1. Introduction
Technological advancements and industrial growth in a nation are accompanied by an increase in energy consumption and have a significant relationship with the expansion of the electrical sector. The generated energy is conveyed to consumers through the electric power system (EPS), a structure composed of interconnected elements for the constitution of a large electrical network, which can contain thousands of buses and equipment such as synchronous generators, power transformers, measuring and protection transformers, circuit breakers, reactors, capacitors, loads, and transmission lines (TL). During transmission, electrical losses occur, and the energy transport capacity is limited based on the number of existing lines. These structures can stretch for hundreds of kilometers and are exposed to weather conditions such as lightning strikes, storms, strong winds, pollution, high insolation, and even vandalism, which are capable of causing short circuits (faults) and disrupting their normal operation, which can lead to instability in the electrical system and even blackouts if a point of equilibrium is not reached.
The occurrence of a fault in a transmission line represents a phenomenon that is difficult to predict. In the event of delayed restoration or unavailability of the line, electric utility companies may be subject to fines from regulatory agencies for the electrical system. Once the situation is diagnosed, it is in the interest of the utility companies to isolate the faulty part of the electrical system, ensure its prompt restoration, and meet the existing load demand quickly and accurately.
The increasing global demand for electrical energy directly influences planning actions for generation expansion and transmission sizing. One strategy to increase transmission capacity and improve the quality of the transmitted energy through the transmission line is series compensation, achieved by inserting a capacitor bank (CB) and its protection elements in series with the line. This approach postpones the need for investments in the construction of new transmission systems in the case of existing lines and brings advantages to the system in which it is implemented [
1], contributing to the increase in power transmission capacity of the line, improvement of stability after a transient, reduction in the need for voltage control equipment such as shunt capacitors, improvement of power division among lines, which can reduce overall system losses, cost savings compared to other technically feasible alternatives, and reduced environmental impact, which is highly relevant in today’s context compared to the construction of a new line. However, the insertion of capacitors and their protective element, the Metal Oxide Varistor (MOV), alters the line impedance [
2,
3], modifying the operation of the electrical system and requiring new methodologies for line protection tasks and fault location [
4,
5], making fault location estimation complex and challenging.
This paper presents results from the application of nonlinear optimization (NLO) using a genetic algorithm for fault location in transmission lines with series compensation (TLC) simulated in the Alternative Transients Program (ATP) [
6]. The objective functions of the problem are formulated considering the presence of the capacitor and MOV. One of the achieved objectives of the proposed methodology is the possibility of practical implementation in electric utility companies using the same data used for a transmission line without series compensation (voltage and current measurements at both terminals and impedance and admittance parameters) as by Johns and Jamali [
7], only adding the capacitance value of the capacitor bank. Therefore, this paper contributes by eliminating the need for an operating curve or parameters of the metal oxide varistor (MOV) and the values of source impedances, which are often not accurately known. This feature reduces the method’s dependence on specific electrical system parameters. Despite this reduction in dependency, the proposed method maintains precision, as demonstrated in the presented results. In addition to simulated cases, a novelty presented is the application to real short-circuit cases, made possible by utilizing oscillograms provided by a Brazilian power utility.
A literature review of the benefits and challenges related to series compensation in transmission lines is presented in [
8]. A significant portion of the research available in the literature addresses the problem analytically, and some studies propose the use of computational intelligence. Only a small number of proposals use optimization techniques. A mathematical analysis of the series compensation model is presented in [
9], considering that at the fault point, the difference in voltages calculated from both terminals is zero. Voltage and current measurements from both ends of the line and the values of source impedances are used to obtain the fault point voltages and estimate the fault location. The proposal used the pattern search method to find the fault distance and a polynomial fit based on the theory presented by Goldsworthy [
10] to represent the capacitor/varistor set. In [
11], a fault location proposal for compensated lines is presented using numerical methods. The source impedances were calculated based on voltage and current values during normal operation (pre-fault). To handle the nonlinearity of the MOV/capacitor set, the linearized model proposed by [
10] was used. In [
12], the differential evolution algorithm was used for the minimization of an objective function formulated based on fault resistance, fault type, fault distance, and transmission line parameters. The source impedances were considered input data, and the linearized model proposed by [
9] was also used. In [
13], fault location in series-compensated lines is approached by solving an optimization problem. The proposed objective functions are formulated to cover fault situations where the MOV does not act and unsynchronized measurements of voltages and currents from the two-line terminals are available. Three population algorithms are used to minimize the simulated data. The determination of the admittance matrix for the system equations requires the values of the source impedances.
2. Series Compensation in Transmission Lines
Series compensation began to be used at higher voltage levels starting in the 1950s when the Swedish State Power Board and the Bonneville Power Administration in the United States installed capacitor banks with a capacity of 25 MVar in 220 kV systems [
14]. From the operational perspective of a transmission line, performance indicators such as transmission losses in joules, reactive power consumption, voltage regulation, and stability margin must be observed [
15]. An important parameter for determining the optimal load on the TL is the surge impedance load (SIL). According to [
16], under operating conditions far from the SIL, the voltage variation along the line is greater, and the transmitted power is related to the length. For short lines up to 80 km, it is possible to operate with powers up to three times higher than the SIL. As the length of the line increases, this value decreases. For example, the limit for lines of 200 km is 1.8 × SIL, and for lines with a length of 300 km, it is recommended to operate with up to 1.4 × SIL. The voltages at the sending (
and receiving (
terminals, as a function of the angular difference between the voltages (δ) at the busbars, are indicated in Equation (1).
Approximating a real transmission line with losses by an ideal lossless line, the transmitted powers can be obtained using Equations (2) and (3), where
represents the nominal series reactance of the line.
The maximum current allowed is related to the thermal limit of the conductors [
16], is estimated in the project, and depends, among other factors, on the ambient temperature, wind speed, and solar radiation. In
Figure 1, the decay of the conductors at a height of D1 is represented, which occurs in a situation of the nominal current. At height D2, due to the increase in transmitted power, the line is at its thermal limit in the condition of maximum current capacity.
The power limits of a transmission line, considering thermal and operational aspects, are described in [
1] and illustrated in
Figure 2.
In short transmission lines, current capacity represents a limiting factor for the load flow. In medium and long transmission lines, the situation is typically delimited by operational limits, with the longitudinal reactance often reaching values that restrict power flow to levels lower than the maximum supported by the cables. It can be observed that as the length of the line increases, the relevance of series compensation expands, which can be evaluated as an alternative to increasing power transfer capacity with reduced environmental impact and avoiding or postponing investments in the construction of new transmission lines.
2.1. Compensation Degree
The compensation degree (k) is defined as the ratio between the capacitive reactance of the bank (
) and the inductive reactance of the transmission line, as stated in Equation (4).
Rewriting Equation (4) in terms of the compensation degree leads to Equation (5).
The effect of the compensation degree on the transmission capacity for a 500 kV line is presented in
Figure 3.
In practice, the series compensation degree typically varies from 25% to 75% [
17,
18]. For a fixed angular difference between the voltages at the line terminals, the transmission capacity increases with the compensation degree. According to [
19], for the same transmitted power value, the angular difference decreases with an increase in the compensation level, resulting in improved dynamic stability of the system.
2.2. Protection of Capacitor Banks
As they are connected in series with transmission lines, capacitor banks are subjected to voltage and current transients related to short circuits. It is essential to use a protection system to limit the voltage at the capacitor terminals and prevent equipment damage. In this regard, the protection system should be specified to limit the maximum overvoltage that each capacitor unit must withstand. For voltages below the determined limit, the CB should operate normally. According to [
1], the metal oxide varistor (MOV) is among the main protection devices, characterized by its high nonlinearity.
Figure 4 shows the operating curve of a MOV with a maximum withstand voltage of 283 kV to protect a 55.62 μF capacitor bank.
A simplified system is represented in
Figure 5. When the overvoltage across the capacitor reaches or exceeds the trigger level, the varistor starts conducting, diverting the current. After the fault is eliminated, the operating conditions are restored. The spark gap starts conducting to protect the varistor when the voltage reaches a predetermined limit, and the circuit breaker is closed if the varistor’s temperature limit is exceeded, preventing equipment damage due to overheating.
2.3. Position of Capacitors in the Transmission Line
The location of the capacitor affects the voltage profile, which smoothly varies along the line with normal load current but undergoes a sudden change at the equipment location. As described in [
5,
20], the capacitor’s position also alters the line segment where a fault would cause voltage inversion, current inversion, or subsynchronous oscillation. The four general positions of the CB are shown in
Figure 6. The equipment can be inserted at one or both ends of the line or along the line, such as at the midpoint or one-third of its length [
21]. From the perspective of the voltage profile, placing capacitors in the middle of the line is more effective. However, in practice, compensation at the line’s ends is more common, as installing capacitors along the line requires the construction of a substation to accommodate the CB and its protection and control equipment.
3. Steps of the Process
Before solving the fault location problem by minimizing the objective function, preprocessing routines are applied to prepare the data.
Figure 7 presents the basic steps involved in the algorithm’s development.
The process begins with reading the voltage and current data from the line’s two terminals, obtained from simulations such as an ATP output file or from recorders installed in substations in COMTRADE format [
22]. Next, the fault instant is determined, allowing the separation of the pre-fault and fault periods. The preconditioning of the signals starts with low-pass filtering, removing higher frequencies using a 2nd order Butterworth filter at 100 Hz. After this step, the sampling frequency of 16 points per cycle at the fundamental frequency is obtained through data interpolation [
23]. The least squares method of [
24] is used to estimate the phasors associated with the fundamental frequency. The next step is fault classification, where the voltage and current phasors involved in the short circuit and the objective function to be used by the algorithm in the fault location are selected. An example of the fault detection and classification steps is presented in [
25]. At the end of the described steps, the voltage and current data are used to minimize the objective function (F), which estimates the distance to the fault point. The steps indicated in
Figure 7 were implemented in the MATLAB software (Version 9.4–R2018a, MathWorks).
5. Fault Location: Simulated Cases
The electrical system used to generate the results of the proposed methodology is based on data from a 256 km transmission line in the central–west region of Brazil. The single-line diagram and the typical tower are represented in
Figure 17 and
Figure 18.
Figure 19 shows the ATP circuit used to generate fault files for the Serra da Mesa 1—Gurupi line, with the aim of applying and evaluating the developed method. Each of the two CBs has a capacitance of 135 µF, allowing for a compensation close to 60%. The transmission line model used was the distributed and frequency-variable parameter model by Martí [
28].
From the ATP, variations in the location of the short circuit were simulated every 10% of the line length, along with the RF value and different types of faults. The fault model is represented in
Figure 20.
In
Table 2, the electrical parameters for the terminal sources and an ideally transposed transmission line with a soil resistivity of 1000 Ωm and a frequency of 60 Hz are presented.
In the event of a fault, after the detection and classification steps, the PRGA is applied to locate the fault.
Figure 21 shows the objective function graph for a simulated AG fault, which occurred at 176.4 km.
Figure 22 shows the percentage location errors obtained for BC, ABC, AG, and ACG faults, simulated every 10% of the 256 km transmission line length, calculated as a function of the total transmission line length, according to Equation (26).
Figure 23 shows the current threshold for MOV—I
th which was obtained in the minimization of the objective function for the AG faults in
Figure 22. The MOV remains the same for all simulations; however, the value of I
th, due to errors in the data preprocessing process and the minimization of the objective function, is estimated with different values, but around 2000 A. The errors obtained in the fault location process are less than 1.3%, even for high fault resistance values, which is acceptable in this engineering problem.
Table 3 shows the average and maximum errors obtained for the faults in
Figure 22.
In
Table 4, the errors for simulated AT faults in ATP are presented for a resistivity of 100 Ωm, but using the transmission line parameters from
Table 2, they are obtained for a resistivity of 1000 Ω. With this change, there was a slight increase in errors, but they remained close to 1%. Regarding data synchronization, with a 30-degree angular difference between the currents and voltages at the line terminals, the errors increased significantly. One solution to this problem is to synchronize the measured values at both terminals using the line parameters and pre-fault measurements. Another approach, as proposed in [
29], also allows for unsynchronized data.
7. Conclusions
This paper presents a new proposal for fault location in transmission lines with series capacitors. Analytically, objective functions dependent on various fault types were developed, considering the modifications caused by the inclusion of series capacitors together with protective varistors, which exhibit nonlinear behavior. When minimized, these functions estimate the fault location and the current at which the varistors initiate conduction to protect the capacitors against overvoltage.
Compared to the fault location process in uncompensated transmission lines, the proposed method only requires additional information on the capacitor bank capacitance value, making it a simple approach regarding input parameters and feasible for field applications. The main contribution of this proposal is that it does not require knowledge of the characteristics or data of the metal oxide varistor (MOV) or the values of source impedances, simplifying the problem formulation and reducing the method’s dependence on electrical system parameters without decreasing precision in the results. The errors obtained for real fault cases indicate the viability of the methodology for applications in electric power companies. Further studies are ongoing, as it has been observed through oscillograms of real analyzed faults that it is necessary to investigate whether there is conduction through the spark gap during the fault, diverting the current from the capacitor and the MOV, which can influence the voltage and current data window used in the problem. The application of the method to systems with thyristor-controlled series capacitors (TCSC) is also being evaluated.