**2. Non-Linear Programming Formulation**

The determination of the point of the voltage collapse in electrical DC networks with constant-power loads is formulated as a non-linear non-convex optimization problem [9]. The non-convexities of this problem are related to the power balance equations in the presence of constant-power loads, as these expressions emerge as a hyperbolic relation between voltage and power that generates non-affine equality constraints [8]. The complete optimization model for analyzing the point of voltage collapse in DC grids is formulated as follows:

*Objective function:*

$$\max z = \sum\_{i=1}^{n} \left( 1 + \lambda\_i \right) p\_i^d \tag{1}$$

where *z* is the value of the objective function related to the maximum power consumption possible in the DC grid; *λi* represents the decision variable associated with the increment in the constant-power load at each node; and *pdi* is the constant power consumption connected to node *i*. Please note that *n* is the total number of nodes in the DC grid. In addition, the objective function (1) is linear, which makes it convex in the solution space.

*Set of constraints:*

$$\left(p\_i^q - (1 + \lambda\_i)\right) p\_i^d = v\_i \sum\_{j=1}^n g\_{ij} v\_{j\prime} \; i \in \mathcal{N}\_\prime \tag{2}$$

$$
\upsilon\_i^{\min} \le \upsilon\_i \le \upsilon\_i^{\max}, \ i \in \mathcal{N}, \tag{3}
$$

$$p\_i^{\text{g,min}} \le p\_i^{\text{g}} \le p\_i^{\text{g,max}}, \; i \in \mathcal{N}, \tag{4}$$

$$
\lambda\_i \succeq 0, \ i \in \mathcal{N} \tag{5}
$$

where *pgi* represents the power generation by the voltage-controlled nodes (i.e., slack nodes) connected to node *i*; *gij* corresponds to the conductance effect that relates nodes *i* and *j* and is considered to be a constant parameter that depends on the node interconnections. *vi* and *vj* are the voltage variables associated with nodes *i* and *j*, respectively, and are lower- and upper-bounded by *v*min *i* and *v*max *i* , respectively. Finally, *pg*,min *i* and *pg*,max *i* correspond to the minimum and maximum power generation bounds in the slack nodes, respectively.

**Remark 1.** *Please note that the decision variables in the problem of determining the stability margin in DC networks shown in (1)–(5) correspond to the loadability parameter λi as well as the voltage profiles in all the nodes, i.e., vi and the power generation in the constant-voltage sources. This suggests that the solution of this problem involves the simultaneous determination of all these variables, which maximizes the chargeability of the grid [12].*

The optimization model defined in (1)–(5) receives the following interpretation: Equation (1) is the objective function that corresponds to the maximization of the total power consumption admissible in all the nodes immediately before reaching the point of the voltage collapse. Equation (2) is the power balance constraint. Equation (3) defines the power capabilities in all generation nodes. Equation (4) determines the voltage bounds admissible for secure operation of the DC grid under normal operative scenarios, and Equation (5) determines the positiveness nature of the loadability variable.

**Remark 2.** *If the variable λi is zero for all the nodes, then the mathematical model (1)–(5) corresponds to the classical power flow problem for DC grids, which can be solved using classical methods, such as the Gauss–Seidel [2], Newton–Raphson [3] or successive approximation methods [15], among others. All these approaches can guarantee the existence and uniqueness of the solution under well-defined voltage conditions through fixed-point theorems.*

Please note that the objective of voltage stability analysis is to determine the maximum power increments in all the constant power loads that carry the DC system to the voltage collapse. Hence, the constraints related to generation capabilities in slack nodes, and voltage bounds in all the nodes are relaxed [12]. Therefore, these constraints are neglected when the objective of the problem is to determine the point at which all the nodes have a voltage collapse [11].

**Remark 3.** *Even though non-linear optimization methods such as the interior-point or gradient-descent methods can solve model (1)–(5), there is no guarantee of reaching the global optimum because of the non-convexity of the power balance constraints.*

The mathematical problem (1)–(5) is transformed into a convex one using semidefinite programming or SOCP to guarantee the uniqueness of the mathematical solution of the voltage stability margin determination in DC grids [9,16]. We used the SOCP to solve the voltage stability determination problem, which represents the main contribution of this study. The SOCP model is described in the next section.
