# Hierarchical Networks for Water Resources Modeling

**Project ID:** 9078

**Principal Investigator:** Nancy Parker

**Research Topic:** Water Operation Models and Decision Support Systems

**Funded Fiscal Years:**
2006 and
2007

**Keywords:** *None*

## Research Question

* Can a general method be developed for imposing a hierarchy of scales on a river system network schematic, based on the order of magnitude of system variables, that will lead to more efficient solutions in simulation-optimization models?

Water systems, unlike transportation or communication networks, typically include large-scale features (e.g., lakes, rivers) with fixed identities whose behavior dominates the system. These features, along with the largest aggregate demands, form the basis of a coarse-scale system representation. Solutions obtained at the coarse scale are then translated to constraints on the finer scale problem. In this way, solutions can be obtained at varying levels of detail and are guaranteed to be consistent.

Our focus is on the mathematical formulation of the problem, not software development.

To demonstrate that the approach is viable the mathematical concepts will be implemented for a schematic of the federal Central Valley Project (CVP) and California's State Water Project (SWP) ystem.

## Need and Benefit

This project addresses a need for tools that help users to organize and manage complex network-optimization modeling tasks. River system planning modeling can be a time-consuming, iterative, and often frustrating process as study goals evolve. The proposed tools will allow more effective use of staff time spent in the development of model applications and thus faster response time to study requests. These benefits are expected to be significant and measurable.

Many mathematical and simulation tools already exist for problems related to storage, allocation and routing in water networks. Systems that cover large geographical areas and/or have complex legal and operational constraints can become difficult to use, document, and explain. This aggravates the problem of meeting stakeholder demands for transparency, robustness, and ease of use. Practically, not every use of a model requires the same level of detail, and users often impose their own informal hierarchy of more versus less important variables in an ad-hoc fashion. There is a clear need for a more systematic approach to the definition of screening levels.

The goal of this Science and Technology (S&T) Program research project is to create a set of rules that allow modelers to specify their systems at different levels of detail and to provide a thorough analysis of how the larger-scale solution interacts with the finer-scale model. This approach could potentially be used by a variety of models that use a simulation-optimization solver. The key point to note is that the computational complexity for network models typically grows exponentially with the number of nodes in the network. Our approach is to find a way to link the fast, cheap, coarse model to the detailed, accurate fine model that is rule-based and can be used in a general setting. Three areas in particular that can benefit from the scaled network approach are dealing with nonlinearities, time-horizon optimization, and isolation of network sub-sets.

Implicit nonlinearities happen where the solution on one part of the network needs to be known before the problem can be fully specified elsewhere. Examples include tiered regulatory environments, weir flows, or unique legislated rules such as California's Coordinated Operations Agreement. This can be handled by iteration, or alternatively by using integer "switches" to choose among possible responses to given conditions. The latter leads to a mixed-integer problem that is, in general, much more costly to solve. An application of our approach would be to use the coarse solution to fix the values of the integers, which could greatly speed up the computational performance of this type of model.

This approach can also be used to tackle optimization over varying time horizons. Travel time of storage releases for specific water rights and integration of changing seasonal forecasts are two examples at different scales of time horizons. For problems like these, a coarse-scale system representation would be the appropriate choice for the long-time analysis, while the finer-scale solution at each time step can be constructed using the methods developed here.

And finally, the approach should allow a sub-set of a large network to be isolated and studied in detail, with the rest of the network represented only at the coarse scale. This type of representation is appropriate to many real situations, where modelers and stakeholders have comprehensive knowledge of a specific area but are less interested in, or have less control over, the details on the rest of the network.

## Contributing Partners

None

## Research Products

Contact the Principal Investigator for information about these documents.

This information was last updated on April 25, 2015

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