Analysis Tools: Short-circuit Analysis

Introduction

The following is a report on the implementation of a computational tool for the simulation of three-phase short circuits in electrical power systems in the DeepEdit program. This work consists of the elaboration of a computational routine written in Java, following the object-oriented programming, which efficiently manipulates the matrices following the existing methodologies in the bibliography and accepted at international level.

The objective of this section is to synthesize the technical bases that support the development of the programming routine. Within the bases are the methods, procedures and, fundamentally, the standardized norms for the calculation of short circuits. Additionally, a simple manual for both novice and advanced users is presented with results of demonstrative numerical applications.

Modeling

Methods

The tool has 2 three-phase short-circuit calculation methods available in the options menu:

• IEC909 Method: According to IEC-60909-0 Standard. Maximums and minimums.

• Full Method: Levels under specific load conditions.

IEC909 method

It is the computational application of the short-circuit calculation method according to the international standards "International Electrotechnical Commission" IEC technical standard number IEC-60909-0 to IEC-80909-7. The next section shows the fundamentals of the method.

Complete method

This method calculates the three-phase short-circuit levels for a specific system operating point. It requires the prior calculation of the system state variables for linear modeling. It models the shunt elements as constant impedances (loads, static compensation, and line susceptance).

Objects

The models chosen for each of the elements of the power system are shown below. In the case of the loads, the model presented is only used when the complete method is employed.

Generators

The equivalent of the generators is simplified to a constant voltage source E and its leakage reactance Xd series. Figure 79 shows a diagram of the model.

Figure 79: Generator model

• E will depend on the pre-fault conditions. (See IEC 909 Method section for E values using the IEC method). The next session shows how to calculate E from the pre-fault load conditions of the circuit).

• The value of the generator leakage reactance Xd, Xd' or Xd''' will depend on the instance of the short circuit the user wishes to analyze.

Lines

The classic “π” model of the line is used. Figure 80 shows the diagram of the model.

Figure 80: Transmission line model

The mean value of the total susceptance will be ignored in the short-circuit calculations by the IEC909 method.

Transformers

Transformers are modeled as a series leakage impedance only. Figure 81 shows the diagram.

Figure 81: Transformer model

Loads (complete method only)

All loads, including the static reactive compensation elements, are modeled as a constant impedance Zc.

The equivalent and constant impedance during short circuit is determined by:

Figure 82: Three phase loads model (full method only)

Developments

General calculation methodology

The program calculates the short-circuit current at a three-phase fault by the Zbus classical method. The short-circuit current for a three-phase fault at bus "k" is:

Where:

If : Post-fault current.

Vkpre : Pre-fault voltage at the faulted bus

Zkk : Element [k,k] of the Zbus impedance matrix.

Zf : Fault impedance.

The calculation of the Zbus matrix, or open-circuit transfer impedance matrix, is performed through a direct construction algorithm. It has been shown that for small networks, where the current vector has only one component (If), direct construction rather than inversion of the admittance matrix is more efficient, even by means of optimal triangular factorization techniques. However, the same reference shows that for very large network problems, triangular factorization has obvious advantages and can effectively and economically solve problems up to 10 times larger than the Zbus construction technique.

The direct construction technique employed follows the algorithms suggested in the abundant literature available for the subject. The following procedure can be used: (the one presented was taken verbatim from Haddad).

Include branches to the reference node:

Conditions:

1. Add reference branch to a new node (q).

2. Branch not mutually coupled.

Rule:

1. Let Zqq = z

z: Self-impedance of the added branch.

Add radial branches to new nodes

Conditions:

1. Add branch from a created node (k) to a new node (q).

2. Branch not mutually coupled.

3. Previously defined nodes (p).

Rule:

1. Let Ziq = Zik for i=1, 2, . . . p

2. Let Zqi = Zki for i=1, 2, . . . p

3. Let Zqq = Zkk + z

z: Eigenimpedance of the added branch.

Close loops to reference by adding an impedance:

Conditions:

1. Add branch between a created node (k) and reference.

2. Branch not mutually coupled.

3. Previously defined nodes (p).

Rule:

1. Let q = p + 1

2. Let Ziq = Zik for i=1, 2, . . . p

3. Let Zqi = Zki for i=1, 2, . . . p

4. Let Zqq = Zkk + z

5. Eliminate row and column “q” by Kron reduction.

z: Eigenimpedance of the added branch.

Close loops between created nodes by adding an impedeance

Conditions:

1. Add branch between a created node (k) and another created node (q).

2. Branch not coupled to each other.

3. Previously defined nodes (p).

Rule:

1. Let q = p + 1

2. Let Ziq = Zik for i=1, 2, . . . p

3. Let Zqi = Zki for i=1, 2, . . . p

4. Let Zqq = Zii + Zkk – Zik – Zki + z

5. Eliminate row and column “q” by Kron reduction.

z: Eigenimpedance of the added branch

For each of the methods mentioned above, certain different conditions are assumed prior to the construction of the Zbus matrix. For example, the IEC method for the calculation of maximum or minimum short-circuit currents assumes that it is valid to disregard all passive shunt elements. This means that loads, offsets, line susceptances, reactors, etc. are ignored. The fundamentals and assumptions of each method are mentioned below.

IEC 909 Method

This is the method approved in IEC 60909-0 standard for the calculation of short-circuit currents. A summary of the main aspects of the method is given below:

Short-circuit current

The effective value of the subtransient current injected into the faulted node is:

where:

Zii: Value [i,i] of the open circuit Zbus impedance matrix,

Vkpre: Value of the voltage before the fault at the faulted bus.

c: Voltage correction factor.

Zkk : Element [k,k] of the Zbus impedance matrix.

Zf : Fault impedance.

According to Haddad, the factor c is determined according to the table below:

Open circuit impedance matrix

The calculation of the open circuit impedance matrix is calculated under the following conditions:

• All load currents are negligible.

• All voltages on all busbars are equal in module and angle according to the table above.

• Positive sequence networks are balanced except at the fault point. In the case under study they will always be balanced.

• All shunt admittances are negligible (compensators, line susceptance, etc).

• The only connection to reference is made through the generators.

Method of calculating the “peak” short-circuit current:

It is the maximum possible value of the short-circuit current. It results from multiplying the peak value of the short circuit subtransient current by a factor "k"

The “k” factor should be calculated by the formula presented below:

One of the following options is usually employed for the estimation of the "k" factor:

Method A: Uniform R/X ratio

It is assumed that the R/X ratio is constant and equal to the smallest of the ratios of the whole network. Based on the determination of this value, the ratio below can be used to estimate the value of the "k" factor.

Method B: R/X ratio inaccuracies method

In simplified form, a correction for inaccuracies in the R/X ratio can be included by multiplying by a correction factor of 1.15 as shown below:

kb is obtained by substituting the R/X ratio determined by the short-circuit impedance. If the R/X ratio in all branches is less than 0.3, the correction factor should be 1.5 as mentioned above. However, it may be to reach 1.8 or a limit of 2.

Method C: Frequency correction method:

K is calculated from the ratio R/X given by:

with: Zc = Rc + jXc

Zc is the equivalent short-circuit impedance, where fc = 20 Hz if the mains frequency is 50 Hz (fc = 24 Hz if the mains frequency is 60 Hz).

Complete method

Performs the short-circuit calculation for a system operating point assuming certain assumptions that will be explained later. The state variables that determine the operating point must be determined by means of a load flow. That is, the method assumes that all the modulus and angle voltages of all the busbars in the system are known. It is assumed, the method should deliver more accurately the value of the short-circuit current for a certain load condition and system configuration.

Short-circuit current

The effective value of the subtransient fault current is:

The prefault voltages are equal in modulus and angle to the value of the voltages resulting from the realization of a previous load flow multiplied by an adjustment coefficient "c". Unlike the IEC909 method where the voltage correction factor c was determined using tabular values (see section IEC 909 Method) , this factor c may be any positive real number assigned by the user to find the desired study conditions.

Open circuit impedance matrix

It is calculated according to the following conditions:

• Loads and compensators can be represented as constant impedances and equal to the value:

• Shunt susceptance of the transmission lines are considered constant and equal to the pre-fault condition.

• The positive sequence network is balanced except at the fault point.

Pre-fault voltages

They are obtained by calculating a load flow and multiplying all of them by the same constant and positive value "c". This value can be any positive real number. The default value will be 1, so that the voltages represent exactly the values of the voltages at the point of operation just before the fault occurred. Any different value may be used to find favorable/unfavorable situations useful for finding maximum and minimum short-circuit contributions.

Short-circuit peak current calculation method

As in the IEC909 method, peak short-circuit currents are estimated by any of the three methods mentioned above.

How to use

The following section is an aid for the user to become acquainted with the basic functions of the short circuit calculation tool incorporated in the Deep Edit program. Any other version of the Deep Edit may differ with the functions detailed in this document. The implemented short-circuit study follows the suggestions set forth in IEC909 and VDE101 standards. However, the authors reserve the right to make modifications and add or remove assumptions to those set forth by the aforementioned standards.

Main window

To start the short-circuit calculation, the configuration window must be accessed. This can be accessed in any of the following ways:

1. Menu: Analysis Tools / Static Tools / Technical Tools / Short Circuit Study

2. By clicking on the short-circuit button in the study toolbar (Figure 83).

Figure 83: Short-circuit studies configuration

The main window contains two tabs called general and advanced.

General tab

Advanced tab

Two cells divide this Tab: One fixed and one dependent on the method chosen in the "method" option of the "general" Tab. The fixed cell is labeled "R/X Method" and the variable cell shows the labels of the advanced options for the IEC909 and Complete methods.

Fixed cell: “R/X Method”:

If the IEC909 option is selected in the "method" choice of the “general” tab, the variable cell shows the advanced options of the IEC909 method.

Variable cell “IEC090”:

Figure 84: Advanced tab showing the IEC909 variable cell

“Complete” variable cell:

Figure 85: Advanced tab showing the “complete” variable cell

Displaying results

The calculation tool delivers results in two different ways: the first, at the user's choice, by means of a printable text file and the second by displaying them through a color scale on the schematic. In the next figure examples of both outputs are shown:

Figure 86: Example of output file in short-circuit calculation

Figure 87: Example of graphical output in the short-circuit calculation

Output file

As seen in the previous section of the manual, the user will have the option to print the results in a file. Figure 86 shows an example of the presentation of these results. There is the option to select all branches as shown below.

All branches selected (all branches?):

Displays the subtransient current Ik" and peak current Ip" information for each of the branches and generators in the system.

All branches deselected:

Prints in the file only the information of the short-circuit contributions of the branches and generators adjacent to the faulted bus.

Graphic output

Allows the user to visualize by means of a color and thickness scale the contributions (in percentage) of the currents by each of the branches to the total short-circuit current Ik Figure 87 shows an example application.

Color scale

Intuitively we can be sure that the current contributions by each of the branches may be relatively small. This is especially true in highly meshed and/or large networks. However, both cases are quite likely to be encountered in real systems. Thus, we can expect the average value of these current inputs to be small compared to the total short-circuit current and, moreover, to be like each other. A preliminary empirical analysis determined that the distribution of the inputs for real system dimensions resembles a log-normal like the one shown in Figure 88.

Figure 88: Distribution of branch currents during a short-circuit

This means that the results are grouped in the neighborhood of the average which, in turn, is close to zero. Then, to disperse these data, a logarithmic scale was used, which transforms the uniform divisions of the current intervals of the same color into short intervals close to the average and increasing as it approaches 100%.

Figure 89: Example of the use of a logarithmic scale for interval determination

In this example, the average of the current inputs by all branches was equal to 0.18 (18.1%) times the total short-circuit current. The defined scale then allows assigning different shades to those contributions between the average and zero that are much finer than a normal scale. Moreover, the scale is "centered" on the average thus achieving an effective dispersion of the data concentrated in its neighborhood.

Once the calculations have been performed, a window will automatically open showing the color scale as described above. Figure 90 shows the window for the above example.

Figure 90: Example of a color scale for an average contribution of 18.1% of the total short-circuit current

Display control window

It also opens automatically at the end of the short-circuit calculations and allows to control the elements to be displayed. Figure 91 shows the content of the window.

Figure 91: Display control window

1. According to IEC 60909-0 standard in force from 2001 to 2007.↩︎

2. Requires a power flow to be performed beforehand.↩︎

3. Notation according to IEC 60909-0 2001 standard. Fault impedance method “B”.↩︎

4. Notation according to IEC 60909-0 2001 standard. Frequency-corrected fault impedance method “C”.↩︎

5. See IEC-60906-0 standard for further details.↩︎