## 3 Simulations and Validations  
In order to verify the capability of separating UHF signals radiated by different PD sources, UHF signals simulated by Ansoft HFSS 13.0 are processed using the algorithm with a 3D coordinate system of the substation space. The simulated antennas used for accepting the signals are located at #1(0.5, 0.5, 1), #2(3.5, 0.5, 1), #3(3.5, 4.5, 1), and #4(0, 4.5, 1) (in meters), while the locations of the three PD sources are \(P_{1}\)(2, 6.5, 3), \(P_{2}\)(3, 6, 3), and \(P_{3}\)(1, 6, 3). The simulated PD signals are stimulated by Gaussian current impulses with amplitudes of 1 A, 1.5 A, and 2 A, and time widths of 4 ns, 6 ns, and 2 ns, respectively. The space for the simulation is shown in Figure 1a. Locations of the simulated antennas and PD sources are shown in Figure 1b. In order to verify the effectiveness of the above method in suppressing Gaussian white noises and frequency-fixed interference, white noises in different SNRs were added into the UHF signals simulated by the software, and interference with amplitude of 0.05 V and frequency of 900 MHz was also mixed into the signals. The signals radiated by PD sources mixed with white noises (when SNR = 25 dB) and frequency-fixed interference are demonstrated in Figure 2a,c,e, and their bispectra in the range of 0-3 GHz are demonstrated in Figure 2b,d,f, respectively. The frequency spectra of the signals are demonstrated in Figure 3a-c, respectively. Figure 1: Schematic diagram of the simulated antenna array and partial discharge (PD) sources. (**a**) The simulated substation. (**b**) The planar diagram of simulated antennas and PD sources. \(P\) means the simulated PD sources, and # means the antennas. Twenty groups of each type of simulated UHF signals, which are mixed with noises are taken as trained samples, and the Fisher-like class separability measures were calculated using Equation (10). The separation measure between simulated signals radiated by sources \(P_{1}\) and \(P_{2}\), and that between \(P_{1}\) and \(P_{3}\) are demonstrated in Figure 4a,b, respectively. Figure 3: Frequency spectra of the simulated ultrahigh frequency (UHF) signals (**a–c**): Frequency spectra of stimulated UHF signal radiated by \(P_{1}\)\(\sim\)\(P_{3}\). Figure 2: Waveform and bispectra of simulated PD signals mixed with noises (**a,b,e**): Signals radiated by simulated PD sources \(P_{1}\)\(\sim\)\(P_{3}\) (**b,d,f**): Bispectra of the simulated signals. Different numbers of bifrequencies at which the Fisher-like class separability measures between the bispectra of simulated signals achieve their locally maximums can be selected as characteristic bifrequencies. Then, RBFNNs used for separating different PD sources are trained. The UHF signals received above are regarded as the observed signals for further analysis. Bispectra of the simulated signals with noise are calculated. Then, the selected bispectra are substituted to the trained RBFNNs. Each PD source in the simulated substation can be obtained by the localization equations established by the estimated time delays [1,4]. The time delays between two signals can be calculated with high order cumulants [7]. This algorithm for estimating time delays has been proved to be insensitive for unknown correlated Gaussian noise [15,16]. The localization results are applied to compare with the separation result, which are obtained by the proposed method. With different SNRs, 50 signals radiated by each simulated PD source can be obtained. Table 1 shows the average results of \(P_{1}\), \(P_{2}\), and \(P_{3}\). The separation rate for identifying the signal type when different characteristic bifrequencies are selected are shown in Table 2. The percent of the signals that are correctly separated is defined as the separation rate. It can be inferred from Table 2 that Gaussian noises as well as the periodic narrow-band interference can be significantly restrained by the application of selected bispectra. Thus, the multiple simulated PD sources can be effectively separated by the proposed method.