Maximum Likelihood Ensemble Filter State Estimation for Power Systems

Abstract

Maximum likelihood ensemble filter (MLEF) is an ensemble-based deterministic filtering method. It optimizes a nonlinear cost function through maximum likelihood and utilizes low-dimensional ensemble space on the calculation of Hessian preconditioning of the cost function. This paper implements the MLEF as a state estimation tool for the estimation of the states of a power system, and presents the first MLEF application study on a power system state estimation. The MLEF methodology is introduced into power systems and the simulations are implemented for a three-node benchmark power system and 68-bus test system which have been employed in several previous studies to address a discontinuous problem where derivative is not defined. This is in contrast to gradient-based methods in the literature that needs gradient and Hessian information which is not defined in jumps. The performance of the filter on the presented problem is analyzed and the results are presented. Results indicate that the estimation convergence is achieved with the MLEF method.

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT 1
Maximum Likelihood Ensemble Filter State
Estimation for Power Systems
Bahri Uzuno˘glu and Muhammed Akif Ülker
Abstract— Maximum likelihood ensemble filter (MLEF) is an
ensemble-based deterministic filtering method. It optimizes a
nonlinear cost function through maximum likelihood and utilizes
low-dimensional ensemble space on the calculation of Hessian
preconditioning of the cost function. This paper implements the
MLEF as a state estimation tool for the estimation of the states of
a power system, and presents the first MLEF application study
on a power system state estimation. The MLEF methodology
is introduced into power systems and the simulations are imple-
mented for a three-node benchmark power system and 68-bus test
system which have been employed in several previous studies to
address a discontinuous problem where derivative is not defined.
This is in contrast to gradient-based methods in the literature
that needs gradient and Hessian information which is not defined
in jumps. The performance of the filter on the presented problem
is analyzed and the results are presented. Results indicate that
the estimation convergence is achieved with the MLEF method.
Index Terms—Control systems, dynamic state estimation, opti-
mization, power system measurements, power systems.
NOMENCLATURE
Pf(k)Forecast error covariance matrix.
Mk−1,kNonlinear model evolution matrix.
(k−1)Model error covariance matrix.
kTime index.
SSize of the ensembles.
NDimension of the state vector.
Pa(k−1)Analysis covariance matrix.
Xk−1Analysis from the previous step.
(Xk−1+pi)One-step control estimate.
xkState vector.
ykObservation vector.
RObservation error covariance matrix.
HNonlinear observation operator.
xbBackground state from the prior step.
CInformation matrix.
ξVector of control variables.
EEigenvector matrix.
Manuscript received July 27, 2017; revised January 22, 2018; accepted
January 28, 2028. This work was supported in part by Computational
Renewables LLC and in part by European Union H2020 ERA-Net Smart-
Grid Project Multi-Input Intelligent Distribution Automation System. The
Associate Editor coordinating the review process was Dr. Roberto Ferrero.
(Corresponding author: Bahri Uzuno ˘glu.)
B. Uzuno˘glu is with the Ångström Laboratory, Division of Electricity,
Department of Engineering Sciences, Uppsala University, SE-751 21 Uppsala,
Sweden, and also with the Department of Mathematics, Florida State Univer-
sity, Tallahassee, FL 32310 USA (e-mail: bahri.uzunoglu@angstrom.uu.se;
bahriuzunoglu@computationalrenewables.com).
M. A. Ülker was with the Ångström Laboratory, Division of Electricity,
Department of Engineering Sciences, Uppsala University, SE-751 21 Uppsala,
Sweden.
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIM.2018.2814066
Eigenvalue matrix.
δmGenerator angle.
ωGenerator angular velocity.
δLoad angle.
VLoad voltage.
MGenerator inertia.
dmGenerator damping.
PmGenerator mechanical power.
nTap ratio.
Q1Constant load reactive power.
P1Constant load active power.
kModel noise.
RkObservation noise.
I. INTRODUCTION
INTEGRATION of new energy sources to the grids with
growing need of energy leads to more complex and larger
power grids. Therefore, it is crucially necessary to have
appropriate monitoring, optimization, and control functions for
the energy management systems (EMSs) or the supervisory
control and data acquisition (SCADA) systems to maintain
reliable and sustainable operation of the power grids. State
estimation is one of the most critical applications to assist
these functions on EMS/SCADA systems.
Since introduction of the state estimation into the power
systems by Schweppe and Wildes [1], it has been a tool to
track the states of the power systems. State estimation for
power systems is mainly classified as static and dynamic.
Static state estimation (SSE) is based on steady-state models.
It is common to experience disconnection issues in power
grids under an emergency event or a power quality require-
ment mismatch which cause to fast grid topology changes.
Human-initiated actions may not be sufficient to deal with
the problem. Therefore, the supervisory systems should be
equipped with active control functions which might operate
autonomously [2], [3]. In order to have a continuous monitor-
ing of the power system, state estimation must be performed
at short intervals of time in view of the fact that the power
systems become extremely larger with the introduction of
new generations and loads to the systems. In addition, it is
needed to have a resource efficient tool with the advent of the
phasor measurement unit (PMU) which has high measurement
frequencies and high data exchange capabilities that produce
massive data synchronized by the global positioning systems.
However, SSE is based on a steady state model, is computa-
tionally heavy, which has a slow data update rate and is not
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2IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT
able to capture power system dynamics accurately [2], [4].
This leads to develop another concept called as dynamic
(forecasting-aided) state estimation (DSE). The idea of DSE
is grounded on recursive update of the estimations by a
forecasting tool that is able to predict the system state at the
next time step. The real-time monitoring can be implemented
with DSE more accurately, and the DSE is able to replace the
missing measurements. Hence, this create significant timing
advantages for EMS/SCADA systems on carrying out the
system analysis and taking control actions for the power
systems via various filtering or smoothing methods [2], [4].
In this paper, the maximum likelihood ensemble filter (MLEF)
technique is employed for the dynamic state estimation of a
benchmark power system which is possible to implement for
autonomous grid operations [3].
There are several existing methods used in DSE and most
of them are based on the Kalman filtering theory introduced
by Kalman and Bucy [5]. The pure form of the Kalman filter
(KF) has been widely employed for the state estimation of
the linear Gaussian systems [6]; however, it is linear and is
not preferred for non-Gaussian and nonlinear systems [6], [7].
For transition of Kalman filtering to the nonlinear and non-
Gaussian systems, techniques such as extended KF (EKF),
ensemble KF (EnKF), unscented KF (UKF), and particle
filter (PF) algorithms are developed [6], [7], and are employed
to wide range of problems from low- to high-dimensional sys-
tems [8]–[13]. The EKF method is implemented by lineariza-
tion of the nonlinearities via using a Jacobian matrix for DSE.
Its limitations are discussed in [14]. The EnKF uses Monte
Carlo methods that help to estimate the error covariances of the
background error, get an approximation to the Kalman–Bucy
filter, and produce an ensemble of initial conditions that can
be utilized in an ensemble forecasting system [12], [15], [16].
The EnKF embeds the nonlinearities into the original linear KF
solution, and it uses an extra covariance inflation to consider
the nonlinearities [17]. The UKF employs the calculation of
an approximate mean and covariance as a linear combination
of a number of propagated points (called as sigma points)
[11], [14], however, it also contains drawbacks addressed
in [18]. The PF and its variants also use the Monte Carlo simu-
lation with sampling method-based approximation of the pos-
terior density of the state vector rather than doing any explicit
functions so it simulates nonlinearity and non-Gaussianity.
Even though PF is a competent tool for the estimation of
the nonlinear and non-Gaussian systems, including power sys-
tems [13], [18]–[23], it needs attention in sampling [24], [25].
The MLEF differs from the EnKF and PF by working on state
space rather than sample space, and it optimizes a nonlinear
cost function through maximum likelihood practice which
reduces the computational time. In essence, it is the dynamic
version of the optimization routines used in SSE and addresses
the stochasticity and the discontinuity while it utilizes the
sampling in low-dimensional space and employs the Hessian
information reported in [17] and [26]–[29]. Several previous
studies in the past employed the MLEF already illustrates its
successful performance in comparison to other methods in
various areas in the context of the data assimilation from low-
to high-dimensional systems [17], [27], [30], [31]. This paper
will be the first application of the MLEF on a power system
state estimation problem.
The remainder of this paper is organized as follows.
In Section II, the MLEF methodology is summarized.
In Section III, the benchmark power system model is
explained. In Section IV, numerical simulations for state
estimation are implemented for the benchmark model and
results are presented. Concluding remarks are in the closing
Section V.
II. MAXIMUM LIKELIHOOD ENSEMBLE FILTER
In this section, the MLEF methodology [17] is shortly
described. The calculation of the maximum likelihood state
estimate is performed using an iterative minimization algo-
rithm as a result the MLEF approach is related to the iterated
KF [6]. The cost function defines an analysis problem that is
nonlinear. This optimization links MLEF with ensemble data
assimilation and a control theory. The MLEF employs an iter-
ative minimization algorithm to obtain a maximum likelihood
state solution. The MLEF consists of two different stages,
called as forecast and analysis steps which is introduced in
Sections II-A and II-B.
A. Forecast Step
The forecast step is connected with the forecast error
covariance evolution regarding the discrete KF [6] is expressed
as
Pf(k)=Mk−1,kPa(k−1)MT
k−1,k+(k−1)(1)
where Pf(k),Mk−1,k,and(k−1)denote the forecast
error covariance matrix, the nonlinear model evolution matrix,
and the model error covariance matrix, respectively. kis the
time index and Pa(k−1)represents the analysis covariance
matrix. Model error is ignored in this paper. With factorization,
the forecast error covariance without time index is expressed
as
(P1/2
f)P1/2
fT=MP1/2
aMP1/2
aT(2)
and the square root of the analysis error covariance matrix is
Pa(k−1)1/2=(p1p2... pS)where pi=⎛
⎜
⎜
⎜
⎝
p1,i
p2,i
.
.
.
pN,i
⎞
⎟
⎟
⎟
⎠(3)
wherein Sis the size of the ensembles and Ndenotes the
dimension of the state vector with SNin practical
implementations. Beside the data assimilation steps, columns
of P1/2
acan also be employed as initial perturbations for
ensemble prediction. In the same way, the square root of the
forecast error covariance matrix is
Pf(k)1/2=(b1b2... bS)(4)
and
bi=M(Xk−1+pi)−M(Xk−1)≈Mpi(5)

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UZUNO ˘
GLU AND ÜLKER: MLEF STATE ESTIMATION FOR POWER SYSTEMS 3
where Xk−1is the analysis from the previous step,
Xk−1+piis the nonlinear ensemble one-step estimate, and
M(Xk−1)is the one-step control estimate which corresponds
to the most likely dynamic state and is obtained from the
maximum likelihood approach.
B. Analysis Step
The maximum likelihood approach is used to derive the
analysis Xk−1that maximizes the posterior probability distri-
bution [32]. In the analysis step of the MLEF, the problem is
transformed into the minimization of a cost function in similar
form with [32] assuming the Gaussian PDF, that is,
J(x)=1
2(x−xb)TP−1
f(x−xb)
+1
2[y−H(x)]TR−1[y−H(x)](6)
where xrepresents the state vector, ydenotes the observation
vector, Ris the observation error covariance matrix, His the
nonlinear observation operator, and xbis the background state
that is obtained from the prior step. Consider that the definition
of Pfis only in the ensemble subspace, correspondingly the
inverse of Pfis also kept in the range of the same subspace.
Thus, the cost function exists effectively also in the range of
Pf. To minimize the cost function effectively in the ensemble
subspace, the Hessian preconditioning is implemented via a
change of variable
x−xb=P1/2
f(I+C)−T/2ξ(7)
where ξdenotes the vector of control variables in the ensemble
subspace and
C=PT
2
fHTR−1HP1
2
f=R−1
2HP1
2
fTR−1
2HP1
2
f.(8)
Here, Cis employed as an information matrix and His in the
Jacobian matrix form. In order to avoid possible analytical
problems to obtain the Cmatrix, it is approximated via the
help of previously known square root forecast error covariance
matrix columns [17] as
zi=R−1
2HP1
2
fi=R−1
2Hbi
≈R−1
2H(x+bi)−R−1
2H(x). (9)
After the preconditioning (7), the nonlinear optimization
problem (6) can be handled via gradient calculation-based iter-
ative minimization in the ensemble subspace. It is important
to note that the adjoint (i.e., transpose) use is avoided via the
help of finite difference approximation on the calculation of
the gradient [17]. If the operators used are linear, the first
minimization iteration via the preconditioned steepest descent
is equivalent to the ensemble-based reduced-rank KF [33] or to
the Monte Carlo-based EnKF [17], [34].
Since zihas the same dimension with observation space
and if a matrix Zis defined as Z=[z1z2... zS],thenit
gives us opportunity to write Cas C=ZTZ. With acquiring
the minimum solution via conjugate-gradient algorithm [35],
[36], the Cmatrix can be updated and then the square root
analysis error covariance P1/2
acan be calculated with
P1/2
a=P1/2
f[I+C(xopt)]−T/2(10)
Fig. 1. Benchmark power system model with tap changer.
where xopt is the minimum of the cost function (6). Thereafter,
the columns of the P1/2
amatrix is utilized for the initial
perturbations for the ensembles for the latter cycle. Here, for
efficient calculation of the inversion and square root contained
in (I+C)−T/2, the eigenvalue decomposition is employed for
the matrix Cand it is obtained as C=EET,whereEis
the eigenvector matrix and is the eigenvalue matrix. Then,
it can be written as
(I+C)−T/2=E(I+)−1/2ET.(11)
The more detailed explanation regarding to the MLEF algo-
rithm can be found in [17]. MLEF is related to ensemble
transform Kalman filter (ETKF) via the definition of matrix C.
The eigenvalue decomposition of Cis equivalent to the matrix
transform of ETKF [37].
III. BENCHMARK POWER SYSTEM MODEL
In this section, the benchmark power system model that is
considered for the application of MLEF state estimation is
summarized. The model is originated from [38], and in this
paper, its extended version in [39] is used which is depicted
in Fig. 1(a) and (b) with referred parameters. The model
has been used in various studies in various forms due to its
dynamic behavior that shows nonlinearity [38]–[41] that can
be observed in Fig. 2 in the form of load voltage and generator
angle states phase diagram from a simulation of the system
with the parameters in [39].
The model considered in this paper, which is depicted
in Fig. 1, can be seen as an equivalent circuit of a local area
connection to a larger external network. The model consists
of a load bus and two generator buses resulting a three-node
system [38] and a tap changer for voltage control [39]. One of
the generator buses is considered as slack bus that represents
the external network. The two-axis model is considered for the
generator in the system. The system load is represented by a
dynamic induction motor and in parallel a constant PQ load
which is treated in terms of power and reactive power demands
by load voltage and frequency. Load bus also includes a
capacitor to keep the voltage magnitude at a proper value.

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4IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT
Fig. 2. Power system model phase space trajectory with initial conditions
[0.20.20.04 0.98]and Q1=11 pu, P1=0 pu, and n=1.
The model here is used as a benchmarking test case and has
some assumptions elaborated introduced as follows.
Generator Model:
˙
δm=ω(12)
M˙ω=−dmω+Pm+nEmYm
n2V
×sin(δ −δm−θm)+(nEm)2Ym
n2sin(θm). (13)
Load Model:
Kqw˙
δ=−Kqv2V2−KqvV+Q
+Q0−Q1(14)
TKpvKqw˙
V=−KpwKqv2V2+(KpwKqv−KqwKpv)V
+Kqw(P−P0−P1)
−Kpw(Q−Q0−Q1)(15)
where M,dm,andPmrepresents the generator inertia, damp-
ing, and mechanical power, respectively. Here, δmis the
generator angle (in radians) and ωdenotes the generator
angular velocity (in radians per second). δrepresents the load
angle (in radians) and Vis the load voltage (in pu). While the
parameters P0and Q0denote the constant real and reactive
powers of the motor, P1and Q1represent the PQ load.
Kcoefficients are arised from the induction motor load model
and are detailed in [38] and [42]–[44]. The static components
of the benchmark model in [38] and [39] are given in the
following.
Network Model:
P=−E
0Y
0Vsin δ+θ
0
−nEmYm
n2Vsin(δ −δm+θm)
+Y
0sin θ
0+Ym
n2sin(θm)V2(16)
Q=E
0Y
0Vcos δ+θ
0
+nEmYm
n2Vcos(δ −δm+θm)
+Y
0cos θ
0+Ym
n2cos(θm)V2.(17)
Here, adjusted values E
0,Y
0,andθ
0of the Thevenin equiva-
lent circuit for accounting the capacitor are as follows and are
explained in [38]:
E
0=E0
1+C2Y−2
0−2CY−1
0cosθ0
(18)
Y
0=Y01+C2Y−2
0−2CY−1
0cosθ0(19)
θ
0=θ0+tan−1CY−1
0sinθ0
1−CY−1
0cosθ0.(20)
The system in differential algebraic equation form is
rearranged to an ordinary differential equation (ODE) formed
problem by substituting (16) and (17) into (14) and (15)
for δand V[38], [39]. In addition, the constant parameters
dm=0.05, M=0.3, Pm=1, P0=0.6, Q0=1.3, T=8.5,
E0=1, Y0=20, θ0=−5, E
0=2.5, Y
0=8, θ
0=−12,
Em=1, Ym=5, θm=−5, Kpw=0.4, Kpv=0.3,
Kpw=−0.03, Kqv=−2.8, and Kqv2=2.1 stated with
the model in [38] are placed in the equations and the system
expressions are resulted as follows:
˙
δm=ω(21)
˙ω=3.3333 0.5642 −0.05ω+5V
nsin(δ −δm+0.0873)
(22)
˙
δ=−33.3333−1.3−Q1+2.8V−V210.0239+4.981
n2
+20Vcos(0.0873 −δ)
+5V
ncos(δm−δ+0.0873)(23)
˙
V=−13.0719
×−1.114V+0.84V2
−0.4−1.3−Q1V27.9239 +4.981
n2
+20Vcos(0.0873 −δ)
+5V
ncos(δm−δ+0.0873)−0.03
×−0.6−P1+V2−1.7431−0.4358
n2
+20Vsin(0.0873 −δ)
+5V
nsin(δm−δ+0.0873) (24)
where Q1,P1,andnare the only left parameters after the
derivations. They are called as bifurcation parameters since the
system shows different chaotic behaviors depending on their
values. After discretization of the above continuous ODEs by
a standard ODE solver [45], the discrete power system state
space model is expressed by
xk=M(xk−1)+k(25)
where the state vector is xk−1=[δk−1
mωk−1δk−1Vk−1]T
and kis the model noise. The model noise is neglected in
this paper.
The measurement model is represented by
yk=H(xk)+Rk(26)

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UZUNO ˘
GLU AND ÜLKER: MLEF STATE ESTIMATION FOR POWER SYSTEMS 5
Fig. 3. Simulation results of MLEF estimation for each state. Simulation is run with 1000 ensembles. All the states have observations with standard deviation
0.01. The assimilation time is 0.1 and the total number of simulation steps are 100. Initial conditions x0=[0.20.20.04 0.98]and Q1=11.285 pu, P1=0
pu, and n=1.001. The background legend defines background without any data assimilation.
where Rkis the measurement noise which can be modeled
with a Gaussian normal distribution with zero mean and
σ2variance following studies [13], [46], [47]. Measurements
are observed on all the states of the benchmark model in this
paper.
IV. NUMERICAL SIMULATIONS AND RESULTS
Numerical results will be presented in this section. For
benchmark studies root-mean-square-error (RMSE) values will
be used. The benchmark RMSE values have the same units as
the states.
A. Three-Bus Benchmark System Simulation
The numerical values are introduced for the estimation
of the states in the state system (25) of the test model to
demonstrate the performance of the MLEF in this section.
The estimation is performed with the initial values of x0=
[0.20.20.04 0.98]for the states xk=[δk
mωkδkVk]
that are assumed the same values with [39]. The system
parameters P1,Q1,andnare fixed at 0, 11.285, and
1.001, respectively. Although no model noise is taken into
account, normal distribution with zero mean and 0.01 standard
deviation is employed for the observation noise for all the
states [46], [48]. Observations are deviated from the true
values of the states in the simulations. The assimilation time
of the measurements is assumed as 0.1 here. Since at the
very start of the simulation there is no previous analysis
cycle, we initiated the ensemble perturbations via multiplying
the square root observation covariance matrix with randomly
chosen 4×10 dimensional values between 0 and 1. Covariance
initiation is directly obtained, no spin-off cycle is used. The
simulations are performed with 10 ensembles, 100 ensembles,
and 1000 ensembles, respectively. For comparison reasons,
the simulations are also run with PF algorithm [19], [24] with
the same number of ensembles. Then, simulations are repeated
for 1000 Monte Carlo runs with 10 ensembles. In addition,
the MLEF estimation is repeated with 10 ensembles with the
assimilation time of 0.04. The simulations are implemented
on a 64-bit operating system supplied with Intel Xeon CPU
E3-1245 3.50 GHz and 32 GB RAM. First, the convergence
of the MLEF estimation for one simulation run on the three-
node power system states is depicted for simulation with
1000 ensembles in Figs. 3 and 4 that illustrate the estimation
comparison between MLEF and PF with 10 ensembles. Then,
the RMSE results for the simulations with both methods with
different numbers of ensembles for each system state are
illustrated in Table I. Finally, the average RMSE for 1000
Monte Carlo runs for both methods are given in Table II
for each system state. It has been observed that MLEF with
10 ensembles with average one-step estimation CPU time
of 0.0399 s has almost the same RMSE result as the PF with
1000 ensembles with average one-step estimation CPU time
of 2.7167 s. The improvement of the CPU time is possible,
since the simulation codes can be optimized. The MLEF
estimation RMSE results with 10 ensembles for different
assimilation time is illustrated in Table III. It is observed that
having more frequent measurements improves the estimation
RMSE. In order to address a disturbance problem, independent
of previous simulations, the MLEF estimation is repeated with
10 ensembles with assimilation time 0.25 for a scenario that
the system parameters P1and nare fixed at 0 and 1 pu,

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6IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT
Fig. 4. Comparison of MLEF versus PF estimation simulation results for each state. Simulation is run with 10 ensembles. All the states have observations
with standard deviation 0.01. The assimilation time is 0.1 and the total number of simulation steps are 100. Initial conditions x0=[0.20.20.04 0.98]and
Q1=11.285 pu, P1=0 pu, and n=1.001.
TAB LE I
ESTIMATION RMSE FOR BOTH METHODS WITH DIFFERENT NUMBERS OF
ENSEMBLES FOR ONE SIMULATIONRUN FOR EACH STATE
TAB LE I I
AVERAGE ESTIMATION RMSE FOR 100 MONTE CARLO RUNS FOR BOTH
METHODS WITH 10 ENSEMBLES FOR EACH STATE
respectively, the parameter Q1is increased from 11 to 11.3pu
at fourth second as a disturbance on the system. The perfor-
mance of the MLEF estimation for one simulation run for the
scenario is depicted in Fig. 5.
B. 68-Bus Test System Simulation
The MLEF method is also tested on reduced version of
the New England–New York interconnection IEEE test sys-
tem [49] which is illustrated in Fig. 6. The system includes
TABLE III
MLEF ESTIMATION RMSE FOR ONE SIMULATION RUN WITH
10 ENSEMBLES FOR EAC H STATE FOR DIFFERENT
ASSIMILATIONTIMES
68 bus and 16 generators. The subtransient model and DC1-
type exciter are used for the generator models. The system
has 160 state variables in total. The parameters of the system
can be found in [49]. The simulation of the power system
is implemented together with a power system toolbox [50].
The simulations are implemented on a 64-bit operating system
supplied with Intel Xeon CPU E3-1245 3.50 GHz and 32 GB
RAM. A total of 16 PMUs are assumed to be installed
at the terminal buses of all generators while there is no
theoretical or numerical limitations on other type of measure-
ment devices such as remote terminal units. Here, the PMU
placement is done randomly, since the optimal PMU placement
is out of the scope of this paper. The assimilation time for
PMU measurements is chosen as 0.04 assumed from [51]. The
number of ensembles for the simulation is determined as 10.
The simulation is run with the scenario of a loss of load at
bus 23. The measurements are considered on terminal voltages
and voltage angles of the generators.
In our test run, terminal voltages and voltage angles of the
generators are readily available to us via a numerical model.

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UZUNO ˘
GLU AND ÜLKER: MLEF STATE ESTIMATION FOR POWER SYSTEMS 7
Fig. 5. Simulation results of MLEF estimation for each state. Simulation is run with 10 ensembles. All the states have observations with standard
deviation 0.01. The assimilation time is 0.25 and the total number of simulation steps are 375. Initial conditions x0=[0.20.20.04 0.98],P1=0 pu,
and n=1, and Q1is increased from 11 to 11.3 pu at the fourth second of the simulation. The background legend defines background without any data
assimilation.
For physical measurement case, in steady state, the rotor angle
of a synchronous generator can be taken from its phasor
diagram using the terminal voltage and current measurements.
During transient, the reactances of the machine change their
effective values while there are analytical methods to estimate
the rotor angle during transient conditions. As in [52]–[55],
for actual measurements, suitable values for the transient
reactance can be projected, and then rotor angle is estimated
by solving the equations describing the dynamics of the
synchronous machine or using the terminal measurements. The
transient reactance can also be estimated, along with the rotor
angle, using a suitable estimation methodology. In both the
methods, the real-time estimation of the rotor angle requires
high refresh rate of the measurements at the terminal of the
machine as in the case of PMUs used in this paper [55].
The assumption that PMUs were installed in the generator
terminal buses in reference to other buses was made. There
was no sensor optimization involved in this paper while this
is not the scope of this research. Any of the 68 buses could
have been chosen to validate the approach while generator bus
measurements were chosen.
Fig. 6. 68-bus test system.
The state estimation employs both the model state and
sparse measurements to estimate full state and as of the nature
of the problem, measurements are always sparse compared to

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8IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT
Fig. 7. Simulation results of MLEF estimation for 68-bus system generator-1 and generator-4 terminal voltages and voltage angles. The simulation is run
with 10 ensembles. The measurements are observed with 16 PMUs on terminal voltages and voltage angles of all generators with standard deviation 0.01.
The assimilation time is 0.04 and the total number of simulation steps are 250.
TAB LE I V
MLEF ESTIMATION RMSE FOR 68-BUS SYSTEM FOR
ONE SIMULATION RUN WITH 10 ENSEMBLES
full model state. The approach herein is general enough to
address any missing PMUs or any other sensors in any of the
buses.
For the example taken, the state estimation CPU time for
single assimilation is within real time scale of measurement
frequency. Since these computing figures are computing hard-
ware and code optimization dependent, it will improve with
improvement in hardware and code optimization.
In order to observe the MLEF performance without mea-
surements, they are stopped after 150th step of the simula-
tion, and the simulation is continued without observations.
The convergence of the MLEF estimation for one simulation
run on the system is illustrated in Fig. 7 for generator-
1 and generator-4 terminal voltages and voltage angles. Then,
the one-simulation run RMSE results for terminal voltages and
voltage angles of all generators are depicted in Table IV.
V. CONCLUSION
The MLEF is introduced for a nonlinear power system.
The MLEF is an algorithm that combines the optimization
with low-dimensional preconditioning in ensemble size sample
space which makes the MLEF low-dimensional ensemble
state estimation tool. It can address discontinuous problems
where derivatives are not defined. This paper demonstrates
the successful performance of the MLEF for the state estima-
tion problem of the benchmark power system model in [39]
and the 68-bus test system in [49]. Results are obtained by
employing algorithms described in [17] and convergence is
achieved with small ensemble size. These successful results on
basic power system model encourages the future investigation
for the MLEF performance on the state estimation of more
complex power systems. In this paper, no model noise is
assumed, but it can be possible to include it in the future
works. Additional issues such as the determination of the size
of ensembles, initiation of the ensembles, and targeting the
observations [26], [27] can be investigated further in the future
studies.
ACKNOWLEDGMENT
B. Uzuno˘glu would like to thank visiting scientist exchange
program granted at Department of Mathematics, Florida State
University with Prof. Y. Hussaini in the context of this paper.

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UZUNO ˘
GLU AND ÜLKER: MLEF STATE ESTIMATION FOR POWER SYSTEMS 9
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Bahri Uzuno˘glu received the Ph.D. degree in
computational engineering from Southampton Uni-
versity, Southampton, U.K.
He is currently a Senior Lecturer with the Electric-
ity Division, Uppsala University, Uppsala, Sweden,
and a Research Faculty with the Department of
Mathematics, Florida State University, Tallahassee,
FL, USA. He is involved in computational engi-
neering and science applied to renewable energy
problems. His current research interests include
optimization and applied statistical methods on
renewable energy, power systems, data assimilation, state and parameter esti-
mation, electricity markets, and weather forecast systems for grid integration.
Muhammed Akif Ülker received the B.Sc. degree
in electrical engineering from Yildiz Technical Uni-
versity, Istanbul, Turkey, and the M.Sc. degree
in wind power from Uppsala University, Uppsala,
Sweden.
He was with Best Transformers and ABB Compa-
nies, Turkey. He was a Research Engineer with Upp-
sala University. His current research interests include
power systems, renewable energy sources, electricity
markets, state estimation, and optimization in power
systems.