Machine learning approaches to modeling interdependent network restoration time

Abstract

The recovery of an infrastructure system after a disruptive event is vital for other systems (and for the community) that require its functionality. Disruptive events occur due to various reasons and for a system to be resilient, it is important to be prepared and ready to respond and restore. Understanding the time required for restoration for different disruptive scenarios enables decision-makers to plan for and schedule resources. In this research work, we explore different machine learning techniques to predict the time taken for an interdependent network to be restored after a disruption. We use as independent variables the restoration rates of disrupted components, and we generate the resulting network restoration time dependent variable from a network restoration optimization model. We illustrate the results of several machine learning techniques with a system of interdependent water, gas, and power utilities in Shelby County, TN and implement two types of disruption: random and spatial. The different predictive techniques used are a linear model, decision trees, gradient boosting, and random forest, which provided consistent predictions. To portray the consistency of prediction, 30 random samples (a widely accepted sample size) were trained, predicted and the results were compared. Linear model provided the best prediction results for both random and spatial disruptions with a mean RMSE of 3.8, mean correlation of 0.92 and mean bias of 0.012 for the random disruption, and mean RMSE of 1.15, mean correlation of 0.99 and mean bias of − 0.002 for the spatial disruption.

Appendix

See Tables

Table 6 Model parameters

6 and

Table 7 Model decision variables

7.

This section includes the modified model focusing on the third level, restoration level, of the existing tri-level formulation proposed by Ghorbani-Renani et al. (2020).

1.1 Model assumptions

The proposed optimization model has the following underlying assumptions associated with the structure and operation of the system of networks and their interdependencies, and recovery, among others.

• Each infrastructure network consists of a set of nodes (including supply, demand, and transshipment nodes) connected by a set of links, such that each supply node, demand node, and link have known supply capacity, demand, and flow capacity, respectively.

• There are work crews (work groups) responsible for repairing disrupted components in each infrastructure network, and this number can vary by network.

• A work crew can only work on a single disrupted component at a time. A disrupted component can be restored by a single work crew (once the disrupted component is assigned to them) until they become operational.

• There is a known restoration rate for each component, $\lambda$, representing the proportion of the component restoration per unit time by each work crew.

• Restoration time for each disrupted component is a function of both its failure and its restoration rate, both of which can vary by component.

• Infrastructure networks are physically interdependent such that every “parent” node must be operational for the dependent “child” nodes to be operational.

1.2 Notation

An undirected network is denoted by $G=\left(N,A\right)$, where $N$ is the set of nodes, and $A$ is the set of links. Assume a set $K$ of networks, each with a set of nodes ${N}^{k}$ such that ${\bigcup }_{k\in K}{N}^{k}=N$ and set of links ${A}^{k}$ such that ${\bigcup }_{k\in K}{A}^{k}=A$. Nodes can consist of supply nodes (${N}_{s}^{k}\subseteq {N}^{k}$), demand nodes (${N}_{d}^{k}\subseteq {N}^{k}$) and transshipment nodes (${N}^{k}\backslash \left\{{N}_{d}^{k},{N}_{s}^{k}\right\})$ such that ${N}_{s}^{k}\cap {N}_{d}^{k}=\varnothing$. Set ${N{^{\prime}}}^{k}\subseteq {N}^{k}$ is disrupted nodes, in network $k\in K$.. Note that this model considers a single commodity flowing through each network, but it could be easily extended to a multicommodity model. $\Psi$ represents interdependency among networks such that $\left(\left(i,k\right),\left(\overline{i },\overline{k }\right)\right)\in \Psi$ denotes node $i\in {N}^{k}$ in network $k\in K$ physically depends on node $\overline{i}\in {N }^{\overline{k} }$ in network $\overline{k }\in K$ where ${N}^{k}\cap {N}^{\overline{k} }=\varnothing$, ${A}^{k}\cap {A}^{\overline{k} }=\varnothing$ and $\forall k,\overline{k }\in K: k\ne \overline{k }$. Set ${R}^{k}$ represents the available work crews in network $k\in K$. Set $T$ provides the set of available time periods. Tables 6 and 7 outline the model parameters and decision variables, respectively.

1.3 Objective function

Define $\zeta (t)$ as the weighted proportion of unmet demand (relative to the met demand before the disruption) at time $t$, as shown in Eq. (3), where ${\eta }_{it}^{k}$ represents demand being met at node $i$ in network $k$ at time $t$, ${\eta }_{{it}_{e}}^{k}$ represents the amount of demand met prior to the disruption at time ${t}_{e}$ (time at which disruption occurred), and ${w}_{i}^{k}$ is the relative importance of node $i$ in network $k$.

$$\zeta (t)=\frac{{\sum }_{i\in {N}_{d}^{k}} \sum_{k\in K}{w}_{i}^{k}\left({\eta }_{{it}_{e}}^{k}-{\eta }_{it}^{k}\right)}{{\sum }_{i\in {N}_{d}^{k}} \sum_{k\in K}{{w}_{i}^{k}\eta }_{{it}_{e}}^{k}}.$$

(3)

Then, the proposed objective function, which seeks to minimize the cumulative weighted fraction of unsupplied demand over the planning horizon for the disrupted scenario, is defined in Eq. (4).

$$\xi =\underset{\eta ,x,\alpha ,,\beta }{\mathrm{min}}\sum_{t\in T}\zeta (t).$$

(4)

1.4 Restoration scheduling

This model is related to the restoration of disrupted components. In this model, flow balance constraints enable decision variables to connect with the objective function. Therefore, restoration scheduling is automatically set to return the system of networks to a stable operation as rapidly as possible.

The restrictions associated with the restoration scheduling comprise constraints (5)–(24). Constraint (5) states that disrupted nodes cannot be functional at period 1, since they require at least one time unit to be reactivated. Constraints (6)–(9) represent the flow balance constraints at node $i\in {N}^{k}$ in infrastructure network $k\in K$ at time $t\in T$. Constraint (10) represents the capacity restriction for each link $(i,j)\in {A}^{k}$. Constraints (11)–(12) ensure that a positive flow through any given link can be attained at a period $t\in T$ only if starting and ending nodes of such a link were already recovered. Constraint (13) ensures that the restoration task of a disrupted node is continued without interruption once it has commenced. Constraints (14) and (15) calculate the total time that a specific work crew should be assigned to restore a disrupted node. Note that constraints (14) and (15) together help to deliver integer value for the total required recovery time of an element. Constraint (16) ensures that once the disrupted node, is fully restored at time $t\in T$, they are labeled reactivated from the next period ($t+1\in T$) to the end of the time horizon of the model. Constraint (17) ensures that once the restoration of a disrupted node commences by a work crew at time $t\in T$, that specific work crew completes restoration of that node. Constraint (18) states that, at the given time $t\in T$, only one work crew can work on the restoration task of a specific disrupted node. Constraint (19) ensures that, at given time $t\in T$, only one disrupted node can be restored by a given work crew. Constraint (20) establishes the interdependency among networks, ensuring that the positive flow through a link can be only available if their corresponding related parent nodes (in other networks) are operational. Finally, constraints (21)–(24) represent the nature of the decision variables for the restoration model

$${\beta }_{i1}^{k}=0\, \forall i\in {{N}^{^{\prime}}}^{k} ,\forall k\in K,$$

(5)

$$\sum_{\left(i,j\right)\in {A}^{k}}{x}_{ijt}^{k}-\sum_{\left(j,i\right)\in {A}^{k}}{x}_{jit}^{k}\le {s}_{i}^{k}\, \forall i\in {N}_{s}^{k} ,\forall t\in T,\forall k\in K,$$

(6)

$$\sum_{\left(i,j\right)\in {A}^{k}}{x}_{ijt}^{k}-\sum_{\left(j,i\right)\in {A}^{k}}{x}_{jit}^{k}=0 \, \forall ~i~ \in N^{k} \backslash ~\left\{ {N_{d}^{k} ,N_{s}^{k} } \right\},\forall ~t \in T,\forall ~k \in K$$

(7)

$$\sum_{\left(i,j\right)\in {A}^{k}}{x}_{ijt}^{k}-\sum_{\left(j,i\right)\in {A}^{k}}{x}_{jit}^{k}\le {s}_{i}^{k}\, \forall i\in {N}_{s}^{k} ,\forall t\in T,\forall k\in K,$$

(8)

$$\sum_{\left(i,j\right)\in {A}^{k}}{x}_{ijt}^{k}-\sum_{\left(j,i\right)\in {A}^{k}}{x}_{jit}^{k}=-{\eta }_{it}^{k}\, \forall i\in {N}_{d}^{k} ,\forall t\in T,\forall k\in K,$$

(9)

$${\eta }_{it}^{k}\le {d}_{i}^{k}\, \forall i\in {N}_{d}^{k} ,\forall t\in T,\forall k\in K,$$

(10)

$${x}_{ijt}^{k}\le {u}_{ij}^{k}\, \forall \left(i,j\right)\in {A}^{k} ,\forall t\in T,\forall k\in K,$$

(11)

$${x}_{ijt}^{k}\le {u}_{ij}^{k}\left( {\beta }_{it}^{k}\right) \forall \left(i,j\right)\in {A}^{k} ,\forall i\in {{N}^{^{\prime}}}^{k}, \forall t\in T,\forall k\in K,$$

(12)

$$\sum \limits_{s = 1}^{t} \alpha_{is}^{kr} \le M \left( {1 - \left( {\alpha_{i,t + 1}^{kr} - \alpha_{i,t}^{kr} } \right)} \right) \forall i \in N^{\prime k} ,\forall t \in T,\forall k \in K,\forall r \in R^{k} ,$$

(13)

$${\sum }_{r \in {R}^{k}} \sum_{t\in T}{\alpha }_{it}^{kr}\ge \frac{{f}_{i}^{k}}{{\lambda }_{i}^{k}}\, \forall i\in {N{^{\prime}}}^{k} , \forall k\in K,$$

(14)

$${\sum }_{r \in {R}^{k}} \sum_{t\in T}{\alpha }_{it}^{kr}<\left(\frac{{f}_{i}^{k}}{{\lambda }_{i}^{k}}+1\right) \forall i\in {N{^{\prime}}}^{k} , \forall k\in K,$$

(15)

$$1-\left(\frac{\frac{{f}_{i}^{k}}{{\lambda }_{i}^{k}} -{\sum }_{r \in {R}^{k}} \sum_{s=1}^{t-1}{\alpha }_{is}^{kr}}{M}\right)\ge {\beta }_{it}^{k}\, \forall i\in {N{^{\prime}}}^{k},\forall t\in T | t\ne 1,\forall k\in K,$$

(16)

$${\sum }_{\begin{array}{c}s \in {R}^{k}\\ s\ne r\end{array}} \sum_{t\in T}{\alpha }_{it}^{ks}\le M\left(1-{\alpha }_{it}^{kr}\right) \forall i\in {N{^{\prime}}}^{k},\forall t\in T,\forall k\in K, \forall r \in {R}^{k},$$

(17)

$${\sum }_{r\in {R}^{k}} {\alpha }_{it}^{kr}\le 1 \forall i\in {N{^{\prime}}}^{k},\forall t\in T ,\forall k\in K,$$

(18)

$$\sum_{i \in {N{^{\prime}}}^{k}}{\alpha }_{it}^{kr}\le 1 \forall t\in T,\forall k\in K,\forall r \in {R}^{k},$$

(19)

$${x}_{ijt}^{k}\le {u}_{ij}^{k}\left({\beta }_{\overline{i}t }^{\overline{k} }\right) \forall \left(i,j\right)\in {A}^{k}, \forall k,\overline{k }\in K ,\forall \overline{i}\in {{N }^{^{\prime}}}^{\overline{k} } |\left(\left(i ,k\right),\left(\overline{i },\overline{k }\right)\right)\in \Psi \mathrm{or} \left(\left(j ,k\right),\left(\overline{i },\overline{k }\right)\right)\in \Psi ,\forall t\in T$$

(20)

$${\eta }_{it}^{k}\ge 0 \forall i\in {N}_{d}^{k} ,\forall t\in T,\forall k\in K,$$

(21)

$${x}_{ijt}^{k}\ge 0 \forall \left(i,j\right)\in {A}^{k} , \forall t\in T,\forall k\in K,$$

(22)

$${\alpha }_{it}^{kr}\in \left\{\mathrm{0,1}\right\}\, \forall i\in {N{^{\prime}}}^{k},\forall t\in T,\forall k\in K, \forall r \in {R}^{k},$$

(23)

$${\beta }_{it}^{k}\in \left\{\mathrm{0,1}\right\}\, \forall i\in {N{^{\prime}}}^{k},\forall t\in T,\forall k\in K.$$

(24)

Parameter $M$ in constraints (13)–(15) only needs to be greater than the maximum required time for restoring the disrupted components.

Note that demand nodes can be prioritized with the aim of emphasizing on their importance in a network. Therefore, parameters ${w}_{it}^{k}$ show the relative importance of node $i$ in network $k$ at time $t$, which can be adjusted based on different aspects including locations of the demand nodes (e.g., near hospitals, shelters, and populated or more vulnerable areas). This prioritization affects the restoration process of the disrupted network by forcing the model to satisfy the demand in high-ranked nodes prior to others.