*Article* **Second-Order Cone Approximation for Voltage Stability Analysis in Direct-Current Networks**

### **Oscar Danilo Montoya 1,2 and Walter Gil-González 3 and Alexander Molina-Cabrera 4,\***


Received: 16 August 2020; Accepted: 22 September 2020; Published: 24 September 2020

**Abstract:** In this study, the voltage stability margin for direct current (DC) networks in the presence of constant power loads is analyzed using a proposed convex mathematical reformulation. This convex model is developed by employing a second-order cone programming (SOCP) optimization that transforms the non-linear non-convex original formulation by reformulating the power balance constraint. The main advantage of the SOCP model is that the optimal global solution of a problem can be obtained by transforming hyperbolic constraints into norm constraints. Two test systems are considered to validate the proposed SOCP model. Both systems have been reported in specialized literature with 6 and 69 nodes. Three comparative methods are considered: (a) the Newton-Raphson approximation based on the determinants of the Jacobian matrices, (b) semidefinite programming models, and (c) the exact non-linear formulation. All the numerical simulations are conducted using the MATLAB and GAMS software. The effectiveness of the proposed SOCP model in addressing the voltage stability problem in DC grids is verified by comparing the objective function values and processing time.

**Keywords:** convex reformulation; direct current networks; non-linear optimization; numerical example; second-order cone programming; voltage stability margin
