Influence of Cable Length on Short-Circuit Calculations

The objective of this article is to establish a comparison between short-circuit levels in medium and low voltage systems, calculated under different scenarios involving utility contributions and variations in the lengths of power supply cables.

Keywords: Short-Circuit Calculation, Selectivity Study

Introduction

Short-circuit calculations and selectivity studies are fundamental activities in industrial electrical design, especially nowadays, when production systems are increasingly dependent on reliable sources of electrical energy.

An important part of short-circuit calculations is related to the mathematical treatment of power cables, whether medium or low voltage. In practice, it is common to disregard short lengths of cable (or use estimated lengths), based on the belief that they do not significantly impact the calculations. On the other hand, some advocate for maximum precision in measuring all cable lengths, a situation which, in certain cases, may bring unnecessary difficulty to calculations and field data collection without resulting in qualitative improvements in the studies.

In this article, Section 1 presents a brief theoretical discussion on short-circuit calculations. Section 2 addresses a simplified methodology for three-phase short-circuit calculations. Section 3 presents two calculation examples for different points in a theoretical system, powered at 13,800 V with low-voltage panels supplied via transformers. Section 4 explores the influence of cable lengths in the system on short-circuit levels, presenting the results of simulations conducted using PTW software, version 6.5.2.7, Dapper module. Variables include the short-circuit level at the service entrance, the length of the main power supply cables, the length of the transformer supply cables and the length of the low-voltage panel supply cables. Finally, Section 5 highlights the main conclusions of this study.

1. Basic Theory on Short-Circuit Calculations in Industrial Electrical Installations

Short-circuit currents mainly depend on the impedances between the power source and the fault location (busbars, cables, machine terminals). Certain types of loads, such as motors, also contribute to the increase in short-circuit levels. However, their influence is generally much smaller than that of the utility provider. In this article, the influence of motors on short-circuit levels will not be addressed. Thus, the short-circuit is considered solely as a function of the impedances between the utility and the fault point.

According to Mamede Filho (2010), points farther from the generation system are strongly influenced by the impedance of transmission lines, as their impedance is significantly higher than that of generators. Therefore, the short-circuit current consists of two main components, illustrated graphically in Figure 1.1:

• Symmetrical Component: As the name suggests, this is the symmetrical part of the current, which predominates in the short-circuit after a few cycles.

• DC Offset Component: This component decreases over time due to the system’s inability to instantly vary magnetic flux.

Figure 1.1 – Current waveform during a short-circuit (Source: SKM, 2006a, 2006b)

Simplified Methodology for Three-Phase Short-Circuit Calculation

The most commonly used method for short-circuit calculations in practice is the per-unit system, also known as “pu”. The value of electrical quantities (voltage, current, power, impedance, etc.) is defined as the ratio between the actual value of the quantity and a selected base value.

The main advantage of this method lies in the fact that different voltage levels exist in our circuits due to the presence of transformers. In this way, by using pu values, transformers can be represented by a single impedance as if their turns ratio were 1:1, simplifying the calculations.

To illustrate the studies in this work, all analyses will be based on the theoretical system shown in the diagram of Figure 2.1.

Figure 2.1 – Single-line diagram of the theoretical system to be studied

2.1 Definition of Base Values

Initially, base values must be selected for the per-unit (pu) calculations. Typically, a base power value (Pb) of 100 MVA is used:

Pb = 100 MVA  (2.1)

For the base voltage (Vb), it’s common to use the highest voltage in the system under analysis. In our case, we will work with a system at 13,800 V:

Vb = 13,800 V  (2.2)

From these values, we can calculate the base current (Ib):

2.2 Utility Company Impedance and Short-Circuit Levels

The impedance may be provided directly by the utility in per unit (pu), or if given in Amperes, it must be converted to pu using the equation:

Z₁(pu)₍utility₎ = Ib / Icc₃ø₍sym, utility₎  (2.4)

Where:

• Icc₃ø₍sym, utility₎ – Symmetrical three-phase short-circuit current at the utility’s point of delivery

• Ib – Base current

• Z₁(pu)₍utility₎ – Positive-sequence impedance at the utility’s point of delivery

2.3 Transformer Impedance

Where:

• Z₁(pu)₍trafo₎, R₁(pu)₍trafo₎, X₁(pu)₍trafo₎ – Positive-sequence impedance, resistance, and inductive reactance of the transformer

• Z% – Percent impedance of the transformer (nameplate data)

• Pt – Transformer power (in VA)

• Pb – Base power (100 MVA)

2.4 Cable Impedance

• Cable R, X, Z – Positive-sequence resistance, inductive reactance, and impedance of the cable

• R per meter and X per meter – Resistance and reactance per meter (manufacturer data)

• Pb – Base power (100 MVA)

• V – Nominal voltage of the cable circuit (in Volts)

• Dist – Distance (in meters)

• ncpf – Number of cables per phase

2.5 – Three-Phase Short-Circuit Currents

• Icc₃ø₍sym₎ – Symmetrical three-phase short-circuit current

• Ib – Base current

• Z(pu) – Equivalent impedance at each point in per unit

• Icc₃ø₍asym₎ – Asymmetrical three-phase short-circuit current

• e – Euler’s number (2.71828)

• π – Pi (3.14159)

• X/R – Ratio between positive-sequence inductive reactance and resistance at the point of interest

Example of Three-Phase Short-Circuit Calculation

To illustrate the equations presented in Section 2 and validate the effectiveness of the calculations made through simulations using the PTW software version 6.5.2.7, we present the short-circuit level calculations performed for the system shown in Figure 2, considering the data from Tables 3.1 to 3.4.

Table 3.1 – Utility Company Data

Table 3.2 – Medium Voltage Cable Impedance

Table 3.3 – Low Voltage Cable Impedance

Source: Ficap catalog, Fipex 0.6/1kV cables, single core, installed in trefoil configuration

Table 3.4 – Transformer Data

3.1 Calculation of System Impedances in pu

Considering base values of Pb = 100 MVA and Vb = 13,800 V, and applying equation 2.4 to the utility company data (Table 3.1), we obtain the following results (Table 3.5):

Table 3.5 – Utility Company Impedance Values in pu

Applying equation 2.8 to the cables, we get the following results (Table 3.6):

Table 3.6 – Cable Impedance Values in pu

Applying equations 2.5, 2.6, and 2.7 to the transformers, we obtain (Table 3.7):

Table 3.7 – Transformer Impedance Values in pu

3.2 Calculation of Equivalent Impedances in pu at Several System Points

The equivalent impedance at SE-Main is given by the vector sum of the utility company impedance and the Main MV cable:

ZeqSE-Main = Zutility + Zcable MT Main = 0.103785 + j 0.830278 + 0.0052 + j0.0016 ZeqSE-Main = 0.108985 + j0.831878 pu ZeqSE-Main = 0.8390 pu (angle = 82.53o)

Using equations 2.11 and 2.12, we get the short-circuit values for SE-Main :

Icc3øsym SE-Main = 4,183.70 / 0.8390 = 4,986A Icc3øasym SE-Main = 6,834A

The equivalent impedance at the primary of TR01 (SE-TR01) is the sum of the utility impedance, Main MV cable, and MV cable TR01:

ZeqSE-TR01 = Zutility + Zcable MT Main + Zcable MT TR01 = 0.103785 + j 0.830278 + 0.0052 + j0.0016 + 0.0070 + j0.0017 ZeqSE-TR0l = 0.115985 + j0.833578 pu ZeqSE-TR0l = 0.8417 pu (angle = 82.07o)

Using equations 2.11 and 2.12, we obtain short-circuit for SE TR-01:

Icc3øsym SE-TR01 = 4,183.70 / 0.8417 = 4,971 A Icc3øasym SE-TR01 = 6,732 A

For the other points of the system, similar calculations are made. The results are presented in Table 3.8, which shows the symmetrical and asymmetrical three-phase short-circuit levels. For low-voltage points, the levels have already been calculated using the nominal voltage of each panel.

Table 3.8 – Calculated Three-Phase Short-Circuit Levels

4. Analysis of the Influence of Medium and Low Voltage Cable Lengths on Short-Circuit Calculations

In order to analyze the variations in short-circuit levels based on different cable lengths and entry short-circuit levels, 150 different simulations were carried out, as follows:

• For each entry three-phase symmetrical short-circuit level of 3,000 A, 4,000 A, 5,000 A, 6,000 A, and 7,000 A, the following simulations were performed:

• Varying the length of the Main MV Cable (0, 20, 40, 60, 80, 100, 120, 140, 160, 180, and 200 m)

• Varying the lengths of MV TR01 and TR02 Cables (same intervals)

• Varying the lengths of LV TR01 and TR02 Cables (same intervals)

Simplified Table 4.1 presents the scenarios in which the simulations were conducted:

Cable sizing is not under discussion in this work, only its influence on short-circuit calculations. Therefore, voltage drop levels under normal operating conditions were not considered.

4.1 Analysis of Short-Circuit Levels by Varying the Length of the Main MV Cable

After simulating the scenarios involving various entry short-circuit levels and variations in the Main MV cable, the results are presented in Figures 4.1 to 4.5.