Power system models are often nonlinear, making analysis tricky. Linearization simplifies these models, allowing us to use linear analysis tools. This process involves approximating the system around an operating point, making it easier to study stability and design controls.

Linearized models help us understand small-signal stability and design power system stabilizers. They're used to study things like subsynchronous resonance and inter-area oscillations. While not perfect, linearization gives us valuable insights into power system behavior.

Linearization Principles for Power Systems

Fundamentals of Linearization

• Linearization approximates nonlinear systems around an operating point, enabling the application of linear analysis tools
• The process involves computing the Jacobian matrix, 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 eigenvalue analysis, frequency response 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 swing equation, 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 automatic voltage regulators (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 DC power flow approximation, 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 load sensitivity coefficients, 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

• Power electronic devices, such as FACTS controllers (SVC, STATCOM) and HVDC converters, 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 PI controllers and phase-locked loops (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 synchronizing torque coefficients and damping coefficients
• 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 system matrix, which are influenced by the operating point
• Operating points close to the stability boundary 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
• Participation factors, derived from the eigenvectors of the linearized model, identify the contribution of each state variable to a particular mode of oscillation
• Sensitivity analysis 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 state-space representation is a compact way to express a linearized power system model using matrix equations
• The state-space model consists of a set of first-order differential equations 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 input matrix (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 pole placement, optimal control, and robust control, 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 (Kalman filters) to estimate the system's state variables based on measured outputs
• State-space models facilitate the analysis of observability and controllability, 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