*Convexity Test*

To demonstrate that the power balance equations in DC networks represent a set of non-linear non-convex constraints, we present a small numerical example as follows: consider a DC power system composed of 3 nodes (see the test feeder presented in Section 4), i.e., one voltage-controlled node and two constant-power loads. For this example, let us rewrite the power balance equation at node 2 using the per-unit (p.u.) representation.

$$-p\_{d\_2} = G\_{20}v\_2v\_0 + G\_{21}v\_1v\_2 + G\_{22}v\_{2'}^2 \tag{6}$$

for simplicity, let us consider that *pd*2 = 1/4 p.u, *v*0 = 1 p.u, *G*20 = −1/2 p.u, *G*21 = −1/2 p.u, *G*22 = 1 p.u, *v*1 = *x*, and *v*2 = *y*, which produces:

$$2y^2 - xy - y + \frac{1}{4} = 0.\tag{7}$$

If we plot the non-linear function (7), the curve illustrated in Figure 1 is obtained. Please note that the solution space is given by the red curve, which implies that it is convex only if a linear combination *tx*1 + (1 − *<sup>t</sup>*)*<sup>x</sup>*2 = *x* is contained in the curve, where *t* is a real number between 0 and 1. In Figure 1, it can be observed that the line points generated (blue line) by the linear combination are outside the red curve (except the extreme points). This means that the constraint (7) is non-convex. Hence, the power balance constraints in power flow analysis generate a non-convex solution space. This implies that it is impossible to ensure a global optimum in power flow analysis.

**Figure 1.** Numerical test to show the non-convexity of the power balance equations.

### **3. Second-Order Cone Programming Formulation**

The SOCP formulation is a component of optimization convex models. This is an approach that has gained considerable importance in engineering because it can solve a family of convex problems reliably and efficiently by guaranteeing a unique solution (global optimum) [17]. The SOCP formulation minimizes a linear function over a convex region, which consists of the intersection of second-order cones with an affine linear space [18].

To transform the problem of the voltage stability margin (formulated from (1)–(5)) into an SOCP model, it is important to mention that the only one non-convex constraint represents the power balance Equation (2). To perform this transformation, we focus on the product between voltage variables, i.e., *vivj*, by redefining a new variable *y* as follows:

$$y\_{ij} = v\_i v\_{j"\prime} \tag{8}$$

Now, if we multiply (8) by *yij*, then, the following result yields

$$y\_{ij}^2 = v\_i^2 v\_j^2 \leftrightarrow \left| \left| y\_{ij} \right| \right|^2 = ||v\_i||^2 \left| \left| v\_j \right| \right|^2,\tag{9}$$

where the pre-multiplication by *vivj* is required to transform the hyperbolic relation between voltages into a conic constraint [19].

Please note that this relaxation is possible because all the voltage variables must be positive for satisfactory operation of DC grid, including in extreme cases, such as the voltage stability margin analyzed in this study.

In (9), it is possible to substitute expression (8), which yields the following result

$$\left|y\_{ij}\right|\Big|^{2} = y\_{ii}y\_{jj}.\tag{10}$$

Please note that (10) is still non-linear non-convex, which implies the need for a relaxation. The first step is to relax the equality constraint using an inequity as follows:

$$\left| \left| \left| y\_{ij} \right| \right| \right|^2 \stackrel{\leftarrow}{=} \mathcal{Y}\_{ii} \mathcal{Y}\_{jj}. \tag{11}$$

**Remark 4.** *Please note that the relaxation of the equality imposition by a lower-equality imposition is required at any conic approximation because equality implies that the solution is only in the contour of the cone. The lower-equal symbol implies that all the points inside the cone are possible solutions, including the contour of the cone, which implies that this relaxation passes the convexity test presented in Section 2 [20].*

**Theorem 1.** *Hyperbolic constraint (11) can be transformed into a conic constraint as follows:*

$$\left| \begin{array}{c} 2y\_{ij} \\ y\_{ii} - y\_{jj'} \end{array} \right| \le y\_{ii} + y\_{jj}. \tag{12}$$

**Proof.** Let us elevate to square both sides of the expression (12), which yields

$$\left| \begin{array}{c} 2y\_{ij} \\ y\_{ii} - y\_{jj'} \end{array} \right|^2 \le \left( y\_{ii} + y\_{jj} \right)^2,\tag{13}$$

This expression can be rewritten as follows:

$$\left| \begin{array}{c} 2y\_{ij} \\ y\_{ii} - y\_{jj'} \end{array} \right|^{T} \left| \begin{array}{c} 2y\_{ij} \\ y\_{ii} - y\_{jj'} \end{array} \right| \right| \leq \left( y\_{ii} + y\_{jj} \right)^{2} \,. \tag{14}$$

Now, if we expand all the components in (14), then the following result is obtained:

$$\begin{aligned} 4y\_{ij}^2 + y\_{ii}^2 - 2y\_{ii}y\_{jj} + y\_{jj}^2 &\le y\_{ii}^2 + 2y\_{ii}y\_{jj} + y\_{jj'}^2\\ 4y\_{ij}^2 &\le 4y\_{ii}y\_{jj} \\ y\_{ij}^2 &\le y\_{ii}y\_{jj'} \end{aligned} \tag{15}$$

Please note that (15) is the same as to (11), and the proof is complete.

Because the power flow constraint can be rewritten as a set of convex restrictions, the equivalent SOCP model representing the problem of maximum loadability in DC networks with constant-power loads can be rewritten as follows:

**Objective function:**

$$\max z = \sum\_{i=1}^{n} \left( 1 + \lambda\_i \right) p\_{i\prime}^d \tag{16}$$

**Set of constraints:**

$$\left(p\_i^{\mathcal{J}} - (1 + \lambda\_i)\right) p\_i^d = \sum\_{j=1}^n g\_{i\bar{j}} y\_{i\bar{j}\prime} \; i \in \mathcal{N} \; \prime \tag{17}$$

$$\left| \begin{array}{c} 2y\_{ij} \\ y\_{ii} - y\_{jj'} \end{array} \right| \le y\_{ii} + y\_{jj'}, i, j \in \mathcal{N} \tag{18}$$

$$v\_i^{\min} v\_j^{\min} \le y\_{ij} \le v\_i^{\max} v\_j^{\max}, \ i, j \in \mathcal{N},\tag{19}$$

$$p\_i^{\text{g,min}} \le p\_i^{\text{g}} \le p\_i^{\text{g,max}}, \ i \in \mathcal{N},\tag{20}$$

$$
\lambda\_i \ge 0, \ i \in \mathcal{N} \tag{21}
$$

**Remark 5.** *Mathematical models (1)–(5) and (17)–(21) are equivalent in (18) if it is guaranteed that the quality characteristic will be maintained in (19).*

**Remark 6.** *To retrieve the original optimization variables in the SOCP model described in (17) to (21), the following expression can be used:*

$$
v\_{i} = \sqrt{y\_{ii}}, \ i \in \mathcal{N},\tag{22}$$
