Fully distributed constrained optimization algorithm over unbalanced network and its application to smart grids

In practical engineering, due to the limited bandwidth and other physical constraints, directed network communication topology among agents is more common. This paper studies the distributed constrained optimization problem over the weight-unbalanced directed network, considering local inequality and coupled equality constraints. First, a distributed consensus scheme with a time-based generator is provided to estimate the left eigenvector of the Laplacian matrix. Building on this estimator, an adaptive proportional–integral-based gradient flow scheme is designed to solve the coupled constraint, introducing a time-varying control parameter to remove the requirement of the Laplacian matrix’s eigenvalues. Subsequently, a fully distributed primal–dual optimization method is proposed based on KKT conditions for the constrained optimization problem with a nonsmooth objective function. The optimality and convergence analysis are conducted through Lyapunov theory under the strongly connected and weight-unbalanced directed network. Finally, the established method is applied to the economic dispatch problem in smart grids.