Summary and Discussion of Lee (1992)

Citation

Lee, HB & Deininger, RA. (1992). Optimal locations of monitoring stations in water distribution system. J. of Env. Eng. 118, 4-16.

Summary

Monitoring drinking water quality is mandated by the USEPA Clean Water Act, to ensure that water is safe for public consumption. In determining where samples should be taken in a distribution system, the goal should be to pick locations with the highest representativeness. This is assessed by the coverage of the node. Coverage is the percentage of the demand at node (d_i) to the overall demand of the system (D). Integer programming can be used to determine the location for water quality sampling that achieves the greatest overall coverage.

In distribution systems, water can travel through different routes and nodes. The water quality measured at a downstream node is a function of the water quality of the nodes contributing to its flow. Thus, water quality at upstream nodes can be inferred by considering flow patterns. These relationships are represented in a water fraction matrix. The water fraction is:

W(i,j) = (flow from node j)/(inflow to sampling station i)

The result is a number between 0 and 1. A decision is required for what fraction of water is required to call a node “covered.” Based on that threshold, the water fraction matrix can be converted into a coverage matrix of binary variables (0 and 1), which represents nodes that are “covered” and “not-covered” by a sampling location.

The integer optimization problem is then takes the following form:

Max ∑d_iy_i

s.t.

∑ x_i < NS

∑ a_i,jx_i – y_i > 0 j = 1,2,….,n

Where x_i, y_i are 0 or 1; NS is number of sampling stations; d_j is demand at node I; a_i,j is the transposed coverage matrix, and n is the number of nodes

The result of a case study on Flint, Michigan, showed that integer programming was able to arrive at a sampling scheme which covered 54% of demand. This was a significant improvement over an existing coverage of 18.5%. The authors also showed that integer programming could account for multiple flow scenarios by developing multiple coverage matrices.

Discussion

I think that this type of problem is an ideal application for optimization analysis, since it avoids many of the complications of “wicked” water problems. In this case, all the choices are known, the results have little impact on human values, and the water managers have control over the distribution system. Given these conditions, the benefits expected from optimization analysis should be able to be fully realized.

I would be interested to see how this type of approach works when the problems become more “messy.” For example, what if the objective is to select water quality sampling locations across an expansive and diverse watershed? How can “upstream” considerations be included when water is traveling overland, as opposed to in pipes? I expect that elevation could be the new measure of “upstream.” It would be interesting to see if the disciplines of GIS and systems analysis could be combined to solve such a complex problem.