Mathematical programming for the support of river water management: water allocation and reservoir location

Surface and ground water availability is variable in space and time and the spatio-temporal pattern of this variability often does not match with the distributed use pattern of sectors and individual consumers. This mismatch can become controversial when overall water availability decreases, e.g., due to climate change, and competition for water increases. It is in this context that the so called WEF-nexus between Water for human consumption and industrial use, water for Energy (hydropower) and water for Food (irrigated agriculture) (WEF) has gained increasing attention in research, business and policy spheres, especially in regions with more arid climate. An additional dimension of this nexus is the water required for sustainable functioning of ecosystems in general and wetlands in particular.

Allocation of scarce water has challenged water managers for decades. The construction and operation of reservoirs is the typical solution put forward.  In this research we addressed the optimization of the allocation of water available in a river-with-reservoir system towards multiple users as a network flow optimization (NFO) problem. There are two classes of methods to tackle NFO problems: heuristic models and mathematical models. Heuristic models are able to provide a feasible solution within reasonable computation time whereas mathematical models are able to come up with the optimal solution often requiring longer computation times. Since for strategic decisions computation times are less crucial, the latter, i.e.  linear programming (LP) models and mixed integer linear programming (MILP) models were the subject of this research. LP and MILP models were formulated to optimize the flow and storage of water through Water Supply Networks (WSN) created from geographic information describing the river basin under study. A WSN encompasses a set of oriented lines connected in georeferenced nodes whereby the lines represent river segments and the nodes represent reservoirs, natural water bodies, inflow points and abstraction points. Whereas inflow and abstraction points are characterized by time series of incoming and required water volumes, the water volume available in river segments, reservoirs and other water bodies, each having predetermined capacities, is updated throughout the simulation period.   

The LP- and MILP-models were first formulated and evaluated for hypothetical river basins characterized by artificial time series. In a next step two real world basins were considered: the Machángara River Basin, located in the Andes region of Ecuador and the Omo River Basin, located in Ethiopia, Kenya and South Sudan. Since for the latter the required time series of water discharge were not operationally available a semi-spatially distributed hydrological model (ARCSWAT) was used to generate the time series based on meteorological archives, digital elevation models and soil and landcover maps. 

The resulting NFO-LP model is meant to optimize water allocation to the different demand nodes assuming that water takes one time step to flow from one node to the next one and that water losses, temporal delays in water availability and water lost during a flood are represented by fixed fractions. The objective function –to be minimized– expressed the sum of monetary penalties related to not meeting or to exceeding the demands, to not meeting the minimum amount of water required to be available in reservoirs and river segments and to the flooding of reservoirs and segments.

The LP model was the basis for an NFO-MILP to find the optimal location of reservoirs in the WSN.  To this end, all nodes present in the WSN are considered to be candidate reservoirs of a predefined capacity. With the MILP model every candidate reservoir is evaluated individually and in combination with other reservoirs in terms of its contribution to the objective function, i.c. the minimization of the considered penalties. Four scenarios were considered: Adding additional reservoirs to the network with the pre-existing reservoirs whereby the reservoirs are either pre-filled or initially empty, and evaluating all nodes for the construction of a reservoir, including those where a reservoir is already present, again prefilled or initially empty. 

The NFO-LP model was found to allow for a quick evaluation of a given water supply network (WSN). With the NFO-MILP it could be verified whether existing reservoirs are located at the most optimal location and whether their capacity is sufficient. Also, the potential location of new reservoirs can be screened. Herewith the huge building and maintenance costs of reservoirs dominate the penalty costs related to the (non-)adequate allocation of water. 

Both the LP and MILP-models can be extended with additional constraints to enhance their real-world application potential. The huge spatio-temporal data requirements remain a hurdle though.