Power quality issues are defined under IEC standards as deviations in the fundamental characteristics of a power supply system that prevent normal, uninterrupted, or interference-free electricity usage under standard operating conditions [15]. Specifically, any deviation in voltage or current parameters—such as amplitude, frequency, or waveform—from their specified values that causes equipment malfunction or failure constitutes a power quality problem.

Current research on DC power quality remains in its early stages, necessitating the adaptation of existing AC power quality frameworks. However, when classifying and defining DC power quality phenomena, it is critical to account for inherent differences between AC and DC systems. Ultimately, a comprehensive DC power quality index system is established, as depicted in Fig. 1.

When formulating DC power quality evaluation standards, indices can be streamlined for specific application scenarios or voltage levels by considering interdependencies among indicators. This approach allows for the consolidation and simplification of metrics. According to the guidelines outlined in GB/T 35727 “Technical Specifications for Medium and Low Voltage DC Distribution Systems,” voltage classifications are categorized as follows: medium-voltage distribution spans ±1.5 kV to ±50 kV (3 kV to 100 kV line-to-line), while low-voltage distribution ranges from ±75 V to ±750 V (110 V to 1500 V line-to-line). The specific voltage classes and corresponding limit values are detailed in Table 1.

In the process of DC distribution, issues related to frequency, phase, or reactive power do not arise. Currently, research on DC power quality problems is still in its infancy, and there is no unified standard for definitions and classifications. Therefore, it is necessary to refer to relevant AC power quality index systems while considering the differences between AC and DC systems to classify and define DC power quality signals [15]. Power quality problems can be categorized into two main types: event-based and variation-based. Event-based power quality problems include voltage sags, swells, and short-duration voltage interruptions. Variation-based issues encompass voltage fluctuations and DC voltage harmonics.

(1) DC voltage sag

Voltage sag is a common power quality phenomenon in DC distribution systems. It refers to a sudden decrease in the DC bus voltage to between 90% and 1% of the nominal voltage, which then returns to normal within a very short period. The primary causes of this phenomenon include DC bus grounding short circuits, abrupt changes in micro-source power, and the switching of DC loads [15].

The severity of voltage sag can be quantified by the magnitude of the voltage drop and its duration. Different electrical environments exhibit varying tolerances to voltage sag. Consequently, the concept of voltage sag depth is introduced for analyzing and evaluating the severity of DC voltage sag. This index can be represented by the following Formula (1): (1)MF% =Ud−Ud′UdcN×100%=usagUdcN×100%where: Ud is the actual voltage measured before the sag occurs, Ud′ is the actual voltage value at the measuring point after the sag occurs, usag is the difference between the actual voltage values at the measuring point before and after the sag occurs, and UdcN is the rated voltage of the DC bus.

(2) DC Voltage Short-Duration Interruption

This phenomenon occurs when the voltage of the DC bus drops to near zero for a short duration. The main causes of this issue include faults in the DC distribution system (e.g., short circuits between terminals), control failures (e.g., protection operation failures or errors during switching), and equipment failures (e.g., faults occurring in DC equipment between the positive and negative poles) [16].

(3) DC Voltage Swell

DC voltage swell refers to a sudden increase in line voltage during DC distribution, rising to more than 1.1 times the system’s rated voltage but typically not exceeding 1.8 times. The voltage quickly returns to normal within a short period, with the phenomenon usually lasting less than one minute. The causes of DC voltage swell include temporary voltage increases in distributed energy sources due to random changes in wind, sunlight, temperature, etc., and voltage transients occurring after fault-induced load shedding during the DC distribution process, followed by a recovery phase [17].

(4) DC Voltage Fluctuation

Given the limited capacity and power of DC distribution systems, which are not infinite sources, the voltage within the DC system can fluctuate due to the influence of distributed energy sources or high-power loads. DC voltage fluctuation refers to the random variation in voltage magnitude within the range of 0.9 to 1.1 times the rated voltage. Under normal conditions, DC voltage fluctuations do not contain fixed frequency components [17].

DC voltage fluctuations are typically characterized by two features: First, the voltage change rate is expressed as the ratio of the amplitude difference to the time difference between them, and the waveform within any sampling period (T_s) is analyzed. This can be represented by the following Formula (2): (2)ε=Udcmax−UdcminΔt×100% =UbΔt×100% where: Udcmax is the peak value of the DC voltage in the cycle, Umin is the valley value of the DC voltage in the cycle, Ub is the difference between the peak and valley of the fluctuation amplitude, and t is the time difference between them.

Second, Relative Voltage Fluctuation is defined as the ratio of the difference between the DC voltage peak and the trough over one period to the rated DC bus voltage, usually expressed as a percentage. It is given by the following Formula (3): (3)δ=Udcmax−UdcminUdcN×100% =UdUdcN×100% where: Udcmax is the peak value of DC voltage in this cycle, Udcmin is the trough value of DC voltage in this cycle, Ud is the difference between the peak and trough of the fluctuation amplitude, UdcN is the rated voltage of the DC bus, corresponding to the voltage level at the measurement place.

Voltage fluctuation primarily originates from four interconnected sources: interference propagating through AC grid interconnections, short-circuit faults occurring in AC-DC hybrid systems, recurrent switching operations of short-duration loads inducing DC-side perturbations, and inherent load variations at the consumer side.

(5) DC Voltage Harmonics

Although DC systems lack a fundamental frequency, studies confirm the presence of high-frequency components in DC distribution. Consequently, DC voltage harmonics are defined as sinusoidal waveforms superimposed on the nominal DC voltage at measurable amplitudes [18].

To measure and assess the magnitude and distortion of DC harmonics, concepts such as Harmonic Rate ((H_{\text{rate}})), Harmonic Amplitude ((H_{\text{amp}})), and Total Harmonic Distortion ((THD)) can be used. The calculation formulas for these concepts are given as follows: (4){HRAh=AhAd×100%AH=∑h=1x(Ah)2THDA=AHAdwhere: A is the voltage, Ah is the root mean square value of the h-order harmonic voltage, and Ad is the root mean square value of the voltage at the test point on the DC side bus.

Compressed sensing (CS), introduced in 2006 by Terence Tao, Donoho D. L., and others [14], is a novel theoretical framework for signal acquisition and processing. This technique enables simultaneous signal acquisition and compression. The theory asserts that a signal compressible or sparsely representable in a specific transform domain can be mapped from high-dimensional to low-dimensional space via an observation matrix—one incoherent with the sparse transformation basis. An optimization algorithm then reconstructs the undersampled signal. The theoretical framework is illustrated in Fig. 2.

Three critical requirements govern the reconstruction of DC power quality disturbances via compressed sensing: (1) selection of an optimal sparse transformation basis for signal representation; (2) deployment of an observation matrix satisfying the Restricted Isometry Property (RIP) to sample the signal and acquire observation vectors from which sparse vectors are reconstructed; and (3) implementation of a suitable optimization algorithm for precise disturbance feature extraction.

The theoretical foundation of compressed sensing and its practical application rely on sparse signal characteristics. Truly sparse signals are uncommon in practice. DC power quality disturbance signals lack inherent sparsity; however, specific transformations can render them sparse, confirming their compressibility. A signal (x) of length (N) can be linearly represented by a set of basis functions as: (5)x=Ψθwhere, Ψ is sparse basis, θ is (N-dimensional vector). Commonly used discrete sparse bases include the Discrete Fourier Transform (DFT), Discrete Wavelet Transform (DWT), and Discrete Cosine Transform (DCT) [19].

In [2], the sparsity characteristics of power quality signals were analyzed across different sparse bases. Experimental results demonstrated that the Discrete Fourier Transform (DFT) provided a more effective sparse representation. Consequently, to reduce computational complexity and leveraging the sparsity of power quality disturbance signals after transformation, this study adopts DFT as the sparse transformation basis.

The observation matrix samples the signal to acquire observation vectors, enabling reconstruction of the original signal or sparse vector as shown in Formula (6).

Crucially, the observation matrix must satisfy the Restricted Isometry Property (RIP). Mathematical proofs establish that the observation matrix and sparse transformation basis must be mutually incoherent—a property fulfilled by matrices such as (6)y=Φx=ΦΨθ=Aθwhere, Ψ is the sparse basis, θ is the sparse coefficient, Φ is the observation matrix (M×N), and A=ΨΦ is the sensing matrix.

The observation matrix must satisfy the Restricted Isometry Property (RIP). Mathematically, the observation matrix and sparse transformation basis must exhibit mutual incoherence—a sufficient condition for RIP compliance. The random Gaussian matrix is mathematically guaranteed to be incoherent with the Discrete Fourier Transform (DFT) basis. Consequently, this paper adopts the random Gaussian matrix as the measurement matrix for DC power quality disturbance signals [20], with the complete workflow illustrated in Fig. 3.

The reconstruction algorithm recovers the disturbance signal using the observation matrix and sparse transformation basis. This paper adopts the Generalized Orthogonal Matching Pursuit (GOMP) algorithm for signal reconstruction. As an enhanced variant of the Orthogonal Matching Pursuit (OMP) algorithm, GOMP’s core innovation lies in drastically reducing iteration counts during sparse reconstruction—particularly effective under sparse sampling conditions—thereby minimizing runtime while accelerating sparse vector acquisition [21].

In MATLAB, five distinct types of DC power quality disturbances were generated, with each signal comprising 1500 samples. The disturbance types include DC voltage swell, DC voltage sag, DC voltage harmonics, DC voltage short-term interruption, and DC voltage fluctuation. For each disturbance type, 1200 samples were allocated to the training set, and the remaining 300 samples served as the test set. Through compressed sensing theory, sparse vectors were derived for all signals. These vectors subsequently formed a training set and a test set with sparse vectors as samples.

A traditional autoencoder (AE) comprises two main components: an encoder and a decoder. It includes an input layer, hidden layers, and an output layer. As illustrated in Fig. 4, the encoder consists of the input layer and the hidden layer, whereas the decoder is composed of the hidden layer and the output layer. During the encoding process, the input signal (x) with dimension (d) passes through the input layer and reaches the hidden layer, where it is transformed into (h). The formula for this operation is as follows:

During encoding, the input signal x of d dimension reaches the hidden layer through the input layer, and the signal becomes h. The expression formula is as follows: (7)h=f(x)=s(Wx+B)where: W is the weight vector, B is the bias value, s is the activation function, and the expression is: (8)s(x)=11+e−y

Decoding is the inverse operation of encoding. The hidden layer maps the signal h to the output layer through the transmission function g(h). The expression formula is as follows: (9)y=g(h)=s(W′h+B′)where, W′ is the weight vector, B′ is the bias value of the decoding process, and W=W′ is usually.

The error between the output layer and the input layer needs to reach the minimum value to extract the characteristics of the power quality signal. The calculation formula is: (10)L(x,y)=L(x,g(fθ(x)))where, L is the minimum mean square error.

Generally, stochastic gradient descent (SGD) is used to adjust network parameters to reduce reconstruction error. As shown in Eq. (11). (11){W←W−l∂L(x,y)∂WB←B−l∂L(x,y)∂B}where, l is the learning rate; ∂L(x,y)/∂W is the partial derivative of L(x,y) with respect to the weight W; ∂L(x,y)/∂B means the partial derivative of L(x,y) with respect to B.

The principle of the denoising autoencoder (DAE) is based on the traditional autoencoder [22]. To prevent overfitting, the original signal x is perturbed, and the corresponding feature vector is extracted. At this point, the input signal becomes x~, and the perturbed signal can be interpreted as adding noise. This enhances the robustness of the encoder. The encoder model is illustrated in Fig. 5.

Then it is encoded and decoded to get the signal: (12)z=g(fθ(x~))

At this point, the error changes from L(x,g(fθ(x))) to L(x,g(fθ(x~))). However, in practical training, a single-layer Denoising Autoencoder (DAE) is insufficient for extracting features from DC power quality disturbance signals [22]. Therefore, it is advisable to stack multiple DAE layers to form a Stacked Denoising Autoencoder (SDAE) for network training. This multi-layer architecture can effectively extract features from the signal, and the extracted features are then used as input to the classifier [23].

In deep learning, it is often necessary to construct a loss function for the original network and then optimize it through algorithms to find the optimal parameter values that minimize the loss function [24]. The traditional Stacked Denoising Autoencoder (SDAE) typically employs Stochastic Gradient Descent (SGD) during the fine-tuning phase. With each iteration, SGD updates the parameters for every sample and generally converges quickly, enabling the model to escape poorer local minima. However, due to the large number of updates, SGD can cause the cost function to fluctuate significantly, leading to unstable convergence and negatively affecting the classification performance of the encoder. Therefore, this paper improves upon the traditional SDAE by replacing SGD with the Root Mean Square Propagation (RMSprop) optimization algorithm during the fine-tuning phase. RMSprop adjusts the learning rate based on the recent gradient history, which helps stabilize the learning process, reduces the likelihood of getting stuck in local minima, and enhances the overall classification accuracy of the encoder.

In this paper, the traditional stacked denoising autoencoder (SDAE) was improved. Instead of using stochastic gradient descent (SGD) to update network parameters, the root mean square propagation (RMSprop) optimization algorithm was employed during the fine-tuning stage to reduce the risk of falling into local minima. The training process is illustrated in Fig. 6.

(1) In MATLAB Simulink, a large number of DC disturbance signals were generated.

(2) After compressed sensing processing, the signal was reconstructed while generating sparse vectors.

(3) To ensure that the signals had the same measurement scale, normalization was applied to all data using Formula (13). Finally, the dataset was divided into two parts: the training set and the test set. (13)xnor=(x−xmin)/(xmax−xmin)where, xnor,xmax,xmin respectively refer to the signal size obtained through normalization and the maximum value of the signal before normalization.

(4) The stacking denoising autoencoder model was constructed, and layer-wise unsupervised pre-training was performed. Due to the large sample size, the training sample M could be divided into several small batch data blocks of m, enabling successive training for improved efficiency. After iterative and layer-by-layer training, the network initialization weights W and bias values B were obtained. The specific training steps are as follows:

1) Assuming there are n hidden layers, the grouped data were trained starting from the first hidden layer;

2) DAE was used to calculate the weight value W and the bias value B, and then the network parameters were initialized;

3) If m<M, proceed to 2). If no, go to the next step;

4) If e<emax (number of iterations set), e=e+1 execute 2), otherwise execute the next step;

5) The weight of the trained NTH layer are saved to obtain the feature vector of the NTH layer, which is used as the training sample of the n + 1 layer.

6) If n<nmax (number of hidden layers), e=e+1 perform 1). If this condition is not met, the training is over.

(5) For parameter fine-tuning, this paper employs a top-down small-batch RMSProp optimization algorithm to fine-tune the weight and bias values of the initial network parameters until the preset number of iterations is reached. The process is as follows:

1) Set the global learning rate l, decay rate ρ, initialize the cumulative variables r1=0,r2=0;

2) Randomly select small batch data of m samples from the training set, and use the error loss function to get the gradient of weight and bias value; (14)w←1m∂∑i=1m(yi−f(W,B)(xi))2∂Wb←1m∂∑i=1m(yi−f(W,B)(xi))2∂B

3) Calculate the cumulative square gradient, as shown in Eq. (15). (15){r1=ρr1+(1−ρ)w⊙wr2=ρr2+(1−ρ)b⊙b

Note: ⊙ is the product symbol.

4) Update the weight and bias values of the encoder, as shown in Eq. (16). (16){W=W−lr1⊙wB=B−lr2⊙b

5) When the number of iterations reaches the set value, the operation will stop. If this condition is not met, perform 2) and continue parameter fine-tuning;

6) Keep the trained network well and test the classification performance of the encoder on the test set. Mean squared error (MSE) is used to evaluate the performance of the encoder. (17)MSE=1m∑i=1m(q(xi)−p(xi))2where: p(xi) is the data distribution predicted by the encoder; q(xi) is the real data distribution; m is the number of samples.

The DC power quality signals obtained in Section 1 are simulated and analyzed using compressed sensing. Figs. 8 and 9 illustrate the reconstructed signals and sparse vectors corresponding to DC voltage fluctuations, DC voltage harmonics, and DC voltage sag. From the waveform graphs of the three sparse vectors, it is evident that the reconstructed signals generated by compressed sensing exhibit a small reconstruction error. Additionally, the sparse vectors, which contain numerous zero elements, significantly reduce the data volume. This clearly demonstrates that the sparse vectors preserve rich features of the original signals, thereby facilitating subsequent disturbance signal classification.

For stacked denoising autoencoders (SDAE), selecting the appropriate number of hidden layers and neurons is critical for achieving optimal disturbance recognition and classification results. In this study, a two-layer stacked denoising autoencoder is employed to train the samples. The number of neurons in each layer is determined based on the evaluation of Mean Squared Error (MSE). Initially, the number of neurons in the first hidden layer is identified by minimizing the MSE. Once the optimal number of neurons is established, the number of neurons in the second hidden layer is similarly determined through optimization of the MSE. When the number of hidden layers exceeds two, the MSE value increases, indicating that adding more layers leads to information loss and an increase in error. Consequently, having too many hidden layers does not yield better training results. Fig. 10 depicts the relationship between the number of hidden layers and neurons. The optimal number of neurons for each layer is found to be 300 and 100, respectively.

In this study, Gaussian white noise with a mean of 0 and variance of 1 is added to the DC disturbance signals without noise. The Signal-to-Noise Ratios (SNR) of the added Gaussian white noise are 20, 30, and 40 dB, respectively, as the “damaged” signals. A two-layer stacked denoising autoencoder model is trained with the first and second hidden layers containing 300 and 100 neurons, respectively, and the number of iterations is set to 1000.

As shown in Fig. 11, when no Gaussian white noise is added, both the original signal and the sparse vector achieve good recognition performance for the DC power quality disturbance signals. After adding Gaussian white noise, due to the robustness of the DAE, the classification accuracy improves. At an SNR of 30 dB, the accuracy of both the original signal and sparse vectors reaches the maximum value, with minimal difference between them. As the SNR of the Gaussian white noise increases, it begins to interfere with the original signal, causing a slight decrease in classification accuracy. However, the accuracy remains comparable to that of traditional encoders. In summary, adding an appropriate amount of Gaussian white noise still results in good recognition performance.

To evaluate the classification performance of the network, a Mean Squared Error (MSE) graph is plotted. A smaller minimum MSE value indicates superior classification performance. As depicted in Fig. 12, as the number of training epochs increases, the MSE value progressively decreases. For the original data, the MSE stabilizes after approximately 30 epochs, reaching its minimum at the 35th epoch. In contrast, the MSE for the sparse vector exhibits minimal change after the 15th epoch and stabilizes by the 20th epoch. Ultimately, both approaches achieve stable error values, with the sparse vector attaining the minimum MSE. This suggests that the sparse vector effectively reduces network complexity and conserves training time.

To further validate the performance of the classification method proposed in this study, its classification accuracy is compared with that of Deep Belief Networks (DBN) [14] and traditional Stacked Denoising Autoencoders (SDAE), as shown in Table 2. From the table, it can be observed that when the input signal corresponds to either the original data or the sparse vector, the classification accuracy of SDAE surpasses that of DBN. Moreover, the improved SDAE achieves higher classification accuracy relative to the conventional SDAE. These results underscore the efficacy of the proposed method in classifying DC power quality disturbance signals. The comparison outcomes are presented in Table 2.

To verify that the improved Stacked Denoising Auto-Encoder (SDAE) has a shorter testing time, three classification methods were also compared. As can be seen from Table 3, when classifying DC power quality disturbance signals, the running time of the Stacked Denoising Auto-Encoder is significantly reduced compared with that of the Deep Belief Network (DBN). The improved encoder also shows a slight speed improvement. Especially when the sparse vector is used as the data sample set, even less time is required. Although under the same conditions, the accuracy is slightly lower than that of the original data, the classification time of the sparse vector is significantly reduced. Considering all factors, the classification method proposed in this paper is feasible for handling DC power quality disturbance signals.