Subsynchronous oscillation monitoring and alarm method based on phasor measurements

1 Introduction

With the large-scale connection of new energy sources in the power grid and the development of high-voltage direct current transmission technology (HVDC),several power electronic devices are installed in the power grid,which changes the characteristics of the traditional power system.Power electronic equipment introduces numerous sub-/supersynchronous harmonic components into the power signal,causing torsional vibration of the generator set shaft system.Subsynchronous oscillations (SSOs) occur frequently,threatening the safe and stable operation of the power system [1-6].Recently,several SSO events,caused by the grid connection of new energy sources and HVDC,have been reported [7-9].Real-time monitoring and timely alarm of SSOs are vital to the safety of power systems.

Torsional vibration protection of the shaft system is widely used in the field,where SSOs are monitored using the mechanical speed signals of rotors;this method serves as the system protection.However,dispatchers are only able to obtain the SSO information subsequent to events,and online real-time monitoring and global alarms have not been achieved [10].Synchronous phasor measurement units (PMUs) provide synchronization,speed,and accuracy,and can realize dynamic real-time monitoring of power systems [11-15].Therefore,PMUs have been widely used to monitor SSOs.For example,when an SSO occurs,the subsynchronous frequency component appears in the amplitude or phase angle of the fundamental phasor measured by a PMU,which can be used to identify the SSO [10,16-18].

At present,PMU-based power system harmonic,interharmonic,and SSO monitoring are primarily based on spectrum analysis,time-frequency analysis,and spectrum estimation methods,such as the fast Fourier transform (FFT) [19-23],wavelet theory [24-25],and the Prony algorithm [26].Of these methods,the FFT-based methods are the most widely used for SSO monitoring,owing to their simple algorithms,fast calculation speeds,and easy implementation in embedded devices.A measurement algorithm based on the FFT and curve fitting is proposed in [27],which can be used to estimate the parameters of subsynchronous components in measured phasors and subsequently monitor the SSOs.This method initially uses the windowing technique and zero-filling method to reduce spectrum leakage and the picket-fence effect;it then uses spectral line curve fitting for inter-harmonic analysis.The method enables a fast measurement speed and high accuracy.

After obtaining the subsynchronous frequency component parameters based on the FFT,it is necessary to set appropriate criteria to realize automatic alarm notifications for SSOs.However,owing to the measurement environment and algorithms,several low amplitude frequency components appear in the signal frequency spectrum.These non-harmful components often have clear characteristics that vary from those that may cause oscillations.These characteristics may be learned using artificial intelligence (AI) to ensure their distinction from those that cause SSOs.However,owing to the difference in frequency susceptibility,trend of divergence or attenuation,and duration,the degree of damage caused by each SSO component also varies,and not all SSO components justify the use of an alarm.

Considering that the FFT algorithm is computationally expensive,and SSOs do not appear frequently,the fast identification of SSOs is necessary.Our previous research proposed an SSO identification method based on a multiple support vector machine (SVM) model,which could identify the PMU data containing SSO behaviors,thereby reducing the computational burden of the FFT.

This study proposes a method for online SSO monitoring and alarm notification on the main station of PMUs,based on phasor measurements.It makes some novel contributions to this issue,which are as follows:

(1) Using SVM learning,adaptive amplitude thresholds are obtained,which can accurately distinguish the nonharmful components from SSO components.

(2) Based on the frequency susceptibility,amplitude attenuation,and energy accumulation,harmful SSO levels are identified.Using this,the adaptive duration thresholds are determined and alarm signals can be timely transmitted.

The remainder of this paper is organized as follows.Section 2 introduces the determination of the adaptive SSO amplitude threshold.The determination of the adaptive SSO duration threshold is described in Section 3.The field data testing and analysis are presented in Section 4.Finally,the conclusion is briefly explained in Section 5.

2 Adaptive SSO amplitude threshold

2.1 Support vector machine algorithm

The SVM algorithm is a supervised machine learning algorithm that can be used for the binary classification of samples [28].For the SSO amplitude threshold problem studied in this section,the SVM algorithm is based on labeled amplitude percentage data.It is used to find a hyperplane in the sample space to separate the samples that require attention from those that do not.In addition,the maximum interval ensures the strongest adaptability to fresh samples.

In the SVM algorithm,the expression of the optimal separating hyperplane is as follows:

where x is the feature vector in the sample space,ω=(ω1;ω2;…;ωd) is the weight vector,and b is the bias.For the amplitude threshold problem,two types of training samples can be linearly separated,and ω,b determines the optimal separating hyperplane.

To identify the most robust hyperplane/interface,the SVM algorithm can be summarized as a conditional extremum problem,the expression of which is

Adding Lagrange multipliers αito each of the above constraints,the problem can be transformed into:

After calculating each Lagrangian multiplier αi,ω and b can also be obtained.Then (1) can be rewritten as

To obtain the optimal interface,the Karush-Kuhn-Tucker conditions should be met:

It can be observed from (5) that for any training sample (xi,yi),αi=0 or yi f(xi)=1 always occurs.If αi=0,the sample will not have any influence on the determination of hyperplane f(x);if αi＞0,then yif(xi)=1 is also necessary.The corresponding sample point is on the maximum interval boundary,and is a support vector.In other words,for the SVM,the final model only depends on a few support vector samples that are nearest to the interface.

2.2 Amplitude thresholds based on SVM algorithm

If the frequency spectrum analysis is performed on continuous PMU voltage/current phasor amplitude data when the SSO occurs,subsynchronous frequency components can be identified.Moreover,noise and spectrum analysis algorithms (such as the spectral leakage generated by the Fourier algorithm) can also produce subsynchronous frequency components.These components tend to have small amplitudes,strong randomness,and account for a large proportion of the total identified components.They appear as “burrs” in the frequency domain and are not harmful to generator shafting and power systems.However,the real SSO frequency components are harmful,often more prominent in the frequency domain,and have relatively larger amplitudes.As shown in Fig.1,it is feasible and necessary to determine the amplitude threshold to distinguish the subsynchronous frequency components caused by SSO from those due to non-harmful causes.The follow-up analysis will focus on crucial frequency components,reducing unnecessary calculation and data storage.

Fig.1 Comparison of subsynchronous frequency components,owing to SSO and non-harmful causes

The method to obtain SSO amplitude thresholds,based on the SVM algorithm,can be described as:

(1) Feature vector sample collection:Using the FFT method proposed in [27],a spectrum analysis on PMU current phasor amplitude historical/real-time data is performed in SSO conditions according to a particular data window length.The frequencies and amplitudes of subsynchronous components are obtained in each window,and the amplitude of each component is calculated as a percentage of the fundamental value (hereinafter referred to as the amplitude percentage).The amplitude percentage is considered as the feature value of each component;thereby forming a one-dimensional feature vector sample.

(2) Sample type determination:The types of all amplitude percentage samples are determined in each window,including the positive class “need to pay attention” and the negative class “no need to pay attention.” The labels are assigned to these samples using the values +1 and -1,respectively.Considering that the amplitudes of SSO frequency components are prominent,the criteria for determining the sample types are shown in Fig.2.

(3) SVM learner training:The SVM learner is trained by labeled feature vector samples of two types,and the interface is identified (for a one-dimensional feature vector,the interface is actually a point).The amplitude percentage value corresponding to this interface is the SSO amplitude threshold.

Accordingly,the key step in calculating the amplitude threshold is to determine the sample types.As shown in Fig.2,using the feature of large amplitudes,a portion of the distinctive SSO and non-harmful components can be identified by Criteria 1 and 2.If an amplitude percentage value exceeds a particular large value M%,determined by the amplitude percentage probability distribution,it is almost certain that the corresponding sample is a real SSO component instead of a non-harmful one;therefore,the label “need to pay attention” can be assigned.Similarly,a sample with an amplitude percentage value lower than a particular low value N% is evidently non-harmful;therefore,the label “no need to pay attention” can be assigned.These two parameters are determined via the analysis of a large set of field PMU data.

Fig.2 Criteria for determining the amplitude percentage sample types

The amplitude percentage values of other samples are neither high nor low;therefore,their labels are determined by the saliency that is Criterion 3.Specifically,if the amplitude percentage value of a particular component exceeds K times the average value of all samples in the same data window,similar to the harmful component in Fig.1,the label “Need to pay attention” can be assigned to the component.The remaining samples are labeled “No need to pay attention”.By analyzing the saliency of SSO components from field PMU data,the value of parameter K is determined.This value can be slightly adjusted to change the amplitude threshold according to the actual demand.

For real-time PMU current phasor amplitude data,FFT spectrum analysis is performed according to the same data window length to obtain the amplitude percentages of subsynchronous frequency components.Thus,two tasks are conducted in parallel.The components with an amplitude percentage value above the threshold are retained for further alarm signaling.Moreover,when a new SSO event occurs,the amplitude percentage samples of the subsynchronous components in the first several data windows are labeled using the method in Fig.2.Then,a new amplitude threshold is adaptively obtained,which will improve the classification accuracy on the following samples;to make the training of the amplitude threshold both effective and efficient,the SVM learner is updated immediately when both types of samples exceed a particular number.

3 Adaptive SSO duration threshold

For the subsynchronous frequency components that are retained after the amplitude threshold filtering,it is necessary to constantly pay attention and transmit a timely alarm signal if the component has a long duration or transmit no alarm signal if it has a high attenuation and short duration.Therefore,it is necessary to determine the adaptive duration threshold as a criterion for transmitting an alarm signal,which depends on three aspects:frequency susceptibility,amplitude attenuation,and energy accumulation.

3.1 Frequency susceptibility

If the current amplitude oscillation frequency is proximal to the torsional vibration frequency of the generator set shaft system,the shaft system will be damaged by subsynchronous torsional vibration.Thus,the susceptible frequency must be set in advance.Considering that the measured frequencies of subsynchronous components typically fluctuate rapidly and irregularly,and the calculation may include a degree of error,a frequency margin fm is determined;the range near the torsional vibration frequency of the generator set ftvshould be set as the susceptible frequency band,as shown in Fig.3.The subsynchronous frequency components in this band require further attention.Therefore,a lower energy accumulation threshold must be set to ensure faster transmission of alarm signals than for components in non-susceptible frequency ranges.

Fig.3 The relation between torsional vibration frequency and the susceptible frequency band

3.2 Amplitude attenuation

The frequency spectrum analysis results of the onsite current amplitude data demonstrate that the measured amplitudes of several SSO frequency components fluctuate with time.Some component amplitudes can gradually attenuate,and do not cause significant harm.When a large amplitude is sustained for a longer duration and does not decay to a lower value,an alarm signal should be transmitted in advance to avoid any further development of hazards,without having to wait until its energy accumulation exceeds the original threshold.Thus,the amplitude attenuation requires real-time monitoring.When effective attenuation cannot be achieved,the energy accumulation threshold must be lowered in time to increase the alarm speed.

For the subsynchronous components continuously appearing in a particular frequency range,the average value of the amplitude percentage during a period before the monitoring time should be calculated at any time.In addition,several different thresholds at varying levels should be set in advance according to a particular gradient.If it is detected that the average value of the amplitude percentages APave is larger than a particular threshold at a particular time,it indicates that the oscillation is severe and will not easily decay.The energy accumulation threshold Thr should also be subsequently lowered on the basis of the present value according to the corresponding gradient,as shown in Fig.4,where th1 ＞ th2 ＞ th3 ＞ th4,and ap1 ＜ ap2 ＜ ap3.

Fig.4 The relation between the energy accumulation
threshold and the average value of the amplitude percentages

3.3 Energy accumulation

The energy accumulation value depends on the amplitude and duration of the subsynchronous frequency component,which is expressed as

where I% is the amplitude percentage of a particular SSO frequency component in the current phasor amplitude data.The energy accumulation value Enr reflects the harmfulness of an SSO frequency component.The SSO will cause stress to the generator set shafting,resulting in fatigue life loss.The higher the energy accumulation value,the greater the fatigue accumulation of the shafting,and the more severe the inflicted damage.Therefore,energy accumulation directly determines the duration threshold of an SSO frequency component.

The steps for adaptively determining the duration threshold based on energy accumulation are as follows:

(1) Energy accumulation calculation:The amplitude percentage of an SSO frequency component monitored in real time is integrated over time to obtain its energy accumulation.

(2) Energy accumulation adjustment:The energy accumulation value must be noted,and its threshold adjusted according to frequency susceptibility and amplitude attenuation in real time.

(3) Duration threshold determination:When the energy accumulation value of a component exceeds the current energy accumulation threshold,the time interval from the start of the SSO to the corresponding moment is obtained,the value of which is equal to the duration threshold.An alarm signal is transmitted at this moment.

During the energy accumulation calculation,a particular frequency margin fm2 should also be set by considering the fluctuation of the measured frequencies of the subsynchronous components.When there are subsynchronous components with frequency differences within a particular range in two adjacent data windows,the energy accumulation is calculated.

4 Field data testing

4.1 Non-harmful frequency component filtering based on amplitude threshold

The proposed SSO monitoring and alarm method is implemented on MATLAB (Mathworks),and a large set of PMU current phasor amplitude field data of a power grid in China is used.The PMU phasor upload frequency is 100 Hz.The window length and the sliding step are both 1 s (100 data points).The corresponding subsynchronous frequency components and amplitude percentage samples are obtained based on FFT analysis of the PMU measurement data in every data window.Labels are then assigned to all samples according to the method in Section II.The values of parameters K,M,and N in the labeling criterion are determined according to the amplitude percentage distribution.The number of positive and negative samples for different K-values are shown in Table1 for 25610 samples,using only Criterion 3 in Fig.2 to determine a label.

Table1 The number of samples for each type for different K-values

Need to pay attention No need to pay attention 4 1196 24414 4.5 1061 24549 5 906 24704 5.5 737 24873 6 595 25015

Therefore,the value of K will affect the proportional relationship between the two types of samples,which can be adjusted according to the actual implementational needs.Fig.5 shows the amplitude percentage value distributions of 906 positive samples and 24704 negative samples for K=5.Among the positive samples,there are 883 with an amplitude percentage higher than 0.2%,accounting for 97.46%;there are 23742 negative samples with an amplitude percentage lower than 1%,accounting for 96.11%.Therefore,M and N can be set as M=1 and N=0.2.

Fig.5 The amplitude percentage value distributions of positive and negative samples for K=5

Based on a larger PMU current phasor amplitude measured dataset from different lines and oscillation events,8613 training samples with the label “need to pay attention” and 92335 training samples with the label “no need to pay attention” are obtained.The SVM learner is trained by these samples to obtain the amplitude threshold,which is utilized to classify these samples in turn.The results are shown in Table2.

Table2 Classification of the two types of samples by amplitude threshold

Need to pay attention No need to pay attention Number of samples correctly classified 8418 89223 Total number of samples 8613 92335 Accuracy 97.74% 96.63%

Using a large field dataset and the labeling criterion in this section,the amplitude threshold of the oscillation alarm is determined as 0.5924%.It can be observed from the results in Table2 that the amplitude threshold determined by the SVM algorithm can distinguish between the subsynchronous components caused by oscillations and by non-harmful causes with a high degree of accuracy.

When a new SSO event occurs,the above threshold can be immediately used to filter out non-harmful components.Meanwhile,subsynchronous component data in new events are collected,and the SVM learner is updated in real time to obtain higher adaptability and accuracy.Because there are significantly more samples labelled “no need to pay attention” than those labelled “need to pay attention”,when a new event occurs,the learner can be updated when the number of samples labelled “need to pay attention” reaches a number T.For T= 5,the new SVM learner training,new amplitude thresholds,and classification results of the two types of samples are shown for six on-site SSO events in Table3 and Table4.

Table3 New SVM learner training for six SSO events

Event Number of training samples Sample collection time (s)FFT analysis time (ms/window)Training time (ms)1(65 s) 67 6 0.385 4.362 2(80 s) 15 2 0.379 3.855 3(55 s) 31 4 0.381 4.109 4(60 s) 26 3 0.353 3.942 5(70 s) 69 6 0.365 4.431 6(85 s) 97 8 0.342 4.805

Table4 New amplitude thresholds and classification results for six SSO events

Event Accuracy of adaptive SVM learner (%)Accuracy of original SVM learner (%)Amplitude threshold (%)Need No need Need No need 1(65 s) 100 100 100 100 0.4153 2(80 s) 100 100 100 92.91 1.0211 3(55 s) 98.73 100 100 93.07 0.9843 4(60 s) 100 99.39 100 97.40 0.8784 5(70 s) 100 99.28 87.30 99.52 0.3779 6(85 s) 100 99.61 86.84 100 0.2902

It can be observed from Tables 3 and 4 that the adaptive amplitude threshold,which is determined by the SVM learner and updated in real time,can be obtained with a few samples within a short time.The algorithm exhibits a fast speed,small number of calculations,and small memory footprint,which is suitable for the occurrence of SSOs.The adaptive amplitude threshold (adaptive SVM learner) can effectively improve the accuracy of classification and correctly filter out non-harmful frequency components.Moreover,the original can be a substitute before the new threshold is obtained.Additionally,when a severe oscillation occurs,the amplitude threshold value is relatively higher,and the time required to collect the “need to pay attention” samples is shorter,which increases the speed of acquiring of the amplitude threshold.

4.2 SSO alarm based on duration threshold

The data from SSO Event 1 and Event 2,used to test the effect of the amplitude threshold in the previous section,are used to conduct an SSO alarm test.The susceptible frequency ftv is set to 34.5 Hz.Considering an error of 1.5 Hz,when the amplitude oscillation frequency fa is between 33 Hz and 36 Hz,the energy accumulation threshold is reduced to 1/n1 (n1 R,n1≥1) of the original.Illustratively,n1 can be expressed as

Several gradients are set by considering the oscillation attenuation.For the amplitude percentages of a particular subsynchronous component,if the average value in three consecutive data windows APave is greater than one particular value,the value of the energy accumulation threshold Thr will be reduced to 1/n2 (n2 R,n2≥1).Illustratively,n2 can be expressed as

where four gradients are set;the greater the value of APave,the greater n2 and the smaller Thr.

For energy accumulation,the initial threshold of a subsynchronous component Thr_ini is set.Illustratively,Thr_ini can be set to 80.The energy accumulation threshold can be expressed as

Fig.6 Frequency,amplitude,and energy accumulation trends of the subsynchronous component near 33.9 Hz

For Event 1,during the period from 1 to 61 s,the current phasor amplitude data contain the prominent subsynchronous component near 33.9 Hz,for which the trends of frequency,amplitude,and energy accumulation are shown in Fig.6.These components are located in the susceptible frequency band and the values of APave are always lower than 5%;thus,n1＞1,n2=1,and the value of Thr is reduced on the basis of Thr_ini.According to Fig.6,the component disappears at 61 s and the total energy accumulation Enrtotal is 61.7148.Therefore,if Thr ＞ Enrtotal,the duration threshold TAis greater than 61 s and no alarm signal will be transmitted;if Thr ＜ Enrtotal,the relation between Thr and the duration threshold can also be described by the curve in Fig.6 (c).

For Event 2,during the time period from 1 to 79 s,the current amplitude data contain the prominent subsynchronous components near 9.6 Hz,19.3 Hz,and 28.9 Hz,for which the trends of frequency,amplitude,and energy accumulation are shown in Fig.7,Fig.8,and Fig.9,respectively.These frequencies are in the non-susceptible frequency band;therefore,the parameter n1=1.

Fig.7 Frequency,amplitude,and energy accumulation trends of the subsynchronous component near 9.6 Hz

According to Fig.7,Fig.8,and Fig.9,for the component near 9.6 Hz and 19.3 Hz,the parameter n2=1;for the component near 28.9 Hz,5% ＜ APave ＜ 8% in the first three windows;therefore,the parameter n2=2 after 3 s.At 79 s,these three components disappear and the total energy accumulations Enrtotal are 162.4542,310.1210,and 329.4576,respectively.For these values,if Thr ＞ Enrtotal,the duration threshold TAis greater than 79 s and no alarm signal will be transmitted;if Thr ＜ Enrtotal,the relation between Thr and the duration threshold can also be described by the curve in Fig.7 (c),Fig.8 (c),and Fig.9 (c),for 9.6 Hz,19.3 Hz,and 28.9 Hz,respectively.

Fig.8 Frequency,amplitude,and energy accumulation trends of the subsynchronous component near 19.3 Hz

Fig.9 Frequency,amplitude,and energy accumulation trends of the subsynchronous component near 28.9 Hz

For parameters n1 and n2,determined according to (7) and (8),respectively,and Thr_ini=80,the duration thresholds of these SSO components in Event 1 and Event 2 are shown in Table5.

Table5 The duration thresholds of SSO components in two SSO events

Event Frequency of SSO (Hz)Energy accumulation threshold Duration threshold (s)Real n1=n2=1 1(65 s) 33.9 40 41 No alarm signal 2(80 s)9.6 80 42 42 19.3 80 22 22 28.9 40 11 19

From Event 1 and Event 2,we can observe that the SSO alarm time can be adjusted on demand by a reasonable setting of the energy accumulation threshold.When a relatively harmful SSO frequency component appears,the energy accumulation value will approach and exceed its threshold faster.The duration threshold value will then be lower and the alarm signals will be transmitted earlier.The criteria for frequency susceptibility and amplitude attenuation accelerates the alarm speed of some severe oscillations and those in the crucial frequency band.

5 Conclusions

This study proposes a method of subsynchronous oscillation real-time monitoring and alarm implementation on a PMU main station.This method is based on the FFT frequency spectrum analysis of PMU current amplitude data and the adaptive setting of amplitude and duration thresholds.

The proposed method uses the SVM algorithm to automatically learn historical and real-time data,adaptively obtain amplitude thresholds,filter out a large number of non-harmful subsynchronous components on the current amplitude data frequency spectrum,and pay attention to oscillation components.Based on frequency susceptibility,amplitude attenuation,and energy accumulation,the degree of harm of different oscillation events are known,and the duration thresholds for different values are determined;the alarm signals are accurately transmitted at the appropriate time.The effectiveness of the proposed method is verified based on the testing of field data from a new energy gathering area in China.In addition,the method can be implemented in the form of software,based on the existing PMU equipment,without the need for large-scale device transformation,which provides strong engineering practical significance.The next research direction is to make a rough estimation of frequency range while quickly identifying SSO before FFT.

Acknowledgements

This work was supported by the National Key R&D Pro gram (2017YFB0902901),National Nature Science Founda tion of China (51725702,51627811,51707064).

Declaration of Competing Interest

We declare that we have no conflict of interest.