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Abstract—Hybrid AC/DC distribution systems are becoming a
popular means to accommodate the increasing penetration of
distributed energy resources and flexible loads. This paper
proposes a decentralized robust state estimation (DRSE) method
for hybrid AC/DC distribution systems using multiple sources of
data. In the proposed decentralized implementation framework, a
unified robust linear state estimation model is derived for each AC
and DC regions, where the regions are connected via AC/DC
converters and only limited information exchange is needed. In
this context, estimation accuracy may be suffering due to
linearization. To enhance the estimation accuracy, a deep neural
network (DNN) based on the smart meter data is used to extract
hidden system statistical information and allow deriving nodal
power injections that keep up with the real-time measurement
update rate. This provides the way of integrating smart meter data,
SCADA measurements and zero injections together for state
estimation. Simulations on two hybrid AC/DC distribution
systems show that the proposed DRSE has only slight accuracy loss
by the linearization formulation but offers robustness of
suppressing bad data automatically, as well as benefits of
improving computational efficiency.
Index Terms—State estimation, hybrid AC/DC distribution
systems, deep neural networks, smart meters.
I. INTRODUCTION
HE increasing penetration of renewable energy sources
(RES), as well as the emerging flexible loads, are leading
to the transformation of the conventional AC distribution
system. The DC-based generations and loads, e.g. photovoltaic
(PV) panels and electric vehicles, call for the resurgence of DC
grids to improve energy efficiency. An inevitable consequence
is the coexistence of the AC and DC grids in the modern
distribution system. The resulting hybrid AC/DC distribution
systems bring up plenty of new challenges in real-time
monitoring, operation, and controlling [1]-[2].
State estimation (SE) is a fundamental tool to extract system
states from raw measurements and to validate the operational
constraints. This concept was first introduced in power systems
by Fred C. Schweppe and one of the representative SE methods
is the weighted least square (WLS) [3]. Other robust SE
methods against bad data, e.g. the least absolute value (LAV)
estimator and the generalized maximum-likelihood estimator
[4]-[5], have been proposed as well. In contrast to the
This work has been supported partially by National Natural Science
Foundation of China under Grant U1966205, and by Fundamental Research
Funds for the Central Universities under Grant B200201067.
M. Huang is with the College of Energy and Electrical Engineering, Hohai
University, Nanjing 210098, China (e-mail: hmy_hhu@yeah.net).
transmission system with good measurement redundancy, the
distribution system has a low coverage of supervisory control
and data acquisition (SCADA) measurements. This can lead to
unsatisfactory estimation accuracy or even algorithm
divergence issue [6]-[7]. A commonly used remedy for that is
to build pseudo measurement via ad hoc modeling approaches
or employ artificial intelligence techniques for distribution
system SE (DSSE) [8]-[10]. On the other hand, the advent of
smart meters brings very useful information (e.g. load power
injection), which allows to enhance the system observability
and to monitor the distribution system [11]. According to the
IoT analytics’ smart meter market report, the penetration of
global smart meters has increased rapidly in recent years and
the number of smart meters is estimated to surpass 1 billion
within the next two years. Although these intelligent meters are
mainly intended for billing purposes, they have the potential to
unlock a series of advanced services such as situation awareness.
Several studies investigate how to incorporate smart meter data
in the DSSE. For example, in [12], the smart meter data are
aggregated from low-voltage distribution systems and used for
medium-voltage DSSE. To further improve the estimation
accuracy, the authors in [13] propose to adjust the variance of
different smart meters in the DSSE formulation. The time-
mismatch between the SCADA measurements (updated every
few minutes) and the smart meter data (updated every half an
hour or hourly) calls for attention [14]-[15]. Note that these
issues for the conventional AC distribution system, i.e., the low
measurement redundancy and the low refresh rate of smart
meters, still exist in the hybrid AC/DC distribution system and
will be considered in this work.
Recently, numerous studies were devoted to investigating the
power flow [16] and SE [17]-[19] of hybrid AC/DC distribution
systems to ensure their secure and economic operation. In the
existing literature, the research of SE for hybrid AC/DC
distribution systems can be classified into two main categories:
centralized and decentralized implementations. The SE solution
proposed in this paper belongs to the second category, which
estimates the states of AC and DC regions separately. Although
the decentralized SE can ensure flexible operating modes and
data privacy, it has to iterate alternately to keep the consistency
of boundary values at the point of common coupling (PCC)
J. Zhao is with the Department of Electrical and Computer Engineering,
Mississippi State University, Starkville, MS, 39759 USA (e-mail:
junbo@ece.msstate.edu).
Z. Wei is with the College of Energy and Electrical Engineering, Hohai
University, Nanjing 210098, China (e-mail: wzn_nj@263.net).
Manyun Huang, Member, IEEE, Junbo Zhao, Senior Member, IEEE, Zhinong Wei, Member, IEEE,
Marco Pau, Member, IEEE, and Guoqiang Sun
Decentralized Robust State Estimation for
Hybrid AC/DC Distribution Systems with
Smart Meters
T
Manuscript File Click here to view linked References
between the AC and DC regions. In [17] and [18], the WLS is
used to solve the hybrid AC/DC SE problem via alternated
iterations. Such sequential iterations may take a long time to
converge due to the nonlinear SE problem. In order to fill the
gap, a distributed SE method with the integration of PMU and
SCADA is proposed in [19] and the performance of both the
centralized and distributed algorithms are compared and
discussed in [20]. Since the deployment of PMUs requires high
economic cost and is not prevalent in the distribution level, this
paper tries to integrate the smart meter data, zero injections and
SCADA measurements together, and proposes a
computationally efficient decentralized robust SE (DRSE)
method for hybrid AC/DC distribution systems.
As discussed above, SE in hybrid AC/DC distribution
systems face a series of challenges and opportunities: 1)
insufficient real-time measurements; 2) the prevalent of smart
meters; 3) rough state estimation based on imprecise pseudo-
measurements; 4) different properties of hybrid networks and
measurements. Note that situations 1)-3) are still remaining in
the conventional AC distribution system, while situation 4) is
the unique property of the hybrid system. Therefore, this paper
tries to integrate smart meter data, zero injections and SCADA
measurements to enhance the measurement redundancy and
designs a decentralized framework with the detailed converter
model for achieving efficient SE and accommodating flexible
operating modes. To summarize, the main contributions of this
paper are as follows.
1) The proposed method technically integrates multi-source
data with different timescales to enhance measurement
redundancy and a robust estimation criterion is employed
against noises and bad measurements;
2) A slow timescale smart meters aided DNN model is
developed to extract hidden system statistical information and
allow deriving nodal power injections that guarantee the
network observability;
3) A decentralized framework with the detailed converter
model is designed for multiple regions and a linear WLAV-
based formulation is derived, to achieve efficient and robust SE.
The remainder of this paper is organized as follows. Section
II describes the AC/DC system state estimation problem. Then,
the decentralized AC/DC SE framework is formulated in
Section III. Numerical results are given and discussed in
Section IV, and finally Section V concludes the paper.
II. AC/DC STATE ESTIMATION PROBLEM STATEMENT
Suppose a hybrid AC/DC distribution system with n nodes,
i.e., N = {1, 2, …, n}. These nodes are composed of both AC
and DC nodes, denoted as NAC N and NDC N respectively.
Each node can only be of one type, and thereby NAC NDC =
and NAC NDC = N. For the model of AC/DC converters given
in Fig. 1, node i belongs to the set of AC nodes (i.e., i NAC)
while node j belongs to the set of DC nodes (i.e., j NDC). Since
the latest converters work without applying the pulse width
modulation technology, the filter between the transformer and
reactor is not necessary [21]. In this case, only the auxiliary
node c is introduced in this paper and assigned as a new AC
node, i.e. NAC c = NAC.
A. State Variables
In the SE of hybrid AC/DC distribution systems, the state
variables of AC nodes are xAC,i = {Vi, θi}, where Vi and θi denote
the voltage magnitude and angle at the AC node i. Analogously,
the state variable at the DC node j is the voltage, i.e., xDC,j = Vj .
To sum up, the state variables x of a hybrid AC/DC
distribution system with node set N can be expressed as:
,
,
,
,
AC i AC
DC j DC
x i N
xx j N
=
(1)
B. Measurement Functions
In hybrid AC/DC distribution systems, measurements may
include power flow, power injection, voltage magnitudes, etc.
The measurement model is described as follows:
()=+z h x e
(2)
where z = [z1, z2, …, zm]T is a m×1 vector, and e is the associated
measurement error vector assumed to be white Gaussian noise.
h(.) represents the nonlinear relationship between the
measurements and states. The measurement functions for a
generic branch power flow and nodal power injections are:
, ' ' ' ' ' '
( cos sin ), { , '}
AC ii i i ii ii ii ii AC
P VV G B i i N
= +
(3)
, ' ' ' ' ' '
( sin cos ), { , '}
AC ii i i ii ii ii ii AC
Q VV G B i i N
= −
(4)
,' { , '}
AC,i AC ii AC
P P i i N=
(5)
,' { , '}
AC,i AC ii AC
Q Q i i N=
(6)
, ' ' ', { , '}
DC jj j j jj DC
P V V Y j j N=
(7)
, , ' { , '}
DC j DC jj DC
P P j j N=
(8)
where PAC, ii’ and QAC, ii’ are the active and reactive power flow
from the AC node i to the AC node i’; PAC, i and QAC, i are the
active and reactive power injection at the AC node i; PDC,jj’ is
the real power flow from the DC node j to the DC node j’; PDC,j
is the real power injection at the DC node j. Parameters G and
B are the conductance and susceptance matrices of the AC
region while Y is the conductance matrix of the DC region.
Here, the topology information and line parameters (i.e.,
matrices G, B and Y) have been obtained in advance [22]-[23],
allowing for accurate state estimation. This is a prerequisite for
employing the proposed SE method. It is worth noting that the
power flow from the converter (omitting the filter) to the AC
node i is as follows.
,0
AC ci VSC
PP−=
(9)
,0
AC ci VSC
QQ−=
(10)
,,VSC VSC loss DC jc
P P P+=
(11)
where PAC,ci and QAC,ci denote the active and reactive power
flow from the AC node c to the AC node i; PVSC and QVSC are
the active and reactive power outputs of the converter;
converter
reactor
transformer
i j
c
Fig. 1 AC/DC converter model
is a quadratic loss of the
converter [24], where
, and PDC,jc represents
the active power flow from DC node j to the converter.
Obviously, the converter loss is associated with the state
variables PVSC, QVSC, Vc and PDC,jc. In addition, constraints on
the related variables (i.e. PVSC, QVSC, VAC, i, VDC, j) of the
converter are determined by the control strategy and described
in detail in [25].
C. Objective Functions
The most widely used WLS aims at minimizing the
following objective function:
2
11( ( ))
m
i i i
i
J w z h x
=
=−
(12)
where zi is the ith measurement and wi = 1/σ2
i is the weight of
measurement zi inversely related to the measurement error
variance. Such an approach has high computational efficiency
but is vulnerable to bad data. Another representative method,
the WLAV, is robust to gross errors and formulated as:
211
( ) ( ), , 0
mm
i i i i i i i i
ii
J w z h x w u l u l
==
= − = +
(13)
where ui−li = zi−hi(x); ui and li are respectively represent the
absolute values of the non-negative and negative measurement
residuals. This reformulation transforms the original WLAV
optimization problem into a linear programming (LP) problem
when all constraints are linear [26].
To this end, the state estimation problem of hybrid AC/DC
distribution systems is described as:
21
T
12
min = ( )
. . ( ) ,
0, 0, {1,2,..., }
[ , ,..., ,...] ,
m
i i i
i
i i i i
ii
j
J w u l
st z h u l
u l i m
x x x j N
=
+
− = −
=
=
xx
x
(14)
III. PROPOSED AC/DC STATE ESTIMATION
To solve the general SE problem of hybrid AC/DC
distribution systems in (14), the decentralized robust state
estimation is proposed (see Fig. 2). In the following, we first
introduce the proposed SE model of hybrid AC/DC distribution
systems and discuss the mathematical model of AC/DC
converters. Then, a unified linear SE algorithm is derived for
AC and DC regions. Finally, a DNN aided method based on the
smart meter data is presented to ensure the system observability.
A. Decentralized SE Model
As a region k with a node set Nk, local state variables xk and
measurements zk, the regional SE is described as:
'
,,
,'
min ( )
. . ( ) 0, '
,
kk
k k k
VSC,k VSC,k k
AC ci VSC,k AC ci VSC,k
DC,jc VSC loss VSC,k
J
st h
P P k D
P P Q Q
P P P
−=
− =
==
−=
xx
z x e
(15)
where k’ denotes the neighbor region of the region k, Dk denotes
the set of neighbor regions (here the regions are partitioned by
AC and DC natures). The active power loss has been considered
and the power flow at the PCC is shown in Fig. 3, where the
AC/DC converter connects node i (i NAC) with node j (j
NDC) and separates region k from region k’. Assuming the same
direction power output of the AC/DC converter in two regions,
the boundary values at the PCC should be equal, i.e.,
, , 'VSC k VSC k
PP=
(16)
,VSC k VSC
QQ=
(17)
Here, the reactive power QVSC is always decided by the control
strategy of the converter (e.g. constant QVSC or constant VAC,i).
It can be observed from Fig. 3 that the converter loss, i.e.
PVSC,loss, is important for ensuring the power balance of AC and
DC regions. Specifically, the converter loss heavily depends on
the converter current as shown below.
2
, 1 2 3
d d d
VSC loss c c
P I I= + +
(18)
22
/3
c ci ci c
I P Q V=+
(19)
where d1, d2 and d3 denote constant loss coefficients; Vc, Ic, Pc,
Qc represent voltage magnitude, current magnitude, active and
reactive power injections at the AC node c. The per-unit
coefficients in the quadratic loss formula are determined with
the data in [21].
To achieve a fully decoupled SE model for hybrid AC/DC
distribution systems, the power flow constraints at the PCC are
relaxed with a Lagrangian form by using a gradient method.
reactor
transformer
i
j
c
PVSC,k
QVSC,k
PVSC,loss
PVSC,k
+PVSC,loss
region kregion k
Fig. 3 Power flow at the PCC between region k and k’.
Power Injection Generating
SCADA
Input Layer Hidde n Layers Output Layer
zy=(P,Q)
y:= f(z)
Error Distribution
f(e/γ):= ωi f (ei |ui,σ i )
-0.01 00.01
power
injections
Wy
weights
of y
Unified Linear
State Estimation
Jk=wk(uk+lk)
+λ(ak+bk)
PVSC
region
Aregion
B
Converter Model
（Boundary values）
PVSC,loss=d1+d2I+d3I2
PVSC,loss
I
DNN-based Method
(for network training)
Historical smart
meter data
Fig. 2 Schematic of the proposed DRSE
'
, , , ,
, , , '
min ( ) | |
. . ( ) ,,'
kk VSC,k VSC,k
k k k
AC ci VSC k AC ci VSC k
DC jc VSC loss VSC k k
J P P
st h
P P Q Q
P P P k D
+−
−=
==
− =
xx
z x e
(20)
where the Lagrange multiplier λ is updated during the iteration.
Finally, the decentralized AC/DC SE model based on the
WLAV is summarized in Algorithm 1, where the regional SE
of all regions is executed in parallel and the boundary values of
each region are determined by the estimation results in the
previous iteration. For example, the converter loss Pl-1
loss,k is
employed in the regional SE (i.e., step 6: xl
k:= argmin Jk), and
then the updated converter loss Pl
loss,k is obtained for the next
iteration l+1. Here, Pl
loss,k denotes the converter loss calculated
by the region k in the lth iteration; the parameter ξ denotes a
positive constant value; the predefined threshold τ is set to
ensure the consistency of boundary values; L denotes the
maximum number of iterations and the parameter R denotes the
number of AC and DC regions in the hybrid AC/DC
distribution system. In this context, only the power flow
information of the converter is exchanged between two regions,
protecting regional information privacy.
B. Unified Linear SE for AC and DC Regions
The SE in AC regions is a nonlinear programming problem,
and it takes time to converge. To enable a computational
efficient decentralized SE algorithm, we extend a linearized
power flow model of AC grids [27]-[28] into our work and
hence convert the nonlinear SE of AC regions to a linear
programming problem. For example, the linear power flow
function of a balanced AC network can be expressed as follows:
22
i j ij ij ij ij
U U R P X Q + +
(21)
i j ij ij ij ij
X P R Q
+ −
(22)
where Ui =V2
i is the square of voltage magnitude Vi; Rij and Xij
denote the resistance and reactance of line (i, j). In (21), the
higher-order term (Vi-Vj)(Vi-Vj)* that is the change in voltage
associated with losses is negligible. The assumption of (22) is
that voltage magnitude is constant and equal to 1 p.u., as well
as the angle difference of two voltage phasors is small enough
to satisfy sin(θi−θj) ≈ θi−θj. In unbalanced conditions, the
voltage magnitude and angle are obtained from three phasors,
i.e. Va,b,c and θa,b,c, and another assumption about the fixed ratio
of voltage phasors is made. The accuracy of such
approximations has been investigated in [27] and [28]. The
validity of the approximation has been demonstrated in our
simulation results as well.
With the linearized model, (3) and (4) can be reformulated as
below:
,22
( ) 2 ( ) , { , }
2( )
i j ij ij i j
AC ij AC
ij ij
U U R X
P i j N
RX
− + −
=
+
(23)
,22
( ) 2 ( ) , { , }
2( )
i j ij ij i j
AC ij AC
ij ij
U U X R
Q i j N
RX
− − −
=
+
(24)
In this context, the measurement function in an AC region can
be expressed as:
new
AC
=+Hz x e
(25)
where z denotes the measurement vector in the AC region; xnew
AC
={Ui, θi, iNAC} consisting of the squared voltage magnitude
and voltage angle represents the state vector in the AC region,
and H is a constant matrix, which depends on the structure and
line parameters of the AC region.
As for DC regions, the voltage magnitude can be denoted as
Vi=1−ΔVi, then the power flow in a DC line is:
,(1 )(1 )
=(1 ) +
(1 )
dc ij i j ij
i j ij i j ij
i j ij
P V V Y
V V Y V V Y
V V Y
= − −
− −
− −
(26)
where the approximation of Pdc,ij neglects the high order term,
i.e., ψ(V)=ΔVi×ΔVj×Y ij. The approximation error associated to
ψ(V) will decrease when V approaches 1 (p.u.). In this way, the
measurement function in a DC region can be also expressed as
a linear measurement function. The effectiveness of such an
approximation has been verified in [29].
learning
distribution of
Pi and Qi
smart
meter data
Monte Carlo
trials
power flow
calculations
DNN training
noise
production
nodal power
injections
DNN model
parameters
(y, z=z,+ e)
(y, z,)
y={P,Q}
Offline learning Online generation
real-time
measurements
Fig. 4 Schematic of the DNN-based method that learns the distribution of
nodal power injections offline and yields nodal power injections online.
Algorithm 1: Decentralized AC/DC SE
1
Relax the regional SE: Jk = J2(xk) + λ|PVSC,k − PVSC,k’|
2
Initialization xk ,λ, ξ, τ, l=0 and L
3
while |PVSC,k − PVSC,k’| > τ and l < L
4
l = l + 1
5
for k=1:D
6
xl
k:= argmin Jk
7
if the region k is an AC region
8
Pl
loss,k is obtained by Eq.(18) and (19)
9
end
10
end
11
λ=λ+ξ×|PVSC,k −PVSC,k’|
12
end
13
Output the final solution
Active power injection
Reactive power injection
AAE
5.29E-04 (p.u.)
2.31E-04 (p.u.)
MAE
0.0058 (p.u.)
0.0030 (p.u.)
Table 1 Errors of power injections by the DNN-based method
Formally, the WLAV-based SE for each region as a linear
programming problem can be shown as follows:
, , ,
1
'
, , , ,
, , , '
,,
min ( ) ( )
..
,
, , , 0 '，
k
m
k k i k i k i k k
i
k k k k k
k k VSC,k VSC,k
AC ci VSC k AC ci VSC k
DC jc VSC loss VSC k
k i k i k k k
J w u l a b
st
a b P P
P P Q Q
P P P
u l a b k D
=
= + + +
− = +
− = −
==
−=
H
x
u l x e
(27)
where xk and zk respectively denote the state vector and
measurement vector in the region k; Hk is a constant matrix,
decided by the structure and line parameters of the region k; ek
denotes the measurement noise vector in the region k; ak, bk, uk,i,
lk,i are non-negative variables associated with the
transformation of the objective function. Notice that if the
region k is an AC region, the state vector is composed by xk =
{Uk, θk}, otherwise, in a DC region, the state vector is xk = {Vk}.
As we discuss in Algorithm 1, the Pvsc,loss of an AC region has
been obtained by the previous iteration and therefore the model
(27) is solved by linear programming.
C. DNN-aided method Using Smart Meter Data
To ensure the network observability and improve the
estimation accuracy of hybrid AC/DC distribution systems,
SCADA measurements, smart meter data and zero power
injections are together employed in the proposed SE model.
However, they are updated at different timescales and the smart
meter data is usually sampled slower than SCADA
measurements. To deal with that, this paper proposes a DNN
aided method to learn the distribution of nodal power injections
from the slow timescale smart meter data (at the offline learning
stage) and later on to yield nodal power injections at a fast time
scale (at the online generation stage). In the proposed
decentralized framework, the DNN model is essential for some
regions with low measurement redundancy. As a representative
method of deep learning, DNN with bulging middle sections
has the ability to extract hidden statistical information from
smart meter data, avoiding the usage of imprecise pseudo
measurements. Moreover, effective nonlinear functions (e.g.,
ReLU and Tanh) could draw complex characteristics and the
fine-tuning process can be achieved by a backpropagation
algorithm in the DNN model.
The schematic of that is given in Fig. 4, where the left hand
of the schematic describes the offline stage. Tracking a large
amount of historical smart meter data, the probability
distribution of nodal power injections could be approximated
by Gaussian mixture modeling (GMM). Then, three
submodules, i.e. Monte Carlo trials, power flow calculations
and noise production, are subsequently adopted to extract a set
of training data (y, z) from the obtained probability distribution
of the nodal power injections. As for a set of the training data,
the output y is constituted by all nodal power injections yielded
by the Monte Carlo trials, while the input z contains some
branch power flows, node voltage magnitudes, etc., corrupted
by simulated noises. It should be noted that the number and
types of elements in the output data z are different from each
test system and determined by the measuring condition. The
parameters of the DNN model can be set by the empirical risk
minimization. On the basis of the training results, the
distribution of power injection errors is approximated by the
GMM model to determine the weights of the associated nodal
power injections in the following SE. In the study of historical
data, some common reconfigurations of distribution systems
have been recognized and well-trained, but the correctness of
the DNN model could not be guaranteed anymore in the case of
the interruption of some unforeseen lines. At the online stage,
the trained DNN model is used for the regression analysis and
to generate nodal power injections associated with the real-time
SCADA measurements at the same update rate. Besides, the
accuracy of the DNN model could be checked through the
deviation of the estimated power injections (calculated by the
estimated states) and the outputs y, and the offline training
would be triggered once the deviation is larger than a predefined
threshold.
IV. CASE STUDY
The estimation performance of the proposed DRSE is tested
on a sample 33-node hybrid AC/DC distribution system [1],
while a mesh 106-node hybrid AC/DC distribution system is
applied to demonstrate its scalability [2]. To illustrate the idea
of DSSE under a low measurement coverage condition, a few
grid
1 3 6 8
=~
19 20
=~
21 22
2324 25 26 2728 29
=~
30 31 32 33
VSC1
VSC2
VSC3 DG3
DG1
DG4 DG5 DG6
PV
Diesel
wind PV
Diesel
AC node
DC node
AC line
DC line
25
49
710 11 12 13 14 15 16 17 18
DG2PV
DC region 1
DC region 2
DC region 3
AC region 1
Fig. 5 Structure of the 33-node hybrid AC/DC distribution system.
0
0.5
1
1.5
2
2.5
node
× 10-3 p.u.
Errors of reactive power injections
26 27 28 2919 20 23 24 25
2 3 4 5 6 7 8 9
0
1
2
3
4
5
6× 10-3 p.u.
26 27 28 2919 20 23 24 25
2 3 4 5 6 7 8 9
Errors of active power injections
node
Fig. 7 Errors of power injections at each node; left: active power injections;
right: reactive power injections.
0100 200 300 400 500 600
0
50
100
150
200
250
300
350
400
Power injection (kW)
Probability density function
Frequency
Gaussian mixture
Statistic results of P5
-250 -200 -150 -100 -50 050 100
0
100
200
300
400
500
600
700
800
900
Probability density function
Frequency
Gaussian mixture
Statistic results of P9
Power injection (kW)
Fig. 6 Distribution learning of the active power injection; (a): at node 5;
(b): at node 9.
power flow measurements of AC lines are assumed to be
available. As for DC regions, we assume that the observability
of the DC network is guaranteed by the real-time measurements
due to the small size of the DC regions and the lower
measurement cost. In the 33-node hybrid AC/DC distribution
system shown in Fig. 5, two types of measurements are
assumed: 1) SCADA measurements that are updated every 15
minutes and placed only at some nodes (the main substation
node and the converter nodes), four AC lines (line 1-2, 2-19, 3-
23 and 6-26), as well as in all DC lines; 2) smart meter data at
consumer nodes. With the similar measurement configuration,
SCADA measurements of the 106-node hybrid AC/DC
distribution system in Fig. 1A consist of power flow
measurements in 12 AC lines and voltage magnitude
measurements in the main substation as well as AC/DC
converter nodes, while smart meters are at consumer nodes. In
our simulations, the smart meter data is assumed to be sampled
hourly at the load/generation nodes and hence updated slower
than the SCADA measurements, calling for the generated
power injections of the proposed DNN-aided method. The
additive measurement noises of SCADA measurements are
assumed to be Gaussian white noise with 1% uncertainty for the
voltage magnitude measurements and 2% for the power flow
measurements [30]-[31], respectively. Besides, the smart meter
data accuracy is assumed to be 2% [32], and all noises are
considered as independent. Note that some communication
issues, e.g. the latency, may result in a larger measurement
uncertainty of smart meter data and our proposed framework is
general to deal with this as well. All simulations were
performed on a computer Intel® CoreTM i7-10710U CPU with
16GB of RAM.
Relying on this simulation setup, two alternative algorithms
are compared to the proposed DRSE. The first one is the
centralized WLS, and the second one is the decentralized WLS
with the same partition principle as the proposed DRSE shown
in Fig. 5. In the decentralized WLS, AC and DC regions are
locally estimated by WLS `with regional measurements, and the
boundary power injections of converters are taken as equivalent
power injection measurements. The uncertainty of the
equivalent power injections is calculated based on the law of
propagation, and the convergence condition of decentralized
WLS is the same as Algorithm 1. The purpose is to assess the
estimation accuracy, the robustness and the computational
efficiency of the proposed DRSE. The indices, including the
average absolute error (AAE) and maximum absolute error
(MAE), are used:
1
1ˆ
An
x i i
ixx
n=
=−
(28)
ˆ
M max , 1,...,
x i i
x x i n= − =
(29)
where Ax and Mx denote the AAE and MAE of the variable x
(i.e., nodal voltage magnitude and voltage angle); i and xi
represent the estimated values and true values obtained by the
power flow calculation. Here, the power flow model of hybrid
AC/DC distribution systems is described in [21] and solved by
the iterative Newton Raphson method.
A. Validation of DNN-aided Power Injection Generation
Since the load/generation information in [1] only contains
one time instant, a set of load/generation profiles collected from
Jiangsu Province, covering January 1st of 2013 to April 30th of
2014 are used. Notice that the data set of 2013 is used for
distribution learning and DNN training, while that of 2014 is
used for testing. During the training process of DNN, the DNN
input and output data in two test systems are listed in Table 2.
We consider GMM to approximate the distribution of power
injections, as shown in Fig. 6. For all nodes, our data analysis
allows for the adoption of the 2-component GMM in
distribution learning. According to the training performance,
the DNN model having four layers with 300 neurons is selected
to generate nodal power injections online. The active function
in the hidden layers is ‘Tanh’ and linear active function is
applied in the output layer. The obtained results for the testing
errors of nodal power injections are given in Table 1. Here the
AAE of active and reactive power injections are respectively
around 0.0005 (p.u.) and 0.0002 (p.u.). It can be found from Fig.
7 that the errors of power injections at some nodes, i.e. nodes 9,
24 and 29, have large uncertainties due to the fluctuation of
distributed energy resources. Again, the distribution of power
injection errors was approximated by the GMM model to
determine the weights of the associated nodal power injections
in the following SE model.
0
0.1
0.2
0.3
0.4
0.5
0.6
in degree
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
AAE of voltage magnitude for AC nodes
in p.u.
p10% p30% dnn p10% p30% dnn p10% p30% dnn
V by CWLS
θby CWLS V by DWLS
θby DWLS V by DRSE
θby DRSE
AAE of voltage angle for AC nodes
Fig. 8 AAE of AC nodes obtained by three algorithms with different
measurement sets.
AAE of voltage magnitude for DC nodes
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2× 10-5 p.u.
p10% p30% dnn p10% p30% dnn p10% p30% dnn
V by CWLS V by DWLS V by DRSE
Fig. 9 AAE of DC nodes obtained by three algorithms with different
measurement sets.
Input data set
Output data set
33-node hybrid
AC/DC system
voltage magnitude at
4 nodes and power
flows at 4 lines
power injections at
17 consumer nodes
106-node
hybrid AC/DC
system
voltage magnitude at
4 nodes and power
flows at 12 lines
power injections at
96 consumer nodes
Table 2 The input and output data of DNN in two test systems
B. Validation of Estimation Accuracy
To evaluate the SE accuracy for each method, all three
methods are tested under the same measurement conditions. 1)
CWLSp: refers to the centralized WLS with pseudo
measurements of nodal power injections; 2) CWLSdnn: refers
to the centralized WLS with nodal power injections obtained by
the proposed DNN-based method; 3) DWLSp: refers to the
decentralized WLS with pseudo measurements of nodal power
injections; 4) DWLSdnn: refers to the decentralized WLS with
nodal power injections obtained by the proposed DNN-based
method; 5) DRSEp: refers to the proposed DRSE with pseudo
measurements of nodal power injections; 6) DRSEdnn: refers
to the proposed DRSE with nodal power injections obtained by
the proposed DNN-based method.
Three SE methods with different accuracy levels of pseudo
measurements and the nodal power injections generated by the
DNN-based method have been tested. Generally, pseudo
measurements are usually obtained by short-term load
forecasting with much larger uncertainty than the SCADA
measurements. Here, it is assumed that the errors of pseudo
measurements follow Gaussian distributions with 10% and 30%
uncertainties. Fig. 8 and Fig. 9 show the voltage magnitude and
angle estimation results of AC and DC nodes obtained by three
algorithms with 1000 Monte Carlo simulations.
It can be observed that the estimation errors of the AC nodes
by the three algorithms vary significantly with different
measurement uncertainties. The estimation errors under the
DNN-generated power injections are smaller than those with
pseudo measurements, showing the benefits brought by the
DNN model. In this regard, the DNN-based method helps
improve the estimation accuracy of all three estimation methods,
not just the proposed DRSE. When using the same
measurement set, the estimation errors of voltage angle
obtained by our proposed DRSE are larger than those obtained
by the CWLS and DWLS due to the linearized process in the
AC region (see Fig. 8). This is a compromise between the
estimation accuracy and computational efficiency during the
linearized process. Furthermore, the estimation errors of
voltage angle by the proposed DRSE with the DNN-generated
power injections are smaller than 0.4-degree, attributing to both
the linearized and the estimated errors. Despite this impact, the
estimation errors of voltage magnitudes at the AC and DC
nodes by the proposed DRSE are smaller enough when the
0
0.5
1
× 10-2 p.u.
AAE at AC nodes
0 5 10 15 20 25 30
Node number
0
5
10
× 10-5 p.u.
AAE at DC nodes
AC nodes DC nodes
0
2
4
× 10-5 p.u.
AAE at DC nodes
0
0.5
1
× 10-2 p.u.
AAE at AC nodes
0 5 10 15 20 25 30
Node number
AC nodes DC nodes
0
2
4
× 10-5 p.u.
AAE at DC nodes
0
0.5
1
× 10-2 p.u.
AAE at AC nodes
0 5 10 15 20 25 30
Node number
AC nodes DC nodes
(a) CWLS with DNN-generated data (b) DWLS with DNN-generated data (c) DRSE with DNN-generated data
0
0.5
1
0 5 10 15 20 25 30
Node number
AC nodes DC nodes
0
1
2
× 10-5 p.u.
× 10-2 p.u.
AAE at AC nodes
AAE at DC nodes
0
2
4
× 10-5 p.u.
AAE at DC nodes
0
0.5
1
× 10-2 p.u.
AAE at AC nodes
0 5 10 15 20 25 30
Node number
AC nodes DC nodes
0
2
4
× 10-5 p.u.
AAE at DC nodes
0
0.5
1× 10-2 p.u.
AAE at AC nodes
0 5 10 15 20 25 30
Node number
AC nodes DC nodes
(d) CWLS with pseudo measurements (e) DWLS with pseudo measurements (f) DRSE with pseudo measurements
Fig. 11 AAE of node voltage magnitude obtained by the three algorithms in Case 2.
V by CWLS
V by DWLS
V by DRSE
V by CWLSdnn
V by DWLSdnn
0
0.005
0.01
0.015
0.02
0.025
MAE of voltage magnitude
Case1 Case2
in p.u.
0
0.16
0.32
0.48
0.64
0.8
Case1 Case2
MAE of voltage angle
in degree
θ by DWLS
θ by DRSE
θ by CWLS
θ by CWLSdnn
θ by DWLSdnn
0
0.5
1
1.5
2
2.5
Case1 Case2
× 10-3 p.u.
V by CWLS
V by DWLS
V by DRSE
V by CWLSdnn
V by DWLSdnn
MAE of voltage magnitude
(a)voltage magnitude of AC nodes (b) voltage angle of AC nodes (c) voltage magnitude of DC nodes
Fig. 10 MAE of AC and DC nodes obtained by the three algorithms in two cases.
voltage magnitudes of AC and DC nodes approaching 1 (p.u.).
Moreover, it can be seen from Fig. 9 that the estimation errors
of the DC nodes under different measurement sets are similar
and small. Such estimation performance at the DC nodes is due
to the same uncertainties of the DC measurements in the three
measuring conditions. To summarize, the proposed DRSE with
the DNN-generated power injections may suffer slight accuracy
loss due to linearization, yet still can provide reliable estimation
results for the hybrid AC/DC distribution system, avoiding the
problem of pseudo measurement definition.
C. Robustness to Bad Data
Another important property affecting the estimation accuracy
is the robust performance that alleviates the estimation errors
brought by bad data. In this regard, two test cases have been
considered. 1) Case 1: one active power output measurement of
a converter is doubled. 2) Case 2: a pair of power flow
measurement of an AC line becomes the opposite number. For
the sake of fairness, the largest normalized residual (NR)-based
statistical test has been used in the centralized and decentralized
WLS to detect and process bad data. Note that the conventional
CWLS and DWLS algorithms with pseudo measurements as
well as those with the DNN-generated power injections, namely
CWLSdnn and DWLSdnn, are compared with the proposed
DRSE. To guarantee the adaptability of the trained DNN model,
the erroneous SCADA measurements that are detected and
rejected by the NR test in the input are substituted with the
0 5 10 15 20 25 30
Node number
0
0.1
0.2 in degree
AAE at AC nodes
AC nodes
0 5 10 15 20 25 30
Node number
0
0.1
0.2 in degree
AAE at AC nodes
AC nodes
0 5 10 15 20 25 30
Node number
0
0.1
0.2 in degree
AAE at AC nodes
AC nodes
(a) CWLS with DNN-generated data (b) DWLS with DNN-generated data (c) DRSE with DNN-generated data
0 5 10 15 20 25 30
Node number
0
0.1
0.2 in degree
AAE at AC nodes
AC nodes
0 5 10 15 20 25 30
Node number
0
0.1
0.2 in degree
AAE at AC nodes
AC nodes
0 5 10 15 20 25 30
Node number
0
0.1
0.2 in degree
AAE at AC nodes
AC nodes
(d) CWLS with pseudo measurements (e) DWLS with pseudo measurements (f) DRSE with pseudo measurements
Fig. 12 AAE of node voltage angle obtained by the three algorithms in Case 2.
Active power flow
Reactive power flow
0 5 10 15
AC Line number
0.025
0.05 p.u.
Active power flow
Reactive power flow
0 5 10 15
AC Line number
0.025
0.05 p.u.
Active power flow
Reactive power flow
0 5 10 15
AC Line number
0.025
0.05 p.u.
(a) CWLS with DNN-generated data (b) DWLS with DNN-generated data (c) DRSE with DNN-generated data
Active power flow
Reactive power flow
0 5 10 15
AC Line number
0.05
0.1 p.u.
Active power flow
Reactive power flow
0 5 10 15
AC Line number
0.05
0.1 p.u.
Active power flow
Reactive power flow
0 5 10 15
AC Line number
0.05
0.1 p.u.
(d) CWLS with pseudo measurements (e) DWLS with pseudo measurements (f) DRSE with pseudo measurements
Fig. 13 AAE of power flow obtained by the three algorithms in Case 2.
values obtained by numerical interpolation. In order to analyze
the robustness of the three algorithms, 1000 simulations have
been performed for each of the three cases, respectively. Fig. 10
gives the MAE estimates under the two different scenarios.
As expected, the proposed DRSE performs more stable than
the other two alternatives in the presence of gross errors. It can
be observed that in Case 1 the MAE of voltage magnitudes at
AC nodes by the proposed DRSE are slightly larger than those
of CWLSdnn and DWLSdnn but are comparable to the
conventional CWLS and DWLS. Recall that, in normal
conditions, the linearized model is in favor of computational
efficiency but yields slightly larger estimation errors, which
could be reduced by the accurate power injections generated by
the DNN model. The MAE of DC nodes by the CWLS in each
case is the largest one, especially in Case 1. The result indicates
that the estimation results of the AC and DC nodes by the
centralized CWLS are affected wherever the gross error appears.
Instead, the MAE for the DC nodes via two decentralized
algorithms remains stable in all cases. Nevertheless, the MAEs
for voltage magnitudes by the DWLS are still larger than those
by the proposed DRSE in the presence of bad data, especially
in Case 2.
To demonstrate the robustness benefits brought by the DNN-
generated power injections, pseudo measurements following
Gaussian distributions with 30% uncertainties and the DNN-
generated power injections are applied for three algorithms in
Case 2, and the detailed MAE of each node is shown in Figs. 11
and 12. It is observed that the estimation errors by each
algorithm with pseudo measurements increase due to the low
coverage of real-time measurements and large uncertainty of
pseudo measurements. Still, the AAE of node voltage
magnitude obtained by the proposed DRSE is smaller than the
CWLS and the DWLS. Although the AAE of node voltage
angle obtained by the proposed DRSE is comparable with the
CWLS and DWLS with DNN-generated data (see Fig. 12), the
AAE of power flows obtained by the proposed DRSE is smaller
(see Fig. 13). These results indicate that the proposed DRSE can
not only obtain reliable node voltages but also provide accurate
power flow estimates, which are essential for the convergence
of power flow calculation. To sum up, the robust performance
of the proposed DRSE method attributes to two factors: 1) the
WLAV-based state estimation model, which could reject gross
errors automatically; 2) the reliable power injections generated
by the DNN-based model.
D. Computational Efficiency Assessment
Additional simulations have been conducted on the two
hybrid AC/DC distribution systems to assess the computational
performance of the proposed DRSE. Here, the larger 106-node
hybrid AC/DC distribution system is given in the Appendix (see
Fig. 1A), where a DC network splits three AC networks. The
computing time includes the execution times of generating the
power injections online and that of state estimation. Notice that,
multiple AC or DC regions are included in the test systems and
hence parallel computing is used to decrease the computing
time when solving the SE problem for these regions. Therefore,
two decentralized algorithms, i.e., the proposed DRSE and
DWLS, are performed 1000 times. The recorded results show
that the power injections generated online take only few
milliseconds, however, the state estimation takes a longer time
and the results are given in Table 3. It could be seen that the
computational performance of the proposed DRSE is better
than the conventional DWLS. As expected, the execution time
of the proposed DRSE is decreased with the linearized model.
As the scale of hybrid AC/DC distribution systems becomes
larger, the computation time increases accordingly but slowly.
Specifically, the number of nodes in the 106-node hybrid
AC/DC distribution system increases from 33 to 106, while the
computation time of the proposed method only increases 1.43
times. This is because the computation time in the decentralized
framework mainly depends on the largest scale of all regions,
and the regional estimation model is linearized. Besides, our
proposed method can also be conducted in various forms of
hybrid AC/DC systems, no matter the AC network is
contiguous or not. If necessary, a contiguous but large-scale
network especially for the AC network could be divided into
multiple regions by some distributed algorithms, e.g. [33]. This
allows the proposed DRSE to provide the latest states in real-
time for the hybrid AC/DC distribution system even when the
SCADA measurements are updated every minute.
V. CONCLUSION
In this paper, a decentralized and robust SE method for
hybrid AC/DC distribution systems has been developed. The
decentralized framework allows conducting the regional SE for
AC and DC regions separately with only limited power flow
Table 3 Execution time of two decentralized algorithms
Sample 33-node system
Mesh 106-node system
DWLS
138 ms
203 ms
DRSE
81 ms
116 ms
AC node
DC node
AC line
DC line
=~
=
~
=
~
grid
8 10 13 15
26 27 28 29
30 3132 3334 3536 37 38 39 40
912
11 1614 17 18 19 20 21 22 23 24 25
grid
41
43
46
48
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
42
45
44
49
47
50
51
52
53
54
55
56
57
58
DG DG
VSC1
VSC3
VSC2
DG
DG
DG
grid
74 76 79 81
92 93 94 95
9697 98
99
100
101
102
103
104
105
106
75 78
77 82
80 83 84 85 86 87 88 89 90 91
DG
DG
DG
DG DG
DG
1
23 4
5
76
AC region
AC region
AC region
DC region
Fig. 1A Structure of the 106-node hybrid AC/DC distribution system.
information exchange requirement. Moreover, a linearized
formulation of the regional SE is derived for AC and DC
regions to improve the computational efficiency at the expense
of slight estimation accuracy. A DNN-based method is further
proposed to tackle the different timescales of SCADA
measurements and smart meter data, avoiding the usage of
imprecise pseudo measurements. Simulation results of two
hybrid AC/DC distribution systems have demonstrated the
effectiveness and scalability of the proposed DRSE. In short,
the proposed DRSE method allows for integrating smart meter
data, SCADA measurements and zero injections in the
estimation model of hybrid AC/DC distribution systems,
protecting regional information privacy, and provide the latest
states in real-time.
APPENDIX A
The topology structure of the 106-node hybrid AC/DC
distribution systems is presented in Figs. 1A. As for the system
partition, the mesh 106-node hybrid AC/DC distribution system
consists of one DC region and three AC regions.
REFERENCES
[1] H. M. A. Ahmed and M. M. A. Salama. “Energy management of AC–
DC hybrid distribution systems considering network reconfiguration,”
IEEE Trans. Power Systems., vol. 34, no. 6, pp. 4583-4594, Nov. 2019.
[2] H. M. A. Ahmed, A. B. Eltantawy and M. M. A. Salama. “A reliability-
based stochastic planning framework for AC-DC hybrid smart
distribution systems,” International Journal of Electrical Power &
Energy Systems, vol. 107, pp. 10-18, May 2019.
[3] F. C.Schweppe and E. J. Handschin. “Static state estimation in electric
power systems,” Proceedings of the IEEE, vol. 62, no. 7, pp. 972-982,
Jul. 1974.
[4] A. Abur and M. K. Celik, “A fast algorithm for the weighted least
absolute value state estimation,” IEEE Trans. Power Systems, vol. 6, no.
2, pp.1-8, Feb. 1991.
[5] J. Zhao, S. Wang, L. Mili, et al. “A robust state estimation framework
considering measurement correlations and imperfect synchronization,”
IEEE Trans. Power Systems, vol. 33, no. 4, pp. 4604-4613, Jul. 2018.
[6] A. Primadianto and C. N. Lu. “A review on distribution system state
estimation,” IEEE Trans. Power Systems, vol. 32, no. 5, pp. 3875-3883,
Sept. 2017.
[7] K. Dehghanpour, Z. Wang, J. Wang, et al. “A survey on state estimation
techniques and challenges in smart distribution systems,” IEEE Trans.
Smart Grid, vol. 10, no. 2, pp. 2312-2322, Mar. 2019.
[8] K. Dehghanpour, Y. Yuan, Z. Wang, et al. “A game-theoretic data-driven
approach for pseudo-measurement generation in distribution system state
estimation,” IEEE Trans. Smart Grid, vol. 10, no. 6, pp. 5942-5951, Nov.
2019.
[9] F. Ahmad, M. Tariq and A. Farooq. “A novel ANN-based distribution
network state estimator,” International Journal of Electrical Power &
Energy Systems, vol. 107, pp. 200-212, May 2019.
[10] J. H. Menke, N. Bornhorst and M. Braun. “Distribution system
monitoring for smart power grids with distributed generation using
artificial neural networks,” International Journal of Electrical Power &
Energy Systems, vol. 113, pp. 472-480, Dec. 2019.
[11] S. Bhela, V. Kekatos and S. Veeramachaneni. “Enhancing observability
in distribution grids using smart meter data,” IEEE Trans. Smart Grid,
vol. 9, no. 6, pp. 5953-5961, Nov. 2018.
[12] Y. Liu, J. Li, L. Wu. “State estimation of three-phase four-conductor
distribution systems with real-time data from selective smart meters,”
IEEE Trans. Power Systems, vol. 34, no. 4, pp. 2632-2643, Jul. 2019.
[13] A. Alimardani, F. Therrien, D. Atanackovic, et al. “Distribution system
state estimation based on nonsynchronized smart meters,” IEEE Trans.
Smart Grid, vol. 6, no. 6, pp. 2919-2928, Nov. 2015.
[14] K. R. Mestav, J. Luengo-Rozas and L. Tong. “Bayesian state estimation
for unobservable distribution systems via deep learning,” IEEE Trans.
Power Systems, vol. 34, no. 6, pp. 4910-4920, Nov. 2019.
[15] S. C. Huang, C. N. Lu and Y. L. Lo. “Evaluation of AMI and SCADA
data synergy for distribution feeder modeling,” IEEE Trans. Smart Grid,
vol. 6, no. 4, pp. 1639-1647, Jul. 2015.
[16] H. M. A. Ahmed, A. B. Eltantawy and M. M. A. Salama. “A generalized
approach to the load flow analysis of AC–DC hybrid distribution
systems,” IEEE Trans. Power Systems, vol. 33, no. 2, pp. 2117-2127,
Mar. 2018.
[17] N. Xia, H. B. Gooi, S. Chen, el al. “Decentralized state estimation for
hybrid AC/DC microgrids,” IEEE Systems Journal, vol. 12, no. 1, pp.
434-443, Mar. 2018.
[18] M. Pau, A. Sadu, S. Pillai, et al. “A state estimation algorithm for hybrid
AC/DC networks with multi-terminal DC grid,” 2016 IEEE PES
Innovative Smart Grid Technologies Conference Europe (ISGT-Europe),
Ljubljana, Slovenia, 2016, pp.1-6.
[19] X. Kong, Z. Yan, R. Guo, et al. “Three-Stage distributed state estimation
for AC-DC hybrid distribution network under mixed measurement
environment,” IEEE Access, vol. 6, pp. 39027-39036, Jul. 2018.
[20] P. Grahn, V. Briggner, L. Johansson, et al. “Centralized versus
distributed state estimation for hybrid AC/HVDC grid,” 2017 IEEE PES
Innovative Smart Grid Technologies Conference Europe (ISGT-Europe),
Torino, Italy, 2017, pp. 1-6.
[21] J. Beerten, S. Cole and R. Belmans. “A sequential AC/DC power flow
algorithm for networks containing Multi-terminal VSC HVDC systems,”
2010 IEEE PES General Meeting, Providence, USA, 2010, 89(1):1 - 7.
[22] J. Zhang, Y. Wang, Y. Weng, et al. “Topology identification and line
parameter estimation for Non-PMU distribution network: a numerical
method,” IEEE Trans. on Smart Grid, vol. 11, no. 5, pp. 4440-4453, Mar.
2020.
[23] R. Dutta, V. S. Patel, S. Chakrabarti, et al. “Parameter estimation of
distribution lines using SCADA measurements,” IEEE Trans. on
Instrumentation and Measurement, 2021, DOI: 10.1109/TIM.2020.
3026116.
[24] A. V. Jaén, E. Acha and A. G. Expósito. “Voltage source converter
modeling for power system state estimation: STATCOM and VSC-
HVDC,” IEEE Trans. Power Systems, vol. 23, no. 4, pp. 1552-1559, Nov.
2008.
[25] J. Beerten, S. Cole, and R. Belmans. “Generalized steady-state VSC
MTDC model for sequential AC/DC power flow algorithms,” IEEE
Trans. Power Systems, vol. 27, no. 2, pp. 821-829, May 2012.
[26] M. Göl and A. Abur. “LAV based robust state estimation for systems
measured by PMUs,” IEEE Trans. Smart Grid, vol. 5, no. 4, pp. 1808-
1814, Jul. 2014.
[27] M. D. Sankur, R. Dobbe, A. von Meier, et al. “Optimal voltage phasor
regulation for switching actions in unbalanced distribution systems,”
2020 IEEE Power & Energy Society General Meeting, Montreal, Canada,
2020, pp. 1-5.
[28] B. A. Robbins and A. D. Domínguez-García. “Optimal reactive power
dispatch for voltage regulation in unbalanced distribution systems,”
IEEE Trans. Power Systems, vol. 31, no. 4, pp. 2903-2913, Jul. 2016.
[29] Y. Zhang and J. Hui. “Towards highly efficient state estimation with
nonlinear measurements in distribution systems,” IEEE Trans. Smart
Grid, vol. 35, no. 3, pp. 2471-2474, May 2020.
[30] M. Farajollahi, A. Shahsavari and H. Mohsenian-Rad. “Topology
identification in distribution systems using line current sensors: an MILP
approach,” IEEE Trans. Smart Grid, vol. 11, no. 2, pp. 1159-1170, Mar.
2020.
[31] J. Liu, J. Tang, F. Ponci, et al. “Trade-offs in PMU deployment for state
estimation in active distribution grids,” IEEE Trans. Smart Grid, vol. 3,
no. 2, pp. 915-924, Jun. 2012.
[32] J. Peppanen, M. J. Reno, M. Thakkar, et al. “Leveraging AMI data for
distribution system model calibration and situational awareness,” IEEE
Trans. Smart Grid, vol. 6, no. 4, pp. 2050-2059, Jul. 2015.
[33] X. Zhou, Z. Liu, Y. Guo, et al. “Gradient-based multi-area distribution
system state estimation,” IEEE Trans. Smart Grid, vol. 11, no. 6, pp.
5325-5338, Nov. 2020.
