# Grid-SiPhyR: An end-to-end learning to optimize framework for combinatorial
problems in power systems

Rabab Haider
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02149
<EMAIL_ADDRESS>
&Anuradha M. Annaswamy
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, MA 02149
<EMAIL_ADDRESS>

###### Abstract

Mixed integer problems are ubiquitous in decision making, from discrete device
settings and design parameters, unit production, and on/off or yes/no decision
in switches, routing, and social networks. Despite their prevalence, classical
optimization approaches for combinatorial optimization remain prohibitively
slow for fast and accurate decision making in dynamic and safety-critical
environments with hard constraints. To address this gap, we propose SiPhyR
(pronounced: cipher), a physics-informed machine learning framework for end-
to-end learning to optimize for combinatorial problems. SiPhyR employs a novel
physics-informed rounding approach to tackle the challenge of combinatorial
optimization within a differentiable framework that has certified
satisfiability of safety-critical constraints. We demonstrate the
effectiveness of SiPhyR on an emerging paradigm for clean energy systems:
dynamic reconfiguration, where the topology of the electric grid and power
flow are optimized so as to maintain a safe and reliable power grid in the
presence of intermittent renewable generation. Offline training of the
unsupervised framework on representative load and generation data makes
dynamic decision making via the online application of Grid-SiPhyR
computationally feasible.

##  1 Introduction

Exactly solving combinatorial optimization problems (also called mixed integer
programs, MIP) is an NP-hard problem that requires exponential time to solve.
Traditional optimization methods become intractable especially when requiring
dynamic decisions (fast + frequent) and high degree of accuracy to remain
within operating constraints or design limits. Traditional solvers also
struggle to take advantage of the structures present in repeatedly solving an
optimization problem, and warm-start techniques may struggle when parameters
vary rapidly – such as with solar generation forecasts or design under time-
varying conditions. Machine learning (ML) algorithms are natural candidates as
function approximators, to learn the underlying problem structure of complex,
high-dimensional, and nonlinear optimization spaces. However, out-of-the-box
implementations typically cannot enforce hard constraints or address mixed
integer variables.

In this work we introduce SiPhyR (Sigmoidal Physics-Informed Rounding;
pronounced as: ‘cipher’), a physics-informed ML framework for end-to-end
learning to optimize for combinatorial problems. We embed within a
differentiable framework the physics equations governing the behaviour of an
arbitrary system in the form of equality and inequality constraints. To tackle
the challenge of mixed integer variables, we introduce a general physics-
informed rounding approach and the concept of a rounding function which is
defined by a set of physics equations of the underlying problem. The physics-
informed approach guarantees certified satisfiability of all equality
constraints, a subset of inequality constraints, and integer variable
constraints. Notably, our SiPhyR framework is an unsupervised method that
accomplishes an end-to-end learning by directly integrating the task of
optimization within the ML framework, without requiring access to the optimal
solution of each training instance and replaces traditional optimization
solvers.

Our key contributions are: (1) A framework for end-to-end learning for
combinatorial problems: We describe a general framework, SiPhyR, with a
physics-informed rounding approach which leverages physics-based system
constraints to guide the neural predictor towards feasible and good solutions
for dynamic decision making (fast + frequent) in the presence of mixed integer
variables. (2) Implementation of SiPhyR on a real-world problem for clean
energy systems: We develop Grid-SiPhyR, an implementation of our end-to-end
framework for dynamic grid reconfiguration, to optimize the topology of and
power flows on the electric grid in the presence of intermittent renewable
generation. (3) Demonstration of the effectiveness of Grid-SiPhyR: We conduct
experiments on two canonical distribution grids and show that Grid-SiPhyR can
learn to optimize while preserving feasibility and enjoying scalability. We
also develop representative datasets for power systems researchers to support
algorithm development and testing in the absence of real-world data.

##  2 Related work

ML for Optimization. The need for fast and repeated solutions to optimization
problems has pushed for ML-based solutions for directly optimizing within a
neural framework [19], learning new algorithms for optimization [34], and
improving the performance of existing solvers. The latter encompasses the vast
majority of efforts, including hyperparameter optimization [38]; identifying
active constraint sets [40, 15]; learning warm start techniques [7, 53]; and
mapping strategies used by optimization solvers [9]. Approaches specifically
for MIPs include the design of primal heuristics [51, 46], including neural
diving [41] and neural branching [23, 25]. Recent surveys on ML for MIPs
includes a review of variable branch selection, cutting plane methods, and
heuristics such as feasibility pump algorithms [58]; and using graph neural
networks (GNN) directly as solvers or to enhance exact solvers [11]. Our
approach falls in the first group: directly solving the MIP using an
unsupervised neural network within an end-to-end learning to optimize
framework. We propose the used of physics-informed ML to significantly improve
prediction performance.

ML for Power Systems. There is a growing body of literature employing ML
techniques for power systems tasks including optimal power flow (OPF) [44, 59,
19, 60, 17, 22, 57, 12], probabilistic power flow [54], security constrained
unit commitment [26, 45, 53], fault isolation [36], and reconfiguration [31,
56, 55, 49]. These works all leverage the significant reduction in
computational runtimes of ML-based methods as compared to classical methods
such as commercial solvers [44]. We argue that physics-informed ML is
necessary to enable high prediction accuracy and satisfy critical physics and
operating constraints in an explainable manner. Prior work has incorporated
power physics models into neural layers [44, 19], embedded into GNN structures
[16, 37, 18, 35], and satisfied voltage and generator limits by using
Lagrangian duals [22, 43], projections [44, 59], or gradient-based algorithms
[19]. Physics-informed ML has also been used to improve computational
performance of weight updates [54]. Our work presents a significant extension
to the prior literature towards solving mixed-integer problems in a neural
framework while satisfying critical physics constraints.

Traditional Methods for Reconfiguration. The reconfiguration problem has been
extensively studied for outage scenarios where switches are operated to
maximally serve loads, and to improve grid efficiency by minimizing line
losses [8]. Classical approaches include single loop optimization [21] and
heuristics for approximating losses without extensive power flow calculations
[13]. With increased computing abilities and successful commercial MIP
software, literature in the 2000’s focused on modeling the power physics of
reconfiguration as a convex problem [50], with a particular focus on radiality
constraints [33, 52, 5, 32]. Despite the advances in MIP solvers,
reconfiguration for realistic power grids remains computationally intractable
[47]; instead literature has proposed using various heuristic methods
including genetic algorithms, constraint elimination, and repeatedly solving
smaller sub-optimization problems [14]. However, these techniques may not
provide optimality or feasibility guarantees, and remain computationally
prohibitive for dynamic applications. We propose physics-informed ML to enable
dynamic decisions for reconfiguration with feasibility guarantees.

##  3 SiPhyR: learning to optimize for combinatorial problems

We propose SiPhyR (pronounced: cipher), a physics-informed machine learning
framework for end-to-end learning to optimize for a class of combinatorial
problems. Consider a mixed integer problem which given input data
$x\in\mathbb{R}^{d}$ finds $\psi$ that minimizes $f(\psi)$ subject to a set of
constraints,

$\displaystyle\min_{\begin{subarray}{c}\psi=[z_{\tau},z_{\setminus\tau},\varphi],\\\
z_{\tau}\in\mathbb{Z}_{2}^{m},[z_{\setminus\tau},\varphi]\in\mathbb{R}^{n}\end{subarray}}f_{x}(\psi),\quad
s.t.\quad g_{x}(\psi)=0,\;\;h_{x}(\psi)\leq 0,\;\;b_{x}(z_{\setminus\tau})=0,$
(1)

where $f,g,h$ are potentially nonlinear and nonconvex, and $b$ has a
particular structure discussed in 3.1. Leveraging variable space reduction
techniques, the decision variable $\psi$ has been separated into independent
$z$ and dependent variables $\varphi$. Variables $z$ are divided into
topological and other variables as $z=[z_{\tau},z_{\setminus\tau}]$. Knowledge
of $z$ and the function $g$ permits the calculation of $\varphi$, i.e.
$\varphi=\tilde{g}_{x}(z)$. In general, this decomposition is non-unique. It
critically depends on the structure of the given problem, which determines the
relationship between the sets of variables, and requires domain knowledge to
exploit the underlying problem structure to produce good solutions. This
optimization problem can be cast as a learning problem with a neural network
parameterized by $\theta$ as $\psi=N_{\theta}(x)$, with suitable
accommodations made for the binary variables and constraints. To solve such a
problem, we propose an unsupervised neural network shown in Fig. 1 composed of
five key components described next.

Lightweight neural network: a neural network with a sigmoidal (Si) output
layer. The neural network predicts the independent variables, denoted as
$\hat{z}$, from the input data $x$. The output of the neural network is
divided into two sets: a vector of probabilities describing the likelihood
that a binary variable takes on the value of 1,
$\hat{z}_{\tau}=\mathbb{P}(z_{\tau}=1)$; and the prediction for the continuous
variables $\hat{z}_{\setminus\tau}$. The sigmoidal function is a cumulative
distribution function of the logistic distribution, and establishes the
interpretation of $\hat{z}$ as a probability.

Physics-informed rounding (PhyR): algorithm to recover integer solutions from
the vector of probabilities. This layer uses the constraints $b$ to design a
rounding function $R_{b}$ that converts probabilities $\hat{z}_{\tau}$ (input)
to binary decisions $z_{\tau}$ (output).

Inequality constraint layers: the prediction from the neural network
$\hat{z}_{\setminus\tau}$ are scaled onto box constraints, using a sigmoidal
function mapping. This ensures that inequality constraints pertaining to the
independent variables are have certified satisfiability. This layer acts in
parallel to the physics-informed rounding. The output of this layer are the
scaled variables $z_{\setminus\tau}$.

Equality constraint layers: leveraging techniques for variable space
reduction, the equality constraints are used to calculate the dependent
variables $\varphi$ from the independent variables
$z=\left[z_{\tau}\;z_{\setminus\tau}\right]$, as $\varphi=\tilde{g}_{x}(z)$.
This ensures that equality constraints have certified satisfiability.

Loss function: the neural network learns to optimize by using an unsupervised
framework. The loss function is composed of the objective function of the MIP
in (1) and regularization via a soft-loss penalty for violating physical
constraints (i.e. inequality constraints of dependent variables). The
resulting loss function is
$l=f_{x}(\psi)+\lambda_{h}\|\text{max}\left\\{0,h_{x}(z,\varphi)\right\\}\|_{2}^{2}$,
where $\lambda_{h}$ is the soft-loss hyperparameter. Note that if $g$ is
nonlinear there may be equality constraint violations during the variable
space completion due to the use of numerical methods. A corresponding penalty
for equality constraints can also be included.

Figure 1: SiPhyR: A physics-informed machine learning framework for end-to-end
learning to optimize for a class of combinatorial problems, employing a
sigmoidal output layer and a physics-informed algorithm to recover integer
solutions

###  3.1 Physics-informed rounding (PhyR)

The underlying concept of the SiPhyR framework is an explicit embedding of
discrete decisions into the neural framework. Traditional optimization
literature deals with MIPs using an array of heuristic methods to develop good
upper and lower bounds, and explore and prune solution branches. We propose a
physics-informed rounding algorithm taking inspiration from the class of
rounding heuristics well-established in the MIP literature ([30], and others
including [24, 39, 29]). We consider the class of MIPs with binary variables
(1) where the function $b$ uniquely defines a cutoff index $L$ such that $L$
variables have value 1, with the remaining $m-L$ variables having value 0. Our
approach is as follows: function $b$ is used to define the cutoff index $L$ of
a rounding function $z_{\tau}=R_{b}(\hat{z}_{\tau})$. This rounding function
determines which $L$ variables are set to 1, and the remaining $m-L$ set to 0
based on the probabilities $\hat{z}_{\tau}$. To ensure gradient information is
retained through PhyR, only $L-1$ variables will be rounded up, and $m-L-1$
variables rounded down. The neural training guides the remaining two variables
to integer solutions. This is formally presented in Algorithm 1, and applied
to power grid reconfiguration in Section 4.2.

Data: Probabilities $\hat{z}_{\tau}=\mathbb{P}(z_{\tau}=1)$

Result: Binary variables $z_{\tau}$

initialization: $R_{b}\Rightarrow L=f(b(\psi))$

Sort $\hat{z}_{\tau}$ in descending order; assign
$\mathbb{I}_{\hat{z}_{\tau}}$ the sorted indices of $\hat{z}_{\tau}$

Assign $L-1$ binary variables to have value 1:
$z_{\tau}\left[\mathbb{I}_{\hat{z}_{\tau}}^{1:L-1}\right]=\max{\left\\{\right.\hat{z}_{\tau}\left[\mathbb{I}_{\hat{z}_{\tau}}^{1:L-1}\right],1}\left.\right\\}$

Assign $m-L-1$ binary variables to be 0:
$z_{\tau}\left[\mathbb{I}_{\hat{z}_{\tau}}^{L+1:m}\right]=\min{\left\\{\right.\hat{z}_{\tau}\left[\mathbb{I}_{\hat{z}_{\tau}}^{L+1:m}\right],0}\left.\right\\}$

Algorithm 1 Physics-Informed Rounding PhyR for binary variables

###  3.2 Extensions to the proposed SiPhyR method

Integer variables: Algorithm 1 can be extended for integer variables
$z_{\tau}\in\mathbb{Z}_{\ell}^{m}$ where the function $b$ defines a set of
cutoff indices $\\{L_{1},L_{2},...,L_{\ell-1}\\}$. The largest $L_{\ell-1}$
probabilities map to the largest integer variable in the set
$\mathbb{Z}_{\ell}$, the smallest $L_{1}$ probabilities map to the smallest
integer variable, and so forth. These cutoff indices define $R_{b}$, which
converts probabilities to integer values on the set $\mathbb{Z}_{\ell}$.

Binary variables in the dependent set: The SiPhyR framework in Fig. 1
restricts binary variables to the independent set $z$ and uses $R_{b}$ to
convert probabilities into binary decisions. This is motivated by the fact
that MIPs are often formulated for sequential decision processes where the
decision over integer variables precedes the decision of continuous variables:
ex. the grid topology is selected before enforcing power flow constraints; a
set of workers are selected before allocating tasks. The SiPhyR framework is
flexible and can be extended to binary variables in the dependent set (i.e.
$\varphi=[\varphi_{\tau},\varphi_{\setminus\tau}$]) by employing the PhyR-
based rounding as a post-processing step, similar to corrective approaches
employed for continuous variables [44, 59, 19]. Note that this approach
changes the interpretation of the variables: $\varphi_{\setminus\tau}$
calculated using the variable space completion must be interpreted as a
probability, which is then passed through the rounding function $R_{b}$ to
recover a binary decision. The use of this extension should be well motivated
by application. It must also be highlighted that the SiPhyR framework
optimizes decisions over both integer and continuous variables simultaneously,
as these decisions are intricately linked through the constraints. Additional
integer variables which impose soft constraints can be included in either the
independent or dependent variable sets. These variables may be driven towards
integer solutions using approximations of rounding functions (see InSi in
Section 5) or penalized in the loss function.

##  4 Grid-SiPhyR: application to power systems for dynamic reconfiguration

In this section we propose Grid-SiPhyR which applies the physics-informed ML
framework SiPhyR to grid reconfiguration. This problem asks the following
question: “Given a distribution grid with a set of switches and distributed
generation, what is the most efficient grid topology and resource dispatch
(power setpoints) that satisfies all loads, subject to power physics,
generator, and topology constraints?” A naive method would conduct an
exhaustive search of the space of switch status permutations and optimize the
resource dispatch upon each topology to find the optimal switch on/off
decisions (topology) and dispatch solution with minimal electrical line
losses. Instead, we propose the use of SiPhyR towards learning to optimize the
reconfiguration problem. The grid reconfiguration problem can be cast as a MIP
as below.

$\displaystyle\min_{\psi\in\mathcal{P}_{x}}$
$\displaystyle\;f_{x}(\psi)=\sum_{(i,j)\in\mathcal{T}_{D}}(P_{ij}^{2}+P_{ji}^{2}+Q_{ij}^{2}+Q_{ji}^{2})R_{ij}$
(2) $\displaystyle
P_{j}^{G}-P_{j}^{L}=\sum_{k:(j,k)}P_{jk}-\sum_{i:(i,j)}P_{ij},\;Q_{j}^{G}-Q_{j}^{L}=\sum_{k:(j,k)}Q_{jk}-\sum_{i:(i,j)}Q_{ij},\quad\forall
j\in\mathcal{B}$ (3) $\displaystyle
v_{j}-v_{i}=-2(R_{ij}(P_{ij}-P_{ji})+X_{ij}(Q_{ij}-Q_{ji})),\quad\forall(i,j)\in\mathcal{T}_{\text{D}}\setminus\mathcal{T}_{\text{D}}^{sw}$
(4) $\displaystyle
z_{ij}+z_{ji}=1,\quad\forall(i,j)\in\mathcal{T}_{D}\setminus\mathcal{T}_{D}^{sw}$
(5) $\displaystyle
z_{ij}+z_{ji}=y_{ij},\quad\forall(i,j)\in\mathcal{T}_{D}^{sw}$ (6)
$\displaystyle\sum_{(i,j)\in\mathcal{T}_{D}^{sw}}y_{ij}=(N-1)-(M-M_{sw})$ (7)

Eq. (2)-(6) describes a general distribution grid as a graph
$\Gamma(\mathcal{B},\mathcal{T}_{D})$, where $\mathcal{B}$ is the set of N
nodes and $\mathcal{T}_{D}\coloneqq\\{(i,j)\\}$ is the set of M edges. The
subset of lines with switches $\mathcal{T}_{D}^{sw}\subset\mathcal{T}_{D}$
contains $M_{sw}$ switches which can be turned on/off to change the grid
topology. The distribution grid is connected to the transmission grid via the
point of common coupling (PCC). The power variables are the real and reactive
power loads $P^{L}_{j},Q^{L}_{j}$ and generation $P^{G}_{j},Q^{G}_{j}$ at
every node $j$, the squared magnitude of nodal voltages $v_{j}$, and directed
power flow through the distribution lines $P_{ij},P_{ji},Q_{ij},Q_{ji}$. The
line parameters of resistance and reactance are denoted as $R_{ij}$ and
$X_{ij}$ respectively. The grid topology variables are the switch status
$y_{ij}\forall(i,j)\in\mathcal{T}_{D}^{sw}\subset\mathcal{T}_{D}$ (where 1 is
closed and 0 is open), and the selection of direction of power flow through
each line $z_{ij},z_{ji}$. These variables are binary variables.

The objective function (2) approximates the electrical line losses which when
minimized ensures efficient grid operations. Constraints (3)-(4) describe
power flow using the Linearized DistFlow model, commonly used for distribution
grid modeling [8]. Constraint (4) is Ohm’s law across all lines which do not
have switches, and constraints (3) are lossless power balance at every node.
Constraints (5)-(7) describe topological constraints and are composed of
integer variables. Notably, constraint (7) describes the radiality constraint
such that the number of total branches is $N-1$, an operating requirement for
US distribution grids. Additional inequality constraints describe the feasible
set $\mathcal{P}_{x}$ of voltage and generator limits, grid connectivity
constraint, and the conditional constraints for power flow across switches.
The full MIQP model is provided in Appendix C for reference.

Next we provide the details of Grid-SiPhyR including the variable space
decomposition into independent and dependent variables and design of the
rounding function $R_{b}$ for topology selection.

###  4.1 Variable space decomposition

The reconfiguration problem includes two sets of binary variables: $y_{ij}$
pertaining to topology selection and $\left\\{z_{ij},z_{ji}\right\\}$
pertaining to power flow direction. The topology selection is a critical
decision imposing hard constraints on the binary nature of switch status,
while the power flow direction is a soft constraint. To accommodate both hard
and soft constraints, the independent variables $z$ are decomposed into
continuous and binary variables, denoted by superscripts C and B,
respectively. We also classify $z_{\tau}$ and $z_{\setminus\tau}$ as the set
of switch variables ($y_{ij}$) and the set of other variables, respectively,
noting that $z_{\tau}\subset z^{B}$. The variable decomposition for grid
reconfiguration is summarized in (8), where the set of independent and
dependent variables are $z=\left[z^{B},z^{C}\right]$ and
$\varphi=\left[\varphi^{B},\varphi^{C}\right]$ respectively.

$\displaystyle x$
$\displaystyle=\left[\left\\{P_{j}^{L},Q_{j}^{L}\;|\;j\in\mathcal{B}\setminus\text{PCC}\right\\}\right],$
$\displaystyle z^{B}$
$\displaystyle=\left[z_{ji},\left\\{y_{ij}\;|\;(i,j)\in\mathcal{T}_{D,1:M_{sw}-1}^{sw}\right\\}\right],$
$\displaystyle z^{C}$
$\displaystyle=\left[\left\\{v_{j}\;|\;j\in\mathcal{B}\setminus\text{PCC}\right\\},P_{ji},P_{ij},Q_{ji},\left\\{Q_{ij}\;|\;(i,j)\in\mathcal{T}_{D}^{sw}\right\\},P_{j^{\\#}}^{L},Q_{j^{\\#}}^{L}\right],$
$\displaystyle\varphi^{B}$
$\displaystyle=\left[z_{ij},\left\\{y_{ij}\;|\;(i,j)\in\mathcal{T}_{D,-1}^{sw}\right\\}\right],\quad\varphi^{C}=\left[P_{j}^{G},Q_{j}^{G},\left\\{Q_{ij}\;|\;(i,j)\in\mathcal{T}_{D}\setminus\mathcal{T}_{D}^{sw}\right\\}\right]$
(8)

The equality constraints governing power flow are used to calculate
$\varphi^{C}$ from the independent variables $z^{C}$. These correspond to
$P_{j}^{G}$ and $Q_{ij}$ for non-switch lines from constraints (3), and
$Q_{j}^{G}$ from constraint (4). The equality constraints governing grid
topology are used to calculate $\varphi^{B}$ from $z^{B}$. In particular, the
corresponding dependent variables $z_{ij}$ are calculated from constraints
(5)-(6).

Implication 1: With the proposed SiPhyR framework the grid topology is
selected, then the power physics are enforced upon the topology via the
variable space completion. This sequential decision making improves prediction
performance by mimicking the simpler OPF problem on a fixed topology, in every
offline training iteration and online prediction. This is corroborated by the
results in Section 6.

Implication 2: A critical feature of the proposed variable decomposition is
the certified satisfiability of inequality constraints describing voltage,
line flow, and generator limits. By selecting voltages $v$ as an independent
variable, these are scaled onto the box constraints describing operating
limits. For a grid operator, this means voltage limits across the grid will
always be satisfied, a critical aspect of power systems operation. This is
inherent in our proposed structure, as compared to other methods which rely on
projections, clipping, or penalties to enforce voltage constraints.

Implication 3: The equality constraints of the Linearized DistFlow model
permit the dependent variables to be determined trivially with zero error. For
problems which involve more complex (potentially nonlinear) equality
constraints, such as nonlinear DistFlow or the full AC power flow, the same
variable space reduction techniques can be used, and programs like Newton’s
method can be leveraged to solve for the dependent variables. In the
backpropagation step through the equality constraint layer, the Jacobians
describing the derivatives can be explicitly written out and the implicit
function theorem used to backpropagate through the dependent variables.

###  4.2 Topology selection using PhyR

Distribution grids in the US are operated with a radial structure. Constraint
(7) restricts the number of closed switches in the grid so it is radial with
$N-1$ total branches, where $L$ switches must be closed and the remaining
$M_{sw}-L$ must be open. This is a necessary condition for radiality. As it
pertains to the PhyR method, the radiality constraint (7) is the function $b$
in Eq. (1). The cutoff index of the rounding function $R_{b}$ is
$L=(N-1)-(M-M_{sw})$. Embedding the above constraints in Grid-SiPhyR permits
the selection of a feasible (i.e. radial) grid topology upon which the power
flow describing the relationship between $z^{C}$ and $\varphi^{C}$ are
satisfied (Implication 1 in the prior section).

##  5 Experiments

We develop and implement multiple neural architectures to compare the
performance of the proposed Grid-SiPhyR (referred to by abbreviations in the
result tables):

  * •

Optimizer: Traditional optimization solver, Gurobi, a state-of-art commercial
solver for MIPs. The optimal solution determined by the optimizer is denoted
by the asterisk, i.e. $z^{\ast},\varphi^{\ast}$.

  * •

InSi: Without the proposed PhyR layer. Integer solutions for the switch status
are encouraged (read: not enforced) by using a differentiable relaxation of
the step function, the integer sigmoid (InSi):
$\sigma_{InSi}(z)=\left[2\frac{1+\mu}{\mu+e^{-\tau z}}-1\right]_{+}$, where
$\tau,\mu$ are free parameters [10]. InSi is used for the binary variables
$\hat{z}^{B}$, while the traditional sigmoid is used for the continuous
variables $\hat{z}^{C}$. The InSi function is parameterized by $\tau$ which
governs the sharpness, i.e. how well it approximates the step function. Larger
values of $\tau$ better discriminate between binary values, but render the
function less differentiable and thus learning more challenging. We set the
parameters $\mu=1$ and $\tau=5$ based on initial tests.

  * •

InSi2R: Use the InSi architecture during training. During testing, the
predictions $z_{\tau}$ are rounded to binary values before using the variable
space completion.

  * •

A set of PhyR-based methods with different output layer functions: ClaPhyR
with a clamp, SiPhyR with a sigmoid, and InSiPhyR with the InSi function. We
enforce the soft constraints on variables $z^{B}\setminus z_{\tau}$ using the
$\sigma_{InSi}$ function.

We compare the performance of the end-to-end framework with supervised
training, wherein the neural network has full knowledge of the optimal
solution during training:

  * •

Supervised-x: method ‘x’ is trained with typical regression loss function:
$l_{sup}(z,\varphi)=\|{(V_{j}-V_{j}^{\ast})^{2}+(P_{j}^{G}-P_{j}^{G\ast})^{2}+(Q_{j}^{G}-Q_{j}^{G\ast})^{2}}\|_{2}^{2}+\|{(y_{ij}-y_{ij}^{\ast})^{2}}\|_{2}^{2}$

  * •

Supervised-x-pen: method ‘x’ is trained with a soft loss penalty on inequality
constraint violation: $l_{sup-
pen}(z,\varphi)=l_{sup}(z,\varphi)+\lambda_{h}\|\text{max}\left\\{0,h_{x}(z,\varphi)\right\\}\|_{2}^{2}$

We also evaluated Grid-SiPhyR as a warm-start technique for traditional
optimizers, but found minimal performance improvement (see Appendix A).
Generally, the benefits seen for warm-start of general optimization are not
easily replicated for MIPs.

###  5.1 Grid reconfiguration datasets

We consider two canonical distribution grids used in literature: BW-33 a
synthetic grid with 33 nodes and 8 switches [8]; and TPC-94 a model of a real
distribution grid with 94 nodes and 14 switches [48]. These present excellent
test cases for grid reconfiguration. The BW-33 is highly lossy with losses up
to 8% of total load and frequently violates grid voltage limits. The TPC-94
has 11 individual distribution feeders which can be connected to one another
via switches (i.e. reconfiguration) to share load across feeders. We develop
datasets by introducing distributed solar generation throughout the grid (to a
penetration of 25% generation-to-peak-load), and introduce load and generation
profiles. We develop five different datasets for TPC-94: DS1 with loads
perturbed about the nominal (typically done in literature [12, 57, 22]); DS2
with residential load profiles; DS3 with residential and commercial load
profiles; DS4 with has solar generation located in different nodes of the
grid; and DS5 with residential load profiles but no solar generation profiles.
Notably DS2 through DS5 are datasets with representative load and generation
profiles. The datasets are presented in detail in Appendix D.

###  5.2 Simulation parameters and hyperparameter configurations

We use a lightweight neural architecture (to which SiPhyR is agnostic) with
two fully connected hidden layers with linear transformation with bias, batch
normalization, and ReLU activation. The size of input and output layers are
determined by $x$ and $\hat{z}$. We backpropagate using ADAM with parameter
$\gamma=0.001$. The soft loss hyperparameter is set as $\lambda_{h}=100$,
chosen to enforce a high penalty on violating inequality constraints while
still allowing the underlying objective function to be improved upon. All
datasets are split as $80\%$ training, and $10\%$ testing/validation. We use
mini-batching with $200$ batch size. The learning rate ($lr=1e-3$ unless
otherwise stated) and width of the neural network (5 for BW-33 and 300 for
TPC-94) are determined through hyperparameter tuning. Training the InSi method
and on representative datasets (DS-2 thru DS-5) requires a lower learning rate
($lr=1e-4$). Details of the hyperparameter tuning is provided in Appendix B. A
committee machine approach is taken to evaluate the neural architectures,
where 10 models are trained with independent weight initialization (all using
He initialization [28]). An ensemble average is used by linearly combining the
predictions across all 10 predictors. All neural architectures were developed
and tested using PyTorch on an Apple M2 Max with 12-core CPU and 96GB RAM.

###  5.3 Neural architecture performance metrics

We assess our method on the following optimality (Opt) and feasibility (Feas)
metrics:

  * •

[Opt] Dispatch error (DispErr): mean-squared error (MSE) in optimal generator
dispatch:
$\frac{1}{N}\sum_{j\in\mathcal{B}}{(P_{j}^{G}-P_{j}^{G\ast})^{2}+(Q_{j}^{G}-Q_{j}^{G\ast})^{2}}$

  * •

[Opt] Voltage error (VoltErr): MSE in nodal voltage prediction:
$\frac{1}{N}\sum_{j\in\mathcal{B}}{(V_{j}-V_{j}^{\ast})^{2}}$

  * •

[Opt] Topology error (TopErr): the Hamming distance [27] between two
topologies, calculated as the ratio of switch decisions not in the optimal
position:
$\frac{1}{M_{sw}}\sum_{(i,j)\in\mathcal{T}_{D}^{sw}}{(y_{ij}-y_{ij}^{\ast})^{2}}$.
For PhyR-based methods the topology error is equivalent to using an indicator
function which returns 1 if the switch status is not optimally selected, and 0
otherwise:
$\frac{1}{M_{sw}}\sum_{(i,j)\in\mathcal{T}_{D}^{sw}}{\mathbb{I}_{y_{ij}\neq
y_{ij}^{\ast}}}$

  * •

[Feas] Inequality violation: magnitude of violations in constraint set,
measuring the mean and maximum as
$\frac{1}{|h_{x}|}\sum_{k}{\max{\\{0,h_{x}^{k}(\psi)\\}}}$ and
$\max_{k}{\\{\max{\\{0,h_{x}^{k}(\psi)\\}}\\}}$ respectively

  * •

[Feas] Number of violations exceeding a threshold: the number of inequality
constraints which are violated by more than an $\epsilon$ threshold:
$\sum_{k}{\mathbb{I}_{\max{\\{0,h_{x}^{k}(\psi)\\}}>\epsilon}}$

##  6 Results

Table 1 compares the performance of Grid-SiPhyR with Optimizer and other
learning approaches for the BW-33 grid. Table 4 scales Grid-SiPhyR to the
larger TPC-94 grid and assesses performance on different datasets. Smaller
values for each metric are better and indicates performance closer to
Optimizer and thus closer to both optimality and feasibility.

Grid-SiPhyR preserves feasibility and achieves reasonable performance on
optimality: We note that from Table 1, Grid-SiPhyR consistently outperforms
other methods by better preserving feasibility with respect to inequality
constraints (5 inequality violations versus 14, with a ten-fold reduction in
both mean and maximum violation magnitude), while achieving reasonable
generator dispatch and nodal voltage predictions compared to optimal. These
results follow from the fact that the neural predictor in our SiPhyR approach
uses the PhyR layer to select a feasible topology for each training instance
upon which it predicts a feasible power flow solution (i.e. equality
constraints are satisfied) and learns to reduce inequality violations while
improving upon optimality objectives. Without PhyR in the training loop (as
with InSi and InSi2R) the neural predictor is unable to preserve feasibility
of inequality constraints: rather than considering a single topology, it must
satisfy power flow across a (potentially large) set of possible topologies
(i.e. $z_{\tau}$ takes on non-binary values when PhyR is not used). Not only
is this a challenging task, but there may not exist a power flow solution
which remains feasible across multiple topologies. Even with explicit
selection of a topology during testing (but not training) as with InSi2R,
neither feasibility nor optimality can be improved upon, as clear by the same
DispErr and VoltErr for InSi and InSi2R (rows 1 and 2 of Table 1). A comment
must be made on the TopErr. Model variance is a challenge in reconfiguration
prediction, with significant spread in TopErr across multiple predictors.
Prediction performance can be improved substantially when considering the best
performing predictor in the ensemble (as shown in brackets): the TopErr for
SiPhyR reduces from 41.5% to 13.7%, and now has comparable performance to the
Supervised-InSi method which has full access to the optimal solution. Grid-
SiPhyR then produces the lowest TopErr of all tested methods.

Grid-SiPhyR can learn to optimize while preserving feasibility: Table 1 shows
that the proposed end-to-end learning framework can obtain comparable
performance to a supervised approach. This ability of Grid-SiPhyR to learn to
optimize is essential to applications in power systems, where the optimal
solution is typically unknown and is computationally prohibitive to obtain for
training data. Clearly, supervised methods have lower optimality errors as
they are given full knowledge of the optimal solution. At this point it is
necessary to distinguish between optimality and feasibility. The power system
is a critical infrastructure and meeting electricity demand with available
generation is a critical action. Thus feasibility is vastly more important
than optimality: only when we have feasibility, can we begin to think about
optimality. This indisputable prioritization gives a clear criterion for
evaluating the performance across different methods. Physics-informed
approaches like Grid-SiPhyR which prioritizes feasibility explicitly in the
design result in the lowest feasibility errors (even compared to Supervised-x-
pen) and has higher prediction accuracy.

Grid-SiPhyR can scale to larger grids and generalize across datasets: Table 2
shows that once again the proposed PhyR-based predictors outperforms InSi2R on
all metrics (particularly for feasibility), especially when tested on
different datasets than training. It is also immediately obvious that the
prediction performance is significantly enhanced by training on representative
load and generation datasets (i.e. DS2 and DS5). Comparing the error metrics
for DS1 on DS3 (top right) with DS2 on DS3 (middle right), the errors are
significantly reduced for the latter: 40% lower DispErr, 95% lower VoltErr,
87% lower Mean ineq, and the number of inequality violations reduces from 8%
to 4%. Since real-world data of network topologies and customer load profiles
are not widely available due to security and privacy concerns, the creation of
synthetic datasets with representative load profiles and DER penetration
requires considerable effort, but is a valuable and necessary task. The
datasets generated in this work are a key contribution towards supporting the
development, testing, and comparison of algorithms for power systems control
and optimization.

Table 1: Results for the BW-33 grid tested on 876 instances. The committee machine performance (i.e. ensemble average over 10 trained models) on each metric averaged over all test instances is shown. The performance of the predictor with the lowest TopErr is shown in brackets for select metrics. Lower values are better for all metrics. Method | Metric
---|---
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 3.24e-2 (3.44e-2) | 2.30e-3 | 49.7% (23.2%) | 1.53e-3 | 0.148 | 16.3 (18.3)
InSi2R | 3.24e-2 (3.44e-2) | 2.30e-3 | 48.6% (19.9%) | 1.02e-3 | 0.114 | 14.1 (15.3)
ClaPhyR | 2.86e-2 (3.29e-2) | 1.12e-3 | 52.4%(42.1%) | 3.80e-4 | 2.40e-2 | 4.42 (2.94)
SiPhyR | 2.89e-2 (8.92e-3) | 1.69e-3 | 41.5%(13.7%) | 4.79e-4 | 4.23e-2 | 5.72 (3.94)
InSiPhyR | 3.18e-2 (3.72e-2) | 1.13e-3 | 44.6%(21.6%) | 5.00e-4 | 4.88e-2 | 6.02 (5.82)
Supervised-InSi | 8.61e-4 | 4.65e-4 | 14.2% | 5.52e-3 | 0.817 | 37.4
Supervised-PhyR | 1.51e-4 | 2.36e-4 | 29.9% | 4.29e-3 | 0.888 | 19.6
Supervised-InSi-pen | 1.00e-3 | 2.80e-3 | 25.8% | 1.55e-3 | 0.173 | 15.7
Supervised-PhyR-pen | 5.78e-4 | 1.35e-3 | 33.6% | 5.49e-4 | 4.84e-2 | 7.26
Table 2: Results for the TPC-94 grid tested on 8640 instances. All networks were trained on the DS1 (perturbed loads), DS2 (residential loads), and DS5 (residential loads without solar profiles) datasets, and are tested on datasets DS2 and DS3 (mixed residential and commercial loads). Lower values are better for all metrics. Method (trained on DS1 | DS2, residential loads | DS3, mixed loads
---|---|---
DispErr | VoltErr | Mean ineq | Num ineq | DispErr | VoltErr | Mean ineq | Num ineq
InSi2R | 1.03 | 1.47e-2 | 3.20e-2 | 165 | 1.10 | 1.02e-2 | 3.69e-2 | 166
ClaPhyR | 1.77e-2 | 3.55e-3 | 4.43e-3 | 107 | 1.99e-2 | 3.41e-3 | 5.01e-3 | 122
SiPhyR | 1.72e-2 | 2.91e-3 | 4.33e-3 | 106 | 1.67e-2 | 2.16e-3 | 4.47e-3 | 118
Method (trained on DS2 | DS2, residential loads | DS3, mixed loads
DispErr | VoltErr | Mean ineq | Num ineq | DispErr | VoltErr | Mean ineq | Num ineq
InSi2R | 1.12e-2 | 3.98e-3 | 1.94e-3 | 71.2 | 2.98e-2 | 2.22e-3 | 5.36e-3 | 108
ClaPhyR | 1.00e-2 | 2.38e-3 | 5.07e-4 | 21.6 | 9.987e-3 | 1.91e-3 | 1.40e-3 | 64.7
Method (trained on DS5 | DS2, residential loads | DS3, mixed loads
DispErr | VoltErr | Mean ineq | Num ineq | DispErr | VoltErr | Mean ineq | Num ineq
InSi2R | 1.05e-2 | 3.86e-3 | 1.45e-3 | 64.3 | 1.29e-2 | 2.22e-3 | 3.04e-3 | 104
ClaPhyR | 9.18e-3 | 2.71e-3 | 8.92e-4 | 39.1 | 9.94e-3 | 2.08e-3 | 1.78e-3 | 79.2

Grid-SiPhyR is an enabling technology for essential grid operations in future
decarbonized energy systems: Simulation results on the two canonical
distribution grids show that the dynamic reconfiguration paradigm enabled by
Grid-SiPhyR contributes to three essential power system design goals: improves
grid efficiency by reducing electrical losses, ensures grid operability by
improving voltage profiles, and supports clean energy directives by increasing
solar energy utilization. Specifically, dynamic reconfiguration can reduce
line losses by 23% (320 MWh, powers an additional 30 US households for a
year). It can also significantly improve voltage profiles in heavily loaded
networks by increasing network-wide voltages and reducing the number of
voltage violations by as much as 66%. Finally, the use of dynamic
reconfiguration allows local solar generation to be directly connected to
loads which increases solar utilization by 6% (250 MWh, powers 23 US
households with clean energy for a year). This increase in local solar
utilization reduces reliance on electricity imported from the bulk system. In
regions where fossil-based generation dominates the energy mix, this
corresponds to reductions in CO2 emissions; for Massachusetts (state in the
US), the reduction is about 107 metric tons of CO2 per year. Detailed
simulation results are presented in Appendix E.

##  7 Conclusion

We have presented a general framework, SiPhyR for end-to-end learning to
optimize for combinatorial problems. Our approach integrates a rounding
function directly within a physics-informed differentiable framework to ensure
certified satisfiability of critical equality constraints and binary
variables. We apply our method to the power grid and show that Grid-SiPhyR can
learn to optimize and provide good and fast solutions while preserving
feasibility of key constraints. Future work will consider formally extending
the SiPhyR framework for integer variables and set constraints. We will also
explore the development of Grid-SiPhyR to generalize to new grid topologies
using GNNs or transfer learning techniques. Finally, we will explore the use
of generative adversarial networks for developing diverse and representative
datasets for power systems optimization and control.

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##  Appendix A Additional experiment results and warm-start results

Extensive evaluation of the proposed physics-informed framework were carried
out with detailed discussion presented in Section 6 of the main paper.
Additional simulation results are presented in this Appendix.

###  A.1 Summary of key results

The key results presented in Section 6 are repeated here for completion.

  1. Result 1:

Physics-informed methods enable higher prediction accuracy in both optimality
and feasibility metrics;

  2. Result 2:

The proposed SiPhyR method can learn to optimize. Supervised learning
outperforms unsupervised learning in optimality metrics, but underperforms in
feasibility metrics;

  3. Result 3:

Datasets used for algorithm development must be representative of real system
load and generation characteristics;

  4. Result 4:

Model variance is a challenge in reconfiguration prediction, with significant
spread in prediction accuracy of topology error across multiple predictors

  5. Result 5:

Warm-start of MIPs is challenging: warm-start with the neural prediction can
improve worst-case performance, but additional tuning is needed to further
reduce solve time. Even then, there are no guarantees on optimizer
performance.

Recall that for all experiments a committee machine approach was taken where
10 models were trained with independent weight initialization (following He
initialization) for each parameter. Unless otherwise indicated, the results
presented are the average over the 10 predictors.

Table 3 presents the performance results for the predictor with the best
performance on topology error for the BW-33 grid. This predictor shows
significant improvement in prediction accuracy across all metrics, and in
particular for the topology error (TopErr), as compared to the ensemble
average over all committee members (see Table 1).

Table 4 thru Table 6 present results for the TPC-94 grid. The predictors are
trained on datasets DS1, DS2, or DS5, and tested on datasets DS1, DS2, DS3,
and DS4. The results showcase the scalability and superior performance of the
proposed SiPhyR method, and indicate the importance of representative training
data, as discussed in Section 6.

Table 3: Results for the BW-33 grid tested on 876 instances. The performance for the predictor with the lowest TopErr is presented (i.e. a single predictor which performs the best among the 10 predictors trained in the committee). Lower values are better for all metrics. Method | Metric
---|---
DispErr | VoltErr | TopErr | Max TopErr | Mean ineq | Max ineq | Num ineq
InSi | 3.44e-2 | 4.01e-3 | 23.2% | 100% | 2.22e-3 | 0.190 | 18.3
InSi2R | 3.44e-2 | 4.01e-3 | 19.9% | 79.5% | 1.34e-3 | 0.147 | 15.3
ClaPhyR | 3.29e-2 | 1.21e-3 | 42.1% | 75.0% | 3.06e-4 | 2.06e-2 | 2.94
SiPhyR | 8.92e-3 | 3.06e-3 | 13.7% | 25.0% | 3.56e-4 | 2.56e-2 | 3.94
InSiPhyR | 3.72e-2 | 6.68e-4 | 21.6% | 50.0% | 5.06e-4 | 5.67e-4 | 5.82
Table 4: Results for the TPC-94 grid tested on 8760 instances. All networks were trained on the DS1 (perturbed loads) dataset and are tested on datasets DS1, DS2, DS3, and DS4. Lower values are better for all metrics. Method (tested on DS1, perturbed) | Metric
---|---
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.41 | 3.31e-2 | 44.3% | 3.21e-2 | 3.01 | 162
InSi2R | 1.41 | 3.31e-2 | 43.6% | 3.20e-2 | 2.98 | 161
PhyR | 1.78e-2 | 1.52e-2 | 45.4% | 8.11e-4 | 5.25e-2 | 40.5
ClaPhyR | 1.15e-2 | 1.31e-2 | 44.1% | 7.67e-4 | 4.63e-2 | 41.1
SiPhyR | 1.12e-2 | 1.51e-2 | 45.4% | 7.27e-4 | 4.25e-2 | 37.5
InSiPhyR | 1.10e-2 | 1.39e-2 | 44.4% | 7.87e-4 | 4.62e-2 | 42.5
Method (tested on DS2, residential) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.03 | 1.47e-2 | 41.6% | 3.22e-2 | 2.61 | 166
InSi2R | 1.03 | 1.47e-2 | 40.9% | 3.20e-2 | 2.61 | 165
PhyR | 1.71e-2 | 4.80e-3 | 43.7% | 4.32e-3 | 0.960 | 107
ClaPhyR | 1.77e-2 | 3.55e-3 | 43.3% | 4.43e-3 | 1.00 | 107
SiPhyR | 1.72e-2 | 2.91e-3 | 46.6% | 4.33e-3 | 0.959 | 106
InSiPhyR | 1.71e-2 | 5.18e-3 | 44.0% | 4.25e-3 | 0.963 | 103
Method (tested on DS3, mixed) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.10 | 1.02e-2 | 42.8% | 3.71e-2 | 3.09 | 167
InSi2R | 1.10 | 1.02e-2 | 41.6% | 3.69e-2 | 3.09 | 166
PhyR | 1.63e-2 | 3.61e-2 | 45.0% | 4.35e-3 | 0.960 | 117
ClaPhyR | 1.99e-2 | 3.41e-3 | 44.4% | 5.01e-3 | 0.988 | 122
SiPhyR | 1.67e-2 | 2.16e-3 | 44.7% | 4.47e-3 | 0.958 | 118
InSiPhyR | 1.64e-2 | 3.87e-3 | 42.4% | 4.31e-3 | 0.962 | 114
Method (tested on DS4, solar error) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.03 | 1.31e-2 | 41.2% | 3.22e-2 | 2.62 | 166
InSi2R | 1.03 | 1.31e-2 | 40.5% | 3.21e-2 | 2.62 | 165
PhyR | 1.71e-2 | 4.06e-3 | 43.7% | 4.33e-3 | 0.960 | 107
ClaPhyR | 1.78e-2 | 3.01e-3 | 43.3% | 4.42e-3 | 0.959 | 107
SiPhyR | 1.73e-2 | 2.38e-3 | 46.5% | 4.33e-3 | 0.959 | 106
InSiPhyR | 1.71e-2 | 4.45e-3 | 44.0% | 4.25e-3 | 0.963 | 103
Table 5: Results for the TPC-94 grid tested on 8760 instances. All networks were trained on the DS2 (residential loads) dataset and are tested on datasets DS1, DS2, DS3, and DS4. Lower values are better for all metrics. Method (tested on DS2, residential) | Metric
---|---
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.12e-2 | 3.98e-3 | 47.8% | 2.29e-3 | 0.324 | 73.1
InSi2R | 1.12e-2 | 3.98e-3 | 45.7% | 1.94e-3 | 0.148 | 71.2
ClaPhyR | 1.00e-2 | 2.38e-3 | 48.6% | 5.07e-4 | 5.57e-3 | 21.6
Method (tested on DS1, perturbed) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 0.449 | 1.80e-2 | 47.5% | 3.80e-2 | 2.95 | 192
InSi2R | 0.449 | 1.80e-2 | 45.5% | 3.77e-2 | 2.95 | 190
ClaPhyR | 0.148 | 2.35e-2 | 46.2% | 2.35e-2 | 2.36 | 168
Method (tested on DS3, mixed) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 2.98e-2 | 2.22e-3 | 46.6% | 5.58e-3 | 0.488 | 110
InSi2R | 2.98e-2 | 2.22e-3 | 44.3% | 5.36e-3 | 0.358 | 108
ClaPhyR | 9.987e-3 | 1.91e-3 | 44.0% | 1.40e-3 | 0.125 | 64.7
Method (tested on DS4, solar error) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.14e-2 | 3.27e-3 | 47.5% | 2.37e-3 | 0.327 | 74.0
InSi2R | 1.14e-2 | 3.27e-3 | 45.4% | 2.02e-3 | 0.153 | 72.0
ClaPhyR | 1.00e-2 | 1.96e-3 | 48.2% | 5.09e-4 | 5.55e-2 | 21.9
Table 6: Results for the TPC-94 grid tested on 8760 instances. All networks were trained on the DS5 (flat solar) dataset and are tested on datasets DS2, DS3, and DS4. Lower values are better for all metrics. Method (tested on DS2, residential) | Metric
---|---
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.05e-2 | 3.86e-3 | 47.8% | 1.80e-3 | 0.314 | 66.2
InSi2R | 1.05e-2 | 3.86e-3 | 44.8% | 1.45e-3 | 0.120 | 64.3
ClaPhyR | 9.18e-3 | 2.71e-3 | 47.9% | 8.92e-4 | 9.26e-2 | 39.1
Method (tested on DS3, mixed) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.29e-2 | 2.22e-3 | 46.5% | 3.33e-3 | 0.340 | 105
InSi2R | 1.29e-2 | 2.22e-3 | 44.3% | 3.04e-3 | 0.230 | 104
ClaPhyR | 9.94e-3 | 2.08e-3 | 46.3% | 1.78e-3 | 0.133 | 79.2
Method (tested on DS4, solar error) | Metric
DispErr | VoltErr | TopErr | Mean ineq | Max ineq | Num ineq
InSi | 1.05e-2 | 3.10e-3 | 47.8% | 1.82e-3 | 0.314 | 67.1
InSi2R | 1.05e-2 | 3.10e-3 | 44.8% | 1.47e-3 | 0.121 | 65.2
ClaPhyR | 9.16e-3 | 2.18e-3 | 47.9% | 9.07e-4 | 9.37e-2 | 39.7

###  A.2 Performance of Grid-SiPhyR as a warm-start technique

Prior work has looked at developing machine learning models to predict warm-
start points for traditional optimization solvers. This section investigates
the ability for SiPhyR to act as a warm-start predictor. Before presenting the
warm-start experiment details and result, it is necessary to note that warm-
start points are generally effective techniques to reduce optimization time.
However, the benefits seen for general linear and nonlinear optimization are
not replicated for mixed integer optimization problems. The techniques
employed to solve the class of mixed integer problems, of which grid
reconfiguration is a member, rely on multiple heuristics such as branch and
bound, cutting planes, node presolve, and symmetry detection (among others).
The selection of which techniques are used, the order in which they are used,
and the techniques themselves are stochastic in nature. In addition, the
optimization solver still needs to prove optimality of any solution (up to a
tolerance level), which is itself a difficult task. For these reasons, there
are no guarantees that providing an initial solution to an MIP solver will
reduce computational time or effort.

####  A.2.1 Experiment Setup

All simulations of the reconfiguration MILP were carried out with Gurobi,
using the Yalmip interface. Different optimization solvers have different
requirements for warm-start points, specifying whether the point must be
feasible or not, and whether a full set of variables must be provided or not.
For Gurobi MIPs the initial point must be feasible, but does not need to be
complete for all variables. Grid-SiPhyR guarantees feasibility of any
prediction for the equality constraints and a subset of the inequality
constraints. Translating the Grid-SiPhyR prediction to a warm-start point can
be done quite easily by simply omitting any variables which violate the
inequality constraints.

Three sets of simulations are carried. Case 1: the reconfiguration problem is
solved without any warm-start. Case 2: the reconfiguration problem is solved
by providing the optimal solution (from Gurobi) as the warm-start point to
Gurobi. This provides a baseline comparison for the performance of warm-start
on the reconfiguration problem. Case 3: the reconfiguration problem is solved
by providing the Grid-SiPhyR prediction as the warm-start point to Gurobi. The
optimization routine metrics are measured for each of these simulations. If
the warm-start approach is effective for the reconfiguration MILP, it’s
expected that case 2 will significantly outperform both cases 1 and 3, and
preferably case 3 will outperform case 1.

The simulations in case 2 were used to determine the set of variables to
provide as a warm-start, denoted with the subscript ‘0’. When all variables
$X_{0}=\left[P^{G}\;Q^{G}\;V\;P_{ij}\;Q_{ij}\;y_{ij}\;z_{ij}\;z_{ji}\right]$
are provided as a warm-start point, the optimization solver is unable to
identify it as a feasible solution. By testing different variables, the
following were selected to provide as a warm-start:
$X_{0}=\left[P^{G}\;Q^{G}\;V\;y_{ij}\;z_{ij}\;z_{ji}\right]$. Any variables
that violated inequality constraints were excluded.

####  A.2.2 Performance metrics: optimization routine metrics

The optimization routine metrics evaluate the performance of the Gurobi
optimization solver when solving the MILP. It must be noted that the proposed
PhyR-based framework is an end-to-end learning to solve the reconfiguration
problem, and does not require an external solver. However, Gurobi is used to
generate the data for the supervised framework. Additionally, a set of
experiments are conducted to evaluate the PhyR-based framework as a warm-start
technique for traditional optimization solvers. This experiment is evaluated
based on the below optimization routine metrics.

Solve time: time in seconds to solve the optimization problem, through the
YALMIP interface

Node count: number of branch-and-cut nodes explored in the most recent
optimization

Iteration count: number of simplex iterations performed during the most recent
optimization. Note that the reconfiguration problem is a linear program (LP)
when the binary variables are fixed. Thus for every node in the branch-and-cut
algorithm, the resulting optimization problem is an LP, efficiently solved
using the simplex method.

####  A.2.3 Experimental results for warm-start

Key takeaway: Warm-start with the neural prediction can improve worst-case
performance, but additional tuning is needed to further reduce solve time.
Even then, there are no guarantees on optimizer performance. Warm-start for
MIPs remains challenging.

Tables 7 and 8 present the results on the BW-33 and TPC-94 grids respectively.
The key result from these results is that warm-start for the reconfiguration
MIP is an ineffective technique, and may even result in poorer optimizer
performance (along all three metrics). Interestingly, on average case 2
slightly increases solve time for the BW-33 grid while decreasing the node and
iteration count, but is otherwise comparable. This difference across the
metrics may be due to the size of the optimization problem and the relative
time taken to reconstruct the full variable space and certify feasibility.
These steps are not explicitly measured by the solver, but it has been noted
by Gurobi developers that these actions can take considerable time. For the
TPC-94 grid, case 2 improved optimizer performance on all metrics, by a larger
margin than the BW-33 grid. Again, this may be due to the relative time taken
in showing feasibility and optimality, versus searching the feasible space.
Further, the search space for the TPC-94 grid is significantly larger than the
BW-33 grid – in topology alone, there are $2^{14}$ possible unique topologies
for TPC-94 as compared to $2^{7}$ for BW-33. So although the optimality errors
in neural prediction are comparable for the two grids, the warm-start in case
2 for the TPC-94 grid is more meaningful by starting closer to the optimal
solution.

The performance of warm-start with case 3 is quite inconclusive. For the BW-33
grid, case 3 outperforms both case 1 and 2 on both mean and median solve
times, which is unexpected. For the TPC-94 grid, case 3 underperforms against
case 2 (expected) and is comparable to case 1. Notably the worst-case solve
time in case 3 is significantly reduced over both cases 1 (desired) and 2
(unexpected). Figure 2 plots the warm-start metrics for all three cases,
evaluated on the TPC-94 grid. The desired shape in the plot is an asymmetrical
‘V’, where the left branch is longer than the right. This plot gives more
insight into the performance of the different warm-start methods. Looking at
the solve time, the desired shape is emerging, with case 2 outperforming the
others and forming the vertex of the ‘V’. The shorter right side shows that
while case 3 does not offer a reduction in solve time on average, it does
permit a smaller in spread in solve times. This can be meaningful in
applications where decisions must be made within a sensitive time window that
is violated when using case 1, such as dynamic reconfiguration or electricity
market clearing. Similar results are also seen for the other two metrics. It
is possible that improving the neural prediction performance will further
improve the warm-start performance of case 3, with a lower bound in solve time
provided by case 2.

Table 7: Warm-start results for the BW-33 grid Metric |  | Case 1
---
Without warm-start
| Case 2
---
Warm-start at optimal
| Case 3
---
Warm-start at Grid-SiPhyR prediction
Solve time |  | Mean: 0.2015
---
Median: 0.1586
Max: 0.7534
| Mean: 0.2057
---
Median: 0.1661
Max: 0.8775
| Mean: 0.1753
---
Median: 0.1371
Max: 1.0414
Node count |  | Mean: 2.2094
---
Median: 1
| Mean: 1.8233
---
Median: 1
| Mean: 2.2215
---
Median: 1
Iteration count |  | Mean: 2781
---
Median: 2629
| Mean: 2770
---
Median: 2656
| Mean: 2796
---
Median: 2663
Table 8: Warm-start results for the TPC-94 grid Metric |  | Case 1
---
Without warm-start
| Case 2
---
Warm-start at optimal
| Case 3
---
Warm-start at Grid-SiPhyR prediction
Solve time |  | Mean: 1.4694
---
Median: 1.3322
Max: 6.6272
| Mean: 0.4516
---
Median: 0.3993
Max: 8.0498
| Mean: 1.4633
---
Median: 1.4157
Max: 3.4846
Node count |  | Mean: 182
---
Median: 137
| Mean: 94.95
---
Median: 51
| Mean: 182.4
---
Median: 140
Iteration count |  | Mean: 11405
---
Median: 10260
| Mean: 5290
---
Median: 3605
| Mean: 11453
---
Median: 10352
Figure 2: Warm-start results on the TPC-94 grid, plotting the optimization
routine metrics. The warm-start experiment included 8640 data points, of which
results for 700 randomly selected data points are plotted here.

##  Appendix B Details on hyperparameter tuning

The following parameters were kept fixed for all neural network models across
all experiments, based on a initial experimentation to ensure convergence and
stability.

  * •

Epochs: 1500 for BW-33, and 2500 for TPC-94

  * •

Batch size: 200

  * •

Depth of network: 2 hidden layers; hidden layer size is kept uniform for both
hidden layers

  * •

Learning rate: $\left\\{0.001,0.0005,0.0001\right\\}$

Based on the universal approximation theorem it is known that a neural network
with a single hidden layer can represent any linear function, and one with two
hidden layers can represent any arbitrary nonlinear function. While the
underlying power flow constraints are linear (the reconfiguration problem is
using the Linear DistFlow model), the presence of binary variables introduces
a nonlinearity to the overall problem. Thus a network with 2 hidden layers is
the smallest network that can be expected to perform well. Initial experiments
were done to compare networks with one and two hidden layers, and corroborate
this statement. A small number of initial experiments were also done to
increase the number of hidden layers or adding 30% dropout on each layer of
the neural network. Overall these architectures provide minimal improvement in
performance, and further tuning was not done.

The key results from the hyperparameter tuning are summarized here:

  * •

Neural network width for BW-33 is 5 neurons per hidden layer, and for TPC-94
is 300 neurons per hidden layer. This captures the tradeoff between training
speed and prediction performance on optimality and feasibility metrics

  * •

InSi-based methods require lower learning rates (1e-4 versus 1e-3)

  * •

Datasets with real load data (DS2 and DS5) require lower learning rates (1e-4
vs 1e-3)

Select results from the hyperparameter tuning are presented next.

Figure 3: Tuning the hidden layer size for the BW-33 grid for the SiPhyR
method. The learning rate is set to be $lr=0.001$.

Fig. 3 shows the validation results during neural training of SiPhyR with two
hidden layers across different depths $N_{nn}$. Figure 4 shows similar results
for the InSi method. It must be noted that the InSi method requires a lower
learning rate than SiPhyR (using $lr=0.0001$ for InSi vs. $lr=0.001$ for
PhyR). Overall the training plots for both SiPhyR and InSi methods are
comparable. The faster convergence in validation loss of SiPhyR may be
attributed to the higher learning rate. The jumping behaviour in the topology
error plot of SiPhyR as compared to the smoother plot of the InSi method can
be explained by the nature of the physics-informed rounding: the SiPhyR method
enforces integer values for switch status predictions, so a change in the
topology prediction may result in step-like behaviour of the corresponding
error metric. In comparison, the InSi method does not enforce integer
solutions, so the corresponding metric is not expected to have a step-like
behaviour. Overall the prediction performance with $N_{nn}=5$ is reasonable
across all three plots, providing a tradeoff between fast convergence, low
error, and having a light-weight small network. This width is selected for the
BW-33 network for all further testing.

Figure 4: Tuning the hidden layer size for the BW-33 grid for the InSi method.
The learning rate is set to be $lr=0.0001$.

Figure 5 shows the hyperparameter tuning plots for SiPhyR on the TPC-94 grid.
The prediction performance of $N_{nn}=\\{300,700\\}$ are best for the
inequality violations, DispErr, and VoltErr, while the largest network
$N_{nn}=1500$ performs the best on TopErr. This wider network however requires
a lower learning rate, and so converges more slowly on the other performance
metrics. The tradeoff between convergence time and overall prediction
performance is shown here. The same experiment was conducted for the InSi
method, showing similar performance across different $N_{nn}$. Of note, the
InSi method required a lower learning rate ($lr=0.0005$ or $lr=0.0001$).
Overall the prediction performance with $N_{nn}=300$ is reasonable, and this
width is selected for all further testing. It should be noted that the InSi
method requires a lower learning rate than the PhyR method. The profile
datasets (DS2 thru DS5) also require lower learning rates than the perturbed
dataset DS1.

Figure 5: Tuning the hidden layer size for the TPC-94 grid for the SiPhyR
method, on DS-1 with 9000 data points. The learning rate is set to be
$lr=0.001$ for all $N_{nn}$ except $N_{nn}=\\{1000,1500\\}$ for which
$lr=0.0001$.

##  Appendix C Grid reconfiguration problem

The grid reconfiguration problem using the Linear DistFlow model of the
distribution grid [8] is formulated as below.

$\displaystyle\min_{\psi}\quad$ $\displaystyle f_{x}(\psi)$ (9a) s.t.
$\displaystyle v_{j}-v_{i}=-2(R_{ij}(P_{ij}-P_{ji})+X_{ij}(Q_{ij}-Q_{ji}))$
$\displaystyle\forall(i,j)\in\mathcal{T}_{\text{D}}\setminus\mathcal{T}_{\text{D}}^{sw}$
(9b) $\displaystyle
v_{j}-v_{i}\leq-2(R_{ij}(P_{ij}-P_{ji})+X_{ij}(Q_{ij}-Q_{ji}))+M(1-y_{ij})$
$\displaystyle\forall(i,j)\in\mathcal{T}_{\text{D}}^{sw}$ (9c) $\displaystyle
v_{j}-v_{i}\geq-2(R_{ij}(P_{ij}-P_{ji})+X_{ij}(Q_{ij}-Q_{ji}))-M(1-y_{ij})$
$\displaystyle\forall(i,j)\in\mathcal{T}_{\text{D}}^{sw}$ (9d) $\displaystyle
P_{j}^{G}-P_{j}^{L}=\sum_{k:(j,k)}P_{jk}-\sum_{i:(i,j)}P_{ij}$
$\displaystyle\forall j\in\mathcal{B}$ (9e) $\displaystyle
Q_{j}^{G}-Q_{j}^{L}=\sum_{k:(j,k)}Q_{jk}-\sum_{i:(i,j)}Q_{ij}$
$\displaystyle\forall j\in\mathcal{B}$ (9f) $\displaystyle z_{ij}+z_{ji}=1$
$\displaystyle\forall(i,j)\in\mathcal{T}_{D}\setminus\mathcal{T}_{D}^{sw}$
(9g) $\displaystyle z_{ij}+z_{ji}=y_{ij}$
$\displaystyle\forall(i,j)\in\mathcal{T}_{D}^{sw}$ (9h)
$\displaystyle\sum_{(i,j)\in\mathcal{T}_{D}^{sw}}y_{ij}=(N-1)-(M-M_{sw})$ (9i)
$\displaystyle\sum_{j:(i,j)}z_{ij}+z_{ji}\geq 1$ $\displaystyle\forall
j\in\mathcal{B}$ (9j) $\displaystyle y_{ij}\in\\{0,1\\}$
$\displaystyle\forall(i,j)\in\mathcal{T}_{D}^{sw}$ (9k) $\displaystyle
z_{ij},z_{ji}\in\\{0,1\\}$ $\displaystyle\forall(i,j)\in\mathcal{T}_{D}$ (9l)
$\displaystyle 0\leq P_{ij}\leq Mz_{ij}$
$\displaystyle\forall(i,j),(j,i)\in\mathcal{T}_{D}$ (9m) $\displaystyle 0\leq
Q_{ij}\leq Mz_{ij}$ $\displaystyle\forall(i,j),(j,i)\in\mathcal{T}_{D}$ (9n)
$\displaystyle\underline{P_{j}^{G}}\leq P_{j}^{G}\leq\overline{P_{j}^{G}}$
$\displaystyle\forall j\in\mathcal{B}$ (9o)
$\displaystyle\underline{Q_{j}^{G}}\leq Q_{j}^{G}\leq\overline{Q_{j}^{G}}$
$\displaystyle\forall j\in\mathcal{B}$ (9p)
$\displaystyle\underline{v_{j}}\leq v_{j}\leq\overline{v_{j}}$
$\displaystyle\forall j\in\mathcal{B}\setminus j^{\\#}$ (9q) $\displaystyle
v_{j^{\\#}}=1$ (9r)

In Eq. 9 a general distribution grid is described as a graph
$\Gamma(\mathcal{B},\mathcal{T}_{D})$, where $\mathcal{B}$ is the set of N
nodes and $\mathcal{T}_{D}\coloneqq\\{(i,j)\\}$ is the set of M edges. The
subset of lines with switches $\mathcal{T}_{D}^{sw}\subset\mathcal{T}_{D}$
contains $M_{sw}$ switches which can be turned on/off to change the grid
topology. The distribution grid is connected to the point of common coupling
(PCC) of the distribution grid to the bulk transmission grid, denoted by node
$j^{\\#}$. The power variables are the real and reactive power loads at every
node $P^{L}_{j},Q^{L}_{j}$, the real and reactive power generation at every
node $P^{G}_{j},Q^{G}_{j}$, the squared magnitude of nodal voltages $v$, and
directed power flow through the distribution lines
$P_{ij},P_{ji},Q_{ij},Q_{ji}$. The line parameters of resistance and reactance
are denoted as $R_{ij}$ and $X_{ij}$ respectively. The grid topology variables
are the switch status
$y_{ij}\forall(i,j)\in\mathcal{T}_{D}^{sw}\subset\mathcal{T}_{D}$ where
$y_{ij}=1$ is closed and $y_{ij}=0$ is open, and the selection of direction of
power flow through each line $z_{ij},z_{ji}$. These variables are binary
variables.

The objective function (9a) approximates the electrical line losses which when
minimized ensures efficient grid operations. Constraints (9b)-(9d) describe
Ohm’s law across all lines in the network, with equality constraints
describing lines without switches, and inequality constraints describing lines
with switches. Notably, the big-M relaxation is used to describe the
conditional constraints describing Ohm’s law for lines
$(i,j)\in\mathcal{T}_{D}^{sw}$ – i.e. Ohm’s law must be binding when the
switches are closed ($y_{ij}=1$) but does not apply when the switches are open
($y_{ij}=0$). Constraints (9e) and (9f) describe lossless power balance at
every node for both real and reactive power.

Constraints (9g)-(9n) describe the topology selection through switch-status,
grid radiality and connectivity constraints, enforce binary constraints on the
$y_{ij},z_{ij},z_{ji}$ variables, and restrict power flow through a line based
on switch-status. In particular, constraint (9i) restricts the number of
closed switches in the grid so it is radial with $N-1$ total branches, where
$L$ switches must be closed and the remaining $M_{sw}-L$ must be open. This
constraint is central to the application of the physics-informed rounding
routine. Constraint (9j) enforces connectivity by requiring power to flow into
or out of a node along at least one line. It should be noted that typical
reconfiguration problem statements also include an arborescence constraint
[50], either explicitly or implicitly in the formulation of the radiality
constraint. However, the increasing penetration of DERs voids this assumption,
and multiple generating sources (roots of the tree) must be permitted. We have
relaxed this arborescence constraint in (9j). Various other mathematical
formulations of radiality and connectivity constraints include constraints on
the determinant of the branch-to-node incidence matrix or spanning tree
constraints and other graph theoretic approaches [33, 52, 5, 32]. However,
many of these suffer from high computational requirements and additional
complexity, and do not leverage the fact that grid connectivity can be ensured
by power flow constraints under normal operation. Our formulation accounts for
this.

Constraints (9o)-(9p) describe generator operating limits, and (9q)-(9r)
describes grid voltage limits where the voltage at the PCC is assumed to be
fixed at 1pu, as is common practice in power systems.

##### Extension for no export limits:

Additional set of constraints described below can be added to describe “no
export” limits on the PCC, where net generation excess of net load in the
distribution grid cannot be injected into the transmission grid. This is
necessary in regions where distribution grids are not permitted to export
power to the bulk grid, or where the amount of power that can be exported is
limited.

$\displaystyle z_{ij^{\\#}}=0$ (no power flow from grid through PCC) (10a)
$\displaystyle\sum_{j:(j^{\\#},j)}P_{j^{\\#},j}=P_{j^{\\#}}^{G}$ (no real
power export) (10b)
$\displaystyle\sum_{j:(j^{\\#},j)}Q_{j^{\\#},j}=Q_{j^{\\#}}^{G}$ (no reactive
power export) (10c)

##### Extension for power outage conditions:

Extensions of the presented model can be made for fault conditions where an
element of the grid has failed (ex. a tree knocks down a power line, or a
generator fails). The connectivity constraint must then be relaxed and
islanding of sections of the grid is permitted, wherein the single
distribution grid is broken down into smaller grids which are not electrically
connected to one another. It must be noted however, that this is an
undesirable phenomenon and only happens in extreme cases of outages and system
failure.

##### Extension for different constraints and applications:

Further extensions to the reconfiguration problem include distinctions between
hard and soft constraints, considering the optimal switch change order to go
from topology A to topology B, and considering grid outage conditions and
subsequent generator restart and load recovery. Note that soft constraints can
include lines which can exceed thermal limits for short periods of time during
a reconfiguration activity.

##### Objective function:

The MILP detailed above is solved to minimize an objective function
$f_{x}(\psi)$. For a modern distribution grid with high DER penetration,
various objectives are sought after by grid operators. Some such objectives
include minimizing electrical line losses (maximizing grid efficiency),
minimizing costs for power generation, minimizing congestion, improving
voltage profiles across the distribution feeder, reducing peak power demand,
ensuring reliability of service (ex. higher capacity margins for feeders and
supply transformers), and balancing load. Depending on the types of switches
in the grid, operators may also minimize the cost incurred by actuating
switches. In general, these objectives can be formulated using a convex
function, thus retaining the uniqueness and global optimality of the optimal
power flow solution. Note that the presence of binary variables means the
search space is not convex nor continuous.

##  Appendix D Datasets

We present the datasets used in the evaluation of the Grid-SiPhyR framework.
Notably, there are few good training datasets available for power system
applications, and no known datasets for the grid reconfiguration problem. The
creation of these datasets is a contribution of this research. There are three
relevant parameters to each dataset: (i) network topology, line parameters,
and location of switches; (ii) location, magnitude, and time-varying profile
of loads; and (iii) location, capacity, and time-varying profile of
distributed generation. Note that (i)-(iii) are the input to the
reconfiguration problem, and determine the optimal configuration of the
network. In the sections that follow, each network dataset and (i)-(iii) will
be presented. The network data (i) are taken from literature, as are the
location and magnitude of loads for a single time period. The time-varying
load profiles (ii) and the solar resource locations and generation profile
(iii) are developed as part of the dataset generation. Table 9 summarizes key
parameters of the two distribution grids (BW-33 and TPC-94) presented in
detail below.

Table 9: Size of reconfiguration problem for two canonical distribution grids | BW-33 | TPC-94
---|---|---
Number of nodes, $|\mathcal{B}|$ | 33 | 94
Number of lines and switches, $|\mathcal{T}_{\text{D}}|,|\mathcal{T}_{\text{D}}^{sw}|$ | 30, 8 | 83, 14
| Number of Discrete variables - topological, $y_{ij}$
---
8 | 14
| Number of Discrete variables - power flow, $z_{ij},z_{ji}$
---
74 | 194
Number of Continuous vars | 248 | 671
Number of Equality constraints | 134 | 369
Number of Inequality constraints | 545 | 1465
Size of Training data, $|x|$ | 64 | 186
Size of Independent variables, $|z|$ | 195 | 510
Size of Dependent variables, $|\varphi|$ | 134 | 369

###  D.1 33-Node Baran-Wu Grid, BW-33

The BW-33 grid presented in [8] is a canonical grid used in the
reconfiguration literature. The grid is very lossy, with losses up to 8% of
total load, and voltage profile violating voltage limits. These
characteristics make the BW-33 grid an excellent test case for dynamic
reconfiguration, with the objective function to reduce line losses. The
network is shown in Fig. LABEL:fig:33node_feeder. Grid data available in [8]
includes the grid topology and line parameters, as presented in Table
LABEL:tab:line_params_33, and location of loads and their nominal power demand
(P and Q) for a single period as presented in Table LABEL:tab:load_data_33.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: BW-33 distribution grid from [8]. The switches are highlighted in
green, sections of the grid highlighted and labelled in orange, and location
of community solar DERs noted in yellow squares.

####  D.1.1 Network Topology and Parameters

The grid consists of 33 nodes ($N=33$), 37 lines ($M=37$) of which 5 are tie
lines (normally open switches, NOS) and the remaining 32 are typically assumed
to be sectionalizing switches (normally closed switches, NCS). Topology and
line parameter data are presented in Table LABEL:tab:line_params_33. To
restrict the problem to a simpler test case, only a subset of the lines are
considered switchable – these include the 5 tie lines (numbered 33 to 37) and
3 NCS lines (line numbers 4, 10, and 26). These are highlighted in green in
Fig. LABEL:fig:33node_feeder, with dashed lines representing normally open
switches and solid lines representing normally closed switches.

####  D.1.2 Load Location and Profiles

The location of loads and their nominal power demand (P and Q) for a single
period is presented in Table LABEL:tab:load_data_33. To develop a diverse set
of training data, the maximum load perturbation at each node is restricted to
70% deviation from nominal value (i.e. $P^{\prime}=\delta
P,\delta=[0.3,1.7]$). The power factor of the loads, which describes the
relationship between the real and reactive power as
$pf=P/(\sqrt{P^{2}+Q^{2}})$, is kept constant to the pf in the nominal data.
This is a common approach used in literature [22, 57, 12].

####  D.1.3 Distributed Generation: Community Solar Dataset

We add a range of community solar facilities (each <5MW), up to a penetration
of 25.3% of nameplate capacity to baseline load. This is a modest DER
penetration compared to that which we would expect in the future grid, and
reflects solar uptake now and over the next few years. We divide the grid into
sections, based on the location of switches, and vary the location of
community solar farms amongst these sections. We denote the distribution of
these DERs (DD) as follows: (i) DD-U: uniform distribution of solar throughout
the grid; (ii-iv) DD-I, DD-II, DD-III: all facilities are in Sections I, II,
or III of the grid respectively; (v) DD-II+III: all facilities are in Sections
II and III of the grid. The DD and location of each community solar facility
is shown in Fig. 6, as indicated by the yellow squares. Different DDs are used
to consider effect on grid reconfiguration, line losses, and voltage profiles.
The location and nameplate capacity of each solar facility is provided in
Table LABEL:tab:solar_locations_33. Figure 7 shows a sample 24-hour generation
profile of a solar facility in the BW-33 dataset, at hourly intervals.

The solar generation data is taken from NREL’s System Advisory Model (SAM)
tool [42]. The data is of a 185kW distributed commercial solar PV facility,
located in Phoenix, AZ, using the SunPower SPR-E19-310-COM module, and SMA
America (STP 60-US-10, 400V) inverter. The DC to AC ratio is set to the
default of 1.2. The desired array size is set to 220kWdc, giving a total AC
capacity of 179.580kWac. All other parameters are left unchanged in the SAM
setup.

Table 10: BW-33 grid topology data and line parameters Branch No. | Upstream Node | Downstream Node | R [ohm] | X [ohm]
---|---|---|---|---
1 | 1 | 2 | 0.0922 | 0.0470
2 | 2 | 3 | 0.4930 | 0.2511
3 | 3 | 4 | 0.3660 | 0.1864
4 | 4 | 5 | 0.3811 | 0.1941
5 | 5 | 6 | 0.8190 | 0.707
6 | 6 | 7 | 0.1872 | 0.6188
7 | 7 | 8 | 0.7114 | 0.2351
8 | 8 | 9 | 1.030 | 0.7400
9 | 9 | 10 | 1.0440 | 0.7400
10 | 10 | 11 | 0.1966 | 0.0650
11 | 11 | 12 | 0.3744 | 0.1238
12 | 12 | 13 | 1.4680 | 1.1550
13 | 13 | 14 | 0.5416 | 0.7129
14 | 14 | 15 | 0.5910 | 0.5260
15 | 15 | 16 | 0.7463 | 0.5450
16 | 16 | 17 | 1.2890 | 1.7210
17 | 17 | 18 | 0.7320 | 0.5740
18 | 2 | 19 | 0.1640 | 0.1565
19 | 19 | 20 | 1.5042 | 1.3554
20 | 20 | 21 | 0.4095 | 0.4784
21 | 21 | 22 | 0.7089 | 0.9373
22 | 3 | 23 | 0.4512 | 0.3083
23 | 23 | 24 | 0.8980 | 0.7091
24 | 24 | 25 | 0.8960 | 0.7011
25 | 6 | 26 | 0.2030 | 0.1034
26 | 26 | 27 | 0.2842 | 0.1447
27 | 27 | 28 | 1.0590 | 0.9337
28 | 28 | 29 | 0.8042 | 0.7006
29 | 29 | 30 | 0.5075 | 0.2585
30 | 30 | 31 | 0.9744 | 0.9630
31 | 31 | 32 | 0.3105 | 0.3619
32 | 32 | 33 | 0.3410 | 0.5302
33 | 8 | 21 | 2.00 | 2.00
34 | 9 | 15 | 2.00 | 2.00
35 | 12 | 22 | 2.00 | 2.00
36 | 18 | 33 | 0.500 | 0.500
37 | 25 | 29 | 0.500 | 0.500
Table 11: BW-33 grid load data $j$ |  $P^{L}$ [kW] |  $Q^{L}$ [kVAR] | $j$ |  $P^{L}$ [kW] |  $Q^{L}$ [kVAR] | $j$ |  $P^{L}$ [kW] |  $Q^{L}$ [kVAR]
---|---|---|---|---|---|---|---|---
2 | 100 | 60 | 13 | 60 | 35 | 24 | 420 | 200
3 | 90 | 40 | 14 | 120 | 80 | 25 | 420 | 200
4 | 120 | 80 | 15 | 60 | 10 | 26 | 60 | 25
5 | 60 | 30 | 16 | 60 | 20 | 27 | 60 | 25
6 | 60 | 20 | 17 | 60 | 20 | 28 | 60 | 20
7 | 200 | 100 | 18 | 90 | 40 | 29 | 120 | 70
8 | 200 | 100 | 19 | 90 | 40 | 30 | 200 | 600
9 | 60 | 20 | 20 | 90 | 40 | 31 | 150 | 70
10 | 60 | 20 | 21 | 90 | 40 | 32 | 210 | 100
11 | 45 | 30 | 22 | 90 | 40 | 33 | 60 | 40
12 | 60 | 35 | 23 | 90 | 50 |  |  |
Table 12: Locations and capacity of community solar facilities under each DD. The generating capacity is in kW DD-U | DD-I | DD-II | DD-III | DD-II+III
---|---|---|---|---
$j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$
4 | 60 | 2 | 185 | 12 | 300 | 29 | 300 | 12 | 170
7 | 100 | 4 | 160 | 14 | 160 | 30 | 160 | 14 | 100
12 | 120 | 7 | 200 | 15 | 200 | 31 | 200 | 15 | 180
14 | 80 | 19 | 185 | 16 | 280 | 32 | 280 | 29 | 160
19 | 110 | 23 | 210 |  |  |  |  | 31 | 190
23 | 80 |  |  |  |  |  |  | 32 | 140
21 | 110 |  |  |  |  |  |  |  |
26 | 70 |  |  |  |  |  |  |  |
29 | 60 |  |  |  |  |  |  |  |
32 | 150 |  |  |  |  |  |  |  |
Figure 7: Sample solar PV generation profile for community PV facility,
queried for May 14, 2019.

####  D.1.4 Dataset

The BW-33 grid dataset consists of 8760 data points, using hourly load and
solar generation over a year.

###  D.2 94-Node Distribution Grid, TPC-94

The TPC-94 grid is presented in [48], and is a practical distribution network
of Taiwan Power Company. The data available in [48]. The network consists of
11 feeders which are able to share load and generation by using tie line
switches. The grid was modified by adding different loads and generation. The
resulting grid is shown in Fig. 8, with the locations of switches, residential
and commercial loads, and distributed PV installations marked.

Figure 8: 94-node distribution grid from [48]. The switches are highlighted in
green. The location of residential and commercial loads are noted by blue and
red circles respectively. The location of community solar DERs are noted in
yellow squares (for the S1 dataset). In the L1 and L2 dataset, all commercial
loads (red) are assumed to follow residential profiles (blue). In the L3
dataset, the residential and commercial loads are located as per this figure.
Note that the selection of locations of residential loads, commercial loads,
and DERs, and their corresponding load/generation profiles are not part of the
original network data.

####  D.2.1 Network Topology and Parameters

The grid consists of 94 nodes ($N=94$), of which 11 are the T-D substations
for feeder heads A thru K, and 83 are the remaining nodes in the network. The
grid has 97 lines ($M=97$) of which 14 are tie lines (NOS) and the remaining
83 are typically assumed to be sectionalizing switches (NCS). Topology and
line parameter data are presented in Table LABEL:tab:line_params_83. To
restrict the problem to that of dynamic reconfiguration during nominal
operation (i.e. full network is connected), only the tie lines are considered
as switches: these include the original 13 tie lines and a new tie line
introduced between nodes 59 and 67. This tie line is introduced to allow the
11 feeders to be connected to one another; in the original network, there are
two sub-networks consisting of feeders A, G and H, and the remaining feeders
B, C, D, E, F, I, J, and K. The new tie line connects feeders H and I.

Table 13: TPC-94 grid topology data and line parameters Branch No. | Upstream Node | Downstream Node | R [ohm] | X [ohm]
---|---|---|---|---
1 | 1 | 12 | 0.0085 | 0.0290
2 | 12 | 13 | 0.0092 | 0.0188
3 | 13 | 14 | 0.0103 | 0.0212
4 | 14 | 15 | 0.0040 | 0.0082
5 | 15 | 16 | 0.0092 | 0.0188
6 | 16 | 17 | 0.0017 | 0.0035
7 | 17 | 18 | 0.0018 | 0.0060
8 | 18 | 19 | 0.0046 | 0.0094
9 | 18 | 20 | 0.0103 | 0.0212
10 | 18 | 21 | 0.0046 | 0.0094
11 | 2 | 22 | 0.0034 | 0.0071
12 | 22 | 23 | 0.0149 | 0.0303
13 | 23 | 24 | 0.0011 | 0.0024
14 | 23 | 25 | 0.0034 | 0.0071
15 | 3 | 26 | 0.0050 | 0.0169
16 | 26 | 27 | 0.0023 | 0.0047
17 | 27 | 28 | 0.0023 | 0.0047
18 | 28 | 29 | 0.0069 | 0.0141
19 | 29 | 30 | 0.0017 | 0.0035
20 | 30 | 31 | 0.0074 | 0.0153
21 | 31 | 32 | 0.0103 | 0.0212
22 | 32 | 33 | 0.0069 | 0.0141
23 | 32 | 34 | 0.0086 | 0.0176
24 | 34 | 35 | 0.0057 | 0.0118
25 | 4 | 36 | 0.0025 | 0.0084
26 | 36 | 37 | 0.0046 | 0.0094
27 | 37 | 38 | 0.0109 | 0.0223
28 | 38 | 39 | 0.0021 | 0.0072
29 | 39 | 40 | 0.0057 | 0.0118
30 | 5 | 41 | 0.0086 | 0.0173
31 | 41 | 42 | 0.0057 | 0.0118
32 | 42 | 43 | 0.0057 | 0.0118
33 | 43 | 44 | 0.0011 | 0.0024
34 | 44 | 45 | 0.0074 | 0.0153
35 | 45 | 46 | 0.0023 | 0.0047
36 | 46 | 47 | 0.0218 | 0.0447
37 | 47 | 48 | 0.0017 | 0.0035
38 | 48 | 49 | 0.0017 | 0.0035
39 | 49 | 50 | 0.0034 | 0.0071
40 | 50 | 51 | 0.0092 | 0.0188
41 | 49 | 52 | 0.0086 | 0.0176
42 | 52 | 53 | 0.0092 | 0.0188
43 | 6 | 54 | 0.0021 | 0.0072
44 | 54 | 55 | 0.0017 | 0.0035
45 | 55 | 56 | 0.0057 | 0.0118
46 | 56 | 57 | 0.0103 | 0.0212
47 | 7 | 58 | 0.0106 | 0.0362
48 | 58 | 59 | 0.0029 | 0.0059
49 | 59 | 60 | 0.0029 | 0.0059
50 | 60 | 61 | 0.0017 | 0.0035
51 | 61 | 62 | 0.0034 | 0.0071
52 | 62 | 63 | 0.0017 | 0.0035
53 | 63 | 64 | 0.0034 | 0.0071
54 | 64 | 65 | 0.0023 | 0.0047
55 | 65 | 66 | 0.0057 | 0.0118
56 | 8 | 67 | 0.0099 | 0.0338
57 | 67 | 68 | 0.0235 | 0.0482
58 | 68 | 69 | 0.0023 | 0.0047
59 | 69 | 70 | 0.0018 | 0.0060
60 | 70 | 71 | 0.0017 | 0.0035
61 | 71 | 72 | 0.0011 | 0.0024
62 | 72 | 73 | 0.0046 | 0.0094
63 | 73 | 74 | 0.0103 | 0.0212
64 | 74 | 75 | 0.0011 | 0.0036
65 | 9 | 76 | 0.0021 | 0.0072
66 | 76 | 77 | 0.0074 | 0.0153
67 | 77 | 78 | 0.0053 | 0.0181
68 | 78 | 79 | 0.0096 | 0.0326
69 | 79 | 80 | 0.0021 | 0.0072
70 | 80 | 81 | 0.0032 | 0.0109
71 | 81 | 82 | 0.0025 | 0.0084
72 | 82 | 83 | 0.0011 | 0.0023
73 | 10 | 84 | 0.0142 | 0.0483
74 | 84 | 85 | 0.0014 | 0.0048
75 | 85 | 86 | 0.0025 | 0.0084
76 | 86 | 87 | 0.0021 | 0.0072
77 | 11 | 88 | 0.0110 | 0.0374
78 | 88 | 89 | 0.0057 | 0.0193
79 | 89 | 90 | 0.0021 | 0.0072
80 | 90 | 91 | 0.0057 | 0.0115
81 | 91 | 92 | 0.0057 | 0.0115
82 | 92 | 93 | 0.0040 | 0.0082
83 | 93 | 94 | 0.0137 | 0.0282
84 | 16 | 66 | 0.0057 | 0.0118
85 | 18 | 71 | 0.0057 | 0.0118
86 | 22 | 54 | 0.0057 | 0.0118
87 | 23 | 83 | 0.0149 | 0.0306
88 | 24 | 87 | 0.0200 | 0.0411
89 | 25 | 29 | 0.0235 | 0.0473
90 | 27 | 37 | 0.0040 | 0.0082
91 | 31 | 94 | 0.0034 | 0.0071
92 | 39 | 43 | 0.0023 | 0.0047
93 | 40 | 50 | 0.0034 | 0.0071
94 | 45 | 57 | 0.0011 | 0.0024
95 | 51 | 53 | 0.0086 | 0.0176
96 | 64 | 75 | 0.0017 | 0.0035
97 | 70 | 78 | 0.0077 | 0.0158
Table 14: TPC-94 grid load data $j$ |  $P^{L}$ [kW] |  $Q^{L}$ [kVAR] | $j$ |  $P^{L}$ [kW] |  $Q^{L}$ [kVAR] | $j$ |  $P^{L}$ [kW] |  $Q^{L}$ [kVAR]
---|---|---|---|---|---|---|---|---
13 | 100 | 50 | 14 | 300 | 200 | 15 | 350 | 250
16 | 220 | 100 | 17 | 1100 | 800 | 18 | 400 | 320
19 | 300 | 200 | 20 | 300 | 230 | 21 | 300 | 260
23 | 1200 | 800 | 24 | 800 | 600 | 25 | 700 | 500
27 | 300 | 150 | 28 | 500 | 350 | 29 | 700 | 400
30 | 1200 | 1000 | 31 | 300 | 300 | 32 | 400 | 350
33 | 50 | 20 | 34 | 50 | 20 | 35 | 50 | 10
36 | 50 | 30 | 37 | 100 | 60 | 38 | 100 | 70
39 | 1800 | 1300 | 40 | 200 | 120 | 42 | 1800 | 1600
43 | 200 | 150 | 44 | 200 | 100 | 45 | 800 | 600
46 | 100 | 60 | 47 | 100 | 60 | 48 | 20 | 10
49 | 20 | 10 | 50 | 20 | 10 | 51 | 20 | 10
52 | 200 | 160 | 53 | 50 | 30 | 55 | 30 | 20
56 | 800 | 700 | 57 | 200 | 150 | 61 | 200 | 160
62 | 800 | 600 | 63 | 500 | 300 | 64 | 500 | 350
65 | 500 | 300 | 66 | 200 | 80 | 68 | 30 | 20
69 | 600 | 420 | 71 | 20 | 10 | 72 | 20 | 10
73 | 200 | 130 | 74 | 300 | 240 | 75 | 300 | 200
77 | 50 | 30 | 79 | 400 | 360 | 82 | 2000 | 1500
83 | 200 | 150 | 86 | 1200 | 950 | 87 | 300 | 180
89 | 400 | 360 | 90 | 2000 | 1300 | 91 | 200 | 140
92 | 500 | 360 | 93 | 100 | 30 | 94 | 400 | 360

####  D.2.2 Load Location and Profiles

The location of loads and their nominal power demand (P and Q) for a single
period is presented in Table LABEL:tab:load_data_83. In the TPC-94 grid
dataset, three different load datasets are generated.

Perturbed dataset, L1: The first load dataset generates random load
perturbations, similar to the BW-33 grid. The maximum load perturbation at
each node is restricted to 70% deviation from nominal value (i.e.
$P^{\prime}=\delta P,\delta=[0.3,1.7]$). The power factor of the loads, which
describes the relationship between the real and reactive power as
$pf=P/(\sqrt{P^{2}+Q^{2}})$, is kept constant to the pf in the nominal data.
Sample data over 6 days is shown in Fig.9(a).

Residential dataset, L2: The second load dataset assumes all loads to be
residential loads, which follow one of six profiles. The power factor of the
loads is kept constant to the pf in the nominal data. The six load profiles
are shown in Fig.9(b). The six residential load profiles are described below:

  * •

Nominal: Typical residential load profile from [20] exhibiting the
characteristic bimodal distribution for residential load. The first mode
occurs in the morning hours between 6-9am, when residential customers wake up
and begin consuming electricity. The second mode occurs in the evening, after
6pm, when residents are returning home from work.

  * •

Early riser: A variation of the nominal profile with both modes shifted
earlier in the day. This profile represents a residential customer which wakes
earlier in the morning, and retires earlier in the evening.

  * •

Weekend/Late night: A variation of the nominal profile for weekend residential
consumption. This profile has higher electricity usage throughout the day,
with a unimodal distribution of afternoon and evening consumption.

  * •

Early-Covid19 (March and April 2020): The residential load profile shifted
visibly during the Covid-19 pandemic, in particular during the early months of
March and April. Analysis on residential electricity demand for the province
on Ontario (Canada) show significant increase in daily electricity
consumption, with a delayed morning peak and a higher maximum peak in the
evening hours. The overnight consumption remains typical to the nominal
residential profile. Residential load profile reported by the Ontario
Independent Electricity System Operator (IESO) is presented in [4], and used
to inform the load profile.

  * •

Massachusetts peak summer and winter load: A report on residential customers
in Massachusetts prepared by Guidehouse Inc. presents the summer and winter
peak day load patterns [2]. The summer peak day is a unimodal distribution
with a steady increase in electricity usage from 5am to 8pm, and a high
afternoon load. In contrast, the winter peak load profile has lower daily
electricity consumption and exhibits two modes, with a peak around 8am, and an
evening peak around 10pm. The primary difference in summer and winter load is
the HVAC (heating, ventilation, and air conditioning) load which increases
substatially in the summer months. Notably, the state of Massachusetts does
not have widespread electrified heating, which would otherwise result in
higher winter loads. These two profiles are adopted here.

Note that while data for the Early-Covid19 profile and the Massachusetts peak
data are from different geographical regions, both of these regions experience
similar climate, with 4 seasons, similar temperatures and precipitation, and
similar solar irradiation. As a result, these load profiles are representative
of a general distribution grid which may be located in a similar climate.

Mixed load dataset, L2 + L3: The third load dataset considers a mix of
residential loads (as in L2) and commercial loads, which are selected from one
of five types. The five commercial loads profiles were selected to be
different from one another and different from the residential load profiles of
L2. In this way, they introduce new load patterns at different nodes in the
grid, and change the net load characteristic as well. The commercial load data
is taken from the NREL ComStockTM dataset [3] for the city of Chicago, which
occupies Weather Zone 4A, similar to most of Massachusetts. The location of
commercial loads was selected by matching the nominal load data from [48] to
the peak hourly load from the commercial load data. Some nodes have multiple
commercial loads of the same type (ex. multiple retail stores, restaurants, or
office buildings at a single node). The location and type of commercial loads
are shown in Table 15. Sample data over the first 6 days of January is shown
in Fig.9(c). The five commercial load profiles are described below:

  * •

Hospital: The load profile has high temporal characteristics, with higher
electricity consumption from 6am to 6pm. The minimum hourly load remains
higher than other commercial facilities, at half the peak load. The peak
hourly load of the hospital is 1700 kW, corresponding to a hospital of
approximately 530,000 sq ft.

  * •

Medium office building: The load profile has a high temporal characteristic
with highest load during the morning hours of 5-7am. This commercial building
has minimum load during weekends and 10pm-5am. The minimum hourly load is
around 20% of peak load. The peak hourly load is 200 kW.

  * •

Quick service restaurant: The load profile has a unique profile in that the
load through the day cyclically increases and decreases. There is significant
difference in electric load throughout the year, likely depending on customer
load through different times of the year. The peak hourly load is 40 kW.

  * •

Stand-alone retail space: The load profile for this commercial building is
complimentary to the residential load profiles. The load during the day is
high for the retail space, corresponding to lower residential loads. The
demand begins reducing earlier in the day, approaching the evening residential
peak. Similar to the quick service restaurant, there is significant difference
in electric load throughout the year. The peak hourly load is 66 kW.

  * •

Warehouse: This commercial building has the fewest hours of load from the
selected profiles, with sustained high load during the middle of the day. The
peak hourly load is 60 kW.

Table 15: Locations and type of commercial loads. The labels (1) thru (5)
represent commercial loads of the following profiles, respectively: hospital,
medium office building, quick service restaurant, stand-alone retail space,
and warehouse.

A | B | C | D | E | F | G | H | I | J | K
---|---|---|---|---|---|---|---|---|---|---
$j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label | $j$ | Label
13 | 4 | 23 | 2 | 28 | 2 | 36 | 5 | 42 | 4 | 55 | 3 | 61 | 2 | 74 | 2 | 77 | 5 | 86 | 3 | 90 | 5
16 | 2 |  |  | 30 | 1 | 37 | 4 | 47 | 4 |  |  |  |  |  |  | 82 | 2 | 87 | 2 | 93 | 4
17 | 1 |  |  | 34 | 5 | 39 | 1 | 52 | 2 |  |  |  |  |  |  |  |  |  |  |  |
|  |  |  | 35 | 5 | 40 | 2 |  |  |  |  |  |  |  |  |  |  |  |  |  |

Figure 9: Load profiles for the perturbed dataset (L1), residential dataset
(L2), and commercial load profiles for the mixed dataset (L3).

####  D.2.3 Distributed Generation: Solar Dataset

We add a range of distributed solar facilities to the network, up to a
penetration of 23% of nameplate capacity to baseline load. This is a modest
DER penetration, which once again reflects the solar uptake now and over the
next few years. The location of solar facilities was selected to encourage
both local utilization of generation, as well as exports from one feeder to
another. Two sets of solar location and capacity data are created: S1
corresponding to the training data (see Table 16) and ‘Solar Error’
corresponding to a set of test data (see Table 17). The location and nameplate
capacity of each solar facility is provided in the corresponding tables, with
each column representing a feeder of the network. Feeders B, H, and K do not
have any solar facilities in the training data, while Feeder K has two solar
facilities in the test data. The ‘Solar Error’ dataset emulates the reality
that system operators have incomplete information of solar PV location and
installed capacity, particularly in regions with lower rates of solar
adoption, or new growth in installed solar. The lack of DER visibility is a
known concern [1]. In the ‘Solar Error’ dataset the location and capacity of
the PV units may be different than what the operator thought they were (i.e.
the training data available). Of note, Feeder G is assumed to have very good
information so there are no errors in solar installation data, while the
operator has no visibility into the two installations at Feeder K. This
dataset can also represent changes in solar adoption over time, and tests the
ability of the machine learning algorithm to perform accurate predictions when
system conditions change. Compared to the $23\%$ penetration in the S1 dataset
used for neural training, the ‘Solar Error’ has a penetration of 26% of
nameplate capacity to baseline load.

To match the load and solar data, the solar generation data is taken from
NREL’s Solar Power Data for Integration Studies, which consist of synthetic
solar PV power plant data points for the United States representing the year
2006. The dataset for Massachusetts was used, for a 12MW distributed PV unit.
The dataset provides solar generation at 5-min intervals. The particular file
used is Actual_42.55_-72.55_2006_DPV_12MW_5_Min.csv. Figure 10 shows a sample
generation profile of the solar facility, over 6 days.

Figure 10: Sample solar PV generation profile for distributed PV located in Massachusetts, across 6 days. Table 16: Locations and capacity of the solar facilities per feeder in the TPC-94 grid in training data. The nameplate generating capacity is in kW. A | C | D | E | F | G | I | J
---|---|---|---|---|---|---|---
$j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$
5 | 150 | 20 | 585 | 40 | 250 | 42 | 450 | 43 | 305 | 48 | 240 | 65 | 180 | 74 | 250
8 | 200 | 21 | 340 |  |  |  |  | 45 | 250 | 50 | 120 | 69 | 340 |  |
10 | 330 | 22 | 750 |  |  |  |  | 46 | 320 | 53 | 430 | 70 | 250 |  |
|  | 24 | 400 |  |  |  |  |  |  |  |  |  |  |  |
|  | 28 | 350 |  |  |  |  |  |  |  |  |  |  |  |
Table 17: Locations and capacity of the solar facilities per feeder in the TPC-94 grid in ‘Solar Error’ test data. The nameplate generating capacity is in kW. A | C | D | E | F | G | I | J | K
---|---|---|---|---|---|---|---|---
$j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$ | $j$ | $\overline{P}^{G}$
3 | 250 | 20 | 585 | 29 | 350 | 36 | 250 | 43 | 325 | 48 | 240 | 65 | 230 | 74 | 250 | 78 | 120
6 | 310 | 21 | 380 |  |  | 42 | 450 | 45 | 310 | 50 | 120 | 69 | 380 |  |  | 81 | 150
8 | 200 | 22 | 750 |  |  |  |  | 46 | 290 | 53 | 430 | 70 | 230 |  |  |  |
9 | 280 | 24 | 450 |  |  |  |  |  |  |  |  |  |  |  |  |  |
|  | 28 | 350 |  |  |  |  |  |  |  |  |  |  |  |  |  |

####  D.2.4 TPC-94 Datasets

Using the load and generation data described above, the following five
datasets were constructed for neural network training and testing. Each
dataset has 17280 data points, generated by using load and solar generation
data at 5-minute intervals over 60 days. It must be noted that the load data
above includes weekend-weekday variations in demand (most notably for the
commercial load profiles), and both load and generation data include seasonal
variations (most notably for the solar generation).

  * •

DS1 (Perturbed): Uses perturbed load dataset L1 and solar generation S1, with
solar resources introduced in the TPC-94 grid as per Table 16

  * •

DS2 (Residential): Uses residential load dataset L2 and solar generation S1,
with solar resources introduced in the TPC-94 grid as per Table 16

  * •

DS3 (Mixed): Uses mixed load dataset L2+L3 and solar generation S1, with solar
resources introduced in the TPC-94 grid as per Table 16

  * •

DS4 (Solar error): Uses residential load dataset L2 and solar generation S1,
with solar resources introduced in the TPC-94 grid as per Table 17

  * •

DS5 (Flat solar): Uses residential load dataset L2, with solar resources
introduced in the TPC-94 grid as per Table 16. Solar resources are assumed to
be generating at nameplate capacity at all times

##  Appendix E Implications of dynamic reconfiguration on power systems

We investigate the power systems implications of using Grid-SiPhyR to enable
dynamic grid reconfiguration. The following three reconfiguration approaches
will be considered:

No reconfiguration: the default grid topology as given in the datasheet is
used. All switches are assumed to be in their default position (normally open
NOS, or normally closed NCS).

Static reconfiguration (StatR): determines a fixed set of switch-states that
optimize losses over a long term, such as a few months or a year. The load and
generation forecasts over the period will be considered, and a robust
optimization approach can be taken to determine the optimal topology across
all grid conditions.

Dynamic reconfiguration (DyR): determines the switch-states which minimizes
losses for a given load and generation condition for a particular period. This
period is shorter in length than for StatR, such as a few minutes, hours, or
days. The introduction of DERs necessitates DyR where local DERs supply loads
in closer proximity to them, thus reducing losses, improving voltage profiles,
and increasing PV (renewable soalr energy) utilization.

The three outcomes (loss reduction, voltage improvement, and increased PV
utilization) are investigated below, on simulations of the BW-33 and TPC-94
grids.

###  E.1 Power systems performance metrics

We assess grid performance in terms of grid efficiency, operability, and clean
energy directives. The corresponding metrics are as follows:

  * •

Grid efficiency: line losses incurred in delivering power to loads, where
power loss across a line $\\{i,j\\}$ is calculated as
$R_{ij}\frac{(P_{ij}-P_{ji})^{2}+(Q_{ij}-Q_{ji})^{2}}{v_{i}}$ [8];

  * •

Grid operability: voltages must remain within operating limits ($\pm 5$% of
the base system voltage in North America). We consider the voltage
distribution, number of undervoltage events
($\sum_{j\in\mathcal{B}}{\mathbb{I}_{\sqrt{V_{j}}<0.95}}$), and average grid
voltage ($\frac{1}{N}\sum_{j\in\mathcal{B}}{\sqrt{V_{j}}}$);

  * •

Clean energy directives: amount of available PV generation that is dispatched
to meet loads. The higher the PV utilization, the more clean energy is used.
PV utilization is measured as the amount of PV generation dispatched as a
ratio of the available PV resource:
$PV_{util}=\frac{P^{G\ast}}{\overline{P}^{G}}$.

###  E.2 Result 1: Dynamic reconfiguration reduces electrical line losses

The following set of results are for the BW-33 grid, a very lossy network:
recall that losses average about 8% of total load. Table 18 shows the
significant loss reduction enabled by dynamic reconfiguration for the BW-33 as
DER locations are varied, upwards of 23% PV penetration. The StatR closes tie
line 35 and opens NCS 10, and keeps this topology fixed. The DyR selects
primarily between two states: (a) closing tie line 35 and opening NCS 10, and
(b) closing tie lines 35 and 36 and opening NCS 10 and 26. For a grid without
any DERs versus a grid with DERs, the loss reduction from reconfiguration is
higher without DERs; this can be attributed to greater losses without
leveraging local generators which are located closer to loads and thus incur
lower losses when supplying those loads. The second column compares DyR to
StatR, showing savings up to 30 MW for a single distribution feeder.

While this is a modest 2.5% improvement of DyR upon StatR, this is a
nontrivial reduction for distribution grid operators and utilities. Current
industry standard methods which aim to reduce the load on a distribution grid,
such as Conservation Voltage Reduction (CVR), typically reduce peak demand by
a modest $2$ to $2.5\%$; for a utility with a peak load of 100 MW, this
translates to savings of US$200,000 per year [6]. Further, it should be noted
that this was obtained with a small test case (33 nodes). As the dimension
increases, with increasing penetration of DERs and switches, with disparate
patterns and topologies, it is expected that this difference may be more
pronounced.

Table 18: Loss reduction using StatR and DyR |  | StatR vs. no reconfig
---
% Loss reduction, MW saved per year *
| Grid-SiPhyR vs. StatR
---
MW saved per year *
No DERs | 23%, 370 MW | 0 MW
DD-U | 20%, 300 MW | 23 MW
DD-I | 20%, 320 MW | 31 MW
DD-II | 19%, 270 MW | 20 MW
DD-III | 21%, 310 MW | 27 MW
DD-II+III | 20%, 280 MW | 28 MW
*The MW (power) saved is equal to the MWh (energy), as the simulation is run for every hour of the year and loss reduction summed for every test case

###  E.3 Result 2: Dynamic reconfiguration improves voltage profile across the
grid

The following set of results are for the BW-33 grid, a grid with significant
voltage violations. Figure 11 plots the voltage distribution across the grid
for an entire year. The ideal distribution is the shape of a short ice cream
cone - wide on top and narrow on the bottom, with the tip above the line
indicating the ANSI minimum voltage limit of 0.95pu. The width of the plot
indicates the total number of voltage observations at the y-axis value, i.e.
wider indicates more voltage observations. On each plot is printed the
percentage of time the voltages are within the ANSI limits, where higher
numbers are better. We make the following key observations: (i) without
reconfiguration, the grid performs very poorly, violating ANSI limits 50-60%
of the time, and voltages drop to 0.88pu (outside of ANSI limits); (ii)
reconfiguration (Stat or Dy) significantly improves the voltages across the
grid, with minimum voltage improving to 0.9pu, and ANSI limits satisfied
77-83% of the time (IEC limits are always satisfied); (iii) DyR reduces the
number of voltage violations throughout the year by 2%, as compared to StatR,
which is a significant improvement as undervoltage can result in brownouts and
even lead to blackouts. We note that since the grid chosen for this test case
is very lossy, in our simulations we enforce a lower voltage limit of 0.87pu
to ensure feasibility of loading conditions (instead of 0.95pu), which our
physics-informed framework always satisfies. While it is interesting to note
the simple case study of the BW-33 grid does not imply a preferred DD over
others, different network topologies and sizes may suggest an optimal DER
distribution.

(a)

(b)

(c)

Figure 11: Voltage distribution over a year (8760 hours). Solid line is ANSI
lower voltage limit.

###  E.4 Result 3: Dynamic reconfiguration enables better solar utilization,
by connecting generation with loads

The following set of results are for the TPC-94 grid. Figure 12 shows the
dispatch results aggregated at the feeder level for the DS2 test case. The
load in each feeder can be met by importing power from the bulk grid at the
local T-D substation (i.e. the PCC of the same feeder), importing power from
neighbouring feeders, or using local PV generation. Most feeders serve the
load using imports from the bulk grid, while Feeders A and E import
significant amounts of power from neighbouring feeders. Feeder F has very high
PV penetration and when solar generation is high, can meet most of its load
locally.

Figure 13 shows a configuration change in the TPC-94 grid, where Feeder G
connects to neighbouring Feeder H when solar generation becomes available.
Power from Feeder G is exported to Feeder H and A. While the remaining network
configuration does not change and the direction of power flow remains the same
(i.e. the same networks export power), the amount of power transferred across
the tie lines reduces substantially. Local PV generation supplies power to
local loads, and the configuration change allows the feeders to meet their
remaining load in an efficient way. Notably, the configuration change allows
PV utilization to increase in the network.

Figure 12: Optimal dispatch results are shown for the TPC-94 network with DS-2
(all residential loads), over a period of 10 days. The optimal reconfiguration
is determined every 5-minutes, resulting in 2880 periods. All plots are in per
unit measurements. (Left) A breakdown of the load served at each feeder, A
thru K. Dark blue: load served by the bulk system at the T-D substation of the
same feeder; Light blue: load served by power imported from neighbouring
feeders; Yellow: load served by local distributed solar generation. (Right)
Power exported to neighbouring feeders. Figure 13: Two configurations for the
TPC-94 network showing the switches connecting neighbouring feeders. These
results correspond to periods 1 and 140 in the dispatch results of Fig. 12.

The PV utilization results are summarized in Table 19, for StatR and DyR
applied to the DS-2 case, and DyR applied to the DS-3 case. The comparison of
StatR and DyR show that the optimal reconfiguration of the TPC-94 grid can
reduce PV curtailment from 23% to 19%. This corresponds to an overall
reduction in PV curtailment by 17%, corresponding to an increase of 250MWh of
solar energy used annually. To put into perspective, this is enough solar
energy to power approximately 23 US households for a year, and results in a
decrease of 107 metric tons of CO2.111This is approximated by 2021 EIA data,
where the average annual electricity consumption for a U.S. residential
utility customer was 10,632 kilowatthours (kWh), averaging around 886 kWh per
month. The CO2 emissions in Massachusetts from electricity generation in 2021
was 974 lbs/MWh. This value was used to approximate the CO2 abatement from
DyR. It must be stressed that this increase in solar energy does not require
new solar installations. Instead, a change in operating paradigm from static
to dynamic permitted by Grid-SiPhyR allows solar PV to be used more
effectively, while simultaneously increasing operating efficiency of the
distribution grid.

Table 19: Summary of PV utilization in TPC-94 | StatR | L2 | DyR | L2 | DyR | L2 + L3
---|---|---|---
Total available PV generation (MWh) | 1042 | 1042 | 880.3
Total utilized PV generation (MWh) | 802.7 | 843.7 | 701.0
Total curtailed PV generation (MWh) | 239.5 | 198.5 | 179.4
Percentage of curtailed PV generation | 23% | 19% | 20%
Average utilized PV generation per hour (kWh) | 557.4 | 585.9 | 486.8
Average curtailed PV generation per hour (kWh) | 166.3 | 137.9 | 124.5
Number of samples (5-min interval) | 17280