Power system consists of several elements such as generators, transformers, power transmission lines, distribution networks, and loads, as well as control elements such as automatic voltage regulators of synchronous machines, automatic load frequency control, protective relays, and circuit breakers. All these elements combine to form a system and every individual element possess their different role and properties to maintain the system.
Under normal conditions, i.e., steady state, the speed of all the connected generators remain the same throughout the system. In the steady-state condition, there is equilibrium between the input mechanical torque and output electrical torque of each machine. This is termed as the synchronous operation of the system.
Consider if a generator losses synchronism and runs faster than another machine, then the rotor angular position of the machine relative to the slower machine will advance. The resulting relative angular difference created will transfers a part of the load from the slow machine to the fast machine, depending on the power–angle relationship, and this tends to reduce the speed difference and hence the angular separation. The power angle characteristic being sinusoidal is non-linear.
And beyond a certain limit, an increase in angular separation will decrease in power transfer capability, which further increases angular separation & finally leads to instability.
The tendency of a power system to develop restoring forces equal to or greater than the disturbing forces to maintain synchronous running of generators is known as stability. Hence, the stability of the system depends on whether or not the deviations in angular positions of the rotors result in sufficient restoring forces.
Therefore, when a machine loses synchronism with other machines of the system, its rotor angle changes, also voltage and frequency may change with larger values from their rated values.
The stability problem in power system is broadly classified into two categories:
- Rotor angle stability
- Voltage stability
The rotor angle stability is further classified into two components i.e.: Transient stability
Steady state stability
TRANSIENT STABILITY
Transient stability concerns with transmission of power from one group of synchronous machines to another. During disturbances, the machines of each group swing together and are said to form a coherent group.
In a two-machine system, when load on one machine is greater than another machine by at least ten times, then it can be represented by an infinite bus (i.e., constant voltage source with zero internal impedance and infinite inertia).
Transient stability is further divided into two categories:
(a). First swing, which lasts for about one second, in which the prime mover input to the generator and its voltage behind transient reactance is assumed to be constant.
(b). Multi swing, which extends over longer period, which affects turbine governor and excitation system.
Swing Equation
Where M = Iω = Angular momentum in J-s mechanical radian.
I = Moment of Inertia.
Pa = Taω = Accelerating power
α = Angular acceleration
ω = Angular velocity
Ps = Shaft power
Pe = Electrical power
δ = Power angle or torque angle
Inertia Constant
Inertia constant 
Stored energy in megajoule = G × H
Inertia constant (H) on a Common Base
Where S = MVA rating
Key Points
The equivalent inertia constant (Heq) of several identical machines working in parallel is the same as that of any one of the machines (Heq = H).
The equivalent inertia constant (Heq) of two machines on a common base swing coherently Heq = H1 + H2
The equivalent inertia constant (Heq) of two synchronous machines which do not swing together is.
METHODS OF IMPROVING TRANSIENT STABILITY
There are various methods of improving transient stability which are as follows:
- By minimizing fault severity and duration
- Increase the restoring synchronizing forces
- Reduce the accelerating torque through prime-mover control
- Reduce the accelerating torque by applying artificial load
- Earthing of transformer neutral through resistance or reactance
- Series compensation of lines
Equal Area Criterion
The accelerating power in swing equation will have sine term. Therefore the swing equation is a non-linear differential equation and obtaining its solution is not simple. For two machine system and one machine connected to infinite m bus bar, it is possible to say whether a system has transient stability or not, without solving the swing equation. Such criteria which decide the stability makes use of the equal area in power angle diagram and hence it is known as EQUAL AREA CRITERION. Thus the principle by which stability under transient conditions is determined without solving the swing equation, but makes use of areas in power angle diagram, is called the EQUAL AREA CRITERION.
STEADY STATE ANALYSIS
Steady state stability relates to the response of synchronous machine to a gradually increasing load. It is basically concerned with the determination of the upper limit of machine loading without losing synchronism, provided the loading is increased gradually.
Power Flow
For lossless line R = 0 and the transmitted power.
Where,
Vs = Sending end voltage
VR = Receiving end voltage
X = Reactance of the line
δ = Load angle
The steady state stability limit of a particular circuit of a power system defined as the maximum power that can be transmitted to the receiving end without loss of synchronism.
IMPROVING STEADY STATE STABILITY LIMIT
The major problem of steady-state stability is due to insufficient damping oscillations in the system. The use of power system stabilizers to control generator excitation is the most cost-effective method for steady-state stability improvement of power systems. In addition, supplementary stabilizing signals, HVDC converter controls, and static var compensators may be used to improve damping of system oscillations.
Steady-state power limit is increased by selecting higher transmission voltages for new systems and upgrading the voltages on the existing systems. Steady- state power limit is also increased by reducing the system transfer reactance by additional parallel transmission lines. The use of bundled phase or oversized conductors in transmission lines reduces the series reactance of transmission lines.
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