Distributed Weighted Gradient Descent Method With Adaptive Step Sizes for Energy Management of Microgrids

Economic dispatch problems for power systems become more challenging as the system sizes, types and data increase significantly. Facing these complex optimization problems, centralized optimization methods may fall short and appear inefficient. Conversely, distributed optimization methods seem to be a potential and powerful tool for dealing with these more complex problems. However, some widely-used distributed methods suffer from their low convergence rates. Therefore, in this paper, a faster distributed optimization method via gradient descent is proposed, where the weighted matrix and adaptive step sizes are the keys of our method. In the distributed gradient method, the weighted matrix is modified and constructed from the out-degrees and upper limits of the second derivatives of objective functions, which boosts gradients reaching consensus more quickly. Further, the adaptive step sizes are presented to overcome the low convergence rate caused by manual settings of step sizes, where the Hessian matrix are used iteratively to obtain the optimal step sizes, which cooperates with the weighted matrix for gradients to lead to a faster convergence rate. Next, two propositions and a theorem are proved to show the linear convergence feature of the proposed method, which guarantees the fast convergence of our method. Finally, the proposed method is applied to solve the economic dispatch problem that consists of economic and environmental objectives in microgrids. Simulation results show that the number of iterations of our method is only one third or even less than those of some widely used methods. Moreover, when solving economic dispatch problems, our method exhibits superior fault tolerance in comparison to centralized methods.