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.
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 [
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 [
Artificial intelligence and soft computing techniques such as neural network, fuzzy logic, and genetic algorithm are quite popular for tracking PQ disturbances [
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 [
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.
In this proposed model, l1 norm penalty is introduced to the quadratic cost function of H∞ filter [
(1) |
where denotes the l1 norm of a vector; 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) [
(2) |
(3) |
(4) |
(5) |
where is the output vector generated through non-sparse filter estimation during the
Two state update equations for convex combination are given in (6) and (7).
(6) |
(7) |
where and are the Kalman gains of individual filtering algorithm which are time-varying parameters to control the weight adjustment in both the equations; and is a value greater than 1. Similarly, the estimation error covariance update equations are given in (8) and (9).
(8) |
(9) |
where is a constant between 0 and 1; is the state transition matrix; and are the initial estimation error covariances used in update equation for filters 1 and 2, respectively; is the process noise covariance; is the conjugate of the measurement matrix; is the updated value of measurement covariance as shown in (10); and .
(10) |
where is the initial value of measurement covariance; is the controlling parameter greater than 1; and is the identity matrix. The net output across convex combination based sparse filtering algorithm is given as:
(11) |
where . The role of the sigmoid function is to restrict the value of between 0 and 1 using and reduce the gradient noise during error convergence [
Mathematically, unbalanced three-phase quantities are expressed in terms of symmetrical components as shown in (12) [
(12) |
where Vak, Vbk, Vck are the voltage magnitudes of phases a, b, c, respectively; , , are the positive, negative, and zero sequence components and , , are their respective phase angles, respectively; and , , are the angular frequency, discrete time instant and sampling time, respectively.
Using the equality, , (12) can be expanded as (13) and the discrete measurement equation used in state space model can be represented as (14).
(13) |
(14) |
where is the state vector. must be included as the overall transmission of the power signal is always corrupted by noise.
By using the Taylor’s series expansion measurement, can be generated as given in (15) and the corresponding state vector can be expressed in (16).
(15) |
(16) |
The state equation for the above model is given by , where is a functional mapping of state variable from the th instant to the 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).
(17) |
(18) |
(19) |
(20) |
(21) |
(22) |
(23) |
To develop an adaptive frequency estimation model, Clarke transformation is used to generate a complex phasor through αβ components [
(24) |
where , is the system frequency; and is the initial phase angle. Using Clarke transformation, αβ components corresponding to (24) can be written mathematically as:
(25) |
The measured complex signal generated through αβ components is corrupted by additive white Gaussian noise and can be expressed as [
(26) |
where , , .
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 . The state vector used in the estimation process is:
(27) |
where ; is the magnitude; and ϕ1 is the initial phase angle of the signal.
The functional mapping of state variable from the th instant to the th instant is:
(28) |
The state transition matrix can be obtained from functional mapping by using Taylor’s series expansion:
(29) |
Thus, the frequency can be estimated from updated state variables by using the mathematical equation as:
(30) |
The state vector is updated by using the implementation steps of H∞ filtering algorithm as given in Section II.
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 . The mixing parameter value 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.
For the purpose of simulation, three-phase voltage equations for a unbalanced system are considered as:
(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 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
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
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
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.
(32) |
The steady-state errors for symmetrical components are shown in

Fig. 2 Estimated steady-state errors. (a) Zero-sequence component. (b) Positive-sequence component. (c) Negative-sequence component.
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 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
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:
(33) |
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.
(34) |
(35) |
Comparison of estimated signals for three phases using different algorithms is shown in

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
Using IEEE-1159-PQE database, three-phase unbalanced signals are generated as shown in

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
To validate the performance of the proposed algorithm for the estimation of harmonics, an experimental setup is taken as shown in

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.

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:
(35) |
The less the value of , the more is the estimation accuracy. The performance of the algorithms is shown in
Along with the above performance index, the mean square estimation error is also calculated to validate the performance of the proposed model in
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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