# Resilience metrics¶

Resilience of water distribution systems refers to the design, maintenance, and operations of that system. All these aspects must work together to limit the effects of disasters and enable rapid return to normal delivery of safe water to customers. Numerous resilience metrics have been suggested [USEPA14]. These metrics generally fall into five categories: topographic, hydraulic, water quality, water security, and economic. When quantifying resilience, it is important to understand which metric best defines resilience for a particular scenario. WNTR includes many metrics to help users compare resilience using different methods.

The following sections outline metrics that can be computed using WNTR, including:

- Topographic metrics (Table 9)
- Hydraulic metrics (Table 10)
- Water quality metrics (Table 11)
- Water security metrics (Table 12)
- Economic metrics (Table 13)

While some metrics define resilience as a single system-wide quantity, other metrics define quantities that are a function of time, space, or both. For this reason, state transition plots [BaRR13] and network graphics are useful ways to visualize resilience and compare metrics, as shown in Figure 11. In the state transition plot, the x-axis represents time (before, during, and after a disruptive incident). The y-axis represents performance. This can be any time varying resilience metric that responds to the disruptive state. State transition plots are often generated to show time varying performance of the system, but they can also represent the time varying performance of individual components, like tanks or pipes. Network graphics are useful to visualize resilience metrics that vary with respect to location. For metrics that vary with respect to time and space, network animation can be used to illustrate resilience.

The example **resilience_metrics.py** demonstrates how to compute these metrics.

## Topographic metrics¶

Topographic metrics, based on graph theory, can be used to assess the connectivity of water distribution networks. These metrics rely on the physical layout of the network components and can be used to understand how the underlying structure and connectivity constrains resilience. For example, a regular lattice, where each node has the same number of edges, is considered to be the most reliable graph structure. On the other hand, a random lattice has nodes and edges that are placed according to a random process. A real world water distribution system probably lies somewhere in between a regular lattice and a random lattice in terms of structure and reliability.

NetworkX includes a wide range of topographic metrics that can be computed using
the WntrMultiDiGraph. WNTR includes additional methods/metrics to help compute
resilience. These methods are in the `WntrMultiDiGraph`

class.
Commonly used topographic metrics are listed in Table 9.
Information on additional topographic metrics supported by NetworkX can be found
at https://networkx.github.io/.

Metric | Description |
---|---|

Node degree | Node degree is the number of links adjacent to a node. Node degree is a
measure of the number of branches in a network. A node with degree 0 is not
connected to the network. Terminal nodes have degree 1. A node connected to every node (including itself)
has a degree equal to the number of nodes in the network.
The average node degree is a system wide metric used to describe the number of
connected links in a network.
Node degree can be computed using the NetworkX method `degree` .
Terminal nodes can be found using the method `terminal_nodes` . |

Link density | Link density is the ratio between the total number of links and the maximum
number of links in the network. If links are allowed to connect a node to
itself, then the maximum number of links is \({n}^{2}\), where \(n\) is the number of nodes.
Otherwise, the maximum number of nodes is \(n(n-1)\). Link density is 0 for a graph without edges
and 1 for a dense graph. The density of multigraphs can be higher than 1.
Link density can be computed using the NetworkX method `density` . |

Eccentricity and diameter | Eccentricity is the maximum number of links between a node and all other nodes
in the graph. Eccentricity is a value between 0 and the number of links
in the network.
Diameter is the maximum eccentricity in the network.
Eccentricity and diameter can only be computed using undirected, connected networks.
Network X includes a method to convert directed graphs to undirected graphs, `to_undirected` , and
to check if graphs are connected, `is_connected` .
Eccentricity and diameter can be computed using the NetworkX methods
`eccentricity` and `diameter` . |

Simple paths | A simple path is a path between two nodes that does not repeat any nodes. NetworkX includes a method, `all_simple_paths` , to compute
all simple paths between two nodes.
The method `links_in_simple_paths`
can be used to extract all links in a simple path along with the number of times each link was used in the paths.
Paths can be time dependent, if related to simulated flow direction. The method `weight_graph` can be used
to weight the graph by a specified attribute. |

Shortest path lengths | Shortest path lengths is the minimum number of links between a source node and all
other nodes in the network. Shortest path length is a value between 0 and
the number of links in the network.
The average shortest path length is a system wide metric used to describe the number
of links between a node and all other nodes.
Shortest path lengths and average shortest path lengths can be computed using the following NetworkX methods
`shortest_path_length` and `average_shortest_path_length` . |

Betweenness centrality | Betweenness centrality is the fraction of shortest paths that pass through each
node. Betweenness coefficient is a value between 0 and 1.
Central point dominance is the average difference in betweenness centrality
of the most central point (having the maximum betweenness centrality)
and all other nodes. These metrics can be computed using the NetworkX methods `betweenness_centrality` and the method `central_point_dominance` |

Closeness centrality | Closeness centrality is the inverse of the sum of shortest path from one node to all other nodes.
Closeness centrality can be computed using the NetworkX method `closeness_centrality` . |

Articulation points | A node is considered an articulation point if the removal of that node
(along with all its incident edges) increases the number of connected
components of a network.
Density of articulation points is the ratio of the number of articulation
points and the total number of nodes.
Density of articulation points is a value between 0 and 1.
Articulation points can be computed using the NetworkX method `articulation_points` . |

Bridges | A link is considered a bridge if the removal of that link increases the number of connected components in the network.
The ratio of the number of bridges and the total number of links in the network is the bridge density. Bridge density is a value between 0 and 1.
The method `bridges` can be used to find bridges in a network. |

## Hydraulic metrics¶

Hydraulic metrics are based upon variable flows and/or pressure. The calculation of these metrics requires simulation of network hydraulics that reflect how the system operates under normal or abnormal conditions. Hydraulic metrics included in WNTR are listed in Table 10.

Metric | Description |
---|---|

Pressure | To determine the number of node-time pairs above or below a specified pressure threshold,
use the `query` method on results.node[‘pressure’]. |

Todini index | The Todini index [Todi00] is related to the capability of a system to overcome
failures while still meeting demands and pressures at the nodes. The
Todini index defines resilience at a specific time as a measure of surplus
power at each node and measures relative energy redundancy.
The Todini index can be computed using the `todini` method. |

Entropy | Entropy [AwGB90] is a measure of uncertainty in a random variable.
In a water distribution network model, the random variable is
flow in the pipes and entropy can be used to measure alternate flow paths
when a network component fails. A network that carries maximum entropy
flow is considered reliable with multiple alternate paths.
Connectivity will change at each time step, depending on the flow direction.
The method `weight_graph` method can be used to weight the graph by a specified attribute.
Entropy can be computed using the `entropy` method. |

Fraction of delivered volume | Fraction of delivered volume is the ratio of total volume delivered to the total volume requested [OsKS02].
This metric can be computed as a function of time or space using the `fdv` method. |

Fraction of delivered demand | Fraction of delivered demand is the fraction of time periods where demand is met [OsKS02].
This metric can be computed as a function of time or space using the `fdd` method. |

Population impacted | Population that is impacted by a specific quantity can be computed using the
`population_impacted` method. For example, this method can be used to compute the population
impacted by pressure below a specified threshold. |

## Water quality metrics¶

Water quality metrics are based on the concentration or water age. The calculation of these metrics require a water quality simulation. Water quality metrics included in WNTR are listed in Table 11.

Metric | Description |
---|---|

Water age | To determine the number of node-time pairs above or below a specified water age threshold,
use the `query` method on results.node[‘quality’] after a simulation using AGE. |

Concentration | To determine the number of node-time pairs above or below a specified concentration threshold,
use the `query` method on results.node[‘quality’] after a simulation using CHEM or TRACE. |

Fraction of delivered quality | Fraction of delivered quality is the fraction of time periods where water quality standards are met [OsKS02].
This metric can be computed as a function of time or space using the `fdq` method |

Average water consumed | Average water consumed is computed at each node, based on node demand and demand patterns [USEPA15].
The metric can be computed using the `average_water_consumed` method. |

Population impacted | As stated above, population that is impacted by a specific quantity can be computed using the
`population_impacted` method. This can be applied to water quality metrics. |

## Water security metrics¶

Water security metrics quantify potential consequences of contamination scenarios. These metrics are documented in [USEPA15]. Water security metrics included in WNTR are listed in Table 12.

Metric | Description |
---|---|

Mass consumed | Mass consumed is the mass of a contaminant that exits the network via node demand at each node-time pair [USEPA15].
The metric can be computed using the `mass_contaminant_consumed` method. |

Volume consumed | Volume consumed is the volume of a contaminant that exits the network via node demand at each node-time pair [USEPA15].
A detection limit can be specified.
The metric can be computed using the `volume_contaminant_consumed` method. |

Extent of contamination | Extent of contamination is the length of contaminated pipe at each node-time pair [USEPA15].
A detection limit can be specified.
The metric can be computed using the `extent_contaminant` method. |

Population impacted | As stated above, population that is impacted by a specific quantity can be computed using the
`population_impacted` method. This can be applied to water security metrics. |

## Economic metrics¶

Economic metrics include network cost and greenhouse gas emissions. Economic metrics included in WNTR are listed in Table 13.

Metric | Description |
---|---|

Network cost | Network cost is the annual maintenance and operations cost of tanks, pipes, vales, and pumps based on the equations from the Battle of
Water Networks II [SOKZ12].
Default values can be included in the calculation.
Network cost can be computed
using the `cost` method. |

Greenhouse gas emissions | Greenhouse gas emissions is the annual emissions associated with pipes based on equations from the Battle of Water Networks II [SOKZ12].
Default values can be included in the calculation.
Greenhouse gas emissions can be computed
using the `ghg_emissions` method. |

Pump operating energy and cost | The energy and cost required to operate a pump can be computed using the `pump_energy` method. This
uses the flowrates and pressures from simulation results to compute pump energy and cost. |