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Ch 2: Power System
Representation
The basic components of a power system are generators,
transformers, transmission lines, and loads.
The interconnections among these components in the power
system may be shown in a so-called one-line diagram.
For analysis, the equivalent circuits of the components are
shown in a reactance diagram or an impedance diagram.
2.1 ONE-LINE DIAGRAMS
Figure 2-1 shows the symbols used to represent the typical
components of a power system.
Figure 2-2 is a one-line diagram for a power system consisting of two generating
stations connected by a transmission line; note the use of the symbols of Fig. 2-1.
The advantage of such a one-line representation is its simplicity:
One phase represents all three phases of the balanced system;
the equivalent circuits of the components are replaced by their standard
symbols; and the completion of the circuit through the neutral is omitted.
2.2 IMPEDANCE AND REACTANCE DIAGRAMS
The one-line diagram may serve as the basis for a circuit
representation that includes the equivalent circuits of the
components of the power system.
Such a representation is called an impedance diagram, or a
reactance diagram if resistances are neglected. The impedance and
reactance diagrams corresponding to Fig. 2-2 are shown in Fig. 23(a) and (b), respectively. Note that only a single phase is shown.
The following assumptions have been incorporated into Fig. 2-3(a):
1. A generator can be represented.by a voltage source in series with an
inductive reactance. The internal resistance of the generator is negligible
compared to the reactance.
2. The loads are inductive.
3. The transformer core is ideal, and the transformer may be represented by a
reactance.
4. The transmission line is a medium-length line and can be denoted by a T
circuit. An alternative representation, such as a ÿ circuit, is equally applicable.
5. The delta-wye-connected transformer T1 may be replaced by an equivalent
wye-wye-connected transformer (via a delta-to-wye transformation) so that
the impedance diagram may be drawn on a per-phase basis.
(The exact nature and values of the impedances or reactances are determined
by methods discussed in later chapters.)
The reactance diagram, Fig. 2-3(b), is drawn by neglecting all resistances, the
static loads, and the capacitance of the transmission line.
2.3 PER-UNIT REPRESENTATION
Computations for a power system having two or more voltage
levels become very cumbersome when it is necessary to convert
currents to a different voltage level wherever they flow through
a transformer (the change in current being inversely proportional
to the transformer turns ratio).
In an alternative and simpler system, a set of base values, or
base quantities, is assumed for each voltage class, and each
parameter is expressed as a decimal fraction of its respective
base.
For instance, suppose a base voltage of 345 kV has been
chosen, and under certain operating conditions the actual
system voltage is 334 kV; then the ratio of actual to base
voltage is 0.97. The actual voltage may then be expressed as
0.97 per-unit. In an equally common practice, per-unit
quantities are multiplied by 100 to obtain percent quantities;
our example voltage would then be expressed as 97 percent.
Per-unit and percent quantities and their bases exhibit the
same relationships and obey the same laws (such as Ohm's law
and Kirchhoff's laws) as do quantities in other systems of units.
A minimum of four base quantities is required to completely define
a per=unit system; these are voltage, current, power, and
impedance (or admittance). If two of them are set arbitrarily, then
the other two become fixed. The following relationships hold on a
per-phase basis:
In a three-phase system, the base kVA may be chosen as the
three-phase kVA, and the base voltage as the line-to-line
voltage; or, the base values may be taken as the phase
quantities.
In either case, the per-unit three-phase kVA and voltage on
the three-phase kVA base and the per-unit per-phase kVA and
voltage on the kVA-per-phase base remain the same.
2.4 CHANGE OF BASE
The per-unit (pu) impedance of a generator or transformer, as
supplied by the manufacturer, is generally based on the rating
of the generator or transformer itself.
However, such a per-unit impedance can be referred to a new
volt-ampere base with the equation:
If the old base voltage and new base voltage are the same,
then (2.6) simplifies to
The impedances of transmission lines are expressed in ohms,
but can be easily converted to pu values on a given volt-ampere
base using (2.1) to (2.5).
2.5 SUMMARY OF THREE-PHASE CIRCUIT RELATIONSHIPS
A three-phase circuit may be connected either in wye or in delta. In
a balanced three-phase circuit the phase and the line values of the
current, power, and voltage are related as follows (the subscripts p
and l refer to phase and line values, respectively):
The delta and wye impedances are related by:
For both types of connections, the apparent and reactive
powers are, respectively,
From the above, it is clear that the phrase angle may be
obtained as:
Gaussian Elimination
https://www.youtube.com/watch?v=2j5Ic2V7wq4
Cramer’s Rule
https://www.youtube.com/watch?v=TtxVGMWXMSE