[IEEE 2014 16th International Conference on Harmonics and Quality of Power (ICHQP) - Bucharest,...

A Direct Approach for Disturbance Detection Basedon Principal Curves

Danton Diego FerreiraEngineering Department

Federal University of Lavras

Lavras, MG, Brazil

[email protected]

Jose Manoel de SeixasFederal University of Rio

de Janeiro, COPPE/Poli

Rio de Janeiro, RJ, Brazil

[email protected]

Carlos Augusto Duqueand Augusto S. Cerqueira

Federal University of Juiz de Fora

Juiz de Fora, MG, Brazil

[email protected]

[email protected]

Paulo Fernando RibeiroFederal University of Itajuba

Itajuba, MG, Brazil

[email protected]

Abstract—This paper proposes a direct approach based onPrincipal Curves for power quality disturbance detection. Themain advantages of the method lie in its low computationalburden in the operational phase, its capability to detect distur-bances when power frequency is time-varying and its good per-formance to signals exhibiting high noise levels. Simulation andexperimental results show that the proposed method outperformsother methods in terms of detection rates and computationalcomplexity.

I. INTRODUCTION

Power Quality (PQ) has emerged as an important research

field in recent years and the main reasons are: (i) the need

to define indices and measures to quantify the quality of the

energy delivered; (ii) the growing development and use of

high power inverters and converters; (iii) the need to localize

the disturbance sources and (iv) the large amount of PQ data

recorded that demands automatic detection [1].

Several methods have been proposed in the literature for

disturbance detection and the most used techniques are based

on wavelet transforms (WT) [2], [3], [4], [5]. However, the

attained results with WT may be seriously affected by system

noise [3]. Other methods that should be mentioned include

S-trasform [6], Hilbert transform [7], fractals [8] and support

vector machines [9]. Each of these techniques have advantages

and disadvantages.

For designing automatic disturbance detection systems in

power systems, some important aspects must be considered:

(i) the sensitivity to noise. The detection system performance

may be strongly corrupted by the presence of power noise; (ii)

the sensitivity to power frequency variations. A 2% variation

in the nominal voltage frequency is typically observed and

this variation can reduce the detection system performance;

(iii) computational cost. For real-time applications, low-cost

techniques should be chosen. For these aims, sophisticated

techniques have recently been proposed [10], [11], [12]. The

aim of these detection techniques is to provide a real-time and

source reliable detection for a variety of disturbances, so that

event classification and underlying identification can be both

achieved.

In this paper, a direct approach for PQ disturbance detection

is proposed, which is based on principal curves (PC) [13].

Principal curves have been defined as “self-consistent” smooth

curves that pass through the “middle” of a D-dimensional

data cloud [13]. Principal curves have the interesting ability

of compact data representation, which is an important issue in

monitoring systems. In a previous work [12], the same authors

exploited principal curves for PQ monitoring showing how to

analyze, extract features, detect and classify PQ disturbances

with principal curves. It was shown that for disturbance detec-

tion, four stages are required (signal filtering, PC-based feature

extraction, neural processing and final logic for decision),

making the method unsuitable for real-time applications.

Here, the nominal class (formed by electrical signals that

are not corrupted by disturbances) is compactly represented

by a principal curve and, hence, a direct approach is obtained,

in which a threshold is used for detection. The stages of

filtering and neural processing are not required. Thus, the main

contribution of this work is the way of monitoring PQ signals

with PC, providing a simple and robust detection system to

noise and power frequency variations.

II. PQ PROBLEM FORMULATION

The discrete version of the monitored voltage signal can be

segmented into non-overlapped frames of N samples, which

are expressed as an additive contribution of several types of

phenomena, as previously formulated in [14]:

v[n] = v(t)|t= nfs

:= f [n] + h[n] + i[n] + t[n] + r[n] (1)

where n = 0, · · · , N − 1, fs is the sampling frequency, the

sequences f [n], h[n], i[n], t[n] and r[n] are the fundamental

component, harmonics, interharmonics, transient and back-

ground noise, respectively. Each of these signals is defined

as follows:

f [n] := A0[n] cos[2πf0[n]

fsn+ θ0[n]], (2)

978-1-4673-6487-4/14/$31.00 c⃝ 2014 IEEE

747

h[n] :=M∑

m=1

hm[n], (3)

i[n] :=J∑

j=1

ij [n], (4)

t[n] := timp[n] + tnot[n] + tosc[n], (5)

were r[n] is an independently and identically distributed (i.i.d.)

normal noise N (0, σ2

n), and independent of f [n], h[n], i[n]and t[n]. In (2), A0[n], f0[n] and θ0[n] refer to the magnitude,

fundamental frequency, and phase of the fundamental compo-

nent, respectively. In (3) and (4), hm[n] and i[n] are the mth

harmonic and the jth interharmonic, respectively, which are

defined as:

hm[n] := Am[n] cos[2πmf0[n]

fsn+

+θm[n]][u[n− nhm,i]− u[n− nhm,f

]], (6)

and

ij [n] := AI,j [n] cos[2πfI,j [n]

fsn+

+θI,j [n]][u[n− nij,i ]− u[n− nij,f ]], (7)

in which u[n] denotes the unit step sequence, nhm,iand

nhm,frefer to the start and end samples of the harmonics,

respectively. Similarly, nij,i and nij,f refer to the start and

end samples of the interharmonics, respectively. In (6), Am[n]is the magnitude and θm[n] is the phase of the mth harmonic.

In (7), AI,j [n], fI,j [n], and θI,j [n] are the magnitude, fre-

quency, and phase of the jth interharmonic, respectively. In

(5), timp[n], tnot[n] and tosc[n] denote impulsive transients,

named spikes, notching and oscillatory transients, which can

be expressed by [14]:

timp[n] :=

Nimp∑

i=1

timp,i[n][u[n−ntimp,i]−u[n−ntimp,f

]], (8)

tnot[n] :=

Nnot∑

i=1

tnot,i[n][u[n− ntnot,i]− u[n− ntnot,f

]], (9)

tosc[n] :=Nosc∑

i=1

Aosc,i[n]exp[−αosc,i[n− nosc,i]][u[n− ntosc,i ]−

−u[n− ntosc,f ]], (10)

where timp,i[n] and tnot,i[n] are the nth samples of the ith

spike and the ith notching, respectively, and ntimp,i, ntnot,i

and ntosc,i denote the start of each impulsive transient and

ntimp,f, ntnot,f

and ntosc,f denote the end of each.In these equations, a nominal voltage waveform is shown

in Figure 1(a), and some corrupted voltage waveforms are

displayed in Figure 1(b)-(h).

0 200 400 600 800 1000−2

0

2

(c)

0 200 400 600 800 1000−2

0

2

(a)

0 200 400 600 800 1000−2

0

2

(b)

0 200 400 600 800 1000−2

0

2

(d)

0 200 400 600 800 1000−2

0

2

Am

plit

ude (

p.u

)

(e)

0 200 400 600 800 1000−2

0

2

Am

plit

ude (

p.u

.)

(f)

0 200 400 600 800 1000−2

0

2

Samples

(g)

0 200 400 600 800 1000−2

0

2

(h)

Samples

Fig. 1. Examples of voltage signals and disturbances: (a) nominal voltagesignal, (b) oscillatory transient, (c) harmonics, (d) notching, (e) sag, (f) swell,(g) spikes and (h) outage.

III. DETECTION PROBLEM FORMULATION

Considering the vector v = [v[n] · · · v[n − N − 1]]T built

from samples of the signal v[n], the detection problem can be

formulated as a hypothesis test problem as:

H0 : v = f + rv

H1 : v = f +∆f + i + t + h + rv (11)

where i = [i[n] · · · i[n−N −1]]T , t = [t[n] · · · t[n−N −1]]T ,

h = [h[n] · · ·h[n − N − 1]]T , re + rf = r = [r[n] · · · r[n −N − 1]]T . The vector ∆f represents a sudden variation in the

fundamental component.

The hypothesis H0 is related to nominal operation and the

hypothesis H1 is associated with abnormal conditions.

IV. PRINCIPAL CURVES

Principal curves were first defined by Hastie and Suetzle

[13] as one-dimensional parameterized curves having the prop-

erty of self-consistency, which pass through the data points in

the original data space. Let X ∈ ℜn be an available random

vector. A principal curve g is a smooth 1-D curve:

g(ℓ) = [g1(ℓ) g2(ℓ) · · · gn(ℓ)]T = E(X|ℓg(X) = ℓ) (12)

where T denotes matrix transposition, ℓg denotes the largest

parameter value ℓ for which the distance between X and g(ℓ)is minimized:

ℓg(X) = sup{ℓ : ∥X − g(ℓ)∥ = inf ∥X − g(µ)∥}, (13)

and is the projection index. Here, ∥ · ∥ denotes the Euclidean

norm in ℜn and µ is an auxiliary variable defined in ℜ.

In this paper, principal curves were extracted using the K-

segments algorithm [15]. This algorithm locates, iteratively,

k different straight-line segments in the data set, then this k

located segments are linked together to form a polygonal line.

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V. PROPOSED METHOD

The design of the proposed PC-based detection system

comprises three stages:

1) PC construction: the principal curve is constructed for

the nominal signal.

2) Feature extraction: computes the square of the Euclidian

distance from the monitored window signal to the PC.

Mathematically, the distance from a given event x to a

principal curve g(ℓ) is computed by

d(x,g) = E{∥x− g(ℓ)∥2}. (14)

3) Set the threshold (dTHR) for disturbance detection.

In the operational stage (see Fig. 2), only the distance from

the monitored event to the extracted PC is measured and

compared to the threshold. The threshold defines a closed

region around the PC, so that events within this region are

assigned to nominal operation (no disturbance) and events

outside this region are assigned to disturbances.

Fig. 2. Operational stage of the proposed method.

VI. DATABASE

For performance evaluation, synthetic voltage signals were

generated with a sampling frequency equal to 15,360 Hz.

Seven classes of PQ disturbances were considered: harmonics,

sags, swells, oscillatory transients, notches, outages and spikes.

A total of 1,000 events were generated for each disturbance

class and 1,000 signals were generated without disturbances.

All signals were generated with an additive Gaussian white

noise with a SNR of 60 dB. Half of data (design data set) was

used for system design (PC extraction and detection threshold

determination) and the other half for performance evaluation.

In order to test the proposed method for different SNR, 500

events from each disturbance class and 500 nominal voltage

signals were generated for the following SNR values: 50 dB,

40 dB, 30 dB and 25 dB.

Analysis with real data were performed on 60 experimental

measurements of PQ disturbances, which were obtained from

the IEEE work group (P1159.3) [16].

VII. SIMULATION RESULTS

Principal curves were extracted from the design data set.

The distances from both windows of (N = 256 samples)

corrupted and non-corrupted voltage signals to the nominal

class PC were measured. The detection and false alarm

probabilities of the PC-based method, considering different

amount of segments is shown as ROC (Receiver Operating

Characteristics) curves (see Figure 3). It can be seen that

when the number of PC segments is increased, a better

discrimination capability is achieved, increasing the detection

probability (Pd) and decreasing false alarm probability (Pf).

The PC extracted with 13 segments was chosen due its good

performance. The detection threshold was chosen for null false

alarm.

Fig. 3. Detection probability (Pd) versus false alarm probability (Pf) fordifferent amount of PC segments, ranging from 3 to 13 segments.

It is important to mention that 13 segments comprise the

maximum number of segments achieved by the K-segments

algorithm for the data set, according to its stopping criteria,

which is to keep inserting segments until the largest possible

data cluster has less than three segments [15].

Table I shows detection rates in terms of Pd and Pf. It

is clearly noted that the noise level does not influence the

detection probability of the proposed method, but slightly

increases the false alarm ratio for signals with SNR up to

30 dB. It is possible to observe that the previous method [12]

and the method in [10] also achieved good performance but

the false alarm ratio increases faster for signals with SNR up

to 30 dB. These methods have achieved a false negative ratio

of 0.5%, which is due to errors in sag and swell detections.

TABLE IDETECTION PERFORMANCE OF DISTURBANCES IN %.

Methods PC-based Previous DFMMDirect Approach PC-based [12] [10]

SNR Pd Pf Pd Pf Pd Pf

60 dB 100 0.0 99.5 0.0 99.5 0.050 dB 100 0.0 99.5 0.0 99.5 0.040 dB 100 0.0 99.5 0.0 99.5 0.030 dB 100 0.5 99.5 0.0 99.5 2.525 dB 100 1.0 99.5 78,5 99.5 99.5

In order to test the sensitivity of the proposed method to

power frequency variations, 500 signals of each class (nominal

and disturbance) were generated with their frequencies varying

uniformly from 57 up to 63 Hz (5 % of variation).

According to [11], the power frequency normally varies

very slowly over a small range. However, for weak power

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systems, the power frequency variation can be large, for

example, in [17] the frequency deviation can reach ±5%of its nominal value. Actually, this variation in fundamental

frequency appears rather unrealistic, depending on the power

system, but we have decided to simulate such situation aiming

at evaluating the proposed method in critical scenarios.

A detection rate of 100 % with null false alarm was achieved

for the proposed method, showing that the power frequency

variations do not influence the detection capability of the

method. The methods in [10] and [12] achieved a detection

rate of 99.5 % with null false alarm.

The proposed method assumes that the differences between

the disturbance and nominal classes can be captured by an

unique 1-dimensional non-linear manifold. This assumption

makes the detection process simpler, since it performs a

dimension reduction from 256 to 1. Thus, in the operational

stage, the detection is performed by simply calculation of

the Euclidean distance from the monitored event to the 1-

dimensional non-linear manifold. This operational cost is

relatively low for implementation in a Digital Signal Processor

(DSP), since the principal curve is extracted only in the design

stage and, for operation, it may be stored in the DSP memory.

VIII. EXPERIMENTAL RESULTS

The application to experimental data was performed by

segmenting the signals into non-overleaping windows of one

cycle.

Figure 4 shows examples of experimental voltage dis-

turbances detected. The left side of the figure shows the

monitored signals, corrupted by: outages, harmonics and short

transients in (a) and (c), and short transients and harmonics in

(e), (g), (i) and (k). The right site of the figure shows the

correspondent cuttings provided by the proposed PC-based

detector. The detector was here set for providing four cycles

of the monitored signal including the disturbance.

From sixty events of different types, only one was not

detected by the proposed method. However, analyzing this

single event, which is depicted in Figure 5, one can see that

no unwanted alterations appears in the signal, and therefore,

the proposed method did not detect this event.

In order to detail the characteristics of this event and

to verify the occurrence of deviations (disturbances), it was

filtered by the wavelet transform, which is commonly used

for power quality event analysis. Here, the Daubechies family

was used. According to [18], the Daubachies family db4 is the

most widely adopted wavelet in power quality applications.

One can see, the approximation, and four levels of details, in

which no single disturbance appeared.

The root mean square (rms) of the non detected event

(rms = 0.70) was also compared to a nominal one (rms =0.71), showing a difference non significant to be considered

a disturbance. The experimental data was processed by the

DFMM method, and the sixty signals were signed as distur-

bances.

Fig. 4. Examples of experimental voltage disturbances detected. The left sideof the figure shows the monitored signals, corrupted by: outages, harmonicsand short transients in (a) and (c), and short transients and harmonics in (e),(g), (i) and (k). The right site shows the correspondent cuttings provided bythe proposed PC-based detector

Fig. 5. Wavelet expansion of the experimental non detected event v[n].

750

IX. CONCLUSION

The proposed method has shown to be effective in detecting

abnormal alterations in the voltage signal. This is achieved

with reduced computational cost in the operational stage.

Taking advantage of data representation of the principal

curve technique, the proposed method has proved to be insen-

sitive to fundamental frequency variations and has presented

good performance to signals exhibiting high noise levels.

Additionally, the distance computed by the method can be

used as a new and general-purpose power quality index, which

gives an idea of how much a given monitored event has

deviated (in terms of a distance) from the nominal operation.

The authors intend to exploit this in future works.

ACKNOWLEDGMENT

The authors would like to thank CNPq/INERGE,

FAPEMIG, FAPERJ and CAPES (Brazil) for support

this work.

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