## 2 Materials and Methodology  
The protection of the distribution networks studied in this work was guaranteed by main protection relays located at the beginning of the line (that is, at the electric substation (S)), and cut-out fuses (F1, F2) at the laterals. The basic topology of this kind of network is shown in Figure 1, where a distributed generation (DG) source has been added. The main feeder is protected by a relay (R), whereas the laterals supplying the loads are protected by means of fuses. Relays are normally equipped with inverse time overcurrent devices, whose performance is defined by the following equation [67]:  
\[t(I)=\frac{a}{\left(\frac{I}{I_{pick-up}}\right)^{b}-1}\text{TMS} \tag{1}\]  
Figure 1: Basic topology of the simple power distribution network studied. A distributed generation (DG) source has been added. where \(t\) is the time of operation of the overcurrent device, \(I\) is the fault current detected by the device, \(I_{pick\_up}\) is the adjustment current (relay pick-up curve) of the device, \(a\) and \(b\) are the control parameters of the actuation curve, and TMS is the Time Multiplier Setting (expressed in seconds) [69]. The characteristic values of the relays used in the present work are included in Table 1.  
Each relay in a protection system has a set of relay settings that determine the primary and backup protection that the relay will provide. The relay settings for each relay are calculated so that the relay fulfills the primary and backup protection requirements of the network it is protecting. Calculations are based on the maximum load current, the maximum and minimum fault currents, and/or the impedance of feeders that the relay is protecting. On the other hand, fuses have an inverse current-time characteristic that is usually plotted as a log-log curve, which is better approximated by a second-order polynomial function. However, it should be underlined that the most interesting part of this curve, in terms of its practical applications, can be approximated by a linear expression [70]. Therefore, it can be assumed that the general equation for the characteristic curve of an expulsion fuse can be expressed as [68; 71; 72; 73; 74]:  
\[\log(t)=m\log(I)+n \tag{2}\]  
where \(t\) is the actuation time of the ejection fuse, \(I\) is the current through it, and \(m\) and \(n\) are fuse constants to be determined as in [75]. The present work was carried out by using computer simulation of a rural/small grid with 3 possible photovoltaic DG sources, selected as a case study test network. To analyze the effect of the photovoltaic DG on the protection devices of such a distribution network, the IEEE-13 bus test feeder system was used (see Figure 2). The simulation was performed with DIgSILENT Power Factory software, already used to study the protection of distribution systems with DG [72; 76; 77; 78; 79]. This distribution network design has been successfully used to study static short-circuit currents [80], the maximum possible photovoltaic power penetration into a network in relation to the demand response [81], voltage regulation strategies in DG [78; 82; 83], or fault ride-through in power networks related to renewable energy (wind and photovoltaic) [79]. Besides, it should be also said that DIgSILENT Power Factory software is a powerful simulation tool that integrates the following standard methodologies for short-circuit calculations: IEC 60909, IEEE 141/ANSI C37, VDE 0102/0103, G74, and IEC 61363. The analyzed distribution network was divided into three Zones of Protection (Zones 1, 2, and 3), assigning a specific area for each DG (starting each one of these zones from the beginning of the output feeder from the electrical substation, see Figure 2). The total load of the distribution network was 4 MW, which was considered constant in all simulations. The connection points of the DG were chosen based on their distance from the main substation.