Swing Equation – Definition and Derivation
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Swing Equation – Definition and Derivation
What is Swing Equation?
The rotor axis and the synchronously rotating stator field axis move in relation to time, according to the generator swing equation. The stability of connected machines—here, a machine is a generator—can be examined very effectively using this equation.
Derivation of Swing Equation:
As is well known, under normal operating conditions, the speed of the rotor axis and stator field axis in a synchronous generator is equal to synchronous speed (N = 120f/P). Simply put, this indicates that there is no relative speed between the rotor axis and the stator field axis. Thus, under typical operating conditions, a constant angle is maintained between the rotor field axis and stator field axis. The term “load angle” or “torque angle” refers to this angle.
The machine’s load determines the load angle’s value. According to the power equation, the load angle δ will increase as the load increases.
P = EfVtSinδ / Xs
where Ef = No Load Excitation Voltage
Vt = Generator Terminal Voltage
Xs = Synchronous Impedance
Additionally, during generator steady state operation, the shaft torque Ts and electromagnetic torque Te have the same value. There is no net torque applied to the rotor because the load torque Ts and electromagnetic torque Te act in opposite directions. For this reason, the generator’s rotor rotates at a constant angular speed during steady state operation.
However, if the load on the machine is suddenly changed, such as by adding or removing weight from the rotor shaft, the rotor will accelerate or decelerate relative to the synchronously rotating stator field. If the steam input to the generator is suddenly increased, the rotor will accelerate because the shaft torque Ts has increased while the electromagnetic torque Te will remain almost constant. The rotor and stator field axes will then begin to move relative to one another. The stator filed axis can be used as a reference for the study of relative motion because generator terminals are assumed to be connected to the grid. Due to this relative motion, the load angle δs will vary with time and can be written as
δs = ωrt + δ ………………(1)
where δs is the angle between the reference stator field axis and rotor axis at any time t and δ is the load angle just before the rotor disturbance.
where wr is the relative angular speed between the rotor axis and stator filed axis.
Also, Ts – Te = Ta
where Ta = Net Accelerating Torque
The above torque can be replaced by power for analysis since a synchronous machine rotates at a constant angular speed. Te and Ts are thus swapped out for electromagnetic power Pe and shaft power Ps, respectively. As a result, the rotor’s accelerating force Pa can be expressed as,
Ps – Pe = Pa
But as per mechanics, power = Torque x Angular Speed
Accelerating power Pa = Taω ………(2)
Since, Torque = Inertia (I) x Angular Acceleration (α)
Therefore from (2),
Pa = Iωα ………………(3)
Let, M = Iω
But Iω is the angular momentum, hence M is called the angular momentum of generator rotor. Therefore from (3),
Rotor / Shaft Accelerating Power Pa = Mα …….(4)
It should be noted that the unit of angular speed w should be taken into consideration in mechanical radians per second when using mechanics to analyse the relative motion between the rotor axis and stator axis. However, since we can use an equation to change this mechanical radian per second into an electrical radian per second
Electrical radian or degree = Mechanical Radian or degree x No. of Pole Pairs
But the angular position of rotor w.r.t is described by (1),
δs = ωrt + δ
Differentiating both side w.r.t time,
dδs / dt = ωr +dδ/dt
Again differentiating w.r.t time,
d2δs / dt2 = d2δ/dt2
But d2δs / dt2 = angular acceleration of rotor i.e. α
Hence from (4),
Pa = Mα = Md2δ/dt2
But Pa = Ps – Pe
⇒ Ps – Pe = Md2δ/dt2 …………..(5)
The above equation is known as the Swing Equation.
Different forms of Swing Equation:
Different ways can be used to express the swing equation mentioned above. It is essential to talk about the terms inertia constant M and H before expressing this equation in terms of other parameters.
Since the machine’s angular momentum M = Iω, M will practically never change as long as the machine maintains a constant synchronous speed. The machine’s inertia constant, M, is referred to as such. The machine’s size affects M’s value. The inertia constant M will increase as the size increases.
Another common constant in stability analysis is H. The ratio of a machine’s kinetic energy to its rated MVA machine capacity is how this constant is defined. Thus
H = Machine Kinetic Energy / Rated MVA Capacity
This constant H is also known as inertia constant. The relationship between M and H can be derived as below.
Let G = Rated MVA capacity of Machine
f = system frequency
As per definition of H,
H = Kinetic Energy / G
Kinetic Energy of Machine = Mω/2
= [M x (2πf)]/2
H = Mπf / G
⇒ M = GH / πf Mega Joule Second / Radian
= GH / 180f Mega Joule Second / electrical radian
Thus from (5) Swing Equation can also be written as,
Ps – Pe = (GH/180f) x d2δ/dt2 ……(6)
In terms of per unit system, the Swing Equation becomes
(Ps – Pe)pu = (H/180f) x d2δ/dt2
Notice that the above equation has been obtained by dividing (6) by G. This is done to convert into per unit system.