Research ArticleMethods of Fault Diagnosis in Fiber Optic CurrentTransducer Based on Allan Variance

Lihui Wang,1,2 Gang Chen,3 Jianfei Ji,3 Jian Sun,3 Jiabin Qian,1 and Xixiang Liu1

1 School of Instrument Science and Engineering, Southeast University, Key Laboratory of Micro-Inertial Instrumentand Advanced Navigation Technology, Ministry of Education, Nanjing 210096, China

2 State Key Laboratory of Transient Optics and Technology, Xian Institute of Optics and Precision Mechanics,Chinese Academy of Sciences, Xian 710119, China

3 Jiangsu Electrical Power Company Research Institute, Nanjing 211103, China

Correspondence should be addressed to Lihui Wang; wlhseu@163.com

Received 24 April 2014; Accepted 31 May 2014; Published 22 June 2014

Academic Editor: Ruqiang Yan

Copyright © 2014 Lihui Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To ensure low failure and high reliability of fiber optic current transducers (FOCTs), it is urgent to study methods of conditionmonitoring and fault diagnosis in FOCT. Faults in FOCT have statistical characteristics. With the analyzing of time domainand frequency domain features in fiber optic current transformers’ measurement data, we establish correspondence between thephysical characteristics of key components in transformer and data features and then build diagnostic analysis model based onAllan variance. According to the Allan variance calculation results, we can diagnose fiber optic current transformer’s health stateand realize faults location. Experiment results show that diagnostic methods based on Allan variance are accurate and effective toidentify fault features.

1. Introduction

Fiber optic current transducers (FOCTs) are achievingincreased acceptance and application in high voltage sub-stations due to their superior accuracy, bandwidth, dynamicrange, and inherent isolation. FOCTs are influenced byvarious factors such as electricity, heat, machinery, and envi-ronment in operation [1, 2], so their performance degradesgradually, which eventually leads to a fault. FOCTs are thebasic components of the power system. Once they fail, thiswill cause local even wide area blackout, resulting in hugeeconomic loss and social impact. At present, domestic andforeign maintenances on power equipment mostly in theregular offline maintenance state, which not only affects thenormal work of the power grid but also causes its relativelylow efficiency [3]. Thus, it is urgent to research on conditionmonitoring and fault diagnosis of FOCT.

According to the information about measured valuesby condition monitoring and their processing results, faultdiagnosis of FOCT reasons, judges, finds out fault’s types,location, and severity, and then puts forward proposals on

equipment repair processing. Condition monitoring is thecollection process of characteristic quantity [4]. It records,classifies, and evaluates the running state of the equipment.And it also provides decisions for equipment maintenanceand repair. Fault diagnosis is the analysis and judgmentprocess after characteristic quantity’s collection, locating theoccurred fault and judging the degree of fault developmentbased on fault tag. Due to imperfect characteristics of theoptical components in FOCT and environmental interfer-ence, there is a series of regular statistical noise in the outputsignal of FOCT. Random noises of FOCT come from lightsource coherent noises, light source intensity noises, pho-todetector shot noises, thermal noises in electronic devices,environmental noises, and time-varying noises because ofdevice aging and other factors. Despite the large number ofnoise sources, output data of FOCT reflect that the character-istics mainly include angle random walk, bias instability, raterandom walk, rate ramp, quantization noise, and sinusoidalnoise.

As a complement to the frequency domain analysis,Allan variance is a time domain analysis technique originally

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 831075, 6 pageshttp://dx.doi.org/10.1155/2014/831075

2 Mathematical Problems in Engineering

SLD

PINdetector

Coupler

Digital signal processing and

convertingCurrent

informationAmplify,

A/D

Mirror

PhasePolarizer

Fiber coilCircuit

Amplify

D/AAmplify,

Quarter-wave retarder

Light sourceHigh voltage areaLow voltage area

modulator45

∘45

∘

1/4𝜆

reflector

filter, filter

Figure 1: Structure of reflective FOCT with reflection mirror.

developed to study the frequency stability of oscillators [5–7].It can be used to determine the character of the underlyingrandom processes that give rise to the data noise. As such, ithelps to identify the source of a given noise term present inthe data, whether it is inherently in the instrument or in theabsence of any plausible mechanism within the instrument,its origin should be sought in the test setup.

In this paper, we analyze the principle of Allan variance,evaluate types of random noise error, identify the statisticalproperties of various types of random errors, and locate noiseerror source. In the application of condition monitoring andfault diagnosis, we establish relational models between thenoise characteristics and fault source with analysis of noisecharacteristics in time and frequency domain, respectively,according to the characteristics and trend of relevant param-eters in FOCT. The model can monitor the health status ofFOCT, judge fault position in fault state, provide the alarmwhennecessary, and finally provide the basis for further steps.

2. Principle of Fault Diagnosis in FOCT

2.1. Introduction of FOCT. FOCT is a kind of optical sensorbased on Faraday magneto optical effects and optical inter-ference theory [8, 9]. It uses closed loop feedback systemto measure the change of light intensity caused by opticalnonreciprocal phase difference in real time, in order toacquire the information of measured current proportional tononreciprocal phase. FOCT is composed of a wide spectrumlight source, a polarization maintaining fiber, a polarizer, anelectrooptical modulator, the signal processing unit, and soforth. There are multiple signal transmission and processinglinks, such as photoelectric conversion and electroopticalconversion. Output data of transducer have the randomnoisecharacteristics of FOCT.

According to the structure of fiber optic sensor head,FOCTs are divided into reflective FOCT and Sagnac FOCT[2, 9].The structure of FOCT with reflection mirror is shownin Figure 1; the structure of FOCTwith fiber loop is shown inFigure 2.

Propagation process of light waves in reflective FOCTcan be described as follows. The waves generated by the lightsource are polarized to linear polarized waves by polarizerafter passing through the coupler. The linear polarized wavesenter the polarization-maintaining fiber at 45∘, transferringevenly into the polarization axis𝑋 fiber (fast axis) and 𝑌-axis

SLD

PINdetector

Amplify,

A/D

Amplify

D/AAmplify,

Light source

Coupler

Digital signal processing and

converting

Current information

Phase modulatorPolarizer

Fiber coil

Circuit

Retarder1Retarder2

High voltage

area

Lowvoltage

area

−45∘

+45∘

filterfilter,

Figure 2: Structure of Sagnac FOCT with fiber loop.

(slow axis).Then, after passing through the 1/4𝜆wave plate at45∘, the two beams of orthogonal mode waves are convertedinto left and right circular polarized waves, respectively,which enter the sensor fiber after that. In the sensor fiber,the two circular polarized waves transfer in different speedsdue to Faraday field effect caused by current. After mirrorreflected by sensor fiber’s end face, they exchange theirpolarization modes (namely, the light of left-handed rotationchanges into the light of right-handed rotation, and the lightof right-handed rotation changes into the light of left-handedrotation.). They pass through the sensor fiber again andinteract with the magnetic field generated by current again,doubling the phase.After passing through the 1/4𝜆wave plate,the two beams of light revert to linear polarized waves. Whilethey entered along the𝑋-axis and 𝑌-axis, they emit out waveplate along the𝑌-axis and𝑋-axis of polarization-maintainingfiber, interfering in the polarizer. In the transmission processof the waves, the two beams of interfered waves pass throughthe 𝑋-axis and 𝑌-axis of polarization-maintaining fiber aswell as left-handed and right-handed rotationmode of sensorfiber, only different in time. Thus, the waves which return todetector only carry the nonreciprocal phase difference causedby Faraday effect.

Propagation process of light waves in Sagnac FOCT canbe described as follows. The waves generated by the lightsource enter the integrated optical chip through the couplerand are polarized to linear polarized waves through thepolarizer. The polarized waves are split into two parts at thebeam splitter in the integrated optical chip, entering the waveplate, respectively, at plus or minus 45∘ (clockwise light is45∘ and counterclockwise light is 45∘). Prior to entering alow birefringence polarization-maintaining fiber optic sensorhead in opposite directions, respectively, the polarized wavesare converted into left and right circular polarized waves by1/4𝜆 wave plate. Faraday field effect caused by current rotatesthe plane of polarization of two circular polarized waves.

Mathematical Problems in Engineering 3

The waves are converted back into linear polarized waveswhen passing through another 1/4𝜆 wave plate and arebrought back to interference. Rotation angles of polarizationplane in two beams of interfered light are equal and opposite,then, interference phase in Sagnac FOCT is twice the Faradayphase shift. According to modulation technology of digitalclosed-loop signals, the information about the magnitude ofcurrent in measured power line can be acquired by detectingthe phase difference of output lights.

2.2. Fault Features in FOCT. Although Sagnac FOCT andreflection FOCT are different in structure, they have thesimilar light waves sensing principle and signal processingsteps. That means they all build digital closed-loop feedbacksystemusing high speed signal processing unit and electroop-tic phase modulator in order to measure the information ofnonreciprocal phase caused by Faraday magnetic field effectin real time and finally acquire the information of externalcurrent. Influenced by environmental factors and inherentfactors, FOCTs reflect the same noise characteristics, whichare mainly divided as the following categories [5, 10].

Bias instability (BI) noises reflect the bias low-frequencyfluctuation of FOCT. BI noises originate from dischargeassembly in FOCTs, plasma discharge, circuit noises, envi-ronmental noises, and many other components which cangenerate random flashing. It is useful to inhibit BI byreliability design of FOCTs and taking corresponding fil-tering method. Angle random walk (ARW) noises showthe ultimate precision of FOCTs and are an importantindicator to measure the IFOG noise level. Photon shotnoises of photoelectric detector (PIN) in FOCT result in theuncertainty of Faraday phase shiftmeasurement, which causea limit of currentmeasurement. Shot noises also cause currentrandom fluctuation of current-voltage feedback impedancein PIN preamplifier, resulting in pseudo-Faraday phase shift.The phase shift influences IFOG minimum bias stabilityand decides FOCT’s precision. ARW noises are the result ofintegrated broadband rate power spectral density, originatingmainly from photodetector shot noises, amplifier noises,electronic device thermal noises, and some high frequencynoises whose relevant time is shorter than sampling time.High frequency noises whose relevant time is shorter thansampling time can be eliminated. It is also efficient to inhibitARW noises by using high-qualified light source and photo-electric detector and improving the stability of environmentaltemperature. Rate ramp (RR) is in essence a definite error,rather than a random error. The strength of light source inFOCT changes monotonously and very slowly and lasts for along time, which causes RR noises. It is useful to reduce RRnoises by ensuring the long-term stability of optoelectronicdevices andworking environment of FOCTs and determiningthe error compensation in the method of establishing themathematical model. Rate randomwalk (RRW) noises reflectindex correlated noises of long correlation time in limitcondition. RRW noises are the integral result of phase valuepower spectral density, associated with long term effects ofresonator. They are generated after white noises pass theintegrator. RRW can be inhibited by reducing the aging

effect of crystal oscillator. Sinusoidal noise (SN) is a kindof systematic error whose power spectral density is pre-sented by several different frequencies.High frequency noisesare generated by laser plasma oscillations in the dischargeprocess; low frequency noises are caused by environmentalperiodic change. When the sinusoidal noises have sinusoidalwaveform with multipeak, it is easier to show SN by theplot of power spectral density. Quantization noises (QN)reflect the minimum resolution of current’s information ofFOCTs. Sampling values of interference signals in FOCT areconverted to digital quantities by A/D and are sent into signalprocessor. In themeasurement of time interval, measurementphase induced by current electromagnetic field is not integertimes of quantified step size, while the amplitude of signalsgets quantified over time, which causes quantization error.In the application environment with requirement of highsampling rate, large QN is caused, which can be reduced byimproving the accuracy of acquisition system and shorteningthe initial sampling time.

2.3. Allan Variance Algorithm and Fault Diagnosis. Sup-posing that the current data with 𝜏

0sampling period in

FOCT is 𝑁-dimensional, we can obtain data collection{𝐼(0), 𝐼(𝜏

0), 𝐼(2𝜏0), . . . , 𝐼(𝑁 ⋅ 𝜏

0)}. Allan variance is a time

domain analysis technique based on function of time length.It can be used to calculate frequency data stability of FOCT inthe time domain, analyze the random noise characteristic ofFOCT, and isolate and identify the random error model andits parameter using the slope of the curve model in log-logplot [5, 6]. The Allan variance is defined as follows:

𝜎2

(𝜏) =1

2⟨(𝐼𝑘+𝑚

(𝜏) − 𝐼𝑘(𝜏))2

⟩ , (1)

where ⟨ ⟩ is the ensemble average. 𝐼𝑘+𝑚

(𝜏) = (𝑤𝑘+2𝑚

−

𝑤𝑘+𝑚

)/𝜏 and 𝐼𝑘(𝜏) = (𝑤

𝑘+𝑚− 𝑤𝑘)/𝜏 are average current.

𝜏 = 𝑚 ⋅ 𝜏0is correlation time. 𝜏

0is the minimum sampling

period. 𝑤𝑘is the value of current for the moment of 𝑘𝜏

0.

Consider

𝜎2

(𝜏) =1

2⟨(𝐼𝑘+𝑚

(𝜏) − 𝐼𝑘(𝜏))2

⟩

=1

2𝜏2⟨(𝑤𝑘+2𝑚

+ 𝑤𝑘− 2𝑤𝑘+𝑚

)2

⟩ .

(2)

The Allan variance can be expressed as follows:

𝜎2

(𝜏) =1

2𝜏2 (𝑁 − 2𝑚)

𝑁−2𝑚

∑

𝑘=1

(𝑤𝑘+2𝑚

+ 𝑤𝑘− 2𝑤𝑘+𝑚

)2

. (3)

The Allan variance obtained by performing the pre-scribed operations is related to the PSD of the noise terms inthe original data set.The relationship between Allan varianceand the two-sided PSD 𝑆

𝐸(𝑓) is given by the following

equation:

𝜎2

(𝜏) = 4∫

∞

0

𝑆𝐸(𝑓) ⋅ 𝐹 (𝑓) 𝑑𝑓 = 4∫

∞

0

𝑆𝐸(𝑓)

sin4 (𝜋𝑓𝑡)(𝜋𝑓𝑡)2

𝑑𝑓.

(4)

4 Mathematical Problems in Engineering

Equation (4) indicates that Allan variance is proportionalto the total energy of random noises when the random noisespass through the filter of function 𝐹(𝑓). Filter of band passdepends on the correlation time 𝜏. By adjusting it (namely,adjusting band filter), different types of stochastic processescan be detected. The credibility of Allan variance estimationimproves with the increasing numbers of independent sets.

Equation (4) is the key result that will be used throughoutto characterize the rate noise PSD from the Allan variancecalculations. Its physical interpretation is that the Allanvariance is proportional to the total noise power of the FOCTcurrent output when passed through a filter with the transferfunction of𝐹(𝑓).This particular transfer function is the resultof the method used to create and operate on the clusters. Itis seen from (4) that the filter of band pass depends on 𝜏.This suggests that different types of random processes canbe examined by adjusting the filter of band pass, namely,by varying 𝜏. Thus the Allan variance provides a means ofidentifying and quantifying various noise terms that existin the data. It is normally plotted as the square root of theAllan variance versus 𝜏[𝜎(𝜏)], on a log-log plot.The followingparagraphs show the application of (4) to a number of noiseterms that are either known to exist in the FOCTor otherwiseinfluence its data. Power spectral density of random noise inFOCT has the following features.

For angle random walk, the associated rate PSD isrepresented by the following equation:

𝑆𝐸(𝑓) = 𝑁

2

, (5)

where𝑁 is the angle random walk coefficient.The following equation is obtained by performing inte-

gration:

𝜎2

(𝜏) =𝑁2

𝜏. (6)

For bias instability, the associated rate PSD is representedby the following equation:

𝑆𝐸(𝑓) = (

𝐵2

2𝜋)1

𝑓, 𝑓 ≤ 𝑓

0,

𝑆𝐸(𝑓) = 0, 𝑓 > 𝑓

0,

(7)

where 𝐵 is the bias instability coefficient and 𝑓0is the cut-off

frequency.For rate random walk, the associated rate PSD is repre-

sented by the following equation:

𝑆𝐸(𝑓) = (

𝐾

2𝜋)

2

1

𝑓2, (8)

where𝐾 is the rate random walk coefficient.For quantization noise, the associated rate PSD is repre-

sented by the following equation:

𝑆𝐸(𝑓) = 𝜏𝑄

2

, (9)

where 𝑄 is the quantization noise coefficient.

FOCT Status Fault feature Comparition

Fault models

Classification

Status trends

Fault mode

Figure 3: Structure of fault diagnosis system in FOCT.

Different error terms of random noises in FOCT appearin different correlation time domain. It is different in thepower spectral density and the function relationship withcorrelation time 𝜏 of different noises. Assuming that the noiseerror term is independent in the statistical sense, integratedAllan variance 𝜎2total(𝜏) is given by the following equation:

𝜎2

total (𝜏) =2

∑

𝑛=−2

𝐴𝑛𝜏𝑛

, (10)

where 𝐴𝑛(𝑛 = −2, −1, 0, 1, 2) correspond to quantization

noise, angle randomwalk, bias stability, rate randomwalk andrate ramp, and many other coefficients of fitted polynomialrelated to noise, respectively. Consider

𝜎2

total (𝜏) =2

∑

𝑛=−2

𝐴𝑛𝜏𝑛

=3𝑄2

𝜏2+𝑁2

𝜏

+2 ln 2𝜋

𝐵2

+𝐾2

3𝜏 +

𝑅2

2𝜏2

.

(11)

In (11), 𝑄 is quantization noise coefficient; 𝑁 is anglerandom walk coefficient, 𝐵 is bias instability coefficient; 𝐾is rate random walk coefficient; 𝑅 is rate ramp coefficient.Each type of noise errors corresponds to different slope inthe Allan variance correlation time log-log plot. The slopeof quantization noise, angle random walk, bias stability,rate random walk, and rate ramp is −1, −0.5, 0, 0.5 and 1,respectively [5, 6].

Allan variance analysis method and the modeling tech-nology can effectively separate and identify several maintypes of randomnoises which influence FOCT’s precision. Byanalyzing the curve, the corresponding noise error values canbe estimated, making a comprehensive evaluation of overallperformance of FOCT. In addition, error source of FOCT’snoise error can be located by analyzing different types of noisevalues. It is not only effective to improve the performanceof FOCT, but also convenient to identify and locate fault inFOCT.

Combining FOCT state parameters and available infor-mation, we can build an intelligent diagnosis system based onsome algorithms, to determine the status of the device withtheoretical derivation and realize fault detection and diag-nosis [11–13]. The structure of intelligent diagnosis system isshown in Figure 3. Figure 4 shows fault diagnosis process inFOCT. We can obtain feature vectors in time domain andfrequency domain with signal processing model and thendetect the failure source by using the mapping relationshipbetween feature vectors and failure source.

Mathematical Problems in Engineering 5

Data processing

Feature extraction

Fault classification

Fault results, fault location

No

OK

Figure 4: Flow chart of fault diagnosis in FOCT.

Figure 5: Experimental environment of current transducer testsystem.

3. Experiments

To verify the performance and fault diagnosis methods offiber optic current transducer, we built a current transformertest system, as shown in Figure 5. Figure 6 shows the principleof current transformer test system. FOCT, standard currenttransducer (CT), and current generator are stringed in thesame current loop; the current generator applies primarycurrent to the two sets of CT, while different error factors areadded to FOCT. The function of transducer calibration partis to process data in FOCT and reference standard CT andto calculate the measurement deviation.The function of faultdiagnosis part is to monitor FOCT’s operation status and todiagnose FOCT’s fault by using Allan variance methods.

We test fault characteristics of several fiber optic currenttransformers, respectively. Figure 7 shows sample of currentmeasurements data in one FOCT. Allan variance log-logcurve expresses the randomerror in fiber current transformeraccurately, including angle random walk, bias instability, raterandom walk, rate ramp, quantization noise, and sinusoidalnoise. Theoretically, the slope of quantization noise, anglerandom walk, bias stability, rate random walk, and rate rampis −1, −0.5, 0, 0.5, and 1, respectively. Based on characteristicsof random error, we can evaluate health state of FOCT;moreover, error source in FOCT is located. Figure 8 showsAllan variance analysis curve of the first FOCT. We can findthat quantization noise with rate ramp of −1, angle randomwalk with rate ramp of −1/2, and bias instability noise with

Signal processing

Part 1. CalibrationPart 2. Fault diagnosis

Signal processing

A

Standard current sensor

FOCT

Current generator

Error factors

Figure 6: Principle block of current transducer test system.

0 1 2 3 4 5 6 7 8499

500

501

Time (s)Cu

rren

t (A

) ×10

4

Figure 7: Current measurements data in FOCT.

Allan correlative time (s)

10−1

10−2

10−3

10−4

100

101

102

103

104

105

Alla

n va

rianc

e𝜎(t

)

Figure 8: Allan variance curve of first FOCT.

10−5

100

Allan correlative time (s)10

010

110

210

310

410

5

Alla

n va

rianc

e𝜎(t

)

Figure 9: Allan variance curve of second FOCT.

rate ramp of 0 are the main errors in FOCT; moreover, wecan locate noise source, mainly from photon shot noise indetector, amplifier noise, thermal noise electronics, and highfrequency noise. Minimum resolution performance of thisFOCT is affected. Figure 9 shows Allan variance analysiscurve of the second FOCT. We can find that angle randomwalk with rate ramp of −1/2 and sinusoidal noise are themainerrors in FOCT, and high-frequency noise dominated systemnoise is the main noise source.

6 Mathematical Problems in Engineering

4. Conclusions

Allan variance can be used to analyze the time domaincharacteristics of the underlying random processes in FOCT.It helps to identify the source of a given noise term present inthe data, whether it is inherently in FOCT or in the absenceof any plausible mechanism within FOCT. In this paper, weverified the validity of theAllan variancemethods in soft faultdiagnosis through theoretical analysis and fault simulationexperiments. Allan variance is suitable for evaluation anddiagnosis of soft fault in FOCT, which can avoid potentialfailures, such as performance fault induced by temperature,light source, and other factors. For abrupt-changing fault, wecan compare and judge the fault characteristics with featuremodel by using wavelet transform methods and then decidewhether the fault is from FOCT or grid failure.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The project is supported by the following funds. Fun-damental Research Funds for the Central Universities(2242013R30016), Natural Science Foundation of JiangsuProvince (BK2012326, BK20130099), National Natural Sci-ence Foundation of China (61203192), Research Fund ofChina Ship 8 Industry (13J3.8.4), and Foundation of KeyLaboratory ofMicro-Inertial Instrument and Advanced Nav-igation Technology, Ministry of Education (201103).

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