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Direct Stability
Methods
Vijay Vittal
Arizona State University
11.1 Review of Literature on Direct Methods ....................... 11 - 11.2 The Power System Model ............................................... 11 - Review of Stability Theory 11.3 The Transient Energy Function...................................... 11 - 11.4 Transient Stability Assessment ....................................... 11 - 11.5 Determination of the Controlling UEP......................... 11 - 11.6 The BCU (Boundary Controlling UEP) Method ....... 11 - 11.7 Applications of the TEF Method and Modeling Enhancements .............................................. 11 -
Direct methods of stability analysis determine the transient stability (as defined in Chapter 7 and described in Chapter 8) of power systems without explicitly obtaining the solutions of the differential equations governing the dynamic behavior of the system. The basis for the method is Lyapunov’s second method, also known as Lyapunov’s direct method, to determine stability of systems governed by differential equations. The fundamental work of A.M. Lyapunov (1857–1918) on stability of motion was published in Russian in 1893, and was translated into French in 1907 (Lyapunov, 1907). This work received little attention and for a long time was forgotten. In the 1930s, Soviet mathematicians revived these investigations and showed that Lyapunov’s method was applicable to several problems in physics and engineering. This revival of the subject matter has spawned several contributions that have led to the further development of the theory and application of the method to physical systems. The following example motivates the direct methods and also provides a comparison with the conventional technique of simulating the differential equations governing the dynamics of the system. Figure 11.1 shows an illustration of the basic idea behind the use of the direct methods. A vehicle, initially at the bottom of a hill, is given a sudden push up the hill. Depending on the magnitude of the push, the vehicle will either go over the hill and tumble, in which case it is unstable, or the vehicle will climb only part of the way up the hill and return to a rest position (assuming that the vehicle’s motion will be damped), i.e., it will be stable. In order to determine the outcome of disturbing the vehicle’s equilibrium for a given set of conditions (mass of the vehicle, magnitude of the push, height of the hill, etc.), two different methods can be used:
- Knowing the initial conditions, obtain a time solution of the equations describing the dynamics of the vehicle and track the position of the vehicle to determine how far up the hill the vehicle will travel. This approach is analogous to the traditional time domain approach of determining stability in dynamic systems.
- The approach based on Lyapunov’s direct method would consist of characterizing the motion of the dynamic system using a suitable Lyapunov function. The Lyapunov function should satisfy certain sign definiteness properties. These properties will be addressed later in this subsection. A natural choice for the Lyapunov function is the system energy. One would then compute the
energy injected into the vehicle as a result of the sudden push, and compare it with the energy needed to climb the hill. In this method, there is no need to track the position of the vehicle as it moves up the hill. These methods are simple to use if the calculations involve only one vehicle and one hill. The complexity increases if there are several vehicles involved as it becomes necessary to determine (a) which vehicles will be pushed the hardest, (b) how much of the energy is imparted to each vehicle, (c) which direction will they move, and (d) how high a hill must they climb before they will go over the top. The simple example presented here is analogous to analyzing the stability of a one-machine-infinite- bus power system. The approach presented here is identical to the well-known equal area criterion (Kimbark, 1948; Anderson and Fouad, 1994) which is a direct method for determining transient stability for the one-machine-infinite-bus power system. For a more detailed discussion of the equal area criterion and its relationship to Lyapunov’s direct method refer to Pai (1981), chap. 4; Pai (1989), chap. 1; Fouad and Vittal (1992), chap. 3.
11.1 Review of Literature on Direct Methods
In the review presented here, we will deal only with work relating to the transient stability analysis of multimachine power systems. In this case the simple example presented above becomes quite complex. Several vehicles which correspond to the synchronous machines are now involved. It also becomes necessary to determine (a) which vehicles will be pushed the hardest, (b) what portion of the disturbance energy is distributed to each vehicle, (c) in which directions the vehicles move, and (d) how high a hill must the vehicles climb before they will go over. Energy criteria for transient stability analysis were the earliest of all direct methods of multimachine power system transient stability assessment. These techniques were extensions of the equal area criterion to power systems with more than two generators represented by the classical model (Anderson and Fouad, 1994, chap. 2). Researchers from the Soviet Union conducted early work in this area (1930s and 1940s). There were very few results on this topic in Western literature during the same period. In the 1960s the application of Lyapunov’s direct method to power systems generated a great deal of activity in the academic community. In most of these investigations, the classical power system model was used. The early work on energy criteria dealt with two main issues: (a) characterization of the system energy, and (b) the critical value of the energy. Several excellent references that provide a detailed review of the development of the direct methods for transient stability exist. Ribbens-Pavella (1971) and Fouad (1975) are early review papers and
FIGURE 11.1 Illustration of idea behind direct methods.
efficiency. Work related to the large-scale demonstration of the TEF method is found in Carvalho et al. (1986). The work dealing with extending the applicability of the TEF method is presented in Fouad et al. (1986). Significant contributions to this aspect of the TEF method can also be found in Padiyar and Sastry (1987), Padiyar and Ghosh (1989), and Abu-Elnaga et al. (1988). In Chiang (1985), Chiang et al. (1987), and Chiang et al. (1988), a significant contribution was made to provide an analytical justification for the stability region for multimachine power systems, and a systematic procedure to obtain the controlling UEP was also developed. Zaborsky et al. (1988) also provide a comprehensive analytical foundation for characterizing the region of stability for multi- machine power systems. With the development of a systematic procedure to determine and characterize the region of stability, a significant effort was directed toward the application of direct methods for online transient stability assessment. This work, reported in Waight et al. (1994) and Chadalavada et al. (1997), has resulted in an online tool which has been implemented and used to rank contingencies based on their severity. Another online approach implemented and being used at B.C. Hydro is presented in Mansour et al. (1995). A recent effort with regard to classifying and ranking contingencies quite similar to the one presented in Chadalavada et al. (1997) is described in Chiang et al. (1998). Some recent efforts (Ni and Fouad, 1987; Hiskens et al., 1992; Jiang et al., 1995) also deal with the inclusion of FACTS devices in the TEF analysis.
11.2 The Power System Model
The classical power system model will now be presented. It is the ‘‘simplest’’ power system model used in stability studies and is limited to the analysis of first swing transients. For more details regarding the model, the reader is referred to Anderson and Fouad (1994), Fouad and Vittal (1992), Kundur (1994), and Sauer and Pai (1998). The assumptions commonly made in deriving this model are: For the synchronous generators
- Mechanical power input is constant.
- Damping or asynchronous power is negligible.
- The generator is represented by a constant EMF behind the direct axis transient (unsaturated) reactance.
- The mechanical rotor angle of a synchronous generator can be represented by the angle of the voltage behind the transient reactance. The load is usually represented by passive impedances (or admittances), determined from the predisturbance conditions. These impedances are held constant throughout the stability study. This assumption can be improved using nonlinear models. See Fouad and Vittal (1992), Kundur (1994), and Sauer and Pai (1998) for more details. With the loads represented as constant impedances, all the nodes except the internal generator nodes can be eliminated. The generator reactances and the constant impedance loads are included in the network bus admittance matrix. The generators’ equations of motion are then given by
M (^) i dvi dt
¼ Pi Pei ddi dt ¼ vi i ¼ 1, 2,... , n (11:1)
where
Pei ¼
Xn j j¼ 6 ¼ (^1) i
C (^) ij sin di dj
þ Dij cos di dj
Pi ¼ Pmi E (^) i^2 Gii Cij ¼ E (^) i EjB (^) ij, Dij ¼ Ei Ej Gij Pmi ¼ Mechanical power input Gii ¼ Driving point conductance Ei ¼ Constant voltage behind the direct axis transient reactance vi, di ¼ Generator rotor speed and angle deviations, respectively, with respect to a synchronously rotating reference frame Mi ¼ Inertia constant of generator Bij (G (^) ij) ¼ Transfer susceptance (conductance) in the reduced bus admittance matrix Equation (11.1) is written with respect to an arbitrary synchronous reference frame. Transformation of this equation to the inertial center coordinates not only offers physical insight into the transient stability problem formulation in general, but also removes the energy associated with the motion of the inertial center which does not contribute to the stability determination. Referring to Eq. (11.1), define
MT ¼
Xn
i¼ 1
Mi
d 0 ¼
MT
Xn
i ¼ 1
Mi
then,
MT vv_ 0 ¼
Xn
i¼ 1
Pi Pei ¼
Xn
i ¼ 1
Pi 2
Xn^1
i ¼ 1
Xn
j ¼i þ 1
Dij cos dij
dd^ _ 0 ¼ v 0
The generators’ angles and speeds with respect to the inertial center are given by
ui ¼ di d 0 vv ~i ¼ vi v 0
i ¼ 1, 2,... , n (11:4)
and in this coordinate system the equations of motion are given by
M (^) i ~vv~vv_i ¼ Pi P (^) mi Mi MT
PCOI
uu^ _i ¼ vv~i i ¼ 1, 2,... , n
11.2.1 Review of Stability Theory
A brief review of the stability theory applied to the TEF method will now be presented. This will include a few definitions, some important results, and an analytical outline of the stability assessment formulation. The definitions and results that are presented are for differential equations of the type shown in Eqs. (11.1) and (11.5). These equations have the general structure given by
xx _(t ) ¼ f (t, x(t)) (11:6)
The system described by Eq. (11.6) is said to be autonomous if f (t, x(t)) f (x), i.e., independent of t and is said to be nonautonomous otherwise.
and determining the one with the lowest level of potential energy with respect to the postdisturbance equilibrium. This value of potential energy then characterized the domain of attraction. In the work that followed, it was found that this approach provided very conservative results for power systems. In Gupta and El-Abiad (1976), it was recognized that the appropriate UEP was dependent on the fault location, and the concept of closest UEP was developed. An approach to determine the domain of attraction was also presented by Kakimoto and colleagues (1978; 1981) based on the concept of the potential energy boundary surface (PEBS). For a given disturbance trajectory, the PEBS describes a ‘‘local’’ approximation of the stability boundary. The process of finding this local approximation is associated with the determination of the stability boundary of a lower dimensional system (see Fouad and Vittal [1992], chap. 4 for details). It is formed by joining points of maximum potential energy along any direction originating from the postdisturbance stable equilibrium point. The PEBS constructed in this manner is orthogonal to the equipotential curves. In addition, along the direction orthogonal to the PEBS, the potential energy achieves a local maximum at the PEBS. In Athay et al. (1979), several simulations on realistic systems were conducted. These simulations, together with the synthesis of previous results in the area led to the development of a procedure to determine the correct UEP to characterize the domain of attraction. The results obtained were much improved, but in terms of practical applicability there was room for improvement. The work presented in Fouad et al. (1981) and Carvalho et al. (1986) made several important contributions to determining the correct UEP. The term controlling UEP was established, and a systematic procedure to determine the controlling UEP was developed. This will be described later. In Chiang et al. (1985; 1987; 1988), a thorough analytical justification for the concept of the controlling UEP and the characterization of the domain of attraction was developed. This provides the analytical basis for the application of the TEF method to power systems. These analytical results in essence show that the stability boundary of the postdisturbance equilibrium point is made up of the union of the stable manifolds of those unstable equilibrium points contained on the stability boundary. The boundary is then approximated locally using the energy function evaluated at the controlling UEP. The conceptual framework of the TEF approach is illustrated in Fig. 11.2.
Controlling UEP
Faulted Trajectory
UEP 1
UEP 2
q s^1
q s^2
UEP 3
Union of Stable Manifolds
Approximation of Stability Boundary Based on Energy Function
Exit Point x e
FIGURE 11.2 Conceptual framework of TEF approach.
11.3 The Transient Energy Function
The TEF can be derived from Eq. (11.5) using first principles. Details of the derivation can be found in Pai (1981), Pai (1989), Fouad and Vittal (1992), Athay et al. (1979). For the power system model considered in Eq. (11.5), the TEF is given by
V ¼
Xn
i¼ 1
Mi vv~^2 i
Xn
i¼ 1
Pi ui us i^2
Xn^1
i¼ 1
Xn
j¼iþ 1
C (^) ij cos uij us ij^2
ui (^) ðþuj
us i 2 þus j^2
Dij cos uij d ui þ uj
where uij ¼ ui uj. The first term on the right-hand side of Eq. (11.10) is the kinetic energy. The next three terms represent the potential energy. The last term is path dependent. It is usually approximated (Uyemura et al., 1996; Athay et al., 1979) using a straight line approximation for the system trajectory. The integral between two points ua^ and ub^ is then given by
I (^) ij ¼ Dij
ubi uai þ ubj uaj ubij uaij
sin ubij sin uaij
In Fouad et al. (1981), a detailed analysis of the energy behavior along the time domain trajectory was conducted. It was observed that in all cases where the system was stable following the removal of a disturbance, a certain amount of the total kinetic energy in the system was not absorbed. This indicates that not all the kinetic energy created by the disturbance, contributes to the instability of the system. Some of the kinetic energy is responsible for the intermachine motion between the generators and does not contribute to the separation of the severely disturbed generators from the rest of the system. The kinetic energy associated with the gross motion of k machines having angular speeds vv~ 1 , vv~ 2 ,... , vv~k is the same as the kinetic energy of their inertial center. The speed of the inertial center of that group and its kinetic energy are given by
vv ~cr ¼
Xk
i¼ 1
Mi ~vvi
Xk
i¼ 1
Mi (11:12)
V (^) KEcr ¼
Xk
i¼ 1
Mi
ðvv ~crÞ^2 (11:13)
The disturbance splits the generators of the system into two groups: the critical machines and the rest of the generators. Their inertial centers have inertia constants and angular speeds Mcr , vv~cr and Msys , vv~sys , respectively. The kinetic energy causing the separation of the two groups is the same as that of an equivalent one-machine-infinite-bus system having inertia constant Meq and angular speed ~vveq given by
Meq ¼
Mcr Msys Meq þ Msys vv ~eq ¼ ~vvcr vv~sys
and the corresponding kinetic energy is given by
V (^) KEcorr ¼
Meq ~vveq
The kinetic energy term in Eq. (11.10) is replaced by Eq. (11.15).
starting point for the exact UEP solution are provided in Fouad and Vittal (1992), section 6.6. An outline of the procedure is provided below:
- Candidate modes to be tested by the MOD test depend on how the disturbance affects the system. The selection of the candidate modes is based on several disturbance severity measures obtained at the end of the disturbance. These severity measures include kinetic energy and acceleration. A ranked list of machines is obtained using the severity measures. From this ranked list, the machines or group of machines at the bottom of the list are included in the group forming the rest of the system and VKEcorr is calculated. In a sequential manner, machines are successively added to the group forming the rest of the system and VKEcorr is calculated and stored.
- The list of VKEcorr calculated above is sorted in descending order and only those groups within 10% of the maximum VKEcorr in the list are retained.
- Corresponding to the MOD for each of the retained groups of machines in step 2, an approxi- mation to the UEP corresponding to that mode is constructed using the postdisturbance stable equilibrium point. For a given candidate mode, where machines i and j are contained in the critical group, an estimate of the approximation to the UEP for an n-machine system is given by ^uuuij
h iT ¼ us 12 , us 22 ,... , p us i^2
,... , p us j^2
h i ,... , us n^2
h i
. This estimate can be further im- proved by accounting for the motion of the COI, and using the concept of the PEBS to maximize the potential energy along the ray drawn from the estimate and the postdisturbance stable equilibrium point us^2.
- The normalized potential energy margin for each of the candidate modes is evaluated at the approximation to the exact UEP, and the mode corresponding to the lowest normalized potential energy margin is then selected as the mode of the controlling UEP.
- Using the approximation to the controlling UEP as a starting point, the exact UEP is obtained by solving the nonlinear algebraic equation given by
fi ¼ Pi Pmi Mi MT
PCOI ¼ 0 i ¼ 1, 2,... , n (11:19)
The solution of these equations is a computationally intensive task for realistic power systems. Several investigators have made significant contributions to determining an effective solution. A detailed description of the numerical issues and algorithms to determine the exact UEP solution are beyond the scope of this handbook. Several excellent references that detail these approaches are available. These efforts are described in Fouad and Vittal (1992), section 6.8.
11.6 The BCU (Boundary Controlling UEP) Method
The BCU method (Chiang et al., 1985, 1987, 1988) provides a systematic procedure to determine a suitable starting point for the controlling UEP solution. The main steps in the procedure are as follows:
- Obtain the faulted trajectory by integrating the equations
Mi vvvv~~_i ¼ P (^) if P (^) eif
Mi MT P (^) COIf
uu^ _i ¼ vv~i , i ¼ 1, 2,... , n
Values of u obtained from Eq. (11.20) are substituted in the postfault mismatch equation given by Eq. (11.19). The exit point xe^ is then obtained by satisfying the condition
Pn i¼ 1
f (^) i vv~i ¼ 0.
- Using ue^ as the starting point, integrate the associated gradient system equations given by
uu^ _i ¼ Pi Pei Mi MT
PCOI , i ¼ 1, 2,... , n 1
un ¼
Xn^1
i¼ 1
Mi ui =Mn
At each step of the integration, evaluate
Pn i¼ 1
jf (^) i j ¼ F and determine the first minimum of F along the gradient surface. Let u* be the vector of rotor angles at this point.
- Using u* as a starting point in Eq. (11.19), obtain the exact solution for the controlling UEP.
11.7 Applications of the TEF Method and Modeling
Enhancements
The preceding subsections have provided the important steps in the application of the TEF method to analyze the transient stability of multimachine power systems. In this subsection, a brief mention of the applications of the technique and enhancements in terms of modeling detail and application to realistic power systems is provided. Inclusion of detailed generator models and excitation systems in the TEF method are presented in Athay et al. (1979), Fouad et al. (1986), and Waight et al. (1994). The sparse formulation of the system to obtain more efficient solution techniques is developed in Bergen and Hill (1981), Abu-Elnaga et al. (1988), and Waight et al. (1994). The application of the TEF method for a wide range of problems including dynamic security assessment are discussed in Fouad and Vittal (1992), chaps. 9–10; Chadalavada et al. (1997); and Mansour et al. (1995). The availability of a qualitative measure of the degree of stability or instability in terms of the energy margin makes the direct methods an attractive tool for a wide range of problems. The modeling enhancements that have taken place and the continued development in terms of computational efficiency and computer hardware, make direct methods a viable candidate for online transient stability assessment (Waight et al., 1994; Chadalavada et al., 1997; Mansour et al., 1995). This feature is particularly effective in the competitive market environment to calculate operating limits with changing conditions. There are several efforts underway dealing with the develop- ment of direct methods and a combination of time simulation techniques for online transient stability assessment. These approaches take advantage of the superior modeling capability available in the time simulation engines, and use the qualitative measure provided by the direct methods to derive preventive and corrective control actions and estimate limits. This line of investigation has great potential and could become a vital component of energy control centers in the near future.
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