
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)11650-7
10.1016/j.heliyon.2024.e35619
e35619
Research Article
Impact of inductive and resistive fault current limiters for transient stability Improvement based on difference between accelerating and decelerating areas
Dehghani-Ashkezari Mahdi m.dehghani@iauashkezar.ac.ir
a
Modaresi Seyed Mahmoud m_modaresi@azad.ac.ir
b⁎
Saied Seyedamin saied@iauyazd.ac.ir
a
Daemi Tahere t.daemi@iauyazd.ac.ir
a
Akbari Hamidreza h.akbari@iauyazd.ac.ir
a
a Department of Electrical Engineering, Yazd Branch, Islamic Azad University, Yazd, Iran
b Department of Electrical Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran
⁎ Corresponding author. m_modaresi@azad.ac.ir
10 8 2024
30 8 2024
10 8 2024
10 16 e3561920 9 2023
23 6 2024
31 7 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The utilization of fault current limiters (FCLs) provides an effective approach to alleviate the negative consequences associated with short circuit currents. This equipment can affect other system specifications. In this paper, the effects of FCL on power system transient stability are investigated by a theoretical-comparative study. In this study, different FCL impedances (inductive and resistive) are analyzed and compared considering two locations for FCL installation and different locations of fault occurrence in a single-machine infinite bus system. To evaluate the power system's transient stability, the authors adopt an approach based on the equal area criterion, which calculates the difference between accelerating and decelerating areas. Another issue addressed in this article is the fault-clearing time effect on the power system's transient stability as FCL is installed in the power grid. The results have been confirmed through the use of MATLAB software.

Keywords

Transient stability
Fault current limiter
Power system stability
Power system dynamic
Short-circuit current
==== Body
pmc1 Introduction

Increasing electricity demand and developing power systems increase the short-circuit current level and damage equipment [1]. In certain conditions, fault currents can escalate levels exceeding 20 times the maximum rated current. This is a significant challenge because it is necessary to increase the interrupting capacity of conventional circuit breakers at a great cost or limit these currents somehow. Failure to quickly limit these fault currents can lead to instability in the power system [2]. Various methods have been proposed to address this issue. Fig. 1 illustrates the classification of these methods, categorized from two perspectives. Firstly, methods are classified as passive or active. From another perspective, the fault current limiting methods are classified into two subcategories, topology-based and equipment-based. Passive methods entail schemes that do not introduce changes to the limiting mechanism before or during a fault. Conversely, active methods are typically triggered during fault conditions, often involving the addition of high impedance to the power system to limit fault currents [3].Fig. 1 Classification of fault current limiting methods [3].

Fig. 1

Fault current limiters (FCLs) have received particular attention recently for their high speed and desirable performance. They are considered cutting-edge devices for mitigating fault currents. This equipment limits the short-circuit current by adding series impedance into the system during a fault [[4], [5], [6]]. In Fig. 1, FCLs are classified into four groups based on their performance: Superconducting Fault Current Limiters (SFCL), Solid-State Fault Current Limiters (SSFCL), Hybrid FCL, and Other Technologies. Among the different types of fault current limiting equipment, only a few are fully commercial equipment. Other technologies are still under development and have been studied by many groups worldwide over the past years or decades [7].

As the FCL resistance increases against short circuit current and it shows low resistance under normal conditions, this equipment can be used in series in network lines to not cause a voltage drop in stable conditions. In addition to reducing short-circuit current, FCLs can have other effects on the power system, such as reducing voltage sag and protecting equipment in the event of a fault. However, this equipment can affect the reliability and the transient stability of the power system by changing the inertia and Thevenin impedance of the system [8,9].

In [10], the effect of using FCL on the transient stability of power generators has been analyzed. In Ref. [11], a specific FCL unit is proposed as an inductive. it consists of a superconducting fault current limiter with a resistor and a Zno device in parallel. The results of this paper show that the proposed FCL unit, in addition to reducing the fault current in the power system, also improves the transient stability of the power system. Reference [12] provides a comparative survey of research activities and emerging technologies concerning FCLs, offering detailed insights into the characteristics of inductive FCLs. In Refs. [13,14], the use of superconductors in FCLs to improve transient stability is explored. Moreover [15], investigates system security and stability enhancement through the application of the Particle Swarm Optimization (PSO) algorithm. Additionally [16], analyzes the stability of a power system with High-Temperature SFCLs installed as part of the technology development process. The effect of the presence of FCL in the distribution network with wind turbines has been evaluated in Ref. [17].

In [18], the utilization of a genetic algorithm (GA) is explored to determine the optimal number and placement of FCLs. Reference [19] presents an iteration-based linear optimization technique aimed at determining the optimal sizing of FCLs within a power system. Additionally [20], employs the capacity constraint method to identify the optimal placement and magnitude of FCLs. In Ref. [21], the investigation focuses on the impact of FCL presence on maintaining current relay coordination within the power network, despite the integration of distributed resources. Furthermore [22], introduces a novel approach for optimizing the quantity and positioning of FCLs to enhance reliability and interrupt short-circuit currents, considering various conflicting objective functions.

In [23], an analysis is performed on the stability of the power system utilizing an innovative switching technique within the bridge-type FCL. In Ref. [24], the installation conditions of the superconducting FCL were examined from a technical and economic point of view. In Ref. [25], the economic benefits of using FCL of superconducting type have been stated and presented in detail.

The published report [26] analyses various types of FCL models in a sample distribution network in Brazil. In Ref. [27], researchers have introduced a solid-state FCL that is effective for connecting upstream and downstream networks in industrial systems. Ref. [28] discussed a superconducting FCL made by Nexan and evaluated its use in a practical network.

The design and placement of the FCL in the ring distribution system have been carried out in Ref. [29]. A practical example of the FCL used in China has been reported in Ref. [30], and the transient state of breakers and power switches has been investigated when FCL is present and without them. A report published in 2014 examined the location and number of FCL in the real network in a part of Iran and optimized the reduction and limitation of the fault current through this FCL by using intelligent algorithms [31].

In [32,33], the optimal placement of the FCL has been chosen based on cost, reliability, and losses by employing the PSO algorithm. Refs. [19,34,35] have combined network protection issues with economic constraints and limited the fault current to improve the previous problems. Also, in Ref. [36], placement and optimal selection of FCL impedance were made using an improved genetic algorithm, fuzzy logic, and PSO algorithm in two stages. Refs. [13,33]show the positive impact of FCL on the power system transient stability; however, in these references, only resistive FCL is used, while the impact of different intervals of impedance values FCL on transient stability is not investigated. Ref. [37] compares the effect of both inductive and resistive FCLs on fault current limiting and power system transient stability.

Based on what has been obtained so far in past studies, both FCLs, inductive, and resistive improve the transient stability of the power system. But in this paper, with a more comprehensive investigation and choosing the range of FCL impedance changes in a larger range and studying stability for larger values of the FCL impedance range, different results have been obtained compared to other papers about resistive FCLs. In addition, the effect of fault-clearing time on the transient stability of the power system is considered. Further, Table 1 provides a detailed comparison between this study and recent papers.Table 1 Comparison between the proposed method and recent papers.

Table 1Ref.	Studying The Effect of FCL Resistance on Stability	Studying The Effect of FCL impedance on Stability	Studying The Effect of Fault Clearing Time on Stability	Studying The Stability of The System in A Wide Range of Impedance and Resistive FCL Values	Examining the simultaneous effect of fault location parameters, FCL installation location, FCL impedance value, FCL impedance type, and fault clearing time on stability	
[1]	✓	✓	✗	✗	✗	
[10]	✓	✗	✗	✗	✗	
[11]	✓	✓	✗	✗	✗	
[13]	✓	✗	✗	✗	✗	
[14]	✓	✗	✗	✗	✗	
[16]	✗	✓	✗	✗	✗	
[23]	✓	✓	✗	✗	✗	
[37]	✓	✓	✗	✗	✗	
This Paper	✓	✓	✓	✓	✓	

1.1 Research Gap and Contribution

The literature review highlights that only a few studies have explored the effects of inductive and resistive FCLs on transient stability. It can also be seen that there is a lack of considering fault clearing time on the power system stability. The present study focuses on comparing the effect of FCL types (inductive or resistive) and variation of FCL impedance on the transient stability of a single-machine infinite bus (SMIB). Furthermore, two locations have been considered to analyze the role of FCL location on transient stability. Also, the proposed method shows the impact of fault clearing time on the power system stability.

Based on what has been obtained so far in past studies, both FCLs, inductive, and resistive improve the transient stability of the power system. But in this paper, with a more comprehensive investigation and choosing the range of FCL impedance changes in a larger range and studying stability for larger values of the FCL impedance range, different results have been obtained compared to other papers about resistive FCLs. In addition, the effect of fault-clearing time on the transient stability of the power system is considered. The primary innovations of this study are delineated as follows.• Investigating the effect of fault clearing time on the transient stability of the power system by installing inductive and resistive FCLs for three different disconnection times of power switches.

• Defining the free range for the FCL impedance value to investigate the effect of FCL on the transient stability more comprehensively.

• Present a method based on the difference between accelerating and decelerating areas using area criterion method to check the stability of the power system. In this method, the index ΔA, which is derived from the equal area criterion, is presented. ΔA provides a measure to calculate and display the amount of stability margin. The use of this index accurately measures the margin of stability/instability of the power system.

• Examining the impact of different parameters on the stability of the system simultaneously, such as the location of the fault, the installation location of the FCL, the type of FCL, the impedance value of the FCL and the fixing time of the fault.

1.2 Organization

Other sections of the paper are organized as follows. In the second section, the mathematical model is presented to investigate the transient stability of the power system. The third section introduces the equivalent circuit of the studied power system along with the numerical results of the simulation. This section also considers the impact of the R/X ratio in transmission lines and fault clearing time on the transient stability. Finally, the conclusions and recommendations for upcoming research are provided in the last section.

2 The impact of FCL on power system transient stability

Generally, Three-phase faults are commonly used to simulate a large disturbance. If the angular oscillations of the rotor drop to zero over time, the system will be stable. The swing equation of the generator rotor is defined for a single machine infinite bus to Eq. (1).(1) d2δdt2=ω2H(Pm−Pe−Ddδdt)

Where δ is the angular difference between the generator bus voltage and the infinite bus voltage, ω is the angular rotor speed, H is the inertia constant, D is the damping coefficient, Pm is the mechanical power received by the generator and Pe is the electrical power transmitted between the synchronous generator and the infinite bus Which is calculated in Eq. (2) [38].(2) Pe=|E|2∙|Y11|cosφ11+|E|.|V|.|Y12|cos(δ−φ12)

Where E represents the synchronous generator voltage, and V denotes the infinite bus voltage, |Y11 | And |Y12 | entries of admittance transfer matrix between the synchronous generator and the infinite bus and φ11 and φ12 are respectively, the angle of the complex term Y11 and Y12.

Fig. 2 shows the curve of the electrical power respected to δ in a SMIB for the three states of the power system: before the fault (Pepre), during the fault (Pefault), and after the fault (Pepos). Before the fault occurs, the generator is operating at point A, in which Pm=Pepre, and the generator rotor angle is δ0. If a symmetric three-phase fault occurs in the system, the generator operating point is transferred from A to B on the Pefault. At this time, if the electrical power is less than the mechanical power, the generator will accelerate; otherwise, the speed of the generator will decrease. According to Fig. 2, after the fault occurs, the generator accelerates, and δ increases. After fault clearing time (tc), the rotor angle reaches δC (fault clearing angle). The breakers operate while the fault is removed from the system. In this way, the system's operating point is transferred from C to D on the curve Pepos. The value of δ is increased at first, then decreased, and finally damped around point G after several oscillations. According to the equal area criterion, the system will remain stable if the decelerating area (DEFD area) is greater than the accelerating area (ABCEA area). Otherwise, the system will be unstable. The critical point is that the value of δC depends on tc. Changes in the value of δC can cause a difference in the accelerating and decelerating areas and change the system's stability. The point-by-point (or step-by-step) method is used to calculate the δC. In this paper, to evaluate the transient stability of the power system, the criterion of difference between decelerating and accelerating areas is employed, which is taken from the equal area criterion. This criterion is defined as follows:ΔA=(deceleratingarea)−(acceleratingarea)

In the above formula, if ΔA is negative, the power system is unstable, and the more negative ΔA is, the more unstable the system will be. If ΔA is positive, the system is stable, and the more positive ΔA is, the greater the margin of system stability will be.Fig. 2 Pe respected to δ for different conditions.

Fig. 2

In Fig. 2 (ΔA= (Area DEFD) - (Area ABCEA)). According to this criterion, the more negative ΔA is, the more unstable the system is, and the higher the ΔA, the higher the stability margin.

2.1 Power system stability analysis with FCL

FCLs connect to the network when the fault occurs and isolate from it immediately after the fault is resolved. Therefore, the system admittance matrix values are fixed before and after the fault and are not dependent on the FCL impedance value. As a result, Pepre and Pepos are constant and not affected by the change in the location or impedance of FCL. However, the value of the system admittance matrix changes during the fault by location and impedance of FCL. In these conditions, Pefault will be a variable. In general, according to different values of FCL impedance, the Pefault curve can be divided into three different states. Then, the transient stability of the system in each state is compared to the state without FCL.

2.1.1 Case A: (PewithFCLfault<PenoFCLfault)

In this case, the value of FCL impedance ends in: Pefault with FCL less than Pefault without FCL (PewithFCLfault<PenoFCLfault). according to Fig. 3, the value of ΔA is decreased for two reasons. One is due to the lower curve PewithFCLfault than PenoFCLfault and the other due to the increase in value δC. The value of δC increased because of the greater difference between electrical and mechanical powers after the fault. As a result, the probability of system stability with FCL is reduced.Fig. 3 Variation of Pe with FCL in case A.

Fig. 3

2.1.2 Case B: PenoFCLfault≤PewithFCLfault≤Pepre

In this case, the FCL impedance value ends in: PenoFCLfault≤PewithFCLfault≤Pepre. Fig. 4 shows that due to shifting the PewithFCLfault curve upwards; the accelerating area ABCEA is lower than in the case without FCL. Also, due to the reduction of the difference between Pe and Pm at the moment of fault, the value of δC has decreased compared to the case without FCL. Therefore, the increase in the value of ΔA leads to an extension of the system's stability limit, thereby expanding the stability region within the power system.Fig. 4 Variation of Pe with FCL in case B.

Fig. 4

2.1.3 Case C: PewithFCLfault>Pepre

In this case, The FCL impedance value ends in: PewithFCLfault>Pepre. According to Fig. 5, the operating point is transferred from A to B after the fault. In this case, because the Pe is more than Pm, the generator speed and consequently δ decreases. The value of δ reduction, depending on the system parameters and the tc, may even be such that δ becomes negative. Due to the reduction of generator torque, even after clearing the fault, the value of δ decrease because of the inertia. Until point G is reached, the generator's speed returns to synchronous speed, resulting in an increase in the generator's speed and consequently in δ. Therefore, in this case, the accelerating and decelerating areas will be equal to the HGEH and EFE areas, respectively. In this case, the value of ΔA can be smaller, decreasing the stability limit and reducing the power stability region.Fig. 5 Variation of Pe with FCL in case C.

Fig. 5

3 Numerical results

Fig. 6-a shows a single-machine infinite bus system (SMIB). In this system, the Pe generated by the synchronous generator is transmitted to an infinite bus represented by its voltage V < 0 by two parallel transmission lines. Two different locations P1 and P2 are provided to install FCL in this network. Point P1 is between the generator and bus 1, and point P2 is at the beginning of line 2. The equivalent circuit of the sample system is shown in Fig. 6b. As can be seen, the location of the fault (point F) has a distance of 10 % from the beginning of bus 3.Fig. 6 single-line diagram(a), the equivalent circuit of the SMIB(b).

Fig. 6

The parameters of the Single-Machine Infinite Bus (SMIB) system are presented in Table 2, with the base power and voltage set at 100 MVA and 110 kV, respectively. For modeling the FCL, a purely inductive or resistive impedance is automatically inserted into the power grid immediately after the fault occurs and returns to a normal state (ZFCL = 0) immediately after the fault is resolved.Table 2 Parameters of SMIB.

Table 2Components	Nominal Power (MVA)	Nominal Voltage (kV)	Impedance Value (p.u)	
Generator	90	22	j0.3	
Transformer	100	22/110	j0.1	
Transmission lines	–	110	ZLine1 = ZLine2 = j0.5	

3.1 The effect of location and impedance type on the power system transient stability

Assuming that the fault occurs at F, to analyze the changes in the operating point of the system, it is necessary to calculate δC. Therefore, first using the point-to-point method, δC at tc = 0.2s is calculated for different values of FCL impedance at two locations P1 and P2. Finally, the variation of δ (delta) for resistive and inductive FCLs in terms of time, from zero to 1.5s is shown in Fig. 7. It is important to highlight that the starting time of the fault is set as zero for the simulations carried out in this study.Fig. 7 Variation of δ for resistive FCL at P1(a), resistive FCL at P2 (b), inductive FCL at P1(c), inductive FCL at P2 (d).

Fig. 7

According to the above curves, when the impedance of FCL is zero (system without FCL), as soon as a fault occurs, δ increases over time, and the system is unstable. For resistive FCL at P1 (RFCLP1) and resistive FCL at P2 (RFCLP2), according to Fig. 7a–b, it is considered that in some range of FCL impedance values, the system is stable and in other values of The FCL impedance system is unstable. However, for inductive FCL (XFCL), if the FCL is at P1 (XFCLP1), the system is unstable for all FCL impedance values, as shown in Fig. 7c. If the FCL is installed at P2 (XFCLP2), as shown in Fig. 7d, when the FCL impedance increases, the δ values decrease and increasing the stability-limit of the system. The ΔA values for inductive and resistive FCLs at P1and P2 at tc = 0.2s are shown in Fig. 8.Fig. 8 Variation of ΔA concerning FCL impedance for inductive and resistive impedances installed at P1and P2

Fig. 8

According to Fig. 8, ΔA for the case ZFCL = 0 is negative; the system is unstable. By installing FCL in the system and increasing its impedance, it is observed that for XFCLP1 case, according to case A in section 2.1, the value of ΔA will be less than the case without FCL, which means reducing the stability margin of the system. However, for XFCLP2 case, according to case B in section 2.1, if the FCL impedance value increases, the value of ΔA increases as a homographic function, and thus the stability margin of the system improves.

It is a bit complicated for resistive FCLs. As shown in Fig. 8, in some ranges of FCL impedance values in both RFCLP1 and RFCLP2 cases, the system is stable, while for other ranges of FCL impedance values, the system will be unstable. However, RFCLP2 case provides system stability over a larger range of impedance values.

Comparing the three cases RFCLP1, RFCLP2, and XFCLP2, it can be seen that each of these cases can have a larger ΔA in a specific value of ZFCL and a larger stability margin than the others. However, in this impedance range, ΔA may not be at its maximum in any of these cases. For example, according to Fig. 8, the RFCLP1 case provides better stability conditions than others in the impedance range (0.13–0.31 p.u) and (0.75–1.5 p.u). While in the impedance range (0.055–0.12 p.u) and (1.6 to infinity), the RFCLP2 case and for the impedance range (0.32–0.74 p.u), the XFCLP2 case will create better stability conditions than the others. These results are shown in Table 3.Table 3 Determining the best stability conditions according to the type, location, and impedance value of FCL.

Table 3Impedance type of FCL	FCL location	FCL impedance range (pu)	
0.055 to 0.12	0.13 to 0.31	0.32–0.74	0.75–1.5	1.6- infinity	
Resistive	FCL at P1	✗	✓	✗	✓	✗	
FCL at P2	✓	✗	✗	✗	✓	
Inductive	FCL at P1	✗	✗	✗	✗	✗	
FCL at P2	✗	✗	✓	✗	✗	

3.2 The effect of fault clearing time on power system stability

Based on Eq. (1) the value of the fault clearing angle (δC) changes according to the fault clearing time (tC). Therefore, according to the position of Pefault in relation to Pepos and Pm, three states may occur at different fault clearing times.

First state: Pefault<Pepos, second state: Pefault>Pepre, third state: PePOS<Pefault<Pm:

In the first case, according to Fig. 3 and the contents of section 2.1.1, the operating point is transferred from Pepre to Pefault after the fault. In this case, the lower tC is, the lower δC is, and the operating point of the system transfers faster to the PePOS curve, which is located at a higher position than Pefault. As a result, the acceleration area (the enclosed area between the electrical power curve and the Pm) decreases. Therefore, the ΔA increases, and the stability margin of the system is improved. The conditions in the second case are almost similar to the first case. But in the third case, according to the type and value of FCL impedance (Fig. 9), after the fault, the operating point transfers from Pepre to Pefault. Here, if tC is lower, the value of δC is also lower, and the operating point transfers to PePOS earlier. Therefore, the accelerating area increases and reduces the stability margin of the system. But if tC is bigger, the value of δC is also increased, and the shift of the operating point from Pefault to Pepos is also delayed, therefore the accelerating area is reduced and the stability margin of the system is improved. The general result is that, if during the fault, Pefault is greater than PePOS and less than Pm, until the position of the curves is unchanged relative to each other, a shorter fault clearing time improves the stability margin.Fig. 9 Variation of Pe with FCL

Fig. 9

To investigate the effect of fault clearing time on the transient stability of the power system, the variations of ΔA for tC = 0.1s, tC = 0.2s, and tC = 0.3s for the system with resistive and inductive FCLs at locations P1 and P2 are shown in the following curve.

As shown in Fig. 10a–c, in cases where inductive FCL is placed at either P1 or P2, the shorter the tc, the more the system transient stability improves. While for resistive FCLs at P1 and P2 locations, the conditions are slightly different. According to Fig. 10b–d, it can be seen that in some ranges of FCL impedance values, even though tc is longer, the ΔA value is also more considerable. As a result, the system stability margin is improved. For example, in Fig. 10b, if the FCL impedance is in the ranges of 0.14–0.18 or 0.95–1.18, the value of ΔA for tC = 0.3s will be greater than the value of ΔA for the tC = 0.1s and 0.2s. Fig. 10d, for FCL impedance values larger than 2, the best result for system stability is always obtained for a tc = 0.3s. Strictly speaking, fault clearing time decreases do not necessarily lead to the best results for transient system stability. In some cases, the fault clearing time increase will improve the system stability margin.Fig. 10 Variation of ΔA for different tc.

Fig. 10

Table 4 Determining the optimal fault clearing time according to the type, location, and impedance value of FCL.

Table 4Impedance type of FCL	FCL location	FCL impedance range (pu)	optimal FCL impedance value (pu)	optimal clearing time (tc) (s)	
Resistive	FCL at P1	0 to 0.13	0.12	0.1	
0.14 to 0.18	0.16	0.3	
0.19 to 0.94	0.94	0.1	
0.95 to 1.18	1.08	0.3	
1.19 to 1.29	1.19	0.2	
1.3 to infinity	1.3	0.1	
FCL at P2	0 to 0.08	0.08	0.1	
0.09 to 0.095	0.095	0.2	
0.096 to 0.11	0.11	0.3	
0.12 to 2.01	2.01	0.1	
2.02 to infinity	4.13	0.3	
Inductive	FCL at P1	0 to infinity	0	0.1	
FCL at P2	0 to infinity	0	0.1	

Based on the information provided in Table 4, let's examine a hypothetical situation where a resistive FCL is positioned at point P1. Within the FCL impedance spectrum ranging from 0.14 to 0.18, an optimal fault clearing time (tc) of 0.3 s is identified. Specifically, when the FCL impedance is set at 0.16, it leads to the attainment of the utmost stability margin for the system.

4 Conclusion

This paper conducts a qualitative analysis with the objective of comparing the effects of two different types of FCLs, specifically inductive and resistive FCLs, on the transient stability of power systems. The results show that inductive and resistive FCLs, depending on their location and impedance value, can improve stability and reduce the transient stability of the power system. However, analysis of different FCL impedance cases shows that one type of FCL cannot be preferred over the other. Depending on the location and type of FCL impedance, in some impedance values, RFCL is the best choice in terms of the maximum transient stability margin of the system, while in some values, XFCL can be the best choice. Although increasing the tc often reduces the transient stability, it can even increase the system stability margins for some impedance values of RFCL. Therefore, the tc, along with parameters such as the type and impedance value of FCL, is effective in the stability condition and the amount of system transient stability margin.

In this study, the ratio of X to R of the transmission line and the location of the fault are assumed to be constant. However, in subsequent studies, the effect of their changes on the power system transient stability studies with FCL can be investigated. It is suggested that in future studies, this analysis should be done by providing a detailed model of different types of resistive and inductive FCLs, especially the inductive type (saturation core, bridge circuit, etc.). Furthermore, the present study will be conducted on generators with the salient rotor.

CRediT authorship contribution statement

Mahdi Dehghani-Ashkezari: Writing – review & editing, Writing – original draft, Visualization, Supervision, Software, Resources, Methodology, Formal analysis, Data curation, Conceptualization. Seyed Mahmoud Modaresi: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software. Seyedamin Saied: Software, Resources, Methodology, Investigation. Tahere Daemi: Writing – review & editing, Writing – original draft, Visualization. Hamidreza Akbari: Visualization, Validation, Supervision, Software.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
==== Refs
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