**4. Graphical Example**

Here, we considered a small DC test feeder composed of three nodes (one of them is the slack node) and two constant-power loads connected to nodes 1 and 2, respectively. This system is used to show the voltage stability margin, i.e., the region of secure operation, graphically. The topology of this test feeder is presented in Figure 2. For this test system, 1000 W and 24 V are considered the power and voltage bases, respectively.

**Figure 2.** Small test feeder with two constant power loads.

Figure 3 shows the numerical behavior of the voltage collapse point when different increments have been used at the point of connection of the constant-power loads. Please note that point **O** is the solution of the classical power flow problem when all *λi* are fixed as zero; this point is (*<sup>v</sup>*1, *<sup>v</sup>*2)=(0.9312, 0.8783). From this initial point, we evaluate the evolution of the voltage collapse in the DC grid when its constant-power loads increase. Trajectory **O**–**A** shows the evolution of the voltage at load nodes when it is increased only at the load connected at node 1, with the load at node 2 being fixed as 0.15 p.u. Please note that point **A** presents the maximum reduction in both load voltages at the same time, i.e., (*<sup>v</sup>*1, *<sup>v</sup>*2)=(0.4802, 0.4265); both voltages are observed to be lower than 0.5000 p.u. In addition, these points represent the maximum objective function possible in this numerical example, which is *z* = 3.4304 p.u, when *λ*1 = 15.4021. Trajectory **O**–**B** shows the evolution of the voltage in the DC system when both loads are increased by the same magnitude, i.e., *λ*1 = *λ*2 = 5.5319. These increments in the loads produce a maximum objective function of 2.2862 p.u, where one voltage is higher than 0.6500 p.u (see node 2), and the other node is lower than 0.4500 p.u (see node 3). This behavior implies that node 3 conditioned the stability margin behavior of this test system since it is more sensitive to load changes than node 2 is. On the other hand, trajectory **O**–**C** presents the voltage evolution of the numerical example when the load connected to node 2 is increased, with the load at node 1 fixed as 0.20 p.u. This trajectory shows that the voltage at node 2 decreases until 0.4534, while the voltage at node 1 remains upper that 0.7600 p.u. This behavior confirms that node 2 has a lower possibility of increasing its load consumption since the voltage collapse point is reached when the total load of the DC system is 1.4611 p.u, which is the minimum objective functions across the three cases analyzed. Please note that in Table 1 the numerical behavior of the voltage stability problem in DC networks resumes when constant-power loads start to increase.

**Figure 3.** Voltage collapse trajectories followed by different increments in the constant power consumptions per node.


**Table 1.** CPL increments for the different simulation cases.

It is important to mention that the results presented in this numerical example have differences lower than 1 % when compared with that in the heuristic approach based on Newton–Raphson sensitivities [11].

**Remark 7.** *The point of voltage collapse in DC radial networks depends on the location of the load node. Hence, it is possible to conclude from the numerical example that loads connected to the final nodes have lower possibilities of incrementing their consumption when compared with loads near the slack source.*

### **5. Test Systems and Simulation Results**

In this section, we present the test system configuration and the numerical results obtained by solving the stability margin calculation problem with different methodologies reported in the specialized literature.
