948 IEEE Transactions on Power Systems, Vol. 10, No. 2, May 1995

CONCEPTS FOR. DESIGN OF FACTS CONTROLLERS TO DAMP POWER SWINGS

Einar V. Larsen (F), Juan J. Sanchez-Gasca (M) Power Systems Engineering Department

GE Industrial and Power Systems Schenectady, NY 12345

Abstract - The design of controllers sited in the transmission network f o r damping interarea power oscillations requires several types of analytical tools and field verification methodr. Probably the most important aspect of such control design is the selection of proper feedback measurements from the network. This paper describes concepts which provide design engineers with the insight to control performance and the understanding needed to ensure the secure operation of the bulk transmission system. Specific attention is directed to procedures for selecting feedback signals. Keywords - Power system stability, Flexible AC Transmission

Systems (FACTS), Control design, Modal decomposition

1. INTRODUCTION

A problem of interest in the power industry is the mitigation of low frequency oscillations which often arise between areas in a large interconnected power network [l-31. These oscillations are due to the dynamics of interarea power transfer and often exhibit poor damping when the aggregate power transfer over a corridor is high relative to the transmission strength. With utilities increasing power exchanges over a fixed network, the use of new equipment in the transmission system to aid the damping of these oscillations is being seriously considered. A number of Static VAr Compensators (SVC) have already been installed to aid power system dynamics [4-61. The Flexible AC Transmission Systems (FACTS) program of EPRI is developing a number of new controllers for this purpose, with Thyristor Controlled Series Compensation (TCSC) now proven in the field and ready for application [7-91.

Such controllers must operate satisfactorily in the presence of many modes of power swings and over a wide range of operation. Given the huge extent and variability of the power system, the control design must be

I I 94 SM 532-2 PWRS by the IEEE Power System mgineering Committee of the IEEE Power Engineering Society for presentation a t the IEEE/PES 1994 Summer Meeting, San Francisco, CA, July 24 - 28, 1994. Manuscript submitted December 30, 1993; made avai lable for print ing June 14, 1994.

A paper recommended and approved

I 1

Joe H. Chow (F) Electrical, Computer & Systems Eng. Dept.

Rensselaer Polytechnic Institute Troy, NY 12180-3590

understandable to engineers in many companies involved witb the design, planning and operation of the power system. Thus, insight to control performance is important to ensure that the needed understanding is derived from field measurements and from simulation results using a large- scale simulation model.

Traditional approaches to aid the damping of power swings include the use of Power System Stabilizers (PSS) to modulate the generator excitation control, for which much experience and insight exist in the industry [10,11]. Unlike PSS control at a generator location, the speed deviations of the machines of interest are not readily available to a FACTS controller sited in the transmission path. Further, since the intent is to damp complex swings involving large numbers of generators, speed signals themselves are not necessarily the best choice for an input signal. For a FACTS controller, it is desired to extract an input signal from the locally-measurable quantities at the controller location. Finding an appropriate combination of measurements is the most important aspect of control design.

The main contribution of this paper is the presentation of design concepts and a systematic approach for the selection of input signals for FACTS damping controller design. The approach is based on an approximate multi- modal decomposition for systems with multiple swing modes and a general input signal, essentially a generalization of the well-known approach applied to a single-machine model in the early days of PSS design [10,11]. The approach takes advantage of a few approximations to provide a computationally-efficient means of displaying key information needed to understand the impact of input-signal selection and control design.

2. APPROXIMATE MULTI-MODAL DECOMPOSITION

The approach described here makes a few approximations to develop engineering insight in the control design process. One key approximation is that the modes of interest exhibit light damping. Another is that the impact of any individual control on the frequency and mode shape of the power swing is small. These assumptions permit breaking the system apart based upon the approximate mode shape information, and determining the incremental effect of controllers on each mode separately.

0885-8950/95/$04.00 0 1994 IEEE

949

ith Modal System - - w I . I I' Swing

Interest . Mode of

Power , System

(Except ith Mode)

i

Effective Control Action

Figure 2.1. Multi-Modal Decomposition Block Diagram.

The effect of the controllers upon themselves also becomes apparent, and the design can proceed using this information to assure minimum self-interaction effects on the final response.

2.1 Formulation

In the single-machine model [lo], the mechanical swing mode is represented in terms of a synchronizing and a damping torque with control loops built around it. In the multi-modal decomposition, we pursue the same representation for each of the swing modes.

Consider the linearized model of a multi-machine power system in the state space form

(2.1) X=Ax+Bu, y=Cx+Du

where x is the vector of state variables, and U and y are the vectors of control and measurement variables, respectively. We assume that in (2.1), the states are arranged in x as

x=[AS1 AS2 ... AS,Ao,~Ao,2 ... Am,, zT]' (2.2)

where the AS's and the Aog's represent the generator angles and speeds, respectively, and z is the vector of all the other state variables. Thus the system matrices A and B are in the form

(2.3)

where I is the identity matrix, o b is the system frequency in radsec, and A21. The matrix A21 relates the generator angles to the derivative of the speeds (acceleration), and represents the network synchronizing effect normalized with respect to the machine inertias. The machine interactions and swing modes are largely determined by A21 and A22, representing the synchronizing and damping effects independent of any other state variables.

The multi-modal decomposition is a transformation based on the modal decomposition of A21 using the eigenvectors of A21. Thus it involves a relatively smaller amount of computations since A21 is n x n where n is the number of generators in the system. The full system matrix A can be more than 10 times larger than A21. Let V be the matrix of the right eigenvectors of A21 such that V-1 A21 V = A, where A is a diagonal matrix whose non-zero entries are the normalized modal synchronizing Coefficients. Then the transformation

v o o x,=u-lx, U = 0 v 0 I. 0 j (2.4)

applied to the system (2.1) results in

(2.5) in =U-' AU x, + U-' BU = A, X, + B,U y=CUx, +Du=C, X, +Du

The angles and speeds in the new system (2.5) represent modal variables and the system. matrix A, has the structure

The concepts developed in this paper can be applied to any subset of the swing-mode set. To illustrate the points, subsequent discussion will focus on a single mode evaluation. This approach can be used to evaluate one mode at a time in any system, and may be adequate when only one swing mode is dominant across a transmission path.

For each swing mode hi with modal frequency Wi, the state variables can be rearranged such that the modal angle A6mi and speed AOmi corresponding to hi become the fnst and second state variables, resulting in the system representation Fj] = [ !lcmi -d,,,i Ob - A d 2 3 ] E j ] + [;;]U

Ad31 Ad32 Ad33

Y Cd2 Cd3] +Du (2.7) [::I where kmi and dmi are the approximate modal synchronizing and damping coefficients, respectively, and zmi consists of all the other state variables. The modal frequency is approximately given by ai = ,/= o k . fad/SeC.

950

System (2.7) now resembles a single-mode system considered in [lo]. As a result, we can construct a block diagram to represent (2.7) using its transfer functions. Using straightforward matrix manipulations, (2.7) is expressed in the frequency domain

S A O ~ ~ ( S ) = -(Ob/S)Kmi(S)A@mi(S) - Kd(S)U(s)

Y(s) = Koi ( s h ~ m i (s) + KIL~ (SI U(S) (2.8)

The forms of the various transfer functions are given in Appendix A. A block diagram of (2.8) is shown in Figure 2.1. We denote Kmi(s). Kci(s), Koi(s) and KIL~(s) as the modal, controllability, observability and inner loop transfer functions, respectively. These transfer functions when evaluated at s = jo, are complex, providing both gain and phase information. This information can be used to select the most effective transfer functions for feedback control. Interpretations of Kmi(s). Kci(s), Koi(s) and KILi(S) are given in subsequent sections.

The system representation (2.8) can also be obtained from the complete eigenspace of A, rather than the transformation (2.4). However, this approach is limited to relatively small size systems [12] and one of the key objectives here is to provide engineering insight in an expeditious manner rather than exact results. At the opposite extreme, only one mode shape need be known in an approximate manner to use the concepts developed here.

2.2 Modal Characteristics and Control Influence

In Figure 2.1, the modal system involving the states Asmi and Aomi represents the lightly-damped swing mode. The modal transfer function Kmi(s) includes the effect of the electrical network, the generator fluxes, the controllers such as the excitation systems and governors, and to a much smaller extent, the other swing modes. This modal transfer function will be predominantly real in the frequency range of interest when analyzing the dominant swing mode, and for s =joi, obKmi(joi) = mi2.

The efective control action (see Figure 2.1)

(2.9)

describes the impact of a given damping controller; KPSDC(S), on the modal system, and is useful in estimating the eigenvalue sensitivity of the ith mode. From Appendix B, the perturbation Ah; = -AOi + jAwi of hi for a controller KPSDC(S) with small gain is

(2.10) AXi = - Kei ( joi)/2

that is,

Aoi = Real(Kei(joi)/2), A o i = -Imag(Kei(joi)/2) (2.11)

This direct relationship of modal sensitivity to the controller and power system characteristics provides the key to achieving the desired insight into control design. Subsequent discussion will focus on each of the terms in (2.9), to show how they relate to control performance and how they can be used to select effective measurements. Future analytical tool developments are expected to focus on numerically-efficient procedures to quantify these terms.

2.3 Interpretations

Controllability

The effect of a FACTS controller on a given swing mode is defined as the controllability function Kci(S), and Kei(s) is directly proportional to K~i(s). In PSS design, Kci(S) depends mostly on the excitation system, generator flux dynamics and network impedances. For network control devices such as TCSC, Kci(s) depends mostly on the network structure and loads. When evaluated at s = joi, Kci(joi) provides a measure of how controllable the ith mode is by the control signal U. If &i(jOi) is zero, then mode i is not affected by U. When more than one damping controller is used, Kci(s) is a vector transfer function.

In general, IGi(jOi) is different for different swing modes. For example, a TCSC sited on a tie line would have significant controllability on the associated inter-area mode, but much smaller controllability over local modes. In cases with multiple interarea modes, the controllability may be nearly 180" out of phase from one mode to the next. Such a situation would mean that if the machine speeds were averaged and transmitted to the controller, then the action of improving the damping on one mode would simultaneously decrease the damping of another mode. This situation must be compensated for by using an appropriate set of measurements so that this inherent adverse impact will be minimized or eliminated.

The quantity Kci(jOi) is a good indicator for evaluating effective locations to apply damping controllers, since the larger this is the greater the leverage on the swing mode. For example, an SVC located on a bus needing voltage support will be more effective for damping control than one close to a generator terminal bus.

Observability

The effective control action Kei(s) is also directly proportional to the observability function Koi(s). The function Koi(s) relates the measured signal y to the i th modal speed Aomi. For a PSS using the machine speed as the input signal, Koi(s) = 1 for the local mode of that machine. When evaluated at s = j o i , Koi(joi) gives an indication of the modal content of the ith swing mode in the measured signal y. Its magnitude can be used to assess the effectiveness of measurements y for damping control applications. In a multi-modal system, since Koi(S) is defined with respect to the modal speed, measurements y directly related to machine speeds will have observability gains that are predominantly real. Signals more closely related to angular separation, such as power flow, will have an integral characteristic, i.e., nearly 90" of lag with respect to the speed. If Koi(joi) is small, then the ith mode is weakly observable from the measurement y. Thus having large Koi(joi) for the dominant modes of interest is one of the criteria in selecting an input signal for a damping controller.

Inner Loop

The control design must also consider the effect of the controller output on its input (i.e., the component of the measured signal y that is due to the control U), other than

95 1

via the swing mode of interest. This effect may be considered a “feed-forward term, but here we have called it the “inner-loop” effect, symbolized by the transfer function KILi(S). This choice is due to its impact on the effective control action (2.9), as illustrated in Figure 2.1.

The inner loop transfer function KILi(s) is extremely important in damping controller design using input signals other than generator speeds. In Section 2.4, we develop simple indices based on the constraint imposed by KIL~(s) to aid the selection of an appropriate measurement signal.

2.4 Damping Controller Design Indices

In this section we define three indices which provide insight to the performance of a damping controller with given measurements. These indices are amendable to efficient computation in a large system.

Controller Phase Index When the controller gain is sufficiently small with

respect to the inner loop (i.e., IKILi(jOi) KPSDCuWi)I<<I), (2.9) reduces to

a i -Kci(joi)KpsDC(joi)Koi(joi )/2 (2.12)

We define the Controller Phase Index (CPI) for the ith

CPI(i)= -(LK, (jo;) + LK,; (joi)) (2.13)

Then the phase of KPSDC(S) at s = joi should be set equal to CPI(i) to achieve a pure damping influence.

The change in the swing mode damping is approximately

mode as

where A$ = LKPSDC(jWi) - CPI(i) is the difference between the actual phase and the pure damping phase for the damping controller.

Similarly, the change in the frequency of the swing mode is

Ami = I K p s x (jwi]lKci (jwi]IKd(jwi)( sin(-A$)/2 (2.15)

For systems with several dominant swing modes, (2.14) can be used to select control channels that are robust with respect to the damping of these swing modes. Since power system swing modes are usually in a small range of frequencies, from 0.3 to 2 Hz, an ideal measurement signal would have the property that the CPI remains roughly constant and in the same quadrant over the frequency range for all of the dominant swing modes.

Once a measurement signal is selected to satisfy this property, selecting the time constants of lead-lag transfer functions for KPS DC(S) is a relatively straightforward exercise. From experience [l 11, matching the phase perfectly is not as crucial as ensuring a good bandwidth, so that higher-frequency interactions will not excessively limit the gain. Some phase lag at the swing mode frequency is acceptable, since it tends to increase the synchronizing torques between the machines, consequently raising the frequency.

Maximum Damping Influence Index

Referring to Figure 2.1, the controller KPSDC(S) forms a feedback loop with KILi(S). For a robust design, we constrain the gain on KPSDC(S) so as to maintain a gain margin in the feedback system, beyond which the inner loop will tend to dominate over the damping control. Using a gain margin of 10 dB as a guide, the quality of a measured signal can be quantified by a Maximum Dumping Influence (MDI) index for each mode hi

Tbe MDI index is a measure of the maximum eigenvalue shift achievable assuming that the largest magnitude of the damping controller is l/&lK,,,(jm,], where the & factor guarantees a gain margin of 1 0 dB. Thus, given two candidate measured signals yl and y2 with similar IKoi(jOi)l for the dominant ith mode, y1 would be preferable if its inner loop gain IKILi(jO)l is less than that of y2, because a higher gain KPSDC(S) can be applied to provide more damping enhancement to the ith mode.

The value of the MDI index is that it indicates the effectiveness of measurements having high observability gain and low inner loop gain. In Section 4, the MDI index is used in a systematic way of scanning impedances for synthesizing the bus voltage angular differences that are effective for damping control design.

For systems with multiple controllers and measured signals, KILi(S) is a matrix. In such cases, MDI may be extended to provide information on potential adverse controller interactions. This is a subject of further research.

Natural Phase Influence Index

In a practical design involving several operating conditions, it is desirable to select a signal whose observability will compensate for the change of the controllability as system condition changes, such that a fixed controller can be used to improve the damping of the dominant swing modes. Such a property requires that the CPI of the dominant modes for the different operating conditions remain approximately the same. Thus, a robust signal should have large MDI, and result in approximately the same phase requirement in compensator design over the operating conditions. However, it may not always be possible to find signals that meet these requirements in a given system, especially when there are several dominant modes. To address this situation, we develop an index utilizing the impact of KILi(s) on the system zeros.

Much discussion exists in recent literature on the “zeros” of the total closed-loop transfer function, from the control U to the measurement y [2,13]. Frequency response measurement of this transfer function in the field will show both peaks and valleys, with the peaks corresponding to the lightly damped poles, which are the swing modes hi, and the valleys corresponding to the complex zeros zi of this particular transfer function.

The complex zeros zi are the locations to which the swing modes hi in the neighborhood of Zi will migrate in the s-plane as the controller gain is increased to infinity, which we call the “natural” behavior of closing a loop with

952

the selected measurement signal. The initial trajectory of this migration is influenced by the phase compensation (2.12) of the controller. However, when the inner-loop impact becomes too large the eigenvalues of the system will tend toward these zeros. If Zi has less damping than the corresponding hi, then the natural behavior of the controller is to decrease the stability of hi - clearly an undesirable situation. However, if Zi has more damping than the corresponding hi, the natural behavior of the controller will be to improve stability - clearly a good situation.

To appreciate the relationship between KILi(s) and system zeros, we let IKpsx(s)I+m in (2.9), resulting in

Kei = -Kci Koi (~)/KIL~ (2.17)

Equation (2.17), which is inversely proportional to KILi(s), represents the effective control action (2.9) with infinite control gain. As IKPSDC(S)I + m, hi + Zi. Using the eigenvalue sensitivity (2.12), Ahni , the "natural" change in the ith swing mode hi due to the controller, is approximately

Ahni = zi -hi = -Kci(jwi)Koi(joi)/(2K1~(jcoi)) (2.18)

Note that this "natural" change will have magnitude and phase characteristics. The magnitude aspect is already captured by the MDI index of (2.16). The phase-related index is quantified by a Natural Phase Influence (NPI) index for each mode hi

WI(i)= -(CPI(i)+ LKIL (jcoi)) (2.19)

A value of NPI(i) = -90" means a pure positive synchronizing influence, while NPI(i) = 0" implies a pure positive damping influence. Thus, the selection of the measured signals should favor those which have NPI(i) near 0". We believe the best region is for NPI(i) to lie in the quadrant O">NPI(i)>-90", for which the natural tendency is toward improving both damping and synchronizing torques.

3. TESTSYSTEM

We will illustrate the design concepts and the signal selection process using the 2-area, 4-machine system proposed in [141 (Figure 3.1). Each area consists of two machines, one small and one large to represent a grouping.

W - E Power Transfer

Local Local ____)

TCSC Mode2

-i XI, 900 MVA 900 MVA p.165

p.165

1550 MVA 1350 MVA

Figure 3. I . Two-Area Four-Muchine System

The system data are the same as those given in [14], except that the machine MVA ratings and the impedances of the tie lines between the coherent areas have been changed (as listed in Figure 3.1) and a TCSC is included in one of the tie lines. The system exhibits an interarea mode near 0.6 Hz and two local modes between the machines within in each area, with damping as noted in Table 3.1. The base system has a total of 600 MW power transfer on three tie lines, while the contingency system has the same power transfer on two tie lines, with tie line 3 not in-service.

TABLE 3.1 SWING MODE FREQUENCY AND DAMPING

Interuea Mode Local Mode 1 Local Mode 2

Base 0.62 Hz, 0.17% 1.15 Hz, 11% 1.02 Hz, 18% Contingency 0.52 Hz, -0.13% 1.13 Hz, loa0 1.00 Hz, 18%

4. MEASUREMENT SELECTION

4.1 Measurement Synthesis

The TCSC control for damping of the interarea mode uses locally-available measurements to synthesize the controller input signals. The synthesized signal should be sensitive to the interarea mode of oscillation, while insensitive to the local modes. In addition, the use of the signals should not cause any adverse interactions between the controllers.

In previous TCSC and TCPR (thyristor controlled phase regulator) controller desigd investigations [7,8,9,15], the angular difference of synthesized remote voltages on each side of the control has shown promise as a good measurement for damping control. This concept is illustrated in Figure 4.1. By measuring the through current IT and the voltages Vmeasi and Vmeas2 at the terminals of a TCSC, the remote complex voltages V s y n l and Vsyn2 can be synthesized as a function of the synthesizing impedances Zsynl and Z3yn2:

The synthesized angular difference is

012 = 4 y n l - 4 y n 2 (4.2)

For a relatively wide range of Z s y n t and Z s y n ~ , the synthesized 012 will consist of mostly the interarea mode with an insignificant amount of the local modes, that is, 012 will mimic the angular difference between the center of machines of the two areas. However, the impact of the TCSC action on this measurement (the inner-loop effect) will be more sensitive to the selection of Zsynl and Z s y n ~ . To a first approximation, setting these impedances close to

Figure 4. I . Synthesized Voltages and Angular Difference.

953

the driving-point impedance seen in both directions is a good starting point. In Section 4.2, we apply the multi- modal decomposition for a systematic selection of Z s y n l and Z s y n ~ , using the MDI and NPI indices described in Section 2.4.

4.2 Synthesizing Impedance Selection

To search for the appropriate Zsynl and Zsyn2, we scan the values of the MDI and NPI indices for a range of Zsynl and Z s y d values. The simulation shown here uses

Zspi = + jXsyni, Zsyn2 = Rsyn2 + j X s y n ~ (4.3)

where the resistances R y n l and Rsyn2 are set at 10% of the reactances Xspi and Xsyn2, respectively.

Figure 4.2 is a three-dimensional plot showing the MDI indices in the '2'-dimension over a (Xsynl , Xsyn2) grid. The plot shows the desirable region where the MDI indices are large, essentially indicating an inverse relationship of the two synthesizing reactances is desired for best damping performance (e.g., if one is large, the other should be small, and visa-versa).

Figure 4.3 is a three-dimensional plot showing the NPI indices in the '2'-dimension over the (Xsynl , Xsyn2) grid.

Figure 4.2. MDI Indices -vs- Synthesizing Reactances for the Interarea Mode of the Base System

0 5

-. 0.5 0

A -'\

I

\ Contingency System I

\r-l MD"l

L

n 0 1.0 0.2 0.3

Xsy"l* P" 0.4

Figure4.4. "Good" MDI Regions for Base and

Note that this has a ridge of similar shape as the MDI index, with the phase being near 270" (= -90") on the side of smaller synthesizing reactances and +90" for larger reactances. Given our desire to keep closer to the former angle, any deviation from the high-MDI synthesizing reactances should be on the low side for as many cases as possible.

Figure 4.4 illustrates the regions of MDb1 for the base and contingency systems. Note that the region of MDbl for the contingency case is wider and at higher synthesizing reactances than for the.base case. The trend toward higher reactances is due to the loss of a portion of the transmission system, thereby increasing impedance as seen from any bus. The wider region of high MDI is due to the contingency removing one of the parallel paths to the line having the TCSC; thus, the TCSC has more leverage on the intertie power swings.

Contingency Cases.

5 . TCSC DAMPING CONTROL PERFORMANCE

To illustrate the previous points, a damping control will be applied with three choices of synthesizing reactances. These choices are labeled as Points A, B and C in Figure 4.4, selected to show optimum and adjacent points for the base case. Point A has an MDI of 0.3, with an NPI on the "good" side (e.g., -90" for improved synchronizing torque). Point C also has an MDI of 0.3, but with an NPI on the "bad" side (e.g., +90" which decreases synchronizing torque). Point B is on the peak of the MDI characteristic.

Based on the CPI = -90" for the interarea mode, the damping controller

1 S S KPSDC(S)=-K~--- 1+0.04s l + s 1+0.04s

Figure 4.3. NPI Indices -vs- Synthesizing Reactances for the Interarea Mode of the Base System.

954

5.5 t 1 1 5 t Zero (A) I x x

2.5 1 i Zero (C)

2 1 I I I I I I I

-1 -0.8 -.06 -.04 6.2 0 0.2 0.4 Real Axis

Figure 5.1. Root Loci of the Interarea Mode for Damping Controllers Using Measurements A, S, or C.

is used, where [s/(l + O.O4s)][l/(l + 0.04s)I is the filtered derivative circuit to convert the angle measurement into a speed measurement, and s/(l + s) is a washout circuit. A root locus analysis on & is shown in Figure 5.1, for each of the three choices of synthesizing impedance.

For synthesis choice B, the controller provides considerable damping with no curl in the locus evident in this scale. For the other two choices, the root loci curve and approach their respective zeros. As expected from the NPI indices, choice A exhibits an increase in frequency, while choice C exhibits a decrease in frequency. Also, the improvement in damping of these non-optimum choices is limited to approximately 0.3 before the effect of the zero becomes significant, as predicted by the MDI index.

6. CONCLUSIONS

The concepts described in this paper can be developed into a set of analytical tools which will provide key insights to aid the task of designing FACTS controllers to damp interarea power oscillations. Several important aspects of controller performance are explained, and quantifiable indices are defined which will permit the evaluation of a large number of operating conditions and contingencies in an expeditious manner. Such a tool is expected to augment existing dynamic simulation methods, so that design engineers can develop a good understanding of the results seen in full-system simulations and can specify field tests to verify the essential aspects of control performance.

7. ACKNOWLEDGMENTS

The support for this work has been provided by EPRI, recently under RP3022-2 plus early development under RP2707-1, and by NSF under grant ECS-9215076. J.H.Chow has also been supported by an RPI sabbatical leave at GE. The support and encouragement of Stig Nilsson, Neal Balu, and Dominic Maratukulam at EPRI, and John Hauer at BPA is greatly appreciated. John Paserba and Ann Hill at GE and Glauco Taranto at RPI have contributed with analysis of example caqes.

8. REFERENCES

J.F. Hauer, “Robust Damping Controls for Large Power Systems,” IEEE Controls Systems Mag., pp. 12- .18. Jan. 1989. J.F. Hauer, “Reactive Power Control as a Means of Enhanced Interarea Damping in the Western Power System - A Frequency Domain Perspective Considering Robustness Needs,” in Applications of Static Var Systems for System Dynamic Pevormance, IEEE Publication 87TH01875-5-PWR, 1987, J.H Chow, Ed., Time-Scale Modeling of Dynamic Networks with Applications to Power System, Lecture Notes in Control and Information Sciences, Vol. 46, Springer-Verlag, 1982. RJ. Piwko, Ed., Applications of Static Var Systems for System Dynamic Performance, IEEE Publication

E.V. Larsen, N. Rostamkolai, D.A. Fisher and A.E. Poitras, “Design of a Supplementary Modulation Control Function for the Chester SVC,” Trans. Power

General Electric Co., “Improved Static Var Compensator Control,” EPRI Report TR-100696, Final Report for Project RP2707-1, June 1992. General Electric Co., “Flexible AC Transmission Systems (FACTS): Scoping Study, Vol. 2, Part 1: Analytical Studies,” EPRI Report EL-6943, Final Report for Project RP3022-2, September 1991. W. A. Mittelstadt, B. Furumatsu, P. Ferron J. Paserba, “Planning and Testing for Thyristor Controlled Series Capacitors,” In Current Activity in Flexible AC Transmission Systems, IEEE Publication 92 TH 0465-5 PWR. April1992. S . Nyati, C.A. Wegner, R.W. Delmerico, R.J. Piwko, D.H. Baker, A. Edris, Effectiveness of Thyrisfor Controlled Series Capacitor in Enhancing Power System Dynamics: An Analog Simulator Study, paper 93 SM 432-5 PWRD, IEEWPES Summer Meeting, Vancouver, B.C., July 18-22, 1993.

87TH01875-5-PWR, 1987.

Deliv., Vol. 8, pp. 719-724, April 1993.

[lo] F.P. DeMello and C. Concordia, “Concepts of Synchronous Machine Stability as Affected by Excitation Control,” IEEE Transactions on Power Apparatus and Systems, Vol. PAS-88, pp. 316-329, 1%9.

[ll] E.V. &en and D.A. Swann, “Applying Power System Stabilizers, Parts I - 111,” IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, pp. 3017-3046, 1981.

[12] P. Kundur, G.J. Rogers, D.Y. Wong, L. Wang and M.G. Lauby, “A Comprehensive Computer Program package for Small Signal Stability Analysis of Power Systems,” IEEE Trans. on Power Systems, Vol. 5, pp.

[13] N. Martins and L.T. G. Lima, “Eigenvalue and Frequency Domain Analysis of Small-Signal Electromechanical Stability Problems,” in Applications of Static Var Systems for System Dynamic Performance, IEEE Publication 87TH01875-5-PWR, 1987.

1076-1083, 1990.

955

M. Klein, GJ. Rogers and P. Kundur, “A Fundamental Study of Inter-Area Oscillations in Power Systems,” IEEE Trans. on Power Systems, Vol. 6, pp. 914-920, 1991. A.T. Hill, et al. Flexible AC Transmission Systems (FACTS): System Studies - Thyristor Controller Retrofit to an Existing Western Area Power Administration Phase Shifter, Final Report, EPRI RP3022-02, October 1993. J.H. Chow and J.J. Sanchez-Gasca, “Pole-Placement Design of Power System Stabilizers.” IEEE Trans. on Power Systems, Vol. 4, pp. 271-277, 1989.

Appendix A

TRANSFER FUNCTIONS IN FIGURE 2.1

The modal controllability &i(S) is the transfer function from U to A h h and is given by

Kci(s)= Ad23(S1-Ad33)-1 Bd3 + Bd2 (A. 1 ) The modal observability of a measurement Koi(s) is the

transfer function from the modal speed to the measurement y, and is computed as

The inner loop gain is given by the transfer function from the control U to the measurement y, with no change in Wmi or &. It is computed as

(A.3)

There is also a transfer function from the modal angle to the modal torque and it is given by

KIL(s )=C~~(SI - Ad33)-l Bd3 + D

[ Ad31 + S Ad32) (A.4)

Appendix B

DERIVATION OF (2.10)

We use an analysis similar to that contained in [16] to show (2.10). The characteristic polynomial of the closed- loop system with the controller KPSDC(S) is

S2 + Kei (S) S + a b Kmi (S) = O (B.1)

We will neglect the real part of h i such that h i = j o i and assume o b K m i ( i o i ) J Oi2. Let KeiQUi) = Kre + jKim, and the roots of (B.l) be s = - AOi + j ( 0 i + AOi). Substituting these quantities into (B.l), we obtain

Multiplying out the terms in (B.2) and neglecting the higher order perturbation terms, (B.2) reduces to two equations, one for the real part and the other for the imaginary part

2A0i0i + Kirnwi J 0, -2A0i0 + Krewi J 0 (B.3)

BIOGRAPHY

Einu V. Lalaea (F‘91) BSEE ‘73, Cal Poly, San Luis Obispo, c.liforni., MSEPE ‘74 Rensselaer Polytechnic Institute, Troy, New York.

Mr. Lusen has been with GE since graduation in 1974. He has spent mart of his career in GE’s Power Systems Engineering Department in Scbenectuly, New York, with a few y w s in G E s HVDC a d SVC product depu tmnt in Philadelphia. He has been involved in many aspects of controls for p o w s system devices, including generator exciters, HVDC systems, SVC system, and power plant steam supply. He developed many new digital-computer tools for analyzing power system dynamics, and led several field tests. He has also been involved with harmonics on power systems, and filter design.

Mr. Lamen is active in IEEE and CIGRE, currently chairing the IEEE Working Group on FACTS.

Juan J. SanchaCasea received his PhD in Electrical Engineering from the University of Wisconsin-Madison in 1983. ?hat year he joined GE where he is a Senior Application Engineer in the Power Systems Engineering Department. He is interested in the dynamic simulation and control of power systems.

Joc H. Chow (F92) received his Ph.D. degree in Electrical Engineering from the University of Illinois in Urbana in 1977. From 1978 to 1987 and in 1993 (sabbatical leave) he worked at General Electric Company in Schenectady. He joined Rensselaer Polytechnic Institute in 1987 and is currently Professor of Electrical, Computer and Systems Engineering, and Electric Power Engineering. His current research interests include power system dynamics and control, large scale systems and robust control.

(-Ao~ + j ( q + Ami)) 2 + (Kre + jKim)(-Ao, + j (Oi + Ami))

956

DISCUSSION E.V. Lawn, JJ. Sanchez-Gasca, J.H. Chow: We wish to D. J. TRUDNOWSKI (Pacific Northwest Laboratory, Richland, thank Dr. Trudnowski for an opportunity to clarify the Washington): The authors are congratulated on providing a valuable perspective on selecting signals for feedback damping control. The assumptions made in deriving (2.19). The potential contributions go beyond power system applications to include many large-scale dynamic systems. approximation (2.10) holds when &i(joi) is small compared

There is some confusion in the derivation of NPI index in equation (2.19) which is derived from (2.18). Equation (2.18) is developed by combining (2.17) and (2.101, but it seems that one cannot use (2.17) and (2.10) together. This is becuase (2.10) is derived assuming a “small” controller gain, and (2.17) is developed assuming the controller gain goes to infinity. One way to aleviate this problem is to remove the “small gain” restriction in deriving (2.10). Could the authors comment on this issue.

Manuscript received August 19, 1994.

to of [see equation @.I)]. ~f the controllef gain is small,

this assumption will be satisfied. The natural phase index

“PI is useful when the mt-locus branch from the lightly

damped pole terminates on a near-by zero. In this case, &i(jq) will be small, and equation (2.19) is valid.

Manuscript received October 26, 1994.