A Power Oscillation Damping Control Scheme Based on ... - IEEE Xplore

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 4, NOVEMBER 2010

A Power Oscillation Damping Control Scheme Based on Bang-Bang Modulation of FACTS Signals Huy Nguyen-Duc, Student Member, IEEE, Louis-A Dessaint, Senior Member, IEEE, Aimé Francis Okou, Member, IEEE, and Innocent Kamwa, Fellow, IEEE

Abstract—In this paper, we develop a new damping control scheme for FACTS devices, using bang-bang modulation of FACTS signals. The scheme is used to attenuate quickly the system’s most dominant mode which is identified using online Prony analysis. An analysis framework to evaluate the robustness of the control schemes to changes in operating conditions and to time-delay is also proposed. Simulation results on various test systems show that following large disturbances, the proposed control scheme is very effective to mitigate the power system critical modes of oscillation. Furthermore, the problem of control interactions is completely avoided with the proposed control scheme. Index Terms— -analysis, damping controller, interarea oscillations, robust control, structured singular values, wide-area control.

I. INTRODUCTION

F

ACTS devices have become more and more popular in power systems. Their primary application is to enhance power transfer capabilities, allow more flexible control of power flows, as well as provide reactive power support. Besides, they can also provide additional oscillation damping control, which improves power system small signal stability. The use of FACTS devices to improve power system stability has long been recognized. The concept of how an SVC adds damping to electromechanical oscillations was presented in some early publications [1]–[3] for single machine–infinite bus systems. For multimachine systems, modal analysis of the linearized power system model is mostly used to analyze the system modes of oscillations. One important concept in designing FACTS based controllers to enhance small signal stability is the Controller Phase Index (CPI) [4], which is the level of phase compensation required to move the open loop poles associated with power oscillations to a desired location. This is in fact very similar to the traditional concept of designing power system stabilizers (PSS) [5]. Manuscript received December 08, 2009; revised February 04, 2010. First published April 26, 2010; current version published October 20, 2010. This work was supported by the Hydro-Québec TransÉnergie Chair on Simulation and Control of Power Systems and the National Science and Engineering Research Council, Canada. Paper no. TPWRS-00949-2009. H. Nguyen-Duc and L.-A. Dessaint are with the Département de génieélectrique, École de Technologie supérieure, Montréal, QC H3C 1K3, Canada (e-mail: [email protected]; [email protected]). A.F. Okou is with the Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, ON K7K 7B4, Canada. I. Kamwa is with the Institut de recherche d’Hydro-Québec (IREQ), Varennes, QC J3X1S1, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2010.2046504

The design of FACTS controllers, however, can be quite more complicated than that of the PSS. A FACTS controller is generally less robust than a PSS and it is difficult to achieve robustness over a wide range of operating conditions [6]. The location of FACTS and the input signals are chosen in order to have maximum controllability/observability of oscillation modes. Besides, the residues of critical modes should not vary too much in different operating conditions, so that a low order, robust controller can be achieved [7]. Another aspect of control synthesis is avoiding interactions. An ideal control loop should only have minimum effect on other system’s noncritical modes [8], [9]. Such demands are often hard to meet using conventional control approach; therefore, a great deal of research efforts have been made to improve the FACTS controller performance, using robust control theories, i.e., [10], -analysis [11], etc. FACTS controllers designed using robust control theory should have better robust performance, but since quite many constraints are often used in the design to ensure good performance of the nominal as well as of the perturbed systems [10], their damping performance may be limited. Moreover, a robust controller is normally of very high order, which complicates its implementation. Besides conventional continuous control methods, there exists a rather simple approach to damping of electromechanical oscillations, based on bang-bang modulation (or reactive power switching) of FACTS input signal. This approach was first used to reduce transient oscillation in power systems [1]–[3]. Reference [12] further discussed how this control method could reduce the critical oscillation mode during a large transient. In fact, an appropriate reactive power switching sequence can produce an oscillation in anti-phase with the observed oscillation (output cancellation strategy). This can be a very effective strategy, since it only deals with the most critical oscillation, and the maximum power rating of FACTS can be used. The effectiveness of this discontinuous control method has long been recognized [13], and can also be applied for active load modulation [14], [15]. The problem of optimal switching time has been studied for simple systems, using nonlinear optimization methods [16], [17]. However, the application of this control method, especially for large, multi-machine power systems, is still an open problem. The main issue that needs to be studied is the robustness of such control scheme to changes in operating condition and other perturbations. This paper is the continuation of our previous work on a bang-bang control scheme based on identification of the critical oscillation [18]. A new control scheme to efficiently dampen electromechanical oscillations is proposed. It is based on identified information of the system critical mode and bang-bang modulation of FACTS input signal. Since control interactions

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are inherently avoided, the control scheme is easily expandable which allows different FACTS devices to act simultaneously, to further improve the overall damping. The paper also provides an analysis framework to evaluate the control scheme performance in large interconnected systems. The robustness is evaluated for a wide range of operating conditions, with varying time delay. The paper is organized as follows: Section II provides some background information on the design of FACTS controllers as well as the control scheme based on bang-bang modulation. Section III presents the structure of the proposed control scheme. A description of the framework for robustness analysis is also presented. Section IV presents some application examples. Section V gives some discussions and conclusions. II. BACKGROUND A. Transfer Function Residue and Location of Zeros To effectively control an oscillation mode using feedback, the chosen input and output signal must have, respectively, large controllability and observability of the mode. These two measures can be combined to give the transfer function residue asbe the residue sociated with the mode of oscillation [19]. Let , a sufof a control loop with respect to a mode ficiently small proportional feedback control causes a change in , as follows:

Fig. 1. Interconnection structure for robust stability analysis.

many robust control design approaches, the analysis using structured singular value theory [21] is an effective method to evaluate the robustness of power system controllers. The general framework for many robust control studies is based on a linear fractional transformation (LFT), depicted in Fig. 1. It represents how the uncertainty affects the input/output relationship of the system under study. is a complex transfer matrix, partitioned as folIn Fig. 1, lows: (2)

respect to

is another complex matrix. The upper LFT with is defined as

(1)

(3)

The magnitude of is an important criterion in selecting control loop. A large magnitude of implies that small control efforts would be required to achieve better damping. The supis the phase compensation required at plementary angle of to achieve a pure damping influence, assuming frequency positive feedback. This phase compensation is mentioned in [4] where the authors introduced the term CPI, which actually . equals to The change of CPI with respect to changes in operating conditions is an important aspect of FACTS control design. For a control loop whose CPI does not vary much in different operating conditions, it is easier to design a robust, low order controller [7]. Roughly speaking, a CPI that remains in the same quadrant in different operating modes implies that a robust controller can be achieved with a simple lead-lag feedback control [4], [20]. Another issue related to FACTS control design is avoiding interactions. While having a positive damping effect on the critical mode, the controller can deteriorate other electromechanical modes, or other system modes. This problem should also be addressed while designing FACTS controllers.

The LFT structure has a useful interpretation: represents the nominal, unperturbed system, and is perturbed by . The transfer matrices , , and reflect a prior knowledge as to how the perturbation affects the nominal transfer matrix. The transfer matrix represents all sources of uncertainties, which could be real parametric uncertainties or neglected dynamics. The structured uncertainty thus may contain blocks of repeated real scalars, or full complex blocks [21]. Hence, we can assume that takes the following form:

(4) Given the interconnection system, as depicted in Fig. 1, the structured singular value (s.s.v. or ) is defined as the smallest structured uncertainty , measured in terms of its maximum singular value which makes :

(5) B. Robustness Evaluation Using -Analysis Conventional power system control designs are based on linearized models of the system about some selected operating points. As a consequence, the performance of the controller may not be robust over a wide range of operating conditions. In recent years, several researches have studied the application of robust control techniques to design power system controllers. Among

If no such structure exists, then . Since is the size of the smallest structured perturbation that makes the system singular, it can be used as a measure of robust stability. For power system application, -analysis can be used to evaluate the system robustness with uncertainty in operating conditions. As the operating condition changes, the state space matrices of the linearized power system also change.

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as follows: If there is a critical oscillation in the measured power transfer in the interconnection line of Fig. 2, such that (8) then the modulation signal for the SVC controller is Fig. 2. Single machine–infinite bus system.

(9)

The dependency of the state matrix coefficients on the parameter variations can be captured to formulate a structured perturbation model. The linear state space model of a power system, subjected to parameter changes , can be written as follows:

(6) , , , and are the state space matrices of the where are determined by nominal system. Matrices , , , and solving a set of overdetermined linear equations. In robust stability studies, the perturbations are normalized so that . The parameter uncertainty representation in (6) can be transformed to an LFT form, suitable for robust analysis. More details on this framework are reported in [22] and [23]. With this framework, changes in system operating conditions are treated as parameter uncertainties. Besides, time delay can be considered as a varying quantity, rather than a fixed value, which is used in many studies [24], [25]. C. Control Using Bang-Bang Modulation For a simple power system model, the concept of FACTS control to improve power swing damping can be derived, without using complicated modal analysis tools. Consider the single machine–infinite bus system, depicted in Fig. 2, where an SVC is placed at the midpoint of the interconnection line. For this power system, the system damping can be enhanced if the midpoint voltage is modulated as a function of [8], as follows: (7) where is a constant. An intuitive explanation of (7) is that it reduces the kinetic energy built up in the rotor during transient [8]. To improve damping of an oscillation , (7) suggests that the modulated SVC voltage should lead the observed oscillation in by 90 degrees at frequency . Alternative signals for can also be used, e.g., or machine speed [2], [3]. Control of is done by reactive power modulation. To achieve maximum damping, a bang-bang control method can be used, in which the SVC reactance is switched to its maximum capacitive/inductive value in synchronization with the rate of change of . In our previous work [18], we have successfully implemented a bang-bang control scheme based on the control rule (7), where the switching time is determined based on the identified frequency and phase of critical power oscillations. The proposed control scheme in [18] can be summarized

which means the switching time is in phase with the rate of change of the power transfer associated with the critical component. In multi-machine systems, large transients may contain several modes. As stated in [18], we propose to use an online identification method to pick up the mode critical oscillation. The controller should react to damp this component only. If there is more than one mode in a transient oscillation, a bang-bang control scheme based solely on derivation of active power transfer [2] would give inaccurate switching time. and the phase Online identification gives the frequency of the most critical component. The only parameter that remains to be determined is the required phase lead. The best phase lead is determined from the residue angle of the open loop transfer function with respect to the critical mode [12]. For the simple power system in Fig. 2, and in fact, for some FACTS control loops using local signal, this angle of residue is close to 90 . This has been proved in [26], by using classical generator model and neglecting transmission line resistance. However, for a large multi-machine system, the control loops are not restricted to using local signals. Thus, it is best to use rigorous modal analysis to determine the required phase lead. Moreover, one must determine if a fixed phase lead can provide damping effect in different operating conditions. The effect of communication delay must also be examined. III. PROPOSED CONTROL SCHEME A. Proposed Control Scheme The proposed control scheme essentially comprises of a special control mode, based on bang-bang modulation of FACTS signal. When a large oscillation is detected in the system, this control mode will be activated to maximize damping effect. When the oscillation amplitude has been reduced significantly, the control scheme will switch back to using a conventional continuous controller. The motivation of this work comes from the fact that critical oscillations in power system often involve one critical mode. To stabilize the system, it is important that this critical mode is effectively damped. When a controller reacts only to this mode, a bang-bang type of control, as discussed in Section II-C, is an effective method, since it utilizes the maximum power rating of FACTS devices. The proposed approach is thus similar to a gain scheduling control scheme, which combines a high gain, narrow bandwidth controller with a small gain, continuous controller to maximize damping, while avoiding control interactions. The bang-bang control mode is demonstrated in Fig. 3. Upon detection of a critical signal, a FACTS device will generate a bang-bang type control signal as follows: (10)


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Fig. 4. Fixed phase controller.

Fig. 3. Proposed control scheme.

The operation principles of the central control unit and local controllers in Fig. 3 are described as follows. • Central controller: The central control unit continuously monitors some input signals and use Prony-analysis [27] to identify the presence of a critical oscillation. The central controller works in two modes: the supervision mode and the active control mode. In the supervision mode, the identification algorithm monitors real parts of oscillation . If a critical mode modes ( ) and their amplitudes (large amplitude and bad damping coefficient) is detected consistently during a specified period, then the active control mode is activated. In the active control mode, Prony analysis is still used to determine oscillation frequencies and damping factors. However, with the presence of a control action, the damping coefficient is modified and is no longer a good critical oscillation index. Therefore the amplitude of the critical component, whose frequency was identified in the supervision mode, is now monitored. When this amplitude is reduced below a certain threshold, the active control mode is disabled, and all FACTS controllers return to their normal working mode. • Local FACTS controllers: Once a dangerous oscillation is detected, the central controller will activate the special control mode and send the critical oscillation information (frequency and phase ) to local controllers. These informations are associated with the central controller’s reference time. Thus, local controllers can compare the received time tag with their own local clocks to generate precise control signals. With this configuration, the control scheme is less affected by communication delay. The output of each local controller is as described in (10), where is appropriately determined based the fixed phase lead on modal analysis. The control scheme in Fig. 3 assumes a generic FACTS model, and thus can be used for any type of FACTS (SVC, STATCOM, TCSC, etc.). For multi-parametric FACTS devices, the control scheme can also be used, simply by modulating one independent component. For example, with the UPFC, the control scheme will only modulate the active component of the

series branch voltage. The shunt voltage controller may still work in its normal working mode. The novelty of this control scheme is that it is a true closed loop identification-control method, as opposed to [28] and [29], where Prony identification is only used to trigger a power swing damping controller. An identification based system protection was also studied in [30], in which a high-resolution Fourier analysis is used to detect generators oscillation coherency. To avoid identification error due to the presences of modes of similar frequency, Prony identification using multiple input signals can be used. The grouping of these signals are based on offline modal analysis, as in [28]. B. Robustness Analysis The performance of this proposed control scheme depends on the identification accuracy of the critical oscillation and the value of the phase lead for each FACTS. To make the control scheme simple, it is desired to use one single value of for all operating modes. As discussed in Section II-C, the best value for is the supplementary angle of the open loop system residue angle at the mode of interest. This phase lead is chosen for the system nominal operating point, and robustness test should be done to verify the control performance in other modes. Since Prony identification is used to pick up the critical oscillation, it is possible for the control scheme to operate in a very narrow bandwidth which would contain only the critical oscillation frequency. In this bandwidth, the controller provides a constant phase lead to the observed oscillation, so as to provide positive damping. Fig. 4 depicts the frequency response of the damping controller for the Kundur power system (which will be presented in Section IV-A), and that of a hypothetical fixed phase controller which represents the bang-bang controller. In the frequency range of the critical oscillation, the two controller responses are essentially similar: both controllers provide a phase lead of 23 at the inter-area mode frequency (3.81 rad/s). On the other hand, the fixed phase controller has a narrow bandwidth, which prevents it from interacting with other system modes. This bandwidth is a design parameter, determined based on the variation of the critical mode frequency in different operating conditions. To evaluate the robustness of the proposed control scheme as well as the possible interactions with other modes, we use the

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Fig. 6. Kundur power system.

Fig. 5. Uncertainty structure for robustness evaluation of the fixed-phase controller.

structured singular value theory framework. Both variations in operating condition and communication delay are considered. The proposed framework for the robustness evaluation is depicted in Fig. 5. Apart from the nominal unperturbed system in Fig. 5, there are three components in this analysis. 1) Structured Uncertainty : The uncertainty block represents the perturbations due to changes in system parameters (e.g., load and generation levels). In order to determine the structure of , one needs to create a grid of operating conditions from the combination of the varying parameters. The construction of has been presented in Section II-B. 2) Time Delay Block: The delay block in Fig. 5 takes into account the possible time lag between the central controller and local controllers, as well as the time lag between the central controller and remote measurement devices. A time lag is represented by the transfer function . The uncertainty block for robustness analysis, which takes into account both parameter uncertainty and communication delay, is (11) In this work, the uncertainty (varying time delay ) is represented this by an input multiplicative uncertainty [31], [32]. The details are presented in Appendix A2. 3) Fixed-Phase Controller: The fixed phase controller has been described at the beginning of this section. The phase lead is an important parameter of this controller, which is determined based on the CPI of the open loop system. In the estimated frequency range of the critical mode, the gain and phase of is strictly constant. For all other frequencies, has zero gain. It is very difficult to find an analytical transfer function for . However, the Analysis and Synthesis Tool Box [33] does not require such an explicit analytical expression. It allows the controller to be represented as complex transfer matrices in function of frequency. It should be noted that the robustness analysis in this work is only used to evaluate the possibility of using a fixed phase lead value for the proposed controller. If the hypothetical fixed phase controller achieves robust stability, then it is possible to

use a fixed value for , with given parameter and time delay uncertainties. On the other hand, the damping performance of the hypothetic fixed-phase controller, as in Fig. 4, does not represent the performance of the proposed controller which uses bang-bang control action. The latter should have better damping effect, since maximum power rating of FACTS is used at each switching instance. IV. CASE STUDIES A. Kundur System The proposed controller is designed for the Kundur power system [5], as shown in Fig. 6. The system has four machines located in two areas. Without PSS installed, the system exhibits one unstable inter-area mode at 3.81 rad/s. A grid of operating conditions is created by varying active load at bus 7 (from 767 MW to 1267 MW) and 9 (from 1167 MW to 1767 MW). We first design a conventional continuous damping controller . The confor SVC at bus 9, with power rating of tinuous controller comprises a wash-out filter and two lead-lag blocks, and is optimized using phase compensation and root locus analysis. Active power measured at bus 7 is chosen as input signal. For this control loop, the required phase compensation at the inter-area mode is 23 . As presented in the previous section, in robust stability study, the bang-bang controller is represented by a fixed phase controller, which has a constant phase lead of 23 for all frequency ranging from 3.5 to 4.2 rad/s (i.e., frequency range of the interarea mode). Fig. 7 shows -analysis results for three systems: 1) the open loop system; 2) the closed loop system with fixedphase controller; and 3) the closed loop system with conventional controller. The communication delay is assumed to vary from 0 to 150 ms. The -analysis shows two distinctive peaks, corresponding to the inter-area mode (near 3.8 rad/s) and the local mode at generator 3 (near 7.5 rad/s). The open loop system is not robustly stable with given uncertainty ranges, as the inter-area mode peak is greater than 1. Both the fixed-phase controller and the conventional controller are able to stabilize the system. The closed loop system singular values with both controllers are similar in the range 3.5 to 4.2 rad/s. On the other hand, the fixed phase controller does not affect the system at other frequencies. The continuous controller tends to make higher frequency modes less stable. Fig. 8 presents simulation results for the worst-case scenario: The system is operating at high power transfer level (450 MW) when a three-phase fault occurs at and clears at at bus 8. The communication delay is 150 ms. The Prony


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Fig. 7. Robust stability analysis for Kundur power system. Fig. 9. System response with a high gain conventional controller.

Fig. 10. System response with SVC and PSS. (a) Tie line power, PSS or SVC alone. (b) Tie line power, with PSS and SVC. Fig. 8. System response to a three-phase fault at bus 8.

identification uses a window length of 6 s, with sampling time . The bang-bang control mode is activated at and disconnected at . As can be seen in Fig. 8, better performance is achieved when the bang-bang control mode is used. In order to evaluate if the continuous controller alone can achieve the same performance, its gain is increased by a factor of three. The simulation result for this case is presented in Fig. 9. While the damping of the inter-area mode is improved, this gain increase also makes a exciter mode near 18 rad/s much more visible. This result confirms the -analysis presented in Fig. 7. In general, the PSS is often more effective than FACTS controllers to damp power oscillations. In this particular system, however, the proposed control scheme using SVC has better performance than a single PSS at generator 1. Fig. 10a compares the damping performance of the PSS at generator 1 and the proposed controller. PSS parameters for the Kundur power system were obtained from [34]. It can be seen that at the first

swings, the PSS has better damping effect. However when the bang-bang mode is activated, the proposed controller catches up with the PSS and finally has better settling time. On the other hand, the two controllers can be used simultaneously to give a much better response [see Fig. 10(b)]. The proposed control scheme also works well in the case of reversed power flow. Simulation results for this case were presented in [18]. B. New England System The proposed control scheme is now applied to the New England power system, depicted in Fig. 11. Complete system description and data can be found in [6]. This system has a weakly damped mode at 2.39 rad/s, in which generators 1–13 oscillate against generators 14–16. The proposed control scheme is thus designed for this mode. All other electromechanical modes of this system are presented in Table I. Mode shape analysis reveals that generators 1–13 are virtually in phase for the 2.39 rad/s mode. For the proposed control scheme, this means any speed signal can be used as

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Fig. 12. Robust stability for various control loops, New England system.

Fig. 11. New England power system.

TABLE I ELECTROMECHANICAL MODES OF NEW ENGLAND SYSTEM

TABLE II PARAMETER VARIATION RANGES

input signal, without having to change the control parameters. This flexibility may also apply to a continuous control scheme, if the chosen loop has no interaction with other system modes. Several control loops have been examined, and for each loop, a continuous controller is derived using standard procedures in [35]. Robust stability analysis for these control loops is performed using the proposed framework in Section II-B. The eigenvalue sensitivity study shows that active loads at bus 9, 33, and 47 have the greatest influence on the system dominant modes. The chosen uncertain ranges for these loads are shown in Table II. For robust stability studies, the time delay is assumed to vary from 0–300 ms. Fig. 12 presents the robust stability analysis

for the open loop system, and for the closed loop systems with several different control loops. The open loop only shows two peaks at 2.39 rad/s and 3.9 rad/s. All closed loop systems have a lower peak at 2.39 rad/s, since the controller is designed to damp this mode. However, the controllers have excited various modes at different frequencies. It can be seen that only the loop (SVC at bus 21, with as input signal) is free from interaction problems. Using different input signals for the SVC results in exciting the inter-area modes at 3.9 and 7.3 rad/s. The TCSC on line 1–47 tends to excite the inter-area modes at 11.34 rad/s and 8.34 rad/s. It should be noted that the large -peaks in Fig. 12 are largely due to the time delay. The peaks for closed loop systems without time delay (not shown here for the sake of brevity) are less pronounced than those in Fig. 12. This result suggests that with a continuous control scheme, it is not possible to change the input signal without having to redesign the controller: Control interactions occur at various frequencies, depending on the input signal. On the contrary, if the proposed control scheme is used, the identification based bang-bang control action will maximize the damping effect. When the oscillation has been reduced considerably, a continuous controller with low feedback gain will be used. This kind of “gain scheduling” control allows to achieve better damping performance while avoiding interaction issues. To validate the above analysis, closed loop performances of these control loops are analyzed with nonlinear simulations, with time delay varying from 0–300 ms. High feedback gains are used with the continuous controllers, in order to match the proposed control scheme damping performance. Fig. 13 presents damping performances of three control in Fig. 13(a) and (b), in loops: Fig. 13(e) and (f), and in Fig. 13(c) and (d), when a three-phase fault at bus 42 is applied at for 200 ms. With each control loop, damping performances of the continuous controller and the proposed control scheme are compared. The continuous controllers’ parameters are given in Appendix B. It can be seen that with the loop , the continuous control scheme achieves a fairly similar damping performance to the proposed control scheme. In fact, in this large


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0

0

0

Fig. 13. Generator speed responses with various control loops. (a) SV C ! , continuous control. b) SV C ! , proposed control scheme. (c) T CSC , continuous control. (d) T CSC ! , proposed control scheme. (e) SV C ! , continuous control. (f) SV C ! , proposed control scheme.

!

0

power system, a single controller could not have as large controllability as in the Kundur system. Therefore, the difference in damping performance between two controllers is smaller. For all other control loops, however, the interaction problems predicted by the -analysis can be easily observed. Using continuous feedback control, the TCSC destabilizes the 11.34 rad/s mode in Fig. 13(c), while the SVC destabilizes the 3.9 rad/s mode in Fig. 13(e). On the other hand, the damping performance of the proposed control scheme is consistent with different control loops in Fig. 13(b), (d), and (f). Therefore, it is possible to change the input signal (e.g., in case of a communication failure/bad measurement), while still maintaining the damping performance. Besides, the damping performance can be further improved if several controllers are activated simultaneously. Fig. 14 presents the system response when three controllers are used simultaneously: the SVC at bus 21, with power rating of , the TCSC on line 1–47, and a UPFC on line 1–27. For damping purpose, the series voltage for UPFC is modulated at a magnitude of . The maximum compensation ratio for the TCSC on line 1–47 is . For the 2.39 rad/s mode, the required phase leads for the two controllers are 5 and 30 , respectively. Compared to the system response with one controller, the system damping has been clearly improved. The performance of the proposed control scheme is also examined for more severe contingencies. Fig. 15 shows the system response with three controllers, with the same fault at bus 42, and lines 47–48 and 30-9-36 tripped out. Compared to

0

0

Fig. 14. Damping performance with three controllers.

the system response in Fig. 14, the generators’ speed tend to increase slightly at steady state. On the other hand, damping performance with respect to the global mode at 2.39 rad/s is still maintained. V. DISCUSSION AND CONCLUSION The proposed controller uses several FACTS controllers to damp one single mode of oscillation. It is thus well suited to

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It has been shown in [15] that the switchings of a small percentage of system load, if well synchronized with the disturbing event, could have highly stabilizing effects. The same conclusion is drawn from this work: In all test cases, the FACTS power ratings required are very small, compared to the system total active power. A large power rating means the scheme can damp the oscillation faster, but also means that it cannot be used when there is a small oscillations (e.g., in smaller contingencies). It would be better to use a small value for FACTS power rating for this type of control, so that it can be used in many contingencies, without violating physical constraints. APPENDIX A MODELING OF VARYING TIME DELAY UNCERTAINTY

Fig. 15. Damping performance with three controllers, lines 47–48, 30–36 out of service.

damp inter-area oscillations, or ideally the global mode of oscillation (where all machines in the system participate), since the number of FACTS that can be used is maximized. The proposed scheme damps one oscillation at a time, but it can be designed for more than one mode. For each mode, there is an associated group of FACTS that can be used. One advantage of the control scheme is the lack of interactions with other system dynamics. As can be seen in all test cases, the proposed control scheme can achieve very good damping performance while a conventional controller using the same input-output signals only achieves moderate damping, due to interactions with other system modes, which can be either another electromechanical mode (New England system), or a exciter mode (Kundur system). A high order controller, designed using a robust control synthesis method, might achieve the same performance as the proposed control scheme. The proposed control scheme, however, presents a simple alternative to various robust control design approaches. It requires knowledge of only the open loop residue angle for the mode of interest. Besides, it has the flexibility of switching easily between different input signals. A controller can also be added or removed from the control scheme without having to change any control parameter. This work reveals that communication delay in the control loop can have a drastic effect on control interactions. Time delay has a similar effect as RHP zeros [21], so that when a mode is stabilized by the control action, another mode will be destabilized. The proposed analysis method in this paper, using structured singular value theory, can reflect very well this problem. More detailed comparative studies between the proposed robust stability analysis framework and conventional modal analysis will be presented in a future publication. One drawback of the proposed approach is that it is not possible to quantify the improvement in damping, since the bang-bang control mode is not a continuous feedback controller. However, the hypothetical “fixed-phase” controller introduced in Section III-B can be used to evaluate the robustness of the proposed control scheme. The fixed phase controller can also provide a lower bound of obtainable performance for the proposed control scheme.

The problem of representing a time delay by an input multiplicative uncertainty is to find an uncertainty weighting function , so that the time delay can be represented as (12) where

, (nominal transfer function). Since , the relative uncertainty should satisfy

(13) Since have

, and

, we

(14) to and in the range of freBy varying from quencies of interest, we can obtain the upper bounds for the lefthand side of (14), in function of frequency. Then a weighting function for can be derived by approximation, using a least square curve fitting routine. APPENDIX B CONTINUOUS CONTROLLER PARAMETERS The structure of the conventional continuous controller in this work is as follows:

(15) Parameters for the New England system controllers: : , , , : , , .

; ,

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Huy Nguyen-Duc (S’06) received the B.S. degree in electrical engineering from Hanoi University of Technology, Hai Ba Trung, Vietnam, in 2001 and the M.S. degree from Sherbrooke University, Sherbrooke, QC, Canada, in 2003. He is currently pursuing the Ph.D. degree in the Electrical Engineering Department of the École de Technologie supérieure, Montréal, QC, Canada. His areas of research interest are simulation and control of power system dynamics.

Louis-A Dessaint (M’88–SM’91) received the B.Ing., M.Sc.A., and Ph.D. degrees in electrical engineering from the École Polytechnique de Montréal, Montréal, QC, Canada, in 1978, 1980, and 1985, respectively. Currently, he is a Professor of electrical engineering at the École de Technologie Supérieure, Montréal. From 1992 to 2001, he was the Director of the Groupe de Recherche en Électronique de Puissance et Commande Industrielle (GREPCI), a research group on power electronics and digital control. Since 2002, he has held the TransEnergie (Hydro-Québec) Chair on Power Systems Simulation and Control. He is an author of the SimPowerSystems simulation software. Dr. Dessaint received the Outstanding Engineer Award from IEEE-Canada in 1997. He is also an associate editor for the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.

Aimé Francis Okou (M’04) received the Dipl.Ing. degree in electrical engineering from the École Supérieure Interafricaine de l’Électricité, Côte d’Ivoire, in 1993, and the M.Eng. degree and the Ph.D. in electrical engineering from the École de Technologie Supérieure (ÉTS), Montreal, QC, Canada, in 1996 and 2001, respectively. Since 2005, he joined The Royal Military College of Canada where he is currently an Assistant Professor in the Electrical and Computer Engineering Department. His research interests include the application of robust and nonlinear control techniques to large-scale systems.

Innocent Kamwa (S’83–SM’98–F’05) received the B.Ing. degree and the Ph.D degree in electrical engineering from Université de Laval, Québec city, QC, Canada, in 1984 and 1988, respectively. Since then, he has been with the Hydro-Québec Research Institute, where at present, he is a Principal Researcher with interests broadly in bulk system dynamic performance. Since 1990, he has held an Associate Professor position in electrical engineering at Université de Laval. Dr. Kamwa is a recipient of the 1998 and 2003 PES Prize Paper Awards and is currently serving on the System Dynamic Performance Committee, AdCom. He is also the acting Standards Coordinator of the PES Electric Machinery Committee and an associate editor for the IEEE TRANSACTIONS ON POWER SYSTEMS. He is a member of CIGRÉ.