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Assessing Placement of Controllers and Nonlinear Behavior Using Normal Form Analysis Shu Liu, Student Member, IEEE, A. R. Messina, Senior Member, IEEE, and Vijay Vittal, Fellow, IEEE
Abstract—Normal form (NF) theory is used to characterize and quantify nonlinear modal interaction near critical equilibria. This study focuses on the analysis of second-order modal interaction and the study of nonlinear aspects of system behavior which are of interest to the design and location of system controllers. A systematic approach to deriving second-order NF representations in the neighborhood of equilibrium points is presented. On the basis of this model, nonlinear interaction measures are then obtained to assess the extent and distribution of nonlinearity in the system. Analytical criteria are developed to predict the existence of nonlinear modal interactions that significantly affect system dynamic performance. To demonstrate the effect of nonlinear interaction, a case study of locating controllers to damp electromechanical oscillations is developed. Examples of application of the developed approaches on a two-area four-machine test system are presented to determine the strength of nonlinear interactions in the system response, and estimating its effects on system dynamic performance and control design. Index Terms—Nonlinear systems, normal forms, power system small-signal stability.
I. INTRODUCTION
T
HE STUDY of nonlinear behavior near critical equilibrium points is of great importance in the study of power system dynamic behavior. Classical approaches to system dynamic analysis have provided reliable methods for designing controllers that rely on linear analysis techniques. Local linearization of the system model at stationary points of vector fields, however, may fail to provide complete characterization of system performance or result in incomplete system information, especially under heavy stress or near the onset of unstable behavior. The analysis of small-signal behavior involves the study of system characteristics in the vicinity of essentially stable equilibrium points. As systems become more stressed, complex phenomena, involving interaction between the fundamental modes of the system may occur. A number of recent studies have shown that under certain ranges of operating conditions the system modes may interact nonlinearly leading to second (or higher) order resonances and hence the onset of more complex dynamical phenomena. In recent work, the problem of near resonance
Manuscript received July 27, 2004; revised January 5, 2005. Paper no. TPWRS-00402–2004. S. Liu and V. Vittal are with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA (e-mail:
[email protected],
[email protected]). A. R. Messina is with the Electrical Engineering Program, Cinvestav, Guadalajara, Jal. 45090, Mexico. (e-mail:
[email protected]). Digital Object Identifier 10.1109/TPWRS.2005.852052
between modes has been addressed by Dobson et al. [1]. Further, the occurrence of second-order near resonance has been evidenced in a number of studies [2]–[4]. Nonlinear modal interaction may play an important role in the dynamic behavior of a power system requiring accurate analytical techniques to be studied in depth [5]. Fundamental to this analysis is the detection of quasiresonance between the modes of oscillation in the system and the development of analytical techniques that allow a more precise characterization of system behavior near singularities. In this paper, we discuss the use of NF theory to characterize and quantify nonlinear modal interaction near critical equilibria. Normal form analysis is utilized to identify nonlinear aspects of the modal interaction phenomenon which are of interest in the analysis and design of system controllers and in the evaluation of stability margins. Analytical procedures for assessing placement of controllers using second-order information are reviewed and generalized. Examples of application of the developed approaches on a twoarea four-machine test system are provided to determine the strength of nonlinear interactions in power system behavior, and estimating its effects on system dynamic performance and control design. Attention is focused on the analysis of critical operating conditions where the system exhibits near first and second-order resonances. It is shown that the use of NF theory in the analysis of nonlinear behavior leads to significant improvement in the description of nonlinear system behavior near resonance conditions and may lead to improved assessment of dynamic performance and control design. II. THE METHOD OF NORMAL FORMS The method of NF provides a simplified representation of local, nonlinear behavior of vector fields in the neighborhood of a stationary or equilibrium point. Consider a general nonlinear system of ordinary differential equations (1) is the vector of system states and is a realwhere valued vector field; the system is supposed to be at rest such . The main idea in the NF method is to introduce that a sequence of nonlinear transformations, in the vicinity of the origin, to transform (1), into its simplest or NF. In practice, this is to be expanded usually done order by order, which requires in a power series.
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LIU et al.: ASSESSING PLACEMENT OF CONTROLLERS AND NONLINEAR BEHAVIOR USING NF ANALYSIS
The Taylor series expansion of (1) up to order 2 about an is equilibrium .. .
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is satisfied. The problem is thus reduced to computing the linear coefficients of the linear mapping in (9); the elements of the are given by [5] second-order transformation,
(2)
where represents the linear part of the vector field, and is a vector of power series in ; the matrices ( ) are the Hessian matrices of second derivatives. When the linear transpart is diagonalizable, the linear transformation forms the system in (2) into the decoupled form
(10) is the corresponding term of in (3). Equation where (10) shows that the mapping (9) is possible if no resonance conare met.1 It then follows that the original ditions system in (3) can be expressed in the NF space as (11)
(3) is the vector of Jordan-form coordinates, and are the matrices of right and left eigenvectors, respec; tively, and contains secondthe polynomial vector order effects in the state representation. From NF theory, (3) can be transformed to its simplest or normal form employing a nonlinear coordinate transformation of the form [6], [7] where
(4) is a complex-valued vector field of polynomial where terms of degree 2 in to be determined, and the vector denotes the new NF coordinate system. By differentiating (4) along the trajectory of the system states, we obtain (5) is the Jacobian of the nonlinear transformation where identity matrix. and is the Making use of (5) and (4) in (3) yields (6) where use has been made of the approximation , for sufficiently small . Straightforward computation shows that (6) can be expressed in the form (7) where
In the more general case, when in (3) leads to the resonant normal form
, use of (4)
(12) where represents resonance terms that can not be removed by any coordinate transformation and remain in the simplified NF. A critical aspect in determining the NF transformation coefficients and hence in linearizing the system is the resonance . condition Two cases are of special interest when the system in (3) cannot be transformed to its simplest expression (11) [8], [9]. a) The resonance case. There is at least one modal combina. In general there is no tion such that formal power series transformation that can linearize the vector field. b) The quasiresonance case. In this case there exists an increasing sequence of modal combinations such that . The vector field can either be linearized by a divergent series, or analytically transformed to a vector field that contains all the quasiresonant monomial terms [9]. This latter condition is of great theoretical and practical interest in the study of power system behavior under stressed operating conditions and will be studied in the following. A. Structural Properties of the NF Transformation Central to the proposed approach are the structural properties of the nonlinear transformation. In order to clarify the importance of this aspect, let the near-identity coordinate change in (4) be expressed as (13)
(8) Given (7), the goal of the nonlinear coordinate transformation such that in the new co(4) is to choose the coefficients of ordinate framework, the second-order terms can be annihilated. From (7) and (8), it is seen that nonlinear terms of degree 2 can be annihilated if the homological equation (9)
Using center manifold theory [6], [7], we can write as (14) 1A local field is said to be resonant if there exist a nontrivial relation of the for all k; l; j ; ; n [8]. In particular, this condition form is referred to as a second-order resonance. The more fundamental case of firstorder resonance arises from the exact coincidence of eigenvalues in frequency and damping [1].
+ 0 =0
= 1 ...
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where is a homogeneous polynomial map of degree and accounts for higher order terms. The significance of these expressions becomes evident when we examine the nature of this transformation around a given initial condition , . To this end, let the th component of (13) be written as (15) Fig. 1.
where
is the th matrix of nonlinear interaction coefficients
.. .
.. .
..
.
.. .
Geometrical illustration of the normal form transformation.
In terms of modal components, the approximate solutions in the Jordan and physical spaces are, respectively
(16)
Matrix is full, complex-valued, and symmetric. 2 Nonlinear coupling between modal and NF coordinates is characterized by , and two key parameters: strength of nonlinearity, coupling density. The former represents the ratio of magnitudes of nonlinear terms; the latter can be conceptually defined by the number of interactions between two coordinate variables with significant magnitudes. For a simple one-dimensional system, (15) defines a nonlinear curve in the – complex plane. For low stress conditions the system behaves linearly and the relationship between coor. At higher loading dinates can be approximated as conditions, the system behaves more nonlinearly, thus requiring a more accurate representation of nonlinear effects. This is illustrated in Fig. 1. Once the geometry of the coordinate transformation is accurately described, we next examine the importance of incorporating nonlinear effects near critical equilibria. B. Closed-Form Analytical Solutions Discarding third-order and higher terms, and assuming that no resonance conditions are found, the system in (11) can be expressed in the uncoupled form
(19)
(20) Equations (19) and (20) allow the computation of the contribution of particular modes of oscillation to the system states and permit direct comparison between linear and nonlinear analysis techniques. As noted in [2], nonlinear effects arise in the NF so, and in the lution both in the linear part in the terms . Large nonlinear part in the terms compared to indicate, in values of the product and principle, strong interaction between mode and modes can be used to assess various aspects of nonlinear system performance. A more meaningful measure of the nonlinear interaction that takes into account both, the structural properties of the nonlinear transformations and the initial conditions excited by the fault is obtained from the analysis of approximate time-domain solutions in (20). Defining the second-order contribution factors as and , and simplifying, the time evolution of the th state becomes (21)
(17) , , which possesses the solutions is the initial condition of the normal form variable where . These equations are the generators of a family of parametric . solutions which depend upon the initial excitation To determine in terms of , we solve the nonlinear set of equations with complex coefficients (18) for a given excitation . The approximate solutions to (17) are then converted back to the physical domain by using the nonlinear transformation (13), and the linear transformation .
H
2By construction matrix has diagonal symmetry so only the upper part is actually required to fully characterize structural properties.
coefficient represents the second-order contribuwhere the tion of the th state to the th single mode. Accordingly, represents the contribution of the th state to the nonlinear mode, . Further simplifications of these formulae follow from a consideration of the structural properties of the system and may be used to assess various aspects of system performance as discussed below. C. Numerical Estimates of Mode-State Participations From the derivation above, nonlinear fault-independent measures are computed to determine nonlinear measures of modestate participations. The key to dealing with nonlinear participations in the suggested approach relies upon finding initial conditions in the normal form space for an arbitrary excitation in real coordinates. This is a problem for which little analytical results have been presented.
LIU et al.: ASSESSING PLACEMENT OF CONTROLLERS AND NONLINEAR BEHAVIOR USING NF ANALYSIS
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A common simplifying assumption in the computation of nonlinear participation factors (PFs) has been obtained in the past by implicitly assuming that [4], [5] (22) It should be observed, however, that (22) is a very special case of the model (14), in that higher order terms are neglected. From (unit (14), we see that when the initial condition vector vector) is applied, the NF initial conditions can be approximated by
Fig. 2. Schematic of the study system.
of a series of complex algebraic equations of the form
Jr 1zr = 0Fr (z)
(23) and where Inserting (23) into (20) and simplifying yields
.
1zr is the th Newton update and Fr (zr ) = (z) 1 1 1 n (z)]T is the vector of residn uals with j ( ) = jo 0 j + n jkl ko lo , k l = 1 2 . . . and Jr = ( Fr (z))jz z is the Jacobian where
r
[ (z)
f2
f1
f
z
f
(24)
j
;
;
z
;n
y
+
( )=
and , . , represents the second-order parThe first term in (24), ticipation of the th single-eigenvalue mode in the th state, and represents the second-order participation the second term of the th state in the mode formed by the combination of the and . Observe that is the linear eigenvalues may be interpreted as a correction PF and term to the linear participation. By selectively including residual terms in (14) up to the desired order of approximation, second and higher order analytical estimates of PFs can be defined. The validity of this approxima, tion is limited by the size of the higher order terms, which may limit the region in the operating space where this approach can be applied. In the context of the analysis of stressed, nonlinear systems, a more precise measure of the nonlinear mode-state participations can be obtained by exciting a given state of interest and solving numerically for the normal form initial conditions, in (13). To achieve this, a Newton-based iterative technique has been implemented that enables the determination of both, the initial conditions arising from a specific fault scenario, as well as the initial conditions associated with exciting a specific set of states (modes). Given a set of states of interest, , the following approach is used to determine nonlinear mode-state relationships. 1. Determine the modes excited in modal coordinates using the relation yo U01 xo 2. Solve (13) using a Newton-based iterative algorithm for zo . This requires the solution
=
z
z
=
matrix with elements n n @ fj z zk h2j @ zj k=1 l=1 k6=j l6=j n n @ fj z h2j zk h2 @ zp k=1 l=1 k6=p l6=p
( ) =1+
where
=1 h2
=1
D
+
z
h2
z
l
l
+2
+2
h2
z
h2
p
j
z ;p
6=
j:
3. With this selection for zo , the approximate solution (20) can be written as x
where
i
( )=
0 ij
p2
n
t
=
j =1 u
0
t ij e
p2
ij zjo and
+
n
n
k=1 l=1
0
( + )t
ikl e
p2
0 ikl = zko zlo [
p2
(25)
]
n u h j =1 ij 2jkl .
The above steps have been implemented in the NF procedure and have the advantage of allowing systematic calculation of various measures of mode-state participations. Thus, for in, where is an appropriate scaling stance, by setting factor, fault-independent numerical approximations to PFs in (24) can be obtained. Conversely, fault-dependent contribution , factors in (21) can be calculated using where is the post disturbance equilibrium point and is the system condition at the end of the disturbance. This latter algorithm is more accurate than current approaches, especially for highly stressed conditions and may be used to define more general measures of nonlinear mode-state participations. III. CASE STUDY The developed procedures are applied to the 2-area, 4-machine test system from [10], modified to consider different operating conditions. Fig. 2 shows a single-line diagram of the system under investigation. For this study, all generators are represented by a fourth-order – axis model and equipped with an exciter. Loads are modeled as constant impedances. The generator data and system parameters are given in the Appendix.
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TABLE I SUMMARY OF OPERATING CONDITIONS
TABLE IV NONLINEAR INTERACTION INDICES FOR KEY MODES – SCENARIO I
TABLE II OSCILLATORY MODES OF THE SYSTEM – OPERATING SCENARIO I
TABLE V NONLINEAR INTERACTION INDICES FOR KEY MODES– SCENARIO II
TABLE III OSCILLATORY MODES OF THE SYSTEM – OPERATING SCENARIO II
weexamine the strength of nonlinear interactionsin power system behavior,and estimate its effects on system dynamic performance and the problems of control placement and design. C. Assessment of Nonlinear Modal Interaction To quantify the extent to which modes interact nonlinearly, we use the interaction index (II), , A. Simulated Conditions An ample range of operating conditions was simulated to fully characterize nonlinear behavior and modal interaction. To stress the system, the load in Area 2 was increased in discrete steps. The load in Area 1 was then modified to achieve a given tie line power transfer. From this analysis, two main operating scenarios to investigate the presence of nonlinear modal interaction were examined: a low-stress case with a 180-MW power transfer between the interconnected areas and a high stress condition obtained by increasing the level of power transference to about 410 MW. Table I summarizes the operating conditions for the above operating conditions. B. Linear Analysis Tables II and III show selected modes together with their associated frequency of oscillation, and the most dominant states. The two-area system exhibits three electromechanical modes of interest for this study; one oscillatory mode involving the exchange of oscillating energy between Areas 1 and 2 (mode 7 for Scenario I, and Mode 9 for Scenario II), and two local modes associated with the local dynamics between generators in each area (modes 1 and 3). Attention in the following analysis is focused on the analysis of nonlinear interaction involving modes 7 and 9. For certain range of operating conditions, nonlinear coupling of these modes gives rise to a complex dynamic behavior involving near first order resonance and second-order resonance conditions. In what follows,
defined in previous work [4]. Tables IV and V show the interaction indices involving the main modes of concern. The analysis of interaction indices indicates that modes 7 and 9 interact strongly with mode (11, 12) and to a lesser extent among themselves. Comparing the results in Table IV with those in Table V, it can be seen that for scenario II the magnitude of nonlinear modal interaction increases significantly indicating the importance of nonlinear effects. Of particular relevance, a more detailed analysis of these indices in Table V shows that the system exhibits multiple near second-order frequency resonance conditions in which the sum ) is close to the frequency of frequency of two modes ( , and a third mode, , namely . In an effort to fully understand the nature of modal interaction leading to strong nonlinear behavior, the trajectory of the system was analyzed for various operating conditions near the operating scenarios. Fig. 3 shows the behavior of mode 9 and mode 7 as a function of system stress. Also of interest, Fig. 4(a) gives the second-order resonance condition for the mode combinations 9,11,11 and 9,7,11 as a function of the intertie real power transfer. For reference and discussion, the operating space is subdivided into two main stability regions. Region 1 starts at 180 MW, whilst Region 2 depicts the nonlinear behavior near the second-order resonance in the vicinity of scenario II. Region 2 is the most relevant to system analysis since it captures both, first order and second-order resonance of the critical modes. Anal
LIU et al.: ASSESSING PLACEMENT OF CONTROLLERS AND NONLINEAR BEHAVIOR USING NF ANALYSIS
Fig. 3. Schematic illustrating near first-order resonance conditions. (a) Eigenvalue movement. (b) Magnitude of right eigenvectors close to first-order resonance.
ysis results in Fig. 3(a) show that as the system is stressed, mode 9 and mode 7 move closer together; a near strong first-order resonance condition is seen to occur at about 350 MW, when the damping and frequency of these modes are very close.3 At this critical point, the right eigenvectors are aligned as shown in Fig. 3(b), thus indicating that both modes have a similar dynamic pattern. The analysis suggests also that nonlinear effects become important when the system approaches the first order resonance. As the system is gradually stressed, the analysis of Region 1 in Fig. 4(a) clearly shows that the frequency resonance conditions , drop sharply until the first order near resonance is met. As the system is stressed further through the resonance, the frequency resonance conditions increase and then decrease sharply near the operating scenario II. The damping of mode 7 3If there are two eigenvectors corresponding to a double eigenvalue, the interaction is called weak. In contrast, if there exists only one eigenvector associated with a double eigenvalue the interaction is called strong [11].
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Fig. 4. Variation of resonance conditions with system stress. (a) Second-order (frequency) resonance condition, ! ! ! . (b) Second-order resonance condition, .
j + 0 j
j + 0 j
increases while that of mode 9 decreases. Accordingly, the , and second-order resonance conditions , begin to decrease slowly in the beginning to then drop rapidly for transmission levels in excess of 410 MW [refer to Fig. 4(b)]. This phenomenon leads to a complex behavior and has a profound implication on the analysis and interpretation of local results near the onset of first and second-order resonances. For operating conditions in Region I far from the first order resonance, linear estimates are consistent with NF results. By contrast, NF results lead to a significant improvement in the description of system dynamic behavior for operating scenarios close to the linear resonance and within Region 2. D. Siting of Controllers; Linear Approach Conventional analysis techniques were used to site power system stabilizers (PSSs) to enhance damping of the inter-area modes 7 and 9 in scenarios I and II, respectively. Figs. 5 and 6 show linear speed-PFs and the mode shapes for the inter-area modes obtained using the SSAT [12] software. Values are normalized with respect to the largest component.
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Fig. 5. Linear PFs and mode shapes for inter-area mode 7 – Operating scenario I. (a) Mode shapes. (b) Linear PFs.
Fig. 6. Linear PFs and mode shapes for inter-area mode 9 – Operating scenario II. (a) Mode shapes. (b) Linear PFs.
The analysis of linear measures for scenarios I and II identifies GEN3 and GEN4 in Area 2 as the best locations to place PSSs. GEN3 has the largest PF for scenario I whilst GEN4 has the largest participation for scenario II . Mode shapes results in Figs. 5(a) and 6(a) are in good agreement with the analysis of PFs.
( ) participating in the state whilst the vertical axis gives the associated magnitude. Further, numerical estimates for nonlinear PFs derived using the procedure outlined in Section II.C were then used to determine the best location for PSSs. Tables VI and VII show the speed-based nonlinear PFs computed using these approximations. For comparison, analytical estimates of nonlinear PFs from (24) are also included. For scenario I, the analysis of PFs in Fig. 7 identifies machines in Area 2 as the best places to install PSSs in order to enhance the damping of mode 7. GEN3 has the largest participation in this mode. For Area 1 the analysis of nonlinear PFs in Fig. 7(a) and (b) shows the presence of the local mode 1 and to a lesser extent the inter-area mode 7. In contrast, the analysis of nonlinear PFs for Area 2 shows that the participation of mode 7 relative to the local mode increases; the results of this analysis suggest that Area 2 is a better option to place PSSs. These results are in good agreement with linear PFs in Fig. 5(b). As the system stress is increased in scenario II, the analysis of nonlinear participations for machines in Area 1 in Fig. 8(a) shows the dominant presence of the inter-area mode 9 suggesting that Area 1 is the best place to install PSSs. By contrast, the analysis of machines in Area 2 indicates the presence of the
E. Placement of Controllers Using Nonlinear Measures The approach outlined above was used to determine nonlinear PFs. In order to provide a basis of comparison with the results obtained from linear analysis, the speed deviations of system generators were expressed in the form
(26) For these states, analytical [from (24)] and numerical estimates [from (25)] of the nonlinear mode-state participations were derived. Figs. 7 and 8 show the top 15 PFs for machines in the system as a function of the interacting modes. In these plots, the horizontal axis gives the mode ( ) or mode combination
LIU et al.: ASSESSING PLACEMENT OF CONTROLLERS AND NONLINEAR BEHAVIOR USING NF ANALYSIS
TABLE VI COMPARISON OF SEED NONLINEAR PFS p
TABLE VII COMPARISON OF SPEED NONLINEAR PFS p
Fig. 7.
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OF
MODE 7 – SCENARIO I
OF
MODE 9-SCENARIO II
Analytical estimates of nonlinear speed PFs – Operating scenario I.
Fig. 9. Block diagram of the PSS (IEEE type PSS1A).
For scenario I, comparison of nonlinear PFs in Table VI enables us to confirm that GEN3 is the best candidate to be equipped with PSS, whilst numerical estimates for scenario II in Table VII single out GEN2 as the best option. Based on these results, detailed nonlinear studies were conducted to identify the underlying mechanism leading to resonance and the assessment of its impact on the location of system controllers. F. Effect of PSSs on Small-Signal Behavior Fig. 8.
Analytical estimates of nonlinear speed PFs – Operating scenario II.
local mode 3 followed by the presence of mode 7 and to a lesser extent the inter-area mode 9. In contrast to linear participations in Fig. 6(b), nonlinear participations suggest that Area 1 is a better alternative to use PSSs. Further, the large participation of modes (3,4), (7, 8) and (11, 12) in the speed deviations of GEN3 and GEN4 for scenario II suggests the potential for undesirable nonlinear modal interaction arising from coupling between control modes and the inter-area mode [see Fig. 8(c) and (d)]. The complexity of this behavior can be appreciated by ) noting that (not shown), for scenario II, the flux deviation ( of GEN4 has the largest linear participation in mode 7. In turn, the analysis of mode 9 in Fig. 6(b) shows that angle deviation of GEN3 has the largest participation (1.00) followed by the flux state of GEN1 (0.984). Further the analysis of mode 11 reveals that the state associated with the second block of the AVR of GEN4 has the largest participation in mode 11. The strong participation of electromechanical and voltage states in mode 9 suggests that both voltage control support and PSSs might be used to enhance damping of the modes and that control action at GEN4 may interact unfavorably with the mode. Attention here is restricted to the use of PSSs.
To verify the effects of nonlinear behavior on the analysis and design of controllers, PSSs were designed using the approach in [13], [14]. In this analysis, a PSS was designed considering a single machine location at a time. Fig. 9 shows the block diagram of the PSS. The parameters for the designed PSSs are shown in Tables VIII and IX. Tables X and XI show the system eigenvalues for the above operating scenarios considered in the study. For scenario I, the results show that the use of PSSs improves significantly the damping of both the local and the inter-area modes for all alternatives considered. The use of a PSS at GEN3 results in the best improvement of damping as suggested by nonlinear and linear PFs. For scenario II, use of PSSs in machines in Area 1 results in more damped behavior. Thus, for instance, for the case with a PSS at GEN2 in scenario II (refer to Table XI), the damping ratio is increased to 10.87%. By contrast, adding PSSs at machines in Area 2 may actually decrease the damping of the inter-area mode 9 from 4.4% to 2.73% and 2.90% for the cases with a PSS added to GEN3 and GEN4, respectively. Of particular interest, analysis results enable us to confirm that use of PSS at GEN2 leads to better dynamic performance as suggested by numerical estimates of nonlinear participations. Increasing the PSS gains in Area 2, three times beyond the values given in Table IX results in an unstable system, and increasing the gains of the PSS in Area 1 results in higher damping. This follows the procedure
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TABLE VIII PARAMETERS FOR THE DESIGNED PSSS – SCENARIO I
TABLE IX PARAMETERS FOR THE DESIGNED PSSS – SCENARIO II
Analysis results show that linear resonances among the fundamental modes may trigger modal interactions that generate significant nonlinear behavior and lead to second-order quasiresonance conditions. For low-stress conditions, linear analysis leads to a good identification of critical machines. In contrast, local linearization of the system’s model for high-stress conditions may preclude an in depth analysis of nonlinearity, and result in an inadequate design of controllers. A more accurate representation of the system that takes into account the effect of the nonlinear modal interaction is therefore needed to fully characterize these phenomena. APPENDIX Data for Case Study in Section III
TABLE X EFFECT OF PSSS ON SYSTEM DAMPING – SCENARIO I
The system data for the various cases studied is given in Tables XII –Tables XVII. TABLE XII GENERATOR DATA (ALL FOUR GENERATORS)
TABLE XIII GENERATOR DAMPING (ON MACHINE BASE)
TABLE XI EFFECT OF PSSS ON SYSTEM DAMPING – SCENARIO II TABLE XIV LINE DATA (ON SYSTEM BASE 100 MVA)
TABLE XV LOAD DATA
outlined in [14]. The values of gains were reduced to make a fair comparison between gains in Areas 1 and 2. Comparing these results with the analysis of nonlinear PFs in Fig. 6, one can see that modal interaction is not properly captured by linear analysis. As a result, conventional techniques do not allow identifying the ideal location for system controllers.
TABLE XVI EXCITER DATA
IV. CONCLUSIONS Normal form analysis has been used to investigate the dynamic behavior of power systems near critical equilibrium points. Attentionhasbeenfocusedontheanalysisofcriticalequilibriawherethe system exhibits first and second-order resonance conditions. The methodofNFprovidesanaccurateframeworkforlocalanalysisof a diverse range of nonlinear behavior near equilibria such as linear resonances and second-order resonances. Such a model can reveal the underlying mechanism that generates nonlinear behavior and provide a basis for a more accurate analysis of the system under study. The extension of these procedures to deal with larger systems remains a challenging problem.
TABLE XVII STEADY-STATE GENERATOR DATA
LIU et al.: ASSESSING PLACEMENT OF CONTROLLERS AND NONLINEAR BEHAVIOR USING NF ANALYSIS
REFERENCES [1] I. Dobson, J. Zhang, S. Greene, H. Engdahl, and P. W. Sauer, “Is modal resonance a precursor to power system oscillations?,” in Proc. Bulk Power System Dynamics and Control-IV Restructuring, Santorini, Greece, Aug. 1998. [2] V. Vittal, N. Bhatia, and A. A. Fouad, “Analysis of the interarea mode phenomena in power systems following large disturbances,” IEEE Trans. Power Syst., vol. PS-6, no. 2, pp. 1515–1521, Apr. 1991. [3] C.-M. Lin, V. Vittal, W. Kliemann, and A. A. Fouad, “Investigation of modal interaction and its effects on control performance in stressed power systems using normal forms of vector fields,” IEEE Trans. Power Syst., vol. 11, no. 2, pp. 781–787, May 2003. [4] G. Jang, V. Vittal, and W. Kliemann, “Effect of nonlinear modal interaction on control performance: Use of normal form techniques in control design,” IEEE Trans. Power Syst., vol. 13, no. 2, pp. 401–407, May 1998. [5] S. K. Starret and A. A. Fouad, “Nonlinear measures of mode-machine participation,” IEEE Trans. Power Syst., vol. 13, no. 2, pp. 389–394, May 1994. [6] D. Arrowsmith and C. M. Place, An Introduction to Dynamical Systems. Cambridge, U.K.: Cambridge Univ. Press, 1990. [7] J. Carr, Centre Manifold Theory. New York: Springer-Verlag, 1981. [8] S. Louies and L. Brenig, “Structure and convergence of Poincaré-like normal forms,” Physics Lett. A, vol. 233, pp. 184–192, 1997. [9] A. Goriely, “Painlevé analysis and normal forms theory,” Physica D, vol. 152–153, pp. 124–144, 2001. [10] M. Klein, G. J. Rogers, and P. Kundur, “A fundamental study of interarea oscillations in power systems,” IEEE Trans. Power Syst., vol. 6, no. 3, pp. 914–921, Aug. 1991. [11] A. P. Seyranian and A. A. Mailybaev, “Interaction of eigenvalues in multi-parameter problem,” J. Sound and Vibration, vol. 267, pp. 1047–1064, 2003. [12] SSAT: Small Signal Analysis Tool. Powertech Labs Inc., Canada. [Online]. Available: www.powertechlabs.com [13] P. Kundur, M. Klein, G. J. Rogers, and M. S. Zywno, “Application of power system stabilizers for enhancement of overall system stability,” IEEE Trans. Power Syst., vol. 4, no. 2, pp. 614–626, May 1989.
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[14] E. V. Larsen and D. A. Swann, “Applying power system stabilizers parts I, II, and III,” IEEE Trans. Power App. Syst., vol. PAS-100, pp. 3017–3046, Jun. 1981.
Shu Liu (S’03) received the B.S. and M.S. degrees in electrical engineering from Tianjin University, Tianjin, China, in 1999 and 2002, respectively. She is currently pursuing the Ph.D. degree in electrical engineering at Iowa State University, Ames.
A. R. Messina (M’85–SM’05) received the M.Sc. degree (Hons.) in electrical engineering from the National Polytechnic Institute of Mexico in 1987 and the Ph.D. degree from Imperial College, London, U.K., in 1991. Since 1997, he has been an Associate Professor at the Center for Research and Advanced Studies, Guadalajara, Mexico. He is currently a Visiting Professor at Iowa State University, Ames.
Vijay Vittal (M’82–F’97) received the B.E. degree in electrical engineering from Bangalore University, Bangalore, India, in 1997, the M. Tech. degree from the Indian Institute of Technology, Kanpur, in 1979, and the Ph.D. degree from Iowa State University, Ames, in 1982. He is a Professor in the Electrical Engineering and Computer Engineering Department, Iowa State University, Ames. Dr. Vittal received the 1985 Presidential Young Investigator Award. He was elected to the U.S. National Academy of Engineering in 2004.
