## 5 Results  
The optimization problem in (9) is solved with a genetic algorithm implemented in the Matlab environment [28]. A transmission band of 1-8 MHz is assumed, because the \(SNR\) is unfavorable outside such interval. A width of 100 kHz has been assigned to each sub-carrier, then 70 sub-carriers are available. In practice, the number of actually available sub-carriers could be much lower, because of the great attenuation of the power line and the presence of noise. In this work, for each channel, the power spectrum of the noise has been obtained with the same NS3 simulator, which is in good agreement with experimental results retrieved from the literature [21; 29]. The number of variables to be optimized is equal to 280 (70 sub-carriers for four channels), and the initial population has been generated using a uniform random number generator in the range {0;1} After 28,000 iterations the algorithm ends producing the matrix \(M\). This allows us to obtain a single frequency response for all the charging stations by multiplying \(M\) with their frequency responses. The values of the genetic algorithm parameters are reported in Table 1. Figure 5 shows the obtained Chimera frequency response assumed to represent all the charging stations in Figure 2. The weight coefficient \(\alpha\) has been set equal to 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Population** & **Initial** & **Elite** & **Crossover** & **Migration** & **Constraint** \\
**Size** & **Population** & **Count** & **Fraction** & **Fraction** & **Initial Penalty** \\ \hline
100 & random & 5 & 0.8 & 0.2 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Genetic algorithm parameters. Figure 5: Power spectral density of the Chimera transmission channel: the different colors refer to the allocation of the 4 charging nodes to the 70 sub-carriers. To perform the offline optimization of \(BR\), \(BER\) and \(P_{tot}\), the function \(BR=f(P_{tot},BER)\) is sampled in a regular \(100\times 100\) grid, within the interval of 1-10 [W] of \(P_{tot}\), and \(10^{-5}\)-\(10^{-3}\) of \(BER\). The upper envelope of the noise PSDs of the four channels has been assumed as unique noise spectrum, as shown in Figure 6. Note that, given the Chimera frequency response and the noise, by applying the water filling algorithm, the power value for each sub-carrier is defined as a function of the total transmitting power \(P_{tot}\). Hence, given the noise, the standard deviation \(\sigma\) is known, then the minimal inter-distance between the constellation points is determined. In this way, the number of points of the constellation is calculated, by obtaining the number of bits associated to each point, then the length of bit frame is obtained, and, finally, the \(BR\). The sampled surface is shown in Figure 7. The non-dominated points are also represented by dots. The relationship between the three objectives allows us to obtain a \(BR\) ranging from 24.3-60 Mb/s. Figure 8 gives an equivalent representation of the Pareto front in the plane (\(P_{tot}\), \(BER\)). It is a contour plot of the front, and the points represent the Pareto-optimal solutions. As it can be noted, in order to have a minimal \(BR\) at least a power of 1 W is needed. Moreover, limiting the available power to about 7 W, a \(BR\) of 52.1 Mb/s can be achieved, with a \(BER\) of the order of \(10^{-5}\). If a higher \(BR\) (e.g., 60 Mb/s) must be achieved, the \(BER\) will strongly affect the transmission. Figure 6: Noise power spectrum. Figure 7: \(BR\) as function of \(BER\) and \(P_{tot}\), and Pareto front (dots). Figure 9 shows the Pareto optimal solutions corresponding to the minimal \(BER=10^{-5}\). Depending on the available total transmitting power, the plot returns the available \(BR\) for each plug in kByte/s. As an example, with an available transmitting power of 5 W, the \(BR\) is equal to about 190 kByte/s for each of the 32 plugs in the four charging stations.