Load balancing of dynamical distribution networks with flow constraints and unknown in/outflows

Abstract

We consider a basic model of a dynamical distribution network, modeled as a directed graph with storage variables corresponding to every vertex and flow inputs corresponding to every edge, subject to unknown but constant inflows and outflows. As a preparatory result it is shown how a distributed proportional-integral controller structure, associating with every edge of the graph a controller state, will regulate the state variables of the vertices, irrespective of the unknown constant inflows and outflows, in the sense that the storage variables converge to the same value (load balancing or consensus). This will be proved by identifying the closed-loop system as a portHamiltonian system, and modifying the Hamiltonian function into a Lyapunov function, dependent on the value of the vector of constant inflows and outflows. In the main part of the paper the same problem will be addressed for the case that the input flow variables are constrained to take value in an interval. We will derive sufficient and necessary conditions for load balancing, which only depend on the structure of the network in relation with the flow constraints.

Disturbance decoupling problem for multi-agent systems: A graph topological approach

“…Now, recall the multi-agent system (4) together with the output (7). Similar to Section 2.2, we first consider the open loop case where no external control input is applied to the agents.…”

“…Another example of dynamical networks in which one seeks for disturbance rejection is balancing demand-supply in distribution networks; see e.g. [6,7]. In this framework, storage variables correspond to vertices, flow inputs corresponding to edges, disturbances amount for inflows and outflows at certain vertices, and the objective is to achieve load balancing, that is convergence of storage variables to the same value by regulating the flow inputs.…”

Dynamic Pricing Control for Constrained Distribution Networks With Storage

“…Under the given assumption that the cost functions P k are of Legendre type, the gradients ∇P k are invertible on the open sets {λ k : −λ k < λ k <λ k } [4]. Since the optimal solution must always be strictly feasible, the two optimality conditions (10) can be combined to…”

“…The aspect of balancing in distribution networks has been studied in [8] using a one-step model predictive control scheme. In [9], [10] distribution networks with constant supply and demand were considered and it was shown that the balancing problem can be formulated in a Hamiltonian framework.…”

Stability of dynamical distribution networks with arbitrary flow constraints and unknown in/outflows

“…In [9], using the same model as in [8], the authors focus on a different problem of driving the state to a small neighborhood of the reference value and relate the control input value at equilibrium to an optimization problem. In the current paper we will generalize most of the results of our previous work [10] to the case of arbitrary constraint intervals, making use of a new technique extending the graph to a graph with a larger number of edges admitting a coverage by nonoverlapping cycles. Section 6 contains the conclusions.…”