## 1 Introduction  
High-voltage direct current (HVDC) systems have played an important role in sub-marine power transmission, due to economic advantages over their alternating current (AC) counterparts [1]. Nowadays, due to the complexity of modern power systems, HVDC systems are adopted in practice, not only for constant power delivery between massive networks, but also to provide grid services for island grids [2; 3; 4; 5; 6; 7; 8]. For example, in Korea, two line-commutated converter (LCC) HVDCs, the 180-kV/300-MW Haenam-Jeju HVDC [9] and \(\pm\)250 kV/400 MW Jindo-Jeju HVDC [10], were constructed by the Korea Electric Power Corporation (KEPCO) for supporting Jeju Island and integrating massive wind power plants. Furthermore, KEPCO is planning to construct one more \(\pm\)150 kV/200 MW voltage-sourced converter (VSC) HVDC between the main grid and Jeju Island [11]. With the technical development of both the LCC and VSC HVDC systems, there are numerous island networks with multi-infeed HVDC (MIDC) systems, encompassing both types of HVDCs, similar to the Jeju Island case. According to trends, many studies have concentrated on the MIDC systems [12; 13; 14; 15]. Major research effort has been focused on the stable operation of the MIDC systems by introducing novel indexes, such as the multi-infeed interaction factor (MIIF), multi-infeed effective short-circuit ratio (MIESCR) [12], apparent increase in short-circuit ratio (AISCR) [13], and improved effective short-circuit ratio (IESCR) [14]. In parallel with such research, some studies have concentrated on the economic operation of MIDC systems, which is represented by optimal power flow (OPF) [17; 18; 19; 20]. In two such studies [17; 18], steady-state VSC-HVDC modeling methods and OPF formulations based on the Newton-Raphson method are proposed. Although only a single HVDC system is considered in these studies [17; 18], the principle can be easily expanded to MIDC systems. In one study [19], a loss minimization method with an AC/DC hybrid grid incorporating only VSC HVDCs is proposed, based on an interior-point method. In another study [20], a similar method adopting both LCC and VSC HVDC is proposed. However, the authors of the above studies only consider the steady-state values of the HVDC systems. Thus, if the load profiles were different from the forecasted value, the previous methods were unable to find the optimal set-point, because the forecasting error was not considered. A common method used to satisfy the power balance between generation and load caused by the forecasting error is frequency-power droop control [21]. Similar to conventional generators, HVDC systems also adopt frequency-power droop controllers [22; 23; 24]. For economic operation of power systems during frequency control, the calculation of droop coefficients is a significant issue, and the coefficients are used to exploit multiple HVDC systems efficiently. It is practically important in the case of an islanded network, because the islanded system has a small load level and high renewable energy penetration, which cause high uncertainties in the power system. Due to those characteristics, off-nominal frequency situations occur more frequently, and the frequency deviation is more severe than the conventional large power network. In one study [25], an optimum calculation method of voltage-power droop coefficients for multi-terminal HVDC (MTDC) systems is proposed. However, the method could only be applied to the specific topology of the DC grid investigated in the paper. Another study proposes an optimization-based droop coefficient calculation method to maximize converter efficiencies for a low-voltage system [26]. A method to minimize DC transmission line loss using voltage-current droop coefficients has also been proposed [27]. The above research suggests that the droop coefficients of converters can be calculated by optimization of the problem. However, to our knowledge, optimization problems have rarely been suggested to calculate frequency-power droop coefficients of MIDC systems.