\section{Discussion}
\label{sec:discussion}


In Section~\ref{subsec:communication}, we analyze how the optimal communication strategy, as determined via Lemma~\ref{thm:restricted_communication} and \eqref{eqn:restricted_communication}, differs between scenario 1 and 2. 
In Section~\ref{subsec:interpretation}, we interpret the resulting controller parameters for inverters located in different parts of the feeder.
Lastly, in Section~\ref{subsec:design}, we discuss practical considerations to integrate the decentralized OPF method across distribution planning and operation. 

\subsection{The Effect of Communication}
\label{subsec:communication}


The ability to fully decentralize OPF problems is surprising.
The information theoretic lens used in this paper complements insights from control-theoretic analysis that show the necessity of communication for a large class of local controllers to ensure that equilibria are feasible with respect to voltage constraints~\cite{cavraro_value_2016}. 
In our setting, the use of exogenous disturbance variables~$d_i$ prevents the need for a dynamic analysis as $d_i$ is not affected by the control signals.
Rather than for the purpose of stability and feasibility, in~\cite{dobbe_fully_2017}, we applied Lemma~\ref{thm:restricted_communication} to the OPF problem for a smaller network \cite{noauthor_ieee_2017}, in order to study the value of communication for improving the reconstruction of $u^*$, yielding an optimal communication strategy. 


In this paper, we also studied the emerging communication topology that arises from maximizing the mutual information between the optimal actions and the combined available measured variables. 
For case 1, where we minimize voltage deviations and losses across the network with DERs spread randomly across all buses, we see no particular patterns in the optimal communication infrastructure.
Some DERs closer to the end of the network tend to communicate with loads nearby, which can be explained by the emerging effect of the OPF problem that supply tends to match demand nearby to minimize losses higher upstream in the radial network.
For case 2, where we are localizing the production of power by minimizing the power delivered from the substation, an interesting pattern arises from the fact that the loads and DER are concentrated in \emph{different} areas.
As a result, if we allow DERs to observe one extra node anywhere in the network, all select a node in the \emph{other} area with concentrated loads. This makes intuitive sense as the optimal control actions are largely responding to the high demand in the other area.  
In addition, across all 20 DERs the selected nodes are all in a set of only 3 nodes, suggesting an efficient sensor implementation.
A relevant question that arises when communication is needed, is whether the different agents have the economic incentives to signal their information truthfully. Here, we have assumed that the inverters are all managed by one entity. However, in the practical scenario where third-parties operate different controllers, a further analysis of how signaling incentives may affect the quality of the decentralized control scheme is relevant, and inspiration may be drawn from the literature on optimal decentralized control~\cite{mahajan_information_2012}.

\subsection{Interpreting the Machine Learning Models}
\label{subsec:interpretation}
In this section, we analyze how the structure of different learned controllers~$i,j \in \C$ differs across the network as reflected by parameters~$\beta^{(i)},\beta^{(j)}$ in~\eqref{eq:MLmodel}. What does this tell us about the network and OPF problem?

In our first case study, we determined regression models for 27 different inverters. Table \ref{tab:InvMdl} shows regression results for inverters A and B (both depicted in Figure~\ref{fig:FeederDiag}). The first two columns present the  features selected by the stepwise regression and the values for the $\beta$-coefficients in~\eqref{eq:MLmodel}. The third column shows the standard error of the estimate, and the fourth lists p-values, which here means the probability that a coefficient is zero. 

Note how the stepwise regression approach results in two clearly different models.
The reactive power output of inverter A depends predominantly on the local reactive power consumption $\varphi^{(A)}_2$, while the output of inverter B is strongly related to the reactive power capacity $\varphi^{(B)}_3$ and $\varphi^{(B)}_2$ is less relevant.
Inverter A's structure can be explained by the fact it is located at the end of the feeder. As a result, the aggregate power flow at A is relatively low and correlated with A's own consumption. As the objective is to minimize losses and voltage deviations, A tries to produce a signal that looks like providing the neighboring demand for reactive power, thereby minimizing current locally \emph{and} upstream.
Inverter B's structure can be explained by its location close to the feeder head. In this area, the aggregate power flow is much larger, and the output of inverter B is needed more as a ``bulk'' product that lowers the loss and voltage drop on the branch between B and the substation.
This causes inverter B to operate at its maximum capacity most of the time, causing correlation with the output.
As the maximum capacity varies with the local real power generation at B (a constraint formulated in~(11) of the main manuscript), the learning algorithm tries to ``listen closely'' to the present maximum capacity.
This example illustrates that optimal reactive power output of inverters tend to have idiosyncratic structures depending on location and the assignment of loads, generation and capacity across a network. Therefore, effective design of controllers based on local measurements is complex, and rather than relying on one-size-fits-all solutions, better and safer results may be achieved with a flexible data-driven approach that respects constraints. 


\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Normalized regression coefficients for inverter A and B, as depicted in Figure~\ref{fig:FeederDiag}.}
\label{tab:InvMdl}
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c c c c || c c c c}
\hline
A & Est. & SE & p-value &  B & Est. & SE & p-value \\
\hline\hline
offset	&	0.02	&	0.01	&	0.01				&	offset	&	0.02	&	0.00	&	$8.2 e^{-4}$ \\
$\varphi_1$	&	0.37	&	0.01	&	$0$		&	$\varphi_1$	&	0.04	&	0.00	&	$0$ \\
$\varphi_2$	&	0.77	&	0.01	&	0				&	$\varphi_3$	&	0.96	&	0.01	&	0 \\
$\varphi_3$	&	-0.21	&	0.02	&	$0$		&	$\varphi_1 \varphi_3$	&	0.06	&	0.01	&	$0 $\\
$\varphi_1 \varphi_2$	&	0.17	&	0.01	&	$0$		&	$\varphi_1^2$	&	-0.03	&	0.00	&	$0$\\
$\varphi_1^2$	&	-0.06	&	0.01	&	$0$		&	$\varphi_3^2$	&	-0.03	&	0.01	&	$0$ \\
\hline
\end{tabular}
}
\end{table}


\subsection{Designing Across Planning and Operation}
\label{subsec:design}
When using supervised learning, some immediate concerns arise around the quality of the training set and the historical data used to construct it.
Firstly, a designer wants to have certainty that the historical data reflect the ``normal'' system behavior expected in future operation. If certain scenarios are not measured historically, the controllers may not \emph{learn} how to respond in an optimal or desired way. 
If future scenarios are not measured, but can be anticipated, is may be possible to instead \emph{simulate} these and augment these to the historical data.
Secondly, a designer will want to analyze how the system behaves under faults and safety-critical scenarios or want to know what happens when changes occur in the system, such as a new street block connecting to a feeder or smaller increments, such as the installation of new electric vehicles. Based on the effects on the behavior of the controllers, it may or may not be needed to retrain all local policies through the central OPF and machine learning steps as covered in Sections~\ref{sec:opf} and~\ref{sec:regression}, and send new controller parameters to the inverters. 
In our future work, we are assessing different learning-based controllers for OPF on their performance across a spectrum of system changes~\cite{karagiannopoulos_data-driven_2019}. For smaller changes, a periodic retraining procedure can update the control parameters incrementally by augmenting the training data with more recent historical measurements.
{"config": "arxiv", "file": "1806.06790/discussion.tex"}