Dual-Frequency Output of Wireless Power Transfer System with Single Inverter Using Improved Di ﬀ erential Evolution Algorithm

: In wireless charging devices, a transmitter that applies a single inverter to output dual-frequency can e ﬀ ectively solve the charging incompatibility problem caused by di ﬀ erent wireless charging standards and reduce the equipment volume. However, it is very di ﬃ cult to solve the switching angle of the modulated dual-frequency waveform, which involves non-linear high-dimensional multi-objective optimization with multiple constraints. In this paper, an improved di ﬀ erential evolution (DE) algorithm is proposed to solve the transcendental equations of switching angle trains of dual-frequency programmed harmonic modulation (PHM) waveform. The proposed algorithm maintains diversity while preserving the elites and improves the convergence speed of the solution. The advantage of the proposed algorithm was veriﬁed by comparing with non-dominated sorting genetic algorithm II (NSGA II) and multi-objective particle swarm optimization (MOPSO). The simulation and experimental results validate that the proposed method can output dual-frequency with a single inverter for wireless power transfer (WPT).


Introduction
The wireless power transfer (WPT) systems transmit electrical power from the power supply to the load without any wired connection [1]. It avoids potential safety hazards such as hot-plugging and wire shorting in wired power transmission, which greatly increases safety. With the continuous development of technology, WPT has been widely used in the fields of portable electronics, implantable medical devices, electric vehicles [2][3][4][5], and so on. The rapid development of WPT technology has inspired an era of wireless charging.
Although WPT technology has the above advantages, there are also some challenges in the field of wireless charging, such as inconsistent charging frequency standards and the need to transmit information in the wireless charging process.
Currently, there are several international standards in the wireless charging field. In February 2017, the Wireless Power Consortium officially published the Qi interface standard and the low-power specification [6]. The Aviation Fuel Alliance gave the wireless power transfer system baseline system specification (BSS) [7] in 2014. This difference in frequency standards will bring great inconvenience to consumers and manufacturers. Wireless charging devices designed by manufacturers according two adjacent switching angles. Since the waveform is a quarter periodic symmetric waveform, even harmonics are automatically eliminated, and only odd harmonic components exist.   can be simplified by analyzing its 1/4 period. The α 1 , α 2 , α 3 , . . . , α N in Figure 2b is the switching angle (0 < α 1 < α 2 < α 3 < . . . < α N < π/2), which is the phase angle sequence of the switches, and t 01 , t 12 , t 23 , . . . , t N /2 represents each pulse width in the figure, and its physical significance is the time interval between two adjacent switching angles. Since the waveform is a quarter periodic symmetric waveform, even harmonics are automatically eliminated, and only odd harmonic components exist. two adjacent switching angles. Since the waveform is a quarter periodic symmetric waveform, even harmonics are automatically eliminated, and only odd harmonic components exist. The output voltage could be represented as a Fourier series, The corresponding frequency component derived from Equation (1) where Vdc is the input DC voltage of the inverter, Vac1 (fundamental frequency) and Vac2 (high frequency) represent the desired magnitudes of two output frequencies.  The output voltage could be represented as a Fourier series, The corresponding frequency component derived from Equation (1) is where V dc is the input DC voltage of the inverter, V ac1 (fundamental frequency) and V ac2 (high frequency) represent the desired magnitudes of two output frequencies. To simplify the subsequent analysis and experiments, the modulation index of the fundamental frequency and high frequency is defined as the value of output AC voltage normalized by DC voltage, It is assumed that V dc = 1 and the output voltage (V ac1 and V ac2 ) are equal to their respective modulation indices. The m is the highest harmonic that can be eliminated. To avoid transcendental equations, m is set to 2N − 1 to form a square matrix, where N is the number of switches in the quarter symmetrical period.
Once the switching angles α 1 , α 2 , α 3 , . . . , α N within the quarter symmetric waveform are found, from the waveform characteristics of Figure 2a, the remaining angle values can be obtained from the following formula: According to Equation (2), if the greatest common divisor of the two frequencies is the fundamental frequency component, it can calculate any harmonic combination. This gives the flexibility of the choice of dual-frequency PHM output frequency. At the same time, the higher the order of the high Energies 2020, 13, 2209 5 of 15 frequency of the modulated output relative to the fundamental frequency, the more switching angles are required.
As for SHEPWM, it eliminates the low-order harmonics, retains and outputs only the fundamental frequency component that is the only nonzero component in the switching angle solution system. However, the dual-frequency PHM requires that two components are nonzero in the switching equations, which greatly increases the difficulty of solving the switching angle. In addition, the two output frequencies should be separated as far as possible to reduce the interference between the two frequencies. Consequently, it increases the order of the high-frequency components, which undoubtedly increases the number of switching angles and the dimension of the equations.
Based on the above analysis, considering the complexity of solving the switching angle of dual-frequency PHM, an improved DE algorithm combined with Newton iteration is adopted to solve the problem.

Proposed Improved DE Algorithm
The problem of solving the switching angle of dual-frequency PHM can be transformed into finding an optimal solution for the following equations, that is, finding a set of switching angles that can satisfy the following constraints simultaneously.
s.t α j,min < α j < α j,max , j = 1, 2, 3, . . . , N where N is the number of switching angles in a quarter cycle of the symmetric square wave, and m is the highest harmonic order that can be eliminated (m = 2N − 1). α j,min and α j,max are the lower and upper bounds of the j-th switching angle, respectively. Considering that the waveform is modulated within a quarter of a period, the range of (α j,min ,α j,max ) is (0, π/2).
The proposed algorithm in this paper introduces the concept of the Pareto frontier on the basis of DE, adds an adaptive mutation operator in the variation link, improves the global search ability of the algorithm, and reduces the possibility of falling into the local optimal solution. After that, the value obtained by evolution is taken as the initial value and iterated into the Newton iterative algorithm. The general structure of the improved DE algorithm is shown in Figure 3.
The cross operation in this algorithm is consistent with the classical DE algorithm, and will not be described here. The termination condition of the algorithm program is that the evolution exceeds the preset maximum evolutionary generation. The remaining main steps to solve the dual-frequency PHM switch angle using the improved DE algorithm are as follows.

Selection
One of the main differences between the proposed algorithm in this paper and the classic DE is the selection. The proposed algorithm uses the concept of the Pareto frontier non-dominated solution to sort the equations corresponding to each set of switching angles of this generation and obtains the Pareto frontier of the equation solution, thus obtaining the Pareto frontier of each set of switching angles ( 0 i P ) as a high-quality individual into the next generation of evolution.
According to the properties of the Pareto front, the number of switching angle groups in the Pareto front must be less than or equal to the population size NP. Assuming that the number of Pareto frontier switching angular groups is Num, then i = 1, 2, 3, ..., Num in 0 i P , where Num ≤ NP.

Newton Iteration
The Newton iteration method has strong convergence, but its limitation is that it needs a suitable initial value. The introduction of Newton's algorithm can speed up the convergence of the proposed algorithm, especially in the later stage of the algorithm. Each set of 0 i P in the Pareto front obtained in Section 4.3 is taken as the initial value to be iterated by the Newton iteration method. The

Parameter Setup
The parameters required by the improved algorithm include solution space dimension (N), population size (NP), mutation factor (F 0 ), crossover probability (CR), boundary constraint condition (α min , α max ), DC input voltage (V dc ) and expected amplitude of two output frequencies V ac1 (fundamental frequency) and V ac2 (High frequency). The boundary must satisfy Equation (5), and F 0 and CR are within the range of [0, 1].

Initial Population
The initial generation population is randomly initialized according to Equation (6).
where the α 0 i represents the initial generation i-th group switching angle in the population, and the α 0 i·j represents the j-th switching angle in α 0 i . The number of groups of switching angles is expressed by NP. rand ( ) is a random number randomly distributed within the interval of (0, 1). Since each set of α 0 i values is the switching angle within a quarter of a week and its angle should increase with time, the switching angle (α 0 i·j ) in each set of α 0 i values should also be sorted from small to large after initializing the population.

Selection
One of the main differences between the proposed algorithm in this paper and the classic DE is the selection. The proposed algorithm uses the concept of the Pareto frontier non-dominated solution to sort the equations corresponding to each set of switching angles of this generation and obtains the Pareto frontier of the equation solution, thus obtaining the Pareto frontier of each set of switching angles (P 0 i ) as a high-quality individual into the next generation of evolution. According to the properties of the Pareto front, the number of switching angle groups in the Pareto front must be less than or equal to the population size NP. Assuming that the number of Pareto frontier switching angular groups is Num,

Newton Iteration
The Newton iteration method has strong convergence, but its limitation is that it needs a suitable initial value. The introduction of Newton's algorithm can speed up the convergence of the proposed algorithm, especially in the later stage of the algorithm. Each set of P 0 i in the Pareto front obtained in Section 4.3 is taken as the initial value to be iterated by the Newton iteration method. The convergence of Newton's iteration is determined by setting the solution accuracy. When the solution accuracy is satisfied, the algorithm converges, and the accuracy is set to 10 −4 (unit: degree) in this algorithm. If the iteration is convergent and the switching angle boundary condition is satisfied, it indicates that a set of switching angles is successfully found. At this time, the set of switching angle values is output and the algorithm is terminated. Otherwise, perform Section 4.5.
Considering that Newton iteration requires initial values with high precision, the initial value obtained by the selection operation may not meet the requirement. Therefore, it is necessary to set an upper limit on the number of iterations to reduce the iteration time and prevent invalid iterations. In this paper, the maximum number of iterations is set to 10 times.

Mutation Operation
The individual mutation in DE is achieved by randomly selecting two different individuals in a population and synthesizing the vectors with the mutated individuals after scaling their vector differences. Evolution in each generation is accompanied by individual mutation.
The mutation operation formula is as follows: where, G stands for evolutionary generation; r 1 , r 2 , and r 3 are three random numbers that are not equal to i and less than or equal to Num; F is a proportional factor with a range of In order to improve the global search ability of the solution and avoid falling into the local optimum, the mutation chain adopts the adaptive mutation operator. The formula is as follows, where G m is the maximum evolutionary generation and G stands for current evolutionary generation.
The mutation operator has a larger value (2F 0 ) in the early evolution stage to maintain individual diversity, thus avoiding premature maturity. With the increase of evolutionary generation, the mutation operator gradually reduces to F 0 to retain good individual information.
The mutated individuals may not meet the preset boundary conditions in which case it is necessary to judge whether M G i,j in the individual M G i after mutation meets the boundary condition. For the M G i,j that does not meet the boundary condition, the method of Section 4.2 shall be adopted to regenerate them until the boundary conditions are satisfied.

Verification
To verify the effectiveness of the improved DE algorithm proposed in this paper for solving the switching angle of a single inverter to achieve dual-frequency output, the improved DE was compared with the existing multi-objective algorithm, and the switching angles obtained by the improved DE was verified by simulation and experiment with the DF inverter.

Algorithm Comparison
In order to illustrate the effectiveness of the improved DE algorithm in solving dual-frequency PHM switching angles and the performance in solving multidimensional multi-objective optimization problems, the proposed algorithm is compared with non-dominated sorting genetic algorithm II (NSGA II) [31] and multi-objective particle swarm optimization (MOPSO) [32] about the solving time of solving 2, 3, 5 and 10 switching angles of dual-frequency PHM with different solving accuracy.
The parameters of the algorithm are set as follows. In the improved DE, NP = 10N, CR = 0.9, F 0 = 0.5. The population size of NSGA II is 200, the crossover probability is 0.9, the mutation probability is 1/N, the simulated binary crossover of η c = 20 and the polynomial mutation strategy of η m = 20 are adopted. The population size of MOPSO is 200, the external population size is 200, and the number of mesh divisions of each dimension in the adaptive mesh is 50. The inertia weight decreases linearly with evaluation times from ω 0 = 0.7 to ω 1 = 0.2 and the two acceleration constants are r 1 = r 2 = 2.0. Each algorithm runs randomly 10 times for each solution condition. When the estimated times of the objective function reaches 10,000 times, the algorithm stops running. Table 1 describes the time taken by the three algorithms to solve 2, 3, 5 and 10 switching angles of dual-frequency PHM with different solution accuracy (10 −3 or 10 −4 ).
From the comparison of the data obtained in Table 1, it can be found that the improved DE is better than NSGA II and MOPSO in terms of solution time and ability to solve more objectives. The improvement of the solution accuracy has almost no impact on the performance of the algorithm. NSGA II has good performance in solving two and three objective problems, but with the increase of the number of solving problems, the performance of NSGA II drops sharply, and five switching angles cannot be solved under the maximum evaluation times. With the increase in the number of objectives, the solution time of MOPSO increases obviously, and it is very sensitive to the solution precision. The improvement of the solution precision will greatly increase the solution time. Under the set parameters, this algorithm can only solve five switching angles. This comparative study shows that the improved DE is more suitable for solving multi-objective problems, especially when the number of objectives is more than five, NSGA II and MOPSO lose the solving ability and the improved DE still has solving capability.

Simulation Verification
In the verification process of simulation and experiments, the number of switching angles in the 1/4 cycle is determined to be 20 (N = 20). The fundamental frequency output is 10 kHz, and the 37th harmonic (370 kHz) is used as the high-frequency output. When the DC voltage is 10 V, the amplitude of the fundamental frequency voltage and the high-frequency voltage is 6 V. Therefore, the parameter in Equation (4) is set to V ac1 = V ac2 = 0.6, V dc = 1. The quarter-cycle switching angle under the solution accuracy of 10 −4 can be obtained, as shown in Table 2. The parameters in the simulation model and experimental circuit are shown in Table 3. MATLAB/Simulink is used as the simulation tool to verify the improved algorithm. The topology of the DF-inverter system simulation circuit in this paper is shown in Figure 4. A full-bridge inverter has been formed using two GaN-based phase-legs. L 1 and C 1 resonate at the fundamental frequency, L 2 and C 2 form high-frequency resonance tank.

R1, R2
Load 10 Ω, 10 Ω MATLAB/Simulink is used as the simulation tool to verify the improved algorithm. The topology of the DF-inverter system simulation circuit in this paper is shown in Figure 4. A full-bridge inverter has been formed using two GaN-based phase-legs. L1 and C1 resonate at the fundamental frequency, L2 and C2 form high-frequency resonance tank. The waveform of the output voltage Vab of the inverter within one cycle is obtained by simulation as shown in Figure 5, which displays the turn-on and turn-off angle of the inverter. The FFT result is shown in Figure 6. The fundamental frequency and the 37th harmonic amplitude are about 6 V, which is basically consistent with the set value, and the harmonics between the two pre-output frequencies are completely eliminated, in accordance with the expected effect.  The waveform of the output voltage V ab of the inverter within one cycle is obtained by simulation as shown in Figure 5, which displays the turn-on and turn-off angle of the inverter. The FFT result is shown in Figure 6. The fundamental frequency and the 37th harmonic amplitude are about 6 V, which is basically consistent with the set value, and the harmonics between the two pre-output frequencies are completely eliminated, in accordance with the expected effect.  The resonant tanks L1-C1-R1 and L2-C2-R2 separate the fundamental and high-frequency components of the inverter output, respectively. The load voltage waveforms of the fundamental and high-frequency are shown in Figure 7. As can be seen from the waveform in the figure above, the waveform is smooth on the whole and the amplitude is relatively stable, but there are some small ripples in the waveform which is caused by other harmonics higher than the 37th harmonic.

Time [ms]
Vab  The resonant tanks L1-C1-R1 and L2-C2-R2 separate the fundamental and high-frequency components of the inverter output, respectively. The load voltage waveforms of the fundamental and high-frequency are shown in Figure 7. As can be seen from the waveform in the figure above, the waveform is smooth on the whole and the amplitude is relatively stable, but there are some small ripples in the waveform which is caused by other harmonics higher than the 37th harmonic. The resonant tanks L 1 -C 1 -R 1 and L 2 -C 2 -R 2 separate the fundamental and high-frequency components of the inverter output, respectively. The load voltage waveforms of the fundamental and high-frequency are shown in Figure 7. As can be seen from the waveform in the figure above, the waveform is smooth on the whole and the amplitude is relatively stable, but there are some small ripples in the waveform which is caused by other harmonics higher than the 37th harmonic.
The resonant tanks L1-C1-R1 and L2-C2-R2 separate the fundamental and high-frequency components of the inverter output, respectively. The load voltage waveforms of the fundamental and high-frequency are shown in Figure 7. As can be seen from the waveform in the figure above, the waveform is smooth on the whole and the amplitude is relatively stable, but there are some small ripples in the waveform which is caused by other harmonics higher than the 37th harmonic.

Experimental Verification
An experimental platform was set up to verify the proposed algorithm, as shown in Figure 8. The experimental system used an FPGA chip EP4CE10F17C8 as the controller to generate dualfrequency PHM signals. After obtaining the predetermined switching angle by off-line calculation of the algorithm, the control resolution of the system hardware was improved by doubling the clock frequency of the FPGA system to 400 MHz, which improved the experimental precision of the dualfrequency PHM signal. Considering that the inverter works under high-frequency switching, a GaN-

Experimental Verification
An experimental platform was set up to verify the proposed algorithm, as shown in Figure 8. The experimental system used an FPGA chip EP4CE10F17C8 as the controller to generate dual-frequency PHM signals. After obtaining the predetermined switching angle by off-line calculation of the algorithm, the control resolution of the system hardware was improved by doubling the clock frequency of the FPGA system to 400 MHz, which improved the experimental precision of the dual-frequency PHM signal. Considering that the inverter works under high-frequency switching, a GaN-based inverter was used, which can operate at frequencies up to several MHz. The component parameters of the experimental system were consistent with the simulation parameters setting.  Experimental voltage waveforms of the full-bridge inverter output Vab, and the load voltage Vac1 and Vac2 of the high and low frequency resonant branch are shown in Figure 9, respectively, where Vab is the dual-frequency PHM waveform which is generated by the system. Its Fourier analysis is shown in Figure 10a, where the amplitude of the residual harmonics between the two frequencies is severely attenuated. In addition to the pre-output frequency, the amplitudes of the 36th and 38th harmonics were observable in the frequency range from the fundamental frequency to the 40th harmonic, but their amplitudes were only 0.611 V and 0.628 V, respectively, and the contents of other harmonics were lower. The amplitudes of the fundamental frequency and the preset high-frequency components were basically consistent with the simulation results, but the harmonics near the preset high-frequency components were not completely eliminated in the experiment, which was caused by the insufficient control resolution of the experimental platform. However, its amplitude was also very low, which was acceptable. Experimental voltage waveforms of the full-bridge inverter output V ab , and the load voltage V ac1 and V ac2 of the high and low frequency resonant branch are shown in Figure 9, respectively, where V ab is the dual-frequency PHM waveform which is generated by the system. Its Fourier analysis is shown in Figure 10a, where the amplitude of the residual harmonics between the two frequencies is severely attenuated. In addition to the pre-output frequency, the amplitudes of the 36th and 38th harmonics were observable in the frequency range from the fundamental frequency to the 40th harmonic, but their amplitudes were only 0.611 V and 0.628 V, respectively, and the contents of other harmonics were lower. The amplitudes of the fundamental frequency and the preset high-frequency components were basically consistent with the simulation results, but the harmonics near the preset Energies 2020, 13, 2209 12 of 15 high-frequency components were not completely eliminated in the experiment, which was caused by the insufficient control resolution of the experimental platform. However, its amplitude was also very low, which was acceptable.
Vab is the dual-frequency PHM waveform which is generated by the system. Its Fourier analysis is shown in Figure 10a, where the amplitude of the residual harmonics between the two frequencies is severely attenuated. In addition to the pre-output frequency, the amplitudes of the 36th and 38th harmonics were observable in the frequency range from the fundamental frequency to the 40th harmonic, but their amplitudes were only 0.611 V and 0.628 V, respectively, and the contents of other harmonics were lower. The amplitudes of the fundamental frequency and the preset high-frequency components were basically consistent with the simulation results, but the harmonics near the preset high-frequency components were not completely eliminated in the experiment, which was caused by the insufficient control resolution of the experimental platform. However, its amplitude was also very low, which was acceptable.  Figure 10b,c shows the spectrum of Vac1 and Vac2, respectively. In Figure 10b, the pre-output highfrequency component in the low-frequency branch accounts for less than 0.5% of the fundamental frequency component. The proportion of the fundamental frequency component in the highfrequency branch in Figure 10c is only 4.613% of the high-frequency component. It can be seen that the corresponding resonant frequency component in the resonance branch is large and the rest harmonic components are small, which indicates that the two frequencies generated by the system are separated very well.
As shown in Table 4, the improved DE was used to obtain the values at bipolar 20 switching  Figure 10b,c shows the spectrum of V ac1 and V ac2 , respectively. In Figure 10b, the pre-output high-frequency component in the low-frequency branch accounts for less than 0.5% of the fundamental frequency component. The proportion of the fundamental frequency component in the high-frequency branch in Figure 10c is only 4.613% of the high-frequency component. It can be seen that the corresponding resonant frequency component in the resonance branch is large and the rest harmonic components are small, which indicates that the two frequencies generated by the system are separated very well.
As shown in Table 4, the improved DE was used to obtain the values at bipolar 20 switching angles when V ac1 = 0.8, V ac2 = 0.4 (normalized), and k = 39. Experiments were performed under the conditions of V dc = 20 V, fundamental frequency 15 kHz, and high-frequency 585 kHz (39th harmonic) to obtain the full-bridge output voltage and its spectrum analysis results, as shown in Figure 11. The experimental results are basically consistent with the expected values, which proves the effectiveness and accuracy of the improved DE once again.

Conclusions
The dual-frequency PHM is essentially a high-dimensional multi-objective problem with complex constraints. The classic DE cannot solve this kind of problem. In this paper, an improved algorithm based on DE is proposed, which provides a solution for solving the switching angle of the dual-frequency output of a single inverter. In this algorithm, the Newton iterative method is combined with DE. By rationally utilizing the strong convergence of Newton's method and the diversity of DE, the efficiency of the solution is improved. In addition, the adaptive mutation operator is used in the mutation process, which can effectively avoid the program falling into local optimum and premature phenomenon. The simulation analysis and experimental results are consistent with the expectation, and the feasibility of the proposed algorithm is verified. The algorithm can be used to solve the dual-frequency and multifrequency switching angles of a single inverter and is also suitable for the optimization of multi-objective and multidimensional problems.

Conclusions
The dual-frequency PHM is essentially a high-dimensional multi-objective problem with complex constraints. The classic DE cannot solve this kind of problem. In this paper, an improved algorithm based on DE is proposed, which provides a solution for solving the switching angle of the dual-frequency output of a single inverter. In this algorithm, the Newton iterative method is combined with DE. By rationally utilizing the strong convergence of Newton's method and the diversity of DE, the efficiency of the solution is improved. In addition, the adaptive mutation operator is used in the mutation process, which can effectively avoid the program falling into local optimum and premature phenomenon. The simulation analysis and experimental results are consistent with the expectation, and the feasibility of the proposed algorithm is verified. The algorithm can be used to solve the dual-frequency and