skip to main content

Title: A New Derivative‐Free Linear Approximation for Solving the Network Water Flow Problem With Convergence Guarantees

Addressing challenges in urban water infrastructure systems, including aging infrastructure, supply uncertainty, extreme events, and security threats, depends highly on water distribution networks modeling emphasizing the importance of realistic assumptions, modeling complexities, and scalable solutions. In this study, we propose a derivative‐free, linear approximation for solving the network water flow problem. The proposed approach takes advantage of the special form of the nonlinear head loss equations, and, after the transformation of variables and constraints, the water flow problem reduces to a linear optimization problem that can be efficiently solved by modern linear solvers. Ultimately, the proposed approach amounts to solving a series of linear optimization problems. We demonstrate the proposed approach through several case studies and show that the approach can model arbitrary network topologies and various types of valves and pumps, thus providing modeling flexibility. Under mild conditions, we show that the proposed linear approximation converges. We provide sensitivity analysis and discuss in detail the current limitations of our approach and suggest solutions to overcome these. All the codes, tested networks, and results are freely available on Github for research reproducibility.

more » « less
Award ID(s):
1847125 1728629
Author(s) / Creator(s):
 ;  ;  ;  ;  
Publisher / Repository:
DOI PREFIX: 10.1029
Date Published:
Journal Name:
Water Resources Research
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    Line coverage is the task of servicing a given set of one‐dimensional features in an environment. It is important for the inspection of linear infrastructure such as road networks, power lines, and oil and gas pipelines. This paper addresses the single robot line coverage problem for aerial and ground robots by modeling it as an optimization problem on a graph. The problem belongs to the broad class of arc routing problems and is closely related to the rural postman problem (RPP) on asymmetric graphs. The paper presents an integer linear programming formulation with proofs of correctness. Using the minimum cost flow problem, we develop approximation algorithms with guarantees on the solution quality. These guarantees also improve the existing results for the asymmetric RPP. The main algorithm partitions the problem into three cases based on the structure of therequired graph, that is, the graph induced by the features that require servicing. We evaluate our algorithms on road networks from the 50 most populous cities in the world, consisting of up to 730 road segments. The algorithms, augmented with improvement heuristics, run within 3 s and generate solutions that are within 10% of the optimum. We experimentally demonstrate our algorithms with commercial UAVs on the UNC Charlotte campus road network.

    more » « less
  2. Summary

    This paper presents a new methodology to determine fleet size and structure for those airlines operating on hub‐and‐spoke networks. The methodology highlights the impact of stochastic traffic network flow effects on fleet planning process and is employed to construct an enhanced revenue model by incorporating the expected revenue optimization model into fleet planning process. The objective of the model is to find a feasible allocation of aircraft fleet types to route legs using minimum fleet purchasing cost, thus ensuring that the expected fleet profit is maximized subject to several critical resource constraints. By using a linear approximation to the total network revenue function, the fleet planning model with enhanced revenue modeling is decomposed into the nonlinear aspects of expected revenue optimization and the linear aspects of determining fleet size and structure by optimal allocation of aircraft fleet types to route legs. To illustrate this methodology and its economic benefits, an example consisting of 6 chosen aircraft fleet types, 12 route legs, and 57 path‐specific origin‐destination markets is presented and compared with the results found using revenue prorated fleet planning formulation. The results show that the fleet size and structure of the methodology proposed in this paper gain 211.4% improvement in fleet profit over the use of the revenue prorated fleet planning approach. In addition, comparison with the deterministic model reveals that the fleet size and structure of this proposed methodology are more adaptable to the fluctuations of passenger demands. Copyright © 2016 John Wiley & Sons, Ltd.

    more » « less
  3. Future wireless networks need to support the increasing demands for high data rates and improved coverage. One promising solution is sectorization, where an infrastructure node (e.g., a base station) is equipped with multiple sectors employing directional communication. Although the concept of sectorization is not new, it is critical to fully understand the potential of sectorized networks, such as the rate gain achieved when multiple sectors can be simultaneously activated. In this paper, we focus on sectorized wireless networks, where sectorized infrastructure nodes with beam-steering capabilities form a multi-hop mesh network for data forwarding and routing. We present a sectorized node model and characterize the capacity region of these sectorized networks. We define the flow extension ratio and the corresponding sectorization gain, which quantitatively measure the performance gain introduced by node sectorization as a function of the network flow. Our objective is to find the optimal sectorization of each node that achieves the maximum flow extension ratio, and thus the sectorization gain. Towards this goal, we formulate the corresponding optimization problem and develop an efficient distributed algorithm that obtains the node sectorization under a given network flow with an approximation ratio of 2/3. Through extensive simulations, we evaluate the sectorization gain and the performance of the proposed algorithm in various network scenarios with varying network flows. The simulation results show that the approximate sectorization gain increases sublinearly as a function of the number of sectors per node. 
    more » « less
  4. Throughout the past decades, many different versions of the widely used first-order Cell-Transmission Model (CTM) have been proposed for optimal traffic control. Highway traffic management techniques such as Ramp Metering (RM) are typically designed based on an optimization problem with nonlinear constraints originating in the flow-density relation of the Fundamental Diagram (FD). Most of the extended CTM versions are based on the trapezoidal approximation of the flow-density relation of the Fundamental Diagram (FD) in an attempt to simplify the optimization problem. However, this relation is naturally nonlinear, and crude approximations can greatly impact the efficiency of the optimization solution. In this study, we propose a class of extended CTMs that are based on piecewise affine approximations of the flow-density relation such that (a) the integrated squared error with respect to the true relation is greatly reduced in comparison to the trapezoidal approximation, and (b) the optimization problem remains tractable for real-time application of ramp metering optimal controllers. A two-step identification method is used to approximate the FD with piecewise affine functions resulting in what we refer to as PWA-CTMs. The proposed models are evaluated by the performance of the optimal ramp metering controllers, e.g. using the widely used PI-ALINEA approach, in complex highway traffic networks. Simulation results show that the optimization problems based on the PWA-CTMs require less computation time compared to other CTM extensions while achieving higher accuracy of the flow and density evolution. Hence, the proposed PWA-CTMs constitute one of the best approximation approaches for first-order traffic flow models that can be used in more general and challenging modeling and control applications. 
    more » « less
  5. Abstract

    Water distribution systems (WDSs) are critical infrastructure used to convey water from sources to consumers. The mathematical framework governing the distribution of flows and heads in extended period simulations of WDSs lends itself to application in a wide range of optimization problems. Applying the classical mixed integer linear programming (MILP) approach to model WDSs hydraulics within an optimization framework can contribute to higher solution accuracy with lower computational effort. However, adapting WDSs models to conform to a MILP formulation has proven challenging because of the intrinsic non‐linearity of system hydraulics and the complexity associated with modeling hydraulic devices that influence the state of the WDS. This paper introduces MILPNet, an adjustable framework for WDSs that can be used to build and solve an extensive array of MILP optimization problems. MILPNet includes constraints that represent the mass balance and energy conservation equations, hydraulic devices, control rules, and status checks. To conform to MILP structure, MILPNet employs piece‐wise linear approximation and integer programming. MILPNet was implemented and tested using Gurobi Python API. Modeling accuracy was shown to be comparable to EPANET, a public domain software for hydraulic modeling, and sensitivity analyses were conducted to examine the impacts of the modeling assumptions on the performance of MILPNet. Additionally, application of the framework was demonstrated using pump scheduling optimization examples in single and rolling horizon scenarios. Our results show that MILPNet can facilitate the construction and solution of optimization problems for a range of applications in WDSs operations.

    more » « less