Module 2 : Equipment and Stability Constraints in System Operation
Lecture 7 : Insert : Numerical Solution of Differential Equations
Objectives
What does one mean by stability?
What is angular stability?
Dynamical equations for a single machine -infinite bus system. Equilibrium points.
What is Stability?
Stability is essentially the ability of a system to recover from disturbances, both large and small, and settle to an
acceptable equilibrium. Randomly occurring load changes, faults resulting in line or generator tripping and
changes in reference values of regulating controllers, are examples of disturbances.
One should note that equipment constraints are distinct from stability constraints in the sense that even though
an equilibrium condition may exist (which is within equipment constraints), a system may not be able to "settle
down" to it if it is perturbed or initially away from it.
If a system is not stable for even small disturbances, it cannot be operated at all since there are always small and
random perturbations in the system due to load variations
If a system is stable for small disturbances but unstable if the disturbances are "large", then the system can be
operated. However, the system may not be secure, i.e., it may be unstable if a large enough disturbance does
actually occur .
The major stability problems which are inherent in AC interconnected grids are discussed next.
Angle Stability
An interesting physical characteristic of interconnected synchronous generators is their ability to generate
restoring torques when disturbed from an equilibrium. These torques ensure that all machines stay in
synchronism --- generator electrical speeds become equal in steady state. Equivalently, the phase angular
differences between ac voltages at various points become constant if machines stay in equilibrium.
However, the restoring torques can become zero or negative for very large disturbances. This can result in
machines falling out of step (i.e., they lose synchronism - the machines do not settle to the same electrical
speed; this makes operation unviable - see Module 1). Sometimes, due to presence of automatic controllers,
damping of the rotor oscillations is inadequate or negative, causing growing or sustained oscillations ('hunting')
and may also lead to loss of synchronism.
The problem of loss of synchronism between synchronous machines is also known as the "angular stability
problem". Note that if machines lose synchronism, then the phase angular difference between ac buses in the
system will not settle to constant values.
To study the angular stability problem, consider the governing equations of the system.
Lecture 7 : Insert : Numerical Solution of Differential Equations
Objectives
What does one mean by stability?
What is angular stability?
Dynamical equations for a single machine -infinite bus system. Equilibrium points.
What is Stability?
Stability is essentially the ability of a system to recover from disturbances, both large and small, and settle to an
acceptable equilibrium. Randomly occurring load changes, faults resulting in line or generator tripping and
changes in reference values of regulating controllers, are examples of disturbances.
One should note that equipment constraints are distinct from stability constraints in the sense that even though
an equilibrium condition may exist (which is within equipment constraints), a system may not be able to "settle
down" to it if it is perturbed or initially away from it.
If a system is not stable for even small disturbances, it cannot be operated at all since there are always small and
random perturbations in the system due to load variations
If a system is stable for small disturbances but unstable if the disturbances are "large", then the system can be
operated. However, the system may not be secure, i.e., it may be unstable if a large enough disturbance does
actually occur .
The major stability problems which are inherent in AC interconnected grids are discussed next.
Angle Stability
An interesting physical characteristic of interconnected synchronous generators is their ability to generate
restoring torques when disturbed from an equilibrium. These torques ensure that all machines stay in
synchronism --- generator electrical speeds become equal in steady state. Equivalently, the phase angular
differences between ac voltages at various points become constant if machines stay in equilibrium.
However, the restoring torques can become zero or negative for very large disturbances. This can result in
machines falling out of step (i.e., they lose synchronism - the machines do not settle to the same electrical
speed; this makes operation unviable - see Module 1). Sometimes, due to presence of automatic controllers,
damping of the rotor oscillations is inadequate or negative, causing growing or sustained oscillations ('hunting')
and may also lead to loss of synchronism.
The problem of loss of synchronism between synchronous machines is also known as the "angular stability
problem". Note that if machines lose synchronism, then the phase angular difference between ac buses in the
system will not settle to constant values.
To study the angular stability problem, consider the governing equations of the system.
