  Motivated by potential applications in power systems, we study a problem of
optimizing a sum of $n$ convex functions on dynamic networks of $n$ nodes when
each function is known to only a single node. The nodes' variables, while
satisfy their local constraints, are coupled through a linear constraint. Our
main contribution is to design a fully distributed primal-dual method for this
problem. Under some fairly standard assumptions on objective functions, strong
convexity and smoothness, we provide an explicit analysis for the convergence
rate of our method on different networks. In particular, the nodes variables
achieve a geometric convergence to the optimal with the associated convergence
time scales quartically in the number of nodes on any sequence of time-varying
undirected graphs satisfying a long-term connectivity condition. Moreover, this
convergence time is constant independent on the number of nodes when the
network is a b-regular simple graph with $b\geq 3$. Finally, to show the
effectiveness of our method we also simulate a number of studies on economic
dispatch problems and demand response problems in power systems.
