp p 3 x
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(4)
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where Δx and Δz are expressed in meters, while Mbehav is given in μH. Model (4) is characterized by low percent errors evaluated on the training dataset T, with μerr ≈ -0.04%, σerr ≈ 2%, errmax ≈ 6.7%. Also, the maximum percent error evaluated on the validation dataset V is limited to errVmax ≈ 4.5%, thus confirming the model generalization capability. Moreover, for all the models of Tab. 5, the maximum errors evaluated on the validation data set are lower than the respective training set errors, thus guaranteeing that the models do not overfit the assigned data. 6
#1
5.5
#2 #3
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#4 #5
#6
#7
4.5
4 0.01
#9
#10
0.015
0.02
0.025
#8
0.03
0.035
RMSE
Fig. 5 Repeatable Pareto-optimal solutions obtained with Approach 1.
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Tab. 4 GP model expressions obtained with Approach 1 Model
Expression
#1
p0 exp p1x z exp p2 z p3 p4 exp p5 x
#2
p0 exp p1x p2 z p3z exp p4 x p0 z z exp p1x p 2 exp p3 x
#3
p0 exp p1x p2 tanh log p3z x
#4
p0 exp p1z p2 x p3x p4
#5 #6
p0 exp p1x p2 tanh p3z 2 exp p4 x
#7
p0 z 2 p1x p2 p3 z exp p4 x
#8
p0 z atan p1z x p2 x p3
#9
p0 x p1 p2 z atan p3 z x
#10
p0 exp p1x p2 p3z
exp p5x
Tab. 5 Metrics values for GP models obtained with Approach 1 Nrun
Ngen
μerr [%]
σerr [%]
errmax [%]
errVmax [%]
Nmon
#1
23
138
-0.01
1.1
4.4
2.2
2.7
#2
15
188
-0.04
2
6.7
4.5
2.5
#3
34
185
-0.04
2
6.7
4.5
2.4
#4
10
238
-0.09
3.1
10.3
6.4
2.5
#5
13
127
-0.1
3.2
10
7.4
4.8
#6
19
90
-0.02
1.3
6.3
2.9
2.4
#7
11
212
-0.09
3
11.3
7.6
2.4
#8
10
159
-0.1
#9
14
98
-0.1
#10
12
110
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Model
-0.06
3.2
10.6
6.2
3
3.2
10.6
6.2
3
2.4
7.6
3.9
3.3
The coefficients pi, i = 0,1,2,3, identified by the GP algorithm for model (4), are shown in Fig. 6 (red circle markers), and can be approximated as cosine functions of α (green curves in Fig. 6):
p i a i ,0 cos 2 a i ,1 , i 0,1, 2,3
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(5)
The values of fitting coefficients ai,0 and ai,1 are provided in Tab. 6.
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Tab. 6 Fitting coefficient values for the behavioral model (4) Coefficient ai,0 ai,1 p0
-42.55e+00
186.77e+00
p1
-322.41e-03
4.94e+00
p2
-4.84e+00
36.66e+00
p3
-221.98e-03
3.37e+00
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240 220
p1
200 180 160 140
0
60
[°]
120
180
42
3.6
36
p3
p2
39
33 30
3.4 3.2
0
60
[°]
120
180
0
60
[°]
120
180
Fig. 6 Behavioral model (4) coefficients pi vs α, i = 0,1,2,3 (red circle markers) and relative cosine fitting curves (green lines).
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In Fig. 7(a) the values of M calculated by using the Neumann formula (blue markers) are compared to the values predicted by the behavioral model (4) over the 300 test conditions of the training data set. In particular, red markers depict the values of M predicted by using model (4) with the original pi coefficients identified by the GP algorithm, whereas green markers represent the values of M predicted by using the cosine approximations (5) of such coefficients.
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Fig. 7(b) depicts the relative percent errors of model (4) with respect to the training set data. It is worth remarking that, even though the function (5) does not fit perfectly some of the coefficients pi (e.g., p1 and p3), the overall impact of such an approximation on the predictions of M by the behavioral model (4) is minor. Indeed, the behavioral model ensures a -0.04% average error with 2% standard deviation, which is sufficiently accurate for fast numerical simulations to analyze the impact of coils misalignment on the overall WPTS performances.
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(a)
(b) 10
Errbehav
5
Errbehav,fit
0 -5 -10
0
50
100
150
200
250
300
Test #
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Fig. 7. Behavioral model predictions: (a) mutual inductance values obtained by using the Neumann formula (blue markers), the behavioral model (4) with the original pi coefficients (red markers) and the behavioral model (4) with the cosine approximations of pi; (b) relative percent errors.
4.2 Approach 2.
The left-hand side of Tab. 7 lists the coils misalignment parameters chosen to assemble the training data set T (the same as in Approach 1), while the right-hand side of Tab. 7 lists the misalignment parameters describing the validation data set V adopted in Approach 2, consisting of 75 test conditions. It should be noted that the values of Δz and α, chosen for the validation data set, are different from the values adopted for the training data set. On the contrary, the values of Δx have been maintained unchanged, so as to use the same model coefficient values p(Δx), discovered during the models identification, also to validate the models.
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Tab. 7 Coils misalignment parameters adopted for training and validation data set in Approach 2 Parameter
Training Values
Validation Values
Δx
{15, 20, 25, 30, 35} cm
{15, 20, 25, 30, 35} cm
Δz
{0, 15, 25, 28, 30, 35} cm
{12, 22, 32} cm
α
{0, 20, 40, 60, 80, 100, 120, 140, 160, 180}°
{10, 50, 90, 130, 170}°
The MOGP algorithm has been executed over 50 runs, with a population of 500 models evolving over 300 generations for each run. Only the non-dominated Pareto-optimal solutions with metrics Nrun ≥ 9 and errmax ≤ 15% have been considered for further comparison. Such solutions are represented as blue markers in Fig. 8 and labelled with increasing solution numbers. The related model expressions are provided in Tab. 8, while the respective values for the metrics {Nrun, Ngen, μerr, σerr, errmax, errVmax , Nmon} are given in Tab. 9. Once again, models labelled as #2 and #3 have been discovered by the GP algorithm under two slightly different formulations, which correspond to the same equivalent model expression given in (6): M behav p 0 exp p1z 2 p 2 p3 z 2 sin 2
(6)
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where Δz is expressed in meters, while Mbehav is given in μH. Model (6) is characterized by low percent errors, with μerr ≈ - 0.01%, σerr ≈ 1.2%, errmax ≈ 6.2% and errVmax ≈ 2.4%. Such model expresses the mutual inductance dependence on the angle α through a sine-squared function, which can be equivalently represented as a cosine function of the double angle 2α, thus correctly modeling the mutual inductance behavior for whatever α value. Indeed, the coils reciprocal position is the same for rotation angles α and 180°+α, resulting in the mutual inductance periodic dependence on α, characterized by a period of 180°. Also, the coils position is symmetrical for rotation angles α and 180°–α, resulting in the same value of the mutual inductance, as correctly predicted by model (6). Hence such model has been herein selected as an optimal behavioral model of mutual inductance for Approach 2.
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Fig. 8 Repeatable Pareto-optimal solutions obtained with Approach 2.
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Tab. 8 GP model expressions obtained with Approach 2 Model #1 #2 #3 #4 #5 #6 #7 #8 #9 #10
Expression
p0 exp p1z p2 sin 2 p3 p4 z 3 2
p0 exp p1z 2 p2 sin 2 p3 p4 z 2 p0 exp p1z 2 p2 sin 2 p3 z 2
p0 exp p1z 2 p2 sin z 2 p3 exp z
p0 exp p1z 2 p2 sin p3 p4 z 2
p0 exp p1z 2 p2 sin z sin p3z p0 p1 sin sin p2 z p3 z
p0 tanh log p1z p2 z 2 sin
p0 exp p1z 2 p2 sin sin p3z
p0 z sin p1 tanh log p2 z
Nrun
Ngen
μerr [%]
σerr [%]
errmax [%]
errVmax [%]
Nmon
#1
9
108
-0.01
1.2
5.1
2.6
1.4
#2
9
130
-0.01
1.2
6.2
2.4
1.4
#3
9
142
-0.01
1.2
6.2
2.4
1
#4
9
99
-0.02
1.5
6.3
2.3
1
#5
28
146
-0.02
1.5
6.9
2.1
1.4
#6
9
132
-0.08
2.8
8.8
6.8
1
#7
19
147
-0.08
2.8
7.9
4.5
1
#8
16
159
-0.09
#9
16
174
#10
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Model
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Tab. 9 Metrics values for GP models obtained with Approach 2
10.4
6.6
1
4
10.9
10.1
1
-0.19
4.4
12.3
10.6
1
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2.9
-0.16
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The coefficients pi, i = 0,1,2,3, identified by the GP algorithm for model (6), are shown in Fig. 9 (red circle markers), and can be approximated as exponential functions of Δx (green curves in Fig. 9):
p i a i ,0 exp a i ,1 x a i ,2 ,
i 0,1, 2, 3
(7)
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where Δx is expressed in meters. The values of fitting coefficients ai,0, ai,1 and ai,2 are provided in Tab. 10.
p0
34.03e+00
-4.20e+00
2.71e+00
p1
15.52e+00
-3.50e+00
2.01e+00
p2
214.93e+00
-9.05e+00
3.91e+00
p3
7.23e-03
3.47e+00
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Tab. 10 Fitting coefficient values for the behavioral model (6) Coefficient ai,0 ai,1 ai,2
64.85e-03
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Fig. 9 Behavioral model (6) coefficients pi vs Δx, i = 0,1,2,3 (red circle markers) and relative fitting curves (green lines).
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(a)
(b)
M Error [%]
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Fig. 10(a) compares the values of M calculated by using the Neumann formula (blue markers) with the values predicted by the behavioral model (6) over the 300 test conditions of the training data set. In particular, red markers depict the values of M predicted by using the model (6) with the original pi coefficients identified by the GP algorithm, whereas green markers represent the values of M predicted by using the exponential approximations (7) of such coefficients. Fig. 10(b) shows the relevant percent errors, highlighting that the behavioral model (6) ensures a -0.01% average error with 1.19% standard deviation over the training data set.
Fig. 10 Behavioral model predictions: (a) mutual inductance values obtained by using the Neumann formula (blue markers), the behavioral model (6) with the original pi coefficients (red markers) and the behavioral model (6) with the exponential approximations of pi; (b) relative percent errors.
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It is worth observing that the two models (4) and (6) discovered by the GP algorithm include the mutual inductance dependence on Δx and α in a dual way. The model (4) contains exponential terms of Δx, while its parameters vary as cosine functions of the double angle 2α. The parameters of the model (6), instead, vary exponentially with Δx, while the mutual inductance dependence on α is expressed through a sine-squared function, which can be equivalently represented as a cosine function of the double angle 2α. Hence the two models (4) and (6) discovered by the GP are coherent between them, have a similar complexity (two terms and four coefficients in each model) and both present a good accuracy. The model (6) outperforms the model (4) in having smoother and quasi-linear (low-order polynomial) coefficient trends. This element can be adopted as a preferential requisite in the evaluation of possible alternative models generated by means of the MOGP approach.
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Eventually, it should be noted that Approach 1 and Approach 2 investigated in this paper are equivalent from the computational viewpoint, in terms of both time employed for the behavioral model generation and number of data points needed to assemble the training data set of the MOGP algorithm. 5. Conclusions and perspectives In this paper, two novel behavioral models of the mutual inductance for rectangular-shaped coils used in Wireless Power Transfer Systems (WPTSs) are proposed, valid for different types of coils misalignment. Such models have been discovered by means of a Multi-Objective Genetic Programming (MOGP) algorithm. One of the two behavioral models expresses the mutual inductance as a function of the axial and lateral misalignments of the coils, parametrized with respect to their rotation angle. The other model describes the mutual inductance as a function of the lateral misalignment and rotation angle, parametrized with respect to the axial misalignment. The two models are coherent between them and present comparable accuracy and complexity. Their resulting mutual inductance values are in good agreement with the literature model predictions and numerical Finite Element simulations.
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Behavioral models, like those ones presented in this paper can provide powerful tools in the design and optimization of static and dynamic WPT systems. First of all, they allow to easily identify the locus of spatial misalignments ensuring a certain value of the mutual inductance M, that is a valuable information to be used, for instance, in the control strategy of the system. In addition, a behavioral model allows an easy implementation of a parametrized equivalent circuit for the coils sub-system, to be embedded in the overall equivalent circuit describing the WPT system, along with the other sub-systems (e.g., the converters and all the other electronic equipment). In this way, it becomes much easier to perform a parametrized system-level circuit simulation, for instance during the design optimization as for the current technology trends [25].
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Conflicts of interest The authors declare no conflict of interest.
Acknowledgements
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The results here presented are developed in the framework of the 16ENG08 MICEV Project. The latter received funding from the EMPIR programme cofinanced by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. The University of Salerno partially supported this work through the project funds “Sistemi di Carica Induttiva di Veicoli Elettrici” (300638FRB17DICAPUA, 300638FRB18DICAPUA). References
Z. Bi, T. Kan, C. Mi, Y. Zhang, Z. Zhao, G. A. Keoleian, “A review of wireless power transfer for electrical vehicles: prospect to enhance sustainable mobility”, Applied Energy, vol. 179, pp. 413-425, 2016.
[2]
V. Cirimele, M. Diana, F. Freschi, M. Mitolo, “Inductive Power Transfer for Automotive Applications: State-ofthe-Art and Future Trends”, IEEE Trans. on Industry Appl., vol. 54, no. 5, pp. 4069-4079, Sep. 2018.
[3]
A. Rakhymbay et al., “Precise Analysis on Mutual Inductance Variation in Dynamic Wireless Charging of Electric Vehicle”, Energies, pp. 624, vol. 11, n. 3, 2018.
[4]
G. Di Capua, N. Femia, K. Stoyka, “Sensitivity Analysis of Inductive Power Transfer Systems for Electric Vehicles Battery Charging”, Proc. of Int. Conf. on Synthesis, Modeling, Analysis and Simulation Methods and Appl. to Circuit Design (SMACD), pp. 173-176, Lausanne, Switzerland, 2019.
[5]
M. Zucca et al., “The Project ‘Metrology for inductive charging of electric vehicles”, Proc of Conf. on Precision Electromagnetic Measurements (CPEM 2018), Paris, France, July 2018.
[6]
M. Zucca et al., “Metrology for Inductive Charging of Electric Vehicles (MICEV)”, Proc. of Int. Conf. of Electrical and Electronic Technologies for Automotive (AEIT AUTOMOTIVE), pp. 1-6, Torino, Italy, 2019.
[7]
Project “Metrology for inductive charging of electric vehicles”, www.micev.eu
[8]
R. Ravaud, G. Lemarquand, V. Lemarquand, C. Akyel, “Cylindrical Magnets and Coils: Fields, Forces, and Inductances,” IEEE Trans. on Magnetics, vol. 46, pp. 3585- 3590, Sept. 2010.
[9]
J. T. Conway, “Inductance Calculations for Noncoaxial Coils Using Bessel Functions”, IEEE Trans. on Magnetics, vol. 43, pp. 1023 - 1034, Feb 2007.
Jo u
rna
lP
[1]
Journal Pre-proof
Babic S., et al, “New Formulas for Mutual Inductance and Axial Magnetic Force Between a Thin Wall Solenoid and a Thick Circular Coil of Rectangular Cross-Section,” IEEE Trans. on Magnetics, pp. 2034 - 2044, vol. 47, Aug 2011.
[11]
A. P. Sample, A. David, D.A. Meyer, J.R. Smith, “Analysis, experimental results, and range adaptation of magnetically coupled resonators for wireless power transfer”, IEEE Transactions on Industrial Electronics, vol. 58, no. 2, pp. 544-554, Feb. 2011.
[12]
Y. Zhang, L. Wang, Y. Guo, C. Tao, “Design and Optimization of Asymmetrical Spiral Rectangular Pads for EV Wireless Charging”, 2018 IEEE 4th Southern Power Electr. Conf. (SPEC), Singapore, Dec. 2018.
[13]
M. S. Carmeli, F. Castelli-Dezza, M. Mauri, G. Foglia, “Contactless energy transmission system for electrical vehicles batteries charging”, 2015 Int. Conf. on Clean Electr. Power (ICCEP), Taormina, June 2015.
[14]
B. Olukotun, J. S. Partridge, R. W. G. Bucknall, “Optimal Finite Element Modelling and 3-D Parametric Analysis of Strong Coupled Resonant Coils for Bidirectional Wireless Power Transfer”, Proc. of 53rd International Universities Power Eng. Conference (UPEC), Glasgow, UK, Sept. 2018.
[15]
J. R. Koza, Genetic Programming: on the programming of computers by means of natural selection. Cambridge, MA, USA: MIT Press, 1992.
[16]
N. Femia, M. Migliaro, C. Pastore, D. Toledo, “In-system IGBT power loss behavioral modeling”, Proc. of Int. Conf. on Synthesis, Modeling, Analysis and Simulation Methods and Appl. to Circuit Design (SMACD), Lisbon, PT, June 2016.
[17]
K. Stoyka, G. Di Capua, N. Femia, “A novel AC power loss model for ferrite power inductors”, IEEE Transactions on Power Electronics, pp. 2680-2692, vol. 34, 2019.
[18]
K. Stoyka, R. A. Pessinatti Ohashi, N. Femia, “Behavioral switching loss modeling of inverter modules”, Proc. of Int. Conf. on Synthesis, Modeling, Analysis and Simulation Methods and Appl. to Circuit Design (SMACD), Prague, CZ, July 2018.
[19]
A. Garg, K. Tai, “Selection of a robust experimental design for the effective modeling of nonlinear systems using genetic programming”, 2013 IEEE Symposium on Computational Intelligence and Data Mining (CIDM), Singapore, Sept. 2013.
[20]
V. Cirimele, “Design and integration of a dynamic IPT system for automotive applications”, PhD Thesis, Politecnico di Torino, Italy, 2017, available at: https://iris.polito.it/handle/11583/2666564#.XXFjEC2B2t8
[21]
R. Albanese, G. Rubinacci, “Finite element methods for the solution of 3D eddy current problems,” Advances in Imaging and Electron Physics, pp. 1-86, vol. 102, 1998.
[22]
R. R. Aubakirov, A. A. Danilov, S. V. Selishchev, “Numerical modeling of inductive transcutaneous energy transfer using coils with square turns”, 2018 IEEE Conf. of Russian Young Researchers in Electrical and Electronic Eng. (EIConRus), Moscow, RU, March 2018.
[23]
W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing. New York, NY, USA: Cambridge University Press, 1992.
[24]
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II”, IEEE Trans. on Evolutionary Computation, pp. 182–197, vol. 6, 2002.
[25]
V. Cirimele, M. Diana, F. Freschi, M. Mitolo, “Inductive power transfer for automotive applications: state-of-theart and future trends”, IEEE Trans. Ind. Appl., pp. 4069-4079, vol. 54, 2018.
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[10]
PII: DOI: Reference:
S0378-4754(20)30005-7 https://doi.org/10.1016/j.matcom.2020.01.004 MATCOM 4928
To appear in:
Mathematics and Computers in Simulation
Received date : 15 October 2019 Revised date : 20 December 2019 Accepted date : 13 January 2020 Please cite this article as: K. Stoyka, G. Di Mambro, N. Femia et al., Behavioral modeling of Wireless Power Transfer System coils, Mathematics and Computers in Simulation (2020), doi: https://doi.org/10.1016/j.matcom.2020.01.004. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2020 International Association for Mathematics and Computers in Simulation (IMACS). ⃝ Published by Elsevier B.V. All rights reserved.
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Behavioral Modeling of Wireless Power Transfer System Coils Kateryna Stoyka1, Gennaro Di Mambro2, Nicola Femia1, Antonio Maffucci2, Salvatore Ventre2, Fabio Villone3 1
DIEM, University of Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano (SA), Italy
2
DIEI, University of Cassino and Southern Lazio, Via G. Di Biasio 43, 03043 Cassino (FR), Italy
3
DIETI, University of Naples Federico II, Via Claudio 21, 80125 Napoli, Italy
Corresponding author: Kateryna Stoyka, [email protected] Highlights
Wireless power transfer system coils were modeled.
An analytical model was identified to describe the mutual inductance of the coils.
A multi-objective genetic programming algorithm was used to identify the model.
The model is able to easily analyze the effects of the coils misalignment on the mutual inductance.
Abstract
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This paper proposes a technique to derive behavioral models for describing the mutual inductance between the coupled coils used in Wireless Power Transfer Systems for the electrical recharging of vehicles. These models describe analytically the dependence of the mutual inductance with respect to geometrical parameters related to the coils misalignments, to take into account the real operating conditions of such recharging systems. A Multi-Objective Genetic Programming (MOGP) algorithm has been adopted to discover behavioral models offering optimal trade-off between accuracy and complexity. The behavioral models are identified from a set of data evaluated by using literature analytical models and are then validated by using another set of such data and also by comparing the results with full 3D Finite Element numerical simulations. Keywords
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Coupled coils; behavioral modeling; electrical mobility; genetic programming; inductive power transfer. 1. Introduction
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Wireless Power Transfer (WPT) is an emerging technology for battery recharging that nowadays is undergoing intense investigations, given its potential breakthrough impact on the electrical mobility [1][2]. Being such a technology based on the inductive coupling, the derivation of suitable models of the coupled coils is essential in view of a reliable design and an effective performance analysis of the overall WPT System (WPTS) [3], including the sensitivity analysis with respect to variations of main operating parameters and coils mutual coupling [4]. Accurate and efficient calculations of the coupled coils are in particular needed to develop reliable techniques for measuring the power transferred on board the vehicle during the inductive charging and evaluate the compliance to human exposure standards, which are the primary goals of the H2020EMPIR project “Metrology for Inductive Charging of Electric Vehicles” (MICEV) [5]-[7].
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The mutual inductance, M, is a key parameter influencing the performance of the WPTs, in terms of efficiency, power transfer, harmonic distortion and so on [2][3]. Therefore the design and the optimization of such systems require the mapping of M in the range of variability of the geometrical and physical parameters. In static recharging systems, for instance, M can be expressed as function of the misalignment of the electrical vehicle with respect to the nominal position, and as function of the ferromagnetic and conducting properties of the materials. In dynamic systems, M should be also mapped in the different points of the vehicle trajectory. The above requests may be easily addressed if an analytical solution can be found for M, expressed in terms of the above geometrical and physical parameters. Unfortunately, analytical models (e.g., based on the classical Biot-Savart law [8]) or semi-analytical ones (e.g., based on Bessel and Struve functions [9] or Heuman’s lambda function [10]) are only available when simple coupling pair structures are analyzed. Instead, when studying real world WPTS, in all the cases of practical interest the coil pair systems are characterized by the presence of complicated 3D geometries including metallic parts (e.g., the vehicle chassis) and ferrite beds for controlling the magnetic flux lines. As a consequence, analytical or semi-analytical expressions of M are not available, and its evaluation requires experimental characterizations and/or numerical full 3D solutions of a magneto-quasi-static (MQS) problem [11]. Finite element solutions of the MQS problem are the most common approach used nowadays for studying these coil systems. For example, this approach is adopted in [12] to analyze different values of coils lateral misalignment and tilt angle, in [13] to optimize the ferrite arrangements and determine the coil turn number for multi-turn spiral coils, and in [14], where a parametric performance evaluation of different resonant coil designs have been performed for different values of axial, longitudinal and
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lateral misalignments of the coils. In all these cases, the mapping is provided by look-up tables of points obtained via full numerical simulation, possibly refined by means of a numerical interpolation of the look-up table values. In this paper, we propose a way to combine the accuracy of the full numerical models to the features of the analytical solutions, by deriving suitable behavioral models, widely adopted in computational electromagnetics. Specifically we propose a technique to identify analytical behavioral models for the WPTS coils mutual inductance, as a function of the coil mismatch, expressed in terms of vertical and horizontal shifts and of the reciprocal rotation. The models are obtained by means of a Multi-Objective Genetic Programming (MOGP) algorithm [15], by imposing trade-off conditions on the metrics associated to all the possible solutions provided by such an algorithm. This evolutionary algorithm was previously used for discovering power loss behavioral models of IGBTs, power inductors and inverter power modules [16]-[18]. An alternative way could be the use of Artificial Neural Networks (ANNs) or Support Vector Machines (SVMs) [19], but both do not provide compact behavioral model expressions and have not been investigated in this study. The paper is organized as follows. In Section 2, a WPTS coils pair studied in MICEV project is briefly presented, and the problem is formulated. In Section 3, two alternative behavioral models are derived expressing the dependence of the mutual inductance on the axial and lateral misalignments of the two coils and on their reciprocal rotation angle. Finally, the results are discussed in Section 4. 2. WPTS coils and problem formulation
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The coils analyzed here are taken from one of the WPTSs studied in MICEV project [20]. The geometry is shown in Fig. 1, highlighting the surfaces associated to the transmitting (TX) and receiving (RX) coils. The model also includes an aluminum structure (which simulates the vehicle chassis) and some ferrite parts. Their main parameters are summarized in Tab. 1, while Tab. 2 lists the measured values of the inductances [20] along with the simulated ones, obtained by means of a full numerical solution of a 3D Magneto-Quasi-Static (MQS) solver (CARIDDI code, [21]). (b)
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x
x
z
z
0.5
1 (m)
y 0
0.5
1 (m)
Fig. 1 The analyzed WPTS coils, with (a) the transmitting coil and (b) the receiver coil highlighted in magenta.
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Tab. 1 Parameters for the analyzed WPTS coils
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Parameter
Value
Frequency
85 kHz
TX and RX coil: wire diameter
6.0 mm
TX and RX coil: number of turns
10
TX coil: inner dimensions
1.5m x 0.5m
RX coil: inner dimensions
0.3m x 0.5m
Vertical distance between coils, Δx
20 cm
Chassis conductivity
22.4 MS/m
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Tab. 2 Computed and measured self and mutual inductance values for the WPTS Inductance
Computed (µH)
Measured (µH)
Self inductance (TX)
278.6
281.4
Self inductance (RX)
118.3
119.8
Mutual inductance, M
17.6
18.2
As previously pointed out, the final goal of this paper is the study of the sensitivity of the mutual inductance M with respect to the reciprocal position between the coils. In particular, we consider different values of the vertical distance, Δx (nominal value Δx = 20cm, see Tab. 1), and a misalignment in the horizontal plane (y, z), described by a displacement (Δy, Δz) and a rotation α, as shown in Fig. 2. The nominal position is given by Δy = Δz = 0cm, α = 0°. As an example, Fig. 3 shows the behavior of the mutual inductance M as a function of Δx, where all the other parameters are set to the nominal values. The meshing requires about 50.000 elements, and the solution of the magneto-static problem for any single combination of parameter values takes about 1.5 hours on a parallel cluster of 25 processors. This makes clear the need of using behavioral models. (b)
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Fig. 2 WPTS coils in: (a) nominal position; (b) misaligned position. The reciprocal position of the coils is given by the coordinates {Δx, Δy, Δz} of the RX coil center point PRX with respect to the TX coil center point PTX (located in the origin), and a by rotation angle .
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Fig. 3 Mutual inductance for the WPTS coils as a function of the vertical distance, Δx, in nominal conditions for the other parameters. The inductance is normalized to its nominal value 17.6 µH (referred to Δx = 20cm).
3. The behavioral model 3.1 Data set generation
In order to build a reliable behavioral model, large data sets are needed for the identification and validation phases, covering the range of interest of the displacement variables Δx, Δz and α with adequate resolution. Let us note that no longitudinal misalignment (Δy) has been considered for the analysis, as only the static WPT process has been simulated (no
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movement of the vehicle along the charging lane). As pointed out, the use of 3D solvers like CARIDDI would require long computation times to generate the required data sets. Thus, since the goal of this paper is that of assessing a technique to derive behavioral models, hereafter, for the sake of simplicity, we will study a simplified system where only the coils are present, without chassis and ferrite beds. For a two coil system in air, the mutual inductance can be computed by the Neumann formula [22], applied to the equivalent filamentary conductors passing along the median lines of the two coils:
nn M 0 1 2 4
l1 l2
dl1 dl 2 Rij
(1)
where dl1 and dl2 are the filaments elements, Rij is the distance between them, μ0 is the vacuum permeability, and n1 and n2 are the turn numbers of the two coils. Equation (1) has been numerically solved and its numerical solutions have been successfully validated by comparing to those obtained for the simplified system by the above-mentioned full 3D code CARIDDI: Fig. 4 shows, for instance, the two solutions as functions of the rotation angle, = {0°,10°,20°,30°,40°}, with the other parameters set to the nominal values (Δx = 20cm and Δy = Δz = 0 cm). The results confirm the reliability of the use of the Neumann model for fast and accurate mutual inductance calculation of the WPTS coils. Hereafter, the data set to be used for training the Genetic Programming (GP) algorithm, as well as that to be used for the model validation, are therefore calculated by means of the Neumann formula (1) under a certain number of test conditions (presented in Section 4). 25
MCARIDDI
10 5 0
10
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0
MNeumann
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20
20
30
40
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[°] Fig. 4 Mutual inductance versus the rotation angle: numerical solution of (1) compared to the reference solution (CARIDDI code).
3.2 Model generation via Genetic Programming
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The GP is a special evolutionary algorithm in which the population is composed of models. During its evolution, the GP algorithm transforms the current population of models into a new population of models by applying classical genetic operations, such as selection, cross-over, mutation, etc. The GP models are generally represented by means of tree structures. To construct such trees, the GP algorithm considers a non-terminal set of functions (sum, multiplication, division, logarithm, arctangent, hyperbolic tangent, sine, exponential and power function) and a terminal set of constant coefficients and input variables. Complexity factors (cf) are usually assigned to all the elements included in these sets. The following values have been herein assumed for the elements of the terminal set: cf = 0.6 for the multiplication of input variables and cf = 1 for all the other operations between input variables and for constant coefficients. Instead, the following values have been considered for the non-terminal set: cf = 1 for sum and/or multiplication of elementary functions, cf = 1.5 for all the other functions. To quantify the global complexity of each GP model, a term Fcomplexity has been introduced and calculated as follows: if a function is the argument of another function, the complexity factors cf of the two functions are multiplied;
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if two functions are multiplied or summed, their complexity factors cf are summed and subsequently multiplied by the complexity factor of a sum or a product, respectively.
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In this paper, the GP goal is the discovery of behavioral models of the mutual inductance as a function of axial (Δx) and lateral (Δz) misalignments and of the rotation angle of the WPTS coils. In order to achieve a kind of separation of the effects of the three geometric variables Δx, Δz and , two variables are selected as the model inputs, whereas the remaining variable is used as the parameter influencing the values of the behavioral model coefficients. Starting from this assumption, two different approaches for the behavioral model Mbehav formulation have been investigated:
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Approach 1: Mbehav(Δx,Δz,p()) is given as a function of Δx and Δz and is parametrized with respect to , where p() is a vector of model coefficients dependent on ;
Approach 2: Mbehav(Δz,,p(Δx)) is given as a function of Δz and and is parametrized with respect to Δx, where p(Δx) is a vector of model coefficients dependent on Δx. Let us note that the scope of investigating the above two approaches consists in the discovery of a compact behavioral model formulation as a function of two geometric variables, whose coefficients are given as functions of the third variable, used as model parameter. To such a scope, the above two approaches have been tested in parallel, so as to discover which of them provides better behavioral models from the point of view of accuracy, complexity and smoothness of the resulting model coefficients. For a more detailed description of the behavioral model identification by the GP algorithm, let us refer to Approach 1, but the same considerations hold also for Approach 2 by interchanging variables Δx and in Mbehav formulation. Let us consider a set of m rotation angle values j, j = 1,…,m, used as the behavioral model parameter. For each j value, n couples of axial and lateral misalignments (Δxi, Δzi) have been analyzed, with i = 1,…,n. For each of the n×m test conditions, a data vector has been created, including the test values (Δxi, Δzi) and the corresponding mutual inductance value Mij=M(Δxi,Δzi,j), evaluated by means of the Neumann formula (1). Such n×m data vectors compose the training data set T of the GP algorithm. The GP algorithm has to identify the behavioral model Mbehav(Δx,Δz,p()), such that the value of the function Mbehav computed for each test condition of the training data set T is as close as possible to the corresponding training value Mij, i {1, …, n} and j {1, …, m}. The structure of the function Mbehav has to be the same for all the rotation angle values j, j = 1,…,m, while the coefficients p vary with . To determine the coefficients p for each j, a Non-Linear Least Squares (NLLS) method, based on Levenberg-Marquardt optimization [23], has been applied to the n experimental data vectors relevant to j. Thus, the values of coefficients p have been determined by minimizing the χ-squared error, evaluated between the training mutual inductance values Mij and the GP-predicted values Mbehav(Δxi,Δzi,p(j)), for i = 1,…,n:
M behav xi , zi , p( j ) M ij M ij i 1
2
n
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(2)
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Then, the interpolating functions p(j) have been determined, as will be discussed in the next Section. To estimate the behavioral model accuracy, the Root Mean Square Error (RMSE) over the whole training data set has been evaluated according to (3): m
1 RMSE m 2j
(3)
j 1
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Eventually, a validation data set V has been created, including test conditions of Δx, Δz and different from the training set conditions, to verify the models generalization capability. A MOGP algorithm has been adopted to discover models offering an optimal trade-off between accuracy (RMSE) and complexity (Fcomplexity). In this paper, an elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) has been used [24], which searches for the models jointly minimizing RMSE and Fcomplexity values. Multiple runs of the MOGP algorithm have been executed, to discover the most repeatable GP models. Specifically, each model generated by the GP has been scored on the basis of the following metrics: - Nrun – the number of runs during which the algorithm has discovered the model; - Ngen – the average number of generations during which the model survived in the population; - {μerr, σerr, errmax} – the mean value, standard deviation and maximum value of the model percent error over the training data set T; - errVmax – the maximum value of the model percent error over the validation data set V; - Nmon – the average number of intervals over which the model coefficients p change their monotonicity with respect to the selected model parameter ( for Approach 1, Δx for Approach 2). The best models are those ones maximizing the first two metrics and minimizing the last five metrics. In general, the use of simpler behavioral models with a lower value of the Nmon metric results in more regular coefficients p, which can be more easily approximated as functions of the selected model parameter.
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4. Results and discussion 4.1 Approach 1 The coils misalignment parameters chosen to assemble the training data set T of the GP algorithm are listed in the left-hand side of Tab. 3: all the combinations of such values have been simulated, resulting in 300 test conditions. The right-hand side of Tab. 3 lists the misalignment parameters describing the validation data set V adopted in Approach 1, consisting of 120 test conditions. It should be noted that the values of Δx and Δz, chosen for the validation data set, are different from the values adopted for the training data set. Tab. 3 Coils misalignment parameters adopted for training and validation data set in Approach 1 Parameter
Training Values
Validation Values
Δx
{15, 20, 25, 30, 35} cm
{18, 23, 28, 33} cm
Δz
{0, 15, 25, 28, 30, 35} cm
{12, 22, 32} cm
{0, 20, 40, 60, 80, 100, 120, 140, 160, 180}°
α
{0, 20, 40, 60, 80, 100, 120, 140, 160, 180}°
On the contrary, the values of have been maintained unchanged, so as to use the same model coefficient values p(), discovered during the models identification, also to validate the models.
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The MOGP algorithm has been executed over 100 runs, with a population of 500 models evolving over 300 generations for each run. Only the non-dominated Pareto-optimal solutions having the repeatability of at least 10 runs and the maximum percent error lower than 15% have been considered for further comparison. Such solutions are represented as blue markers in Fig. 5 and labelled with increasing solution numbers. The related model expressions are provided in Tab. 4, while the respective values for the metrics {Nrun, Ngen, μerr, σerr, errmax, errVmax , Nmon} are given in Tab. 5. Let us note that Fig. 5 contains several model solutions discovered by the GP algorithm during certain runs, which are dominated by some other solutions discovered in the other independent runs. Since such models still present high repeatability and good error metrics, they have also been considered for the final comparison. In particular, models #2 and #3 have been discovered by the GP algorithm under two different formulations, which correspond to the same equivalent model expression, thus increasing the overall model repeatability. In principle, model #6 dominates models #2 and #3 in terms of better RMSE and Fcomplexity values, but has lower values of Nrun and Ngen metrics. The other models have a lower repeatability, or higher Nmon values (i.e., more irregular coefficient trends), or both. Hence models #2 and #3, reformulated as given in (4), present a good trade-off among all the metrics and has been therefore selected as an optimal behavioral model of the mutual inductance for Approach 1:
M behav p 0 z 1.5 exp p1x p 2 exp p 3 x
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(4)
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where Δx and Δz are expressed in meters, while Mbehav is given in μH. Model (4) is characterized by low percent errors evaluated on the training dataset T, with μerr ≈ -0.04%, σerr ≈ 2%, errmax ≈ 6.7%. Also, the maximum percent error evaluated on the validation dataset V is limited to errVmax ≈ 4.5%, thus confirming the model generalization capability. Moreover, for all the models of Tab. 5, the maximum errors evaluated on the validation data set are lower than the respective training set errors, thus guaranteeing that the models do not overfit the assigned data. 6
#1
5.5
#2 #3
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5
#4 #5
#6
#7
4.5
4 0.01
#9
#10
0.015
0.02
0.025
#8
0.03
0.035
RMSE
Fig. 5 Repeatable Pareto-optimal solutions obtained with Approach 1.
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Tab. 4 GP model expressions obtained with Approach 1 Model
Expression
#1
p0 exp p1x z exp p2 z p3 p4 exp p5 x
#2
p0 exp p1x p2 z p3z exp p4 x p0 z z exp p1x p 2 exp p3 x
#3
p0 exp p1x p2 tanh log p3z x
#4
p0 exp p1z p2 x p3x p4
#5 #6
p0 exp p1x p2 tanh p3z 2 exp p4 x
#7
p0 z 2 p1x p2 p3 z exp p4 x
#8
p0 z atan p1z x p2 x p3
#9
p0 x p1 p2 z atan p3 z x
#10
p0 exp p1x p2 p3z
exp p5x
Tab. 5 Metrics values for GP models obtained with Approach 1 Nrun
Ngen
μerr [%]
σerr [%]
errmax [%]
errVmax [%]
Nmon
#1
23
138
-0.01
1.1
4.4
2.2
2.7
#2
15
188
-0.04
2
6.7
4.5
2.5
#3
34
185
-0.04
2
6.7
4.5
2.4
#4
10
238
-0.09
3.1
10.3
6.4
2.5
#5
13
127
-0.1
3.2
10
7.4
4.8
#6
19
90
-0.02
1.3
6.3
2.9
2.4
#7
11
212
-0.09
3
11.3
7.6
2.4
#8
10
159
-0.1
#9
14
98
-0.1
#10
12
110
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Model
-0.06
3.2
10.6
6.2
3
3.2
10.6
6.2
3
2.4
7.6
3.9
3.3
The coefficients pi, i = 0,1,2,3, identified by the GP algorithm for model (4), are shown in Fig. 6 (red circle markers), and can be approximated as cosine functions of α (green curves in Fig. 6):
p i a i ,0 cos 2 a i ,1 , i 0,1, 2,3
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(5)
The values of fitting coefficients ai,0 and ai,1 are provided in Tab. 6.
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Tab. 6 Fitting coefficient values for the behavioral model (4) Coefficient ai,0 ai,1 p0
-42.55e+00
186.77e+00
p1
-322.41e-03
4.94e+00
p2
-4.84e+00
36.66e+00
p3
-221.98e-03
3.37e+00
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240 220
p1
200 180 160 140
0
60
[°]
120
180
42
3.6
36
p3
p2
39
33 30
3.4 3.2
0
60
[°]
120
180
0
60
[°]
120
180
Fig. 6 Behavioral model (4) coefficients pi vs α, i = 0,1,2,3 (red circle markers) and relative cosine fitting curves (green lines).
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In Fig. 7(a) the values of M calculated by using the Neumann formula (blue markers) are compared to the values predicted by the behavioral model (4) over the 300 test conditions of the training data set. In particular, red markers depict the values of M predicted by using model (4) with the original pi coefficients identified by the GP algorithm, whereas green markers represent the values of M predicted by using the cosine approximations (5) of such coefficients.
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Fig. 7(b) depicts the relative percent errors of model (4) with respect to the training set data. It is worth remarking that, even though the function (5) does not fit perfectly some of the coefficients pi (e.g., p1 and p3), the overall impact of such an approximation on the predictions of M by the behavioral model (4) is minor. Indeed, the behavioral model ensures a -0.04% average error with 2% standard deviation, which is sufficiently accurate for fast numerical simulations to analyze the impact of coils misalignment on the overall WPTS performances.
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M [ H]
(a)
(b) 10
Errbehav
5
Errbehav,fit
0 -5 -10
0
50
100
150
200
250
300
Test #
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Fig. 7. Behavioral model predictions: (a) mutual inductance values obtained by using the Neumann formula (blue markers), the behavioral model (4) with the original pi coefficients (red markers) and the behavioral model (4) with the cosine approximations of pi; (b) relative percent errors.
4.2 Approach 2.
The left-hand side of Tab. 7 lists the coils misalignment parameters chosen to assemble the training data set T (the same as in Approach 1), while the right-hand side of Tab. 7 lists the misalignment parameters describing the validation data set V adopted in Approach 2, consisting of 75 test conditions. It should be noted that the values of Δz and α, chosen for the validation data set, are different from the values adopted for the training data set. On the contrary, the values of Δx have been maintained unchanged, so as to use the same model coefficient values p(Δx), discovered during the models identification, also to validate the models.
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Tab. 7 Coils misalignment parameters adopted for training and validation data set in Approach 2 Parameter
Training Values
Validation Values
Δx
{15, 20, 25, 30, 35} cm
{15, 20, 25, 30, 35} cm
Δz
{0, 15, 25, 28, 30, 35} cm
{12, 22, 32} cm
α
{0, 20, 40, 60, 80, 100, 120, 140, 160, 180}°
{10, 50, 90, 130, 170}°
The MOGP algorithm has been executed over 50 runs, with a population of 500 models evolving over 300 generations for each run. Only the non-dominated Pareto-optimal solutions with metrics Nrun ≥ 9 and errmax ≤ 15% have been considered for further comparison. Such solutions are represented as blue markers in Fig. 8 and labelled with increasing solution numbers. The related model expressions are provided in Tab. 8, while the respective values for the metrics {Nrun, Ngen, μerr, σerr, errmax, errVmax , Nmon} are given in Tab. 9. Once again, models labelled as #2 and #3 have been discovered by the GP algorithm under two slightly different formulations, which correspond to the same equivalent model expression given in (6): M behav p 0 exp p1z 2 p 2 p3 z 2 sin 2
(6)
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where Δz is expressed in meters, while Mbehav is given in μH. Model (6) is characterized by low percent errors, with μerr ≈ - 0.01%, σerr ≈ 1.2%, errmax ≈ 6.2% and errVmax ≈ 2.4%. Such model expresses the mutual inductance dependence on the angle α through a sine-squared function, which can be equivalently represented as a cosine function of the double angle 2α, thus correctly modeling the mutual inductance behavior for whatever α value. Indeed, the coils reciprocal position is the same for rotation angles α and 180°+α, resulting in the mutual inductance periodic dependence on α, characterized by a period of 180°. Also, the coils position is symmetrical for rotation angles α and 180°–α, resulting in the same value of the mutual inductance, as correctly predicted by model (6). Hence such model has been herein selected as an optimal behavioral model of mutual inductance for Approach 2.
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Fig. 8 Repeatable Pareto-optimal solutions obtained with Approach 2.
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Tab. 8 GP model expressions obtained with Approach 2 Model #1 #2 #3 #4 #5 #6 #7 #8 #9 #10
Expression
p0 exp p1z p2 sin 2 p3 p4 z 3 2
p0 exp p1z 2 p2 sin 2 p3 p4 z 2 p0 exp p1z 2 p2 sin 2 p3 z 2
p0 exp p1z 2 p2 sin z 2 p3 exp z
p0 exp p1z 2 p2 sin p3 p4 z 2
p0 exp p1z 2 p2 sin z sin p3z p0 p1 sin sin p2 z p3 z
p0 tanh log p1z p2 z 2 sin
p0 exp p1z 2 p2 sin sin p3z
p0 z sin p1 tanh log p2 z
Nrun
Ngen
μerr [%]
σerr [%]
errmax [%]
errVmax [%]
Nmon
#1
9
108
-0.01
1.2
5.1
2.6
1.4
#2
9
130
-0.01
1.2
6.2
2.4
1.4
#3
9
142
-0.01
1.2
6.2
2.4
1
#4
9
99
-0.02
1.5
6.3
2.3
1
#5
28
146
-0.02
1.5
6.9
2.1
1.4
#6
9
132
-0.08
2.8
8.8
6.8
1
#7
19
147
-0.08
2.8
7.9
4.5
1
#8
16
159
-0.09
#9
16
174
#10
41
188
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Model
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Tab. 9 Metrics values for GP models obtained with Approach 2
10.4
6.6
1
4
10.9
10.1
1
-0.19
4.4
12.3
10.6
1
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2.9
-0.16
lP
The coefficients pi, i = 0,1,2,3, identified by the GP algorithm for model (6), are shown in Fig. 9 (red circle markers), and can be approximated as exponential functions of Δx (green curves in Fig. 9):
p i a i ,0 exp a i ,1 x a i ,2 ,
i 0,1, 2, 3
(7)
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where Δx is expressed in meters. The values of fitting coefficients ai,0, ai,1 and ai,2 are provided in Tab. 10.
p0
34.03e+00
-4.20e+00
2.71e+00
p1
15.52e+00
-3.50e+00
2.01e+00
p2
214.93e+00
-9.05e+00
3.91e+00
p3
7.23e-03
3.47e+00
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Tab. 10 Fitting coefficient values for the behavioral model (6) Coefficient ai,0 ai,1 ai,2
64.85e-03
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Fig. 9 Behavioral model (6) coefficients pi vs Δx, i = 0,1,2,3 (red circle markers) and relative fitting curves (green lines).
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(a)
(b)
M Error [%]
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Fig. 10(a) compares the values of M calculated by using the Neumann formula (blue markers) with the values predicted by the behavioral model (6) over the 300 test conditions of the training data set. In particular, red markers depict the values of M predicted by using the model (6) with the original pi coefficients identified by the GP algorithm, whereas green markers represent the values of M predicted by using the exponential approximations (7) of such coefficients. Fig. 10(b) shows the relevant percent errors, highlighting that the behavioral model (6) ensures a -0.01% average error with 1.19% standard deviation over the training data set.
Fig. 10 Behavioral model predictions: (a) mutual inductance values obtained by using the Neumann formula (blue markers), the behavioral model (6) with the original pi coefficients (red markers) and the behavioral model (6) with the exponential approximations of pi; (b) relative percent errors.
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It is worth observing that the two models (4) and (6) discovered by the GP algorithm include the mutual inductance dependence on Δx and α in a dual way. The model (4) contains exponential terms of Δx, while its parameters vary as cosine functions of the double angle 2α. The parameters of the model (6), instead, vary exponentially with Δx, while the mutual inductance dependence on α is expressed through a sine-squared function, which can be equivalently represented as a cosine function of the double angle 2α. Hence the two models (4) and (6) discovered by the GP are coherent between them, have a similar complexity (two terms and four coefficients in each model) and both present a good accuracy. The model (6) outperforms the model (4) in having smoother and quasi-linear (low-order polynomial) coefficient trends. This element can be adopted as a preferential requisite in the evaluation of possible alternative models generated by means of the MOGP approach.
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Eventually, it should be noted that Approach 1 and Approach 2 investigated in this paper are equivalent from the computational viewpoint, in terms of both time employed for the behavioral model generation and number of data points needed to assemble the training data set of the MOGP algorithm. 5. Conclusions and perspectives In this paper, two novel behavioral models of the mutual inductance for rectangular-shaped coils used in Wireless Power Transfer Systems (WPTSs) are proposed, valid for different types of coils misalignment. Such models have been discovered by means of a Multi-Objective Genetic Programming (MOGP) algorithm. One of the two behavioral models expresses the mutual inductance as a function of the axial and lateral misalignments of the coils, parametrized with respect to their rotation angle. The other model describes the mutual inductance as a function of the lateral misalignment and rotation angle, parametrized with respect to the axial misalignment. The two models are coherent between them and present comparable accuracy and complexity. Their resulting mutual inductance values are in good agreement with the literature model predictions and numerical Finite Element simulations.
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Behavioral models, like those ones presented in this paper can provide powerful tools in the design and optimization of static and dynamic WPT systems. First of all, they allow to easily identify the locus of spatial misalignments ensuring a certain value of the mutual inductance M, that is a valuable information to be used, for instance, in the control strategy of the system. In addition, a behavioral model allows an easy implementation of a parametrized equivalent circuit for the coils sub-system, to be embedded in the overall equivalent circuit describing the WPT system, along with the other sub-systems (e.g., the converters and all the other electronic equipment). In this way, it becomes much easier to perform a parametrized system-level circuit simulation, for instance during the design optimization as for the current technology trends [25].
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Conflicts of interest The authors declare no conflict of interest.
Acknowledgements
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The results here presented are developed in the framework of the 16ENG08 MICEV Project. The latter received funding from the EMPIR programme cofinanced by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. The University of Salerno partially supported this work through the project funds “Sistemi di Carica Induttiva di Veicoli Elettrici” (300638FRB17DICAPUA, 300638FRB18DICAPUA). References
Z. Bi, T. Kan, C. Mi, Y. Zhang, Z. Zhao, G. A. Keoleian, “A review of wireless power transfer for electrical vehicles: prospect to enhance sustainable mobility”, Applied Energy, vol. 179, pp. 413-425, 2016.
[2]
V. Cirimele, M. Diana, F. Freschi, M. Mitolo, “Inductive Power Transfer for Automotive Applications: State-ofthe-Art and Future Trends”, IEEE Trans. on Industry Appl., vol. 54, no. 5, pp. 4069-4079, Sep. 2018.
[3]
A. Rakhymbay et al., “Precise Analysis on Mutual Inductance Variation in Dynamic Wireless Charging of Electric Vehicle”, Energies, pp. 624, vol. 11, n. 3, 2018.
[4]
G. Di Capua, N. Femia, K. Stoyka, “Sensitivity Analysis of Inductive Power Transfer Systems for Electric Vehicles Battery Charging”, Proc. of Int. Conf. on Synthesis, Modeling, Analysis and Simulation Methods and Appl. to Circuit Design (SMACD), pp. 173-176, Lausanne, Switzerland, 2019.
[5]
M. Zucca et al., “The Project ‘Metrology for inductive charging of electric vehicles”, Proc of Conf. on Precision Electromagnetic Measurements (CPEM 2018), Paris, France, July 2018.
[6]
M. Zucca et al., “Metrology for Inductive Charging of Electric Vehicles (MICEV)”, Proc. of Int. Conf. of Electrical and Electronic Technologies for Automotive (AEIT AUTOMOTIVE), pp. 1-6, Torino, Italy, 2019.
[7]
Project “Metrology for inductive charging of electric vehicles”, www.micev.eu
[8]
R. Ravaud, G. Lemarquand, V. Lemarquand, C. Akyel, “Cylindrical Magnets and Coils: Fields, Forces, and Inductances,” IEEE Trans. on Magnetics, vol. 46, pp. 3585- 3590, Sept. 2010.
[9]
J. T. Conway, “Inductance Calculations for Noncoaxial Coils Using Bessel Functions”, IEEE Trans. on Magnetics, vol. 43, pp. 1023 - 1034, Feb 2007.
Jo u
rna
lP
[1]
Journal Pre-proof
Babic S., et al, “New Formulas for Mutual Inductance and Axial Magnetic Force Between a Thin Wall Solenoid and a Thick Circular Coil of Rectangular Cross-Section,” IEEE Trans. on Magnetics, pp. 2034 - 2044, vol. 47, Aug 2011.
[11]
A. P. Sample, A. David, D.A. Meyer, J.R. Smith, “Analysis, experimental results, and range adaptation of magnetically coupled resonators for wireless power transfer”, IEEE Transactions on Industrial Electronics, vol. 58, no. 2, pp. 544-554, Feb. 2011.
[12]
Y. Zhang, L. Wang, Y. Guo, C. Tao, “Design and Optimization of Asymmetrical Spiral Rectangular Pads for EV Wireless Charging”, 2018 IEEE 4th Southern Power Electr. Conf. (SPEC), Singapore, Dec. 2018.
[13]
M. S. Carmeli, F. Castelli-Dezza, M. Mauri, G. Foglia, “Contactless energy transmission system for electrical vehicles batteries charging”, 2015 Int. Conf. on Clean Electr. Power (ICCEP), Taormina, June 2015.
[14]
B. Olukotun, J. S. Partridge, R. W. G. Bucknall, “Optimal Finite Element Modelling and 3-D Parametric Analysis of Strong Coupled Resonant Coils for Bidirectional Wireless Power Transfer”, Proc. of 53rd International Universities Power Eng. Conference (UPEC), Glasgow, UK, Sept. 2018.
[15]
J. R. Koza, Genetic Programming: on the programming of computers by means of natural selection. Cambridge, MA, USA: MIT Press, 1992.
[16]
N. Femia, M. Migliaro, C. Pastore, D. Toledo, “In-system IGBT power loss behavioral modeling”, Proc. of Int. Conf. on Synthesis, Modeling, Analysis and Simulation Methods and Appl. to Circuit Design (SMACD), Lisbon, PT, June 2016.
[17]
K. Stoyka, G. Di Capua, N. Femia, “A novel AC power loss model for ferrite power inductors”, IEEE Transactions on Power Electronics, pp. 2680-2692, vol. 34, 2019.
[18]
K. Stoyka, R. A. Pessinatti Ohashi, N. Femia, “Behavioral switching loss modeling of inverter modules”, Proc. of Int. Conf. on Synthesis, Modeling, Analysis and Simulation Methods and Appl. to Circuit Design (SMACD), Prague, CZ, July 2018.
[19]
A. Garg, K. Tai, “Selection of a robust experimental design for the effective modeling of nonlinear systems using genetic programming”, 2013 IEEE Symposium on Computational Intelligence and Data Mining (CIDM), Singapore, Sept. 2013.
[20]
V. Cirimele, “Design and integration of a dynamic IPT system for automotive applications”, PhD Thesis, Politecnico di Torino, Italy, 2017, available at: https://iris.polito.it/handle/11583/2666564#.XXFjEC2B2t8
[21]
R. Albanese, G. Rubinacci, “Finite element methods for the solution of 3D eddy current problems,” Advances in Imaging and Electron Physics, pp. 1-86, vol. 102, 1998.
[22]
R. R. Aubakirov, A. A. Danilov, S. V. Selishchev, “Numerical modeling of inductive transcutaneous energy transfer using coils with square turns”, 2018 IEEE Conf. of Russian Young Researchers in Electrical and Electronic Eng. (EIConRus), Moscow, RU, March 2018.
[23]
W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing. New York, NY, USA: Cambridge University Press, 1992.
[24]
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II”, IEEE Trans. on Evolutionary Computation, pp. 182–197, vol. 6, 2002.
[25]
V. Cirimele, M. Diana, F. Freschi, M. Mitolo, “Inductive power transfer for automotive applications: state-of-theart and future trends”, IEEE Trans. Ind. Appl., pp. 4069-4079, vol. 54, 2018.
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[10]