Total Weighted Value (*TWVSI*)

The total weighted value *TWVSI* related to every node *v* in a standalone infrastructure can be defined as follows:

$$TWV\_{SI}(v) = \sum\_{n=1}^{m} Parm\_n(v) \tag{8}$$

where *Parmn* 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<sup>l</sup>* nodes with {*l* = 1, 2, ... *L*}. The geographic space is divided into *K* zones denoted by *Geoi* with {*i* = 1, 2, ... *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.

*Fi* denotes the price of building and running a microgrid in *Geoi*. Determining the total project cost *Fi* 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 *Fi* is pre-computed by employing a relevant model [8], including the scaling toward the range of *TWV*. Let the decision variable *xi* denote a binary variable equivalent to one if area *Geoi* is selected as the location for a microgrid. We define *Si* as the power production of a microgrid at position *i*. Let *PCivl* denote the price regarding energy provided through a microgrid at position *i* passed to node *v<sup>l</sup>* of the *l*th infrastructure. The decision variable *yivl* implies the portion of power node *<sup>v</sup><sup>l</sup>* of infrastructure *<sup>l</sup>* obtained from a microgrid at location *i*. Given the notation above, the microgrid location problem is to be expressed as an optimization problem *P*1 as follows:

$$P1: \quad Min \quad \text{a} \sum\_{i=1}^{K} \sum\_{l=1}^{L} \sum\_{\mathbf{v}^{l} \in \text{Gco}\_{i}} -\mathbb{C}\_{p}(\mathbf{v}^{l}) y\_{i\mathbf{v}^{l}} P\mathbb{C}\_{i\mathbf{v}^{l}} T\mathcal{W}V(\mathbf{v}^{l}) + \boldsymbol{\beta} \sum\_{i=1}^{K} F\_{i} \* \mathbf{x}\_{i} \tag{9}$$

$$\sum\_{l=1}^{L} \sum\_{\substack{\mathbf{z}^l \in \mathsf{G}co\_i}} \mathsf{C}\_p(\mathbf{z}^l) y\_{i:v^l} \le S\_{\bar{i}} \ast \mathsf{x}\_{\bar{i}} \qquad \forall \ \mathsf{i} \tag{10}$$

$$\sum\_{\boldsymbol{\nu}^{l} \in \mathbf{C} \boldsymbol{\nu} \boldsymbol{\nu}\_{i}} \mathbb{C}\_{p}(\boldsymbol{\nu}^{l}) \boldsymbol{y}\_{i \boldsymbol{\nu}^{l}} \leq \boldsymbol{\eta}^{l} \mathbb{S}\_{i} \* \boldsymbol{\chi}\_{i} \qquad \forall \; I \tag{11}$$

$$\sum\_{i=1}^{K} x\_i = 1\tag{12}$$

$$0 \le y\_{i\nu^l} \le 1 \quad \forall i, \; \upsilon^l; \qquad \quad \quad \upsilon\_i \in \{0, 1\}, \tag{13}$$

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<sup>l</sup>* from a microgrid placed in *Geoi* weighted by the importance *TWV*(*v<sup>l</sup>* ) of the node. The *TWV*(*v<sup>l</sup>* ) 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 *Geoi*. Finally, *α* and *β* 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 *Geoi*. The second constraint seeks to enforce the community/shared nature of the microgrid by ensuring that a single infrastructure can receive a maximum of *η<sup>l</sup>* percent of the power produced by the microgrid in *Geoi*. The *η<sup>l</sup>* 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 *Geoi* can only power infrastructure nodes *v<sup>l</sup>* in *Geoi*. Instead, it assumes that the potential location of a microgrid is the center of each *Geoi* and definesd the distance from a microgrid placed in *Geoi* to node *<sup>v</sup><sup>l</sup>* as *divl* . Furthermore, *dmax* 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 *P*2.

$$P2: \quad Min \quad \mathfrak{a} \sum\_{i=1}^{K} \sum\_{l=1}^{L} \sum\_{\upsilon^l=1}^{N^l} -\mathbb{C}\_p(\upsilon^l) y\_{i\upsilon^l} P C\_{i\upsilon^l} T\mathcal{W}V(\upsilon^l) + \beta \sum\_{i=1}^{K} F\_i \* x\_i \tag{14}$$

$$\sum\_{l=1}^{L} \sum\_{v^l=1}^{N^l} \mathbb{C}\_p(v^l) y\_{iv^l} \le \mathbb{S}\_i \* \mathbb{x}\_i \qquad \forall \ i \tag{15}$$

$$\sum\_{v^l=1}^{N^l} \mathsf{C}\_p(v^l) y\_{iv^l} \le \eta^l S\_i \ast \varkappa\_i \qquad \forall \ I \tag{16}$$

$$\sum\_{i=1}^{K} x\_i = 1\tag{17}$$

$$
\partial\_i y\_{i\nu^l} d\_{i\nu^l} \le d\_{\text{max}} \qquad \forall \ i, \ \upsilon^l \tag{18}
$$

$$0 \le y\_{iv^l} \le 1 \quad \forall i, \ v^l; \qquad \quad \quad \mathbf{x}\_i \in \{0, 1\}, \tag{19}$$

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 *P*1, 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 *P*1 and *P*2 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 *P*1 and *P*2 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<sup>l</sup>* are arranged in descending order based on the *TWV*(*v<sup>l</sup>* ) values. The nodes with the largest *TWV*(*v<sup>l</sup>* ) 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 *Cd*, *Cb*, *Cc*, and *Cp*. In the standalone infrastructures, the lists have been generated using the outcomes value of *Parmn* 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 *Geoi* is taken as a square of 3 km × 3 km. The optimization models *P*1 or *P*2 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.

**Figure 3.** Critical node geographic representation.

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

$$TI\_i = \sum\_{l=1}^{L} \sum\_{v^l \in \text{Groy}} \left( TWW\_{II}(v^l) + TWW\_{SI}(v^l) \right) \quad . \tag{20}$$

Since node *v<sup>l</sup>* is a part of a single infrastructure, just one of *TWVI I*(*v*) or *TWVSI*(*v*) remains non-zero. *Geoi* with the most significant *T Ii* 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 *CP*(*v*) of each node *v* were created by sampling a uniform [0, 1] random variable. Table 1 lists the twenty most critical nodes using the combined metric ranking. Table 2 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 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.


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

### 3.1.1. Healthcare

The healthcare infrastructure data are presented in Table 4, which were normalized and applied to evaluate the *TWVSI* 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.


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

**Table 3.** Comparison between node ranking in both approaches.


### 3.1.2. Cellular Network

Table 6 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.


**Table 4.** Healthcare.

### 3.1.3. Water System

Table 5 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.

**Table 5.** Water treatment plants (WTP).


### **Table 6.** LTE base stations.



**Table 6.** *Cont.*

### 3.1.4. Emergency Shelter

Table 7 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.

**Table 7.** Emergency shelter.


Given the infrastructure data above in Tables 4–7, we made an initial analysis of the city, considering a grid of 3 km × 3 km squares as *Geoi*. 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 shows the overall importance *T Ii* value for each *Geoi*. 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, the power demand of each square is broken down by individual infrastructure. Note that the power is included in scaled form in the *T I* 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 *P*1 and *P*2 require estimating the total cost *Fi* for microgrid to construct and operate in a specfic location *Geoi*. 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 *Geoi* Considering a lifetime of 23 years and a net present cost value of *Fi* covering the capital cost, replacement, salvage, operating and fuel, and repair, a discount value of 6% was found as *Fi* = {6, 6.2, 7.4, 7.2, 11, 11.5, 5.4, 5.6, 4.4}. The difference in *Fi* values is the real estate cost. The additional *P*1 optimization model

variables were *Si* = 4*MW* ∀*i* and *α* = *β* = 0.5. Table 10 shows the outcomes of solving optimization model *P*1 and the heuristic based on *T Ii* while varying the number of sequentially built microgrids. Table 10 clearly shows that *Geo*<sup>5</sup> 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.


**Table 8.** Total importance value for every zone.

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


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


The trade-offs were also investigated through the infrastructure and cost weight by modifying the value of *α* and the value of *β* considering one microgrid, with the results presented in Table 11. The table shows that geographical zone number five is again selected, excluding the case where *α* = 0 shows that the cost is only minimized.

**Table 11.** Optimization outcomes varying weights.


**Figure 4.** Critical node heat map for each individual infrastructure.

**Figure 5.** Heat map of critical nodes across all infrastructures.

The solution of optimization model *P*2 was studied next. As noted earlier, *P*2 allows a microgrid to supply important power nodes outside of the *Geoi* 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 *P*2, using the same cost values *Fi* 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 *Geo*<sup>5</sup> powers two nodes (cellular base stations) in *Geo*1. Similarly, the microgrid in *Geo*<sup>6</sup> connects to nodes in *Geo*<sup>5</sup> and *Geo*<sup>3</sup> as well as nodes in *Geo*6. Table 12 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 *P*2 with the results of problem *P*1, the total cost of three microgrids will be less than *P*2 (29.9 vs. 33.5 million).

**Figure 6.** Microgrid locations from solution of *P*2.

**Table 12.** Percentage of power generated.


Furthermore, the effects of varying the capacity of the microgrids *Si* 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 *Fi*. The Pittsburgh case study's solution to both optimization algorithms does not change. For the solution to problem *P*2, the location of three microgrids is *Geo*5, *Geo*6, and *Geo*3, regardless of the microgrid capacity. Similarly, the solution to *P*1 is *Geo*5, *Geo*5, and *Geo*<sup>6</sup> 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 communitybased, 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.

**Author Contributions:** Formal analysis, A.A. and D.T.; Methodology, A.A. and D.T.; Validation, S.F.A.-G.; Writing—original draft, A.A. and D.T.; Writing—review & editing, S.F.A.-G. and R.E.-S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding

**Institutional Review Board Statement:** Not applicable

**Informed Consent Statement:** Not applicable

**Data Availability Statement:** Not applicable

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


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