Impact of the Short-Circuit Current Value on the Operation of Overhead Connections in High-Voltage Power Stations

Abstract

This paper focuses on determining the effects of short-circuit current values on the operation of high-voltage substations. The analyses performed focused on the dynamic and thermal effects on the short-circuit current for different input data configurations. The analyses of the dynamic effects of short-circuit current were performed in two ways. First, calculations were executed using the procedures contained in “IEC 60865-1:2012 Short-circuit currents, calculations of the effects of short-circuit currents. Part 1: Definitions and calculation methods”. Analytic calculations were accomplished for different values of the initial three-phase short-circuit current, the type of short circuit, the static tension force of the conductors, the stiffness of the structure and the methods of attaching the lashing chains. Secondly, the same analysis was performed using PRIMTECH 3D software. The connections were modeled in computer software, for which the calculations were implemented according to normative recommendations. After obtaining the results, an analysis was performed using the model based on the input data. Performing the analysis in two ways made it possible to compare the results with each other and develop conclusions. An analysis of the thermal effects of the short-circuit current was also provided. The results of the calculations showed that the choice of conductor cross sections according to the value of the initial short-circuit current was correct. Particular attention has been paid to rigid conductors and flexible conductors, which are directly affected by the short circuit and have an impact on the selection of the complete equipment for the high-voltage switchgear.

1. Introduction

It is necessary to perform an in-depth analysis of the operation of any device, both in normal and emergency conditions. This analysis should be carried out during the design phase of any new device but also during the analysis of the usability of an existing device. Emergency conditions are those in which the values of the parameters defining normal operating conditions are exceeded for a certain period of time [1,2,3,4]. In the case of electrical power systems, short circuits are the typical emergency conditions considered, and devices are tested for their resistance during such emergency conditions. Short circuits occurring in an electrical network can be classified into different categories according to defined criteria. Taking into account the number of simultaneously short-circuited points, single and multiple short circuits are distinguished. On the other hand, the location of the short circuits should be taken into consideration, in order to categorize them into internal and external short circuits. Short circuits which occur, for example, inside a transformer or power devices are considered internal short circuits, while those which occur in the busbars or between them and the support structures are considered external short circuits [5,6,7,8]. It is also possible to establish another classification of short circuits in relation to the distance between the short circuit and the generator: in this case, short circuits far from the generator and near the generator are distinguishable. Despite the different classification criteria listed above, the main short-circuit classification is based on the number of short-circuited phases. From this point of view, the following are distinguished:

• Three-phase short circuits;
• Two-phase short circuits;
• Two-phase short circuits with earth;
• Single-phase short circuits.

The causes of short circuits can also be classified into two categories, namely, electrical and non-electrical. Among the electrical causes are atmospheric and switching overvoltages, switching errors, but also overloading of equipment and isolated power grids, which lead to overheating and, eventually, insulation breakdown. The non-electrical causes are also multiple, among which it is viable to enumerate the manufacturing defects of the equipment, the mechanical damage to the equipment, the structures, and the insulation, the fouling of the insulators of overhead electric lines, the mutual proximity of the conductors of the overhead lines caused by wind action and animal interference. By analyzing the list of possible causes of short circuits, it is easy to come to the conclusion that they are unavoidable. It is necessary to ask what the effects that could be caused by the passage of a short-circuit current of too high a value for a duration greater than the permitted duration are. Some examples of the effects of a short circuit occurring in the power network are presented below [9,10,11,12,13,14]:

• Strong heating of power system components;
• Fire;
• The generation of large dynamic forces, the impact of which may lead to deformation or damage to, for example, buses, support insulators, equipment or support structures;
• Exceeding the permissible values of touch and step voltages;
• The formation of electromagnetic interference;
• The possibility of a loss of stability of the power system.

In view of the fact that emergency operating states of the power system cannot be avoided, the effects of their impact should be properly minimized. To this end, the duration of the short circuit and the values of the short-circuit currents must be reduced to a minimum, and the maximum must be carried out so that the effects of its impact are only local. The means of reducing short-circuit currents are presented below:

• Increasing the short-circuit impedance by introducing additional elements into the network or by shaping the network accordingly;
• Rapid switching off of the short circuit;
• Limiting the duration of a short circuit by means of power automation.

However, power system components should be designed in such a way that the possible flow of a short-circuit current of a certain value does not damage them. In a three-phase power system, a three-phase short circuit or a three-phase-to-earth short circuit is classified as a symmetrical short circuit, which means that all phases are equally loaded [15,16,17,18]. All the other aforementioned types of short circuits are classified as asymmetrical short circuits. In practice, three-phase symmetrical short circuits practically do not happen.

Procured work concerns the results of the analysis of the effects of the short-circuit current impact on flexible busbars located in the highest voltage substations. The analysis was executed in three ways. Novelties that applied to this work can be distinguished below:

• Correcting the normative calculations (IEC 60865-1:2012 standard) and taking into account the two-point attachment of the haul-off chains;
• Enhancement and impact on the method of calculating electrodynamic forces in flexible wires on high voltage lines were obtained;
• Enhancement and impact on the calculations of mechanical stresses in supporting structures and poles of the highest voltage lines were obtained;
• Scalability of the calculations for lower voltage lines, if it is possible to use two-point fastening of the haul-off chains.

2. The Effect of the Short-Circuit Current on Individual Elements of High Voltage System

Power installations and equipment must be designed, built, commissioned, and operated in such a way as to ensure the safety of users while meeting the condition of reliability of their operation, even under unusual operating conditions [19]. Therefore, during the design phase, extreme short-circuit conditions where, for example, a very high short-circuit current value appears, as in the case of a three-phase short circuit with a longer duration than normal, can be considered as a basis for determining the operating parameters of the designed equipment. In the case of facilities such as substations, short-circuit conditions affect the selection of virtually all equipment, including the selection of structures and foundations, which may not seem so obvious, since these are not electrical devices [20,21,22,23,24]. A general short-circuit current waveform is shown in Figure 1 [25]. Short-circuit analyses are conducted for extreme cases that may arise during the use of the installation or equipment, for example, for the maximum value of the non-periodic component of the short-circuit current or for its greater large initial value. However, it should be borne in mind that too pessimistic an approach to the calculations will result in the oversizing of equipment, which in turn will affect the total cost of the device or investment.

Short-circuit analyses are usually executed for three-phase symmetrical short circuits, and therefore without taking into account the non-periodic component. In reality, symmetrical short circuits are rare in power networks, although the calculation procedures in IEC 60865-1:2012 do not take them into account. It is worth remembering that the non-periodic component will have a significant impact on the results of calculations when the duration of the short circuit is less than 0.1 s. Formulas to determine the values of individual currents are contained in “IEC 60909-0:2016-09 Short-circuit currents in three-phase AC networks—Part 0: Calculation of currents”. The initial short-circuit current is the rms value of the periodic component at the first moment of the short circuit [25]. The operator of the transmission network provides this value as the rated short-circuit current, which is the basis for short-circuit analyses. The value of the initial three-phase short-circuit current is described by the following expression:

$${\mathrm{I}}_{\mathrm{K}3}^{″}=\frac{\mathrm{c}·{\mathrm{U}}_{\mathrm{N}}}{\sqrt{3}·\left|{\mathrm{Z}}_{\mathrm{K}}\right|}$$

(1)

It is necessary to know the impedance of the short circuit to determine the expression (1). In the case of a three-phase short circuit, the impedance for the symmetrical component is taken into account. In the case of a three-phase short circuit, the impedance of the symmetrical component of the appropriate equivalent circuit is considered. The voltage factor c of (1) is selected according to the voltage of the power grid. In networks with voltages above 110 kV, this factor takes the value of 1.1. In short-circuit analyses, the value of the two-phase short-circuit current is used. The value of the initial two-phase short-circuit current can be determined from the value of the initial three-phase short-circuit current according to the following expression:

$${\mathrm{I}}_{\mathrm{K}2}^{″}=\frac{\sqrt{3}}{2}·{\mathrm{I}}_{\mathrm{K}3}^{″}$$

(2)

The surge short-circuit current is the highest instantaneous value of the short-circuit current, including the non-periodic component; it is determined from relation (3) [25,26,27]. Its value is about 2–2.5 times the initial value. To determine the surge value, it is necessary to know the surge factor κ, which depends on the ratio R/X. For far-from-generator short circuits occurring deep in the network, it is assumed that this ratio is equal to 0.07 Ω, which gives a time constant of 45 ms and a surge factor of 1.8. For substations in the immediate vicinity of power plants, the time constant can be in the order of about 50–70 ms, in which case the surge factor will be equal to 1.9 [4]:

$${\mathrm{i}}_{\mathrm{p}}=\mathsf{\kappa }·\sqrt{2}·{\mathrm{I}}_{\mathrm{K}3}^{″}$$

(3)

$$\mathsf{\kappa }=1.02+0.98·{\mathrm{e}}^{\raisebox{1ex}{$-3·\mathrm{R}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{X}$}\right.}$$

(4)

The flow of the short-circuit current is associated with a rapid increase in temperature. In short-circuit analysis, the concept of equivalent thermal short-circuit current is introduced. This quantity is described as a sinusoidal current of constant amplitude or such an effective value, so that the flow of this current will give off the same amount of heat as the short-circuit current [13]. The determination of the equivalent thermal current value makes it possible to check whether the flow of a short-circuit current of a certain value and for a certain period of time will not cause the allowable temperatures of the conductors to be exceeded. When calculating the equivalent thermal current value, the values of the m and n coefficients, used to consider the thermal effect of the non-periodic and periodic components of the short-circuit current, are expressed in a formula as follows:

$${\mathrm{i}}_{\mathrm{t}\mathrm{h}}={\mathrm{I}}_{\mathrm{K}3}^{″}·\sqrt{\mathrm{m}+\mathrm{n}}$$

(5)

There are two types of short-circuit current interactions, dynamic and thermal. Dynamic interactions, often referred to as mechanical interactions, are related to the electrodynamic forces created between conductors due to the flow of the short-circuit current. Thermal effects are related to the intense heating of the current paths during a short circuit. Factors affecting the magnitude of short circuit effects are numerous and can be summarized into three main groups:

−

Short-circuit current parameters:

• The value of the initial short-circuit current;
• The time constant of the short circuit depending on the R/X ratio;
• The surge factor, which depends on the R/X ratio.

−

Span parameters:

• The wire cross section;
• The conductor mass;
• The cable length;
• The elasticity of the conductor;
• The static tension of the conductors;
• The distance between axes of supports;
• The length of the chains;
• The distance between phases;
• The number of wires in a bundle;
• The distance between conductors in a bundle;
• The number of spacers;
• The type of spacers.

−

Parameters of support structures:

• The rigidity of support structures.

The number of parameters affecting the scale of the short circuit effects presented above is large. Moreover, there is a certain interdependence between the parameters because each of them can have an impact on the others, so the possible calculation configurations are numerous. Some of the above parameters have a greater influence on the calculation results, and others have a lesser one. In addition, some of the above parameters can be freely modified by the substation designer, while others are pre-imposed values, and the designer has no influence on them at all. When performing an analysis of the effects of short-circuit currents on individual components, connections should be configured in such a way as to minimize the negative effects of the short circuit. The rating of the short-circuit current and its duration are among the main parameters that determine the effects of the short-circuit current flow. The time during which the device can conduct a current equal to the rated short-endurance current for a given device is called the rated short-circuit time [13]. For the highest voltage networks, the rated duration of a short circuit is 1 s. However, this does not mean that each short circuit will have a duration of 1 s. The actual value of the duration of a short circuit is much smaller; it depends on the protection systems used and their settings and also on the type of circuit breakers installed at the substation. In general, it can be assumed that for the highest-voltage switchgear, the time for tripping the protections and opening the circuit by the circuit breaker is about 120 ms. Nevertheless, the assumption that a short circuit will last for 1 s represents the worst case in which, for example, the protection automation does not operate or the circuit breaker fails. This makes it possible to determine the maximum effects that will result from the flow of the short-circuit current; it is, therefore, possible to properly design equipment or power infrastructure facilities.

2.1. Mechanical Impact of Short-Circuit Current

Mechanical interactions of short-circuit currents can be classified into two categories: interfacial interactions and intra-bundle interactions. Interfacial interactions occur regardless of whether the connections are made with flexible or rigid conductors, while intra-bundle interactions occur only when the connections are made with a bundle of flexible conductors or with multi-bundle rigid conductors. The flow of short-circuit current through parallel wires creates an electrodynamic force. This force acts directly on the current paths, causing their deformation and elongation, and also set them in motion by attracting or repelling each other, depending on whether the direction of current flow in adjacent conductors is compatible or opposite. The resulting electrodynamic force affects the switchgear rail, apparatuses, support structures and also the foundations. If the non-periodic component is neglected, the electromagnetic force oscillates at a frequency of 100 Hz and is proportional to the product of the currents flowing in the conductors. If the values of the currents in the two conductors are equal, then it is proportional to the square of their value [13]. For rigid rails, the highest forces are generated in the case of a three-phase short circuit. Since most of the analyses performed to determine the mechanical impact of the short-circuit current are performed for a symmetrical three-phase short-circuit current, the determined values of the forces are not maximum values, although this is in accordance with the calculation procedure in the standard. The situation is slightly different for switchgear with flexible rails. From the point of view of the value of the force associated with the bending and dropping of the conductors, the maximum angle of bending of the conductors and the observance of minimum interfacial distances, this two-phase short circuit is a worse case than a three-phase short circuit.

Figure 2 below presents a section of the line field cross section of a 400 kV switchyard operating in a 3/2W system with two busbars. From the cross section of the entire branch, a section from the point of entry of the overhead line to the end of the first bridge of the branch was selected. As a result, the presented cross section of the field shows all typical connections to be analyzed for the impact of the short-circuit current. Among the typical connections are the following (marked in Figure 2):

• The busbar system made with rigid conductors with a tubular profile (A);
• The top rail of the switchgear made with a bundle of flexible conductors (B);
• Discharge wires (C);
• Connections between apparatuses made with flexible conductors (D);
• Connections between apparatuses made with rigid conductors (E).

The busbar system shown in this section is made with AR250 × 8 mm pipe conductor. The distance between the conductors is 5 m, and this is a typical phase-to-phase distance for a 400 kV switchgear. Visible in the section, the connections between the apparatuses made with rigid conductors are made with AR120 × 12 pipes; the distance between the apparatuses is also 5 m. The top railings of the switchgear made with a bundle of AFL-8 525 mm² wires are attached to high steel structures with the lashing chains fixed with two-point attachment. The distance between phases for the top rail, in this case, is 6 m. However, it happens that this distance is reduced to 5.5 or even to 5 m. The drain wires shown are used to connect the top rail to the apparatuses. At the switchgear site, connections between apparatuses made by flexible conductors are also distinguished. Usually, on switchgears, the same type of cable is used for the top rail, discharge wires and connections between apparatuses, but there are differences in the number of wires in the bundle.

2.2. Electrical Connections

Rigid connections were described in detail by the authors and analyzed in accordance with the standard IEC 60865-1:2012, which also indicates the calculation procedures when rigid conductors were made as bundled, although this type of solution is not used in the highest voltage switchgear [14]. Hence, here the description is not included, and the emphasis is on flexible connections and physical phenomena associated with the flow of short-circuit current. Two types of interfaces are distinguished in switchgears in which connections are made with a bundle of flexible conductors; these are interfacial and intra-bundle interactions. Two types of forces are associated with interfacial interactions. Intra-bundle interaction results in an additional force. These forces and the reasons for their occurrence are detailed below:

• The force associated with the deflection of the cable during the short circuit, denoted as F_t,d and arising from interfacial interaction;
• The force associated with the descent of the wire after the short circuit has disappeared, denoted as F_f,d and arising from interfacial interaction;
• The force associated with bringing the wires together in the bundle, denoted as F_pi and resulting from intra-bundle interaction.

These forces, as in the case of rigid rails, arise from the acting electrodynamic force. The formula to determine the value of the electrodynamic force acting per unit length of the cable depends on the construction of the span and the type of short circuit. If the current flows through the entire length of the span and the short circuit that occurs is a three-phase short circuit, the resulting electrodynamic force is determined by the following relationship:

$${\mathrm{F}}^{\prime }=\frac{{\mathsf{\mu }}_0}{2\mathsf{\pi }}·0.75·\frac{{\left({\mathrm{I}}_{\mathrm{K}3}^{″}\right)}^2}{\mathrm{a}}·\frac{{\mathrm{l}}_{\mathrm{c}}}{\mathrm{l}}$$

(6)

If, on the other hand, the span has discharge wires connecting the top rail to the apparatus and the current flows through only half of the span, the value of the force is calculated as follows:

$${\mathrm{F}}^{\prime }=\frac{{\mathsf{\mu }}_0}{2\mathsf{\pi }}·0.75·\frac{{\left({\mathrm{I}}_{\mathrm{K}3}^{″}\right)}^2}{\mathrm{a}}·\frac{\frac{{\mathrm{l}}_{\mathrm{c}}}{2}+\frac{{\mathrm{l}}_{\mathrm{v}}}{2}}{\mathrm{l}}$$

(7)

In the case of a two-phase short circuit, the value of the two-phase current should be substituted in the above relations instead of the value of the three-phase current, and the factor of 0.75 also disappears. Compared to Formula (6), in the case of flexible rails, there is a parameter labeled l_c, which describes the length of the wire in the span. In the case where the current does not flow through the entire span, there is also a parameter l_v, which denotes the length of the discharge wire. The value of the current is also different, which is the basis for determining the force. For rigid rails, the value of the surge short-circuit current is used, and for flexible conductors, the value of the initial short-circuit current is used due to the short duration of the surge.

The above figures show a typical response of a span made with a bundled wire to a short circuit created in the network with a duration of about 0.6 s. In Figure 3, three characteristic points and their corresponding occurrence times are indicated. Point No. 1 is related to the bundle force; point No. 2 is another peak in the tension of the wires caused by the deflection of the wires while the short circuit is still in progress. Point No. 3 corresponds to the tension created by the dropping of the wires just after the short circuit was deactivated. Figure 4 shows the curve of wire movement in one of the phases affected by the short circuit. Point 0.0, marked as 4, is the point of the suspension of insulators. The path of wire movement is surrounded by a solid blue line. It begins about 1.5 m below the 0.0 point because that is the sag level in the center of the span for the case. Points 1–3 have a link to the force diagram. From the figures provided, it is very easy to see the order of occurrence of the various phenomena during a short circuit. Intra-bundle phenomena take place practically immediately after the short circuit occurs. Under normal operating conditions, the harness wires are separated from each other by rigid or flexible spacers. Only a few milliseconds after the disturbance occurs, the bundle force causes the wires in the bundle to attract each other and eventually, if the geometric dimensions of the bundle allow it, to brace wires together and remain in this state until the end of the disturbance. The sudden sticking of the wires causes a sudden increase in the tension force. At the first moment, the force F_pi can reach up to six times the static tension force of the wires [5]; this is shown in Figure 5a,b. However, it should be noted that at the maximum value of the force F_pi, the displacement of the structure is practically zero. This is because the support structures and power apparatuses have high inertia, so such momentary loads do not affect them. Therefore, during the analysis, it should be analyzed whether it is the switching equipment that should be characterized by resistance to such large forces or also the support structures [5,6].

The pinch force depends primarily on the geometric dimensions of the bundle, for which the spacers are responsible. Specifically, the bundle force depends on the number of spacers, their type, and also their dimensions. This force is approximately proportional to the distance between the wires. The wires in the bundles must be installed so that they are relatively close to each other. This distance, however, cannot be too small because it will have negative consequences. Among other things, there may be a higher transfer possibility of contacting certain wires to others during normal operating conditions, which could consequently have a reduction in the load capacity of the entire bundle. One can also imagine that in winter, there can be a situation in which the wires stick together when icing occurs. In the highest voltage switchgear, the distance between the wires in the bundle in the top rail is usually 20 cm. For connections between apparatuses, this distance is usually reduced to 10 cm. The distance between different spacers is also important. In the bays constituting the top rail of the switchgear, we prefer relatively important distances between the spacers. Moreover, flexible spacers are used to allow the movement of the wires in the bundle, so that the resulting forces are reduced. In drain connections and between apparatuses, the number of spacers is greater. The spacers are often installed near the apparatus terminals, so that a larger part of the load is transferred to the spacers and the apparatus terminal is protected. The interfacial action is slightly delayed with respect to the in-bond action.

The movement of the dangling wires occurs with a certain delay, but it continues for a period relatively long after the disturbance disappears. The F_t,d force is related to the maximum swing of the wires during the period where the short circuit is still present. It is characterized by a very small kinetic and potential energy since, at maximum deflection, the wire velocity is close to zero. An important part of this force is converted into deformation energy. The maximum angle of deflection of the conductor and the angle of deflection when the short circuit is deactivated depend primarily on the construction of the span. The presence of discharge wires in the span can significantly limit the range of movement of the overhead rail wires, while a situation may arise in which the discharge wires are torn from the terminals of the apparatuses. The duration of the short circuit, the value of the short-circuit current, the presence of autoreclosing, the ratio of the electromagnetic force to the gravitational force and also the tension force of the conductors are also significant. Determining the value of the maximum conductor angle is necessary to determine whether the minimum phase-to-phase distance will be maintained. In a situation where a two-phase short circuit occurs in the summer when the overhangs of the conductors are much greater than in the winter, the oscillating conductor may approach the adjacent phase close enough for a jump to occur. According to IEC 619361:2011/A1:2014-10 AC electrical installations for voltages higher than 1 kV—Part 1: General provisions, the minimum insulation clearance between phases during a short circuit must be at least 50% of the rated clearance. Once the short circuit is switched off, the phase-to-phase interaction disappears. This involves a rapid fall of the conductor, which takes place when the short circuit fades. In the vast majority of cases, it is this type of force that is decisive in the dimensioning of tall structures. It is worth noting that according to PN-EN 60865-1:2012, the maximum values of the forces F_f,d, and the maximum horizontal displacement of the conductors should be determined during a two-phase short-circuit test. This is because the standard assumes that the three-phase short circuit that occurs will be symmetrical. In this case, the middle phase would remain unloaded, and the determined values of forces and displacements would not reach their maximum values [4,5]. Based on the analysis of the mechanical effects of the short-circuit current, it is necessary to determine which of the forces that occur will have the highest value. This value will be necessary for the proper design of support structures. Factors affecting the results of the calculations are numerous, and they were listed at the beginning of the section. One of the most important factors is the static tension of the conductors. From one side, its value must be large enough to obtain optimal wire overhangs that ensure the maintenance of minimum phase-to-phase distances; on the other hand, the smaller the initial static tension, the smaller the values of forces generated by the flow of the short-circuit current. Other equally important factors are the value of the initial short-circuit current, the stiffness of the support structures, the geometric dimensions of the bundle and the span.

The characteristics above show the values of individual forces as a function of span length. Those were determined for spans of 20 to 80 m in length for three short-circuit currents: 40 kA, 50 kA and 63 kA. It was assumed that the connection would be made with a bundle of AFL-8 525 mm² wires with a tension of 20 kN per phase, which corresponds to the actual tension of the wires during the winter. The number of spacers varied from one to four so that the distance between adjacent spacers in each span length configuration was similar. The resultant stiffness value of the two tall structures forming the span was assumed to be 440⋅103 N/m. From the curves in Figure 6 and Figure 7, it can be easily deduced that as the span length increases, the value of the individual force increases. The higher the value of the short-circuit current, the greater this increase. In Figure 7, it is noticeable that with a current of 40 kA and a span length of 20 m, the force associated with the descent of the conductor does not occur, or rather is not taken into account because the requirements in the standard for the angle of deflection and the ratio of electrodynamic force to gravitational force are not met. The characteristics in Figure 8 show a plot of the value of the beam force as a function of span length. The characteristics shown for the three values of short-circuit current overlap. This is because the value of the beam force mainly depends on the geometric dimensions of the beam, and the characteristics were presented for a T_pi time of 50 ms, so peak force values were ignored. By varying the number of spacers in the spans, the distances between the spacers were kept similar so that the value of the F_pi force is also more or less at the same level, around 30 kN, regardless of the span length. Comparison of the graphs also gives us the opportunity to see that for short spans, it is the F_pi force that will be the decisive force, dimensioning the substructures. Hundreds of similar characteristics could be determined. Tension values, interfacial distances, structural stiffness values and the like could change.

3. Analysis of the Dynamic Effects of Short-Circuit Current

Analyses of the dynamic effects of the short-circuit current can be performed for various connection configurations. Parameters such as the value of the initial three-phase short-circuit current, the duration of the short circuit, the value of the static tension force of the conductors, the rigidity of the structure, the type of conductors used, the span length, the distance between phases, the number of spacers, the location of spacers, the method of attachment of rigid pipe conductors to support insulators and lashing insulators to high support structures and the like can be changed. Changing any of the above-mentioned parameters more or less affects the final calculation results. In addition, all these parameters are interrelated so that there are thousands of possible configurations. The analysis of the dynamic effects resulting from the flow of short-circuit current was carried out for rigid and flexible rail in two ways. First, calculations were carried out based on IEC 60865-1:2012 Short-circuit currents. Calculation of the effects of short-circuit currents. Part 1: Definitions and calculation methods. PRIMTECH 3D software was used to perform the second version of the calculations. In the program, individual connections were modeled, and calculations were made in accordance with normative recommendations. Then, for the same input data, an analysis was performed to compare the results. The IEC 60865-1:2012 standard describes the methods to be followed to determine the value of forces generated by the flow of short-circuit current. In the case of connections made with rigid conductors, the standard allows us to perform calculations for many configurations. It is possible to perform calculations taking into account the presence of reclosing automation and the natural frequency of the conductors’ own vibrations, and also taking into account the different ways of attaching the conductors to support insulators. Calculations executed for railings made with flexible conductors should be executed for extreme operating temperatures of the conductor, that is, for those at which static tension forces reach extreme values. In addition, calculations to determine the maximum value of the force associated with the cable drop and the maximum displacement of the conductors should be executed for the lowest value of cable tension and using the value of two-phase short-circuit current rather than three-phase. This is due to the fact that the IEC 60865-1:2012 standard has introduced a simplification. According to a note in the standard, the course of the short-circuit current in the first 100 ms containing non-periodic components is completely ignored. This simplification makes it possible to assume that the disturbance occurring will be a three-phase symmetrical short circuit, so that the inter-phase interaction makes the movement of the central phase negligible. Therefore, performing calculations for the aforementioned quantities should be carried out, assuming that a two-phase short circuit will occur in the network. The calculation procedures contained in the standard have some limitations. In the case of connections made with flexible conductors, according to the standard, we can perform analyses only for spans whose length does not exceed 120 m, and whose sag is no greater than 8%. In addition, if the difference in height between the conductors’ points of attachment differs by more than 25%, then the calculations for such a connection should already be executed as for discharge, vertical conductors. Spans equipped with drainage conductors for connecting the top rail of the switchgear with the apparatuses below cannot always be fully mapped. The standard allows consideration of drainage conductors in the span only if they are located in the center of the span, or if they are offset by 10% of the length of the conductor. However, it may not be possible to perform calculations for such a connection because there are further limitations this time for the discharge conductors. According to the standard, calculations for discharge conductors can be made only if the length of the discharge conductor is less than twice the length of the section connecting in a straight line the points of attachment of the conductor, and if it is within the range of 1.4–3.3⋅w, where “w” means the horizontal distance between the points of attachment of the discharge conductor. Within the framework of this thesis, an analysis of the effect of short-circuit current on the top rail of the switchgear made with flexible conductors and on the busbar system made with rigid conductors with a tubular profile was performed. For each variant of the analysis, the parameters of individual connections and short-circuit data are presented. Then, for one of the variants, sample calculations in accordance with the normative recommendations are presented, and the numerical results obtained using PRIMTECH 3D software are also included. Several criteria guided the development of input data for the calculation variants. Analyses of the dynamic effects of the short-circuit current were performed to determine the maximum values of the forces that will be generated by the flow of the short-circuit current. Therefore, in all variants of the analyses, 1 s was used as the duration of the short circuit. This made it possible to avoid situations in which the determined values of forces would not be maximum values. Adopting a short-circuit duration that was too short, for example, 0.2 s, could be associated with a situation in which the swinging wires would not have time to reach the maximum swing, and so the value of the force associated with the swinging and falling wires would not be the largest possible either. Analyses were performed for both three-phase and two-phase short-circuit currents. Performing the analysis for a two-phase short circuit made it possible to see the differences in the values of the individual forces and also to check whether, indeed, the forces associated with the pivoting and dropping of the conductors reach higher values with a two-phase short circuit. One variant of the analysis was executed in a slightly different way than it is presented in the standard recommendations. Some of the formulas in the standard were modified to take into account the two-point attachment of the lashing chains.

3.1. Connections Made with Flexible Wires

The analysis for connections made with flexible conductors was derived for the top rail of the switchgear. It was assumed that, in the analyzed span, there would be no discharge wires, so that the current would flow through the entire span and the discharge wires would not restrict the sway of the wires. One variant of the analysis was executed in a slightly different way than is presented in the standard recommendations. The calculation methods in the standard do not provide an opportunity to take into account how the lashing chains are attached. Therefore, in one variant of the analysis, the formulas presented in the standard will be modified to take into account the two-point attachment of the lashing chains. Below is a list of the parameters to be changed in each analysis variant.

• The value of the initial three-phase short-circuit current;
• The time T_pi;
• The type of short circuit;
• The static tension force of the wire harness;
• The rigidity of the structure;
• The method of attachment of the lashing chains.

3.2. Input Data for Analytical Calculations

Table 1. Calculation of dynamic effects of short-circuit current. Connection made with flexible conductors; input data values.

Input Data
Parameter	Designation	Value	Measuring Unit
Physical constants
Magnetic permeability in a vacuum	${\mathsf{\mu }}_0$	${1.25·10}^{-6}$	[H/m]
Earth acceleration	${\mathrm{g}}_{\mathrm{n}}$	9.81	[${\mathrm{m}/\mathrm{s}}^2$]
Electrical parameters
R/X ratio	R/X	0.07	[Ω]
Frequency of network	f	50	[Hz]
Impact factor	k	1.81	[-]
Initial value of three-phase short circuit current	${\mathrm{I}}_{\mathrm{k}}^{″}$	40	[kA]
Short circuit duration	${\mathrm{T}}_{\mathrm{k}}$	1	[s]
Time to beam force	${\mathrm{T}}_{\mathrm{p}\mathrm{i}}$	50	[ms]
Wire parameters
Unit weight of AFL-8 525 ${\mathrm{m}\mathrm{m}}^2$ cable	${\mathrm{m}}_{\mathrm{s}}$	1.939	[kg/m]
Young’s modulus	E	${62.00·10}^9$	[${\mathrm{N}/\mathrm{m}}^2$]
Outer diameter of the conduit	d	32.2	[mm]
Cross section of aluminum in the conduit	${\mathrm{A}}_{\mathrm{A}\mathrm{l}}$	525.89	[${\mathrm{m}\mathrm{m}}^2$]
Cross section of steel in the conduit	${\mathrm{A}}_{\mathrm{F}\mathrm{e}}$	64.39	[${\mathrm{m}\mathrm{m}}^2$]
Span parameters
Span parameters	S	440$·$103	[N/m]
Static tension of the wire harness	${\mathrm{F}}_{\mathrm{s}\mathrm{t}}$	20	[kN]
Distance between axes of supports	l	67	[m]
Insulator length	${\mathrm{l}}_{\mathrm{i}}$	5.15	[m]
The length of the cable in the span	${\mathrm{l}}_{\mathrm{c}}$	56,7	[m]
Inter-phase distance	a	6	[m]
Number of wires in the harness	${\mathrm{n}}_{\mathrm{s}}$	2	[pcs]
Spacing between wires in the harness	${\mathrm{a}}_{\mathrm{s}}$	0.2	[m]
Number of spacers	${\mathrm{k}}_{\mathrm{s}}$	3	[pcs]
Distance between spacers	${\mathrm{l}}_{\mathrm{s}}$	14.18	[m]
Calculation criteria
Type of short circuit	Three-Phase
Method of fixing support insulators	Single

below shows the input data on the basis of which the first variant of the analysis was carried out for the top rail of the switchyard.

This variant assumes that the span will be 67 m, and the top rail of the switchgear will be made with an AFL-8 525 mm² cable bundle. The tension value assumed in the variant corresponds to some approximation of the actual tension value of a span of such a span in the winter period. It was determined that three spacers would be placed in the span at an equal distance from each other. The resultant stiffness value of the two tall structures forming the span was assumed to be 440⋅10³ N/m. As in the case of connections made with a rigid conductor, the first variant of calculations was carried out for a three-phase short-circuit current, the initial value of which is 40 kA, and the duration of the short circuit is 1 s.

3.3. Analytical Calculations According to Normative Recommendations

As in the case of connections made with a rigid conductor, the first step that begins the analysis is to determine the electrodynamic force acting per unit length of the conductor. Since there are no discharge wires in the span and the current flows through the entire length of the span, the resulting electrodynamic force is described by the dependency already mentioned in the third section:

$${\mathrm{F}}^{\prime }=\frac{{\mathsf{\mu }}_0}{2\mathsf{\pi }}·0.75·\frac{{\left({\mathrm{I}}_{\mathrm{K}3}^{″}\right)}^2}{\mathrm{a}}·\frac{{\mathrm{l}}_{\mathrm{c}}}{\mathrm{l}}=\frac{4\mathsf{\pi }·{10}^{-7}}{2\mathsf{\pi }}·0.75·\frac{{\mathrm{40,000}}^2}{6.00}·\frac{56.70}{67.00}=33.85074 [\frac{\mathrm{N}}{\mathrm{m}}]$$

(8)

where the length of the cable l_c is determined from the following formula:

$${\mathrm{l}}_{\mathrm{c}}=\mathrm{l}-2·{\mathrm{l}}_{\mathrm{i}}=67.00-2\cdot 5.15=56.70 [\mathrm{m}]$$

(9)

Knowing the value of the electrodynamic force acting on the wires allows determining the value of the coefficient r, which determines the ratio of electromechanical force on a short-circuiting conductor to the gravitational force:

$$\mathrm{r}=\frac{{\mathrm{F}}^{\prime }}{\mathrm{n}·{\mathrm{m}}_{\mathrm{s}}^{\prime }·\mathrm{g}}=\frac{33.85}{2·1.939·9.81}=0.8977 [-]$$

(10)

Then, based on the coefficient r, we determine the value of the directional angle of the force:

$${\mathsf{\delta }}_1=\arctan \left(\mathrm{r}\right)=\arctan \left(0.8977\right)=0.7315\left[\mathrm{r}\mathrm{a}\mathrm{d}\right]=41.91 [°]$$

(11)

In the next step, the value of static cable sag was determined in the center of the span. This value is the basis for determining in further steps the dynamic sag of the cable resulting from the flow of short-circuit current and, consequently, is also the basis for determining whether the minimum phase-to-phase distances will be maintained as a result of the movement of the wires during a short circuit:

$${\mathrm{f}}_{\mathrm{e}\mathrm{s}}=\frac{\mathrm{n}·{\mathrm{m}}_{\mathrm{s}}^{\prime }·\mathrm{g}·{\mathrm{l}}^2}{8·{\mathrm{F}}_{\mathrm{s}\mathrm{t}}}=\frac{2·1.939·9.81·{67.00}^2}{8·\mathrm{20,000}}=1.0673 [\mathrm{m}]$$

(12)

In the proceeding steps, the parameters necessary for the final results are determined. The free oscillation period of the cable, under normal operating conditions, is given by the expression:

$$\mathrm{T}=2\mathsf{\pi }·\sqrt{0.8·\frac{{\mathrm{f}}_{\mathrm{e}\mathrm{s}}}{\mathrm{g}}}=2\mathsf{\pi }·\sqrt{0.8·\frac{1.07}{9.81}}=1.85 [\mathrm{s}]$$

(13)

The random period of oscillation of the wire during a short circuit can be calculated as follows:

$${\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}=\frac{\mathrm{T}}{\sqrt[4]{1+{\mathrm{r}}^2}·\left[1-\frac{{\mathsf{\pi }}^2}{64}\cdot {\left[\frac{{\mathsf{\delta }}_1}{90°}\right]}^2\right]}=\frac{1.85}{\sqrt[4]{1+{0.8977}^2}·\left[1-\frac{{\mathsf{\pi }}^2}{64}\cdot {\left[\frac{41.91°}{90°}\right]}^2\right]}=1.651 [\mathrm{s}]$$

(14)

Additionally, the actual value of Young’s modulus is given by the following formula:

$${\mathrm{E}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\left\{\begin{array}{cc}\mathrm{E}\cdot \left[0.3+0.7·\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{{\mathrm{F}}_{\mathrm{s}\mathrm{t}}}{{\mathrm{n}·\mathrm{A}}_{\mathrm{s}}·{\mathsf{\sigma }}_{\mathrm{f}\mathrm{i}\mathrm{n}}}·90°\right)\right]& \mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{F}}_{\mathrm{s}\mathrm{t}}}{\mathrm{n}\cdot {\mathrm{A}}_{\mathrm{s}}}\le {\mathsf{\sigma }}_{\mathrm{f}\mathrm{i}\mathrm{n}}\\ \mathrm{E}& \mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{F}}_{\mathrm{s}\mathrm{t}}}{\mathrm{n}\cdot {\mathrm{A}}_{\mathrm{s}}}>{\mathsf{\sigma }}_{\mathrm{f}\mathrm{i}\mathrm{n}}\end{array}\right.$$

(15)

where σ_fin is the smallest value of stress at which Young’s modulus becomes constant; according to the standard, it is 50⋅10⁶ [N/m²]. The cross section of the cable is the sum of the aluminum and steel parts of the cable and can be expressed as follows:

$${\mathrm{A}}_{\mathrm{s}}={\mathrm{A}}_{\mathrm{a}\mathrm{l}}+{\mathrm{A}}_{\mathrm{s}\mathrm{t}}=\left(525.89\cdot {10}^{-6}\right)+\left(64.39\cdot {10}^{-6}\right)=590.28\cdot {10}^{-6}\left[{\mathrm{m}}^2\right]$$

(16)

To determine the actual value of Young’s modulus, in this case, the following expression is used:

$$\begin{array}{c}{\mathrm{E}}_{\mathrm{e}\mathrm{f}\mathrm{f}}=\mathrm{E}\cdot \left[0.3+0.7·\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{{\mathrm{F}}_{\mathrm{s}\mathrm{t}}}{{\mathrm{n}·\mathrm{A}}_{\mathrm{s}}·{\mathsf{\sigma }}_{\mathrm{f}\mathrm{i}\mathrm{n}}}·90°\right)\right]=\\ =62.00\cdot {10}^9\cdot \left[0.3+0.7·\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\mathrm{20,000}}{2\cdot 590.28\cdot {10}^{-6}·50.00\cdot {10}^6}·90°\right)\right]=40.623{\cdot 10}^9\left[\frac{\mathrm{N}}{{\mathrm{m}}^2}\right]\end{array}$$

(17)

The correctly selected value of the actual Young’s modulus and the properly adopted value of the stiffness of the structure affect the result of the stiffness coefficient. This coefficient, on the other hand, affects the values of the determined forces; the smaller its value, the higher the determined force values. The issue of adopting an appropriate value for the stiffness of the structure has already been addressed. It is possible to determine the stiffness coefficient N from the following relationship:

$$\mathrm{N}=\frac{1}{\mathrm{S}·\mathrm{l}}+\frac{1}{\mathrm{n}·{\mathrm{E}}_{\mathrm{e}\mathrm{f}\mathrm{f}}·{\mathrm{A}}_{\mathrm{s}}}=\frac{1}{440·{10}^3·67.00}+\frac{1}{2·40.623{\cdot 10}^9·590.28\cdot {10}^{-6}}=54.77\cdot {10}^{-9} [\frac{1}{\mathrm{N}}]$$

(18)

In the next step, the so-called cable stress intensity factor ξ is determined.

$$\mathsf{\xi }=\frac{{\left(\mathrm{n}·\mathrm{g}·{\mathrm{m}}_{\mathrm{s}}^{\prime }·\mathrm{l}\right)}^2}{24·{\mathrm{F}}_{\mathrm{s}\mathrm{t}}^3·\mathrm{N}}=\frac{{\left(2·9.81·1.939·67.00\right)}^2}{24·{\mathrm{20,000}}^3·54.77\cdot {10}^{-9}}=0.6178 [-]$$

(19)

In all variants of the analysis, the adopted short circuit duration would be 1 s. This was to ensure that the determined force values would be maximum values. Accordingly, the time of 1 s was assumed to be the time until the first tripping of the circuit breaker; in the standard, this time is denoted as T_k1. However, in the calculation procedure outlined in the standard, there is an assumption that if the time T_k1 is unknown or its value is greater than the value of 0.4⋅T, as in our case, then in further calculations, the value of 0.4⋅T should be taken as the duration of the short circuit to the first tripping of the circuit breaker. This means that the wire already reaches its maximum deflection after a time of 0.4⋅T.

$${\mathrm{T}}_{\mathrm{k}1}=0.4\cdot \mathrm{T}=0.4\cdot 1.85=0.74 [\mathrm{s}]$$

(20)

$$\frac{{\mathrm{T}}_{\mathrm{k}1}}{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}=\frac{0.74}{1.66}=0.45 [\mathrm{s}]$$

(21)

$${\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}=\left\{\begin{array}{cc}{\mathsf{\delta }}_1\cdot \left[1-\cos \left(360°\cdot \frac{{\mathrm{T}}_{\mathrm{k}1}}{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}\right)\right]& \mathrm{f}\mathrm{o}\mathrm{r}0\le \frac{{\mathrm{T}}_{\mathrm{k}1}}{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}\le 0.5\\ 2{\mathsf{\delta }}_1& \mathrm{f}\mathrm{o}\mathrm{r}\frac{{\mathrm{T}}_{\mathrm{k}1}}{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}>0.5\end{array}\right.$$

(22)

The value of the wire angle at the end of the short circuit is determined depending on the ratio of time T_k1 to time T_res. In this case, the ratio is 0.45; it is possible to use the dependency (22) to determine the end angle.

$${\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}={\mathsf{\delta }}_1\cdot \left[1-\cos \left(360°\cdot \frac{{\mathrm{T}}_{\mathrm{k}1}}{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}\right)\right]=41.91\cdot \left[1-\cos \left(360°\cdot \frac{0.74}{1.66}\right)\right]=81.41\left[°\right]$$

(23)

$${\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}=1.4209 [\mathrm{r}\mathrm{a}\mathrm{d}]$$

(24)

The maximum angle of deflection of conductors that can occur during or after a short circuit is determined as follows:

$$\mathsf{\chi }=\left\{\begin{array}{cc}1-\mathrm{r}\cdot \sin {\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}& \mathrm{f}\mathrm{o}\mathrm{r} 0\le {\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}\le 90°\\ 1-\mathrm{r}& \mathrm{f}\mathrm{o}\mathrm{r} {\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}>90°\end{array}\right.$$

(25)

$${\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\left\{\begin{array}{cc}1.25\cdot \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\chi }& \mathrm{f}\mathrm{o}\mathrm{r} 0.766\le \mathsf{\chi }\le 1\\ 10°+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\chi }& \mathrm{f}\mathrm{o}\mathrm{r} -0.985\le \mathsf{\chi }<0.766\\ 180°& \mathrm{f}\mathrm{o}\mathrm{r} \mathsf{\chi }<-0.985\end{array}\right.$$

(26)

Due to the fact that the ${\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}$ value is less than 90°:

$$\mathsf{\chi }=1-\mathrm{r}\cdot \sin {\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}=1-0.8977\cdot \sin \left(1.4209\right)=0.11236 [-]$$

(27)

$${\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}={10}^{°}+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\chi }={10}^{°}+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}0.11236=93.55\left[°\right]=1.6328 [\mathrm{r}\mathrm{a}\mathrm{d}]$$

(28)

The angle of deflection of the conductor at the end of the short circuit allows it to be known if it is possible to determine the value of the force F_t,d associated with the deflection of the conductors. The calculation of said force begins with the determination of the load parameter φ and the coefficient ψ. The value of the load parameter depends on the coefficient r or on the value of the angle on the deflection of the conductors at the end of the short circuit, depending on the times T_k1 and T_res.

$$\mathsf{\phi }=\left\{\begin{array}{cc}3·\left(\sqrt{1+{\mathrm{r}}^2}-1\right)& \mathrm{f}\mathrm{o}\mathrm{r} {\mathrm{T}}_{\mathrm{k}1}\ge \frac{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}{4}\\ 3·\left(\mathrm{r}\cdot \sin {\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}+\cos {\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}-1\right)& \mathrm{f}\mathrm{o}\mathrm{r} {\mathrm{T}}_{\mathrm{k}1}<\frac{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}{4}\end{array}\right.$$

(29)

$$\mathsf{\phi }=3·\left(\sqrt{1+{\mathrm{r}}^2}-1\right)=3·\left(\sqrt{1+{0.8977}^2}-1\right)=1.0315\left[-\right]$$

(30)

It is recommended to derive the coefficient of ψ in two ways. The first way is to read the value of the coefficient from the graph in Figure 9 below.

The ψ coefficient takes values from 0 to 1 stress intensity of the cable (Figure 9). Determining the value of the ψ coefficient from the graph each time would be inefficient, and when performing a number of analyses, a situation could arise in which a change in the value of this parameter was omitted, which would lead to incorrect results. In addition, reading the value from the graph will never achieve sufficiently high accuracy. Therefore, in this analysis, the value of the coefficient was determined based on the equation shown below:

$${\mathsf{\phi }}^2{\mathsf{\psi }}^3+\mathsf{\phi }\left(2+\mathsf{\xi }\right){\mathsf{\psi }}^2+\left(1+2\mathsf{\xi }\right)\mathsf{\psi }-\mathsf{\xi }\left(2+\mathsf{\phi }\right)=0$$

(31)

$${\mathrm{F}}_{\mathrm{t},\mathrm{d}}={\mathrm{F}}_{\mathrm{s}\mathrm{t}}·\left(1+\mathsf{\phi }·\mathsf{\psi }\right)=\mathrm{20,000}·\left(1+1.0315·0.49069\right)=\mathrm{30,122.94}\left[\mathrm{N}\right]$$

(32)

One of the three values of the forces due to the mechanical action of the short-circuit current on the connections made by the bundle of flexible wires is shown in Formula (32). Its value is over 30 kN and thus is 10 kN higher than the assumed static tension of the wires under normal operating conditions. In the following steps, the other two forces were determined, and the condition of maintaining the minimum interfacial distance was checked. The second type of force is related to the descent of the cable just after the short circuit is switched off. According to IEC 60865-1:2012, the force F_f,d should be taken into account only if the following is true:

• The value of r is greater than 0.6;
• The angle δ_max is greater than or equal to 70°;
• The determined value of the angle δ in the span, in which there are discharge wires, is greater than 60°—this does not apply to the mentioned case.

The standard does not allow for the determination of the force F_f,d for very short spans, the length of which does not exceed 100 times the diameter of a single cable. In the analyzed variant, all the above conditions are met.

$$\begin{array}{c}{\mathrm{F}}_{\mathrm{f},\mathrm{d}}=\mathrm{1,2}\cdot {\mathrm{F}}_{\mathrm{s}\mathrm{t}}·\sqrt{1+\left(8\cdot \mathsf{\xi }\cdot \frac{{\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{180}\right)}=\\ =1.2\cdot \mathrm{20,000}·\sqrt{1+\left(8\cdot 0.6178\cdot \frac{93.55}{180}\right)}=\mathrm{45,338.25} [\mathrm{N}]\end{array}$$

(33)

The determined value of the force created by the cable’s fall is just over 45 kN. This is more than twice the value of the initial tension of the wires. The last force to be determined as part of the analysis of the mechanical effects of the short-circuit current is the bundle force, denoted as F_pi. The bundle force very quickly reaches its maximum value, which can be up to six times greater than the tension of the conductors under normal operating conditions. In the standard, a relation is presented that allows iteratively determining the time T_pi, after which the maximums of the bundle force occur. Nevertheless, these maxima do not occur for the entire duration of the disturbance because the bundle force varies over time. This is perfectly illustrated in Figure 4. provided in the third section. Hence, the value of time T_pi adopted should be relatively larger than the value at which the beam strength maximum occurs. In the present analysis, a value of 50 ms was adopted as the T_pi time. The following is the procedure for determining the beam force with the set time T_pi. The first thing to be performed is to verify that the configuration of the wiring harness will allow the harness wires to collide effectively. It can be considered that the wires in the bundle will collide effectively if the distance between the bundle wires and the distance between the spacers l_s meet the following conditions:

$$\begin{array}{ccc}\frac{{\mathrm{a}}_{\mathrm{s}}}{\mathrm{d}}\le 2& \mathrm{a}\mathrm{n}\mathrm{d}& {\mathrm{l}}_{\mathrm{s}}\ge 50\cdot \end{array}{\mathrm{a}}_{\mathrm{s}}$$

(34)

$$\begin{array}{ccc}\frac{{\mathrm{a}}_{\mathrm{s}}}{\mathrm{d}}\le 2.5& \mathrm{o}\mathrm{r}\mathrm{a}\mathrm{z}& {\mathrm{l}}_{\mathrm{s}}\ge 70\cdot \end{array}{\mathrm{a}}_{\mathrm{s}}$$

(35)

In the analyzed span, these conditions are not met, so the beam force is calculated according to the following guidelines. Calculations are initiated by determining the coefficients v₁, v₂ and v₃:

$$\begin{array}{c}{\mathsf{\nu }}_1=\mathrm{f}·\frac{1}{\mathrm{s}\mathrm{i}\mathrm{n}\frac{180°}{\mathrm{n}}}\sqrt{\frac{{(\mathrm{a}}_{\mathrm{s}}-\mathrm{d})·{\mathrm{m}}_{\mathrm{s}}^{\prime }}{\frac{{\mathrm{\mu }}_{\mathrm{o}}}{2\mathsf{\pi }}·{\left(\frac{{\mathrm{I}}_{\mathrm{K}}^{″}}{\mathrm{n}}\right)}^2·\frac{\left(\mathrm{n}-1\right)}{{\mathrm{a}}_{\mathrm{s}}}}}=\\ =50·\frac{1}{\mathrm{s}\mathrm{i}\mathrm{n}\frac{180°}{2}}\sqrt{\frac{\left(0.20-32.20\cdot {10}^{-3}\right)·1.939}{\frac{4\mathsf{\pi }·{10}^{-7}}{2·\mathsf{\pi }}·{\left(\frac{\mathrm{40,000}}{2}\right)}^2·\frac{\left(2-1\right)}{0.20}}}=1.43\left[-\right]\end{array}$$

(36)

$${\mathsf{\nu }}_2={\left(\frac{{\mathsf{\nu }}_1}{\mathrm{f}·{\mathrm{T}}_{\mathrm{p}\mathrm{i}}}\right)}^2={\left(\frac{1.43}{50·0.050}\right)}^2=0.33 [-]$$

(37)

$$\begin{array}{c}{\mathsf{\nu }}_3=\frac{\frac{\mathrm{d}}{{\mathrm{a}}_{\mathrm{s}}}}{\mathrm{s}\mathrm{i}\mathrm{n}\frac{180°}{\mathrm{n}}}\cdot \frac{\sqrt{\frac{{\mathrm{a}}_{\mathrm{s}}}{\mathrm{d}}-1}}{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\sqrt{\frac{{\mathrm{a}}_{\mathrm{s}}}{\mathrm{d}}-1}}=\\ =\frac{\frac{32.20\cdot {10}^{-3}}{0.20}}{\mathrm{s}\mathrm{i}\mathrm{n}\frac{180°}{2}}\cdot \frac{\sqrt{\frac{0.20}{32.20\cdot {10}^{-3}}-1}}{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\sqrt{\frac{0.20}{32.20\cdot {10}^{-3}}-1}}=0.32 [-]\end{array}$$

(38)

In the next step, the values of two deformation coefficients of the wire bundle related to the tension force of the wires and the force occurring between the wires in the bundle are determined according to the expression

$${\mathsf{\epsilon }}_{\mathrm{p}\mathrm{i}}=0.375·\mathrm{n}·\frac{{{\mathrm{F}}_{\mathsf{\nu }}·\mathrm{l}}_{\mathrm{s}}^3·\mathrm{N}}{{\left({\mathrm{a}}_{\mathrm{s}}-\mathrm{d}\right)}^3}{\left(\mathrm{s}\mathrm{i}\mathrm{n}\frac{180°}{\mathrm{n}}\right)}^3=$$

$$=0.375·2·\frac{5814.1·{14.18}^3\cdot 54.77\cdot {10}^{-9}}{{\left(0.20-32.20\cdot {10}^{-3}\right)}^3}·{\left(\mathrm{s}\mathrm{i}\mathrm{n}\frac{180°}{2}\right)}^3=144.07 [-]$$

(39)

Based on the above coefficients, another coefficient, j, is determined by using the Formula (40). If its value is greater than or equal to 1, it means that the wires in the bundle will collide. Otherwise, the wires in the bundle will not collide.

$$\mathrm{j}=\sqrt{\frac{{\mathsf{\epsilon }}_{\mathrm{p}\mathrm{i}}}{1+{\mathsf{\epsilon }}_{\mathrm{s}\mathrm{t}}}}=\sqrt{\frac{144.07}{1+11.73}}=3.3641 [-]$$

(40)

Knowing the value of the j factor, one can proceed to determine the beam force from the corresponding formula, which is given below:

$${\mathrm{F}}_{\mathrm{p}\mathrm{i},\mathrm{d}}={\mathrm{F}}_{\mathrm{s}\mathrm{t}}·\left(1+\frac{{\mathsf{\nu }}_{\mathrm{e}}}{{\mathsf{\epsilon }}_{\mathrm{s}\mathrm{t}}}·\mathsf{\xi }\right)$$

(41)

The missing elements of the above formula are the ν_e coefficient and the ζ coefficient. The ζ coefficient, like the ψ coefficient, can be determined in two ways: by reading the value from a graph or by solving a third-degree equation. In any case, as in the previous case, the value of the ζ coefficient was determined from the equation shown below, which was solved using the script:

$${\mathsf{\zeta }}^3+{\mathsf{\epsilon }}_{\mathrm{s}\mathrm{t}}\cdot {\mathsf{\zeta }}^2-{\mathrm{j}}^2\cdot \left(1+{\mathsf{\epsilon }}_{\mathrm{s}\mathrm{t}}\right)=0$$

(42)

$${\mathsf{\nu }}_{\mathrm{e}}=2.05 [-]$$

(43)

By knowing all of the above parameters, it is possible to determine the final value of the force associated with the effect of the short-circuit current on the top rail of the switchgear:

$${\mathrm{F}}_{\mathrm{p}\mathrm{i},\mathrm{d}}={\mathrm{F}}_{\mathrm{s}\mathrm{t}}·\left(1+\frac{{\mathsf{\nu }}_{\mathrm{e}}}{{\mathsf{\epsilon }}_{\mathrm{s}\mathrm{t}}}·\mathsf{\xi }\right)=\mathrm{20,000}·\left(1+\frac{2.0544}{11.7336}·3.11514\right)=\mathrm{30,908.41} [\mathrm{N}]$$

(44)

All three force values have already been determined; however, an equally important point to be realized during the analysis is to verify that the phase-to-phase distances present in the normal operating state will allow the minimum phase-to-phase distances to be maintained during a short circuit. The analysis should take into account the fact that during the flow of the short-circuit current, there will be an elongation of the cable. Therefore, in order to determine the maximum horizontal displacement of the conductors, it is necessary to determine the values of the coefficients that account for the elastic elongation of the conductors and the thermal elongation, and also take into account the increase in the overhang of the conductors. Coefficients responsible for the elastic and thermal elongation of cables can be written down as dependency:

$${\mathsf{\epsilon }}_{\mathrm{e}\mathrm{l}\mathrm{a}}=\mathrm{N}·\left({\mathrm{F}}_{\mathrm{t},\mathrm{d}}-{\mathrm{F}}_{\mathrm{s}\mathrm{t}}\right)=54.77\cdot {10}^{-9}·\left(\mathrm{30,122.94}-\mathrm{20,000}\right)=0.55443\cdot {10}^{-3} [-]$$

(45)

$${\mathsf{\epsilon }}_{\mathrm{t}\mathrm{h}}=\left\{\begin{array}{cc}{\mathrm{C}}_{\mathrm{t}\mathrm{h}}·{\left(\frac{{\mathrm{I}}_{\mathrm{k}}^{″}}{{\mathrm{n}\cdot \mathrm{A}}_{\mathrm{s}}}\right)}^2·\frac{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}{4}& \mathrm{f}\mathrm{o}\mathrm{r} {\mathrm{T}}_{\mathrm{k}1}\ge \frac{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}{4}\\ {\mathrm{C}}_{\mathrm{t}\mathrm{h}}·{\left(\frac{{\mathrm{I}}_{\mathrm{k}}^{″}}{{\mathrm{n}\cdot \mathrm{A}}_{\mathrm{s}}}\right)}^2·{\mathrm{T}}_{\mathrm{k}1}& \mathrm{f}\mathrm{o}\mathrm{r} {\mathrm{T}}_{\mathrm{k}1}<\frac{{\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}}{4}\end{array}\right.$$

(46)

The value of the c_th coefficient is selected depending on the conductor material. The standard distinguishes three values of this coefficient depending on whether the conductor is copper or aluminum–steel. For aluminum–steel conductors, the value of the coefficient also depends on the ratio of aluminum to steel. In the analyzed variant, the top rail is made with a bundle of AFL-8 525 mm² conductors, whose ratio of aluminum to steel is greater than 6. Therefore, the value of 0.27⋅10⁻¹⁸ m⁴/(A²⋅s) is taken as the c_th coefficient. Since the time T_k1 is greater than the time T_res divided by 4, it is possible to determine the value of the coefficient ε_tn as follows:

$$\begin{array}{c}{\mathrm{C}}_{\mathrm{D}}=\sqrt{1+\frac{3}{8}\cdot {\left(\frac{\mathrm{l}}{{\mathrm{f}}_{\mathrm{e}\mathrm{s}}}\right)}^2\cdot \left({\mathsf{\epsilon }}_{\mathrm{e}\mathrm{l}\mathrm{a}}+{\mathsf{\epsilon }}_{\mathrm{t}\mathrm{h}}\right)}=\\ =\sqrt{1+\frac{3}{8}\cdot {\left(\frac{67.00}{1.0673}\right)}^2\cdot \left(0.55443\cdot {10}^{-3}+0.1279\cdot {10}^{-3}\right)}=1.4171 [-]\end{array}$$

(47)

The coefficient, taking into account the change in wire sag under the influence of wire deformation during a short circuit, depends on the coefficient r determined at the beginning:

$${\mathrm{C}}_{\mathrm{F}}=\left\{\begin{array}{cc}1.05& \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{r}\le 0.8\\ 0.97+0.1\cdot \mathrm{r}& \mathrm{f}\mathrm{o}\mathrm{r} 0.8<r<1.8\\ 1.15& \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{r}\ge 1.8\end{array}\right.$$

(48)

$${\mathrm{C}}_{\mathrm{F}}=0.97+0.1·\mathrm{r}=0.97+0.1·0.8977=1.05977 [-]$$

(49)

It is possible to determine the value of the increased overhang as follows:

$${\mathrm{f}}_{\mathrm{e}\mathrm{d}}={\mathrm{C}}_{\mathrm{F}}·{\mathrm{C}}_{\mathrm{D}}·{\mathrm{f}}_{\mathrm{e}\mathrm{s}}=1.05977\cdot 1.4171·1.0673=1.6029 [\mathrm{m}]$$

(50)

The sag value increased by more than 50 cm, more exactly 53 cm; this is a significant difference. The last step to be taken is to determine whether, as a result of the increase in the sag of the wires, the minimum phase-to-phase distances will be maintained. The value of the maximum horizontal displacement of the conductors for spans in which there are no discharge conductors is determined from the value of the directional angle of force and the maximum angle of deflection of the conductor. It was also possible to derive the value of the minimum spacing to be maintained (a_min):

$${\mathrm{b}}_{\mathrm{h}}=\left\{\begin{array}{cc}{\mathrm{f}}_{\mathrm{e}\mathrm{d}}·\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\delta }}_1& \mathrm{d}\mathrm{l}\mathrm{a} {\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\ge {\mathsf{\delta }}_1\\ {\mathrm{f}}_{\mathrm{e}\mathrm{d}}·\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}& \mathrm{d}\mathrm{l}\mathrm{a} {\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}<{\mathsf{\delta }}_1\end{array}\right.$$

(51)

$${\mathrm{b}}_{\mathrm{h}}={\mathrm{f}}_{\mathrm{e}\mathrm{d}}·\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\delta }}_1=1.6029\cdot \sin \left(41.91°\right)=1.07068 [\mathrm{m}]$$

(52)

$${\mathrm{a}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{a}-\left(2{·\mathrm{b}}_{\mathrm{h}}\right)=6.00-\left(2\cdot 1.07068\right)=3.85864 [\mathrm{m}]$$

(53)

According to IEC 61936-1:2011, the minimum phase-to-phase distance for 400 kV is 3.60 m, so for the given short-circuit conditions and for a given span length, the minimum phase-to-phase distance was maintained.

3.4. PRIMTECH 3D Program Numerical Calculations

As in the case of rigid connections, this subsection presents the results of calculations performed for an identical connection in PRIMTECH 3D software. In the case of the analysis for rail made with rigid conductors, the results obtained are similar to those obtained in the analysis based on the calculation procedure contained in the standard. The results are graphically presented in Figure 10. The occurrence of slight differences in the determined value of the electromagnetic force acting per unit length of the cable entails minimal discrepancies in the values of the other parameters. It is worth noting the values of the parameters responsible for the value of the bundle force F_pi; the differences here are significant, making the value of the bundle force about 10 kN higher than the value determined in accordance with the standard. The advantage of PRIMTECH 3D software is that, in addition to generating calculation results, the program allows the user to graphically represent the movement of the wires. This makes it perfectly clear whether the minimum phase-to-phase distances will be maintained and whether there will be a collision between conductors of adjacent phases. Below are graphics showing the movement of the range of conductor motion for this analysis variant, and for an example variant in which collision of conductors of adjacent phases would occur. The results of the numerical calculations are presented in

Table 2. Calculation of dynamic effects of short-circuit current; connection made with flexible conductors. Variant I. Results of calculations by the PRIMTECH 3D program.

Calculation Results
Parameter	Designation	Value	Measuring Unit
Intermediate results
Unit electromagnetic force	${\mathrm{F}}^{\prime }$	34	[N/m]
Ratio of electromagnetic force to gravitational force	r	0.89	[-]
Angular direction of force	${\mathsf{\delta }}_1$	41.8	$[°]$
Static cable sag in the middle of the span	${\mathrm{f}}_{\mathrm{e}\mathrm{s}}$	1.06	[m]
Free vibration period of the wire	T	1.85	[s]
Period of vibration of the wire during a short circuit	${\mathrm{T}}_{\mathrm{r}\mathrm{e}\mathrm{s}}$	1.65	[s]
Short circuit duration used in calculations	${\mathrm{T}}_{\mathrm{k}\mathrm{l}}$	0.74	[s]
Actual Young’s modulus	${\mathrm{E}}_{\mathrm{e}\mathrm{f}\mathrm{f}}$	${40.62·10}^9$	$[{\mathrm{N}/\mathrm{m}}^2$]
Norm stiffness	N	${54.92·10}^{-9}$	[1/N]
Tension coefficient of the cable	$\zeta$	0.61	[-]
Coefficient Χ	Χ	0.12	[-]
Coefficient Φ	Φ	1.02	[-]
Coefficient Ψ	Ψ	0.49	[-]
Elastic expansion	${\mathsf{\epsilon }}_{\mathrm{e}\mathrm{l}\mathrm{a}}$	${0.55·10}^{-3}$	[-]
Dilatation factor	CD	1.42	[-]
Form factor	CF	1.06	[-]
Deformation coefficients of conductors related to force ${\mathrm{F}}_{\mathrm{s}\mathrm{t}}$$\mathrm{and} {\mathrm{F}}_{\mathsf{\gamma }}$	${\mathsf{\epsilon }}_{\mathrm{s}\mathrm{t}}{\mathsf{\epsilon }}_{\mathrm{p}\mathrm{i}}$	11.76 1144.52	[-] [-]
Parameter that determines the configuration of the wire harness	j	9.47	[-]
Coefficient ζ	ζ	7.67	[-]
Force between bundle wires	${\mathrm{F}}_{\gamma }$	46,089.65	[N]
$\mathrm{Coefficient} {\mathsf{\gamma }}_1$	${\mathsf{\gamma }}_1$	1.43	[-]
$\mathrm{Coefficient} {\mathsf{\gamma }}_2$	${\mathsf{\gamma }}_2$	2.58	[-]
$\mathrm{Coefficient} {\mathsf{\gamma }}_3$	${\mathsf{\gamma }}_3$	0.32	[-]
$\mathrm{Coefficient} {\mathsf{\gamma }}_4$	${\mathsf{\gamma }}_4$	5.21	[-]
$\mathrm{Coefficient} {\mathsf{\gamma }}_e$	${\mathsf{\gamma }}_e$	1.58	[-]
Final results
Final angle of cable swing	${\mathsf{\delta }}_{\mathrm{e}\mathrm{n}\mathrm{d}}$	81.38	$[°]$
Maximum angle of cable deflection	${\mathsf{\delta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$	93.34	$[°]$
Dynamic cable overhang in the middle of the span	${\mathrm{f}}_{\mathrm{e}\mathrm{d}}$	1.59	[m]
Maximum horizontal displacement of the cable	${\mathrm{b}}_{\mathrm{h}}$	1.06	[m]
Minimum insulation distance	${\mathrm{a}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$	3.87	[m]
The force due to deflection of the cable	${\mathrm{F}}_{\mathrm{t},\mathrm{d}}$	30,008.56	[N]
The force caused by the descent of the cable	${\mathrm{F}}_{\mathrm{f},\mathrm{d}}$	45,102.48	[N]
The force of the pinch effect	${\mathrm{F}}_{\mathrm{p}\mathrm{i}}$	40,622.87	[N]

below.

4. Analysis Summary

The analyses of the mechanical effects and thermal effects of the short-circuit current were accomplished as part of this work, which was to determine the effect of the value of the short-circuit current on the operation of the extra-high voltage substation. The analyses of the mechanical effects of short-circuit current contained in the fourth section were performed in two ways, both for the rail made with rigid conductors and for the rail made with flexible conductors. All analyses were performed in several variants, allowing us to see the influence of individual parameters on the calculation results. For both types of rail, six variants of calculations were performed for each. Below is a list of the individual criteria for performing calculations and the parameters that changed, along with the values adopted.

• The value of the initial three-phase short-circuit current: 40 kA, 50 kA or 63 kA;
• The type of short circuit: three-phase or two-phase;
• The type of conductor pipe: AR120 × 12 or AR250 × 8;
• Consideration of the presence of protection automation: YES or NO;
• Consideration of natural frequency of conductors: YES or NO;
• The method of attachment of conductors: FLEXIBLE or COMBINED.

Below is a similar list of conditions for the pipe rail:

• The value of the initial three-phase short-circuit current: 40 kA, 50 kA or 63 kA;
• The Tpi time: 10 ms, 12 ms or 50 ms;
• The type of short circuit: three-phase or two-phase;
• The static tension force of the wire harness: 10 kN or 20 kN;
• The construction stiffness: 440⋅103 N/m, 880⋅103 N/m or 1320⋅103 N/m;
• The attachment method of the lashing chains: SINGLE-POINT or TWO-POINT.

Considering the results of the analyses of the dynamic effects of the short-circuit current, it is worth noting that the calculation results obtained by two independent routes are almost identical. The values of individual parameters are overwhelmingly equal to each other. Any discrepancies are negligible. In some cases, the obtained values differ from each other by a few hundredths or a few tenths. The differences that occur mean that the final calculation results representing the values of the individual forces generated by the flow of short-circuit current also differ from each other by a few hundredths or a few tens of newtons. Nevertheless, in the case of forces with values of tens of kilo newtons, the difference at such a level is negligibly small. The resulting discrepancies in the obtained analysis results are related to the input data adopted by the PRIMTECH 3D program, over which the user has no influence. For example, the value of the R/X parameter for in-system stations adopted by the program is slightly higher than the value adopted for calculations according to normative recommendations. This causes the value of the F_m force determined with the software to be smaller, and consequently, the determined value of the dynamic force acting on the support insulators is also smaller than the values obtained from the analyses accomplished according to the normative recommendations. The case is different about the parameters to determine the value of the beam forces. In the case of these parameters, the discrepancies that occur are significant, and the determined values of forces differ from each other by several kilo newtons. This is due to the fact that the PRIMTECH 3D program determines the maximum value of the beam force and therefore assumes a much shorter T_pi time. This is confirmed by the fifth and sixth variants of the analysis, in which a five-times smaller value of the T_pi time is assumed, as a result of which the beam force values determined by the two ways are close to each other. Determining the maximum value of the beam force may be superfluous, as the structures will be oversized. On the other hand, selecting a T_pi time that is too short, and thus determining a smaller value of beam force, may mean that the designed structures will be too weak.

Analyzing the different variants of the calculations performed for rigid rails, it is easy to see that the value of the resulting dynamic forces acting on the support insulators is greatly influenced not only by the value of the initial three-phase short-circuit current, but also by the criteria adopted for the calculations. In both variants, the presence of automatic reclosing was taken into account, as well as that the short circuit that occurs will be a three-phase short circuit. In the second variant, it was assumed that the initial value of the three-phase short-circuit current would be 50 kA and that the natural frequency of the conductors would not be included in the calculations. In the other variant, the frequency of conductor vibration was taken into account, and a value of 63 kA was assumed as the short-circuit current. The difference between the determined values of dynamic forces is about 1.5 kN with an increase in short-circuit current of 13 kA. This comparison shows directly how important it is to include the f_cm frequency in the calculations. Equally important in the calculations is to take into account the presence of automatic reclosing. This can be shown by comparing the second, third, and fourth variants with each other. In all three variants, identical short-circuit conditions were assumed. The different input configurations differ only in that they alternately take into account or not the presence of automatic reclosing and the f_cm frequency. The compilation of the results for all three variants illustrates how large an error can arise if the analysis for the actual switchgear is accomplished, taking into account the f_cm frequency, but without taking into account the presence of reclosing automation. The results obtained in the fourth variant are more than three times smaller than the others. This is because, in the absence of the presence of automatic reclosing, the stresses created during the short circuit will have time to disappear, as there will be no second flow of short circuit current.

The third and final revised criterion referred to the method of attaching the conductors to the support insulators. The last variant, unlike the others, was accomplished for the case in which flexible attachment of the tubular conductors on both sides was taken into account. When analyzing the presented results, it should be noted that both support insulators are equally loaded. In the case of fixing the conductors in a combined manner, the load on the insulators was not evenly distributed. The issue of how the conductors are attached is of great importance in the analyses performed for connections between apparatuses, which are characterized by much lower strengths against dynamic forces.

It is worth noting that for the correct selection of rigid conductors and the correct selection of support insulators, which will be characterized by the appropriate force in short-circuit calculations, several additional issues must be taken into account. We can enumerate, among others, the atmospheric conditions, the presence of vibration-damping conductors that are placed inside the conductors and the like. Since short-circuit analyses should be performed for extreme cases, it would be necessary to assume that the disturbance will arise, for example, in winter, when the conductors are additionally loaded with a layer of ice or snow. A sufficiently thick layer of ice or snow, combined with the effect of wind on the conductor, could lead to a situation in which conductors characterized by adequate strength for specific short-circuit conditions in summer would not be sufficient under such weather conditions. Analyses for railings made with flexible conductors are much more complicated than those for rigid connections. This is due to the fact that flexible conductors move during a short circuit, which affects the tension forces on the conductors and the change in the interfacial distances.

Considering the short-circuit analyses performed in this work for flexible rail and Figure 6, Figure 7 and Figure 8 showing the relationship between the values of the forces F_t,d, F_f,d and F_pi and the length of the span, it can be concluded that the value of the individual forces depends equally on the short-circuit parameters and the physical dimensions of the span or the wire harness. In the finalized analyses, the values of the forces range from 30 kN up to 90 kN. According to IEC 60865-1:2012, the designed support structures for flexible conductors should be able to withstand the highest force among the three determined values. It is worth mentioning that the maximum force acting on equipment terminals is determined as the largest of the values of 1.5 x F_t,d, 1.0 x F_f,d and 1.0 x F_pi. The factor of 1.5, in this case, is intended to account for the oscillation energy absorbed by the mass of the insulator.

One variant of the analysis was finalized in a slightly different way than is presented in the standard’s recommendations. The calculation procedures contained in the standard do not provide an opportunity to take into account how the lashing chains are attached. In a sense, it is assumed that the lashing chains are fixed single-pointed; that is, they are an extension of the cable, so to speak, and have the ability to perform movement along with the oscillating cable. In reality, station lashing chains are attached two-pointedly, and they do not have the ability to move horizontally. This results in a decrease in the total length of the cable that can swing. Therefore, to compare the results, two variants of calculations were achieved for identical input data. The third variant was finalized as the other variants, according to the normative recommendations, while the fourth variant was implemented on the basis of modified formulas allowing to take into account the two-point attachment of the lashing chains. By analyzing the results, it can be seen that although the maximum and final angle of deflection of the conductors in the case of two-point attachment of insulators is greater, the value of the force associated with the descent of the conductor after the disappearance of a short circuit is smaller by about 10 kN. There is also a significant difference in the value of wire sag in the middle of the span. For the two-point attachment of insulators, the value of sag decreased by about 50 cm, which also reduced the maximum horizontal displacement of the conductors despite the increase in the maximum angle of deflection of the conductors during a short circuit by 20°. It is worth noting that the results of the PRIMTECH 3D program analysis are identical in both variants.

5. Conclusions

The finalized analyses confirm the theoretical assumptions, according to which the maximum values of the forces associated with the tilt and sag of the wires and the maximum horizontal displacement of the wires should be determined on the assumption that the resulting short circuit will be a two-phase and that the short circuit will occur in the summer. During the summer, the static tension forces of the wires are the smallest, and the sags of the wires in the middle of the span are the largest due to atmospheric conditions. With the same input data, the value of the force F_f,d for a two-phase short circuit increases by about 16 kN, and the preserved phase-to-phase distance, which in one of the variants was almost 2 m, decreased to 0.37 m.

The present work was conducted following the flow chart that is shown in Figure 11 below. First, the calculations were executed on the basis of the calculation procedure contained in the IEC 60865-1:2012 standard. Then, the calculations were made using PRIMTECH 3D substation design software. The program modeled the connection for which calculations were performed according to the normative recommendations, and then, for identical input data, simulations of the short-circuit current flow were launched. Finally, the analysis of the effects of the short-circuit current impact on the flexible busbars was performed by modifying the calculation procedure presented in the standard. The mentioned modification made it possible to take into account the two-point fastening of the chains, which is much more common in power substations than single-point fastening. In connection with the above, the mathematical description of the occurring physical phenomena has become closer to reality. The highest voltage line was selected (line operating voltage—400 kV). Calculations were made for different distances between columns. This is in line with practice, as at the main supply point, these distances vary from 30 to almost 90 m between poles.

Analyses of the thermal effects of the short-circuit current were performed for two steel–aluminum flexible conductors, AFL-6 240 mm² and AFL-8 525 mm², as well as for two types of copper and galvanized steel. The values of the initial three-phase short-circuit current were changed in each variant: 40 kA, 50 kA and 63 kA. The calculations show that in most cases, the conductors used in the highest voltage switchgear AFL-8 525 mm² are able to withstand the thermal effects of the short-circuit current. The exception is the case where the short-circuit current was 63 kA. The situation is different for the AFL-6 240 mm² conductor; already at a current of 40 kA, this conductor does not meet the condition. The calculations performed for grounding connections made with coopers show us how large the differences in the required minimum cross sections of grounding conductors are. In the case of a short-circuit current of 63 kA, the difference in the cross section between copper and galvanized steel copper is almost 500 mm². Therefore, there may be a situation where it is more cost-effective to perform grounding with copper.