Glossary

8.1 Swing equation and power-angle characteristics

4 min read•august 1, 2024

The and power-angle characteristics are crucial for understanding rotor angle stability in power systems. These concepts describe how synchronous machines respond to disturbances, linking mechanical power input to electrical power output through the rotor angle.

By exploring these relationships, we can analyze a machine's stability limits and its ability to maintain synchronism. This knowledge forms the foundation for applying the equal area criterion and developing strategies to enhance power system stability.

Swing equation for a single machine

Derivation and components

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  • The swing equation is a second-order nonlinear differential equation that describes the rotational dynamics of a synchronous machine
  • It relates the (H), mechanical power input (Pm), electrical power output (Pe), and the rotor angle (δ) of the machine
  • The swing equation is derived by applying Newton's second law of motion to the rotor of the synchronous machine, considering the accelerating torque as the difference between mechanical and electrical torques
  • The swing equation is expressed as: (2H/ωs)×(d2δ/dt2)=Pm−Pe(2H/ω_s) × (d²δ/dt²) = Pm - Pe, where ω_s is the synchronous angular velocity

Assumptions and damping

  • The swing equation assumes that the mechanical power input remains constant during the transient period following a disturbance
  • Damping is often included in the swing equation to account for the damping torque provided by the machine's damper windings and the system's load
  • Damping helps to reduce oscillations and improve the stability of the machine
  • The presence of damping torque is represented by an additional term in the swing equation, proportional to the derivative of the rotor angle (dδ/dt)

Power-angle characteristics of synchronous machines

Relationship between electrical power output and rotor angle

  • The power-angle characteristic curve represents the relationship between the electrical power output (Pe) and the rotor angle (δ) of a synchronous machine
  • The power-angle characteristic is derived from the phasor diagram of a synchronous machine connected to an infinite bus, considering the machine's internal voltage (E), the infinite bus voltage (V), and the reactance (X) between them
  • The electrical power output is given by: Pe=(EV/X)×sin(δ)Pe = (EV/X) × sin(δ), where δ is the angle between the internal voltage and the infinite bus voltage

Nonlinearity and stability limit

  • The power-angle characteristic is nonlinear, with the electrical power output increasing sinusoidally with the rotor angle up to the stability limit
  • The maximum electrical power output occurs at a rotor angle of 90 degrees, known as the steady-state stability limit
  • Operating the machine beyond the stability limit leads to a decrease in electrical power output and potential loss of synchronism
  • The slope of the power-angle characteristic at the operating point determines the synchronizing power coefficient, which indicates the machine's ability to maintain synchronism after a small disturbance

Rotor angle and power output relationship

Steady-state and disturbance conditions

  • The rotor angle (δ) represents the angular position of the rotor with respect to a synchronously rotating reference frame
  • In a steady-state condition, the rotor angle remains constant, and the mechanical power input equals the electrical power output
  • When a disturbance occurs, the rotor angle changes due to the imbalance between mechanical and electrical power, causing the machine to accelerate or decelerate

Impact on stability

  • An increase in the rotor angle results in an increase in the electrical power output, as per the power-angle characteristic, until the stability limit is reached
  • If the rotor angle exceeds the stability limit, the electrical power output decreases with further increase in the rotor angle, potentially leading to loss of synchronism
  • The relationship between rotor angle and electrical power output is crucial in understanding the stability of the synchronous machine and its response to disturbances
  • Maintaining the rotor angle within the stable operating range is essential for ensuring the stable operation of the synchronous machine

Stability factors of synchronous machines

Machine parameters and excitation system

  • The inertia constant (H) of the machine determines its ability to resist changes in rotor speed during disturbances. A higher inertia constant improves stability (e.g., machines with larger rotors)
  • The reactance (X) between the machine's internal voltage and the infinite bus voltage affects the power-angle characteristic. A lower reactance results in a steeper curve and a higher maximum power output, but reduces the stability margin
  • The excitation system of the synchronous machine influences its ability to maintain the internal voltage during disturbances. A fast-acting excitation system (e.g., static excitation systems) enhances stability

Operating conditions and disturbances

  • The initial operating point of the machine on the power-angle characteristic determines its proximity to the stability limit. Operating closer to the stability limit reduces the stability margin
  • The nature and severity of the disturbance, such as a sudden change in mechanical power input or a fault in the system, affect the machine's ability to maintain stability
  • Faults near the synchronous machine have a more severe impact on stability compared to distant faults

Supplementary stabilizing devices

  • The presence of power system stabilizers (PSS) can enhance the damping of low-frequency oscillations and improve the overall stability of the synchronous machine
  • PSS provide an additional input signal to the excitation system, modulating the machine's internal voltage to dampen oscillations
  • Other stabilizing devices, such as braking resistors or fast valving, can also be employed to improve the stability of synchronous machines during severe disturbances

Key Terms to Review (17)

Angle Stability Margin: Angle stability margin refers to the range of power-angle conditions under which a synchronous machine can maintain its synchronism and not lose stability following a disturbance. This concept is crucial in understanding how variations in load or generation affect the rotor angle of a generator, and ultimately how these variations impact system stability and reliability.
Automatic Generation Control: Automatic Generation Control (AGC) is a system used in power generation to maintain the balance between electrical supply and demand by automatically adjusting the output of generators. This technology plays a critical role in ensuring system frequency stability, supporting overall power system stability, and achieving control objectives that align with operational requirements.
Critical Angle: The critical angle is the maximum angle of inclination at which a generator can operate without losing synchronism with the power system. It is crucial in understanding how the mechanical and electrical powers interact during system disturbances and is directly related to the stability of synchronous machines under varying load conditions. When the power angle exceeds this critical angle, the system may experience instability and potential loss of synchronism.
Damping Factor: The damping factor is a measure of how oscillations in a system decay after a disturbance. It indicates the degree to which a system can return to its stable state after being disturbed, which is crucial in maintaining stability in power systems. A higher damping factor leads to quicker stabilization and less sustained oscillation, making it an essential aspect in both generator performance and the analysis of dynamic behavior in systems.
Dynamic stability: Dynamic stability refers to the ability of a power system to maintain equilibrium during and after disturbances, ensuring that the system can return to a stable operating condition. This concept is crucial for understanding how power systems react to changes, such as faults or load variations, and is closely linked to control objectives, power flow formulations, and the response of the system's components over time.
Frequency Response: Frequency response is the measure of a system's output spectrum in response to an input signal, reflecting how the system reacts at different frequencies. It helps in understanding the dynamic behavior of power systems and is crucial for designing controllers and stabilizers to ensure system stability and performance across varying operational conditions.
Inertia Constant: The inertia constant is a crucial parameter in power systems that quantifies the amount of kinetic energy stored in a rotating mass, typically measured in seconds. It plays a significant role in determining how a generator responds to changes in mechanical power input and electrical load, influencing the stability and dynamic behavior of the system. Understanding this concept is essential for analyzing the behavior of various prime movers, the dynamics captured by swing equations, transient stability assessments through the equal area criterion, and the rotor angle stability in multi-machine systems.
Inter-area oscillations: Inter-area oscillations refer to the low-frequency oscillations that occur between different groups of generators in a power system, often as a response to disturbances or changes in load. These oscillations typically involve the synchronized movement of large areas of the grid, and understanding them is crucial for maintaining stability and ensuring reliable operation of interconnected power systems.
Local mode oscillations: Local mode oscillations refer to the dynamic behavior of power systems where generators oscillate at specific frequencies due to small disturbances, leading to stable or unstable oscillatory motion. These oscillations occur primarily in localized areas of the system and can significantly affect system stability, making their analysis crucial for ensuring reliable operation.
Phase margin: Phase margin is a measure of the stability of a control system, defined as the difference between the phase angle of the open-loop transfer function and -180 degrees at the frequency where the gain crosses unity (0 dB). A positive phase margin indicates stability, while a negative phase margin suggests potential instability. This concept is crucial for understanding how systems respond to changes and disturbances, which is essential for assessing small-signal stability, designing controls, and analyzing system dynamics.
Power System Stabilizer: A power system stabilizer (PSS) is a control device used in electric power systems to enhance the stability of the system by providing supplementary damping to oscillations in the rotor angle of synchronous machines. It helps improve the overall performance of the power system, especially during disturbances, by adjusting the excitation of generators based on measured system parameters. This device plays a crucial role in ensuring that the power system remains stable during both small-signal and large-signal disturbances.
Power-angle curve: The power-angle curve illustrates the relationship between the electrical power output of a synchronous generator and the angular displacement (or angle) between the rotor's magnetic field and the stator's magnetic field. This curve is crucial for understanding how changes in angle affect power generation, stability, and system performance during different operating conditions.
Small-signal stability analysis: Small-signal stability analysis is a method used to evaluate the stability of a power system under small disturbances, focusing on the system's response near an equilibrium point. This analysis helps in understanding how changes in load or generation can affect system performance, providing insights into the power-angle characteristics and the dynamics of synchronous machines. It uses linearized models around an operating point to assess the system's ability to return to equilibrium after minor disturbances.
Swing Equation: The swing equation is a fundamental equation used in power system stability analysis that describes the dynamics of a synchronous machine's rotor angle in relation to mechanical and electrical power. This equation is crucial for understanding the behavior of generators during disturbances, as it relates changes in rotor angle to the difference between generated and consumed electrical power. Analyzing the swing equation helps in assessing transient stability and provides insights into how power systems respond to various operating conditions.
System damping ratio: The system damping ratio is a dimensionless measure that indicates how oscillations in a dynamic system decay after a disturbance. It provides insight into the stability and response of the system, particularly in power systems where oscillatory behavior can affect performance. A higher damping ratio means quicker decay of oscillations, leading to more stable operation, while a lower damping ratio suggests sustained or increasing oscillations, which can indicate instability.
Transient Stability: Transient stability refers to the ability of a power system to maintain synchronism when subjected to a disturbance, such as a fault or sudden change in load. It focuses on the immediate response of the system after such disturbances and how well it can return to a stable operating condition. This concept is crucial in understanding system behavior during and after transient events, particularly in multi-machine environments.
Voltage Stability: Voltage stability refers to the ability of a power system to maintain steady voltage levels at all buses in the system after being subjected to a disturbance. This concept is crucial because voltage instability can lead to voltage collapse, where voltages drop significantly, causing widespread outages and affecting system reliability.
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