The introduction of bi-directional flow between nodes also makes it possible to consider loops, which are usually present in various infrastructure networks (e.g., electricity, water, transport). The loops are essential for the functioning of a network when the latter is damaged by a hazard. An issue that may arise from the inclusion of loops into a network is a possibility that the network flow optimisation problem has more than one solution. However, this usually does not occur when the costs of commodity flows are proportional to the lengths of the physical connections (e.g., electricity transmission and distribution lines, water pipes) represented by the links; even if this does occur, it can easily be fixed by assigning slightly different flow costs to two opposite direction links, which connect a pair of nodes in the loop (and by that to define a preferable direction of the commodity flow in the undamaged network).
2.1.1. Extended Network Flow Model
The nodes represent the infrastructure components/assets associated with the productions, consumption, trans-shipment, and storage of commodities; these include electricity substations, water, and sewage plants, as well as hospitals or residential buildings. In general, the infrastructure components can be in physical, cyber, geographical, or logical, and can interact with each other. The nodes are connected by links/edges, by which the interaction between the infrastructure components, usually in the form of the flow of commodities, is modelled. A network can be either balanced (when demand for commodities is equal to their supply) or unbalanced (when the demand exceeds the supply). In the latter case, unmet demand for a commodity or commodities occurs initially in one or more nodes, which then may have a cascading effect on other parts of the network.
The traditional network flow model can be used to optimise the performance of an undamaged balanced infrastructure system, which is presented as a network with
V nodes and
E edges, by solving the following linear programming problem that minimises the total operational cost of the system,
C [
15]:
where
and
are the flow rate of community on edge
e and the maximum flow capacity of this edge, respectively;
is the cost associated with this flow;
and
are the in-flow and out-flow rates at node
i, respectively;
ki is the demand/supply rate of the commodity at node
i. This traditional model cannot deal with unbalanced infrastructure systems, as well as accounting for such essential features/functions of infrastructure systems as storage and production. To deal with unbalanced (i.e., damaged) infrastructure systems and simulate more realistically their operation, including interdependencies, additional variables, representing unmet demands, storage, production, and trans-shipment, were introduced into the network flow model, and then a linear programming optimisation problem, which included these variables, was formulated in [
5]. The formulation was also time-variant, which allowed us to simulate the performance of infrastructure systems from the moment of their damage by a particular hazard until their complete recovery. The full formulation of this extended model (see [
5]) will not be reproduced herein; instead, we will concentrate on the elements of the model which have been changed/improved in the present study.
To illustrate the model changes made herein, we considered an electricity network in a small coastal town in the UK, shown in
Figure 1. As can be seen, the network includes several loops, which need to be kept in the model since they ensure the network redundancy. The latter is essential for the network functioning in the case of damage. To account for the redundancy provided by the loops, two opposite direction edges need to be introduced in the model for each connected pair of nodes in the loops. In this way, the reverse of the flow of electric current in the loops can be modelled. It is important to note that all nodes in the network operate at the voltage of 11 kV, except for the node in the lower-right quarter of the map, which represents a 132 kV substation (
Figure 1). For this network, the optimisation problem formulation from [
5] can be significantly simplified and expressed as:
where
Ui is the unmet demand in
i node;
cU,i is the cost of this unmet demand per unit.
2.1.2. Network with Loops
We began the consideration of the changes to the model in more detail with the analysis of networks with loops. Initially, we considered very simple networks with three to four nodes, as shown in
Figure 2. In this figure, circles with numbers inside represent nodes; arrows—directional links/edges, which are denoted as X1, X2, …; two links in the opposite directions between a pair of nodes will be further referred to as a bidirectional arc (e.g., links X3 and X4 between Nodes 2 and 3 in
Figure 2a, respectively). We considered only unbalanced (i.e., damaged) networks. In all three networks, Node 1 is the source of the commodity; there are unmet demands in other nodes, which will be denoted as U1, U2, … (e.g., U1 denotes unmet demand in Node 2). The costs of flow in the links and unmet demand in the nodes will be denoted as C(name of link/unmet demand), e.g., C(X1)–cost of flow in link X1, C(U1)–cost of unmet demand U1.
Using the three simple networks shown in
Figure 2, we will illustrate how the costs of flow in the edges and the costs of the unmet demands in the nodes affect the distribution of a commodity in an unbalanced network and the flow directions in bidirectional arcs. For example, in the network shown in
Figure 2a, when the network is undamaged and the cost of flow in its edges is simply proportional to the length of the physical connections modelled by these edges, the commodity will flow from Node 1 (source) to Nodes 2 and 3 (consumers) via edges X1 and X2. However, if edge X1 is damaged and the flow through it is either impossible or its costs significantly increase, so that C(X1) > C(X2) + C(X3), then edge X3 in the bidirectional arc becomes active (i.e., the pathway from Node 1 to Node 2 is via edges X2 and X3). In a similar way, if edge X2 is damaged, then the commodity will flow through edges X1 and X4 (i.e., edge X4 becomes active). The costs of unmet demands do not affect the flow pathways; however, they affect the distribution of a commodity between the consumers. For example, if the costs of unmet demands U1 and U2 are the same (i.e., C(U1) = C(U2)) and edge X1 is damaged, then the shortage of the commodity will always be more severe in Node 2. This means that as long as the supply from Node 1 exceeds the demand in Node 3, there will be no unmet demand in Node 3, while the amount of the commodity received by Node 2 will be the supply minus the demand in Node 3; when the supply is less than or equal to the demand in Node 3, Node 2 will stop receiving any commodity. In order to avoid that, the costs of unmet demand should be different. Selecting C(U1) > C(U2), we can redistribute unmet demand equally between Nodes 2 and 3; by further increasing C(U1), we can prioritise Node 2 so that its demand is fully met until it is possible and then it receives all available supply, while Node 3 receives nothing.
In the 4-node network in
Figure 2b, the edges in the two bidirectional arcs will not be activated as long as the costs of flow in the edges are proportional to the actual lengths of the connections (i.e., the commodity flow from Node 1–source will be via edges X1, X2, and X3). In this case, if C(U1) = C(U2) = C(U3), the distribution of unmet demand will depend on the costs of flow: C(X1), C(X2), and C(X3). For example, if C(X1) > C(X2) > C(X3), then U1 will increase first, followed by U2, and then U3. To change the distribution of unmet demand between the nodes, the costs C(U1), C(U2), and C(U3) need to be unequal. For example, to prioritise the supply of Node 2 C(U1) should be set higher than C(U2) and C(U3). Damage to the network can lead to activation of the edges in the bidirectional arcs. For example, if X2 is damaged (i.e., fails completely or C(X2) becomes very high), then the flow pathway from Node 1 to Node 3 may be either via X1 and X5 when C(X1) + C(X5) > C(X3) + C(X7), or via X3 and X7 otherwise. Eventually, the activation of the edges in the bidirectional arcs and resulting flow pathways depend on damage caused to the network and the costs of flow in the edges.
The 4-node network shown in
Figure 2c can illustrate the case when a single solution does not exist. If C(X1) + C(X3) = C(X2) + C(X5) and the flow capacities of the edges are large enough so that the corresponding constraints remain inactive, then the optimisation problem of minimising the flow cost has two possible solutions. To avoid such a situation, one of the costs of flow in the edges should be slightly changed. It is worth noting that the possibility of such a problem’s occurrence in networks representing real infrastructure systems, especially when the flow costs are proportional to the lengths of the physical connections, is very low. To supply a commodity to Node 4 when C(X1) + C(X3) < C(X2) + C(X5), edge X3 will be activated, and X5 otherwise. If edge X1 is completely damaged, the flow of the commodity will be through edges X2, X5, and X4. As in the other two networks, to control the distribution of the commodity in the unbalanced network (i.e., supply of the commodity is less than its total demand) the costs of unmet demands, C(U1), C(U2), and C(U3), should be set in accordance to specified priorities.
2.1.3. Extension of the Model Capability
As is clear from the examples of the simple networks in the previous section, the activation of bidirectional arcs depends on the flow costs of edges, while the distribution of commodities between consumers in unbalanced networks can be controlled by the costs of unmet demands. To further illustrate the extended network flow model, in particular, its coupling with a flood model, we considered an electricity network in a small town in the UK (and further in the paper, its interaction with a sewage network). This is a low-voltage distribution network with a nominal voltage of 11 kV. The town suffered from flooding in the past and is susceptible to flooding in the future. Since this is a coastal town with a river flowing through it, the town is in danger of both river and coastal flooding; the lowland areas are particularly at risk, where a large caravan park and a major electricity substation (see
Figure 1) are located. The actual electricity network has been simplified—several distribution lines are assumed to be straight, but the loops appearing in the network have been mainly preserved (see
Figure 1 and
Figure 3). The lengths of the straight lines have been assigned in such a way that the relative differences between these lines remain the same as such differences between the corresponding lines in the original layout. Hence, the simplifications should not have any noticeable effects on the results of the network analyses. The notation in
Figure 3 is similar to that in
Figure 2, except that the unmet demands in the consumer nodes (from 2 to 17) are denoted as X41, X42, …, X56. The major electricity substation, treated as the source of the electrical power in the network, is represented by Node 1. In order to consider redundancy in the network provided by the loops, several bidirectional arcs have been introduced, which allow switching the direction of the electric current flow in the loops.
Initially, we assumed that the costs of flow in all edges were proportional to the lengths of the edges and the costs of all unmet demands were the same (scenario S0 in
Table A1 in
Appendix A). The solution for this case (i.e., the minimum cost flow pattern) is shown in
Figure 3 by red lines, while the red numbers near the nodes along the lines indicate the sequence in which the nodes stop receiving electricity, as the supply in Node 1 gradually decreases at the rate of 10% per hour due to an accident/hazard (see
Figure 4). The results show that Node 15, which is the farthest node from the electricity substation (i.e., Node 1–source), will be the first one without electricity in 3 h after the accident (see
Figure 3 and
Figure 4). It will be followed by Node 12—the second farthest node from the source (in 4 h after the accident), and so on. The last nodes left without electricity, in 11 h after the accident, are Nodes 4, 9, and 7, which are the closest nodes to the source. Thus, it can be seen that the loss of the electricity supply in the consumer nodes occurs consecutively, i.e., node by node, and the sequence depends on the length of the links between the consumer nodes and the source. This means some consumers will be without electricity much longer than others (e.g., the consumers represented by Node 15 will be without electricity 8 h longer than those related to Nodes 4, 7 and 9); this may be unacceptable if Node 15 is an important asset.
When the costs of unmet demands among the consumer nodes are different, the order in which these nodes stop receiving the electrical power, as its supply decreases, can be changed. For example, when the cost of unmet demand for Node 15 (X54) is set at a higher value than such costs for the other nodes, Node 12 may become the first one to be left without electricity. However, the sequential character of the process in which the consumer nodes lose their electricity supply does not change. Therefore, if we simply modify the costs of unmet demands for different nodes, there are still nodes without electricity much longer than others following the one-by-one pattern, although the order is changed.
Each node in the network model contains real-life consumers (e.g., residential houses, shops, hospitals, water or wastewater pumping stations, etc.). It is considered to be highly undesirable for some consumers to be prioritised over others because of their locations. Moreover, some nodes may comprise critical infrastructure assets (e.g., hospitals, water/wastewater pumping stations, petrol stations), which need electricity to maintain at least some basic operations as long as it is still possible. For example, Node 15 includes a sewage treatment plant (see
Figure 1 and
Figure 3), which uses about 80% of the total electricity consumption at this node. Since functioning of this sewage treatment plant is critical for the whole town, electricity should be supplied to the node for maintaining at least the basic operations of this plant as long as it is still possible. There are critical infrastructure facilities also in some other nodes of the network. Node 13 contains a hospital, which consumes about 40% of the total demand for electricity in this node, while Node 14 comprises a sewage pumping station, which consumes about 10% of the electricity demand in this node. It is essential to supply electricity to these facilities, for as long as possible, when the electricity network is unbalanced (e.g., due to damage caused by flooding). This means it is necessary to be able to control the distribution of electricity between the nodes, maintaining its supply at certain levels for each node, when the network becomes unbalanced. As shown previously, this cannot be done by simply setting different but constant costs of unmet demands for different nodes.
To solve the above problem, we describe the cost of unmet demand by a stepwise function; i.e., we divide unmet demand in node
i,
, into
J(
i) sub-total unmet demands
so that
, and for each sub-total unmet demand
, we introduce its individual cost
The optimisation problem given by Equations (4)–(6) is then reformulated in the following way:
where
are the relative bounds of the sub-total unmet demands
at node
i, e.g.,
means that
(i.e.,
does not exceed 20% of the demand rate
of the commodity at node
i),
then means that
, and so on; the sum of
should be equal to
, i.e.,
.
Using this new formulation of the network flow problem, we can control the distribution of a commodity between different nodes over time, as the available supply of the commodity decreases. The selection of the number of sub-total demands and their bounds for each node depends on what needs to be achieved. For example, we may aim to ensure that critical infrastructure facilities in various nodes of the network receive a sufficient amount of a commodity for their operation as long as this is still possible, i.e., as long as the remaining supply of the commodity is not less than its demand by these critical facilities. In the network shown in
Figure 3, Node 15 contains a sewage treatment plant, which consumes about 80% of the electricity demand of this node. The unmet demand at this node can then be divided into two sub-totals—
and
, with
and
. If the cost of
is set noticeably higher than that of
and of the unmet demands at other nodes, this means that after the unmet demand at this node exceeds 20% of the node’s demand, the electricity supply to this node will be prioritised when compared to the other nodes, i.e., the sewage treatment plant will continue to receive sufficient electricity for its operation (of course, until the total electricity supply in the network drops to the demand of this plant). The unmet demand at Node 13, which comprises a hospital that requires about 40% of the node’s electricity demand, can be treated in a similar way. It can be divided into two sub-totals—
and
, with
and
. If the hospital and the sewage treatment plant are considered equally important to the town, then the cost of
should be similar to that of
, and higher than the costs of the unmet demands at other nodes; however, if it is decided that the hospital is more important than the sewage treatment plant, then the cost of
should be higher than that of
.
In the following, we examine how the distribution of electricity in the network shown in
Figure 3 can be controlled by selecting different sub-total numbers of unmet demands and their bounds when the network becomes unbalanced. We started with a simple case, in which we divided the unmet demand at each consumer node into two sub-totals—U1 and U2. The costs of the first sub-total are set up one order lower than that of the second sub-total, so that the electricity supply to the most important infrastructure assets in each node, which are related to U2, is prioritised (it is assumed that each node contains critical infrastructure assets, which have higher priority than other consumers). For example, the most important infrastructure assets in the first-interrupted node (e.g., Node 15) are always supplied, while the relatively less important consumers in that node are in shortage. The two-layer profile scenarios are shown in
Figure 5. It was assumed that total electricity supply drops by 10% hourly, which means that all nodes will be without electricity after 11 h, as shown in
Figure 4. For scenario S14 (
Figure 5a), the first-interrupted node (Node 15) could maintain 60% of its electricity demand for an extra 3 h, while the other nodes are interrupted partially when compared with the normal one-by-one case (
Figure 4). This means, for example, under the interruption, the sewage treatment plant can operate for extra 3 h, even when the total supply is in shortage continuously. Then, other CIs in Node 12 could have the electricity supply for 7 h rather than 4 h as in the baseline case (
Figure 4). This is similar for scenarios S15 (
Figure 5b), but with different distributed weights between the two sub-total costs. Therefore, the introduction of stepwise costs for unmet demand at each node could control the redistribution of the commodities in the unbalanced network system, which is in analogue to a division of multiple layers in neural network. This could increase the capability of controlling the system by human intervention or hazards, for example, by asking the residents to reduce less important consumptions if feasible; or the demands of one community would be reduced due to flooding.
Therefore, we could expect that the more layers we introduce for the cost profiles, the slower the supply to each node with higher priority assets is interrupted, to ensure that these higher priority assets remain in basic operations as long as possible. All other scenarios representing different cost step profiles are listed in
Table A1. The selected solutions are shown in
Figure 5, in which we explore how the changes of cost profile affect the entire CIs’ system. We introduced several scenarios by dividing costs into 2, 3, 4, and 5 sub-total costs in percentage (
Table A1) and the sum of these percentages should be 1. For example, for S15 scenario, 20% for first cost, 80% for second cost (
Figure 5b); for S32 scenarios (4 sub-costs), 20% for first cost, 40% for second cost, 20% for third cost, and 20% for fourth cost (
Figure 5e). Therefore, by changing these cost profiles, the performance of each node would be changed accordingly.
Generally, it is shown that the supply-to-demand ratio at each node is proportional to the cost allocated to each sub-total of unmet demand. For example, the four sub-cost scenario (S3) leads to the supply reduction at each node within four steps, rather than the dramatic one-step drop as seen in the one-by-one case (i.e., S0). This gradual reduction in the electricity supply enables to support the basic functioning of critical infrastructure assets at each node for a few extra hours, which gives more time for responses, to mitigate consequences of a hazard. This also improves the resilience of CI assets during extreme events.
In S32 scenario (
Figure 5e), the electricity supply at most nodes drops initially by 20% of their demand because the first sub-cost is allocated to 20% of the total demand in the nodes. After that, the supply starts dropping to 40% once all nodes reach 80% of demand, since 40% of second sub-cost is allocated, and this lasts longer than the previous one due to the larger sub-cost (40% > 20%). Then, as the total electricity supply reduces continuously, the nodes start to drop to 20% one by one after all nodes reach 20% of their functions, assuming the third sub-cost is 20% of the total cost of unmet demand in each node. Then, all drop to 0% once all of them reach 20%, according to the 20% of fourth cost (small figure in
Figure 5e). It is noted that the length of sub-total cost profile (scale of horizontal axis in the sub-figure in
Figure 5e) is proportional to the length of extra time that each node could gain, as indicated by the horizontal length of the large figure in
Figure 5e.
The step number of cost profiles could determine the reduction step of each node. For example, the two-stepwise profile makes the solution drop twice (
Figure 5a,b) while the three-stepwise profiles cause the solution to reduce three times, gradually (
Figure 5c,d). Then, the length to each step contributes to the length of extra hours in each node. For example, in S32 scenario shown in
Figure 5e, the first 20% sub-total cost could let the supply of each node reduce by 20% for 2 h, as the production is assumed to reduce 10% each hour. Then, the second 40% sub-total cost contributes to an extra 4 h after they drop a further 40%. After that, the further drop of 20% for 2 h is due to the next step (20%), and the final drop to 0% is due to last step of 20%.
While setting up the upper bounds of sub-total unmet demands, it is necessary to make these as percentages of the total upper bounds. For example, Xxub1 = 0.6Xxub; Xxub1 = 0.4Xxub. In this case, the model results may deviate from the solution of the case without subdividing the unmet demands where the results do not change. But if one only changes the upper bounds without considering the stepwise of sub-total unmet demands, the results are always the same as the original solution. Therefore, the constraints of the equation should introduce the percentage of maximum demands . Therefore, by considering the stepwise of cost profile, the corresponding upper bounds should also be modified (Equations (8) and (9) are necessary).
By applying this model, it is possible to redistribute the commodity in the system network when the supply is partially interrupted. Keeping CI assets at nodes on minimum supply can improve the resilience of the town. For example, there is a sewage treatment plant connecting Node 15. If only one constant cost is employed for the unmet demand in this node, the sewage treatment plant cannot be functional, even if the total supply is not very low (
Figure 4). However, if we consider this cost as a stepwise function, e.g., as shown in
Figure 5e, then the plant could be kept functional for at least an extra hour before reducing electricity supplement of the connected node to 60%, and extra 3 hours before reducing to 40%. This could prevent the spillage of untreated sewage during the hazard, while not affecting the minimum electricity supply requirements of other critical infrastructure components such as hospitals or home cares. This is also the case for the sewage pumping station at Node 14, which indicates that the assumed stepwise cost of the unmet demand enables the pumping station to be in operation for a longer period.
In all scenarios in
Table A1, due to the lack of specific data, it is assumed that all nodes follow the same cost profile shown in the small graphs. However, this can be changed according to the situation in each node, if needed. Additionally, the model can distinguish the steps only because of the 10th order difference between each sub-total step. This is similar to the process of making the entire system with several virtual layers (4 virtual layers for a 4-step cost profile); in each layer, the optimisation with one unmet demand coefficient is simulated before the final combination of all of these layers.
The question is then how to build up the stepwise profile for each node, and whether these profiles are interdependent. If it is connected to flood conditions, it can be assumed that the probability of failure of a specific infrastructure asset (e.g., electricity substation) is proportional to the flood depth at the location of this asset. Additionally, the rate of the supply reduction may also be related to the pace of the evolution of the hazard itself: flooding to a specific depth at the location of an infrastructure facility can interrupt its functioning partially or fully. The geographic connection with a flood model is a promising method. Puno et al. [
16] used a GIS spatial technique to connect the flood risk with a hydraulic flood model. Therefore, we introduce a numerical flood model next to consider the probability of failure for each node based on a temporal–spatial distribution of flooding conditions. This type of a system-modelling approach is applicable to other infrastructure systems as long as a proper connection between the two models is established.