A solution to the single-snapshot non-linear L1 estimation of the power transmission network is presented. The non-linear L1 estimation problem is formulated as a non-linear program and solved using a primal-dual interior-point approach. The efficiency of this approach is dependent on the numerical procedure used to solve the reduced Karush-Kuhn-Tucker (KKT) system of equations. It is shown that two mathematically equivalent formulations of the non-linear programming problem can be obtained. These formulations lend themselves to fundamentally different numerical procedures to solve the reduced KKT system. Numerical testing on IEEE systems is used to quantify the performance of the interior-point approach on both formulations. Comparisons are also carried out with a recent implementation of an iteratively reweighted least-squares method for non-linear L1 regression.