Objective The accurate assessment of sensitive equipment fault probability caused by voltages sags is an important reference to precisely mitigate voltage sags. Currently, the fault probability assessment of sensitive equipment due to voltage sags faces two major problems: small samples and insufficient a priori knowledge. A stochastic modeling method for fault probability assessment with small samples of sensitive equipment due to voltage sag was proposed by using autoencoder technology and maximum entropy principle. Methods Firstly, the fault samples of sensitive equipment would be constrained into the same uncertainty area in voltage tolerance curve (VTC). Meanwhile, considering the fact that the sensitive equipment is mainly sensitive to the voltage sag magnitude and duration, and has the uncertainty of VTC, the adaptive Kmeans clustering algorithm was utilized to cluster the magnitudes and durations of voltage sag samples respectively to find out the center vector not only representing VTC uncertainty constraints but also neglecting the influence of samples of outlier, and then added it to the loss function of the sparse autoencoder (SAE) for better sample feature learning. The modified SAE was used to produce new samples with the input of the processed fault samples, so that a fault sample augmentation method based on SAE–Adaptive Kmeans was proposed. Secondly, in view of the problem of insufficient a priori knowledge, a maximum entropy modeling method for the fault probability assessment of sensitive equipment based on the augmented samples was proposed. Finally, taking personal computers (PCs) as examples, simulation verifications were carried out in the cases that the VTC probability density function obeys uniform distribution, normal distribution, different exponential distribution and the sample number was only 5, and the proposed method was compared with the traditional maximum entropy method and the method with SAE sample augmentation that introduced the constraints of the uncertain region of the VTC but didn’t introduce adaptive Kmeans clustering. In the same time, the proposed method was compared with the assessment methods based on subjective assumption under insufficient a priori knowledge. Results and Discussions From the distribution of the 5 augmented PC’s fault samples of Case 1 to 4 produced by the proposed SAE-Adaptive Kmeans augmentation method, the augmented fault samples were always distributed into the uncertainty area of VTC, which meant the SAE-Adaptive Kmeans augmentation method assured the constraint of VTC uncertainty. Meanwhile, on the basis of conforming to the characteristics of the original sample distribution and neglecting the outlier samples, it realized the effective supplementation of the original sample space. The methods to be compared under small sample circumstances included the proposed assessment method (Method 1), the assessment method based on the maximum entropy principle (Method 2) and the proposed assessment method only constraining the VTC uncertainty boundaries without using adaptive Kmeans clustering algorithm (Method 3). The assessment results of PC’s voltage sag fault probability from Cases 1 to 4 under small sample circumstances showed that: The single largest errors for Methods 1 to 3 in Case 1 were 52.76%, 41.87%, and 20.36%; The single largest errors for Methods 1 to 3 in Case 2 were 20.72%, 32.99%, and 41.98%; The single largest errors for Methods 1 to 3 in Case 3 were 7.54%, 32.15%, and 13.37%; The single largest errors for Methods 1 to 3 in Case 4 were 102.67%, 918.67%, and 197.90%. At the same time, the mean errors for Methods 1 to 3 in Case 1 were 33.89%, 21.00%, and 12.86%; and the mean errors for Methods 1 to 3 in Case 2 were 8.60%, 15.60%, and 21.56%; and the mean errors for Methods 1 to 3 in Case 3 were 3.72%, 14.04%, and 5.58%; and the mean errors for Methods 1 to 3 in Case 4 were 30.05%, 203.60%, and 92.81%. It can be seen that Method 1 minimizes both single and average errors, except for Case 1 which is not obvious enough. The fault frequency assessment results of PC showed that the assessment errors for Method 1 from the four cases were 0.63%, 6.03%, 2.28%, and 3.11%, which were all lower than those for Method 2. The methods to be compared under insufficient a priori knowledge circumstances included Method 1, the method assuming VTC obeys a uniform distribution in the uncertain area (Method 4), the method assuming VTC obeys a normal distribution in the uncertain area (Method 5), the method assuming VTC obeys an exponential distribution in the uncertain area (Method 6), and the method assuming VTC obeys an inverse exponential distribution in the uncertain area (Method 7). The assessment errors of PC’s voltage sag fault probability from the four cases under insufficient a priori knowledge circumstances showed that: Methods 1, 5 to 7 had single maximum errors of 52.78%, 43.50%, 222.04%, and 97.00% in Case 1, with the mean errors of 33.89%, 23.12%, 92.76%, and 82.83%; Methods 1, 4, 6, and 7 had single maximum errors of 20.72%, 30.31%, 267.70%, and 96.57% in Case 2, with the average errors of 8.60%, 19.41%, 88.19%, and 85.55%; Methods 1, 4, 5, and 7 in Case 3 had single maximum errors of 7.54%, 68.95%, 72.80%, and 99.07%, and the average errors of 3.72%, 44.17%, 34.43%, and 89.04%; and Methods 1, 4 to 6 had single maximum errors in Case 4 of 102.67%, 3228.00%, 2814.70 %,10617.00%, and the average error of 30.05%, 864.24%, 863.69%, 2199.80%. It can be seen that Method 1 still minimizes both single and average errors except for Case 1 which is not obvious enough. The fault frequency assessment results of PC showed that: Method 1 has the lowest error in evaluating the frequency of failures in each case after removing the most desirable method in every case, while the average of the total errors for the four cases of Method 1, Methods 4 to 7 are 3.01%, 114.01%, 135.25%, 225.30%, and 62.88%, respectively, which further proves the validity and accuracy of the methodology of this paper. Conclusions The results show that the proposed method is applicable to the small samples and different distributions, and the errors of assessment results are lower than those of the traditional maximum entropy method and those of the methods based on subjective assumption, which verifies the effectiveness, rationality and feasibility of the SAE sample augmentation and the maximum entropy modeling for the probability assessment due to voltage sags of small-sample equipment failures, and insures the accurate further analysis of voltage sag corresponding problems. © 2024 Sichuan University. All rights reserved.
