Swing equation

It's about the Swing Equation which is define for the Transmission lines.

1. GANDHINAGAR INSTITUTE OF TECHNOLOGY ELECTRICAL DEPARTMENT Interconnected Power System (2170901) ALA Presentation on Topic: Swing Equation Guided By: Prof. Piyush Pandya Prepared By: Darshil Shah (140120109050)

2. CONTENTS Introduction What is Power System Stability Stability Limit and Why Transient Stability limit is lower the Steady State stability limit? Rotor Dynamics of Power System Swing Equation Swing Curves Equal Area Criteria Conclusion Reference

3. INTRODUCTION In the Power System Analysis, approaches says that the Power system consists lots of devices interconnected to each other. Although Stability at every point must be common for all those devices. Here, each and every devices have them own protection from the transients. Stability of any system can easily be calculated from the generation or load side also.

4. POWER SYSTEM STABILITY Power System is a very large complex network consisting of synchronous generators, transformers, switch gears etc. Basically, if the fault is occurs in the system the synchronism may break and it will affect the whole line from Generation to the Load side. So, the stability criteria is most important here for remove the fault condition and make the suitable and appropriate operation for the all machines and whole power system. Power System Stability is the ability of the power system to return to steady state without losing synchronism. Usually Power System Stability is categorized into Steady State, Transient and Dynamic Stability.

5. STABILITY LIMIT The Stability limit is the maximum power that can be transferred in a network between source and load without loss of synchronism. The steady state limit is maximum power that can be transferred without the system becoming unstable. The transient stability limit is the maximum power that can be transferred without the system becoming unstable when sudden or large disturbance occurs. If in increase in field current or adjustments in speed settings occurs simultaneously with an increase of load from the use of automatic voltage regulators & speed governmenors, the stability limit would be increased significantly.

6. WHY TRANSIENT STABILITY IS LOWER THAN STEADY STATE STABILITY? If the system experiences a shock by sudden and large power changes and violent fluctuations of voltage occurs. Consequently machines or group of machines may go out of step. The rapidly of the application of the large disturbances is responsible for the loss of stability, otherwise it may be possible to maintain stability if the same large load is applied gradually. Thus, the transient stability limit is lower than the steady state limit.

7. ROTOR DYNAMICS OF POWER SYSTEM A synchronous machine is a rotating body, the laws of machines of rotating bodies are also applicable to it. Kinetic energy of the rotor at synchronous machine is, J = rotor moment of inertia in kg- m synchronous speed in rad (mech)/sec   MJ   But, = rotor speed in rad (elect)/sec P = number of machine poles MJ = where, = moment of inertia in MJ-sec/elect rad. We shall define the inertia constant H such that, MJ It immediately follows that, MJ-sec/elect rad.  

8. SWING EQUATION Figure shows the torque, speed and flow of mechanical and electrical power in synchronous machine. It is assumed that the voltage, friction and iron loss torque is negligible. The differential equation governing the rotor dynamics can then be written as: Where, = angle in radian ( mech ) = turbine torque in Nm; it acquires a negative value for a motoring machine. = electromagnetic torque developed in Nm While the rotor undergoes dynamics as per equation, the rotor speed changes by insignificant magnitude for the time period of interest (1sec). This equation can be converted into its more convenient power form by assuming the rotor speed to constant at synchronous speed ( ). Multiply bothe side of equation by so,   Nm   MW   Where, = mechanical power input in MW = electrical power output in MW Rewriting this equation, MW Or . It is more convenient to measure the angular position of the rotor with respect to synchronously rotating frame of reference. ; rotor angle displacement from sync. Rotating reference frame called Torque angle or Power angle.  

9. SWING EQUATION From the equation, Hence, the equation can be written in terms of, MW With M as defined in equation, we can write MW Dividing through G, the MVA rating in machine,   ; in pu of machine rating as base   or, pu This equation is called as “Swing Equation” . It is a second order differential equation where the damping term is absent because of the assumption of a lossless machine & the fact that the torque of damper winding has been ignored. Since, the electrical power depends upon the sine of angle . Swing equation is non-linear second order diff. equation.  

10. SWING CURVE

11. EQUAL AREA CRITERIA This is a simple graphical method to predict the transient of two machine system or a single machine against infinite bus. This criterion does not required Swing Equation or solution or Swing Equation to determine the stability condition. The stability condition are determined by equating the areas of segments on Power angle diagram. The equation as follows: Consider the relationship between load angle and power derived from Swing equation, Here, Angular momentum is denoted as M , Load angle is denoted as δ L , and Actual power is denoted as P A .  

12. EQUAL AREA CRITERIA Consider the relation between electrical and mechanical angles, Therefore, the equation becomes ,  

13. CONCLUSION Hence, this equation is very helpful to determine the stability condition of any complex power system through calculating the angle δ and also having the equal area criteria. So, any kind of transients can be eliminated from this.

14. REFERENCE www.en.Wikipedia.org/Swing-Equation https:// www.electrical4u.com/transient-stability-and-swing-equation www. circuitglobe.com/swing-equation Power Systems and Analysis by D P Kothari and I J Nagrath

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