# Small-Signal Dynamics of Lossy Inverter-Based Microgrids for Generalized
Droop Controls

Abdullah Al Maruf1, Anamika Dubey2, Sandip Roy3 1California State University,
Los Angeles<EMAIL_ADDRESS>Washington State University.
<EMAIL_ADDRESS>Texas A&M University<EMAIL_ADDRESS>
###### Abstract

A network-level small-signal model is developed for lossy microgrids, which
considers coupled angle and voltage dynamics of inverter-based microgrids and
uses a more general framework of droop controls in the inverter. It is shown
that when relative resistances of the lines in the microgrid are reasonably
consistent and differences of voltage angles across the lines are small at the
operating point, the generalized droop controls can be designed to enforce
decoupling between angle dynamics and voltage dynamics. Next, structural
results for the asymptotic stability of small-signal angle and voltage
dynamics are given for the case when generalized droop control achieves
decoupling. Simulated transient responses of a modified IEEE 9-bus system are
presented to validate the theoretical findings which show the effectiveness of
generalized droop controls in independently shaping the settling times of the
angle and voltage responses of the lossy microgrid system.

##  I Introduction

A microgrid is an interconnected low/medium-voltage power distribution network
primarily supplied by inverter-based distributed energy resources (DERs) [1,
2]. Droop controls are popularly used in inverter-based microgrids to achieve
power-sharing between multiple DERs by mimicking the inertial dynamics and
governor controls of generators in the bulk grid. Droop control does not
require any communication to operate, which makes it more appealing to use
[3]. Traditionally, two separate droop controls, i.e., $P-\omega$ and $Q-V$
droop controls, are used in the microgrid to regulate the voltage angles and
voltage magnitudes of the grid based on the standard decoupling assumption [3,
4, 5]. While this decoupling assumption simplifies analysis (which mimics the
analysis for the bulk grid) as undertaken in [6, 7, 8, 9], it is often not
accurate for lossy microgrids since lines in power distribution networks have
substantially large resistance to inductance ratio and also may exhibit
voltage magnitude/angle differentials across them [4, 10]. This poses a
significant concern when using traditional droop-controls in microgrids to
achieve stability and the desired power sharing.

In this study, we consider a generalization of the conventional droop
controller where the grid-side voltage frequency and magnitude are both set in
a co-dependent way based on the locally measured real and reactive power [4,
10]. This generalized droop scheme was proposed with the idea that it can
approximately decouple the angle (i.e., voltage angle) and voltage (i.e.,
voltage magnitude) dynamics in lossy microgrids, and hence should allow
systematic shaping of these two dynamics and power sharing. However, a
network-level model and analysis of microgrids with generalized droop have yet
to be undertaken in the literature. Furthermore, it is necessary to develop
structural analyses of the microgrid dynamics – i,e., ones that are phrased in
terms of the graph topology of the microgrid and other general characteristics
of the system rather than the specific parameter values because of high
variability in operating conditions for microgrids. We point out that the
important study of Schiffer et al [11] captures the coupling of the voltage
and angle dynamics for traditional droops, but it does not obtain structural
characterizations of stability or design rubrics for the lossy case.

One main focus of this study is to develop an accurate model and structural
analysis for lossy microgrids that use generalized droop controls. Our main
contributions in this study are the following:

1) A complete signal-signal model for an islanded microgrid’s network-level
dynamics is developed, which captures resistive losses, generalized droop
controls, smoothing/filtering used in droop controls, and multiple bus types.

2) A design of generalized droop is presented, which can approximately
decouple the voltage magnitude and angle dynamics and thus achieve desirable
small-signal properties, provided that the $R/X$ ratios of lines are
reasonably consistent across the microgrid and bus angle differences across
the lines are small at the microgrid’s operating point.

3) For the droop control that achieves the decoupling, structural small-signal
stability results are presented. These results mainly follow from our previous
studies [8, 9], but these studies pre-supposed decoupling and did not consider
the generalized droop controls.

4) Simulated small-disturbance responses of a modified IEEE 9-bus system are
presented to verify our results. We specifically show how the settling time of
small disturbance responses of voltage angles and magnitudes can be shaped by
appropriately designing generalized droop controls.

##  II Small-Signal model for A Lossy Microgrid with Generalized Droop
Controls

The transient and small-signal dynamics of a lossy microgrid operating in
islanded mode are modeled in this section. The model presented here
incorporates 1) a generalized droop scheme for the primary controller at
inverter-based generators, 2) low-pass filtering of measured local powers
implemented at the droop controls, 3) heterogeneous (i.e., inverter and non-
inverter) bus types, and 4) resistive losses in network lines. First, a
nonlinear differential-algebraic-equation (DAE) model is developed. Then, two
simplifications – an approximation of the algebraic equations as dynamic ones
via singular perturbation, and linearization – are undertaken to obtain a
network-level small-signal model. The derived small-signal model is suitable
for the analysis and control of small-signal properties of microgrids.

###  II-A Microgrid Network: Nonlinear DAE Model

A nonlinear DAE model is developed for a lossy microgrid that uses generalized
droop controls at inverter-based generators, which also generalizes
previously-developed network-level models for conventional droop-controlled
microgrids [8, 9, 11]. We present the microgrid network model in three steps:
1) the model for the generalized droop control (Section II-A1); 2) the network
power-flow model (Section II-A2); and 3) the full DAE model which combines the
network power-flow models with dynamic models of the DERs (Section II-A3).

####  II-A1 Droop-Controlled Inverter Model

A grid-forming inverter operating in an islanded microgrid acts as an ideal AC
voltage source, where its voltage (both magnitude and frequency) can be
assigned at will by the fast inner control systems. To support power sharing
among multiple inverters, conventional droop controls are usually used as a
primary controller, which mimics the governor control of synchronous
generators in the bulk grid in the sense that a $P-\omega$ droop control
regulates frequency (i.e., frequency of voltage) based on the measured active
power while in a similar way a $Q-V$ droop control regulates voltage (i.e.
voltage magnitude) based on the measured reactive power at the connected bus.
These independent controls of frequency and voltage are ineffective for lossy
microgrids as the decoupling assumption between angle dynamics and voltage
does not hold. A generalization of conventional droop controls has been
proposed to achieve better performance for lossy microgrids, where both
frequency and voltage are set in a co-dependent way based on the measured real
and reactive powers [4]. This generalized control offers an additional design
parameter to modify the governing equations for setting the frequency and
voltage of an inverter. In this study, we consider the generalized droop
control scheme in the inverters and develop a network-level model of a
microgrid that employs such a scheme.

Droop controls are incorporated with low pass filters to suppress the high-
frequency variations in measured powers. Here, in our framework, the filter
and the time delay due to fast inner control loops are abstracted as a first-
order low pass filter. The model of an inverter at bus $i$ with a generalized
droop control scheme is depicted in Fig. 1. Mathematically, for the
generalized droop control, the frequency and voltage at the inverter can be
written in terms of the real and reactive power in the Laplace domain as
follows:

Figure 1: Block diagram of generalized droop control with filtering.

$\displaystyle\big{(}(P_{G_{i}}(s)-P^{0}_{G_{i}})\sin(\alpha_{i})-(Q_{G_{i}}(s)-Q^{0}_{G_{i}})\cos(\alpha_{i})\big{)}$
$\displaystyle\times\frac{-1/D_{1i}}{1+T_{1i}s}=\omega_{i}(s)-\omega^{0}$ (1)

and,

$\displaystyle\big{(}(P_{G_{i}}(s)-P^{0}_{G_{i}})\cos(\alpha_{i})+(Q_{G_{i}}(s)-Q^{0}_{G_{i}})\sin(\alpha_{i})\big{)}$
$\displaystyle\times\frac{-1/D_{2i}}{1+T_{2i}s}=V_{i}(s)-V_{i}^{0}$ (2)

where, $\omega^{0}$ is the common nominal angular frequency111$\omega^{0}$ is
ideally set to $0$ since droop control operates relative to the nominal
rotational frequency e.g. 377 rad/s in North America.; $\omega_{i}(s)$ is the
Laplace transform of frequency $\omega_{i}$; $V_{i}^{0}$ is the nominal
voltage; $V_{i}(s)$ is the Laplace transform of voltage $V_{i}$;
$P^{0}_{G_{i}}$ and $Q^{0}_{G_{i}}$ are the reference active power and
reactive power respectively; $P_{G_{i}}(s)$ and $Q_{G_{i}}(s)$ are the Laplace
transform of generated active power and generated reactive power respectively;
$D_{1i},D_{2i}\geq 0$ are the reciprocals of droop gains of the inverter;
$T_{1i},T_{2i}$ are the time constants of the low pass filters and
$\alpha_{i}$ is an additional design parameter in this droop law which
corresponds to an angle. This angle specifies the co-dependence between two
droop equations and is introduced to reduce the coupling between the angle and
voltage dynamics. We note that the traditional droop law is recovered when
this angle is selected as $\alpha_{i}=\pi/2$ in (II-A1) and (II-A1). Equations
(II-A1) and (II-A1) can be rewritten in the time domain as:

$\displaystyle(P_{G_{i}}-P^{0}_{G_{i}})\sin(\alpha_{i})-(Q_{G_{i}}-Q^{0}_{G_{i}})\cos(\alpha_{i})$
$\displaystyle=D_{1i}\omega^{0}-D_{1i}\dot{{\theta}_{i}}-D_{1i}T_{1i}\ddot{{\theta}_{i}}$
(3)

and,

$\displaystyle(P_{G_{i}}-P^{0}_{G_{i}})\cos(\alpha_{i})+(Q_{G_{i}}-Q^{0}_{G_{i}})\sin(\alpha_{i})$
$\displaystyle=D_{2i}V_{i}^{0}-D_{2i}{V_{i}}-D_{2i}T_{2i}\dot{V_{i}}$ (4)

Equations (II-A1) and (II-A1) govern the voltage and angle dynamics of
inverter-based buses with generalized-droop control.

####  II-A2 Network Model

Here we derive the standard network model for power flow, which includes lossy
or resistive lines. We consider a connected microgrid network with $n$ buses,
labeled $i=1,\ldots,n$. The admittance of the line between buses $i$ and $k$
obtained from the bus-admittance matrix is denoted by
$Y_{ik}\angle\phi_{ik}=G_{ik}+jB_{ik}$ where $G_{ik}$ and $B_{ik}$ are
respectively conductance and susceptance of the line according to the bus-
admittance matrix. Note for regular power system, $\pi/2\leq\phi_{ik}\leq\pi$
for $i\neq k$. The active and reactive power flow equations for bus $i$ are
given as the following [12]:

$\displaystyle P_{G_{i}}=P_{L_{i}}$ $\displaystyle+$ $\displaystyle
V_{i}^{2}G_{ii}$ (5) $\displaystyle+$ $\displaystyle\sum_{k\in
adj(i)}{V_{i}V_{k}Y_{ik}~{}\cos(\theta_{i}-\theta_{k}-\phi_{ik})}$

and,

$\displaystyle Q_{G_{i}}=Q_{L_{i}}$ $\displaystyle-$ $\displaystyle
V_{i}^{2}B_{ii}$ (6) $\displaystyle+$ $\displaystyle\sum_{k\in
adj(i)}{V_{i}V_{k}Y_{ik}~{}\sin(\theta_{i}-\theta_{k}-\phi_{ik})}$

where, $P_{G_{i}}$ and $P_{L_{i}}$ are the active power generation and demand
at bus $i$, similarly $Q_{G_{i}}$ and $Q_{L_{i}}$ are the reactive power
generation and demand at bus $i$. Here $adj(i)$ refers to the set of buses
that are adjacent to bus $i$ (or equivalently, the set of buses connected to
bus $i$ via a distribution line).

In this study, we consider the constant power model for the loads, which is
commonly used in the stability analysis of bulk-grid and microgrid [12, 11].
According to the constant power load model, the active and reactive power for
the load at bus $i$ can be expressed as: $P_{L_{i}}=P^{0}_{L_{i}}$ and
$Q_{L_{i}}=Q^{0}_{L_{i}}$ where both $P^{0}_{L_{i}}$ and $Q^{0}_{L_{i}}$ are
arbitrary constants.

####  II-A3 DAE Model for the Microgrid Network

The DAE model for the microgrid is developed by integrating the network model
and the generalized-droop-control model. Let us first denote the set of buses
that have inverters as $\mathcal{V}_{A}$. Using (II-A1), (II-A1), (5), (6) and
the load model, the governing differential equations for the bus
$i\in\mathcal{V}_{A}$ can be written as:

$\displaystyle
D_{1i}T_{1i}\ddot{{\theta}_{i}}=\big{(}(P^{0}_{G_{i}}-P^{0}_{L_{i}})\sin(\alpha_{i})-(Q^{0}_{G_{i}}-Q^{0}_{L_{i}})\cos(\alpha_{i})$
$\displaystyle~{}~{}~{}~{}+D_{1i}\omega^{0}\big{)}-D_{1i}\dot{\theta}_{i}-V_{i}^{2}\big{(}G_{ii}\sin(\alpha_{i})+B_{ii}\cos(\alpha_{i})\big{)}$
$\displaystyle~{}~{}~{}-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\cos(\theta_{i}-\theta_{k}-\phi_{ik}+\pi/2-\alpha_{i})}$
(7)

and,

$\displaystyle
D_{2i}T_{2i}\dot{V_{i}}=\big{(}(P^{0}_{G_{i}}-P^{0}_{L_{i}})\cos(\alpha_{i})+(Q^{0}_{G_{i}}-Q^{0}_{L_{i}})\sin(\alpha_{i})$
$\displaystyle~{}~{}~{}~{}+D_{2i}V_{i}^{0}\big{)}-D_{2i}V_{i}+V_{i}^{2}(-G_{ii}\cos(\alpha_{i})+B_{ii}\sin(\alpha_{i}))$
$\displaystyle~{}~{}~{}~{}-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\sin(\theta_{i}-\theta_{k}-\phi_{ik}+\pi/2-\alpha_{i})}$
(8)

We next present the governing equations for the buses that do not have
inverter-based generators $(P_{G_{i}}=0,~{}Q_{G_{i}}=0)$. It is convenient to
denote the set of these buses as $\mathcal{V}_{B}$. Using (5), (6) and the
load model, the governing equations for the bus $i\in\mathcal{V}_{B}$ can be
written as:

$\small 0=-P^{0}_{L_{i}}-V_{i}^{2}G_{ii}-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\cos(\theta_{i}-\theta_{k}-\phi_{ik})}$ (9)

and,

$\small 0=-Q^{0}_{L_{i}}+V_{i}^{2}B_{ii}-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\sin(\theta_{i}-\theta_{k}-\phi_{ik})}$ (10)

Equations (II-A3), (II-A3), (9) and (10) in together describe nonlinear DAE
model of microgrid.

###  II-B Simplified Models for Analysis

We develop a linear differential-equation approximation of the nonlinear DAE
to enable formal analysis of small-signal properties. First, a singular
perturbation argument is applied to obtain a differential-equation
approximation of the DAE in a way that maintains the topological structure of
the network (hence, we refer to this as a structure-preserving model). Then,
the nonlinear differential-equation model is linearized to obtain the small-
signal model of the microgrid dynamics.

####  II-B1 Structure-Preserving Model

Here we apply a singular perturbation argument to approximate the algebraic
equations as fast dynamics yielding a differential-equation model. The
singular-perturbation approach has the advantage of preserving the topological
structure of the microgrid, which is helpful for structural analysis of the
microgrid. Before applying singular perturbation approximation, we undertake a
transformation on the set of algebraic equations of the DAE to match
expressions of the differential equations. This will allow us to obtain a
compact and suitable expression for the eventual model, while sufficing that
the dynamical properties of the original DAE model are maintained in the
approximated differential equation model (with the undertaken transformation).

In analogy with the generalized droop controls of inverter-based bus, for each
bus $i\in\mathcal{V}_{B}$ we consider an angle $\alpha_{i}$ where
$0\leq\alpha_{i}\leq\pi/2$ to undertake the following transformation on (9)
and (10).

$\displaystyle\small
0=(-P^{0}_{L_{i}}\sin(\alpha_{i})+Q^{0}_{L_{i}}\cos(\alpha_{i}))$
$\displaystyle-
V_{i}^{2}\big{(}G_{ii}\sin(\alpha_{i})+B_{ii}\cos(\alpha_{i})\big{)}$
$\displaystyle-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\cos(\theta_{i}-\theta_{k}-\phi_{ik}+\pi/2-\alpha_{i})}$
(11)

and,

$\displaystyle\small
0=(-P^{0}_{L_{i}}\cos(\alpha_{i})-Q^{0}_{L_{i}}\sin(\alpha_{i}))$
$\displaystyle+V_{i}^{2}(-G_{ii}\cos(\alpha_{i})+B_{ii}\sin(\alpha_{i}))$
$\displaystyle-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\sin(\theta_{i}-\theta_{k}-\phi_{ik}+\pi/2-\alpha_{i})}$
(12)

Now we apply singular perturbation approximations directly on (II-B1) and
(II-B1). Upon appropriate selection of $\alpha_{i}$, based on the singular-
perturbation argument we can say that each $i\in{\mathcal{V}}_{B}$ bus
approximately has the following dynamics:

$\displaystyle\small\epsilon_{1}\ddot{\theta_{i}}=(-P^{0}_{L_{i}}\sin(\alpha_{i})+Q^{0}_{L_{i}}\cos(\alpha_{i}))-\epsilon_{2}{\dot{\theta}}_{i}$
$\displaystyle-
V_{i}^{2}\big{(}G_{ii}\sin(\alpha_{i})+B_{ii}\cos(\alpha_{i})\big{)}$
$\displaystyle-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\cos(\theta_{i}-\theta_{k}-\phi_{ik}+\pi/2-\alpha_{i})}$
(13)

and,

$\displaystyle\small\epsilon_{3}\dot{V_{i}}=(-P^{0}_{L_{i}}\cos(\alpha_{i})-Q^{0}_{L_{i}}\sin(\alpha_{i}))$
$\displaystyle+V_{i}^{2}(-G_{ii}\cos(\alpha_{i})+B_{ii}\sin(\alpha_{i}))$
$\displaystyle-\sum_{k\in
adj(i)}{V_{i}V_{k}{Y}_{ik}~{}\sin(\theta_{i}-\theta_{k}-\phi_{ik}+\pi/2-\alpha_{i})}$
(14)

where $\epsilon_{1},\epsilon_{2}$ and $\epsilon_{3}$ are sufficiently small
positive numbers. Some discussions on these factors are worthwhile. First,
note that the approximated differential equation system and the original DAE
system have the same equilibrium solutions. Specifically in our case, the
system given by (II-A3), (II-A3), (9), (10) and the system given by (II-A3),
(II-A3), (II-B1), (II-B1) have the same equilibrium solutions. Additionally,
provided that the factors $\epsilon_{1}$, $\epsilon_{2}$, $\epsilon_{3}$ are
chosen appropriately (specifically, $\epsilon_{2}$ and $\epsilon_{3}$ are
sufficiently small and $\epsilon_{1}$ is on the order of $\epsilon_{2}^{2}$)
and noting that both systems depend on the relative angles of bus voltage
rather than their actual values (thus, a common zero eigenvalue is maintained
by the Jacobians of both systems at equilibrium), the local stability of the
equilibrium of approximate model (II-A3), (II-A3), (II-B1), (II-B1) suffices
the local stability of the DAE model (II-A3), (II-A3), (9), (10) according to
the singular perturbation theory [13, 14]. Therefore, we will use differential
equation set (II-A3), (II-A3), (II-B1), (II-B1) for deriving the structural
stability conditions given that the above factors are selected appropriately.
The appropriate bounds for these factors can be derived from the singular-
perturbation literature [13].

To enable structural analysis for the stability of microgrid, it is convenient
to present a compact form of the approximated model based on a graph-theoretic
description of the microgrid. First, we consider a directed graph
$\mathcal{G}$ defined for the microgrid, where a vertex represents each bus,
and the line between buses $i$ and $k$ corresponds to two directed edges
$(i,k)$ and $(k,i)$. Hence, the vertex set $\mathcal{V}$ has cardinality $n$
and edge set $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ has
cardinality $2l$, where $l$ is the number of lines in the microgrid. For
convenience, we arbitrarily label and order the edges, with edge $e_{m}$
($m=1,\ldots,2l$) representing edge $(i,k)$ of the graph. Then, the incidence
matrix $\boldsymbol{E}\in\mathbb{R}^{n\times 2l}$ of $\mathcal{G}$ is defined
as $E_{im}=1$ and $E_{km}=-1$ for each $e_{m}\in\mathcal{E}$, with all other
entries being zero. Furthermore, the orientation matrix is defined for the
directed graph (see [15]) as a matrix $\boldsymbol{C}\in\mathbb{R}^{n\times
2l}$ with entries $C_{im}=1$ if $E_{im}=1$ and zero otherwise. We also use the
following notations: $\boldsymbol{x}=[x_{i}]$ denotes a vector
$\boldsymbol{x}=[x_{1},x_{2},\cdots,x_{n}]^{T}\in\mathbb{R}^{n}$, and
$diag(\boldsymbol{x})$ denotes a diagonal matrix in $\mathbb{R}^{n\times n}$
whose diagonal entries are $x_{1},x_{2},\cdots,x_{n}$ respectively. Using
these notations, we can write (II-A3), (II-A3), (II-B1), (II-B1) as vector
equations as follows.

$\displaystyle\boldsymbol{M}_{P}\ddot{\boldsymbol{\theta}}=\boldsymbol{P}^{0}-\boldsymbol{D}_{P}\dot{\boldsymbol{\theta}}-(diag(\boldsymbol{V}))^{2}\hat{\boldsymbol{G}}$
$\displaystyle-\boldsymbol{C}\boldsymbol{U}\cos(\boldsymbol{E}^{T}\boldsymbol{\theta}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha})$
(15)
$\displaystyle\boldsymbol{M}_{Q}\dot{\boldsymbol{V}}=\boldsymbol{Q}^{0}-\boldsymbol{D}_{Q}\boldsymbol{V}+(diag(\boldsymbol{V}))^{2}\hat{\boldsymbol{B}}$
$\displaystyle-\boldsymbol{C}\boldsymbol{U}\sin(\boldsymbol{E}^{T}\boldsymbol{\theta}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha})$
(16)

In above the vectors/matrices represent corresponding physical quantities or
parameters of the microgrid. Specifically,
$\boldsymbol{\theta}=[\theta_{i}]\in\mathbb{R}^{n}$ and
$\boldsymbol{V}=[V_{i}]\in\mathbb{R}^{n}$ denote the vectors of bus voltage
phase angles and magnitude respectively.
$\boldsymbol{\alpha}=[\alpha_{i}]\in\mathbb{R}^{n}$ whereas
$\boldsymbol{\phi}=[\phi_{ik}]\in\mathbb{R}^{2l}$ is the vector containing
phase angles of the admittance of each directed lines.
$\boldsymbol{M}_{p}~{}\in\mathbb{R}^{n\times n}$ is a diagonal matrix
imitating inertia, where $M_{p_{ii}}=D_{1i}T_{2i}$ if $i\in\mathcal{V}_{A}$,
otherwise $M_{P_{ii}}=\epsilon_{1}$. Similarly,
$\boldsymbol{M}_{Q}~{}\in\mathbb{R}^{n\times n}$ is also a diagonal matrix,
where $M_{Q_{ii}}=D_{2i}T_{2i}$ if $i\in\mathcal{V}_{A}$, otherwise
$M_{Q_{ii}}=\epsilon_{3}$.
$\boldsymbol{P}^{0}=[P^{0}_{i}]~{}\in\mathbb{R}^{n}$ denotes nominal injected
(transformed) active power in each bus, where
$P^{0}_{i}=(P^{0}_{G_{i}}-P^{0}_{L_{i}})\sin(\alpha_{i})-(Q^{0}_{G_{i}}-Q^{0}_{L_{i}})\cos(\alpha_{i})$
$+D_{1i}\omega^{0}$ if $i\in\mathcal{V}_{A}$, otherwise
$P^{0}_{i}=(-P^{0}_{L_{i}}\sin(\alpha_{i})+Q^{0}_{L_{i}}\cos(\alpha_{i}))$. In
the same manner, $\boldsymbol{Q}^{0}=[Q_{i}^{0}]~{}\in\mathbb{R}^{n}$ denotes
nominal (transformed) injected reactive power in each bus, where
$Q^{0}_{i}=(P^{0}_{G_{i}}-P^{0}_{L_{i}})\cos(\alpha_{i})+(Q^{0}_{G_{i}}-Q^{0}_{L_{i}})\sin(\alpha_{i})+D_{2i}V_{i}^{0}$
if $i\in\mathcal{V}_{A}$, otherwise
$Q^{0}_{i}=-P^{0}_{L_{i}}\cos(\alpha_{i})-Q^{0}_{L_{i}}\sin(\alpha_{i})$.
$\boldsymbol{D}_{P}~{}\in\mathbb{R}^{n\times n}$ is a diagonal matrix
representing damping coefficients, where $D_{P_{ii}}=D_{1i}$ if
$i\in\mathcal{V}_{A}$, otherwise $D_{P_{ii}}=\epsilon_{2}$. Similarly,
$\boldsymbol{D}_{Q}~{}\in\mathbb{R}^{n\times n}$ is also a diagonal matrix,
where $D_{Q_{ii}}=\ D_{2i}$ if $i\in\mathcal{V}_{A}$, otherwise
$D_{Q_{ii}}=0$. $\boldsymbol{U}\in\mathbb{R}^{2l\times
2l}=diag(V_{i}V_{k}Y_{ik})$ is a diagonal matrix in above whose diagonal
entries represent the magnitudes of active/reactive powers flow through the
edges. $\hat{\boldsymbol{G}}$ and $\hat{\boldsymbol{B}}$ are vectors in
$\mathbb{R}^{n}$ where
$\hat{G}_{i}=G_{ii}\sin(\alpha_{i})+B_{ii}\cos(\alpha_{i})$ and
$\hat{B}_{i}=-G_{ii}\cos(\alpha_{i})+B_{ii}\sin(\alpha_{i})$ respectively.
$\boldsymbol{1}_{n}$ denotes a vector in $\mathbb{R}^{n}$ with all entry equal
to 1. It should be noted that in (II-B1) and (II-B1) the functions $\cos(.)$
and $\sin(.)$ are meant in the Hadamard sense (i.e. component-wise).

Next, we use equations (II-B1) and (II-B1) to develop the small-signal model.

####  II-B2 Small-Signal Model

Linearization is undertaken around an arbitrary equilibrium to obtain a small-
signal differential-equation model for the microgrid network, which allows the
study of small signal properties at an equilibrium of interest. Now, we point
out an interesting characteristic of the equilibrium of microgrid systems. We
note that each equilibrium of a microgrid system corresponds to a manifold
since the microgrid’s dynamics only depend on the differences of phase angles
of bus voltages (see the model given by (II-B1) and (II-B1) or equivalently
(II-A3), (II-A3), (II-B1), (II-B1)). Therefore, if
${\boldsymbol{x}^{s}=\begin{bmatrix}\boldsymbol{\theta}^{s}&\boldsymbol{\dot{\theta}}^{s}&\mathbf{V}^{s}\end{bmatrix}}^{T}$
is an equilibrium of a microgrid system, then
$\boldsymbol{x}^{s}+k\boldsymbol{v}_{0}$ is also an equilibrium of the system
(or vice versa), where
${\boldsymbol{v}_{0}=\begin{bmatrix}\boldsymbol{1}_{n}&{\bf 0}_{n}&{\bf
0}_{n}\end{bmatrix}}^{T}$ (note, $\boldsymbol{0}_{n}$ is a vector in
$\mathbb{R}^{n}$ with all entries equal to $0$) and $k\in\mathbb{R}$. From the
power system viewpoint, these are physically the same equilibrium as only the
relative phase angles among the buses are of interest. Therefore, from this
point, by an equilibrium $\boldsymbol{x}^{s}$ of a microgrid system, we refer
to the manifold $\boldsymbol{x}^{s}+k\boldsymbol{v}_{0}$. Now, let us define
the state vector as
${\boldsymbol{x}=\begin{bmatrix}\boldsymbol{\theta}&\boldsymbol{\dot{\theta}}&\mathbf{V}\end{bmatrix}}^{T}$
and denote an equilibrium point (or the manifold) as
$\boldsymbol{x}^{s}=\begin{bmatrix}\boldsymbol{\theta}^{s}&\boldsymbol{\dot{\theta}}^{s}&\mathbf{V}^{s}\end{bmatrix}^{T}$.
By linearizing (II-B1) and (II-B1) around the equilibrium point
$\boldsymbol{x}^{s}$ (or more accurately around the equilibrium manifold
$\boldsymbol{x}^{s}+k\boldsymbol{v}_{0}$ as discussed before) we get:

$\boldsymbol{\Delta}\dot{\boldsymbol{x}}=\begin{bmatrix}\boldsymbol{\Delta\dot{\theta}}\\\
\boldsymbol{\Delta\ddot{\theta}}\\\
\boldsymbol{\Delta\dot{V}}\end{bmatrix}=\boldsymbol{J}(\boldsymbol{x}^{s})\begin{bmatrix}\boldsymbol{\Delta\theta}\\\
\boldsymbol{\Delta\dot{\theta}}\\\ \boldsymbol{\Delta
V}\end{bmatrix}=\boldsymbol{J}(\boldsymbol{x}^{s})\boldsymbol{\Delta x}$ (17)

The Jacobian matrix $\boldsymbol{J}\left({\boldsymbol{x}}^{s}\right)$ of the
nonlinear model evaluated at the equilibrium is given in the appendix. In this
Jacobian,
$\boldsymbol{W}_{1}(\boldsymbol{x}^{s})=\frac{\partial(\boldsymbol{U}^{s}\cos(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha}))}{\partial(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha})}=diag\big{(}-\boldsymbol{U}^{s}\sin(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha})\big{)}$
$\in\mathbb{R}^{2l\times 2l}$ and
$\boldsymbol{W}_{2}(\boldsymbol{x}^{s})=\frac{\partial(\boldsymbol{U}^{s}\sin(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha}))}{\partial(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha})}=diag\big{(}\boldsymbol{U}^{s}\cos(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha})\big{)}$
$\in\mathbb{R}^{2l\times 2l}$. Here, $\boldsymbol{0}_{n\times n}$ and
$\boldsymbol{I}_{n\times n}$ denote $n\times n$ zero matrix and $n\times n$
unit matrix respectively.

Here we emphasize that the linearization undertaken in (II-B1) and (II-B1)
should be understood as a linearization over the manifold
$\boldsymbol{x}^{s}+k\boldsymbol{v}_{0}$. Therefore the linearized system
holds an invariant manifold where all the angles are synchronized and all the
frequency and voltage deviations are zero. In analogy with [7], we define
$\boldsymbol{W}_{1}(\boldsymbol{x}^{s})$ and
$\boldsymbol{W}_{2}(\boldsymbol{x}^{s})$ as the edge weights of two graphs
$\mathcal{G}_{1}$ and graph $\mathcal{G}_{2}$ respectively. This definition
allows us to rewrite the dynamics in terms of the (directed) Laplacian matrix
of these graphs, which is helpful for the analysis of the stability. Following
the existing literature [7], we consider the weighted directed graphs
$\mathcal{G}_{1}(\boldsymbol{x}^{s})=(\mathcal{V},\mathcal{E},\boldsymbol{W}_{1}(\boldsymbol{x}^{s}))$
and
$\mathcal{G}_{2}(\boldsymbol{x}^{s})=(\mathcal{V},\mathcal{E},\boldsymbol{W}_{2}(\boldsymbol{x}^{s}))$.
We note that the edge weights in the graphs
$\mathcal{G}_{1}(\boldsymbol{x}^{s})$ and
$\mathcal{G}_{2}(\boldsymbol{x}^{s})$ are different and depend on the
operating point $\boldsymbol{x}^{s}$ and the parameter $\boldsymbol{\alpha}$,
but the vertex and edge sets of both graphs are same as graph $\mathcal{G}$.
The Laplacian matrix for these directed weighted graphs can be written as
$\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))=\boldsymbol{C}\boldsymbol{W}_{1}(\boldsymbol{x}^{s}){\boldsymbol{E}}^{{T}}$
and
$\boldsymbol{L}(\mathcal{G}_{2}(\boldsymbol{x}^{s}))=\boldsymbol{C}\boldsymbol{W}_{2}(\boldsymbol{x}^{s}){\boldsymbol{E}}^{{T}}$.
Therefore, we can simplify the Jacobian as:

$\displaystyle\scriptsize\boldsymbol{J}(\boldsymbol{x}^{s})=\begin{bmatrix}\boldsymbol{0}_{n\times
n}&\boldsymbol{I}_{n\times n}&\boldsymbol{0}_{n\times n}\\\
-\boldsymbol{M}_{P}^{-1}\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))&-\boldsymbol{M}_{P}^{-1}\boldsymbol{D}_{P}&-\boldsymbol{M}_{P}^{-1}\Big{(}-\boldsymbol{L}(\mathcal{G}_{2}(\boldsymbol{x}^{s}))+2~{}diag(\hat{\boldsymbol{P}}^{s})\Big{)}(diag(\boldsymbol{V}^{s}))^{-1}\\\
-\boldsymbol{M}_{Q}^{-1}\boldsymbol{L}(\mathcal{G}_{2}(\boldsymbol{x}^{s}))&\boldsymbol{0}_{n\times
n}&-\boldsymbol{M}_{Q}^{-1}\Big{(}\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))+2~{}diag(\hat{\boldsymbol{Q}}^{s})+\boldsymbol{D}_{Q}~{}diag(\boldsymbol{V}^{s})\Big{)}(diag(\boldsymbol{V}^{s}))^{-1}\end{bmatrix}$
(18)

Here
$\hat{\boldsymbol{P}}^{s}=diag(\hat{\boldsymbol{G}})(diag(\boldsymbol{V}^{s}))^{2}+diag(\boldsymbol{C}\boldsymbol{U}^{s}$
$\cos(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha}))$
and
$\hat{\boldsymbol{Q}}^{s}=-diag(\boldsymbol{\hat{B}})(diag(\boldsymbol{V}^{s}))^{2}$
$+diag(\boldsymbol{C}\boldsymbol{U}^{s}$
$\sin(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha})$
denote the vectors whose entries are related to injected powers of the buses
at equilibrium. Specifically,
$\hat{P}_{i}^{s}=(V_{i}^{s})^{2}\big{(}G_{ii}\sin(\alpha_{i})$
$+B_{ii}\cos(\alpha_{i})\big{)}$ $+\sum_{k\in
adj(i)}{V_{i}^{s}V_{k}^{s}{Y}_{ik}~{}\cos(\theta_{i}^{s}-\theta_{k}^{s}-\phi_{ik}+\pi/2-\alpha_{i})}$
and $\hat{Q}_{i}^{s}=(V_{i}^{s})^{2}(G_{ii}\cos(\alpha_{i})$
$-B_{ii}\sin(\alpha_{i}))$ $+\sum_{k\in
adj(i)}{V_{i}^{s}V_{k}^{s}{Y}_{ik}~{}\sin(\theta_{i}^{s}-\theta_{k}^{s}-\phi_{ik}+\pi/2-\alpha_{i})}$.

Note that the parameter $\alpha_{i}$ for $i\in{\mathcal{V}}_{B}$ is arbitrary
given that it is appropriately chosen following the singular perturbation
argument. Also note that by substituting $\alpha=\pi/2$ in (18), we obtain the
small-signal model of the microgrid for the traditional $P-\omega$ and $Q-V$
droop control. Therefore, (18) provides a more general form for the small-
signal model of a droop-controlled inverter-based microgrid. Next, we present
a specific droop control design that can enforce decoupling between angle and
voltage dynamics.

##  III Design of Generalized Droop Control to Achieve Decoupling

In this section, we discuss the design of generalized droop controls so that
decoupling can be achieved between angle and voltage dynamics. Specifically,
we want to select the parameter $\alpha_{i}$, which will enforce decoupling in
dynamics in a lossy microgrid. Recall that $\alpha_{i}$ represents a
physically meaningful angle for inverter buses; for non-inverter buses,
$\alpha_{i}$ is an arbitrary angle which merely introduces a mathematical
transformation.

We select the parameter $\alpha_{i}$ leveraging some practical facts. As
distribution lines of a typical microgrid are usually manufactured from same
constituent materials, $R/X$ ratios of the lines lie in a close range of
values. Based on this fact it is reasonable to assume that $\phi_{ik}$ of each
line $(i,k)$ is approximately equal to a fixed value $\phi_{0}$ i.e.
$\phi_{ik}\approx\phi_{0}$ for each $(i,k)\in\mathcal{E}$. (Note, impedance
angle $\beta_{ik}=\tan^{-1}(R_{ik}/X_{ik})$ is related to the admittance angle
in Y-bus matrix as: $\beta_{ik}=\pi-\phi_{ik}$.) Additionally, it is not too
restrictive to assume that the differences of bus angles across the lines at
equilibrium (i.e. $|\theta_{i}^{s}-\theta_{k}^{s}|$) are small enough. This is
a standard assumption in bulk grid which is typically true for microgrid when
the operating point of the inverters is around the rated condition. Besides,
microgrids typically have radial structures and the differences of bus angles
at equilibrium are known to be small for radial networks. Under these
assumptions here we show that it is possible to select the parameter
$\alpha_{i}$ in a way so that decoupling can be achieved. Specifically, we
show that setting $\alpha_{i}=\pi-\phi_{0}=\beta_{0}$ for each bus $i$ leads
to decoupling where $\beta_{0}$ is the impedance angle of all the lines (i.e.
$\phi_{ik}=\phi_{0}$ for all $(i,k)\in\mathcal{E}$).

First note that by substituting $\phi_{ik}=\phi_{0}$ and
$\alpha_{i}=\pi-\phi_{0}$, the edge weights
$\boldsymbol{W}_{1}(\boldsymbol{x}^{s}))$ and
$\boldsymbol{W}_{2}(\boldsymbol{x}^{s}))$ of the graphs
$\mathcal{G}_{1}(\boldsymbol{x}^{s})$ and
$\mathcal{G}_{2}(\boldsymbol{x}^{s})$ can be written as:
$\boldsymbol{W}_{1}(\boldsymbol{x}^{s}))=diag\big{(}\boldsymbol{U}^{s}\cos(\boldsymbol{E}^{T}\boldsymbol{\theta}^{s})\big{)}$
and
$\boldsymbol{W}_{2}(\boldsymbol{x}^{s}))=diag\big{(}\boldsymbol{U}^{s}\sin(\boldsymbol{E}^{T}\boldsymbol{\theta}^{s})\big{)}$.
Note that the edge weights of both $(i,k)$ and $(k,i)$ edges in
$\mathcal{G}_{1}(\boldsymbol{x}^{s})$ become equal to
$V_{i}^{s}V_{k}^{s}Y_{ik}\cos(\theta_{i}^{s}-\theta_{k}^{s})$. Hence
$\mathcal{G}_{1}(\boldsymbol{x}^{s})$ can be considered as an undirected graph
where each line between bus $i$ and $k$ corresponds to an undirected edge
$(i,k)$. For undirected graph, the Laplacian matrix is symmetric and thereby
$\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))$ can be written as
$\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))=\boldsymbol{E}_{u}\boldsymbol{W}_{u}(\boldsymbol{x}^{s})\boldsymbol{E}_{u}^{T}$
where $\boldsymbol{E}_{u}\in\mathbb{R}^{n\times l}$ is the incidence matrix
and
$\boldsymbol{W}_{u}(\boldsymbol{x}^{s})=diag(V_{i}^{s}V_{k}^{s}Y_{ik})\cos{(\boldsymbol{E}_{u}^{T}\boldsymbol{\theta}^{s})}\in\mathbb{R}^{l\times
l}$ is the weights of the edges in the undirected graph
$\mathcal{G}_{1}(\boldsymbol{x}^{s})$. Following the existing literature [7,
16], we refer this weighted undirected graph
$\mathcal{G}_{1}(\boldsymbol{x}^{s})$ as active power flow graph. Now,
applying small angle difference assumption,
$\boldsymbol{L}(\mathcal{G}_{2}(\boldsymbol{x}^{s}))$ can be approximated as
zero since
$\boldsymbol{W}_{2}(\boldsymbol{x}^{s}))=diag\big{(}\boldsymbol{U}^{s}\sin(\boldsymbol{E}^{T}\boldsymbol{\theta}^{s})\big{)}\approx\mathbf{0}$.
Furthermore, noting that $\frac{B_{ii}}{G_{ii}}=\tan(\phi_{0})$, we find
$\hat{\boldsymbol{P}}^{s}\approx\mathbf{0}$ as
$\hat{P}_{i}^{s}=(V_{i}^{s})^{2}\big{(}G_{ii}\sin(\alpha_{i})$
$+B_{ii}\cos(\alpha_{i})\big{)}$ $+\sum_{k\in
adj(i)}{V_{i}^{s}V_{k}^{s}{Y}_{ik}~{}\sin(\theta_{i}^{s}-\theta_{k}^{s})}$. By
substituting these results we can approximate eqn. (18) as eqn. (19).

From the block form of (19) it is obvious that when the above assumptions hold
and $\alpha_{i}$ in each bus is chosen as $\alpha_{i}=\pi-\phi_{0}=\beta_{0}$,
the angle and voltage dynamics become decoupled. Thus, by selecting
$\alpha_{i}$ in each generalized droop control the same as the impedance angle
of the lines, we can achieve decoupling. This result, in fact, has been
recognized for a single-bus system [4] but has not been formally analyzed for
a network consisting of multiple inverters. Thus, we have extended and
verified the decoupling-achieving design of generalized droop controls for a
network. Note, for non-inverter buses $\alpha_{i}$ is arbitrary and thus we
can free to select $\alpha_{i}=\pi-\phi_{0}=\beta_{0}$ for non-inverter buses
also in our model.

Remark: When the above assumptions (i.e. impedance angle is constant
throughout the network and bus angle differences are sufficiently small) do
not hold, then the edge weights $\boldsymbol{W}_{2}(\boldsymbol{x}^{s}))$ can
not be approximated to zero. As a consequence, we can not conclude decoupling
between the angle and voltage dynamics for the proposed selection of
$\alpha_{i}$. In that case, the norm of
$\boldsymbol{W}_{2}(\boldsymbol{x}^{s})$ provides us a measure of coupling
between angle and voltage dynamics of microgrid. However, in practice these
assumptions are not required to be satisfied strictly. Specifically, when
$R/X$ ratios of lines lie within a small range and bus angle differences are
relatively small, we can use an average of the $R/X$ ratios in the design of
generalized droop control to achieve decoupling in an approximate sense and
thereby predict stability or shape small signal response reasonably (see
simulation examples).

$\displaystyle\scriptsize\boldsymbol{J}(\boldsymbol{x}^{s})=\begin{bmatrix}\boldsymbol{0}_{n\times
n}&\boldsymbol{I}_{n\times n}&\boldsymbol{0}_{n\times n}\\\
-\boldsymbol{M}_{P}^{-1}\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))&-\boldsymbol{M}_{P}^{-1}\boldsymbol{D}_{P}&\boldsymbol{0}_{n\times
n}\\\ \boldsymbol{0}_{n\times n}&\boldsymbol{0}_{n\times
n}&-\boldsymbol{M}_{Q}^{-1}\Big{(}\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))+2~{}diag(\hat{\boldsymbol{Q}}^{s})+\boldsymbol{D}_{Q}~{}diag(\boldsymbol{V}^{s})\Big{)}(diag(\boldsymbol{V}^{s}))^{-1}\end{bmatrix}$
(19)

##  IV Small-Disturbance Analysis under decoupling-Achieving Design

The decoupling achieving droop control simplifies the analysis of the model
significantly as the analysis of angle and voltage dynamics can be undertaken
separately in an independent manner. It also enables us to shape the small
disturbance responses of a lossy microgrid effectively. To begin with our
analysis, we make the following assumptions to guarantee that the model (19)
validly represents the small signal model of a lossy microgrid for the
selected generalized droop control.

Assumptions 1:

  1. 1.

Phase angles differences (i.e. $|\theta_{i}^{s}-\theta_{k}^{s}|$) across the
lines are sufficiently small.

  2. 2.

The impedance angles (or equivalently $R/X$ ratios) of all lines are
approximately same.

  3. 3.

The parameter $\alpha_{i}$ is chosen as $\alpha_{i}=\pi-\phi_{0}=\beta_{0}$
for each inverter bus where $\beta_{0}$ is the common impedance angle of the
lines.

The above assumptions ensure that the selected droop control achieves
decoupling and (19) represents a valid small-signal model. Now we provide
separate structural analyses for the angle dynamics and voltage dynamics of
the model (19) with a particular focus on stability.

We begin by noting that from the block form of (19) it is obvious that the
small-signal angle dynamics of the microgrid is dictated by the following
state matrix:

$\displaystyle\boldsymbol{J}_{A}(\boldsymbol{x}^{s})=\begin{bmatrix}\boldsymbol{0}_{n\times
n}&\boldsymbol{I}_{n\times n}\\\
-\boldsymbol{M}_{P}^{-1}\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))&-\boldsymbol{M}_{P}^{-1}\boldsymbol{D}_{P}\end{bmatrix}$
(20)

These dynamics are identical to the small signal angle dynamics derived for
conventional $P-\omega$ droop under the decoupling assumption [8]. In [8], we
showed that when the Laplacian matrix
$\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))$ satisfies the conditions
– i.e., the matrix is positive semidefinite and has a non-repeated eigenvalue
at zero, the angle dynamics is asymptotically stable irrespective of filtering
constant or droop gain. The Laplacian is guaranteed to be positive semi-
definite and has a non-repeated eigenvalue at zero, if its off-diagonal
entries are nonnegative and the matrix is irreducible (equivalently, its
associated graph is connected and has positive edge weights). For the
Laplacian matrix in our formulation, the weight of the edge between vertices
$i$ and $k$ is given by
$V_{i}^{s}V_{k}^{s}Y_{ik}\cos{(\theta_{i}^{s}-\theta_{k}^{s})}$. This weight
is positive if the difference between the power-flow bus angle across the
corresponding line is less than $\pi/2$, or more generally
$|\theta_{i}^{0}-\theta_{k}^{0}|~{}mod~{}2\pi<\pi/2$. Since this condition is
automatically satisfied in our case due to the small angle difference
assumption, the small signal model (20) is readily asymptotically stable.
These observations are summarized in the following theorem (presented without
proof since this is an automatic consequence of the results derived in [8]).

Theorem 1: Consider the microgrid model governed by (II-A3), (II-A3), (9) and
(10). Given the conditions mentioned in Assumptions 1 are satisfied, the
small-signal angle dynamics of the microgrid (20) is asymptotically stable for
any positive droop gain and filtering time constant.

Now we we turn our attention to the analysis of voltage dynamics of small-
signal model given by (18). First note that from the block form of (18) that
the small-signal voltage dynamics of the microgrid is dictated by the
following state matrix:

$\displaystyle\small\boldsymbol{J}_{V}(\boldsymbol{x}^{s})=-\boldsymbol{M}_{Q}^{-1}\Big{(}\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))+2~{}diag(\hat{\boldsymbol{Q}}^{s})$
$\displaystyle+\boldsymbol{D}_{Q}~{}diag(\boldsymbol{V}^{s})\Big{)}(diag(\boldsymbol{V}^{s}))^{-1}$
(21)

These dynamics are also identical to the small signal voltage dynamics derived
for conventional $Q-V$ droop under the decoupling assumption [9]. Following
the same approach in [9],
$\boldsymbol{L}(\mathcal{G}_{1}(\boldsymbol{x}^{s}))+2~{}diag(\hat{\boldsymbol{Q}}^{s})$
can be seen as the Laplacian of an enhanced version of the graph
$\mathcal{G}_{1}(\boldsymbol{x}^{s})$ where each vertex has a self-loop with
weight $2\hat{Q}_{i}^{s}$ $=2(V_{i}^{s})^{2}(G_{ii}\cos(\alpha_{i})$
$-B_{ii}\sin(\alpha_{i}))$ $-2\sum_{k\in
adj(i)}{V_{i}^{s}V_{k}^{s}{Y}_{ik}~{}\cos(\theta_{i}^{s}-\theta_{k}^{s})}$.
Thereby for the purpose of structural insights here we introduce a self-looped
version of active power flow graph $\mathcal{G}_{lp}(\mathbf{x}^{s})$
$=(\mathcal{V},\mathcal{E},\mathbf{W}_{u}(\mathbf{x}^{s}),diag(2\hat{\mathbf{Q}}^{s}))$
which we call loopy active power flow graph. Therefore we can write
$\mathbf{L}(\mathcal{G}_{lp}(\mathbf{x}^{s}))=\mathbf{L}(\mathcal{G}_{1}(\mathbf{x}^{s}))+diag(2\mathbf{Q}^{s})=\boldsymbol{E}_{u}\boldsymbol{W}_{u}(\boldsymbol{x}^{s})\boldsymbol{E}_{u}^{T}+2~{}diag(\mathbf{Q}^{s})$.
Substituting this in (IV) we get:

$\small\mathbf{J}_{V}(\mathbf{x}^{s})=-\mathbf{M}_{Q}^{-1}\big{(}\mathbf{L}(\mathcal{G}_{lp}(\mathbf{x}^{s}))+\mathbf{D}_{Q}~{}diag(\mathbf{V}^{s})\big{)}(diag(\mathbf{V}^{s}))^{-1}$
(22)

Now we can readily apply the graph-theoretic stability conditions for the
small-signal model (22) from our previous study. Following Theorem 1 of [9] we
can state the following result, which gives conditions on the Laplacian of the
loopy active power flow graph under which the voltage dynamics are
asymptotically stable for any droop gains and filter time constants (presented
without proof since this is an automatic consequence of the results derived in
[9]).

Theorem 2 Consider the microgrid model governed by (II-A3), (II-A3), (9) and
(10). Given the conditions mentioned in Assumptions 1 are satisfied, the
small-signal voltage dynamics of the microgrid (22) is asymptotically stable
for any positive droop gain and filtering time constant if the Laplacian
$\mathbf{L}(\mathcal{G}_{lp}(\mathbf{x}^{s}))$ is positive definite.

We refer readers to [9] for further discussion on structural results on
voltage stability. It is important to note that the derived stability
conditions in the Theorems 1 and 2 are independent of droop and filtering
parameters. Although these conditions do not depend on the droop gain $D$ and
filtering time constant $T$, these parameters can affect other performance
metrics such as power sharing and settling time of the angle and voltage
response. Since the developed stability conditions are independent of droop
and filtering parameters, our results allow the design of these parameters at
will to achieve other desired performance goals (e.g., power sharing and
settling time) while ensuring stability.

##  V Simulation Example

Here we simulate transient response of an example inverter-based microgrid
employing generalized droop controls. For this, we use a modified IEEE 9-bus
system (adapted to capture characteristics of a typical microgrid). The
network diagram of the modified IEEE 9-bus network and model parameters for
this example are detailed in [17] due to space constraint. We note that the
modified microgrid network is made to be radial (the line between Bus 5 and
Bus 6 was removed from the IEEE 9-bus network) to make angle differences at
equilibrium small. We consider lossy lines where $R/X$ for each line follows a
normal distribution with mean $0.7$ and standard deviation $0.02$. The network
is operating at equilibrium A as detailed in [17]. The maximum angle
difference across lines at this equilibrium is obtained as $3.2265^{0}$. For
the generalized droop, we select $\alpha_{i}=\tan^{-1}(0.7)$ as suggested in
our analysis. We consider a small disturbance to the bus voltages away from
the equilibrium A and simulate the small disturbance response of the microgrid
for different filter time constants.

Figure 2: Small-disturbance angle and voltage response of modified IEEE 9-bus
lossy system with radial topology for generalized droop with $T_{1i}=0.01,0.5$
and $2$ sec in the droop controls. Figure 3: Small-disturbance angle and
voltage response of modified IEEE 9-bus lossy system with radial topology for
generalized droop with $T_{2i}=0.1,2$ and $10$ sec in the droop controls.

Since bus voltage angles are small and $\alpha_{i}=\tan^{-1}(0.7)$, Theorem 1
implies that the small-signal angle dynamics of the system is asymptotically
stable regardless of the filter time constant. We also note that the Laplacian
matrix $\mathbf{L}(\mathcal{G}_{lp})$ is positive definite. Therefore,
according to Theorem 2, the small-signal voltage dynamics of the system is
asymptotically stable regardless of the filter time constant. Figures 2 and 3
show the angle and voltage response of an inverter bus (Bus 2) and non-
inverter bus (Bus 7) for different time filter constants. We notice that for
all these time constants, both angle and voltage dynamics are asymptotically
stable, as implied by the theorems.

Now we turn our attention to the settling times of the responses. In Fig. 2,
$T_{1i}$ is varied to $0.01$s, $0.5$s and $2$s whereas $T_{2i}$ is kept fixed
to $10$s in the generalized droop controls of the inverters. Similarly, in
Fig. 3, $T_{2i}$ is varied to $0.1$s, $2$s and $10$s whereas $T_{1i}$ is kept
fixed to $0.01$s in the generalized droop controls of the inverters. In Fig.
2, we see that settling times of angle responses are increased with the
increase of filtering time constant $T_{1i}$ whereas voltage responses remain
almost the same. Similarly, in Fig. 3 we see that settling times of voltage
responses are increased with the increase of filtering time constant $T_{2i}$
whereas angle responses remain exactly the same. This example suggests that
the designed generalized droop control enforces decoupling between angle and
voltage dynamics effectively and allows us to independently shape settling
times of both angle and voltage responses via selecting appropriate filtering
time constants $T_{1i}$ and $T_{2i}$.

##  VI Conclusions

A network-level structure preserving small-signal model has been developed for
generalized droop control. It has been verified that for arbitrary but
relatively fixed $R/X$ ratios of lines decoupling can still be achieved by
appropriate selection of droop law. For the decoupling-achieving design,
several structural conditions have been presented for the asymptotic stability
of both angle and voltage regardless of the filtering and droop parameters of
controls. Simulation results have been presented that support the derived
theoretical results and show generalized droop control’s ability to shape
small disturbance responses of microgrid using our model. We stress that the
presented research represents only an initial step toward a comprehensive
stability analysis of microgrids; key next steps include analysis for more
general microgrid models with variable $R/X$ ratio, and application of the
analyses in the design of larger systems.

##  VII Appendix

$\displaystyle\tiny\begin{bmatrix}\boldsymbol{0}_{n\times
n}&\boldsymbol{I}_{n\times n}&\boldsymbol{0}_{n\times n}\\\
-\boldsymbol{M}_{P}^{-1}\boldsymbol{C}\boldsymbol{W}_{1}(\boldsymbol{x}^{s})\boldsymbol{E}^{T}&-\boldsymbol{M}_{P}^{-1}\boldsymbol{D}_{P}&-\boldsymbol{M}_{P}^{-1}\Big{(}-\boldsymbol{C}\boldsymbol{W}_{2}(\boldsymbol{x}^{s})\boldsymbol{E}^{T}+2~{}diag(\hat{\boldsymbol{G}})(diag(\boldsymbol{V}^{s}))^{2}+2~{}diag(\boldsymbol{C}\boldsymbol{U}^{s}\cos(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha}))\Big{)}(diag(\boldsymbol{V}^{s}))^{-1}\\\
-\boldsymbol{M}_{Q}^{-1}\boldsymbol{C}\boldsymbol{W}_{2}(\boldsymbol{x}^{s})\boldsymbol{E}^{T}&\boldsymbol{0}_{n\times
n}&-\boldsymbol{M}_{Q}^{-1}\Big{(}\boldsymbol{C}\boldsymbol{W}_{1}(\boldsymbol{x}^{s})\boldsymbol{E}^{T}-2~{}diag(\boldsymbol{\hat{B}})(diag(\boldsymbol{V}^{s}))^{2}+2~{}diag(\boldsymbol{C}\boldsymbol{U}^{s}\sin(\boldsymbol{E}^{T}\boldsymbol{\theta^{s}}-\boldsymbol{\phi}+\frac{\pi}{2}\boldsymbol{1}_{n}-\boldsymbol{\alpha}))+\boldsymbol{D}_{Q}~{}diag(\boldsymbol{V}^{s})\Big{)}(diag(\boldsymbol{V}^{s}))^{-1}\end{bmatrix}$

Figure 4: Diagram of the modified IEEE 9-bus lossy test system. TABLE I: Bus parameters of modified IEEE 9-bus lossy test system Bus |  $P_{G_{i}}^{0}-P_{L_{i}}^{0}$ (p.u.) |  $Q_{G_{i}}^{0}-Q_{L_{i}}^{0}$ (p.u.) | $D_{1i}$ | $D_{2i}$
---|---|---|---|---
1 |  |  | 5 | 10
2 | 0.3260 |  | 5 | 10
3 | 0.1700 |  | 5 | 10
4 | 0 | 0 | $\epsilon_{2}$ | 0
5 | -0.18 | -0.12 | $\epsilon_{2}$ | 0
6 | 0 | 0 | $\epsilon_{2}$ | 0
7 | -0.2 | -0.4 | $\epsilon_{2}$ | 0
8 | 0 | 0 | $\epsilon_{2}$ | 0
9 | -0.25 | -0.6 | $\epsilon_{2}$ | 0
TABLE II: Line parameters of modified IEEE 9-bus lossy test system Line |  $R_{i}$ (p.u.) |  $X_{i}$ (p.u.)
---|---|---
(1,4) | 0.0387 | 0.0576
(4,5) | 0.0648 | 0.0920
(3,6) | 0.0412 | 0.0586
(6,7) | 0.0703 | 0.1008
(7,8) | 0.0517 | 0.0720
(8,2) | 0.0433 | 0.0625
(8,9) | 0.1100 | 0.1610
(9,4) | 0.0600 | 0.0850
TABLE III: Voltage magnitudes and phase angles at equilibrium A Bus |  $V_{i}^{s}=V_{i}^{0}$ (p.u.) |  $\theta_{i}^{s}$ (degee)
---|---|---
1 | 1 | 0
2 | 1 | 5.1802
3 | 1 | 5.5607
4 | 0.9780 | 0.0791
5 | 0.9542 | -0.4604
6 | 0.9932 | 4.9809
7 | 0.9818 | 3.9652
8 | 0.9869 | 3.9639
9 | 0.9673 | 0.7374

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