This paper delves into a class of distributed nonconvex optimization control (DNOC) problems for multiagent systems (MASs), aiming to attain consensus among agents while minimizing the collective sum of local cost functions, referred to as the global cost function, through local information exchange. To accomplish this objective, it is introduced a novel distributed predefined-time optimal (DPTO) control algorithm. We prove that the proposed DPTO control algorithm guides the system states towards convergence to the global optimal solution within a user-defined time frame, provided that the global cost function satisfies the Polyak-Lojasiewicz condition, which is less stringent than the standard strong convexity condition. Finally, theoretical findings are substantiated through simulation studies.

In this paper, an optimal retail market pricing design for demand response in day-ahead scheduling of smart distribution networks is investigated. Through the well-designed retail market electricity price, the profit of each user is maximized; while the profit of the Distribution Network Operator (DNO) is guaranteed and the risk management model based on the Information Gap Decision Theory (IGDT) method is proposed to protect the DNO from financial risk arising out of the uncertainty of wholesale market prices. Because of the convexity of the problems of users and the DNO, the demand response program is developed based on the Predictor Corrector Proximal Multiplier, which is a distributed algorithm and guaranteed to converge to a global optimal solution, i.e. the retail market electricity price. The numerical simulation is given to show the effectiveness of the proposed pricing method.