# Two-level Robust State Estimation for Multi-Area Power Systems Under Bounded
Uncertainties

Shiva Moshtagh and Mehdi Rahmani Shiva Moshtagh and Mehdi Rahmani are with
Department of Electrical Engineering, Imam-Khomeini International University,
Qazvin, Iran (email<EMAIL_ADDRESS>
###### Abstract

This paper introduces a two-level robust approach to estimate the unknown
states of a large-scale power system while the measurements and network
parameters are subjected to uncertainties. The bounded data uncertainty (BDU)
considered in the power network is a structured uncertainty which is
inevitable in practical systems due to error in transmission lines, inaccurate
modelling, unmodeled dynamics, parameter variations, and other various
reasons. In the proposed approach, the corresponding network is first
decomposed into smaller subsystems (areas), and then a two-level algorithm is
presented for state estimation. In this algorithm, at the first level, each
area uses a weighted least squares (WLS) technique to estimate its own states
based on a robust hybrid estimation utilizing phasor measurement units (PMUs),
and at the second level, the central coordinator processes all the results
from the subareas and gives a robust estimation of the entire system. The
simulation results for IEEE 30–bus test system verifies the accuracy and
performance of the proposed multi-area robust estimator.

###### Index Terms:

State estimation, weighted least squares, phasor measurement unit,
uncertainty, multi-area power system

##  I Introduction

State estimation (SE) determines the most likely states of a power system
using the measurements collected by SCADA system via remote terminal units
(RTUs). Since measurements are not available exactly, the state variables are
estimated based on the statistical methods. In this regard, weighted least
squares (WLS) is one of the most common criteria that minimizes the sum of the
squares of weighted differences between the estimated and actual variables. As
the current electrical power systems continue to become larger and more
complex, there has been a rising demand for computationally efficient robust
SE algorithms. Also, to complete the process of estimation, information is
collected from smaller and often competing companies. These companies usually
employ distinctive algorithms for SE, and may be reluctant to share the
detailed information with their rivals. Therefore, this estimation is
performed in a distributed paradigm for multi-area power systems.

Research on multi-area SE can be traced back to the 1970s [1], right after the
introduction of SE in power systems by Schweppe [2]. In multi-area SE, the
network should first be broken down into individual areas. There are different
strategies for network decomposition. In [3, 4], the system is decomposed such
that the areas (subsystems) are overlapping; while in [5, 6, 7], they are non-
overlapping. The latter kind of decomposition can speed up and simplify the
estimation process because the estimation is separately performed at each area
in a parallel manner. Most of multi-area SE techniques utilize a coordinator
to handle the interaction of local estimators [8, 9, 10, 11]; however, some
others do not require a coordinator [12, 13]. In this regard, a two-stage
multi-area approach is presented in [14] in which at the first stage, local
estimators at each area operate separately. The second stage integrates the
solutions from all local estimators along with other available measurements to
give a central estimation of the whole system via coordinator. In [15], a
taxonomy of multi-area SE has been proposed based on multiple classification
criteria.

In the recent years, the advent of phasor measurement unit (PMU) has
revolutionized power system SE due to their high speed and accuracy. Before
these digital devices, SE relied on the measurements obtained from a SCADA
system which includes traditional measurements such as bus power injections
and line power flows. It is shown that with the aid of enough PMUs, the
performance of multi-area SE will improve [14]. However, Since it is not
economically and technically possible to have enough number of PMUs in a
large-scale network, the optimal PMU placement has also been a favorite
research topic recently [8]. By using only PMU measurements, a linear SE can
be utilized to determine all states of a observable system. In [16, 17], a
multi-area robust estimation has been addressed using only PMUs through a
linear model. PMUs are generally utilized along with the traditional
measurements. This scheme is so-called hybrid estimation method. This
combination can either be directly [18], or indirectly [19]. As a two-step
hybrid method, [20] proposed an algorithm in which a traditional SE is first
executed via SCADA measurements, then the results of the previous step as well
as PMU measurements will be incorporated in a linear post processing
estimator.

It is very important to note that error in transmission of information through
the network, inaccurate modeling, etc cause uncertainty in the network model.
It can affect the accuracy of the estimation significantly [21]. Since the WLS
method is sensitive to inaccurate data, it is not a robust method for
estimation. In the literature, several methods have been studied towards
developing robust alternatives to this non-robust estimator. Among all, the
least absolute value (LAV) estimator could be a practical and more robust
approach providing that there are not leverage measurements [22]. In [23], the
problem of leverage measurements has been solved by using only PMU
measurements which could not be affordable. Several multi-area robust
approaches have been provided in [24, 25, 26] by handling bad data related to
the outlier measurements which deteriorate the performance of the estimator.
Model validation can also be efficient along to safeguarding the estimation
process against uncertainty, yet is challenging [27]. Accordingly, it is a
more viable way to make estimators more robust to model uncertainty. As far as
we know, the robust SE problem for power systems subject to norm-bounded
uncertainties using PMUs has not been studied in the literature yet, in
particular, in a distributed (two-level) approach for multi-area power
systems.

This paper presents a novel algorithm for multi-area robust SE to estimate the
states of an uncertain large-scale power system using SCADA and PMU
measurements. The contribution of this paper is primarily considering bounded
structured uncertainty in both measurements and network parameters of the
entire power system and then, implementing a multi-area estimation through a
two-level approach integrating hybrid linear method. Initially, the entire
system is decomposed into a specific number of non-overlapping areas. Then, at
the first level of the algorithm, the individual areas separately perform
their estimators in parallel based on their internal measurements. The central
coordinator marks the second level of the estimation process by receiving
estimated values of states and boundary measurements as well, and finally
gives a robust SE for the power system. It is remarkable that this idea has
not been applied to power systems’ SE so far. The reason for employing such a
technique arises from firstly, the accuracy and secondly, the fast speed of
the proposed distributed algorithm which does not require iteration between
the local estimators and the coordinator. This multi-area algorithm does not
enforce any particular requirements on the boundary measurements. Moreover,
each area is empowered to have its own estimation algorithm, network database,
and measurement without affecting the performance of the other areas’
estimators since the local estimators do not exchange or interact data. It is
assumed that there is enough redundancy to ascertain the observability, and
bad data have been eliminated; however, bounded data uncertainty as another
major source of misestimating is imposed on the measurements.

The remainder of this paper is structured as follows. Section II introduces
the formulation of multi-area SE. Section III addresses a brief review on WLS
and robust estimation separately. Section IV focuses on a framework which
presents the algorithm and structure of the proposed method. In Section V
simulation results on IEEE 30–bus test system are discussed and in Section VI
conclusion remarks are presented.

Notations: The notations used throughout the paper are fairly standard which
are explained as follows

Notation | Meaning
---|---
$A^{-1}$ | Inverse of matrix A
$A^{{\dagger}}$ | Moore-Penrose pseudoinverse of matrix A
$\mathbb{E}(a)$ | Mathematical expectation of variable a
$\text{cov}(A)$ | Covariance of matrix A
$\text{diag}(A)$ | Diagonal elements of matrix A
$\|a\|$ | Norm of variable a
$\mathbb{R}^{+}$ | Coordinate space over the positive numbers

##  II Multi-area SE Modeling

In the proposed multi-area SE, the large-scale power network should be
decomposed into non-overlapping local areas. The partitioned zones are
connected to each other by tie-lines. The state variables in a power system
are bus voltage magnitude and phase angle that are needed to be estimated
through some measurements including bus power injections, and line power
flows.

For multi-area decomposition, a power network consisting of $n$ buses is
considered to be decomposed into $r$ areas. This division is done in such a
way that the minimum boundary tie-lines exist in the decomposed network.
Accordingly, at $i$th area, each bus can be one of these types:

  * •

Internal bus: buses that are completely within area $i$ and their neighbors
are members of the same area.

  * •

Boundary bus: buses that have at least one connection to buses from other
areas.

  * •

External bus: buses that are located at another area and have at least one
link to a boundary bus at area $i$.

Figure 1: Decomposed power network in $r$ non-overlapping areas

Fig. 1 shows the sketch of a decomposed power system along with three types of
buses for area 1. The state variables of these three types of bus, for
$i=1,2,\cdots,r$, are represented by ${x_{i}}^{\text{int}}$,
${x_{i}}^{\text{bnd}}$ and ${x_{i}}^{\text{ext}}$ that contain the voltage
magnitude and phase angle of internal buses, boundary buses, and external
buses, respectively.

Consequently, The state vector of $i$th area is introduced as follows

$\displaystyle
x_{i}=[{x_{i}}^{\text{int}},{x_{i}}^{\text{bnd}},{x_{i}}^{\text{ext}}]^{T}$
(1)

The dimension of state vector $x$ at area $i$ is $n_{i}$. Under the above
definition, the state vector of each area contains not only its variables, but
also parts of the state variables of other areas. Therefore, some state
variables will be estimated simultaneously by several neighboring estimators.

By decomposition of a power network into separate areas, every subsystem has
its own measurements and also there would be some boundary measurements. Fig.
2 shows the boundary measurements in buses $m$ and $k$ in two different areas.
In fact, the boundary between two areas determines which measurement belongs
to which area. As illustrated in Fig. 2, the power injection to bus $k$ and
power flow from this bus at area $i$, cannot be available at area $j$ and the
same concept is established for area $j$. Hence, the boundaries are very
deciding in each subsystems’ measurements.

Figure 2: Boundary measurements in decomposed power network

Each area has a reference bus with an arbitrary value and therefore, other
phase angles are determined with respect to this reference bus. One bus at
area 1 is usually chosen as the global reference bus for the entire system.
Note that the voltage phase angle of the reference buses will not be estimated
unless they have been equipped with PMUs.

##  III Least Square Based SE

In this section, we concentrate on the basics of the least square based
estimation that is used in the proposed approach. Consider a nonlinear model
as follows

$z=h(x)+e=\ \begin{bmatrix}z_{1}\\\ z_{2}\\\ \vdots\\\ z_{m}\end{bmatrix}=\
\begin{bmatrix}h_{1}(x_{1},x_{2},...,x_{n})\\\ h_{2}(x_{1},x_{2},...,x_{n})\\\
\vdots\\\ h_{m}(x_{1},x_{2},...,x_{n})\end{bmatrix}+\ \begin{bmatrix}e_{1}\\\
e_{2}\\\ \vdots\\\ e_{m}\end{bmatrix}$ (2)

where $z$ is the measurement vector, $x$ is the unknown state variable vector,
$h$ is the vector of nonlinear functions that represent the relationship
between the measurements and the states of the system, and $e$ is the
measurement error vector that indicates the mismatch between the measured
vector $z$, and the vector $h(x)$. It is assumed that the measurement errors
are independent variables, with zero mean and a diagonal covariance matrix as
follows

$W=\text{cov}(e)=\mathbb{E}(e.e^{T})=\text{diag}\\{\sigma_{1}^{2},\sigma_{2}^{2},...,\sigma_{m}^{2}\\}$
(3)

where $\sigma_{i}$ is the standard deviation of errors and $i$ refers to the
measurement number.

The WLS method tries to estimate the state variables by minimizing the
following performance index.

$J(x)=(z-h(x))^{T}W^{-1}(z-h(x))$ (4)

where $W\in\mathbb{R}^{m\times m}$ is a weighting matrix.

By applying a first-order Taylor series expansion and using the Gauss-Newton
iterative solution for (4), the SE can be calculated from the following
recursive equation

$x^{k+1}=x^{k}+\Delta x^{k}$ (5)

The iterations will continue until the alteration in state variables has
fallen within a satisfactory range as

$\max\leavevmode\nobreak\ \|\Delta x^{k}\|<\epsilon$ (6)

where $\epsilon$ is a small positive number and indicates the accuracy of the
estimation.

By defining $H(x)=\frac{\partial h(x)}{\partial x}$, the covariance matrix of
these estimated variable can be achieved as follows

$\text{cov}(\hat{x})=(H^{T}(\hat{x})W^{-1}H(\hat{x}))^{-1}$ (7)

###  III-A Robust SE

Although power systems are nonlinear, with some approximations they can be
modeled as a linear system. In the following, we investigate a robust
estimation based on a linear model as follows

$z=Hx+e$ (8)

The above-mentioned model is a linear version of (2) in which the Jacobian
matrix $H$ includes all derivatives of measurement equations with respect to
state variables.The nominal data $\\{z,H\\}$ in (8) are often subjected to
disturbances and uncertainties. Such errors can deteriorate the performance of
the estimation process. To solve this issue, the robust least squares problem
is presented as a min–max optimization that is known as bounded data
uncertainty (BDU) problem [28]. The BDU problem is trying to obtain an
estimation that minimizes the worst case possible of the cost function in
presence of uncertainty. Consider the model presented in (8) subject to
uncertainties as follows

$(z+\delta z)=(H+\delta H)x+e$ (9)

where $\delta z$ denotes an $m\times 1$ perturbation vector, and $\delta H$ is
an $m\times n$ perturbation matrix. Note that these quantities are considered
as unknown bounded values which are different from $\sigma$, the standard
deviation of meters.

In this model, both the measurements and the network parameters are assumed to
be uncertain with the following structure

$[\begin{matrix}\delta H&\delta z\end{matrix}]=\
S\Delta[\begin{matrix}E_{h}&E_{z}\end{matrix}]$ (10)

where $S$, $E_{h}$ and $E_{z}$ are known parameters with appropriate
dimensions, and $\Delta$ represents an unknown bounded uncertainty that
satisfies

$\|\Delta\|\leq 1$ (11)

######  Remark 1.

Perturbation model like (10) is common in robust estimation and control. It
could arise from tolerance specifications on physical parameters [29].

Considering the model in (9), the cost function given in (4) is changed to the
following robustified version

$J(x,y)=\min_{x}\max_{\parallel
y\parallel\leq\phi(x)}[(Hx-z+Sy)^{T}R(Hx-z+Sy)]$ (12)

where $R=W^{-1}$ is the weighting matrix.

Here, $y$ is a $q\times 1$ unknown perturbation vector and $S$ is an $m\times
q$ known matrix, where $q$ is the dimension of uncertainty parameter $\Delta$.
Since $y$ itself is not known as a set of uncertainty, we consider a bound on
its Euclidean norm. The upper bound of $y$ is
${\textstyle{\phi(x)=\|E_{h}x-E_{z}\|}}$, that is a nonnegative function of
$x$. We can also define

$Sy\triangleq{\delta Hx-\delta z}$ (13)

The estimated values of (12) can be obtained as

$\hat{x}=\arg\min_{x}\max_{\parallel y\parallel\leq\phi(x)}J(x,y)$ (14)

Consequently, we have a constrained two-player game problem which needs to be
solved once for any fixed $x$ and once for any fixed $y$. The solution of this
problem is achieved in [30] as follows

$\hat{x}=(H^{T}\hat{R}H)^{-1}{(H^{T}\hat{R}z+\hat{\lambda}E_{h}^{T}E_{z})}$
(15)

where the modified weighting matrix $\hat{R}$ are derived from $R$ via

$\hat{R}\triangleq{R+RS(\hat{\lambda}I-S^{T}RS)^{\dagger}S^{T}R}$ (16)

The nonnegative scalar parameter $\hat{\lambda}$ is obtained from the
following optimization problem λ≥∥ ST R S ∥G(λ) ^λ= where the function
$G(\lambda)$ is defined as (17), in which
${\textstyle{\|Hx(\lambda)-z\|}_{\scriptscriptstyle{R(\lambda)}}}$ is the
weighted norm of vector ${\textstyle(Hx-z)}$.

$G(\lambda)=\lambda\|E_{h}x(\lambda)-E_{z}\|+{\|Hx(\lambda)-z\|}_{R(\lambda)}$
(17)

where

$R(\lambda)=R+RS(\lambda I-S^{T}RS)^{\dagger},$ (18)

and

$x(\lambda)\triangleq{(H^{T}R(\lambda)H)^{-1}(H^{T}R(\lambda)z+\lambda
E_{h}^{T}E_{z})}.$ (19)

We thus see that solving (14) requires an optimal nonnegative scalar parameter
$\hat{\lambda}$, and it needs to minimize $G(\lambda)$ over the semi-open
interval ${[\parallel S^{T}RS\parallel,\infty)}$. Despite the difficulties of
solving the optimization problem (1), it has a unique solution [30].
Furthermore, in [31] a practical approximation for $\lambda$ is given as

$\lambda=(1+\mu)\|S^{T}RS\|$ (20)

where $\mu\in\mathbb{R}^{+}$ is an arbitrary value to attain a suitable
performance.

##  IV Framework for Multi-Area Robust SE

In this section, we first present the robust hybrid SE to discuss using both
SCADA and PMU measurements in SE, and then the proposed two-level robust SE
approach for multi-are power systems is investigated.

###  IV-A Robust Hybrid SE

In the proposed method, the hybrid SE consists of two steps in which
traditional and PMU measurements are processed in individual state estimators,
and then joined in a linear one. By this way, the difference between SCADA and
PMU’s sampling rate will not be problematic; furthermore, there is no need to
change the SCADA-based estimators to incorporate PMU measurements as they are
employed in separate estimators. At the first step, we estimate the states
using the traditional SCADA measurements via the nonlinear model described at
the beginning of the Section III. At the second step, we solve a robust SE
problem using phasor measurements along with the results of the first step.
This combination leads to more accurate estimations. The second step’s robust
linear SE is discussed in Section III-A. Note that one PMU can measure not
only the voltage phasor, but also the line’s current phasors. Therefore, a PMU
can give more than one measurement.

######  Remark 2.

The results of the first step of estimation are in polar coordinates. Prior to
be used at the second step, the obtained state vector of the first step should
be transformed from polar to rectangular coordinates for the linearity
requirement to be satisfied.

The complete measurement model is shown in (21), in which $z$ is the vector
containing both traditional and PMU measurements and its elements are in
rectangular coordinates. $V_{R}$, $I_{R}$, $V_{I}$ and $I_{I}$ are the real
and imaginary parts of bus voltage and current, respectively. The subscripts
TSE and PMU refer to values obtained from the traditional SE and PMU
measurements, respectively. Since the state variables are the real and
imaginary parts of voltage, the partial derivatives in Jacobian matrix $H$ are
with respect to $V_{R}$ and $V_{I}$. Also, $e$ is the measurement error vector
in which related elements to TSE are obtained from (7). Likewise, since all
the amounts in this technigue are indicated in rectangular coordinates, the
standard deviations of variables should also be converted to rectangular
format.

$\scriptsize\begin{multlined}z\\!=\\!\begin{bmatrix}{\begin{bmatrix}V_{R}\\\
V_{I}\end{bmatrix}}_{\text{TSE}}\\\\[7.0pt] {\begin{bmatrix}V_{R}\\\
V_{I}\end{bmatrix}}_{\text{PMU}}\\\\[7.0pt] {\begin{bmatrix}I_{R}\\\
I_{I}\end{bmatrix}}_{\text{PMU}}\end{bmatrix}\\!=\\!\begin{bmatrix}\frac{\partial
V_{R,\text{TSE}}}{\partial V_{R}}&\frac{\partial V_{R,\text{TSE}}}{\partial
V_{I}}\\\\[5.0pt] \frac{\partial V_{I,\text{TSE}}}{\partial
V_{R}}&\frac{\partial V_{I,\text{TSE}}}{\partial V_{I}}\\\\[5.0pt]
\frac{\partial V_{R,\text{PMU}}}{\partial V_{R}}&\frac{\partial
V_{R,\text{PMU}}}{\partial V_{I}}\\\\[5.0pt] \frac{\partial
V_{I,\text{PMU}}}{\partial V_{R}}&\frac{\partial V_{I,\text{PMU}}}{\partial
V_{I}}\\\\[5.0pt] \frac{\partial I_{R,\text{PMU}}}{\partial
V_{R}}&\frac{\partial I_{R,\text{PMU}}}{\partial V_{I}}\\\\[5.0pt]
\frac{\partial I_{I,\text{PMU}}}{\partial V_{R}}&\frac{\partial
I_{I,\text{PMU}}}{\partial
V_{I}}\end{bmatrix}\\!.\\!\begin{bmatrix}V_{R}\\\\[5.0pt]
V_{I}\end{bmatrix}\\!+\\!\begin{bmatrix}e_{V_{R,\text{TSE}}}\\\\[4.0pt]
e_{V_{I,\text{TSE}}}\\\\[4.0pt] e_{V_{R,\text{PMU}}}\\\\[4.0pt]
e_{V_{I,\text{PMU}}}\\\\[4.0pt] e_{I_{R,\text{PMU}}}\\\\[4.0pt]
e_{I_{I,\text{PMU}}}\end{bmatrix}\end{multlined}z\\!=\\!\begin{bmatrix}{\begin{bmatrix}V_{R}\\\
V_{I}\end{bmatrix}}_{\text{TSE}}\\\\[7.0pt] {\begin{bmatrix}V_{R}\\\
V_{I}\end{bmatrix}}_{\text{PMU}}\\\\[7.0pt] {\begin{bmatrix}I_{R}\\\
I_{I}\end{bmatrix}}_{\text{PMU}}\end{bmatrix}\\!=\\!\begin{bmatrix}\frac{\partial
V_{R,\text{TSE}}}{\partial V_{R}}&\frac{\partial V_{R,\text{TSE}}}{\partial
V_{I}}\\\\[5.0pt] \frac{\partial V_{I,\text{TSE}}}{\partial
V_{R}}&\frac{\partial V_{I,\text{TSE}}}{\partial V_{I}}\\\\[5.0pt]
\frac{\partial V_{R,\text{PMU}}}{\partial V_{R}}&\frac{\partial
V_{R,\text{PMU}}}{\partial V_{I}}\\\\[5.0pt] \frac{\partial
V_{I,\text{PMU}}}{\partial V_{R}}&\frac{\partial V_{I,\text{PMU}}}{\partial
V_{I}}\\\\[5.0pt] \frac{\partial I_{R,\text{PMU}}}{\partial
V_{R}}&\frac{\partial I_{R,\text{PMU}}}{\partial V_{I}}\\\\[5.0pt]
\frac{\partial I_{I,\text{PMU}}}{\partial V_{R}}&\frac{\partial
I_{I,\text{PMU}}}{\partial
V_{I}}\end{bmatrix}\\!.\\!\begin{bmatrix}V_{R}\\\\[5.0pt]
V_{I}\end{bmatrix}\\!+\\!\begin{bmatrix}e_{V_{R,\text{TSE}}}\\\\[4.0pt]
e_{V_{I,\text{TSE}}}\\\\[4.0pt] e_{V_{R,\text{PMU}}}\\\\[4.0pt]
e_{V_{I,\text{PMU}}}\\\\[4.0pt] e_{I_{R,\text{PMU}}}\\\\[4.0pt]
e_{I_{I,\text{PMU}}}\end{bmatrix}$ (21)

The measurement Jacobian coefficient matrix $H$, can be rewritten in the
following form

$H=\ \begin{bmatrix}\textrm{I}&0\\\\[5.0pt] 0&\textrm{I}\\\\[5.0pt]
\textrm{II}&0\\\\[5.0pt] 0&\textrm{II}\\\\[5.0pt] \frac{\partial
I_{R,\text{PMU}}}{\partial V_{R}}&\frac{\partial I_{R,\text{PMU}}}{\partial
V_{I}}\\\\[5.0pt] \frac{\partial I_{I,\text{PMU}}}{\partial
V_{R}}&\frac{\partial I_{I,\text{PMU}}}{\partial V_{I}}\end{bmatrix}$ (22)

where I is an identity matrix of dimension $n\times n$ because TSE
measurements are directly related to state variables, and II denotes a
$p\times n$ sparse matrix which $p$ refers to the number of PMUs. Each element
of this matrix is zero except those are related to the buses which have a PMU.
In other words, depending on the placement of PMUs, the zero elements of this
matrix are replaced by 1. In addition, the covariance matrix $W$, has been
expanded to include the standard deviations of the TSE and PMU measurement
values as follows

$\scriptsize W_{h}=\begin{bmatrix}{{\sigma^{{\scriptscriptstyle 2}}}_{{\tiny
V_{R,\text{TSE}}}}}&0&0&0&0&0\\\ 0&{{\sigma^{\scriptscriptstyle 2}}_{{\tiny
V_{I,\text{TSE}}}}}&0&0&0&0\\\ 0&0&{{\sigma^{\scriptscriptstyle 2}}_{{\tiny
V_{R,\text{PMU}}}}}&0&0&0\\\ 0&0&0&{{\sigma^{\scriptscriptstyle 2}}_{{\tiny
V_{I,\text{PMU}}}}}&0&0\\\ 0&0&0&0&{{\sigma^{\scriptscriptstyle 2}}_{{\tiny
I_{R,\text{PMU}}}}}&0\\\ 0&0&0&0&0&{{\sigma^{\scriptscriptstyle
2}}_{{\scriptscriptstyle I_{I,\text{PMU}}}}}\\\ \end{bmatrix}$ (23)

In (23), each element, $\sigma^{2}$, is a diagonal matrix containing the
individual variances within the distinct measurement type. Now, to solve the
hybrid estimation problem, we can apply a cost function as follows

$J_{\text{hybrid}}(x)=(z-Hx)^{T}W_{h}^{-1}(z-Hx)$ (24)

Therefore, the weighted least squares solution of this model is non-iterative
and is given by

$\hat{x}_{\text{hybrid}}=(H^{T}W_{h}^{-1}H)^{-1}H^{T}W_{h}^{-1}z$ (25)

This two-step estimation leads to faster process due to less dimension, and no
need to run further iterations at the second step, as well as more accurate
results owing to inclusion of high-precision PMU measurements. The linearity
of the second step is the motivation of consolidation of structured
uncertainty (10) into existing hybrid method. In the following subsection, we
will propose a framework for multi-area robust SE by adding uncertainty into
the model of the decomposed power system. In fact, we will solve a distributed
hybrid SE with this difference that the matrices i.e. the network parameters
and the measurements, are implicitly uncertain.

######  Remark 3.

Hybrid estimation is used so as to make the estimation process linear to pave
the way for utilizing structured uncertainty which has not been applied to
power system so far. In view of the fact that linearization of power system
has difficulties and also needs the operating point which is unknown, using
two-step hybrid estimation comes across as the best way.

###  IV-B Proposed Two-level Robust SE for Multi-Area Power Systems

In this section, we propose a framework for multi-area robust SE in a large-
scale power system in which both measurements and network parameters are
implicitly uncertain. The main goal is to find a robust solution for voltage
phasors in all buses. The proposed multi-area approach consists of the two
following levels.

####  IV-B1 Level 1 – Local Area SE

At the first level, each subsystem uses a robust hybrid estimator to estimate
its own state variables by minimizing the following cost function as the first
step of hybrid technique.

$\min_{z_{i}=h_{i}(x_{i})+e_{i}}J_{i}={e_{i}}^{T}{W_{i}}^{-1}e_{i}$ (26)

where

  * •

$z_{i}$ is the measurement vector at area $i$ which has $m_{i}$ elements. It
includes not only internal measurements but also boundary measurements such as
power injections and power flows in boundary buses of area $i$.

  * •

$e_{i}$ is the measurement error vector at area $i$.

  * •

$W_{i}$ is the covariance matrix at area $i$

  * •

$h_{i}(x_{i})$ is the nonlinear measurement function at area $i$.

The results of the first level should be unbaised estimations of internal
states of each area, $\hat{x}_{i}^{\text{int}},\leavevmode\nobreak\
(i=1,\cdots,r)$. Otherwise, it is required to augment the number of meters or
revise their placement.

####  IV-B2 Level 2 – Central Coordinator

At the second level, the central coordinator tries to obtain an accurate
estimation of the boundary state variables by processing the data received
from all the individual areas along with the boundary and PMU measurements.
Therefore, the following optimization problem is considered

$\min_{z_{c}=h_{i}(x_{c})+e_{c}}J_{c}={e_{c}}^{T}{W_{c}}^{-1}e_{c}$ (27)

where
${\textstyle{z_{c}=[z_{b}^{T},z_{\scriptstyle\text{PMU}}^{T},\hat{x}^{\text{bnd}^{T}},\hat{x}^{\text{ext}^{T}}]^{T}}}$
is the measurement vector of the central coordinator, $z_{b}$ is the boundary
measurement vector including power injections in boundary buses and power
flows in boundary tie-lines, $z_{\scriptstyle\text{PMU}}$ is the PMU
measurement vector,
${\textstyle{\hat{x}^{\text{bnd}}=[\hat{x}_{1}^{\text{bnd}^{T}},\hat{x}_{2}^{\text{bnd}^{T}},...,\hat{x}_{r}^{\text{bnd}^{T}}]^{T}}}$
and
${\textstyle{\hat{x}^{\text{ext}}=[\hat{x}_{1}^{{\text{ext}}^{T}},\hat{x}_{2}^{{\text{ext}}^{T}},...,\hat{x}_{r}^{\text{ext}^{T}}]^{T}}}$
are the estimated boundary and external state variables vector by local
estimators, respectively. These estimated values are considered as pseudo-
measurements in the central coordinator because they have priorly been
estimated at level 1 by local estimators. The cost function (27) tries to
estimate ${\textstyle{x_{c}=[x^{\text{bnd}^{T}},u^{T}]^{T}}}$ in which $u$ is
the reference bus vector with respect to the global reference bus at area 1.

As a whole, at the first level, we use a robust hybrid technique at each area,
then at the second level by using the results of the previous level as pseudo-
measurements and other available boundary and PMU measurements, a global
robust hybrid SE is performed by the central coordinator. Note that every
estimator uses a hybrid-based model in estimation process as introduced in
Section IV-A. The structure of the method is shown in Fig. 3. Moreover,
details of the proposed multi-area robust estimation approach is presented in
Algorithm 1.

######  Remark 4.

The advantages of the proposed multi-area hybrid estimation can be listed as
follows:

  * •

Increasing estimation accuracy by incorporating both unsynchronized
traditional and synchronized phasor measurements.

  * •

No requirements for changing the existing SCADA software package owing to
utilizing a non-invasive manner.

  * •

The difference between the sampling rates of SCADA and PMU measurements will
not be troublesome.

  * •

Reducing computational time due to less dimension, no need for iterations at
the second linear step, and the capability of parallel processing at areas at
the first level of proposed two-level algorithm.

######  Remark 5.

If the model does not have uncertainty, the proposed method will be reduced to
a non-robust multi-area hybrid estimation by setting the uncertainty
parameters to zero in the algorithm. Furthermore, if the real system does not
have uncertainties and we run a robust algorithm for SE, the results will be
sub-optimal because of the conservatism of the robust methods.

Figure 3: The structure of the proposed multi-area robust SE in power system
Algorithm 1 Multi-area robust SE in a power system

Level 1

Decompose the power system into $r$ non-overlapping areas and do the following
items for each area in parallel.

Initial condition: Set $x_{0}=0$ rad, for the voltage phase angle, and
$x_{0}=1$ p.u, for the voltage magnitude at all buses. Set $k=0$ as the
iteration index

  1. 1.

Input $z_{{\scriptstyle\text{RTU}}}$ , $W$, $k_{\scriptstyle\text{limit}}$,
$\epsilon$

  2. 2.

Determine $\hat{x}_{i}$ by minimizing $J(x)$ in (26)

  3. 3.

Convert the estimates and estimation errors obtained from (5) and (7),
respectively, from polar coordinates to rectangular coordinates

  4. 4.

Input PMU measurements in rectangular coordinates,
$(z_{\scriptstyle\text{PMU}})$

  5. 5.

Construct a new hybrid measurement vector,
$z_{\scriptstyle\text{hybrid}}=(\hat{x}_{r},\leavevmode\nobreak\
\hat{x}_{i},\leavevmode\nobreak\ z_{\scriptstyle\text{PMU}})^{T}$

  6. 6.

Construct $H_{\scriptstyle\text{hybrid}}$ as in (22)

  7. 7.

Add uncertainty to $z_{\scriptstyle\text{hybrid}}$ and
$H_{\scriptstyle\text{hybrid}}$
${(\textstyle{H_{\scriptstyle\text{hybrid}}\longrightarrow
H_{\scriptstyle\text{hybrid}}+\delta_{H_{\scriptscriptstyle\text{hybrid}}}\hskip
4.5pt,\hskip 4.5ptz_{\scriptscriptstyle\text{hybrid}}\longrightarrow
z_{\scriptscriptstyle\text{hybrid}}+\delta_{z_{\scriptscriptstyle\text{hybrid}}}})}$

  8. 8.

For the uncertain model (9)–(10), choose an appropriate value for $\lambda$ as
in (20)

  9. 9.

Determine $\hat{x}_{\scriptstyle\text{robust}}$ by solving the min-max
optimization problem in (12)

  10. 10.

Convert the estimates from rectangular coordinates to polar coordinates

Level 2

  1. 1.

Receive the pseudo-measurements from the previous level
$(\hat{x}_{\text{bnd}})$

  2. 2.

Input boundary measurements from each area $(z_{\text{bnd}})$

  3. 3.

Construct the central coordinator measurement vector
${(z_{\text{coordinator}}=\begin{bmatrix}\hat{x}_{\text{bnd}}\\\
z_{\text{bnd}}\end{bmatrix})}$

  4. 4.

Determine $\hat{x}_{c}$ by minimizing $J(x)$ in (27)

  5. 5.

Convert the estimates from rectangular coordinates to polar coordinates

  6. 6.

Input PMU measurements in rectangular coordinates,
$(z_{\scriptstyle\text{PMU}})$

  7. 7.

Construct the central coordinator hybrid measurement vector
${\textstyle{(z_{c_{\text{hybrid}}}=\begin{bmatrix}\hat{x}_{c_{\text{real}}}\\\
\hat{x}_{c_{\text{imag}}}\\\ z_{c_{\text{PMU}}}\end{bmatrix})}}$

  8. 8.

Construct $H_{c_{\text{hybrid}}}$ as in (22)

  9. 9.

Add uncertainty to $z_{c_{\text{hybrid}}}$ and $H_{c_{\text{hybrid}}}$
${(\textstyle{H_{c_{\text{hybrid}}}\longrightarrow
H_{c_{\text{hybrid}}}+\delta_{H_{c_{\text{hybrid}}}}\hskip 1.79997pt,\hskip
1.79997ptz_{c_{\text{hybrid}}}\longrightarrow
z_{c_{\text{hybrid}}}+\delta_{z_{c_{\text{hybrid}}}}})}$

  10. 10.

Choose an appropriate value for $\lambda$ as in (20)

  11. 11.

Determine $\hat{x}_{c_{\text{robust}}}$ by solving the min-max optimization
problem in (12) for the central coordinator

  12. 12.

Convert the estimates from rectangular coordinates to polar coordinates

##  V Simulation Results

To evaluate the performance of the proposed multi-area robust estimation
approach, the simulations are carried out with MATLAB software for the IEEE
30–bus test system. The diagram of this network is depicted in Fig. 4. As
shown, the network has been decomposed into 3 non-overlapping areas which are
connected together by multiple tie-lines. In this network, buses 4, 6, 9, 10,
12, 15, 22, 23, 24 and 28 are boundary buses. Buses 4, 15 and 24 are chosen as
slack buses at each area and equipped with PMUs. Detailed information about
the type and number of buses at each area are presented in Table I.

Table I: IEEE 30–bus system area composition

Area | 1 | 2 | 3
---|---|---|---
Internal buses | 5 | 9 | 5
Boundary buses | 3 | 5 | 3
External buses | 4 | 4 | 4

To produce realistic noisy measurements, a load flow is first ran to generate
true data, then the measurement error variance, $\sigma^{2}$, corresponding to
each measurement type is added to them since the existing meters are not
accurate. By defining enough measurements, we intend to estimate the unknown
variables of this network, including the magnitude and phase angle of all
buses. The quantities for standard deviations of measurements from [32, 33]
are tabulated in Table II. Note that a PMU has a smaller error deviation than
other conventional measurements.

Figure 4: IEEE 30–bus test system diagram Table II: Standard Deviations of the
Measurements

Power injection | Power flow | PMU
---|---|---
0.01 | 0.008 | 0.001

Table III: Measurement type and numbers in IEEE 30–bus system

Area | 1 | 2 | 3
---|---|---|---
Power injection pairs | 3 | 5 | 3
Power flow pairs | 15 | 21 | 12

It should be pointed out that in the network decomposition, every area must be
observable. Let us define $m$ as the number of measurements, $n$ as the number
of variables, and $\eta$ as the ratio of the number of measurements per the
number of variables. In this test, we consider this ratio as 1.3. Table III
has more detailed information regarding the measurement numbers for the test.
One power injection pairs contains active and reactive parts of that. To
clarify the performance of the proposed approach, the results are compared
with the two-level version of hybrid estimation presented in Section IV-A.
Note that both cases are exposed to uncertainty in measurements and network
parameters. Fig. 5 shows the estimation errors of voltage magnitude and phase
angle at each buses at area 1 for the IEEE 30–bus system with two methods.
Figs. 6 and 7 show the same results at area 2 and 3, respectively. The error
denotes the absolute difference between estimated state variables and true
values which are obtained from a load flow run. Given that the slack buses are
equipped with PMUs in this network, the phase angle of these buses is also
estimated. Each area has to estimate its internal buses correctly; therefore,
the inaccuracy in estimated boundary and external buses can be disregarded.
These buses will be re-estimated in the central coordinator as shown in Fig. 8
where the central coordinator estimates the boundary states. It can be seen
from the results that the proposed multi-area robust SE has the ability to
limit the uncertainty and can provide more accurate results, i.e. less
estimation error than the WLS method.

Figure 5: Estimation errors of voltage magnitude and phase angle at area 1
Figure 6: Estimation errors of voltage magnitude and phase angle at area 2
Figure 7: Estimation errors of voltage magnitude and phase angle at area 3
Figure 8: Estimation errors of voltage magnitude and phase angle by the
central coordinator

##  VI Conclusions

In this paper, a novel algorithm for multi-area robust SE in large-scale power
systems with bounded uncertainties is proposed. For this purpose, the power
network is first decomposed into several subsystems. Then, a two-level
algorithm is proposed to find a robust solution for estimating the unknown
states of the system. In the proposed approach, at the first level, a
nonlinear SE is performed based on the traditional SCADA measurements and the
network topology. After that, by using the PMU results as a new measurement
set, a robust hybrid SE is done at each area. At the second level, the central
coordinator gives a robust central estimation of the entire system by
receiving the results of estimated states of each area and other available
boundary and PMU measurements. The privilege of this approach is the ability
to do a distributed robust estimation in a large-scale power system while both
the measurements and network parameters are subjected to structured bounded
data uncertainty. By applying the proposed method on the IEEE 30–bus test
system, we showed that it can limit the impact of the uncertainties well and
provide a better solution compared with the hybrid method.

## References

  * [1] K. Clements, “A multi-area approach to state estimation in power system networks,” in _IEEE PES Summer Meeting, 1972_ , 1972.
  * [2] F. C. Schweppe and J. Wildes, “Power system static-state estimation, part I: Exact model,” _IEEE Transactions on Power Apparatus and systems_ , no. 1, pp. 120–125, 1970.
  * [3] D. M. Falcao, F. F. Wu, and L. Murphy, “Parallel and distributed state estimation,” _IEEE Transactions on Power Systems_ , vol. 10, no. 2, pp. 724–730, 1995.
  * [4] C. Muscas, M. Pau, P. A. Pegoraro, S. Sulis, F. Ponci, and A. Monti, “Multiarea distribution system state estimation,” _IEEE Transactions on Instrumentation and Measurement_ , vol. 64, no. 5, pp. 1140–1148, 2015.
  * [5] T. Van Cutsem, J.-L. Horward, and M. Ribbens-Pavella, “A two-level static state estimator for electric power systems,” _IEEE transactions on power apparatus and systems_ , no. 8, pp. 3722–3732, 1981.
  * [6] J. Le, X. Li, Q. Zhou, and L. Zhao, “Multi-area distributed state estimation strategy for large-scale power grids,” _IEEE Access_ , vol. 7, pp. 117 580–117 590, 2019.
  * [7] N. Rathod, H. Patel, and S. Joshi, “Distributed hybrid state estimation with ill conditioning of sub area,” in _2019 8th International Conference on Power Systems (ICPS)_. IEEE, 2019, pp. 1–6.
  * [8] W. Jiang, V. Vittal, and G. T. Heydt, “A distributed state estimator utilizing synchronized phasor measurements,” _IEEE Transactions on Power Systems_ , vol. 22, no. 2, pp. 563–571, 2007.
  * [9] A. Gomez-Exposito and A. de la Villa Jaen, “Two-level state estimation with local measurement pre-processing,” _IEEE Transactions on Power Systems_ , vol. 24, no. 2, pp. 676–684, 2009.
  * [10] G. N. Korres, “A distributed multiarea state estimation,” _IEEE Transactions on Power Systems_ , vol. 26, no. 1, pp. 73–84, 2010.
  * [11] Y. Guo, L. Tong, W. Wu, H. Sun, and B. Zhang, “Hierarchical multi-area state estimation via sensitivity function exchanges,” _IEEE Transactions on Power Systems_ , vol. 32, no. 1, pp. 442–453, 2016.
  * [12] A. J. Conejo, S. de la Torre, and M. Canas, “An optimization approach to multiarea state estimation,” _IEEE Transactions on Power Systems_ , vol. 22, no. 1, pp. 213–221, 2007.
  * [13] Q. Li, L. Cheng, W. Gao, and D. W. Gao, “Fully distributed state estimation for power system with information propagation algorithm,” _Journal of Modern Power Systems and Clean Energy_ , 2020.
  * [14] L. Zhao and A. Abur, “Multi area state estimation using synchronized phasor measurements,” _IEEE Transactions on Power Systems_ , vol. 20, no. 2, pp. 611–617, 2005.
  * [15] A. Gómez-Expósito, A. de la Villa Jaén, C. Gómez-Quiles, P. Rousseaux, and T. Van Cutsem, “A taxonomy of multi-area state estimation methods,” _Electric Power Systems Research_ , vol. 81, no. 4, pp. 1060–1069, 2011.
  * [16] C. Xu and A. Abur, “Robust linear state estimation for large multi-area power grids,” in _2016 IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT)_. IEEE, 2016, pp. 1–5.
  * [17] ——, “Robust linear state estimation using multi-level power system models with different partitions,” in _2017 IEEE Manchester PowerTech_. IEEE, 2017, pp. 1–5.
  * [18] R. Zivanovic and C. Cairns, “Implementation of pmu technology in state estimation: an overview,” in _Proceedings of IEEE. AFRICON’96_ , vol. 2. IEEE, 1996, pp. 1006–1011.
  * [19] R. Nuqui and A. G. Phadke, “Hybrid linear state estimation utilizing synchronized phasor measurements,” in _2007 IEEE Lausanne Power Tech_. IEEE, 2007, pp. 1665–1669.
  * [20] M. Zhou, V. A. Centeno, J. S. Thorp, and A. G. Phadke, “An alternative for including phasor measurements in state estimators,” _IEEE transactions on power systems_ , vol. 21, no. 4, pp. 1930–1937, 2006.
  * [21] G. D’Antona, “Power system static-state estimation with uncertain network parameters as input data,” _IEEE transactions on Instrumentation and Measurement_ , vol. 65, no. 11, pp. 2485–2494, 2016.
  * [22] A. Abur and M. K. Celik, “A fast algorithm for the weighted least absolute value state estimation (for power systems),” _IEEE Transactions on Power Systems_ , vol. 6, no. 1, pp. 1–8, 1991.
  * [23] M. Göl and A. Abur, “Lav based robust state estimation for systems measured by pmus,” _IEEE Transactions on Smart Grid_ , vol. 5, no. 4, pp. 1808–1814, 2014.
  * [24] V. Kekatos and G. B. Giannakis, “Distributed robust power system state estimation,” _IEEE Transactions on Power Systems_ , vol. 28, no. 2, pp. 1617–1626, 2012.
  * [25] X. Li and A. Scaglione, “Robust decentralized state estimation and tracking for power systems via network gossiping,” _IEEE Journal on Selected Areas in Communications_ , vol. 31, no. 7, pp. 1184–1194, 2013.
  * [26] C. Lin, W. Wu, and Y. Guo, “Decentralized robust state estimation of active distribution grids incorporating microgrids based on pmu measurements,” _IEEE Transactions on Smart Grid_ , vol. 11, no. 1, pp. 810–820, 2019.
  * [27] J. Qi, A. F. Taha, and J. Wang, “Comparing kalman filters and observers for power system dynamic state estimation with model uncertainty and malicious cyber attacks,” _IEEE Access_ , vol. 6, pp. 77 155–77 168, 2018.
  * [28] A. H. Sayed, V. Nascimento, and S. Chandrasekaran, “Estimation and control with bounded data uncertainties,” _Linear Algebra and its Applications_ , vol. 284, no. 1-3, pp. 259–306, 1998.
  * [29] C. Yi and B. L. De Moor, “Robustness analysis and control system design for a hydraulic servo system,” _IEEE Transactions on Control Systems Technology_ , vol. 2, no. 3, pp. 183–197, 1994.
  * [30] A. H. Sayed and V. H. Nascimento, “Design criteria for uncertain models with structured and unstructured uncertainties,” in _Robustness in identification and control_. Springer, 1999, pp. 159–173.
  * [31] A. H. Sayed _et al._ , “A framework for state-space estimation with uncertain models,” _IEEE Transactions on Automatic Control_ , vol. 46, no. 7, pp. 998–1013, 2001.
  * [32] I. PES, “Ieee standard for scada and automation systems,” _vol. IEEE Std C_ , vol. 37, 2008.
  * [33] K. E. Martin, G. Benmouyal, M. Adamiak, M. Begovic, R. Burnett, K. Carr, A. Cobb, J. Kusters, S. Horowitz, G. Jensen _et al._ , “Ieee standard for synchrophasors for power systems,” _IEEE Transactions on Power Delivery_ , vol. 13, no. 1, pp. 73–77, 1998.