Distributed Optimization Of Multi-Agent Systems And Its Applications In Dynamic Economic Dispatch Problem

By coordinating between multiple agents,distributed optimization effectively solves the problem of large-scale optimization that many centralized algorithms are difficult to accomplish.In this paper,we study two distributed optimization problems using multi-agent systems.For the first problem,we study a distributed convex optimization problem with inequality constraints.Each agent is associated with its own cost function,and can only exchange information with its neighbors.It is assumed that each cost function is convex and subject to an inequality constraint.The objective is to make all the agents reach consensus,and meanwhile converge to the minimum point of the sum of local cost functions.A distributed protocol is proposed to guarantee that all the agents can reach consensus in finite time and converge to the optimal point within the inequality constraints.Based on the ideas of parameter projection,the protocol includes two decent directions.One makes cost function decrease,and the other makes agents step forward to the constraint set.It is shown that the proposed protocol solves the problem under undirected connected graphs without using Lagrange multiplier technique.Especially,all the agents could reach the constraint sets in finite time and stay in there after.For the second problem,we study the dynamical economic dispatch problem.In this problem,the supply demand balance,capacity constraints and ramp-rate constraints need to be considered.In order to accommodate varying power load and power storage in reality,we introduce two variables where each variable is time-varying and has its own dynamics.The renewable power is also considered.Barrier functions are introduced to deal with the local constraints by means of imposing penalty terms into the objective function to ensure that the optimal solution satisfies the corresponding constraints.Based on the Langrange dual theory,the primal optimization problem is transformed into the dual problem,which is solved by the primary-dual algorithm proposed in this paper.Under the assumption that the communication graph is undirected and connected,we prove the convergence of the proposed algorithm.Finally,simulation examples are given to illustrate the effectiveness of the proposed optimization algorithms.