Lossy DC Power Flow
  The DC Power Flow approximation has been widely used for decades in both industry and academia due to its computational speed and simplicity, but suffers from inaccuracy, in part due to the assumption of a lossless network. Here we present a natural extension of the DC Power Flow to lossy networks. Our approach is based on reformulating the lossy active power flow equations into a novel fixed-point equation, and iterating this fixed-point mapping to generate a sequence of improving estimates for the active power flow solution. Each iteration requires the solution of a standard DC Power Flow problem with a modified vector of power injections. The first iteration returns the standard DC Power Flow, and one or two additional iterations yields a one or two order-of-magnitude improvement in accuracy. For radial networks, we give explicit conditions on the power flow data which guarantee (i) that the active power flow equations possess a unique solution, and (ii) that our iteration converges exponentially and monotonically to this solution. For meshed networks, we extensively test our results via standard power flow cases. 
John W. Simpson-Porco
math.OC