The limitations of centralized optimization methods in managing power distribution systems operations motivate distributed control and optimization algorithms. However, the existing distributed optimization algorithms are inefficient in managing fast varying phenomena, resulting from highly variable distributed energy resources (DERs). Related online distributed control methods are equally limited in their applications. They require thousands of time-steps to track the network-level optimal solutions, resulting in slow performance. We have previously developed an online distributed controller that leverages the system's radial topology to achieve network-level optimal solutions within a few time steps. However, it requires solving a nodelevel nonlinear programming problem at each time step. This paper analyzes the solution space for the node-level optimization problem and derives the analytical closed-form solutions for the decision variables. The theoretical analysis of the node-level optimization problem and obtained closed-form optimal solutions eliminate the need for embedded optimization solvers at each distributed agent and significantly reduce the computational time and optimization costs.

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