Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Lyapunov Methods in Power System Stability

  • Hsiao-Dong ChiangEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_266-1

Keywords

Power System Equilibrium Point Energy Function Stability Boundary Electric Power System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction

Energy functions, an extension of the Lyapunov functions, have been practically used in electric power systems for several applications. A comprehensive energy function theory for general nonlinear autonomous dynamical systems along with its applications to electric power systems will be summarized in this article.

We consider a general nonlinear autonomous dynamical system described by the following equation:
$$(t) = f(x(t))$$
(1)
We say a function V : R n R is an energy function for the system (1) if the following three conditions are satisfied (Chiang et al. 1987):
  1. (E1):

    The derivative of the energy function V (x) along any system trajectory x(t) is nonpositive, i.e., \(\dot{V }(x(t)) \leq 0\).

     
  2. (E2):

    If x(t) is a nontrivial trajectory (i.e., x(t) is not an equilibrium point), then along the nontrivial trajectory x(t) the set \(\{t \in R :\dot{ V }(x(t)) = 0\}\) has measure zero in R.

     
  3. (E3):

    That a trajectory x(t) has a bounded value of V (x(t)) for tR +implies that the trajectory x(t) is also bounded.

     

Condition (E1) indicates that the value of an energy function is nonincreasing along its trajectory, but does not imply that the energy function is strictly decreasing along any trajectory. Conditions (E1) and (E2) imply that the energy function is strictly decreasing along any system trajectory. Property (E3) states that the energy function is a proper map along any system trajectory but need not be a proper map for the entire state space. Obviously, an energy function may not be a Lyapunov function.

As an illustration of the energy function, we consider the following classical transient stability model and derive an energy function for the model. Consider a power system consisting of n generators. Let the loads be modeled as constant impedances. Under the assumption that the transfer conductance of the reduced network after eliminating all load buses is zero, the dynamics of the ith generator can be represented by the equations
$$\displaystyle\begin{array}{rcl} \dot{\delta }_{i}& =& \omega _{i} \\ M_{i}\dot{\omega }_{i}& =& P_{i} - D_{i}\omega _{i} -\displaystyle\sum _{j=1}V _{i}V _{j}B_{ij}\mathrm{sin}(\delta _{i} -\delta _{j})\end{array}$$
(2)
where the voltage at node i+1 is served as the reference, i.e., \(\delta _{i+\mathbf{1}} :\,= \mathbf{0}\). This is a version of the so-called classical model of the power system. It can be shown that the following function is an energy function V (δ, ω) which satisfies conditions (E1)–(E3) for the classical model (2).
$$\displaystyle\begin{array}{rcl} V (\delta ,\omega )& =& \frac{1} {2}\displaystyle\sum \nolimits _{i=\mathrm{\mathbf{1}}}^{n}M_{ i}\omega _{i}^{\mathrm{\mathbf{2}}} -\displaystyle\sum \nolimits _{ i=\mathbf{1}}^{n}P_{ i}\left (\delta _{i} -\delta _{j}^{s}\right ) \\ & & -\displaystyle\sum \nolimits _{i=\mathbf{1}}^{n}\displaystyle\sum \nolimits _{ j=i+\mathbf{1}}^{n+\mathbf{1}}V _{ i}V _{j}B_{ij}\{\mathbf{cos}(\delta _{i} -\delta _{j}) -\mathbf{cos}(\delta _{i}^{s} -\delta _{ j}^{s})\}\end{array}$$
(3)
where x s = (δ s , 0) is the stable equilibrium point under consideration.

Energy Function Theory

In general, the dynamical behaviors of trajectories of general nonlinear systems can be very complicated. The asymptotical behaviors (i.e., the ω-limit set) of trajectories can be quasiperiodic trajectories or chaotic trajectories. However, as shown below, every trajectory of system (1) having an energy function has only two modes of behaviors: its trajectory either converges to an equilibrium point or goes to infinity (becomes unbounded) as time increases. This result is explained in the following theorem:

Theorem 1 (Global Behavior of Trajectories) 

If there exists a function satisfying condition (E1) and condition (E2) of the energy function for system (1) , then every bounded trajectory of system (1) converges to one of the equilibrium points.

Theorem 1 asserts that there does not exist any limit cycle (oscillatory behavior) or bounded complicated behavior such as almost periodic trajectory, chaotic motion, etc. in the system. We next show a sharper result, asserting that every trajectory on the stability boundary must converge to one of the unstable equilibrium points (UEPs) on the stability boundary. Recall that for a hyperbolic equilibrium point, it is an (asymptotically) stable equilibrium point if all the eigenvalues of its corresponding Jacobian have negative real parts; otherwise it is an unstable equilibrium point. Let \(\hat{x}\) be a hyperbolic equilibrium point. Its stable and unstable manifolds, \({W}^{s}(\hat{x})\) and \({W}^{u}(\hat{x})\), are well defined. There are many physical systems such as electric power systems containing multiple stable equilibrium points. A useful concept for these kinds of systems is that of the stability region (also called the region of attraction). The stability region of a stable equilibrium point x s is defined as
$$A(x_{s}) := \left \{x \in {R}^{n} :\mathop{\mathbf{lim}}\limits _{ t\rightarrow \infty }\boldsymbol{\Phi }_{t}(x) = x_{s}\right \}$$
The boundary of stability region A(x s ) is called the stability boundary of (x s ) and will be denoted by ∂A(x s ).

Theorem 2 (Trajectories on the Stability Boundary (Chiang et al.  1987)) 

If there exists an energy function for system (1) , then every trajectory on the stability boundary ∂A(x s ) converges to one of the equilibrium points on the stability boundary ∂A(x s ).

The significance of this theorem is that it offers an effective way to characterize the stability boundary. In fact, Theorem 2 asserts that the stability boundary ∂A(x s ) is contained in the union of stable manifolds of the UEPs on the stability boundary, i.e.,
$$\partial A(x_{s}) \subseteq \displaystyle\bigcup \limits _{x_{i}\in \{E\cap \partial A(x_{s})\}}{W}^{s}(x_{ i})$$

The following two theorems give interesting results on the structure of the equilibrium points on the stability boundary. Moreover, it presents a necessary condition for the existence of certain types of equilibrium points on a bounded stability boundary.

Theorem 3 (Structure of Equilibrium Points on the Stability Boundary (Chiang and Thorp  1989)) 

If there exists an energy function for system (1) which has an asymptotically stable equilibrium point x s (but not globally asymptotically stable), then the stability boundary (x s ) must contain at least one type one equilibrium point. If, furthermore, the stability region is bounded, then the stability boundary ∂A(x s ) must contain at least one type one equilibrium point and one source.

Theorem 4 (Sufficient Condition for Unbounded Stability Region (Chiang et al.  1987)) 

If there exists an energy function for system (1) s which has an asymptotically stable equilibrium point x s (but not globally asymptotically stable) and if ∂A(x s ) contains no source, then the stability region A(x s ) is unbounded.

A direct application of this is that the stability boundary ∂A(x s ) of an (asymptotically) stable equilibrium point of the classical power system stability model (2) is unbounded.

Optimally Estimating Stability Region Using Energy Functions

In this section, we focus on how to optimally determine the critical level value of an energy function for estimating the stability boundary ∂A(x s ). We consider the following set:
$$S_{v}(k) =\{ x \in {R}^{n} : V \mbox{ (}x\mbox{ )} < k\}$$
(4)
where V (. ) : R n R is an energy function. We shall call the boundary of set (2) \(\partial S(k) :=\{ x \in {R}^{n} : V (x) = k\}\) the level set (or constant energy surface) and k the level value. Generally speaking, this set S(k) can be very complicated with several connected components even for the 2-dimensional case. We use the notation S k (x s ) to denote the only component of the several disjoint connected components of S k that contains the stable equilibrium point x s .

Theorem 5 (Optimal Estimation) 

Consider the nonlinear system (1) which has an energy function V (x). Let x s be an asymptotically stable equilibrium point whose stability region A(x s ) is not dense in R n . Let E 1 be the set of type one equilibrium points and \(\hat{c} =\min _{x_{i}\in \partial A(x_{s})\mathop{\cap} \nolimits _{\mathrm{\mathbf{1}}}^{E}}\,V \mbox{ (}x_{i}\mbox{ )}\) , and then

  1. 1.

    \(S_{\hat{c}}(x_{s}) \subset A\mbox{ (}x_{s}\mbox{ )}\) .

     
  2. 2.

    The set \(\{S_{b}(x_{s}) \cap \bar{ {A}}^{c}(x_{s})\}\) is nonempty for any number b > c.

     
This theorem leads to an optimal estimation of the stability region A(x s ) via an energy function V (.) (Chiang and Thorp 1989). For the purpose of illustration, we consider the following simple example:
$$\displaystyle\begin{array}{rcl} \dot{x}_{1}& =& -\sin x_{i} - 0.5\sin (x_{1} - x_{2}) + 0.01 \\ \dot{x}_{2}& =& -0.5\sin x_{2} - 0.5\sin (x_{2} - x_{1}) + 0.05\end{array}$$
(5)
It is easy to show that the following function is an energy function for system (5):
$$V (x_{1},x_{2}) = -2\cos x_{1} -\cos x_{2} -\cos (x_{1} - x_{2}) - 0.02x_{1} - 0.1x_{2}$$
(6)
The point x s x 1 s , x 2 s = (0. 02801, 0. 06403) is the stable equilibrium point whose stability region is to be estimated. Applying the optimal scheme to system (5), we have the critical level value of − 0.31329. The Curve A in Fig. 1 is the exact stability boundary ∂A(x s ) while Curve B is the stability boundary estimated by the connected component (containing the s.e.p. x s ) of the constant energy surface. It can be seen that the critical level value, − 0.31329, is indeed the optimal value.
Fig. 1

Curve A is the exact stability boundary ∂A(x s ) of system (5), while Curve B is the stability boundary estimated by the constant energy surface (with level value of − 0.31329) of the energy function

Constructing Analytical Energy Functions for Transient Stability Models

The task of constructing an energy function for a (post-fault) transient stability model is essential to direct stability analysis of power systems. The role of the energy function is to make feasible a direct determination of whether a given point (such as the initial point of a post-fault power system) lies inside the stability region of post-fault SEP without performing numerical integration. It has been shown that a general (analytical) energy function for power systems with losses does not exist (Chiang 1989). One key implication is that any general procedure attempting to construct an energy function for a lossy power system transient stability model must include a step that checks for the existence of an energy function. This step essentially plays the same role as the Lyapunov equation in determining the stability of an equilibrium point.

Several schemes are available for constructing numerical energy functions for power system transient stability models expressed as a set of general differential-algebraic equations (DAEs) (Chu and Chiang 19992005).

Applications

After decades of research and development in the energy-function-based direct methods and the time-domain simulation approach, it has become clear that the capabilities of direct methods and that of the time-domain approach complement each other. The current direction of development is to include appropriate direct methods and time-domain simulation programs within the body of overall power system stability simulation programs (Chiang et al. 1995; Chiang 19992011; Fouad and Vittal 1991; Sauer and Pai 1998). For example, the direct method provides the advantages of fast computational speed and energy margins which make it a good complement to the traditional time-domain approach. The energy margin and its functional relations to certain power system parameters are an effective complement to develop tools such as preventive control schemes for credible contingencies which are unstable and to develop fast calculators for available transfer capability limited by transient stability.

An effective, theory-based methodology for online screening and ranking of a large set of contingencies at operating points obtained from state estimators has been developed in Chiang et al. (2013). A set of improved BCU classifiers, along with their analytical basis, has been developed. Extensive evaluation of the improved BCU classifiers on a large test system and on the actual PJM interconnection system for a fast screening has been performed. This evaluation study is the largest in terms of system size, 14,500 buses and 3,000 generators, for a practical online transient stability assessment application. The evaluation results, performed on a total number of 5.3 million contingencies, were very promising in terms of speed, accuracy, reliability, and robustness (Chiang et al. 2013). This study also confirms the practicality of theory-based methodology for online transient stability assessment of large-scale power systems; in particular, theory-based methods are suitable for power system online applications which demand speed, accuracy, reliability, and robustness.

Cross-References

Recommended Reading

A recent book which contains a comprehensive treatment of energy functions theory and applications is Chiang (2011).

Bibliography

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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.School of Electrical and Computer EngineeringCornell UniversityIthacaUSA