The University of Adelaide Press publishes peer-reviewed academic books by University of Adelaide staff. List of symbols, acronyms and abbreviations. i,j)th element, respectively the ith row and the ith column vector in matrix A.
Introduction
Why analyse the small-signal dynamic performance of power sys- tems?
1 and, more importantly, Henri Poincaré [1] showed that if the linearized form of the nonlinear system is stable, then the nonlinear system is also stable in the stationary state where the system is linearized. In practice, if the modeling of the devices is adequate, small signal tests with generator controls, for example, have shown close agreement between simulation and test results.
The purpose and features of the book
Examples are: (a) the performance of the “Integral Acceleration Power PSS” in Chapter 8; (b) the characteristics of two tools, Mode Shapes and Participation Factors in Chapter 9 (these are used in the analysis of the performance of multi-machine systems). A comprehensive set of the various small-signal models of synchronous generators and FACTS devices is provided in Chapter 4.
Synchronizing and damping torques
The net torque acting on the generator shaft will cause an acceleration relative to the system. The angle of the rotor of the generator immediately begins to increase according to the latter characteristic, which increases the flow of electricity from the generator.
Definitions of power system stability
Small-disturbance (or small-signal) rotor angle stability is concerned with the power system's ability to maintain synchronism under small disturbances. Voltage stability refers to the ability of a power system to maintain stable voltages across all buses in the system after being subjected to an outage.
Types of modes
This is an oscillating electromechanical mode and is associated with a group of generating stations in one area of the system swinging against a group of stations in one or more other areas of the system. Modes between areas are usually accompanied by (possibly) weak interconnections between geographically separated areas of the system.
Synchronous generator and transmission system controls
1 of the modal behavior as revealed by small-signal analysis provides a synoptic view of system characteristics that would require many large-signal studies of faults and other disturbances at different locations to obtain similar but not identical information. correctly. When investigating the small-signal performance of a system, not only its behavior under normal conditions is of interest, but also its performance in the immediate post-fault condition before the tap-changer and switching operations have had time to occur. reactive and also when all these operations have been completed after the disturbance.
Power system and controls performance criteria and measures
- Power system damping performance criteria
- Control system performance measures
Damping of power system fluctuations should be evaluated for design purposes according to design criteria that state that power system damping is considered adequate if, after the most critical credible contingency, simulations calibrated against past performance show that the half-life is at least the damped electromechanical mode of oscillation is no longer than five seconds. which is a guide presenting dynamic performance criteria, definitions and test objectives for excitation control systems used in power systems; (ii) Australian National Electricity Clause S5.2.5.13 “Voltage and Reactive Power Control”.
Validation of power system models
Robust controllers
1 synchronizes torques on the generators' shafts so that the power system is stable subject to the relevant stability margins. See section 1.2, point 3) (Note: as shown in chapter 13, the stabilizer for some inter-area modes may degrade the damping moments of other generators).
How small is ‘small’ in small-signal analysis?
1 Based on our criterion for a steady-state oscillatory peak-peak angular oscillation around the steady-state angle ; the 300 MW peak-to-peak power fluctuation is "linearly related" to the angular fluctuation. Similarly, at steady state, the maximum power swing angle is limited to 90 MW peak-to-peak and is "linearly related" to the smaller peak-to-peak angular swing.
Units of Modal Frequency
Advanced control methods
It is clear that under more loaded conditions, the range of disturbances over which the operation of the system can be considered more or less linear is much smaller. Nevertheless, combining ideas in advanced control methods and using the "inherent" characteristics of the system and its devices may not only be a fruitful direction of research, but may also lead to practical results.
Vowles, “Coordinating compensation methods for PSS in multi-machine systems,” Power Systems, IEEE Transactions on, vol. Angell, “Damping of System Oscillations with a Hydrogen Generating Unit,” Power Apparatus and Systems, IEEE Transactions on, vol.
Control systems techniques for
Introduction
- Purpose and aims of the chapter
2 on the analysis of the linearized system relates to those aspects of the nonlinear system. As also mentioned earlier, a theorem by Poincaré states that information about the stability of the nonlinear system, based on a stability analysis of the linearized equations, is exact at the selected steady-state operating condition.
Mathematical model of a dynamic plant or system
2 (ii) Derive an expression that describes the behavior of the shaft speed as the motor torque varies. Since the field current is independent of the load current, the electric torque is proportional to the load current.
The Laplace Transform
The first term, called the forced response, is determined by the nature of the forcing function, the input U(s). Applying these results directly to (2.8), we find the general form of the differential equation that describes the plant.
The poles and zeros of a transfer function
For example poles Note that poles and zeros can be real or complex. The meaning of poles and zeros will be examined in more detail in the following sections.
The Partial Fraction Expansion and Residues
- Calculation of Residues
- A simple check on values of the residues
When the transfer function is expressed as a summation of first-order transfer functions, as in (2.14), the constant is also known as the remainder, , of the pole at . The factor in in (2.15) will be small or negligible, and so will the rest.
Modes of Response
In the following cases, find the time response of the plant to a unit input step of magnitude A. The transient terms are determined by the initial magnitude of the input function (at time t(0+).
The block diagram representation of transfer functions
Similarly, Figure 2.3(c) shows that parallel blocks can be represented by the sum or difference of individual transfer functions. The purpose of automatic control of stable closed-loop systems is to reduce the error, i.e.
Characteristics of first- and second-order systems
- First-order system
- The second-order system
The second form is the ideal or classical form of the second-order transfer function, which has complex poles. It is often useful to refer to the properties of the step response of the ideal second-order system with a complex pair of poles.
The stability of linear systems
If any of the poles lie on the imaginary axis and all others lie on the left half of the s-plane, the system is said to be marginally stable. However, in practical linear control systems it is not possible to locate and maintain the poles exactly on the imaginary line, so the limit stability is only of academic interest.
Steady-state alignment and following errors
The closed loop system's response to a step change in the reference output is shown in figure 2.10. In the following, we will derive the alignment errors for a unity-feedback closed-loop system, i.e.
Frequency response methods
2 Frequency response methods assume that a sinusoidal signal of constant amplitude is applied to the system. Typically, the frequency response is plotted in two forms, the polar plot and the Bode plot; we will use Bode-type responses in later chapters.
The frequency response diagram and the Bode Plot
- Plotting the frequency response of the open-loop transfer function
Plot the straight-line approximations to the frequency response of the following open-loop transfer function of a unity feedback control system. The transfer function of the closed-loop system shown in Figure 2.9 was derived in Section 2.7; is the transfer function.
The Q-filter, a passband filter
An example of such a case is the model of a Francis turbine in a hydroelectric plant. The frequency response characteristics of the filter are shown in Figure 2.21 as a function of the damping ratio; note that the phase responses go through zero degrees at resonance.
State equations, eigen-analysis and applications
Introduction
- Example 3.1
If we eliminate , , , and from the above equations and insert the derivatives on the left side of the corresponding equations, we get There are two independent energy storage elements, the inductance of the field circuit and the inertia of the rotating system, L and J.
The concept of state and the state equations [1]
The linearized model of the non-linear dynamic system
- Linearization procedure
The dynamic performance of the linearized system can be characterized by the location of its poles in the complex s-plane. The modularity and sparsity of the system equations can be exploited when calculating eigenvalue sensitivities.
Solution of the State Equations
- The Natural Response
- Example 3.3
- Example 3.4: Natural response
- The Forced Response
- Example 3.4 (continued)
Elimination of the algebraic variables from the DAEs yields the conventional 'ABCD' form of the equations of state, ie it is also the characteristic equation of the system; the zeros of (3.25) are the poles of the system defined by the state matrix.
Eigen-analysis
The -element column vector, , is called the right eigenvector of the matrix corresponding to the eigenvalue. As described in Section 3.5.1, the eigenvalue of the system matrix is a real or complex number.
Decoupling the state equations
The unforced response of a time-invariant linear system is a (weighted) superposition of the response of each of the system's modes. It was noted in Section 2.9 that the system is unstable if a pole or a pair of complex poles lie in the right half of the s-plane.
Determination of residues from the state equations
Based on the concepts of controllability, the mode of care is 'highly' controllable from the input of the state space model of the system if the magnitude of is large relative to that evaluated at all other inputs - subject to a caveat 1. Therefore, the size of the residual must be relatively large at the modal frequency for a candidate controller or a stabilizer in the path from the output to the input to be effective.
Determination of zeros of a SISO sub-system
The expansion of the above determinant results in a polynomial in and the associated roots correspond to the zeros of the SISO subsystem. At the heart of the QZ algorithm is the definition of unitary matrices and such that both and are upper diagonal.
Mode shapes
For a given mode, the relative amplitudes or shapes of the responses are determined by the associated right-hand eigenvector. Consequently, we can plot the mode shapes of selected modes, these shapes reveal not only the relative amplitude of the modes within the mode, but also the relative phase between the responses of the modes.
Participation Factors
- The relative participation of a mode in a selected state
- The relative participation of a state in a selected mode
Therefore, in (3.49) provides a measure of the relative extent to which the htheigenvalue participates in the state at time; is therefore known as the participation factor of the h-th eigenvalue in the state. Therefore, when each state is excited in turn by a unit vector, the participation factor also provides a measure of the relative extent to which each of the n states participates at an eigenvalue at a given instant.
Eigenvalue sensitivities
In its simplest form, and the sensitivity is the product of the elements of the left and right eigenvectors, . The above analysis can then be extended to evaluate the sensitivity of to changes in the parameter.
Lima, "Efficient methods for finding zeros of the transfer function of power systems," Power Systems, IEEE Transactions on, vol. Rommes, "Computation of Dominant Zeros of the Transfer Function With Applications to Oscillation Damping Control of Large Power Systems," Power Systems, IEEE Transactions on, vol.
Small-signal models of synchronous generators, FACTS devices and the power system
Introduction
90 Generators, FACTS Devices, and System Models Small-signal models of the FACTS device class are formulated in Section 4.3 and include Static Var Compensator (SVC), Voltage Converter (VSC), Static Synchronous Compensator (STATCOM) models. , and HVDC transmission links. The general VSC model formulated in Section 4.3.3 is used as a component in the simplified STATCOM model in Section 4.3.4 and for the rectifier and converter in the VSC HVDC transmission link model in Section 4.3.7.
Small-signal models of synchronous generators
- Structure of the per-unit linearized synchronous generator models
- Generator modelling assumptions
- Alternative d- and q- axis rotor structures
- Per-unit rotor equations of motion
- Non-reciprocal definition of the per-unit field voltage and current
- Modelling generator saturation
- Generator parameter conversions
100 Generators, FACTS Devices & System Models Table 4.3 Summary of parameters in the coupled circuit per unit. The initial values of the saturation demagnetizing currents in the non-reciprocal system per unit are .
Small-signal models of FACTS Devices
- Linearized equations of voltage, current and power at the AC terminals of FACTS Devices: general resultsFACTS Devices: general results
- Model of a Static VAR Compensator (SVC)
- Model of a Voltage Sourced Converter (VSC)
So per unit on the base quantities of the device's k-th connection. All quantities are per unit based on the base quantities of either the kth terminal of the device (d,k) or that of the network bus to which the kth terminal is connected (n,k).
