Performing short-circuit calculations requires an understanding of various system components and their interaction.
In Part 1 of this article, which was featured in the June 1995 issue, we discussed the types of networks to calculate short-circuit current (i.e., symmetrical rms current). In Parts 2 and 3 (April 1996 issue), we'll describe the per-unit method of performing short-circuit calculations in accordance with ANSI/IEEE 141-1993, IEEE Recommended Practice for Electric Power Distribution for Industrial Plants (the Red Book). We'll use a simple example of an industrial power system to show the data preparation steps necessary in determining the appropriate per-unit reactances and resistances of power system equipment for first-cycle (momentary) and contact-parting (interrupting) networks.
Overview of per-unit analysis
Per-unit analysis is based on "normalized" representations of the electrical quantities (i.e., voltage, current, impedance, etc.). The per-unit equivalent of any electrical quantity is dimensionless and is defined as the ratio of the actual quantity in units (i.e., volts, amperes, ohms, etc.) to an appropriate base value of the electrical quantity. This is expressed by the following equation.
per-unit value = actual value [divided by] base value (eq. 1)
The actual value can be a phasor or complex number (i.e., magnitude and phase) in units, whereas the base value is simply a real number in units.
The key factor of a per-unit normalization procedure is the selection of the base values of the electrical quantities. In practice, the base values of 3-phase apparent power in MVA (i.e. base MVA) and line-to-line voltage in kV (i.e. base kV) are assigned on each side of every 3-phase power transformer in accordance with the following simple rules.
First, a convenient base MVA is chosen (e.g., base MVA = 1, 10, or 100) and is common throughout the entire power system. Second, base kV on any side of a 3-phase power transformer is designated to be equal to the nominal, line-to-line nameplate kV rating of the transformer.
The base values of per-phase base impedance in ohms (i.e., base Z) and line current in kiloamperes (i.e., base kA) are then derived from base MVA and base kV, according to the following equations. base Z = [(base kV).sup.2] [divided by] base MVA (eq. 2) base kA = base MVA [divided by] ([square root of 3] x base kV) (eq. 3)
You gain an important advantage by assigning the base values per the above selection rules and using them to normalize (i.e., "per-unitize") the electrical quantities. This advantage can be seen in the per-phase equivalent circuit model of any 3-phase transformer connection: It is simply a per-unit series impedance that accounts for the conductor losses and leakage fluxes.
The following equation is often used in [TABULAR DATA FOR TABLE 1 OMITTED] the data preparation stage to adjust the per-unit impedance of power system apparatus whenever the 3-phase nameplate ratings are different than the power system's 3-phase base quantities.
adjusted [Z.sub.pu] = unadjusted [Z.sub.pu] x (base MVA [divided by] MVA rating) x [(kV rating [divided by] base kV).sup.2](eq. 4)
Data preparation for a simplified example In the absence of nameplate information or data from equipment manufacturers, typical data can be referenced from tables and figures included in the IEEE Red Book. The data preparation steps to determine the appropriate per-unit reactances and resistances for first-cycle (momentary) and con-tacting-parting (interrupting) networks for the simplified industrial power system shown in Fig. 1 are as follows.
Base values of voltage. The base values of line-to-line voltage are simply the nominal nameplate, line-to-line voltage ratings of the 3-phase transformers. Base MVA is chosen to be 10 MVA and is constant throughout the system.
Utility. The per-phase equivalent circuit model of the utility tie at the plant is a voltage source in series with an impedance. This source is the nominal per-unit line-to-neutral voltage at the service entrance point. The per-unit impedance is the same for both first-cycle and interrupting networks; thus, no superscript "f" or "I" is necessary for its symbol. (In the equations that follow, superscripts "f" and "I" refer to the per-unit reactance and resistance, respectively, for the first-cycle and interrupting networks.)
The magnitude of the impedance ([Z.sub.u] = 0.01 per unit) is calculated by using the following equation.
[Z.sub.u] = base MVA [divided by] (SCA MVA)(eq. 5)
Here, SCA MVA is equal to 1000 and represents the available short-circuit apparent power delivered by the utility from all sources outside the plant.
To resolve [Z.sub.u] into reactive ([X.sub.u]) and resistive ([R.sub.u]) components, the following equations are used.
X = Z sin ([tan.sup.-1] X/R)(eq. 6a)
R = Z cos ([tan.sup.-1] X/R)(eq. 6b)
In this event, a conservative assumption is to let [X.sub.u] = [Z.sub.u] = 0.01 per unit and [R.sub.u] = 0.
In our example, as shown in Fig. 1, the short-circuit X/R ratio at the utility tie is unavailable. Thus, we assume [X.sub.u] = 0.01 per unit and [R.sub.u] = 0.
Transformers. The per-phase equivalent circuit model of any 3-phase transformer [TABULAR DATA FOR TABLE 2 OMITTED] connection is simply a per-unit series impedance. This impedance is the same for both first-cycle and interrupting networks; thus, no superscript "f" or "I" is necessary for its symbol.
The unadjusted impedance of the transformer is provided on its nameplate and is expressed as a percentage of rated impedance. In other words, you simply divide it by 100% to arrive at its unadjusted per-unit value. This per-unit impedance is adjusted with respect to the system base quantities per Equation 4, and the adjusted per-unit impedance ([Z.sub.t]) is resolved into reactive ([X.sub.t]) and resistive ([R.sub.t]) components per Equations 6a and 6b. (The typical short-circuit X/R ratios of the transformers in Fig. 1 are taken from Fig. 4A-1 of the 1993 IEEE Red Book.) The results for the transformers in Fig. 1 are shown in Table 1.
Cable. The per-phase equivalent circuit model of the cable in Fig. 1 is simply a per-unit series impedance. This impedance of the cable is the same for both first-cycle and interrupting networks; thus, no superscript "f" or "I" is necessary for its symbol.
To find the cable's per-unit reactance ([X.sub.c]) and resistance ([R.sub.c]), the following equations are used.
[X.sub.c] = [(X ohms per 1000ft) x (length of run in ft)] [divided by] [number of parallel conductors per phase x base Z in ohms] (eq. 7a)
[R.sub.c] = [(R ohms per 1000ft) x (length of run in ft)] [divided by] [no. of parallel conductors per phase x base Z in ohms] (eq. 7b)
Using the above equations, [X.sub.c] is 0.0030 per unit and [R.sub.c] = 0.0043 per unit.
(The approximate reactance and resistance data (in ohms per 1000 ft) noted alongside the cable in Fig. 1 are taken from Table 4A-7 in the 1993 IEEE Red Book.)
Turbine-generator. In general, the per-phase equivalent circuit model of a rotating machine is a voltage source in series with an impedance that varies with time during the fault. Based on Table 4-1, Chapter 4 of the 1993 IEEE Red Book, the unadjusted per-unit reactance of the turbine-generator for both the first-cycle and interrupting networks is 1.0 [X.sub.d]", where [X.sub.d]" is the saturated direct-axis subtransient reactance of the generator in per-unit. You adjust this reactance with respect to the system base quantities by using Equation 9, with the corresponding adjusted resistance calculated by using Equation 8.
adjusted R = adjusted X [divided by] short-circuit X/R (eq. 8) [X.sup.f]=(1.0 [X.sub.d]") x (base MVA [divided by] MVA rating) x [(kV rating [divided by] base kV).sup.2] (eq. 9)
The calculation results are shown in Table 2. (The typical machine reactance and short-circuit X/R data are from Table 4A-1 and Figs. 4A-2 and 4A-3 of the 1993 IEEE Red Book. Incidentally, there is a typographical error in Table 4A-1 of the first printing of this book: the left-most column of Table 4A-1 should be labeled [X.sub.d]" and the right-most column [X.sub.d]'.)
Large motors. Based on Table 4-1, Chapter 4 of the 1993 IEEE Red Book, the unadjusted per-unit reactances for the first-cycle and interrupting networks are 1.0 [X.sub.d]" and 1.5 [X.sub.d]" respectively.
Equation 9 is used to adjust the first-cycle reactance, where the 3-phase kVA rating is approximately equal to the horsepower (hp) rating for induction motors and 0.8 power factor (PF) synchronous motors. The corresponding adjusted first-cycle resistance is calculated by using Equation 8. The following equations are then used to calculate the adjusted interrupting reactance and resistance from the corresponding first-cycle values.
[X.sub.[M.sup.I]] = 1.5 x [X.sub.[M.sup.f]] (eq. 10a)
[R.sub.[M.sup.I]] = 1.5 x [R.sub.[M.sup.f]] (eq. 10b)
The results of these calculations also are shown in Table 2.
Small horsepower induction motors. The unadjusted per-unit reactances of the small horsepower (less than 250 hp) induction motors shown in Fig. 1 for the first-cycle [TABULAR DATA FOR TABLE 3 OMITTED] and interrupting networks are taken from the footnotes of Table 4-2 in the 1993 IEEE Red Book. Specifically, you should refer to the footnotes of the row entitled "All others, 50 HP and above" for the 150 hp induction motor. Also refer to the footnotes of the row entitled "All smaller than 50 HP" for the group of small-horsepower induction motors whose ratings are less than 50 hp. The following equation is then used to adjust both the per-unit first-cycle and interrupting reactances, where the 3-phase kVA rating is approximately equal to the hp rating for an individual induction motor or the sum total of hp ratings for a group of motors.
[X.sub.M] = (unadjusted [X.sub.M]) x (base MVA [divided by] MVA rating) x [(kV rating [divided by] base kV).sup.2] (eq. 11)
The corresponding adjusted first-cycle and interrupting resistances are calculated by using Equation 8. The results of the calculations are listed in Table 3.