Robust estimation of power quality disturbances using unscented H∞ filter

Electrical Power and Energy Systems 73 (2015) 438–447

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Robust estimation of power quality disturbances using unscented H1 filter Harish K. Sahoo a,⇑, P.K. Dash b a b

Department of Electronics and Telecommunication, IIIT Bhubaneswar, A University Established by Govt. of Odisha, India MDRC, SOA University, Bhubaneswar, India

a r t i c l e

i n f o

Article history: Received 31 August 2014 Received in revised form 8 May 2015 Accepted 9 May 2015

Keywords: Unscented transform H1 filter Kalman filter RLS Additive white Gaussian noise State space model

a b s t r a c t This paper proposes a novel adaptive filtering method for tracking the power quality disturbances present in distorted power signals. The proposed filter known as unscented H1 filter (UHF) is the robustification of unscented Kalman filter (UKF) and is based on state space modeling. The performance of unscented H1 filter has been compared with other adaptive filters considering signals, which can represent worst case measurement and network conditions in a typical power system. State space modeling is used to estimate power quality disturbances like sag, swell, notch in presence of additive white Gaussian noise (AWGN). Also the amplitudes and phases of different harmonics under high noisy conditions and decaying DC are also estimated which shows the robustness of the filter. Comparison results demonstrate that under identical conditions, the performance of UHF is better compared to UKF. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction Power quality has been an issue of growing concern amongst a broad spectrum of power customers over the past few years. Electric utilities are becoming more concerned about power system harmonics and voltage distortion. This increased concern is due to the increase in the application of power electronic devices in almost all kind of operation. The power system components continuously inject varying harmonics in the system giving rise to non stationary harmonics voltages and currents in the distribution system. These disturbances cause problems, such as overheating, equipment failures, inaccurate metering and malfunctioning of protective equipments. Some of the major factors, which contribute towards deteriorating power quality, are voltage sags, swells and the presence of harmonics. A voltage dip implies that the required energy is not being delivered to the load and this can have serious consequences depending on the type of the load involved, which sometimes lead to power service outage. The presence of harmonics often cause interference in communication circuits, over heating of magnetic portions of electrical systems, resonance of mechanical devices etc. Detection and subsequently elimination of harmonics using suitable harmonic filter as well as prediction of voltage dips has therefore been a major research

⇑ Corresponding author. E-mail address: [email protected] (H.K. Sahoo). http://dx.doi.org/10.1016/j.ijepes.2015.05.031 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

concern of power engineers in recent years. Adaptive parametric estimation techniques [29] are popular approaches that have been widely used not only for estimation of non-stationary signal parameters but also to identify unknown nonlinear dynamic plants. Various estimation methods have also been proposed in each field such as linear-prediction (LP)-based methods [1] like forward–backward LP (FBLP) method, singular value decomposition (SVD), iterative filtering algorithm (IFA). All the above-mentioned methods have been basically developed only for frequency estimation. Therefore, they have not been able to directly estimate a sinusoidal signal itself (or amplitude and phase of each component) from the observed data [2,3]. In order to provide the quality of the delivered power, it is imperative to know the harmonics parameters such as magnitude and phase. This is essential for designing filter for eliminating or reducing the effects of harmonics in a power system. Adaptive algorithms based on Fourier analysis and linear combiners are proposed to evaluate the harmonics and flicker magnitude [4–6] in a power system. Online tracking of harmonics in power systems using variable gain filtering approach was described by Girgis et al., which was still unable to track abrupt changes of signals. Recently developments of soft computing techniques [13,14] have encouraged the researchers to use these methods for harmonic estimation. Since estimation of harmonic parameters is a nonlinear problem, Genetic Algorithm (GA), being a heuristic and stochastic global searching algorithm [7,8] was used for estimation. But GA suffers from larger time

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requirement for convergence when estimating multiple frequency components because of large number of parameters should be identified simultaneously. Due to different quantitative values of amplitudes and phases of harmonics, it is difficult to get a homogeneous genetic pool with respect to the final solution. LAV state estimation based algorithm [9] is also proposed for measurement of voltage flicker. Wavelet and S-transform [10–12] are effectively used to detect and classify power quality events Forgetting Factor Recursive Least Square (FFRLS) approach [15] is also used for estimating not only voltage sag, swell, momentary interruption but also the amplitudes and phases of harmonics in case of time varying power signals. But RLS has poor convergence properties in case of time-varying and non-stationary environments. On the other hand, extended Kalman filter (EKF) has often been applied to frequency estimation. Although it can estimate the amplitudes and phases of frequency components, the non- linearity could cause the EKF to diverge in some poorly initial conditions. However, the EKF approach [16–20] has the advantages that the estimates are computed recursively and that it can cope easily with time variation in the signal parameters, whereas the SVD or MLE approach assumes that the parameters remain constant in time. Unscented Kalman filter [24–26] is based on the unscented transformation to propagate the sigma points and the calculation of jacobian matrix can be avoided as done in case of EKF. In contrast to Kalman filter which requires exact and accurate system model as well as perfect knowledge of the noise statistics, the H1 filter [21–23] requires no aprior information of the noise statistics but finite bounded energies. Unscented H1 filter (UHF) [27,28] is based on state-space representation of noisy sinusoidal signal and passing the sigma points through unscented transformation. But the filter performs better than UKF under high noisy condition due to auto tuning of measurement error covariance when state variables of the signal model are updated. The paper is organized as follows: Section ‘Signal modeling for power quality estimation’ develops signal model in state space for power quality estimation. The nonlinear unscented H1 filtering algorithm is presented in Section Unscented H1 filtering algorithm. Section ‘Stabil ity analysis of unscented filter’ demonstrates the stability of the proposed filtering approach. The simulation results of the proposed filter are presented in Section ‘Simulation results’ and conclusion is presented in Section ‘Conclusion’.

Rk ¼ E½v k v T k , where ⁄ means the complex conjugate and T is the transpose. For notice of the fundamental frequency, here we consider a single complex sinusoid zk with angular frequency x1 in the presence of white noise. The state variables used for noisy signal modeling are given as:

x1k ¼ cosðkxk T s þ /k Þ x2k ¼ sinðkxk T s þ /k Þ x3k ¼ xk T s x4k ¼ Ak The measured value of the signal can then be written in state space form as:

State equation xkþ1 ¼ f ðxk Þ þ xk

ð3Þ

Measurement equation yk ¼ hðxk Þ þ v k

ð4Þ

where state vector is given by

xk ¼ ½ x1k

x2k

x3k

x4k T

ð5Þ

The functions used for unscented transformation of sigma points are given as

2

x1k cosðx3k Þ  x2k sinðx3k Þ

3

6 x sinðx Þ  x cosðx Þ 7 3k 2k 3k 7 6 1k f ðxk Þ ¼ 6 7 5 4 x3k

ð6Þ

x4k hðxk Þ ¼ x1k x4k þ v k

ð7Þ

Signal model for harmonic estimation Harmonics are electric voltages and currents that appear on the electric power system as a result of non-linear electric loads. Three test signals are considered for estimation, which contains higher order harmonics and also slowly decaying dc component. The test signal 2 containing the third and fifth order harmonics is considered as:

zk ¼ a1 sinðkw1 T s þ /1 Þ þ a3 sinðkw3 T s þ /3 Þ þ a5 sinðkw5 T s þ /5 Þ ð8Þ

Signal modeling for power quality estimation

The test signal 3 is given by:

Two types of signal models are proposed to estimate power quality disturbances like voltage sag and swell, notch, momentary interruption as well as amplitudes and phases of different harmonics like fundamental, third and fifth harmonics.

ð9Þ

where

a3 ðkÞ ¼ 0:05 sin 2pf 3 kT s þ 0:02 sin 2pf 5 kT s

Consider a signal yk at time k is a sinusoid zk in the presence of white Gaussian noise v k :

ð1Þ

a5 ðkÞ ¼ 0:025 sin 2pf 1 kT s þ 0:005 sin 2pf 5 kT s The test signal 4 is given by

zk ¼ a1 ðkÞ sinðkxT s þ /1 Þ þ a3 ðkÞ sinð3kxT s þ /3 Þ þ a5 ðkÞ  sinð5kxT s þ /5 Þ þ aDC expðaDC kT s Þ

where

zk ¼ a1 sinðkx1 T s þ /1 Þ

þ ð0:2 þ a5 ðkÞÞ sinð5kwT s þ /5 Þ þ aDC expðaDC kT s Þ

a1 ðkÞ ¼ 0:15 sin 2pf 1 kT s þ 0:05 sin 2pf 5 kT s

Signal model for power quality disturbances

y k ¼ zk þ v k

zk ¼ ð1:5 þ a1 ðkÞÞ sinðkwT s þ /1 Þ þ ð0:5 þ a3 ðkÞÞ sinð3kwT s þ /3 Þ

ð2Þ

and

x1 = fundamental of angular frequency. /1 = fundamental of phase angle. a1 = fundamental amplitude of the signal. The observation noise vk is a Gaussian white noise with zero mean and variance r2v . The covariance of measured errors is

ð10Þ

where

a1 ðkÞ ¼ 8 expða1 kT s Þ a3 ðkÞ ¼ 1:5 expða1 kT s Þ a5 ðkÞ ¼ 0:75 expða1 kT s Þ The distorted power signal in presence of harmonics as mentioned in Eq. (8) can be modeled in a state space form in different ways. But here the main concern is to estimate the amplitude as

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well as the phase of fundamental, third and fifth harmonic components. The state variables used in state space modeling are given by:

x1k ¼ a1 ðkÞ x2k ¼ a3 ðkÞ x3k ¼ a5 ðkÞ x4k ¼ kx1 T s x5k ¼ kx3 T s

ð11Þ

x6k ¼ kx5 T s

X 0k1jk1 ¼ ^xk1jk1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X sk1jk1 ¼ ^xk1jk1 þ ð ðn þ kÞPk1jk1 Þs; s ¼ 1; . . . ; n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X sk1jk1 ¼ ^xk1jk1  ð ðn þ kÞPk1jk1 Þs; s ¼ n þ 1; . . . ; 2n

ð15Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where k 2 R a scaling is factor and ðn þ kÞP k1jk1 is the sth row or column of the matrix square root of ðn þ kÞPk1jk1 . ws is the normalized weight associated with the sth sigma point. Note that Cholesky decomposition is required for the computation of the matrix square root. Step-2: Computation of predicted mean and covariance

x7k ¼ /1 ðkÞ x8k ¼ /3 ðkÞ x9k ¼ /5 ðkÞ The sigma points are passed through nonlinear functions f ð:Þ and hð:Þ; which can be expressed as:

3 x1k 7 6x 7 6 2k 7 6 7 6 x3k 7 6 7 6 6 x4k þ x1 T s 7 7 6 7 f ðxk Þ ¼ 6 6 x5k þ x3 T s 7 7 6 6 x6k þ x5 T s 7 7 6 7 6 x7k 7 6 7 6 5 4 x8k x9k 2

X sk1jk1 ¼ f ð^xk1jk1 Þ

^xkjk1 ¼

2n X ws X skjk1

ð17Þ

ð12Þ Pkjk1 ¼

2n X T ws ½X skjk1  ^xkjk1 ½X skjk1  ^xkjk1  þ Q k

ð18Þ

s¼0

where Q k is the process noise covariance matrix. The weights are defined as:

w0 ¼

k nþk

ws ¼

k ; s ¼ 1; . . . ; 2n 2ðn þ kÞ

ð13Þ

The modeling of signal given in Eqs. (9) and (10) are generalized taking eleven states considering decaying DC component. The state variables can be represented as:

ð19Þ

Step-3: Computation of measurement and cross-correlation covariance

x1k ¼ a1 ðkÞ x2k ¼ a3 ðkÞ x3k ¼ a5 ðkÞ

The updated equations can be reformulated by using the statistical linear error propagation method. In other words, the measurement covariance and its cross-correlation covariance can be approximated by

x4k ¼ kx1 T s x5k ¼ kx3 T s x6k ¼ kx5 T s

ð16Þ

s¼0

hðxk Þ ¼ x1k sinðx4k þ x7k Þ þ x2k sinðx5k þ x8k Þ þ x3k sinðx6k þ x9k Þ þ v k

The predicted mean and covariance can be obtained by implementing the unscented transform [14]. The unscented transformation function is applied on the prior calculated sigma points to obtain the transformed sigma points that help in calculation of predicted mean and covariance.

ð14Þ

x7k ¼ /1 ðkÞ x8k ¼ /3 ðkÞ

Pyy kjk1 ¼

x11k ¼ aDC kT s Unscented H‘ filtering algorithm Unscented H1 filter is the modification of unscented Kalman filter (UKF) and updating the estimation error covariance through auto adjustment of measurement error covariance by selecting a proper scalar parameter c as shown in Eqs. (24) and (25). Step-1: Selection of sigma points A set of sigma points are generated based on the state estimates at time k  1, considering the assumption that the state estimate ^ xk1jk1 and the associated covariance Pk1jk1 have been derived at the time k  1 .

ð20Þ

s¼0

x9k ¼ /5 ðkÞ x10k ¼ aDC

2n X ^kjk1 ½hðX sk1jk1 Þ  y ^kjk1 T ws ½hðX sk1jk1 Þ  y

Pxy kjk1 ¼

2n X ^kjk1 T ws ½hðX sk1jk1 Þ  ^xkjk1 ½hðX sk1jk1 Þ  y

ð21Þ

s¼0

^kjk1 ¼ with y

2n X ws hðX skjk1 Þ

ð22Þ

s¼0

Step-4: Formulating the filtered estimates Finally the posterior state estimate and its error covariance matrix can be given by the same function as that in UKF algorithm: 1 yy ^xkjk ¼ ^xkjk1 þ Pxy ^ kjk1 ðR þ P kjk1 Þ ðyk  ykjk1 Þ

2 Pkjk ¼ Pkjk1 

1 4 ½Pxy kjk1 P kjk1 Re;k

½P xy kjk1  PTkjk1

T

ð23Þ

3 5

ð24Þ

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H.K. Sahoo, P.K. Dash / Electrical Power and Energy Systems 73 (2015) 438–447

2

where

" Re;k ¼

R þ Pyy kjk1

Pxy kjk1

Pxy kjk1

c2k I þ Pkjk1

# ð25Þ

Note that the parameter c should be set carefully to guarantee the positiveness of P kjk and hence the existence of H1 filter. By applying the matrix inversion lemma, it can be easily shown that

P1 kjk

¼

P 1 kjk1

þ

HTk R1 HK

 c2 K I > 0

ð26Þ

Rearranging above terms yields

n

1

T 1 c2k > max eigðP1 kjk1 þ Hk R H K Þ

o

ð27Þ

where maxfeigðAÞ1 g denotes the maximum eigen value of the matrix A. Thus ck can be chosen as:

n o 1 T 1 c2k ¼ a max eigðP1 kjk þ Hk R HK Þ

and the transformed equation is:



r^ 2kjk1 ð1Þ ckjk1 ð1Þ ckjk1 ð2Þ 6 7 7 ^2 ¼6 4 ckjk1 ð1Þ rkjk1 ð2Þ ckjk1 ð3Þ 5 2 ckjk1 ð2Þ ckjk1 ð3Þ r^ kjk1 ð3Þ

h

1 xy 1 P1 P xy c2k ¼ a max eig P1 kjk kjk þ P kjk P kjk1 R kjk1

iT 

ð29Þ

ð35Þ

The filter gain of the proposed filter from Eq. (23) can be given as 1

yy K k ¼ Pxy kjk1 ðRk1 þ P kjk1 Þ

ð36Þ

Using matrix inversion lemma, the gain can also be expressed in an alternative form as

b kjk1 HT ÞðHk P b kjk1 HT þ Rk1 Þ K k ¼ ðP k k

ð37Þ

From

2

^xkjk ð2Þr ^ 2kjk1 ð1Þ þ ^xkjk ð1Þckjk1 ð1Þ  ckjk1 ð2Þ

3

7 6 7 ^ ^ ^2 K k ¼ g2k 6 4 xkjk ð2Þckjk1 ð1Þ þ xkjk ð1Þrkjk1 ð2Þ  ckjk1 ð3Þ 5

ð38Þ

^xkjk ð2Þckjk1 ð2Þ þ ^xkjk ð1Þckjk1 ð3Þ  r ^ 2kjk1 ð3Þ

ð28Þ

xy According to the above equations, HTk ¼ P 1 kjk P kjk1 is substituted



b kjk1 P

3

^ 2kjk1 ðÞ and ckjk1 ðÞ represent the diagonal and off-diagonal where r ^ kjk1 ; respectively elements of R

1 ! ^ 2kjk1 ð1Þ þ ^x2kjk ð1Þr ^ 2kjk1 ð2Þ 1 þ ^x2kjk ð2Þr

g2k ¼

ð39Þ

þ2^xkjk ð1Þ^xkjk ð2Þckjk1 ð1Þ  z Stability analysis of unscented filter

and g2k > 0

In this section, the stability of the proposed nonlinear filter is analyzed considering the case of three state variables. The estimation error covariance matrix dimension will increase if more number of state variables is used for signal modeling. If the covariance matrix due to estimation error converges to zero as time index k tends to infinity, then the proposed filter can be considered to be stable. Considering nonlinear discrete time systems given in Eqs. (3) and (4) representing state and measurement vectors at time k, with xk and v k as the uncorrelated zero mean white Gaussian noise having covariances Q k and Rk respectively. The estimation and prediction errors are given as

^ 2kjk1 ð3Þ z ¼ 2^xkjk ð2Þckjk1 ð2Þ þ 2^xkjk ð1Þckjk1 ð3Þ  r

ð40Þ

From Eq. (35)

2

3

r^ 2kjk ð1Þ ckjk ð1Þ ckjk ð2Þ 6 7 b kjk1 ¼ 6 c ð1Þ r ^ 2kjk ð2Þ ckjk ð3Þ 7 P 4 kjk 5 ckjk ð2Þ ckjk ð3Þ r^ 2kjk ð3Þ

ð41Þ

This leads to

^ 2kjk ð1Þ

r

^ 2kjk1 ð1Þ  g2k a2 ¼r

ð42Þ

~xk ¼ xk  ^xk

ð30Þ

r^ 2kjk ð2Þ ¼ r^ 2kjk1 ð2Þ  g2k b2

ð43Þ

~xkjk1 ¼ xk  ^xk

ð31Þ

r^ 2kjk ð3Þ ¼ r^ 2kjk1 ð3Þ  g2k c2

ð44Þ

xk1 gives Expanding xk by a Taylor series expansion about ^

1 xk ¼ f ð^xk1 Þ þ rf ð^xk1 Þ~xk1 þ r2 f ð^xk1 Þ~x2k1 þ . . . . . . þ xk 2

ð32Þ

^ Similarly expanding ^ x k by Taylor series about xk1 gives



 L h X pffiffiffiffiffiffiffiffiffi i L k f ð^xk1 Þ þ  f ^xk1 þ c Pk1 i Lþk 2ðL þ kÞ i¼1

þ

1.4 1.2

2L h X pffiffiffiffiffiffiffiffiffi i k f ^xk1 þ c Pk1  iL 2ðL þ kÞ i¼Lþ1

1 ¼ f ð^xk1 Þ þ r2 f ð^xk1 ÞP k1 þ . . . . . . : 2

ð33Þ

In case of UHF algorithm, predicted error covariance can be calculated as

b kjk1 ¼ P

2n X T ws ½X skjk1  ^xkjk1 ½X skjk1  ^xkjk1  þ Q k1

Original RLS EKF UKF UHF

1.6

Amplitude [p.u]

^xk ¼

Estimated Amplitude for sag & swell

1.8

1 0.8 0.6 0.4

ð34Þ

s¼0

where Q k1 is the covariance term due to process noise. As the error covariance matrix is a positive definite matrix of dimension 3  3 considering three state variables, the matrix can be represented as

0.2 0 0

100

200

300

400

500

600

700

800

900

1000

No.of iteration (N) Fig. 1a. Estimated amplitude with 50% voltage sag and 50% voltage swell.

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H.K. Sahoo, P.K. Dash / Electrical Power and Energy Systems 73 (2015) 438–447

ckjk ð1Þ ¼ ckjk1 ð1Þ  g2k ab

ð45Þ

ckjk ð2Þ ¼ ckjk1 ð2Þ  g2k ac

ð46Þ

The error covariance matrix for k þ 1 th instant can be given as

b kþ1jk P

2 k bc

ckjk ð3Þ ¼ ckjk1 ð3Þ  g

3 r^ 2kþ1jk ð1Þ ckþ1jk ð1Þ ckþ1jk ð2Þ 7 6 7 ^2 ¼6 4 ckþ1jk ð1Þ rkþ1jk ð2Þ ckþ1jk ð3Þ 5 2 ^ ckþ1jk ð2Þ ckþ1jk ð3Þ rkþ1jk ð3Þ 2

ð47Þ

ð48Þ

where,

^ 2kjk1 ð1Þ þ ^xkjk ð1Þckjk1 ð1Þ  ckjk1 ð2Þ a ¼ ^xkjk ð2Þr ^ 2kjk1 ð2Þ þ ^xkjk ð2Þckjk1 ð1Þ  ckjk1 ð3Þ b ¼ ^xkjk ð1Þr ^ 2kjk1 ð3Þ: c ¼ ^xkjk ð2Þckjk1 ð2Þ þ ^xkjk ð1Þckjk1 ð3Þ  r

^ 2kþ1jk ð1Þ ¼ r ^ 2kjk ð1Þ where r

ð49Þ

r^ 2kþ1jk ð2Þ ¼ ^xkjk ð2Þð^xkjk ð2Þr^ 2kjk ð1Þ þ ^xkjk ð1Þckjk ð1Þ  ckjk ð2ÞÞ ^ 2kjk ð2Þ  ckjk ð3ÞÞ þ^xkjk ð1Þð^xkjk ð2Þckjk ð1Þ þ ^xkjk ð1Þr ^ 2kjk ð3ÞÞ ð^xkjk ð2Þckjk ð2Þ þ ^xkjk ð1Þckjk ð3Þ  r

ð50Þ

Absolute Error Plot For Sag & swell 0.9

Absolute Error Plot for Sag & Notch 1.2

0.8

RLS RLS

EKF

0.7

1

UKF

EKF UKF

UHF

0.6

UHF

0.5

Absolute Error

Absolute Error

0.8

0.4 0.3

0.6

0.4

0.2 0.2

0.1 0

0

-0.1 0

100

200

300

400

500

600

700

800

900

1000

-0.2

No. of iterations (N)

0

100

200

300

400

500

600

700

800

900

1000

No. of iterations (N)

Fig. 1b. Absolute error plot.

Fig. 1d. Absolute error plot.

Estimated amplitude for Sag & Notch 3

Estimated amplitude for Swell & Momentary Interuption 1.8

Original RLS

1.6

EKF

2.5

UKF

1.4

UHF 1.2

Amplitude [p.u]

Amplitude [p.u]

2

1.5

1 0.8 0.6

Original

1

0.5

0.4

RLS EKF

0.2

UKF UHF

0 0 0

100

200

300

400

500

600

700

800

900

No.of iteration (N) Fig. 1c. Estimated amplitude with 50% voltage sag and notch.

1000

-0.2 0

100

200

300

400

500

600

700

800

900

1000

No.of iteration (N) Fig. 1e. Estimated amplitude with 50% voltage swell and momentary interruption.

443

H.K. Sahoo, P.K. Dash / Electrical Power and Energy Systems 73 (2015) 438–447

Therefore

Absolute Error for Swell & Momentary Interruption 1.2

2

RLS

b kþ1jk P

EKF

1

UKF

3 0 0 0 6 ^ 2kþ1jk ð2Þ ckþ1jk ð3Þ 7 ¼ 40 r 5: ^ 2kþ1jk ð3Þ 0 ckþ1jk ð3Þ r

UHF

^ 2kþ1jk ð3Þ from Eq. (51): Now we focus on r

Absolute Error

0.8

r^ 2kþ1jk ð3Þ ¼ r^ 2kjk ð2Þ ¼ r^ 2kjk1 ð2Þ  g2k b2 r^ 2kjk ð2Þ  r^ 2kjk1 ð2Þ ¼ g2k b2 6 0

0.6

^ 2kjk1 ð2Þ ¼ 0 i:e:; r ^ 2kþ1jk ð3Þ ¼ 0 As k ! 1; r

0.4

ð58Þ

^ 2kjk1 ð3Þ ¼ 0 implies that b ¼ 0 and r 0.2

^xkjk ð2Þckjk1 ð1Þ þ ^xkjk ð1Þr ^ 2kjk1 ð2Þ  ckjk1 ð3Þ ¼ 0

ð59Þ

From Eq. (59):

0

0 þ 0  ckjk1 ð3Þ ¼ 0 ) ckþ1jk ð3Þ ¼ 0: -0.2

0

100

200

300

400

500

600

700

800

900

1000

No. of iterations (N)

ð60Þ

Eqs. (59) and (60) lead to

2

0 0 0

3

7 b kþ1jk ¼ 6 P 4 0 0 0 5 as k ! 1

Fig. 1f. Absolute error plot.

0 0 0 Simulation results

^ 2kþ1jk ð3Þ

r

^ 2kjk ð2Þ

¼r

ð51Þ

ckþ1jk ð1Þ ¼ ^xkjk ð2Þr^ 2kjk ð1Þ þ ^xkjk ð1Þckjk ð1Þ  ckjk ð2Þ

ð52Þ

ckþ1jk ð2Þ ¼ ckjk ð1Þ

ð53Þ

ckþ1jk ð3Þ ¼ ^xkjk ð2Þckjk ð1Þ þ ^xkjk ð1Þr^ 2kjk ð2Þ  ckjk ð3Þ

ð54Þ

b kþ1jk , we have Taking the (1, 1) component of P

r^ 2kþ1jk ð1Þ  r^ 2kjk1 ð1Þ ¼ g2k :a2 ¼ g

2 k

^xkjk ð2Þr ^ 2kjk1 ð1Þ þ ^xkjk ð1Þckjk1 ð1Þ ckjk1 ð2Þ

!2 ð55Þ

^ 2kþ1jk ð1Þ continues to monoFrom Eq. (55), it can be seen that r tonically decrease with k until |a| = 0 as k ! 1:jaj ¼ 0 ¼> r^ 2kþ1jk ð1Þ converges to a certain value that is greater than or equal to zero

jaj ¼ 0 )

j^xkjk ð2Þ ^ 2kjk1 ð1Þ

r

The performance of UHF is presented through MATLAB simulations and comparison plots with RLS, EKF and UKF are shown. For this purpose four test signals are considered. First test signal is taken to show the tracking capability of the filter when the signal is distorted with sag, swell, momentary interruption and notch. The second, third and fourth test signals are considered to show the tracking capability of the filter when the signal is distorted with higher order harmonics. For UHF and UKF the initial values of measurement and process noise co-variances are taken as 0.01 and 0.001 and for RLS algorithm the value of forgetting factor is chosen to be 0.996. Sampling rate of 1.5 kHz and SNR of 30 dB in presence of white Gaussian noise is considered for simulation. Fundamental frequency of 50 Hz and amplitude 1 p.u. are considered for the power signals. Test signal 1 is considered to estimate power quality disturbances like sag, swell, notch and momentary interruption. Test signals 2, 3 and 4 are used for estimation of time varying harmonic parameters.

Test Signal 1 ¼ Ak sinðkxT s þ /k Þ þ v k

þ ^xkjk ð1Þckjk1 ð1Þ  ckjk1 ð2Þj ¼ 0

Original amplitude1 vs Estimated amplitude1 3.5

^ 2kþ1jk ð1Þ ¼ 0 ) r ^ 2kjk1 ð1Þ ¼ 0 and r ^ 2kjk ð1Þ ¼ 0. as r

Original

Then

EKF

ð56Þ

UKF 2.5

Considering the (1, 2) or (2,1) element, from Eqs. (46) and (47):

ckþ1jk ð1Þ  ckjk1 ð1Þ ¼ ^xkjk ð2Þr^ 2kjk ð1Þ þ ^xkjk ð1Þckjk ð1Þ  ckjk ð2Þ  ckjk1 ð1Þ ¼ 0 þ ^xkjk ð1Þckjk1 ð1Þ  ckjk1 ð2Þ  ckjk1 ð1Þ:

Amplitude (p.u.)

j^xkjk ð1Þckjk1 ð1Þ  ckjk1 ð2Þj ¼ 0

RLS

3

UHF

2

1.5

1

From Eq. (56) 0.5

ckþ1jk ð1Þ  ckjk1 ð1Þ ¼ 0  ckjk1 ð1Þ ) ckþ1jk ð1Þ ¼ 0 i.e., ckjk ð1Þ ¼ 0 and ck|k-1(1) = 0 From Eq. (53)

ckþ1jk ð2Þ ¼ ckjk ð1Þ ¼ 0:

0

0

100

200

300

400

500

600

700

800

900

No. of iterations (N)

ð57Þ

Fig. 2a. Estimated amplitude of fundamental harmonic.

1000

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H.K. Sahoo, P.K. Dash / Electrical Power and Energy Systems 73 (2015) 438–447

Estimated phase1

Original amplitude3 vs Estimated amplitude3 1.5

1.5

Original

Original

1.4

RLS

RLS

EKF

EKF

UKF

1.3

UKF

UHF

UHF

Amplitude (p.u.)

Phase (radian)

1

0.5

1.2

1.1

1

0.9

0.8 0

0

100

200

300

400

500

600

700

800

900

0

100

200

300

400

500

600

700

800

900

1000

No. of iterations (N)

1000

No. of iterations(N)

Fig. 2d. Estimated phase of fundamental harmonic.

Fig. 2b. Estimated amplitude of third harmonic.

Estimated phase3 1.2

Estimated amplitude5 1.2 estimated amp5(rls) estimated amp5(ekf) estimated amp3(ukf) estimated Amp5(uhf)

1

Original

1

RLS EKF 0.8

0.8

UKF

Phase (radian)

Amplitude (p.u.)

UHF 0.6

0.4

0.2

0.6

0.4

0.2

0 0 -0.2 -0.2 -0.4

0

100

200

300

400

500

600

700

800

900

1000

No. of iterations (N)

0

100

200

300

400

500

600

700

800

900

1000

No. of iterations (N) Fig. 2e. Estimated phase of third harmonic.

Fig. 2c. Estimated amplitude of fifth harmonic.

Condition A: signal with voltage sag and swell Sag occurs from 100 to 500 iterations with amplitude 0.5 p.u. and swell occurs from 600 to 800 iterations with amplitude 1.5 p.u. For the rest of the iterations the value of amplitude is 1 p.u. The estimated amplitude with 50% voltage sag and 50% voltage swell is shown in Fig. 1a and corresponding absolute error plot is shown in Fig. 1b. Condition B: signal with voltage sag and notch Sag occurs from 100 to 500 iterations with value at 0.5 p.u. Notch occurs at the 600th iteration with value at 2.5 p.u. The estimated amplitude with 50% voltage sag and notch is shown in Fig. 1c and corresponding absolute error plot is shown in Fig. 1d.

Condition C: signal with swell and momentary interruption Swell occurs from 100 to 500 iterations with value at 1.5 p.u. and momentary Interruption occurs from 600 to 800 iterations. The estimated amplitude with 50% voltage swell and momentary interruption is shown in Fig. 1e and corresponding absolute error plot is shown in Fig. 1f. For harmonic estimation, test signal 2, test signal 3 and test signal 4 given in Eqs. 8–10 are considered in presence of white Gaussian noise. For test signal 2, fundamental, third and fifth harmonic amplitudes are considered to be 2 p.u., 0.8 p.u. and 0.3 p.u. respectively. The estimated plots for amplitudes and phases of fundamental, third and fifth harmonics considering test signal 2 as given in Eq. (8) are shown in Fig. 2a–f. The estimated time varying amplitudes of fundamental, third and fifth harmonics considering test signals 3 and 4 as given in Eqs. (9) and (10) are shown in

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H.K. Sahoo, P.K. Dash / Electrical Power and Energy Systems 73 (2015) 438–447

Estimated phase5

Estimated amplitude3 0.8

2

estimated amp3(rls) estimated amp3(ekf) estimated Amp3.(ukf) abs err in amp3(uhf)

Original 0.6

RLS 1.5

EKF 0.4

UKF UHF

Amplitude (p.u.)

Phase (radian)

1

0.5

0.2 0 -0.2

0 -0.4 -0.5

-0.6 -0.8

-1

0

100

200

300

400

500

600

700

800

900

1000

900

1000

No. of iterations (N) -1.5

0

100

200

300

400

500

600

700

800

900

Fig. 3b. Estimated amplitude of third harmonic.

1000

No. of iterations (N) Original amplitude5 vs Estimated amplitude5

Fig. 2f. Estimated phase of fifth harmonic.

1

2 Original

0.8

RLS 1.5

EKF UKF UHF

1

Amplitude (p.u.)

AMPLITUDE (P.U.)

0.6

0.5

0.4

0.2

0 0 -0.5 -0.2 -1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 -0.4

TIME IN SECOND

0

100

200

300

400

500

600

700

800

No. of iterations (N)

Fig. 3. Test signal 3 with harmonics.

Fig. 3c. Estimated amplitude of fifth harmonic.

Original amplitude1 vs Estimated amplitude1

1.2

12 Original RLS EKF UKF UHF

1 0.8

8

AMPLITUDE (P.U.)

Amplitude (p.u.)

0.6

10

0.4 0.2 0 -0.2

6 4 2 0 -2

-0.4

-4

-0.6 0

100

200

300

400

500

600

700

800

900

No. of iterations (N) Fig. 3a. Estimated amplitude of fundamental harmonic.

1000

-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TIME IN SECOND Fig. 4. Test signal 4 with harmonics.

0.8

0.9

1

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H.K. Sahoo, P.K. Dash / Electrical Power and Energy Systems 73 (2015) 438–447

Original amplitude1 vs Estimated amplitude1

Estimated amplitude

4

12

Original 3.5

Original

RLS EKF

RLS

10

UKF

3

EKF

UHF

UKF UHF

Amplitude [p.u]

8

Amplitude [p.u]

2.5

6

2

1.5

1 4 0.5 2

0

-0.5 0

0

100

200

300

400

500

600

700

800

900

0

100

200

300

1000

400

500

600

700

800

900

1000

No of iteration

No of iteration Fig. 4c. Estimated amplitude of fifth harmonic. Fig. 4a. Estimated amplitude of fundamental harmonic.

Table 1 MSE comparison for different adaptive filtering algorithms for PQ disturbances estimation (Test signal 1).

Original amplitude3 vs Estimated amplitude3 6

Original RLS

5

EKF UKF

Amplitude [p.u]

4

Types of distortion

RLS

EKF

UKF

UHF

Sag and swell Sag with notch Swell with momentary interruption

0.01267 0.01417 0.03104

0.00013 0.00012 0.00015

7.0260  105 6.0806  105 4.4747  105

2.7101  105 6.0806  105 4.2007  105

UHF

3

Table 2 MSE comparison for different adaptive filtering algorithms for harmonic estimation (Test signal 2).

2

1

Amplitudes

RLS

EKF

UKF

UHF

Fundamental Third harmonic Fifth harmonic

0.00379 0.00289 0.00284

0.00338 0.00383 0.00238

0.0038 0.0037 0.0025

0.0021 0.0029 0.0014

0

-1

0

100

200

300

400

500

600

700

800

900

1000

No of iteration Fig. 4b. Estimated amplitude of third harmonic.

Figs. 3a–c and 4a–c respectively. The mean square error (MSE) due to estimation of different power quality disturbances is calculated and the comparisons between different adaptive filters are presented in Table 1. MSE due to harmonic estimation are calculated and comparisons between different adaptive filters are presented in Table 2. The results presented in the tables clearly demonstrate that MSE due to estimation is quite small in case of UHF for different power quality disturbances. MSE due to amplitude and phase estimation in case of harmonics can be calculated as:

ek ¼

N 1X ^ðkÞj2 jaðkÞ  a N k¼1

ð61Þ

ek ¼

N 1X 2 ^ j/ðkÞ  /ðkÞj N k¼1

ð62Þ

^ ^ðkÞ; /ðkÞ are where aðkÞ; /ðkÞ are the actual amplitude and phase, a the estimated amplitude and phase respectively, k is the number of iterations.

Conclusion The proposed unscented H1 filtering approach is robust not only for the estimation of power quality disturbances but also for the tracking of higher order harmonics. The estimation accuracy of different adaptive filtering techniques is also compared in terms of mean square error. The performance of UHF algorithm is tested under different abnormal changing conditions in power systems and the comparison results prove the better performance of the filter over the RLS, EKF and UKF when system dynamics is highly nonlinear.

H.K. Sahoo, P.K. Dash / Electrical Power and Energy Systems 73 (2015) 438–447

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