7.1 Linearization of power system models
5 min read•august 1, 2024
Power system models are often nonlinear, making analysis tricky. simplifies these models, allowing us to use linear analysis tools. This process involves approximating the system around an , making it easier to study stability and design controls.
Linearized models help us understand and design . They're used to study things like subsynchronous resonance and . While not perfect, linearization gives us valuable insights into power system behavior.
Linearization Principles for Power Systems
Fundamentals of Linearization
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- Linearization approximates nonlinear systems around an operating point, enabling the application of linear analysis tools
- The process involves computing the , which contains the partial derivatives of the system equations with respect to state variables and inputs
- Linearized models are valid only in the vicinity of the operating point and may not capture the full behavior of the nonlinear system under large disturbances
- Linearization simplifies the analysis of power system stability, allowing the use of , techniques, and linear control theory
Applications of Linearization in Power Systems
- Linearized models facilitate the study of small-signal stability, which assesses the system's ability to maintain synchronism under small disturbances
- Linearization enables the design of linear controllers (power system stabilizers) to enhance system damping and improve stability
- Linearized models are used in the analysis of subsynchronous resonance (SSR) in power systems, which can lead to shaft damage in generators
- Linearization is applied in the study of inter-area oscillations, where multiple generators in different areas oscillate against each other
Linearized Models of Power System Components
Synchronous Generators
- Synchronous generators can be linearized around an operating point, resulting in a linear model that relates small changes in rotor angle, speed, and electrical power to changes in mechanical power and field voltage
- The linearized model typically includes the , which describes the rotor dynamics, and the voltage equation, which relates the field voltage to the generator's internal voltage
- The linearized generator model incorporates the effects of (AVRs) and power system stabilizers (PSSs) on the generator's dynamic behavior
- The model parameters, such as inertia constant, damping coefficient, and synchronizing torque coefficient, are obtained from the generator's physical characteristics and operating conditions
Transmission Lines and Loads
- Transmission lines can be linearized using the , which neglects reactive power flow and assumes small voltage angle differences between buses
- The DC power flow model represents the transmission network as a set of linear equations relating power injections to voltage angles
- Linearized models of power system loads often employ the concept of , which relate changes in load power to changes in voltage and frequency
- Load models can include static load characteristics (constant impedance, current, or power) and dynamic load characteristics (induction motors, thermostatic loads)
Power Electronic Devices
- , such as (SVC, STATCOM) and , can be linearized around their operating points to obtain small-signal models suitable for stability analysis
- The linearized models capture the dynamic behavior of the power electronic devices and their interaction with the power system
- Linearized models of FACTS controllers include the control systems, such as and (PLLs), which regulate the device's output
- HVDC converter models incorporate the dynamics of the converter control systems, such as the firing angle controller and the extinction angle controller
Operating Point Impact on Linearized Models
Influence of Operating Point on Model Parameters
- The operating point of a power system, determined by the system's state variables (voltage magnitudes, voltage angles, generator rotor angles), affects the linearized model
- Changes in the operating point lead to changes in the elements of the Jacobian matrix, resulting in different linearized models
- The operating point influences the values of the linearized model parameters, such as the and
- The operating point determines the initial conditions for the linearized model, which affect the system's dynamic response to disturbances
Stability Assessment using Linearized Models
- The stability of the linearized model depends on the eigenvalues of the , which are influenced by the operating point
- Operating points close to the may result in linearized models with eigenvalues near the imaginary axis, indicating a higher risk of instability
- Eigenvalue analysis of the linearized model provides insights into the system's small-signal stability, including the damping and frequency of oscillatory modes
- , derived from the eigenvectors of the linearized model, identify the contribution of each state variable to a particular mode of oscillation
- can be performed to assess the impact of small changes in the operating point on the stability of the linearized model
State-Space Representation of Linearized Power Systems
Formulation of State-Space Models
- The is a compact way to express a linearized power system model using matrix equations
- The state-space model consists of a set of that relate the state variables' derivatives to the current state variables and system inputs
- The state variables in a power system model typically include generator rotor angles, generator speeds, and other relevant dynamic variables (flux linkages, controller states)
- The system matrix (A) in the state-space representation contains the coefficients that relate the state variables' derivatives to the current state variables
- The (B) relates the system inputs, such as changes in mechanical power or control signals, to the state variables' derivatives
Applications of State-Space Models
- The state-space representation allows for the application of various linear control techniques, such as , , and , to enhance power system stability
- State-space models are used in the design of power system stabilizers (PSSs) to provide supplementary damping control to generators
- The state-space formulation enables the development of state estimators () to estimate the system's state variables based on measured outputs
- State-space models facilitate the analysis of and , which determine the feasibility of state estimation and control, respectively
- The state-space representation is used in the study of multi-machine power systems, where the models of individual components are combined to form a comprehensive system model
Key Terms to Review (33)
Automatic Voltage Regulators: Automatic voltage regulators (AVRs) are devices used in electrical systems to maintain a constant voltage level automatically. They play a crucial role in ensuring the stability and reliability of power systems by adjusting the output voltage based on variations in load and other operating conditions, helping to achieve control objectives related to voltage stability, power quality, and system reliability.
Controllability: Controllability refers to the ability of a system to be controlled in such a way that it can be driven to a desired state using external inputs. In power systems, this concept is crucial as it determines whether the system can respond effectively to disturbances and maintain stability. A system that is fully controllable can be manipulated to reach its operational targets, while systems with limited controllability may struggle under certain conditions, impacting overall performance and stability.
Damping Coefficients: Damping coefficients are parameters used in power system stability analysis to quantify the amount of damping present in a system's oscillatory behavior. These coefficients help in determining how quickly oscillations will decay after a disturbance, influencing the overall stability and performance of the power system. Higher damping coefficients indicate that oscillations will dissipate more rapidly, leading to a more stable system, while lower values suggest prolonged oscillations that may compromise stability.
Dc power flow approximation: The dc power flow approximation is a simplified model used in power systems to analyze the flow of direct current in a network while assuming constant voltage magnitudes and neglecting losses. This approximation allows for easier analysis and computation of power system behavior, especially when examining the relationship between power injections and network flows without getting into complex alternating current calculations.
Eigenvalue Analysis: Eigenvalue analysis is a mathematical technique used to study the stability and dynamic behavior of systems by evaluating the eigenvalues of their linearized models. It helps in understanding how small perturbations affect system performance and aids in the design of control strategies to enhance stability. This analysis plays a critical role in assessing small-signal stability, optimizing system responses, and tuning stabilizers in power systems.
FACTS Controllers: FACTS (Flexible AC Transmission Systems) controllers are advanced power electronics devices designed to enhance the controllability and stability of power transmission systems. By dynamically managing power flow and voltage levels, FACTS controllers can improve system reliability, increase transfer capacity, and help stabilize the grid during disturbances or fluctuations. They play a crucial role in linearizing power system models, enabling better analysis and control of complex electrical networks.
First-Order Differential Equations: First-order differential equations are equations that involve the derivatives of a function with respect to one independent variable, where the highest derivative is first order. These equations are essential in understanding dynamic systems, particularly when approximating nonlinear systems around an operating point, making them relevant for linearization in power system models.
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.
HVDC Converters: HVDC converters are devices that convert alternating current (AC) into direct current (DC) and vice versa, playing a crucial role in high-voltage direct current (HVDC) transmission systems. These converters allow for efficient long-distance power transmission and the interconnection of different power grids, helping to stabilize and optimize power flow.
Input matrix: An input matrix is a mathematical representation that relates the input variables of a system to its output variables, often used in the context of linearized power system models. It serves as a key component in analyzing how changes in control inputs, like voltage or power generation, affect the overall behavior of the power system. Understanding the input matrix helps in designing effective control strategies to enhance system stability.
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.
Jacobian Matrix: The Jacobian matrix is a mathematical representation that describes how a vector-valued function changes as its input variables change. It consists of all first-order partial derivatives of the function and is essential for analyzing the behavior of nonlinear systems, especially in power system stability and control.
Kalman Filters: Kalman filters are mathematical algorithms used for estimating the state of a dynamic system from a series of incomplete and noisy measurements. They are particularly useful in the context of linearization in power systems, where they help improve the accuracy of system state estimates by minimizing the mean of the squared errors.
Linearization: Linearization is the process of approximating a nonlinear system by a linear model around a specific operating point. This technique simplifies the analysis and control design of complex systems, allowing for easier understanding of their behavior under small perturbations. By transforming nonlinear equations into linear ones, it becomes possible to use established methods for stability analysis and control system design.
Load Sensitivity Coefficients: Load sensitivity coefficients are numerical values that represent how changes in load affect the system's operating conditions, particularly in the context of power system stability. These coefficients help quantify the relationship between load variations and system responses, making it easier to analyze and predict system behavior under different conditions. By utilizing load sensitivity coefficients, engineers can make informed decisions to enhance stability and control within power systems.
Observability: Observability refers to the ability to infer the internal states of a system based on its external outputs. In power systems, this concept is crucial for monitoring and controlling system dynamics, allowing operators to understand the behavior of the system based on measurable data. A system is considered observable if all internal states can be determined using the outputs over time, which is essential for effective control and stability analysis.
Operating Point: An operating point refers to a specific set of conditions under which a power system operates, characterized by variables such as voltage, current, power flow, and frequency. This point is crucial for analyzing system stability and performance because small deviations from this point can significantly impact system behavior and response. Understanding the operating point is essential for effective linearization of power system models, as it provides the baseline from which perturbations and dynamics are evaluated.
Optimal Control: Optimal control refers to the process of determining a control policy that minimizes (or maximizes) a certain performance criterion over time, often applied in dynamic systems to achieve desired performance while considering constraints. This concept is essential in designing efficient and effective control strategies for power systems, ensuring stability and performance under varying conditions. By applying optimal control methods, engineers can fine-tune their systems for better response, stability, and efficiency, directly impacting the overall reliability of power generation and distribution.
Participation Factors: Participation factors are numerical values that indicate the contribution of each state variable to the overall system response in a power system. These factors help understand how changes in one part of the system affect the entire system, making them crucial for analyzing stability and control. They also play a key role in identifying which generators or loads have the most influence on system behavior during disturbances.
Phase-locked loops: Phase-locked loops (PLLs) are control systems that generate a signal with a fixed relationship to the phase of an input signal. They are widely used in power systems for synchronization, ensuring that the output frequency matches the input frequency, which is crucial for maintaining stability and control in power networks. PLLs play a key role in various applications, including voltage control, frequency modulation, and the synchronization of generators with the grid.
Pi Controllers: Pi controllers are a type of control strategy used in power systems that combine proportional and integral control actions to regulate system variables effectively. These controllers adjust the output based on the current error and the accumulated past errors, allowing for improved steady-state performance and reduced offset in system responses. This makes pi controllers particularly valuable for achieving desired control objectives in power systems, such as maintaining voltage levels and frequency stability.
Pole Placement: Pole placement is a control technique used to determine the dynamic behavior of a system by strategically placing the poles of its transfer function in desired locations within the complex plane. This method ensures that the system meets specific performance criteria, such as stability and response time, by influencing the characteristics of the closed-loop system. In power systems, pole placement plays a crucial role in linearization processes and tuning control strategies to improve system stability and performance.
Power Electronic Devices: Power electronic devices are electronic components that manage and control electrical power through the process of converting, switching, and conditioning electrical energy. These devices play a crucial role in enhancing the efficiency and performance of power systems, especially when dealing with varying loads and renewable energy sources, which require precise control for stability and reliability.
Power System Stabilizers: Power system stabilizers are control devices used in electrical power systems to enhance the stability of synchronous machines by damping oscillations in rotor speed and improving overall system performance. These stabilizers play a crucial role in maintaining the balance between generation and load, ensuring that the system remains stable during disturbances and variations in operating conditions.
Robust Control: Robust control refers to a control strategy designed to maintain performance and stability of a system in the presence of uncertainties and variations in system parameters. This approach is crucial for ensuring that power systems can effectively handle disturbances and changes without compromising reliability. It emphasizes the ability to cope with worst-case scenarios, making it particularly relevant in the context of linearized power system models and coordinated control mechanisms.
Sensitivity analysis: Sensitivity analysis is a technique used to determine how different values of an independent variable impact a particular dependent variable under a given set of assumptions. This approach helps in understanding the behavior of power system models and their stability, revealing how sensitive these systems are to changes in parameters such as load, generation, or network topology.
Small-signal stability: Small-signal stability refers to the ability of a power system to maintain its equilibrium under small disturbances or fluctuations, ensuring that the system returns to its original state without experiencing significant oscillations or instability. This concept is crucial for analyzing and designing control strategies in power systems, as it involves understanding how changes in load, generation, and system parameters affect the overall stability.
Stability boundary: The stability boundary refers to the delineation between stable and unstable operating conditions in a power system. It defines the limits within which a power system can operate without losing stability, often represented graphically to show the threshold where small disturbances may lead to significant changes in system behavior. This concept is crucial when linearizing power system models, as it helps in determining the operational limits and assessing the impact of various control strategies.
State-space representation: State-space representation is a mathematical modeling framework that describes a dynamic system by using a set of first-order differential equations. This approach captures the internal state of the system at any given time and relates it to its inputs and outputs, allowing for the analysis and control of complex systems in various fields, including power systems.
Subsynchrnous resonance: Subsynchrnous resonance refers to a condition in power systems where the interaction between generator oscillations and the system's electrical network creates oscillatory behavior at frequencies below the synchronous speed of the generators. This phenomenon can lead to instability and can result in damage to equipment due to excessive vibrations or torque fluctuations. The occurrence of subsynchrnous resonance highlights the importance of understanding system dynamics and modeling to mitigate these risks.
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.
Synchronizing torque coefficients: Synchronizing torque coefficients refer to the measure of the torque required to maintain the synchronization between generators in a power system. This concept is crucial for understanding how different machines or generators can stay in phase while operating together, ensuring stability and reliability in power delivery. These coefficients help quantify the relationship between rotor speed and torque, which is essential for linearizing power system models when analyzing system dynamics.
System Matrix: A system matrix is a mathematical representation that encapsulates the dynamics of a system, often used in control theory and engineering to analyze system behavior. It organizes the relationships between different state variables and inputs in a structured way, enabling the linearization of complex nonlinear models around an operating point. Understanding the system matrix is essential for designing controllers and predicting system responses in power systems.