Overview of water supply reliability methods

Adequate water supplies can be assured (i.e. made reliable) via an optimum design programme involving: robust hardware, alternative flexible emergency response resources, redundant supplies and networks, and real-time control systems.

Although urban water supply systems are notable for the high degree of redundancy of their distribution networks, they are also, in a larger sense, significantly serial in nature. That is, many urban water supply systems are typically serially connected sub-systems, with many cities having only one major supply/watershed, connected by transmission along only one right-of-way, and with only one or a few treatment plants prior to connection to terminal reservoirs and the highly interconnected distribution system. Of course, each of these sub-systems (especially the distribution system) is more complex. For the overall serially connected system, or any of the sub-systems, two types of analysis can be performed:

(a) Connectivity analyses measure post-earthquake completeness, "connectedness," or "cut-ness" of links and nodes in a network. Such analyses ignore link, node, or system capacities and seek only to determine whether, or with what probability, a path remains operational between given sources and given destinations.

(b) Serviceability analyses seek an additional valuable item of information: if a path or paths connect selected nodes following an earthquake, what is the remaining, or residual, capacity between these nodes? The residual capacity is found mathematically by convolving link and node capacities with network "connectedness."

Thus, serviceability analyses seek to determine adequacy. For the simplest series system (assuming each element is independent),

the probability that Start will be connected to End (denoted P_s = probability of survival) is simply:


where
is the probability of survival of link A, etc. The probability that flow greater than x will be transmitted from Start to End, or the probability that the system will be serviceable for flow x, is similarly the product of the probabilities that each link A, B,..., n will transmit flow greater than x or be serviceable to that level (note that this treats the links as independent). For parallel and especially for more complex networks, Monte Carlo techniques are typically employed, especially for serviceability analyses (see Scawthorn et al., 1993, for a review of the methods employed in these types of analysis).

Using this type of analysis, the probability of non-serviceability or failure, P_F = 1 - P_s, can be determined, and combined with the consequences of failure to determine the expected cost of failure. For example, analyses may show that a number of ignitions will occur as a result of an earthquake, and that these ignitions will grow into a conflagration of dimensions such that N buildings and property will be destroyed by the fire. If the analysis also determines that x flow of water is required to prevent these ignitions from growing into a conflagration, then the expected cost of failure given the event is

where
is the probability of failure to provide x flow of water. The expected cost of failure can be compared with the cost of stronger or additional pipe, or other mitigation measures, in a cost-benefit analysis to determine the optimum reliability-based design. Note that in this cost-benefit analysis, the avoided cost of failure is the "benefit" (which is derived by investing in the improvements of additional pipe, etc.).