#  Observers for Differential Algebraic Equation Models of Power Networks:
Jointly Estimating Dynamic and Algebraic States

Sebastian A. Nugroh$\text{o}^{\star}$, Ahmad F. Tah$\text{a}^{\dagger}$,
Nikolaos Gatsi$\text{s}^{\ddagger}$, and Junbo Zha$\text{o}^{\diamond}$
⋆Department of Electrical Engineering and Computer Science, University of
Michigan, 1301 Beal Ave., Ann Arbor, MI 48109 (snugroho@umich.edu).
†Department of Civil and Environmental Engineering, Vanderbilt University,
2201 West End Ave., Nashville, TN 37235 (ahmad.taha@vanderbilt.edu).
‡Department of Electrical and Computer Engineering, The University of Texas at
San Antonio, 1 UTSA Circle, San Antonio, TX 78249 (nikolaos.gatsis@utsa.edu).
⋄Department of Electrical and Computer Engineering, University of Connecticut,
Storrs, CT 06269 (junbo@uconn.edu). †Corresponding author. This work is
partially supported by The Valero Energy Corporation and National Science
Foundation (NSF) under Grants 1719043, 1847125, 2013786, and 2044430.

###### Abstract

Phasor measurement units (PMUs) have become instrumental in modern power
systems for enabling real-time, wide-area monitoring and control. Accordingly,
many studies have investigated efficient and robust dynamic state estimation
(DSE) methods in order to accurately compute the dynamic states of generation
units. Nonetheless, most of them forego the dynamic-algebraic nature of power
networks and only consider their nonlinear dynamic representations. Motivated
by the lack of DSE methods based on power network’s differential-algebraic
equations (DAEs), this paper develops a novel observer-based DSE framework in
order to perform simultaneous estimation of the dynamic and algebraic states
of multi-machine power networks. Specifically, we leverage the DAE dynamics of
a power network around an operating point and combine them with a PMU-based
measurement model capable of capturing bus voltages and line currents. The
proposed $\mathcal{H}_{\infty}$ observer, which only requires detectability
and impulse observability conditions which are satisfied for various power
networks, is designed to handle various noise, unknown inputs, and input
sensor failures. The results obtained from performing extensive numerical
simulations on the IEEE $9$-bus and $39$-bus systems showcase the
effectiveness of the proposed approach for DSE purposes.

###### Keywords:

Transmission power networks, dynamic state estimation, differential-algebraic
equation, robust $\mathcal{H}_{\infty}$ observer.

##  I Introduction

With the vast utilization of Phasor Measurement Units (PMUs), power systems
monitoring can be conveniently performed in real-time, thereby paving the
possibility of implementing estimation-based feedback controls on power
networks [1]. Consequently, this results in the prevalence of research on
dynamic state estimation (DSE) in the past two decades. Numerous approaches
have been proposed accordingly. For instance, Kalman Filter (KF)-based DSE and
its variants, such as Extended Kalman Filter, Unscented Kalman Filter,
Ensemble Kalman Filter, as well Particle Filter-based approach have been
investigated for centralized/decentralized DSE [2]. In addition to these
stochastic approaches, deterministic observers and their applications in DSE
have also been recently proposed and studied. For example, the work in [3]
presents a comparative study between KF-based methods with a state observer
designed for nonlinear systems, while [4] focuses on the estimation of a
synchronous generator’s internal states based on a nonlinear ordinary
differential equation (ODE) model.

In spite of these advances, the majority of literature on DSE overlook the
dynamic-algebraic structure of power networks in their investigations,
especially in a centralized DSE framework for multi-machine power networks.
For example, the works in [5, 6, 3, 7] consider nonlinear differential
representations of power networks—usually obtained by means of the Kron
reduction technique—which only consist of generators’ dynamics. Although the
resulting ODE models have a smaller number of states, the elimination of
algebraic constraints and variables introduces several downsides. First of
all, it is argued in [8] that the nonlinear ODE model may fail to accurately
capture the power network’s dynamics with respect to topological changes,
which can be triggered by faults. Secondly, it is not clear how more detailed
loads’ dynamics can be incorporated on the basis of ODE models. Thirdly, if
nonlinear ODEs of power networks are employed, it is difficult to study the
DSE in which the PMUs are also installed on non-generator buses (such as load
buses and non-generator unit buses) as the relations between bus voltages at
non-generator buses and generators’ states need to be established.

In addition to the above arguments, PMUs are initially designed to measure
voltage and current phasors in power systems [9] and some literature focusing
on the state estimation problem in power systems are considering that a PMU
installed on a bus is able to measure the complex quantities of bus voltage
and line currents [10, 11, 12, 13]. Hence, the use of such measurement model
in DSE analysis should have been widely considered. Indeed, there exist
approaches for DSE based on such measurement model. One of the prominent
approaches is referred to as the two-stage method [14, 15, 16]. In the first
stage, the algebraic variables (e.g. bus voltages) are estimated based on
information collected from PMUs and in the second stage, the estimated bus
voltages are used by the state estimator to estimate the actual generators’
variables. For instance, the works in [14, 15] implement the Least Absolute
Value (LAV) method in the first stage and Unscented KF in the second stage,
whereas in [16], the authors develop an adaptive KF to deal with unknown
dynamics and measurement errors in the first stage while utilizing the
Extended KF to estimate the dynamic states. It is important to note that,
albeit this two-stage approach retains the dynamic-algebraic structure of the
power network, two state estimation algorithms have to be performed
separately, thereby increasing the computational burden required for
performing DSE.

Motivated by the aforementioned limitations, in this work, we exploit the
simultaneous dynamic and algebraic nature of power networks to perform DSE
from a descriptor system and control-theoretic perspective. Indeed, this
particular research direction is less prominent in power systems literature.
One of the earliest works in this direction is found in [17], where the
authors study the observability of power networks based on its DAE model using
a linearization technique. Recently, a DSE approach in which distributed
sliding-mode and algebraic observers are employed simultaneously to estimate
each generator’s electrical frequency is proposed in [18] whereas in [19],
high-order sliding mode observers are developed to estimate generators’ rotor
speeds and complex bus voltages based on simple nonlinear, power network
differential-algebraic equations (DAEs)—both works become the basis of [20].
These studies, however, are conducted based on an overly simplified power
network model using some assumptions including the use of a 2-order swing
equation to represent generators’ dynamics, the negligence of transmission
lines’ resistances and reactive power flows, the consideration of a uniformly-
flat bus voltage profile, and PMUs measuring generator’s rotor angle installed
only on generator buses. Such simplification may not be sufficient to aid
frequency regulation in power systems through an output-feedback control
framework—for instance, to perform load following control [21]. A new approach
to estimate both dynamic and algebraic states of generators in a decentralized
framework is recently proposed in [22], where an algebraic observer is
developed to estimate the load angle and quadrature-axis internal voltage of
each generator. In order to estimate the relative rotor speed, the authors in
[22] combine the Immersion and Invariance technique [23] with the Dynamic
Regressor and Mixing [24] in their observer design. Due to the decentralized
fashion however, this DSE approach relies on PMUs placed on generator
terminals.

To that end, in this paper we construct a novel DSE framework built upon DAE
representations of power networks which dynamics are studied around an
operating point. This framework considers the aforementioned measurement model
in which PMUs are capable of measuring bus voltage and line current
phasors—such model allows PMUs to be placed arbitrarily: on generator, load,
or even non-generator unit buses. Unlike other DSE methods, e.g. in [3, 25, 6,
7] and also others that are based on decentralized DSE framework such as [22],
this feature allows for flexible PMU placements since every generator terminal
may not be equipped with a PMU [14]. In contrast to [18, 19, 20], our work
herein utilizes a DAE representation of power networks with (i) more detailed
and comprehensive 4-order generator’s transient dynamics, stator’s algebraic
constraints, generator’s real and reactive power, and the network’s complex
power balance equations, and (ii) more realistic and practical PMU-based
measurement equations.

To perform the DSE, a new robust $\mathcal{H}_{\infty}$ observer design for
linear DAEs is introduced. Indeed, there exist several observer designs
developed in the literature especially for linear DAEs and descriptor systems
[26, 27]. For example, to provide a more robust state estimates in the
presence of disturbance, numerous $\mathcal{H}_{\infty}$ observers are
proposed in [28, 29, 30]. Although the observers developed in [28, 29, 30] do
consider disturbance, due to their relatively complex linear matrix inequality
(LMI) formulations, their applicability and effectiveness to perform DSE in
power networks have never been studied. Our $\mathcal{H}_{\infty}$ observer,
on the other hand, has much simpler LMI formulations and this allows it to be
employed to perform DSE with minimal computational efforts. The main
contributions of in this paper include:

  * •

The unification of an in-depth differential-algebraic model of power
networks—encompassing generators’ transient models, stator algebraic
constraints, generators’ complex power, and power balance equations—together
with a PMU-based measurement model to allow a more realistic approach for
performing DSE in power networks. The DAEs capture the power network’s
dynamics around a certain operating point and become the foundation for the
observer-based DSE. A careful analysis on the regularity, index
characterization, impulsiveness, detectability, and impulse observability of
the linearized power network’s DAEs is also presented.

  * •

A novel LMI-based $\mathcal{H}_{\infty}$ DAE observer is introduced—this
$\mathcal{H}_{\infty}$ observer is then employed to perform robust DSE for the
linearized dynamics of power networks. The key features of this observer are
attributed to the more straightforward design as well as the simplicity of the
associated LMI formulations used to compute the observer gain matrix. In order
to tackle unknown inputs, a robust proportional-integral (PI) version of the
$\mathcal{H}_{\infty}$ DAE observer—developed by augmenting the original DAE
dynamics with an unknown inputs model—is also proposed. Since we utilize a
static-gain observer, the complexity of performing DSE is considerably much
lower than using the two-stage approach as in [14]. One primary advantage of
the robust $\mathcal{H}_{\infty}$ DAE observer—as opposed to KF-based
filters—is that it requires no information regarding the statistical
properties of the noise.

  * •

The effectiveness of the proposed approach to address DSE in power networks is
demonstrated through the IEEE $9$-bus and $39$-bus systems, in which the
proposed $\mathcal{H}_{\infty}$ DAE observer is tested for estimating
generators’ internal states and unmeasured bus voltages following a three-
phase fault under various scenarios, including the presence of Gaussian and
non-Gaussian process and measurement noise as well as unknown inputs.
Numerical advantages of the developed observer over the two-stage approach
[14, 15], which combines LAV with KF in performing DSE are also showcased.

A preliminary version of this paper was published in [31]. The paper’s
organization is as follows. Section II, presents the linearized semi-explicit
DAEs of power networks with a PMU-based measurement models while Section III
provides a brief study on the regularity, index characterization, and
impulsiveness of the linear DAEs as well as some conditions to ensure
detectability and impulse observability. In Section IV, simple Luenberger and
robust $\mathcal{H}_{\infty}$ DAE observers are designed. Section V presents a
way to modify the observers to handle unknown inputs and input sensor failures
based on a PI framework. Thorough numerical studies are presented out in
Section VI and the results are discussed accordingly. Finally, the paper is
summarized in Section VII.

Notation. The notations $\mathbb{R}$ and $\mathbb{R}_{++}$ denote the set of
real and positive real numbers, while the notation $\mathbb{C}$ denotes the
set of complex numbers. The notations ${\mathbb{C}}_{-}$ and
$\bar{\mathbb{C}}_{+}$ denote the set of all complex numbers such that the
real parts are on the open left-half and closed right-half of the complex
plane. The symbols $\mathbb{R}^{n}$ and $\mathbb{R}^{p\times q}$ denote the
sets of row vectors with $n$ elements and matrices with size $p$-by-$q$ with
elements in $\mathbb{R}$. The matrix $\bm{O}$ denotes the zero matrix of
appropriate dimension. The operator $\mathrm{Blkdiag}(\cdot)$ constructs a
block diagonal matrix while the symbol $*$ is used to represent symmetric
entries in symmetric matrices.

##  II The Linearized DAE Model of Power Networks

We consider a power network consisting $N$ number of buses, modeled by a graph
$(\mathcal{N},\mathcal{E})$, where $\mathcal{N}=\\{1,\ldots,N\\}$ is the set
of nodes and $\mathcal{E}\subseteq\mathcal{N}\times\mathcal{N}$ is the set of
edges. It is then straightforward to define
$\mathcal{N}_{i}\subseteq\mathcal{N}$ as the set of buses that are connected
to bus $i$. The set of buses equipped with PMUs are denoted by
$\mathcal{N}_{M}\subseteq\mathcal{N}$. Note that $\mathcal{N}$ consists of
generator and load buses, i.e., $\mathcal{N}=\mathcal{G}\cup\mathcal{L}$ where
$\mathcal{G}=\\{1,\ldots,G\\}$ collects the buses containing $G$ synchronous
generators while $\mathcal{L}=\\{1,\ldots,L\\}$ collects the $L$ buses that
contain loads only. Note that for bus $i\in\mathcal{G}$, it might also contain
a load. Throughout the paper, complex bus voltages are expressed in
rectangular forms—that is, $\bm{v}_{i}:=[v_{Ri}\;\;v_{Ii}]^{\top}$ for
$i\in\mathcal{N}$.

###  II-A Synchronous Generator Model & Power Flow Equations

In this work, we leverage the standard 4-order dynamics of a two-axis,
transient model of synchronous generators, which excludes turbine governor and
exciter dynamics. The dynamics of a synchronous generator $i\in\mathcal{G}$
can be written as [9]

$\displaystyle\dot{\delta}_{i}$ $\displaystyle=\omega_{i}-\omega_{0}$ (1a)
$\displaystyle\begin{split}M_{i}\dot{\omega}_{i}&=T_{{M}i}-\left(e^{\prime}_{qi}-x^{\prime}_{di}i_{di}\right)i_{qi}-\left(e^{\prime}_{di}-x^{\prime}_{qi}i_{qi}\right)i_{di}\\\
&\quad-D_{i}(\omega_{i}-\omega_{0})\end{split}$ (1b) $\displaystyle
T^{\prime}_{d0i}\dot{e}^{\prime}_{qi}$
$\displaystyle=-e^{\prime}_{qi}-\left(x_{di}-x^{\prime}_{di}\right)i_{di}+E_{fdi}$
(1c) $\displaystyle T^{\prime}_{q0i}\dot{e}^{\prime}_{di}$
$\displaystyle=-e^{\prime}_{di}+\left(x_{qi}-x^{\prime}_{qi}\right)i_{qi}.$
(1d)

The time-varying parts in (1) include $\delta_{i}:=\delta_{i}(t)$ denotes the
generator rotor angle, $\omega_{i}:=\omega_{i}(t)$ denotes the generator rotor
speed, $e^{\prime}_{qi}:=e^{\prime}_{qi}(t)$ and
$e^{\prime}_{di}:=e^{\prime}_{di}(t)$ are the transient voltages along
$\mathrm{q}$ and $\mathrm{d}$ axes, $T_{{M}i}:=T_{{M}i}(t)$ denotes the
generator’s mechanical input torque, $E_{fdi}:=E_{fdi}(t)$ denotes the
internal field voltage, while $i_{qi}$ and $i_{di}$ are generator’s stator
currents at $\mathrm{q}$ and $\mathrm{d}$ axes. The constants in (1) are as
follows: $M_{i}$ is the rotor’s inertia constant
($\mathrm{pu}\times\mathrm{s}^{2}$), $D_{i}$ is the damping coefficient
($\mathrm{pu}\times\mathrm{s}$), $x_{qi}$ and $x_{di}$ are the direct-axis
synchronous reactance ($\mathrm{pu}$), $x^{\prime}_{qi}$ and $x^{\prime}_{di}$
are the direct-axis transient reactance ($\mathrm{pu}$), $T^{\prime}_{q0i}$
and $T^{\prime}_{d0i}$ are the open-circuit time constants ($\mathrm{s}$), and
$\omega_{0}$ denotes the rotor’s synchronous speed ($\mathrm{rad/s}$). Note
that each synchronous generator has a total of four dynamic states, given by
$\tilde{\bm{x}}_{{i}}:=[\delta_{i}\;\;\omega_{i}\;\;e^{\prime}_{qi}\;\;e^{\prime}_{di}]^{\top}$,
whereas generator’s mechanical input torque and internal field voltage are
considered as inputs, i.e.,
$\tilde{\bm{u}}_{{i}}:=[T_{{M}i}\;\;E_{fdi}]^{\top}$. Note that these inputs
may not be available for measurement. The relations among generator’s rotor
angle, transient voltage, stator’s current, and terminal voltage are
represented by the stator electrical circuit, which can be described by the
following algebraic constraints [9]

$\displaystyle
e^{\prime}_{di}-v_{Ri}\sin\delta_{i}+v_{Ii}\cos\delta_{i}-R_{si}i_{di}+x^{\prime}_{qi}i_{qi}$
$\displaystyle=0$ (2a) $\displaystyle
e^{\prime}_{qi}-v_{Ri}\cos\delta_{i}-v_{Ii}\sin\delta_{i}-R_{si}i_{qi}-x^{\prime}_{di}i_{di}$
$\displaystyle=0,$ (2b)

where $R_{si}$ indicates stator’s resistance ($\mathrm{pu}$). We define
$\bm{i}_{gi}(t)$ as generator’s stator currents constructed as
$\bm{i}_{gi}:=[i_{di}\,\,i_{qi}]^{\top}$. Together, (1) and (2) form the
generator’s DAE.

The power flow equations resembling the power balance between generators and
loads can be expressed as [9]

$\displaystyle\begin{split}&P_{G,i}+P_{L,i}=G_{ii}\left(v_{Ri}^{2}+v_{Ii}^{2}\right)\\\
&+\sum_{j\neq
i}^{N}G_{ij}\left(v_{Ri}v_{Rj}+v_{Ii}v_{Ij}\right)+B_{ij}\left(v_{Ii}v_{Rj}-v_{Ri}v_{Ij}\right)\end{split}$
(3a)
$\displaystyle\begin{split}&Q_{G,i}+Q_{L,i}=-B_{ii}\left(v_{Ri}^{2}+v_{Ii}^{2}\right)\\\
&-\sum_{j\neq
i}^{N}G_{ij}\left(v_{Ri}v_{Ij}-v_{Ii}v_{Rj}\right)+B_{ij}\left(v_{Ri}v_{Rj}+v_{Ii}v_{Ij}\right),\end{split}$
(3b)

for all generator bus $i\in\mathcal{G}$ whereas $G_{ij}$ and $B_{ij}$
respectively denote the conductance and susceptance between bus $i$ and $j$.
In the left-hand sides of (3a) and (3b), $P_{L,i}$ and $Q_{L,i}$ denote the
real and reactive load power, which can be time-varying and nonlinearly
related with the bus voltage $\bm{v}_{i}$[9], where $P_{G,i}$ and $Q_{G,i}$
are generator’s real and reactive power, computed as

$\displaystyle P_{G,i}$
$\displaystyle=\left(v_{Ri}i_{di}+v_{Ii}i_{qi}\right)\sin\delta_{i}+\left(v_{Ri}i_{qi}-v_{Ii}i_{di}\right)\cos\delta_{i}$
(4a) $\displaystyle Q_{G,i}$
$\displaystyle=\left(v_{Ri}i_{di}+v_{Ii}i_{qi}\right)\cos\delta_{i}+\left(v_{Ii}i_{di}-v_{Ri}i_{qi}\right)\sin\delta_{i}.$
(4b)

The power flow equations for load buses $i\in\mathcal{L}$ are the same as in
(3) but with an exception that $P_{G,i}=0$ and $Q_{G,i}=0$. The multi-machine
power networks (1)-(4) can be lumped into

$\displaystyle\dot{\tilde{\bm{x}}}(t)$
$\displaystyle=\bm{f}(\tilde{\bm{x}}(t),\bm{i}_{g}(t),\tilde{\bm{u}}(t)),\;\;\bm{0}=\bm{g}(\tilde{\bm{x}}(t),\bm{i}_{g}(t),\bm{v}(t)),$
(5)

where the nonlinear mapping
$\bm{f}:\mathbb{R}^{4G}\times\mathbb{R}^{2G}\times\mathbb{R}^{2G}\rightarrow\mathbb{R}^{4G}$
and
$\bm{g}:\mathbb{R}^{4G}\times\mathbb{R}^{2G}\times\mathbb{R}^{2N}\rightarrow\mathbb{R}^{2G+2N}$
represent the differential and algebraic equations for the power networks. The
vectors
$\tilde{\bm{x}}:=\\{\tilde{\bm{x}}_{i}\\}_{i\in\mathcal{G}}\in\mathbb{R}^{4G}$,
${\bm{i}_{g}}:=\\{\bm{i}_{g,i}\\}_{i\in\mathcal{G}}\in\mathbb{R}^{2G}$,
$\tilde{\bm{u}}:=\\{\tilde{\bm{u}}_{i}\\}_{i\in\mathcal{G}}\in\mathbb{R}^{2G}$,
${\bm{v}}:=\\{{\bm{v}}_{i}\\}_{i\in\mathcal{N}}\in\mathbb{R}^{2N}$ represent
generators’ internal states, stator currents, inputs, and power network’s bus
voltages. In (5), $\bm{i}_{g}$ and $\bm{v}$ act as algebraic variables.

###  II-B Power Network’s Linearized DAE Model

The linearization of power networks (5) around a particularly stable operating
point $\delta_{i}^{0}$, $\omega_{i}^{0}$, $e^{\prime 0}_{qi}$ $e^{\prime
0}_{di}$, $i_{di}^{0}$, $i_{qi}^{0}$, $T_{{M}i}^{0}$, $E_{fdi}^{0}$ for
$i\in\mathcal{G}$, $P_{L,i}^{0}$ and $Q_{L,i}^{0}$ for $i\in\mathcal{L}$, and
$\bm{v}_{i}^{0}$ for $i\in\mathcal{N}$ are performed as follows. First, the
synchronous generator’s linearized dynamics model can be computed and given as

$\displaystyle\bm{E}_{D}\Delta\dot{\tilde{\bm{x}}}(t)=\bm{A}_{D}\Delta\tilde{\bm{x}}(t)+\bm{D}_{D}\Delta{\bm{i}}_{g}(t)+\bm{B}_{D}\Delta\tilde{\bm{u}}(t).$
(6)

In (6), $\Delta{\tilde{\bm{x}}}:=\tilde{\bm{x}}-\tilde{\bm{x}}^{0}$,
$\Delta{\bm{i}}_{g}:={\bm{i}}_{g}-{\bm{i}}_{g}^{0}$, and
$\Delta\tilde{\bm{u}}:=\tilde{\bm{u}}-\tilde{\bm{u}}^{0}$. The linearized
stator’s algebraic equations, with
$\Delta{\bm{v}}_{g}:={\bm{v}}_{g}-{\bm{v}}_{g}^{0}$ in which
${\bm{v}}_{g}:=\\{\bm{v}_{i}\\}_{i\in\mathcal{G}}$ collects the voltages of
all generator buses, can be expressed as

$\displaystyle
0=\bm{A}_{A}\Delta\tilde{\bm{x}}(t)+\bm{D}_{A}\Delta{\bm{i}}_{g}(t)+\bm{G}_{A}\Delta{\bm{v}}_{g}(t).$
(7)

Finally, the linearized power flow equations are given as

$\displaystyle\begin{split}0&=\bm{A}_{G}\Delta\tilde{\bm{x}}(t)+\bm{D}_{G}\Delta{\bm{i}}_{g}(t)+\bm{G}_{GG}\Delta{\bm{v}}_{g}(t)\\\
&\quad+\bm{G}_{GL}\Delta{\bm{v}}_{l}(t)\end{split}$ (8a) $\displaystyle 0$
$\displaystyle=\bm{G}_{LG}\Delta{\bm{v}}_{g}(t)+\bm{G}_{LL}\Delta{\bm{v}}_{l}(t),$
(8b)

where $\Delta{\bm{v}}_{l}:={\bm{v}}_{l}-{\bm{v}}_{l}^{0}$ and
${\bm{v}}_{l}:=\\{\bm{v}_{i}\\}_{i\in\mathcal{L}}$ populates the voltages of
all load buses. Readers are referred to Appendix A for the details on the
matrices above. It is worth noting that the form of the matrices in (8)
depends on the type of the loads—whether they are of constant impedance,
current, or power [9]. Now, from (7), we have
$\Delta{\bm{i}}_{g}(t)=-\bm{D}_{A}^{-1}\bm{A}_{A}\Delta\tilde{\bm{x}}(t)-\bm{D}_{A}^{-1}\bm{G}_{A}\Delta{\bm{v}}_{g}(t)$.
Note that the premise on the nonsingularity of $\bm{D}_{A}$ is not restrictive
since, by the construction of $\bm{D}_{A}$ provided in Appendix A, it follows
that $\bm{D}_{A}^{-1}$ exists if and only if

$\displaystyle R_{si}^{2}+x^{\prime}_{di}x^{\prime}_{qi}$ $\displaystyle\neq
0,\quad\forall i\in\mathcal{G}.$

Substituting the expression for $\Delta{\bm{i}}_{g}$ into (6) and (8) yields

$\displaystyle\underbrace{\begin{bmatrix}\bm{E}_{D}\;\bm{O}\\\
\bm{O}\;\;\;\bm{O}\end{bmatrix}}_{\bm{E}}\underbrace{\begin{bmatrix}\Delta\dot{\tilde{\bm{x}}}(t)\\\
\Delta\dot{\bm{v}}(t)\end{bmatrix}}_{\dot{\bm{x}}(t)}=\underbrace{\begin{bmatrix}\bm{A}_{1}\;\;\bm{A}_{2}\\\
\bm{A}_{3}\;\;\bm{A}_{4}\end{bmatrix}}_{\bm{A}}\underbrace{\begin{bmatrix}\Delta\tilde{\bm{x}}(t)\\\
\Delta{\bm{v}}(t)\end{bmatrix}}_{{\bm{x}}(t)}+\underbrace{\begin{bmatrix}\bm{B}_{D}\\\
\bm{O}\end{bmatrix}}_{\bm{B}_{u}}{\bm{u}}(t),$ (9)

where in (9),
$\Delta\bm{v}^{\top}:=\begin{bmatrix}\Delta{\bm{v}}_{g}^{\top}\;\;\Delta{\bm{v}}_{l}^{\top}\end{bmatrix}$,
$\bm{u}:=\Delta\tilde{\bm{u}}$, and

$\displaystyle\bm{A}_{1}$
$\displaystyle=\bm{A}_{D}-\bm{D}_{D}\bm{D}_{A}^{-1}\bm{A}_{A},\;\bm{A}_{2}=\begin{bmatrix}-\bm{D}_{D}\bm{D}_{A}^{-1}\bm{G}_{A}&\bm{O}\end{bmatrix},$
$\displaystyle\bm{A}_{3}$
$\displaystyle=\begin{bmatrix}\bm{A}_{G}-\bm{D}_{G}\bm{D}_{A}^{-1}\bm{A}_{A}\\\
\bm{O}\end{bmatrix},\;\bm{A}_{4}=\begin{bmatrix}\bar{\bm{G}}&\bm{G}_{GL}\\\
\bm{G}_{LG}&\bm{G}_{LL}\end{bmatrix},$

with $\bar{\bm{G}}$ in $\bm{A}_{4}$ is equal to
$\bm{G}_{GG}-\bm{D}_{G}\bm{D}_{A}^{-1}\bm{G}_{A}$. From (9), a compact linear
DAE for power networks can be written in a descriptor form
$\bm{E}\dot{\bm{x}}(t)=\bm{A}{\bm{x}}(t)+\bm{B}_{u}\bm{u}(t)$, where
$\bm{E}\in\mathbb{R}^{n}$, $\bm{A}\in\mathbb{R}^{n}$,
$\bm{B}_{u}\in\mathbb{R}^{m}$ are constant matrices with $n=4G+2N$ and $m=2G$,
$\bm{E}$ is singular with $\operatorname{rank}(\bm{E}):=r$ where $r$ equals to
$\operatorname{rank}(\bm{E}_{D})=4G$, ${\bm{x}}\in\mathbb{R}^{n}$ comprises
differential and algebraic variables $\Delta\tilde{\bm{x}}$ and
$\Delta{\bm{v}}$ such that
${\bm{x}}^{\top}=\begin{bmatrix}\Delta\tilde{\bm{x}}^{\top}\;\;\Delta{\bm{v}}^{\top}\end{bmatrix}$,
and $\bm{u}\in\mathbb{R}^{m}$ denotes the deviation of control inputs around
their steady-state values. The measurement model for linear DAE (9) with PMUs
is provided in the next section.

###  II-C Power Network’s Measurement Model with PMUs

For a PMU installed on bus $j\in\mathcal{N}_{M}$, it is practical to consider
that it has the capability to measure (i) bus voltage $\bm{v}_{j}$ and (ii)
line currents for all branches that are connected to bus $j$ [10, 11, 32, 33].
Define $\mathcal{N}_{j}\subseteq\mathcal{N}$ as the set of buses connected to
bus $j$. A line current $\bm{I}_{jk}=I_{Rjk}+jI_{Ijk}$ constitutes the current
flowing on a branch originating from bus $j$ to bus $k$. The set of line
currents measured by a PMU at bus $j\in\mathcal{N}_{M}$ is defined as
$\mathcal{I}_{j}:=\\{\bm{I}_{jk}\\}_{k\in\mathcal{N}_{j}}$ such that
$|\mathcal{I}_{j}|=|\mathcal{N}_{j}|$. To obtain the expressions for line
currents, we need to utilize the branch admittance matrix $\bm{Y}_{ft}$ which
is constructed as

$\displaystyle\bm{Y}_{ft}:=\begin{bmatrix}\bm{Y}_{f}\\\
\bm{Y}_{t}\end{bmatrix}\in\mathbb{C}^{2|\mathcal{E}|\times|\mathcal{N}|},$
(10)

where $\bm{Y}_{f}\in\mathbb{C}^{|\mathcal{E}|\times|\mathcal{N}|}$ and
$\bm{Y}_{t}\in\mathbb{C}^{|\mathcal{E}|\times|\mathcal{N}|}$ are the from and
to branch admittance matrices [34]. As such, all line currents can be computed
using the method provided in [11]. That is

$\displaystyle\begin{bmatrix}\mathcal{I}_{R}\\\
\mathcal{I}_{I}\end{bmatrix}=\begin{bmatrix}\mathrm{Re}(\bm{Y}_{ft})&-\mathrm{Im}(\bm{Y}_{ft})\\\
\mathrm{Im}(\bm{Y}_{ft})&\mathrm{Re}(\bm{Y}_{ft})\end{bmatrix}\tilde{\bm{v}},$
(11)

where
$\tilde{\bm{v}}^{\top}:=\begin{bmatrix}\\{v_{Ri}\\}_{i\in\mathcal{N}}^{\top}\;\;\\{v_{Ii}\\}_{i\in\mathcal{N}}^{\top}\end{bmatrix}$
is the rearranged network’s bus voltages (since $\tilde{\bm{v}}$ has different
ordering from ${\bm{v}}$ given in Section II-A). This configuration allows a
PMU installed on bus $j\in\mathcal{N}_{M}$ to measure the following quantities

$\displaystyle\bm{y}_{j}^{\top}=\begin{bmatrix}v_{Rj}&v_{Ij}&\\{I_{Rjk}\\}_{k\in\mathcal{N}_{j}}^{\top}&\\{I_{Ijk}\\}_{k\in\mathcal{N}_{j}}^{\top}\end{bmatrix},$

where each $I_{Rjk}$ and $I_{Ijk}$ can be linearly obtained from bus voltages
via the utilization of $\bm{Y}_{ft}$, thanks to (11), provided as
$\bm{y}_{j}(t)=\tilde{\bm{C}}_{j}\tilde{\bm{v}}(t)$ where $\tilde{\bm{C}}_{j}$
is given as [11]

$\displaystyle\tilde{\bm{C}}_{j}:=\begin{bmatrix}\bm{e}_{j}^{\top}&\bm{0}\\\
\bm{0}&\bm{e}_{j}^{\top}\\\
\bm{S}_{j}\mathrm{Re}(\bm{Y}_{ft})&-\bm{S}_{j}\mathrm{Im}(\bm{Y}_{ft})\\\
\bm{S}_{j}\mathrm{Im}(\bm{Y}_{ft})&\bm{S}_{j}\mathrm{Re}(\bm{Y}_{ft})\end{bmatrix}.$

In the above, $\bm{e}_{j}$ is the vector of standard basis in $\mathbb{R}^{N}$
with $1$ at row $j$ and zero otherwise and
$\bm{S}_{j}\in\mathbb{R}^{|\mathcal{N}_{j}|\times 2|\mathcal{E}|}$ is a
binary, selection matrix that selects the corresponding row of $\bm{Y}_{ft}$
which are originating from bus $j$. The overall measurement model for power
networks with PMUs can then be expressed as

$\displaystyle\bm{y}(t)=\tilde{\bm{C}}\bm{C}_{M}{\bm{v}}(t),$ (12)

where in (12), the vector $\bm{y}\in\mathbb{R}^{p}$ lumps the measured
outputs, $\tilde{\bm{C}}=\\{\tilde{\bm{C}}_{i}\\}_{i\in\mathcal{N}_{M}}$, and
$\bm{C}_{M}$ is an orthogonal matrix such that
$\tilde{\bm{v}}(t)=\bm{C}_{M}{\bm{v}}(t)$. The branch admittance matrix
$\bm{Y}_{ft}$ in (10) can be constructed from MATPOWER [34].

##  III Regularity, Index Characterization, and Observability of Power
Networks

The linearized power network’s DAE (9) with measurement (12) can be lumped
into the following state-space equations

$\displaystyle\bm{E}\dot{\bm{x}}(t)$
$\displaystyle=\bm{A}{\bm{x}}(t)+\bm{B}_{u}\bm{u}(t),\;\quad\bm{y}(t)=\bm{C}{\bm{x}}(t),$
(13)

where
$\bm{C}=\begin{bmatrix}\bm{O}\;\;\;\tilde{\bm{C}}\bm{C}_{M}\end{bmatrix}$. As
opposed to ODEs, the existence and uniqueness of solutions for DAEs are not
always guaranteed for any initial conditions. On that regard, one of the most
important property for DAEs is regularity. That is, a linear DAE is said to be
regular if and only if it has a unique solution for every consistent initial
condition [35]. The regularity of DAE can be characterized from the matrix
pair $(\bm{E},\bm{A})$, i.e., the DAE (13) is regular if and only if there
exists $s\in\mathbb{C}$ such that $\det{(s\bm{E}-\bm{A})}\neq 0$. For the case
of power networks (13), first define $\tilde{\Lambda}(\cdot)$ as the set of
finite poles of a linear DAE pair

$\displaystyle\tilde{\Lambda}(\bm{E},\bm{A}):=\\{s\in\mathbb{C}\,|\,\det(s\bm{E}-\bm{A})=0\\}.$

Now, the regularity condition of power networks becomes

$\displaystyle\det{(s\bm{E}-\bm{A})}$
$\displaystyle=\det\left(\begin{bmatrix}s\bm{E}_{D}-\bm{A}_{1}&-\bm{A}_{2}\\\
-\bm{A}_{3}&-\bm{A}_{4}\end{bmatrix}\right).$ (14)

By considering a finite $s\notin\tilde{\Lambda}(\bm{E},\bm{A})$ such that
$s\bm{E}_{D}-\bm{A}_{1}$ is nonsingular, the right-hand side of (14) is
equivalent to

$\displaystyle\det\left(\bm{A}_{4}+\bm{A}_{3}\left(s\bm{E}_{D}-\bm{A}_{1}\right)^{-1}\bm{A}_{2}\right)\neq
0.$ (15)

Hence, the DAE (13) is regular if and only if (15) holds for some finite
$s\notin\tilde{\Lambda}(\bm{E},\bm{A})$. It is difficult to assess the
regularity of DAE (13) due to weak structure in (13) or lack thereof. However,
several researches have been conducted to analyze the solvability of
simplified power network’s DAEs—see [36, 8]. It is discovered therein that,
for a lossless power network with unity magnitude on all bus voltages, the
regularity of the linearized DAE model is ensured if and only if there exist
paths such that every load is connected to a generator bus.

Another important property in linear DAE is the differentiation index, which
refers to the number of differentiations that need to be executed in order to
transform the DAE into an ODE. It is understood that the DAE model of the
aforementioned simplified power networks is of index one if only if it is
regular [36, 8]. For power networks model considered in this paper, it is
revealed that if $\bm{A}_{4}$ is of full rank, then DAE (13) is impulse-
free***Linear DAEs that are impulse-free do not contain impulsive terms in
their solutions for given arbitrary initial conditions., of index one, and
regular—shown in the result below (the proof is provided in Appendix B).

######  Theorem 1.

Suppose that the matrix $\bm{A}_{4}$ in DAE (13) is of full rank. Then, the
power networks DAE (13) is impulse-free, of index one, and regular.

The above proposition provides another alternative other than (15) to
determine the regularity of the power networks model. Because of its useful
implications, the following statement is assumed throughout the paper.

######  Assumption 1.

The matrix $\bm{A}_{4}$ in DAE (13) is of full rank.

The matrix $\bm{A}_{4}$ describes the relation among loads and generators via
power flow equations. Thus, the nonsingularity of $\bm{A}_{4}$ simply
translates to the connectivity among generators and loads, which is similar to
the results presented in [36, 8] on the regularity of the DAE. Note that
Assumption 1 is mild and holds for the IEEE test cases considered in the
numerical study section.

Before a state observer can be designed, it is crucial to assess whether the
DAE’s internal states can be estimated via limited measurements. There exist
numerous concepts for observability in DAEs, in addition to detectability—see
[35]. However, as we are looking to design observers with the least possible
assumptions, we choose herein to consider detectability and impulse
observability—discussed as follows. The DAE (13) is said to be detectable if
and only if there exists a matrix $\bm{L}\in\mathbb{R}^{n\times p}$ such that
the DAE described by the matrix pair $(\bm{E},\bm{A}-\bm{L}\bm{C})$ is stable,
that is, $\tilde{\Lambda}(\bm{E},\bm{A}-\bm{L}\bm{C})\in{\mathbb{C}}_{-}$ or
equivalently, all finite eigenvalues of $(\bm{E},\bm{A}-\bm{L}\bm{C})$ are on
open left-half of complex plane. The detectability of DAE (13) can be ensured
if the following rank condition holds [35]

$\displaystyle\operatorname{rank}\left(\begin{bmatrix}s\bm{E}-\bm{A}\\\
\bm{C}\end{bmatrix}\right)=n,\;\;\;\;\forall s\in\bar{\mathbb{C}}_{+},$ (16)

which is akin to the PBH test for detectability of linear differential
systems. On the other hand, the impulse observability (or I-observability)
condition concerns with the ability to observe impulsiveness of the actual
states from the impulsive behavior of the measurements. The condition that
ensures I-observability of a linear DAE is given by [35]

$\displaystyle\operatorname{rank}\left(\begin{bmatrix}\bm{E}&\bm{A}\\\
\bm{O}&\bm{E}\\\ \bm{O}&\bm{C}\end{bmatrix}\right)$
$\displaystyle=n+\operatorname{rank}(\bm{E}).$ (17)

The following result presents a reduced equivalent condition to achieve
I-observability for power network’s DAE.

######  Theorem 2.

The DAE (13) is I-observable if and only if

$\displaystyle\operatorname{rank}\left(\begin{bmatrix}\;\bm{A}_{3}\;\;\;\;\,\bm{A}_{4}\;\;\;\\\
\bm{E}_{D}\;\;\;\;\;\bm{O}\;\;\;\;\\\
\;\bm{O}\;\;\;\;\tilde{\bm{C}}\bm{C}_{M}\end{bmatrix}\right)=\operatorname{rank}\left(\begin{bmatrix}\bm{A}_{4}^{-1}\bm{A}_{3}\;\;\bm{C}_{M}^{\top}\\\
\;\;\;\,\bm{I}\;\;\;\;\;\;\;\,\bm{O}\;\;\;\\\ \
\;\;\bm{O}\;\;\;\;\;\;\;\tilde{\bm{C}}\;\;\;\end{bmatrix}\right)$
$\displaystyle=n.$ (18)

Please refer to Appendix C for the proof of Theorem 2. On that regard, the
following assumption is considered.

######  Assumption 2.

The PMUs are distributed in such a way that the power network’s DAE (13) is
detectable and I-observable, i.e., (16) and (18) are satisfied.

The above assumption is crucial for enabling power system’s DSE using
observers—detailed in Section IV. It is worth noting that this assumption is
also mild and easily satisfied since we use similar number and location of
PMUs as in the literature. Regardless, the problem of placing PMUs in order to
achieve detectability and I-observability is not covered in this work as this
problem deserves a standalone dedication and an in-depth analysis which are
beyond the scope of this paper and as such, is left for future research. In
the following section, we explore several DAE state observer designs that can
be utilized to perform DSE based on DAE (13).

##  IV Synthesis of Luenberger DAE State Observers

There exist two different realizations of state observer for DAE: DAE observer
and ODE observer. The former is considered herein mainly due to (i) its
simplicity as no equivalence transformation required and (ii) it only assumes
detectability and I-observability. This is in contrast to many ODE observers
where the original DAE needs to be transformed into a specific restricted
system equivalent while requiring more stricter conditions, e.g. in [37]. With
that in mind, two kinds of DAE observer are presented. in this section: (a) a
simple Luenberger observer and (b) a robust $\mathcal{H}_{\infty}$ observer.
First, based on [38], the next Luenberger DAE state observer for (13) is
proposed

$\displaystyle\bm{E}\dot{\hat{\bm{x}}}(t)$
$\displaystyle=\bm{A}\hat{\bm{x}}(t)+\bm{B}_{u}\bm{u}(t)+\bm{L}(\bm{y}(t)-\hat{\bm{y}}(t))$
(19a) $\displaystyle\hat{\bm{y}}(t)$ $\displaystyle=\bm{C}\hat{\bm{x}}(t),$
(19b)

where $\hat{\bm{x}}\in\mathbb{R}^{n}$ denotes the estimate of the actual
system’s state ${\bm{x}}$, $\hat{\bm{y}}\in\mathbb{R}^{p}$ is the output
estimate, and $\bm{L}\in\mathbb{R}^{n\times p}$ is the observer gain matrix.
Our goal is finding $\bm{L}$ such that
$\lim_{t\rightarrow\infty}(\bm{x}(t)-\hat{\bm{x}}(t))=\bm{0}$ for arbitrary
initial states $\bm{x}_{0}$, $\hat{\bm{x}}_{0}$ and inputs $\bm{u}(t)$ for all
$t\geq 0$. If the state estimation error is defined as
$\bm{e}(t):=\bm{x}(t)-\hat{\bm{x}}(t)$, the error dynamics can be written as

$\displaystyle\bm{E}\dot{{\bm{e}}}(t)$
$\displaystyle=(\bm{A}-\bm{L}\bm{C})\bm{e}(t).$ (20)

Note that the existence of $\bm{L}$ such that (20) is stable is guaranteed due
to the detectability condition in Assumption 2. A linear DAE that is impulse-
free, regular, and stable is called admissible [39, 40]. To that end, we seek
for a systematic way to compute $\bm{L}$ that makes the error dynamics in (20)
to be admissible. The following result from [39] provides a sufficient and
necessary condition for the admissibility of DAE (20).

######  Lemma 1 ([39]).

A linear DAE denoted by $(\tilde{\bm{E}},\tilde{\bm{A}})$ is admissible if and
only if there exist $\bm{X}\in\mathbb{S}^{n}_{++}$ and
$\bm{Y}\in\mathbb{R}^{(n-r)\times n}$ such that the following LMI is feasible

$\displaystyle\tilde{\bm{A}}^{\top}(\bm{X}\tilde{\bm{E}}+\tilde{\bm{E}}^{\perp\top}\bm{Y})+(\bm{X}\tilde{\bm{E}}+\tilde{\bm{E}}^{\perp\top}\bm{Y})^{\top}\tilde{\bm{A}}\prec
0,$ (21)

where $\tilde{\bm{E}}^{\perp}\in\mathbb{R}^{(n-r)\times n}$ is an orthogonal
complement of $\tilde{\bm{E}}$.

Using (21), a condition for admissibility of state estimation error dynamics
is obtained—see Appendix D for the proof.

######  Theorem 3.

The state estimation error dynamics in (20) is admissible if and only if there
exist $\bm{X}\in\mathbb{S}^{n}_{++}$, $\bm{Y}\in\mathbb{R}^{(n-r)\times n}$,
and $\bm{W}\in\mathbb{R}^{n\times p}$ such that the following LMI is feasible

$\displaystyle\bm{A}^{\top}\bm{X}\bm{E}+\bm{E}^{\top}\bm{X}\bm{A}+\bm{A}^{\top}$
$\displaystyle\bm{E}^{\perp\top}\bm{Y}+\bm{Y}^{\top}\bm{E}^{\perp}\bm{A}$
$\displaystyle-\bm{C}^{\top}\bm{W}-\bm{W}^{\top}\bm{C}\prec 0,$ (22)

where the gain matrix $\bm{L}$ can be recovered as
$\bm{L}=\left(\bm{W}\bm{P}^{-1}\right)^{\top}$ and
${\bm{E}}^{\perp}\in\mathbb{R}^{(n-r)\times n}$ is an orthogonal complement of
${\bm{E}}$.

######  Remark 1.

Although the satisfaction of LMI (22) ensures the stability of error dynamics
(20), the convergence rate to which $\hat{\bm{x}}(t)$ approaches ${\bm{x}}(t)$
as $t\rightarrow\infty$ can be relatively poor. As a potential remedy, one can
minimize the maximum eigenvalue of $\bm{E}^{\top}\bm{X}\bm{E}$ by solving the
following convex optimization problem

$\displaystyle{\mathbf{P1}}\min_{\kappa,\bm{X},\bm{Y},\bm{W}}\kappa\;;\;\operatorname*{s.to}\,\eqref{eq:admissibility_LMI},\,\kappa>0,\bm{X}\succ
0,\kappa\bm{I}-\bm{E}^{\top}\bm{X}\bm{E}\succeq 0.$

Theorem 3 provides a systematic method to find $\bm{L}$ that makes (20)
admissible as the LMI in (22) can be solved using standard semidefinite
programming (SDP) solvers.

The above Luenberger DAE state observer (19) assumes an ideal operating
condition of power networks in the sense that no disturbance is present.
Nevertheless, in a more practical set up, disturbances can manifest in various
forms: from unknown inputs, process and measurement noise, to cyber-attacks.
Hence, in addition to the the Luenberger observer (19), here we also design a
robust $\mathcal{H}_{\infty}$ DAE observer to minimize the effect of
disturbance to the state estimation error. The power network’s DAE subject to
external disturbances can be written as

$\displaystyle\bm{E}\dot{\bm{x}}(t)$
$\displaystyle=\bm{A}{\bm{x}}(t)+\bm{B}_{u}\bm{u}(t)+\bm{B}_{w}\bm{w}(t)$
(23a) $\displaystyle\quad\bm{y}(t)$
$\displaystyle=\bm{C}{\bm{x}}(t)+\bm{D}_{w}\bm{w}(t),$ (23b)

where $\bm{w}\in\mathbb{R}^{q}$ lumps all disturbances into a single vector
and matrices $\bm{B}_{w}$ and $\bm{D}_{w}$ describe how the external
disturbances are distributed in the system. Specifically, $\bm{B}_{w}$ takes
into account any disturbances affecting the system’s dynamics whereas
$\bm{D}_{w}$ considers disturbances affecting the measurements. It is presumed
from now on that matrices $\bm{B}_{w}$ and $\bm{D}_{w}$ are known. By
utilizing the same Luenberger state observer as in (19), the estimation error
dynamics can be derived as follows

$\displaystyle\bm{E}\dot{{\bm{e}}}(t)$
$\displaystyle=(\bm{A}-\bm{L}\bm{C})\bm{e}(t)+(\bm{B}_{w}-\bm{L}\bm{D}_{w})\bm{w}(t)$
(24a) $\displaystyle\quad\bm{\epsilon}(t)$
$\displaystyle=\bm{\Gamma}\bm{e}(t),$ (24b)

where $\bm{\epsilon}\in\mathbb{R}^{n}$ is the performance of error dynamics
with respect to the user-defined performance matrix
$\bm{\Gamma}\in\mathbb{R}^{n\times n}$. The next result summarizes the design
of a robust $\mathcal{H}_{\infty}$ DAE observer.

######  Theorem 4.

For the state estimation error dynamics in (24), it holds that (a) it is
stable whenever $\bm{w}(t)=\bm{0}$ for all $t\geq 0$ and (b)
$\left\lVert\bm{\epsilon}(t)\right\rVert^{2}_{L2}<\gamma\left\lVert\bm{w}(t)\right\rVert^{2}_{L2}$
where $\gamma\geq 0$ for every bounded disturbance $\bm{w}(t)$ and zero
initial error $\bm{e}_{0}=\bm{0}$, if there exist
$\bm{X}\in\mathbb{S}^{n}_{++}$, $\bm{Y}\in\mathbb{R}^{(n-r)\times n}$, and
$\bm{W}\in\mathbb{R}^{n\times p}$ such that

$\displaystyle\begin{bmatrix}\bm{A}^{\top}\bm{X}\bm{E}+\bm{E}^{\top}\bm{X}\bm{A}+\bm{\Xi}+\bm{\Gamma}^{\top}\bm{\Gamma}&*\\\
\bm{B}_{w}^{\top}\bm{X}\bm{E}+\bm{B}_{w}^{\top}\bm{E}^{\perp\top}\bm{Y}-\bm{D}_{w}^{\top}\bm{W}&-\gamma\bm{I}\end{bmatrix}\prec
0,$ (25)

is feasible where the matrix $\bm{\Xi}$ is defined as

$\displaystyle\bm{\Xi}:=\bm{A}^{\top}$
$\displaystyle\bm{E}^{\perp\top}\bm{Y}+\bm{Y}^{\top}\bm{E}^{\perp}\bm{A}-\bm{C}^{\top}\bm{W}-\bm{W}^{\top}\bm{C}.$

Moreover, the matrix $\bm{L}$ can be computed as
$\bm{L}=\left(\bm{W}\bm{P}^{-1}\right)^{\top}$ and
${\bm{E}}^{\perp}\in\mathbb{R}^{(n-r)\times n}$ is an orthogonal complement of
${\bm{E}}$.

The proof of Theorem 4 can be obtained from Appendix E. The LMI (25)
guarantees the boundedness of estimation error performance in the sense that
$\left\lVert\bm{\epsilon}(t)\right\rVert^{2}_{L2}<\gamma\left\lVert\bm{w}(t)\right\rVert^{2}_{L2}$
for every bounded disturbance $\bm{w}(t)$ and and zero initial error. To
minimize the impact of $\bm{w}(t)$, the following convex problem can be
considered

$\displaystyle\mathbf{P2}\;\min_{\gamma,\bm{X},\bm{Y},\bm{W}}\;\gamma\;;\;\operatorname*{s.to}\;\eqref{eq:H_infinity},\,\gamma>0,\,\bm{X}\succ
0,$

while the multi-objective optimization problem below can be considered to
improve the estimation error convergence rate

$\displaystyle\mathbf{P3}\;\min_{\gamma,\kappa,\bm{X},\bm{Y},\bm{W}}\;\;$
$\displaystyle c_{1}\kappa+c_{2}\gamma\;$
$\displaystyle\operatorname*{subject\ to}\;$
$\displaystyle\eqref{eq:H_infinity},\,\gamma,\kappa>0,\,\bm{X}\succ
0,\,\kappa\bm{I}-\bm{E}^{\top}\bm{X}\bm{E}\succeq 0,$

where $c_{1},\,c_{2}\in\mathbb{R}_{++}$ are predefined constants.

##  V Tackling Unknown Inputs and Sensor Failures

In generators particularly equipped with brushless excitation systems, it is
difficult to measure the exciter’s field current and voltage [41].
Consequently, the field voltage $E_{fd}$ is not always available for the
observer [42, 43, 44]. To that end, this section focuses on extending the DAE
state observers introduced in Section IV to consider unknown inputs through a
proportional-integral (PI) framework. To start, consider the disturbed power
networks DAE (23) with unknown inputs

$\displaystyle\bm{E}\dot{\bm{x}}(t)$
$\displaystyle=\bm{A}{\bm{x}}(t)+\bm{B}_{u}\bm{u}(t)+\bm{B}_{\nu}\bm{\nu}(t)+\bm{B}_{w}\bm{w}(t)$
(26a) $\displaystyle\quad\bm{y}(t)$
$\displaystyle=\bm{C}{\bm{x}}(t)+\bm{D}_{w}\bm{w}(t),$ (26b)

where $\bm{\nu}\in\mathbb{R}^{v}$ represents unknown inputs and
$\bm{B}_{\nu}\in\mathbb{R}^{n\times v}$ is a known matrix dictating how the
unknown inputs are distributed in the network. The idea here is to eliminate
the effect of unknown inputs (as well as input sensor failures) by estimating
the dynamic behavior of the unknown inputs and exploiting the estimate to
compensate the state estimation error caused by unknown inputs. An estimate of
unknown inputs’ dynamics is provided as
$\dot{\bm{\nu}}(t)=\bm{\Psi}\bm{\nu}(t)$ for some matrix $\bm{\Psi}$ of
appropriate dimension [45]. The matrix $\bm{\Psi}$ is typically designed based
on the knowledge of unknown inputs’ dynamics. However, when the dynamics are
unknown, the choice $\bm{\Psi}=\bm{O}$ can provide a sufficient estimate [46].
The augmented linearized DAEs, which is a reformulation of (26), can be
expressed as

$\displaystyle\underbrace{\begin{bmatrix}\bm{E}\;\;\bm{O}\\\
\bm{O}\;\;\bm{I}\;\end{bmatrix}}_{\bm{E}_{\xi}}\underbrace{\begin{bmatrix}\dot{{\bm{x}}}(t)\\\
\dot{\bm{\nu}}(t)\end{bmatrix}}_{\dot{\bm{\xi}}(t)}$
$\displaystyle=\underbrace{\begin{bmatrix}\bm{A}\;\;\bm{B}_{\nu}\\\
\bm{O}\;\;\,\bm{O}\;\;\end{bmatrix}}_{\bm{A}_{\xi}}\underbrace{\begin{bmatrix}{\bm{x}}(t)\\\
{\bm{\nu}}(t)\end{bmatrix}}_{{\bm{\xi}}(t)}+\underbrace{\begin{bmatrix}\bm{B}_{D}\\\
\bm{O}\end{bmatrix}}_{\bm{B}_{u,\xi}}{\bm{u}}(t)+\underbrace{\begin{bmatrix}\bm{B}_{w}\\\
\bm{O}\end{bmatrix}}_{\bm{B}_{w,\xi}}{\bm{w}}(t),$ $\displaystyle\bm{y}(t)$
$\displaystyle=\underbrace{\begin{bmatrix}\bm{C}\;\;\,\bm{O}\end{bmatrix}}_{\bm{C}_{\xi}}\underbrace{\begin{bmatrix}{\bm{x}}(t)\\\
{\bm{\nu}}(t)\end{bmatrix}}_{{\bm{\xi}}(t)}+\bm{D}_{w}\bm{w}(t),$

with $\operatorname{rank}\left(\bm{E}_{\xi}\right)=r+\nu$. The above DAE can
be reduced into

$\displaystyle\bm{E}_{\xi}\dot{\bm{\xi}}(t)$
$\displaystyle=\bm{A}_{\xi}\bm{\xi}(t)+\bm{B}_{u,\xi}\bm{u}(t)+\bm{B}_{w,\xi}\bm{w}(t)$
(27a) $\displaystyle\quad\bm{y}(t)$
$\displaystyle=\bm{C}_{\xi}\bm{\xi}(t)+\bm{D}_{w}\bm{w}(t),$ (27b)

where
$\bm{\xi}:=\begin{bmatrix}{\bm{x}}^{\top}\;\bm{\nu}^{\top}\end{bmatrix}^{\top}\in\mathbb{R}^{\varsigma}$
is the state vector of the augmented DAE. The linear DAE (27) retains the
properties of the power networks DAE (13), which is shown in the result below.

######  Theorem 5.

The augmented linear DAE (27) is impulse-free, of index one, and regular.
Moreover, the augmented linear DAE (27) is detectable and I-observable if and
only if the power networks DAE in (13) is detectable and I-observable.

The implication of Theorem 5 ( Appendix F has the proof) is the existence of
an impulse-free DAE state observer for the augmented system (27). If
$\hat{\bm{\xi}}\in\mathbb{R}^{\varsigma}$ is an estimate of $\bm{\xi}$ where
$\hat{\bm{\xi}}^{\top}:=\begin{bmatrix}\hat{\bm{x}}^{\top}\;\hat{\bm{\nu}}^{\top}\end{bmatrix}$
and $\hat{\bm{x}},\,\hat{\bm{\nu}}$ are estimates of ${\bm{x}},\,{\bm{\nu}}$
respectively, the PI Luenberger DAE state observer can be constructed as

$\displaystyle\underbrace{\begin{bmatrix}\bm{E}\;\;\bm{O}\\\
\bm{O}\;\;\bm{I}\;\end{bmatrix}}_{\bm{E}_{\xi}}\underbrace{\begin{bmatrix}\dot{\hat{\bm{x}}}(t)\\\
\dot{\hat{\bm{\nu}}}(t)\end{bmatrix}}_{\dot{\hat{\bm{\xi}}}(t)}$
$\displaystyle=\underbrace{\begin{bmatrix}\bm{A}\;\;\bm{B}_{\nu}\\\
\bm{O}\;\;\,\bm{O}\;\;\end{bmatrix}}_{\bm{A}_{\xi}}\underbrace{\begin{bmatrix}\hat{\bm{x}}(t)\\\
\hat{\bm{\nu}}(t)\end{bmatrix}}_{\hat{\bm{\xi}}(t)}+\underbrace{\begin{bmatrix}\bm{B}_{D}\\\
\bm{O}\end{bmatrix}}_{\bm{B}_{u,\xi}}{\bm{u}}(t)+\underbrace{\begin{bmatrix}\bm{L}_{\xi,P}\\\
\bm{L}_{\xi,I}\end{bmatrix}}_{\bm{L}_{\xi}}\Delta\bm{y}(t),$
$\displaystyle\hat{\bm{y}}(t)$
$\displaystyle=\underbrace{\begin{bmatrix}\bm{C}\;\;\,\bm{O}\end{bmatrix}}_{\bm{C}_{\xi}}\underbrace{\begin{bmatrix}\hat{\bm{x}}(t)\\\
\hat{\bm{\nu}}(t)\end{bmatrix}}_{\hat{\bm{\xi}}(t)},$

where $\Delta\bm{y}(t):=\bm{y}(t)-\hat{\bm{y}}(t)$ and
$\bm{L}_{\xi}\in\mathbb{R}^{\varsigma\times p}$ is the observer gain matrix
and comprised of proportional and integral gains. The above observer dynamics
can be simplified into

$\displaystyle\bm{E}_{\xi}\dot{\hat{\bm{\xi}}}(t)$
$\displaystyle=\bm{A}_{\xi}\hat{\bm{\xi}}(t)+\bm{B}_{u,\xi}\bm{u}(t)+\bm{L}_{\xi}\left(\bm{y}(t)-\hat{\bm{y}}(t)\right)$
(28a) $\displaystyle\quad\hat{\bm{y}}(t)$
$\displaystyle=\bm{C}_{\xi}\hat{\bm{\xi}}(t).$ (28b)

Setting $\bm{e}_{\xi}:=\bm{\xi}-\hat{\bm{\xi}}$, the state estimation error
dynamics can be derived from (27) and (28) and given as

$\displaystyle\bm{E}_{\xi}\dot{{\bm{e}}}_{\xi}(t)$
$\displaystyle=(\bm{A}_{\xi}-\bm{L}_{\xi}\bm{C}_{\xi})\bm{e}_{\xi}(t)+(\bm{B}_{w,\xi}-\bm{L}_{\xi}\bm{D}_{w})\bm{w}(t)$
(29a) $\displaystyle\bm{\epsilon}_{\xi}(t)$
$\displaystyle=\bm{\Gamma}_{\xi}\bm{e}_{\xi}(t).$ (29b)

where $\bm{\epsilon}_{\xi}\in\mathbb{R}^{\varsigma}$ is the performance of
augmented error dynamics and
$\bm{\Gamma}_{\xi}\in\mathbb{R}^{\varsigma\times\varsigma}$ is the user-
defined performance matrix. Since the state estimation error dynamics (29)
share a similar structure to that of (24), the LMI in (25) can be utilized.

##  VI Numerical Results and Discussions

In this section, we perform numerical simulations to study the applicability
and performance of the proposed DAE observers in estimating the trajectories
of both algebraic and dynamic states of selected IEEE test networks under
various conditions around a certain operating point. All simulations are
performed using MATLAB R2020a running on 64-bit Windows 10 with a 3.4GHz
IntelR CoreTM i7-6700 CPU and 16 GB of RAM, whereas all convex linear programs
(LPs) and SDPs are solved through YALMIP [47] optimization interface along
with MOSEK [48] solver. The dynamical simulations for all DAEs are carried out
using MATLAB’s ODEs solver ode15i.

(a)  (b)

Figure 1: State estimation results for Case-9 with Gaussian noise: (a) the
internal states of Generator-$2$ and (b) complex voltage at Bus-$8$.

###  VI-A Power Networks Test Cases: Parameters and Setup

Herein we consider two power networks of contrasting size: the Western System
Coordinating Council (WSCC) $3$-machine, $9$-bus system (referred to as
Case-9) and the New England $9$-machine, $39$-bus system (referred to as
Case-39). The single-line diagrams for Case-9 and Case-39 are available in [9]
and [49, Appendix A]. The steady-state operating points utilized to construct
the matrices in (6)-(8) are acquired from solving the power flow equations (3)
using MATPOWER [34]. Each synchronous generator parameters appearing in (1)
and (2) are obtained from Power System Toolbox [9] case files d3m9bm.m for
Case-9 and datane.m for Case-39. The power base for both systems is chosen to
be $100$ MVA and synchronous speed $\omega_{0}:=2\pi 60\;\mathrm{rad/sec}$.
All loads are assumed to be of constant power type. For Case-9, the loads at
buses number $5$, $7$, and $9$ are specified as $0.9+j0.3\;\mathrm{pu}$,
$1.0+j0.35\;\mathrm{pu}$, and $1.25+j0.5\;\mathrm{pu}$. Since Case-39 consists
of $21$ loads, they are not specified here—the details can be seen from
MATPOWER data file named case39.m.

Figure 2: The comparison of estimation error norm for Case-9 and Case-39 with
Gaussian noise.

(a)  (b)

Figure 3: State estimation results with non-Gaussian measurement noise: (a)
transient voltage at Generator-$3$ with Cauchy noise (Case-9) and (b) rotor
angle and speed at Generator-$10$ with Laplace noise (Case-39).

###  VI-B DSE Under a Three-Phase Fault Contingency

To examine the performance of the proposed observers in performing DSE under a
contingency event, a dynamic response is generated by applying a three-phase
fault, where for Case-9, the fault occurs at Bus-$4$ of Line $4$-$5$ at
$t=25\,\mathrm{sec}$ within a $60\,\mathrm{sec}$ simulation window and then
cleared at the near and remote ends after $50\,\mathrm{msec}$ and
$200\,\mathrm{msec}$. The three-phase fault for Case-39 occurs at Bus-$3$ of
Line $3$-$4$ at $t=20\,\mathrm{sec}$, which is cleared at the near and remote
ends after $50\,\mathrm{msec}$ and $70\,\mathrm{msec}$. It is assumed herein
that the input $\bm{u}(t)$ is known to the observers at all times. We also
impose a Gaussian process noise assuming a diagonal covariance matrix which
entries are the square of $10$% of the largest state changes [44] as well as a
Gaussian measurement noise with variance $0.01^{2}$ for Case-9 and $0.005^{2}$
for Case-39. Particularly for Case-9, two PMUs are installed with
configuration $\mathcal{N}_{M}=\\{4,6\\}$, where for Case-39, we follow [50]
to install $9$ PMUs such that
$\mathcal{N}_{M}=\\{2,6,10,19,20,22,23,25,29\\}$. It is worth noting that
these PMU locations render the power networks to be detectable and
I-observable—according to (16) and (18). The two observers presented in
Section IV are implemented: the gain for the standard Luenberger DAE observer
is obtained from solving P1 while the one for the $\mathcal{H}_{\infty}$
observer is computed from solving P2, from which we get $\kappa=2.2\times
10^{-3}$, $\gamma=5.78\times 10^{-2}$ for Case-9 and $\kappa=1$,
$\gamma=5.24\times 10^{-2}$ for Case-39. The initial conditions for observer’s
states are randomized with $10$% maximum deviation from the actual states’
steady-state values, except for generators’ rotor speed, which are set to be
equal to the synchronous speed $\omega_{0}$. The disturbance matrices are
chosen to be
$\bm{B}_{w}:=\begin{bmatrix}\bm{I}\;\;\bm{O}\end{bmatrix}\in\mathbb{R}^{n\times
q}$ and
$\bm{D}_{w}:=\begin{bmatrix}\bm{O}\;\;\bm{I}\end{bmatrix}\in\mathbb{R}^{p\times
q}$ where $q=n+p$.

The results of this numerical study on Case-9 are provided in Fig. 1. Despite
Bus-$2$ and Bus-$8$ are not equipped with PMUs, their terminal voltages and
Generator-$2$’s internal states can be estimated. It is worth noting the high
noise attenuation provided by the $\mathcal{H}_{\infty}$ DAE observer,
compared to the standard Luenberger DAE observer. Similar results are also
observed for Case-39, which are not shown here for conciseness. The comparison
of estimation error norm is shown in Fig. 2. It can be seen that the
$\mathcal{H}_{\infty}$ observer provides a superior state tracking—this is
indicated by the low estimation error norm. Motivated by the work in [51], we
briefly test the DAE observers in handling non-Gaussian measurement noise such
as Cauchy and Laplace noise. Following [4], the Cauchy noise is generated by
setting $\bm{w}_{mi}=a+b\cdot\mathrm{tan}\big{(}\pi(R_{2}-0.5)\big{)}$, where
$a=0$, $b=5\times 10^{-4}$, and $R_{2}$ is a random number inside $(0,1)$ and
implemented on Case-9, while the Laplace noise is characterized by the signal
$\bm{w}_{mi}=m-s\cdot\mathrm{sgn}(R_{1})\cdot\mathrm{ln}(1-2|R_{1}|)$, where
$m=0$, $s=1\times 10^{-3}$, and $R_{1}$ is a number randomly chosen inside the
set $(-0.5,0.5]$ and implemented on Case-39. The results are presented in Fig.
3, from which it can be claimed the superiority of the $\mathcal{H}_{\infty}$
DAE observer over the Luenberger observer.

TABLE I: The corresponding RMSE (30) for various configurations of PMU on
Case-9 with Luenberger and $\mathcal{H}_{\infty}$ DAE observers.

Remark | PMU conf. $\mathcal{N}_{M}$ | RMSE
---|---|---
Luenberger Obs. |  $\mathcal{H}_{\infty}$ Obs.
$2$ PMUs: $3$-branch bus only | $\\{4,6\\}$ | $0.906128$ | $0.149777$
$\\{4,8\\}$ | $0.857766$ | $0.131873$
$\\{6,8\\}$ | $0.586053$ | $0.128455$
$3$ PMUs: two $3$-branch bus, one load bus | $\\{4,5,8\\}$ | $0.906862$ | $0.128968$
$\\{4,7,8\\}$ | $0.833687$ | $0.130704$
$\\{4,8,9\\}$ | $0.680171$ | $0.118765$
$4$ PMUs: two generator bus, two load bus | $\\{1,2,5,7\\}$ | $0.734701$ | $0.141977$
$\\{1,3,7,9\\}$ | $0.764526$ | $0.129512$
$\\{2,3,5,9\\}$ | $0.686932$ | $0.126877$
multiple PMUs: two $3$-branch bus, multiple load bus | $\\{4,5,6\\}$ |  $0.757261$$\dagger$ |  $0.153421$$\dagger$
$\\{4,5,6,7\\}$ |  $0.746314$$\dagger$ |  $0.145797$$\dagger$
$\\{4,5,6,7,9\\}$ |  $0.644191$$\dagger$ |  $0.127354$$\dagger$

  * $\dagger$

this value corresponds to the average of RMSE after performing the DSE five
times to compensate for noise variability.

TABLE II: The comparison of RMSE (30) for DSE with unknown inputs using five different types of observers. Case |  RMSE & $\gamma$ (for $\mathcal{H}_{\infty}$ Observers only)
---|---
Luenb. Obs. |  $\mathcal{H}_{\infty}$ Obs. | PI Luenb. |  S-PI $\mathcal{H}_{\infty}$ |  O-PI $\mathcal{H}_{\infty}$
9 | $0.9100$ |  $0.5030$ $\gamma=1.444$ | $0.6733$ |  $0.4396$ $\gamma=0.968$ |  $0.1890$ $\gamma=1.068$
39 | $3.6294$ |  $1.1531$ $\gamma=1.318$ | $2.3175$ |  $0.2085$ $\gamma=1.318$ |  $0.0779$ $\gamma=1.318$

(a)  (b)

Figure 4: State estimation results for Case-9 with unknown inputs: (a) the
rotor angle and speed of Generator-$2$ and (b) complex voltage at Bus-$2$.

###  VI-C Effect of PMU’s Configuration on Estimation Quality

This section studies the influence of varying number of PMU together with
different configuration towards the quality of DSE and as such, we limit our
study to Case-9 only. The root-mean-square error (RMSE) [4, 14, 52], a metric
used in the DSE literature, is utilized to quantify the estimation quality

$\displaystyle\mathrm{RMSE}$
$\displaystyle=\sum_{i=1}^{n}\sqrt{\frac{1}{k_{f}}\sum_{k=0}^{k_{f}}(e_{i}[k])^{2}},$
(30)

where $\bm{e}[k]$ is the sampled estimation error and $k_{f}$ denotes the
final time such that $k_{f}T=60\,\mathrm{sec}$ where $T=0.05\;\mathrm{sec}$ is
a constant period such that $\bm{e}[k]=\bm{x}(kT)-\hat{\bm{x}}(kT)$. The
results of this study are shown in Tab. I. The first column in Tab. I
describes different scenarios pertaining to the number and location of PMUs,
whereas the second column shows the PMU configurations. It can be seen that,
for the same number of PMUs, both observers yield varying RMSE. The
$\mathcal{H}_{\infty}$ DAE observer produces smaller RMSE compared to the
Luenberger observer. It is also observed that different placement of PMUs
produces different RMSE. There is no clear pattern to indicate that particular
PMU locations return better or worse estimation error and/or noise
attenuation. To that end, a further study is required to reveal the relation
between PMU locations and estimation quality in a dynamic-algebraic state
estimation framework. Nonetheless, we observe that adding more PMUs generally
reduces the RMSE in both observers. This observation is expected since
increasing the number of PMUs implies that more states are measured (and
therefore, less states are needed to be estimated), thereby decreasing the
total estimation error.

(a)  (b)

Figure 5: State estimation results for Case-39 with unknown inputs: (a) the
transient voltage of Generator-$2$ and (b) complex voltage at Bus-$8$.

###  VI-D Estimation Performance with Unmeasurable Inputs

As mentioned in Section V, $T_{Mi}$ and $E_{fdi}$ are difficult to measure
practically. To that end, here we consider the case when the actual control
inputs are unknown to the observers. Instead, the observers are given static,
steady-state control inputs [9]. Note that, due to the three-phase fault, the
actual $T_{Mi}$ and $E_{fdi}$ for all $i\in\mathcal{G}$ are fluctuating and
settle to a slightly different steady-state value. For this purpose we choose
$\bm{B}_{\nu}:=\bm{B}_{u}$ such that $\nu=m$. The PMU locations and Gaussian
noise for both power networks are set to remain the same as in Section (VI-B),
with an exception that the variance of Gaussian measurement noise is increased
to $0.05^{2}$ for Case-9. Five different observers are benchmarked:

  * •

Luenb. Obs.: the Luenberger observer (19) with P1.

  * •

$\mathcal{H}_{\infty}$ Obs.: the robust $\mathcal{H}_{\infty}$ observer (19)
with P2.

  * •

PI Luenb. Obs.: the PI Luenberger observer (27) with P1.

  * •

S-PI $\mathcal{H}_{\infty}$ Obs.: the robust PI $\mathcal{H}_{\infty}$
observer (27) with P2.

  * •

O-PI $\mathcal{H}_{\infty}$ Obs.: the robust PI $\mathcal{H}_{\infty}$
observer (27) with P3.

The performance matrices for the robust observers are chosen to be
$\bm{\Gamma}:=0.5\bm{I}$ and $\bm{\Gamma}_{\xi}:=0.1\bm{I}$, whereas the
constants in P3 for the O-PI $\mathcal{H}_{\infty}$ observer are set to be
$c_{1}=c_{2}=1$. The state estimation results are given in Fig. 4 for Case-9.
Specifically, in Fig. 4a, we show the comparison of three PI observers in
estimating some internal states of Generator-$2$. It is clear that the
estimation from PI Luenberger observer is the most vulnerable to noise. Strong
noise attenuation is demonstrated by the S-PI $\mathcal{H}_{\infty}$ observer.
However, it is relatively inaccurate in tracking the actual states trajectory
for $25\;\mathrm{sec}\leq t\leq 40\;\mathrm{sec}$. This is presumed to be
caused by the assumed steady-state unknown input dynamics [46]. The O-PI
$\mathcal{H}_{\infty}$ observer provides a sufficient noise attenuation at
steady-state (although not as good as the S-PI $\mathcal{H}_{\infty}$
observer) as well as small tracking error when the states are fluctuating.
Fig. 4b depicts the estimates of complex voltage at Bus-$2$ with the three
$\mathcal{H}_{\infty}$ observers. It is seen that the standard
$\mathcal{H}_{\infty}$ observer provides the least accurate estimation
compared to the robust PI observers. Despite the fact that O-PI
$\mathcal{H}_{\infty}$ observer is able to properly track the actual states
especially for $25\;\mathrm{sec}\leq t\leq 40\;\mathrm{sec}$ with a relatively
small error, at steady-state, it is observed from a detailed inspection that
its state estimates are experiencing more fluctuations compared to the
resulting state estimates from the S-PI $\mathcal{H}_{\infty}$ observer, which
is a consequence of higher performance index $\gamma$ from the O-PI
$\mathcal{H}_{\infty}$ observer—as seen from Tab. II. Note that higher
$\gamma$ means lower noise attenuation. Similar results are also obtained from
Case-39—see Fig. 5. The summary of RMSE for both cases is presented in Tab.
II, where it is seen that the O-PI $\mathcal{H}_{\infty}$ observer has the
smallest RMSE. Thus, it can be concluded that the O-PI $\mathcal{H}_{\infty}$
observer is able to provide superior state estimates, relative to the other
observers, when the inputs are unknown.

(a)  (b)  (c)

Figure 6: The estimation of Generator-$3$ rotor speed and Bus-$3$ real voltage
in Case-9: (a) the $\mathcal{H}_{\infty}$ observer and LAV-KF with noise only,
(b) the O-PI $\mathcal{H}_{\infty}$ observer and LAV-KF with noise and unknown
inputs (UI), and (c) the corresponding estimation error norm.

###  VI-E Comparison with A Robust Two-Stage DSE Method

In this section we finally compare the proposed DSE framework utilizing the
standard and O-PI $\mathcal{H}_{\infty}$ observers with the two-stage approach
introduced in [14] to perform DSE on Case-9. In the first stage of the latter
approach, a linear phasor estimator is utilized to estimate bus voltages based
on measurements provided by the PMUs. The estimated bus voltage values are
later used to aid DSE in estimating generators’ states in the second stage.
The linear phasor estimator implements the Least Absolute Value (LAV) method
pioneered in [52] for the purpose of power system’s state estimation. It is
demonstrated therein that the LAV method can provide better estimates than the
weighted least square method in some instances. Due to the two-stage nature,
this approach can only be implemented in discrete-time. In each iteration
corresponding to a time step $k$, the noisy measurements from PMUs—denoted by
$\tilde{\bm{y}}[k]$—are sampled. Afterwards, the following convex LP is solved
[52]

$\displaystyle\mathbf{P4}\;\min_{\hat{\bm{r}},\hat{\bm{v}}}\;\;$
$\displaystyle\sum_{i=1}^{p}\left\lvert\hat{r}_{i}\right\rvert;\;\;\operatorname*{subject\
to}\;\tilde{\bm{y}}[k]=\tilde{\bm{C}}\bm{C}_{M}\hat{\bm{v}}+\hat{\bm{r}},$

where vectors $\hat{\bm{v}}\in\mathbb{R}^{2N}$ and
$\hat{\bm{r}}\in\mathbb{R}^{r}$ denote the estimates of bus voltages and
measurement noise, respectively. Using $\hat{\bm{v}}[k]$ and our knowledge on
$T_{Mi}$ and $E_{fdi}$ for all $i\in\mathcal{G}$, the dynamic states can be
estimated by implementing a simple discrete-time KF [53].

In the first instance of this comparative simulation, the presence of Gaussian
process and measurement noise are assumed. Three PMUs are installed such that
$\mathcal{N}_{M}=\\{4,6,8\\}$. The estimators also have the perfect
information of $T_{Mi}$ and $E_{fdi}$ at all times. In the two-stage approach
(coined as LAV + KF), the LAV solves P4 in every iteration whereas the KF
utilizes the discrete-time dynamics of the power network discretized using the
forward Euler method. The time period for this case is chosen to be
$0.05\;\mathrm{sec}$. The transient response is generated by introducing a
fault at $t=15\;\mathrm{sec}$. The comparison results are presented in Fig.
6a. It is apparent that the $\mathcal{H}_{\infty}$ observer provides better
estimates than the two-stage approach: the LAV-KF yields less accurate
estimates during transient periods, in addition to the noisy estimates. Next,
in the second instance, we assume that the estimators do not have access to
the actual time-varying values of $T_{Mi}$ and $E_{fdi}$ and instead, they are
only fed with their estimated steady-state values. Fig. 6b shows the
comparison of estimation results between the O-PI $\mathcal{H}_{\infty}$
observer with the two-stage approach. The superiority of the standard and O-PI
$\mathcal{H}_{\infty}$ observers is also evident from the much smaller
estimator error norm—see Fig. 6c. On the matter of simulation running time,
the two-stage approach is also considerably much slower: $644.9\;\mathrm{sec}$
against $15.7\;\mathrm{sec}$ for the case with noise; $629.3\;\mathrm{sec}$
against $14.6\;\mathrm{sec}$ for the case with noise and unknown inputs.

##  VII Concluding Remarks and Future Work

This paper develops a novel DSE framework on the basis of a detailed,
linearized DAE model of power systems—considering a high-order of generator’s
dynamics, stator’s algebraic constraints, generator’s complex power, and the
network’s power balance equations—with PMU-based measurements sensing only bus
voltages and line currents. A novel robust $\mathcal{H}_{\infty}$ DAE observer
is proposed, which only requires detectability and impulse observability to
operate—these conditions are easily satisfied for the proposed power network’s
model with a few PMUs. The numerical test results showcase the performance of
several observers in estimating generators’ internal states and unmeasured bus
voltages under various conditions. In particular, it is revealed that the
standard and O-PI robust $\mathcal{H}_{\infty}$ observers are superior
compared to the Luenberger observers and the two-stage approach that is based
on LAV and KF in performing DSE in the presence of noise and unknown inputs.

The utilization of a linear DAE representation of transmission power networks
inevitably becomes the major limitation in this approach. On that regard,
extending the proposed approach based on a nonlinear, DAE representation of
power networks is a worthy direction for future research. We also plan to
study the effectiveness of the proposed approach in managing sensor failures
from PMUs as well as potential cyber-attacks.

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##  Appendix A The Description of Matrices in (6), (7), and (8)

The matrices in generator’s linear dynamics (6) can be written as
$\bm{E}_{D}=\mathrm{blkdiag}\left(\\{\bm{E}_{Di}\\}_{i\in\mathcal{G}}\right)$,
$\bm{A}_{D}=\mathrm{blkdiag}\left(\\{\bm{A}_{Di}\\}_{i\in\mathcal{G}}\right)$,
$\bm{D}_{D}=\mathrm{blkdiag}\left(\\{\bm{D}_{Di}\\}_{i\in\mathcal{G}}\right)$,
and
$\bm{B}_{D}=\mathrm{blkdiag}\left(\\{\bm{B}_{Di}\\}_{i\in\mathcal{G}}\right)$,
where the matrices for each $i$ are described as follows

$\displaystyle\bm{E}_{Di}$
$\displaystyle=\mathrm{Diag}(1,M_{i},T^{\prime}_{d0i},T^{\prime}_{q0i}),$
$\displaystyle\bm{A}_{Di}$ $\displaystyle=\begin{bmatrix}0&1&0&0\\\
0&-{D_{i}}&-{i^{0}_{qi}}&-{i^{0}_{di}}\\\ 0&0&-1&0\\\
0&0&0&-1\end{bmatrix},\bm{B}_{Di}^{\top}=\begin{bmatrix}0&1&0&0\\\
0&0&1&0\end{bmatrix},$ $\displaystyle\bm{D}_{Di}$
$\displaystyle=\begin{bmatrix}0&0\\\
{i^{0}_{qi}(x^{\prime}_{di}-x^{\prime}_{qi})-e^{\prime
0}_{di}}&{i^{0}_{di}(x^{\prime}_{di}-x^{\prime}_{qi})-e^{\prime 0}_{qi}}\\\
-{\left(x_{di}-x^{\prime}_{di}\right)}&0\\\
0&{x_{qi}-x^{\prime}_{qi}}\end{bmatrix},$

where the corresponding matrices in stator’s algebraic equations (7) can be
constructed as
$\bm{A}_{A}=\mathrm{blkdiag}\left(\\{\bm{A}_{Ai}\\}_{i\in\mathcal{G}}\right)$,
$\bm{D}_{A}=\mathrm{blkdiag}\left(\\{\bm{D}_{Ai}\\}_{i\in\mathcal{G}}\right)$,
and
$\bm{G}_{A}=\mathrm{blkdiag}\left(\\{\bm{G}_{Ai}\\}_{i\in\mathcal{G}}\right)$.
The matrices for each $i$ are given as

$\displaystyle\bm{A}_{Ai}$
$\displaystyle=\begin{bmatrix}-v_{Ri}^{0}\cos\delta_{i}^{0}-v_{Ii}^{0}\sin\delta_{i}^{0}&0&0&1\\\
v_{Ri}^{0}\sin\delta_{i}^{0}-v_{Ii}^{0}\cos\delta_{i}^{0}&0&1&0\end{bmatrix},$
$\displaystyle\bm{D}_{Ai}$
$\displaystyle=\begin{bmatrix}-R_{si}&x^{\prime}_{qi}\\\
-x^{\prime}_{di}&-R_{si}\end{bmatrix},\bm{G}_{Ai}=\begin{bmatrix}-\sin\delta_{i}^{0}&\cos\delta_{i}^{0}\\\
-\cos\delta_{i}^{0}&-\sin\delta_{i}^{0}\end{bmatrix}.$

The matrices in (8) are obtained from linearizing (3) for all generator and
load buses. The details on these matrices are not provided here for brevity,
as similar procedures can be followed from [9, Section 8.2] to compute them.

##  Appendix B Proof of Theorem 1

Let $\bm{M},\;\bm{N}\in\mathbb{R}^{n\times n}$ be two nonsingular matrices
such that $\bm{M}=\operatorname{Blkdiag}(\bm{E}_{D}^{-1},\bm{I})$ and
$\bm{N}=\bm{I}$. By defining $\tilde{\bm{E}}$, $\tilde{\bm{A}}$, and
$\tilde{\bm{B}}_{u}$ respectively as $\tilde{\bm{E}}:=\bm{M}\bm{E}\bm{N}$,
$\tilde{\bm{A}}:=\bm{M}\bm{A}\bm{N}$, and
$\tilde{\bm{B}}_{u}:=\bm{M}\bm{B}_{u}\bm{N}$, the DAE (13) is a restricted
system equivalent to

$\displaystyle\tilde{\bm{E}}\dot{\bm{x}}(t)=\tilde{\bm{A}}{\bm{x}}(t)+\tilde{\bm{B}}_{u}\bm{u}(t),\;\quad\bm{y}(t)=\bm{C}{\bm{x}}(t),$
(31)

where the matrices $\tilde{\bm{E}}$, $\tilde{\bm{A}}$, and
$\tilde{\bm{B}}_{u}$ are detailed as

$\displaystyle\tilde{\bm{E}}=\begin{bmatrix}\,\bm{I}\;\;\,\bm{O}\\\
\bm{O}\;\;\bm{O}\end{bmatrix},\,\tilde{\bm{A}}=\begin{bmatrix}\bm{E}_{D}^{-1}\bm{A}_{1}\;\;\bm{E}_{D}^{-1}\bm{A}_{2}\\\
\bm{A}_{3}\;\;\;\;\;\;\;\bm{A}_{4}\end{bmatrix},\,\tilde{\bm{B}}_{u}=\begin{bmatrix}\bm{E}_{D}^{-1}\bm{B}_{D}\\\
\bm{O}\end{bmatrix}.$

Since $\bm{A}_{4}$ is nonsingular, then according to [39, Lemma 2.3], the
transformed DAE (31) is impulse-free, implying that the DAE is of index one
and regular [40]. Nonetheless, as (31) is a restricted system equivalent to
DAE (13), the considered power networks model is also impulse-free, of index
one, and regular. This completes the proof.  $\blacksquare$

##  Appendix C Proof of Theorem 2

Substituting the matrices $\bm{E}$, $\bm{A}$, and $\bm{C}$ in DAE (13) to
(17), then due to $\bm{E}_{D}$ being a full rank matrix, we obtain

$\displaystyle\operatorname{rank}\left(\begin{bmatrix}\bm{E}_{D}&\bm{A}_{1}&\bm{A}_{2}\\\
\bm{O}&\bm{A}_{3}&\bm{A}_{4}\\\ \bm{O}&\bm{E}_{D}&\bm{O}\\\
\bm{O}&\bm{O}&\tilde{\bm{C}}\bm{C}_{M}\end{bmatrix}\right)-\operatorname{rank}(\bm{E}_{D})$
$\displaystyle=n$
$\displaystyle\Leftrightarrow\operatorname{rank}(\bm{E}_{D})+\operatorname{rank}\left(\begin{bmatrix}\;\bm{A}_{3}\;\;\;\;\,\bm{A}_{4}\;\;\;\\\
\bm{E}_{D}\;\;\;\;\;\bm{O}\;\;\;\;\\\
\;\bm{O}\;\;\;\;\tilde{\bm{C}}\bm{C}_{M}\end{bmatrix}\right)-\operatorname{rank}(\bm{E}_{D})$
$\displaystyle=n$
$\displaystyle\Leftrightarrow\operatorname{rank}\left(\begin{bmatrix}\bm{A}_{4}^{-1}&\bm{O}&\bm{O}\\\
\bm{O}&\bm{E}_{D}^{-1}&\bm{O}\\\
\bm{O}&\bm{O}&\bm{I}\end{bmatrix}\begin{bmatrix}\;\bm{A}_{3}\;\;\;\;\,\bm{A}_{4}\;\;\;\\\
\bm{E}_{D}\;\;\;\;\;\bm{O}\;\;\;\;\\\
\;\bm{O}\;\;\;\;\tilde{\bm{C}}\bm{C}_{M}\end{bmatrix}\right)$
$\displaystyle=n$
$\displaystyle\Leftrightarrow\operatorname{rank}\left(\begin{bmatrix}\bm{A}_{4}^{-1}\bm{A}_{3}&\bm{I}\\\
\bm{I}&\bm{O}\\\
\bm{O}&\tilde{\bm{C}}\bm{C}_{M}\end{bmatrix}\begin{bmatrix}\bm{I}&\bm{O}\\\
\bm{O}&\bm{C}_{M}^{\top}\end{bmatrix}\right)$ $\displaystyle=n,$

which is equivalent to (18), thereby completing the proof.  $\blacksquare$

##  Appendix D Proof of Theorem 3

By substituting $\tilde{\bm{E}}=\bm{E}$ and
$\tilde{\bm{A}}=\bm{A}-\bm{L}\bm{C}$ to the matrix inequality (21) in Lemma 1,
one can immediately obtain

$\displaystyle\bm{A}^{\top}\bm{X}\bm{E}+\bm{E}^{\top}\bm{X}\bm{A}+$
$\displaystyle\bm{A}^{\top}\bm{E}^{\perp\top}\bm{Y}+\bm{Y}^{\top}\bm{E}^{\perp}\bm{A}$
$\displaystyle\;-\bm{C}^{\top}\bm{L}^{\top}\bm{P}-\bm{P}^{\top}\bm{L}\bm{C}\prec
0.$

By defining a new matrix $\bm{W}\in\mathbb{R}^{n\times p}$ such that
$\bm{W}:=\bm{L}^{\top}\bm{P}$, the LMI in (22) is established. This ends the
proof.  $\blacksquare$

##  Appendix E Proof of Theorem 4

Suppose that the LMI (25) holds. Then the $(1,1)$ block matrix in (25) implies
that the LMI (22) is satisfied, thereby indicating that the undisturbed error
dynamics in (24) is stable. Next, consider a Lyapunov function candidate
$V:\mathbb{R}^{n}\rightarrow\mathbb{R}_{+}$ where
$V(\bm{e}):=\bm{e}(t)^{\top}\bm{E}^{\top}\bm{P}\bm{e}(t)$ with
$\bm{P}\in\mathbb{R}^{n\times n}$ nonsingular and
$\bm{E}^{\top}\bm{P}=\bm{P}^{\top}\bm{E}\succeq 0$. Then, the time derivative
of $V(\cdot)$ along its trajectory can be written as follows

$\displaystyle\dot{V}(\bm{e})$
$\displaystyle=\bm{e}(t)^{\top}\left(\bm{A}^{\top}\bm{P}+\bm{P}^{\top}\bm{A}-\bm{C}^{\top}\bm{L}^{\top}\bm{P}-\bm{P}^{\top}\bm{L}\bm{C}\hskip
1.42271pt\right)\bm{e}(t)$
$\displaystyle\quad+\bm{w}(t)^{\top}\left(\bm{B}_{w}^{\top}\bm{P}-\bm{D}_{w}^{\top}\bm{L}^{\top}\bm{P}\right)\bm{e}(t)$
$\displaystyle\quad+\bm{e}(t)^{\top}\left(\bm{P}^{\top}\bm{B}_{w}-\bm{P}^{\top}\bm{L}\bm{D}_{w}\right)\bm{w}(t).$
(32)

For any bounded disturbance $\bm{w}(t)$, then integrating the condition
$\dot{V}(\bm{e})+\bm{\epsilon}^{\top}\bm{\epsilon}-\gamma\bm{w}^{\top}\bm{w}<0$
from $0$ to $t$ reads to

$\displaystyle\int_{0}^{t}\dot{V}(\bm{e}(\tau))d\tau<-\int_{0}^{t}\bm{\epsilon}(\tau)^{\top}\bm{\epsilon}(\tau)d\tau+\int_{0}^{t}\gamma\bm{w}(\tau)^{\top}\bm{w}(\tau)d\tau$
$\displaystyle\Leftrightarrow
V(e(t))-V(e_{0})<-\int_{0}^{t}\left\lVert\bm{\epsilon}(\tau)\right\rVert^{2}_{2}d\tau+\int_{0}^{t}\gamma\left\lVert\bm{w}(\tau)\right\rVert^{2}_{2}d\tau.$

Taking the limit $t\rightarrow\infty$ on both sides of the above inequality
while realizing that $\lim_{t\rightarrow\infty}V(e(t))=0$ due to the stability
of error dynamics and $V(e_{0})=0$ due to zero initial error, one can obtain
the following result

$\displaystyle\int_{0}^{\infty}\left\lVert\bm{\epsilon}(\tau)\right\rVert^{2}_{2}d\tau<\int_{0}^{\infty}\gamma\left\lVert\bm{w}(\tau)\right\rVert^{2}_{2}d\tau\Leftrightarrow\left\lVert\bm{\epsilon}(t)\right\rVert^{2}_{L2}<\gamma\left\lVert\bm{w}(t)\right\rVert^{2}_{L2}.$

Now, from (32), the condition
$\dot{V}(\bm{e})+\bm{\epsilon}^{\top}\bm{\epsilon}-\gamma\bm{w}^{\top}\bm{w}<0$
can be shown to be equivalent to the following matrix inequality

$\displaystyle\begin{bmatrix}\bm{A}^{\top}\bm{P}+\bm{P}^{\top}\bm{A}-\bm{C}^{\top}\bm{L}^{\top}\bm{P}-\bm{P}^{\top}\bm{L}\bm{C}+\bm{\Gamma}^{\top}\bm{\Gamma}&*\\\
\bm{B}_{w}^{\top}\bm{P}+-\bm{D}_{w}^{\top}\bm{L}^{\top}\bm{P}&-\gamma\bm{I}\end{bmatrix}\prec
0.$

Finally, by constructing $\bm{P}$ such that
$\bm{P}:=\bm{X}\bm{E}+\bm{E}^{\perp\top}\bm{Y}$ for some $\bm{X}$ and $\bm{Y}$
where ${\bm{E}}^{\perp}$ being an orthogonal complement of $\bm{E}$ [39] and
defining a new matrix $\bm{W}$ as $\bm{W}:=\bm{L}^{\top}\bm{P}$, the LMI (25)
is established. This concludes the proof.  $\blacksquare$

##  Appendix F Proof of Theorem 5

Firstly, observe that for the pair $(\bm{E}_{\xi},\bm{A}_{\xi})$ we have

$\displaystyle\det\left(s\bm{E}_{\xi}-\bm{A}_{\xi}\right)=\det\left(\begin{bmatrix}s\bm{E}-\bm{A}&-\bm{B}_{\nu}\\\
\bm{O}&s\bm{I}\end{bmatrix}\right).$ (33)

Since it is assumed that $\bm{A}_{4}$ in DAE (13) is a full rank matrix, then
due to Theorem 1, we know that the pair $(\bm{E},\bm{A})$ is regular and of
index one. Thus, without loss of generality, there exists $s\in\mathbb{C}$
where $s\neq 0$ such that $s\bm{E}-\bm{A}$ is nonsingular. As such, also from
the right-hand side of (33), it can be shown that

$\displaystyle\deg\left(\det\left(s\bm{E}_{\xi}-\bm{A}_{\xi}\right)\right)$
$\displaystyle=\deg\left(\det(s\bm{E}-\bm{A})\det(s\bm{I})\right)$
$\displaystyle=\deg\left(s^{\nu}\det(s\bm{E}-\bm{A})\right)$
$\displaystyle=\nu+\deg\left(\det(s\bm{E}-\bm{A})\right)$
$\displaystyle=\nu+\operatorname{rank}(\bm{E})=\operatorname{rank}(\bm{E}_{\xi}),$

hence suggesting that the pair $(\bm{E}_{\xi},\bm{A}_{\xi})$ is impulse-free,
which further implies that $(\bm{E}_{\xi},\bm{A}_{\xi})$ is also of index one
and regular [40]. Secondly, based on the detectability condition in (16), it
is straightforward to show from (27) that

$\displaystyle\operatorname{rank}\left(\begin{bmatrix}s\bm{E}_{\xi}-\bm{A}_{\xi}\\\
\bm{C}_{\xi}\end{bmatrix}\right)$
$\displaystyle=\operatorname{rank}\left(\begin{bmatrix}s\bm{E}-\bm{A}&-\bm{B}_{\nu}\\\
\bm{O}&s\bm{I}\\\ \bm{C}&\bm{O}\end{bmatrix}\right)$
$\displaystyle=\operatorname{rank}\left(\begin{bmatrix}s\bm{E}-\bm{A}\\\
\bm{C}\end{bmatrix}\right)+\nu$ $\displaystyle=n+\nu=\varsigma.$ (34)

Thirdly, it also follows from I-observability condition (17) and the augmented
linear DAE (27) that

$\displaystyle\operatorname{rank}\left(\begin{bmatrix}\bm{E}_{\xi}&\bm{A}_{\xi}\\\
\bm{O}&\bm{E}_{\xi}\\\ \bm{O}&\bm{C}_{\xi}\end{bmatrix}\right)$
$\displaystyle=\operatorname{rank}\left(\begin{bmatrix}\bm{E}&\bm{O}&\bm{A}&\bm{B}_{\nu}\\\
\bm{O}&\bm{I}&\bm{O}&\bm{O}\\\ \bm{O}&\bm{O}&\bm{E}&\bm{O}\\\
\bm{O}&\bm{O}&\bm{O}&\bm{I}\\\
\bm{O}&\bm{O}&\bm{C}&\bm{O}\end{bmatrix}\right)$
$\displaystyle=\operatorname{rank}\left(\begin{bmatrix}\bm{E}&\bm{A}\\\
\bm{O}&\bm{E}\\\ \bm{O}&\bm{C}\end{bmatrix}\right)+2\nu$
$\displaystyle=n+\operatorname{rank}(\bm{E})+2\nu$
$\displaystyle=\varsigma+\operatorname{rank}(\bm{E}_{\xi}).$ (35)

Therefore, from (34) and (35), detectability and I-observability of DAE (27)
imply the detectability and I-observability of power networks DAE (13) and
vice versa. This ends the proof.  $\blacksquare$