Swing equation

A synchronous generator is driven by a prime mover. The equation governing the rotor motion is given by:{{sfn|Grainger|Stevenson|1994|pp=700-702}}

$$J\backslash frac\{d^2\{\backslash theta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; T\_a\; =\; T\_\backslash text\{m\}\; -\; T\_\backslash text\{e\},$$

where:

• $J$ is the total moment of inertia of the rotor mass in kg-m²
• $\backslash theta\_\backslash text\{m\}$ is the angular position of the rotor with respect to a stationary axis in radians (rad)
• $t$ is time in seconds (s)
• $T\_a$ is the net accelerating torque, in N-m
• $T\_\backslash text\{m\}$ is the mechanical torque supplied by the prime mover in N-m
• $T\_\backslash text\{e\}$ is the electrical torque output of the alternator in N-m

Neglecting losses, the difference between the mechanical and electrical torque gives the net accelerating torque $T\_a$. In the steady state, the electrical torque is equal to the mechanical torque and hence the accelerating power is zero. During this period the rotor moves at synchronous speed $\backslash omega\_s$ in rad/s. The electric torque $T\_\backslash text\{e\}$ corresponds to the net air-gap power in the machine and thus accounts for the total output power of the generator plus Electric_power#Resistive_circuits in the armature winding.{{sfn|Umans|2013|pp=263-264}}

The angular position $\backslash theta$ is measured with a stationary reference frame. Representing it with respect to the synchronously rotating frame gives:

$$\backslash theta\_\backslash text\{m\}\; =\; \backslash omega\_\backslash text\{s\}t\; +\; \backslash delta\_\backslash text\{m\},$$

where $\backslash delta\_m$ is the angular position with respect to the synchronously rotating reference frame. The derivative of the above equation with respect to time is:

$$\backslash frac\{d\backslash theta\_\backslash text\{m\}\}\{dt\}\; =\; \backslash omega\_\backslash text\{s\}\; +\; \backslash frac\{d\backslash delta\_\backslash text\{m\}\}\{dt\}.$$

The above equations show that the rotor angular speed is equal to the synchronous speed only when $d\backslash delta\_m\; /dt$ is equal to zero. Therefore, the term $d\backslash delta\_m\; /dt$ represents the deviation of the rotor speed from synchronism in rad/s.

By taking the second order derivative of the above equation it becomes:

$$\backslash frac\{d^2\backslash theta\_\backslash text\{m\}\}\{dt^2\}\; =\; \backslash frac\{d^2\backslash delta\_\backslash text\{m\}\}\{dt^2\}.$$

Substituting the above equation in the equation of rotor motion gives:

$$J\backslash frac\{d^2\{\backslash delta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; T\_a\; =\; T\_\backslash text\{m\}\; -\; T\_\backslash text\{e\}.$$

Multiplying both sides by the angular velocity of the rotor, given by

$$\backslash omega\_\backslash text\{m\}\; =\; \backslash frac\{\{d\backslash theta\_\backslash text\{m\}\}\}\{dt\},$$

results in

$$J\backslash omega\_\backslash text\{m\}\backslash frac\{d^2\{\backslash delta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; P\_a\; =\; P\_\backslash text\{m\}\; -\; P\_\backslash text\{e\},$$

where $P\_a$, $P\_\backslash text\{m\}$ and $P\_\backslash text\{e\}$ respectively are the accelerating, mechanical and electrical (active) power in Watt (W). Intuitivley, the equation can also be derived by taking the time derivative of the rotational energy.

The coefficient $J\; \backslash omega\_\{\backslash text\{m\}\}$ is the angular momentum of the rotor at synchronous speed $\backslash omega\_\{\backslash text\{s\}\}$. In machine data supplied for stability studies this coefficient is often denoted by $M$ and called the inertia constant of the machine. In practice, $\backslash omega\_\{\backslash text\{m\}\}$ does not differ significantly from synchronous speed when the machine is in steady state; allowing for another constant of inertia:

$$H\; =\; \backslash frac\{\backslash frac\{1\}\{2\}J\backslash omega\_\backslash text\{s\}^2\}\{S\_\{\backslash text\{rated\}\}\}=\; \backslash frac\{\backslash frac\{1\}\{2\}M\; \backslash omega\_\backslash text\{s\}\}\{S\_\{\backslash text\{rated\}\}\}=\; \backslash frac\{\backslash text\{stored\; kinetic\; energy\; in\; mega\; joules\; at\; synchronous\; speed\}\}\{\backslash text\{machine\; rating\; in\; MVA\}\},$$

where $S\_\{\backslash text\{rated\}\}$ is the three phase rating of the machine in MVA. Substituting in the above equation

$$2H\backslash frac\{S\_\{\backslash text\{rated\}\}\}\{\backslash omega\_\backslash text\{s\}^2\}\backslash omega\_\backslash text\{m\}\backslash frac\{d^2\{\backslash delta\_\backslash text\{m\}\}\}\{dt^2\}\; =\; P\_\backslash text\{m\}\; -\; P\_\backslash text\{e\}\; =\; P\_a.$$

Since $P\_\backslash text\{m\}$, $P\_\backslash text\{e\}$ and $P\_a$ in the machine data are given in MW, dividing them by the generator MVA rating gives these quantities in per unit. Dividing the above equation on both sides by $S\_\{\backslash text\{rated\}\}$ gives

{{Equation box 1

|indent = :

|equation = $\backslash frac\{2H\}\{\backslash omega\_\backslash text\{s\}\}\backslash frac\{d^2\{\backslash delta\}\}\{dt^2\}\; =\; P\_\backslash text\{m\}\; -\; P\_e\; =\; P\_a$ per unit

|cellpadding= 5

|border

|border colour = #0073CF

|bgcolor=#F9FFF7

}}

The above equation describes the behaviour of the rotor dynamics and hence is known as the swing equation. The angle $\backslash delta$ is the angle of the internal EMF of the generator and it dictates the amount of power that can be transferred. This angle is therefore called the power angle. When considering a network without resistance, the corresponding power angle equation is:{{sfn|Schavemaker|van der Sluis|2008|pp=71-74}}{{sfn|Grainger|Stevenson|1994|pp=709-712}}

$$P\_e\; =\; P\_\{max\}\; \backslash sin\; (\backslash delta).$$

Hence, the swing equation is non-linear and can be solved numerically using, e.g., the fourth-order Runge-Kutta algorithm. When $\backslash delta$ is small, the equation can be linearized as $P\_e\; \backslash approx\; P\_\{max\}\backslash delta$.{{sfn|Guru|Hiziroǧlu|2001|pp=648-651}}

• {{Slink|Electric motor|Operating principles}}
• Synchronverter

{{reflist}}

• {{cite book|first1=John J. |last1=Grainger|first2=William D.|last2= Stevenson|title=Power system analysis|url={{google books |plainurl=y |id=NBIoAQAAMAAJ}}|date=1994|publisher=McGraw-Hill|isbn=978-0-07-061293-8}}
• {{cite book | last=Guru | first=Bhag S. | last2=Hiziroǧlu | first2=Hüseyin R. | title=Electric machinery and transformers | publisher=Oxford University Press | publication-place=New York | year=2001 | isbn=978-0-19-513890-0 | oclc=43286555}}
• {{cite book | last1=Schavemaker | first1=P. | last2=van der Sluis | first2=L.| title=Electrical Power System Essentials | publisher=Wiley | publication-place=Chichester, England ; Hoboken, NJ | date=2008 | isbn=978-0-470-51027-8 | oclc=193911699}}
• {{cite book | last=Umans | first=Stephen D. | title=Fitzgerald & Kingsley's Electric Machinery | publisher=McGraw-Hill Education | publication-place=New York | date=2013-01-28 | isbn=978-0-07-338046-9}}

Category:Equations

Category:Electric power transmission systems