Abstract
Purpose
The purpose of this paper is to show how the EStra-Many method works on optimization problems characterized by high-dimensionality of the objective space. Moreover, a comparison with a more classical approach (a constrained bi-objective problem solved by means of NSGA-II) is done.
Design/methodology/approach
The six reactances of a compensation network (CN) for a wireless power transfer system (WPTS) are synthesized by means of an automated optimal design. In particular, an evolutionary algorithm EStra-Many coupled with a sorting strategy has been applied to an optimization problem with four objective functions (OFs). To assess the obtained results, a classical genetic algorithm NSGA-II has been run on a bi-objective problem, constrained by two functions, and the solutions have been analyzed and compared with the ones obtained by EStra-Many.
Findings
The proposed EStra-Many method identified a solution (CN synthesis) that enhances the WPTS, considering all the four OFs. In particular, to assess the synthesized CN, the Bode diagram of the frequency response and a circuital simulation were evaluated a posteriori; they showed good performance of the CN, with smooth response and without unwanted oscillations when fed by a square wave signal with offset. The EStra-Many method has been able to find a good solution among all the feasible solutions, showing potentiality also for other fields of research, in fact, a solution nondominated with respect to the starting point has been identified. From the methodological viewpoint, the main finding is a new formulation of the many-objective optimization problem based on the concept of degree of conflict, which gives rise to an implementation free from hierarchical weights.
Originality/value
The new approach EStra-Many used in this paper showed to properly find an optimal solution, trading-off multiple objectives. The compensation network so synthesized by the proposed method showed good properties in terms of frequency response and robustness. The proposed method, able to deal effectively with four OFs, could be applied to solve problems with a higher number of OFs in a variety of applications because of its generality.
Keywords
Citation
Bertoluzzo, M., Di Barba, P., Forzan, M., Mognaschi, M.E. and Sieni, E. (2022), "Multiobjective optimization of compensation networks for wireless power transfer systems", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 41 No. 2, pp. 674-689. https://doi.org/10.1108/COMPEL-06-2021-0204
Publisher
:Emerald Publishing Limited
Copyright © 2021, Manuele Bertoluzzo, Paolo Di Barba, Michele Forzan, Maria Evelina Mognaschi and Elisabetta Sieni.
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Wireless power transfer systems (WPTSs) operations rely on the inductive coupling between a transmitting and a receiving coil (Bi et al., 2016; Campi et al., 2020; Covic and Boys, 2013a, 2013b; Di Capua et al., 2021; Feng et al., 2018; Kindl et al., 2020; Lukic and Pantic, 2013; Jha et al., 2018; Choi et al., 2015; Siqi and Mi, 2015). Suitable compensation networks (CNs) are connected to the coils to increase the power transferred to the load and the efficiency of the device (Bertoluzzo, Forzan, et al., 2020; Chen et al., 2020; Feng et al., 2018; Villa et al., 2012; Kindl et al., 2020; Li et al., 2015; Mohamed et al., 2016; Pantic and Lukic, 2012; Qu et al., 2017; Zhang and Mi, 2016). Optimization methods were used in the design of WPTSs and CNs for car supply (Cirimele et al., 2018; Cruciani et al., 2020a; Li et al., 2018; Orasanu et al., 2018; Winges et al., 2019; Yakala et al., 2021; Zhang and Mi, 2016) or to size the electromagnetic shield (Arduino et al., 2020; Campi et al., 2020; Cruciani et al., 2020a, 2020b). The optimization of WPTS is largely explored to design the wireless recharge of other vehicles as e.g. electric railway (Lee et al., 2014) or other electrical devices, as e.g. in Ali et al. (2019, 2020), Shen and Clerckx (2021) and Tan et al. (2019). In Chen et al. (2020), evolutionary algorithms were used to optimize CNs, whereas in Kindl et al. (2020) and Li et al. (2015, 2018), the improvement of the devices is based on the search for the best efficiency related to the load.
In Bertoluzzo et al. (2020), Bertoluzzo and Sieni (2019) authors presented automatic methods to design the reactances of generic CNs for automotive applications. In those papers, bi-objective, efficiency and power-to-load, the problem is solved. Usually, the CNs are designed considering only the nominal supply angular frequency ω_{0}; however, their impedance at higher and lower angular frequency ω must be high to filter out the high-frequency current-harmonics and to prevent the circulation of continuous-current components. The first of these occurrences arise because usually the WPTSs are supplied by a square wave modulated voltage that encompasses all the odd harmonics of the supply frequency; the second derives from the not ideal matching between the amplitude and the duty cycles of the positive and the negative semi periods of the supply voltage. From this background, a many-objective problem arises i.e. maximize the efficiency and the transferred power and maximize the impedance at ω_{100}=100·ω_{0} and ω_{0.01}= ω_{0}/100, where ω_{100} and ω_{0.01} are chosen as representative of the low and high-frequency bands, respectively. In this paper, to solve a many-objective problem with four objective functions (OFs), an algorithm presented by two of the authors in Di Barba et al. (2020) is applied. Results were compared with the ones found using a constrained-version of the NSGA-II algorithm (Deb, 2001; Lahanas et al., 2003; Deb and Jain, 2014; Qingfu Zhang, 2009; D’Souza et al., 2010).
2. The many-objective problem
Figure 1 shows the CN that connects the coupling inductances L_{t} and L_{pu} and the WPTS system to the power supply (left) and to the load (right). The power circuit works at 85 kHz, ω_{0,} as prescribed by the society of automotive engineers (SAE) regulation (“SAE J 2954 Wireless Power Transfer for Light-Duty Plug-in/Electric Vehicles and Alignment Methodology”, n.d.). The considered CN has a T topology at both sides of the WPTS. In this problem, the CN is characterized by six design variables (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}), i.e. the reactances of the CN components represented in Figure 1.
They can vary in the range [–500; 500] Ω: if the optimal value is positive, an inductance is identified; otherwise, it is a capacitance. Four OFs have been considered: function f_{1} is the efficiency of the WPTS, η, function f_{2} is the power transferred to the load, P_{L}. Because they were minimized, the following functions were considered in the optimization problem as follows:
In computing f_{1} and f_{2} the inductive reactances are modeled considering their inductance and an equivalent series-connected parasitic resistance (ESR). The ESRs cause unwanted losses in the CN and should be reduced as much as possible by adopting suitable technologies in building the inductive elements of the CN. In setting the many-objective problem, it is not possible to know in advance the actual values of the ESRs, and consequently, it has been hypothesized that the quality factor Q = ω_{0}L/ESR of the inductive element is equal to 130, i.e. the same as the coupling coil available in the laboratory device, which has the parameters reported in Table 1.
During the optimization process, the ESRs of the inductive elements are adjusted to their optimized inductances to maintain the given Q for each of them.
The OFs f_{3} and f_{4} are expressed as follows:
3. The optimization methods
3.1 The many-objective method
The idea behind this paper is to exploit a single-objective optimization algorithm for minimizing a suitable preference function, which takes into account all the OFs of the optimization problem. The chosen method is EStra, a lowest-order evolutionary algorithm, which has proven to be effective and reliable (Di Barba and Mognaschi, 2009).
Traditionally, preference functions are defined as weighted sums of individual objectives, and the choice of individual weights is the bias. In contrast, in the proposed method, the aim is to define a weight-free preference function, in view of this, the key idea is the degree of conflict among solutions because, in Pareto-like optimality, the nondominated solutions exhibit the lowest degree of conflict in the set of solutions.
Hence, the road map turns out to be as follows:
define a preference function, modeling the degree of conflict among objectives, whatever the number of objectives;
minimize the preference function by means of an evolution strategy algorithm, i.e. a derivative-free and global-optimum oriented algorithm; and
identify a set of least-conflict solutions, approximating a Pareto-optimal set.
The core operation is to compute the degree of conflict between m ≥ 2 OFs (f_{1}, f_{2},…, f_{m}) that are assumed to be simultaneously minimized. After sorting in ascending order, the rank of a solution is defined as the sorting index of the relevant OF values.
The computation of the degree of conflict of a solution, called score, is based on the sorting indices of the OFs; its computing procedure is shown in Figure 2.
Eventually, the goal is to minimize s with respect to any x in the design space Ω. In view of this, suppose that the preference function s has been linked with a zero-order optimization algorithm; whenever a new candidate solution is created by the algorithm, the set S is expanded by incorporating the (n + 1)th solution (growing set scheme). The whole procedure is repeated up to convergence. For the algorithm to work, an initial set S_{0} has to be supplied.
It has to be noted that the ranking value assigned to each solution is relevant to the current set S of solutions; whenever a new solution is added to or removed from the set S, the solution score of the individual solution has to be recomputed. Hence, the sorting (or ranking) operation of Figure 2 must be done at each iteration of the optimization, considering the current set of solutions S.
Specifically, the EStra method, a zero-order optimization algorithm, is chosen. If the method converged, it identified a single Pareto-optimal solution. The EStra method coupled with the sorting strategy explained above is called EStra-Many.
3.2 Constrained classical bi-objective method
To compare the results obtained by the EStra-Many method with a well-known and assessed method, the non-dominated sorting genetic algorithm (NSGA-II) is chosen. An interesting alternative to NSGA-II could be NSGA-III (Deb and Jain, 2014), which is more recent and able to deal with more than two OFs at the same time, on the other hand; however, NSGA-III does not seem to be so far as popular as NSGA-II, probably because of the need of providing reference solution points in the search space, which might be cumbersome for the user to provide.
In general, NSGA-II, the classical nondominated sorting genetic algorithm, developed by Deb (D’Souza et al., 2010; Qingfu Zhang, 2009; Deb and Jain, 2014), is used to solve optimization problems from the multiobjective viewpoint, which means to approximate the whole Pareto front; however, the higher dimension of the objective space the less selective the Pareto front because there are very many in different solutions and so the classical criterion is loosely selective. In this paper, NSGA-II was used to optimize the efficiency of the WPTS and the power transferred to the load, f_{1} and f_{2}, respectively, whereas f_{3} and f_{4} are considered as constraints. In practice, a threshold level for f_{3} and f_{4} was set, and for all the solutions with f_{3} or f_{4} larger than the prescribed threshold value, an out-of-range value for f_{1} and f_{2} is assigned. In this respect, a penalty technique is applied for handling constraints. In practice, the solutions are discarded when not fulling the constraints. In the paper, depending on the considered case, different constraint levels were considered. In particular, the level was set between 1.5·10^{−4} and 1 because in this problem, f_{3} and f_{4} must be lower than 1.
3.3 Classical multiobjective method
A further comparison between the proposed EStra-Many method and the classical multiobjective approach is carried out by means of NSGA-II applied to four OFs f_{1}, f_{2}, f_{3} and f_{4}.
4. Optimization results
The optimization algorithm has been run for 700 iterations. Because EStra is a 1 + 1 algorithm, one individual is processed at each run. The starting point and the improved solution are in Table 2.
NSGA-II method was run many times. In particular, the thresholds for f_{3} and f_{4} were different for each run: in general, the evolutionary algorithms have a poor behavior when constraints are handled, the threshold values have been tightened more and more at each run to make the constraints progressively more severe. In each run, 50 individuals and 250 generations are considered. The results are shown in Figure 3; elements highlighted with green point-down triangles represent the NSGA-II solutions with maximum transferred power, green point-up triangles denote the NSGA-II solutions with maximum efficiency, and red stars are relevant to the solutions chosen by the designer as a good trade-off between the values taken by the OFs.
Looking at Figure 3, it can be noted that the fronts found by NSGA-II are wider when the threshold of the constraints is higher; the lower the constraint values, the narrower the Pareto fronts (black triangles in Figure 3).
All these solutions are characterized by function f_{3} equal to or higher than 1.68·10^{−4} and f_{4} higher than 1.5·10^{−4}. All in one, it can be noted that when the threshold values of f_{3} and f_{4} are comparable to the relevant values independently found by EStra-Many, then the front recovered by NSGA-II degenerates to a single solution point. The design variables and the values of the OF f_{1} and f_{2} reached at the end of the optimization with constraints set to 1.5·10^{−4} and 1.8·10^{−4} are listed in Table 3.
It can be noted that these solutions are Pareto-indifferent with respect to the EStra-Many solution (blue diamond in Figure 3) because they worsen function f_{1}, whereas they are better for function f_{2}. This is also true if all the four OFs are considered at a time; for the sake of an example, let us consider Solutions 1 and 2 characterized by k > 2 OFs. It is enough that just one OF of Solution 1 is better than the corresponding OF of Solution 2, and another OF of Solution 1 is worse than the corresponding of Solution 2 that the two solutions are Pareto-indifferent. No matter the behavior of the remaining OFs.
Figure 4 shows the same Pareto fronts shown in Figure 3, but in Figure 4, the points are represented with different colors. Each color represents the range of values of the functions f_{3} [Figure 4(a)] and f_{4} [Figure 4(b)], which characterizes each solution. For instance, a green point in Figure 4(a) has an f_{3} value between 10^{−2}<f_{3}<10^{−1}, whereas a black point has 10^{−4}<f_{3}<10^{−3}. The same colors are used to represent the range of values for f_{4} in Figure 4(b).
Tables 4 and 5 show a selection of solutions related to constrained NSGA-II optimization considering different constraint levels for f_{3} and f_{4}. The considered levels are 1.5·10^{−4}, 1.8·10^{−4}, 10^{−3}, 10^{−2}, 10^{−1} and 1. The selected solutions are those for which f_{3} and f_{4} are lower than 5·10^{−4}, respectively. It can be noted that even if f_{3} and f_{4} are constrained to be lower than a given threshold (e.g. 1), in the optimal solutions, f_{3} and f_{4} can assume lower values (e.g. close to 10^{−4}).
Table 6 shows a set of optimal solutions found with NSGA-II using four different constraint levels from 1 to 10^{−3}. In particular, the chosen solutions are those for which f_{1} and f_{2} are maximum, respectively (green triangles in Figure 3 and focus on point up and down, respectively) and others named design solutions, chosen a posteriori by the designer. These solutions were chosen as a trade-off between f_{1} and f_{2} objectives, also considering the value of the function f_{3} and f_{4} and the shape of the Bode diagram of the impedance at the input of the WPTS. The chosen solutions show improved values for all four OFs.
Figure 5 shows the Bode diagram of the impedance
The last comparison is performed between the EStra-Many solution, and the NSGA-II applied to four OFs. NSGA-II has been run with 50 individuals and 250 generations. The results are shown in Figure 7, where the projections of f_{1} versus f_{2}, f_{3} and f_{4}, respectively, and f_{3} versus f_{4} are shown. The EStra-Many solution is represented by means of a blue diamond, whereas the NSGA-II results are represented with crosses; the red crosses are solutions fulfilling the constraints f_{3}<1 or f_{4}<1, whereas the black crosses are unfeasible solutions.
After Figure 7, the following remarks can be put forward; the Estra-Many solution is located in the feasible region of the objective space i.e. in the region the red cross symbols belong to.
Moreover, the Estra-Many solution is close to the NSGA-II solutions; this shows the capability of the proposed method to find a globally non-dominated solution. Hence, the two methods exhibit similar performance.
As far as the cost is concerned, we also note that EStra-Many finds one solution with 100–150 OF calls, whereas NSGA-II finds a set of 50 solutions with 250 generations i.e. up to 50 × 250 = 6,250 OF calls.
5. Performance of the optimized compensation networks
The analysis Figures 5 and 6 allow drawing some general considerations about the behavior of the synthesized CNs. A common feature that appears in nearly all the diagrams is a resonance at the supply frequency ω_{0}. This characteristic is common also to conventional CNs, such as those built with the series-series or the series-parallel configurations. In fact, to maximize the transferred power with a given supply voltage, as the minimization of f_{1} requires, the impedance at the output of the supply inverter must be minimized to obtain a high current. This remark helps also in explaining why the diagrams on the left column of Figure 6, characterized by low transferred power, have a poorly defined peak of resonance at ω_{0}, or no peak at all, like the first diagram.
Thanks to the requirement of minimizing both Z_{ω}_{0}/Z _{ω}_{100} and Z_{ω}_{0}/Z_{ω}_{0.01}, all the diagrams exhibit a more or less defined wings-like shape and achieve high impedance values at high and low angular frequencies with respect to the supply one. Another common feature is the presence of a pair of spurious antiresonance-resonance peaks at angular frequencies higher than ω_{0}. The antiresonance peak is always set at a lower frequency than the resonance one, and their distance decreases as the requirements about the impedance ratios are tightened. It can also be found that when these requirements are loose, like it happens in the first two rows of Figure 6, some harmonics of the supply frequency are near to the spurious resonance peak, whereas in the other cases, they are far from it.
To make easier the comparison between the optimized solution reached by EStra-Many and those found by NSGA-II, the three plots of Figure 5 have been superimposed to obtain Figure 8.
The red plot refers to the optimized solution obtained by EStra-Many and is characterized by the reactances listed in Table 2. The dashed blue and green plots are relevant to the optimized solutions reported in Table 3 and computed by NSGA-II. The parameters of the inductances and the equivalent load R_{L} used in computing the OFs are shown in Table 1. They are taken from a prototypal low-power WPTS device (Buja et al., 2015).
The analysis of the plots in Figure 8 shows that both the optimization methods successfully prevent the flowing of unwanted current components in the supply inverter and in the transmitting CN. Indeed, for EStra-Many the ratios Z_{ω}_{0}/Z_{ω}_{100} and Z_{ω}_{0}/Z_{ω}_{0.01} are 75.5 dB and 72.9 dB, respectively, whereas for both the Bode diagrams relevant to NSGA-II, the ratios are 76.5 dB and 74.8 dB.
The pair of antiresonant-resonant peaks is between three and four times the supply frequency. From this point of view, EStra-Many provides better results than NSGA-II because the spurious resonance peaks have lower amplitude and are located at frequencies further away from the third harmonic of ω_{0}. Therefore, it is less likely that the resonance peaks affect the current of the supply inverter even if it generates a square wave voltage with a frequency a little higher than the nominal one. More precisely, the peak of minimum impedance coming from EStra-Many is set at a normalized frequency of 3.85 and has a magnitude of 37.19 dB, whereas both the peaks coming from NSGA-II are set at a normalized frequency of 3.47, and their magnitude is 16.45 dB, so that corresponding impedance is about 10 times lower than that obtained with EStra-Many.
At frequencies a little higher than the supply one, the Bode diagrams of the impedance exhibit a small distortion, as shown in Figure 8(b). Also, in this case, the solution coming from EStra-Many performs better than the other two because the distortion is smaller and happens at an angular frequency further away from the supply one.
A second verification of the optimization performance of EStra-Many has been carried out by a circuital simulation. Substituting in the circuit of Figure 1 the optimized values of the impedances, the circuit of Figure 9 has been obtained. It has been implemented in the Simulink environment to check the ability of the optimized CNs to reject both high and low-frequency components. To this end, the circuit has been supplied with a square-wave voltage with an offset equal to the voltage amplitude. The plot of the supply voltage is shown in the upper half of Figure 10(a). The waveform of the corresponding supply current is reported in the lower half of the same figure.
It clearly appears that the current does not exhibit any continuous component, and it is slightly distorted only in the very first supply periods while any harmonics disappear after a very short time interval. Figure 10(b) demonstrates that, despite the sudden application of the supply voltage, the current reaches smoothly its steady-state condition, without discontinuities or undue oscillations of its amplitude.
6. Conclusions
The comparison of the values taken by the OFs before and after the optimization shows that the EStra-Many algorithm effectively enhanced the performance of the CNs and their impedance to the unwanted current harmonics, as confirmed by the magnitude Bode diagram of Figure 7 and by the results coming from the simulations reported in Figure 9.
The EStra-Many method has been able to find a good solution among all the feasible solutions, showing potentiality also for other fields of research, in fact, a solution non-dominated with respect to the starting point has been identified. From the methodological viewpoint, the main finding is a new formulation of the many-objective optimization problem based on the concept of degree of conflict, which gives rise to an implementation free from hierarchical weights.
Figures
Parameters of the coupling inductances
L_{t} | 120 µH |
---|---|
L_{pu} | 120 µH |
M | 30 µH |
Q | 130 |
R_{L} | 5.6 Ω |
Starting and final points of the optimization: design variables and OF values found with EStra-Many method
x_{1} [Ω] | x_{2} [Ω] | x_{3} [Ω] | x_{4} [Ω] | x_{5} [Ω] | x_{6} [Ω] | |
---|---|---|---|---|---|---|
Starting point | –69.7 | 223 | –457 | 247 | 427 | 179 |
Final point | 188 | –380 | –451 | –57.2 | –295 | –22.4 |
η [NU] | P [W] | Z_{ω}_{0}/Z_{ω}_{100} (f_{3} [NU]) | Z_{ω}_{0}/Z_{ω}_{0.01} (f_{4} [NU]) | |
---|---|---|---|---|
Starting point | 6.32·10^{–4} | 1.61·10^{–8} | 9.05·10^{–2} | 6.42·10^{–2} |
Final point | 4.48·10^{–1} | 7.11·10^{–2} | 1.68·10^{–4} | 1.53·10^{–4} |
Final points of the NSGA-II optimizations: design variables and OF values
x_{1} [Ω] | x_{2} [Ω] | x_{3} [Ω] | x_{4} [Ω] | x_{5} [Ω] | x_{6} [Ω] | |
---|---|---|---|---|---|---|
NSGA-II 1.5·10^{–4} | 122.18 | –375.32 | –248.76 | 112.08 | –82.62 | 125.35 |
NSGA-II 1.8·10^{–4} | 119.96 | –369.60 | –246.21 | 108.72 | –79.30 | 123.41 |
η [NU] | P_{L} [W] | Z_{ω}_{0}/Z_{ω}_{100} (f_{3} [NU]) | Z_{ω}_{0}/Z_{ω}_{0.01} (f_{4} [NU]) | |
---|---|---|---|---|
NSGA-II 1.5 10^{–4} | 0.286 | 7.81·10^{–2} | 1.5·10^{–4} | 1.2·10^{–4} |
NSGA-II 1.8 10^{–4} | 0.369 | 8.55·10^{–2} | 1.8·10^{–4} | 1.5·10^{–4} |
Improved solutions for which f_{3} < 5·10^{−4}
x_{1} [Ω] | x_{2} [Ω] | x_{3} [Ω] | x_{4} [Ω] | x_{5} [Ω] | x_{6} [Ω] | η [NU] | PL [W] | f_{3} [NU] | f_{4} [NU] | |
---|---|---|---|---|---|---|---|---|---|---|
EStra-Many | 188 | –380 | –451 | –57.2 | –295 | –22.4 | 0.448 | 0.0711 | 1.68·10–4 | 1.53·10–4 |
NSGA-II constraint level | ||||||||||
1·10^{0} | 40 | –56 | –182 | 1 | –471 | –67 | 0.73 | 0.19 | 4.8·10^{–4} | 4.5·10^{–4} |
1·10^{–1} | NA | |||||||||
1·10^{–2} | NA | |||||||||
1·10^{–3} | NA | |||||||||
1.8·10^{–4} | 120 | –370 | –246 | 109 | –79 | 123 | 0.37 | 0.09 | 1.8·10^{–4} | 1.5·10^{–4} |
1.5·10^{–4} | 122 | –375 | –249 | 112 | –83 | 125 | 0.29 | 0.08 | 1.5·10^{–4} | 1.2·10^{–4} |
NA: not available solution for the selected constraint level, NU: not unit
Improved solutions for which f_{4}<5·10^{−4}
x_{1} [Ω] | x_{2} [Ω] | x_{3} [Ω] | x_{4} [Ω] | x_{5} [Ω] | x_{6} [Ω] | η [NU] | P_{L} [W] | f_{3} [NU] | f_{4} [NU] | |
---|---|---|---|---|---|---|---|---|---|---|
EStra-Many | 188 | –380 | –451 | –57.2 | –295 | –22.4 | 0.448 | 0.0711 | 1.68·10^{–4} | 1.53·10^{–4} |
NSGA-II constraint level | ||||||||||
1·10^{0} | 40 | –57 | –179 | 0 | –470 | –66 | 0.75 | 0.17 | 5.3·10^{–2} | 5.0·10^{–4} |
1·10^{–1} | 2 | –376 | –62 | –8 | 15 | –11 | 0.77 | 0.15 | 1.4·10^{–2} | 4.7·10^{–4} |
1·10^{–2} | 23 | –226 | –100 | –86 | –349 | 1 | 0.75 | 0.12 | 1.4·10^{–3} | 4.7·10^{–4} |
1·10^{–3} | 124 | –298 | –282 | –98 | –49 | 18 | 0.76 | 0.06 | 4.8·10^{–4} | 4.1·10^{–4} |
1.8·10^{–4} | 120 | –370 | –246 | 109 | –79 | 123 | 0.37 | 0.09 | 1.8·10^{–4} | 1.5·10^{–4} |
1.5·10^{–4} | 122 | –375 | –249 | 112 | –83 | 125 | 0.29 | 0.08 | 1.5·10^{–4} | 1.2·10^{–4} |
NU: not unit
Improved solutions for which f_{1} and f_{2} assume the maximum value, and the trade-off solutions at design point of view
Constr. level | Criteria | x_{1} [Ω] | x_{2} [Ω] | x_{3} [Ω] | x_{4} [Ω] | x_{5} [Ω] | x_{6} [Ω] | η [NU] | P_{L} [W] | f_{3} [NU] | f_{4} [NU] |
---|---|---|---|---|---|---|---|---|---|---|---|
10^{0} | max(f_{1}) | 5 | –500 | –263 | –30 | –419 | –36 | 0.894 | 0.001 | 8.37·10^{–3} | 3.14·10^{–1} |
10^{–1} | max(f_{1}) | 2 | –373 | –64 | –46 | 30 | –13 | 0.930 | 0.029 | 2.92·10^{–3} | 9.83·10^{–2} |
10^{–2} | max(f_{1}) | 23 | –317 | –104 | –70 | –349 | 1 | 0.892 | 0.019 | 2.96·10^{–3} | 9.91·10^{–3} |
10^{–3} | max(f_{1}) | 126 | –300 | –281 | –72 | –66 | 7 | 0.840 | 0.033 | 8.71·10^{–4} | 1.00·10^{–3} |
10^{0} | max(f_{2}) | 36 | –43 | –258 | 0 | –471 | –67 | 0.602 | 0.331 | 2.45·10^{–4} | 2.51·10^{–4} |
10^{–1} | max(f_{2}) | 2 | –375 | –62 | 7 | 16 | −6 | 0.497 | 0.231 | 2.02·10^{–4} | 7.23·10^{–3} |
10^{–2} | max(f_{2}) | 22 | –307 | –93 | –132 | –349 | 18 | 0.456 | 0.187 | 1.70·10^{–4} | 5.48·10^{–4} |
10^{–3} | max(f_{2}) | 118 | –249 | –288 | –97 | –26 | 21 | 0.478 | 0.114 | 1.57·10^{–4} | 1.78·10^{–4} |
10^{0} | design | 41 | –55 | –211 | –5 | –468 | –67 | 0.602 | 0.331 | 2.45·10^{–4} | 2.51·10^{–4} |
10^{–1} | design | 1 | –376 | –61 | –7 | 18 | –13 | 0.805 | 0.135 | 5.67·10^{–4} | 2.74·10^{–2} |
10^{–2} | design | 22 | –308 | –94 | –124 | –349 | 17 | 0.520 | 0.183 | 1.97·10^{–4} | 6.34·10^{–4} |
10^{–3} | design | 124 | –298 | –283 | –93 | –36 | 16 | 0.759 | 0.065 | 3.95·10^{–4} | 4.62·10^{–4} |
NU: not unit
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Further reading
SAE (n.d.). “SAE J 2954 wireless power transfer for Light-Duty plug-in/electric vehicles and alignment methodology”, available at: www.sae.org/standards/content/j2954_201605/