M.Tech (Power System )

M. TECH. I-SEMESTER

POWER SYSTEM ANALYSIS
Fault Analysis: Positive, Negative and Zero sequence equivalent circuits of lines, two and
three winding transformers and synchronous machines. Analysis of shunt and series faults,
effect of neutral grounding.
Admittance and Impedance Model and Network Calculations: Calculation of Z-bus, Y-bus.
Algorithm for the formation of bus admittances and impedance matrices, Fault calculation
using Z-bus.
Load Flow Studies: Formulation of load flow problem. Various types of buses. Gauss-Siedel,
Newton-Raphson and Fast Decoupled Algorithms.
Calculation of reactive power at voltage controlled buses in the Gauss-Siedel interactive
method using Y-bus. Representation of transformers-Fixed tap setting transformer, Tap
changing under load transformers, Phase shifting transformers, Comparison of methods for
load flow.
Power System Security and State Estimation: Concepts of security states and security analysis
in power system, State estimation in power system.

SYMMETRICAL COMPONENTS AND REPRESENTATION OF FAULTED NETWORKS

An unbalanced three-phase system can be resolved into three balanced systems in the sinusoidal steady state. This method of resolving an unbalanced system into three balanced phasor system has been proposed by C. L. Fortescue. This method is called resolving symmetrical components of the original phasors or simply symmetrical components. In this chapter we shall discuss symmetrical components transformation and then will present how unbalanced components like Y- or D-connected loads, transformers, generators and transmission lines can be resolved into symmetrical components. We can then combine all these components together to form what are called sequence networks.

SYMMETRICAL COMPONENTS

A system of three unbalanced phasors can be resolved in the following three symmetrical components:

·         Positive Sequence: A balanced three-phase system with the same phase sequence as the original sequence.

·         Negative sequence: A balanced three-phase system with the opposite phase sequence as the original sequence.

·         Zero Sequence: Three phasors that are equal in magnitude and phase.

Fig.  depicts a set of three unbalanced phasors that are resolved into the three sequence components mentioned above. In this the original set of three phasors are denoted by V_a, V_b and V_c, while their positive, negative and zero sequence components are denoted by the subscripts 1, 2 and 0 respectively. This implies that the positive, negative and zero sequence components of phase-a are denoted by V_a1, V_a2 and V_a0 respectively. Note that just like the voltage phasors given in Fig.  we can also resolve three unbalanced current phasors into three symmetrical components.

Fig.  Representation of (a) an unbalanced network, its (b) positive sequence, (c) negative sequence and       (d) zero sequence.

Symmetrical Component Transformation

Before we discuss the symmetrical component transformation, let us first define the a-operator. 

Note that for the above operator the following relations hold

Also note that we have

 Using the a-operator we can write from Fig. (b)

Similarly from Fig. (c) we get

Finally from Fig.  (d) we get

The symmetrical component transformation matrix is then given by

Defining the vectors V_a012 and V_abc as

we can write 

where C is the symmetrical component transformation matrix and is given by

The original phasor components can also be obtained from the inverse symmetrical component transformation, i.e.,

Finally, if we define a set of unbalanced current phasors as I_abc and their symmetrical components as I_a012, we can then define