Electrical Power and Energy Systems 32 (2010) 236–242

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Digital frequency relaying based on the modified least mean square method

Daniel Barbosa *, Renato M. Monaro, Denis V. Coury, Mário OleskoviczDepartment of Electrical Engineering, Engineering School of São Carlos, University of São Paulo, Av. Trabalhador Sancarlense, 400 – CEP 13566-590, São Carlos, SP, Brazil

a r t i c l e i n f o

Article history:Received 28 August 2008Received in revised form 30 June 2009Accepted 3 July 2009

Keywords:Frequency estimationAdaptive filterDigital protectionPower systemLeast mean square

0142-0615/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijepes.2009.07.009

* Corresponding author. Tel.: +55 16 3373 8147.E-mail addresses: dbarbosa@usp.br, dbarbosa0@gm

a b s t r a c t

This research presents a method for frequency estimation in power systems using an adaptive filter basedon the Least Mean Square Algorithm (LMS). In order to analyze a power system, three-phase voltageswere converted into a complex signal applying the ab-transform and the results were used in an adaptivefiltering algorithm. Although the use of the complex LMS algorithm is described in the literature, thispaper deals with some practical aspects of the algorithm implementation. In order to reduce computingtime, a coefficient generator was implemented. For the algorithm validation, a computing simulation of apower system was carried out using the ATP software. Many different situations were simulated for theperformance analysis of the proposed methodology. The results were compared to a commercial relay forvalidation, showing the advantages of the new method.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In order to protect Electrical Power Systems (EPS), a precisedetection of faulty or abnormal situations is necessary so that theycan be eliminated and consequently return to the normal operationcondition as soon as possible. Taking this into account, protectiverelays constantly monitor the three-phase voltage and current sig-nals, including their frequency.

Frequency is one of the main parameters to be monitored in anEPS as during a faulty or undesired situation, it will experience asignificant alteration. In practice, a range of frequency variationsfor a system operation is established as 60 � 0:5 Hz [1]. Variationson these limits are constantly observed as a consequence of the dy-namic unbalance between the generation and the load. However,larger variations may indicate faulty situations, as well as a systemoverload. Considering the latter, the frequency relay can help in theload shedding decision process [2,3].

The importance of a correct frequency estimation for EPS is thenobserved, especially if the established limits for its normal opera-tion are not reached. This can cause serious problems for theequipment connected to the power utility, such as capacitor banks,generators and transmission lines, affecting the power balance.Therefore, frequency relays are widely used in the system to detectpower oscillations outside the acceptable operation levels of theEPS.

It must also be remembered that due to technological advancesand a considerable increase in the use of electronic devices

ll rights reserved.

ail.com (D. Barbosa).

throughout the last decades, the concern for frequency variationin EPS has significantly grown. This is because these modern com-ponents are more sensitive to frequency variation.

Taking this into account, the study of new techniques for a bet-ter and faster estimation of the system frequency has become ex-tremely important for a power system operation. Having this inmind, various techniques were proposed, mainly based on the pha-sor estimation, using the Least Mean Square (LMS) Method, theFast Fourier Transform (FFT), intelligent techniques, as well asthe Kalman Filter [4–9]. The adaptive filter based on the LMS pro-posed by Pradhan [10] should be outlined. The LMS was first intro-duced by Widrow and Holf [11] for digital signal processing andhas been widely used since because of its simplified structure, effi-ciency and computing robustness.

In this research, the LMS was used in its complex form [12] withan adaptive step-size [13]. Although the use of the complex LMSalgorithm is described in the literature, this paper deals with somepractical aspects of the algorithm implementation. The algorithmlogic used an one cycle moving window, making it more feasiblefor commercial applications. In order to reduce computing time,a coefficient generator was implemented.

It should also be highlighted that in order to test the proposedalgorithm, computing simulations were performed using the ATPsoftware [14]. A fairly complex power system was implementedwhere its dynamic behavior could be observed. The simulation in-cludes: a synchronous generator with speed and voltage control,transmission lines using frequency dependent parameters andpower transformers. Extreme operational situations were testedin order to observe the behavior of the proposed technique, tocompare it to the performance of commercial relays and finally

Fig. 1. Adaptive filter based on LMS.

Fig. 2. Basic relay algorithm.

D. Barbosa et al. / Electrical Power and Energy Systems 32 (2010) 236–242 237

to validate the obtained results. This way of testing is not foundeasily in the literature.

2. The algorithm based on LMS

The algorithm based on the LMS method, presented in Fig. 1, is acombination of the adaptive process with digital filtering. In thisFigure, �uðn� 1Þ ¼ ½uðn� 1Þ uðn� 2Þ . . . uðn�MÞ � is the vectorwith M past values; �wðnÞ ¼ ½w1ðnÞ w2ðnÞ . . . wNðnÞ �T is thevector with the filter coefficients; yðnÞ is the desired signal (theoutput signal) and eðnÞ is the error associated to the filterapproximation.

The input signal of the filter can be estimated by minimizing thesquared error by the coefficient adaptations ð�wðnÞÞ, which arerecursively adjusted to obtain optimal values. At each iteration,the coefficients can be calculated by:

�wðnþ 1Þ ¼ �wðnÞ þ l �rðnÞð Þ ð1Þ

where l is the convergence parameter and r is the gradient of theerror performance surface and is responsible for determining thenecessary adjustments to the coefficients.

The LMS algorithm is very sensitive to l. This can be mainly ob-served by the speed of the estimation and processing time. Thelower the l value, the greater time to achieve the desired errorand vice-versa.

However, it is important to respect the convergence interval gi-ven by [15]:

0 < l <1

NSmaxð2Þ

where N is the filter size and Smax is the maximum power spectraldensity value of the input signal.

3. The adaptive algorithm and the frequency estimation

Regarding digital protection, digital filters provide the fre-quency component estimation used in digital relay algorithms.The information contained in the input data from a three-phasesystem can be processed simultaneously, making it possible to ob-tain more precise results, if compared to conventional methods.

It must be highlighted that the proposed algorithm, called theFrequency Estimation Algorithm by the Least Mean Square(FEALMS), uses three-phase signals as inputs. Considering that,the three-phase voltages from EPS can be represented by:

VaðnÞ ¼ AmaxcosðxnDt þ /Þ þ nna

VbðnÞ ¼ Amaxcos xnDt þ /� 2p3

� �þ nnb

VcðnÞ ¼ Amaxcos xnDt þ /þ 2p3

� �þ nnc

ð3Þ

where, Amax is the peak value; x is the signal angular frequency 1; nis the sample number of the discrete signal; Dt is the time betweentwo consecutive samples; / is the signal phase and nn is the noise.

1 x ¼ 2pf , where f is a power system frequency. Fig. 3. Data acquisition flowchart.

Fig. 2 illustrates the proposed relay algorithm.

3.1. Data acquisition

All the stages in the data acquisition are taken into account,which enables a more realistic analysis of the results. The inputsimulated voltage signals are characterized by a high sampling rateto represent the analog signals better.

Fig. 3 shows the data acquisition flowchart. A second order lowpass Butterworth filter with a cut-off frequency of 200 Hz wasused. A sample rate of 1920 Hz and an analog-to-digital converter(ADC) of 16 bits were also used.

The low pass filter (LPF) was used to prevent spectral spreadingand to ensure that the digital signal after ADC conversion repre-sented the original signal. It is important to comment that a lowcut-off frequency of 200 Hz was used to stabilize the method andensure that the LMS algorithm converges. For this reason, mostof the harmonic components were eliminated, increasing the pre-cision of the proposed method.

The data acquisition was carried out in a moving window withone sample step. All the filter processing should be executed in thetime between two consecutive samples, taking into account themoving window process and the available time for processing.Fig. 4 illustrates this process.

3.2. Normalization

Normalization enables the input power system data to be stan-dardized, regardless of the voltage level. Consequently, if either asag or a swell occurs in any phase, the algorithm will maintainits estimation without loss of precision or speed. Fig. 5 illustratesthe normalization process which was implemented.

Fig. 4. Moving window process.

Fig. 5. Data normalization flowchart.

Fig. 6. Coefficient generator flowchart.

238 D. Barbosa et al. / Electrical Power and Energy Systems 32 (2010) 236–242

3.3. Pre-processing

After normalization, a pre-processing stage was executed,obtaining the signal in its complex form for the digital filter. Thiswas obtained by applying the ab-transform on the three-phasevoltages, as represented in the equation below [16].

VaðnÞVbðnÞ

� �¼

ffiffiffi23

r1 � 1

2 � 12

0ffiffi3p

2 �ffiffi3p

2

" #VðnÞ½ � ð4Þ

where VðnÞ ¼ ½VaðnÞ VbðnÞ VcðnÞ �T . After the pre-processingstage, the complex voltage is defined as:

uðnÞ ¼ VaðnÞ þ jVbðnÞ ð5Þ

3.4. Coefficient generator

The adaptation of the filter coefficients is simple and inherentto the LMS algorithm. This adjustment is performed sample bysample minimizing the mean squared error. However, to im-prove the algorithm performance and minimize processing time,the estimation filter coefficients are initialized, �wðnÞ, with theestimation of the previous window [17]. The first window isstarted using the fundamental power system frequency. Themain objective of this procedure is to increase the speed ofthe estimation process. The coefficient generator flowchart isshown in Fig. 6.

3.5. Adaptive filter

In the adaptive filter, the coefficients are updated recursively tominimise the squared error. The error is calculated as thedifference between the desired and estimated values, given by:

eðnÞ ¼ uðnÞ � yðnÞ ð6Þ

where yðnÞ represents the estimated value, and is calculated by:

yðnÞ ¼ �wHðnÞ�uðn� 1Þ ð7Þ

where H is the conjugate transpose.It is important to emphasize that the LMS task is to find the fil-

ter coefficients which minimize the error. Following this proce-dure, the filter coefficients are updated until the error issufficiently small. The complex weight vector at each samplinginstant is given by [12]:

�wðnþ 1Þ ¼ �wðnÞ þ lke�k�uðn� 1Þ ð8Þ

where the (*) symbol denotes the complex conjugate and l is theconvergence factor controlling the stability and rate of convergenceof the algorithm.

The step-size lk is modified for better convergence in thepresence of noise and its equation is given by [13]:

lkþ1 ¼ klk þ cpkp�k ð9Þ

where pk represents the error autocorrelation and is calculated as:

pk ¼ qpk�1 þ ð1� qÞekek�1 ð10Þ

In the equation, q is the exponential weighting parameter. Theq ð0 < q < 1Þ; k ð0 < k < 1Þ and c ð0 < cÞ constants control the con-vergence time and they are determined by statistical studies [18].

3.6. Frequency estimation

The frequency was estimated according to [19]. To find thephase difference, the complex variable C was defined as:

C ¼ yðnÞyðn� 1Þ� ð11Þ

The frequency of the estimated signal yðnÞ was calculated in func-tion of the phase difference between two consecutive samples,and the latter was given by the equation below:

fest ¼fs

2parctan

IðCÞRðCÞ

� �ð12Þ

where fest is the estimated frequency, fs is the sample frequency andRðÞ and IðÞ are the real and imaginary parts, respectively.

3.7. Convergence

The stop criteria adopted was the maximum number of itera-tions (1000) or absolute error smaller than 10�5. This error canbe estimated by:

erelat ¼ abs yðnÞ � uðnÞð Þ ð13Þ

D. Barbosa et al. / Electrical Power and Energy Systems 32 (2010) 236–242 239

where erelat is the relative error between samples, absðÞ is the abso-lute value, yðnÞ is the estimate value and uðnÞ is the desired value orinput sample.

3.8. Post-processing of the output signal

The output signal (estimated frequency) is additionally filteredby a second order low pass Butterworth digital filter with a cut-offfrequency of 5 Hz. This procedure reduces the oscillation in theproposed method output, avoiding errors due to abrupt frequencyvariations of the signal. It is important to observe that the delay ofthe low pass filter does not influence the algorithm performancenegatively, as can be seen in the results.

4. The simulated electrical system

The electrical system was simulated using ATP (AlternativeTransients Program) software. Fig. 7 shows the representation ofthe simulated power system, taking into account load switchingand permanent faults in order to evaluate the frequency estimationtechnique proposed in this work.

The simulated power system consists of a 13.8 kV and 76 MVAsynchronous generator, 13.8:138 kV/138:13.8 kV and 25 MVAthree-phase power transformers, transmission lines between 80and 150 km in length and loads between 5 and 25 MVA with a0.92 inductive power factor. Power transformers have a delta con-nection in the high voltage winding and a star connection in thelow voltage winding. Tables 1 and 2 show the parameters usedin order to simulate the power system components using ATPsoftware.

In Table 1, S is the total three-phase volt-ampere rating of themachine, Np is the number of poles which characterize the ma-chine rotor, VL is the rated line-to-line voltage of the machine, f

Fig. 7. Power system representation using ATP software.

is the electrical frequency of generator, IFD is the field current, Ra

is the armature resistance, Xl is the armature leakage reactance,Xo is the zero-sequence reactance, Xd is the direct-axis synchro-nous reactance, Xq is the quadrature-axis synchronous reactance,X0d is the direct-axis transient reactance, X00d is the direct-axis sub-transient reactance, X00q is the quadrature-axis subtransient reac-tance, s0do is the direct-axis open-circuit transient time constant,s00do is the direct-axis open-circuit subtransient time constant ands0qo is the quadrature-axis open-circuit transient time constant.

It is important to emphasize that the transmission line modelused was JMARTI from ATP. This model assumes line parametersvarying with the frequency and consequently better representsthe system’s behavior when facing disturbances resulting fromunbalance between generation and load.

It must also be emphasized that the synchronous generator wassimulated with an automatic speed control for hydraulic systems[20] and automatic voltage regulation (AVR) [21–23], consideringvarious electrical and mechanical parameters from the generator.Eq. (14) shows the transfer function of the speed regulator used:

gðsÞDFðsÞ ¼ �

1R� 1þ sTr

1þ sTg� �

1þ s rR Tr

� � ð14Þ

where gðsÞ is the servomotor position, DFðsÞ is the frequencydeviation, R is the steady-state speed droop, r is the transient speeddroop, Tg is the main gate servomotor time constant and Tr is thereset time. Table 3 presents the parameters concerning the speedregulator.

Fig. 8 shows the block diagram of the excitation control systemwhich was used. The basic function of the excitation control sys-tem automatically adjusts the magnitude of the DC field currentof the synchronous generator to maintain the terminal voltageconstant as the output varies according to the capacity of thegenerator [24].

The field voltage control can improve the transient stability ofthe power system after a major disturbance. However, the extentof the field voltage output is limited by the exciter’s ceiling voltage,which is restricted by generator rotor insulation [24,25].

5. Simulation results

The purpose of this section is to present some results regard-ing the proposed algorithm, comparing them to a commercial re-lay, as well as the actual frequency variation curve given by theATP software. Various different tests were performed in the sys-tem shown in Fig. 7. It should be emphasized that the sample

Table 1Synchronous generator data used in the simulation.

Description Value (unit) Description Value (unit)

S 76 (MVA) Np 8VL 13.8 (kVrms) f 60 (Hz)IFD 250 (A) Ra 0.004 (p.u.)Xl 0.175 (p.u.) Xo 0.132 (p.u.)Xd 1.150 (p.u.) Xq 0.685 (p.u.)X0d 0.310 (p.u.) X00d 0.210 (p.u.)

X00q 0.182 (p.u.) s0do 5.850 (s)

s00do 0.036 (s.) s00qo 0.073 (s)

Table 2Power transformer data used in the simulation.

Element RþðXÞ LþðmHÞ

Primary impedance of transformer 1.7462 151.37Secondary impedance of transformer 0.0175 1.514

Table 3Parameters concerning the speed regulator.

Description Value (unit)

Main gate servomotor time constant ðTgÞ 0.600 (s)Reset time ðTrÞ 0.838 (s)Transient speed droop ðrÞ 0.279Steady-state speed droop ðRÞ 0.100Moment of inertia ðMÞ 1.344 (s)Water starting constant ðTW Þ 0.150 (s)

Fig. 8. Block diagram of the excitation control system.Fig. 9. Connection of load blocks on the BGCH3 busbar at 2 s.

240 D. Barbosa et al. / Electrical Power and Energy Systems 32 (2010) 236–242

rates of 1920 Hz and 1000 Hz were used in the FEALMS and acommercial frequency relay (IEEE device function number 81[26]), respectively.

Due to a great influence from the adjustment of the filterparameters in the results, these parameters were selectedaccording to [18], and they are: lmax ¼ 0:18;lmin ¼ 0:001; pinicial

¼ 0; k ¼ 0:97; c ¼ 0:01 and q ¼ 0:99. Based on Fig. 7, the simulatedsituations were:

� a sudden connection of load blocks;� a permanent fault involving phase A and ground (AG) on the

BGER busbar at 2s;� a sudden disconnection of TR1E and TR3E transformers at 1s;� a permanent fault at 50% of line 1;� the generator overexcitation;� the TR3E transformer energization with full load.

5.1. A sudden connection of load blocks

Fig. 9 shows the estimation of the synchronous generator fre-quency using a commercial frequency relay, the ATP softwarereference curve and the FEALMS considering the connection ofload blocks in the BGCH3 busbar. In the figure, a slight delay inthe frequency estimation by the relay can be observed, whencompared to the correct result given by the ATP software curve.In this situation, very good precision concerning the FEALMScan be observed, even in critical points of the system’s behavior.The error concerning the application of the proposed techniqueis also presented.

Fig. 10. AG fault on BGER busbar at 2 s.

5.2. A permanent fault involving phase A and ground (AG) on the BGERbusbar at 2 s

Fig. 10 shows the estimation given by the proposed technique,the ATP reference curve, as well as the commercial frequency relayperformances for an AG fault on the BGER busbar at 2 s.

It can be seen that the commercial frequency relay testedloses the reference voltage, hindering the frequency estimation.It should be emphasized that even in this unfavorable condi-tion, the FEALMS estimated the power system frequencysatisfactorily.

5.3. A sudden disconnection of TR1E and TR3E transformers at 1 s

Fig. 11 also illustrates the ATP reference curve, as well as thecommercial frequency relay and the FEALMS responses for a dis-connection of TR1E and TR3E transformers. The analysis of thiscondition is fundamental to test the algorithm’s robustness facingpractical field situations, taking into account the high level of dis-tortion presented in the input signals.

It can be clearly observed that the commercial relay (function81), as well as the FEALMS, presented similar satisfactoryresponses. Once more, a delay can be observed for the commercialrelay (function 81), possibly due to the filtering process.

5.4. A permanent fault at 50% of line 1

In Fig. 12, a frequency variation is observed considering an AGfault at 50% of the transmission length of line 1. In this situation,

Fig. 11. A sudden disconnection of TR1E and TR3E transformers at 1 s.

Fig. 12. AG fault at 50% of the line 1 length.

Fig. 13. Frequency estimation for generator overexcitation.

Fig. 14. TR3E transformer energization with full load.

D. Barbosa et al. / Electrical Power and Energy Systems 32 (2010) 236–242 241

the correspondent three-phase circuit breaker opened after 80 msfrom the fault inception.

It should also be observed that the recovery of the machine syn-chronization was achieved without the need of the relay action. Inthe Brazilian power system, it should happen when the frequencygoes outside the range of 60 � 0:5 Hz [1].

5.5. The generator overexcitation

Fig. 13 illustrates the FEALMS and commercial relay (function81) responses, as well as the ATP reference curve for the generatoroverexcitation. This situation illustrates the ability of the algorithmto estimate the frequency in the presence of harmonics in the inputsignals.

242 D. Barbosa et al. / Electrical Power and Energy Systems 32 (2010) 236–242

5.6. The TR3E transformer energization with full load

Fig. 14 illustrates the frequency estimation of the proposedtechnique together with the commercial frequency relay perfor-mance, as well as the ATP reference curve for a case of frequencydrop concerning TR3E transformer energization with full load.

It can be seen from Fig. 14 that most of the estimation errors forFEALMS are below 0.05%.

6. Conclusions

This work presented an alternative method for the power sys-tem frequency estimation using the modified LMS algorithm. Theimplementation of digital filter and the ab-transform in the pro-posed technique made the simultaneous use of the three-phasepower system voltages for the estimation purpose possible.

In the complex LMS algorithm considered, the adjustment ofthe step-size was used as being adaptive based on the estimationsof the error.

The simulations used for testing the proposed algorithm wereobtained using the ATP software. The adaptive filter theory appliedto the digital protection was fast and reliable. Some points shouldbe observed:

1. The paper deals with practical aspects of the FEALMS includingoptimization of the computing time.

2. The FEALMS can be applied to various situations and differentvoltage levels.

3. The three-phase voltages were analyzed by the FEALMS con-trary to a single phase (used in commercial relays), makingthe proposed algorithm more robust.

4. The technique is easy to implement and does not need adjust-ments and knowledge of further functions, as is the case ofthe commercial frequency relay, which was used.

5. The average error in all cases studied was 0.08%.

It is also important to highlight the feasibility and computingefficiency of this method, making it suitable for commercialapplications.

Acknowledgements

The authors would like to acknowledge the Department of Elec-trical Engineering, Engineering School of São Carlos, University ofSão Paulo (Brazil), for research facilities provided to conduct thisproject. Our thanks also to the financial support received fromCAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Supe-rior), CNPQ (Conselho Nacional de Desenvolvimento Científico eTecnológico) and FAPESP (Fundação de Amparo á Pesquisa do Esta-do de São Paulo).

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