PU Method of Representing Quantities | Power System | Electrical Engineering

The quantities involved in power system are kVA, voltage, current and impedance of the equivalent circuits of the various system components. The equivalent circuits are at different voltages and are connected together in the system by means of transformers and interconnections.

Each apparatus is rated in kVA and its impedance in actual ohms or in percentage value referred to its rated kVA and rated voltage. In power system analysis, it is usual to express voltage, current, kVA and impedance in per unit of base or reference values of these quantities. Such a method simplifies the calculations.

Advantages and Drawbacks of Per Unit (PU) System:

Advantages:

1. Calculations are simplified.

2. The characteristics of machines (generators, transformers, motors etc.) when described in per unit system are specified by almost the same number, regardless of the rating of the machines. In other words, the characteristics (or parameters) tend to fall in relatively narrow range, making erroneous values conspicuous. Thus per unit system provides a method of comparison.

3. For circuits connected by transformers, per unit system is particularly suitable. By choosing suitable base kV’s for the circuits the per unit reactance remains the same, referred to either sides of the transformer. Therefore, the various circuits can be connected in the reactance diagram.

4. This method is useful to eliminate ideal transformers as circuit components since the typical power system contains hundreds, if not thousands of transformers, and this is a non-trivial savings.

Drawbacks:

1. Some equations that hold in the unsealed cases are modified when scaled into per unit. Factors such as JH and 3 are removed or added by this method.

2. Equivalent circuits of the components are modified, making them somewhat more abstract. Sometimes phase shifts that are clearly present in the unsealed circuit vanish in the per unit circuit.

Selection of Bases:

For a common representation, base kVA and base voltage are to be chosen.

Then the base current and base impedance can be expressed as follows:

For a single phase, phase-to-neutral voltage, kVA per phase are taken as bases while in a three-phase system, three-phase line-to-line voltage and three-phase kVA are used as bases. This simplifies the calculations.

The base values in a system are so selected that the per unit voltages and currents in the system are approximately unity. Sometimes the base kVA is chosen equal to the sum of the ratings of the various equipment’s in the system or equal to the kVA rating of the largest unit connected in the system.

If in the network there is no transformer present, the same base voltage is used throughout, but if the transformers are present, the rule is to change the base voltage in proportion to the transformation ratio of the transformer when transformer is reached.

Thus, all impedances in the network before the transformer is reached, including the transformer primary leakage impedance, are expressed in per unit, to the voltage base chosen for the primary side; all impedances beyond the transformer, including the transformer secondary leakage impedance, are expressed in per unit to a new base voltage which is the primary side base voltage multiplied by the transformer transformation ratio. This is very important.

Determination of Base Impedance:

Voltage, current, volt-amperes and impedances are so related that selection of base values for any two of them determines the base values of the remaining two.

For example for a single phase circuit we have the following relations:

where S_B, V_B, I_B and Z_B represent the base power, base voltage, base current and base impedance expressed in volt-amperes, volts, amperes and ohms respectively. Thus from above two equations, if any two of the four quantities are specified, the remaining two may be determined without any problem. Base kVA and base kV are usually specified. Let kVA_B and kV_B be the base kVA and base kV, then –

In a 3-phase system rather than obtaining the per unit values using per phase quantities, the per unit values can be obtained directly by using three-phase base quantities (total kVA, line-to-line voltage and line currents). Let kVA_B be the base kVA (total output of three phases) and kV_B be the base line-to-line voltage in kV.

Assuming star-connection (equivalent star can always be found) –

It should be noted that the same expression is obtained for a single phase and three-phase systems. In power system problems the data are given in terms of 3-phase kVA, line-to-line kV. The machine kVA and machine phase-to-phase (or line-to-line) kV rating. Further the direct-axis synchronous reactance is also known as positive sequence reactance of the machine.

Change of Base:

Normally the per unit impedance of various components corresponding to its own rating voltage and kVA are given and since we choose one common base kVA and base kV for the whole system, therefore, it becomes imperative to determine the per unit impedance of the various equipment’s corresponding to the common base kV and base kVA.

If the individual quantities are Z_{pu old}, kVA_old, kV_old and the common base quantities are Z_{pu new}, kVA_new and kV_new, then making use of Eq. (2.24), according to which per unit impedance is directly proportional to the base kVA and inversely proportional to the square of the base kV, we have-

This is very important relation used in power system analysis.

Per Unit Impedance of a Two Winding Transformer:

The approximate equivalent circuit of a two winding transformer with all impedances referred to primary (low-voltage) side is illustrated in Fig. 2.10:

According to Eq. (2.24)-

where Z₀₁ is the total impedance of the transformer referred to primary side in ohms, kVA_B is the rated kVA of the transformer and kV_B1 is the rated voltage of the transformer on the primary side in kV.

Now let us consider the transformer with all its impedances referred to secondary side. The total impedance of the transformer referred to secondary (high voltage side).

Thus the impedance of the transformer in per unit viewed from the secondary (high voltage) side,

where Z₀₂ is the total impedance of the transformer referred to secondary side in ohms and kV_B2 is the rated voltage of the transformer on the secondary side in kV.

Substituting the value of Z₀₂ from Eq. (2.29) in Eq. (2.30) we have –

Comparing Eqs. (2.28) and (2.31) we conclude that the per unit impedance is the same regardless of the side from which it is viewed. Thus the simple equivalent circuit for the two-winding transformer in which Z_{01 pu} = Z_{02 pu} = Z_pu is depicted in Fig. 2.11.

Mutual Impedance in Per Unit between Lines of Different Voltage Levels:

Let us consider two three-phase lines of different voltage levels running together, with mutual inductance X_m present, as depicted in Fig. 2.12:

There is one value X_{m pu} (mutual reactance in per unit), which will serve both lines.

Thus in terms of line 2, we have –