  The optimal power flow (OPF) problem is an important mathematical program
that aims at obtaining the best operating point of an electric power grid. The
optimization problem typically minimizes the total generation cost subject to
certain physical constraints of the system. The so-called linearized
distribution flow (LinDistFlow) model leverages a set of linear equations to
approximate the nonlinear AC power flows. In this paper, we consider an OPF
problem based on the LinDistFlow model for a single-phase radial power network.
We derive closed-form solutions to the marginal values of both real and
reactive power demands. We also derive upper bounds on the congestion price
(a.k.a. `shadow price'), which denotes the change in marginal demand prices
when the apparent power flow limits of certain lines are binding at optimum.
Various cases of our result are discussed while simulations are carried out on
a $141$-bus radial power network.
