On the Asymptotic Behavior and Parameter Estimation of a Double-Sided LCC-Compensated Wireless Power Transfer System

: This study investigates the statistic behavior and parameter estimation problems of a double-sided, LCC-compensated, wireless power transfer system. Based on the commonly used wireless charging circuit model, this study proposes a ﬁve-step parameter estimation method, which is applicable to automotive static wireless charging systems. The eight parameters in the circuit model of this study are the most important key components of the wireless charging system. The study also found that, under certain conditions, the statistic mode of wireless charging systems has a speciﬁc distribution. However, the current status of these eight components for wireless charging of electric vehicles will have complex parameter drift problems. These drift problems will deteriorate the performance of the vehicle power systems. This study probes these factors and proposes some related mathematical theories. The noted factors can be applied to the analysis of the wireless charging system and provide alternative solutions to explain the deteriorations from coil misalignments. Both simulations and experiments are given to show the evaluated issues of the proposed study.


Introduction
A wireless power transfer (WPT) system is a resonant circuit consisting of transmitting and receiving coils and a compensation circuit. The compensation circuit determines the system performance, so its topology and parameter design have become the most important part of WPT systems. Regarding the wireless power transfer, many approaches are referenced and studied, such as magnetic flux induction, two-coil loosely coupled transformer, microwaves, and lasers [1,2]. Herein, the two-coil loosely coupled transformer, which can transfer power wirelessly with high efficiency across large air gaps, is of note. Four basic topologies are often referenced. Depending on how the compensation capacitors are added to the transmitting and receiving coils, they are simply named as series-series (SS), series-parallel (SP), parallel-series (PS), and parallel-parallel (PP) topologies [3,4]. Compared with the conventional topologies, the double-sided LCC compensation network is shown in Figure 1. It works as a current source to both the input and output [5][6][7][8][9], which is suitable for light and heavy load conditions. Even when no load is placed in the load side, this topology can maintain suitable power distributions. This feature is very suitable for dynamic WPT because misalignment will commonly appear during the dynamic charging. The compensation capacitor is determined by the self-inductance, so coil misalignment does not significantly affect the resonance. Moreover, the circuit can realize unity power factor at the input, so there is no reactive power in the system. Furthermore, the circuit can be designed to provide zero-voltage-switching (ZVS) for the input inverter and eliminate the reverse recovery loss in the rectifier, so the system efficiency can reach 96%. Also, two of the inductors can be designed as planar inductors, integrated with Machines 2021, 9,287 2 of 13 the main coils to save space [10,11]. As illustrated in Figure 1a, this topology consists of one inductor and two capacitors at both the primary and secondary sides. Note that in Figure 1a, the double-sided LCC compensation topology was applied in the WPT unit of this paper. Its applications are highly suitable for both dynamic and static WPT of electric vehicles [12,13]. Additionally, due to the coils, the system can be placed under the chassis. Such structure of WPT is also easy to integrate into autonomous vehicles [14,15]. Eight parameters in the LCC circuit model are applied in the wireless charging system. These factors affect the system performance such as power, efficiency, and safety [16]. Basically, temperature, DC bias, and stochastic process may affect the parameter drift significantly. In practice, these drift affections will deteriorate the output performance of the vehicle systems. Clearly, parameter drift problems in static WPT and misalignment issues in dynamic WPT reveal similar effects to unsteady output, which are both important issues that could be investigated. input inverter and eliminate the reverse recovery loss in the rectifier, so the system efficiency can reach 96%. Also, two of the inductors can be designed as planar inductors, integrated with the main coils to save space [10,11]. As illustrated in Figure 1a, this topology consists of one inductor and two capacitors at both the primary and secondary sides. Note that in Figure 1a, the double-sided LCC compensation topology was applied in the WPT unit of this paper. Its applications are highly suitable for both dynamic and static WPT of electric vehicles [12,13]. Additionally, due to the coils, the system can be placed under the chassis. Such structure of WPT is also easy to integrate into autonomous vehicles [14,15]. Eight parameters in the LCC circuit model are applied in the wireless charging system. These factors affect the system performance such as power, efficiency, and safety [16]. Basically, temperature, DC bias, and stochastic process may affect the parameter drift significantly. In practice, these drift affections will deteriorate the output performance of the vehicle systems. Clearly, parameter drift problems in static WPT and misalignment issues in dynamic WPT reveal similar effects to unsteady output, which are both important issues that could be investigated.
where From Figure 1a, the equivalent circuit can be obtained, as Figure 1b. The circuit in Figure 1b can be described by the following equations: where a = jωL f 1 + 1 Machines 2021, 9, 287 3 of 13 Note that L M = k √ L 1 L 2 stands for the mutual inductance and k is the coupling coefficient. The resonance can be obtained in many ways. The useful conditions are obtained when all the diagonal elements are real and no imaginary part is present. Thus, the following resonant conditions can be derived: Clearly, the resonant frequency ω should be controlled under the condition of Accordingly, in order to obtain all the diagonal elements being real, with no imaginary parts, the following assumption is imposed in the sequel:

Problem Formulations
In this study, we consider a stochastic model V 1 (t), I L 1 (t), I L 2 (t), I 2 (t) of the wireless power transfer as follows: where {B(t)} t≥0 is a one dimensional Brownian motion, and σ > 0, ρ > 0, e = −1 . σ, ρ, L M , and R L are given parameters; and , and C f 2 are parameters that need to be estimated. For sim- From Formula (6), we have This gives For simplicity, denote

Steady Behavior of Primary Current I 1 (t)
In order to clarify the asymptotic behavior of {I 1 (t)} t≥0 , we propose the following definition at first: for any Borel set A and each fixed t > 0. From Formula (5), we obtain As a matter of fact, it is well known that {I 1 (t)} t≥0 is the Ornstein-Uhlenbeck process. Thus, we point out that Bhattacharya and Waymire's study [17] is a readable literature devoted to the proof.
Result 3.1. {I 1 (t)} t≥0 has the following probability density function: Result 3.2. {I 1 (t)} t≥0 has the following invariant measure: Proof. By Result 3.1, if we take lim t→∞ on both sides of Formula (7), we obtain the following: This yields that is the invariant measure of {I 1 (t)} t≥0 . This completes the proof. Accordingly, the random variable I 1 (∞) has the probability density function √ 2ρ √ 2πσ 2 exp − ρy 2 σ 2 .

Result 4.1.
Under Hypothesis 1, for any x 1 , x 2 , x 3 , x 4 ∈ R, we obtain the following: Proof. It is trivial from the statement of this section above. This completes the proof.

Estimation Algorithm
For sake of estimating (θ 1 , θ 2 , · · · , θ 8 ) ∈ V, we refer to the assumptions of Result 3.1 and [18] via Metropolis-Hastings algorithm as follows. Result 4.2 ensures the convergence of the following estimation algorithm.
Step 5: Repeat the previous steps until we obtain a convergent chain.

Simulations and Experiments
Under Hypothesis 1 and ρ = 0.5, σ = 1, R L = 0, for the sake of simplifying a simulation, we set  (396, 396, 66, 66, 12, 12, 60, 60). Then, proceed the estimation algorithm as mentioned in Section 4.3. For i from 1 to 10,000, the synthesized values were revealed in Figure 2. In Figure 2, all tests are done by 10,000 iterations. As can be seen in Machines 2021, 9, 287 8 of 13 this figure, the sample path is reasonably well converged, which satisfies the conditions that an asymptotic stable circuit should have. Under the condition of Hypothesis 1, the convergent value of capacitance reaches its steady state quickly. However, the inductance has a slower convergent rate. The convergent speed of steady value is different between inductor and capacitor because of the dynamical behavior of LCC topology. Note that only static WPT fits this conclusion. Misalignment in static WPT can be applied to simulate the approaching or leaving statuses (or both) of dynamic WPT. In dynamic WPT, parameter drift affects the transmission power and efficiency, respectively. Basically, the whole process of "approachingcentral-leaving" of the dynamic WPT can be treated as a series of parameter varying, which can be modeled by stochastic process. Hence, using the stochastic model to probe the system performance is an appropriate solution. Via simulation, engineers can use the stochastic model to preview the system performance before the activation of dynamic WPT process. For further demonstrating the LCC-compensated performance in dynamic charging, a small scale of experiment was employed to show its feasibility. The verification experiments are carried out on the test bench, as shown in Figure 3. Table 1 illustrates the relevant specifications of test bench. Its rated DC voltage is 10V, which is a small scale of WPT for demonstration. Figure 4 is the measurement data for coupling coefficient. Figure 5 shows the resonant conditions. Clearly, the resonant frequency exactly falls on 85 kHz, which satisfies the requirement from WPT. To test the affections of misalignment, as shown in Figure 6, a scenario of dynamic movement of receiving coils are scheduled. The experimental result is revealed in Figure 7. Clearly, the received energy is sensitive to the misalignment, which is relevant to the coupling coefficient, as shown in Figure 4. Misalignment in static WPT can be applied to simulate the approaching or leaving statuses (or both) of dynamic WPT. In dynamic WPT, parameter drift affects the transmission power and efficiency, respectively. Basically, the whole process of "approaching-centralleaving" of the dynamic WPT can be treated as a series of parameter varying, which can be modeled by stochastic process. Hence, using the stochastic model to probe the system performance is an appropriate solution. Via simulation, engineers can use the stochastic model to preview the system performance before the activation of dynamic WPT process. For further demonstrating the LCC-compensated performance in dynamic charging, a small scale of experiment was employed to show its feasibility. The verification experiments are carried out on the test bench, as shown in Figure 3. Table 1 illustrates the relevant specifications of test bench. Its rated DC voltage is 10V, which is a small scale of WPT for demonstration. Figure 4 is the measurement data for coupling coefficient. Figure 5 shows the resonant conditions. Clearly, the resonant frequency exactly falls on 85 kHz, which satisfies the requirement from WPT. To test the affections of misalignment, as shown in Figure 6, a scenario of dynamic movement of receiving coils are scheduled. The experimental result is revealed in Figure 7. Clearly, the received energy is sensitive to the misalignment, which is relevant to the coupling coefficient, as shown in Figure 4.

Conclusions
Considering the misalignment in the WPT process, this study investigated the statistic model for double-sided, LCC-compensated, wireless power transfer system. The main contributions are considered as: For the initial value I 1 (0) = x, {I 1 (t)} t≥0 has the following probability density function: In addition, {I 1 (t)} t≥0 has the following invariant measure: Under Hypothesis 1, for any x 1 , x 2 , x 3 , x 4 ∈ R, we obtain Additionally, if ρ = 0.5, σ = 1, and R L = 0, then the Markov chains {Θ n } ∞ n=0 with the state space V = (c 1 , d 1 ) × (c 2 , d 2 ) × · · · × (c 8 , d 8 ) have the following invariant measure: The statistic mode of the wireless charging system constructed in this paper is suitable for dynamic wireless power transfer, based on the LCC topology. The stochastic model can be applied to simulate the deteriorations from misalignment of the static WPT. Therefore, it can be employed to evaluate the system's features, such as efficiency in advance. After the simulation, engineers can apply the stochastic model to preview the power or the efficiency (or both) before the activation of the dynamic WPT process. This study analyzed eight key components on the circuit. When it is used for dynamic wireless charging scenarios, there will be complex parameter drift problems, which lead to the deterioration of the power systems. In consequence, from the perspective of statistic model, this paper discussed the performance deteriorations and put them into practice in WPT applications of electric vehicles. Additionally, for further evaluations, simulations and experiments were given to show the noted factors of the proposed study.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.