# Equidistant Sets on Alexandrov Surfaces

Logan S. Fox  Fariborz Maseeh Dept. of Math. & Stat., Portland State
University<EMAIL_ADDRESS>and  J.J.P. Veerman  Fariborz Maseeh Dept. of Math.
& Stat., Portland State University<EMAIL_ADDRESS>
###### Abstract.

We examine properties of equidistant sets determined by nonempty disjoint
compact subsets of a compact 2-dimensional Alexandrov space (of curvature
bounded below). The work here generalizes many of the known results for
equidistant sets determined by two distinct points on a compact Riemannian
2-manifold. Notably, we find that the equidistant set is always a finite
simplicial 1-complex. These results are applied to answer an open question
concerning the Hausdorff dimension of equidistant sets in the Euclidean plane.

##  1\. Introduction

Given two nonempty sets, say $A$ and $B$, in a metric space $X$, we will
denote by $E(A,B)$ the set of points whose distance to $A$ is the same as
their distance to $B$. That is, $E(A,B)$ is the set of points _equidistant_ to
$A$ and $B$. Perhaps unsurprisingly, the structure of the set $E(A,B)$ depends
not only on qualities of the sets $A$ and $B$, but also the properties of the
ambient space $X$. The goal of this paper is to extend work done in [VB06],
[BV07], and [HPV17], where many results are given for equidistant sets
determined be pairs of points on compact Riemannian 2-manifolds. We generalize
many of these results in two directions: (i) we consider equidistant sets
determined by nonempty disjoint compact subsets; and (ii) the ambient space is
a compact Alexandrov surface.

Our work is aided in part by the observation that every point of the
equidistant set (determined by $A$ and $B$) admits at least two distinct
shortest paths to $A\cup B$ (one to $A$ and one to $B$). As such, the
equidistant set is contained in the _cut locus_ to $A\cup B$. This allows us
to make use of results concerning the cut locus to a compact set on an
Alexandrov surface – particluarly, those developed by Shiohama and Tanaka in
[ST96]. Although equidistant sets are closely related to cut loci, we find
that equidistant sets are in general much ‘nicer’ (see Corollary 4.14 and its
preceding paragraph).

This paper is organized as follows: In Section 2, we review important
properties of length spaces and Alexandrov spaces. In Section 3, we prove some
basic facts concerning equidistant sets, particularly when $X$ is a proper
length space with no branching geodesics. Section 4 combines techniques from
[ST96], [VB06], and [BV07], to characterize equidistant sets on compact
Alexandrov surfaces. Finally, in Section 5 we address the two open questions
found in [PS14] concerning equidistant sets in the Euclidean plane.

##  2\. Preliminaries

This section collects necessary definitions and results concerning the
fundamentals of Alexandrov spaces. With the exception of the doubling theorem,
the content of this section can be found in the standard survey articles
[BGP92], [Pla02], and [Shi93], as well as the text [BBI01].

###  2.1. Length Spaces

A _path_ (or _curve_) in a metric space $X$ is a homeomorphism
$\gamma:[a,b]\to X$, where $[a,b]$ is an interval of $\mathbb{R}$ (in other
words, by path or curve we mean a Jordan arc). The _length_ of any path
$\gamma$ is denoted $L(\gamma)$ and is the supremum of the distance along
finite partitions of the path:

$L(\gamma)=\sup\left\\{\sum_{k=1}^{n-1}d\big{(}\gamma(t_{k}),\gamma(t_{k+1})\big{)}:a=t_{1}<t_{2}<\cdots<t_{n}=b\right\\}.$

If the length is finite, then the path is said to be _rectifiable_.

A path $\gamma$ is a _shortest path_ if $d(\gamma(s),\gamma(t))=|s-t|$ for all
$s,t\in[a,b]$. Note that shortest paths are distance minimizing and
parametrized by arc length. A path is a _geodesic_ if it is locally a shortest
path. Typically, we take the domain of a shortest path $\gamma$ to be $[0,T]$
so that $t=d\big{(}\gamma(0),\gamma(t)\big{)}$ for all $t\in[0,T]$. However,
it can be convenient to reparametrize a path to some other domain. Therefore,
we say that a shortest path $\gamma:[c,d]\to X$ is _linearly parametrized_ if
$d(\gamma(s),\gamma(t))=\lambda|s-t|$ for all $s,t\in[c,d]$, where $\lambda$
is the constant

$\lambda=\frac{d(\gamma(c),\gamma(d))}{d-c}.$

A _length space_ is a metric space such that the distance between any pair of
points is equal to the infimum of lengths of paths connecting those points. In
an arbitrary length space, there may be no shortest path connecting a given
pair of points; however, as longs as the space is complete and locally
compact, the Hopf–Rinow theorem guarantees a shortest path always exists.

######  Theorem 2.1 (Hopf–Rinow).

If $X$ is a complete and locally compact length space, then every closed and
bounded subset of $X$ is compact, and any two points in $X$ can be connected
by a shortest path.

In addition to the Hopf–Rinow theorem, we will also make use of a well-known
consequence of the Arzelá–Ascoli theorem, which tells us that every sequence
of shortest paths in a compact length space contains a convergent subsequence.

######  Theorem 2.2 (Arzelà–Ascoli).

If $X$ is a separable metric space and $Y$ is a compact metric space, then
every uniformly equicontinuous sequence of functions $f_{n}:X\to Y$ has a
subsequence that converges uniformly to a continuous function $f:X\to Y$.

######  Corollary 2.3.

If $X$ is a compact length space and $\\{\gamma_{n}\\}_{n}$ is a sequence of
linearly parametrized shortest paths, $\gamma_{n}:[0,1]\to X$, then
$\\{\gamma_{n}\\}_{n}$ contains a subsequence which converges uniformly to a
linearly parametrized shortest path.

###  2.2. Alexandrov Spaces

We will denote by $\mathbb{M}^{2}_{k}$ the complete $2$-dimensional Riemannian
manifold of constant sectional curvature $k$. Letting $X$ be a length space
and $x,y,z\in X$ be three distinct points satisfying

$d(x,y)+d(x,z)+d(y,z)\leq
2\operatorname{diam}\mathbb{M}^{2}_{k}=\begin{cases}2\pi/\sqrt{k}&\text{ if
}k>0\\\ \infty&\text{ if }k\leq 0\end{cases}$

we can isometrically embed these points as the vertices of a geodesic triangle
in $\mathbb{M}^{2}_{k}$, which we will denote $\Delta^{k}(x,y,z)$. We then
denote by $\angle^{k}(y,z)$ the angle at $x$ in $\Delta^{k}(x,y,z)$.

Given two shortest paths emanating from the same point, say $\gamma:[0,T]\to
X$ and $\eta:[0,S]\to X$ with $\gamma(0)=\eta(0)$, the _upper angle_ between
$\gamma$ and $\eta$ is defined by fixing a $k$ and taking the limit superior
of angles $\angle^{k}\big{(}\gamma(t),\eta(s)\big{)}$ as $s$ and $t$ vanish,

$\angle^{+}(\gamma,\eta)=\limsup_{s,t\to
0^{+}}\angle^{k}\big{(}\gamma(t),\eta(s)\big{)}.$ (1)

One similarly defines the lower angle by taking the limit inferior in (1). If
the upper and lower angles are the same, then we say that the angle exists and
simply denote it $\angle(\gamma,\eta)$.

If there is a $k$ such that for any $s,t>0$ and any shortest paths $\gamma$
and $\eta$ as above, we find
$\angle^{+}(\gamma,\eta)\geq\angle^{k}\big{(}\gamma(t),\eta(s)\big{)}$, then
we say that $X$ is a space of _curvature bounded below_ (or curvature $\geq
k$).111On the other hand, if
$\angle^{+}(\gamma,\eta)\leq\angle^{k}\big{(}\gamma(t),\eta(s)\big{)}$, then
we say that $X$ is a space of curvature bounded above, but we will not
consider such a space in this article. It is well known that the bounded
curvature condition guarantees the existence of angles between shortest paths.

For the purposes of this article, we define Alexandrov spaces as follows.

######  Definition 2.4.

An _Alexandrov space_ is a complete and locally compact length space of
curvature bounded below.

In general it is not required that Alexandrov spaces are locally compact.
However, we will rely heavily on the Hopf–Rinow theorem, so we define
Alexandrov spaces accordingly. In fact, the later sections further assume that
the spaces in question are $2$-dimensional, which we will call _Alexandrov
surfaces_.

When referring to the dimension of an Alexandrov space, we generally mean the
Hausdorff dimension (although this is known to be equivalent to topological
dimension [Pla02, Theorem 156]). See [BBI01, Chapter 10] – the source of each
item of the following proposition – for a more complete discussion.

######  Proposition 2.5 ([BBI01]).

Let $X$ be an Alexandrov space.
(i) The Hausdorff dimension of $X$ is a nonnegative integer or infinity.
(ii) All open subsets of $X$ have the same Hausdorff dimension.
(iii) If $X$ is $2$-dimensional, then it is a topological $2$-manifold,
possibly with boundary.

###  2.3. No Branching Geodesics

One important consequence of the lower curvature bound condition, is that it
removes the possibility of branching geodesics. Intuitively, two geodesics
which emanate from the same point are said to _branch_ if they initially
overlap for some time interval, but then become disjoint at some point on the
interior of both paths.

######  Definition 2.6.

Let $X$ be a length space and let $\gamma,\eta:[0,1]\to X$ be linearly
parametrized shortest paths with $\gamma(0)=\eta(0)$. If there are values
$t,s\in(0,1)$ such that

$\gamma\big{(}[0,t]\big{)}=\eta\big{(}[0,s]\big{)}\quad\text{and}\quad\gamma\big{(}[t,t+\varepsilon]\big{)}\cap\eta\big{(}[s,s+\varepsilon]\big{)}=\gamma(t)$

for some $\varepsilon>0$, then $\gamma$ and $\eta$ are said to _branch_. The
point $\gamma(t)$ is the _branch point_ between $\gamma$ and $\eta$.

######  Example 2.7.

Consider a length space formed by two cones with their vertices identified.
Any pair of shortest paths which begin at the same point on one cone and end
at distinct equidistant points on the other cone will branch at the shared
vertex.

######  Lemma 2.8 ([Shi93, Lemma 2.4]).

If $X$ is an Alexandrov space, then there are no branching geodesics in $X$.

######  Corollary 2.9.

Let $X$ be an Alexandrov space. For any shortest paths $\gamma:[0,T]\to X$ and
$\eta:[0,S]\to X$, with $\gamma(0)=\eta(0)$, $\angle(\gamma,\eta)=0$ if and
only if one path is a subpath of the other.

Note that the converse of Lemma 2.8 is not necessarily true, even for compact
length spaces, as the following example illustrates.

######  Example 2.10.

Let $X$ be the surface of revolution in $\mathbb{R}^{3}$ obtained by rotation
the graph of $z=\sqrt{x}$, for $0\leq x\leq 1$, around the $z$-axis. In other
words,

$X=\left\\{(x,y,z)\in\mathbb{R}^{3}:z=\sqrt[4]{x^{2}+y^{2}}\ ,\
x^{2}+y^{2}\leq 1\right\\}.$

Since $X$ is the continuous image of a compact set, it is compact. First
equipping $X$ with the subspace metric induced by the Euclidean norm, we can
then equip $X$ with an intrinsic metric by defining $d(x,y)$ to be the infimum
of lengths of paths from $x$ to $y$. This space does not have branching
geodesics (and no geodesic contains the singular point in its interior);
however, there is no lower curvature bound, which we can see by examining a
triangle with one vertex at the singular point $(0,0,0)$.

###  2.4. The Space of Directions

Let $X$ be an Alexandrov space. For any fixed $x\in X$, let $\Gamma_{x}$ be
the set of all shortest paths emanating from $x$. We define an equivalence
relation on $\Gamma_{x}$ by

$\gamma\sim\eta\iff\angle(\gamma,\eta)=0$

and let $\Sigma_{x}$ be the set of equivalence classes of $\Gamma_{x}/\sim$.
It is straightforward to verify that the angle is a metric on $\Sigma_{x}$.
The _space of directions_ , $S_{x}X$, is then defined as the completion of
$(\Sigma_{x},\angle)$. For any shortest path $\gamma$, we will denote by
$[\gamma]$ (or simply $\gamma$ when there is no ambiguity) its equivalence
class in $S_{x}X$.

If $X$ is an $n$-dimensional Alexandrov space (with $n\geq 2$) then for every
$x\in X$, the space of directions $S_{x}X$ is itself a compact
$(n-1)$-dimensional Alexandrov space of curvature $\geq 1$ [BBI01, Theorem
10.8.6]. For $2$-dimensional Alexandrov spaces, this tells us that the space
of directions at every point is either a line segment or a circle.
Furthermore, by the radius sphere theorem [GP93],
$\operatorname{diam}S_{x}X\leq\pi$.

The _tangent cone_ (or tangent space) at $x$, which we denote $T_{x}X$, is the
Euclidean cone over $S_{x}X$:

$T_{x}X=\big{(}S_{x}X\times[0,\infty)\big{)}/\big{(}S_{x}X\times\\{0\\}\big{)}.$

We equip the tangent cone with the metric induced by the law of cosines,

$d\big{(}(\gamma,t),(\eta,s)\big{)}=\sqrt{t^{2}+s^{2}-2st\cos\big{(}\angle(\gamma,\eta)\big{)}}.$

When $X$ is an Alexandrov space of curvature $\geq k$, $T_{x}X$ is an
Alexandrov space of curvature $\geq 0$ for all $x\in X$. Furthermore, if $X$
is $n$-dimensional, then so is $T_{x}X$ (this follows from the dimension of
$S_{x}X$).

There is a notion of exponential map, $\exp_{x}:T_{x}X\to X$, defined simply
by

$\exp_{x}(\gamma,t)=\gamma(t).$

However, this assumes there is in fact a shortest path $\gamma$ at $x$ with
direction $[\gamma]$ and $L(\gamma)\geq t$. In general, there may be no
neighborhood of the origin such that $\exp_{x}$ is defined for all points in
the neighborhood (see Example 4.7). We will make limited use of the
exponential map in Section 4.

###  2.5. The Doubling Theorem

Let $X$ be an $n$-dimensional Alexandrov space with nonempty boundary
$\partial X$, and let $\phi:X\to Y$ be an isometry. We define the doubling of
$X$, which we denote $\widetilde{X}$, as the gluing of $X$ with itself (i.e.
with $Y$) along the boundary,

$\widetilde{X}=X\cup_{\phi|_{\partial X}}Y.$

After identifying $\partial X$ with $\phi(\partial X)$, we can equip the
doubled space with the metric

$\tilde{d}(x,y)=\begin{cases}d_{X}(x,y)&\text{ if }x,y\in X\\\
d_{Y}(x,y)&\text{ if }x,y\in Y\\\ \inf\\{d(x,z)+d(y,z):z\in\partial
X\\}&\text{ if }x\in X\text{ and }y\in Y.\end{cases}$

It is straightforward to see that $\widetilde{X}$ is a length space, but it is
in fact also an $n$-dimensional Alexandrov space of curvature $\geq k$.

######  Theorem 2.11 (Doubling Theorem [Per91]).

Given an $n$-dimensional Alexandrov space $X$ of curvature $\geq k$ (for some
$k\in\mathbb{R}$) with nonempty boundary, the doubled space $\widetilde{X}$ is
an $n$-dimensional Alexandrov space of curvature $\geq k$ with empty boundary.

Note that Perelman’s proof of the doubling theorem was never formally
published.222Anton Petrunin kindly keeps a copy of Perelman’s preprint
available on his website: https://anton-petrunin.github.io/papers/ However, it
is a direct consequence of the gluing theorem [Pet97], which generalizes the
doubling theorem to any two Alexandrov spaces with isometric boundaries.

##  3\. Equidistant Sets

######  Definition 3.1.

Let $(X,d)$ be a metric space. Given two nonempty sets $A,B\subseteq X$, the
_equidistant set_ (also _mediatrix_ or _midset_) determined by $A$ and $B$ is
the set of points of equal distance to $A$ and $B$;

$E(A,B)=\\{x\in X:d(x,A)=d(x,B)\\}.$

In the case of singleton sets, say $\\{a\\}$ and $\\{b\\}$, we use
$E(a,b)=E(\\{a\\},\\{b\\})$ for simplicity of notation.

Although equidistant sets are defined for any metric space, properties of
equidistant sets can vary greatly depending on the metric.

######  Example 3.2.

Consider the real line with the following metrics

$d_{1}(x,y)=\frac{|x-y|}{1+|x-y|}\quad\text{and}\quad
d_{2}(x,y)=\begin{cases}|x-y|,&\text{ if }|x-y|<1\\\ 1,&\text{
otherwise.}\end{cases}$

The metrics $d_{1}$ and $d_{2}$ are strongly equivalent with $d_{1}(x,y)\leq
d_{2}(x,y)\leq 2d_{1}(x,y)$. It is easy to verify that for any
$p,q\in(\mathbb{R},d_{1})$, we have $E(p,q)=\\{\frac{p+q}{2}\\}$. However, in
$(\mathbb{R},d_{2})$, if $p=2$ and $q=-2$,

$E(p,q)=(-\infty,-3]\cup[-1,1]\cup[3,\infty).$

It should be further observed that one can easily construct a metric space for
which the equidistant set determined by two points is actually empty. However,
this is easily avoided by assuming the space is path connected (which length
spaces always are).

######  Lemma 3.3.

Let $X$ be a path connected metric space. If $A$ and $B$ are nonempty subsets
of $X$, then $E(A,B)$ is nonempty.

###### Proof.

Define the function $f_{AB}:X\to\mathbb{R}$ by

$f_{AB}(x)=d(x,A)-d(x,B)$

and notice that $E(A,B)=f^{-1}_{AB}(\\{0\\})$. Fixing some $a\in A$ and $b\in
B$ and letting $\sigma:[0,1]\to X$ be a path from $a$ to $b$, the intermediate
value theorem tells us that $f_{AB}(\sigma(t))=0$ for some $t\in[0,1]$. ∎

In order to ensure that our equidistant sets are ‘well behaved’ in some sense,
we will follow [VB06] and assume the ambient metric space is always a proper
length space with no branching geodesics. In particular, this includes
Alexandrov spaces.

######  Definition 3.4.

Given a point $x$ and a set $A$, the _metric projection_ $P_{A}(x)$ is the set
of points of $A$ which realize the distance $d(x,A)$,

$P_{A}(x)=\\{a\in A:d(x,a)=d(x,A)\\}.$

Whenever we say that $\gamma:[0,T]\to X$ is a shortest path from a point $x$
to a set $A$, we mean that $\gamma(0)=x$ and $\gamma(T)\in P_{A}(x)$, assuming
$P_{A}(x)$ is nonempty.

The following proposition, and its corollaries, extend properties of
equidistant sets determined by distinct points found in [VB06, §2] to
equidistant sets determined by disjoint closed sets.

######  Proposition 3.5.

Let $X$ be a proper length space with no branching geodesics, $A$ and $B$ be
nonempty disjoint closed subsets of $X$, and $x\in E(A,B)$ be given.
(i) If $\gamma:[0,T]\to X$ is a shortest path from $x$ to $A$, then
$d(\gamma(t),A)<d(\gamma(t),B)$ for all $t\in(0,T]$.
(ii) There is a path connecting some $a\in A$ to some $b\in B$ which does not
intersect $E(A,B)\setminus\\{x\\}$.

###### Proof.

For (i): Let $x\in E(A,B)$ and $\gamma$ with $\gamma(T)=a\in P_{A}(x)$ be
given. For the sake of contradiction, suppose that there is a $t_{0}\in(0,T)$
such that $d(\gamma(t_{0}),B)\leq d(\gamma(t_{0}),A)$. Fix $b_{0}\in
P_{B}(\gamma(t_{0}))$. Since $d(x,a)\leq d(x,b_{0})$ and
$d(\gamma(t_{0}),b_{0})\leq d(\gamma(t_{0}),a)$,

$\displaystyle d(x,\gamma(t_{0}))+d(\gamma(t_{0}),b_{0})$
$\displaystyle=d(x,a)-d(\gamma(t_{0}),a)+d(\gamma(t_{0}),b_{0})$
$\displaystyle\leq d(x,b_{0})-d(\gamma(t_{0}),a)+d(\gamma(t_{0}),a)$
$\displaystyle=d(x,b_{0})$

so $d(x,b_{0})=d(x,\gamma(t_{0}))+d(\gamma(t_{0}),b_{0})$. Given that $a\neq
b_{0}$, this implies that $\gamma$ is branching, which contradicts our
hypothesis. Thus, $d(\gamma(t),A)<d(\gamma(t),B)$ for all $t\in(0,T]$.

For (ii): Given any $x\in E(A,B)$, if $\gamma$ if a shortest path from $a\in
P_{A}(x)$ to $x$, and $\eta$ is a shortest path from $x$ to $b\in P_{B}(x)$,
then by part (i), the concatenation of $\gamma$ and $\eta$ connects $a$ to $b$
without intersecting $E(A,B)\setminus\\{x\\}$. ∎

######  Corollary 3.6.

If $X$ is a proper length space with no branching geodesics, then for any
nonempty disjoint closed sets $A,B\subseteq X$, the equidistant set $E(A,B)$
has empty interior.

###### Proof.

Let $x\in E(A,B)$ be given. By Proposition 3.5, an open ball of any radius
centered at $x$ cannot be contained in $E(A,B)$ since any shortest path from
$x$ to $A$ (or $B$) immediately leaves $E(A,B)$. ∎

Our next corollary is related to idea of _minimal separating_ introduced in
the study of Brillouin spaces and equidistant sets determined by points
[VPRS00]. In our case, $E(A,B)$ separates $X$ into two sets

$\\{x\in X:d(x,A)<d(x,B)\\}\quad\text{and}\quad\\{x\in X:d(x,B)<d(x,A)\\}.$

but each of these sets may consist of more than one component of $X\setminus
E(A,B)$. However, if we additionally assume that $A$ and $B$ are connected,
then this separation is minimal.

######  Definition 3.7.

Let $X$ be a connected metric space. A set $E\subseteq X$ is _separating_ if
$X\setminus E$ consists of more than one component. If $E$ is separating, but
no proper subset of $E$ is separating, then $E$ is _minimal separating_.

######  Corollary 3.8.

Let $X$ be a proper length space with no branching geodesics. If $A,B\subseteq
X$ are nonempty disjoint closed and connected, then $E(A,B)$ is minimal
separating.

###### Proof.

For simplicity of notation, define the sets

$X_{A}=\\{x\in X:d(x,A)<d(x,B)\\}\quad\text{and}\quad X_{B}=\\{x\in
X:d(x,B)<d(x,A)\\}$

so that $X$ is the union of the disjoint sets $X_{A}$, $X_{B}$, and $E(A,B)$.

First, we will show that $X_{A}$ is in fact a single component of $X\setminus
E(A,B)$. Certainly $A\subseteq X_{A}$, and given that $A$ is connected, $A$
must lie in one component of $X_{A}$. Letting $x_{0}$ be any element of
$X_{A}$ and $\gamma:[0,T]\to X$ be a shortest path from $x_{0}$ to $A$, the
same reasoning as Proposition 3.5(i) shows that the image of $\gamma$ is
contained in $X_{A}$. In particular, $x_{0}$ is in the same component as $A$.
Therefore, $X_{A}$ (and subsequently, $X_{B}$) is a single component.

Applying Proposition 3.5(ii), we see that removing any point from $E(A,B)$
allows for a path from $X_{A}$ to $X_{B}$. Thus, $E(A,B)$ is minimal
separating. ∎

###  3.1. Directions to the Equidistant Set

Given an Alexandrov space $X$, a compact set $A\subseteq X$, and a point $x\in
X\setminus A$, we define $\Theta_{A}$ to be the set of directions of shortest
paths from $x$ to $A$:

$\Theta_{A}=\\{[\gamma]\in S_{x}X:\gamma\text{ is a shortest path from
}x\text{ to }A\\}.$

######  Lemma 3.9.

Let $X$ be an Alexandrov space and let $A$ and $B$ be disjoint compact subsets
of $X$. For any $x\in E(A,B)$, the sets $\Theta_{A}$ and $\Theta_{B}$ are
disjoint compact subsets of $S_{x}X$.

###### Proof.

First, we verify that $\Theta_{A}$ and $\Theta_{B}$ are disjoint. By way of
contradiction, suppose that there are paths $\gamma_{a}$ and $\gamma_{b}$ from
$x$ to $a\in P_{A}(x)$ and $b\in P_{B}(x)$, respectively, such that
$\angle(\gamma_{a},\gamma_{b})=0$. By Corollary 2.9, one of these paths is a
subpath of the other. But since they have the same length, we must have
$\gamma_{a}=\gamma_{b}$, which is a contradiction since $A$ and $B$ are
disjoint.

To show that $\Theta_{A}$ (and subsequently, $\Theta_{B}$) is compact, let
$\\{[\gamma_{n}]\\}_{n}$ be a sequence in $\Theta_{A}$. For each $n$ let
$\gamma_{n}:[0,d(x,A)]\to X$ be a shortest path in the equivalence class
$[\gamma_{n}]$. Since $L(\gamma_{n})=d(x,A)$ for each $n$, by Corollary 2.3
and the compactness of $A$, $\\{\gamma_{n}\\}_{n}$ contains a subsequence
which converges uniformly to a shortest path $\gamma:[0,d(x,A)]\to X$ which
connects $x$ to $A$. Therefore, $\\{[\gamma_{n}]\\}_{n}$ contains a
subsequence which converges to $[\gamma]\in\Theta_{A}$. Thus, $\Theta_{A}$ is
compact. ∎

###  3.2. Connections to the Cut Locus

We close this section with a quick detour to establish some connections
between equidistant sets and the cut locus. See [ST96, §§1,2] for further
discussion on the cut locus for Alexandrov surfaces.

######  Definition 3.10.

Let $X$ be an Alexandrov space and $K\subseteq X$ be compact. A point $x\in X$
is a _cut point_ to $K$ if there is a shortest path $\gamma$ from $x$ to $K$
such that $\gamma$ is not properly contained in any other shortest path to
$K$. The _cut locus_ to $K$, denoted $C(K)$, is the set of all cut points to
$K$.

Let $A$ and $B$ are disjoint nonempty compact subsets of an Alexandrov space.
Since there are no branching geodesics, and each $x\in E(A,B)$ admits at least
two shortest paths to $A\cup B$, we immediately see that

$E(A,B)\subseteq C(A\cup B)$

(this was similarly noted in [Zam04, p. 378]). This observation is beneficial
as it allows us to apply much of the preliminary work done in [ST96] – which
was originally developed for the cut locus – to equidistant sets (in
particular, see Lemmas 4.3 and 4.4).

##  4\. Equidistant Sets on Alexandrov Surfaces

We will now exclusively assume that $X$ is a compact $2$-dimensional
Alexandrov space, and $A$ and $B$ are nonempty disjoint closed subsets of $X$.
We additionally assume that $X$ is without boundary, although the doubling
theorem will allow us to extend Theorem 4.13 to spaces with boundary.

The discussion in this section expands upon [VB06, §4], [BV07, §1], and
[HPV17, §3], as we consider Alexandrov spaces instead of Riemannian manifolds,
and disjoint nonempty compact sets instead of distinct points.

###  4.1. Sectors and Wedges

The following definition gives a construction of ‘wedge-shaped’ neighborhoods
which we will use to describe the local behavior of $E(A,B)$. We assume the
radius $\rho$ always satisfies $\rho<\inf\\{d(a,b):a\in A,b\in B\\}$, as well
as being sufficiently small so that the open ball $B_{\rho}(x)$ is
homeomorphic to a disk, and the boundary

$\partial B_{\rho}(x)=\\{y\in X:d(x,y)=\rho\\}$

is homeomorphic to a circle. Given that $A$ and $B$ are disjoint and compact,
and $X$ is a $2$-manifold (Proposition 2.5(iii)), such a $\rho$ always exists.

######  Definition 4.1.

Let $X$ be an Alexandrov surface and $K\subseteq X$ be nonempty and compact. A
_sector_ of radius $\rho$ at $x\in C(K)$ is a component of

$B_{\rho}(x)\setminus\\{\gamma(t):[\gamma]\in\Theta_{K},t\in[0,\rho)\\}.$

Now let $A$ and $B$ be disjoint nonempty compact sets. We define a _wedge_ of
radius $\rho$ at $x\in E(A,B)\subseteq C(A\cup B)$, which we will denote
$W_{\rho}(x)$, to be any sector at $x$ such that one side is bounded by a
shortest path to $A$, and the other is bounded by a shortest path to $B$.

See Figure 1 for clarification of sectors and wedges. Note that wedges and
sectors are open sets. Furthermore, every point of the equidistant set admits
at least two wedges (and always an even number), but only finitely many (see
Lemma 4.2).

Figure 1. A point $x\in E(A,B)$ which admits five sectors, but only two
wedges.

######  Lemma 4.2.

Let $X$ be a compact Alexandrov surface and let $A,B\subseteq X$ be disjoint
closed sets. For any $x\in E(A,B)$, there are only finitely many wedges at
$x$.

###### Proof.

By way of contradiction, suppose there are infinitely many wedges at $x\in
E(A,B)$. Then there must be infinitely many distinct directions
$\\{\theta_{A,n}\\}_{n=1}^{\infty}\subseteq\Theta_{A}$ and
$\\{\theta_{B,n}\\}_{n=1}^{\infty}\subseteq\Theta_{B}$ which form the wedges.
Furthermore, we can assume that these sequences are alternating in the sense
that $\theta_{B,n}$ is always between $\theta_{A,n}$ and $\theta_{A,n+1}$
(i.e., if we start from $\theta_{A,1}$ we can move along a geodesic in
$S_{x}X$ which passes through $\theta_{B,1}$, then $\theta_{A,2}$, then
$\theta_{B,2}$, and so on). Since $\Theta_{A}$ is compact, there is a
convergent subsequence

$\theta_{A,n_{k}}\to\bar{\theta}\in\Theta_{A}.$

The subsequence $\\{\theta_{B,n_{k}}\\}_{k}$ of directions between
$\theta_{A,n_{k}}$ and $\theta_{A,n_{k}+1}$ also converges to $\bar{\theta}$.
Thus, $\bar{\theta}$ is the direction of a shortest path to both $A$ and $B$,
which contradicts Lemma 3.9. ∎

Given that $E(A,B)\subseteq C(A\cup B)$, and a wedge at $x\in E(A,B)$ is
simply a special case of a sector at $x\in C(A\cup B)$, one can use the exact
same proofs found in [ST96] for the following lemmas to arrive at Corollary
4.5.

######  Lemma 4.3 ([ST96, Basic Lemma]).

Let $R_{\rho}(x)$ be a sector at $x\in C(K)$. Then there exists a point $y\in
C(K)\cap R_{\rho}(x)$ and a Jordan arc $\sigma:[0,1]\to
C(K)\cap\overline{R_{\rho}(x)}$ such that $\sigma(0)=x$ and $\sigma(1)=y$.

######  Lemma 4.4 ([ST96, Lemma 2.3]).

Every Jordan arc $\sigma:[0,1]\to C(K)$ constructed in Lemma 4.3 is
rectifiable.

######  Corollary 4.5.

Let $W_{\rho}(x)$ be a wedge at $x\in E(A,B)$. Then there exists a point $y\in
E(A,B)\cap W_{\rho}(x)$ and a rectifiable Jordan arc $\sigma:[0,1]\to
E(A,B)\cap\overline{W_{\rho}(x)}$ such that $\sigma(0)=x$ and $\sigma(1)=y$.

###  4.2. The Direction of the Equidistant Set

Although the equidistant set is not generally a smooth curve, given any point
in the equidistant set, we can at least predict the directions from the
starting point which stay in the equidistant set. These directions are
precisely the bisectors of any wedge at that point (Theorem 4.10). To see why
this is true, first recall our function $f_{AB}$ defined by

$f_{AB}(x)=d(x,A)-d(x,B).$

Since $f_{AB}$ is a linear combination of distance functions (to compact
sets), it admits a one-sided directional derivative (see [FOV21] for a
detailed exposition of this derivative).

######  Theorem 4.6.

Let $X$ be an Alexandrov space, $\gamma:[0,T]\to X$ a shortest path, and $K$ a
compact set not containing $\gamma(0)$. Then

$\lim_{t\to 0}\frac{d(\gamma(t),K)-d(\gamma(0),K)}{t}=-\cos(\angle_{\min})$

where $\angle_{\min}$ is the infimum of angles between $\gamma$ and any
shortest path connecting $\gamma(0)$ to $K$.

Letting $x$ be a point of $E(A,B)$, and $W_{\rho}(x)$ be a wedge at $x$, if
$\gamma$ is a shortest path emanating from $x$, Theorem 4.6 gives us

$\lim_{t\to
0^{+}}\frac{f_{AB}(\gamma(t))-f_{AB}(\gamma(0))}{t}=-\cos\big{(}\angle(\gamma,\Theta_{A})\big{)}+\cos\big{(}\angle(\gamma,\Theta_{B})\big{)}.$

Since $E(A,B)$ is the level set $f_{AB}=0$, the directional derivative in the
equidistant set should be $0$. Therefore, we expect the equidistant set to be
locally well approximated by shortest paths satisfying

$\cos\big{(}\angle(\gamma,\Theta_{A})\big{)}=\cos\big{(}\angle(\gamma,\Theta_{B})\big{)}.$

Now let $\gamma_{A}$ and $\gamma_{B}$ be the shortest paths to $A$ and $B$,
respectively, which bound our wedge $W_{\rho}(x)$. Given that there are no
other shortest paths from $x$ to $A$ or $B$ in the wedge,

$\angle(\gamma,\Theta_{A})=\angle(\gamma,\Theta_{B})\iff\angle(\gamma,\gamma_{A})=\angle(\gamma,\gamma_{B}).$

Thus, we expect the direction of the equidistant set to be exactly halfway
between the directions which determine the wedge. Note however, there may not
be an actual shortest path which realizes the desired direction, as the
following example shows.

######  Example 4.7.

Let $C$ be the compact Alexandrov surface obtained by doubling the closed unit
disk. If $a$ is the point at the center of one disk and $b$ is the point at
the center of the other disk, then $E(a,b)$ is precisely the identified
boundary of the disks. At any point of $E(a,b)$, the space of directions is a
unit circle, but there are no shortest paths which realize the two directions
along the identified boundary.

Although there may not be a shortest path which admits the desired direction,
we can always achieve the desired direction as a limit of shortest paths.
Given a sequence $\\{x_{n}\\}_{n}\subseteq E(A,B)\cap W_{\rho}(x)$ such that
$x_{n}\to x$, if $\gamma_{n}$ is the shortest path from $x$ to $x_{n}$, we
expect to find

$\lim_{n\to\infty}\Bigg{(}\lim_{t\to
0^{+}}\frac{f_{AB}\big{(}\gamma_{n}(t)\big{)}-f_{AB}(x)}{t}\Bigg{)}=0$

since $f_{AB}(x_{n})=f_{AB}(x)=0$. This idea is formalized by the Theorem
4.10, but first we establish a preliminary lemma.

######  Lemma 4.8.

Let $x\in E(A,B)$ be a wedge at $x$, and let $W_{\rho}(x)$ be a wedge at $x$
bounded by shortest paths $\gamma_{A}$ and $\gamma_{B}$ to $A$ and $B$
respectively. If there exists a shortest path
$\eta:[0,T]\to\overline{W_{\rho}(x)}$ with $\eta(0)=x$ such that the image of
$\eta$ intersects $E(A,B)$ in an infinite sequence of distinct points
converging to $x$, then

$\angle(\eta,\gamma_{A})=\angle(\eta,\gamma_{B})=\begin{cases}\frac{1}{2}\angle(\gamma_{A},\gamma_{B})&\text{
if }\angle(\eta,\gamma_{A})\leq\frac{1}{2}\operatorname{diam}S_{x}X\\\
\operatorname{diam}S_{x}X-\frac{1}{2}\angle(\gamma_{A},\gamma_{B})&\text{ if
}\angle(\eta,\gamma_{A})>\frac{1}{2}\operatorname{diam}S_{x}X.\end{cases}$

######  Remark.

When $\operatorname{diam}S_{x}X=\pi$, the angle formula matches our Euclidean
intuition.

###### Proof.

Let $\\{t_{n}\\}_{n=1}^{\infty}$ be the sequence such that $\eta(t_{n})\in
E(A,B)$ for all $n$ and $\eta(t_{n})\to x$ (clearly $t_{n}\to 0$). By Theorem
4.6,

$\displaystyle 0$
$\displaystyle=\lim_{n\to\infty}\frac{f_{AB}(\eta(t_{n}))-f_{AB}(x)}{t_{n}}$
$\displaystyle=\lim_{n\to\infty}\frac{d(\eta(t_{n}),A)-d(x,A)}{t_{n}}-\lim_{n\to\infty}\frac{d(\eta(t_{n}),B)-d(x,B)}{t_{n}}$
$\displaystyle=\lim_{t\to 0^{+}}\frac{d(\eta(t),A)-d(\eta(0),A)}{t}-\lim_{t\to
0^{+}}\frac{d(\eta(t),B)-d(\eta(0),B)}{t}$
$\displaystyle=-\cos\big{(}\angle(\eta,\gamma_{A})\big{)}+\cos\big{(}\angle(\eta,\gamma_{B})\big{)}.$

Thus, $\angle(\eta,\gamma_{A})=\angle(\eta,\gamma_{B})$, so $[\eta]$ is one of
(up to) two directions in $S_{x}X$ equidistant to $[\gamma_{A}]$ and
$[\gamma_{B}]$. If the wedge is acute, then
$\angle(\eta,\gamma_{A})=\frac{1}{2}\angle(\gamma_{A},\gamma_{B})$.
Conversely, if the wedge is obtuse, then
$\angle(\eta,\gamma_{A})=\operatorname{diam}S_{x}X-\frac{1}{2}\angle(\gamma_{A},\gamma_{B})$.
∎

######  Definition 4.9.

Let $\sigma:[0,T]\to X$ be a path in an Alexandrov space. If for every
sequence $t_{n}\to t^{+}$, the sequence of directions
$\\{\gamma_{n}\\}_{n}\subseteq S_{x}X$ from $\sigma(t)$ to $\sigma(t_{n})$
converges, then we call the limiting direction the _right tangent_ at
$\sigma(t)$ and denote it $\sigma^{+}(t)$.

The following theorem shows that the the equidistant set has a (one-sided)
tangent direction and it is the bisector of the directions which determine the
wedge. Compare to [HPV17, Theorem A] or [ST96, Lemma 2.1].

######  Theorem 4.10.

Let $W_{\rho}(x)$ is a wedge at $x\in E(A,B)$, bounded by shortest paths
$\gamma_{A}$ and $\gamma_{B}$, to $A$ and $B$ respectively. For any curve

$\sigma:[0,1]\to E(A,B)\cap\overline{W_{\rho}(x)}$

such that $\sigma(0)=x$, the right tangent $\sigma^{+}(0)$ exists and is the
bisector of the directions $[\gamma_{A}]$ and $[\gamma_{B}]$.

###### Proof.

Let $\theta^{*}\in S_{x}X$ be the direction which is the desired bisector of
$[\gamma_{A}]$ and $[\gamma_{B}]$, the directions determining our wedge. For
any given $\varepsilon>0$, let $B_{\varepsilon}(\theta^{*})$ be the open
interval of radius $\varepsilon$ around $\theta^{*}\in S_{x}X$, and define the
set

$C_{\pm\varepsilon}=\exp_{x}\Big{(}B_{\varepsilon}(\theta^{*})\times(0,\delta)\Big{)}$

for some fixed $\delta>0$. In order to prove the theorem, we will show that
$\sigma(s)\in C_{\pm\varepsilon}$ for all sufficiently small $s>0$.

Seeking a contradiction, we suppose the alternative possibilities:
(i) there is a constant $c>0$ such that $\sigma(s)$ is outside of
$C_{\pm\varepsilon}$ whenever $0<s<c$; or
(ii) there is a sequence $\\{s_{n}\\}_{n=1}^{\infty}$ with $s_{n}\to 0^{+}$
such that infinitely many $\sigma(s_{n})$ are in $C_{\pm\varepsilon}$ and
infinitely many $\sigma(s_{n})$ are outside of $C_{\pm\varepsilon}$.

If case (i), then without loss of generality, we can assume that
$d(z,A)>d(z,B)$ for all $z$ in $C_{\pm\varepsilon}$. Letting $\gamma$ be a
shortest path with $[\gamma]\in B_{\varepsilon}(\theta^{*})$ and
$\angle(\gamma,\gamma_{A})<\angle(\gamma,\gamma_{B})$, we have

$\displaystyle 0$ $\displaystyle=\lim_{t\to
0^{+}}\frac{f_{AB}(\sigma(t))-f_{AB}(x)}{d(\sigma(t),x)}$
$\displaystyle\leq\lim_{t\to 0^{+}}\frac{f_{AB}(\gamma(t))-f_{AB}(x)}{t}$
$\displaystyle=-\cos\big{(}\angle(\gamma,\gamma_{A})\big{)}+\cos\big{(}\angle(\gamma,\gamma_{B})\big{)}$
$\displaystyle<0$

which is a contradiction.

If case (ii), then we can find a shortest path $\gamma$ with
$0<\angle(\gamma,\theta^{*})\leq\varepsilon$ such that $\sigma$ intersects
$\gamma$ infinitely many times in a sequence which converges to $x$. However,
this contradicts Lemma 4.8.

We conclude that $\sigma(s)\in C_{\pm\varepsilon}$ for all $s$ sufficiently
small. Since $\varepsilon$ can be taken arbitrarily close to $0$, we find
$\sigma^{+}(0)$ exists and is the bisector direction $\theta^{*}$. ∎

###  4.3. The Equidistant Set is a Simplicial 1-Complex

For sufficiently small radius, a wedge will only contain a single Jordan arc
segment of the equidistant set.

######  Lemma 4.11.

Let $x\in E(A,B)$ be given and let $W_{\rho}(x)$ be a wedge at $x$. There is a
sufficiently small $\delta$ such that the intersection of $E(A,B)$ and
$\overline{W_{\varepsilon}(x)}$ is a Jordan arc for all
$0<\varepsilon<\delta$.

###### Proof.

By Corollary 4.5, there is a Jordan arc $\sigma:[0,1]\to
E(A,B)\cap\overline{W_{\rho}(x)}$ with $\sigma(0)=x$. We assume that $\rho$ is
taken sufficiently small so that $d(\sigma(0),\sigma(1))=\rho$. By way of
contradiction, suppose that $W_{1/n}(x)$ contains a point $x_{n}\in
E(A,B)\setminus\sigma([0,1])$ for all $n$. Passing to a subsequence if
necessary, we can assume that $\\{x_{n}\\}_{n\geq 1/\rho}$ is contained in one
of the two components of $W_{\rho}(x)\setminus\sigma([0,1])$, so we will
assume it is in the component whose boundary contains $\gamma_{A}$, a shortest
path from $x$ to $A$. For each $n$, let $\eta_{n}$ be a shortest path from
$x_{n}$ to $B$. Since $x_{n}\to x$, we must have a subsequence of
$\\{\eta_{n}\\}$ which converges uniformly to a shortest path from $x$ to $B$.
However, this is impossible since $\eta_{n}$ cannot cross $\gamma_{A}$ or
$\sigma$, and $W_{1/n}(x)$ contains no shortest paths from $x$ to $B$. ∎

######  Corollary 4.12.

There are only finitely many $x\in E(A,B)$ which admit more than two wedges.

###### Proof.

If there was an infinite sequence of distinct points
$\\{x_{n}\\}_{n=1}^{\infty}\subseteq E(A,B)$ which admit more than two wedges,
then by the compactness of $E(A,B)$, there is a subsequence which converges to
some $\bar{x}\in E(A,B)$. By Lemma 4.2 there are only finitely many wedges at
$\bar{x}$, so there is a wedge $W_{\rho}(\bar{x})$ containing infinitely many
of these points for any $\rho>0$, which contradicts Lemma 4.11. ∎

######  Theorem 4.13.

Let $X$ be a compact Alexandrov surface (possibly with boundary). For any pair
of disjoint nonempty compact subsets $A,B\subseteq X$, the equidistant set
$E(A,B)$ is homeomorphic to a finite closed simplicial 1-complex.

###### Proof.

First, suppose that $X$ has no boundary. At each $x\in E(A,B)$ which admits
more than two wedges (of which there are finitely many), we place a vertex at
$x$. By compactness of $E(A,B)$ and Lemma 4.11, there is a natural
concatenation of finitely many Jordan arcs, which forms an edge between
appropriate vertices. In the case that a point $x$ belongs to a component of
$E(A,B)$ that is a simple closed curve (so there are no points with more than
two wedges) we simply place a vertex at $x$ and then create our edge of Jordan
arcs as before. Thus, $E(A,B)$ is homeomorphic to a finite simplicial
1-complex.

In the case that $X$ has nonempty boundary, we apply the doubling theorem to
get a space without boundary. By our discussion in the previous paragraph,
$E(A,B)$ is a finite simplicial 1-complex in this space. By compactness, the
boundary $\partial X$ can only intersect $E(A,B)$ finitely many times. To see
this, first note that $\partial X\cap E(A,B)$ cannot contain a Jordan arc: If
there is a wedge which is bisected by the boundary (Theorem 4.10), then the
directions that determine the wedge, $\gamma_{A}$ and $\gamma_{B}$, must
terminate at the same point, which is a contradiction. Therefore, the
intersection happens at Jordan arcs in $E(A,B)$ passing through the boundary.
If we assume there are infinitely many such intersections, then there must be
a cluster point, but there can be no wedge at such a cluster point. Thus,
removing the part of $E(A,B)$ contained in the ‘doubled’ part of the space and
placing vertices at the finitely many points in $\partial X$ leaves us with a
finite simplicial 1-complex. ∎

We can conclude from [ST96, Theorem A] that Hausdorff dimension of equidistant
sets (with the conditions of the above theorem) is always 1. However, it is
possible to produce a cut locus with infinite 1-dimensional measure on a
compact surface (see [Ito96, Theorem B] or [ST96, Example 4]). Equidistant
sets on compact surfaces, on the other hand, will always have finite measure.

######  Corollary 4.14.

Let $X$ be a compact Alexandrov surface. If $A,B\subseteq X$ are nonempty
disjoint and compact, then the $1$-dimensional Hausdorff measure of $E(A,B)$
is nonzero and finite.

###### Proof.

Let $\\{W_{i}\\}_{i\in I}$ be an open cover of $E(A,B)$ by wedges which admit
a unique rectifiable Jordan arc in their interior (see Lemma 4.11).
Compactness of $E(A,B)$ implies we can find a finite subcover
$\\{W_{k}\\}_{k=1}^{n}$. Let $J_{k}$ be the unique rectifiable segment of
$E(A,B)$ contained in the closure of each wedge $W_{k}$ of this finite
subcover. Then we have

$0<\mathscr{H}^{1}(J_{1})\leq\mathscr{H}^{1}(E(A,B))\leq\sum_{k=1}^{n}\mathscr{H}^{1}(J_{k})<\infty.\qed$

###  4.4. A Bound on the Homology of the Equidistant Set

Now that we know the equidistant set is a $1$-complex, it is appropriate to
ask if we can provide any further details on the structure. While the
specifics of the equidistant set will depend greatly on the structure and
placement of the sets $A$ and $B$, we can at least provide an upper bound on
the dimension of the first homology group (Theorem 4.15). This bound depends
on both the genus of the surface as well as the number of disconnected pieces
of $A$ and $B$.

The following theorem is a generalization of [VB06, Theorem 4.2] for $n=2$.

######  Theorem 4.15.

Let $X$ be a compact Alexandrov surface (without boundary). If $A,B\subseteq
X$ are nonempty disjoint and compact, then $H_{1}(E(A,B))=\mathbb{Z}^{k}$ for
some positive integer $k$ satisfying

$1\leq k\leq\dim H_{1}(X;\mathbb{Z}_{2})+\dim H_{0}(A)+\dim H_{0}(B)-1.$

######  Remark.

Note that the $k$ in the statement of Theorem 4.15 is in fact finite, even
though $\dim H_{0}(A)+\dim H_{0}(B)$ could certainly be infinite.

###### Proof.

For simplicity of notation, let $E$ be the equidistant set $E(A,B)$. Since the
equidistant set is a simplicial $1$-complex (Theorem 4.13), $\dim
H_{1}(E)=\dim H_{1}(E;\mathbb{Z}_{2})$, so we will use $\mathbb{Z}_{2}$
coefficients to avoid issues of orientability. Consider the following portion
of the long exact sequence for the pair $(X,E)$:

${\cdots}$${H_{2}(E;\mathbb{Z}_{2})}$${H_{2}(X;\mathbb{Z}_{2})}$${H_{2}(X,E;\mathbb{Z}_{2})}$${H_{1}(E;\mathbb{Z}_{2})}$${H_{1}(X;\mathbb{Z}_{2})}$${\cdots.}$$\scriptstyle{\partial_{3}}$$\scriptstyle{i_{2}}$$\scriptstyle{p_{2}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{p_{1}}$

First, we observe that $E$ has empty interior (Corollary 3.6), so
$H_{2}(E;\mathbb{Z}_{2})=0$. Furthermore, $X$ is a manifold without boundary
(Proposition 2.5(iii)), so $H_{2}(X;\mathbb{Z}_{2})=\mathbb{Z}_{2}$. We claim
that $H_{2}(X,E;\mathbb{Z}_{2})=(\mathbb{Z}_{2})^{\ell}$ where $\ell$ is a
positive integer satisfying

$2\leq\ell\leq\dim H_{0}(A;\mathbb{Z}_{2})+\dim H_{0}(B;\mathbb{Z}_{2}).$ (2)

Before proving the claim, we show how it proves the theorem. Filling in the
known homology groups, our portion of the long exact sequence becomes

${\cdots}$${0}$${\mathbb{Z}_{2}}$${(\mathbb{Z}_{2})^{\ell}}$${H_{1}(E;\mathbb{Z}_{2})}$${H_{1}(X;\mathbb{Z}_{2})}$${\cdots.}$$\scriptstyle{\partial_{3}}$$\scriptstyle{i_{2}}$$\scriptstyle{p_{2}}$$\scriptstyle{\partial_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{p_{1}}$

Exactness of this sequence tells us that
$\ker(\partial_{2})=\operatorname{im}(p_{2})=\mathbb{Z}_{2}$, so we must have
$\operatorname{im}(\partial_{2})=(\mathbb{Z}_{2})^{\ell-1}$. Given that
$\ell\geq 2$, this gives us the lower bound for $\dim
H_{1}(E;\mathbb{Z}_{2})$. To get the upper bound, notice that
$\ker(i_{1})=\operatorname{im}(\partial_{2})=(\mathbb{Z}_{2})^{\ell-1}$, and
$\operatorname{im}(i_{1})$ is contained in $H_{1}(X;\mathbb{Z}_{2})$, so we
have $\dim H_{1}(E;\mathbb{Z}_{2})\leq\dim H_{1}(X;\mathbb{Z}_{2})+\ell-1$.

To prove $H_{2}(X,E;\mathbb{Z}_{2})=(\mathbb{Z}_{2})^{\ell}$ with $\ell$ as in
(2), we define the open sets

$X_{A}=\\{x\in X:d(x,A)<d(x,B)\\}\quad\text{and}\quad X_{B}=\\{x\in
X:d(x,B)<d(x,A)\\}.$

Let $\ell_{A}$ and $\ell_{B}$ be the number of components of $X_{A}$ and
$X_{B}$ respectively. First, we will show that $\ell_{A}$ (and subsequently,
$\ell_{B}$) is finite. Let

$\varepsilon=\inf\\{d(a,b):a\in A,b\in B\\}.$

Since $A$ and $B$ are compact and disjoint, $\varepsilon>0$. By compactness of
$A$, we can find finitely many $a_{i}\in A$ such that
$A\subseteq\bigcup_{i=1}^{N}B_{\varepsilon/3}(a_{i})$. Furthermore, by the
definition of $\varepsilon$,

$\bigcup_{i=1}^{N}B_{\varepsilon/3}(a_{i})\subseteq X_{A}.$

By Proposition 3.5(ii), $X_{A}$ can have no more than $N$ components (one for
each $a_{i}$). Thus $\ell_{A}$ is finite. To be more precise, one can use the
same reasoning as Corollary 3.8 to see that if $B_{\varepsilon/3}(a_{i})\cap
B_{\varepsilon/3}(a_{j})$ is nonempty, then these balls lie in the same
component of $X_{A}$. It follows that $1\leq\ell_{A}\leq\dim H_{0}(A)$.
Applying the same reasoning for $\ell_{B}$, we see that $X\setminus E$ has
$\ell_{A}+\ell_{B}$ components. We conclude, via Lefschetz duality, that

$\dim H_{2}(X,E;\mathbb{Z}_{2})=\dim H^{0}(X\setminus
E;\mathbb{Z}_{2})=\ell_{A}+\ell_{B}$

where $2\leq\ell_{A}+\ell_{B}\leq\dim H_{0}(A)+\dim H_{0}(B)$. ∎

If we additionally assume that $A$ and $B$ are connected, then Corollary 3.8
gives the following corollary to Theorem 4.15. This bound for the homology is
precisely that found in [VB06, Theorem 4.2] (for $n=2$) when $A$ and $B$ are
singleton sets.

######  Corollary 4.16.

Let $X$ be a compact Alexandrov surface (without boundary). If $A,B\subseteq
X$ are nonempty disjoint compact and connected, then

$1\leq\dim H_{1}(E(A,B))\leq\dim H_{1}(X;\mathbb{Z}_{2})+1.$

##  5\. Equidistant Sets in the Plane

We now use the techniques developed in previous sections to answer two prompts
for equidistant sets in the plane given in [PS14].

###  5.1. Equidistant Sets Determined by Connected Sets

First, we address the following problem:

> [PS14, p. 32] Problem: Characterize all closed sets of $\mathbb{R}^{2}$ that
> can be realized as the equidistant set of two connected disjoint closed
> sets.

Interestingly, the answer to this question has been known since 1975: In
[Bel75, Theorem 1] it is shown that the equidistant sets determined by two
nonempty disjoint closed connected subsets of the Euclidean plane is a
connected topological 1-manifold (i.e. the equidistant set is homeomorphic to
a circle or the real line). However, the result is misquoted in [Lov76] (and
subsequently in [PS14]) as being true only for _compact_ subsets.333This is
likely due to Bell’s own abstract, which claims the result for “mutually
disjoint plane continua” (a continuum being generally understood to be a
connected compact set), but the statement of the theorem, and its proof, is
for connected closed sets. In order to illustrate the strength of some of the
techniques developed in Section 4, we give a quick alternative proof of Bell’s
theorem.

######  Theorem 5.1.

If $A$ and $B$ are nonempty disjoint closed connected subsets of the Euclidean
plane, then $E(A,B)$ is a connected topological $1$-manifold.

###### Proof.

First, the fact that $E(A,B)$ is connected is essentially a consequence of
Corollary 3.8. If $E(A,B)$ is not connected, then it has at least two
connected components. As such, we could find one connected component of
$E(A,B)$ which separates the plane, but this would be a proper subset of
$E(A,B)$, so it is not minimal separating. See also [Wil75, Theorem 4] for an
alternative proof of connectedness.

Now, let $x\in E(A,B)$ be given. In order to prove the theorem, we will show
that for some $\varepsilon>0$, $E(A,B)\cap B_{\varepsilon}(x)$ is the interior
of a Jordan arc, and therefore every point of $E(A,B)$ is contained in a
neighborhood homeomorphic to an open interval of the real line.

Fix a real number $r>d(x,A)$, and note that $\overline{B}_{3r}(x)$ is a
compact Alexandrov surface (with boundary) of curvature $\geq 0$. We define
the compact sets

$A_{x}=A\cap\overline{B}_{3r}(x)\quad\text{and}\quad
B_{x}=B\cap\overline{B}_{3r}(x)$

and consider $E(A_{x},B_{x})$ in the space $\overline{B}_{3r}(x)$. Note that
for every $y\in\overline{B}_{r}(x)\cap E(A,B)$, if $a_{y}\in P_{A}(y)$, then

$d(x,a_{y})\leq d(x,y)+d(y,A)\leq 2d(x,y)+d(x,A)<3r$

so $P_{A}(y)\subseteq\overline{B}_{3r}(x)$ (and the same is true for
$P_{B}(y)$). As such,

$E(A_{x},B_{x})\cap\overline{B}_{r}(x)=E(A,B)\cap\overline{B}_{r}(x).$

Given that $A$ and $B$ are each connected, every $x\in E(A,B)$ admits exactly
two wedges (if there were more than two wedges at a point, we could use the
set $A$ and two distinct paths from $x$ to $A$ to separate $B$ from itself),
so let $W^{1}_{\rho}(x)$ and $W^{2}_{\rho}(x)$ be two such wedges at $x$ (for
some sufficiently small $\rho>0$). Applying Lemma 4.11 shows that we can find
an $\varepsilon$ with $0<\varepsilon<\rho$ such that

$E(A,B)\cap\overline{B}_{\varepsilon}(x)=\big{(}E(A,B)\cap\overline{W^{1}_{\varepsilon}(x)}\big{)}\cup\big{(}E(A,B)\cap\overline{W^{2}_{\varepsilon}(x)}\big{)}$

is the image of a Jordan arc. ∎

###  5.2. Hausdorff Dimension of the Equidistant Set

We now answer the following question (the term _focal sets_ refers to the sets
$A$ and $B$ which determine the equidistant set):

> [PS14, p. 32] Question: Does there exist an equidistant set in the plane
> with connected disjoint focal sets, having Hausdorff dimension greater than
> 1? What about other notions of dimension? How does the dimension of an
> equidistant set depend on the dimension of its focal sets?

Note that the question does not require the _closure_ of the sets be disjoint.
This allows for interesting constructions, such as the example in [Wil75],
which shows an equidistant set (with disjoint focal sets) can have Hausdorff
dimension 2. This is achieved by using two ‘interlocking combs’ as the sets
$A$ and $B$. In fact, as the following example illustrates, we can exploit
overlapping closures to produce equidistant sets in the plane of any Hausdorff
dimension between $1$ and $2$.

######  Example 5.2.

Let $K$ be the (boundary of the) Koch snowflake. Define $A$ as the bounded
subset of $\mathbb{R}^{2}\setminus K$ and $B$ as the unbounded subset of
$\mathbb{R}^{2}\setminus K$. Then $A\cap B=\varnothing$, but $E(A,B)$ is
precisely $K$, which has Hausdorff dimension $\log_{3}(4)\approx 1.26186$.

On the other hand, using two sets with overlapping closure is ‘cheating’ in
some sense when working with equidistant sets, as we can have
$d(x,A)=d(x,B)=0$ even when $x$ is outside of $A$ and $B$. For this reason, it
is more appropriate to ask the question for sets with disjoint closure. In
this case, we find the equidistant set is always 1-dimensional, even if $A$
and $B$ are not connected.

######  Theorem 5.3.

If $A$ and $B$ are nonempty disjoint closed subsets of the Euclidean plane,
then the Hausdorff dimension of $E(A,B)$ is $1$.

###### Proof.

By the countable stability of the Hausdorff dimension [Fal90, p. 29],

$\dim_{\mathscr{H}}\bigcup_{n=1}^{\infty}F_{n}=\sup_{1\leq
n<\infty}\dim_{\mathscr{H}}F_{n}$

so it is sufficient to show that the equidistant set is the union of countably
many $1$-dimensional sets. Letting $x\in E(A,B)$ and $n\geq 1$ be given, we
will show that $\overline{B}_{n}(x)\cap E(A,B)$ is $1$-dimensional, which
implies that

$\displaystyle\dim_{\mathscr{H}}E(A,B)$
$\displaystyle=\dim_{\mathscr{H}}\bigcup_{n=1}^{\infty}\Big{(}\overline{B}_{n}(x)\cap
E(A,B)\Big{)}$ $\displaystyle=\sup_{1\leq
n<\infty}\dim_{\mathscr{H}}\overline{B}_{n}(x)\cap E(A,B)$ $\displaystyle=1.$

First, following the same reasoning as in Theorem 5.1, we will consider the
compact subspace $\overline{B}_{R}(x)$, where $R>2n+d(x,A)$, so that the
equidistant set determined by the disjoint compact sets

$A_{x}=A\cap\overline{B}_{R}(x)\quad\text{and}\quad
B_{x}=B\cap\overline{B}_{R}(x)$

satisfies
$E(A,B)\cap\overline{B}_{n}(x)=E(A_{x},B_{x})\cap\overline{B}_{n}(x)$. It
follows from Corollary 4.14 that $E(A_{x},B_{x})$ has Hausdorff dimension $1$
and finite measure. Given that $E(A,B)\cap\overline{B}_{n}(x)$ contains some
rectifiable curve,

$0<\mathscr{H}^{1}\big{(}E(A,B)\cap\overline{B}_{n}(x)\big{)}\leq\mathscr{H}^{1}\big{(}E(A_{x},B_{x})\big{)}$

so $E(A,B)\cap\overline{B}_{n}(x)$ has Hausdorff dimension $1$. ∎

## References

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# Small-Signal Stability of Load and Network Dynamics on Grid-Forming
Inverters ††thanks: This research was supported by the U.S. Department of
Energy by the Advanced Grid Modeling program within the Office of Electricity,
under contract DE-AC02-05CH1123, and by the Solar Energy Technologies Office
through award 38637 (UNIFI Consortium).

Rodrigo Henriquez-Auba  Department of Electrical Engineering and Computer
Sciences
University of California Berkeley
Berkeley, USA
<EMAIL_ADDRESS>Jose Daniel Lara & Duncan S. Callaway  Energy and
Resources Group
University of California Berkeley
Berkeley, USA
{jdlara<EMAIL_ADDRESS>
###### Abstract

This paper presents several stability analyses for grid-forming inverters and
synchronous generators considering the dynamics of transmission lines and
different load models. Load models are usually of secondary importance
compared to generation source models, but as the results show, they play a
crucial role in stability studies with the introduction of inverter-based
resources. Given inverter control time scales, the implications of considering
or neglecting electromagnetic transients of the network are very relevant in
the stability assessments. In this paper, we perform eigenvalue analyses for
inverter-based resources and synchronous machines connected to a load and
explore the effects of multiples models under different network
representations. We explore maximum loadability of inverter-based resources
and synchronous machines, while analyzing the effects of load and network
dynamic models on small-signal stability. The results show that the network
representation plays a fundamental role in the stability of the system of
different load models. The resulting stability regions are significantly
different depending on the source and load model considered.

###### Index Terms:

Grid-forming inverters, small-signal analysis, line dynamics, load dynamics.

##  I Introduction

As power systems move towards a larger share of Renewable Energy Sources
generation, more Inverter-Based Resources (IBRs) replace traditional
generation interfaced via synchronous machines (SMs), creating new challenges
to the understanding of system stability, and dynamic behavior [1]. There have
been significant efforts to understand the effects of control strategies of
IBRs on system stability, e.g., grid-following (GFL) and grid-forming (GFM)
using Quasi-Static Phasors (QSP) and Electromagnetic Transients (EMT)
approaches. Recent studies have uncovered interactions between excitation
systems and inverter controllers using small-signal analyses [2], as well as
effects of QSP network modeling in the global and small-signal stability of
GFM inverters [3, 4].

However, most of IBR system-wide analyses commonly use constant impedance load
models despite the fact that load representation has an important influence on
dynamic behavior and system stability [5]. For example, [6] identifies that
accurate load modeling is an important issue in replicating measurements after
disturbances. As discussed by Charles Concordia in the 80’s and corroborated
in a survey in 2010 by the CIGRE working group C.4605 [7], the industry
acknowledges the importance of adequate load modeling. Yet, the emphasis is
mostly placed on the accurate modeling of generating units, such as detailed
control structures of IBRs, while load models are regarded as of secondary
importance. Further, as power electronics also become commonplace on the load
side, the commonly used assumptions about load model structures also need to
be revised [8, 9].

The main objective of this work is to explore the interaction of static and
dynamic load representations and network dynamics on the small-signal
stability of GFM inverters. In this paper, we present a detailed study of the
effects of modeling assumptions on grid-forming stability and dynamic behavior
of GFM IBRs. We focus our analysis in three GFM control strategies: droop,
virtual synchronous machine (VSM) and dispatchable virtual oscillator control
(dVOC). The results showcase the control interactions depending on the network
model and load representations in the resulting local stability
characteristics.

The main contributions of this work are:

  * •

A stability analysis for network dynamics represented in QSP/EMT domain
against constant power and ZIP loads.

  * •

A thorough small-signal stability and bifurcation analysis for maximum
loadability scenarios for GFM controls considering different load models.

  * •

A discussion of the increased modeling details required to capture constant
power load effects and composite loads in the EMT domain.

Notation: The complex imaginary unit is represented by $j=\sqrt{-1}$. We
denote the synchronous rotating reference frame (SRF) of the network as
$\mathsf{DQ}$, rotating at a fixed frequency $\omega_{s}=1$ p.u. Bold
lowercase symbols are used to represent complex variables in the $\mathsf{dq}$
or $\mathsf{DQ}$ reference frames,
$\boldsymbol{x}=x_{\mathsf{d}}+jx_{\mathsf{q}}$. Bold capital symbols are used
to denote complex matrices $\boldsymbol{Y}=G+jB$. Finally, we use
$\dot{x}=dx/dt$ and $f_{x}=\partial f/\partial x$.

$v$$i$$(i_{0},v_{0})$$\Delta i$SourceCPL$\bullet$$\bullet$$\bullet$ Figure 1:
$v$-$i$ curves for common voltage sources and ideal CPLs.

##  II Instability aspects for Constant Power Loads

To give intuition about the stability challenges in supplying Constant Power
Loads (CPLs), we start our analysis for DC systems, on which a voltage source
is supplying a CPL through an inductor. This intuition is similar for dynamic
phasor analysis in AC systems, as we discuss later. As mentioned in [9], ideal
CPLs satisfy instantaneously at all times $v\cdot i=\text{constant}$; thus, if
a voltage across a CPL increases/decreases, then the current must
decrease/increase respectively to satisfy the constant power condition. This
behavior can be visualized in Figure 1. Most generation sources, such as GFM
IBRs or generators whose Automatic Voltage Regulators (AVRs) are set to
voltage control, describe a concave $v$-$i$ curve that decreases the voltage
when the current increases. The intersection $x_{0}=(i_{0},v_{0})$ (black
circle) determines the operating point of the source connected to the load.
Suppose that an external perturbation increases the current $\Delta i$ through
the inductor, then the load voltage (in red) is lower than the source voltage
(in blue), resulting in an increase of the current provided by the source.
Similarly, if a disturbance decreases the current $\Delta i$, the source
voltage dips below the load’s voltage, decreasing the current. In both cases,
the system moves away from $x_{0}$, resulting in an unstable operating
condition.

Network dynamics also play a role in the stability assessment of AC systems
with CPLs. When network dynamics are neglected, as in standard QSP modeling,
the circuit dynamics are represented by the algebraic relationship using the
admittance matrix $\boldsymbol{i}=\boldsymbol{Y}\boldsymbol{v}$. In this case,
CPLs do not necessarily induce instability in the dynamic model of a
generation system. However, using EMT dynamic phasor modeling, Allen & Ilić
[10] show that small-signal stability for a source connected to a CPL
($S=\boldsymbol{v}\boldsymbol{i}^{*}=$ constant) via a series resistor and
inductance requires the condition:
$\left|\boldsymbol{v}_{\text{source}}-\boldsymbol{v}_{\text{load}}\right|>\left|\boldsymbol{v}_{\text{load}}\right|$.
This condition is highly unlikely to occur in AC transmission systems, where
$|\boldsymbol{v}_{\text{source}}|\approx|\boldsymbol{v}_{\text{load}}|\approx
1.0$ p.u. Thus, the operating points over the desired part of the P-V curve on
typical load conditions are always unstable. In practice this is not observed,
since load models and composition changes the stability regions, as we discuss
next.

##  III Maximum loadability analysis

To study the effect of load modeling and GFM controls on system stability, we
consider a two-bus source vs load balanced system as depicted in Figure 2. For
GFM inverters, we use a detailed outer control model and inner control models
taken from [11, 12, 2], and tune the outer control parameters to have a 2%
$P$-$\omega$ p.u./p.u. and 5% $Q$-$v$ p.u./p.u. droop steady-state response
using the procedure described in [13]. Since the focus is on small-signal
stability, we ignore hybrid dynamical effects of limiters.

For comparison purposes, the simulation also includes results for SM models.
We utilize the GENROU machine model for QSP studies and the Marconato machine
model for EMT studies [14]. To study voltage instabilities it is common to
analyze the power/voltage (P-V) curves to understand if the theoretical
maximum value of power can be delivered through a single line. From a static
load flow perspective, this maximum value is attained at the nose of the upper
part of the P-V curve. However, as our results will show, network and load
dynamics reduce the stability regions.

The simulations and analyses are implemented using the Julia package
PowerSimulationsDynamics.jl. The parameters and study cases to replicate the
results are available in the Github repository https://github.com/Energy-
MAC/GFM-LoadModeling. Summary of the main results of the following subsections
are presented in Table I.

TABLE I: Summary of Maximum Loadability results Case |  | Source
---
Model
$P^{\star}$ [p.u.] | Bifurcation |  | Participating
---
States
QSP CPL | GENROU | 1.169 | Hopf |  $e_{\mathsf{q}}$, $E_{\text{fd}}$
Droop/VSM | 1.242 |  | Singularity-
---
Induced
$\boldsymbol{\xi}_{\mathsf{dq}}$
dVOC | 1.241 |  | Singularity-
---
Induced
$\boldsymbol{\xi}_{\mathsf{dq}}$
EMT CPL | All | >0 | Unstable |  | See
---
Sec. II
| QSP/EMT
---
CIL or CCL
All | 4.5 | Stable |
EMT Single -Cage IM | Marconato | 0.688 | Hopf |  $e_{\mathsf{q}}$, $E_{\text{fd}}$
Droop/VSM | 1.015 | Transcritical | $\boldsymbol{\psi}_{\mathsf{dq}}$
dVOC | 1.049 | Transcritical | $\boldsymbol{\psi}_{\mathsf{dq}}$
EMT Active Load | Marconato | 1.075 | Hopf |  $e_{\mathsf{q}}$, $E_{\text{fd}}$
Droop/VSM | 1.458 | Hopf |  $\zeta$, $v_{\text{DC}}$
dVOC | 1.430 | Hopf |  $\zeta$, $v_{\text{DC}}$
Figure 2: Problem set-up for load studies.

###  III-A QSP P-V curves for CPLs

A QSP analysis ignores network dynamics and assumes that circuit dynamics
evolve to a stable equilibrium. Thus, filters and transmission lines are
modeled using the algebraic admittance matrix representation
$\boldsymbol{i}=\boldsymbol{Y}\boldsymbol{v}$.

Figure 3-(a) depicts our analysis in the QSP domain assuming a single source
connected to a CPL. The results only show the P-V curves for GENROU and
Droop/VSM for clarity, and P-V curves for all models are available in the
repository. The curves were constructed using load power from 0 to 4.5 p.u.,
the nose of the P-V curve, using a discrete step of 0.01 p.u. We examined the
eigenvalues of the linearized reduced system $A=f_{x}-f_{y}g_{y}^{-1}g_{x}$
(see Section IV) at each operating point. For all source models, the colored
regions depict stable operating points, while empty regions showcase unstable
operating conditions.

We can observe that the instability boundary occurs at similar loading levels
for all generation sources $P\approx 1.2$ p.u. (see Table I); however, the
bifurcation characteristics are significantly different between GFM inverters
and the GENROU model. In the GENROU case, the critical eigenvalues are a
complex pair that crosses the $j\omega$ axis when the load reaches the
critical $P^{\star}$. This behavior corresponds to a subcritical Hopf
bifurcation for SMs [15], where the $\mathsf{q}$-axis voltage
$e_{\mathsf{q}}^{\prime}$ and the field voltage $E_{\text{fd}}$ states have
most significant participation factors in the unstable eigenvalues.

The bifurcation characteristics of the GFM inverters are significantly
different than the SM case. Specifically, we observe a singularity-induced
bifurcation (SIB) as the load value increases [16]. When the system reaches
$P^{\star}$ one eigenvalue, associated with the integrator of the GFM voltage
controller $\boldsymbol{\xi}_{\mathsf{dq}}$, crosses ±$\infty$ to the other
side of the complex plane, changing the stability of the system (see Table I).

Furthermore, the stability regions of the Droop/VSM are equivalent since the
outer control parameters achieve the same steady-state response, and the inner
control parameters are the same. The case of the dVOC is slightly different
since the droop behavior depends on $e_{0}$, which changes at different load
levels [13]. When tuning the dVOC parameters to match the steady state droop
response, we assume $e_{0}=1$ p.u. (i.e., no load).

###  III-B EMT P-V curves for ZIP and IM models

The analysis in this section has the same setup as Subsection III-A, but now
including detailed circuit models for the AC filters and transmission lines in
a SRF, described by equations (1) (see Section IV). Ideal CPL models result in
a unstable system as the eigenvalues associated with the transmission become
positive, as discussed in Section II. On the contrary, algebraic models for
Constant Current Load (CCL) and Constant Impedance Load (CIL) result in stable
regions for all operating conditions and all source models considered.

The single-cage induction machine (IM) load uses 5th-order dynamic phasor EMT
model [17] including an algebraic capacitor model in parallel to ensure
unitary power factor of the IM device. A similar P-V curve is constructed
(available in the repository) with the IM parameters. Table I showcases that
in this case the system exhibits a similar Hopf bifurcation for the states
$E_{\text{fd}}$ and $e_{\mathsf{q}}^{\prime}$ for the SM case. These results
are similar to the QSP-CPL case, in which the critical eigenvalues are
associated with the excitation system. This reveals that AVR re-tuning or the
addition of a power system stabilizer can enhance stability regions.

In contrast to the QSP-CPL setting, there are no algebraic states when the IM
is supplied through GFM, so no SIB can occur. However, for GFM inverters at
the critical load $P^{\star}$, a single real eigenvalue crosses to the right
hand side causing a transcritical bifurcation. The states associated with this
eigenvalue, according to its participation factors, are the IM rotor flux
linkages $\boldsymbol{\psi}_{\mathsf{dq}}^{r}$. Hence, in this case the
stability limits comes from the the IBR and load models interactions.

###  III-C EMT P-V curves for an active load

As discussed in Section II, ideal CPL models are not suitable for EMT studies,
so in this section we employ a 12-state model _Active Load_ model that
measures the AC side using a Phase-Lock-Loop (PLL) and regulates a DC voltage
to supply a resistor $r_{L}$. This model induces a CPL-_like_ behavior as it
tries to maintain a fixed DC voltage to supply $P=v_{\text{DC}}^{2}/r_{L}$
[18]. Figure 2 showcases the _Active Load_ model, on which the reference term
$i_{\mathsf{q}}^{\star}$ from the DC voltage controller is chosen to regulate
minimum reactive power consumed from the active load.

Figure 3: P-V curves for (a) QSP/CPL model, (b) EMT/active load model.

The resulting P-V curves are depicted in Figure 3-(b). The GFM inverters case
shows two critical eigenvalues associated with the load’s model DC voltage
controller integrator state $\zeta$ and the DC Voltage $v_{\text{DC}}$. As the
load power is increased, two complex conjugate eigenvalues move from the left-
hand to right-hand side complex plane generating a Hopf bifurcation as
depicted in Figure 4. The phase portrait showcases the states $v_{\text{DC}}$
and $\zeta$, as we introduce different perturbations in the value of
$v_{\text{DC}}$ to assess the system stability. At the critical load
$P^{\star}$ there an is unstable limit cycle around a locally stable
equilibrium point, which is consistent with a subcritical Hopf bifurcation. A
re-tuning in the active load DC voltage PI controller is a future direction to
explore to enhance system stability, changing their dynamic characteristics if
needed.

Figure 4: Phase portrait $v_{\text{DC}}$–$\zeta$ for droop GFM supplying
active load at $P^{\star}$. Figure 5: ZIP Load example.

##  IV Importance of Composite Load Modeling in GFM dominated systems

The ZIP model consists of three loads in parallel, a CPL, a CCL, and a CIL,
and it is ubiquitous in QSP power system studies. A ZIP load can capture the
static and dynamic behavior of many aggregated composite loads in power
systems with the appropriate proportion assignment to each submodel.
Unfortunately, there are no closed-form results (such as those presented in
[10]) on small-signal stability for the dynamic phasor domain. However, it is
possible to obtain results in some cases; for example, in the system depicted
by Figure 5, the electric current dynamic phasor (in p.u.) flowing through the
line are given by:

$\displaystyle\frac{\ell}{\Omega_{b}}\frac{d\boldsymbol{i}_{\mathsf{dq}}}{dt}$
$\displaystyle=\boldsymbol{v}_{\mathsf{dq}1}-\boldsymbol{v}_{\mathsf{dq}L}-r\boldsymbol{i}_{\mathsf{dq}}-jx\boldsymbol{i}_{\mathsf{dq}},$
(1)

in which $\Omega_{b}=2\pi 60$ rad/s the base frequency and $x=\omega_{s}\ell$
is the line reactance, where $\omega_{s}=1.0$ p.u. Given an operating point,
we are looking to compute the eigenvalues of the linearized system. To do so
we solve for the term $\boldsymbol{v}_{L}$ given the load parameters of the
resistor $r_{L}$ and CPL values ($P,Q$):

$\displaystyle\boldsymbol{i}=\boldsymbol{i}_{1}+\boldsymbol{i}_{2},\qquad\boldsymbol{v}_{L}\boldsymbol{i}_{1}^{*}=P+jQ,\qquad\boldsymbol{i}_{2}=\frac{\boldsymbol{v}_{L}}{r_{L}}$
(2)

Equations (2) are a system of non-linear equations where is possible to solve
$\boldsymbol{v}_{L}$ as a function of $\boldsymbol{i}$, resulting in two
possible solutions. Given the complexity of symbolic solution, there is no
closed form relationship for assessing stability as a function of the
proportion between CIL $\boldsymbol{i}_{1}$ and CPL $\boldsymbol{i}_{2}$ in
the ZIP load model. However, it is possible to analyze the stability by
constructing the following system of Differential-Algebraic (DAE) equations:

$\displaystyle\dot{x}=f(x,y,\eta),\qquad 0=g(x,y,\eta)$ (3)

where $f(x,y,\eta)$ represents the line current equations (1) and
$g(x,y,\eta)$ are the algebraic equations of the ZIP load (2). In here,
$x=\boldsymbol{i}$ are differential states and
$y=[\boldsymbol{v}_{L},\boldsymbol{i}_{1},\boldsymbol{i}_{2}]$ are algebraic
states and the parameter $\eta$ is used to control the proportion of CPL to
CIL. Assuming no reactive power load (i.e. $Q=0$) in this system, we obtain
the determinant of the Jacobian of the algebraic constraints with respect to
the algebraic states as follows:

$\displaystyle\det(g_{y}(x,y,\eta))=|\boldsymbol{i}_{1}|^{2}-\frac{|\boldsymbol{v}_{L}|^{2}}{r_{L}^{2}}=|\boldsymbol{i}_{1}|^{2}-|\boldsymbol{i}_{2}|^{2}$
(4)

Equation (4) becomes zero when the magnitude of the currents flowing through
the CPL and the impedance are equal introducing a singularity-induced
bifurcation [16].

We illustrate the bifurcation with the following numerical example: consider
the system presented in Figure 5 with $r=0.01,~{}x=0.1,~{}P_{\text{cpl}}=1$
p.u., and $\boldsymbol{v}_{1}=1.0+0j$ p.u. The proportion of current flowing
through $r_{L}$ and the CPL can be adjusted using the parameter $\eta\in(0,1)$
as follows:

$\displaystyle P_{\text{zip}}=\eta
P_{\text{cpl}}+P_{r_{L}}=1~{}\to~{}P_{r_{L}}$ $\displaystyle=1-\eta
P_{\text{cpl}}$ (5a) $\displaystyle
P_{r_{L}}=\frac{|\boldsymbol{v}_{L}|^{2}}{r_{L}}=1-\eta
P_{\text{cpl}}~{}\to~{}r_{L}$
$\displaystyle=\frac{|\boldsymbol{v}_{L}|^{2}}{1-\eta P_{\text{cpl}}}$ (5b)

In this fashion, we ensure the total load to be always 1 p.u. (reactive power
is set to zero); hence, steady-state voltage is constant at
$|\boldsymbol{v}_{L}|=0.985$ p.u. Equations (5) allow us to traverse through
the singularity manifold $S\equiv\\{(x,y,\eta)\mid\det(g_{y}(x,y,\eta))=0\\}$,
to analyze the trajectory of the line current $i_{\mathsf{d}},i_{\mathsf{q}}$
eigenvalues.

As we change the parameter $\eta$ from $0\to 1$, one of the eigenvalues of the
reduced system ($A=f_{x}-f_{y}g_{y}^{-1}g_{x}$) moves towards –$\infty$, and
changes sign as we cross the singularity manifold at $\eta^{\star}=0.5$, while
the second eigenvalue remains on the left side of the complex plane. The line
current states remain small-signal stable if $\eta<0.5$, a condition where the
current magnitude through the resistor is larger than the CPL current
magnitude. Figure 6 showcases the root-locus of this eigenvalue as we increase
$\eta$, on which we observe that diverges through –$\infty$, changing sign
when $\eta\geq 0.5$.

Figure 6: Root locus as $\eta$ changes from $0\to 1$.

The example highlights the importance of load model composition on network
stability. As we move towards a system with larger shares of IBRs and fast
dynamics become more relevant, the results show the dependence between current
stability and different proportions of a ZIP load model. The analysis
showcases the potential inadequacies of modeling only CIL or CPL and the need
for detailed composite load models as we increase the shares of IBR
generation.

##  V Conclusions

This paper demonstrates that small-signal stability conclusions for IBRs are
heavily dependent on modeling assumptions regarding electromagnetic network
phenomena and load representation. The results indicate that considering line
and voltage dynamics significantly affects which load models are suitable to
use in dynamic studies. As power systems integrate increasing amounts of
electronics-based loads that behave as CPLs and IBRs, more advanced CPL models
are needed to assess stability.

However, we have only explored single-line systems and a specific set of
parameters and load models. More comprehensive studies are needed using
composite load models in more extensive systems to properly assess the
changing stability regions as we increase system loading. Future work should
focus on exploring different load model distribution scenarios using a
combination of SMs and IBRs models in the system. Further, the inclusion of
limiters for SMs controllers and current limits in IBRs can significantly
affect system stability and introduce other kinds of bifurcations. Exploring
more precise control and protection effects in IBR-dominated systems in
combination with load recovery models is also a relevant research direction.

## References

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