# Optimization of Multiresonant Wireless Power Transfer Network Based on Generalized Coupled Matrix

Authors: Qiang Zhao; Anna Wang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

Magnetic coupling resonant wireless power transfer network (MCRWPTN) system can realize wireless power transfer for some electrical equipment real-time and high efficiency in a certain spatial scale, which resolves the contradiction between power transfer efficiency and the power transfer distance of the wireless power transfer. A fully coupled resonant energy transfer model for multirelay coils and ports is established. A dynamic adaptive impedance matching control based on fully coupling matrix and particle swarm optimization algorithm based on annealing is developed for the MCRWPTN. Furthermore, as an example, the network which has twenty nodes is analyzed, and the best transmission coefficient which has the highest power transfer efficiency is found using the optimization algorithm, and the coupling constraints are considered simultaneously. Finally, the effectiveness of the proposed method is proved by the simulation results.


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## Body

## 1. Introduction

Since wireless power transfer was firstly proposed by Nikola Tesla 100 years ago [1], much attempts have been devoted to this field. But these attempts did not get much breakthrough for several decades. Until 2006, a MIT research team found that efficient mid-range wireless power transfer could be achieved by utilizing near field magnetic coupling between two identical resonators [2, 3]. This progress excited renewed interest in this subject.The magnetic coupling resonant wireless power transmission network (MCRWPTN) is to achieve wireless energy transmission through the resonant coils. In order to improve the transmission efficiency and power and adapt to the needs of the system, the MCRWPT has a variety of advanced multiresonator transmission structures, including relay resonator structure, multitransmit coil, or transmit coil array structure, multireceive coil, or multiple load structure. With the increase of the complexity of the multiresonator structure, a series of new design problems have been introduced in the MCRWPT system, which is a hot research direction in the field of MCRWPT technology.From the mutual inductance formula, the transmission distance can be increased by increasing the size of the coil, but it is not feasible to apply the method to the limit of the coil geometry. Reducing the load of the transmitter and receiver can also increase the transmission distance of [4], but the actual application of the load cannot be arbitrarily changed; this method also has a lot of limitations. The increase of relay resonator is an effective way to improve the transmission distance.The concept of relay coil was put forward in [5] based on the induction of electromagnetically induced transparency in quantum interference. By using the coupled mode theory, the energy transfer efficiency of the system was greatly improved by adding a coil with the same resonant frequency between the transmitting coil and the receiving coil. Further study in [6–10] showed that the higher transmission efficiency and power level can be achieved at a larger distance by increasing the relay resonator.The effect of the addition of 1 relay coil on the transmission efficiency of the MCRWPT system with four coils was studied in [7], and the coaxial relay coil placed on the system to enhance the transmission efficiency was greater, but the vertical display had wider practical application prospects. The [8] proposed MCRWPT array resonator based on N resonator through the relay placed level could achieve N times the diameter of the resonator of wireless power transmission (WPT) distance, when using the 10-array resonator, the efficiency of energy transmission was still close to 85%. An efficient WPT system with a relay coil was designed in [10]. Based on the analysis and simulation of the circuit model, the appropriate operating frequency was selected to achieve a transmission power of 6.6 kW at a distance of 200 mm and transmission efficiency was 95.57%.In the practical application of radio energy transmission, such as charging for mobile electronic devices, there was a large demand for the supply of multiple devices simultaneously. With the development of WPT system, more and more attention had been paid to the research of multireceiving coil and multiple load structure.The basic transmission characteristics such as transmission efficiency and power of multireceiving coil in [11–13] were studied by using circuit model or coupled mode theory. The research on the multireceiver coil structure in WPT was carried out in [11], which verified the feasibility of transmitting the power to a plurality of receiving coils. In a single transmitting coil to the two receiving coils power supply research in [12], the results show that although the transmission efficiency of each single coil is relatively not high, it overall can achieve higher transmission efficiency of the system. Based on the coupled mode theory, [13] was used to study the transmission of multiple load coils. With the same conclusion as in [12], with the two receiving coils, the overall transmission efficiency of multiple receive coils is higher than that of a single receiver coil, but as the number of receivers increases, the overall efficiency of the system tends to saturate and the efficiency of each individual receive coil becomes very low. In [14], the cross-coupling of multireceiving coils was considered, and the optimal design of multiload reception was realized by impedance matching of the transmitter. The paper [15] presented a novel analysis method using band-pass filter model, which made the circuit model with multireceiving coil system more simple and easier to analyze, but its use conditions were relatively harsh.In [16], the concept of time-sharing transmission in communication system was used to study the multiload charging in MCRWPT system. It was found that time-sharing wireless transmission can achieve efficient and uniform power transmission. The literature [17] was further extended to N resonant coils. The simulation results showed that when there were N resonant coils in the system and the frequency splitting occurred, there would be N efficiency peak points. When there were odd resonant coils, the system transmission efficiency could reach the peak at the original resonant frequency of the coil, and when the number of resonant coils was even, the transmission efficiency was shifted at the original resonant frequency.The impedance matching method on the load side can simultaneously realize the compensation of the frequency splitting phenomenon and the control of the power distribution. Considering the limitation of the frequency tracking method, the impedance matching is the main control strategy in the multicoil system. The impedance matching strategy for power allocation was studied in [15, 18, 19]. In the actual system, the coupling cannot be ignored when the number of receiving coils is very close [15, 18], considering the cross-coupling between multiple receiving coils. The impedance matching and power division method utilizing impedance inverters only at receiver sides was proposed. The mathematical equations in the proposed method were then generalized for arbitrary number of receivers and arbitrary number of repeaters. The coupling of the driving coil and the multiple receiving coils was studied in [19], and it was pointed out that the coupling will change the optimal impedance matching condition given the design method of matching impedance at this time.Using the scattering parameters as the objective function of the fully coupling matrix extraction and combining the particle swarm optimization algorithm with simulated annealing technique to optimize impedance matching are the goal of this work. The control strategy must overcome some difficulties such as the highly nonlinear and coupled dynamics more over the dynamic, complex, and unstructured environments which may cause unpredictable disturbances to the impedance matching. So dealing with some states in each step will improve matching speed and accuracy. Using the coupling analysis and extraction condition of fully coupling matrix to design an optimal impedance matching strategy for MCRWPTN, it has been shown that the resulting optimization has excellent performance, as demonstrated by simulations.The paper is outlined as follows. The model of multirelay MCRWPTN is established, and the coupling relation between multiple coils is analyzed in Section2. The generalized coupling matrix of the system is extracted and an optimal design method is given in Sections 3 and 4. Simulations using FEKO and MATLAB and experiments are performed to validate the new method in Section 5. Finally, Section 6 presents some conclusions.


## 2. Problem Formulation

Wireless power transfer network (WPTN) is an energy-centric network, which includes an exciting coil, a transmitting coil, a relay coil, a receiving coil, and a pickup coil. The generalized coupling model of the single energy transmission link composed ofn relay energy nodes in the WPTN network is shown in Figure 1.Figure 1
Schematic of a wireless power transfer route.An internal resonance network is composed of a transmitting, a relay, and a receiving coil, and the resonant frequency and the structure of the transmitting coil, the relay coil, and the receiving coil are the same. The exciting coil and the pickup coil are connected to the power and load ports, respectively, and form an external resonant network with the internal resonant network. The magnetic coupling of the internal resonant network forms a contactless energy transmission link, and multiple transmission links constitute the entire WPTN.In Figure1, Cp, Lp, and Rp(p=S,L) are power and load port parameters; Ci, Li, and Ri are the parameters of each coupling node. Mip is the mutual inductance between the port and the node; Mi,i+1 is the mutual inductance between adjacent nodes.TheZ parameter equation of the two-port system can be expressed as(1)Z∗I=jmn-λr-jp∗I=V,(2)ZSSZS1⋯ZSnZSLZS1Z11⋯Z1nZ1L⋮⋮⋱⋮⋮ZSnjZ1n⋯ZnnZnLZSLZ1L⋯ZnLZLLISI1⋮InIL=Vs0⋮00,where [Z] is the system equivalent impedance matrix and [m] is a fully coupling matrix. For systems with i(i=1,…,n) resonant networks and p ports, the order of the equivalent impedance matrix [Z] is n+2. λ is the normalized frequency factor; r is the unit matrix. p is the port matrix, except for p11=ZSS and pn+2,n+2=ZLL and other elements are zero.The impedance matrix of the system is shown in (3). Matrix [Z] is derived from the loop equation formulation of the source and load, external and inner-resonator couplings, and resonator impedances. (3)Z=RS+jωLS-j1ωCSjω0MS1jω0MS2⋯jω0MS,n-1jω0MSnjω0MSLjω0MS1R1+jωL1-j1ωC1jω0M12⋯jω0M1,n-1jω0M1njω0M1Ljω0MS2jω0M12R2+jωL2-j1ωC2⋯jω0M2,n-1jω0M2njω0M2L⋮⋮⋮⋱⋮⋮⋮jω0MS,n-1jω0M1,n-1jω0M2,n-1⋯Rn-1,n-1+jωLn-1,n-1-j1ωCn-1,n-1jω0Mn-1,njω0Mn-1,Ljω0MSnjω0M1njω0M2n⋯jω0Mn-1,nRn+jωLn-j1ωCnjω0MnLjω0MSLjω0M1Ljω0M2L⋯jω0Mn-1,Ljω0MnLRL+jωLL-j1ωCL.Without loss of generality, we can assume that all nodes are fixed and stationary randomly distributed within the network, and each node has a unique number. All nodes have the same structure and circuit topology, and the coupling between nonadjacent nodes is negligible. Each node has the same function and status equivalence as the source node, relay node, and destination node.The impedance matrix[Z] is normalized as shown in (4)Z~=×RS-1×ω0L1-1×ω0L2-1⋮⋮×ω0Ln-1×RL-1×RS-1ZSSZS10⋮000×ω0L1-1ZS1Z11Z12⋮000×ω0L2-10Z12Z22⋮000⋯⋯⋯⋯⋮⋱⋯⋯⋯⋯000⋮Zn-1,n-1Zn-1,n0×ω0Ln-1000⋮Zn-1,nZnnZnL×RL-1000⋮0ZnLZLL,where 1/Rp is an impedance scaling factor (p=S,L), 1/ω0Li  (i=1,…,n) is a resonator scaling factor, and ω0=1/LiCi.Generalize matrix[Z] such that the resultant matrix can be normalized in terms of couplings and port impedances only. The resultant matrix is the normalized impedance matrix [Z~], as shown in (5)Z~=1+jXSRSjωMS1ω0RSL10⋯000jωMS1ω0RSL11Q1+jλjM12L1L2⋯0000jM12L1L21Q2+jλ⋯000⋮⋮⋮⋱⋮⋮⋮000⋯1Qn-1+jλjMn-1,nLn-1Ln0000⋯jMn-1,nLn-1Ln1Qn+jλjωMnLω0LnRL000⋯0jωMnLω0LnRL1+jXLRL.Normalized variables are used in representing the impedance matrix as follows. Qi=ω0Li/Ri is the unloaded Q factor of resonator i,  λ=ω/ω0-ω0/ω is the normalized frequency factor, which reflects the relationship between the resonant network and the frequency in the frequency response of MCRWPT. The coupling coefficient of the internal resonator is defined as ki,i+1=Mi,i+1/LiLi+1, and the external coupling between resonator i and port p for i=1,n and p=S,L is defined as ES1=ωMS1/ω0RSL1 and EnL=ωMnL/ω0LnRL. The normalized reactance of port p for p=S,L is xS=XS/RS and xL=XL/RL.This normalized impedance matrix will be referred to as[Z~]. Substituting in these variables allows us to make the decomposition of the matrix seen below:(6)Z~=jxSEs10⋯000Es1-jQ1k12⋯0000k12-jQ2⋯000⋮⋮⋮⋱⋮⋮⋮000⋯-jQn-1kn-1,n0000⋯kn-1,n-jQnEnL000⋯0EnLxL+100⋯000000⋯000000⋯000⋮⋮⋮⋱⋮⋮⋮000⋯000000⋯000000⋯001+jλ000⋯000010⋯000001⋯000⋮⋮⋮⋱⋮⋮⋮000⋯100000⋯010000⋯000.Using algebraic manipulation, [Z~] is separated into constitutive components into the form given by(7)Z~=jm+jλr+p.For the purposes of WPT optimization, the variableλ simply reflects the frequency dependency of the MRCWPT frequency response. The unit matrix [r] of the resonant network is n∗n identity matrix, the port normalized impedance matrix [p] is zero except for the power and load port elements, the coupling matrix [m] is a reciprocal matrix of n∗n, and the other nondiagonal elements represent the coupling between the two resonators. If it is zero, the corresponding two resonators are not coupled.


## 3. Fully Coupling Matrix Extraction

From (1) we can see that the current vector can be expressed as(8)I=-jZ-1V.Thus, the scattering parameter S of the entire coupling circuit can be expressed as(9)S21=2ZSSZLLIn=-2jZSSZLLZSL-1,S11=1-2ZSSI1=1+2jZSSZSS-1.We can get from (7) that(10)A=-jZ~=m-jp+λr,where the matrix [A] is built from the coupling matrix [m], the source matrix [p], and the resonator matrix [r]. It can be shown that, from inverting the [A] matrix, the scattering parameters can be found.(11)S21=-2jASL-1,S11=1+2jASS-1.The objective function K is uniquely determined by the zero position, the pole position, and the in-band return loss.(12)K=∑i=1nS11ωzi2+∑i=1pS21ωpi2+S11ω=-1-ε1+ε22+S11ω=1-ε1+ε22,p≤n,where ωpi is artificially specified quantity and ωzi is a function of ωpi.  ωzi and ωpi are the zero and pole of the characteristic function Cn(ω), respectively. The latter two are used to determine the in-band return loss. When K=0, the target is satisfied and the optimization is finished. Set the following expression:(13)Cnω=12Gnω+Gn′ω∏i=1n1-ω/ω0,(14)Gnω=∏i=1nω-1ω0+ω2-11-1ω02,(15)Gn′ω=∏i=1nω-1ω0-ω2-11-1ω02,(16)Gnω=Unω+Vnω,(17)Gn′ω=Unω-Vnω,(18)Unω=u0+u1ω+u2ω2+···+unωn,(19)Vnω=ω2-1v0+v1ω+v2ω2+···+vnωn.G1(ω) can be obtained as substituting the first pole ωp1 into (14), and the values of u0,u1,v0,v1 are obtained by comparing the coefficients, and then determine U1(ω), V1(ω). (20)Gnω=Un-1ω+Vn-1ωω-1ω0+ω2-11-1ω02.G2(ω) can be obtained as substituting the first pole ωp2, and so on, until the N pole is replaced. From formulas (16) and (17), the molecules of Cn(ω) can be obtained by iteration.(21)NUMCnω=12Gnω+Gn′ω=Unω.Solving the roots of Un(ω)=0 can obtain the N zeros of the characteristic function. The parameters of the objective function can be determined after substituting the zero point, the pole, and the return loss into (12). The relation between generalized coupling matrix [m] and objective function K can be established by the above derivation, and the optimization model of the coupling matrix extraction is shown in formula (22).(22)minK=∑i=1nS11ωzi2+∑i=1pS21ωpi2+S11ω=-1-ε1+ε22+S11ω=1-ε1+ε22,p≤ns.t.Cnω=12Gnω+Gn′ω∏i=1n1-ω/ω0Gnω=∏i=1nω-1ω0+ω2-11-1ω02Gn′ω=∏i=1nω-1ω0-ω2-11-1ω02.


## 4. Impedance Matching Optimization

In order to improve the extraction speed of generalized coupling matrix, a particle swarm optimization algorithm based on simulated annealing is presented. Fitness function can be expressed as follows:(23)fittness=Gnω=∏i=1nω-1ω0+ω2-11-1ω02.Firstly, the initial population is generated randomly, and a new group of individuals is generated by the basic particle swarm optimization algorithm.(24)Vi+1=w∗Vi+c1∗rand∗pbest-xi+c2∗rand∗gbest-xixi+1=xi+Vi,where pbest is the best location of the particle itself and gbest is the best location for all particles in the population.Then, the crossover and the Gaussian mutation operation are carried out independently. Simulated annealing is performed for each individual, and the result is used as an individual in the next generation.In each evolution, crossover operation selects a specified number of particles into a pool according to crossing probability, which are randomly crossed to produce the same number of children, instead of parents to keep the number of particles unchanged. The position of the child particle is calculated by the weighted sums of the parent particles.(25)child1x=p∗parent1x+1-p∗parent2x,child2x=p∗parent2x+1-p∗parent1x,where x is the D-dimension position vector and childk(x) and parentk(x),  k=1,2, respectively, are the location of the child or parent particles. p is a uniform random number vector of D-dimension, with each component in [0,1] value. The speed of the child particles is given by the following formula:(26)child1v=parent1v+parent2vparent1v+parent2vparent1v,child2v=parent1v+parent2vparent1v+parent2vparent2v,where v is the velocity vector of the D-dimension.In each evolution, the specified number of particles is selected by mutation operation according to the mutation probability, which was mutated by Gaussian mutation operator, and the mutated particles are used instead of the original particles.(27)mutationx=x∗1+Gaussianδ.The implementation of the algorithm is composed of two parts. Firstly, the evolution of the basic particle swarm optimization algorithm (global search) is used to produce a better group, and then the crossover and mutation operations are applied, and the particles are further optimized by simulated annealing (local search). The evolution process is iterated until the termination condition is satisfied. The algorithm flow is as follows:(1)
Initialization parameters.(2)
The population of randomly generatedN particles.(3)
Using (24) to manipulate the particles in the population.(4)
The population generated by (3) select the particle to form the subpopulation by the crossover probability and carries out the following operations to produce a new species population: from the subpopulation randomly selected individuals xj and xk according to (25) for cross-operation to produce two new individuals xj′, xk′, calculate the adaptive function value fxj,fxk,fxj′,fxk′; if min1,exp-fxj′-fxj′/T>random, then xj′ is a new individual; if min⁡1,exp⁡-fxk′-fxk′/T>random, then xk′ is a new individual, where random is a random number on the [0,1] interval.(5)
The population generated by crossing selects the particle to form the subpopulation by the mutation probability and carries out the following operations to produce a new species population. An individual is selected from the subpopulation to perform a Gaussian mutation operation to produce a new individualxj′, and calculate the adaptive function value fxj and fxj′; if min⁡1,exp⁡-fxj′-fxj′/T>random, then xj′ is a new individual.(6)
If the current optimal individual satisfies the convergence condition, the evolution process ends successfully and returns the global optimal solution.(7)
If the number of evolutions is less than the maximum number of times of evolution, the annealing temperature of the population is modified, and then return to step (3).


## 5. Simulation Results

In order to optimize the extraction of generalized coupling matrix parameters fast convergence, the initial value should be set close to the actual value. The parameters of the MCRWPTN system are shown in Table1. The optimized and simulated S parameters are shown in Figures 2–4.Table 1
The system parameters configuration.
Parameters
Values
R
S
50 Ω
R
L
50 Ω
Port coils
Radiusr
100 mm
Number of turnsN
1 turn
Resonant coils
RadiusR
100 mm
Number of turnsN
5 turns
Resonant frequencyf0
12.85 MHz
Coupling coefficient between resonatorski,i+1
0.2
Coupling coefficient between nonadjacent resonators
0Figure 2
The curves of scattering parameters of WPTN.Figure 3
The curves of the scattering parameters of external coupling network of power port.Figure 4
The curves of the scattering parameters of external coupling network of load port.When the number of ports isp=2, the number of coupling relay coils is n=20, and the system scattering parameter curve is shown in Figures 2–4. These three curves represent the specific distribution of the scattering parameters of the total transmission network and the two transmission subnetworks of the MCRWPTN, respectively, which can reflect the law of the scattering parameters of the system with frequency, where S21 is forward transmission coefficient and S11 is reflection coefficient of port 1. They can respond to the transmission characteristics of the system, the higher S21 is better, and the opposite of S11 is as low as possible.As can be seen from Figure2, the frequency division of the system has low frequency resonance and high frequency resonance. At the resonant frequency, S21 reaches the maximum, and S11=0.1 is the smallest. So the maximum power transmission efficiency is obtained. The response curve obtained after optimization is in good agreement with the S parameters obtained from the simulation. From Figures 3 and 4, the existence of external coupling network makes the port impedance match well.Figure5 shows the simulated S11 and S22 responses of the complex load impedance accommodated response measured at the interresonator coupling location kST=0.01. As shown, the resonance dip of S22 response approaches the center of the Smith chart. This indicates that the output impedance of the impedance matching optimized system is the complex conjugate of Zp=Rp+jXp.Figure 5
Simulation validation of complex load impedance accommodation.Figure6 is the modulus value of the reflection coefficient obtained by different algorithms. By comparing the three different algorithms of reflection coefficient magnitude, the particle swarm optimization combined with simulated annealing algorithm is a new type of impedance matching algorithm, and the reflection coefficient tends to zero to achieve impedance matching finally, which is better than the improved particle swarm algorithm in speed.Figure 6
The modulus value of the reflection coefficient obtained by different algorithms.The relationship between the number of coupled nodes and the number of iterations is shown in Figure7. As the number of nodes increases, the algorithm is superior to other algorithms and is easy to extend to multirelay MCRWPTN.Figure 7
Multicoupled node network optimization.


## 6. Conclusion

A fully coupling matrix for multirelay MCRWPTN is proposed and theory to impedance matching optimization algorithm is developed. This combination will emphasize not only the energy transfer but also the matching speed and scalability. The optimization algorithm combines the ability of the simulated annealing algorithm to jump out of the local optimal solution and the global optimization ability of the particle swarm optimization algorithm, which avoids the shortcomings of falling into the local extreme point and improves the algorithm convergence speed and accuracy. Theoretical results are supported by numerical simulations with impedance matching of the proposed fully coupled matrix and optimization algorithm.



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