# Singular Signal Measurement Based on Bidirectional Recursive Complex-Valued Wavelet Algorithm

Authors: Weiping Lv; Jinshan Xie; Jing Huang
Journal: Security and Communication Networks
 (2022)

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## Abstract

The normal operation fault of the power system is usually caused by a short-circuit fault. At this time, the system changes drastically from one state to another, accompanied by complex transient phenomena. Therefore, the measured signal contains a large number of transient components. How to effectively analyze such signals, extract their characteristics, and develop new protection devices has always been an important research field in power system protection technology. The protection of the power system is to achieve the purpose of correct action and elimination of faults by quickly detecting and locating faults. At present, the power signal analysis tools used in microcomputer protection include FFT, Kalman filter, and finite impulse response filter. They are efficient for the analysis of stationary signals, but have their limitations in analyzing nonstationary signals; especially it is difficult to identify nonlinear faults, such as the detection of high-impedance nonlinear short-circuit faults, which is a long-term unsolved problem in power systems. Based on wavelet transform, this paper selects complex-valued wavelet algorithm, analyzes a real-time recursive wavelet algorithm, and deduces the realization process of the algorithm in detail. The algorithm greatly reduces the computational complexity of the existing two-way recursive algorithm, can be used for real-time detection of fault signals in various fields of power system, and can be extended to realize other fast recursive algorithms of wavelet functions. Based on the sensitivity of complex-valued wavelet transform phase information to singularity, a method for real-time monitoring of power system fault mutation signals using the phase information of complex-valued wavelet fast recursion algorithm to assist amplitude information is proposed. The validity and practicability of this complex-valued wavelet and its real-time recursive algorithm for fault detection are demonstrated by an example.


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## Body

## 1. Introduction

A singularity signal refers to a signal in which the signal itself or a certain order of its derivative has a sudden change at a certain moment, and the singularity detection is to identify the singularity of the signal and determine the degree of singularity. For a long time, the Fourier transform has been the main tool for studying the singularity of functions (signals). However, due to the lack of spatial local characteristics of the Fourier transform, it can only determine the overall nature of the singularity of a function, and it is difficult to determine the location and location of the singularity in space. Distribution and the singularity or sudden change of the signal is the most critical and important property of nonstationary signals in many cases. For example, mechanical failures, power system failures, abnormalities in the electrocardiogram of the EEG, and so on, all correspond to the mutation points of the test signal. Although they occur in different backgrounds, if the measured data is viewed as a signal, they are all reflected in the how to extract the location of the mutation point in the signal and determine its singularity (or smoothness). Therefore, the detection of signal singularity has particularly important practical significance. Fourier transform is an ideal tool for studying stable signals. For a long time, Fourier transform is also the main tool to study the singularity of signals. Transformation lacks spatial locality. It can only determine the overall nature of a signal’s singularity, but it is difficult to determine the location and distribution of singular points in space. In fact, the singular points and irregular sudden changes in the signal often carry more important signals, which are one of the important characteristics of the signal. Wavelet analysis is an emerging discipline that has developed rapidly in recent years. It breaks through the Fourier transform without any resolution limitation in the time domain and can analyze the signal components in a specified frequency band and time period. It has good localization properties in both the time domain and the frequency domain. Wavelet transform can accurately capture the characteristics of transient signals and uses gradually fine time domain or spatial sampling steps for frequency components so that any detail of the signal can be focused on. Vividly speaking, it is to analyze the signal in a “slice type” and observe the evolution and characteristics of the signal in different frequency bands [1–10]. For example, detecting high-impedance faults (low-current arc faults) is a long-term unsolved problem in the power system. Traditional overcurrent protection devices simply cannot be detected; and some faults include arcs, which cannot form a fixed ground, resulting in a very small current, and sometimes there is no current at all.A notable feature of power system failures is the sharp increase in current, which damages electrical equipment in terms of force and heat. Fuse protection and overcurrent protection are the original protection principles that reflect the characteristics of the sharp increase in current. Another feature of the fault is the sharp drop in voltage and the corresponding low-voltage protection. A protection principle that reflects both the voltage drop and the current increase is impedance (distance) protection. It reflects the distance to the fault by the decrease in impedance. Decide on the protection action: in order to more accurately distinguish between normal operating conditions and fault (or abnormal) conditions, there are no or few fault conditions during normal operation but there are large electrical quantities, such as negative sequence or zero sequence current, voltage, and power judgment: the use of power system fault protection is not limited to electrical quantities, but also other physical quantities, such as a large amount of gas and an increase in oil flow rate or oil pressure caused by a fault in the transformer tank [11–15]. The fault diagnosis is shown in Figure 1. This will cause the fault to last for hours or even days. In many cases, it can only be found by observation. This type of fault signal is irregular in time. The fault level may be quite high in a few cycles, but it drops to a normal load level in many cycles.Figure 1
Fault diagnosis.However, they generally have problems in identifying nonstationary time-varying signals and abrupt signals caused by nonlinear faults, which are difficult or even misidentified. As a result, fires, electrical breakdowns, personal accidents, burnout of switchgear, and damage to other devices are caused [16–22]. Therefore, in order to ensure the safe and reliable operation of the power system, it is necessary to conduct state monitoring and fault diagnosis of the power equipment. According to the various signals measured during the operation of the power equipment, the operating status can be determined through signal analysis [23–25]. The moment of the sudden change of the equipment fault information and the magnitude of the sudden change are captured, which helps to take early measures to restore the system to normal at the beginning of the fault. For fault identification, it is often through the analysis of information such as the amplitude and frequency components of the fault signal, combined with the existing fault signs of the power equipment to make judgments, which is convenient for fault location and rapid maintenance. The phase information of wavelet transform is often more sensitive to singularity, easy to capture singular points, and accurately detect signal mutations. Therefore, complex-valued wavelets can be used to calculate wavelet coefficients, but each calculation needs to complete a complete integral. The amount of calculation is with the rapid increase in the amount of data. Therefore, seeking a fast algorithm that satisfies the real-time requirements and retains the original calculation accuracy is extremely important for quickly capturing abnormal information of equipment, issuing forecasts in the early stage of equipment failure, and improving equipment operation reliability.


## 2. Real-Time Two-Way Recursive Complex Wavelet Algorithm

Singularities and irregular mutations in signals often contain important signals, which are one of the important characteristics of signals, such as spikes and spikes in eeg, edges of objects in images, rock ultrasonic detection signals with time-varying spectral characteristics and mechanical fault detection signals, and so on. The application of wavelet in power system can be expressed in the following aspects.


### 2.1. Status Monitoring and Fault Diagnosis of Power Equipment

Power equipment status monitoring and fault diagnosis is to decompose and process various electromagnetic, mechanical, and other physical signals generated by the basic equipment of the power system during operation and to determine its status in real time in order to issue an alarm at the initial stage of the fault or in the event of a fault. The electromagnetic signals emitted by power equipment are relatively stable during normal operation, and abnormal conditions in a day must contain singularities.


### 2.2. Harmonic Analysis of Power System

When a fault occurs in the power system, it is accompanied by the generation of high-order harmonics; in the high-voltage direct current transmission system, both the AC side and the DC side of the converter station produce high-order harmonics. In order to avoid the adverse effects of these harmonics, it is necessary to analyze and suppress them. Wavelet analysis transforms and projects this type of signal to different scales, which will clearly show the characteristics of these high-frequency and singular high-order harmonic signals. In particular, the wavelet packet has the characteristics of further subdividing the frequency space, which will be very good. Suppress high-order harmonics and provide a reliable basis.


### 2.3. Power System Transient Stability

When the power system is subject to large disturbances, various electromagnetic signal parameters that characterize the operating state of the system will undergo rapid changes and oscillations. To analyze such a sudden and localized signal, wavelet analysis is undoubtedly a good choice. The ability of wavelet analysis to capture and process weak mutation signals can be used in power system transient stability prediction research based on weak signals. Its “local refinement and amplification” feature can identify and track weak mutations in system variables, and then infer that mutations are caused. The time and location of local failures can improve the real-time and accuracy of power system transient stability prediction.


### 2.4. Dynamic Safety Analysis of Power System

When the power system is disturbed, it will cause the system voltage to fluctuate. Too low voltage will endanger the stability of the power system and may cause “voltage avalanche” in severe cases. Therefore, it is increasingly important to study the dynamic response of voltage. When the system is disturbed, it generates a voltage mutation signal. Apply wavelet analysis to decompose this abrupt signal into different scales, and then analyze the magnitude and phase of the abrupt signal on these different scales to determine the dynamic and safe operation of the power system.


### 2.5. Antielectromagnetic Interference

The large number of electromagnetic interference signals generated by the power grid makes it difficult to extract the characteristic signals of the operation behavior of power equipment. Wavelet analysis can apply wavelet transform to decompose the mixed signal containing the desired signal and electromagnetic interference signal to different scales. The wavelet coefficients associated with the signal are set to zero (to clear the interference signal), and then the reconstruction formula is used to construct the required signal, which also realizes the separation of the required signal and the interference signal and achieves the purpose of antielectromagnetic interference. The use of wavelet analysis to filter out the white noise in the signal has been successfully applied.


### 2.6. Fault Location of Transmission Line

Reliable operation of the power system requires timely and accurate knowledge of the fault location. Existing fault location methods and fault locators have been able to achieve this function, but there are still some problems in the processing of fault signals. If the current and voltage signals are obtained through fault recording, wavelet transform is used to decompose such singular and instantaneous fault signals, and the fault signals are clearly reflected on different scales, thus constructing the distance function (distance function). function), and then infer the time and location of the fault that caused the sudden change signal, and finally reflect the fault distance to achieve the purpose of fault location, which will improve the accuracy of fault location.


### 2.7. Short-Term Load Forecasting of Power System

Short-term load forecasting of the power system is a prerequisite for the economical and reliable operation of the power system. The various existing forecasting methods mainly include time series neural network method. The electric load has a special periodicity. The load fluctuates in cycles of days, weeks, and years, and small cycles are nested in large cycles. Wavelet transform can decompose various interleaved mixed signals composed of different frequencies into block signals on different frequency bands. Therefore, the wavelet transform of the load sequence can project the load sequence on different scales, and each scale can be approximated; the ground is regarded as different “frequency bands” so that the subsequences on each scale represent the components of different “frequency domains” in the original sequence, and they more clearly show the periodicity of the load sequence.


### 2.8. HVDC Transmission System

The analysis and detection of the bridge arm short circuit and commutation failure of the HVDC transmission system can still be realized by means of wavelet analysis, which is sensitive to singular, weak, and transient fault signals. In addition, wavelet analysis has equally important applications in HVDC system fault diagnosis, main equipment status monitoring, HVDC transmission line fault location, antielectromagnetic interference, and many other aspects.Generally speaking, the expression of complex wavelet is(1)ψt=1+σt+σ22t2e−σtejw0t,where w0=2π, σ = 2π/3. According to this, a two-way recursive algorithm (recursive wavelet transform) can be constructed. For a certain scale s, only 5 complex coefficients such as δ1 and λ1 and their conjugates need to be calculated once, and then only the wavelet coefficients need to be calculated recursively. With the small increase in data, each calculation of a wavelet coefficient requires 36 real number multiplications and 35 real number additions. Compared with direct integral transformation, the amount of calculation is greatly reduced, so it can be used for fast calculation. But it is composed of two parts, causal and noncausal, which need to calculate backward and forward recursively, respectively. The result of wavelet transform is not easy to be stable, and the calculation speed is reduced. Therefore, this paper has made further improvements on this basis and established a lower-order wavelet function, which only needs to perform backward one-way recursive calculation, which overcomes the deficiencies of the above methods. The repeated and irregular occurrence of these sudden fault currents results in a very low average RS current level. Even if the general protection device is set low enough, it will be affected by the abovementioned irregular characteristics and refuse to operate. However, a large amount of electric energy flows continuously to the point of failure. There are functions:(2)ψ1t=−65σt−65σ2t2−σ3t3eσtejw0tu−t.Then, the wavelet function can be obtained:(3)ψt=65σt−65σ2t2+σ3t3e−σtejw0tut,where it satisfies(4)ψt=ψ¯1−t.Its Fourier transform is(5)ψ^w=65⋅4σ3−σw−w02σ+jw−w04,where w0=2π, f = 1/s, σ = π. The wavelet function ψ(t) satisfies the admissibility condition, and its time-domain and frequency-domain waveforms are shown in Figure 2.(6)ψ^0=0.Figure 2
Time domain, frequency domain waveform.For the selected waveletψ(t), the integral wavelet transform of the continuous signal f(t) is(7)Wfs,x=1s∫−∞∞ftψx−tsdt.From the above formula, the discrete wavelet transform of the digital signal is shown in the following formula:(8)Wfs,kTs=TssfkTψkTss,where f(nTs) is the digital signal sequence (Ts is the sampling period) and ψ is the mother wavelet discrete sequence.From theZ-transform of the sequence and its time-domain convolution properties, the Z-transform of the signal sequence, mother wavelet sequence, and wavelet coefficient sequence is(9)Wz=Tssfzψz.For wavelet sequence,(10)ψkTss=6σkTs5s−65σkTss2+σkTss3e−σTsk/sejw0Tsk/s.Let(11)β=σTss,A=−σTss−jw0Tss.Finally, the recurrence formula obtained by the inverseZ-transformation is(12)Ws,k=Tssδ1fk−1Ts+δ2fk−2T2+δ3fk−3Ts−λ1Ws,k−1Ts−λ2Ws,k−2Ts−λ3Ws,k−3Ts−λ4Ws,k−4Ts.On the basis of this algorithm, only 4 wavelet transform initial values need to be calculated, and then all wavelet coefficients can be calculated by backward one-way recursion, which avoids forward recursive operations and reduces multiplication and addition operations by half. The required coefficients (δ, λ, etc.) only need to be calculated once and have a lower power calculation; this algorithm can meet the needs of real-time calculation. Use wavelet analysis theory to do multiresolution analysis (MRA) on the obtained singular electromagnetic signal, and decompose the signal into different scales.Different wavelets show different characteristics in terms of orthogonality, tight support, smoothness, and even symmetry, but it is difficult to construct a wavelet function with four characteristics at the same time. Daubechies proved that when the scale function and wavelet function determined by a multiresolution analysis are real functions and both have compact support, the wavelet function no longer has symmetry or antisymmetric (except for Haar wavelet). In practical applications, only according to the needs of different signal decomposition, a compromise is made between several characteristics, and the wavelet that meets the needs is selected for decomposition. Qualitatively speaking, when the oscillation frequency of the detected signal is close to the oscillation frequency of the wavelet function of the corresponding scale, the signal obtains a wavelet decomposition with larger coefficients. This is the reason why wavelet analysis can extract different frequency components of the signal at multiple scales. In other words, the “entropy” value is usually used to measure the distance between the signal and the wavelet basis. The smaller the distance (i.e., the smaller the entropy value), the smaller the difference between the signal and the basis, and the signal gets the maximum decomposition. Therefore, different wavelets need to be selected for different transient signals. By comparing the entropy value of different wavelet decompositions, the wavelet with smaller entropy value is selected to obtain larger decomposition and achieve better detection effect.The idea of the above recursive algorithm can also be extended to other wavelets with the same form. As long as the selected wavelet satisfies the form of the product of polynomial and exponential decay, itsZ-transform can be transformed into a rational function of z-1, and the corresponding structure is constructed according to the aforementioned method. For the real-time recursive algorithm, the difference lies in the calculation of the coefficients. The predicted data is shown in Figure 3. The components on each scale reflect the different frequency components of the original signal, which can show the fault signal to reach the state the purpose of monitoring or fault diagnosis. This method has been successfully applied in the diagnosis of broken rotor bars in electric motors.Figure 3
Predicted data.


## 2.1. Status Monitoring and Fault Diagnosis of Power Equipment

Power equipment status monitoring and fault diagnosis is to decompose and process various electromagnetic, mechanical, and other physical signals generated by the basic equipment of the power system during operation and to determine its status in real time in order to issue an alarm at the initial stage of the fault or in the event of a fault. The electromagnetic signals emitted by power equipment are relatively stable during normal operation, and abnormal conditions in a day must contain singularities.


## 2.2. Harmonic Analysis of Power System

When a fault occurs in the power system, it is accompanied by the generation of high-order harmonics; in the high-voltage direct current transmission system, both the AC side and the DC side of the converter station produce high-order harmonics. In order to avoid the adverse effects of these harmonics, it is necessary to analyze and suppress them. Wavelet analysis transforms and projects this type of signal to different scales, which will clearly show the characteristics of these high-frequency and singular high-order harmonic signals. In particular, the wavelet packet has the characteristics of further subdividing the frequency space, which will be very good. Suppress high-order harmonics and provide a reliable basis.


## 2.3. Power System Transient Stability

When the power system is subject to large disturbances, various electromagnetic signal parameters that characterize the operating state of the system will undergo rapid changes and oscillations. To analyze such a sudden and localized signal, wavelet analysis is undoubtedly a good choice. The ability of wavelet analysis to capture and process weak mutation signals can be used in power system transient stability prediction research based on weak signals. Its “local refinement and amplification” feature can identify and track weak mutations in system variables, and then infer that mutations are caused. The time and location of local failures can improve the real-time and accuracy of power system transient stability prediction.


## 2.4. Dynamic Safety Analysis of Power System

When the power system is disturbed, it will cause the system voltage to fluctuate. Too low voltage will endanger the stability of the power system and may cause “voltage avalanche” in severe cases. Therefore, it is increasingly important to study the dynamic response of voltage. When the system is disturbed, it generates a voltage mutation signal. Apply wavelet analysis to decompose this abrupt signal into different scales, and then analyze the magnitude and phase of the abrupt signal on these different scales to determine the dynamic and safe operation of the power system.


## 2.5. Antielectromagnetic Interference

The large number of electromagnetic interference signals generated by the power grid makes it difficult to extract the characteristic signals of the operation behavior of power equipment. Wavelet analysis can apply wavelet transform to decompose the mixed signal containing the desired signal and electromagnetic interference signal to different scales. The wavelet coefficients associated with the signal are set to zero (to clear the interference signal), and then the reconstruction formula is used to construct the required signal, which also realizes the separation of the required signal and the interference signal and achieves the purpose of antielectromagnetic interference. The use of wavelet analysis to filter out the white noise in the signal has been successfully applied.


## 2.6. Fault Location of Transmission Line

Reliable operation of the power system requires timely and accurate knowledge of the fault location. Existing fault location methods and fault locators have been able to achieve this function, but there are still some problems in the processing of fault signals. If the current and voltage signals are obtained through fault recording, wavelet transform is used to decompose such singular and instantaneous fault signals, and the fault signals are clearly reflected on different scales, thus constructing the distance function (distance function). function), and then infer the time and location of the fault that caused the sudden change signal, and finally reflect the fault distance to achieve the purpose of fault location, which will improve the accuracy of fault location.


## 2.7. Short-Term Load Forecasting of Power System

Short-term load forecasting of the power system is a prerequisite for the economical and reliable operation of the power system. The various existing forecasting methods mainly include time series neural network method. The electric load has a special periodicity. The load fluctuates in cycles of days, weeks, and years, and small cycles are nested in large cycles. Wavelet transform can decompose various interleaved mixed signals composed of different frequencies into block signals on different frequency bands. Therefore, the wavelet transform of the load sequence can project the load sequence on different scales, and each scale can be approximated; the ground is regarded as different “frequency bands” so that the subsequences on each scale represent the components of different “frequency domains” in the original sequence, and they more clearly show the periodicity of the load sequence.


## 2.8. HVDC Transmission System

The analysis and detection of the bridge arm short circuit and commutation failure of the HVDC transmission system can still be realized by means of wavelet analysis, which is sensitive to singular, weak, and transient fault signals. In addition, wavelet analysis has equally important applications in HVDC system fault diagnosis, main equipment status monitoring, HVDC transmission line fault location, antielectromagnetic interference, and many other aspects.Generally speaking, the expression of complex wavelet is(1)ψt=1+σt+σ22t2e−σtejw0t,where w0=2π, σ = 2π/3. According to this, a two-way recursive algorithm (recursive wavelet transform) can be constructed. For a certain scale s, only 5 complex coefficients such as δ1 and λ1 and their conjugates need to be calculated once, and then only the wavelet coefficients need to be calculated recursively. With the small increase in data, each calculation of a wavelet coefficient requires 36 real number multiplications and 35 real number additions. Compared with direct integral transformation, the amount of calculation is greatly reduced, so it can be used for fast calculation. But it is composed of two parts, causal and noncausal, which need to calculate backward and forward recursively, respectively. The result of wavelet transform is not easy to be stable, and the calculation speed is reduced. Therefore, this paper has made further improvements on this basis and established a lower-order wavelet function, which only needs to perform backward one-way recursive calculation, which overcomes the deficiencies of the above methods. The repeated and irregular occurrence of these sudden fault currents results in a very low average RS current level. Even if the general protection device is set low enough, it will be affected by the abovementioned irregular characteristics and refuse to operate. However, a large amount of electric energy flows continuously to the point of failure. There are functions:(2)ψ1t=−65σt−65σ2t2−σ3t3eσtejw0tu−t.Then, the wavelet function can be obtained:(3)ψt=65σt−65σ2t2+σ3t3e−σtejw0tut,where it satisfies(4)ψt=ψ¯1−t.Its Fourier transform is(5)ψ^w=65⋅4σ3−σw−w02σ+jw−w04,where w0=2π, f = 1/s, σ = π. The wavelet function ψ(t) satisfies the admissibility condition, and its time-domain and frequency-domain waveforms are shown in Figure 2.(6)ψ^0=0.Figure 2
Time domain, frequency domain waveform.For the selected waveletψ(t), the integral wavelet transform of the continuous signal f(t) is(7)Wfs,x=1s∫−∞∞ftψx−tsdt.From the above formula, the discrete wavelet transform of the digital signal is shown in the following formula:(8)Wfs,kTs=TssfkTψkTss,where f(nTs) is the digital signal sequence (Ts is the sampling period) and ψ is the mother wavelet discrete sequence.From theZ-transform of the sequence and its time-domain convolution properties, the Z-transform of the signal sequence, mother wavelet sequence, and wavelet coefficient sequence is(9)Wz=Tssfzψz.For wavelet sequence,(10)ψkTss=6σkTs5s−65σkTss2+σkTss3e−σTsk/sejw0Tsk/s.Let(11)β=σTss,A=−σTss−jw0Tss.Finally, the recurrence formula obtained by the inverseZ-transformation is(12)Ws,k=Tssδ1fk−1Ts+δ2fk−2T2+δ3fk−3Ts−λ1Ws,k−1Ts−λ2Ws,k−2Ts−λ3Ws,k−3Ts−λ4Ws,k−4Ts.On the basis of this algorithm, only 4 wavelet transform initial values need to be calculated, and then all wavelet coefficients can be calculated by backward one-way recursion, which avoids forward recursive operations and reduces multiplication and addition operations by half. The required coefficients (δ, λ, etc.) only need to be calculated once and have a lower power calculation; this algorithm can meet the needs of real-time calculation. Use wavelet analysis theory to do multiresolution analysis (MRA) on the obtained singular electromagnetic signal, and decompose the signal into different scales.Different wavelets show different characteristics in terms of orthogonality, tight support, smoothness, and even symmetry, but it is difficult to construct a wavelet function with four characteristics at the same time. Daubechies proved that when the scale function and wavelet function determined by a multiresolution analysis are real functions and both have compact support, the wavelet function no longer has symmetry or antisymmetric (except for Haar wavelet). In practical applications, only according to the needs of different signal decomposition, a compromise is made between several characteristics, and the wavelet that meets the needs is selected for decomposition. Qualitatively speaking, when the oscillation frequency of the detected signal is close to the oscillation frequency of the wavelet function of the corresponding scale, the signal obtains a wavelet decomposition with larger coefficients. This is the reason why wavelet analysis can extract different frequency components of the signal at multiple scales. In other words, the “entropy” value is usually used to measure the distance between the signal and the wavelet basis. The smaller the distance (i.e., the smaller the entropy value), the smaller the difference between the signal and the basis, and the signal gets the maximum decomposition. Therefore, different wavelets need to be selected for different transient signals. By comparing the entropy value of different wavelet decompositions, the wavelet with smaller entropy value is selected to obtain larger decomposition and achieve better detection effect.The idea of the above recursive algorithm can also be extended to other wavelets with the same form. As long as the selected wavelet satisfies the form of the product of polynomial and exponential decay, itsZ-transform can be transformed into a rational function of z-1, and the corresponding structure is constructed according to the aforementioned method. For the real-time recursive algorithm, the difference lies in the calculation of the coefficients. The predicted data is shown in Figure 3. The components on each scale reflect the different frequency components of the original signal, which can show the fault signal to reach the state the purpose of monitoring or fault diagnosis. This method has been successfully applied in the diagnosis of broken rotor bars in electric motors.Figure 3
Predicted data.


## 3. Real-Time Two-Way Recursive Complex Wavelet Algorithm in Power System Singular Signal Measurement

EMTP (Electromagnetic Transients Program) is an electromagnetic transient analysis program. It was originally created by Professor HW Dommel of the University of British Columbia (UBC) in Canada, and then it has been improved by the joint efforts of many experts. The mathematical model of EMTP is based on the phase coordinate system. It has a basic idea, which is to treat the three-phase components widely existing in the power system as three single-phase components with electromagnetic coupling. Such a processing method makes the simulation program have great flexibility. The program is mainly used for computer simulation of the electromagnetic transient process of the power system. The program can solve linear and nonlinear resistance, inductance, and capacitance circuits including lumped parameters, multiphaseπ-type circuits, multiphase distributed parameter circuits, various types of switches, the steady-state or transient process of a large-scale power system composed of transformers, power supplies, and control systems. In addition, all other systems that can be simulated by circuits, such as the steady-state or transient processes of mechanical systems, can be calculated with this program. The EMTP program has the advantages of large scale, strong function, and real simulation. It has been widely used in my country’s institutions of higher learning, scientific research, design, and manufacturing and has been used in the research of some national key projects. On this basis, different subload sequences are respectively predicted. Since the periodicity of each subsequence is more obvious, it is obvious that the prediction results of the periodic autoregressive model (PAR) are more accurate.Now the power system electromagnetic transient program (EMTP) based on offline calculation is becoming more and more powerful. The scale of the computing system network is getting larger and larger, and it can simulate the power system with thousands of nodes; there are many components that can be simulated, such as various power electronic devices such as SVC, SVG, TCSC, and AC/DC converter station; it can also simulate the internal fault of the transformer, the inrush current of the transformer, and the transient process of the current transformer (TA) and saturation voltage transformer (TV). Compared with the all-digital simulator based on real-time calculation, offline EMTP can not only simulate all the situations that the all-digital real-time simulator can simulate, but also can simulate some situations that it cannot simulate, and has some all-digital real-time simulators that do not have. Advantages include low price and low hardware requirements. Therefore, it is very meaningful if you can make full use of the powerful functions of offline EMTF to test relay protection. Finally, through sequence reconstruction, a complete hourly load forecast result is obtained, and its accuracy is greatly improved compared to directly using the original load sequence to predict.Figure4 (left one) shows the measured a-phase electric music waveforms when the a-phase grounding short circuit via linear resistance and the nonlinear resistance grounding short-circuit occur at the midpoint of the system transmission line. Figure 4 (right one) shows the three-phase voltage waveforms measured when a three-phase voltage short-circuited by linear resistance and nonlinear resistance occurs at the midpoint of the system transmission line. It also shows the measured voltage waveforms of the two phases ab when the two-phase ab is short-circuited to the ground via a linear resistance and the two-phase ab is short-circuited to the ground via a nonlinear resistance at the midpoint of the transmission line of the system. Besides, it also shows the measured three-phase voltage waveforms when a three-phase short-circuit to the ground via a nonlinear resistance occurs at the midpoint of the transmission line of the system.Figure 4
Measureda-phase electric music waveforms when the a-phase grounding short-circuit.Traditionally, Fourier analysis is used for dynamic and transient signals in power systems. But with the deepening of research, the Fourier transform can no longer meet the requirements of the sudden change of the signal. The emerging wavelet transform provides a powerful tool for sudden singularity signals and has been successfully applied to many fields. In the fault diagnosis, the identification of the fault is often through the analysis of the fault signal amplitude and frequency components and other information, combined with the existing fault signs of the fault system to make judgments, which is convenient for fault location and rapid maintenance. The phase information of wavelet transform is often more sensitive to singular signals, easy to capture singular points, and accurately detect signal mutations, so complex wavelets can be used to calculate wavelet coefficients. Complex wavelet transform can not only provide amplitude information, identify feature scale and its position on the time axis, but also provide phase information, which can be used to locate the singular point of the process. Therefore, the use of complex wavelet will help with the fault diagnosis rate. For the measured three-phase voltage signal, this paper uses wavelet transform to extract its characteristics.The so-called feature extraction is to find indicators that can effectively reflect the signal so as to further identify and classify the signal. The wavelet analysis method has a strong feature extraction function, which can extract the fault features of the fault phase. As long as the various current and voltage signals are collected at a suitable sampling frequency, and each sampled value is subjected to wavelet transformation, it can be found that at the moment of the fault, the wavelet transformation of the fault phase current and the fault phase voltage has a modulus maximum value. Apparently, the characteristics of the fault are obvious; the wavelet transform of the nonfault phase current and the nonfault phase voltage has no sudden modulus maximum. Therefore, the wavelet transform of current or voltage can be used to constitute a phase selection element reflecting the transient component of the fault because the modulus maximum value of the wavelet transform only appears at the moment of the fault. The wavelet component obtained by the dyadic wavelet transform of the signal cannot be used as the characteristic quantity because although it contains all the information of the signal, the amount of data is too large to facilitate the online identification of the signal, and the edge of the signal wavelet transform often contains the most important part of the signal. Important features, and the amount of data is small, so it is more suitable for signal recognition. The algorithm is shown in Figure5.Figure 5
The algorithm.In the process of feature extraction, there is one point that needs special explanation. Both theory and actual decomposition prove that the position of the modulus maximum point of wavelet analysis is not constant on adjacent scales. However, if two modulus maximum points on two adjacent scales belong to the same modulus maximum line in the scale space (S, x), then it is considered that the modulus maximum points propagate along the scale. At the same time, the choice of wavelet base is also very important. The evaluation is shown in Figure 6.Figure 6
Evaluation.To illustrate the effectiveness of the above real-time algorithm for detecting signal mutations, the following takes the voltage amplitude mutation shown in Figure4 and the voltage harmonic distortion signal shown in Figure 7 as examples for analysis (fs = 2000 Hz). The wavelet and one-way recursive algorithms of this paper are used to decompose the wavelet, respectively, and the amplitude and phase information of the wavelet transform are extracted, and the percentage errors of the calculation results of the two methods are compared. The abrupt amplitude signal shown in Figure 7 decreased by 2% in amplitude at 55 points, recovered in amplitude at 100 points, decreased by 1% at 150 points, and increased by 1% at 200 points, accompanied by 1% frequency deviation. Due to the small amplitude change, several signal amplitude change points cannot be detected from the original signal and its Fourier transform. Using the method in this paper and the literature to extract the amplitude and phase information of the wavelet transform, only two high-frequency scales with obvious effects a = 0 are listed in the figure. It can be clearly seen from the figure that the one-way recursive algorithm in this paper can still accurately locate the singularity of the signal from the two aspects of the wavelet transform amplitude and phase information. Compared with the wavelet transform in the previous literature, the suberrors can be controlled in the range of 5% or even lower, to achieve similar detection accuracy, and reduce the multiplication and addition operations by half compared with the method in the previous literature, and the decomposition result quickly reaches a stable value, so this recursive algorithm can be used to detect singularities in real time. As can be seen, the wavelet transform is very effective in signal localization and singularity analysis.Figure 7
Voltage harmonic distortion signal.The harmonic distortion signal is that the power frequency signal contains high-order harmonics between 50 and 100 points, the harmonic distortion is 1.41%, and there are slight disturbances at 80 and 150 points. Due to the small harmonic content and disturbance, the sudden change information cannot be detected from the time domain signal and the Fourier spectrum. Also use the aforementioned two methods to perform wavelet transform on it, and extract the amplitude and phase information. It can also be clearly seen from the figure that the recursive algorithm in this paper can also accurately detect the singularity of the harmonic distortion signal from the wavelet transform amplitude and phase information and can also distinguish the disturbance in the harmonic component. At 80 points, the percentage error of the method in the previous literature can still be controlled in a low range, which once again reflects that the recursive algorithm in this paper not only greatly reduces the amount of calculation, but the accuracy of detecting distortion can also reach the previous literature. The predicted results are shown in Figure8.Figure 8
Predicted results.It should be noted that the wavelet and one-way recursive algorithm in this article can be widely used in other fields in addition to the above examples of detecting singularities. The author applies it to extract the wavelet ridgeline of the stator current when the rotor bar of the asynchronous motor is broken to distinguish the fault and obtains good results.


## 4. Conclusion

Based on the integral wavelet transform, this paper analyzes a real-time recursive wavelet algorithm based on the selected complex wavelet as an example and derives the implementation process of the algorithm in detail. The algorithm greatly reduces the amount of calculation of the existing two-way recursive algorithm; thus, it can be used in real-time detection of fault signals in various fields of power system and can be extended to realize other fast recursive algorithms of wavelet functions. Based on the sensitivity of complex wavelet transform phase information to singularity, a method for real-time monitoring of power system fault mutation signals using phase information auxiliary amplitude information of complex wavelet fast recursive algorithm is proposed. An example is used to demonstrate the effectiveness and practicality of this complex wavelet and its real-time recursive algorithm to detect faults.The work of this paper still needs to be further deepened. On the one hand, the selected complex wavelet has a two-way recursive algorithm. Although it can be calculated quickly, the algorithm includes two parts of forward and reverse recursive calculations. The reverse recursive part needs to wait until the data of the entire analysis window is completely obtained.It actually can be carried out, which is not conducive to the real-time requirements of fault diagnosis. In order to remove the backward recursive calculation and improve the real-time performance of diagnosis, it can be considered that only the positive time function is used to form the mother wavelet function.



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*Source: 9013770-2022-05-14.xml*