**1. Introduction**

Direct current (DC) electrical networks are promising grids capable of supplying multiple users at different voltage levels from high-voltage DC (HVDC) to low-voltage DC (LVDC) in monopole or bipole configurations [1,2]. The implementation of DC technologies avoids the need for managing reactive power or frequency, in contrast to their alternating current (AC) counterparts. This is an important advantage that makes DC grids easily controllable and operable. Additionally, power losses are lower, and voltage profiles are better in DC grids than in AC systems. Hence, DC networks are more efficient than AC networks [3,4].

Two types of strategies are used to analyze DC electrical networks: dynamical and static approaches. The first strategy is executed in the time domain and is used for developing primary and secondary controllers in power electronic DC-DC converters [5,6]. The second type of analysis, i.e., static studies, is used to determine all the state variables under stationary conditions. The most typical types of analysis are power flow analysis [3], optimal power flow studies [2], economic dispatch approaches [7], and voltage stability analysis [8,9]. In addition, these approaches are combined with the optimal sizing and location of distributed generators for DC grids [10].

In this study, we focus on the voltage stability calculation for DC grids. This is a non-linear non-convex optimization problem recently analyzed in specialized literature, and few approaches for this task have been reported. In [9], a semidefinite programming (SDP) model was proposed by guaranteeing a global and unique solution. Nevertheless, the complexity of this model is mainly due to the quadratic increase in the number of variables and the semidefinite requirements of the matrix that contains all the voltage variables; this causes longer computational times when the number of nodes in the DC system increases. The authors of [8] presented a linear matrix inequalities formulation to determine the maximum load increments in a small DC grid composed of two constant-power loads. However, this approach cannot be extended for multiple loads, because the resulting equations are unsolvable using analytical methods. The authors of [11] employed the classical Newton–Raphson method in conjunction with a linear search to determine the maximum load increment. This is performed by observing the sign variation in the Jacobian matrix in the power flow equations. This method is easily implementable; however, the selection of the step size has an undesirable influence on the final solution. In [12], the voltage stability margin problem in DC grids was solved by incorporating the non-linear formulation into an optimization package known as the general algebraic modeling system (GAMS). Even though this software is efficient for solving non-linear problems, it is not possible to guarantee a global solution to the problem, because the calculations are usually stuck in local solutions.

Unlike in previous works, in this study, a second-order cone programming (SOCP) model is proposed to address the voltage stability margin calculation in DC grids. The main advantage of this approach is that it guarantees a global optimum and unique solution by transforming the exact non-linear non-convex optimization problem into a convex problem [13,14]. In addition, this approach has not been previously proposed for the analysis of voltage stability in DC networks. Therefore, there is a gap in the literature that this study tries to fulfill. The convex approach has lower computational requirements than SDP approaches because it avoids semidefinite matrices in its formulation.

Even if recently reported approaches typically use the exact non-linear formulation (GAMS solvers) and heuristic searches (Newton-Raphson) [11,12] because of the non-convexities introduced by the power balance constraints, it is not possible to ensure a global optimum, even if for both test feeders these solutions coincide with convex approaches, i.e., the semidefinite programming model [9] and the newly proposed SOCP model.

The remainder of this paper is organized as follows: Section 2 presents the classical non-linear non-convex formulation of the voltage stability calculation in DC grids. Section 3 shows the proposed second-order cone programming reformulation and its main assumptions for developing a convex mathematical model. Section 4 presents a small numerical example with three nodes to demonstrate the effect of the load increment and the voltage collapse problem. Section 5 shows the numerical implementation of the proposed SOCP model in two test systems, namely an HVDC network and a medium-voltage DC (MVDC) grid. Section 6 presents the main conclusions drawn from this research.
