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Economic dispatch problem for a networked power system has been considered. The objective is to minimize the total generation cost while meeting the overall supply-demand balance and generation capacity. In particular, a more practical scenario has been studied by considering the power losses. A non-convex optimization problem has been formulated where the non-convexity comes from the nonlinear equality constraint representing the supplydemand balance with the power losses. It is shown that the optimization problem can be solved using convex relaxation and dual decomposition. A simple distributed algorithm is proposed to solve the optimization problem. Specifically, the proposed algorithm does not require any initialization process and hence robust to various changes in operating condition. In addition, the behavior of the proposed algorithm is analyzed when the problem is infeasible.

In this paper, we propose a continuous-time distributed algorithm for the dynamic quantile problem. The problem is to find the quantile of time-varying signals in a network of agents, each of which having the signal of its own. For example, this problem includes finding the median, maximum, or the second largest value of the signals. The proposed algorithm guarantees convergence from arbitrary initial conditions and does not use the decaying gains. Hence our algorithm is suitable for plug-and-play operation, where agents may freely join or leave the network during the operation. An application to a simplified electricity market problem is presented to show the effectiveness of the design.INDEX TERMS Blended dynamics, consensus protocols, continuous time systems, distributed algorithms, multi-agent systems.