8.3 Multi-machine rotor angle stability analysis

Rotor angle stability in multi-machine systems is crucial for power system reliability. It involves analyzing how generators maintain synchronism during disturbances. This topic extends single-machine concepts to complex networks with interacting generators.

Multi-machine analysis considers the coupled dynamics of all generators in the system. It uses swing equations and power flow relationships to model system behavior. Understanding these interactions is key to assessing stability and preventing widespread outages.

Swing Equation for Multi-Machine Systems

Extension of Single-Machine Swing Equation

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• The swing equation for a single machine can be extended to a multi-machine system by considering the interactions between the machines and the network
• In a multi-machine system, each machine's swing equation is coupled with the others through the network admittance matrix and the power flow equations
• The multi-machine swing equation is a set of coupled nonlinear differential equations that describe the dynamics of the rotor angles and speeds of all machines in the system
  • It takes into account the individual machine's inertia constants, damping coefficients, mechanical power inputs, and electrical power outputs
  • The electrical power output of each machine depends on the voltages and angles at its terminals, which are determined by the network equations and the states of the other machines

Formulation of Multi-Machine Swing Equation

• The multi-machine swing equation can be expressed in matrix form as:
  • $M\ddot{\delta} + D\dot{\delta} = P_m - P_e(\delta, V)$
  • Where $M$ is the diagonal matrix of inertia constants, $D$ is the diagonal matrix of damping coefficients, $\delta$ is the vector of rotor angles, $P_m$ is the vector of mechanical power inputs, and $P_e$ is the vector of electrical power outputs
• The electrical power output vector $P_e$ is a nonlinear function of the rotor angles $\delta$ and the bus voltages $V$, which can be computed from the power flow equations:
  • $P_e = \operatorname{Re}(diag(V)(Y_bV)^*)$
  • Where $Y_b$ is the bus admittance matrix and $^*$ denotes the complex conjugate
• The multi-machine swing equation, together with the power flow equations, forms a complete model of the electromechanical dynamics of the power system

Multi-Machine Rotor Angle Stability

Stability Assessment of Equilibrium Points

• The multi-machine rotor angle stability problem involves determining the stability of the equilibrium points of the multi-machine swing equation
• The equilibrium points correspond to the steady-state operating conditions of the power system, where the mechanical power input of each machine equals its electrical power output
  • They can be found by solving the power flow equations and setting the derivatives in the swing equation to zero
• The stability of an equilibrium point can be assessed by linearizing the multi-machine swing equation around that point and analyzing the eigenvalues of the resulting state matrix
  • If all eigenvalues have negative real parts, the equilibrium point is stable, and the system can maintain synchronism following a small disturbance
  • If any eigenvalue has a positive real part, the equilibrium point is unstable, and the system may lose synchronism even under small disturbances

Nonlinear Dynamical System Perspective

• The multi-machine rotor angle stability problem can be formulated as a nonlinear dynamical system with multiple equilibrium points, some of which may be stable while others are unstable
• The stability region of an equilibrium point is the set of initial conditions from which the system will converge to that equilibrium point
  • The stability regions of different equilibrium points are separated by the stable manifolds of the unstable equilibrium points, which act as boundaries
• The system may have multiple stable equilibrium points, each with its own stability region, and the final state of the system depends on the initial conditions and the disturbances
  • Large disturbances may cause the system to cross the stability boundaries and move from one stability region to another, resulting in a different steady-state condition or even instability

Factors Affecting Rotor Angle Stability

Network and Machine Parameters

• The stability of a multi-machine system depends on various factors, such as the network topology, the machine parameters, the operating conditions, and the disturbances
• A stronger transmission network with higher admittance values tends to enhance the synchronizing torque between the machines and improve stability
  • This is because a higher admittance reduces the electrical distance between the machines and facilitates the exchange of synchronizing power
• Machines with higher inertia constants and lower reactances are generally more stable, as they are less sensitive to disturbances and can maintain synchronism more easily
  • Higher inertia provides more stored kinetic energy to resist changes in speed, while lower reactance allows for more power transfer and synchronizing torque

Operating Conditions and Disturbances

• Operating conditions with higher power transfers and lower voltage profiles are more prone to instability, as they reduce the stability margins and increase the stress on the system
  • Higher power transfers mean that the machines are operating closer to their stability limits, and any disturbance can more easily push them out of synchronism
  • Lower voltage profiles indicate a weak transmission system and reduced power transfer capability, which makes the system more vulnerable to disturbances
• The location, type, and severity of disturbances play a crucial role in determining the stability of the system
  • Faults near critical machines or in weak areas of the network are more likely to cause instability, as they have a greater impact on the power balance and the synchronizing torque
  • The severity of the disturbance, such as the duration of a fault or the amount of load loss, also affects the stability margin and the ability of the system to recover

Control Systems and Stability Enhancement

• The actions of control systems, such as excitation systems, power system stabilizers, and FACTS devices, can significantly influence the stability of the multi-machine system by modulating the machine voltages, damping the oscillations, and controlling the power flows
• Excitation systems regulate the voltage of the generators and can provide fast-acting voltage support during disturbances, which helps maintain synchronism and prevent voltage collapse
  • Modern excitation systems with high gain and fast response can greatly enhance the transient stability of the system
• Power system stabilizers (PSS) are supplementary control devices that modulate the excitation of the generators to damp out the low-frequency oscillations in the system
  • They use signals such as rotor speed, frequency, or power to create a damping torque that counteracts the oscillations and improves the small-signal stability of the system
• Flexible AC Transmission Systems (FACTS) devices, such as static var compensators (SVC), static synchronous compensators (STATCOM), and thyristor-controlled series capacitors (TCSC), can control the power flows and improve the stability of the system by injecting or absorbing reactive power, regulating voltage, and modulating the impedance of the transmission lines
  • They can provide fast and continuous control of the power system parameters and enhance the transient and dynamic stability of the system

Stability Assessment of Multi-Machine Systems

Numerical Simulation Techniques

• Numerical simulations are essential tools for assessing the stability of multi-machine power systems, as they allow the modeling of complex nonlinear dynamics and the analysis of various scenarios
• The multi-machine swing equation can be solved numerically using methods such as Runge-Kutta or trapezoidal integration, which discretize the continuous-time differential equations into discrete-time algebraic equations
  • These methods approximate the solution of the differential equations by iteratively computing the state variables at discrete time steps based on the initial conditions and the system model
• The power flow equations can be solved using iterative methods such as Newton-Raphson or fast-decoupled load flow, which find the bus voltages and angles that satisfy the power balance equations
  • The power flow solution provides the initial conditions for the dynamic simulation and the reference values for the machine powers and voltages
• The numerical simulation typically involves setting up the initial conditions based on the steady-state power flow solution, applying the disturbances at specific times, and observing the transient response of the system over a certain period
  • The disturbances can be modeled as changes in the machine powers, bus voltages, or network topology, such as faults, line trips, or load changes

Stability Assessment Criteria and Techniques

• The stability of the system can be assessed by examining the time-domain waveforms of the rotor angles, speeds, voltages, and powers of the machines, as well as the frequency and damping of the oscillations
• Unstable cases are characterized by growing oscillations, diverging angles, or loss of synchronism, while stable cases show damped oscillations and convergence to a new equilibrium point
  • The stability margin can be quantified by the maximum allowable disturbance that the system can withstand without losing stability, such as the critical clearing time of a fault or the maximum power transfer limit
• Numerical simulations can be used to determine the critical clearing time of faults, the maximum power transfer capacity of the system, and the effectiveness of various control measures in enhancing stability
  • The critical clearing time is the maximum duration of a fault that the system can sustain without losing synchronism, and it is a key indicator of the transient stability of the system
  • The maximum power transfer capacity is the highest amount of power that can be transferred through the system without violating the stability limits, and it depends on the network topology, the machine parameters, and the control settings
• Modal analysis techniques, such as eigenvalue analysis and participation factors, can be used to identify the critical modes of oscillation and the key factors affecting the stability of the system
  • The eigenvalues of the linearized system provide information about the frequency and damping of the oscillatory modes, while the eigenvectors and participation factors indicate the relative contribution of each machine or state variable to the modes
  • Modal analysis can help design and tune the control systems, such as PSS and FACTS, to improve the damping of the critical modes and enhance the overall stability of the system

Probabilistic Stability Assessment

• Monte Carlo simulations can be performed to assess the probability of instability under various uncertainties, such as load variations, renewable generation intermittency, and equipment failures
  • Monte Carlo methods involve running multiple simulations with randomly sampled input parameters from their probability distributions and collecting statistics on the output variables of interest
• Probabilistic stability assessment provides a more comprehensive and realistic evaluation of the system stability, as it accounts for the stochastic nature of the power system and the variability of the operating conditions
  • It can help identify the most likely and the worst-case scenarios, as well as the expected frequency and duration of instability events
• Probabilistic stability indices, such as the probability of instability, the expected energy not served, and the expected cost of blackouts, can be computed from the Monte Carlo simulation results and used to quantify the risk and the impact of instability
  • These indices can inform the decision-making process for planning, operation, and maintenance of the power system, such as the allocation of resources for stability enhancement measures, the setting of reliability standards, and the pricing of ancillary services
• Probabilistic stability assessment can also be used to evaluate the effect of different uncertainties and risk factors on the stability of the system, such as the penetration of renewable energy sources, the aging of equipment, and the occurrence of extreme events
  • It can help identify the critical uncertainties and the most effective mitigation strategies, such as the addition of storage devices, the reinforcement of the transmission network, or the implementation of adaptive protection schemes