Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Small Signal Stability in Electric Power Systems

  • Vijay VittalEmail author
Living reference work entry
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.

The power system representation is described by a set of nonlinear differential algebraic equations shown in (1)
$$\begin{array}{l} \dot{x} = f\left (x,z\right ) \\ 0 = g\left (x,z\right )\\ \end{array}$$
(1)
where 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:
$$\left [\begin{array}{*{20}c} \Delta \dot{x}\\ 0\\ \end{array} \right ] = \left [\begin{array}{*{20}c} J_{1} & J_{2} \\ J_{3} & J_{4}\\ \end{array} \right ]\left [\begin{array}{*{20}c} \Delta x\\ \Delta z\\ \end{array} \right ]$$
(2)
The power system state matrix can be obtained by eliminating the vector of algebraic variables\(\Delta z\) in (2)
$$\Delta \dot{x} = \left (J_{1} - J_{2}J_{4}^{-1}J_{ 3}\right )\Delta x = A\Delta x$$
(3)
where 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
$$Au_{i} =\lambda _{i}u_{i}$$
(4)
There also exists a row vector v i known as the left eigenvector of A which satisfies
$$v_{i}A =\lambda _{i}v_{i}$$
(5)
For a system which has distinct eigenvalues, the right and left eigenvectors form an orthogonal set governed by
$$\begin{array}{l} v_{i}u_{j} = k_{ij} \\ \mbox{ where}\\ k_{ ij}\neq 0\mbox{ }i = j \\ k_{ij} = 0\mbox{ }i\neq j\\ \end{array}$$
(6)
One set (either right or left) of eigenvectors are usually scaled to unity and the other set obtained by solving (6) with 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
$$V = {U}^{-1}$$
(7)
and
$$V AU = \Lambda $$
(8)
where\(\Lambda \) is a diagonal matrix with the distinct eigenvalues as the diagonal entries. The relationship in (8) is a similarity transformation and in the case of distinct eigenvalues provides a pathway to obtain solutions to the linear system of equations (3). Applying the following similarity transformation to (3)
$$\Delta x = Uz\mbox{ } \rightarrow\Delta x_{i}\left (t\right ) =\displaystyle\sum \limits _{ j=1}^{n}u_{ ij}z_{j}{e}^{\lambda _{j}t}$$
(9)
$$U\dot{z} = AUz$$
(10)
$$\dot{z} = {U}^{-1}AUz = V AUz = \Lambda z$$
(11)
$$\dot{z}_{i}\left (t\right ) =\lambda _{i}z_{i} \Rightarrow z_{i}\left (t\right ) = z_{i}\left (0\right ){e}^{\lambda _{i}t}$$
(12)
$$z_{i}\left (0\right ) = v_{i}^{T}\Delta x\left (0\right )$$
(13)
$$z_{i}\left (t\right ) = v_{i}^{T}\Delta x\left (0\right ){e}^{\lambda _{i}t}$$
(14)
From (9) and (14), it can be observed that the right eigenvector describes how each mode of the system is distributed throughout the state vector (and is referred to as the mode shape), and the left eigenvector in conjunction with the initial conditions of the system state vector determines the magnitude of the mode. The right eigenvector or the mode shape has been often used to identify dynamic patterns in small signal dynamics. One problem with the mode shape is that it is dependent on the units and scaling of the state variables as a result of which it is difficult to compare the magnitudes of entries that are disparate and correspond to states that impact the dynamics differently. This resulted in the development of the participation factors (Pérez-Arriaga et al. 1982) which are dimensionless and independent of the choice of units. The participation factor is expressed as
$$p_{ik} = v_{ik}u_{ik}$$
(15)
The magnitude of the participation factor measures the relative participation of the ith state variable in the kth mode and vice versa.

Small Signal Stability Analysis Tools for Large Power Systems

Efficient software tools exist that facilitate the application of the methods in section “Small Signal Rotor Angle Stability in Power Systems” to large power systems (Powertech 2012; Martins 1989). These tools incorporate detailed models of power system components and also leverage the sparsity in power systems. The building of the 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. 1.

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

     
  2. 2.

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

     
In addition to the features described above, commercial software packages also provide features to evaluate:
  1. 1.

    Frequency response plots

     
  2. 2.

    Participation factors

     
  3. 3.

    Transfer functions, residues, controllability, and observability factors

     
  4. 4.

    Linear time response to step changes

     
  5. 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. 1.

    Power system stabilizer design

     
  2. 2.

    Automatic voltage regulator tuning

     
  3. 3.

    Governor tuning

     
  4. 4.

    DC link current control

     
  5. 5.

    Small signal stability analysis for subsynchronous resonance

     
  6. 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. 1.

    Power system stabilizer design

     
  2. 2.

    HVDC link modulation

     
  3. 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

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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Arizona State UniversityTempeUSA