Power Systems : Power System Stability

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.

If the system is occurred by any disturbance, either small or large, it affects the synchronous operation of the system. This occurrence creates the disturbance between mechanical input torque and electrical output torque of each machine, resulting in acceleration or deceleration of the machine.

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:

1. Rotor angle stability
2. Voltage stability

The rotor angle stability is further classified into two components i.e.

(1). Steady state stability is the ability of a system to restore the initial condition after a small disturbance or to reach a condition very close to the initial condition when the disturbance is still present. These small changes in load or generation is termed as disturbances.

(2). Transient stability is a condition that characterizes the dynamics of a power system subjected to a fault, the initial state preceding the fault being a balanced one. Power system faults, which result in a sudden dip in bus voltages and require immediate remedial action in the form of clearing of the fault, can be termed as large disturbances. A system is said to possess transient stability if after the fault it is capable of maintaining synchronous operation and returning to its initial state or a state close to it.

Transient stability of a system depends on various factors such as time of fault clearance, type and location of fault, time of reclosing after fault clearance.

The maximum amount of steady-state power that the system can be transmitted for specified operating conditions without losing synchronism after being subjected to a fault is known as the transient stability limit of the system.

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.

Assumptions made in Transient stability Analysis:

1. The synchronous machine is represented by a constant voltage E’ behind direct axis transient reactance X’_d.
2. Power input P_i from governor and turbine remains constant, i.e. equal to the pre-fault value during complete analysis.
3. The mechanical angle δ_i of each rotor coincides with the electrical phase angle of voltage behind the transient reactance.
4. Damping or asynchronous power is negligible.
5. The network parameters are modelled as lumped circuits and remain fixed during stability analysis, even in the variation in frequency during transient period which is taken to be negligible.
6. Loads are converted into equivalent admittances based on the voltages at the buses calculated from the pre-fault analysis.
7. Synchronous power may be calculated from the steady-state solution of network to which the machines are connected.
   
   Swing Equation

   

   Where M = Iω = Angular momentum in J-s mechanical radian.

   I = Moment of Inertia.

   P_a = Iαω = T_aω = Accelerating power

   α = Angular acceleration

   ω = Angular velocity

   P_s = Shaft power

   P_e = 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 (H_eq) of several identical machines working in parallel is the same as that of any one of the machines (H_eq = H).
     
     
   • The equivalent inertia constant (H_eq) of two machines on a common base swing coherently H_eq = H₁ + H₂
     
     
   • The equivalent inertia constant (H_eq) of two synchronous machines which do not swing together is.
     
     

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