Rectifier Fault Diagnosis Based on Euclidean Norm Fusion Multi-Frequency Bands and Multi-Scale Permutation Entropy

3.1. The Mentioned Fault Diagnosis Method

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Fault signal collection. We collected input voltage Uab signals of 15 fault modes of the rectifier under different working conditions by a voltage sensor to form the original fault sample set.

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Data processing includes signal decomposition and optimal frequency band selection. Signal decomposition. We selected the db series wavelet functions suitable for signal decomposition and reconstruction, such as dbN (N = 2, 3, … 10). Using 9 wavelet functions, respectively, the fault signal was decomposed by wavelet packet multi-resolution, and 9 groups of multi-frequency bands were obtained. We calculated the energy–information entropy ratio of 9 groups of frequency bands, respectively, and selected a group of multi-frequency bands corresponding to the maximum ratio as the optimal multi-frequency band information of the fault signal. At this point, the optimal multi-frequency band information of each fault signal after wavelet packet decomposition reached a total of 2ⁿ frequency bands (n is the decomposition layers number of the wavelet packet), including the high-frequency and low-frequency information of fault signals.

(3)

Feature extraction. We calculated the multi-scale permutation entropy of each frequency band to form a multi-scale feature vector ${{\displaystyle F}}_i$:

$${{\displaystyle F}}_i={{\displaystyle [P{F}_1,P{F}_2,\cdots ,P{F}_m]}}^T$$

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where ${{\displaystyle F}}_i$ represents the multi-scale permutation entropy feature of the i-th frequency band, m represents the scale factor, and PF stands for permutation entropy.

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Feature fusion. The Euclidean norm is used to fuse multi-scale feature vectors of each frequency band, which can combine multiple entropies into a single entropy that characterizes the fault feature. The feature vector after the fusion of the i-th frequency band is expressed as ${{\displaystyle L2F}}_i$. The final fault feature vector of each fault signal is as follows:

$$signal={{\displaystyle [L2F}}_1,{{\displaystyle L2F}}_2,\cdots {{\displaystyle ,L2F}}_{\delta }]$$

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where $\delta$ indicates the number of frequency bands.

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Pattern recognition. The SVM classifier was used for the fault identification mode. The feature was divided into a training set and a test set, and the mapping relationship between the fault mode, feature and fault was completed by training the model several times. The trained model will diagnose real-time or new test data, and output results such as whether fault, fault type and severity.

3.2. Signal Decomposition and Optimal Multi-Frequency Band Selection

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Assuming the fault signal, 9 Daubechies (dbN, N = 2, 3, … 10) wavelet functions, the fault signal is decomposed by the wavelet packet. After m layer decomposition, W frequency bands are obtained, and the coefficient of the r-th frequency band is ${{\displaystyle C}}_m^W(r)/\mathrm{db}N$ and $r=0,1,2,\cdots ,(W-1)$.

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After the fault signal is decomposed by the wavelet packet, 9 groups of frequency band information are obtained.

$$\left\{\begin{array}{c}{{\displaystyle f}}_{\mathrm{db}2}(m,W,r)=\left\{{{\displaystyle C}}_m^W(r)/\mathrm{db}2\right\}\\ \begin{array}{l}{{\displaystyle f}}_{\mathrm{db}3}(m,W,r)=\left\{{{\displaystyle C}}_m^W(r)/\mathrm{db}3\right\}\\ ⋮\end{array}\\ {{\displaystyle f}}_{\mathrm{db}9}(m,W,r)=\left\{{{\displaystyle C}}_m^W(r)/\mathrm{db}9\right\}\end{array}\right.$$

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where ${{\displaystyle f}}_{\mathrm{db}N}(m,W,r)$ represents the frequency band information corresponding to different wavelet functions.

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Energy $E({{\displaystyle f}}_{\mathrm{db}N})$ and information entropy $IE({{\displaystyle f}}_{\mathrm{db}N})$ for each group of frequency bands are calculated as follows:

$$E({{\displaystyle f}}_{\mathrm{db}N})={\displaystyle \sum _{r=1}^W{{\displaystyle \left|{{\displaystyle C}}_m^W(r)/\mathrm{db}N\right|}}^2}$$

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$$IE({{\displaystyle f}}_{\mathrm{db}N})=-{\displaystyle \sum _r^W{p}_i{\log }_2{p}_i}$$

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where $p_i={\left|{{\displaystyle C}}_m^W(r)/\mathrm{db}N\right|}^2/E({{\displaystyle f}}_{\mathrm{db}N})$ represents the energy probability distribution.

(4)

Calculate the energy–entropy ratio of the frequency band coefficient.

$$\eta ={\displaystyle \frac{E({{\displaystyle f}}_{\mathrm{db}N})}{IE({{\displaystyle f}}_{\mathrm{db}N})}}$$

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3.3. Multi-Scale Permutation Entropy and the Euclidean Norm

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Obtain the subsequence of the phase space by reconstructing the frequency band signal phase space, expressed as:

$$S(i)=\{s(i),s(i+\delta ),\cdots ,,x(i+(d-1)\delta )\},i=1,2,\cdots ,L-(d-1)\delta$$

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where d and $\delta$, respectively, represent the embedding dimension and time delay.

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Sort the subsequences in the phase space from smallest to largest.

$$q(i+({{\displaystyle k}}_1-1)\delta )\le q(i+({{\displaystyle k}}_2-1)\delta )\le \cdots \le q(i+({{\displaystyle k}}_m-1)\delta )$$

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where ${{\displaystyle k}}_1,{{\displaystyle k}}_2,{{\displaystyle \cdots ,k}}_m$ is the element position index in $S(i)$ after reordering, and $({{\displaystyle k}}_1,{{\displaystyle k}}_2,{{\displaystyle \cdots ,k}}_m)$ is called a permutation. In this way, each $S(i)$ is mapped to a matching sequence with a total of $d!$ permutations.

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The occurrence probability of each permutation is ${{\displaystyle G}}_1,{{\displaystyle G}}_2,{{\displaystyle \cdots ,G}}_k$, and ${{\displaystyle G}}_k$ is defined as:

$${{\displaystyle G}}_k={\displaystyle \frac{f(\alpha )}{L-(d-1)\delta }}$$

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where $f(\alpha )$ represents the number of occurrences of the $\alpha -\mathrm{th}$ symbol sequence.

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According to information entropy, the permutation entropy of time series $s(t)$ is obtained.

$${{\displaystyle G}}_{PE}(d,\delta )=-{\displaystyle \sum _{k=1}^{\alpha }-}{{\displaystyle G}}_k\ln {{\displaystyle G}}_k$$

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When calculating the permutation entropy, only the numerical arrangement relationship of elements in the phase point is considered, and the key factor of numerical size is ignored. Therefore, the coarse process can be introduced to obtain the multi-scale permutation entropy. The coarse-graining process is:

$${{\displaystyle y}}_{\omega }(i)={\displaystyle \frac{1}{\omega }}{\displaystyle \sum _{k=(i-1)\omega +1}^{i\omega }s(k)}i=1,2,\cdots ,{\displaystyle \frac{L}{\omega }}$$

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where $\omega$ expresses the coarse-graining factor, ${{\displaystyle y}}_{\omega }$ expresses the coarse-graining signal when the coarse-graining factor is $\omega$, and $s(k)$ is the k-th value of the time series.

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The Euclidean norm, also known as the L2 norm, has a certain smoothing and equalizing effect, which can weaken the entropy fluctuations caused by various disturbance and other abnormal factors on individual scales. After the fusion of multi-scale permutation entropy through the Euclidean norm, the entropy information of different scales is no longer an isolated individual, but can be organically combined to form a more comprehensive feature description. Euclidean norm fusion multi-scale permutation entropy:

$$L2={{\displaystyle ‖x‖}}_2=\sqrt{{\displaystyle \sum _{k=1}^K{{\displaystyle x}}_k^2}}$$

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4.1. Construct the Fault Sample

4.2. Decomposition of Fault Signals and Selection of Optimal Multi-Frequency Bands

4.3. Feature Extraction for Permutation Entropy

4.4. Feature Extraction and Fusion for Multi-Scale Permutation Entropy

4.5. Evaluation of Analysis Using Different Indexes