Linear and Non-Linear Models for Optimal Water Allocation Based on Emergy Accounting

Allocation of water resources can be represented as a mathematical model. In this model, available water is transferred from its source to several users throughout a network, which typically involves a cost per transported volume. Thus, it is desired to use the network efficiently, that is, to supply the demand with the minimum possible cost. This constitutes a network flow problem.

Network flow problems usually consider constant unit costs, generally expressed in economic terms. Both considerations overlook important aspects of water distribution. On one hand, energy consumption incurred in water distribution increases non-linearly with the volume transported because of groundwater pumping. On the other hand, evaluating energy costs in economic terms tends to disregard environmental aspects, such as the value of natural resources. Additionally, economic value fluctuates over time.

Because of this model features, non-linear programming (NLP) would be needed to minimize energy costs, especially when there exists water table depletion due to groundwater overexploitation. In NLP optimization problems, models composed by a great number of variables and/or elements may require a large amount of computational resources. Thus, the aim of the work is solving this network flow model by means of linear programming (LP) using an equivalent cost slope, providing shorter run times and less performed iterations than NLP with similar results.

Regarding water volume, emergy accounting provides a basis for assessing it, as well as products and services in common solar energy units. Therefore, the scale used in this work is emergy (spelled with an “m”), whose units are solar emjoules (sej).

A water capacitated network flow, which encompasses 71 components (subgraphs of a disconnected network), has been built taking as study zone the basin of the Upper Course of the Lerma River (UCLR), Mexico. NLP and LP optimization algorithms were created, and their results contrasted to modify ad hoc the LP algorithm to improve its accuracy.

LP’s accuracy, measured as the difference between LP and NLP’s objective function, showed relative errors varying from 9% to 88%. Another way to assess LP’s accuracy is by its number of prediction errors. If there is a volume of water being conducted from a source to a user in NLP, but there exists no such flow in LP, the prediction error committed is called false negative. The causes of this errors are discussed. By identifying renewable and non-renewable water resources in the model, almost 50% of prediction errors were corrected.