# Data-Efficient Power Flow Learning for Network Contingencies

Parikshit Pareek, Deepjyoti Deka and Sidhant Misra  Theoretical Division, Los
Alamos National Laboratory, NM, USA
<EMAIL_ADDRESS>
###### Abstract

This work presents an efficient data-driven method to learn power flows in
grids with network contingencies and to estimate corresponding probabilistic
voltage envelopes (PVE). First, a network-aware Gaussian process (GP) termed
Vertex-Degree Kernel (VDK-GP), developed in prior work, is used to estimate
voltage-power functions for a few network configurations. The paper introduces
a novel multi-task vertex degree kernel (MT-VDK) that amalgamates the learned
VDK-GPs to determine power flows for unseen networks, with a significant
reduction in the computational complexity and hyperparameter requirements
compared to alternate approaches. Simulations on the IEEE 30-Bus network
demonstrate the retention and transfer of power flow knowledge in both N-1 and
N-2 contingency scenarios. The MT-VDK-GP approach achieves over 50% reduction
in mean prediction error for novel N-1 contingency network configurations in
low training data regimes (50-250 samples) over VDK-GP. Additionally, MT-VDK-
GP outperforms a hyper-parameter based transfer learning approach in over 75%
of N-2 contingency network structures, even without historical N-2 outage
data. The proposed method demonstrates the ability to achieve PVEs using
sixteen times fewer power flow solutions compared to Monte-Carlo sampling-
based methods.

###### Index Terms:

Power Flow, Gaussian Process, Transfer Learning.

The authors acknowledge the funding provided by LANL’s Directed Research and
Development (LDRD) project: “High-Performance Artificial Intelligence”
(20230771DI) and and The Department of Energy (DOE), USA under the Advanced
grid Modeling (AGM) program. The research work conducted at Los Alamos
National Laboratory is done under the auspices of the National Nuclear
Security Administration of the U.S. Department of Energy under Contract No.
89233218CNA000001

##  I Introduction

The steady-state of a power grid is determined by solving a set of nonlinear
equations known as Alternating Current Power Flow (ACPF), which relates nodal
voltages (states) to input nodal power injections. These equations lack
closed-form expressions for nodal voltages and are typically solved
iteratively using methods like the Newton-Raphson load flow (NRLF) [1].
Dealing with injection uncertainties complicates the problem, often leading to
the use of the various model-based [1] and data-based [2, 3, 4] approximations
to solve the ACPF. Linear models, while interpretable, struggle to capture the
full non-linearity of power flow, and lifted approaches tend to sacrifice
interpretability and require extensive samples [5].

A significant limitation of data-driven models lies in their lack of
topological awareness, resulting in limited adaptability to out of sample
data. This problem is particularly accentuated when testing on grids with
sparse topology changes from the base case. Practical power grids often
experience contingencies or undergo regular maintenance, leading to occasional
changes in topology [6]. The number of potential power network configurations
is combinatorial in nature, presenting substantial computational challenges in
learning, particularly when coupled with load uncertainties [7]. Moreover,
power flow models under topological uncertainties are required to efficiently
test system feasibility under contingency and aid in security constrained
decision-making [8]. To tackle this challenge and address the variability in
topology, topology-aware data-driven methods have emerged as a promising
solution. These methods either aim to learn a single model suitable for
various network topologies [7, 9, 10] or acquire an average model that can be
efficiently adapted to different topologies [11]. The incorporation of
topology into these methods is primarily achieved by encoding it as input
features [9, 10] or embedding it into a neural network (NN) structure [12]. In
previous approaches focusing on topology encoding, features such as voltage
differences between different topologies [10] and diagonal elements of the
susceptance matrix [13] have been employed. While these methods can handle
multiple topologies, their adaptability to untrained topologies is limited due
to potential shortcomings in fully capturing topology properties during input
vector construction. Another line of research integrates power grid topology
within the forward-propagation process of Graph Convolutional Networks (GCNs),
inherently equipped with topology features. These studies utilize constant
graph convolution kernels derived from the Laplacian matrix [14] and the
adjacency matrix of transmission lines [15, 12, 16] to construct GCNs and
approximate power flow mappings. In [11], a meta-learning approach is
presented, where an average model for optimal power flow over multiple
topologies is learned and can be subsequently updated to achieve desired
performance with new topology data. Despite enhancing GCN adaptability, these
methods may not fully consider the physical relationships between nodes,
limiting their applicability across various topological scenarios. Recently,
[17] provided a critical overview of these methods and introduced a physics
embedding approach using neighborhood aggregation in GCN. However, this GCN
architecture requires a substantial amount of training data and has shown
limited robustness against uncertain unseen topologies, along with acceptable
error margins. Additionally, approaches aiming to create a single model for
different network topologies implicitly assume that voltage is a continuous
function of the network graph for a given load, that may not hold.

In this paper, we attempt to develop a framework for the efficient learning of
voltage-power functions for power grid with topology changes. This work
focuses on reducing the training data requirement for learning power flow over
a new topology by leveraging existing models learned for prior topologies.
This is in line with operational realities where limited data is available for
a newly realized network topology due to contingencies, while only a few prior
network structures can be deeply analyzed. We build a set of transfer learning
(TL) methods based on the recently proposed vertex degree kernel (VDK) [18]
and its ability to incorporate network structures into GP learning. A multi-
task VDK (MT-VDK) is designed by combining VDKs from different existing models
with a parameterized VDK for the new topology. Existing or learned VDKs have
fixed hyperparameters, and a weight parameter is used over the sum of these
VDKs. This means that the growth in the number of hyperparameters is linear
with system size and is independent of the number of existing (or source)
models used. Furthermore, by separating the source and new model training, the
proposed MT-VDK-GP has lower learning complexity than MT-GP [19]. Moreover, we
develop a probabilistic voltage envelope (PVE) construction method for
different grid buses using learned MT-VDK-GP model for a given new topology.
This allows us to obtain PVEs with very few samples compared to ACPF sample-
based envelop construction methods, while maintaining similar probabilistic
bounds.

Main contributions of this paper can be summarized as:

  * •

Designing a multi-task vertex degree kernel (MT-VDK) to utilize available
data/models in learning voltage-power function, in data-efficient manner, for
new network structures. A simple hyperparameter transfer learning (HTL)
mechanism has also been designed exploiting network structure-based VDK [18].

  * •

Developing a computationally efficient method to obtain probabilistic voltage
envelope (PVE) for different network topologies and input distributions. The
MT-VDK-GP is used for numerical evaluation of voltage envelopes with
probabilistic guarantees.

  * •

Benchmarking the proposed transfer learning mechanisms for N-1 and N-2
contingencies with $\pm 10\%$ uncertainty in all node injections, for IEEE
30-Bus system. In low data regime, proposed methods show more than two fold
reduction in errors compared to cold-start VDK-GPs.

##  II Topology Aware Power Flow Learning

This section begins with an overview of power flow in functional form, from GP
learning perspective. Subsequently, we explore the intricacies of the
topology-aware kernel design. The alternating current power flow (ACPF) model
is used in polar coordinate form with node voltage referred to as
$v_{j}=V_{j}\angle{\theta_{j}}$ ($V_{j}$ is magnitude, and $\theta_{j}$ is the
phase angle). Hereafter, voltage is used to refer voltage magnitude. Injection
refers to generation minus demand, while load is equal to negative injection.
The injection vector is denoted as $s_{j}=[p_{j};q_{j}]$. Here, $p_{j}$
($q_{j}$) represents real (reactive) load/injection at the $j$-th node. The
set of all nodes is denoted as $\mathcal{N}$ with $|\mathcal{N}|$ being the
number of nodes. The set of branches for a given network structure or topology
is denoted by $\mathcal{E}$. Total number of branches are denoted by $N$ with
$N-1$ for single and $N-2$ for double branch outage contingencies. Standard
ACPF equation for a given network structure represented by admittance matrix
$\rm Y$ (Y-Bus). Our goal is to represent ACPF as a map between a node voltage
magnitude and injection vector [20, 18], as

$\displaystyle V(\mathbf{s})=f(\mathbf{s})+\varepsilon$ (1)

where, $V(\mathbf{s})$ is node voltage magnitude measurement or NRLF solution
at any target node of the network (to enhance brevity, the subscript ’$j$’ is
omitted hereafter). In (1), $f(\mathbf{s})$ is unknown and $\varepsilon$ is
i.i.d Gaussian noise $\varepsilon\sim\mathcal{N}(0,\sigma^{2}_{n})$. Note that
ACPF is solved for a given Y-Bus i.e. for a given network topology, and thus
(1) is valid for that network structure only. Now, we utilize GP to model a
node voltage function, denoted as $f(\mathbf{s})$ and defined as a zero-mean
Gaussian process, [21]:

$\displaystyle f(\mathbf{s})\sim\mathcal{GP}\Big{(}\mathbf{0},K(S,S)\Big{)}$
(2)

here, $K(\cdot,\cdot)$ represents the covariance, or kernel matrix, computed
over the training samples, $K_{i,j}=k(\mathbf{s}^{i},\mathbf{s}^{j})$. The
design matrix $S=[\mathbf{s}^{1}\dots\mathbf{s}^{i}\dots\mathbf{s}^{N_{s}}]$
comprises $N_{s}$ i.i.d injection vector samples forming the training set
$\\{\mathbf{s}^{i},V^{i}\\}^{N_{s}}_{i=1}$. See [21, 20, 18] for more details.
This kernel function operates on injection vectors and incorporates
hyperparameter vector $\boldsymbol{\theta}$. These hyperparameters can be
estimated (optimally) using maximum log-likelihood estimation (MLE) for exact
inference [21]. Once learned, the GP provides predictions of the function’s
mean and variance, represented as:

$\displaystyle\mathbb{E}[f(\mathbf{s})]$
$\displaystyle=\mu_{f}(\mathbf{s})=\mathbf{k}^{T}\boldsymbol{\alpha}$ (3)
$\displaystyle\mathbb{V}[f(\mathbf{s})]$
$\displaystyle=\sigma^{2}_{f}(\mathbf{s})=k(\mathbf{s},\mathbf{s})-\mathbf{k}^{T}\rm
A\mathbf{k}$ (4)

with $\rm A=[K(S,S)+\sigma^{2}_{n}I]^{-1}$ and
$\boldsymbol{\alpha}=\rm{A}\mathbf{V}$ being constant after learning for a
given hyperparameter vector $\boldsymbol{\theta}$. The variance (4) is a
direct indicator of quality of learning by quantifying the confidence in
predicted voltage for an injection111A GP model, provides a predicted point,
denoted as $\mu_{f}(\mathbf{s})$ in (3), and also offers a measure of
confidence through the predictive variance, $\sigma^{2}_{f}(\mathbf{s})$. Put
simply, this confidence can be interpreted as indicating a 95% likelihood that
the actual voltage solution obtained by solving the power flow at a specific
injection point will fall within the range of $\mu_{f}(\mathbf{s})\pm
2\sigma_{f}(\mathbf{s})$. Therefore, a lower value of
$\sigma^{2}_{f}(\mathbf{s})$ corresponds to a more reliable GP model.[21].

Referring to (2), it is clear that the learning process is directly influenced
by the choice of the kernel function. If the chosen Kernel is topology-
agnostic and valid only for the topology of the training dataset, any network
change or contingency would require relearning the entire model from scratch
and require sufficient data from the new topology. The first step towards
working with multiple network structures involves making the GP learning model
aware of network topology. The following subsection introduces the recently
developed vertex degree kernel (VDK) [18] and elucidates methods for adapting
VDK to various network structures.

$k_{1}([{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{}\mathbf{s}_{1}};\mathbf{s}_{2};{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\pgfsys@color@rgb@stroke{.75}{0}{.25}\pgfsys@invoke{}\pgfsys@color@rgb@fill{.75}{0}{.25}\pgfsys@invoke{}\mathbf{s}_{3}}],\cdot)$$k_{2}([\mathbf{s}_{1};{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{}\mathbf{s}_{2}};\mathbf{s}_{12}],\cdot)$$k_{3}([{\color[rgb]{.75,0,.25}\definecolor[named]{pgfstrokecolor}{rgb}{.75,0,.25}\pgfsys@color@rgb@stroke{.75}{0}{.25}\pgfsys@invoke{}\pgfsys@color@rgb@fill{.75}{0}{.25}\pgfsys@invoke{}\mathbf{s}_{1}};{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{}\mathbf{s}_{3}};\mathbf{s}_{5};\mathbf{s}_{12}],\cdot)$$k_{1}([{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{}\mathbf{s}_{1}};\mathbf{s}_{2}],\cdot)$$k_{2}([\mathbf{s}_{1};{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{}\mathbf{s}_{2}};\mathbf{s}_{12}],\cdot)$$k_{3}([{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{}\pgfsys@color@rgb@fill{0}{0}{1}\pgfsys@invoke{}\mathbf{s}_{3}};\mathbf{s}_{5};\mathbf{s}_{12}],\cdot)$

Figure 1: Different sub-kernels of VDK for different topologies of a part of
118-Bus system. Right hand side network shows the changes in kernels due to
line 1-3 outage under N-1 contingency. Purple colored injections on the left
are absent from sub-kernels on right hand side network.

###  II-A Network Structured Kernel Design

Recent work [18] exploits the network graph to design an additive kernel by
utilizing nodal sub-kernels that focus on neighborhood injections.
Intuitively, the VDK assumes that injections from nodes which are far away
from each other do not have a correlated effect on any node voltage in the
network [18]. For a given node $n$, neighborhood injections come from the
nodes directly connected to $n$. The set of variables for neighborhood
injections is defined as
$\\{\mathbf{x}_{n}=\\{\mathbf{s}_{m}|(n,m)\in\mathcal{E}\\}\\}$. A VDK is

$\displaystyle
k_{\mathcal{E}}(\mathbf{s}^{i},\mathbf{s}^{j})=\sum^{|\mathcal{N}|}_{n=1}k_{n}(\mathbf{x}^{i}_{n},\mathbf{x}^{j}_{n})$
(5)

where, $k_{n}(\mathbf{x}^{i}_{n},\mathbf{x}^{j}_{n})$ is a neighborhood kernel
working over $\mathbf{x}_{n}\subset\mathbf{s}$. Using the power flow
function’s smoothness property, we select square exponential kernel with
hyperparameters $\boldsymbol{\theta}_{n}=\\{\ell_{n},\tau_{n}\\}$ [21, 20].
The VDK in (5) incorporates the network topology by operating over the edge
set $\mathcal{E}$. Consequently, for each distinct network topology or graph,
a unique VDK construction $k_{\mathcal{E}}(\cdot,\cdot)$ exists, derived from
the neighborhood connections in the network graph. It’s important to note that
both real and reactive power injections are accounted for at each node in both
$\mathbf{s}$ and $\mathbf{x}_{n}$’s. Fig. 1, for instance, illustrates the
construction of sub-kernels for VDK for two different network configurations.
Likewise, $N-2$ or $N-3$ contingencies can be modeled in the VDK design.
Moreover, controlled alterations in the power network resulting from
reconfiguration can also be incorporated into the VDK design.

Next section presents a transfer learning framework which exploit the VDK’s
ability to incorporate different network structures, appearing due to branch
outages or network reconfiguration by operator.

##  III Transfer Learning Strategies

As discussed in the introduction, learning power flow with network
contingencies is computationally hard, due to the large number of possible
network graph configurations and the combinatorial nature of contingencies.
For a 30-Bus network having a total of 41 branches, there are 41 and 820
possible network structures for $N-1$ and $N-2$ contingencies, respectively.
These large numbers of graph possibilities, coupled with injection
uncertainties, makes it hard to accurately model the voltage-power
relationships under network contingency conditions.

This section presents two strategies for efficiently learning power flow with
a new network structure by leveraging existing knowledge about voltage-power
relationships. We use the term ‘source’ to refer to the dataset available with
the operator and ‘target’ to indicate training a model for a new network
structure, which is not present in the source. This situation aligns with
practical scenarios where operators have power flow data for only a few
contingencies, often referred to as critical contingencies [22]. Our idea
involves utilizing the source dataset to efficiently learn the voltage-power
function $f(\mathbf{s})$ for a new network structure. First, we suggest using
the hyperparameters of the source power flow models222Here, ‘source power flow
models’ refer to having comprehensive data needed to calculate the predictive
mean (3) and variance (4) for various network structures when dealing with
load uncertainties. This data encompasses the VDK $k_{\mathcal{E}}$ and
corresponding optimal hyperparameters $\boldsymbol{\theta}_{\mathcal{E}}$,
acquired during the training of the VDK-GP model for each source network
structure $\mathcal{E}$. to hot start the learning process for target function
hyperparameter learning. Then a subsection presents strategy to use source
VDKs along with hyperparameter hot starting to learn target voltage-power
function in data efficient manner.

Employing a hyperparameter hot-starting approach for GP learning, is well-
founded as GP regression performance is kernel function dependent, which is
parameterized using hyperparameters $\boldsymbol{\theta}$. A VDK, denoted as
$k_{\mathcal{E}}(\cdot,\cdot)$, captures the underlying assumptions about the
voltage function’s behavior and its dependence on power injections for a
network configuration $\mathcal{E}$. Further,as MLE aims to find optimal
hyperparameters for model fitting, it is apparent that transferring a
hyperparameter vector will imply having a starting point with higher
likelihood during optimization. Moreover, from the structure of VDK (5), the
total number of hyperparameters and their positioning in the kernel structure
is network topology independent. This is because the set of graph
branches/lines, denoted as $\mathcal{E}$, only affects kernel inputs in the
form of $\mathbf{x}_{n}$ and not the structure of hyperparameter vector. We
initialize the hyperparameters of the target function VDK with the average of
source VDK hyperparameters as

$\displaystyle\boldsymbol{\theta}^{o}=\\{\boldsymbol{\theta}^{\star}_{1}+\dots+\boldsymbol{\theta}^{\star}_{M_{s}}\\}\big{/}{M_{s}}$
(6)

here, $M_{s}$ represents the number of source models available with different
network configurations. $\boldsymbol{\theta}^{\star}_{m}$ denotes the optimal
VDK hyperparameters for the $m$-th network configuration, while
$\boldsymbol{\theta}^{o}$ represents the initial hyperparameters for the
target VDK. This HTL strategy is designed to provide advantages in terms of
faster convergence in both sample and iteration senses, relying on knowledge
transfer related to power flow physics. This transfer predominantly involves
the ’relative contribution’ of different node injections to node voltage.

The proposed HTL has limitations in transferring power flow knowledge because
the hyperparameter vector does not directly capture topological information.
In the next section, we introduce Multi-Task VDK-GP, which leverages both
source kernels and HTL to enhance performance when dealing with unsampled
network topologies.

###  III-A Multi-Task VDK-GP (MT-VDK-GP)

The concept behind Multi-Task VDK-GP involves designing a combined kernel, a
strategy also explored with various kernel structures in [23, 19]. In standard
multi-task GPs [19], coregionalization matrices are employed to establish
relationships within and between different data sets (source and target),
where these data sets represent various contingency cases in our context.
However, it’s essential to note that the number of coregionalization
hyperparameters grows cubically with the number of sources used for transfer
learning, leading to a significant increase in the total number of
hyperparameters (cubic matrix+linear noise+kernel hyperparameters) [19]. This
growth is a limitation for efficient hyperparameter learning.

We design the MT-VDK, where the total number of hyperparameters for target
learning is $2|\mathcal{N}|+2$. Here $2|\mathcal{N}|$ corresponds to VDK
hyperparameters from (5) and square exponential kernel for an
$|\mathcal{N}|$-Bus network. The number of hyperparameters grows linearly with
the network’s size, and reduction strategies, as detailed in [18], can be
applied to further decrease the hyperparameter vector’s size. To reduce
training complexity, we propose designing the MT-VDK-GP in alignment with
standard system operation practices. Operators can analyze a few source
contingencies in detail using a large number of samples to learn the voltage-
power functions for these scenarios. However, the target function learning
process must be efficient in terms of sample requirements and complexity.
Hence, we introduce MT-VDK-GP, enabling distinct learning for both source and
target functions. The proposed MT-VDK is given as

$\displaystyle
k_{\mathcal{E}_{T}}(\mathbf{s}^{i},\mathbf{s}^{j})=\sum^{|\mathcal{N}|}_{n=1}k_{n}(\mathbf{x}^{i}_{n},\mathbf{x}^{j}_{n})+\omega\sum^{M_{s}}_{m=1}k_{\mathcal{E}_{m}}(\mathbf{s}^{i},\mathbf{s}^{j})$
(7)

here, neighborhood kernel $k_{n}(\cdot,\cdot)$ has structure dependent on
target network structure $\mathcal{E}_{T}$
($\\{\mathbf{x}_{n}=\\{\mathbf{s}_{m}|(n,m)\in\mathcal{E}\\}$) while
$k_{\mathcal{E}_{m}}$ for $m=1\dots M_{s}$ are source kernels having constant
structure (based on $\mathcal{E}_{m}$) and constant (optimal) hyperparameters
$\boldsymbol{\theta}^{\star}_{m}$ based on the source learning stage. The
constant source kernel means that hyperparameters associated with these
kernels does not change during the target function learning process. The
target hyperparameters vector consist of $\tau_{n}$’s, $\ell_{n}$’s of
neighborhood kernels $k_{n}(\cdot,\cdot)$ in (7), noise variance $\sigma_{n}$
from A in (3) and source kernel aggregated weight $\omega$ taking total
hyperparameters to $2|\mathcal{N}|+2$.333Multi-task GP model training
complexity [19] scales as $\mathcal{O}((N_{t}+N_{s})^{3})$, where $N_{t}$
($N_{s}$) represents the number of training samples used for target (source)
function learning. As a result of using (7) in MT-VDK-GP, the training
complexity for learning the target function is
$\mathcal{O}(N_{t}^{3}+N_{s}^{3})$. It’s important to note that this
complexity is significantly lower than that of multi-task GP learning, as the
“cube of the sum” grows much faster than the “sum of cubes.”

Below, in a remark, we describe a variation of the MT-VDK (7) that offers
higher expressiveness by increasing the number of hyperparameters. The next
section presents an application of the proposed MT-VDK-GP for developing
Probabilistic Voltage Envelopes (PVE) with varying topologies.

######  Remark.

In the multi-task VDK-GP scheme, source kernels can also be included with
individual tuneable weight parameter $\omega_{m}$ i.e.
$k_{\mathcal{E}_{T}}(\mathbf{s}^{i},\mathbf{s}^{j})=\sum^{|\mathcal{N}|}_{n=1}k_{n}(\mathbf{x}^{i}_{n},\mathbf{x}^{j}_{n})+\sum^{M_{s}}_{m=1}\omega_{m}k_{\mathcal{E}_{m}}(\mathbf{s}^{i},\mathbf{s}^{j})$.
This will increase number of hyperparameters for target function learning, and
will provide more flexibility for model learning.

##  IV Probabilistic Voltage Envelopes (PVE)

We consider the problem of constructing PVEs for a power network for all $N-k$
contingencies and for a given load distribution. We define PVE as the maximum
(and minimum) value of a node voltage for various network topologies and a
load distribution with a given probability [24]. Due to the large number of
possible contingencies and load scenarios involved, computing these PVEs is
highly computationally burdensome. The proposed MT-VDK-GP models can alleviate
this computational burden through: i) data-efficient learning of voltage-power
functions and ii) faster evaluation of voltage for a large number of power
injections to compute PVEs. These envelopes can then be directly used to
assess critical contingencies for a node voltage or a set of node voltages. A
straightforward but computationally intensive method to construct PVEs is to
solve a large number of power flows for each network topology and then develop
probabilistic guarantees based on the number of power flows solved. One such
statistical guarantee comes from the worst-case performance theorem [25, 26],
in power system’s context:

######  Theorem 1.

For i.i.d. samples $\mathbf{s}=\\{\mathbf{s}^{1}\dots\mathbf{s}^{T}\\}$ drawn
from the same net-load sub-space as per a probability distribution
$\mathbb{P}_{s}$, let the estimated maximum voltage be defined as

$\displaystyle\widehat{\beta}=\max_{i=1\dots
T}f(\mathbf{s}^{i}).\,\,\text{Then
if}\,\,T\geq\frac{\ln{{1}/{\delta}}}{\ln{\frac{1}{1-{\epsilon}/{2}}}},$

ACPF solutions (voltage samples) are used, we have
$\mathbb{P}_{T}\big{\\{}\mathbb{P}_{s}\big{\\{}f(\mathbf{s})<\widehat{\beta}\big{\\}}\geq
1-{\epsilon}\big{/}{2}\big{\\}}\geq 1-\delta,$ with $\delta$ being confidence
and $\epsilon$ being accepted violation probability.

###### Proof.

The proof follows directly from the worst-case performance analysis result
from [26, 25]. ∎

By constructing similar result for lower voltage limits, according to Theorem
1, $T$ i.i.d. samples can be used to obtain PVE, for a fixed network topology,
with $\epsilon$ accepted violation probability and $\delta$ confidence. With
this approach, a total of ${TN!}/{k!(N-k)!}$ power flow solutions will be
required for all $N-k$ contingency situations where $N$ is number of branches.

We propose to use MT-VDK-GP to reduce the computational burden by training
power flow models for different network topologies and then using those
trained models for evaluation of power flow. Moreover, to accommodate
approximation error, we use predictive variance and $\beta$ estimation in
Theorem 1 is performed as

$\displaystyle\beta=\max_{i=1\dots
T}\,\mu(\mathbf{s}^{i})+\kappa\sigma_{f}(\mathbf{s}^{i})$ (8)

Now considering that GP has been able to model the power flow correctly,
$\kappa$ will determine the confidence level adjustment required in Theorem 1.
Below we present a corollary, using tail probability or quantile
$\gamma(\kappa)$, to adjust it as

######  Corollary 1.

If estimation of maximum voltage is done using $\beta$ (8) instead of
$\widehat{\beta}$ from Theorem 1, then

$\mathbb{P}_{T}\Big{\\{}\mathbb{P}_{s}\big{\\{}f(\mathbf{s})<\beta\big{\\}}\geq
1-\dfrac{\epsilon}{2}\Big{\\}}\geq(1-\delta)(1-\gamma(\kappa))^{T}$

###### Proof.

The violation probability
$\mathbb{P}_{s}\big{\\{}f(\mathbf{s})<\widehat{\beta}\big{\\}}\geq
1-{\epsilon}/{2}$ in Theorem 1 is true with respect to $\beta$ if
$\widehat{\beta}\leq\beta$. Now, $\widehat{\beta}\leq\beta$ if true voltage
solution $V(\mathbf{s}^{i})$ is be bounded by
$\mu(\mathbf{s}^{i})+\kappa\sigma_{f}(\mathbf{s}^{i})$ for every injection
sample $\mathbf{s}^{i}$ when $i=1\dots T$. Further, assuming that true power
flow function follows Gaussian distribution, probability of
$V(\mathbf{s}^{i})\leq\mu(\mathbf{s}^{i})+\kappa\sigma_{f}(\mathbf{s}^{i})$ is
$1-\gamma(\kappa)$ for an injection vector $\mathbf{s}^{i}$. Here,
$\gamma(\kappa)$ is tail probability from normal distribution With i.i.d
injection samples, the probability that the above inequality holds for samples
all $i=1\dots T$ can be given as $(1-\gamma(\kappa))^{T}$. Therefore,
confidence in
$\mathbb{P}_{s}\big{\\{}f(\mathbf{s})<\widehat{\beta}\big{\\}}\geq
1-{\epsilon}/{2}$ using GP-based $\beta$ is
$(1-\delta)(1-\gamma(\kappa))^{T}$. ∎

Now, using (8) and Theorem 1 (as Corollary 1), the number of power flow
solutions required will will be ${N_{s}N!}/{k!(N-k)!}$ where $N_{s}<<T$ is the
number of power flow samples required to train the models. Evaluation of these
models $T$ times requires negligible additional time since they involve direct
function evaluation. Further, as GP model evaluation does not depend on
distribution information, a single model can be used to evaluate PVEs for
different injection distributions[20]. This will provide additional
computational benefits as sampling methods (like Monte-Carlo Simulations) will
require again to solve at least $T$ power flows for each possible injection
distribution. See [20, 27] for more on advantages of non-parametric nature of
GP-based power flow and optimal power flow models. Note that this sample
complexity benefit comes at the cost of confidence reduction by a factor of
$(1-\gamma(\kappa))^{T}$. However, the Gaussian distributions have short tail
and $\gamma(\kappa)$ decrease very fast with increase in $\kappa$. As an
example for $\kappa\geq 3.5$ and $T=1000...$ we get
$(1-\gamma(\kappa))^{T}=0.9157$.

##  V Results and Discussion

In this section we use IEEE 30-Bus system [28] to perform all the training and
testings. The 30-Bus system used has total 41 lines and out of which only 38
N-1 contingency scenarios are base case feasible. We restrict this section to
analyze these 38 different topologies for N-1 contingencies while 356
different network topologies (out of total possible 820) are used for N-2
contingency training-testing. For load uncertainty, a $\pm 10\%$ hypercube is
considered. This implies that real and reactive power injection at each node
lies within $\pm 10\%$ of its base-case value [20, 18]. All testing is done
using out-of-sample analysis for power injection inputs and number of training
samples, testing samples and iterations for MLE optimization are given with
respective results. All models are implemented in Julia and ACPF is solved
using PowerModels.jl for sample generation. We use mean absolute error (MAE)
to evaluate performance as.

MAE
$\displaystyle=\frac{1}{M}\sum_{i=1}^{M}\Big{|}\mu(\mathbf{s}^{i})-\widehat{V}(\mathbf{s}^{i})\Big{|}$
(9)

where, $\mu(\mathbf{s}^{i})$ is mean voltage prediction of that particular
node voltage (3), $\widehat{V}(\mathbf{s}^{i})$ represents the true ACPF
solution values and $M$ is number of testing samples. Fig. 2 shows voltage
sample densities for 38 different N-1 topologies and base network structure
when 2000 input injections are used within $\pm 10\%$ (uniformly distributed).
It is evident that contingencies lead to varying voltage spreads among
different node voltages relative to the base network case. In next two
subsections, we use these voltages to benchmark proposed methods against
baseline approaches. For baseline performance, we use two cold starting GP
approaches, i) Full GP to represent standard GP learning using square
exponential kernel with both hyperparameters as $\exp\\{1\\}$ and
$\sigma_{n}=\exp\\{-5\\}$ [20], ii) VDK-GP for GP using VDK without any
transfer learning and initial hyperparameters for all sub-kernels is same i.e.
$\tau_{n}=\ell_{n}=\exp\\{1\\}$ and $\sigma_{n}=\exp\\{-5\\}$. The proposed
methods are referred as HTL and MT-VDK-GP as discussed in Section III and
III-A respectively. Source sets contains the base topology power flow dataset
and the same for a set of chosen contingency topologies as mentioned in the
respective figure captions.

Figure 2: Different voltage distributions across all total 38 different N-1
topologies for IEEE 30-Bus system. Total 2000 power flow solutions, at
different nodal load vectors within $\pm 10\%$ load hypercube, are considered
at each topology as well as with base case (complete 41-line structure).
Figure 3: Minimum, mean and maximum error variations with number of training
samples for $V_{4}$ in 30-Bus system. Here, Minimum, mean and maximum is
calculated across 38 different network topologies obtained after a single line
outage at a time (N-1 contingency). Number of training iterations are same as
training samples and 500 data points are used for testing with each network
topology in out-of-sample manner. All y-axis scales are in $\log$ and note
that right-most sub-figure (maximum) as different y-axis range. Figure 4: Mean
absolute error (MAE) densities for 24 node voltages for 38 different network
topologies (N-1 contingencies). Different source sets are used for transfer
learning. Indices of branches removed for source set $\rm A$ are 1,5 and 10;
for $\rm B$ are 1,5,8 and 10 while in source set $\rm C$ branch 12, 15, 18,
22, 35 are removed to construct source dataset as discussed in Section III-A.
For each network topology and each node, 60 training samples are used with 50
training iterations while 1000 out-of-sample data points are used for testing.
Right most sub-figure shows violin curve of MAE values when training is done
using VDK-GP without any transfer learning for all nodes with all network
topologies. Figure 5: Relative MAE variation with respect to training samples–
defined as the ratio of MAE obtained using MT-VDK-GP to VDK-GP. A value of 1.0
on the y-axis indicates similar performance by both methods. The training and
testing conditions are the same as in Fig. 2. Valleys in the error curves
reflect the superior performance of the proposed MT-VDK-GP at low data
regimes.

###  V-A N-1 Benchmarking

To gauge the performance of the proposed transfer learning (TL) methods
comprehensively, we start this sub-section by describing benchmarking
experiments conducted on $V_{4}$ learning, as it has the maximum variation
among all node voltages in the 30-Bus system, across different N-1 network
topologies. Fig. 3 shows the comparative performance in terms of MAE (pu) for
four different GP learning methods. Starting from the left, the sub-figures
display the minimum, mean, and maximum MAE performance across 38 distinct
contingencies. To clarify, the maximum curves represent the least favorable
performance observed among the 38 N-1 contingency power flow models.The first
observation is that Full GP is the worst-performing approach, which is in line
with findings presented in [18]. Moreover, the rate of decrease in MAE with
sample increase is higher using both TL methods compared to the VDK-GP
baseline performance. This results in a faster convergence of the error curve
for both HTL and MT-VDK-GP compared to VDK-GP. Hence, Fig. 3 aids in drawing
the conclusion that source models do contain information on power flow
physics, which can be beneficial in decreasing the MAE at a higher rate than
training fresh and individual power flow models for each new network topology
that has not been previously encountered. Fig. 5 shows the ratio of MAE
achieved using MT-VDK-GP with VDK-GP. It depicts a two-fold advantage in mean
performance by the proposed approach over VDK-GP when the number of training
samples is relatively low, lying between 50-250. The valleys establish the
data efficiency of the proposed MT-VDK-GP. It can be observed in Fig. 3 that
the blue curve representing VDK-GP eventually coincides with that of the
proposed methods. We conclude that with enough data all methods will
eventually achieve the same MAE, whereas for the low sample regime (5-250 for
the chosen example) MT-VDK-GP offers superior performance.

Figure 6: Densities of MAE (first two sub-figures from left) using proposed
transfer learning mechanisms and VDK-GP for 356 different network topologies
arising due to simultaneous two branch outages (N-2 contingency). For each
voltage, $V_{6}$ and $V_{25}$, 60 training sample per network topology are
used along with 50 training iterations for all three different GP settings and
source set for TL has branch 1 and 10 outage data along with base topology.
Rightmost sub-figure depicts densities of MAE difference (MAE with MT-VDK-GP -
MAE with HTL) for different node voltage functions. Higher density on the left
of zero line (more than 75% in both cases) shows superior MT-VDK-GP
performance over HTL. Figure 7: Probabilistic Voltage Envelopes (PVEs) for
three node voltages and 38 different network topologies arising due to N-1
contingencies in 30-Bus network. Violins represent node voltage distribution
at each topology for 2000 power injections samples, out of which 60 are used
for training, 1000 for $\beta$-Upper and $\beta$-lower calculation while rest
are unseen injection samples. Training conditions (60 samples, 50 iterations)
are same as in Fig. 4 and MT-VDK-GP models are used with source set B.

With different topologies, some node voltages can change significantly. Thus,
it is important to measure the performance of TL methods on all node voltages.
Fig. 4 shows density plots of MAEs obtained for all node voltages (24
different non-generator nodes) for 38 different topologies (N-1 contingencies)
in the 30-Bus system. Also, from left, the first three sub-figures are
obtained when different topologies are part of the source set, i.e., available
for pre-training, while the right-most figure shows the MAE distribution when
VDK-GP is trained for each voltage and each topology from the baseline (no
transfer learning). We observe that the proposed methods have effectively
halved the mean of MAE density, shifting it from $\sim 10\times 10^{-4}$ to
$\sim 5\times 10^{-4}$ when compared to the VDK-GP baseline. Also, the tail of
MAE densities is shorter with the proposed method and does not extend much
beyond $10\times 10^{-4}$, as is the case with VDK-GP. The left-shifting of
MT-VDK-GP-associated MAE density (green) compared to HTL MAE density (red
dashed) indicates that due to source kernel addition in MT-VDK, the power flow
knowledge retention is better than using only hyperparameters in HTL. Table I
highlights the comparative performance shown in Fig. 4 by calculating the area
under different density curves for different MAE cut-offs. It validates the
observations made above regarding MAE density being below $10\times 10^{-4}$
and the majority of voltage-topology instances having error less than $5\times
10^{-4}$ with proposed methods. An important observation from Table I is that
HTL method performance is nearly source-set-agnostic. This is because
hyperparameters do not contain network topology information directly.
Therefore, the proposed MT-VDK-GP has better performance compared to HTL, and
performance improvement can be observed with an increase in source-set size
going from set A to C in Table I. Note that source set elements, along with
size, will also impact the performance of MT-VDK-GP. The selection of a source
set can be done based on critical contingency (or branches connected to the
largest source/load), if the operator has the option of collecting data by
choice. An intelligent source set selection method and its impact on the
performance of MT-VDK-GP require a separate work and will be part of the
future studies in this direction.

TABLE I: Area Under Density Curves for Different MAE Cut-Offs |  | Source Set
---|---|---
MAE (pu) | VDK-GP | A | B | C
$\times 10^{-4}$ |  | MT | HTL | MT | HTL | MT | HTL
$<10.0$ | 0.68 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00
$<5.00$ | 0.00 | 0.69 | 0.60 | 0.71 | 0.60 | 0.74 | 0.59
$<2.50$ | 0.00 | 0.15 | 0.05 | 0.17 | 0.04 | 0.18 | 0.05
MT := MT-VDK-GP & Train-Test Settings same as in Fig. 4.

###  V-B N-2 Benchmarking

We analyze the performance of proposed methods for network topologies arising
due to simultaneous double branch outage (N-2 contingencies). In Fig. 6, first
two sub-figures from left shows MAE density curves for total 356 different
network structures arising due to double line outage (N-2) contingencies. It
is clear that proposed TL methods achieve lesser error for both voltages. Here
note that source set does not contain any data on N-2 topologies. Therefore,
the transfer capability is shown in Fig. 6 is for completely out-of-sample
network structures. Also, similar to the observations made with Fig. 4, the
proposed methods also reduces tail of MAE distributions which implies lower
worst case error. The right most sub-figure in Fig. 6 compares the performance
of proposed TL methods with each other by calculating the difference in MAE.
As for majority of network topologies the MT-VDK-GP outperforms the HTL,
density of difference MAE444Difference MAE =
$\text{MAE}_{\texttt{MT}}-\text{MAE}_{\texttt{HTL}}$ is higher on left hand
side of zero line. Among 356 topology instances, MT-VDK-GP has lesser error
compared to HTL in 276 (77.52%) and 268 (75.28%) instances for node 6 and node
25 voltage respectively. These results shows that providing higher and direct
topological information leads to better learning performance for N-2
contingencies, even when only N-1 contingency data is available for source
training.

###  V-C Probabilistic Voltage Envelopes (PVEs)

The goal of PVEs is to provide limits within which the voltages will remain,
with a given probability and confidence. Fig. 7 shows PVEs for 38 different
network topologies, constructed using MT-VDK-GP. We considered $\kappa=3.75$
in Corollary 1, which provides $\gamma=8.8\times 10^{-5}$ and $\delta$ is
taken as $10^{-4}$. Thus, from Corollary 1 using 1000 samples for $\beta$
estimation, we obtain a violation probability of less than 1.9% with more than
90% confidence. Moreover, from Fig. 7, it is clear that the proposed method
has been able to get upper ($\beta$-Upper) and lower ($\beta$-Lower) estimates
of the maximum voltage, which bounds all 2000 true voltage solutions, in all
network structure cases. Also, it takes only $3.3\pm 0.6$ sec. to develop PVEs
for one node voltage on a given contingency along with the time to generate 60
samples. The total number of power flow solutions required will be $60\times
38$ using the proposed method, which is more than sixteen times less than
$1000\times 38$ samples required using brute force MCS. Further, by increasing
training samples for target functions in MT-VDK-GP beyond 60, we can tighten
the bounds shown in Fig. 7. Moreover, with a change in injection distribution,
the MT-VDK-GP model can be used without re-training (only evaluation) while
the brute force MCS method will require resolving 1000 power flows again.

##  VI Conclusions

In summary, this work presents data-efficient power flow learning techniques
designed for network contingencies. Leveraging the vertex-degree kernel (VDK)
for Gaussian process (GP) learning, we harness existing power flow data and
models to effectively learn voltage-load functions in previously unencountered
network configurations. Our proposed multi-task vertex degree kernel (MT-VDK)
combines historical VDKs with parameterized VDKs for unseen networks, offering
a computationally efficient alternative to multi-task GP (MTGP) methods.
Additionally, we introduce a simple hyperparameter transfer learning (HTL)
approach for employing historically-trained power flow models when
encountering previously unencountered network contingencies. Using MT-VDK-GP
models for voltage-power functions, we have developed a sample-efficient
mechanism for probabilistic voltage envelopes (PVEs). Simulation results on
the IEEE 30-Bus network demonstrate the ability of historically-trained models
to retain and transfer power flow insights in N-1 and N-2 contingency
scenarios. Future directions for this research could include extending the
application of MT-VDK-GP to larger power networks and exploring its potential
in power dispatch adjustment after contingency realization under load
uncertainty.

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$\displaystyle\coloneqq\\{(\mathcal{C},(x,y)):(x,y)\text{ is the maximum of a
maximal horizontal subchain in }\mathcal{C}\\},$
$\displaystyle\mathcal{E}^{V}$
$\displaystyle\coloneqq\\{(\mathcal{C},(x,y)):(x,y)\text{ is the maximum of a
maximal vertical subchain in }\mathcal{C}\\},$ $\displaystyle\mathcal{E}^{S}$
$\displaystyle\coloneqq\\{(\mathcal{C},(x,y)):(x,y)\text{ is the maximum of a
maximal strictly monotone subchain in }\mathcal{C}\\}.$

###### Proof.

$(x,y)\in\mathcal{C}$ is a closed condition. Then $(x,y)$ is the maximum of a
horizontal subchain in $\mathcal{C}$ if and only if $(x+1/n,y)\in\mathcal{C}$
for some $n\in\mathbb{N}$, which is a $\mathbf{\Sigma}^{0}_{2}$ condition.
Finally, it is the maximum of a _maximal_ horizontal subchain in $\mathcal{C}$
if and only if moreover $(x-1/n,y)\not\in\mathcal{C}$ for all
$n\in\mathbb{N}$, which is a $\mathbf{\Pi}^{0}_{2}$ condition, showing that
$\mathcal{E}^{H}$ is Borel. The argument for $\mathcal{E}^{V}$ is exactly the
same so it remains to show that $\mathcal{E}^{S}$ is Borel. Indeed, $(x,y)$ is
the maximum of a maximal strictly monotone subchain in $\mathcal{C}$ if and
only if the following conditions are all satisfied:

  1. (a)

$(x,y)\in\mathcal{C}$, which is a closed condition,

  2. (b)

there exists $n\in\mathbb{N}$ such that for all $m\in\mathbb{N}$ there are no
$(x_{1},y_{1})$ and $(x_{2},y_{2})\in\mathcal{C}$ such that for
$i\in\\{1,2\\}$, $x\leq x_{i}\leq x+1/n$, $y\geq y_{i}\geq y-1/n$ and either
$x_{1}=x_{2}$ and $y_{1}=y_{2}+1/m$ or $y_{1}=y_{2}$ and $x_{1}=x_{2}+1/m$,
which is a $\mathbf{\Sigma}^{0}_{3}$ condition,

  3. (c)

for all $n\in\mathbb{N}$ there exists $m\in\mathbb{N}$ and there exist
$(x_{1},y_{1})$ and $(x_{2},y_{2})\in\mathcal{C}$ such that for
$i\in\\{1,2\\}$, $x\geq x_{i}\geq x-1/n$, $y\leq y_{i}\leq y+1/n$ and either
$x_{1}=x_{2}$ and $y_{1}=y_{2}+1/m$ or $y_{1}=y_{2}$ and $x_{1}=x_{2}+1/m$,
which is a $\mathbf{\Pi}^{0}_{3}$ condition,

showing that $\mathcal{E}^{S}$ is Borel.

∎

######  Lemma A.3.

The subspace $\Phi^{\infty}([0,1])\subseteq\Phi([0,1])$ consisting of all
chains containing infinitely many maximal subchains is Borel. The subspace
$\Phi^{0}([0,1])$ of chains with no horizontal or vertical subchains is
$\mathbf{\Pi}^{0}_{2}$.

###### Proof.

Note that a chain contains finitely many maximal subchains if and only if it
contains finitely many horizontal subchains and finitely many vertical
subchains, since between any two distinct maximal strictly monotone subchains
there must be a maximal horizontal or vertical subchain. Therefore, to prove
that $\Phi^{\infty}([0,1])$ is Borel, it suffices to show that, for each
$n\in\mathbb{N}$, the sets $\Phi^{\geq n}_{H}([0,1])$ and $\Phi^{\geq
n}_{V}([0,1])$ consisting respectively of chains containing at least $n$
horizontal subchains and chains containing at least $n$ vertical subchains are
Borel. Note that, for all $n,m\in\mathbb{N}$ the set $\Phi^{\geq
n,m}_{H}([0,1])$ consisting of all chains that contain at least $n$ horizontal
subchains, which are each of length at least $1/m$ and are pairwise at
distance at least $1/m$ is closed, and that every element of $\Phi^{\geq
n}_{H}([0,1])$ is contained in $\Phi^{\geq n,m}_{H}([0,1])$ for some
$n,m\in\mathbb{N}$. This shows that

$\Phi^{\geq n}_{H}([0,1])=\bigcup_{m\in\mathbb{N}}\Phi^{\geq n,m}_{H}([0,1])$

is $\mathbf{\Sigma}^{0}_{2}$. By an analogous argument $\Phi^{\geq
n}_{V}([0,1])$ is also $\mathbf{\Sigma}^{0}_{2}$, showing that
$\Phi^{\infty}([0,1])$ is $\mathbf{\Pi}^{0}_{3}$.

Moreover, $\Phi^{0}([0,1])$ is the complement of the union of $\Phi^{\geq
1}_{H}([0,1])$ and $\Phi^{\geq 1}_{H}([0,1])$, and thus is
$\mathbf{\Pi}^{0}_{2}$. ∎

We are now able to show that there is generic chain on the interval.

######  Proposition A.4.

The chain $\mathcal{C}_{0}\coloneqq\\{[1/2-\alpha,1/2+\alpha]:0\leq\alpha\leq
1/2\\}$ is a generic chain on $[0,1]$.

###### Proof.

We prove that
$\operatorname{Homeo}([0,1])\cdot\mathcal{C}_{0}=\Phi^{0}([0,1])$ and that
$\Phi^{0}([0,1])$ is dense in $\Phi([0,1])$, and conclude by Lemma A.3.

The chain $\mathcal{C}_{0}$ is the graph of the function $f_{0}(x)\coloneqq
1-x$ from $[0,1/2]\to[1/2,1]$. Clearly $\Phi^{0}([0,1])$ is invariant under
the action of $\operatorname{Homeo}([0,1])$, so fix
$\mathcal{C}\in\Phi^{0}([0,1])$ with the objective of finding
$h\in\operatorname{Homeo}([0,1])$ such that
$h\cdot\mathcal{C}=\mathcal{C}_{0}$. Up to precomposing with a
$g\in\operatorname{Homeo}_{+}([0,1])$ which maps the minimum of $\mathcal{C}$
to $(1/2,1/2)$, we can suppose that also $\mathcal{C}$ has minimum
$(1/2,1/2)$. Then $\mathcal{C}$ is the graph of a continuous strictly
decreasing function $f\colon[0,1/2]\to[1/2,1]$ such that $f(0)=1,f(1/2)=1/2$.
Let $h\in\operatorname{Homeo}([0,1])$ be the identity on $[0,1/2]$ and
$h(y)\coloneqq 1-f^{-1}(y)$ for $1/2\leq y\leq 1$. Then

$hf(x)=1-x=f_{0}(x),$

that is, $h\cdot\mathcal{C}=\mathcal{C}_{0}$.

To check for density, notice that given any finite sequence
$(x_{0},y_{0})\sqsubseteq\cdots\sqsubseteq(x_{n-1},y_{n-1})$ of points in the
interior of $T$, and any $\varepsilon>0$, there is a continuous strictly
decreasing function $f\colon[0,1]\to[0,1]$ such that $f(0)=1$ and $\lvert
f(x_{i})-y_{i}\rvert<\varepsilon$, for all $i<n$, and that the intersection of
the graph of $f$ with $T$ is a chain in $\Phi^{0}([0,1])$. We conclude by
noticing that the above condition determines a basic open set in
$\Phi([0,1])$. ∎

######  Theorem A.5.

Chains on the interval $[0,1]$ are classifiable by countable structures.

###### Proof.

Since $\operatorname{Homeo}_{+}([0,1])$ is a clopen subgroup of
$\operatorname{Homeo}([0,1])$, it is enough to prove that
$\operatorname{Homeo}_{+}([0,1])\curvearrowright\Phi([0,1])$ is classifiable
by countable structures, and apply [hjorth2000classification]*Lemma 4.7.

We concentrate on $\Phi^{\infty}([0,1])$; the case for chains with finitely
many maximal subchains can be shown to be classifiable by countable structures
with an analogous, but easier, argument, which we leave to the reader. Since
$\Phi^{\infty}([0,1])$ is Borel by Lemma A.3, there is no harm in treating the
two cases separately. Let
$\mathcal{E}^{\infty}\coloneqq(\Phi^{\infty}([0,1])\times
T)\cap(\mathcal{E}^{H}\cup\mathcal{E}^{V}\cup\mathcal{E}^{S})$, which is Borel
by Lemma A.2. Since, for every $\mathcal{C}\in\Phi^{\infty}([0,1])$, the
section $\\{(x,y)\in T:(\mathcal{C},(x,y))\\}\in\mathcal{E}^{\infty}$ is
countably infinite, it follows from the Lusin-Novikov uniformization theorem
[KecBook]*Exercise 18.12 that there exists a sequence
$e_{n}\colon\Phi^{\infty}([0,1])\to T$ of Borel functions such that for every
$\mathcal{C}\in\Phi^{\infty}([0,1])$,
$\\{e_{n}(\mathcal{C}):n\in\mathbb{N}\\}$ is an injective enumeration of
$\\{(x,y)\in T:(\mathcal{C},(x,y))\in\mathcal{E}^{\infty}\\}$. In addition to
the quantity and type of maximal subchains occurring in $\mathcal{C}$ we only
need to code their relative position in a countable structure in order to
determine $\mathcal{C}$ up to homeomorphism. Let
$\mathcal{L}\coloneqq\\{P_{H},P_{V},P_{S},<\\}$ be the language containing the
three unary predicates $P_{H},P_{V}$ and $P_{S}$ together with a binary
relation $<$. Given a chain $\mathcal{C}$, consider the structure
$\mathcal{M}(\mathcal{C})$ on $\mathbb{N}$ defined as follows:

  1. (1)

$P_{H}^{\mathcal{M}(\mathcal{C})}(n)$ holds if and only if
$(\mathcal{C},e_{n}(\mathcal{C}))\in\mathcal{E}^{H}$,

  2. (2)

$P_{V}^{\mathcal{M}(\mathcal{C})}(n)$ holds if and only if
$(\mathcal{C},e_{n}(\mathcal{C}))\in\mathcal{E}^{V}$,

  3. (3)

$P_{S}^{\mathcal{M}(\mathcal{C})}(n)$ holds if and only if
$(\mathcal{C},e_{n}(\mathcal{C}))\in\mathcal{E}^{S}$,

  4. (4)

$n<^{\mathcal{M}(\mathcal{C})}m$ holds if and only if
$e_{n}(\mathcal{C})\sqsubseteq e_{m}(\mathcal{C})$.

Since the $\mathcal{E}^{H}$, $\mathcal{E}^{V}$, $\mathcal{E}^{S}$, all of the
$e_{n}$’s, and the partial order $\sqsubseteq$ (as a subset of $T^{2}$) are
Borel, the assignment $\mathcal{C}\mapsto\mathcal{M}(\mathcal{C})$ is Borel.

It remains to verify that two chains
$\mathcal{C},\mathcal{C}^{\prime}\in\Phi^{\infty}([0,1])$ are in the same
$\operatorname{Homeo}_{+}([0,1])$-orbit if and only if
$\mathcal{M}(\mathcal{C})\cong\mathcal{M}(\mathcal{C}^{\prime})$.

Suppose first that $\mathcal{C}^{\prime}=h\cdot\mathcal{C}$ for some
$h\in\operatorname{Homeo}_{+}([0,1])$. Then the function
$\psi\colon\mathcal{M}(\mathcal{C})\to\mathcal{M}(\mathcal{C}^{\prime})$,
defined by letting $\psi(n)$ be the unique $m$ such that
$e_{m}(\mathcal{C}^{\prime})=h(e_{n}(\mathcal{C}))$, is the desired
isomorphism.

Conversely suppose that $\mathcal{C}$ and $\mathcal{C}^{\prime}$ are such that
the associated structures are isomorphic and fix an isomorphism
$\psi\colon\mathcal{M}(\mathcal{C})\to\mathcal{M}(\mathcal{C}^{\prime})$. Let
$x_{0}$ and $x_{0}^{\prime}$ be the roots of $\mathcal{C}$ and
$\mathcal{C}^{\prime}$ respectively. We enumerate the maximal subchains of
$\mathcal{C}$ as $\\{\mathcal{I}_{n}\\}_{n\in\mathbb{N}}$, where
$\mathcal{I}_{n}$ is such that its maximum is $e_{n}(\mathcal{C})$.
Analogously we enumerate the maximal subchains of $\mathcal{C}^{\prime}$ as
$\\{\mathcal{I}^{\prime}_{n}\\}_{n\in\mathbb{N}}$ so that
$e_{n}(\mathcal{C}^{\prime})$ is the maximum of $\mathcal{I}^{\prime}_{n}$.
For all $n$ we define a pair of order-preserving homeomorphisms
$h_{n}^{x}\colon\mathrm{dom}(\mathcal{I}_{n})\to\mathrm{dom}(\mathcal{I}^{\prime}_{\psi(n)})$
and
$h_{n}^{y}\colon\mathrm{codom}(\mathcal{I}_{n})\to\mathrm{codom}(\mathcal{I}^{\prime}_{\psi(n)})$.
We will then verify that $h^{x}=\bigcup_{n}h_{n}^{x}$ is a densely defined,
order-preserving partial homeomorphism $[0,x_{0}]\to[0,x_{0}^{\prime}]$ and
that $h^{y}$ defined analogously has the same properties as a function
$[x_{0},1]\to[x_{0}^{\prime},1]$ so that $h^{x}\cup h^{y}$ can be extended to
the desired homeomorphism of $[0,1]$.

Given $n\in\mathcal{M}(\mathcal{C})$, suppose first that
$P^{\mathcal{M}(\mathcal{C})}_{H}(n)$ holds, so that $\mathcal{I}_{n}$ is a
maximum horizontal subchain of $\mathcal{C}$, with maximum
$e_{n}(\mathcal{C})=(x_{1},y_{1})$ and minimum $(x_{2},y_{1})$. Similarly
$\mathcal{I}_{\psi(n)}^{\prime}$ is a maximal horizontal subchain of
$\mathcal{C}$ with maximum
$e_{\psi(n)}(\mathcal{C}^{\prime})=(x_{1}^{\prime},y_{1}^{\prime})$ and
minimum $(x_{2}^{\prime},y_{1}^{\prime})$. Notice that if
$y_{1}\in[x_{1},x_{2}]$, then $y_{1}=x_{2}$, since $(x_{2},y_{1})\in T$. But,
in this case, $n$ is the minimum element of $<^{\mathcal{M}(\mathcal{C})}$,
and thus also $\psi(n)$ is the minimum element of
$<^{\mathcal{M}(\mathcal{C}^{\prime})}$, so $y_{1}^{\prime}=x_{2}^{\prime}$.
In any case, define
$h_{n}^{x}\colon\mathrm{dom}(\mathcal{I}_{n})\to\mathrm{dom}(\mathcal{I}^{\prime}_{\psi(n)})$
to be any order preserving homeomorphism and define $h_{n}^{y}(y_{1})\coloneqq
y_{1}^{\prime}$. Note that

(A.1)
$\mathcal{I}^{\prime}_{\psi(n)}=\\{(h_{n}^{x}(x),h_{n}^{y}(y)):(x,y)\in\mathcal{I}_{n}\\}.$

The case in which $P_{V}^{\mathcal{M}(\mathcal{C})}(n)$ holds is analogous, so
it remains to deal with the case in which
$P_{S}^{\mathcal{M}(\mathcal{C})}(n)$ holds. By definition this means that
$\mathcal{I}_{n}$ is a maximal strictly monotone subchain in $\mathcal{C}$,
with maximum $e_{n}(\mathcal{C})=(x_{1},y_{1})$ and minimum $(x_{2},y_{2})$.
Similarly $\mathcal{I}^{\prime}_{\psi(n)}$ is a maximal strictly monotone
subchain of $\mathcal{C}^{\prime}$ with maximum
$e_{\psi(n)}(\mathcal{C}^{\prime})=(x^{\prime}_{1},y^{\prime}_{1})$ and
minimum $(x^{\prime}_{2},y^{\prime}_{2})$. Now $\mathcal{I}_{n}$ is the graph
of a continuous, strictly decreasing function
$f_{1}\colon\mathrm{dom}(\mathcal{I}_{n})\to\mathrm{codom}(\mathcal{I}_{n})$,
and similarly we obtain from $\mathcal{I}^{\prime}_{\psi(n)}$ a continuous
strictly decreasing function
$f_{2}\colon\mathrm{dom}(\mathcal{I}^{\prime}_{\psi(n)})\to\mathrm{codom}(\mathcal{I}^{\prime}_{\psi(n)})$.
If
$\mathrm{dom}(\mathcal{I}_{n})\cap\mathrm{codom}(\mathcal{I}_{n})\neq\emptyset$,
then they meet only in $x_{2}=y_{2}$, since $(x_{2},y_{2})\in T$. As above, in
this case, $n$ is the minimum element of $<^{\mathcal{M}(\mathcal{C})}$, and
thus also $\psi(n)$ is the minimum element of
$<^{\mathcal{M}(\mathcal{C}^{\prime})}$, so $x_{2}^{\prime}=y_{2}^{\prime}$.

In any case, let
$h_{n}^{x}\colon\mathrm{dom}(\mathcal{I}_{n})\to\mathrm{dom}(\mathcal{I}^{\prime}_{\psi(n)})$
be any orientation preserving homeomorphism, and
$h_{n}^{y}\colon\mathrm{codom}(\mathcal{I}_{n})\to\mathrm{codom}(\mathcal{I}^{\prime}_{\psi(n)})$
be $h_{n}^{y}\coloneqq f_{2}h_{n}^{x}f_{1}^{-1}$. Notice that also in this
case:

(A.2) $\displaystyle\mathcal{I}^{\prime}_{\psi(n)}$
$\displaystyle=\\{(x^{\prime},f_{2}(x^{\prime})):x^{\prime}\in\mathrm{dom}(\mathcal{I}^{\prime}_{\psi(n)})\\}$
$\displaystyle=\\{(h_{n}^{x}(x),h_{n}^{y}(f_{1}(x))):x\in\mathrm{dom}(\mathcal{I}_{n})\\}$
$\displaystyle=\\{(h_{n}^{x}(x),h_{n}^{y}(y)):(x,y)\in\mathcal{I}_{n}\\}.$

Finally let $h^{x}=\bigcup_{n\in\mathbb{N}}h_{n}^{x}$ and
$h^{y}=\bigcup_{n\in\mathbb{N}}h_{n}^{y}$. Note that if $\mathcal{I}_{n}$ and
$\mathcal{I}_{m}$ are two maximal subchains of $\mathcal{C}$ whose domains
$[x_{1}^{n},x_{2}^{n}]$ and $[x_{1}^{m},x_{2}^{m}]$ have nonempty
intersection, then they meet in exactly one point and we must have either
$x_{2}^{n}=x_{1}^{m}$ or $x_{1}^{n}=x_{2}^{m}$. Suppose without loss of
generality that the former holds, which implies that $m$ is the
$<^{\mathcal{M}(\mathcal{C})}$-immediate successor of $n$. We must have that
$h_{n}^{x}(x_{2}^{n})$ is the minimum of the domain of
$\mathcal{I}^{\prime}_{\psi(n)}$, while $h_{m}^{x}(x_{1}^{m})$ must be the
maximum of the domain of $\mathcal{I}^{\prime}_{\psi(m)}$, but since $\psi$ is
an isomorphism, $\psi(m)$ is the
$<^{\mathcal{M}(\mathcal{C}^{\prime})}$-immediate successor of $\psi(n)$,
showing that

$h_{n}^{x}(x_{2}^{n})=h_{m}^{x}(x_{1}^{m})=h_{m}^{x}(x_{2}^{n}),$

that is, $h^{x}$ is well defined. Since $h^{x}$ is the union of a family of
injective functions that cohere on their common domains, $h^{x}$ is injective
as well. With a similar argument $h^{y}$ is shown to be well defined and
injective too.

######  Claim A.5.1.

$h^{x}$ is a densely defined, order-preserving, partial homeomorphism
$[0,x_{0}]\to[0,x_{0}^{\prime}]$ with dense image. Similarly $h^{y}$ is a
densely defined, order-preserving, partial homeomorphism
$[x_{0},1]\to[x_{0}^{\prime},1]$ with dense image.

###### Proof of the Claim..

We only verify the claim for $h^{x}$, since the case of $h^{y}$ is analogous.
The domain of $h^{x}$ is dense in $[0,x_{0}]$ because $h^{x}$ is defined on
the union of the domains of all maximal subchains of $\mathcal{C}$, while its
image is dense in $[0,x_{0}^{\prime}]$, because $h^{x}$ has as image the union
of the domains of the maximal subchains of $\mathcal{C}^{\prime}$.

In order to show that $h^{x}$ is order-preserving fix $z<z^{\prime}$ in its
domain and let $n,m\in\mathbb{N}$ be such that $z$ belongs to the domain of
$\mathcal{I}_{n}$, while $z^{\prime}$ belongs to the domain of
$\mathcal{I}_{m}$. If $n=m$,
$h^{x}(z)=h^{x}_{n}(z)<h^{x}_{n}(z^{\prime})=h^{x}(z^{\prime})$, since
$h^{x}_{n}$ is order preserving. Otherwise, by definition of
$<^{\mathcal{M}(\mathcal{C})}$, we have $m<^{\mathcal{M}(\mathcal{C})}n$, so
that $\psi(m)<^{\mathcal{M}(\mathcal{C}^{\prime})}\psi(n)$. This implies that
$h^{x}(z)\leq h^{x}(z^{\prime})$, since the former is in the domain of
$\mathcal{I}^{\prime}_{\psi(n)}$ while the latter is in the domain of
$\mathcal{I}^{\prime}_{\psi(m)}$. Since $h^{x}$ is injective, we have
$h^{x}(z)<h^{x}(z^{\prime})$, as desired.

Since each $h_{n}^{x}$ is a homeomorphism from its domain onto its image,
$h^{x}$ is a homeomorphism, e.g., by the pasting lemma [MR3728284]*Exercise
2.9(c). ∎

By the previous claim $h^{x}\cup h^{y}$ extends to an order-preserving
homeomorphism $h\colon[0,1]\to[0,1]$, and it only remains to verify that
$h\cdot\mathcal{C}=\mathcal{C}^{\prime}$. But
$h\cdot\mathcal{I}_{n}=\mathcal{I}^{\prime}_{\psi(n)}$ by (A.1) and (A.2), and
$\bigcup_{n}\mathcal{I}_{n},\bigcup_{n}\mathcal{I}^{\prime}_{\psi(n)}$ are
dense in $\mathcal{C},\mathcal{C}^{\prime}$, respectively, so we conclude by
continuity of $h$. ∎

###  A.2. Chains on $S^{1}$

In this section we reduce to the results in the previous section to prove
analogous statements for the circle.

######  Theorem A.6.

The orbit equivalence relation of
$\operatorname{Homeo}(S^{1})\curvearrowright\Phi(S^{1})$ is continuously
reducible to that of $\operatorname{Homeo}([0,1])\curvearrowright\Phi([0,1])$.
Therefore, chains on the circle $S^{1}$ are classifiable by countable
structures. Moreover, there is a generic chain in $S^{1}$.

###### Proof sketch..

Let $S^{1}=[0,1]/\text{ mod}\,1$, and denote by $\pi$ be quotient map. Since
the closure of the complement of a subcontinuum of $S^{1}$ is itself a
subcontinuum of $S^{1}$, to every chain $\mathcal{C}\in\Phi(S^{1})$ we can
associate not only its root, $\bigcap\mathcal{C}$ but also its _endpoint_ :

$e(\mathcal{C})\coloneqq\bigcap_{C\in\mathcal{C}}\operatorname{Cl}(S^{1}\setminus
C),$

which is a point, since the diameters of the complements go to zero.

For each $z\in S^{1}$, let $\Phi_{z}(S^{1})$ be the closed subset of chains
whose endpoints is $z$, and fix the clockwise $\rho_{z}$ rotation taking $z$
to $\pi(0)$. Let $\mathcal{R}\colon\Phi(S^{1})\to\Phi_{\pi(0)}(S^{1})$ be the
continuous map:

$\mathcal{C}\mapsto\rho_{e(\mathcal{C})}\cdot\mathcal{C}.$

Then $h\cdot\mathcal{C}=\mathcal{C}^{\prime}$ if and only if
$\rho_{e(\mathcal{C}^{\prime})}h\rho_{e(\mathcal{C})}^{-1}\cdot\mathcal{R}(\mathcal{C})=\mathcal{R}(\mathcal{C}^{\prime})$,
showing that $\mathcal{R}$ is a continuous reduction from the orbit
equivalence relation of
$\operatorname{Homeo}(S^{1})\curvearrowright\Phi(S^{1})$ to that of
$\operatorname{Homeo}_{\pi(0)}(S^{1})\curvearrowright\Phi_{\pi(0)}(S^{1})$,
where $\operatorname{Homeo}_{\pi(0)}(S^{1})$ is the stabilizer of $\pi(0)$.

But $\pi$ induces an isomorphism $\pi^{*}$ of flows between
$\operatorname{Homeo}([0,1])\curvearrowright\Phi([0,1])$ and
$\operatorname{Homeo}_{\pi(0)}(S^{1})\curvearrowright\Phi_{\pi(0)}(S^{1})$,
since $\pi$ is injective on proper non-trivial subcontinua of $[0,1]$. We
conclude by Theorem A.5.

For the moreover, notice that $\mathcal{R}^{-1}(\pi^{*}[\Phi^{0}([0,1])])$ is
comeager and an orbit. ∎

##  Appendix B The Sierpiński carpet

This appendix is devoted to proving that the action
$\operatorname{Homeo}(S)\curvearrowright\Phi(S)$ is minimal, for the
Sierpiński carpet $S$, and thus concluding that $\operatorname{Homeo}(S)$ does
not have the generic point property and its universal minimal flow is not
metrizable. Since, as we will see, the Sierpiński carpet is not homogeneous,
and thus not strongly locally homogeneous, the minimality of
$\operatorname{Homeo}(S)\curvearrowright\Phi(S)$ does not follow from Theorem
7.7. We do not settle whether this action is turbulent.

An _$S$ -curve_ is a one-dimensional planar Peano continuum such that its
complement in the plane consists of countably many connected open sets whose
boundaries are mutually disjoint simple closed curves. [whyburn]*Theorem 4
states there exists a unique one-dimensional planar Peano continuum $S$ with
no locally separating points, and that all $S$-curves are homeomorphic to such
$S$. The space $S$ is called the _Sierpiński carpet_.

As we can see, $S$ satisfies condition (2) of Theorem 1.2.

######  Corollary B.1.

There is no generic chain on the Sierpiński carpet.

The Sierpiński carpet has two types of points: _rational_ and _irrational_.
The collection of rational points is given by the union of all the boundaries
of the maximal connected components of the complements of $S$ in the plane or,
equivalently and more abstractly, it is the union of all non-trivial non
locally separating proper subcontinua of $S$ (see [MR0200910]*§8). There are
countably many maximal such subcontinua, called _rational boundary curves_ ,
they are all simple closed curves, and their union is dense and co-dense in
the space. Since it is the union of countably many non locally separating
subcontinua, the set of rational points is itself non locally separating, by
[MOT]*§2.2. The set of irrational points is the complement. A subcontinuum of
$S$ is said to be rational/irrational if all of its points are
rational/irrational.

The action $\operatorname{Homeo}(S)\curvearrowright S$ has two orbits, the
rational and irrational points [krasinkiewicz], thus it is minimal, but not
locally transitive.

We need a couple of preliminary statements before showing that the action
$\operatorname{Homeo}(S)\curvearrowright\Phi(S)$ is minimal. The following
proposition mirrors Facts 4.2 and 4.3.

######  Proposition B.2.

Let $\mathcal{W}$ be a brick partition of $S$ and $W\in\mathcal{W}$.

  1. (1)

For each $W\mathbin{E}W^{\prime}$ there is an irrational point
$x\in\operatorname{Cl}(W)\cap\operatorname{Cl}(W^{\prime})\cap(W\vee
W^{\prime})$.

  2. (2)

For any $x,y\in\operatorname{Cl}(W)$, there is an arc $\gamma\subseteq
W\cup\\{x,y\\}$ connecting $x,y$, which is irrational except possibly at
$x,y$.

  3. (3)

For any $A,B\subseteq\operatorname{Cl}(W)$, disjoint and containing exactly
two points each, there are two disjoint arcs $\gamma_{0},\gamma_{1}\subseteq
W\cup A\cup B$, connecting $A$ to $B$ which are irrational except possibly at
their endpoints.

###### Proof.

(1). If $W=W^{\prime}$, take any irrational point in $W$. Otherwise, suppose
towards contradiction that each point in
$C\coloneqq\operatorname{Cl}(W)\cap\operatorname{Cl}(W^{\prime})\cap(W\vee
W^{\prime})$ is rational in $S$ and thus also in $\operatorname{Cl}(W\cup
W^{\prime})$, by [MR0749904]*Lemma 2. Since it is the countable union of non
locally separating subcontinua, $C$ is non locally separating in
$\operatorname{Cl}(W\cup W^{\prime})$. But $(W\vee W^{\prime})\setminus
C=W\cup W^{\prime}$, which is disconnected, a contradiction.

(2). If $x,y$ are in $W$ and are irrational, then there is an irrational arc
connecting $x,y$ because the set of rational points is non locally separating.
By following the strategy laid out in [nadler2017continuum]*8.32(a), we then
obtain the general statement.

(3). By 4.3.(2), there are two disjoint arcs $\lambda_{0},\lambda_{1}$
connecting $A$ to $B$ in $W\cup A\cup B$. Let $\mathcal{V}\preceq\mathcal{W}$
be fine enough that $\operatorname{Star}(\lambda_{0},\mathcal{V})$ and
$\operatorname{Star}(\lambda_{1},\mathcal{V})$ are disjoint, and call
$P_{i}\coloneqq\bigvee\operatorname{Star}(\lambda_{1},\mathcal{V})$, $i=0,1$.
By item (2) of the current proposition, there are $\gamma_{i}$ in $P_{i}\cup
A\cup B$, with the same endpoints as $\lambda_{i}$, and which are irrational
except possibly at such endpoints. Since the $P_{i}$’s are disjoint, so are
the $\gamma_{i}$’s. ∎

We recall that $C(S)$ is the space of all closed connected subsets of $S$,
with the Vietoris topology (see Section 2.1).

######  Proposition B.3.

Irrational simple closed curves are dense in $C(S)$.

###### Proof.

A basis of open sets in $C(S)$ is given by sets of the form
$O(\mathcal{W}_{0})$, for an $E$-connected subset
$\mathcal{W}_{0}\subseteq\mathcal{W}$ of some brick partition $\mathcal{W}$ on
$S$. Let $w$ be a reduced walk on $\mathcal{W}$, which is confined to and
visits all vertices of $\mathcal{W}_{0}$.

By  (Upgrading Walks to Paths). (Theorem 6.2) and  (Path Doubling).
(Proposition 6.1), there are a brick partition $\mathcal{V}\preceq\mathcal{W}$
and two paths $v_{0},v_{1}$ on $\mathcal{V}$ which are disjoint except for
their endpoints and such that $\rho^{\mathcal{V}}_{\mathcal{W}}\cdot
v_{i}\sqsubseteq w$, for $i<2$. Let $v$ be the a circular path which follows
$v_{0}$ forward and then $v_{1}$ backwards. For each $i<\ell$, let $x_{i}$ be
an irrational point in
$\operatorname{Cl}(v(i))\cap\operatorname{Cl}(v(i+1\text{
mod}\,\ell))\cap(v(i)\vee v(i+1\text{ mod}\,\ell))$, which exists by
Proposition B.2.(1) . By point (2) of the same proposition, there are
irrational arcs $\gamma_{i}$ in $v(i)\cup\\{x_{i},x_{i+1\text{ mod}\,\ell}\\}$
connecting $x_{i}$ to $x_{i+1\text{ mod}\,\ell}$. Let $\gamma$ be the union of
such arcs. It is an irrational simple closed curve which meets each $v(i)$,
for $i<\operatorname{lh}(v)$, and is contained in
$\bigvee_{i<\operatorname{lh}(v)}v(i)$. Since
$\rho^{\mathcal{V}}_{\mathcal{W}}\cdot v\sqsubseteq w$, it follows that
$\gamma$ meets each $W\in\mathcal{W}_{0}$ and is contained in
$\bigvee\mathcal{W}_{0}$, that is, $\gamma\in O(\mathcal{W}_{0})$. ∎

The proof of Theorem B.6 builds upon the following fundamental homogeneity
result for the Sierpiński carpet from [whyburn] (see also [MR0200910]*(7.3)).

######  Theorem B.4 ([whyburn]).

Let $\gamma$ be a rational boundary curve of $S$ and let $h_{0}$ be a
homeomorphism of $\gamma$ onto itself. Then there is
$h\in\operatorname{Homeo}(S)$ which extends $h_{0}$.

Since $S$ is planar, each simple closed curve $\gamma\subseteq S$ separates
$S$ into two components $W_{0},W_{1}$ (one of which could be empty). As noted
in [whyburn]*§2, if every rational maximal subcontinuum of $S$ that is
contained in $W_{0}\cup\gamma$ is either disjoint from $\gamma$ or coincides
with $\gamma$, then $W_{0}\cup\gamma$ is an $S$-curve (and similarly for
$W_{1}\cup\gamma$), and hence it is homeomorphic to $S$. This is the case for
instance if $\gamma$ is irrational. A fact that we will often use implicitly
is that a point in $W_{0}$ is irrational in $W_{0}\cup\gamma$ if and only if
it is irrational in $S$.

The last ingredient is the following technical lemma.

######  Lemma B.5.

Let $\gamma$ be a rational boundary curve of $S$ and let $U_{0},U_{1}\subseteq
S$ be disjoint nonempty open connected subsets such that
$U_{0}\cap\gamma\neq\emptyset\neq U_{1}\cap\gamma$. Then there are open sets
$W_{0}\subseteq W^{\prime}_{0}$, with
$\operatorname{Cl}(W^{\prime}_{0})\subseteq U_{0}\setminus\gamma$, such that
for any $x\in S\setminus\operatorname{Cl}(W^{\prime}_{0})$ there are three
disjoint arcs $\theta,\zeta,\eta$ whose endpoints belong to $\gamma$ and which
are irrational everywhere else, and which are such that:

  1. (1)

$\theta,\zeta\subseteq U_{0}$, one of the connected components of
$S\setminus\theta$ is contained in $U_{0}$, and the same is true of one of the
connected components of $S\setminus\zeta$,

  2. (2)

$\eta\subseteq U_{1}$, and one of the connected components of $S\setminus\eta$
is contained in $U_{1}$,

  3. (3)

$\zeta$ separates $\theta$ from $x$ and $\eta$,

  4. (4)

$\theta$ separates $W_{0}$ from $\zeta$.

###### Proof.

Apply Proposition B.3 to find an irrational simple closed curve
$\gamma^{\prime}_{0}$ which is entirely contained in $U_{0}$. It separates $S$
into two nonempty components, one of which is contained in $U_{0}$. Call this
latter component $W^{\prime}_{0}$. In particular,
$\operatorname{Cl}(W^{\prime}_{0})\subseteq U_{0}\setminus\gamma$.

Repeat this operation to find an irrational simple closed curve $\gamma_{0}$
contained in $W^{\prime}_{0}$. It separates $S$ in two components, one of
which, call it $W_{0}$, is contained in $W^{\prime}_{0}$.

Now, let $x\in S\setminus\operatorname{Cl}(W^{\prime}_{0})$ be given. Notice
that $\Sigma\coloneqq S\setminus W^{\prime}_{0}$ is an $S$-curve and that
$\gamma,\gamma^{\prime}_{0}$ are rational boundary curves in it. By Lemma 6.8,
there are a brick partition $\mathcal{V}^{\prime}$ of $\Sigma$ and two
disjoint spaced circular paths $c,c^{\prime}_{0}$ on $\mathcal{V}^{\prime}$
which are $\gamma$-compatible and $\gamma^{\prime}_{0}$-compatible,
respectively. We can moreover suppose that $\mathcal{V}^{\prime}$ is fine
enough so that:

  1. (i)

there is $i<\operatorname{lh}(c)$ such that
$c(i)\in\mathcal{V}^{\prime}(U_{1})$, and

  2. (ii)

there is a path $P^{\prime}$ from $c$ to $c^{\prime}_{0}$ in
$\mathcal{V}^{\prime}(U_{0})$ which is disjoint from
$\operatorname{Star}(x,\mathcal{V}^{\prime})$,

which is possible by  (Path Doubling). (Proposition 6.1). Since $c$ is
$\gamma$-compatible, $\lvert\gamma\cap c(i)\rvert\geq 2$, for each
$i<\operatorname{lh}(c)$, and the same is true for $c^{\prime}_{0}$ and
$\gamma^{\prime}_{0}$.

Using Proposition B.2.(2), we find an arc $\eta\subseteq c(i)$ whose endpoints
belong to $\gamma$ and whose other points are irrational. In particular,
$\eta\subseteq U_{1}$, and since $\gamma\cap c(i)$ is connected, one the
connected components of $S\setminus(\gamma\cup\eta)$ is contained in $U_{1}$.

By Proposition B.2.(3), there are two disjoint arcs $\zeta_{0},\zeta_{1}$ in
$\bigvee_{i\in\operatorname{lh}(P^{\prime})}P^{\prime}(i)$ which are
irrational (in $\Sigma$) except at their endpoint and connect
$\gamma^{\prime}_{0}$ to $\gamma$. Now, $\gamma^{\prime}_{0}$ is split into
two arcs by the endpoints of $\zeta_{0},\zeta_{1}$. One of these arcs, call it
$\zeta_{2}$, is such that $\zeta\coloneqq\zeta_{0}\cup\zeta_{2}\cup\zeta_{1}$
separates $S$ into two components, one of which contains $W^{\prime}_{0}$ and
is contained in
$W^{\prime}_{0}\cup\bigvee_{i\in\operatorname{lh}(P^{\prime})}P^{\prime}(i)\subseteq
U_{0}$. Call this component $V^{\prime}_{0}$. Then $W_{0}\subseteq
V^{\prime}_{0}$, while $x$ and $U_{1}$ are in $S\setminus V^{\prime}_{0}$.

By the same reasoning as above, there are two disjoint arcs
$\theta_{0},\theta_{1}$ in $V^{\prime}_{0}\cup\gamma$ which are irrational (in
$S\setminus W_{0}$) except at their endpoint and connect $\gamma_{0}$ to
$\gamma$. Now, $\gamma_{0}$ is split into two arcs by the endpoints of
$\theta_{0},\theta_{1}$. One of these arcs, call it $\theta_{2}$, is such that
$\theta\coloneqq\theta_{0}\cup\theta_{2}\cup\theta_{1}$ separates $S$ into two
components, one of which contains $W_{0}$ and is contained in
$V^{\prime}_{0}$. Call this component $V_{0}$.

Then $\theta\subseteq V^{\prime}_{0}$, while $x$ and $U_{1}$ are in
$S\setminus V^{\prime}_{0}$, so point (3) is satisfied. On the other hand,
$W_{0}\subseteq V_{0}$ and $\zeta\subseteq S\setminus V_{0}$, so also point
(4) is taken care of. ∎

######  Theorem B.6.

The action $\operatorname{Homeo}(S)\curvearrowright\Phi(S)$ is minimal, for
the Sierpiński carpet $S$.

###### Proof.

Since $\operatorname{Homeo}(S)\curvearrowright S$ is minimal, it suffices to
verify the conditions of Proposition 7.1. Let $\mathcal{C}\in\Phi(X)$ with
root $x_{0}$ be given, together with open connected sets $U_{0},U_{1},V$ such
that $x_{0}\in U_{0}$ and $U_{0},U_{1}\subseteq V$. In case $U_{0}\cap
U_{1}\not=\emptyset$, by minimality of
$\operatorname{Homeo}(S)\curvearrowright S$ we can assume that $x_{0}\in
U_{0}\cap U_{1}$, so the conclusion follows. We henceforth consider the case
$U_{0}\cap U_{1}=\emptyset$.

By Proposition B.3, there is a simple closed irrational curve $\gamma\in
O(U_{0},U_{1},V)$. Such curve $\gamma$ separates $S$ into two nonempty
components, one of which is contained in $V$ and meets $U_{0}$ and $U_{1}$.
Call this component $Z$, and its closure $\Sigma\coloneqq Z\cup\gamma$. Call
the other component $Z^{\prime}$, with closure $\Sigma^{\prime}\coloneqq
Z^{\prime}\cup\gamma$. Up to shrinking $U_{0}$ and $U_{1}$, we can assume that
$U_{0}\cap\Sigma$ and $U_{1}\cap\Sigma$ are connected.

Let $W_{0}\subseteq W^{\prime}_{0}\subseteq U_{0}\setminus\gamma$ be given by
Lemma B.5 applied to $\gamma$ and $\Sigma$. By minimality of
$\operatorname{Homeo}(S)\curvearrowright S$, we can assume that $x_{0}\in
W_{0}$, hence by maximality of $\mathcal{C}$, there is $C\in\mathcal{C}$ such
that $C\subseteq Z$ and $C\nsubseteq\operatorname{Cl}(W^{\prime}_{0})$. Let
$x_{1}$ be a point of $C\setminus\operatorname{Cl}(W^{\prime}_{0})$.

By Lemma B.5, there are three disjoint arcs $\theta,\zeta,\eta$ whose
endpoints belong to $\gamma$ and which are irrational everywhere else, and
which are such that:

  1. (1)

$\theta,\zeta\subseteq U_{0}$, one of the connected components of
$S\setminus\theta$ is contained in $U_{0}$, and the same is true of one of the
connected components of $S\setminus\zeta$,

  2. (2)

$\eta\subseteq U_{1}$, and one of the connected components of $S\setminus\eta$
is contained in $U_{1}$,

  3. (3)

$\zeta$ separates $\theta$ from $x_{1}$ and $\eta$,

  4. (4)

$\theta$ separates $W_{0}$ from $\zeta$.

The endpoints of $\theta,\zeta,\eta$ separate $\gamma$ into $6$ arcs
$\gamma_{0},\dots,\gamma_{5}$, such that

$\displaystyle\sigma_{0}$ $\displaystyle\coloneqq\theta\cup\gamma_{0},$
$\displaystyle\sigma_{1}$
$\displaystyle\coloneqq\theta\cup\gamma_{1}\cup\zeta\cup\gamma_{5},$
$\displaystyle\sigma_{2}$
$\displaystyle\coloneqq\zeta\cup\gamma_{2}\cup\eta\cup\gamma_{4},\text{ and}$
$\displaystyle\sigma_{3}$ $\displaystyle\coloneqq\eta\cup\gamma_{3}$

are all simple closed curves. Also, let
$\sigma_{<0,1>}\coloneqq\gamma_{0}\cup\gamma_{1}\cup\zeta\cup\gamma_{5}$, and
$\sigma_{<2,3>}\coloneqq\zeta\cup\gamma_{2}\cup\gamma_{3}\cup\gamma_{4}$;
these too are simple closed curves. Moreover, all of them are irrational
curves in $S$, and they each separate $S$ into two components, one of which is
contained in $\Sigma$. Call these components $Z_{i}$, and their closures
$\Sigma_{i}=Z_{i}\cup\sigma_{i}$, for $i\in\\{0,1,2,3,<0,1>,<2,3>\\}$. Then
all the $\Sigma_{i}$’s are $S$-curves, and
$\Sigma_{<0,1>}=\Sigma_{0}\cup\Sigma_{1}$ and
$\Sigma_{<2,3>}=\Sigma_{2}\cup\Sigma_{3}$.

[scale=1]img/sierpi

Figure 3. The construction in the proof of Theorem B.6.

By condition (4), $x_{0}\in\Sigma_{0}$; by condition (1),
$\Sigma_{<0,1>}\subseteq U_{0}$; by condition (3), $x_{1}\in\Sigma_{2}$; and
by condition (2), $\Sigma_{3}\subseteq U_{1}$.

Let $\tilde{h}_{2,3}\colon\sigma_{<2,3>}\to\sigma_{3}$ be any orientation
preserving homeomorphism which maps $\zeta$ onto $\eta$. By Theorem B.4,
$\tilde{h}_{2,3}$ extends to a homeomorphism
$h_{2,3}\colon\Sigma_{<2,3>}\to\Sigma_{3}$.

Let $\tilde{h}_{1}\colon\sigma_{1}\to\sigma_{2}$ be any orientation preserving
homeomorphism which maps $\theta$ onto $\zeta$ and agrees with
$\tilde{h}_{2,3}$ on $\zeta$. By Theorem B.4, $\tilde{h}_{1}$ extends to a
homeomorphism $h_{1}\colon\Sigma_{1}\to\Sigma_{2}$.

Finally, let $\tilde{h}_{0}\colon\sigma_{0}\to\sigma_{<0,1>}$ be any
orientation preserving homeomorphism which agrees with $\tilde{h}_{1}$ on
$\theta$. By Theorem B.4, $\tilde{h}_{0}$ extends to a homeomorphism
$h_{0}\colon\Sigma_{0}\to\Sigma_{<0,1>}$.

The homeomorphisms $h_{0},h_{1},h_{2,3}$ all agree on their common domains so
they give rise to a homeomorphism $h\colon\Sigma\to\Sigma$ that fixes set-wise
$\gamma$. We can use one last time Theorem B.4 on $\tilde{h}\coloneqq
h\upharpoonright\gamma$ to obtain a homeomorphism
$h^{\prime}\colon\Sigma^{\prime}\to\Sigma^{\prime}$ which agrees with $h$ on
$\gamma$. Their union $g$ is a homeomorphism of $S$ such that $g(x_{0})\in
U_{0}$, $g(x_{1})\in U_{1}$ and $g[Z]=Z\subseteq V$, so in particular
$g[C]\subseteq V$, as desired. ∎

Therefore we conclude that:

######  Corollary B.7.

$\operatorname{Homeo}(S)$, for the Sierpiński carpet $S$, does not have the
generic point property. In particular, its universal minimal flow is not
metrizable.

## References