Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Adaptive Three-phase Estimation of Sequence Components and Frequency Using H Filter Based on Sparse Model  PDF

  • Umamani Subudhi 1
  • Harish Kumar Sahoo 2
  • Sanjeev Kumar Mishra 1
1. Department of Electronics and Telecommunication Engineering, Indian Institute of Information Technology (IIIT), Bhubaneshwar, India; 2. Department of Electronics and Telecommunication Engineering, Veer Surendra Sai University of Tecnology (Burla), Sambalpur, India

Updated:2020-09-28

DOI:10.35833/MPCE.2018.000440

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Abstract

The estimation of sequence or symmetrical components and frequency in three-phase unbalanced power system is of great importance for protection and relay. This paper proposes a new H filter based on sparse model to track the sequence components and the frequency of three-phase unbalanced power systems. The inclusion of sparsity improves the error convergence behavior of estimation model and hence short-duration non-stationary PQ events can easily be tracked in the time domain. The proposed model is developed using l1 norm penalty in the cost function of H filter, which is quite suitable for estimation across all the three phases of an unbalanced system. This model uses real state space modeling across three phases to estimate amplitude and phase parameters of sequence components. However, frequency estimation uses complex state space modeling and Clarke transformation generates a complex measurement signal from the unbalanced three-phase voltages. The state vector used for frequency estimation consists of two state variables. The proposed sparse model is tested using distorted three-phase signals from IEEE-1159-PQE database and the data generated from experimental laboratory setup. The analysis of absolute and mean square error is presented to validate the performance of the proposed model.

I. Introduction

THE steady-state operation condition of the three-phase balanced power system is disrupted due to several reasons such as single-phase loading, unbalanced impedance of transmission lines and transformers, incomplete transposition of lines, lightning, line breaks, etc. Unbalance is considered as one of the PQ issues which can be measured through sequence components [

1]. Due to the unbalanced situation of voltage across the phases, the system performance degrades and shortens the life of the equipment used in industry. Similarly, any deviation of system frequency from the normal value is a clear indicator of unbalanced condition between load and generation, which may lead to the grid failure by inadequate load shedding.

Therefore, the detection and subsequent mitigation of faults by designing protective equipment are crucial for a healthy power grid. To protect the grid, sequence components across all the phases and system frequency have to be estimated correctly. The amplitude and phase information of all the three phases in an unbalanced system can be extracted by analyzing the sequence components in terms of zero, positive, and negative sequence components. Adaptive filters are efficient to estimate the sequence components and frequency by modeling the three-phase voltage and current equations in a parametric form.

Several techniques have been proposed to estimate the symmetrical components and frequency from three-phase voltage and current signals. Discrete Fourier transform (DFT) is a popular nonparametric method used for frequency estimation, but this method suffers from inaccuracies due to the presence of system noise, inter- and intra-harmonics and spectral leakage [

2], [3]. However, the errors due to the effect of spectral leakage and picket-fence are mitigated by modulated sliding discrete Fourier transform (msDFT) which is proposed for the measurement of harmonic and sequence components [4]. Reference [5] proposes an adaptive method using DFT and online algorithm, which is based on converting signal into phasor for symmetrical component estimation. Reference [6] also proposes a DFT-based technique for frequency estimation, which has good resistance to harmonics.

Artificial intelligence and soft computing techniques such as neural network, fuzzy logic, and genetic algorithm are quite popular for tracking PQ disturbances [

7], [8]. These methods can track the signal parameters accurately with longer tracking time and slower convergence. Adaptive linear combiner structure is a single-layer model frequently used for harmonic and frequency estimation for single-phase power system [9]. However, the model based on multi-output can be used to track the symmetrical components of a three-phase system. Least mean square (LMS) is used for the update of weight vector, which requires a proper choice of step size and the overall convergence rate is quite slow [10].

Compared to Kalman filter, extended Kalman filter (EKF) is an effective approach to estimate nonstationary PQ disturbances with increased noise level. Although EKF shows good estimation performance, it does not have stable convergence in the case of abrupt frequency change. Reference [

11] proposes unscented Kalman filter (UKF) which is a derivative-free filter for the estimation of frequency in balanced and unbalanced power systems with the reduced complexity of computation [11]. However, the estimation accuracy of UKF is affected by the factors such as the divergence of estimation error due to some poorly-chosen initial conditions and the nature of noise. To overcome the above limitations, H filter is proposed, which is a robust form of Kalman filter to estimate the parameters of the sinusoid corrupted by white Gaussian noise [12], [13]. However, the filter performance is again restricted due to computation complexity and stable convergence for short-duration time-varying parameters.

The research work presented in this paper is based on a convex combination of H filters with the introduction of l1 norm to the quadratic cost function for the estimation of both sequence components and frequency in an unbalanced three-phase power system. The proposed model is tested for wide categories of time-varying and short-duration PQ disturbances and the respective simulation results are presented.

The remainder of this paper is organized as follows. Section II explains the implementation steps of the H filter based on the sparse model. Section III and Section IV describe the state space models used to estimate the symmetrical components and frequency. Simulation results are presented in Section V to test and validate the proposed method considering different test cases such as frequency drift, harmonics, fault condition, etc. Section VI presents the conclusion of the results.

II. Filtering Algorithm of Sparse H

In this proposed model, l1 norm penalty is introduced to the quadratic cost function of Hfilter [

14], [15]. The advantage of l1 relaxation to the cost function lies in the extraction of the sparse solution compared to l0 and l2 norms with faster convergence. Moreover, in the case of Lasso regularized regression, if the budget used in the inequality is small enough, the solution vector is sparse but non-convex. In the case of higher order norms, the sparsity is lost which leads to more computation complexity [16]. The modified cost function with norm penalty Jk can be expressed as [17]:

Jk=12ek2+δx1 (1)

where 1 denotes the l1 norm of a vector; ek is the estimation error used in the cost function; x is the state vector; and δ is the weight assigned to the penalty term. Sparse H filter is based on two equations, namely, measurement equation and state equation which are given in (2)-(5) [

12]. Measurement matrices and state transition matrices are generated using Taylor’s series expansions of measurement and state equations. The information about measurement matrices and state matrices for sequence components, as well as frequency estimation, are provided in the subsequent sections.

ŷ1,k/k=Hkx̂1,k/k+vk (2)
ŷ2,k/k=Hkx̂2,k/k+vk (3)
x̂1,k+1/k=f(x̂1,k/k)+wk (4)
x̂2,k+1/k=f(x̂2,k/k)+wk (5)

where ŷ1,k/k is the output vector generated through non-sparse filter estimation during the kth instant using the predicted state vector of the kth instant x̂1,k/k; subscripts 1,k and 2,k are used to represent the convex combination using non-sparse and sparse adaptive filtering algorithms, respectively; Hk is the measurement matrix; vk and wk are the measurement noise and process noise, respectively, which are both white Gaussian noise process with zero mean; and f() is the supply frequency. As convex combination is formed using sparse and non-sparse adaptive filtering algorithms, x1,k and y1,k correspond to non-sparse adaptive filtering algorithm while x2,k and y2,k correspond to sparse adaptive filtering algorithm, respectively.

Two state update equations for convex combination are given in (6) and (7).

x̂1,k/k=x̂1,k/k-1+K1,k(yk-Hkx̂1,k/k-1) (6)
x̂2,k/k=x̂2,k/k-1-ρsgn(x̂2,k/k-1)+K2,k(yk-Hkx̂2,k/k-1) (7)

where K1,k and K2,k are the Kalman gains of individual filtering algorithm which are time-varying parameters to control the weight adjustment in both the equations; and ρ is a value greater than 1. Similarly, the estimation error covariance update equations are given in (8) and (9).

p̂1,k+1/k=Fkp̂1,k/k-1Fk*T+β2GkGkT-Fkp̂1,k/k-1[Hk*THk*T]Re,k-1HkHkp̂1,k/k-1Fk*T+Qk (8)
p̂2,k+1/k=Fkp̂2,k/k-1Fk*T+β2GkGkT-Fkp̂2,k/k-1[Hk*THk*T]Re,k-1HkHkp̂2,k/k-1Fk*T+Qk (9)

where β is a constant between 0 and 1; Fk is the state transition matrix; p̂1,k/k-1 and p̂2,k/k-1 are the initial estimation error covariances used in update equation for filters 1 and 2, respectively; Qk is the process noise covariance; Hk* is the conjugate of the measurement matrix; Re,k is the updated value of measurement covariance as shown in (10); and Gk=Hk*T.

Re,k=Rk+HkHkp̂1,k/k-1[Hk*THk*T]Rk=I00-γf2I (10)

where Rk is the initial value of measurement covariance; γf is the controlling parameter greater than 1; and I is the identity matrix. The net output across convex combination based sparse filtering algorithm is given as:

ŷk=λkŷ1,k/k+(1-λk)ŷ2,k/k (11)

where λk=sgm ak=1/(1+e-ak). The role of the sigmoid function is to restrict the value of λk between 0 and 1 using ak and reduce the gradient noise during error convergence [

18].

III. Model of Symmetrical Component Estimation

Mathematically, unbalanced three-phase quantities are expressed in terms of symmetrical components as shown in (12) [

10], [19]-[21].

vak=V0sin(ωkTs+ϕ0)+Vpsin(ωkTs+ϕp)+Vnsin(ωkTs+ϕn)vbk=V0sin(ωkTs+ϕ0)+Vpsin(ωkTs+ϕp-120°)+Vnsin(ωkTs+ϕn+120°)vck=V0sin(ωkTs+ϕ0)+Vpsin(ωkTs+ϕp+120°)+Vnsin(ωkTs+ϕn-120°) (12)

where Vak, Vbk, Vck are the voltage magnitudes of phases a, b, c, respectively; Vp, Vn, V0 are the positive, negative, and zero sequence components and ϕp, ϕn, ϕ0 are their respective phase angles, respectively; and ω, k, Ts are the angular frequency, discrete time instant and sampling time, respectively.

Using the equality, sinA+B=sin Acos A+cos Asin B, (12) can be expanded as (13) and the discrete measurement equation used in state space model can be represented as (14).

Vak=V0sin(ωkTs)cosϕ0+V0cos(ωkTs)sinϕ0+Vpsin(ωkTs)cos ϕp+Vpcos(ωkTs)sinϕp+Vnsin(ωkTs)cosϕn+Vncos(ωkTs)sinϕnVbk=V0sin(ωkTs)cosϕ0+V0cos(ωkTs)sinϕ0+Vpsin(ωkTs-120°)cosϕp+Vpcos(ωkTs-120°)sinϕp+Vnsin(ωkTs+120°)cosϕn+Vncos(ωkTs+120°)sinϕnVck=V0sin(ωkTs)cosϕ0+V0cos(ωkTs)sinϕ0+Vpsin(ωkTs+120°)cosϕp+Vpcos(ωkTs+120°)sinϕp+Vnsin(ωkTs-120°)cosϕn+Vncos(ωkTs-120°)sinϕn (13)
yk=Hkxk+vk (14)

where xk is the state vector. vk must be included as the overall transmission of the power signal is always corrupted by noise.

By using the Taylor’s series expansion measurement, Hk can be generated as given in (15) and the corresponding state vector xk can be expressed in (16).

Hk=sin(ωkTs)cos(ωkTs)sin(ωkTs)cos(ωkTs)sin(ωkTs)cos(ωkTs)sin(ωkTs)cos(ωkTs)sin(ωkTs-120°)cos(ωkTs-120°)sin(ωkTs+120°)cos(ωkTs+120°)sin(ωkTs)cos(ωkTs)sin(ωkTs+120°)cos(ωkTs+120°)sin(ωkTs-120°)cos(ωkTs-120°) (15)
xk=V0cos ϕ0V0sin ϕ0Vpcos ϕpVpsin ϕpVnsin ϕnVnsin ϕnT (16)

The state equation for the above model is given by xk+1=f(xk)+wk, where f(xk) is a functional mapping of state variable from the kth instant to the (k+1)th instant and can be represented as given in (17). This functional mapping is required to generate a state transition matrix, which is a derivative based matrix obtained by using Taylor’s series expansions. The amplitudes and phases of symmetrical components can be estimated by (18)-(23).

f(xk)=xk(1)xk(2)xk(3)xk(4)xk(5)xk(6) (17)
V̂0k=x̂k2(1)+x̂k2(2) (18)
V̂pk=x̂k2(3)+x̂k2(4) (19)
V̂nk=x̂k2(5)+x̂k2(6) (20)
ϕ̂0k=arctanx̂k2(2)x̂k2(1) (21)
ϕ̂pk=arctanx̂k2(4)x̂k2(3) (22)
ϕ̂nk=arctanx̂k2(6)x̂k2(5) (23)

IV. Frequency Estimation Model

To develop an adaptive frequency estimation model, Clarke transformation is used to generate a complex phasor through αβ components [

22]-[24]. The discrete form of voltages across three phases of unbalanced system can be expressed as:

vak=Vaksin(ωkTs+θ)vbk=Vbksin(ωkTs+θ-2π/3)vck=Vcksin(ωkTs+θ+2π/3) (24)

where ω=2πf, f is the system frequency; and θ is the initial phase angle. Using Clarke transformation, αβ components corresponding to (24) can be written mathematically as:

VαkVβk=231-1/2-1/203/2-3/2VakVbkVckT (25)

The measured complex signal generated through αβ components is corrupted by additive white Gaussian noise and can be expressed as [

25]:

yk=Vαk+jVβk=(A+jB)cos(ωkTs+θ)+(B+jC)sin(ωkTs+θ)+vk (26)

where A=2/3Vak+(Vbk+Vck)/26, B=(Vbk-Vck)/22, C=3/2(Vbk+Vck)/2.

To model the complex phasor, measurement equation of sparse H filtering algorithm can be used as (14) where the corresponding measurement matrix is given by Hk=0, 1. The state vector used in the estimation process is:

xk=xk(1)xk(2) (27)

where xk(1)=ejωTs; xk(2)=αkejωkTs+ϕ1; αk is the magnitude; and ϕ1 is the initial phase angle of the signal.

The functional mapping of state variable from the kth instant to the (k+1)th instant is:

f(xk)=xk(1)xk(1)xk(2) (28)

The state transition matrix can be obtained from functional mapping by using Taylor’s series expansion:

Fk=10xk(2)xk(1) (29)

Thus, the frequency can be estimated from updated state variables by using the mathematical equation as:

f̂k=Imlog xk(1)2πTs (30)

The state vector is updated by using the implementation steps of H filtering algorithm as given in Section II.

V. Simulation Results and Discussions

In the following analysis, different cases of unbalanced power signals are simulated and the corresponding comparison results are presented using sparse and non-sparse adaptive filtering algorithms including EKF, sparse EKF, H filtering algorithm and proposed sparse H filtering algorithm. For the simulation purpose, the controlling parameter δ and the parameter β in (8) are 0.98 and 0.10, respectively. The estimation error covariance matrix is also initialized as P0=ρI. The mixing parameter value λk is judiciously chosen between 0 and 1 to develop a convex combination. The response of the sparse H filtering algorithm is studied for different categories of PQ disturbances and comparison results are generated using MATLAB/Simulink.

A. Frequency Estimation of Three-phase Unbalanced Signals

For the purpose of simulation, three-phase voltage equations for a unbalanced system are considered as:

vak=Vakcos(2πfkTs+θ)vbk=Vbkcos(2πfkTs+θ-2π/3)vck=Vckcos(2πfkTs+θ+2π/3) (31)

The amplitudes of unbalanced three-phase signals are 1, 1.2 and 1.1 p.u. for phases a, b, and c, respectively. The signal frequency is chosen as 50 Hz. The complex phasor generated through αβ transform is corrupted by white Gaussian noise of 30 dB signal noise ratio (SNR). The comparison is made to test the estimation accuracy between different adaptive filtering algorithms and the frequency estimation plot is shown in Fig. 1(a). Figure 1(b) represents a comparison of mean square estimation error between the filtering algorithms. The results indicate that the sparse model-based H filtering algorithm has a faster error convergence compared to other filtering algorithms.

Fig. 1 Frequency estimation results. (a) Estimated frequency comparison. (b) Mean square error in frequency estimation. (c) Estimated frequency of power signal with step change in frequency. (d) Estimated frequency of power signal with ramp change in frequency.

Absolute frequency error is obtained by calculating the difference between the desired value and estimated filter output. Absolute frequency errors at different noise levels using different algorithms and the execution time are listed in Table I. Although EKF, sparse EKF and H filtering algorithms are capable of estimating frequency components, their performance deteriorates under high-noise conditions.

Table I Performance Comparison of Different Algorithms
AlgorithmFrequency estimation error (Hz)Execution time (ms)
20 dB30 dB40 dB
EKF 0.2020 0.151 0.0832 1.530
Sparse EKF 0.1425 0.107 0.0741 0.732
H 0.0860 0.035 0.0120 0.596
Sparse H 0.0310 0.008 0.0010 0.646

A step change in frequency is introduced between 0.02 s and 0.04 s to the power signal of fundamental frequency of 50 Hz. In Fig. 1(c), it is observed that sparse H filtering algorithm tracks the frequency variation more accurately than other algorithms.

Further, a ramp change in frequency from 55 Hz to 45 Hz is considered between 0.3 s to 0.5 s and then the frequency remains constant at 45 Hz for the rest of the time in the unbalanced three-phase signal which is expressed in (31). It is observed that the proposed method is efficiently tracking the ramp change of signal with convergence time less than a half cycle and has good estimation accuracy as shown in Fig. 1(d). In the case of Hfiltering algorithm, the tracking starts beyond 0.3 s, i.e., around 0.48 s, which leads to the wrong result. Similarly, sparse EKF tracks the ramp variation from 40 Hz instead of 55 Hz, which is clearly visible from the figure.

B. Symmetrical Component Estimation of Unbalanced Signals

For the estimation of sequence components, the voltage signals for an unbalanced three-phase system are considered in (32). The estimation of zero, positive and negative sequence components are being carried out using state space modeling as explained in Section III.

vak=100sin(ωkTs+30°)vbk=50sin(ωkTs+300°)vck=30sin(ωkTs+180°) (32)

The steady-state errors for symmetrical components are shown in Fig. 2. It is observed from Fig. 2 that the convergence time of sparse H filter is less than 2.5 ms, which is the least.

Fig. 2 Estimated steady-state errors. (a) Zero-sequence component. (b) Positive-sequence component. (c) Negative-sequence component.

Table II gives the absolute information of steady-state errors under different noise conditions. It is observed that the proposed model has lesser error value even under high-noise conditions such as 20 dB SNR.

Table II Absolute Steady-state Error
AlgorithmZero-sequence estimation error (V)Positive-sequence estimation error (V)Negative-sequence estimation error (V)
20 dB30 dB40 dB20 dB30 dB40 dB20 dB30 dB40 dB
EKF 0.0140 0.0139 0.0089 0.0151 0.0055 0.00380 0.0162 0.0146 0.0144
Sparse EKF 0.0136 0.0092 0.0051 0.0046 0.0037 0.00210 0.0149 0.0118 0.0071
H 0.0137 0.0109 0.0061 0.0087 0.0053 0.00230 0.0158 0.0121 0.0065
Sparse H 0.0102 0.0032 0.0022 0.0033 0.0021 0.00018 0.0073 0.0035 0.0011

C. Symmetrical Component Estimation of Unbalanced Signals with Voltage Sag

The three-phase unbalanced signal is now considered with sag condition as shown in (32). The sag is introduced in a non-uniform manner from 0.050 s to 0.065 s across all the phases with 30% in phase a, 20% in phase b and 66% in phase c, respectively, which strictly follows IEEE standard (10%-90%).

In the case of zero--sequence and positive-sequence components, the disturbance caused by the sag is tracked with greater accuracy with the proposed sparse H filtering algorithm as shown in Fig. 3(a) and (b).

Fig. 3 Estimated amplitude. (a) Zero-sequence component. (b) Positive-sequence component. (c) Negative-sequence component.

The same has also been reflected through the estimation of negative-sequence components shown in Fig. 3(c), but the magnitude is less than the other two sequence components. In this case, the proposed algorithm has a superior performance.

D. Symmetrical Component Estimation of Harmonic Distorted Unbalanced Signals

The proposed model is tested considering unbalanced three-phase voltages which are contaminated with the 3rd harmonics. Its magnitudes equal to half those of the corresponding fundamental frequency components shown as:

vak=100sin(ωkTs+30°)+50sin(ωkTs+20°)vbk=50sin(ωkTs+300°)+25sin(ωkTs+100°)vck=30sin(ωkTs+180°)+15sin(ωkTs+70°) (33)

Table III shows the estimated fundamental and third harmonic magnitudes along with their exact values. It is quite apparent from the results that the estimation errors lie in the range of 10-3, whereas for other filters, the difference between the actual value and estimated value lies in the range of 10-2. Also, it is observed from the results that the symmetrical components of fundamental and harmonics are estimated independently, i.e., the 3rd harmonics have no effect on the estimations.

Table III Sequence Components Estimation with Harmonics
AlgorithmFundamental amplitude (V)The 3rd harmonic amplitude (V)
Zero-sequencePositive-sequenceNegative-sequenceZero-sequencePositive-sequenceNegative-sequence
EKF 27.2774 57.9842 18.9857 24.4484 13.0195 18.6931
Sparse EKF 27.2759 57.9822 18.9832 24.4518 13.0145 18.6910
H 27.2756 57.9802 18.9841 24.4533 13.0143 18.6908
Sparse H 27.2828 57.9719 18.9845 24.4610 13.0100 18.6881
Actual 27.2926 57.9709 18.9872 24.4894 12.9682 18.6735

E. Unbalanced Signal Estimation of Unbalanced Signals with Time-varying Amplitude

Three-phase unbalanced signals with time-varying amplitudes are taken into consideration in (34). The time variation in amplitudes is incorporated through slowly-varying sinusoidal components of frequency 1, 3 and 6 Hz, respectively.

vak=A1(kTs)sin(ωkTs+π/6)vbk=A2(kTs)sin(ωkTs+π/3)vck=A3(kTs)sin(ωkTs+π) (34)
A1(kTs)=0.15sin(2πf1kTs)+0.05sin(2πf5kTs)A2(kTs)=0.05sin(2πf3kTs)+0.02sin(2πf5kTs)A3(kTs)=0.025sin(2πf1kTs)+0.005sin(2πf5kTs) (35)

Comparison of estimated signals for three phases using different algorithms is shown in Fig. 4(a)-(c).

Fig. 4 Estimated signal and mean square error. (a) Phase a. (b) Phase b. (c) Phase c. (d) Mean square error.

Though all algorithms track the time-varying signals, the proposed algorithm estimates the amplitude parameters more accurately than others. Mean square error plot is shown in Fig. 4(d). Average mean square error values of all phases for all algorithms are tabulated in Table IV. It is observed that mean square error is in the order of 10-7 with the proposed algorithm, which is much less than other algorithms.

Table IV Average Mean Square Estimation Error
AlgorithmMean square estimation error (p.u)
EKF 1.06×10-2
Sparse EKF 2.60×10-3
H 9.53×10-4
Sparse H 6.44×10-7

F. Estimation of Unbalanced Signals with IEEE Database

Using IEEE-1159-PQE database, three-phase unbalanced signals are generated as shown in Fig. 5(a) [

26]. For the purpose of testing, reconstructed signals obtained by all algorithms are compared with the original signals.

Fig. 5 Signals. (a) From IEEE database. (b) Estimated in phase a. (c) Estimated in phase b. (d) Estimated in phase c.

The respective comparison plots for the individual phases are shown in Fig. 5(b)-(d). Also, the mean square value of the difference between the desired response and estimated filter output is calculated and tabulated in Table V. In Table V, it is quite apparent that the proposed model convergence behavior has substantially improved.

Table V Mean Square Estimation Error of Signals in Different Phases
AlgorithmMean square estimation error (p.u.)
Phase aPhase bPhase c
EKF 2.5000×103 1.3000×103 2.4000×103
Sparse EKF 1.1000×103 3.3031×103 5.2126×103
H 9.4723×106 6.5344×105 5.3988×105
Sparse H 2.4672×108 4.0371×107 5.0840×107

G. Experimental Studies and Results

To validate the performance of the proposed algorithm for the estimation of harmonics, an experimental setup is taken as shown in Fig. 6 in the laboratory. The setup uses the voltage source inverter (VSI) with three-phase 180° conduction mode.

Fig. 6 Experimental setup for real data generation.

Unbalanced resistive loads of 100 W, 40 W, 200 W are connected to phases a, b, and c, respectively. The specifications of the system are: ① DC supply: 75 V; ② inverter: three-phase, 180° mode VSI (Frax); ③ load: 100W , 40 W, 200 W; ④ digital storage oscilloscope: 100 MHz, sample rate 50 ks/s, four channels, personal computer (PC) connectivity-universal serial bus port; ⑤ PC: 2.4 GHz, 2 GB RAM.

The load voltage waveforms are stored in a digital storage oscilloscope (Gwinstek) across the load resistors and then acquired data are transferred to PC through communication software.

Figure 7 represents the signal generated in all three phases of the experimental setup. Figure 8 shows the estimation of voltage signals across all the phases employing different algorithms using the data obtained from the setup. It is observed that the proposed algorithm estimates very close to the actual signal than other algorithms.

Fig. 7 Signal generated from experimental setup.

Fig. 8 Estimation of voltage signals. (a) Phase a. (b) Phase b. (c) Phase c.

The performance of the algorithms which gives the accuracy of estimation is estimated by:

p=k=1N(yk-ŷk)2k=1Nŷk2×100% (35)

The less the value of p, the more is the estimation accuracy. The performance of the algorithms is shown in Table VI.

Table VI Comparison of Performance Indices of Practical Data
AlgorithmParameter p
EKF 1.5900
Sparse EKF 0.3200
H 0.0301
Sparse H 0.0118

Along with the above performance index, the mean square estimation error is also calculated to validate the performance of the proposed model in Table VII. It is seen from the values that the proposed method achieves more significant estimation accuracy than others.

Table VII Mean Square Estimation Error
AlgorithmMean square estimation error (p.u.)
Phase aPhase bPhase c
EKF 6.2130×104 1.196×105 1.2443×104
Sparse EKF 3.0234×106 1.357×106 1.5700×105
H 1.9262×107 1.710×107 2.3300×107
Sparse H 2.1630×109 1.710×107 8.9829×109

VI. Conclusion

The proposed sparse model is tested using three-phase unbalanced voltages to estimate sequence components and frequency similar to practical systems. The estimation error is bounded due to norm penalty introduced in the cost function. The most important advantage of this method lies in developing a general-purpose processing unit for three-phase unbalanced conditions with less computation burden. The proposed model is also able to resolve the three-phase voltages in terms of sequence components and the estimated parameters across all the phases can provide accurate information about the dominant harmonics presenting in three-phase power system. With the proper knowledge about the order of harmonics, harmonic elimination filter can be designed for power system protection.

Acknowledgements

The authors acknowledge the support of Indian Institute of Information Technology, Bhubaneswar, India, and Veer Surendra Sai University of Tecnology (Burla), Sambalpur, India, in terms of Laboratory and online Journal facilities to carry out this research work.

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