A Novel Model for Enhancing the Resilience of Smart MicroGrids’ Critical Infrastructures with Multi-Criteria Decision Techniques

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This paper is concerned with a multi-criteria technical and economical allocation procedure of microgrids in smart cities.

Abstract

Microgrids have the potential to provide reliable electricity to key components of a smart city’s critical infrastructure after a disaster, hence boosting the microgrid power system’s resilience. Policymakers and electrical grid operators are increasingly concerned about the appropriate configuration and location of microgrids to sustain post-disaster critical infrastructure operations in smart cities. In this context, this paper presents a novel method for the microgrid allocation problem that considers several technical and economic infrastructure factors such as critical infrastructure components, geospatial positioning of infrastructures, power requirements, and microgrid cost. In particular, the geographic allocation of a microgrid is presented as an optimization problem to optimize a weighted combination of the relative importance of nodes across all key infrastructures and the associated costs. Furthermore, the simulation results of the formulated optimization problem are compared with a modified version of the heuristic method based on the critical node identification of an interdependent infrastructure for positioning microgrids in terms of the resilience of multiple smart critical infrastructures. Numerical results using infrastructure in the city of Pittsburgh in the USA are given as a practical case study to illustrate the methodology and trade-offs. The proposed method provides an effective method for localizing renewable energy resources based on infrastructural requirements.

1. Introduction

The portion of the world population inhabiting an urban environment has grown in the last decade from about 33% to 55% [1]. That growth has produced enormous demand and stress on the infrastructures and systems that deliver essential city services, resulting in significant interest in developing smart cities. The main purpose of smart city schemes is to create intelligent infrastructures for cities by harnessing innovations in cyber-physical systems, data science, and information and communication technology (ICT). Moreover, smart infrastructures are more dependent on both ICT and electrical power for proper operation. This increased dependence can introduce new vulnerabilities and lower infrastructure resilience [2]. In particular, severe weather (e.g., snow/ice storms, typhoons, tornadoes, drought-induced wildfires, etc.) is a growing vulnerability concern as the frequency, intensity, and geographic scope of severe weather events are predicted to increase with climate change [3]. Currently, severe weather events [4] are the number one reason for power outages in the United States, which are in turn, the top reason for outages in ICT services. For that, intelligent infrastructures provide more consistent and reliable system performance, new features/functions, and increased sustainability.

Industrial scale microgrids are basically self-supporting power systems typically in the 1.5–5 MW range. They have been advocated as a mechanism to improve the availability of power to significant societal and business facilities such as hospitals, military bases, and factories. Additionally, microgrids are promoted as an approach to incrementally incorporate shared renewable power production, such as wind and solar, into the bulk power grid in the case of disaster [5]. Mircrogrids are also proposed in the literature as a solution to achieve climate adaptation and mitigation goals [6].

Figure 1 gives a typical microgrid architecture. As shown in Figure 1, the building blocks of the microgrid are the controller, electrical power switches, local energy supplies (e.g., solar cells, wind turbines, and diesel generators), energy storage, and various loads. Microgrids are designed to operate in standalone mode and joined mode to the primary grid. In the joined mode scenario, the microgrid serves as additional energy back up to the bulk power system, decreasing peak loads, enhancing power stability, and reducing harmful emissions [7]. In island mode, the microgrid disconnects from the bulk grid and functions as a standalone power supply. The microgrid controller manages the transitions between modes seeking to maintain voltage and synchronization while minimizing load dropping and disruption.

Since the available power is limited in island mode, the power loads are grouped into their significance categories: mission critical, mission priority, and non-critical. Mission critical loads are given the highest priority and consists of the essential components of the critical infrastructures of interest (e.g., hospitals, water treatment plants, and cellular network base stations). Mission priority loads are given second priority and would include loads that are important to society but not essential to the functioning of smart critical infrastructures (e.g., drug store, gas station, etc.). Lastly, non-critical loads would include residential and non-essential businesses (e.g., movie theaters).

Generally, microgrids have fixed geographical limits and are planned in island mode to produce power sufficient to sustain mission-critical loads inside the geographic boundaries. Thus, based on the power accessible in island mode, the microgrid controller may implement load shedding, dropping non-critical loads and a portion of the mission priority loads [5]. Furthermore, microgrids are required to possess the capability to shift efficiently from island mode to grid-connected operation, providing re-synchronization with minimal consequence to significant loads through the transition phases.

Unlike the nanogrids used in residential settings, a significant obstacle to the implementation of industrial size microgrids is the high cost in constructing, operating, and maintaining a microgrid. Currently, industrial capacity microgrids are mostly owned by an individual private organization. Recognizing the non-linear economic costs of implementing microgrids [8], the authors in [9] advocated for shared mid-size microgrids, with the expense borne by both the vital infrastructure proprietors with critical loads (e.g., cellular network operators and hospitals) and the government organizations that will utilize the infrastructures during disaster recovery. This shared community use approach could be facilitated by government-sponsored financing and tax credits. Furthermore, it may justify either re-insurance or bonding mechanisms to help in reducing the cost. Here, the goal of this work is to design and place microgrids based on minimizing the overall expenses and to ensure power flow connected over the most vital critical infrastructure parts. Therefore, the proposed design is able to achieve significant techno-economical merits for microgrids.

Related work on microgrids includes using microgrids to improve radial distribution power grid restoration after a natural disaster [5,10] and dynamically forming local microgrids around distributed generation sources after a disaster [11]. Kelly-Pitou [6] introduced the notion of employing a microgrid for the purpose of enhancing both power resilience and for alleviating climate change impacts. Nevertheless, this early work did not propose a method for determining the location of microgrids to improve the resilience of different critical infrastructures viewed as a group. In [12], the authors suggested using an algorithm to achieve multi-agent resource allocation in distributed scenarios through a shared microgrid, including residential and commercial buildings, with the least amount of information exchange between the users. However, their study lacks the commercial and residential segments by not assigning the priorities to critical nodes in the network.

The bulk of the research literature on critical infrastructures within smart-city schemes has focused on optimizing performance and providing new functionalities. Previous work covering smart-city resilience has focused specifically on developing frameworks [13] or “solidifying” critical infrastructures. Traditionally, policy-makers mandated or supported hardening techniques such as constructing flood barriers and rebuilding levees according to the probability of 1 in 100-year situations. However, when considering smart critical infrastructures and the increasing weather variability, planners need to move beyond physical hardening techniques, adopting new preparedness methods and policies that acknowledge the dependence on both power and ICT.

In [9], a structure for providing power and ICT is developed to enhance smart city critical infrastructure services in post-disaster conditions. The proposed method in this work employs multi-user microgrids to generate electricity concurrently with cellular-based communications, which are dynamically re-adjusted into a mesh network along with local edge computing to control/operate smart critical infrastructures. The main aim of the framework is to construct districts within a mid size zone that act as secure area, including essential critical infrastructure functions operating at limited but acceptable levels. Guaranteeing that the combination of microgrids, cellular-based catastrophe recovery mesh network, and edge computing are geographically located in this “socially planned” fashion will assist in reducing at-risk districts and bolster the economic argument for microgrids.

Related work on enhancing critical infrastructure resilience has focused on hardening [14,15] the essential elements in every infrastructure. Various techniques have been introduced in the literature for discovering the most vital components in a critical infrastructure, such as graph theoretic analysis, simulation-based analysis, stochastic modeling, and expert judgment [14]. Graph centrality measures have been utilized in developing critical infrastructure protection strategies, including vulnerability studies of power systems. A heuristic approach introduced in [16] uses five graph centrality measures, namely degree centrality, between centrality, centered centrality, eccentricity centrality, and radially. The authors evaluated the nodes with each of the five centrality measures. If a node is highly ranked by at least two measures, it is considered a critical node. The model was applied to assess the impact of potential attacks on the Swiss power network. However, the author utilized the proposed model considering the impact on the power network’s infrastructure, neglecting the dependency on other infrastructures that could potentially have a higher effect. In [17], the authors introduced a scheme to determine the critical nodes of a smart power network according to the highest power flow in the system. However, their investigation concentrated on the betweenness centrality as the primary graph measure, which could lead to a bias in the outcome by neglecting other essential centrality measures in networks, such as node degrees and closeness. The authors of [18] considered the centrality metrics of a dependency risk graph, exploring the connection between dependency risk paths and graph centrality. They mapped different critical infrastructures into one graph and applied the centrality metrics on each node, assuming that the links are represented by the escalating failure values between nodes from different infrastructures. The primary motivation for that was to identify the critical infrastructure nodes between interdependent critical infrastructures that noticeably impact the essential routes of risk in the network and then to analyze cascading failures to different nodes or links in the network.

In [19], the authors proposed a model to identify the most critical nodes in interdependent critical infrastructures. They developed an integer linear programming optimization formulation that models the approach of an attacker who targets a collection of nodes with the intention of compromising or damaging them. They assumed that the attacker is motivated by three objectives: (i) minimizing the size of the largest connected component, (ii) maximizing the number of disconnected components, and (iii) minimizing the cost of an attack. All three objectives are based on graph theory metrics and can be used to determine where to hardened the infrastructure. Here, the problem of where to harden multiple infrastructures as a group using a microgrid is considered.

Noting that microgrids are the most costly element in our framework, our focus is on where to place a microgrid in order to promote smart critical infrastructure operations post-disaster. An holistic approach is adopted for the microgrid location problem, considering multiple critical infrastructures at once, and focuses on factors such as component importance within a critical infrastructure, the geospatial placement of infrastructures, power requirements, and microgrid cost. Optimization problems are formulated to determine the location of a microgrid in a geographic space that optimizes a weighted combination of the relative importance of nodes across all critical infrastructures and the cost. Furthermore, a simple heuristic method for positioning microgrids is presented and demonstrated. This method is compared with the optimization problem. Numerical results using Pittsburgh as a case study are given to illustrate the effectiveness of the methodology and its trade-offs.

This paper is structured as follows. Section 2 presents the proposed methodology, which determines the placement of microgrids. In Section 3, the numerical results and a discussion of implementing the proposed method are presented. Section 4 provides the findings of the study and future work.

2. Materials and Methods

2.1. Microgrid Location Methodology

Consider a neighborhood or section of a city where multiple infrastructures geographically overlaid co-exist, as illustrated in Figure 2. For instance, the three infrastructures shown could include an intelligent water network, a natural gas pipeline system, and a healthcare system consisting of a hospital and community health clinics. The proposed approach to determining the location of a microgrid essentially has four steps as follows:

• First, each smart critical infrastructure is analyzed individually to determine the relative importance of each node/component of the infrastructure.
• Second, a geographic grid of the city is defined, and the infrastructures are aligned spatially.
• An optimization problem is formulated to determine the place in a geographic grid that optimizes a combination of the importance of nodes/components across infrastructures and the cost of the microgrid.
• Following the three-step procedure, the analysis of critical infrastructures is considered to define the comparable significance of elements.

2.2. Critical Infrastructure Analysis

Critical infrastructures are grouped into two classes according to the applicability of modeling the infrastructure with a network graph: (1) interconnected infrastructures and (2) standalone infrastructures.

2.2.1. Interconnected Infrastructure

Network science methods based on graph theory have been applied to analyze critical infrastructures that include the interconnected elements or operations, such as power grids [20], transportation networks [21], water systems [22], and optical backbone communication networks [23]. Interconnected infrastructures are modeled with a graph $G=(V,E)$, where V is the set of vertices or network nodes (e.g., power plants and substations, pumps and pipe junctions, and optical switches), and E is the set of edges or links or connections (e.g., power lines, water pipes, and optical fibers) joining the nodes. Given a graph model of an interconnected infrastructure, one can use network science methods in part to determine the relative importance of nodes in the infrastructure based on graph metrics. This paper adopted centrality metrics similar to [16], as explained in the following.

Degree Centrality

Node degree denotes the total number of neighbor nodes to which a node is immediately attached. The degree of centrality is a primary graph analysis measure that can be calculated as the total of edges connected to node v, which is called degree $deg\left(v\right)$, and identified as centrality degree $C_d\left(v\right)$, which is provided as follows:

$$C_d\left(v\right)=deg\left(v\right)$$

(1)

Betweenness Centrality

Betweenness centrality is a benchmark of centrality measure used in graph networks that reflects the shortest paths within couples of vertices s and t. It can be calculated as the percentage of shortest paths that cross through a vertex $v\in V\left(G\right)$. Hence, $C_b\left(v\right)$ can be written as follows:

$$C_b\left(v\right)=\sum _{s\ne v\ne t}\frac{{\sigma }_{st\left(v\right)}}{{\sigma }_{st}},$$

(2)

where ${\sigma }_{st}$ signifies the number of shortest routes from node s to node t, and ${\sigma }_{st\left(v\right)}$ equals the total number of these routes that cross within v.

Closeness Centrality

In any connected graph, the normalized closeness centrality $C_c\left(v\right)$ of a particular node $v\in V\left(G\right)$ equals the average distance of the shortest routes connecting node v to all additional nodes in the graph. The closeness can be defined as follows:

$$C_c\left(v\right)=\frac{1}{{\sum }_{s}dist(s,v)},$$

(3)

In the above equation $dist(s,v)$ denotes the length connecting vertices s and v.

Power Requirements

The power $C_p\left(v\right)$ required for vertex $v\in V\left(G\right)$ is one of the factors considered and is assumed to be given.

Total Weighted Value ($TW{V}_{II}$)

Combining the three centrality metrics forward beside the power requirements, we define a total weight value for interconnected infrastructure $TW{V}_{II}\left(v\right)$ for each node $v\in V\left(G\right)$. In calculating $TWV$, feature scaling [24] is used to bring all values into a common range by applying the z-$score$ normalization approach based on the mean and standard deviation of each node metric across all nodes v [25], as shown for metric $C_d\left(v\right)$:

$$C_{d_{norm}}=\frac{C_d\left(v\right)-mean\left(C_d\left(v\right)\right)}{standarddeviation(C_d\left(v\right)}$$

(4)

We apply the normalization method to each metric, resulting in the following:

$$TW{V}_{II}\left(v\right)=C_{d_{norm}}\left(v\right)+C_{b_{norm}}\left(v\right)+C_{c_{norm}}\left(v\right)+C_{p_{norm}}\left(v\right).$$

(5)

2.2.2. Standalone Infrastructure

Many critical infrastructures are not typically presented as a graph (e.g., factories and healthcare). Instead, as standalone infrastructures, we utilize a weighted mixture of m context-dependent factors to discover the related importance of infrastructure elements. For instance, the characteristic factors of one healthcare element, such as a hospital, are the capacity of that hospital in terms of power requirements, bed number, and the population inhabited around the hospital.

Four critical infrastructures have been measured in this paper, as follows:

Healthcare

A healthcare infrastructure analysis has shown that hospitals are the most critical components. Therefore, ensuring that hospitals are running is crucial to disaster response and resilience. Accordingly, three parameters were employed to describe hospitals: (1) power consumption, (2) capacity, and (3) population surrounding a hospital area. Hospital size or capacity is typically determined through a specific quantity of beds that can be obtained from publicly available information. Employing the same hospital capacity further points to a helpful parameter called energy consumption over applying the formula conducted by Schneider Electric [26], as

$$P_H=BD\ast \overline{U}$$

(6)

This equation represents the hospital energy use as $P_H$, $BD$ denotes the number of operated beds in the same hospital, and $\overline{U}$ signifies the bed’s power use in kWh (assumed as 30,000 kWh/year in [26]). The population surrounding the hospital is another parameter assigned to measure the importance of healthcare nodes. It can be assigned using the density data from [27] and a one km radius circular space throughout that node (hospital).

Water System

Through natural disasters, ensuring a reserve of drinking water is crucial. Therefore, water treatment plants have been selected to be critical in this model to ensure the supply of such infrastructures. Their size can also be characterized by millions of gallons per day (MGD). Thus, the power consumption $P_W$ was also estimated utilizing the following formula:

$$P_W=G\ast J$$

(7)

G in this equation reflects the portion of water processed in the MGD unit, and J represents the power required to process a million gallons [28], where the average water planet use around 1470 kWh/MG.

Cellular Network

The communication infrastructure is considered one of the most vital infrastructures to a society remaining functional. Due to the ubiquity of cell phones, cellular base stations are considered significant elements in the communication infrastructure. Furthermore, the authors of [9] discussed how the cellular network could be reconfigured to be used to provide disaster recovery communications. This study identified the most critical cellular network base stations by employing three factors: (1) geographic coverage, (2) population covered, and (3) power requirements. The geographic coverage for each base station was classified into short, medium, and long in terms of distance in miles.

Emergency Shelter

Through natural disasters, it is critical to provide emergency shelters with power. Governments commonly utilize event centers as a shelter when emergencies occur. For instance, the George R. Brown Convention Center in Houston, TX, covered thousands of people throughout hurricane Harvey [29]. Three inputs were selected to classify emergency shelters: energy usage, size, and capacity. Both data regarding the capacity and size of emergency shelters are openly accessible online. Nevertheless, power consumption was determined by calculating the size in $f{t}^2$ and then by multiplying it by the power average required for universal non-residential property space, which is 14 kWh per $f{t}^2$ [30].

Total Weighted Value ($TW{V}_{SI}$)

The total weighted value $TW{V}_{SI}$ related to every node v in a standalone infrastructure can be defined as follows:

$$TW{V}_{SI}\left(v\right)=\sum _{n=1}^{m}Par{m}_n\left(v\right)$$

(8)

where $Par{m}_n$ is the context-dependent factor for that infrastructure. As in the interconnected infrastructure case, feature scaling is used to put the parameters on the same scale.

2.3. Optimization Model Formulation

Consider a set of L smart infrastructures in an area such as a city. Each infrastructure include $N^l$ nodes with $\{l=1,2,\dots L\}$. The geographic space is divided into K zones denoted by $Ge{o}_i$ with $\{i=1,2,\dots K\}$. The size and configuration of the geographic zones can be based on various properties of the region investigated (e.g., political boundaries, squares, etc.). Note that the geographical range of microgrid elements are normally limited to within 10 km² [31]. Here, for simplicity, the geographic space is spilt into K equal size squares, as shown in Figure 2.

$F_i$ denotes the price of building and running a microgrid in $Ge{o}_i$. Determining the total project cost $F_i$ will be based on many determinants, such as the volume of power produced, the combination of power sources (i.e., fuel cells, wind turbines, solar cells, and batteries), the value of property, financing, and the construction and maintenance cost. Here, we assume that $F_i$ is pre-computed by employing a relevant model [8], including the scaling toward the range of $TWV$. Let the decision variable $x_i$ denote a binary variable equivalent to one if area $Ge{o}_i$ is selected as the location for a microgrid. We define $S_i$ as the power production of a microgrid at position i. Let $P{C}_{i{v}^l}$ denote the price regarding energy provided through a microgrid at position i passed to node $v^l$ of the lth infrastructure. The decision variable $y_{i{v}^l}$ implies the portion of power node $v^l$ of infrastructure l obtained from a microgrid at location i. Given the notation above, the microgrid location problem is to be expressed as an optimization problem $P1$ as follows:

$$P1:Min\alpha \sum _{i=1}^K\sum _{l=1}^L\sum _{v^l\in Ge{o}_i}-C_p\left(v^l\right)y_{i{v}^l}P{C}_{i{v}^l}TWV\left(v^l\right)+\beta \sum _{i=1}^K{F}_i\ast x_i$$

(9)

$$\sum _{l=1}^L\sum _{v^l\in Ge{o}_i}{C}_p\left(v^l\right)y_{i{v}^l}\le S_i\ast x_i\forall i$$

(10)

$$\sum _{v^l\in Ge{o}_i}{C}_p\left(v^l\right)y_{i{v}^l}\le {\eta }^l{S}_i\ast x_i\forall l$$

(11)

$$\sum _{i=1}^K{x}_i=1$$

(12)

$$0\le y_{i{v}^l}\le 1\forall i,v^l;x_i\in \{0,1\},$$

(13)

The objective (9) is to find the minimum cost location for the microgrid while powering the most critical infrastructure nodes. The objective function represents, in the first term, the expense of transferring power to a node $v^l$ from a microgrid placed in $Ge{o}_i$ weighted by the importance $TWV\left(v^l\right)$ of the node. The $TWV\left(v^l\right)$ values are conducted using Equations (5) or (8) depending on the type of infrastructure. The second term in the objective function is the total microgrid project cost if installed in $Ge{o}_i$. Finally, $\alpha$ and $\beta$ in the objective function are weights that can be customized to trade-off infrastructure importance versus cost of building and operating. The first constraint ensures that the power transferred to the infrastructure loads is less than or equal to the production of the microgrid if it is located in $Ge{o}_i$. The second constraint seeks to enforce the community/shared nature of the microgrid by ensuring that a single infrastructure can receive a maximum of ${\eta }^l$ percent of the power produced by the microgrid in $Ge{o}_i$. The ${\eta }^l$ values are assumed and could reflect the infrastructure’s financial contribution of infrastructure l to the cost of constructing and operating the microgrid. The constraint in (12) guarantees that only a single microgrid is built, and the constraints in (19) ensure the boundaries of the decision variables.

Note that several alternate formulations and extensions to the optimization model are possible. For example, one can relax the constraint that a microgrid in $Ge{o}_i$ can only power infrastructure nodes $v^l$ in $Ge{o}_i$. Instead, it assumes that the potential location of a microgrid is the center of each $Ge{o}_i$ and definesd the distance from a microgrid placed in $Ge{o}_i$ to node $v^l$ as $d_{i{v}^l}$. Furthermore, $d_{max}$ is defined as the maximum distance that a node can be located from a microgrid. In this case, the microgrid placement problem can be formulated as problem $P2$.

$$P2:Min\alpha \sum _{i=1}^K\sum _{l=1}^L\sum _{v^l=1}^{N^l}-C_p\left(v^l\right)y_{i{v}^l}P{C}_{i{v}^l}TWV\left(v^l\right)+\beta \sum _{i=1}^K{F}_i\ast x_i$$

(14)

$$\sum _{l=1}^L\sum _{v^l=1}^{N^l}{C}_p\left(v^l\right)y_{i{v}^l}\le S_i\ast x_i\forall i$$

(15)

$$\sum _{v^l=1}^{N^l}{C}_p\left(v^l\right)y_{i{v}^l}\le {\eta }^l{S}_i\ast x_i\forall l$$

(16)

$$\sum _{i=1}^K{x}_i=1$$

(17)

$$y_{i{v}^l}{d}_{i{v}^l}\le d_{max}\forall i,v^l$$

(18)

$$0\le y_{i{v}^l}\le 1\forall i,v^l;x_i\in \{0,1\},$$

(19)

The cost of the microgrid is minimized while connecting nodes with greater importance in the infrastructures considered to the microgrid. The first three sets of constraints serve the same function as in model $P1$, that is, (12) ensures that the capacity of the microgrid is not exceeded, (13) limits the fraction of power that a single infrastructure can use, and (14) requires that only a single location for the microgrid is selected. In addition, the constraints defined by (15) limit the maximum distance that a node can be from the microgrid, thereby ensuring a practical geographic span for the system. Lastly, constraints (16) define the restrictions on the decision variables.

The optimization models $P1$ and $P2$ are mixed integer linear programming (MILP) problems that the bound and branch model can solve for undersized problem instances using standard optimization software (e.g., CPLEX, Gurobi). The outcomes of the models show the optimal location for a microgrid and have the benefit of selecting which infrastructure nodes attach to the pre-selected microgrid. In general, given the regulatory constraints on the size and ownership of microgrids, we expect that if multiple microgrids are deployed, they are built sequentially and support different consortia of infrastructure owners and community groups. In this scenario, the optimization models can be applied iteratively by re-running the model while modifying the power requirement and value of TWV, excluding nodes linked to a previously deployed microgrid.

2.4. Critical Node Identification

The branch and bound algorithm used to solve $P1$ and $P2$ is known to have scalability issues as the fundamental problem is NP-hard. Furthermore, the number of nodes/components in critical infrastructures can be quite large in a city. For example, consider the core (9 km × 9 km) area of the Pittsburgh, Pennsylvania metro area, which has a population of 2.3 million. According to the US Department of Homeland Security, the core area contains over 2700 critical infrastructure nodes, which include 80 water infrastructure nodes and 530 communication infrastructure nodes. Hence, optimizing all nodes/components in several infrastructures will be computationally complex. Here, the microgrid location optimization models are scaled by reducing the number of nodes in each infrastructure to only the most critical nodes determined by the TWV values. In effect, this reduces the search space over which the optimization models are solved, significantly speeding up the computation but at the expense of loss of global optimality guarantees. Various approaches targeting the selection of the most critical or essential nodes in each infrastructure have been introduced in the literature. For that, two methods are considered, as follows:

2.4.1. Combined Metric

In this method, the nodes $v^l$ are arranged in descending order based on the $TWV\left(v^l\right)$ values. The nodes with the largest $TWV\left(v^l\right)$ values are considered the highest critical nodes. For simplicity, a size of 20 nodes has been selected to show the model’s top nodes.

2.4.2. List of Lists

An alternate method is to rank the nodes according to each parameter/metric and then to combine the lists for an overall ranking. Hence, a prioritized list following descending order has been generated toward interdependent infrastructures using the values of $C_d,C_b,C_c$, and $C_p$. In the standalone infrastructures, the lists have been generated using the outcomes value of $Par{m}_n$ for every infrastructure. The positional ranking value in each list is taken as the score for that list. Next, all positional rank values are summed into a total score and sorted in ascending order from below to most crucial to discover the critical nodes (a low score implies a more important node).

2.5. Microgrid Location Heuristics

With subsequent determination of the critical nodes concerning every infrastructure, the geographic space for the smart city is aligned before the optimization implementation. Figure 3 illustrates an example where $Ge{o}_i$ is taken as a square of 3 km × 3 km. The optimization models $P1$ or $P2$ can then be solved over this reduced set of infrastructure nodes. Note that the set size selection can control the optimization model solution’s computational run time. The larger the set size, the larger the search space, resulting in longer solution times and being closer to a global optimal.

Alternatively, we propose a simple heuristic based entirely on the $TWV$ values (i.e., ignoring the cost). Let $T{I}_i$ denote the total importance value of area $Ge{o}_i$. Then, $T{I}_i$ is determined by adding the $TWV$ regarding every node positioned inside $Ge{o}_i$ as revealed below.

$$T{I}_i=\sum _{l=1}^L\sum _{v^l\in Ge{o}_i}(TW{V}_{II}\left(v^l\right)+TW{V}_{SI}\left(v^l\right)).$$

(20)

Since node $v^l$ is a part of a single infrastructure, just one of $TW{V}_{II}\left(v\right)$ or $TW{V}_{SI}\left(v\right)$ remains non-zero. $Ge{o}_i$ with the most significant $T{I}_i$ value is chosen as the most optimal microgrid position. If many microgrids happen to be discovered, one reproduces the heuristic sequentially concerning every microgrid and coordinates the power and TWV rates in every repetition. Note that the heuristic is computationally simple and can be solved over the entire set of critical infrastructures nodes.

3. Results and Discussions

First, the two critical node selection methods of Section 2.4 are compared by developing a random graph of N nodes and E edges. The edges in this scenario can link to a node using a probability p. For simplicity in the calculation, parameter values $N=200$, $E=238$, and $p=0.01$ were selected by drawing on previous work that used random graphs to model transportation networks and smart power grid networks [32]. The power requirements $C_P\left(v\right)$ of each node v were created by sampling a uniform [0, 1] random variable.

Table 1. Top 20 most critical nodes utilizing the Combined Metric approach.

Node ID	${\mathit{C}}_{{\mathit{b}}_{\mathbf{norm}}}$	${\mathit{C}}_{{\mathit{c}}_{\mathbf{norm}}}$	${\mathit{C}}_{{\mathit{d}}_{\mathbf{norm}}}$	${\mathit{C}}_{{\mathit{p}}_{\mathbf{norm}}}$	Sum	Rank (Sum)
172	2.737	0.525	3.553	−0.023	6.792	1
197	3.874	0.537	2.075	0.130	6.616	2
133	2.881	0.529	2.814	−0.227	5.998	3
49	4.075	0.513	2.075	−0.872	5.791	4
144	2.461	0.493	1.337	1.450	5.740	5
160	1.390	0.465	2.075	1.416	5.346	6
1	2.481	0.548	1.337	0.680	5.046	7
182	1.548	0.496	2.075	0.750	4.870	8
32	2.739	0.502	1.337	0.209	4.786	9
5	1.878	0.533	2.075	0.276	4.762	10
10	1.024	0.455	1.337	1.468	4.284	11
188	1.177	0.450	2.075	0.556	4.258	12
195	1.135	0.442	1.337	1.289	4.203	13
162	1.952	0.500	2.075	−0.532	3.995	14
142	1.287	0.498	1.337	0.735	3.857	15
137	1.249	0.479	1.337	0.699	3.765	16
130	1.412	0.493	0.598	1.150	3.653	17
86	1.350	0.496	1.337	0.358	3.541	18
116	0.284	0.488	1.337	1.384	3.493	19
30	1.084	0.512	0.598	1.244	3.438	20

lists the twenty most critical nodes using the combined metric ranking.

Table 2. Top 20 most critical nodes utilizing the List of Lists approach.

Node ID	${\mathit{C}}_{{\mathit{b}}_{\mathbf{norm}}}$	$\mathit{R}\left({\mathit{C}}_{\mathit{b}}\right)$	${\mathit{C}}_{{\mathit{c}}_{\mathbf{norm}}}$	$\mathit{R}\left({\mathit{C}}_{\mathit{c}}\right)$	${\mathit{C}}_{{\mathit{d}}_{\mathbf{norm}}}$	$\mathit{R}\left({\mathit{C}}_{\mathit{d}}\right)$	${\mathit{C}}_{{\mathit{p}}_{\mathbf{norm}}}$	$\mathit{R}\left({\mathit{C}}_{\mathit{p}}\right)$	Sum	Rank (Sum)
144	2.461	7	0.493	28	1.337	11	1.450	7	53	1
1	2.481	6	0.548	1	1.337	11	0.680	64	82	2
160	1.390	22	0.465	48	2.075	3	1.416	12	85	3
182	1.548	20	0.496	23	2.075	3	0.750	55	101	4
197	3.874	2	0.537	2	2.075	3	0.130	97	104	5
10	1.024	34	0.455	53	1.337	11	1.468	6	104	5
30	1.084	31	0.512	13	0.598	34	1.244	26	104	5
5	1.878	13	0.533	5	2.075	3	0.276	85	106	8
116	0.284	54	0.488	31	1.337	11	1.384	14	110	9
142	1.287	25	0.498	21	1.337	11	0.735	57	114	10
130	1.412	21	0.493	29	0.598	34	1.150	34	118	11
32	2.739	4	0.502	18	1.337	11	0.209	89	122	12
176	0.758	40	0.500	20	0.598	34	1.202	29	123	13
172	2.737	5	0.525	8	3.553	1	−0.023	110	124	14
50	0.883	37	0.519	9	1.337	11	0.628	67	124	14
195	1.135	29	0.442	63	1.337	11	1.289	23	126	16
4	0.996	35	0.482	35	0.598	34	1.251	25	129	17
133	2.881	3	0.529	6	2.814	2	−0.227	120	131	18
132	0.303	52	0.508	15	0.598	34	1.156	32	133	19
137	1.249	26	0.479	38	1.337	11	0.699	62	137	20

shows the twenty most important nodes of the same infrastructure employing the list of lists ranking. As discussed earlier, the z-$score$ normalization method was used for scaling the terms throughout our numerical results.

Table 3. Comparison between node ranking in both approaches.

Node Ranking	Combined	List
1	172	144
2	197	1
3	133	160
4	49	182
5	144	197
6	160	10
7	1	30
8	182	5
9	32	116
10	5	142
11	10	130
12	188	32
13	195	176
14	162	172
15	142	50
16	137	195
17	130	4
18	86	133
19	116	132
20	30	137

contrasts the results of the two critical node identification methods for an individual network. Observe the variations within each rank concerning the most critical nodes; the collection of nodes in the highest twenty possess ≥ 80% equivalent. Since both ranking methods produce comparable outcomes, the combined metric has been utilized for the rest of our study.

3.1. Case Study

As a case study illustrating the location problem, critical infrastructures in the city of Pittsburgh, Pennsylvania, were analyzed, which has a metropolitan area population of 2.3 million. We concentrate on the center section regarding the metro area, studying a 9 km × 9 km section centered on downtown. In addition, the four infrastructures of water, cellular communications, healthcare, and emergency shelter were studied and discussed.

3.1.1. Healthcare

The healthcare infrastructure data are presented in

Table 4. Healthcare.

Hospital	Power Consumption (kWh)	Population	Size (Beds)	${\mathit{TWV}}_{\mathit{SI}}$
West Penn Hospital	9,510,000	12,915	317	−0.034
UPMC Montefiore	7,500,000	24,080	250	0.976
UPMC Mercy	14,460,000	15,714	482	2.361
Allegheny Hospital	11,430,000	5949	381	−0.402
UPMC St. Margaret	7,470,000	5200	249	−2.078
Children’s Hospital	8,880,000	15,500	296	0.136
LifeCare Hospitals	7,080,000	9555.00	236	−1.529
UPMC Presbyterian	23,850,000	24,080.00	795	7.396
VA Healthcare	4,380,000	14,117	146	−1.854
UPMC Shadyside	15,600,000	15,916	520	2.841
Magee’s Hospital	10,800,000	10,850	360	0.140
St. Clair Hospital	9,840,000	4569	328	−1.249
Children’s Genetics	7,500,000	10,850	250	−1.156

, which were normalized and applied to evaluate the $TW{V}_{SI}$ from (8). Three parameters were selected to represent each hospital’s importance: capacity, power consumption, and population using Equation (6). The table shows that the UPMC Presbyterian, UPMC Shadyside, and UPMC Mercy Hospitals have the highest total weighted values.

3.1.2. Water System

Table 5. Water treatment plants (WTP).

Facility	Capacity (Million G/Day)	Power Consumption (kWh)	${\mathit{TWV}}_{\mathit{SI}}$
Pittsburgh WTP	70,000,000	327,600,000	3.029
Brush WTP	1,500,000	7,020,000	−1.387
Plum Creek WTP	2,234,669	10,458,250.9	−1.340
Harmar Twp WTP	1,500,000	7,020,000	−1.387
Westview WTP	39,850,000	186,498,000	1.085

delivers Pittsburgh’s City water treatment plants, using public data, including the power consumption for each water treatment facility, as calculated using the Equation (7). Then, Equation (8) is applied for the total weighted value for standalone infrastructure. The table shows that two of the five tested water treatment planets are considered critical: Pittsburgh WTP and Westview WTP.

3.1.3. Cellular Network

Table 6. LTE base stations.

eNB ID	Coverage Index	Population Covered	Power Consumption (kWh)	${\mathit{TWV}}_{\mathit{SI}}$
780206	3	4475	25	2.156
780007	2	16,227	20	1.848
780165	2	5338	20	−0.141
780059	3	7119	25	2.639
780213	1	15,500	15	−0.739
780108	2	15,500	20	1.715
780399	1	12,915	15	−1.211
780184	1	5096	15	−2.639
780017	3	843	25	1.492
780527	1	8424	15	−2.032
780037	1	8424	15	−2.032
780364	1	24,080	15	0.828
780154	2	6446	20	0.061
780178	2	6446	20	0.061
780560	3	6446	25	2.516
780225	2	2710	20	−0.621
780163	3	7622	25	2.731
780032	1	2688	15	−3.079
780540	3	5948	25	2.425
780167	2	4593	20	−0.277
780477	2	4593	20	−0.277
780169	1	3316	15	−2.965
780218	1	6078	15	−2.460

shows the three parameters selected for measuring the importance of LTE base stations: coverage index, population, and power consumption. The coverage index has been assigned to a specific value from the high (3) to low (1). For simplicity purposes, the base stations were considered from a single operator (i.e., AT&T). The geographic coverage range of a base station and the population density around that station determine the population included. Finally, applying the energy model from [33], we measure the power requirement of each base station. The table displays the data obtained in the specified area of central Pittsburgh, USA.

3.1.4. Emergency Shelter

Table 7. Emergency shelter.

Facility	Power Consumption (kWh)	Size ${\mathit{ft}}^2$	Occupancy	${\mathit{TWV}}_{\mathit{SI}}$
Petersen Center	224,000	16,000	12,508	−0.198
PPG Paints Arena	10,080,000	720,000	19,758	−0.017
Convention Center	21,000,000	1,500,000	109,445	2.225
Irish Centre	168,000	12,000	315	−0.503
Sigmas Center	140,000	10,000	150	−0.507
Sherwood Center	252,000	18,000	500	−0.499

lists the $TWV$ regarding every emergency shelter in Pittsburgh city. Power usage and shelter capacities are the selected parameters, as explained in Section 2, to measure the shelter’s total weighted value. The table also shows that the Convention Center has the highest values applying such parameters.

Given the infrastructure data above in Table 4, Table 5, Table 6 and Table 7, we made an initial analysis of the city, considering a grid of 3 km × 3 km squares as $Ge{o}_i$. Figure 4 displays heat maps of the top significant nodes in each infrastructures. Figure 5 exhibits the heat map when all four infrastructures are overlain and considered a group. The results present the outputs of applying the proposed heuristic and optimization models to the data.

3.2. Heuristic

Table 8. Total importance value for every zone.

${\mathit{Geo}}_{\mathit{i}}$	Healthcare	Water	Cellular	Shelter	TI	Rank
1	0.5793	0.5598	0	1.1171	2.2562	5
2	0	0	0	0.0527	0.0527	8
3	0.3497	1	3.0117	0	4.3614	3
4	0	0	0	0	0	9
5	8.7257	0	3.5511	4.0278	16.3046	1
6	3.2216	0	4.3177	0.1063	7.6456	2
7	0.5609	0	0	0	0.5609	7
8	0	0	1.2411	0	1.2411	6
9	0	0.0107	2.9535	0	2.9642	4

shows the overall importance $T{I}_i$ value for each $Ge{o}_i$. The table also shows the fifth, sixth, and third squares are in the top of the list with the highest values at $16.3046$, $7.6456$, and $4.3614$. In

Table 9. Total power demand for every zone (kWh).

${\mathit{Geo}}_{\mathit{i}}$	Healthcare	Water	Cellular	Shelter	Total
9	0	3285	15	0	3300
8	0	0	45	0	45
7	27,333.3	0	0	0	27,333
6	94,416.7	0	95	44,467	138,979
5	189,000	0	90	88,336	277,426
4	0	0	0	0	0
3	20,750	98,280	75	0	119,105
2	0	0	0	389	389
1	33,000	55,949	0	22,400	111,349
Total	364,500	157,514	320	155,592	677,926

, the power demand of each square is broken down by individual infrastructure. Note that the power is included in scaled form in the $TI$ values of Table 8. Interestingly, when the squares are ranked by power requirements only, they do not match the ranking based on total importance score except for the first and last position (squares 5 and 4, respectively). Considering both tables, the fifth, sixth, and third squares are the most important compared to the rest and thus are the most desirable for locating a microgrid.

3.3. Optimization Model

The optimization models $P1$ and $P2$ require estimating the total cost $F_i$ for microgrid to construct and operate in a specfic location $Ge{o}_i$. HOMER design software [34] was used to determine the cost of a 4 MW microgrid consisting of a mix of diesel generators, DC/AC converters, flat panel photovoltaic cells, 1.5 kW wind turbines, and 1 kW lithium acid batteries for storage. Furthermore, the cost of real estate assuming a greenfield deployment of the microgrid was estimated for each $Ge{o}_i$ Considering a lifetime of 23 years and a net present cost value of $F_i$ covering the capital cost, replacement, salvage, operating and fuel, and repair, a discount value of 6% was found as $F_i=\{6,6.2,7.4,7.2,11,11.5,5.4,5.6,4.4\}$. The difference in $F_i$ values is the real estate cost. The additional $P1$ optimization model variables were $S_i=4MW\forall i$ and $\alpha =\beta =0.5$.

Table 10. Optimization vs. Heuristic represented by square numbers.

Microgrid Quantity	1	2	3
Heuristic $Ge{o}_i$	5	5, 5	5, 5, 6
$P1$ Optimization Model $Ge{o}_i$	5	5, 5	5, 5, 6

shows the outcomes of solving optimization model $P1$ and the heuristic based on $T{I}_i$ while varying the number of sequentially built microgrids. Table 10 clearly shows that $Ge{o}_5$ is the preferred location for a microgrid due to the high number of critical nodes in that geographic space. Furthermore, Table 10 shows that both the optimization model and heuristic implemented sequentially favor the critical squares with high total weight value, even in the case of multiple microgrids.

The trade-offs were also investigated through the infrastructure and cost weight by modifying the value of $\alpha$ and the value of $\beta$ considering one microgrid, with the results presented in

Table 11. Optimization outcomes varying weights.

${\mathit{Geo}}_{\mathit{i}}$	5	5	5	5	5	5	5	5	5	5	9
$\alpha$	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1	0
$\beta$	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1

. The table shows that geographical zone number five is again selected, excluding the case where $\alpha =0$ shows that the cost is only minimized.

The solution of optimization model $P2$ was studied next. As noted earlier, $P2$ allows a microgrid to supply important power nodes outside of the $Ge{o}_i$ it is located in, as long as the distance to the node is less than a given maximum (e.g., 6 km). The results of solving $P2$, using the same cost values $F_i$ as discussed above, to place three microgrids sequentially are shown in Figure 6. In Figure 6, the squares identify the microgrid locations, and the circles denote the nodes connected to the microgrid. Notice that the microgrid in $Ge{o}_5$ powers two nodes (cellular base stations) in $Ge{o}_1$. Similarly, the microgrid in $Ge{o}_6$ connects to nodes in $Ge{o}_5$ and $Ge{o}_3$ as well as nodes in $Ge{o}_6$.

Table 12. Percentage of power generated.

${\mathit{Geo}}_{\mathit{i}}$	Hospital	Water	Cellular	Shelter	Total
5	68.13%	0.00%	0.03%	31.84%	100.00%
6	99.41%	0.00%	0.10%	0.49%	100.00%
3	17.42%	82.52%	0.06%	0.00%	100.00%

shows how the power created by each microgrid is shared among the four infrastructures for each microgrid. Notice that it varies with location. Comparing the solution of $P2$ with the results of problem $P1$, the total cost of three microgrids will be less than $P2$ (29.9 vs. 33.5 million).

Furthermore, the effects of varying the capacity of the microgrids $S_i$ in the optimization models were considered. Specifically, we varied the microgrid capacity over $3,4,5$, and 10 MW and determined each case’s corresponding cost $F_i$. The Pittsburgh case study’s solution to both optimization algorithms does not change. For the solution to problem $P2$, the location of three microgrids is $Ge{o}_5,Ge{o}_6$, and $Ge{o}_3$, regardless of the microgrid capacity. Similarly, the solution to $P1$ is $Ge{o}_5,Ge{o}_5$, and $Ge{o}_6$ for the location of three microgrids for all microgrid capacities considered.

For this paper, which focuses on showing the applicability of shared Microgrid among interdependent infrastructure, we did not consider the existence of emergency generation assets required by regulation at critical infrastructure. However, such work can be extended in future work by addressing all the possible scenarios and applying all required policies.

4. Conclusions

In this work, it has been proven that emerging smart critical infrastructures will need disaster resilience that includes continuity of power and ICT support in addition to traditional infrastructure-specific methods. Furthermore, we advocated for using community-based, multi-ownership microgrids and studied where to locate microgrids to enhance the resilience of smart critical infrastructures. The suggested method takes a holistic view of considering multiple critical infrastructures and incorporates several factors, such as the component importance within critical infrastructure, the geospatial placement of infrastructures, power requirements, and microgrid cost. Furthermore, optimization models were proposed to determine the microgrid location to optimize a weight combination of cost and infrastructure node criticality. Additionally, a heuristic for determining the microgrid location based on infrastructure node importance was proposed. A case study demonstrating our method was presented for the city of Pittsburgh. From a resilience viewpoint, quantifying and perceiving which geographic zones in a city would most benefit from a microgrid will help provide a community-wide justification for microgrids. Future avenues of work include studying economic and regulatory models to improve the microgrid cost.