Structure of Optimal Transport Networks Subject to a Global Constraint

We derive a linear system of ordinary differential equations (ODEs) to approximate the dynamics of natural gas in pipeline networks. Although a closed-form expression of the eigenvalues of the state matrix does not generally exist, the poles of an irrational transfer function corresponding to the linearized partial differential equations are used to approximate the eigenvalues of the ODE system. Our analysis qualitatively demonstrates that the eigenvalues of the state matrix ...

Despite its importance for practical applications, not much is known about the optimal shape of a network that connects in an efficient way a set of points. This problem can be formulated in terms of a multiplex network with a fast layer embedded in a slow one. To connect a pair of points, one can then use either the fast or slow layer, or both, with a switching cost when going from one layer to the other. We consider here distributions of points in spaces of arbitrary dimens...

The fundamental theory of energy networks in different energy forms is established following an in-depth analysis of the nature of energy for comprehensive energy utilization. The definition of an energy network is given. Combining the generalized balance equation of energy in space and the Pfaffian equation, the generalized transfer equations of energy in lines (pipes) are proposed. The energy variation laws in the transfer processes are investigated. To establish the equati...

We show that suitable convex energy functionals on a quadratic Wasserstein space satisfy a maximum principle on minimal networks. We explore consequences of this maximum principle for the structure of minimal networks.

Understanding the interactions among nodes in a complex network is of great importance, since they disclose how these nodes are cooperatively supporting the functioning of the network. Scientists have developed numerous methods to uncover the underlying adjacent physical connectivity based on measurements of functional quantities of the nodes states. Often, the physical connectivity, the adjacency matrix, is available. Yet, little is known about how this adjacent connectivity...

Existing techniques for the cost optimization of water distribution networks either employ meta-heuristics, or try to develop problem-specific optimization techniques. Instead, we exploit recent advances in generic NLP solvers and explore a rich set of model refinement techniques. The networks that we study contain a single source and multiple demand nodes with residual pressure constraints. Indeterminism of flow values and flow direction in the network leads to non-linearity...

The aim of this paper is a short survey of models and methods that developed by the authors. These models and methods are used to optimize general networks with nonlinear non-convex restrictions and objectives possessing mixed continuous-discrete optimization variables. There are discussed the problem formulations and solution methods for simulation, optimization, sensitivity and stability analysis for flow with nonlinear potential in general networks. These problems and the ...

We define a \emph{thermal network}, which is a network where the flow functionality of a node depends upon its temperature. This model is inspired by several types of real-life networks, and generalizes some conventional network models wherein nodes have fixed capacities and the problem is to maximize the flow through the network. In a thermal network, the temperature of a node increases as traffic moves through it, and nodes may also cool spontaneously over time, or by emplo...

We investigate the electrical current and flow (number of parallel paths) between two sets of n sources and n sinks in complex networks. We derive analytical formulas for the average current and flow as a function of n. We show that for small n, increasing n improves the total transport in the network, while for large n bottlenecks begin to form. For the case of flow, this leads to an optimal n* above which the transport is less efficient. For current, the typical decrease in...

The steady-state solution of fluid flow in pipeline infrastructure networks driven by junction/node potentials is a crucial ingredient in various decision support tools for system design and operation. While the non-linear system is known to have a unique solution (when one exists), the absence of a definite result on existence of solutions hobbles the development of computational algorithms, for it is not possible to distinguish between algorithm failure and non-existence of...