Analysis and Control of Optimal Power Distribution for Multi-Objective Wireless Charging Systems

Zhen Zhang 1, Ruilin Tong 1, Zhenyan Liang 1, Chunhua Liu 2,* and Jiang Wang 1 1 School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China; zhangz@tju.edu.cn (Z.Z.); tongrl@tju.edu.cn (R.T.); liangzy@outlook.com (Z.L.); jiangwang@tju.edu.cn (J.W.) 2 School of Energy and Environment, City University of Hong Kong, Hong Kong, China * Correspondence: chunliu@cityu.edu.hk


Introduction
Recently, wireless power transfer (WPT) provides a brand new method for power transmission with salient advantages of cordless energization, multiple-channel transmission and flexibility by comparing with the traditional charging way.In particular, the inductive power transfer (IPT) utilizes the induced electromagnetic field to make the energy receptor harness the power over the air [1][2][3][4], which has been widely adopted for various applications, such as portable consumer electronics [5], implantable devices [6,7] etc.
Electric vehicle (EV), which has higher efficiency and higher stability, is getting increasing attention [8,9].In Reference [10], a gearless motor with smoother torque and better torque density for EV is analyzed and designed, which makes EV driving more viable.In Reference [11], electromagnetic compatibility on charging EV is discussed and well concerned, which secures the charging procedure.The combination of EV and WPT combine the advantages of both EV and WPT, and further eliminate the limitation of battery energy storage and charging [12,13].
As one of its salient advantages, the power can be transmitted to multiple objectives wirelessly and simultaneously [14,15]; problems of cable entanglement, cable corrosion and number of charging ports are eliminated, which greatly improves charging flexibility and reduces system cost.In multi-objective conditions, however, the fairness of power distribution should be taken into account according to various power demands of loads.In previous studies, a circuit model was proposed for the design procedure of WPT systems and the analysis of the transmission efficiency [16][17][18].In Reference [19], a maximum power transfer method for WPT systems was proposed to ensure the maximum power transmission.In order to optimize the efficiency for multi-objective transmission, a selective WPT

Mathematic Model
In multi-objective WPT systems, the flux equation of one excitation coil and multiple receptor coil can be expressed as: where ψ ex represents the flux linkage of the excitation coil, ψ rei represents the flux linkage of the ith receptor coil, L ex represents the self-inductance of the excitation coil, L rei represents the self-inductance of the ith receptor coil and M rei represents the mutual inductance between the excitation coil and the ith receptor.The "ex" in the subscript means "excitation" and the "re" in the subscript means "receptor".
The "i" in the subscript means the index of the receptor.To achieve maximum current, there should be capacitors to match reactive impedance in excitation and receptor circuits.The voltage equation of the excitation circuit and the receptor circuit can be expressed as: where u ex , r ex , i ex and u rei , r rei , i rei are voltage, resistance, current of excitation circuit and the ith receptor, respectively.By substituting (1) into (2), it yields: In WPT systems, the supply voltage of the excitation circuit is a sinusoidal wave.Besides, in the excitation and receptor circuit, the inductor and capacitor connect in series.The current gain of the excitation and receptor circuit in frequency domain can be calculated as: where G ex and G rei mean the current of the excitation and receptor circuit, respectively.ω means the working angular frequency.When the circuits work in resonance mode, namely: G ex and G rei can meet the maximum value.The resonance frequency ω 0 is selected as the working frequency for the WPT system.Besides, the receptors are usually passive, namely u rei equals 0. Therefore, the circuit equation can be transformed to the symbolic-complex model as: .
Then, the current of the excitation and receptor circuits can be obtained as: Since the reactive impedance can be eliminated as the circuits works on resonance mode, the solution can be simplified as: From ( 8), the total equivalent impedance R se and individual equivalent impedance R reie can be calculated as: As shown in (9), the equivalent impedance is connected in series in the excitation circuit.As shown in (10), equivalent impedance can be increased or decreased by reducing or enlarging r rei .
According to (6), the multi-receptor charging system is equivalent to the circuit that is shown in Figure 1a, both excitation circuit and receptor works in resonance mode, namely the reactive impedance equals zero.According to (8), the circuit in Figure 1a can be furtherly simplified to the circuit in Figure 1b in which the equivalent impedance of different loads are connected in series.The output power Pex, efficiency e and individual load power Prei can be given by: The output power P ex , efficiency e and individual load power P rei can be given by: where I ex mean the amplitude of the excitation current and U ex means the amplitude of the excitation voltage.The relationship between R se , P ex , e and P rei is depicted in Figure 2. As shown in Figure 2, when R se equals r ex , P rei can reach its maximum value and the efficiency is 50%.When R se is larger than r ex , P rei decreases as the increment of R se .Besides, the larger R se is, the higher efficiency is.The output power Pex, efficiency e and individual load power Prei can be given by: where Iex mean the amplitude of the excitation current and Uex means the amplitude of the excitation voltage.The relationship between Rse, Pex, e and Prei is depicted in Figure 2. As shown in Figure 2, when Rse equals rex, Prei can reach its maximum value and the efficiency is 50%.When Rse is larger than rex, Prei decreases as the increment of Rse.Besides, the larger Rse is, the higher efficiency is.

Analysis of Power Distribution
In public application, the receptor is not identical.Besides, the number of load and individual load impedance are both random and vary unpredictably.As shown in Figure 1b, equivalent impedance of a different load is connected in the series in an equivalent circuit.As a result, the individual load with a larger equivalent impedance will be able to attain a larger portion of total power supply, and vice versa.Power of the ith load can be calculated as:

Analysis of Power Distribution
In public application, the receptor is not identical.Besides, the number of load and individual load impedance are both random and vary unpredictably.As shown in Figure 1b, equivalent impedance of a different load is connected in the series in an equivalent circuit.As a result, the individual load with a larger equivalent impedance will be able to attain a larger portion of total power supply, and vice versa.Power of the ith load can be calculated as: For the individual load, once the power supply is deficient, the larger equivalent impedance ensures a larger portion of total power, while a smaller equivalent impedance will lead to a smaller portion of total power supply.Thus, the best and simplest strategy for individual load whose power is deficient is to enlarge its equivalent impedance.In the case of power surplus, the smaller equivalent impedance leads to a smaller portion of total power supply, while the larger equivalent impedance will lead to a larger portion of total power supply.As a result, the best and simplest strategy for individual load whose power is surplus is to reduce its equivalent impedance.
The ith Load power after adjustment is given by: Energies 2018, 11, 1726 5 of 16 where ∆R reie represents the increment of R reie after adjustment.Power of total equivalent impedance P t can be calculated by: There is a special case that the sum of ∆R reie equals 0. In this case, P t equals total power demand and remains constant, the value of R se in this case is the optimal value and remains constant.The adjustment of individual equivalent impedance R reie is equivalent to adjust the equivalent impedance portion of a different load.
When P t is lower than total power demand, adjusting a portion of individual equivalent impedance R reie cannot achieve the optimal power distribution, namely, not all power demand will be satisfied.Some loads whose power is lower than demand will enlarge their equivalent impedance to acquire a larger portion of total power supply, which will cause some other loads to obtain deficient power and start enlarging their equivalent impedance.Therefore, more and more loads enlarge their impedance.As analyzed in Section 2, the larger total equivalent impedance is, the lower power all loads will share.Finally, all loads enlarge their equivalent impedance to the maximum value and the current of the excitation circuit meet its minimum value.
In the case of power surplus, adjusting a portion of individual equivalent impedance R reie cannot satisfy all power demand either.In this case, some loads whose power is higher than demand will reduce their equivalent impedance to obtain a smaller portion of total power supply, which will cause some other loads to obtain surplus power and start reducing their equivalent impedance.Therefore, more and more loads reduce their equivalent impedance.The smaller total equivalent impedance is, the higher power all loads will share.Finally, all loads reduce their equivalent impedance to the minimum value and the current of the excitation circuit meet its maximum value.
According to the above analysis, optimal power distribution need not only the adjustment of individual equivalent impedance, but also the adjustment of total equivalent impedance.And only when total power supply satisfies load power demand, will the balance be achieved.In this case, loads whose power is higher than power demand will reduce their impedance and other loads whose power is lower than power demand will enlarge their impedance.When all load power demand is satisfied, load impedance adjustment stops and the appropriate power distribution is achieved.
In the case that the charging object is moving such as EV, the effect of mutual inductance should be taken into consideration.As deduced from (10), the smaller load impedance r rei is, the larger equivalent impedance r eie is, the larger mutual inductance M rei is, and the larger equivalent impedance is.Actually, there exists a minimum value of r rei , the maximum equivalent impedance can be calculated as: where r reimin means the minimum value of load impedance r rei , and r reimax means the maximum value of equivalent impedance r reie .There is also a maximum amplitude of the excitation current I ex .
If load impedance meets its minimum value r reimin and I ex meets its maximum value, the power of the individual load depends on the mutual inductance.And only when the mutual inductance is large enough, individual load can enlarge its equivalent impedance and make the excitation circuit reduce total equivalent impedance.As a result, the power demand cannot be satisfied in the case that mutual inductance is too small.In the case of charging batteries and super capacitors, the power demand is not constant and varies as the state of charge changes.Therefore, optimal power distribution should be achieved dynamically with fast response and high stability.
Practically, the speed of equivalent impedance adjustment of a different load is not identical, which causes the sum of the increment of the individual load equivalent impedance ∆R reie unequal to 0. According to the analysis of Section 2, once the total equivalent impedance is larger or smaller than the optimal value of R se , the power of the total equivalent impedance is lower or higher than Energies 2018, 11, 1726 6 of 16 total power demand, which will furtherly cause the total equivalent impedance to increase or decrease sharply.In this case, even the total equivalent impedance R se equals the optimal value, the appropriate power distribution cannot be achieved.To ensure the robustness of the system, the power of total equivalent impedance P t should be able to endure the fluctuation.As long as the power of total equivalent impedance P t remains constant, the optimal power distribution will finally be achieved.

Principle of Load Adjustment
An efficient way to adjust individual load is to use a rectifier and a DC-DC converter to change the equivalent impedance by adjusting its duty ratio.The relation between the input and the output impedance in Cuk circuit is given by: where α means the duty ratio of Cuk circuit, R li represents the real load impedance.According to the power of the ith load, input voltage of the DC-DC converter u rei can be calculated as: where U rei means the DC input voltage of the DC-DC converter.According to the input voltage of the DC-DC converter of the ith load u rei and the power demand of the ith load P refi , the duty ratio α can be calculated as: In Reference [23], a total equivalent impedance adjusting method was proposed, a capacitor C p is connected in parallel with the excitation current.The total equivalent impedance can be adjusted by changing the capacitor C p . the total equivalent impedance can be adjusted to: The relationship between capacitance of C p , total equivalent impedance R se , output power P ex and efficiency e is depicted in Figure 3.As shown in Figure 3, the total equivalent impedance can be adjusted between the maximum value and the minimum value by selecting the appropriate value of C p .Besides, the larger total equivalent is, the higher the efficiency is.The circuit of the multi-objective WPT system is shown in Figure 4.Each receptor has an impedance regulator to adjust the equivalent impedance, aiming to meet various demands of the load power.The equivalent impedance of each load is adjusted according to the obtained power by the receptors.In the excitation circuit, a current sensor is used to detect load variation and the controller alters the capacitor array Cp to alter the total equivalent impedance.A smart injection circuit is used to inject a current to compensate the voltage of Cex to match the reactive impedance of excitation The circuit of the multi-objective WPT system is shown in Figure 4.Each receptor has an impedance regulator to adjust the equivalent impedance, aiming to meet various demands of the load power.The equivalent impedance of each load is adjusted according to the obtained power by the receptors.In the excitation circuit, a current sensor is used to detect load variation and the controller alters the capacitor array C p to alter the total equivalent impedance.A smart injection circuit is used to inject a current to compensate the voltage of C ex to match the reactive impedance of excitation circuit to ensure the circuit works in resonant mode.A couple of capacitors and inductors are used at the input port of the inverter to ensure power fluctuation tolerance.The circuit of the multi-objective WPT system is shown in Figure 4.Each receptor has an impedance regulator to adjust the equivalent impedance, aiming to meet various demands of the load power.The equivalent impedance of each load is adjusted according to the obtained power by the receptors.In the excitation circuit, a current sensor is used to detect load variation and the controller alters the capacitor array Cp to alter the total equivalent impedance.A smart injection circuit is used to inject a current to compensate the voltage of Cex to match the reactive impedance of excitation circuit to ensure the circuit works in resonant mode.A couple of capacitors and inductors are used at the input port of the inverter to ensure power fluctuation tolerance.

Methodology of Load Adjustment
As analyzed in Section 3.1, only when the total equivalent impedance equals the optimal value, can the appropriate power distribution be achieved.If the total equivalent impedance is larger than the optimal value, the total equivalent impedance will increase sharply.While if the total impedance is smaller than the optimal value, total equivalent impedance will decrease sharply.An optimal power distribution control method was proposed according to the analysis in Section 3.1.
The main idea of the control method is to find the optimal value of the total equivalent impedance.The variation of the excitation current, which reflects the variation total equivalent impedance caused by individual load adjustment, can be used as the indicator of total equivalent impedance adjustment.When the optimal value of the total equivalent impedance is found, the total equivalent impedance remains constant, after the adjustment of the power portion of the different load, appropriate power distribution is achieved.The optimal total equivalent impedance is solved by the dichotomy.Namely, find the medium value between the maximum value and the minimum value, and set the current value as the minimum value when the optimal value is larger than the current value or set the current value as the maximum value when the optimal value is smaller than the current value.The variation of Iex reflects whether the optimal total equivalent impedance is larger

Methodology of Load Adjustment
As analyzed in Section 3.1, only when the total equivalent impedance equals the optimal value, can the appropriate power distribution be achieved.If the total equivalent impedance is larger than the optimal value, the total equivalent impedance will increase sharply.While if the total impedance is smaller than the optimal value, total equivalent impedance will decrease sharply.An optimal power distribution control method was proposed according to the analysis in Section 3.1.
The main idea of the control method is to find the optimal value of the total equivalent impedance.The variation of the excitation current, which reflects the variation total equivalent impedance caused by individual load adjustment, can be used as the indicator of total equivalent impedance adjustment.When the optimal value of the total equivalent impedance is found, the total equivalent impedance remains constant, after the adjustment of the power portion of the different load, appropriate power distribution is achieved.The optimal total equivalent impedance is solved by the dichotomy.Namely, find the medium value between the maximum value and the minimum value, and set the current value as the minimum value when the optimal value is larger than the current value or set the current value as the maximum value when the optimal value is smaller than the current value.The variation of I ex reflects whether the optimal total equivalent impedance is larger or smaller than the current total equivalent impedance.The time complexity of the solving process is given by the following equation: where N represents the number of the available value of capacitor C p .From formula (22), the time cost is greatly reduced and thus the speed of response is greatly improved.Figure 5 portraits the procedure of impedance adjustment.The case of power surplus is shown in Figure 5a, and the case of power deficiency is depicted in Figure 5b.As shown in Figure 5a, in the first step, all loads reduce their equivalent impedance and their power is much higher than their demand.In the second step, the controller alters the total equivalent impedance to the medium value between the maximum value and the minimum value and the total equivalent impedance in step 1 is set as the minimum value, and load power decreases correspondingly.After the adjustment of step 2, total equivalent impedance is still smaller than the optimal value.In the third step, the individual loads reduce their equivalent impedance for a smaller portion of total power, and the total equivalent impedance decreases sharply.According to the variation of the total equivalent impedance, the controller altered the total equivalent impedance to the medium value between the maximum value and the minimum value in the fourth step.At this time, total equivalent impedance is close to the optimal total equivalent impedance.In the fifth step, a different load with a different power demand alters their equivalent impedance to achieve the appropriate power distribution.The adjustment procedure in the case of power deficiency is depicted in Figure 5b.In the first step, due to power deficiency, all loads enlarge their equivalent impedance to the maximum value.In the second step, the controller reduces the total equivalent impedance to the medium value between the maximum value and the minimum value, and the total equivalent impedance in step 1 is set as the maximum value, and load power increases correspondingly.Due to the total equivalent impedance still being larger than the optimal value, individual loads enlarge their equivalent impedance for a larger power portion in the third step.In the fourth step, total equivalent impedance is altered to the medium value between the maximum and the minimum value, at this time, total equivalent impedance is close to the optimal value.In the fifth step, a different load with a different power demand adjusts their equivalent impedance to achieve the appropriate power distribution.
The pseudo code of the total equivalent impedance adjustment is shown in Algorithm 1.

Algorithm 1
Initialize(R exmax , R exmin , i exmax ) while(ture) do // find the optimal value by dichotomy.input(I ex ) // use the excitation current to detect total equivalent impedance variation.if(I ex > I exmax ) break // if there is no charging load or the charging load is to heavy, close the system.end if if I ex > 1.5 × I ex0 // total equivalent impedance decreases sharply.
R se ←U ex /I ex − r ex R' se ←(R exmax + R exmin )/2 // set the total equivalent impedance to the medium value.R exmin ←R se // set the current value as the minimum value.
ex ) input(I ex ) I ex0 ←I ex // record the excitation current to detect if total equivalent impedance changes.else if I ex < 0.066 × I ex0 // total equivalent impedance increases sharply.
R se ←U ex /I ex − r ex R' se ←(R exmax + R exmin )/2 // set the total equivalent impedance to the medium value.

Simulated Results
To verify the validity of the proposed scheme, a computational simulation is carried out by using three receptors.In order to further demonstrate the feasibility, stability, accuracy and response speed of the proposed power distribution scheme in different cases, four case studies are given in this paper, including Case 1 to verify the effectiveness of the proposed scheme; Case 2 to illustrate the multi-channel transmission performance by taking into account the variation of power demand; Case 3 to verify the response speed, accuracy and stability for charging moving EVs and Case 4 to demonstrate the effectiveness of the proposed scheme for charging batteries and super capacitors.Key parameters are listed in Table 1.Case 1-The comparative simulation is firstly carried out for charging potable electronics as shown in Figure 6.During t = 0~3 s, the impedance regulators of the excitation circuit and receptor circuit are inactivated.In such a case, the power of Load 1, Load 2 and Load 3 equal 32.7 W, which causes power deficiency of load 2. At t = 3 s, the impedance regulator of receptor 1 and receptor 2 are activated while the impedance regulator of receptor 3 remains inactive.The power deficiency of Load 2 results in the increasing of the equivalent impedance of Load 2, thus leading to the power deficiency of Load 1 and Load 3. Due to the increased equivalent impedances of Load 1 and Load 2, the total supply power reduces to 8.8 W, while Load 1 and Load 2 obtain a larger portion of power supply.At t = 7 s, when the impedance regulator of excitation circuit is activated, the total equivalent impedance is adjusted to satisfy the power demand of loads.The power of Load 1 and Load 2 can meet the minimum demand, namely P re1 = 21.7 W and P re2 = 42.3W, by adopting the proposed control scheme; while the impedance regulator is inactivated, the power of Load 3 (64.8W) exceeds the maximum power demand (40 W).Thus, the proposed control scheme can effectively ensure the load power demand.Case 2-By adding and removing loads, a test is carried out by taking into account the variation of power demand in charging portable electronics, which aims to further illustrate the effectiveness of the proposed control scheme.All impedance regulators are activated through the whole testing process.The charging procedure is depicted in Figures 7 and 8. Case 2-By adding and removing loads, a test is carried out by taking into account the variation of power demand in charging portable electronics, which aims to further illustrate the effectiveness of the proposed control scheme.All impedance regulators are activated through the whole testing process.The charging procedure is depicted in Figures 7 and 8. Case 2-By adding and removing loads, a test is carried out by taking into account the variation of power demand in charging portable electronics, which aims to further illustrate the effectiveness of the proposed control scheme.All impedance regulators are activated through the whole testing process.The charging procedure is depicted in Figures 7 and 8.As shown in Figure 7, Load 1 and Load 2 are in the charging state during t = 0~2 s, where Rre1e = 200 Ω, Rre2e = 400 Ω, Rre3e = 0 Ω, and Pre1 = 18.9 W, Pre2 = 37.9 W and Pre3 = 0 W. At t = 2 s, Load 3 is added, which results in an immediate power deficiency of all loads as shown in Figure 8, power deficiency incurs larger impedance as shown in Figure 7.By adjusting the total impedance equaling 375 Ω, the load power meets the minimum demand, namely Pre1 = 18.5 W, Pre2 = 36.9W, and Pre3 = 27.7 W. At t = 6 s, Load 2 is removed, which causes an immediate power surplus.The total impedance is adjusted to 703.2 Ω.The impedance and power of loads remain Rre1e = 281.3Ω, Rre2e = 0 Ω, Rre3e = 421.9Ω, Pre1 = 19.8W, Pre2 = 0 W and Pre3 = 27.9W.Then, as shown in Figure 8, the loads power meets the demand again.Based on the relationship as shown in Figure 2, the proposed control scheme can ensure the power supply meets the minimum power demand, which means the multi-objective WPT system can work at the maximum efficiency point even if there is a variation of power demand.Case 3-The charging demand of the moving objects is also taken into consideration to further verify the validity and effectiveness of the proposed method.Figure 9 portraits the charging procedure of moving EVs.There are in total, three EVs passed by.As shown in Figure 9a.At t = 1 s, EV 3 moved into the charging area, during t = 1~2.5 s, power of EV 3 increased as EV 3 was approaching the excitation coil.During t = 2.5~4 s, power demand of EV 3 was satisfied and the power of EV 3 was kept stable at 1921 W. At t = 4 s, EV 2 which has a higher speed and higher power demand moved into the charging area, which caused an immediate change of total power demand and total equivalent impedance.After the adjustment, the power of EV 3 was kept stable at 2148 W. during t = 4~5 s power of EV 2 increased as EV 2 approached the excitation coil.At t = 4.5 s, the power of EV 3 started decreasing due to the decrement of mutual inductance and EV 1 moved into the charging area.Due to the higher speed and higher power demand of EV 1, total power demand and total equivalent impedance varied at a higher speed, which caused the overshoot of the adjustment to be much larger, and EV 1 and EV 2 obtained the maximum power of Pre1 = 6797 W and Pre2 = 5215 W. At t = 5 s, the power of EV 1 and EV 2 met the power demand and were kept stable at Pre1 = 4786 W and Pre2 = 2874 W. At t = 6 s, the power of EV 1 and EV 2 started decreasing.After t = 7.3 s, there was no EV to charge and the total power was kept to zero.As shown in Figure 7, Load 1 and Load 2 are in the charging state during t = 0~2 s, where R re1e = 200 Ω, R re2e = 400 Ω, R re3e = 0 Ω, and P re1 = 18.9 W, P re2 = 37.9 W and P re3 = 0 W. At t = 2 s, Load 3 is added, which results in an immediate power deficiency of all loads as shown in Figure 8, power deficiency incurs larger impedance as shown in Figure 7.By adjusting the total impedance equaling 375 Ω, the load power meets the minimum demand, namely P re1 = 18.5 W, P re2 = 36.9W, and P re3 = 27.7 W. At t = 6 s, Load 2 is removed, which causes an immediate power surplus.The total impedance is adjusted to 703.2 Ω.The impedance and power of loads remain R re1e = 281.3Ω, R re2e = 0 Ω, R re3e = 421.9Ω, P re1 = 19.8W, P re2 = 0 W and P re3 = 27.9W.Then, as shown in Figure 8, the loads power meets the demand again.Based on the relationship as shown in Figure 2, the proposed control scheme can ensure the power supply meets the minimum power demand, which means the multi-objective WPT system can work at the maximum efficiency point even if there is a variation of power demand.
Case 3-The charging demand of the moving objects is also taken into consideration to further verify the validity and effectiveness of the proposed method.Figure 9 portraits the charging procedure of moving EVs.There are in total, three EVs passed by.As shown in Figure 9a.At t = 1 s, EV 3 moved into the charging area, during t = 1~2.5 s, power of EV 3 increased as EV 3 was approaching the excitation coil.During t = 2.5~4 s, power demand of EV 3 was satisfied and the power of EV 3 was kept stable at 1921 W. At t = 4 s, EV 2 which has a higher speed and higher power demand moved into the charging area, which caused an immediate change of total power demand and total equivalent impedance.After the adjustment, the power of EV 3 was kept stable at 2148 W.During t = 4~5 s power of EV 2 increased as EV 2 approached the excitation coil.At t = 4.5 s, the power of EV 3 started decreasing due to the decrement of mutual inductance and EV 1 moved into the charging area.Due to the higher speed and higher power demand of EV 1, total power demand and total equivalent impedance varied at a higher speed, which caused the overshoot of the adjustment to be much larger, and EV 1 and EV 2 obtained the maximum power of P re1 = 6797 W and P re2 = 5215 W. At t = 5 s, the power of EV 1 and EV 2 met the power demand and were kept stable at P re1 = 4786 W and P re2 = 2874 W. At t = 6 s, the power of EV 1 and EV 2 started decreasing.After t = 7.3 s, there was no EV to charge and the total power was kept to zero.Case 3-The charging demand of the moving objects is also taken into consideration to further verify the validity and effectiveness of the proposed method.Figure 9 portraits the charging procedure of moving EVs.There are in total, three EVs passed by.As shown in Figure 9a.At t = 1 s, EV 3 moved into the charging area, during t = 1~2.5 s, power of EV 3 increased as EV 3 was approaching the excitation coil.During t = 2.5~4 s, power demand of EV 3 was satisfied and the power of EV 3 was kept stable at 1921 W. At t = 4 s, EV 2 which has a higher speed and higher power demand moved into the charging area, which caused an immediate change of total power demand and total equivalent impedance.After the adjustment, the power of EV 3 was kept stable at 2148 W. during t = 4~5 s power of EV 2 increased as EV 2 approached the excitation coil.At t = 4.5 s, the power of EV 3 started decreasing due to the decrement of mutual inductance and EV 1 moved into the charging area.Due to the higher speed and higher power demand of EV 1, total power demand and total equivalent impedance varied at a higher speed, which caused the overshoot of the adjustment to be much larger, and EV 1 and EV 2 obtained the maximum power of Pre1 = 6797 W and Pre2 = 5215 W. At t = 5 s, the power of EV 1 and EV 2 met the power demand and were kept stable at Pre1 = 4786 W and Pre2 = 2874 W. At t = 6 s, the power of EV 1 and EV 2 started decreasing.After t = 7.3 s, there was no EV to charge and the total power was kept to zero.The detailed charging procedure in the case of power fluctuation is depicted in Figure 9b.During t = 6.2~6.5 s, EV 1 and EV 2 cannot obtain enough power.Thus they dynamically adjust their The detailed charging procedure in the case of power fluctuation is depicted in Figure 9b.During t = 6.2~6.5 s, EV 1 and EV 2 cannot obtain enough power.Thus they dynamically adjust their equivalent impedance for a larger power portion.Due to the adjustment of EV 1, EV 2 and the excitation circuit, power of EV 1 and EV 2 fluxed in a limited range and continued decreasing, which proved the rapidity and stability of the proposed method.
The charging detail of EV 1 in the case of power variation is shown in Figure 9c.During t = 1.85~2.1 s, power demand of EV 1 was not satisfied.During t = 1.9~2.05s, as the mutual inductance increased, the equivalent impedance increased at t = 2.03 s, which caused a power decrease, and the excitation circuit altered the total equivalent impedance to a smaller value and the power increased.At t = 2.07 s, the power of the load kept stable but did not satisfy the power demand due to the limitation of mutual inductance.
According to the charging procedure depicted in Figure 9, the ability for charging moving objects with high speed and high power demand is well proved.Namely, the proposed method can charge multiple moving EVs dynamically with fast response and high stability.
Case 4-The case of power demand variation is also considered and simulated.The charging procedure of batteries and super capacitors is depicted in Figure 10, it depicts the procedure of charging batteries and super capacitors, respectively.As shown in Figure 10a, at t = 0.5 h, battery 1 started charging and charged in trickle mode, the charging power P re1 equaled 1441 W and was kept constant.At t = 1.2 h, battery 1 shifted into constant current mode, and the charging power demand increased to 7044 W. At t = 1.5 h, battery 2 started charging in trickle mode with P re2 = 1961 W, the sudden change of power demand and equivalent impedance caused a power fluctuation with the maximum overshoot of 0.76 on battery 1.At t = 1.9 h, battery 2 shifted to constant current mode and kept the charging power to 11047 W. At t = 2.2 h, battery 1 shifted to constant voltage mode and power demand started decreasing continuously.At t = 2.5 h, battery 2 shifted to constant voltage mode and power started decreasing continuously.Total demanded energy and charged energy of battery 1 and battery 2 equaled 36 GJ, 36.42GJ, 29.77 GJ and 28.62 GJ, respectively, which implies that the energy accuracy is in the range of 95~105%.
At t = 2.07 s, the power of the load kept stable but did not satisfy the power demand due to the limitation of mutual inductance.
According to the charging procedure depicted in Figure 9, the ability for charging moving objects with high speed and high power demand is well proved.Namely, the proposed method can charge multiple moving EVs dynamically with fast response and high stability.
Case 4-The case of power demand variation is also considered and simulated.The charging procedure of batteries and super capacitors is depicted in Figure 10, it depicts the procedure of charging batteries and super capacitors, respectively.As shown in Figure 10a, at t = 0.5 h, battery 1 started charging and charged in trickle mode, the charging power Pre1 equaled 1441 W and was kept constant.At t = 1.2 h, battery 1 shifted into constant current mode, and the charging power demand increased to 7044 W. At t = 1.5 h, battery 2 started charging in trickle mode with Pre2 = 1961 W, the sudden change of power demand and equivalent impedance caused a power fluctuation with the maximum overshoot of 0.76 on battery 1.At t = 1.9 h, battery 2 shifted to constant current mode and kept the charging power to 11047 W. At t = 2.2 h, battery 1 shifted to constant voltage mode and power demand started decreasing continuously.At t = 2.5 h, battery 2 shifted to constant voltage mode and power started decreasing continuously.Total demanded energy and charged energy of battery 1 and battery 2 equaled 36 GJ, 36.42GJ, 29.77 GJ and 28.62 GJ, respectively, which implies that the energy accuracy is in the range of 95~105%.The simulation well proved that the proposed control method can achieve optimal power distribution with fast response, high accuracy and high stability.

Experimental Results
In order to further demonstrate the validity of the proposed scheme, an experimental prototype was carried out as shown in Figure 11, where the demand power range of Load 1 and Load 2 were set as 1.5~2.5 W and 3~4 W, respectively.
Accordingly, the power of Load 1 and Load 2 meet the demand, namely 2 W and 3 W. Optimal power distribution is achieved through a four-step adjustment.At this time, the output power of DC power supply is 6.64 W. Total efficiency of the charging system is 75.3%.Thus, the experimental results well agree with the theoretical analysis and simulated results, which has verified that the proposed scheme can ensure the optimal power distribution for multi-objective WPT systems with fast response, high stability and high accuracy.As shown in Figure 12a, the procedure stated in power surplus condition, each load adjusted their equivalent impedance to its minimum value, namely R e1 = 2.4 Ω and R e2 = 3.7 Ω.The voltages of Load 1 and Load 2 are 8 V and 9 V, respectively.The power of Load 1 and Load 2 reached the maximum value, namely 4.3 W and 5.4 W. Due to the sharp increment of the excitation current, the capacitor C p is adjusted to 47 nF, aiming to reducing the load power immediately.As shown in Figure 12b, due to the adjustment of the total equivalent impedance, the voltage and the power of Load 1 and Load 2 decrease to 4.9 V, 5.1 V, 1.6 W and 1.7 W, respectively, load power demand is not satisfied.In the case of power deficiency, the proposed impedance regulator increases the equivalent impedance of Load 1 and Load 2 to 7.2 Ω and 11 Ω for a larger portion of total power supply, the excitation current decreases correspondingly.As shown in Figure 12c, the voltage of each load decreases to 2.6 V and 2.2 V. Additionally, the load power reduces to the minimal values, namely 1.3 W and 1 W. Since total impedance increases sharply, namely the excitation current increases sharply, the capacitor array C p changes the capacitance to 16.8 nF.The optimal value of the total equivalent impedance is found and total supply power increases and matches the power demand.As shown in Figure 12d, the impedance of Load 1 and Load 2 are adjusted in the proportion of the power demand.Accordingly, the power of Load 1 and Load 2 meet the demand, namely 2 W and 3 W. Optimal power distribution is achieved through a four-step adjustment.At this time, the output power of DC power supply is 6.64 W. Total efficiency of the charging system is 75.3%.Thus, the experimental results well agree with the theoretical analysis and simulated results, which has verified that the proposed scheme can ensure the optimal power distribution for multi-objective WPT systems with fast response, high stability and high accuracy.

Conclusions
This paper proposed an optimal power distribution scheme for multi-objective WPT systems, which can effectively satisfy various power demands from multiple loads without any communication networks.By adopting the DC/DC-converter-based load impedance adjustment and the capacitor-array-based total impedance adjustment schemes, the proposed power distribution scheme can successfully satisfy different power demands for multiple loads with fast response, high stability and high accuracy.Thus, this can be used for charging portable electronics, implantable devices and EVs.The corresponding simulated and experimental results well agree with the theoretical analysis, which has illustrated the enhanced transmission performance for multi-objective WPT systems.

Conclusions
This paper proposed an optimal power distribution scheme for multi-objective WPT systems, which can effectively satisfy various power demands from multiple loads without any communication networks.
By adopting the DC/DC-converter-based load impedance adjustment and the capacitor-array-based total impedance adjustment schemes, the proposed power distribution scheme can successfully satisfy different power demands for multiple loads with fast response, high stability and high accuracy.Thus, this can be used for charging portable electronics, implantable devices and EVs.The corresponding simulated and experimental results well agree with the theoretical analysis, which has illustrated the enhanced transmission performance for multi-objective WPT systems.

Figure 2 .
Figure 2. Relationship between R se , P ex , e and P rei .

Figure 3 .
Figure 3. Relationship among equivalent impedance, power efficiency and Cp.

Figure 3 .
Figure 3. Relationship among equivalent impedance, power efficiency and C p .

Figure 3 .
Figure 3. Relationship among equivalent impedance, power efficiency and Cp.

Figure 9 .
Figure 9. EV charging procedure.(a) Whole charging procedure.(b) Detailed charging procedure of power fluctuation.(c) Detailed procedure in power demand variation.

Figure 9 .
Figure 9. EV charging procedure.(a) Whole charging procedure.(b) Detailed charging procedure of power fluctuation.(c) Detailed procedure in power demand variation.

Figure 10 .
Figure 10.Charging procedure of batteries and super capacitors.(a) Charging procedure of batteries.(b) Charging procedure of super capacitors.Figure 10.Charging procedure of batteries and super capacitors.(a) Charging procedure of batteries.(b) Charging procedure of super capacitors.

Figure 10 .
Figure 10.Charging procedure of batteries and super capacitors.(a) Charging procedure of batteries.(b) Charging procedure of super capacitors.Figure 10.Charging procedure of batteries and super capacitors.(a) Charging procedure of batteries.(b) Charging procedure of super capacitors.

Figure 10b depicts the
Figure 10b depicts the charging procedure of two super capacitors.Power demand varies continuously.Maximum overshoot of super capacitor 1 and super capacitor 2 equals 0.89 and 0.75 respectively.Total demanded energy and charged energy of super capacitor 1 and super capacitor 2 equals 19.9 MJ, 19.15 MJ, 7.99 MJ and 7.56 MJ, respectively.The energy accuracy is in the range of 95-105%.The simulation well proved that the proposed control method can achieve optimal power distribution with fast response, high accuracy and high stability.
record the excitation current to detect if total equivalent impedance changes.end if if t%50 = 0 Initialize(Rexmax, Rexmin) //every 50 steps reinitialize Rexmax and Rexmin.end if t←t + 0.001 //each step cost 1 ms.end record the excitation current to detect if total equivalent impedance changes.The pseudo code of the individual equivalent impedance adjustment is shown in Algorithm 2.