# A Review of Symmetry-Based Open-Circuit Fault Diagnostic Methods for Power Converters

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## Abstract

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## 1. Introduction

- 1.
- Introduction and classification of the symmetry-based OC fault diagnostic methods for power converters;
- 2.
- The performance of some symmetry-based OC fault diagnostic methods are compared, and the advantages and disadvantages of distance, entropy, and similarity/correlation are summarized;
- 3.
- Two factors that may have an influence on the symmetry of power converters are analyzed, and other functions or methods that may be feasible for measuring symmetry are pointed out.

## 2. Topology and Symmetry Analysis of Power Converters

#### 2.1. Symmetry—A Common Property of Power Converters

#### 2.2. Symmetry Analysis in Different Situations—Taking a Two-Level Three-Phase Voltage-Source-Inverter as an Example

## 3. Symmetry-Based Fault Diagnostic Methods for Power Converter

#### 3.1. Distance-Based Diagnostic Methods

- Euclidean Distance. Euclidean Distance (ED) is the most intuitive and widely used calculation formula for distance. Based on Equation (1), The ED between two current time series at instant t is$$\begin{array}{cc}E{D}_{mn}\left(t\right)=\sqrt{{\displaystyle \sum _{k=1}^{L}}{({I}_{mk}\left(t\right)-{I}_{nk}\left(t\right))}^{2}},& m,n=a,b,c\end{array}$$Wavelet transform can remove noise from the signal, reduce the length of the sequence, and make the features more prominent. In [27], the phase current signals of inverters are decomposed by a wavelet into several layers. In layer k, after decomposition, the approximate component ${A}_{mk}$ and the detail component ${D}_{mk}$ were obtained, and they were both time series, containing a series of decomposed coefficients. Then ${A}_{m3}^{2}$ was adopted as a measure of the energy in current signals. Finally, the ED between the energy sequences of each phase was calculated, and the fault was diagnosed by comparing this distance with the threshold. Under healthy conditions, although there are phase differences in the signals of each phase, due to symmetry, they have the same amplitude and period, and therefore contain the same energy. Therefore, the ED of the energy sequence is 0. When an OC fault occurs in a leg, compared to the healthy phase, the energy contained in the current in the corresponding phase will decrease due to a decrease in conduction time. Thanks to the excellent property of Wavelet transform, this paper achieved an accurate diagnosis of nine types of faults solely through ED.In addition, due to the limitation that ED is related to the amplitude of the signal, the standardized Euclidean distance (SED) has been proposed,$$\begin{array}{cc}SE{D}_{mn}\left(t\right)=\sqrt{{\displaystyle \sum _{k=1}^{L}}{(\frac{{I}_{mk}\left(t\right)}{{S}_{m}\left(t\right)}-\frac{{I}_{nk}\left(t\right)}{{S}_{n}\left(t\right)})}^{2}},& m,n=a,b,c\end{array}$$Among them, ${S}_{m}\left(t\right)$ is the standard deviation of the sequence ${I}_{m}\left(t\right)$, sometimes replaced by the amplitude of the signal for convenience. In [12], the “Euclidean similarity function” is used to measure the similarity between currents, which is also a variant of ED. ED and its variants have a wide range of applications.
- Manhattan Distance. Manhattan distance (MhtD), also known as the ${L}_{1}$ norm, is a measure of the distance between two points on the plane. The MhtD between time series ${I}_{m}\left(t\right)$ and ${I}_{n}\left(t\right)$ is$$\begin{array}{c}Mht{D}_{mn}\left(t\right)={\displaystyle \sum _{k=1}^{L}}\left(\right)open="|"\; close="|">{I}_{mk}\left(t\right)-{I}_{nk}\left(t\right)\\ m,n=a,b,c\end{array}$$In [28], an effective diagnostic method based on the MhtD of armature current was proposed to diagnose the fault in the rotating rectifier. A similar method was also adopted in [29] and a reasonable comparison threshold for MhtD was selected through circuit model and symmetry analysis. In [15], the “similarity” between phase currents in three-phase inverters is analyzed and adopted to determine the location of the faulty leg. The “similarity” is calculated with the aid of the MhtD between normalized phase currents. The MhtD- and similarity-based fault diagnosis algorithm was implemented on the TMS320F2806 board. This paper demonstrates the significant advantages of a symmetry-based fault diagnosis method, which is fast, computationally efficient, and easy to implement in engineering. Since the similarity in the text is a reflection of symmetry, we also point out two properties of similarity:
- 1.
- symmetry$${I}_{m}\left(t\right)\approx {I}_{n}\left(t\right)\to {I}_{n}\left(t\right)\approx {I}_{m}\left(t\right)$$
- 2.
- transitivity$$\left(\right)open\; close="\}">\begin{array}{c}{I}_{m}\left(t\right)\approx {I}_{n}\left(t\right)\\ {I}_{m}\left(t\right)\approx {I}_{h}\left(t\right)\end{array}$$

- Cosine Distance. Cosine distance (CD) is the cosine value of the angle between two vectors in the same dimensional space. For time series ${I}_{m}\left(t\right)$ and ${I}_{n}\left(t\right)$,$$\begin{array}{cc}C{D}_{mn}\left(t\right)=\frac{{\displaystyle \sum _{k=1}^{L}}{I}_{mk}\left(t\right){I}_{nk}\left(t\right)}{\sqrt{{\displaystyle \sum _{k=1}^{L}}{I}_{mk}{\left(t\right)}^{2}}\sqrt{{\displaystyle \sum _{k=1}^{L}}{I}_{nk}{\left(t\right)}^{2}}},& m,n=a,b,c.\end{array}$$In [26], the final fault location is achieved by combining CD and ED. The range of CD is [−1,1]. A larger distance indicates a smaller angle between two vectors, while a smaller distance indicates a larger angle between two vectors. When the directions of two vectors coincide, $CD=1$, and when the directions of two vectors are completely opposite, $CD=-1$. CD is independent of the modulus of the vector; that is, it is independent of the amplitude of the time series and is an excellent indicator for symmetry measurement.
- Mahalanobis Distance. Mahalanobis distance (MalD) was proposed by Indian statistician P. C. Mahalanobis as an effective method for calculating the distance between two multidimensional samples. Similar to standardized ED, MalD not only is scale-independent but also takes into account the connections between components. Consider the time series collected at instant t as a multidimensional sample, for time series ${I}_{m}\left(t\right)$ and ${I}_{n}\left(t\right)$, there are$$\begin{array}{cc}Mal{D}_{mn}\left(t\right)=\sqrt{({I}_{m}\left(t\right)-{I}_{n}\left(t\right)){S}^{-1}{({I}_{m}\left(t\right)-{I}_{n}\left(t\right))}^{\top}},& m,n=a,b,c.\end{array}$$

#### 3.2. Entropy-Based Diagnostic Methods

- Information entropy. Information entropy is the basic form of entropy, and its calculation formula is Equation (12). In [35], based on topological symmetry, the information entropy of current in Neutral Point Clamped Asymmetric-Half-Bridge converter under normal and fault conditions was analyzed, and a normalized symmetry index was proposed to diagnose SC and OC faults. In [36], wavelet packet decomposition and empirical mode decomposition (EMD) were used to transform the current signals of high-speed railway traction inverters. Then, the information entropy of the decomposed coefficient sequence was extracted. Combining information entropy with some small improvements, the fault is accurately diagnosed within one current period. Similarly, in [37], discrete wavelet transform was applied to inverter bus voltage signals and the information entropy was calculated for diagnosis. While in [38], the short-term wavelet packet was adopted, and then the information entropy was calculated as the feature for further classification. Equation (12) is the first-order form of information entropy. As a higher-order extension, the Renyi entropy was introduced with an order parameter, as is shown below.$${H}_{\alpha}\left(X\right)=\frac{1}{1-\alpha}\mathrm{log}(\sum _{i}{p\left({x}_{i}\right)}^{\alpha}).$$In [17], the Renyi entropy is adopted in the feature selection process to deal with the hard fault and soft fault diagnosis in a superbuck converter circuit (SCC). The “signal decomposition+information entropy” based methods were also adopted in [39,40,41], demonstrating the simplicity and effectiveness of information entropy in fault diagnosis within multi-phase symmetric systems.
- Fuzzy Entropy. Fuzzy entropy is a method based on the concept of approximate entropy and sample entropy which can be to measure the complexity of a time series [34]. For a given time series composed of L data, ${X}^{L}=[x\left(1\right),x\left(2\right),\dots ,x\left(L\right)]$, its Fuzzy entropy can be calculated as follows [42].Step 1. Divide ${X}^{L}$ into a group of sub-sequences with a length of $m(m\le L-2)$.$$\begin{array}{c}{X}_{i}^{m}=[x\left(i\right),x(i+1),\dots ,x(i+m-1)]-\frac{1}{m}{\displaystyle \sum _{k=0}^{m-1}}x(i+k),\\ i=1,2,\dots ,L-m+1.\end{array}$$Step 2. Calculate the Chebyshev Distance between two sub-sequences ${X}_{i}^{m}$ and ${X}_{j}^{m}$.$${d}_{ij}^{m}=\underset{k\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0,\dots ,m-1}{\mathrm{max}}\left(\right)open="|"\; close="|">x(i+k)-x(j+k)$$Step 3. Calculate the similarity between ${X}_{i}^{m}$ and ${X}_{j}^{m}$ based on the Chebyshev Distance. Give parameters n and r. The similarity is defined as$${D}_{ij}^{m}(n,r)={e}^{-ln\left(2\right){\left({d}_{ij}^{m}\left./\phantom{{d}_{ij}^{m}r}\right)\phantom{\rule{0.0pt}{0ex}}r\right)}^{2}},i,j=1,2,\dots ,L-m+1,i\ne j.$$Step 4. Define the function ${\mathrm{\Phi}}^{m}(n,r)$$${\mathrm{\Phi}}^{m}(n,r)=\frac{1}{L-m+1}\sum _{i=1}^{L-m+1}(\frac{1}{L-m}\sum _{j=1,j\ne i}^{L-m+1}{D}_{ij}^{m}(n,r))$$Step 5. The Fuzzy entropy of the time series ${X}^{L}$ is defined as:$$FuzEn(m,r,n)=\mathrm{log}{\mathrm{\Phi}}^{m}(n,r)-\mathrm{log}{\mathrm{\Phi}}^{m+1}(n,r).$$In [43], the phase voltage signal of the grid-connected inverter is decomposed by EMD, and the Fuzzy entropy of the decomposed signal is extracted. Then it is used as the input feature for support vector machine for fault classification, ultimately achieving high diagnostic accuracy. The drawback of Fuzzy entropy is that it requires extra parameters and is sensitive to parameters.
- Joint Entropy and Conditional Entropy. Entropy is also known as self-information. The information entropy and Fuzzy entropy of each phase only reflect the uncertainty/symmetry of its own phase current. But the converter is a whole composed of interconnected phases. As is shown in Figure 5, the fault in phase-a not only leads to the distortion of ${i}_{a}$, but also causes distortion in ${i}_{b}$ and ${i}_{c}$. And the degree of fault in phase-a varies, so does the impact on phase-b and phase-c. Considering these, conditional entropy, joint entropy, mutual information, etc. have been proposed. Their relationship is shown in Figure 6.Assuming X and Y are two correlated random events, their information entropy is $H\left(X\right)$ and $H\left(Y\right)$, respectively. Then, their joint entropy is$$H(X,Y)=-\sum _{x,y}p(x,y)\mathrm{log}p(x,y)=-\sum _{i=1}^{n}\sum _{j=1}^{n}p({x}_{i},{y}_{j})\mathrm{log}p({x}_{i},{y}_{j}).$$$$H\left(Y\right|X)=H(X,Y)-H(X)=H(Y)-I(X;Y),$$$$H\left(X\right|Y)=H(X,Y)-H(Y)=H(X)-I(X;Y).$$In [44], “mean conditional entropy” is proposed, which is actually an improved form of conditional entropy. The fault is diagnosed by extracting the conditional entropy between the two-phase currents of the inverter. Compared to information entropy, conditional entropy reflects the relationship between two phases and is a very effective measure of symmetry between phases.
- Mutual information. As is shown in Figure 6, mutual information is a kind of entropy, which defines the degree of dependence between two almost random variables. It can be regarded as the amount of information about another random variable contained within a random variable, or in other words, the reduced uncertainty of a random variable due to the knowledge of another random variable. For two discrete random variables X and Y, their mutual information is$$I(X,Y)=\sum _{i=1}^{n}\sum _{j=1}^{n}p({x}_{i},{y}_{j})\mathrm{log}\frac{p({x}_{i},{y}_{j})}{p\left({x}_{i}\right)p\left({y}_{j}\right)}.$$By analyzing the properties of conditional probability and combining Equations (19)–(22), it can be concluded that$$I(X,Y)=H\left(X\right)+H\left(Y\right)-H(X,Y).$$In [45], the mutual information between the phase currents was extracted, and then it was used as a feature dataset to train the classifier. This method achieves an effective diagnosis of motor bearing faults through mutual information of motor current signals, which is highly innovative and practical. One reason for the good performance of this study is that the motor itself and the inverter are both symmetrical structures, and both mechanical and electrical faults can cause symmetry to be disrupted. Mutual information effectively measures the symmetry of the system.
- Relative entropy. Relative entropy, also known as Kullback–Leibler Divergence, is used to measure the difference between the probability distributions of two random events. The larger the value, the greater the difference between the two probability distributions; when two probability distributions are completely equal, the relative entropy value is 0. For two random variables X and Y, the relative entropy between them is defined as$$KL\left(X\right|\left|Y\right)=\sum _{i}p\left({x}_{i}\right)\mathrm{log}\frac{p\left({x}_{i}\right)}{p\left({y}_{i}\right)}=\sum _{i}p\left({x}_{i}\right)\mathrm{log}p\left({x}_{i}\right)-\sum _{i}p\left({x}_{i}\right)\mathrm{log}p\left({y}_{i}\right).$$In [46], a relative entropy-based fault prognostic method was proposed for photovoltaic inverters. The relative entropy in this paper can be seen as a measure of the overall symmetry between inverters. For power transistor faults inside the converter, due to the fact that under healthy conditions, the current not only has a different phase but the same distribution, it is easy to infer that $KL\left(m\right|\left|n\right)=0$ ($m,n$ indicate the phase-m and phase-n of the converter). When there is a malfunction, the normal phase and the faulty phase are different, $KL\left(m\right|\left|n\right)>0$. Thus, compared to other forms of entropy, relative entropy has a unique advantage. In addition, the latter term $\sum _{i}p\left({x}_{i}\right)\mathrm{log}p\left({y}_{i}\right)$ in Equation (17) is usually known as “Cross Entropy”, which is often used as the cost function in machine learning. For two time series composed of phase currents, cross entropy can reflect the differences between them, indirectly reflecting the symmetry of power converters. Therefore, it can also be used for fault diagnosis.

#### 3.3. Similarity/Correlation-Based Diagnostic Methods

- Correlation. In statistics, various coefficients have been proposed to reflect the relationships between variables. For power converters, some coefficients such as Pearson’s correlation coefficient not only reflect the connections between signals, but also serve as a mapping of the symmetry of the signal source. Therefore, various coefficients have been adopted for fault diagnosis.For power converters, if it is symmetrical between each phase, the correlation between phase currents must be high. On the contrary, the converter may experience a decrease or even no correlation between the phases due to faults. In [48], an improved variational mode decomposition (VMD) method was proposed for the three-phase current signals to obtain an elementary function called the band-limited intrinsic mode functions (BLIMFs). Then the Pearson correlation coefficients between the original signals and theirs BLIMFs are utilized to detect and locate fault phase, ultimately promoting effective fault diagnosis. The Pearson correlation coefficient between two variables x and y is defined as the quotient of the covariance and standard deviation between the two variables [49]:$${\rho}_{xy}=\frac{{\displaystyle \sum _{i=1}^{n}}({x}_{i}-\overline{x})({y}_{i}-\overline{y})}{\sqrt{{\displaystyle \sum _{i=1}^{n}}{({x}_{i}-\overline{x})}^{2}}\sqrt{{\displaystyle \sum _{i=1}^{n}}{({y}_{i}-\overline{y})}^{2}}}$$Similarly to [48], the VMD + Pearson correlation coefficient-based method was also adopted in [50]. A variant form of the Pearson correlation coefficient was adopted in [20,26]. In [51], the correlations of voltage signals between various sub-modules of Modular multilevel converters (MMCs) were calculated to measure the symmetry between sub-modules. In the paper, the residual voltage was used for fault detection and the correlation coefficients were used for localization. The two-stage method is simple and practical.In [52], the gray relation analysis (GRA) theory was introduced to the field of inverter fault diagnosis and the “gray correlation” based on current signals was proposed. For converter power transistor OC fault diagnosis, set the current reference sequence as ${x}_{0}\left(k\right)$ and the current comparison sequence as ${x}_{i}\left(k\right)$, then the formula for the calculation of gray correlation $\gamma ({x}_{0},{x}_{i})$ is:$$\gamma ({x}_{0}\left(k\right),{x}_{i}\left(k\right))=\frac{\underset{i}{\mathrm{min}}\underset{k}{\mathrm{min}}\left(\right)open="|"\; close="|">{x}_{0}\left(k\right)-{x}_{i}\left(k\right)}{+}$$$$\gamma ({x}_{0},{x}_{i})=\frac{1}{n}\sum _{k=1}^{n}\gamma ({x}_{0}\left(k\right),{x}_{i}\left(k\right)).$$By comparing the gray correlation coefficients of the reference sequence and the comparison sequence, the fault was diagnosed. Compared to the Pearson correlation coefficient, the gray correlation coefficient is susceptible to the influence of outliers and extreme values. Therefore, from the results of this paper, it can be seen that the coefficients calculated under various faults are very similar, which may affect the accuracy of diagnosis. Various other coefficients have also been proposed in research [14,21,53,54], but they are not widely applied.
- Structural Similarity. In [55], structural similarity (SSIM) and contour similarity were successfully used for fault diagnosis in a T-type rectifier. SSIM was originally proposed for image similarity evaluation and has achieved great success in measuring image quality and classifying images [56]. Given two digital images x and y, the SSIM between the two images is calculated as$$SSIM(x,y)=\frac{(2{\mu}_{x}{\mu}_{y}+{c}_{1})(2{{\sigma}_{x}}_{y}+{c}_{2})}{({{\mu}_{x}}^{2}+{{\mu}_{y}}^{2}+{c}_{1})({{\sigma}_{x}}^{2}+{{\sigma}_{y}}^{2}+{c}_{2})}.$$
- Distribution Similarity. In Section 3.2, the current generated at instant t is assumed to be a random variable, and therefore ${I}_{m}\left(t\right)$ is a set of random variables with a one-dimensional distribution. In fact, information entropy can be considered as a statistical value of the distribution of random variables. In [57], the time series ${I}_{m}\left(t\right)$ was transformed and expanded through Wigner–Ville distribution analysis, obtaining the two-dimensional distribution of the signal in the time-frequency domain. Then the two-dimensional distribution similarity between phase-a and phase-b was defined as Equation (29),$${S}_{mn}=\frac{{\displaystyle \sum _{n=0}^{{N}^{\prime}-1}}{\displaystyle \sum _{k=0}^{N-1}}{M}_{A}(n,k)\xb7{M}_{B}(n,k)}{{\displaystyle \sum _{n=0}^{{N}^{\prime}-1}}{\displaystyle \sum _{k=0}^{N-1}}[{M}_{A}(n,k){]}^{2}\xb7{\left[{M}_{B}(n,k)\right]}^{2}},$$

## 4. Discussion

- Symmetry under different time series lengths. In previous studies, the length of time series is usually chosen as the amount of current data sampled within a fundamental current period. However, analyzing the characteristics of the output signal of the power converter, it was found that the sequence length can actually be shorter. Taking the current of the inverter under healthy conditions as an example, redraw Figure 5a as shown in the Figure 7.Divide the current in a period T into four sub-intervals I, II, III, and IV, with each interval length $T/4$. Dashed lines ${l}_{l}$, ${1}_{2}$, and ${1}_{3}$ are the dividing lines of the interval. For current signals in phase-a, the waveforms located in sub intervals I and II are symmetrical with symmetry axis ${l}_{l}$. While the current waveform in intervals II and III is centrally symmetrical about $P(T/2,0)$, and so is the waveform in the former half and the latter half in a period. This paper refers to them as “quarter-wave symmetry (QWS)” and “half-wave symmetry (HWS)”, respectively. Moreover, in the healthy condition, in terms of the time dimension, the current waveform in the latter period is the same as or symmetrical to the previous one. Therefore, not only are the three-phase currents symmetrical, but each phase current itself is also symmetrical. Previous studies have mostly focused on the symmetry between currents in two phases, with less research on the symmetry of the current itself, which may be a worthwhile direction for further investigation.
- The influence of reference frame on symmetry. In the time–current coordinate system, as shown in Figure 5, the current waveform is half-wave symmetrical under healthy conditions but asymmetric when a fault occurs. But if the three-phase currents are transformed into the $\alpha $-$\beta $ coordinate through Clark transformation, things will be different. The formula of Clark transformation is given in Equation (31), where k is a proportional coefficient. Clark transformation can transform the current in the three-phase abc coordinate system into the Cartesian coordinate system. Figure 8 shows the current trajectories in $\alpha $-$\beta $ coordinate under different conditions [58,59]. It can be seen that the waveforms of current trajectories under fault conditions are still symmetrical. Fault diagnosis can be performed through the axis of symmetry/center of symmetry. In the two-dimensional current–current coordinate system, the current trajectories are symmetrical for some types of faults, while others are not [60]. Therefore, it can be concluded that the symmetry is related to the reference frame, and the symmetry under different reference frames is also worth further research.$$\left(\right)open="["\; close="]">\begin{array}{c}{i}_{\alpha}\\ {i}_{\beta}\end{array}\xb7\left(\right)open="["\; close="]">\begin{array}{c}{i}_{a}\\ {i}_{b}\\ {i}_{c}\end{array}$$

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A topological structure of variable frequency speed control system for induction motor, including a rectifier and an inverter.

**Figure 6.**The relationship between information entropy, joint entropy, conditional entropy, and mutual information.

**Figure 8.**The current trajectories in $\alpha $-$\beta $ coordinate of a two-level three-phase inverter under four types of faults. (

**a**) Single-transistor OC fault. (

**b**) Double-transistor OC fault on the same leg. (

**c**) Double-transistor OC fault on different leg (one is in the upper, the other in the lower). (

**d**) Double-transistor OC fault occur on different leg (both are in the upper or lower).

Diagnostic Feature | Diagnostic Time ^{1} | Computational Cost | Robustness ^{2} | Diagnosable Fault Type ^{3} |
---|---|---|---|---|

ED [21] | $T/4-T/2$ | medium | medium | 21 |

WT + ED [27] | T | medium | high | 9 |

CD + ED [26] | $T/4$ | low | high | 21 |

MhtD [15] | T | low | high | 9 |

Normalized covariance [30] | $T/2$ | low | high | 9 |

^{1}The time from the occurrence of the fault to it is located, measured in current cycles T.

^{2}The independence to transient disturbance such as load fluctuation.

^{3}The OC fault types that the method can diagnose.

**Table 2.**Comparison of the advantages and disadvantages of distance-based, entropy-based, and similarity-based diagnosis methods.

Methods | Computational Cost | Robustness to Change of Signal Amplitude | Robustness to Change of Signal Seriod |
---|---|---|---|

distance-based | low | low | low |

entropy-based | medium | medium | high |

similarity/correlation-based | low | high | medium |

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**MDPI and ACS Style**

Zhou, Y.; Zhao, J.; Wu, Z.
A Review of Symmetry-Based Open-Circuit Fault Diagnostic Methods for Power Converters. *Symmetry* **2024**, *16*, 204.
https://doi.org/10.3390/sym16020204

**AMA Style**

Zhou Y, Zhao J, Wu Z.
A Review of Symmetry-Based Open-Circuit Fault Diagnostic Methods for Power Converters. *Symmetry*. 2024; 16(2):204.
https://doi.org/10.3390/sym16020204

**Chicago/Turabian Style**

Zhou, Yang, Jin Zhao, and Zhixi Wu.
2024. "A Review of Symmetry-Based Open-Circuit Fault Diagnostic Methods for Power Converters" *Symmetry* 16, no. 2: 204.
https://doi.org/10.3390/sym16020204