# Differential Evolution in Waveform Design for Wireless Power Transfer
## 4 Results
### Performance Comparison  
The average rectified DC power for the waveforms designed by the algorithms under test is listed in Table 2. We can easily derive that all algorithms yield very similar results. Considering the performance of SCP-QCLP as the reference, we will compare the other algorithms with it. In DE, we observe a maximum percentage decrease of 2.96% in channel 1 for \(T_{o}=320\) ns. This is a slight performance deterioration that could be explained by the fact that the tuning of \(F\) and \(CR\) parameters was based only on the characteristics of the second propagation channel. However, taking into consideration the overall outcome, the selected values of \(F\) and \(CR\) give satisfactory results for all the channels under test. Jaya's performance is just slightly deteriorated for \(N=32\) if we take into account the maximum percentage decrease of 0.66% in channel 4. CoDE and L-SHADE generally score values of a very slight improvement in the rectified DC power compared to SCP-QCLP, with a maximum of around 0.06% in channel 4 for \(T_{o}=40\) ns. However, this improvement is trivial and it can be attributed to the value of the termination criterion applied in SCP-QCLP mode (it was set to \(\in=10^{-3}\) as in [10]). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(T_{o}\) **(ns)** & \(N\) & \(FE_{max}\) & **(F \& CR)** \\ \hline
20 & 2 & 500 & (1 \& 0.9) \\ \hline
40 & 4 & 5000 & (0.4 \& 0.8) \\ \hline
80 & 8 & 20,000 & (0.3 \& 0.7) \\ \hline
160 & 16 & 50,000 & (0.3 \& 0.9) \\ \hline
320 & 32 & 200,000 & (0.1 \& 0.7) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal \(F\), \(CR\) values of DE algorithm and termination criteria for waveforms with \(B=100\) MHz bandwidth. Table 3 lists the standard deviation values of the rectified DC power over 50 runs for each algorithm as they presented in Table 2. We can easily conclude that these values are quite low compared to the corresponding average ones. Note that SCP-QCLP should normally demonstrate zero standard deviation in any of the cases since it is not a stochastic algorithm and it is developed to converge always to the same solution. However, the observed values for SCP-QCLP are not exactly zero, due to the roundoff error of floating-point arithmetic. Tables 4 and 5 present the best and worst values of rectified DC power detected by each algorithm. We can derive that the best values are quite similar to the worst, as expected from the corresponding low standard deviation. Also, we should remark that EAs did not detect any solution in any of the cases to demonstrate more than 0.06% increase in the rectified DC power compared to SCP-QCLP. We should point out that the SCP methods used in [9, 10] make use of some techniques to transform the original non-convex objective function into a convex one that approximates the problem and can be solved iteratively until convergence. Also, in these methods, the search starts from only one point in the solution space, which could easily lead to obtaining a locally optimal solution. On the other hand, in this paper, we employ evolutionary algorithms using the original non-convex objective function. EAs initially sample randomly the solution space and utilize a population of solutions to guide the search. Furthermore, they are equipped with mechanisms, like the crossover, to enhance the population's diversity and prevent convergence to a local optimum. However, convergence to a globally optimal solution is still not guaranteed. We should also note that the four EAs tested in this paper have different characteristics and are suitable for different types of problems. Considering all the above, the results indicate that the obtained solutions may be the globally optimal ones. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Ch. No. ** & \(T_{o}\) **(ns)** & \(N\) & **DE** & **CODE** & **L-SHADE** & **Jaya** & **SCP-QCLP** \\ \hline