impact of anti-aliasing filter and mimic filter on … · the power system time constant. this...

IMPACT OF ANTI-ALIASING FILTER AND MIMIC FILTER ON DIGITALPROTECTION RELAYING

Joao Tiago Campos∗, Washington Neves†, Damasio Fernandes Jr.†, Flavio Costa‡

∗Post-Graduate Program in Electrical Engineering - PPgEE - COPELE

†Electrical Engineering Department (DEE) Federal University of Campina Grande (UFCG) 58429-900,Campina Grande - PB - Brazil

‡ Federal University of Rio Grande do Norte (UFRN), School of Science and Technology (ECT),Campus Universitario Lagoa Nova, Natal - RN,Brazil CEP: 59078970

Emails: [email protected], [email protected], [email protected],

[email protected]

Abstract— This paper presents a study about the impact that the anti-aliasing and mimic filter have in thedigital relay response in the frequency domain. In addition, an evaluation of Butterworth filter response time isshown. At last, a detailed analysis of the mimic filter is shown. The frequency response of the mimic filter aswell as a strategy to decrease the limitation of using fixed time constant are also evaluated in this work. For themimic filter evaluation, several fault simulations were performed using ATP (Alternative Transients Program)and compared the time response between adaptive and fixed mimic filter. It is concluded that the adaptivemimic demands high power computing and the introduced time delay on fundamental wave depends of chosentime constant for mimic filter. From the analog filter analysis, it is concluded that the choice of the order andcutt-off frequency of the analog filter has a large influence on the time delay introduced into the signal of thefundamental frequency.

Keywords— Phasor estimation, Anti-Aliasing, Fourier, Wavelet, Digital Mimic Filter.

Resumo— Neste artigo, o impacto do filtro anti-aliasing e do filtro mımico na resposta em frequencia de relesdigitais e estudado. Uma avaliacao do impacto na resposta no tempo do filtro butterworth analogico e mostrada,bem como sao analisadas a resposta em frequencia e uma estrategia para diminuir a limitacao da constante detempo fixa do filtro mımico. Para provar os resultados para o filtro analogico, recorreu-se a computacao dasfuncoes transferencia, variando os parametros de frequencia de corte e ordem do filtro, e analisou-se a influenciana frequencia de 60 hz. No caso da analise do filtro mımico, foram feitas varias simulacoes de faltas num sistemade potencia no ATP(Alternative Transients Program) e realizada uma comparacao de desempenho. Conclui-seque a escolha da ordem e da frequencia de corte do filtro analogico influencia na resposta no tempo e que odimensionamento precisa ser executado com atencao. No que se refere a analise do filtro mımico, verificou-se quea variante adaptativa demanda muito esforco computational e que o desempenho do filtro mımico de constantefixa depende do valor da constante do sistema, atentando-se para o fato de que quanto maior a constante escolhidado filtro, maior e o atraso no tempo introduzido.

Palavras-chave— Estimacao de Fasores, Anti-Aliasing, Fourier, Wavelet, Filtro Mımico Digital.

1 Introduction

Since, Rockefeller (1969) idealized the digital re-lays theoretically. Nowadays, such relays arethe most used in comparison to the electrome-chanical relays for power system protection forvarious reasons, such as the flexibility in im-plementing multiple protection functions, cost,implementation of sequence of events and oscil-lography, self-monitoring and self-testing and soon (Committee, 2009).

Digital relays use analog and digital filters toprocess voltage and current signals from the elec-trical system. The signals processed by the relayare used, for example, to measure a trip situationin case of fault. An important advantage of digitalrelays compared to electromechanical is the possi-bility to compute the frequency and time responsein order to know the signal behaviour which inelectromechanical is only possible via testing.

It is well-known that the digital relay usesan analog low-pass filter (Horowitz and Phadke,

2008) to avoid the phenomenon of aliasing whichis influenced by the choice of the order and cutt-offrequency chosen. The knowledge about the anti-aliasing filter is important to the engineer becauseit introduces delays to the digital relay phasorsthat can be not supported by the goals of relaytime response. However, for a good relay time re-sponse it compromises the quality of the filteredsignal and this choice of the filters parameters de-pends of the main goal of the project. In thispaper, it is shown the impact that the cutt-off fre-quency and order filter plays in the quality andresponse time required.

Another important issue is the digital re-lay filters that can suffers with DC exponentialdecay which appears in faults during the tran-sients (Phadke and Thorp, 2008). Good digi-tal filters most overcome the DC component, har-monics and be simple to design (E.O. Schweitzerand Hou, 1993). Therefore, to overcome the DCexponential decay effects in digital filters havebeen proposed many papers using Kalman filter

Anais do XX Congresso Brasileiro de Automática Belo Horizonte, MG, 20 a 24 de Setembro de 2014

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(Girgis, 1982), using Fourier manipulations (Hartet al., 2000; Guo et al., 2003; Kang et al., 2009) be-cause it is well-known that the Fourier algorithmthat is more used in digital signal processing isaffected by DC exponential decay (Phadke andThorp, 2009). In addition, also wavelet trans-form does not filter the DC exponential decay(Silva et al., 2008). The frequency response of theFourier and wavelet-based algorithms are showedand clearly denotes the issue of the DC exponen-tial decay component.

In order to overcome the drawbacks of Fourierand wavelet-based algorithms, auxiliary filters areneeded to diminish the effects of the DC compo-nent. For this reason, Benmouyal (1995) proposesthe digital mimic filter which filter the DC com-ponent when the mimic time constant is equal tothe power system time constant.

This paper shows that the choice of the timeconstant influences the performance of the digitalfilters. For this reason, Silva and Kausel (2012)proposed a filter that adaptively changes the timeconstant. In this paper, the performance of themimic adaptive filter is compared to the mimicfilter with time constant fixed.

2 Anti-Aliasing Filters

The anti-aliasing filtering is of crucial impor-tance in signal processing within the digital re-lay. Therefore, aliasing must be avoided, and theButterworth filters are the most used (Phadkeand Thorp, 2009) because they have passband re-sponse with constant amplitude. The anti-aliasinglow-pass filter filters the high frequency compo-nents to avoid overlapping of the spectrum withlow frequency signals, thus preventing the anti-aliasing. The anti-aliasing filtering is performedbefore the signal sampling, and cut-off frequencyhas to comply the Nyquist sampling theorem.This ensure the reconstruction of the original sig-nal without loss.

In the analog filters design must be consid-ered:

• the type of the filter (Butterworth, Ellip-tic,...),

• order (1, 2.,..),

• cutoff frequency (450 Hz, 540 Hz,...).

The answers to these three parameters willdepend on the type of performance desired for thedigital relay. Fast or quality signal? Avoid surelyalmost 100%, the influence of high frequency sig-nals? These questions are answered according tothe performance that is demanded to the digitalrelay, and the purpose for which it is the sampledsignal.

Fig. 1 depicts the Butterworth filter fre-quency and phase response to a 60 Hz signal vary-ing the cuttoff frequency. The filters were com-pared to the five order and the sampling frequencyis 960 Hz (16 samples per cycle). To comply theNyquist theorem the filter must have a cutoff fre-quency up to 480 Hz to avoid aliasing. The mod-ule of the Butterworth anti-aliasing filter (Sedraand Smith, 2007) is given by:

|H| = 1√1 + ε2(

ω

ωp)2N

, (1)

where ε is the maximal variation in the passband,ωp is cuttoff frequency, N is the number of samplesper cycle and |H| is the filter magnitude response.

Figure 1: Amplitude response for a 60 Hz signalvarying the filter order.

According to Fig. 1, the fundamental fre-quency amplitude is dependent on the cutoff fre-quency adopted as well as the filter order. Theunity gain is achieved with the highest filter orderand cutt-off frequency.

Fig. 2 depicts that the lower the cut-off fre-quency and the higher the order of the filter morethe high-frequency signals are attenuated.

Figure 2: Amplitude response for a 480 Hz signalvarying the filter order.

The increasing of the filter order ensures unitygain and greater security regarding the filtering of


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high-frequency signals but introduces time delay.However, with higher filter order and lower cut-off frequency, greater is the delay introduced inthe fundamental frequency. This behaviour is de-picted in the Fig. 3.

Figure 3: Time delay in the fundamental fre-quency as a function of the cut-off frequency.

For this reason, the choice of cut-off frequencyand filter order of the anti-aliasing filter will de-pend on the constraints introduced in the time lagas well as the desired signal quality. A second or-der filter with cutoff frequency of 300 Hz could bea good choice in relation to the delay, amplitudeand high frequency components filtering. Accord-ing to the chosen parameters, this filter has a gainamplitude at the fundamental frequency of 0.9992,time delay of 0.76 ms and gain of 0.36 for frequen-cies above 480 Hz.

3 Digital Filter Formulation andFrequency Response

To prove the influence of DC component in theFourier and wavelet-based algorithms, it is shownthe frequency response using 16 samples per cycle(960 Hz).

3.1 One-Cycle Fourier Algorithm (OCF)

The phasor for fundamental frequency is obtainedwith Fourier Transform as follows:

Ycp =2

N

N∑n=1

yn cos(pnθ), (2)

Ysp =2

N

N∑n=1

yn sin(pnθ), (3)

where Ycp and Ysp are the Fourier series coeffi-cients, N is the samples per cycle and θ = 2π

N , ynis the sampled signal. For the fundamental phasorestimation the p mus be equal to one. The Fourieralgorithm frequency response is shown in Fig. 4.The OCF does not filter the DC offset, which isone of the major problems of this classical algo-rithm.

Figure 4: Frequency response of the one-cycleFourier algorithm.

3.2 Wavelet-Based (WB) Algorithm

The wavelet-based algorithm proposed in Silvaet al. (2008) uses the filtering properties of thewavelet transform combined with the propertiesof the least squares filter as follows:

Y = A+B, (4)

where A+ is the pseudo-inverse matrix. Combin-ing the properties of wavelet and least squares:

Y = A+[Mj ]X = GX, (5)

being [Mj ] a matrix composed by M consecutive

rows of the matrix Mj . For more detailed of theseequations see (Silva et al., 2008).

The frequency response of the wavelet-base al-gorithm (Silva et al., 2008) is depicted in Fig. 5:

Figure 5: Frequency response of the wavelet-basedalgorithm.

According to the frequency response shown inFig. 5, this algorithm is not immune to DC com-ponent although the algorithm has good responseto sub-harmonics. Therefore, the performance ofthis algorithm will depend of auxiliary filters tofilter the DC exponential decay component.

4 Power System Modelling and FaultSimulations

In this paper, fault simulations were performedin a power system composed of two equivalentsources connected through a transmission line toprove that the time constant of the power system


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is not fixed. Therefore, the power system equiva-lent sources were modelled with lumped parame-ters calculated at the frequency of 60 Hz and thetransmission line was modelled with distributedparameters dependent of frequency and assumingideal transposition.

Tables 1 and 2 summarizes the system param-eters, where V1 and V2 are the initial power flowconditions in bus 1 and 2, Z0 and Z1 are the zeroand positive sources impedance, Y0 and Y1 are thezero and positive line shunt admittance.

Table 1: Sources Parameters.Source 1 Source 2

V1 1.0506 0 pu V V2 0.956 −10 pu VZ0 1.041 + i18.754 Ω Z0 1.127 + i20.838 ΩZ1 0.871 + i25.661 Ω Z1 0.968 + i28.513 Ω

Table 2: Transmission Line Data.Z0 0.053 + i1.000 Ω/ kmY0 2.293 Ω−1/kmZ1 0.098 + i0.510 Ω /kmY1 3.252 Ω−1/km

length 180 km

The applied faults were summarize in table 3,where AG, AB, ABG and ABC are respectively,phase-ground fault, phase-phase fault, phase-phase-to-ground fault and three-phase fault. Rgis the ground resistance fault used in ground faulttypes such as AG and ABG. Rab is the phase-phase resistance between phase A and B used inphase-phase faults such as AB, ABG and ABC.Rbc and Rca were used in the ABC three-phasefaults and are respectively the phase-phase resis-tance between phases B and C and, phases C andA.

Table 3: Faults data.Fault type AG, AB, ABG and ABC

Fault inception angle() 0, 30, 45 , 60, 90Fault distance (Km) 20, 40, 60, 80, 100,

120, 140, 160Rg , Rab, Rbc, Rac (Ω) 1, 5, 10, 50, 100

5 Evaluation of the Mimic Filter (MFI)performance

In this section, the importance of setting appro-priately the mimic time constant (τ) is studiedand it is show the importance of using adaptive fil-ters to overcome the problem of using a fixed timeconstant. Therefore, the mimic filter with timeconstant fixed is compared against the adaptivemimic filter (Silva and Kausel, 2012) which variesthe time constant along the time. For this reason,the parameter overshoot is used. The overshoot

is the overreach experienced by the digital filterduring transients caused by the DC exponentialdecay as depicted in Fig. 6.

Figure 6: Overshoot definition.

The digital mimic filter is proposed by Ben-mouyal (1995) to filter the DC exponential decaycomponent. In addition, when setting the timeconstant in the mimic filter equal to the powersystem line time constant all the DC componentwill be filtered because the digital mimic filter isderived from a RL filter that has this properties.A detailed explanation about how the RL filterfilters the DC exponential decay can be read inBenmouyal (1995).

This filter is important to algorithms that donot have the ability to filter the DC componentsuch as 3.1 and 3.2.

In the fault simulations, the τ which give thelowest overshoot performed by the phasor estima-tion varies between 0 and 7 cycles as depicted inFig. 7. The Fig. 7 was created by an iterativeway using the OCF combined with MFI and test-ing varying τ by 0 to 10 cycles by an interval of 0.1for each fault, the τ which gives the lowest over-shoot is saved. For example, an AG fault withmimic filter time constant set to 1 cycle experi-ence a 5% overshoot, in the same AG fault themimic filter is then set to 2 cycle and experience a6% overshoot, for this fault the ideal mimic filtertime constant is 1 cycle.

Figure 7: Ideal τ value.

The major limitation of the digital mimic fil-ter is that the time constant has to be fixed butthis time constant varies with the fault locationand fault resistance as depicted in Fig. 8.

In order to overcome the problem with themimic filter time constant fixed, the algorithm


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Figure 8: τ variation depending of faults variables.

proposed by Silva and Kausel (2012) adaptivelycalculates the mimic filter time constant from thesignal samples. This algorithm needs a fault de-tector, which can be a limitation, to start cal-culate the mimic filter time constant and in thebeginning of the fault the time constant is setto a initial value which depends of the engineerchoice. At the beginning of the fault, the adap-tive mimic filter(AMF) estimates the time con-stant from the available samples, then when theAMF has only samples post-fault it accurately es-timates the time constant. For a more comprehen-sive about the steps for the adaptive time constantcalculation read Silva and Kausel (2012). In Ta-ble 4 is showed the overshoots experienced by theFourier and wavelet-based algorithms with adap-tive mimic are close to the optimal value obtainedin 8 for the same simulated case, in this case thepower system time constant is equal to 0.75 cycles.

Table 4: Overshoot experienced for τ power sys-tem equal to 0.75 cycles

Filters τ=0.7500

Fourier + adaptive mimic 0.7280Wavelet-Based + adaptive mimic 0.8713

Fig. 9 depicts that the overshoot experiencedby the mimic filter depends on the chosen τ , in thiscase the power system time constant is 1.2 cycles.This figure show the concerns about which timeconstant is more suitable for the digital mimic fil-ter, using the adaptive mimic filter this issue dis-appears.

Fig. 10 depicts that increasing the τ , morehigher is the time delay introduced in the signal bythe mimic filter which is undesirable. Therefore,the choice of time constant is to be made carefuland it has to be set to a recommended value of0.75-1.25 cycles.

Figure 9: Overshoot variation depending of τ .

Figure 10: Lag introduced by the mimic filter de-pending of τ .

In the performed simulations, it is concludedthat a value of τ equal to 1 cycle results in op-timal response to mimic filter because the algo-rithms for values smaller than 1 cycle time con-stant of the system experience the highest over-shoot. With a choice of time constant near 1 cy-cle it is assured that the overshoot obtained byfixed mimic filter is not too severe for this eval-uated power system. To solve the limitation offixed time constant setting a mimic adaptive filterwhich uses the signal samples to estimate a valuefor the time constant closer to real value was pro-posed by Silva and Kausel (2012). Despite theadaptive filter mimic demonstrates good resultscompared to mimic without adaptive features, theadded complexity is not worth the improvementsachieved.

6 Conclusion

In this paper, the impact of the anti-aliasing fil-ters in digital relaying is shown as well as theissue about how to optimal set the mimic filtertime constant. It is concluded that the choice ofanti-aliasing filter has to be carefully and dependsof goals of the project. A compromise has to beachieved between signal quality and time delay.In addition, a study about the variation of thepower system time constant and their concerns indigital filtering are shown. The mimic filter withfixed time constant is compared with a adaptive


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mimic filter using various fault simulations. Itis concluded if the time constant of the digitalmimic filter is setting to one cycle of the powersystem the impact of the DC component is mini-mized.Therefore, if a more reliable phasor value isneeded the adaptive mimic filter is the best solu-tion.

Acknowledgment

The authors would like to thank to CAPES andCNPq for the financial support.

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