## 1 Introduction  
Over the last decades, technological development and research have aimed for the enhancement of the performance, robustness, and security of power grids, which have grown in size and complexity due to the inclusion of new sources of power generation [1] and recent trends in electricity markets [2]. This growth makes power grids more difficult to control [3], and makes them prone to succumb to power outages due to operational failures. Currently, the interest on the reduction of failures has been a priority of the utmost importance, as they cause interruptions of service to clients of the grid. Operational failures can appear at any stage of power supply, from generation, passing through transmission, and up to distribution to consumers. Therefore, every element of the power grid is vulnerable to faults. When a portion of an electric network goes out of service because of a power outage, this portion can be brought back online using top-down restorations with the aid of the surrounding elements of the grid [4], however, when the outage encompasses great portions of the network, power has to be brought back bottom-up [5].This kind of restoration is called black start and, as it is notably more complex and difficult than the other kind, efforts are constantly undertaken in order to develop new optimization schemes [6], performance evaluation methods [7], and alternative and innovating solutions that can provide modern power grids with increased robustness and security in this aspect. Massive blackouts can have devastating consequences if they are not mitigated in a fast, efficient way, and, as we learned from the 2003 blackout in the USA and others [8], every possible action should be undertaken in order to mitigate their duration and effects. Conventional black start schemes use generating units that can start by themselves (without the need of auxiliary power supply), some of which can be hydro, diesel, and gas units [9; 10]. In microgrids, the black start schemes include energy storage systems (ESS) which lead with the issues generated by overcharge and over-discharge from the penetration of renewable energy sources or cyber-attacks [11; 12]. In order to reduce the variability generated by the renewable energy and achieve a faster restoration of the black-start system, the ESS have been widely used [11; 12; 13; 14; 15]. Additionally, different strategies and schemes of control are used to improve the black-start system at smart grids for medium voltage levels. The strategies of control used during the black start requires maintain system frequency stability, system balance, and even facing the unavailability of communication networks [14; 15]. In both schemes, the restoration of power back to the system is done in an orderly manner, supplying power from the black start unit to the rest of the grid, and can happen simultaneously in various parts [16] by separating the affected zones into islands that will later be synchronized, forming bigger and bigger portions of the grid until total restoration is reached. Nevertheless, as any proposal that can enhance and speed up this process should be explored [17]. Due to the importance of achieving an appropriate response in this kind of event, the use of other elements and equipment within the grid has been explored as a possible way to develop better black start schemes. A great number of these proposals are based on the use of High Voltage Direct Current (HVDC) links, especially those working through Voltage-Source Converters (VSC) [18; 19; 20; 21; 22]. VSC-HVDC links, besides transmitting power over long distances, have a big potential for mitigating the adverse effects that faults have on grid performance, increasing a grid's flexibility, resilience, and robustness when they are present [23], as they provide reactive power support, AC voltage control [24], and frequency control [25] while \"firewalling\" one AC system so that disturbances do not spread to adjacent zones [26].