*5.2. Numerical Validation*

The proposed SOCP formulation was validated by being compared with approaches reported in the specialized literature. The interior point was used for solving the exact non-linear programming formulation (IP-NLP), the Newton–Raphson formulation based on determinants of Jacobian matrices (NR-DJM), and SDP formulations [9,11]. These methodologies were implemented in MATLAB and GAMS.

Table 2 presents the maximum loadability factor, i.e., *λ*, for the HVDC and MVDC systems. Please note that the proposed SOCP model allows for reaching the optimal global solution of this problem for both test systems since it is a convex transformation of the exact non-linear non-convex problem.

**Table 2.** Voltage stability index for the HVDC and MVDC test feeders.


In the case of the HVDC system, the proposed solution technique, i.e., the SOCP model, as well as the comparative methods, reach the same loadability factor (*λ* = 5.6588) for all the nodes. In contrast, for the MVDC system, the SDP model presents an underestimation of the loadability factor when compared with the global optimum, i.e., 3.0200. This error is approximately 0.44% when the SDP is compared with NR-DJM, IP-NLP, and the proposed approach.

In terms of the computational performance with respect to the processing times required to solve the voltage stability margin problem, all the methodologies listed in Table 2 require between 0.25 s and 30 s in the case of SOCP. In the case of the proposed approach, for the HVDC system, the processing time is 0.28 s, while for the MVDC, the time is 4.611 s. These results confirm that the proposed SOCP model is more efficient when compared with the NR-DJM (4.19 s and 21.11 s), SDP (0.30 s and 12.44 s), and the IP-NLP (0.28 s and 3.57 s) models reported in [11].

In the case of the NR-DJM, it is difficult to select the heuristic parameter *α* reported in [11] to determine the convergence of the algorithm. In case of the 69-node test feeder, before the voltage collapse ( *λ* = 3.01), the DJM is approximately 3.1856 × 10268, and at the point of the voltage-collapse, it is approximately −7.9939 × 10265. This implies that the tuning of this heuristic search requires multiple power flow evaluations. An additional complication of the NR-DJM approach is the selection of the step *δ*, because large values of this parameter make the algorithm faster but sacrifice precision, while small values increase the precision of the method. Large values also increase the computational time required for the solution of the problem. In other words, even if the NR-DJM method is intuitive and easy to implement, it requires adequate parametrization of the algorithm, which makes it highly dependent on the programmer. However, this is not the case with the proposed SOCP approach; this approach does not require any adjustment parameter.

Figure 5 depicts the voltage profile for the power flow problem considering all the chargeability factors as zero and the voltage collapse point when all the constant-power loads increase in the same magnitude.

**Figure 5.** Voltage behavior in the test system for the initial state of load and voltage collapse: (**a**) HVDC test system, and (**b**) MVDC test feeder.

From Figure 5, it is possible to extract the following behaviors:

