Distributed Widely Linear Kalman Filtering for Frequency ...€¦ · Distributed Widely Linear Kalman Filtering ... Kronecker delta function. A. Distributed Complex Kalman ... LINEAR

IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS, VOL. 1, NO. 1, MARCH 2015 45

Distributed Widely Linear Kalman Filteringfor Frequency Estimation in Power Networks

Sithan Kanna, Student Member, IEEE, Dahir H. Dini, Yili Xia, Member, IEEE, S. Y. Hui, Fellow, IEEE,and Danilo P. Mandic, Fellow, IEEE

Abstract—Motivated by the growing need for robust andaccurate frequency estimators at the low- and medium-voltage dis-tribution levels and the emergence of ubiquitous sensors networksfor the smart grid, we introduce a distributed Kalman filteringscheme for frequency estimation. This is achieved by using widelylinear state space models, which are capable of estimating thefrequency under both balanced and unbalanced operating con-ditions. The proposed distributed augmented extended Kalmanfilter (D-ACEKF) exploits multiple measurements without impos-ing any constraints on the operating conditions at different parts ofthe network, while also accounting for the correlated and noncir-cular natures of real-world nodal disturbances. Case studies overa range of power system conditions illustrate the theoretical andpractical advantages of the proposed methodology.

Index Terms—Adaptive networks, frequency estimation,Kalman filters, sensor fusion, smart grid.

I. INTRODUCTION

T HE MODERN power network, commonly known as the“smart grid,” aims to be more reliable by incorporating

distributed generation, typically from renewable sources (wind)and often in a local fashion (solar, micro-wind turbines). Therapid growth in renewable energy sources in the smart grid alsogives rise to challenges in maintaining the stability of the net-work, as these renewable sources are connected to low-voltagedistribution networks through grid-interfaced power electronicsinverters.

The balance between power generation and consumption isa pre-requisite for stable operation of the power network. Animbalance between generation and load causes a frequencydeviation and grid operators require fast and accurate estimatesof the frequency to control the stability of the network [1],[2]. In addition, power inverters themselves require accuratefrequency estimates for stable grid-interfacing under noisy con-ditions. As inverters are switching circuits, they themselvesintroduce impulsive noise to the voltage signal [3], which, in

Manuscript received December 10, 2014; revised April 10, 2015; acceptedMay 22, 2015. Date of publication June 23, 2015; date of current version June29, 2015. The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Isao Yamada.

S. Kanna, D. H. Dini, and D. P. Mandic are with the Department of Electricaland Electronic Engineering, Imperial College London, London SW7 2AZ, U.K.(e-mail: [email protected]; [email protected]).

Y. Xia is with the School of Infirmation Science and Engineering, SoutheastUniversity, Nanjing 210096, China.

S. Y. (Ron) Hui is with Hong Kong University, Hong Kong, and alsowith Imperial College London, London SW7 2AZ, U.K. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TSIPN.2015.2442834

turn, requires frequency estimation algorithms with enhancedrobustness against noise and voltage spikes.

Currently used frequency estimation techniques include:i) Fourier transform approaches [4]–[7], ii) gradient decent andleast squares adaptive estimation [8], and iii) state space meth-ods and Kalman filters [9]–[12]. However, these are typicallydesigned for single-phase systems and often assume balancedoperating conditions (equal voltage amplitudes and equallyspaced phases) and are therefore inadequate for the demands ofmodern three–phase and dynamically optimised power systems.

To avoid the loss of information and the compromise inaccuracy by estimating the frequency from only one of thethree phases, the Clarke transform is used to jointly represent athree-phase voltage signal as a complex-valued scalar variableknown as the αβ voltage [13]–[15]. Our earlier work showedthat this αβ voltage admits a widely linear autoregressivemodel under both balanced and unbalanced system conditions[16]. It was also shown that standard, strictly linear, complex-valued estimators applied to the complex αβ voltage introducebiased estimates for unbalanced system conditions, widely lin-ear estimators (also known as “augmented” estimators) areable to provide optimal and consistent estimates of the systemfrequency over a range of operating conditions [16]–[19].

The aim of this work is to extend the single-node widely lin-ear state space frequency estimator in [18] to the distributedscenario to suit the requirements of the smart grid. Distributedestimation has already found application in both military andcivilian scenarios [20]–[24], as cooperation between the nodes(sensors) provides more accurate and robust estimation over theindependent nodes, while approaching the performance of cen-tralised systems at much reduced communication overheads.Recent distributed approaches include diffusion least-mean-square estimation [25]–[27] and Kalman filtering [22], [24],[28], however, these consider circular measurement noise with-out cross-nodal correlations, which is not realistic in real-worldpower systems.

To this end, we propose the diffusion augmented (widelylinear) complex Kalman filter (D-ACKF) and the diffusion aug-mented extended Kalman filter (D-ACEKF) by adapting thediffusion scheme in [29] and the Kalman filtering models in[18], [30] to suit our problem where the system frequency isidentical over a certain geographical area at the distributionlevel while the voltage imbalances and cross-nodal correlationscan be different. In particular, extending the widely linear fre-quency estimator to the distributed case is non-trivial as thestates that are to be estimated are not-identical in the network,as required by the classical diffusion scheme [29].

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46 IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS, VOL. 1, NO. 1, MARCH 2015

The proposed D-ACEKF is able improve the frequency esti-mates at the low- and medium-voltage distribution levels ofthe electricity grid by exploiting a diversity of measurementswithout imposing any constrains on the operating conditions.Moreover, the proposed D-ACEKF accounts for the correla-tion between the observation noises at neighbouring nodes,typically encountered when node signals are exposed to com-mon sources of interference (harmonics, fluctuations of reactivepower), which is not catered for in current estimation algo-rithms. Case studies using synthetic and real world signalssupport the claims.

II. BACKGROUND ON WIDELY LINEAR MODELLING

Statistical signal processing in the complex-domain under-pins a number of disciplines, including wireless communica-tions [31], [32] and power systems [16]. Although it may beconvenient to process complex-valued data by representing thereal and imaginary parts as a bivariate signal in the real domain,any intuition and physical meaning inherent in processing in thecomplex domain would be obscured. However, the statisticaltools for complex random variables were derived by assum-ing (often implicitly) that these complex random variables weresecond order circular (proper)1 [33]–[35].

Consider the problem of estimating a real-valued vectorx ∈ R

L from the data vector y ∈ RK . According the linear

minimum mean square error (LMMSE) estimation theory, theoptimal estimate (in terms of second order statistics) is2

x = E {x|y} = RxyR−1y y (1)

where Rxy = E{xyT

}is the cross-covariance matrix, Ry =

E{yyT

}is the autocovariance matrix, and (·)T denotes the

transpose operator. When the vectors x and y are complex-valued, the so-called strictly linear estimation assumes thesame model in (1) with complex valued correlation matricesRxy = E

{xyH

}and Ry = E

{yyH

}where the transpose

operator is replaced with the Hermitian operator (·)H [30], [36].It has been shown that the strictly linear model in (1) is

sub-optimal for a general class of complex-valued signals.The widely linear minimum mean square error (WLMMSE)solution exploits the full second order statistics of the signalcontained within the augmented autocovariance matrix

Ray = E

[yy∗

] [yH yT

]=

[Ry Py

P∗y R∗

y

](2)

and augmented cross-covariance matrix

Raxy = E

[xyH xyT

]=[Rxy Pxy

](3)

1Proper signals have real and imaginary parts with equal variance and areuncorrelated. Circular signals have rotationally invariant probability densityfunctions (pdf). We use the terms “circular” and “proper” interchangeably forGaussian signals, as they have pdfs that can be fully characterised by the firstand second order statistics.

2This problem is sometimes posed as that of estimating vector y from obser-vations x. To be consistent with the Kalman filtering literature, however, wechose to represent the state with variable x and observation with y.

Fig. 1. An example of a distributed network topology.

which not only contain the autocovariance and cross-covariancematrices Ryy and Rxy but also the pseudocovariance matri-ces Pxy = E

{xyT

}and Py = E

{yyT

}. The WLMMSE

solution is given by

x = E {x|y,y∗} = By +Cy∗ (4)

where the B =[Rxy −PxyR

−∗y P∗

y

]D−1

y , C = [Pxy −RxyR

−1y Py

]D−∗

y and Dy =[Ry −PyR

−∗y P∗

y

]is the

Schur’s complement of the augmented auto-covariance matrixRa

y in (2). The WLMMSE solution in (4) is equivalent to theLMMSE estimator derived from treating the real and imagi-nary parts of the signals as bivariate real-valued vectors [37]but provides much more physical insight in the analysis. Forjointly-circular signals that have vanishing pseudocovariancematrices Pxy = 0 and Py = 0 (referred to as circular or propersignals), the widely linear model in (4) degenerates to thestrictly linear model in (1).

III. DIFFUSION KALMAN FILTERING

For a standard linear state space model, every node i in adistributed system (see Fig. 1) is given by [38]

xn = Fn−1xn−1 +wn

yi,n = Hi,nxn + vi,n

(5)

where xn ∈ CL and yi,n ∈ C

K are respectively the state vectorat time instant n and observation (measurement) vector at nodei. The symbol Fn−1 denotes the state transition matrix, wn ∈C

L white state noise, Hi,n the observation matrix, and vi,n ∈C

K white measurement measurement noise (both at node i).Standard state space models assume the noises wn and vi,n

to be uncorrelated and zero-mean, with their covariance andpseudocovariance matrices described by

E

[wn

vi,n

] [wH

k vHi,k

]=

[Qn 00 Ri,n

]δnk

E

[wn

vi,n

] [wT

k vTi,k

]=

[Pn 00 Ui,n

]δnk

(6)

where δnk is the standard Kronecker delta function.

A. Distributed Complex Kalman Filter

The distinguishing feature of the proposed class of dis-tributed Kalman filters is that we generalise the diffusion

KANNA et al.: DISTRIBUTED WIDELY LINEAR KALMAN FILTERING FOR FREQUENCY ESTIMATION IN POWER NETWORKS 47

strategy in [22] by equipping it with state and noise mod-els that do not impose any restrictions on: i) the correlationproperties of the cross-nodal observation noises, or ii) the sig-nal and noise circularity at different nodes. This also allowsdistributed Kalman filtering algorithms [22], [24], [39] to beused in wider application scenarios. Fig. 1 illustrates the dis-tributed estimation scenario; the highlighted neighbourhood ofnode i compromises the set of nodes, denoted by Ni, that com-municate with the node i (including Node i itself). The stateestimate at node i with a complex Kalman filter (CKF) is thenbased on all the data from the neighbourhood Ni consisting ofM = |Ni| nodes, and is denoted by xi,n|n, where the symbol|Ni| denotes the number of nodes in the neighbourhood Ni.Finally, the collective neighbourhood observation equation atnode i is given by

yi,n

= Hi,nxn + vi,n (7)

while the collective (neighbourhood) variables are defined as

yi,n

=[yTi1,n

,yTi2,n

, . . . ,yTiM ,n

]THi,n =

[HT

i1,n,HT

i2,n, . . . ,HT

iM ,n

]Tvi,n =

[vTi1,n

,vTi2,n

, . . . ,vTiM ,n

]Twhere {i1, i2, . . . , iM} are the nodes in the neighbourhood Ni.The covariance and pseudocovariance matrices of the collectiveobservation noise vector are given by

Ri,n = E{vi,nv

Hi,n

}=

⎡⎢⎢⎢⎣Ri1,n Ri1i2,n · · · Ri1iM ,n

Ri2i1,n Ri2,n · · · Ri2iM ,n

......

. . ....

RiM i1,n RiM i2,n · · · RiM ,n

⎤⎥⎥⎥⎦

Ui,n = E{vi,nv

Ti,n

}=

⎡⎢⎢⎢⎣Ui1,n Ui1i2,n · · · Ui1iM ,n

Ui2i1,n Ui2,n · · · Ui2iM ,n

......

. . ....

UiM i1,n UiM i2,n · · · UiM ,n

⎤⎥⎥⎥⎦where Ria,n = E

{via,nv

Hia,n

}, Riaib,n = E

{via,nv

Hib,n

},

Uia,n = E{via,nv

Tia,n

}and Uiaib,n = E

{via,nv

Tib,n

}, for

a, b ∈ {1, 2, . . . ,M}.Diffusion step. The local neighbourhood state estimates are

followed by the diffusion (combination) step, given by

xi,n|n =∑

�∈Ni

c�ix�,n|n (8)

which produces the diffused state estimates xi,n|n as a weightedsum of the estimates from the neighbourhood Ni, where c�i ≥ 0are the weighting coefficients satisfying

∑�∈Ni

c�i = 1. Thecombination weights c�i used by the diffusion step in (8)can obey a number of rules, including the Metropolis [25],Laplacian [40] or the nearest neighbour [22] rules, however,finding the set of optimal weights remains an open issue thoughprogress has been made in important cases[29], [41], [42].

The distributed complex Kalman filter (D-CKF) aims toapproach the performance of a centralised Kalman filter (accessto data from all the nodes) via neighbourhood collaborations

and diffusion, and is summarised in Algorithm 1. Each nodewithin D-CKF forms a collective observation, as in (7), usinginformation from its neighbours; thereafter, each node com-putes a neighbourhood state estimate which is again shared withneighbours in order to be used for the diffusion step.

Algorithm 1. Diffusion Complex Kalman Filter (D-CKF)

Initialisation: for Nodes i = {1, 2, . . . , N}1: xi,0|0 = E

{xT0

}2: Mi,0|0 = E

(x0 − xi,0|0

) (x0 − xi,0|0

)HAt each time instant n > 0:

1: for Nodes i = {1, 2, . . . , N} do2: xi,n|n−1 = Fn−1xi,n−1|n−1

3: Mi,n|n−1 = Fn−1Mi,n−1|n−1FHn−1 +Qn

4: Gi,n=Mi,n|n−1HHi,n

(Hi,nMi,n|n−1H

aHi,n +Ra

i,n

)−1

5. xi,n|n = xi,n|n−1 +Gi,n

(yi,n

−Hi,nxi,n|n−1

)6: Mi,n|n = (I−Gi,nHi,n)Mi,n|n−1

7: Diffuse the states from the network:xi,n|n =

∑�∈Ni

c�ix�,n|n8: end for

Remark 1: The D-CKF algorithm3 given in Algorithm 1 is avariant of that proposed in [22]. It employs the standard (strictlylinear) state space model (5), and thus does not cater for widelylinear complex state space models or noncircular state andobservation noises (Pn �= 0 and Ui,n �= 0 for i = 1, . . . , N ).Unlike existing distributed complex Kalman filters, the D-CKFpresented in Algorithm 1 caters for the correlations between theneighbourhood observation noises. When no such correlationsexits, the D-CKF is identical to Kalman filter given in [22].

B. Distributed Augmented Complex Kalman Filter

We next employ the widely linear model in (4) in conjunc-tion with D-CKF in Algorithm 1 to cater for widely linearstate and observation models, and for improper measurements,states, and state and observation noises. The widely linear ver-sion of the distributed state space model in (5) (see also [30],[43]) is given by

xn = Fn−1xn−1 +An−1x∗n−1 +wn

yi,n = Hi,nxn +Bi,nx∗n + vi,n.

(9)

The compact, “augmented” representation, of this model is

xan = Fa

n−1xan−1 +wa

n

yai,n = Ha

i,nxan + va

i,n

(10)

where xan = [xT

n ,xHn ]T and ya

n = [yTn ,y

Hn ]T , while

Fan =

[Fn An

A∗n F∗

n

]and Ha

i,n =

[Hi,n Bi,n

B∗i,n H∗

i,n

].

For strictly linear systems, An = 0 and Bi,n = 0, so thatthe widely linear (augmented) state space model degenerates

3The matrices Mi,n|n and Mi,n|n−1 do not represent the covariances ofxi,n|n and xi,n|n−1, as is the case for the standard Kalman filter operating onlinear Gaussian systems. This is due to the use of the suboptimal diffusion step,which updates the state estimate and not the covariance matrix Mi,n|n.


into a strictly linear one. However, the augmented state spacerepresentation is still preferred in order to account for the pseu-docovariances of the noise vectors which reflect the improprietyof the noise.

The augmented covariance matrices of the process noisevector wa

n = [wTn ,w

Hn ]T and observation noise vector va

i,n =

[vTi,n,v

Hi,n]

T are then given by

Qan = E

{wa

n waHn

}=

[Qn Pn

P∗n Q∗

n

]Ra

i,n = E{vai,nv

aHi,n

}=

[Ri,n Ui,n

U∗i,n R∗

i,n

].

(11)

Neighbourhood variables. To perform collaborative estimationof the state within distributed networks, neighbourhood obser-vation equations use all available neighbourhood observationdata, to give

yi,n

= Hi,nxn +Bi,nx∗n + vi,n (12)

where the symbol Bi,n =[BT

i1,n,BT

i2,n, . . . ,BT

iM ,n

]Tdenotes

the conjugate observation matrix, and {i1, i2, . . . , iM} ∈ Ni.The augmented neighbourhood observation equations nowbecome

yai,n

= Hai,nx

an + va

i,n (13)

with the augmented neighbourhood variables defined as

yai,n

=

[yi,n

y∗i,n

], Ha

i,n =

[Hi,n Bi,n

B∗i,n H∗

i,n

], va

i,n =

[vi,n

v∗i,n

].

(14)

Consequently, the covariance of the augmented neighbourhoodobservation noise va

i,n takes the form

Rai,n = E

{vai,nv

aHi,n

}=

[Ri,n Ui,n

U∗i,n R∗

i,n

]. (15)

Remark 2: The augmented second order statistics in (15)caters for both the covariances E

{vi,nv

Hi,n

}and cross-

covariances E{vi,nvH�,n}, i �= � between the nodal observation

noises. This is achieved through the standard covariance matrixRi,n and the pseudocovariances E

{vi,nv

Ti,n

}, while the cross-

pseudocovariances E{vi,nvT�,n} are accounted for through the

pseudocovariance matrix Ui,n. Finally, the augmented diffusedstate estimate becomes

xai,n|n =

∑�∈Ni

c�ixa�,n|n (16)

and represents a weighted average of the augmented (neigh-bourhood) state estimates. The proposed distributed augmentedcomplex Kalman filter (D-ACKF), employs the widely linearstate space model in (9), and is given in Algorithm 2.

For strictly linear systems (An = 0, Bi,n = 0, ∀ n, i) withcircular state and observation noises (Pn = 0, Ui,n = 0, ∀n, i), the D-ACKF and D-CKF algorithms yield identical stateestimates for all time instants n. Notice that the D-ACKF is

more general than the D-CKF, since it also caters for the non-circular natures of data and noise, together with correlated stateand observation noises.

Algorithm 2. Diffusion Augmented Complex Kalman Filter(D-ACKF)

Initialisation: for Nodes i = {1, 2, . . . , N}1: xa

i,0|0 = E[xT0 , xH

0

]T2: Ma

i,0|0 = E(xa0 − xa

i,0|0)(

xa0 − xa

i,0|0)H

At each time instant n > 0 :1: for Nodes i = {1, 2, . . . , N} do2: xa

i,n|n−1 = Fan−1x

ai,n−1|n−1

3: Mai,n|n−1 = Fa

n−1Mai,n−1|n−1F

aHn−1 +Qa

n

4: Gai,n=M

ai,n|n−1H

aHi,n

(Ha

i,nMai,n|n−1H

aHi,n+R

ai,n

)−1

5: xai,n|n = xa

i,n|n−1 +Gai,n

(yai,n

−Hai,nx

ai,n|n−1

)6: Ma

i,n|n = (I−Gai,nH

ai,n)M

ai,n|n−1

7: Diffuse the states from the network:xai,n|n =

∑�∈Ni

c�ixa�,n|n

8: end for

IV. DISTRIBUTED AUGMENTED COMPLEX EXTENDED

KALMAN FILTER

Consider a nonlinear state space model of the form

xn = f [xn−1] +wn

yi,n = hi [xn] + vi,n

(17)

where the nonlinear functions f [·] and hi[·] are respectively the(possibly time varying) process model and observation modelat node i, the remaining variables are as defined above. Withinthe extended Kalman filter (EKF) framework, the nonlinearstate and observation functions are approximated by their firstorder Taylor series expansions (TSE) about the state estimatesxi,n−1|n−1 and xi,n|n−1 for each node i, so that [44]

xn ≈ Fi,n−1xn−1 +Ai,n−1x∗n−1 + ri,n−1 +wn

yi,n ≈ Hi,nxn +Bi,nx∗n + ui,n + vi,n

(18)

where the Jacobians of functions f [·] and hi[·] are defined as

Fi,n =∂f [x]

∂x|x=xi,n|n , Ai,n =

∂f [x]

∂x∗ |x∗=x∗i,n|n

,

Hi,n =∂hi[x]

∂x|x=xi,n|n−1

and Bi,n =∂hi[x]

∂x∗ |x∗=x∗i,n|n−1

and the vectors

ri,n = f [xi,n−1|n−1]− Fi,n−1xi,n−1|n−1 −Ai,n−1x∗i,n−1|n−1

ui,n = hi[xi,n|n−1]−Hi,nxi,n|n−1 −Bi,nx∗i,n|n−1

are deterministic inputs calculated from the state space modeland state estimate. The full second order statistics in the lin-earised state space in (18) is accounted for by its widely linearversion, for which the augmented form is given by

xan ≈ Fa

i,n−1xan−1 + rai,n−1 +wa

n

yai,n ≈ Ha

i,nxan + ua

i,n + vai,n

(19)


where rai,n = [rTi,n, rHi,n]

T , uai,n = [uT

i,n,uHi,n]

T , while

Fai,n =

[Fi,n Ai,n

A∗i,n F∗

i,n

]and Ha

i,n =

[Hi,n Bi,n

B∗i,n H∗

i,n

].

The collective neighbourhood augmented observation equationfor node i now takes the form

yai,n

= hai [xn] + va

i,n (20)

while the collective observation function is defined as

hai [xn] =

[hTi [xn],h

Hi [xn]

]Thi[xn] =

[hTi1 [xn],h

Ti2 [xn], . . . ,h

TiM [xn]

]Twhere i ∈ {i1, i2, . . . , iM} spans all the nodes in the neighbour-hood Ni. The first order approximation of (20) is then

yai,n

≈ Hai,nx

an + ua

i,n + vai,n (21)

while the Jacobian of the collective observation becomes

Hai,n =

[Hi,n Bi,n

B∗i,n H∗

i,n

]with Hi,n = [HT

i1,n,HT

i2,n, . . . ,HT

iM ,n]T and Bi,n = [BT

i1,n,

BTi2,n

, . . . ,BTiM ,n]

T , calculated as

Hik,n=∂hik [x]

∂x

∣∣∣∣x=xi,n|n−1

and Bik,n =∂hik [x]

∂x∗ |x∗=x∗i,n|n−1

.

Algorithm 3 summarises the proposed distributed augmentedcomplex extended Kalman filter, where each node i shares its(nonlinear) observation model hi[·] with its neighbours. Thefunction Jacobian (f ,x) for steps 3 and 5 in Algorithm 3 com-putes the Jacobian matrix of the function f evaluated at thepoint x.

Algorithm 3. Diffusion Augmented Complex ExtendedKalman Filter (D-ACEKF)

Initialisation: for Nodes i = {1, 2, . . . , N}1: xa

i,0|0 = E[xT0 , xH

0

]T2: Ma

i,0|0 = E(xa0 − xa

i,0|0)(

xa0 − xa

i,0|0)H

At each time instant n > 0 :1: for Nodes i = {1, 2, . . . , N} do

2: xai,n|n−1 =

[fT[xi,n−1|n−1

], fH

[xi,n−1|n−1

]]T3: Fa

i,n−1 = Jacobian(fa, xa

i,n−1|n−1

)4: Ma

i,n|n−1 = Fai,n−1M

ai,n−1|n−1F

aHi,n−1 +Qa

n

5: Hai,n = Jacobian

(hai , xa

i,n|n−1

)6: Ga

i,n=Mai,n|n−1H

aHi,n

(Ha

i,nMai,n|n−1H

aHi,n+R

ai,n

)−1

7: xai,n|n = xa

i,n|n−1 +Gai,n

(yai,n

− hai

[xi,n|n−1

])8: Ma

i,n|n = (I−Gai,nH

ai,n)M

ai,n|n−1

9: Diffuse the states from the network:xai,n|n =

∑�∈Ni

c�ixa�,n|n

10: end for

Remark 3: The D-ACEKF algorithm in Algorithm 3 extendsthe Distributed Extended Kalman filter in [45] to widely linearstate spaces, and caters for the improper second order statisticalmoments of the state and noise models, together with the cor-relations present between the nodal observation noises. This isa perfect match for distributed estimation in unbalanced smartgrids.

V. DISTRIBUTED WIDELY LINEAR FREQUENCY

ESTIMATION

The proposed augmented state space models are particularlysuited for frequency estimation in power grid, as due to sys-tem inertia, the frequency can be assumed identical over thenetwork of measurement nodes, while unbalanced systems anddistributed nodes generate noncircular and noisy measurements[17], [19]. For a three phase system, the instantaneous voltagesat a node i are given by⎡⎣va,i,nvb,i,n

vc,i,n

⎤⎦︸︷︷︸

vi,n

=

⎡⎣ Va,i,n cos(ωnT + φa,i)Vb,i,n cos(ωnT + φb,i − 2π

3 )Vc,i,n cos(ωnT + φc,i +

2π3 )

⎤⎦︸︷︷︸

si,n

+

⎡⎣za,i,nzb,i,nzc,i,n

⎤⎦︸︷︷︸

zi,n

(22)

where Va,i,n, Vb,i,n, and Vc,i,n are the amplitudes of thethree-phase voltages at time instant n, ω = 2πf the angularfrequency, f the system frequency, T the sampling interval,while za,i,n, za,i,n, and za,i,n are zero-mean observation noiseprocesses. The term φa,i is used to denote the phase of thefundamental component, while φb,i = φa,i +Δb,i and φc,i =φa,i +Δc,i are used to indicate the phase distortions relative toa balanced three-phase system. Notice that the only commonparameter for all the nodes in the neighbourhood of node i isthe digital frequency 2πfT .

A. Clarke Transform for Dimensionality Reduction

Although, the frequency of the system can be estimateddirectly from any one of the three-phases, utilising the infor-mation from all three phases to estimate the frequency is morerobust to noise [46]. We would like to estimate the frequency ofthe signal in (22) by first reducing the dimensionality of the sig-nal from R

3 to C. Instead of using any single-phase voltage, wejointly-represent the three phases by firstly projecting the signalon to an orthogonal basis using the Clarke (or αβ) transform[

vα,i,nvβ,i,n

]=

√2

3

[1 − 1

2 − 12

0√32 −

√32

]︸︷︷︸

ClarkeMatrix

⎡⎣va,i,nvb,i,nvc,i,n

⎤⎦ (23)

thus converting the αβ voltage vector on the left side of (23)into a complex-valued scalar vi,n = vα,i,n + jvβ,i,n. The trans-formation of the three-phase voltage vector vi,n in (22) into acomplex-valued scalar can be summarized as

vi,n = eHvi,n = eHsi,n + eHzi,n

= si,n + zi,n(24)


where eH =√

23

[1 ej

2π3 e−j 2π

3

]and si,n and zi,n are

complex-valued scalars corresponding to the signal and thenoise respectively. Using Euler’s formula, the noise-free signalsi,n in (22) can be expressed as

si,n =1

2

⎡⎣Va,i,n V ∗a,i,n

Vb,i,n V ∗b,i,n

Vc,i,n V ∗c,i,n

⎤⎦[ ejωnT

e−jωnT

](25)

where the phasors are given by Va,i,n = Va,i,nejφa,i ,

Vb,i,n = Vb,i,nej(φb,i− 2π

3 ), and Vc,i,n = Vc,i,nej(φc,i+

2π3 ).

Substituting (25) into (24) gives

si,n =eHsi,n = Ai,nejωnT +Bi,ne

−jωnT (26)

where the scalar phasors Ai,n and Bi,n are

Ai,n =

√6

6

[Va,i,ne

jφa,i + Vb,i,nejφb,i + Vc,i,ne

jφc,i]

Bi,n =

√6

6

[Va,i,ne

−jφa,i + Vb,i,ne−j(φb,i+

2π3 )

+ Vc,i,ne−j(φc,i− 2π

3 )] (27)

and the complex-valued observation noise is

zi,n =eHzi,n =

√2

3

[za,i,n + ej

2π3 zb,i,n + e−j 2π

3 zc,i,n

].

Therefore, under general operating conditions, the complex-valued voltage in (24) takes the form

vi,n =Ai,nejωnT +Bi,ne

−jωnT + zi,n. (28)

For a balanced system under nominal conditions, Va,i,n =Vb,i,n = Vc,i,n and φa,i = φb,i = φc,i, so that the coefficientBi,n vanishes, see (27), and the complex-valued voltage is

vi,n =si,n + zi,n = Ai,nejωnT + zi,n. (29)

Moreover, the signal si,n in (29) can be expressed as anautoregressive (AR) model4 sn = ejωT sn−1 such that at eachnode i, the voltage measurement is

vi,n = ejωT si,n−1 + zi,n. (30)

Fig. 2 illustrates that for a balanced system, for which Va,i,n =Vb,i,n = Vc,i,n and φa,i = φb,i = φc,i, the Clarke’s voltage vi,nin (30) has a circular trajectory, thus making the signal circular.The system in (30) that generates the circular signal is strictlylinear because the output of the system is only a function of theinput si,n−1 and not its conjugate s∗i,n−1.

Strictly Linear State Space Model 1. It is based on the AR(1)voltage evolution model in (30) whose strictly linear state spacemodel for the balanced system system at a node i is given in

(32a) and (32b), where the state variables are hndef= ejωT and

4The usual assumption in this type of estimation is that for a samplingfrequency >> 50Hz, we have An ≈ An−1.

Fig. 2. For a balanced system, characterised by Va,i,n = Vb,i,n = Vc,i,n andΔb,i = Δc,i = 0, the trajectory of Clarke’s voltage vi,n is circular (blue line).For unbalanced systems, the voltage trajectories are noncircular (red and greenlines).

the noise-free signal si,n, while wn and zi,n are respectivelythe state and observation noises. The system frequency is thenderived from the state variable hn as

fn =1

2πTarctan

(Im {hn}Re {hn}

)(31)

where Re {·} and Im {·} are respectively the real and imaginaryparts of a complex variable.

State Space Model 1. Strictly Linear State Space (StSp-SL)

1: State Equation[hn

si,n

]=

[hn−1

si,n−1hn−1

]+wn (32a)

2: Observation Equation

vi,n =[0 1

] [ hn

si,n

]+ zi,n (32b)

The strictly linear system model in (30) is inaccurate whenthe system is operating under unbalanced conditions, such aswhen the voltage amplitudes Va,i,n, Vb,i,n and Vc,i,n are nolonger equal or if the condition Δb,i = Δc,i = 0 is not satis-fied. In these cases, Bi,n in (27) is not zero and the signalbecomes noncircular (ellipse in Fig. 2). Therefore, this stan-dard strictly linear estimator introduces biased estimates thatoscillate at twice the system frequency for unbalanced systemconditions [17].

Strictly Linear AR(2) based State Space Model. Withoutassuming a balanced operating condition, the αβ voltage in (26)can be expressed as a strictly linear AR(2) model [47]–[50]

si,n = 2 cos(ωT )si,n−1 − si,n−2 (33)

The system frequency can be derived by defining the stateas 2 cos(ωT ) and estimating it using a complex Kalman fil-ter. This strictly linear AR(2) model is referred to as thestrictly linear “three-point” (SL3PT) model and serves as abenchmark for our proposed method, as it can be applied


to both any single-phase voltage and the complex-valued αβvoltage. A major disadvantage to the three-point model in(33) is that it produces biased estimates in the presence ofnoise [51].

Widely Linear AR(1) based State Space Model 2. Observethat the αβ voltage in (26) can be interpreted as the sum of twophasors, one rotating clockwise and the other rotating counterclockwise at the same frequency.

If we were to express the signal (26) in an autoregressiveform, it is only natural and intuitive to consider the previousvalue si,n−1 and its conjugate s∗i,n−1, where the conjugate rep-resents the phasor rotating in the opposite direction. The widelylinear model for the complex-valued αβ voltage at any node iis therefore given by [16]–[18]

si,n =hi,n−1si,n−1 + gi,n−1s∗i,n−1. (34)

This is a first-order widely linear autoregressive model withcoefficients hi,n and gi,n. By substituting the values of si,n−1

and s∗i,n−1 from (26) into (34) and some algebraic manipula-tion, we obtain [17]

hi,n = ejωT − B∗i,n

Ai,ngi,n, gi,n =

Bi,n

A∗i,n

(e−jωT − hi,n

).

(35)

Solving the two equations in (35) for the system frequencyyields

fn =1

2πTarctan

⎛⎝√

Im2{hi,n} − |gi,n|2Re {hi,n}

⎞⎠ . (36)

From (35), notice that when the system is balanced (Bi,n =0), the coefficient gi,n = 0, and the widely linear frequencyestimate in (36) is identical to its strictly linear counterpart in(31). While the strictly linear AR(2) model in (33) is iden-tical for both balanced and unbalanced voltages, the widelylinear model provides an intuitive advantage as the coefficientgi,n represents the negative sequence which characterises theimbalance of the system voltage.

The state space model in (37a) and (37b) provides a real-istic and robust characterisation of real world power systems,as it represents both balanced and unbalanced systems, whileits nonlinear state equation also models the coupling betweenstate variables. State Space Model 2 can be implemented usingthe proposed distributed augmented complex extended Kalmanfilter in Section IV.

State Space Model 2 Widely Linear State Space (StSp-WL)

1: State Equation⎡⎢⎢⎢⎢⎢⎢⎢⎣

hi,n

gi,nsi,nh∗i,n

g∗i,ns∗i,n

⎤⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

hi,n−1

gi,n−1

si,n−1hi,n−1 + s∗i,n−1gi,n−1

h∗i,n−1

g∗i,n−1

s∗i,n−1h∗i,n−1 + si,n−1g

∗i,n−1

⎤⎥⎥⎥⎥⎥⎥⎦+wn (37a)

2: Observation Equation

[vi,nv∗i,n

]=

[0 0 1 0 0 00 0 0 0 0 1

]⎡⎢⎢⎢⎢⎢⎢⎣

hi,n

gi,nsi,nh∗i,n

g∗i,ns∗i,n

⎤⎥⎥⎥⎥⎥⎥⎦+

[zi,nz∗i,n

](37b)

B. Diffusion-step

The classical diffusion scheme in (16) obtains the weightedaverage of the estimated states xa

i,n|n in the neighbourhood Ni

under the assumption that each node is estimating the sameoptimal state. However, the state estimate of the State SpaceModel 2, which is given by

xai,n|n =

[hi,n|n, gi,n|n, si,n|n, h∗

i,n|n, g∗i,n|n, s

∗i,n|n

]T(38)

is unique at each node since the coefficients hi,n and gi,n arefunctions of both system frequency ω (which is identical inthe network) and the level of imbalance, Bi,n

Ai,n(which is not

necessarily identical in the network) – see (35). Applying thediffusion scheme without considering this phenomenon willtherefore result in spurious frequency estimates when the levelof system imbalance is different at each node.

This problem is circumvented by the fact that the state xai,n|n

also includes the noise-free signal estimate si,n|n. When all thenodes are connected and weighed equally, each node effectivelyestimates the frequency from the diffused signal estimate si,n|nformed by diffusing the signal estimates s�,n|n from each node

si,n|n =1

|Ni|∑�∈Ni

s�,n|n = Ai,nejωnT + Bi,ne

−jωnT (39)

where Ai,n = 1|Ni|

∑�∈Ni

A�,n and Bi,n = 1|Ni|

∑�∈Ni

B�,n

are the diffused phasors in the neighbourhood Ni.Remark 4: Since the frequency is derived from the diffused

signal estimate in (39), which is now identical throughout thenetwork, our proposed D-ACEKF yields the correct frequencyestimate even with different levels of imbalance at differentnodes.

VI. FREQUENCY ESTIMATION EXAMPLES

A. Experiment (Synthetic Data) Set-Up

The simulations were based on a network of 5 substations(nodes) where each substation has access to three-phase volt-age measurements via transformers with metering capabilities.Without loss in generality, we used the distributed networktopology shown in Fig. 3. The power system under consid-eration had a nominal frequency of 50Hz, and was sampledat a rate of 5kHz while the signal to noise ratio (SNR) wasdetermined by the metering accuracy class of the potentialtransformer. The BS EN 61869-1:2009 standard for the meter-ing accuracy of potential transformers classifies six separate


Fig. 3. A distributed power network with N = 5 nodes (Sub-stations) used inthe simulations.

classes for metering requirements, which translates to an SNRrange of 30 dB to 60 dB [52]. To illuminate the robustness ofour proposed augmented diffusion Kalman filters, we have cho-sen an SNR level of 25 dB in all our simulations, unless statedotherwise.

B. Algorithms

The proposed D-ACEKF (Algorithm 3) was benchmarkedagainst the D-CEKF which uses State Space Model 1 andthe distributed strictly linear three point (D-SL3PT) algorithmwhich uses the signal model in (33) with the D-CKF inAlgorithm 2. The D-SL3PT was chosen since it is a well-knowclassical algorithm which employs a similar principle to the D-ACEKF (i.e. estimating the frequency with an autoregressivemodel of the signal). Single node (uncooperative) estimates ofthe algorithms are also presented for completeness.

Case Study #1: Voltage sags. In the first set of simula-tions, the performances of the algorithms were evaluated foran initially balanced system which became unbalanced afterundergoing a Type C voltage sag starting at 0.1s, characterisedby a 20% voltage drop and 10o phase offset on both the vb andvc channels, followed by a Type D sag starting at 0.3s, char-acterised by a 20% voltage drop at line va and a 10% voltagedrop on both vb and vc with a 5o phase angle offset. The pha-sor trajectories and degrees of noncircularity of these systemimbalances are illustrated in Fig. 4.

Fig. 5 shows that, conforming with the analysis, the widelylinear algorithms, ACEKF and D-ACEKF, were able to con-verge to the correct system frequency for both balanced andunbalanced operating conditions, while the strictly linear algo-rithms, CEKF and D-CEKF, were unable to accurately estimatethe frequency during the voltage sag due to under-modeling ofthe system (not accounting for its widely linear nature) – see(34). As expected, the widely linear and strictly linear algo-rithms had similar performances under balanced conditions, asillustrated in the time interval 0–0.1 s. The distributed algo-rithms, D-CEKF and D-ACEKF, outperformed their uncoop-erative counterparts, CEKF and ACEKF, owing to the sharingof information between neighbouring nodes.

Case Study #2: Presence of switching noise. Fig. 6 illus-trates frequency estimation in the presence of random spikenoise, which models the switching noise from the invert-ers which interface the renewable sources to the grid. The

Fig. 4. Circularity (left) and phasor (right) views of Type C and D unbalancedvoltage sags. The real-imaginary phasor plots illustrate the noncircularity ofClarke’s voltage in unbalanced conditions, indicated by the elliptical shapes ofcircularity plots. The eccentricity of this ellipse (degree of noncircularity) serveto identify the type of fault (in this case a voltage sag).

distributed algorithms, D-CEKF and D-ACEKF, outperformedtheir uncooperative counterparts, CEKF and ACEKF, becauseneighbouring nodes were able to share information to facilitatebetter estimation performances.

Case Study #3: Bias and variance of the proposed estima-tors. Fig. 7 illustrates the bias and variance for the proposeddistributed frequency estimators. The steady state frequencyestimate at a node i for the trial m is given by fi,ss[m]. Thebias and variance of the frequency estimators were calculatedover 2000 independent trials using

Bias =1

2000 ·N2000∑m=1

N∑i=1

fi,ss[m]− f0

Variance =1

2000 ·N2000∑m=1

N∑i=1

(fi,ss[m]− fi,ss

)2 (40)

where N = {1, 5} for single-node and distributed cases respec-tively, while f0 = 50Hz is the fundamental frequency. Thesample mean was evaluated as fi,ss =

12000

∑2000m=1 fi,ss[m].

The algorithms were evaluated at different SNR levels for anunbalanced system undergoing a Type D voltage sag. Observethat both the single- and multiple-node widely linear algo-rithms, ACEKF and D-ACEKF, were asymptotically unbiased(left panel) while both the single- and multiple-node strictlylinear algorithms were biased. In terms of the variance of theestimators (right panel), both the distributed estimation algo-rithms outperformed their non-cooperative counterparts, whilethe only consistent estimator was the proposed D-ACEKF.

Case Study #4: Benchmark against classical methods. TheD-ACEKF was compared to the single-node (uncooperative)


Fig. 5. Frequency estimation performance of single node (CEKF and ACEKF) and distributed (D-CEKF and D-ACEKF) algorithms for a system at 25dB SNR.The system is balanced up to 0.1s, it then undergoes a Type C voltage imbalance followed by a Type D voltage imbalance at 0.3s.

Fig. 6. Frequency estimation performance of single node (CEKF and ACEKF) and distributed (D-CEKF and D-ACEKF) algorithms when the phase voltages ofthree nodes in the network are contaminated with random spike noise.

Fig. 7. Bias and variance analysis of the proposed distributed state space frequency estimators for an unbalanced system undergoing a Type D voltage sag. Left:Estimation bias. Right: Estimation variance.

Fig. 8. Frequency estimation performance of single node (SL3PT and ACEKF) and distributed (D-SL3PT and D-ACEKF) algorithms for a system at 50dB SNR.The system is balanced up to 0.1s, it then undergoes a Type C voltage imbalance followed by a Type D voltage imbalance at 0.3s.

and distributed implementations of the classical strictly linearthree-point (SL3PT) model given in (33). The set-up in CaseStudy #1 was used with the exception that the SNR wasincreased to 50 dB as the SL3PT was performing poorly atlower SNRs. For a fair comparison, the D-SL3PT was con-figured to have the same convergence speed as the D-ACEKF.From Fig. 8, we can observe that the D-SL3PT produced nois-ier estimates compared to the D-ACEKF when their adaptationspeeds were equalised.

Next, the bias and variance of the D-ACEKF were bench-marked against the D-SL3PT using the methodology in (40).

As seen in Fig. 9, while the variance of the frequency estimatesdecreased with the diffusion scheme, the bias of the SL3PTmodel did not. As mentioned in Section V, the SL3PT and D-SL3PT both exhibit high bias due to modelling inaccuracies inthe presence of noise [51]. This shows the markedly increasedrobustness to noise of the widely linear model compared to thestrictly linear AR(2) model.

Case Study #5: Different Voltage Sags. Fig. 10 shows theprofile of the voltages at different nodes in the network. Eachsubstation underwent different faults, which is reflected bythe different relative amplitudes and phase shifts of the αβ


Fig. 9. Bias and variance analysis of the proposed distributed state space frequency estimators against classical strictly linear three-point (SL3PT) models for anunbalanced system undergoing a Type D voltage sag. Left: Estimation bias. Right: Estimation variance.

Fig. 10. Each substation (node) has a different αβ voltages including caseswhere the voltage drops to zero (line cut).

Fig. 11. The D-ACEKF is able to estimate the frequency in a distributed settingwith different types of faults at each node, see Fig. 10 for voltage profiles.

voltages. In addition Substations 2 and 4 underwent total linefailures from 0.3s to 0.5s and 0.1s to 0.3s respectively.

Fig. 11 shows that due to its unique state space structure, theproposed D-ACEKF was able to estimate the frequency of thenetwork even with different levels of imbalance at each node,see Remark #4 in Section V-B.

Case Study #6: Frequency tracking. Fig. 12 illustrates theperformance the D-ACEKF when a power network was con-taminated with white noise at 25dB and 60dB SNR and under-went a gradual drop and increase in frequency from 0.1s to 0.3s.This is a typical scenario when generation does not match theload and system inertia keeps the frequency from changing tooquickly. From 0.3s to 0.5s the system undergoes a step-changein frequency; this is the scenario when there is not enough

Fig. 12. Frequency tracking performance of D-ACEKF at 25dB and 60dBSNR, which experiences a gradual change fro, 0.1s to 0.3s and a step changein system frequency to 48Hz at 0.3s. The D-ACEKF is able to track both slowand rapid changes in frequency.

Fig. 13. The mean square error (MSE) of the D-ACEKF which shares only thestate estimates (standard diffusion scheme) and full D-ACEKF which sharesboth state estimates and observations (proposed diffusion scheme) with variousdegrees of correlation between three nodes in the network.

inertial response in the system caused by the influx of elec-tronic inverters that are replacing synchronous machines. TheD-ACEKF was able to track the frequency in both cases illus-trating that its suitability for both the current electricity grid andfuture smart grids.

Case Study #7: Noise Correlation. Next, the D-ACEKF wasevaluated for the case where the observation noises betweenthree nodes were correlated, exhibiting various degrees ofcorrelation. The proposed D-ACEKF was compared to the D-ACEKF with the classical diffusion scheme where only the


Fig. 14. Real world case study. Left: The αβ voltages at Sub-station 1 before and during the fault event. Right: Frequency estimation using the proposed algorithmsbefore and during the fault event.

states were shared between the nodes. As seen in (20), the pro-posed diffusion scheme shares both observations vi,n and stateestimates xi,n|n, which allows the proposed diffusion schemeto exploit cross-nodal correlations of the observation noise.Fig. 13 shows the mean square error (MSE) of the frequencyestimates where MSE = Bias2 +Variance, and the Bias andVariance terms are defined in (40). The proposed D-ACEKFoutperformed the classical diffusion scheme especially whenthe cross-nodal correlations of the noise were large.

Real World Case Study. We next assessed the performanceof the proposed algorithms on a real world case study usingthree-phase voltage measurements from two adjacent sub-stations in Malaysia during a brief line-to-earth fault on the29th June 2014. This caused voltage sags, similar to those inCase Study #1. The three-phase measurements were sampled at5 kHz and the voltage values were normalized. The left panelin Fig. 14 shows the normalized αβ voltages at one of the sub-stations. The fault that occurs in phase A around 0.1s is reflectedin the voltage dip in vα. The right panel in Fig. 14 shows the fre-quency estimate from the D-ACEKF and D-CEKF. Conformingwith the analysis and the single node scenario in Fig. 5, the col-laborative widely linear D-ACEKF was able to track the realworld frequency of a power network under both balanced andunbalanced conditions, whereas the strictly linear D-CEKF wasunable to track the frequency after 0.1s when the line-to-earthfault (non-circularity) occurred.

VII. CONCLUSION

We have proposed a novel class of diffusion based distributedcomplex-valued Kalman filters for cooperative frequency esti-mation in power networks. To cater for the general case ofimproper states, observations, and state and observation noises,we have introduced the distributed (widely linear) augmentedcomplex Kalman filter (D-ACKF) and its nonlinear version,the distributed augmented complex Kalman filter (D-ACEKF).These have been shown to provide sequential state estimation ofthe generality of complex signals, both circular and noncircular,within a general and unifying framework which also caters forcorrelated nodal observation noises. This novel widely linearframework has been applied for distributed state space basedfrequency estimation in the context of three-phase power sys-tems, and has been shown to be optimal for both balanced andunbalanced operating conditions. Simulations over a range ofbalanced and unbalanced power system conditions and for bothsynthetic and real world measurements have illustrated that the

proposed distributed state space algorithms are consistent esti-mators, offering accurate and fast frequency estimation in bothbalanced an unbalanced system conditions.

APPENDIX

A. Bias Analysis of the D-ACKF Estimates

To analyse the mean performance of the D-ACKF(Algorithm 2), we employ the methodologies in [22][30]. Firstconsider the local (non-diffused) error at node i ∈ [1, N ]

eai,n|ndef=xa

n − xai,n|n. (41)

Substituting Step 2.5 in Algorithm 2, into the non-diffused error(41) gives

eai,n|n =xan − xa

i,n|n−1 −Gai,n

(yai,n

−Hai,nx

ai,n|n−1

). (42)

Using the optimal observation equation yan in (10), this non-

diffused error can be expressed as

eai,n|n =(I−Ga

i,nHai,n

)eai,n|n−1 −Ga

i,nvai,n. (43)

Now consider the diffused a priori error eai,n|n−1

def= xa

n −xai,n|n−1. Subtracting xa

n in (10) from Step 2.2 in Algorithm2 yields the expression for the diffused a priori error as

eai,n|n−1 = Fan−1e

ai,n−1|n−1 +wa

n. (44)

Finally, consider the diffused a posteriori error eai,n|ndef= xa

n −xai,n|n which can be expressed in terms of the local (non-

diffused) errors in (41) using the diffusion step in (16) as

eai,n|n =∑

�∈Ni

c�iea�,n|n. (45)

Substituting (43) and (44) into (45) and usingMa

�,n|n(Ma�,n|n−1)

−1 = I−Ga�,nH

a�,n from the covariance

update in Step 2.6 in Algorithm 2, we have

eai,n|n =∑�∈Ni

c�i

[Ma

�,n|n(Ma�,n|n−1)

−1Fan−1e

a�,n−1|n−1

+Ma�,n|n(M

a�,n|n−1)

−1wan −Ga

�,nva�,n

]. (46)

Upon taking the statistical expectation E {·}, the recursion in(46) leads to a closed form expression for the mean error of theD-ACKF algorithm, given by

E{eai,n|n

}=∑�∈Ni

c�iMa�,n|n(M

a�,n|n−1)

−1Fan−1E

{ea�,n−1|n−1

}= 0. (47)


Remark 5: The D-ACKF is an unbiased estimator of bothproper and improper complex random signals.

ACKNOWLEDGMENT

The authors would like to thank G. K. Supramaniam fromTNB Malaysia for his insights and for providing us with thereal-world measurements.

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Sithan Kanna (S’13) received the M.Eng. degree inelectrical and electronic engineering with manage-ment from Imperial College London, London, U.K.,in 2012. His research interests include complex-valued statistical signal processing and frequencyestimation in the smart grid. He was awarded a fullscholarship by The Rector’s Fund at Imperial CollegeLondon to pursue the Ph.D. degree in adaptive signalprocessing.

Dahir H. Dini received the M.Eng. degree in electri-cal and electronic engineering and the Ph.D. degreein adaptive signal processing from Imperial CollegeLondon, London, U.K., in 2009, and 2014, respec-tively. His work on frequency estimation for thesmart grid has been the subject of several papers andpatents.

Yili Xia (M’11) received the B.Eng. degree ininformation engineering from Southeast University,Nanjing, China, in 2006, the M.Sc. (Hons.) degreein communications and signal processing from theDepartment of Electrical and Electronic Engineering,Imperial College London, London, U.K., in 2007, andthe Ph.D. degree in adaptive signal processing fromthe same college in 2011.

He has been a Research Associate with ImperialCollege London since graduation, and is currently anAssociate Professor with the School of Information

and Engineering, Southeast University, Nanjing, China. His research interestsinclude linear and nonlinear adaptive filters, and complex valued and quaternionvalued statistical analysis.

S. Y. (Ron) Hui (M’87–SM’94–F’03) receivedthe B.Sc. (Hons.) degree in engineering from theUniversity of Birmingham, Birmingham, U.K., in1984, and the D.I.C. and Ph.D. degree from ImperialCollege London, London, U.K., in 1987. Currently,he is the Philip Wong Wilson Wong Chair Professorwith the University of Hong Kong, Hong Kong, anda part-time Chair Professor with Imperial CollegeLondon. He has authored over 300 technical papers,including more than 190 refereed journal publica-tions. Over 55 of his patents have been adopted

by industry. Dr. Hui is an Associate Editor of the IEEE TRANSACTIONS

ON POWER ELECTRONICS and the IEEE TRANSACTIONS ON INDUSTRIAL

ELECTRONICS, and an Editor of the IEEE JOURNAL OF EMERGING AND

SELECTED TOPICS IN POWER ELECTRONICS. He is a Fellow of the AustralianAcademy of Technological Sciences and Engineering. He was the recipientof the IEEE Rudolf Chope R&D Award from the IEEE Industrial ElectronicsSociety and the IET Achievement Medal (The Crompton Medal) in November2010 and the 2015 IEEE William E. Newell Power Electronics Award.

Danilo P. Mandic (M’99–SM’03–F’12) is aProfessor of Signal Processing with Imperial CollegeLondon, London, U.K., where he has been involvedin nonlinear adaptive signal processing and nonlineardynamics. He has been a Guest Professor withKatholieke Universiteit Leuven, Leuven, Belgium,the Tokyo University of Agriculture and Technology,Tokyo, Japan, and Westminster University, London,U.K., and a Frontier Researcher with RIKEN, Wako,Japan. He has two research monographs titledRecurrent Neural Networks for Prediction: Learning

Algorithms, Architectures and Stability (West Sussex, U.K.: Wiley, 2001) andComplex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linearand Neural Models (West Sussex, U.K.: Wiley, 2009), an edited book titledSignal Processing Techniques for Knowledge Extraction and InformationFusion (New York, NY, USA: Springer, 2008), and more than 200 publicationson signal and image processing. Prof. Mandic has been a member of theIEEE Technical Committee on Signal Processing Theory and Methods, and anAssociate Editor of the IEEE SIGNAL PROCESSING MAGAZINE, the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS II, the IEEE TRANSACTIONS

ON SIGNAL PROCESSING, and the IEEE TRANSACTIONS ON NEURAL

NETWORKS. He has produced award winning papers and products resultingfrom his collaboration with the industry.

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