**3. Results**

The proposed MOTS algorithm has been tested with a modified IEEE 300-bus power system network [81] with 69 PV nodes whose data are in Table 2. Voltage limits between 0.9 and 1.1 of the nominal voltage have been added in the nodes, and the thermal limit has been placed at 1.2 of the nominal current in the lines and transformers. In addition, DC lines and associated converters have been removed.

For the initial solution, *TIC* is zero, *TGC* is 4,541,173.09 USD/year and there are 45 violations of the proposed constraints (18 bus minimum voltage, 26 maximum and one minimum reactive power). Results were obtained using an AMD AM3+ FX 6300 CPU with 24 GB RAM and MATLAB 2019a.


**Table 2.** PV nodes data from the initial solution.

N: bus location; Cap: capacity of generator in p.u.

Three strategies were tested with the MOTS algorithm to improve the original case: the first strategy consists of adding only DG units, the second is to add only FACTS, and the third is a combined strategy that allows adding both DG units and FACTS. The comparison between these three strategies is analysed in this study by means of the Pareto optimal frontier, as shown in Figures 1–3. The algorithm minimises two objectives: *TIC* and *TGC*. "Black dots" are the solutions that are part of the Pareto front and the "star dot" is the initial solution. The range of variation of the *TGC* is similar, but not equal, for the three strategies, however there are significant differences when the *TIC* is compared. The DG installation is the cheapest strategy and the FACTS installation is the most expensive, while the combined strategy has intermediate costs.

**Figure 2.** Pareto optimal frontier for the FACTS devices case.

First, considering the Pareto front for the DG units case in Figure 1, it can be seen that the generation costs are low with relatively small investments, in the range of 0 to 2 × 10<sup>4</sup> USD/year. However, from that point on, strong investments, about 14 × 10<sup>4</sup> USD/year, are needed to only slightly reduce the cost of generation. In other words, the installation of DG is cheap, and reduces losses in the network, but eventually the network is saturated. Furthermore, in the DG units case, it is possible to keep the voltages within the allowed limits, as can be seen in Figure 4b. In the FACTS devices case, as shown in Figure 4c, it is possible to obtain generation costs similar to those obtained in the DG units case, although higher investments are necessary. In this case, there is also the disadvantage that it is not possible to maintain the voltages at all buses within their limits, as shown in Figure 4c; i.e., the voltage profile is improved, but not enough at all buses. Finally, in the Pareto front for the Combined devices case in Figure 3, it can be observed that the cost of generation may be reduced to a point below the minimum obtained in the previously mentioned cases. The solution for this case requires greater investments than in the DG units case, but much smaller than in the FACTS devices case, in the range of 2 × 10<sup>4</sup> to 8 × 10<sup>4</sup> USD/year. That is, the joint installation of DG and FACTS allows

to solve the problems that a high penetration of DG in the network produces, decreasing the losses in the network and keeping the voltage profile within the permissible limits, as presented in Figure 4d.

The algorithm monitors two technical indices during the search, shown in Figure 5, to determine if the technical characteristics of the solutions found with the different strategies are similar. These indices are: the L-index [82], which is a parameter that indicates the proximity of the node to voltage collapse, and the other is the difference between the reference voltage and the real voltage.

The L-index for add-only DG or add-only FACTS strategies approaches one, meaning that such solutions are near to the point of collapse. In contrast, for the combined strategy, a value further from the point of collapse is obtained. Thus, the inclusion of both DG and FACTS allows for the system to approach the limit operation point without collapse problems.

**Figure 3.** Pareto optimal frontier for the combined units case.

All the solutions of the Pareto curve are non-dominated, so to choose one, one must use an external criteria for the optimization to be carried out. The solution for each curve is the one with the best L-index and smallest voltage variation. These solutions are marked with a blue square in the Pareto curves and the description of location and characteristics of the installed devices are detailed in Table 3.

The *TIC* plus the *TGC* for the three strategies (DG, FACTS and combination) are presented in Table 4. The solution for only DG units has zero bus voltage violations and 38 lines are less saturated, obtaining a reduction of 437,064.48 USD/year compared to the initial solution. The solution with only FACTS devices obtains a 221,462.40 USD/year reduction in comparison with the initial solution, 47 lines are less saturated and the number of bus voltage violations is reduced to nine. The combination of DG and FACTS devices achieves a reduction of 497,787.92 USD/year in comparison to the initial solution, with zero bus voltage violations, and 53 lines less saturated.

Figure 4 depicts the bus voltage profiles comparison between the selected solutions for each strategy analysed in this work, where it can be seen that the first and third strategies provide better results. The second strategy (FACTS addition) is not able to reach an acceptable bus voltage profile improvement. The combination of DG and FACTS results in the best solution, finishing with more non-expenses and better bus voltage profile for all buses.

**Figure 4.** Bus voltage profile for the selected solutions on each strategy.


**Table 3.** Location and characteristics of installed devices.

Dev.: device type; N: bus or line location; Observ.: characteristics of device; *α*: SVC firing angle; *vm*: SVC measured voltage; *b*: SVC susceptance; *x*: TCSC series reactance; *vcs*: SSSC voltage in quadrature with the line current; *vp*: UPFC series voltage in phase with the line current; *vq*: UPFC series voltage in quadrature with the line current; *iq*: UPFC shunt current wich is in quadrature with the bus voltage in the line.

**Table 4.** Costs results for selected solutions in [USD/year].


(**a**) L-index

**Figure 5.** Evolution of technical parameters in the MOTS search.
