Swing Equation: Know The Definition, Derivation, And Curve!
The swing equation holds tremendous importance in analyzing the transient
stability of synchronous generators in power systems. It is a nonlinear second-
order differential equation that describes the rotor swing dynamics of synchronous
machines.
This equation helps determine how the rotor angle (δ) of a synchronous generator
changes with respect to time under disturbances. It relates the acceleration of the
rotor to the mechanical input power and electrical output power of the generator.
In this comprehensive guide, we will discuss the swing equation in detail. We will
also explore the difference between steady-state and transient stability.
What Is Swing Equation?
The swing equation in power system represents the motion dynamics of the rotor
of a synchronous generator. It describes the oscillatory relative motion between the
rotor and synchronously rotating magnetic field with time in case of any
disturbance.
Mathematically, it is represented as:
Where,
M = Moment of inertia of the rotating masses of the machine (kg-m2)
δ = Rotor angle (electrical radians)
Pm = Mechanical input power to the generator (W)
Ps = Electrical output power from the generator (W)
d2δ/dt2 = Angular acceleration of the rotor (rad/s2)
Fig- Angular Position Of The Rotor
In simple terms, the swing equation equates the acceleration of the rotor (left side)
to the difference between the mechanical input power and electrical output power
(right side). It is a second order nonlinear differential equation that models the
oscillatory behavior of synchronous machines during small disturbances.
Derivation of Swing Equation
Let's see the step by step of the swing equation derivation:
Consider a generator supplying an active power P and reactive power Q
from its terminals.
The mechanical power input to the generator is Pm.
By the law of mechanics, the torque balance equation for the generator rotor
is:
Tm - Te = J(dω/dt)
Where,
Tm = Mechanical torque
Te = Electromagnetic torque
J = Moment of inertia of rotor
ω = Angular speed
The system's transient stability can be assessed using the swing equation. Consider
θ as the rotor's angular position at a given moment, t. The value of θ changes over
time, and for ease of measurement, it is referenced to the axis indicated in the
diagram provided.
The rotor's angular position is defined by the following equation:
………..(1)
Where,
θ is the angle between the rotor field and the reference axis
ws is synchronous speed
δ is the angular displacement
Taking the derivative of equation (1), we will get
………..(2)
Taking the derivative of equation (2), we will get
………..(3)
The rotor’s angular acceleration
………..(4)
The diagram shows the power flow within a synchronous generator. Ignoring
damping effects, the accelerating torques (Ta) in a synchronous generator can be
determined by calculating the difference between the mechanical torque input at
the shaft and the electromagnetic torque output, specifically,
Fig- Power Flow Diagram
………..(5)
Where,
Ta is accelerating torque
Ts is shaft torque
Te is electromagnetic torque
The angular momentum of the rotor is expressed by the equation
………..(6)
Where,
W - the synchronous speed of the rotor
J - moment of inertia of the rotor
M - angular momentum of the rotor
Multiplying both sides of equation (5) by w we will get
………..(7)
Where,
Ps - mechanical power input
Pe -electrical power output
Pa - accelerating power
But,
………..(7)
Equation (8) establishes the correlation between accelerating power and angular
acceleration, known as the swing equation. This equation portrays the dynamics
of the rotor in synchronous machines and plays a critical role in system
stabilization.
Importance Of Swing Equation
The swing equation plays a vital role in power system stability studies.
Some of its main significance include:
It aids in transient stability analysis to determine the ability of synchronous
machines to remain in synchronism after being subjected to a severe
disturbance.
The oscillatory behavior of synchronous generators during small
disturbances can be modeled using this equation.
The critical clearing time of faults can be estimated using the swing curve
derived from the swing equation.
It helps evaluate how angular separation between generators builds up post
fault-clearing and if the machines can withstand angular swings.
Parameters like inertia constant (H) can be determined from the swing
equation which characterizes the stored kinetic energy and transient response
of machines.
This equation forms the basis for simulation tools used in offline transient
stability assessment like electro-mechanical simulations.
Thus, a comprehensive understanding of the swing equation is necessary for power
system planning and operation involving transient stability aspects.
Swing Curve In Power System
The swing curve is a plot between the rotor angle (δ) and time (t) obtained by
solving the swing equation numerically or graphically. It provides useful insights
into the transient behavior of synchronous machines.
Fig- Swing Curve
The typical swing curve consists of three stages:
o Accelerating Swing: During this initial stage after fault, the accelerating
power is positive and the rotor accelerates.
o Steady Swing: Further acceleration reduces and the rotor settles to an
oscillatory motion at a constant speed.
o Diverging Swing: If fault clearing is delayed, accelerating power turns
negative and results in rotor deceleration leading to loss of synchronism.
By analyzing the swing curve, we can determine the critical clearing time:- the
maximum time by which a fault must be cleared to ensure that the rotor angle
swing does not increase beyond the stable limits. This aids in selecting proper
protection schemes.
Power Angle Equation
The power angle (δ) is defined as the angular separation between the rotational
position of the rotor flux vector and the stator flux vector of a synchronous
generator. It is a very important parameter in power system analysis.
The active power (P) output of a synchronous generator can be represented as a
function of power angle (δ) using the power angle equation
Fig- Swing Curve
The typical swing curve consists of three stages:
o Accelerating Swing: During this initial stage after fault, the accelerating
power is positive and the rotor accelerates.
o Steady Swing: Further acceleration reduces and the rotor settles to an
oscillatory motion at a constant speed.
o Diverging Swing: If fault clearing is delayed, accelerating power turns
negative and results in rotor deceleration leading to loss of synchronism.
By analyzing the swing curve, we can determine the critical clearing time:- the
maximum time by which a fault must be cleared to ensure that the rotor angle
swing does not increase beyond the stable limits. This aids in selecting proper
protection schemes.
Power Angle Equation
The power angle (δ) is defined as the angular separation between the rotational
position of the rotor flux vector and the stator flux vector of a synchronous
generator. It is a very important parameter in power system analysis.
The active power (P) output of a synchronous generator can be represented as a
function of power angle (δ) using the power angle equation
Fig- Power Angle curve
Difference between Steady State Stability and Transient Stability
The major differences between steady state and transient stability are:
Steady State Stability:
o Deals with the ability to maintain synchronism under small disturbances
o The system settles to a new steady operating state post-disturbance
o Analyzed through power flow solutions using swing equation linearized
Transient Stability:
o Concerned with maintaining synchronism during large disturbances
o The system may experience oscillatory swings before settling into a new
state
o Relatively longer settling time, analyzed through exact nonlinear swing
equation
While steady state stability focuses on maintaining synchronism in case of minor
faults, transient stability examines the capability to withstand major disturbances
like three phase faults and line outages from a dynamic perspective using swing
equation based simulations.
Conclusion
In this comprehensive guide, we discussed the swing equation in detail - its
derivation, meaning of terms, derivation of swing curve, significance, and
application in power system transient stability assessment.
The swing equation acts as the foundation for modeling synchronous generator
dynamics and serves as an important analytical tool. Mastery of this concept is
necessary for power engineers to analyze system behavior during disturbances and
ensure uninterrupted power supply.
