## 4 Model Construction and Simulation Experiment Results
### Model Construction  
In order to increase the number of training samples and make the established GMM more representative, add the characteristic values above to Gaussian white noise of 40 dB, 45 dB, 50 dB, 55 dB, 60 dB, 65 dB, 70 dB, 75 dB, and 80 dB. The preliminary data are expanded to 400 groups. The 400 sets of data are used as training samples to construct a training model for SaDE-ELM. Then, on the basis of the training model, test samples are input to determine the situation of the proposed method of fault diagnosis for marine fuel systems. The descriptive analysis of the pressure wave characteristics of the initial 400 sets of data used in the research is shown in Figure 8 below. One of the common faults of the fuel system is the poor quality fuel when the ship is refilling, which has caused some problems in the fuel system. According to the combination of different pressure waveform characteristics, the following types of marine fuel system states are formed: normal state, insufficient fuel supply (75% oil, 25% oil, idle oil volume), stuck needles to plug holes (small oil quantity, standard oil quantity), needle valve leakage, and oil valve failure. That is, eight sub-states, normal injection, 75% oil volume, 25% oil volume, idle oil volume, injector needle stuck 1, injector needle stuck 2, injector needle leakage, and high-pressure pump outlet valve failure [33]. Where the system state except the normal state belongs to the type of faults of the marine fuel system. In the condition of insufficient oil supply, the waveform will decrease with the decrease of oil supply, the waveform area will decrease, and the average waveform will also decrease. When the needle valve is located in faults, the pressure wave rises very quickly and the pressure wave curve is sharp. And the reflected wave is quite intense in the pipe, the durable time is longer, and the residual pressure is higher than the normal pressure. Compared the standard oil volume with the small oil volume, the amount of oil and residual pressure is greater, and the attenuation is faster. The leakage of the needle valve will cause the fuel pressure to drop; when the outlet valve fails, the residual pressure of the fuel is high, the pressure fluctuation is obvious, and the attenuation is slow. Figure 9 below shows the pressure waveforms of different states of the marine fuel system. Figure 8: Pressure wave characteristics of training data. In the process of constructing a model through SaDE-ELM, the objective function \(E_{0}\) gradually iterates to obtain the minimum value. First, judge the effect of different hidden layer numbers on the fault diagnosis of EL algorithm and SaDE-ELM algorithm and select 20 times of the default number of iterations, as shown in Table 1.  
The experimental results show that, with the increase of the number of hidden layers, the error of the algorithm gradually decreases, but the time cost of the training model increases slightly. The SaDE-ELM algorithm has a better performance for the fault diagnosis of marine fuel systems. All test samples can be classified into the correct state types, while the ELM algorithm also increases with the number of hidden layers. The training and testing errors gradually decrease, but the fault diagnosis effect is worse than the SaDE-ELM algorithm. SaDE-ELM is 25 in hidden layers, and the training error is 0, so the subsequent comparison experiment chooses 25 hidden layers to construct the fault diagnosis model of the marine fuel system.