# Small Signal Stability in Electric Power Systems

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_260-1

## Abstract

Small signal rotor angle stability analysis in power systems is associated with insufficient damping of oscillations under small disturbances. Rotor angle oscillations due to insufficient damping have been observed in many power systems around the world. This entry overviews the predominant approach to examine small signal rotor angle stability in large power systems using eigenvalue analysis.

## Keywords

Small signal rotor angle stability Oscillatory modes Low-frequency oscillations Eigenvalues Eigenvectors Mode shape Participation factors## Small Signal Rotor Angle Stability in Power Systems

As power system interconnections grew in number and size, automatic controls such as voltage regulators played critical roles in enhancing reliability by increasing the synchronizing capability between the interconnected systems. As technology evolved the capabilities of voltage regulators to provide synchronizing torque following disturbances were significantly enhanced. It was, however, observed that voltage regulators tended to reduce damping torque, as a result of which the system was susceptible to rotor angle oscillatory instability. An excellent exposition of the mechanism and the underlying analysis is provided in the textbooks (Anderson and Fouad 2003; Sauer and Pai 1998; Kundur 1993), and a number of practical aspects of the analysis are detailed in Eigenanalysis and Frequency Domain Methods for System Dynamic Performance (1989) and Rogers (2000). Two types of rotor angle oscillations are commonly observed. Low-frequency oscillations involving synchronous machines in different operating areas are commonly referred to as inter-area oscillations. These oscillations are typically in the 0.1–2 Hz frequency range. Oscillations between local machines or a group of machines at a power plant are referred to as plant mode oscillations. These oscillations are typically above the 2 Hz frequency range. The modes associated with rotor angle oscillations are also termed inertial modes of oscillation. Other modes of oscillations associated with the various controls also exist. With the integration of significant new wind and photovoltaic generation which are interconnected to the grid using converters, new modes of oscillation involving the converter controls and conventional synchronous generator states are being observed.

The basis for small signal rotor angle stability analysis is that the disturbances considered are small enough to justify the use of linear analysis to examine stability (Kundur et al. 2004). As a result, Lyapunov’s first method Vidyasagar (1993) provides the analytical underpinning to analyze small signal stability. Eigenvalue analysis is the predominant approach to analyze small signal rotor angle stability in power systems. Commercial software packages that utilize sophisticated algorithms to analyze large-scale power systems with the ability to handle detailed models of power system components exist.

*x*is the state vector and

*z*is a vector of algebraic variables. Small signal stability analysis involves the linearization of (1) around a system operating point which is typically determined by conducting a power flow analysis:

*A*represents the system state matrix. Based on Lyapunov’s first method, the eigenvalues of

*A*characterize the small signal stability behavior of the nonlinear system in a neighborhood of the operating point around which the system is linearized. The eigenvectors corresponding to the eigenvalues also provide significant qualitative information. For each eigenvalue

*λ*

_{ i }, there exists a vector

*u*

_{ i }known as the right eigenvector of

*A*which satisfies the equation

*v*

_{ i }known as the left eigenvector of

*A*which satisfies

*k*

_{ ij }= 1. The right eigenvectors can be assembled together as columns of a square matrix

*U*, and the corresponding left eigenvectors can be assembled as rows of a matrix

*V*; then

*i*th state variable in the

*k*th mode and vice versa.

## Small Signal Stability Analysis Tools for Large Power Systems

*A*matrix is a complex task for large power systems with a multitude of dynamic components. The approach in Powertech (2012) utilizes a technique where state space equations are developed for each dynamic component in the system using a solved power flow solution and the dynamic data description for a given system. These state space equations are then coupled based on the system topology, and the system

*A*matrix is derived as in (3). Reference Martins (1989) takes advantage of the sparsity of the Jacobian matrix in (2) and develops efficient algorithms to determine the eigenvalues and eigenvectors. The software tools also provide the flexibility of a number of different options with regard to eigenvalue computations:

- 1.
Calculation of a specific eigenvalue at a specified frequency or with a specified damping ratio

- 2.
Simultaneous calculation of a group of relevant eigenvalues in a specified frequency range or in specified damping ratio range

- 1.
Frequency response plots

- 2.
Participation factors

- 3.
Transfer functions, residues, controllability, and observability factors

- 4.
Linear time response to step changes

- 5.
Eigenvalue sensitivities to changes in specified parameters

## Applications of Small Signal Stability Analysis in Power Systems

Small signal stability analysis tools are used for a range of applications in power systems. These applications include:

*Analysis of local stability problems*– These types of stability problems are primarily associated with the tuning of control associated with the synchronous generator, converter interconnected renewable resources, and HVDC link current control. In certain cases analysis of local stability problems could also involve design of supplementary controllers which enhance the stability region. Since the stability problem pertains to a local portion of the power system, there is significant flexibility in modeling the system. In many instances local stability problems facilitate the use of a simple representation of a power system which could include the particular machine or a local group of machines in question together with a highly equivalenced representation of the rest of the system. In cases where controls other than generator controls influence stability, e.g., static VAr compensators or HVDC links, the system representation would need to be extended to include portions of the system where these devices are located. Typical small signal stability problems that are analyzed include:

- 1.
Power system stabilizer design

- 2.
Automatic voltage regulator tuning

- 3.
Governor tuning

- 4.
DC link current control

- 5.
Small signal stability analysis for subsynchronous resonance

- 6.
Load modeling effects on small signal stability

References Eigenanalysis and Frequency Domain Methods for System Dynamic Performance (1989) and Rogers (2000) provide comprehensive examples of the analysis conducted for each of the problems listed above.

*Analysis of global stability problems*– These types of stability problems are associated with controls that impact generators located in different areas of the power systems. The analysis of these inter-area problems requires a more systematic approach and involves representation of the power system in greater detail. The problems that are analyzed under this category include:

- 1.
Power system stabilizer design

- 2.
HVDC link modulation

- 3.
Static VAr compensator controls

References Eigenanalysis and Frequency Domain Methods for System Dynamic Performance (1989); Rogers (2000) again provide details of the analysis conducted for each of the problems listed under this category.

## Cross-References

## Bibliography

- Anderson PM, Fouad AA (2003) Power system control and stability, 2nd edn. Wiley Interscience, HobokenGoogle Scholar
- Eigenanalysis and Frequency Domain Methods for System Dynamic Performance (1989) IEEE Special Publication, 90TH0292-3-PWRGoogle Scholar
- Kundur P (1993) Power system stability and control. McGraw Hill, San FranciscoGoogle Scholar
- Kundur P, Paserba J, Ajjarapu V, Andersson G, Bose A, Canizares C, Hatziargyriou N, Hill D, Stankovic A, Taylor C, Van Cutsem T, Vittal V (2004) Definition and classification of power system stability. IEEE/CIGRE joint task force on stability terms and definitions report. IEEE Trans Power Syst 19:1387–1401CrossRefGoogle Scholar
- Martins N (1989) Efficient eigenvalue and frequency response methods applied to power system small signal stability studies. IEEE Trans Power Syst 1:74–82Google Scholar
- Pérez-Arriaga IJ, Verghese GC, Schweppe FC (1982) Selective modal analysis with applications to electric power systems, part 1: Heuristic introduction. IEEE Trans Power Appar Syst 101:3117–3125CrossRefGoogle Scholar
- Powertech (2012) Small signal analysis tool (SSAT) user manual. Powertech Labs Inc, SurreyGoogle Scholar
- Rogers G (2000) Power system oscillations. Kluwer Academic, DordrechtCrossRefGoogle Scholar
- Sauer PW, Pai MA (1998) Power system dynamics and stability. Prentice Hall, Upper Saddle RiverGoogle Scholar
- Vidyasagar M (1993) Nonlinear systems analysis, 2nd edn. Prentice Hall, Englewood CliffszbMATHGoogle Scholar