Equal Area Criteria MCQ Quiz - Objective Question with Answer for Equal Area Criteria - Download Free PDF

Explanation:

Equal Area Criterion in Power-Angle Curve

Definition: The equal area criterion is a method used to analyze the stability of a synchronous machine under transient conditions. It is based on the principle that the area of acceleration (A₁) and the area of deceleration (A₂) in the power-angle curve should be equal for the system to remain stable after a disturbance. The power-angle curve represents the relationship between the electrical power output of a synchronous machine and the rotor angle (δ).

Working Principle: When a disturbance occurs, the rotor angle changes, causing a temporary imbalance between the mechanical input power (P_m) and the electrical output power (P_e). The rotor accelerates or decelerates depending on whether P_m is greater than or less than P_e. The equal area criterion ensures that the energy gained during acceleration is equal to the energy lost during deceleration, allowing the rotor to settle at a new equilibrium point.

Correct Option Analysis:

The correct option is:

Option 3: \(\rm A_{1}=\int_{\delta_{o}}^{\delta_{c}}\left(P_{m}-P_{e}\right) d \delta=0\)

This option accurately defines the area of acceleration (A₁) in the power-angle curve. The integral represents the energy gained by the rotor during acceleration, which occurs when the mechanical input power (P_m) exceeds the electrical output power (P_e). The limits of integration are δ_o (the initial rotor angle at the start of the disturbance) and δ_c (the critical rotor angle where the system transitions to deceleration). The condition for stability is that the total energy gain (A₁) during acceleration equals the total energy loss (A₂) during deceleration.

Mathematical Representation:

The energy gained during acceleration is given by:

\(A_{1} = \int_{\delta_{o}}^{\delta_{c}} (P_{m} - P_{e}) \, d\delta\)

The energy lost during deceleration is given by:

\(A_{2} = \int_{\delta_{c}}^{\delta_{max}} (P_{e} - P_{m}) \, d\delta\)

For stability:

\(A_{1} = A_{2}\)

This ensures that the rotor angle returns to a stable operating point after the disturbance.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm A_{1}=\int_{\delta_{c}}^{\delta_{o}}\left(P_{m}-P_{e}\right) d \delta=0\)

This option incorrectly reverses the limits of integration. The area of acceleration (A₁) is always defined from the initial rotor angle (δ_o) to the critical rotor angle (δ_c), not vice versa. Reversing the limits would result in a negative value for A₁, which is physically incorrect in the context of energy gained during acceleration.

Option 2: \(\rm A_{1}=\int_{\delta_{o}}^{\delta_{c}}\left(P_{e}-P_{m}\right) d \delta=0\)

This option swaps the terms inside the integral, using (P_e - P_m) instead of (P_m - P_e). This representation corresponds to the area of deceleration (A₂), not acceleration (A₁). The area of acceleration must be defined by the difference (P_m - P_e), which represents the net power causing the rotor to accelerate.

Option 4: \(\rm A_{1}=\int_{\delta_{c}}^{\delta_{o}}\left(P_{e}-P_{m}\right) d \delta=0\)

This option combines the errors from options 1 and 2. It reverses the limits of integration and swaps the terms inside the integral. As explained earlier, the limits of integration for A₁ must be from δ_o to δ_c, and the integrand must be (P_m - P_e). This option is therefore doubly incorrect.

Conclusion:

The equal area criterion is a vital tool for analyzing the transient stability of synchronous machines. By ensuring that the areas of acceleration (A₁) and deceleration (A₂) are equal, we can determine whether the system will return to a stable operating point after a disturbance. The correct representation of A₁ is:

\(A_{1} = \int_{\delta_{o}}^{\delta_{c}} (P_{m} - P_{e}) \, d\delta\)

This integral accurately describes the energy gained during acceleration, and its equality with the energy lost during deceleration ensures stability. Understanding the limits of integration and the terms inside the integral is crucial for correctly applying the equal area criterion in power system analysis.

Explanation:

Transient Stability Analysis and Equal Area Criterion:

Definition: Transient stability analysis is a crucial aspect of power system stability that evaluates the ability of a power system to maintain synchronism when subjected to a large disturbance, such as a short circuit or sudden load change. The rotor angle stability is an essential parameter in this analysis, as it represents the relative angular displacement of the synchronous machine rotors.

Equal Area Criterion: The equal area criterion is a graphical method used to assess transient stability. It states that for a system to remain stable following a disturbance, the area representing accelerating power (the excess mechanical input power over electrical output power) should be equal to the area representing decelerating power (the excess electrical output power over mechanical input power). These areas are plotted on the power-angle curve, which depicts the relationship between electrical power and rotor angle.

Correct Option Analysis:

The correct option is:

Option 3: Greater than 90˚

In transient stability analysis, the maximum rotor angle to which the rotor can oscillate is determined by the equal area criterion. The rotor angle can exceed 90˚ depending on the disturbance magnitude, system parameters, and power-angle characteristics. While small disturbances generally result in rotor angles below 90˚, large disturbances may cause the rotor angle to oscillate beyond 90˚ without losing synchronism, provided the accelerating and decelerating areas on the power-angle curve are equal.

This behavior occurs due to the nonlinear nature of the power-angle relationship, where stability depends not only on the magnitude of the rotor angle but also on the system's ability to balance accelerating and decelerating powers. Therefore, when the equal area criterion is satisfied, the rotor angle can oscillate to values greater than 90˚ without losing stability.

Additional Considerations:

It is important to note that the rotor angle exceeding 90˚ does not necessarily imply instability. Stability is determined by the ability of the system to return to synchronism after the disturbance. As long as the equal area criterion is satisfied, the system remains stable even if the rotor angle exceeds 90˚ momentarily during oscillations.

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 1: 0˚ to 20˚

This option is incorrect because it severely underestimates the range of rotor angle oscillation during transient stability analysis. Rotor angles typically oscillate to values much higher than 20˚, especially for large disturbances. The power-angle curve and equal area criterion allow for a much broader range of oscillation, including values exceeding 90˚.

Option 2: 45˚ to 50˚

While rotor angles may oscillate within this range under specific conditions, this option is not universally applicable. The transient stability analysis considers a wide range of rotor angle oscillations, and the maximum angle depends on the system parameters and disturbance magnitude. In many cases, the rotor angle can exceed 50˚ or even 90˚ without losing stability.

Option 4: 65˚ to 85˚

This option is closer to the typical range observed in transient stability analysis but is still not universally applicable. The rotor angle can exceed 85˚ in scenarios involving large disturbances, provided the equal area criterion is satisfied. Therefore, limiting the maximum angle to this range does not encompass all possible stable conditions.

Conclusion:

The transient stability analysis is a complex evaluation of power system behavior under disturbances. The correct understanding of rotor angle oscillations, as governed by the equal area criterion, is crucial for ensuring system stability. While rotor angles within certain ranges may be typical, the ability of the system to remain stable depends on the balance of accelerating and decelerating powers, not the specific value of the rotor angle. As explained, the rotor angle can oscillate beyond 90˚ without losing stability, making Option 3 the correct answer.

Synchronous Machine Dynamics During Disturbances:

In synchronous machines, the rotor and stator magnetic fields lock together to maintain a constant speed called synchronous speed. During disturbances such as sudden changes in load, the rotor can experience oscillations before settling back to a steady state. These oscillations are characterized by changes in rotor angle and speed.

Explanation of the Concept:

• The power angle curve represents the relationship between the electrical power output of the synchronous machine and the rotor angle. Points A, B, and C on this curve indicate different states during the machine's dynamic response to a disturbance.
• Point A: The initial operating point before the disturbance.
• Point B: The maximum rotor angle deviation during oscillation. At this point, the machine's speed is not synchronous due to the inertia of the rotor, leading to acceleration or deceleration.
• Point C: The final steady-state operating point after the oscillation settles.

Detailed Solution:

• During the oscillation, the rotor speed will vary above and below the synchronous speed. The rotor accelerates when the electrical power is greater than the mechanical power and decelerates when the electrical power is less than the mechanical power.
• At points A and C, the machine is in a steady state, meaning the rotor speed is synchronous. This is because there are no unbalanced forces acting on the rotor, and it is locked with the stator magnetic field.
• At point B, the rotor speed is not synchronous as it represents the maximum deviation from the steady state due to the disturbance. The rotor will have either accelerated or decelerated during its swing from A to B.

Conclusion:

• The correct answer is option 3: A and B. At points A and C, the rotor speed is synchronous, but at point B, it is not.

Equal area criteria:

• It is a graphical method that allows assessing the transient stability of electric power systems in a simple and comprehensive way.
• It is also a graphical solution to the basic swing equation.
• Its use eliminates the need of computing the swing curves of the system, thus saving a considerable amount of work.

For transient stability:

• For the system to be stable the accelerating area should be less than decelerating area.
• For the system to be critically stable the accelerating area is equal to the decelerating area.
• If the accelerating area is more than decelerating area then the system is unstable.
• During the swinging of the rotor, the maximum may exceed 90°. It is only area stability, but not angle stability.
• For stability, it only considers the first swing of the rotor after disturbance. 

Assumptions to evaluate the transient stability: 

• The resistances, the shunt capacitance of generator and transmission lines are ignored. The shunt capacitor or shunt inductor at load Bus or generator bus is ignored.
• The mechanical input is assumed to be constant.
• There is no change in the speed of the alternator.
• The damping force provided by damper winding is ignored.
• The voltages behind the reactances of the Machines are ignored.

Equal Area Criteria:

• The equal-area criterion is a simple graphical method for concluding the transient stability.
• This principle does not require the swing equation for the determination of stability conditions.
• The stability conditions are recognized by equating the areas of segments on the power angle diagram.

​Starting with the swing equation:

\(M\left( {\frac{{{d^2}δ }}{{d{t^2}}}} \right) = {P_s} -P_e\)

Where,

M is the moment of inertia or angular momentum

δ is the power angle between rotor rotating field and stator rotating field

Ps is a mechanical input power

P_e is the electrical output

Multiply the above equation with dδ / dt

we get,

\(\frac{1}{2}M\frac{d}{{dt}}{\left( {\frac{{dδ }}{{dt}}} \right)^2} = \left( {{P_s} - {P_e}} \right)\frac{{dδ }}{{dt}}\)

Rearranging, multiplying by dt, and integrating, we have

\(\frac{{dδ }}{{dt}} = \sqrt {\mathop \smallint \limits_{{δ _0}}^δ \frac{{2\left( {{P_s} - {P_m}} \right)}}{M}dδ } \)

At steady-state condition, the torque angle was not changing i.e. before the disturbance.

dδ / dt = 0

Also, if the system has transient stability the machine will again operate at synchronous speed after the disturbances, i.e.,

dδ /dt = 0

Hence the condition for stability is dδ / dt =0

Equal area criterion:

• It is used to determine the limit on the load which can be acquired by the system without crossing the stability limit.
• The principle of this method consists of the basis that when δ oscillates around the equilibrium point with constant amplitude, transient stability will be maintained.
• It is employed for the determination of the transient stability limit of the single-machine-infinite bus system.
• It is also used for determining the maximum limit on the load that the system can take without exceeding the stability limit.
• In a single machine infinite bus system, if the system is unstable after a fault is cleared, δ(t) increases indefinitely with time, till the machine loses synchronism. In contrast, in a stable system, δ(t) reaches a maximum and then starts reducing.

Equal area criterion:

• It is used to determine the limit on the load which can be acquired by the system without crossing the stability limit.
• The principle of this method consists of the basis that when δ oscillates around the equilibrium point with constant amplitude, transient stability will be maintained.
• It is employed for the determination of the transient stability limit of the single-machine-infinite bus system.
• It is also used for determining the maximum limit on the load that the system can take without exceeding the stability limit.
• In a single machine infinite bus system, if the system is unstable after a fault is cleared, δ(t) increases indefinitely with time, till the machine loses synchronism. In contrast, in a stable system, δ(t) reaches a maximum and then starts reducing.

The bus whose voltage and frequency remains constant even after the variation in the load is known as the infinite bus. It has infinite inertia.

Equal area criterion:

• It is used to determine the limit on the load which can be acquired by the system without crossing the stability limit.
• The principle of this method consists of the basis that when δ oscillates around the equilibrium point with constant amplitude, transient stability will be maintained.
• It is employed for the determination of the transient stability limit of the single-machine-infinite bus system.
• It is also used for determining the maximum limit on the load that the system can take without exceeding the stability limit.
• In a single machine infinite bus system, if the system is unstable after a fault is cleared, δ(t) increases indefinitely with time, till the machine loses synchronism. In contrast, in a stable system, δ(t) reaches a maximum and then starts reducing.

The bus whose voltage and frequency remains constant even after the variation in the load is known as the infinite bus. It has infinite inertia.

Equal area criterion:

• It is used to determine the limit on the load which can be acquired by the system without crossing the stability limit.
• The principle of this method consists of the basis that when δ oscillates around the equilibrium point with constant amplitude, transient stability will be maintained.
• It is employed for the determination of the transient stability limit of the single-machine-infinite bus system.
• It is also used for determining the maximum limit on the load that the system can take without exceeding the stability limit.
• In a single machine infinite bus system, if the system is unstable after a fault is cleared, δ(t) increases indefinitely with time, till the machine loses synchronism. In contrast, in a stable system, δ(t) reaches a maximum and then starts reducing.

Therefore, both Statement I and Statement II are individually true but Statement II is not the correct explanation of Statement I.

Explanation:

Equal Area Criterion in Power-Angle Curve

Definition: The equal area criterion is a method used to analyze the stability of a synchronous machine under transient conditions. It is based on the principle that the area of acceleration (A₁) and the area of deceleration (A₂) in the power-angle curve should be equal for the system to remain stable after a disturbance. The power-angle curve represents the relationship between the electrical power output of a synchronous machine and the rotor angle (δ).

Working Principle: When a disturbance occurs, the rotor angle changes, causing a temporary imbalance between the mechanical input power (P_m) and the electrical output power (P_e). The rotor accelerates or decelerates depending on whether P_m is greater than or less than P_e. The equal area criterion ensures that the energy gained during acceleration is equal to the energy lost during deceleration, allowing the rotor to settle at a new equilibrium point.

Correct Option Analysis:

The correct option is:

Option 3: \(\rm A_{1}=\int_{\delta_{o}}^{\delta_{c}}\left(P_{m}-P_{e}\right) d \delta=0\)

This option accurately defines the area of acceleration (A₁) in the power-angle curve. The integral represents the energy gained by the rotor during acceleration, which occurs when the mechanical input power (P_m) exceeds the electrical output power (P_e). The limits of integration are δ_o (the initial rotor angle at the start of the disturbance) and δ_c (the critical rotor angle where the system transitions to deceleration). The condition for stability is that the total energy gain (A₁) during acceleration equals the total energy loss (A₂) during deceleration.

Mathematical Representation:

The energy gained during acceleration is given by:

\(A_{1} = \int_{\delta_{o}}^{\delta_{c}} (P_{m} - P_{e}) \, d\delta\)

The energy lost during deceleration is given by:

\(A_{2} = \int_{\delta_{c}}^{\delta_{max}} (P_{e} - P_{m}) \, d\delta\)

For stability:

\(A_{1} = A_{2}\)

This ensures that the rotor angle returns to a stable operating point after the disturbance.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: \(\rm A_{1}=\int_{\delta_{c}}^{\delta_{o}}\left(P_{m}-P_{e}\right) d \delta=0\)

This option incorrectly reverses the limits of integration. The area of acceleration (A₁) is always defined from the initial rotor angle (δ_o) to the critical rotor angle (δ_c), not vice versa. Reversing the limits would result in a negative value for A₁, which is physically incorrect in the context of energy gained during acceleration.

Option 2: \(\rm A_{1}=\int_{\delta_{o}}^{\delta_{c}}\left(P_{e}-P_{m}\right) d \delta=0\)

This option swaps the terms inside the integral, using (P_e - P_m) instead of (P_m - P_e). This representation corresponds to the area of deceleration (A₂), not acceleration (A₁). The area of acceleration must be defined by the difference (P_m - P_e), which represents the net power causing the rotor to accelerate.

Option 4: \(\rm A_{1}=\int_{\delta_{c}}^{\delta_{o}}\left(P_{e}-P_{m}\right) d \delta=0\)

This option combines the errors from options 1 and 2. It reverses the limits of integration and swaps the terms inside the integral. As explained earlier, the limits of integration for A₁ must be from δ_o to δ_c, and the integrand must be (P_m - P_e). This option is therefore doubly incorrect.

Conclusion:

The equal area criterion is a vital tool for analyzing the transient stability of synchronous machines. By ensuring that the areas of acceleration (A₁) and deceleration (A₂) are equal, we can determine whether the system will return to a stable operating point after a disturbance. The correct representation of A₁ is:

\(A_{1} = \int_{\delta_{o}}^{\delta_{c}} (P_{m} - P_{e}) \, d\delta\)

This integral accurately describes the energy gained during acceleration, and its equality with the energy lost during deceleration ensures stability. Understanding the limits of integration and the terms inside the integral is crucial for correctly applying the equal area criterion in power system analysis.

Explanation:

Transient Stability Analysis and Equal Area Criterion:

Definition: Transient stability analysis is a crucial aspect of power system stability that evaluates the ability of a power system to maintain synchronism when subjected to a large disturbance, such as a short circuit or sudden load change. The rotor angle stability is an essential parameter in this analysis, as it represents the relative angular displacement of the synchronous machine rotors.

Equal Area Criterion: The equal area criterion is a graphical method used to assess transient stability. It states that for a system to remain stable following a disturbance, the area representing accelerating power (the excess mechanical input power over electrical output power) should be equal to the area representing decelerating power (the excess electrical output power over mechanical input power). These areas are plotted on the power-angle curve, which depicts the relationship between electrical power and rotor angle.

Correct Option Analysis:

The correct option is:

Option 3: Greater than 90˚

In transient stability analysis, the maximum rotor angle to which the rotor can oscillate is determined by the equal area criterion. The rotor angle can exceed 90˚ depending on the disturbance magnitude, system parameters, and power-angle characteristics. While small disturbances generally result in rotor angles below 90˚, large disturbances may cause the rotor angle to oscillate beyond 90˚ without losing synchronism, provided the accelerating and decelerating areas on the power-angle curve are equal.

This behavior occurs due to the nonlinear nature of the power-angle relationship, where stability depends not only on the magnitude of the rotor angle but also on the system's ability to balance accelerating and decelerating powers. Therefore, when the equal area criterion is satisfied, the rotor angle can oscillate to values greater than 90˚ without losing stability.

Additional Considerations:

It is important to note that the rotor angle exceeding 90˚ does not necessarily imply instability. Stability is determined by the ability of the system to return to synchronism after the disturbance. As long as the equal area criterion is satisfied, the system remains stable even if the rotor angle exceeds 90˚ momentarily during oscillations.

Important Information:

To further understand the analysis, let’s evaluate the other options:

Option 1: 0˚ to 20˚

This option is incorrect because it severely underestimates the range of rotor angle oscillation during transient stability analysis. Rotor angles typically oscillate to values much higher than 20˚, especially for large disturbances. The power-angle curve and equal area criterion allow for a much broader range of oscillation, including values exceeding 90˚.

Option 2: 45˚ to 50˚

While rotor angles may oscillate within this range under specific conditions, this option is not universally applicable. The transient stability analysis considers a wide range of rotor angle oscillations, and the maximum angle depends on the system parameters and disturbance magnitude. In many cases, the rotor angle can exceed 50˚ or even 90˚ without losing stability.

Option 4: 65˚ to 85˚

This option is closer to the typical range observed in transient stability analysis but is still not universally applicable. The rotor angle can exceed 85˚ in scenarios involving large disturbances, provided the equal area criterion is satisfied. Therefore, limiting the maximum angle to this range does not encompass all possible stable conditions.

Conclusion:

The transient stability analysis is a complex evaluation of power system behavior under disturbances. The correct understanding of rotor angle oscillations, as governed by the equal area criterion, is crucial for ensuring system stability. While rotor angles within certain ranges may be typical, the ability of the system to remain stable depends on the balance of accelerating and decelerating powers, not the specific value of the rotor angle. As explained, the rotor angle can oscillate beyond 90˚ without losing stability, making Option 3 the correct answer.

Equal area criterion:

• It is used to determine the limit on the load which can be acquired by the system without crossing the stability limit.
• The principle of this method consists of the basis that when δ oscillates around the equilibrium point with constant amplitude, transient stability will be maintained.
• It is employed for the determination of the transient stability limit of the single-machine-infinite bus system.
• It is also used for determining the maximum limit on the load that the system can take without exceeding the stability limit.
• In a single machine infinite bus system, if the system is unstable after a fault is cleared, δ(t) increases indefinitely with time, till the machine loses synchronism. In contrast, in a stable system, δ(t) reaches a maximum and then starts reducing.

• The synchronous machine swings from A to B due to disturbance and settles at a point ‘C’.
• During this process, the rotor angle ‘δ’ increases until the synchronous speed are achieved, and the mechanical input and electrical output are balanced
• The rotor will be having synchronous speed at point ‘A’ and point ‘B’ before it finally settles.

Equal Area Criteria:

• The equal-area criterion is a simple graphical method for concluding the transient stability.
• This principle does not require the swing equation for the determination of stability conditions.
• The stability conditions are recognized by equating the areas of segments on the power angle diagram.

​Starting with the swing equation:

\(M\left( {\frac{{{d^2}δ }}{{d{t^2}}}} \right) = {P_s} -P_e\)

Where,

M is the moment of inertia or angular momentum

δ is the power angle between rotor rotating field and stator rotating field

Ps is a mechanical input power

P_e is the electrical output

Multiply the above equation with dδ / dt

we get,

\(\frac{1}{2}M\frac{d}{{dt}}{\left( {\frac{{dδ }}{{dt}}} \right)^2} = \left( {{P_s} - {P_e}} \right)\frac{{dδ }}{{dt}}\)

Rearranging, multiplying by dt, and integrating, we have

\(\frac{{dδ }}{{dt}} = \sqrt {\mathop \smallint \limits_{{δ _0}}^δ \frac{{2\left( {{P_s} - {P_m}} \right)}}{M}dδ } \)

At steady-state condition, the torque angle was not changing i.e. before the disturbance.

dδ / dt = 0

Also, if the system has transient stability the machine will again operate at synchronous speed after the disturbances, i.e.,

dδ /dt = 0

Hence the condition for stability is dδ / dt =0

Equal area criteria:

• It is a graphical method that allows assessing the transient stability of electric power systems in a simple and comprehensive way.
• It is also a graphical solution to the basic swing equation.
• Its use eliminates the need of computing the swing curves of the system, thus saving a considerable amount of work.

For transient stability:

• For the system to be stable the accelerating area should be less than decelerating area.
• For the system to be critically stable the accelerating area is equal to the decelerating area.
• If the accelerating area is more than decelerating area then the system is unstable.
• During the swinging of the rotor, the maximum may exceed 90°. It is only area stability, but not angle stability.
• For stability, it only considers the first swing of the rotor after disturbance. 

Assumptions to evaluate the transient stability: 

• The resistances, the shunt capacitance of generator and transmission lines are ignored. The shunt capacitor or shunt inductor at load Bus or generator bus is ignored.
• The mechanical input is assumed to be constant.
• There is no change in the speed of the alternator.
• The damping force provided by damper winding is ignored.
• The voltages behind the reactances of the Machines are ignored.