\section{Introduction}


Within the United States and many other areas of the world, remote communities are disconnected from bulk transmission systems. Given the economic hurdles associated with connecting remote communities to these systems, many will remain isolated for the foreseeable future. However, it is important that these communities have the same level of reliability afforded by the bulk transmission systems \cite{DoeromdstRC2015}. To address this need, we develop an expansion planning model for off-grid microgrids that balances the costs of designing the system with the cost of operating these grids under N-1 reliability criteria. This model includes all three major decisions associated with the design and operation of off-grid microgrids: identifying the installation locations of power sources, determining capacity and power dispatch of those resources, and prescribing the network topology \cite{lasseter2004microgrid}. Though critically important, this problem is very difficult to solve given the non-convexities in discrete installation choices and power flow physics.


To address this problem, we use the mixed-integer, quadratically constrained, quadratic programming (MIQCQP) resource planning model of \cite{chalil2017remote,Mashayekh2016} and generalize it to support expansion planning with N-1 security constraints on lines. The resulting model is significantly more challenging to solve (the methods of \cite{chalil2017remote,Mashayekh2016} do not directly scale to this problem) and we develop a novel rolling-horizon (RH) algorithm to solve this problem.
In short, the key contributions  of this paper are:
\begin{itemize}
    \item To the best of our knowledge, this is the first  distribution  systems planning model with topology decisions and N-1 reliability on lines  that incorporates nonlinear ac physics,
    time-extended operations, distributed energy resource planning decisions (including conventional generators and batteries), and power-device efficiencies. We refer to this problem as the ac integrated resource planning problem for microgrids (ACIRPM).
    \item An algorithm which decomposes the problem by time and scenario that efficiently solves this problem.
    \item A demonstration on real system data and empirical validation of the results when compared with state-of-the-art MIQCQP solvers.
\end{itemize}


\begin{figure*}
\vspace{0.2cm}
\centering
\includegraphics[width=\textwidth]{FlowDiagram3.pdf}
\caption{Modified flowchart of the microgrid resource planning model described in \cite{chalil2017remote}. 
This flow chart expresses the stages and time-scales of the decisions modeled in our expansion planning model. The top level includes all planning decisions. This level includes our contribution, the inclusion of topology investment decisions. These investment decisions are used to determine time-extended operating decisions (second level). The operating decisions are constrained by N-1 security constraints (third level) and are applied at all operating points. Here, our contribution is the modeling of line contingencies.}
\label{Fig:SchematicDiag-P2}
\end{figure*}

\noindent


\paragraph{Literature Review} 
The importance of topology designs in power systems are discussed extensively in many research papers. An early study of the investment and operation of multiple energy systems along with their topology is presented in \cite{bakken2007etransport}. This paper examines the design and operation of multiple energy carriers within a locality and considers suggestions for alternate locations that satisfy pre-defined future demands. The authors recommend ranking the installation of various energy carriers over a defined time line. Their solution selects the network topology that minimizes the total cost of installation and operation. Even though \cite{bakken2007etransport} considers topological design decisions, no contingency analysis was performed on the system.  Furthermore, \cite{crucitti2004topological} studies an estimation for the vulnerability of an electric grid using topological analysis.   The authors model cascading failures of power systems via dynamic load redistribution on the network and observe that the system is highly vulnerable when heavily loaded nodes are removed from the system.

In many places in the literature, the optimization of microgrid topology design and operations is discussed as one of the main research needs in power systems, i.e. \cite{perez2016review, mashayekh2017mixed}. This observation has driven a number of studies on how topology impacts system security 


\cite{albert2004structural,crucitti2004topological,bakken2007etransport}. As noted in \cite{lasseter2002microgrid}, typical microgrid architectures 
are organized in groups of radial feeders that are part of either a distribution network or in independent, remote, off transmission locations. Under these architectures, the removal of sets of nodes (network disruptions, generator breakdowns or line failures), can lead to cascading failures.  Given this observation, \cite{albert2004structural} considers the robustness of power systems from a topological perspective and verifies correlations between reliability and redundancy of network structure and emphasizes that a redundant network enhances reliability.   In all of these papers, focus is placed on analyzing existing topology choices and these papers do not discuss how to design the topology of the network. 


Some of the techniques for enhancing microgrid reliability using topological designs include interconnected microgrids and network redundancy \cite{erol2011reliable,kahveci2016optimization,zinchenko2016optimal}. Works by \cite{erol2011reliable} and \cite{kahveci2016optimization} discuss the role of interconnected microgrids.  Kahveci \etal \cite{kahveci2016optimization} present topology layouts derived from heuristics that employ clustering and graph theoretic methods. The authors discuss a heuristic approach to topology design for both ``greenfield'' sites and the expansion of existing military distribution networks. The algorithm first identifies a minimum spanning tree between various nodes and next identifies clusters that are electro-mechanically stable during islanding conditions. Unfortunately, this paper considers only topology design and did not consider the operational aspects of the microgrid, technology citing, and capacity design. Zinchenko \etal \cite{zinchenko2016optimal} solve the transmission expansion planning problem with line redundancies as a two-path problem using a variant of Dijkstra's algorithm to find the shortest path between origin and destination. They conclude that redundancy may be the only option to ensure resiliency in power systems. Hence in our paper, we consider installation of parallel, redundant lines to ensure that the network is N-1 secure for line contingencies. To the best of our knowledge, no other previous research efforts consider N-1 security analysis on line contingencies in microgrids.


The most closely related work to this paper is found in \cite{chalil2017remote,Mashayekh2016}. These papers develop a resource planning model for optimizing the placement of generation resources that enforces N-1 generator reliability on microgrids.  Though these papers also consider time-extended operations, the algorithms presented do not scale and can take up to 24 hours to solve the problem (on easier instances). Further, they do not consider expansion planning or N-1 reliability on lines.  These two modeling details, in combination with time-extended operation, can significantly increase the complexity of the problem and necessitate the development of new algorithmic approaches.
More generally, the power engineering community has developed a number of techniques for solving problems with multiple time periods like ACIRPM. These methods include Benders decomposition \cite{alguacil2000multiperiod,fortenbacher2018optimal}, rolling-horizon (RH) methods \cite{palma2013microgrid}, graph partitioning \cite{che2016optimal}, and branch-and-bound algorithms coupled with Langrangian-dual relaxation \cite{gopalakrishnan2013global}. Considering the strength of recent RH methods in industrial domains, such as supply chain optimization \cite{zamarripa2016rolling} and  microgrids \cite{palma2013microgrid}, we developed a modified RH approach for solving the ACIRPM. Uniquely, we consider different approaches, such as scenario-based decomposition (SBD), for solving the sub problems constructed by the proposed RH approaches.


The rest of the paper is organized as follows: Section II proposes a mathematical formulation for the resilient design and operation of off-grid microgrids  with N-1 security constraints on generators and lines. Section III presents a rolling-horizon algorithm for solving the model efficiently and compares its results with scenario-based decomposition method. Next, numerical results on two case studies are discussed in Section IV. Finally, Section V presents  conclusions and future research directions.
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