# Detection and classification of multiple power signal patterns with Volterra series and interval type-2 fuzzy logic system

- Rahul
^{1}Email author, - Rajiv Kapoor
^{1}and - M. M. Tripathi
^{2}

**2**:9

**DOI: **10.1186/s41601-017-0039-z

© The Author(s) 2017

**Received: **3 September 2016

**Accepted: **28 February 2017

**Published: **29 March 2017

## Abstract

The paper deals with the application of Volterra bound Interval type −2 fuzzy logic techniques in power quality assessment. This work proposes a new layout for detection, localization and classification of various types of power quality events. The proposed method exploits Volterra series for the extraction of relevant features, which are used to recognize different PQ events by Interval type-2 fuzzy logic based classifier. Numerous single as well as multiple powers signal disturbances have been simulated to testify the efficiency of the proposed technique. This time–frequency analysis results in the clear visual detection, localization, and classification of the different power quality events. The simulation results signify that the proposed scheme has a higher recognition rate while classifying single and multiple power quality events unlike other methods. Finally, the proposed method is compared with SVM, feed forward neural network and type −1 Fuzzy logic system based classifier to show the efficacy of the proposed technique in classifying the Power quality events.

### Keywords

Non-stationary power signals Power quality (PQ) Volterra series Interval type-2 fuzzy logic system (IT2FLS) Power Spectral Entropy (PSE) Standard Deviation (SD)## 1 Introduction

In the new era of power systems, power quality (PQ) issues have attained considerable attention in the last few decades due to increased demand of power electronics and/or microprocessor based non-linear controlled loads. While these devices create power quality problems, at the same time, devices may also malfunction due to the severe power quality problems [1]. Electricity is now treated as commercial product that is evaluated not only by its reliability but also by its quality. The customer will choose the supplier providing electrical energy having better power quality, at lower cost and acceptable reliability that meet his load needs. The utilities or other electric power providers have to ensure a high quality of power delivery to remain competitive and to retain/attract the customers in new electricity market scenario [2].

PQ disturbances/events cover a broad frequency range with significantly different magnitude variations and can be stationary or non-stationary signals [3]. The on-line detection and identification of PQ disturbances are essential so that source and cause of such disturbances is known for taking appropriate mitigation/corrective actions. A feasible approach to achieve this goal is to incorporate detection capabilities into monitoring equipment so that the events of interest can be recognized, captured and classified automatically. This is done in a sequential manner by detecting the disturbance, then localizing it and finally classifying the various PQ events [4]. To carry out this task a tool is required which has both the capability to analyze different PQ events and classify them as well.

This paper proposes a tool which is the combination of Volterra series and Type-2 fuzzy logic. Volterra series does signal decomposition to extract valuable information from the signal in order to detect power quality disturbances. The proposed type-2 fuzzy logic method classifies the power quality disturbances even in the presence of higher level of uncertainty hence it is very useful in real time scenario. The general structure of fuzzy reasoning handles much of the uncertainties and fuzzy systems utilize type-1 fuzzy sets (which express uncertainty by numbers in the range [0, 1]) for classification.

This paper has following 5 sections. First, the Volterra series expansion for nonlinear systems explained in Section II; Application of Volterra series in power quality events’ detection presented in Section III; in Section IV power spectral entropy for analysis of power quality events, the utilization of the Interval type-2 fuzzy logic system (IT2FLS) for classification PQ events is presented in Section V, Proposed methodology for classification of PQ events given in Section VI, results and discussion included in section VII and Section VIII concludes the paper.

## 2 Volterra series expansion for nonlinear systems

_{p}[m

_{1}, m

_{2,}…, m

_{p}] is known as the p-the order Volterra kernel of the system, Without any loss of generality, the kernels can be assumed to be symmetric. In general any kernel \( {\mathrm{h}}_{\mathrm{p}} \) [m

_{1}, m

_{2}. …, m

_{p}] can be replaced by symmetric one by simple setting.

_{1}, m

_{2,}.... m

_{p}. The Volterra series is a power series with memory. This can be checked by changing the input by a gain factor d so that the new input is dx(t). By using (2), the new output is

This is a power series with amplitude factor d. The integrals are convolutions it shows that series having memory. As an effect of its power series features, it has some limitations associated with the application of the Volterra series to nonlinear problems. The convergence of Volterra series is one of the major limitations. One can think of the Volterra series expansion as a Taylor series expansion with memory. The limitations of the Volterra series expansion are similar to those of the Taylor series expansion both expansions do not do well when there are discontinuities in the system description [5].

### 2.1 Volterra kernels estimation by exponential method

where r under the summation sign indicates that the sum includes all the distinct vectors \( \Big({r}_1,{r}_2,\dots .{r}_q \)) such that \( \sum_{\mathrm{i}=1}^{\mathrm{q}}{\mathrm{r}}_{\mathrm{i}}=\mathrm{p} \) . If \( {r}_1={r}_2=\dots ={r}_q=1 \) then the amplitude associated with the exponential term \( {\mathrm{e}}^{\left({\mathrm{s}}_{{\mathrm{q}}_1+\dots }{\mathrm{s}}_{{\mathrm{q}}_{\mathrm{m}}}\right)\mathrm{t}} \) is q \( !{\mathrm{H}}_{\mathrm{q}}\left({\mathrm{s}}_1,\dots .{\mathrm{s}}_{\mathrm{q}}\right) \). Now just calculating the transfer function of the system, we can calculate Volterra series kernel.

Volterra series having application in various fields of engineering and physics and can be practically classified into two distinct categories. In the first classification, a model of an observed dynamical phenomenon is build using Volterra series and the estimation of Volterra frequency response function requires experimental and numerically generated data. The second classification comprises the analysis of dynamical systems that are already represented by an analytic model, such as differential equation in mathematics. The harmonically excited nonlinear systems’ behavior can be investigated better using Volterra series representation [6].

Volterra series expansion exists for nonlinearity and non-stationarity. Even though PQ events are non-stationary in nature so this technique is applicable to power quality disturbances, Volterra system models have been successfully employed in PQ events detection and localization applications in this paper and such models can also be implemented in real time PQ events monitoring system which can change the present scenario of PQ events monitoring system.

## 3 Application of Volterra series

## 4 Power spectral entropy

## 5 Interval type-2 fuzzy logic system (IT2FLS)

*Ã*, can be calculated as

Interval type-2 FLS provides the capability of handling a higher level of uncertainty and provides a number of missing components that have held back successful deployment of fuzzy systems in human decision making.

## 6 Proposed methodology for classification

The Proposed state of art uses limiting method to handle the lengthy calculations of type-2 FLS. Now type-2 fuzzy converted into type −1. The obtained output of IT2FLS provides more feasible result in comparison of other classification techniques.

## 7 Results and Discussion

Rule Editor for IT2FLS

SD\PSE | Low | Moderate | High |
---|---|---|---|

Low | Swell | Swell+Harmonics | None |

Moderate | Harmonics | Tansient | Sag+Harmonics |

High | None | None | Sag |

The rules designed for classification of multiple events such as sag plus harmonics and swell plus harmonics show in Table 1, if SD is moderate and PSE is high then multiple events sag plus harmonics occur as disturbance in power quality. In another case when SD is low and PSE is moderate then disturbance classified will be swell + harmonics.

Classification results under ideal signal condition

Events | Sag | Swell | Harmonics | Transcient | Sag+Har | Swell+Har |
---|---|---|---|---|---|---|

Sag | 50 | |||||

Swell | 50 | |||||

Harmonics | 49 | |||||

Transcient | 48 | |||||

Sag+Har | 47 | |||||

Swell+Har | 47 | |||||

Calssification efficiency in % | 100 | 100 | 98 | 96 | 94 | 94 |

Classificatione error in % | 0 | 0 | 2 | 4 | 6 | 6 |

Overall efficiency | 97 |

When classification of power quality events done under ideal condition, then no noise in present in the system then IT2FLS system classify sag and swell events with 100% classification efficiency but harmonics and Transient events having classification efficiency 98 and 96% respectively. When multiple events occur such as sag plus harmonics and swell plus harmonics then efficiency comes out to be 94%. Overall efficiency in ideal signal condition is 97% which shows that this technique can classify most of the events with more than 95% efficiency respectively as shown in Table 2.

Classification results under SNR 30 db

Events | Sag | Swell | Harmonics | Transcient | Sag+Har | Swell+Har |
---|---|---|---|---|---|---|

Sag | 50 | |||||

Swell | 48 | |||||

Harmonics | 48 | |||||

Transcient | 45 | |||||

Sag+Har | 43 | |||||

Swell+Har | 44 | |||||

Classification efficiency in % | 100 | 96 | 96 | 90 | 86 | 88 |

Classification error in % | 0 | 4 | 4 | 10 | 14 | 12 |

Overall Efficiency | 93 |

Classification results under SNR 20 db

Events | Sag | Swell | Harmonic | Transcient | Sag+Har | Swell+Har |
---|---|---|---|---|---|---|

Sag | 48 | |||||

Swell | 47 | |||||

Harmonic | 47 | |||||

Transcient | 42 | |||||

Sag+Har | 42 | |||||

Swell+Har | 41 | |||||

Classification efficiency in % | 96 | 94 | 94 | 84 | 84 | 82 |

Classification error in % | 6 | 6 | 6 | 16 | 16 | 18 |

Overall efficiency | 89 |

All these classification results show that in ideal and practical situation this technique performed well for with overall efficiency more than 85%. We can apply this technique in real time monitoring, this novel method definitely perform well in detection and classification of power quality events in real time scenario.

## 8 Conclusions

This paper proposes a PQ event detection and classification scheme utilizing a Volterra series based feature extractor and a classifier based on interval type-2 fuzzy logic system. The proposed method can reduce the quantity of extracted features of distorted signal without losing its characteristics and thus, requires less memory space and computation time. The performance of classifier is test under three conditions i.e. ideal, SNR 30 dB and SNR 20 for effective classification of PQ events. It is observed that IT2FLS correctly classifies the PQ event with high accuracy and IT2FLS gives the best performance as compared to neural network based classifier and SVM [12]. Therefore, the proposed method can be used as the PQ event classifier in real time system. The overall classification efficiency of IT2FLS is 93% if take average of all three condition mention in this paper. The simulations result show that IT2FLS has higher performance than ANN with feed forward multilayer back propagation (FFML), learning vector quantization (LVQ), probabilistic neural network (PNN) [13, 14].

## Declarations

### Acknowledgment

This work was supported by my mentors Prof. Rajiv Kapoor & Dr. M M Tripathi. I am grateful for their guidance, feedback and advice. I would like to express my deepest thanks to my mentors. They are supportive in their feedback and always motivated me to do work hard.

### Authors’ contributions

R - Analysis for detection of power quality with volterra series and feature based on volterra series extracted by this author. RK - Interval Type-2 fuzzy logic for classification of power quality is applied for power quality signals by this author. MMT analyzed the results of classification technique and calculated efficiency of classification in different noisy condition and concluded this research article. All authors read and approved the final manuscript.

### About the authors

Rahul having B.Tech, M.Tech Degree in electronics & communication and pursuing PhD in electronics & communication department of Delhi technological university, Delhi, India his areas of interest in Power quality, Power line communication, Digital communication and Digital signal processing.

Rajiv Kapoor having BE, ME & PhD degree in electronics & communication and working as a Professor in EC department of Delhi technological university, Delhi India, his areas of interest in Power Quality, Image Processing, Computer Vision, Signal Processing, Cognitive Radio.

MM Tripathi having B. Tech and PhD degree in Electrical Engineering and working as Associate Professor in EE Department of Delhi Technological University, Delhi, India and his area of interest are Power quality, Artificial Intelligence (AI) application in power system and Embedded Systems.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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