Toward A Globally Robust Decentralized Control ... - Semantic Scholar
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Toward a Globally Robust Decentralized Control for Large-Scale Power Systems Haibo Jiang, Hongzhi Cai, John F. Dorsey, Senior Member, IEEE, and Zhihua Qu, Senior Member, IEEE Abstract—A robust control scheme is presented that stabilizes a nonlinear model of a power system to a very large class of disturbances that includes any disturbances causing the system to exhibit sustained oscillation. The disturbance can be anywhere in the power system. The fact that the improvement in stability is significant and system wide leads to the name globally robust control. The control is local or decentralized in the sense that the control of each generator depends only on information available at that generator, and is derived using Lyapunov’s direct method. The derivation is quite general, permitting a second-order representation of the turbine/governor and any generator model. Simulation results are presented which show the effectiveness of the proposed control against instabilities of current importance including sustained oscillations following a major system disturbance such as a fault or major line outage. The control is also effective for steady-state operation. Index Terms—Control, Lyapunov direct method, power system, robust control, stability, steady state.
I. INTRODUCTION
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VER the last decade, the interconnected power system in the United States has become less stable. This is a trend that will continue as deregulation of the energy industries continues. As a result, the large interconnected subsystems that coordinate their activities have begun to detect sustained oscillations in their simulation analysis. Real oscillations have also been observed. These oscillations usually follow a major disturbance, for instance removal of a fault or following the loss of a major transmission line. One of the more prominent examples of this phenomenon is the 0.7-Hz oscillation that arises in the Western System Coordinating Council (WSCC) following the loss of one of the ac or dc interties between the Pacific Northwest and California [6]. Preventing this sustained oscillation is of great interest since the potential for an oscillation causes lower operational limits to be set on transfer levels on the interties. Oscillations can also arise during “normal” steady-state operation. This is a phenomenon that will only increase in frequency as deregulation proceeds. Thus, there is a need to formulate a control strategy that will work effectively both during transient periods following major interruptions and at steady state. Further, if deregulation proceeds as proposed, Manuscript received September 19, 1995. Recommended by Associate Editor, S. Kumagai. H. Jiang is with the Robinson Humphries Co., Atlanta, GA 30332 USA. H. Cai is with the Foxboro Co., Foxboro, MA 02035 USA. J. F. Dorsey is with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. Z. Qu is with the Department of Electrical Engineering, University of Central Florida, Orlando, FL 32816-0450 USA. Publisher Item Identifier S 1063-6536(97)03275-2.
and the suppliers of power are determined on the basis of a regular “auction,” then the configuration of the interconnected grid will routinely be in a state of change. In this situation it will be increasingly difficult to predict the strength or location of disturbances. Thus, the distinction between transient and steady-state stability will be moot. This paper considers the damping of oscillations because they are forerunners to more complex disturbances that will be an inevitable byproduct of deregulation. The proposed control, however, is very general and is formulated to be effective against the expected control problems of the foreseeable future. We have given the proposed control the name globally robust decentralized control. The term globally robust is one that requires careful explanation. No power system control strategy can claim to be globally stable. There is always a disturbance waiting in the wings that will defeat any reasonable control strategy. On the other hand, control strategies can be distinguished by the class of disturbances against which they are effective. Some control strategies are designed to work against specific disturbances. Indeed, this was the normal procedure before deregulation since in the regulated environment, the topology of the interconnected grid was reasonably well fixed, and certain contingencies (disturbances) could be identified as the most destabilizing. Under deregulation what is required is a control strategy that will counteract a wide variety of disturbances that may occur anywhere is the system. The proposed control is globally robust in this sense. The damping of sustained oscillations has been studied by a number of investigators. However, the approach has been to use a linearized model of the power system as the basis for designing a power system stabilizer (PSS) which augments existing excitation systems to improve the damping of the electric power system during low-frequency oscillations. The basic idea is to add PSS to specific machines to damp oscillations due to a specific disturbance, or set of disturbances, at particular points in the system. As we have already indicated, this is not a strategy that will be effective in the future. The existing literature does not, with a few exceptions, address the problem of finding a globally robust control that will protect the power system against a large class of disturbances, anywhere in the system. Further, since the existing analysis is based almost exclusively on a linearized model of the power system, it is not applicable to cases where the system suffers a major disturbance, such as a fault. In such cases a nonlinear model of the power system is required. What is really needed under deregulation is a control, based on a nonlinear model of the power system that is effective
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against system wide disturbances and can be used for control both during the transient period following a major disturbance and at steady state. There is some work on nonlinear stabilizing control of power systems [5], [8], [9]. In [5], the major drawback is that the control requires global feedback information and exact knowledge about the power system. In our earlier papers [8] and [9], we first began the development of a decentralized robust control that stabilized a power system for disturbances anywhere in the power system. However, in [8] and [9], we used the classical generator model and a first-order model of the turbine/governor. The excitation system was not modeled and consequently not included in the derivation of the control or the proof of stability by Lyapunov’s direct method. Also the effect of the simplified modeling on local stability was not considered. The motivation of this paper is to present a scheme for designing power system control (PSC), which uses a nonlinear model of the generator and a second-order model of the turbine/governor, making it distinctly different from conventional designs of PSS’s. The excitation system is included implicitly. PSC is more general than PSS. It is designed not only to damp out sustained oscillations, but also to stabilize a power system against almost all reasonable disturbances. It cannot, of course, be effective against unforseen catastrophic contingencies, such as earthquakes, where no amount of control can hold the system together. However, even in these extreme cases, the proposed control strategy will have a strong mitigating effect that can help the graceful islanding of the system. As with our previous efforts, the control is based on local linear feedback at each generator, and thus is completely decentralized. We note that currently every major generator in a power system has integral feedback through the turbine governor to maintain frequency, and excitation control fed back through the electrical side to prop up the internal voltage of the generators in the hope of improving stability. What we are proposing is a straightforward modification or augmentation of the existing control. The payoff is high, because we obtain a control that is very robust, in the sense that it stabilizes the power system to a much wider class of disturbances. In establishing PSC, we combine the swing equations with a second-order model of the turbine/governor. In addition, we make the following observations. 1) For all kinds of generator models, the swing equations are essentially the same with the electrical power being a function of the model chosen. 2) For any excitation model, the internal voltage is bounded both by saturation effects within the generator and by stability limitations in the magnitude and angle differences between buses in the power system. Based on the above observations, we can design a PSC which is independent of the internal voltages, and then apply the PSC to any kind of generator model. This makes the PSC more realistic than our earlier results. The control presented in this paper is derived from a proof of stability based on Lyapunov’s direct method. The proposed control is linear and local, avoiding the shortcomings of [5]. A
judicious choice of the Lyapunov function allows us to show that the control is robust to parameter and load variations, and to changes in the power system topology. In the next section we give a detailed formulation of the problem discussed in this paper. In Section III a transient control and corresponding stability result are investigated under different type of disturbances. In this same section we also discuss the general form of electrical power functions under different disturbances. Section IV provides simulation results that validate the control scheme. The derivation of the Lyapunov function is in the Appendix. II. PROBLEM FORMULATION In formulating the power system model for constructing and analyzing a robust control scheme, one would obviously want as detailed a machine model as possible. In particular, for 20-s response times following a major disturbance, or for steady-state analysis, we would like to consider the effects of excitation systems and turbine/governor control. Choosing a more detailed generator model and including the dynamics of the excitation system and turbine/governor leads to extremely complex algebraic expressions that obscure the design of the control. In the next section we will introduce general forms of electrical power functions that depend on the type of disturbance, and which implicitly include all generator and excitation models in the Lyapunov analysis that leads to the control and the proof of stability. With the help of the new electrical power functions, we can avoid the complex algebraic expressions of the generator models, and use only the swing equations and the turbine/governor equations. The electrical dynamics of the generator is included through the electrical power functions. A final observation is the following. Although any kind of turbine/governor model can, at least theoretically, be used in the formulation, the use of a model of order more than two leads to algebraic expressions that will be too complicated to make it easy to find a suitable Lyapunov function. This causes difficulty in proving stability. It should be pointed out that “the difficulty in proof of the stability” does not mean the proposed control will fail for the high-order turbine/governor model. Simulations show the control is effective when used with higher order turbine/governor models. Our feeling is that a second-order turbine/governor model combined with the swing equations captures all the essential dynamics of the mechanical side of the turbine/generator system. We plan to address the issue of higher order turbine/governor models in a paper in the very near future. For the present analysis, we chose the second-order turbine/governor model given in [2]. In this paper we consider a power system of generators, given by with the swing equation of machine , (1) where is the angle of machine relative to the synchronous angle of the system, and , , , , and are the inertia constant, damping coefficient, electrical output power, prefault steady-state mechanical input power, and mechanical power deviation, respectively, of machine .
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Fig. 1. Block diagram of a second-order turbine/governor.
The block diagram of a second-order turbine/governor model is shown in Fig. 1, and yields the following differential equation:
(2) , , , , and are the turbine time where constant, governor time constant, governor gain, frequency regulation, and control signal, respectively, of machine . The frequency feedback included in the proposed control, effectively replaces the standard frequency feedback. Hence, in order to keep the system equations simple, we remove the standard frequency feedback in Fig. 1. Equation (2) then becomes (3) Rewriting (1) and (3) as first-order differential equations yields
(4) and are the frequency deviation and mechanwhere ical power derivative of the machine , respectively. Let the equilibrium points of (4) be characterized by , , , , , and , which are the solutions of the following equations:
The problem now becomes how to design the control such that the equilibrium points of (4) are the steady-state operating points of the power system. Since the system is operating under excitation control and the system parameters are not known exactly, the equilibrium points cannot be found explicitly. In the next section, we design a controller which controls the power system smoothly to a steady-state operating point. If the postfault system configuration remains the same as that of the prefault system, the steady-state operating point will be close, if not identical, to the prefault steady-state operating point. Otherwise, the post fault steady-state operating point will be different, depending on the change in power system topology. We emphasize that the actual steady-state operating point does not need to be known to implement the proposed control. It is introduced as a mathematical convenience to facilitate the proof of robustness. It is worth noting that not knowing the precise steady-state operating point is not a practical limitation. In practice, once the disturbance has been controlled, the system operators can move the system to the exact operating point they desire, just as they routinely move this operating point many times per day. III. A ROBUST CONTROL The control developed in this paper will stabilize the power system to almost any major disturbance anywhere in the power system. As discussed earlier, contingencies can occur that call for more control effort than any reasonable control strategy can supply. However, the proposed control represents a significant improvement in stability over that presently available. One of the system disturbances that we correct in the simulation is a sustained oscillation that arises after the disturbance is removed. However, we emphasize that this is not the only potential postdisturbance system instability that the control can operate against. To that end we also consider a case where the system is unstable without control. The overall control consists of local controllers, one at each machine, with each local controller using only information available locally, that is at the machine. We emphasize that the proposed control represents a straightforward augmentation of the control already in place on most machines. The proposed robust controller for the th machine is given by
(6) or
(7) (5)
, , , and The differences are the errors between the states of the power system
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and the desired steady states. The variable is the transient control input into the th machine. It should be noted that the desired steady states are arbitrary, local references, and that , , , and are local measurements of the machine . The control does not require the knowledge of the states of the other machines. The choices for the desired states do not change the stability of the overall power system, and therefore can be made arbitrarily; but they will affect the actual post fault steady state. Specifically, post fault steady state is the solution of (5) after replacing the variables in (7) by their corresponding ones with superscript and then substituting (7) into (5). To study the stability of power system (4) under control (6), we need to derive the error system. We first define the error states as
and
(10) (8) , and are the steady-state solutions where , , of (4). Combining (4)–(6), we have
The term is the electrical power deviation from the steady-state value of the th bus. Defining , we can write the error system as follows: (11) where
diag diag
Writing the above equations in matrix form yields (9) where
The function is the electrical power deviation at bus . It is a function of the angles and the internal voltages , . is tailored to the generator model chosen. Before proceeding, we first define two different types of disturbances. Type 1 disturbance: After the fault is removed, the system exhibits a sustained oscillation under conventional turbine/governor and excitation control but without the proposed PSC. Type 2 disturbance: After the fault is removed, the system will collapse under conventional control. We define these disturbances in terms of faults, since faults are usually the most pronounced insults the system receives. There are certainly other sources of both types of disturbances. The analysis presented here would go through in the same way for all but the most pathological of contingencies. In the following discussion, we assume that the system suffers a Type 1 Disturbance. We will discuss the case of a Type 2 Disturbance later. We note here that for a Type 2 disturbance the best that the proposed PSC can do is increase the fault clearing time significantly. If the fault duration is long enough,
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no control can keep the machine angles from separating too far with a consequent loss of synchronism. In a power system under sustained oscillation, the following facts are worth noting. 1) is a combination of sinusoidal functions of . 2) The internal voltage is always bounded. Its exact value, however, depends on the generator model chosen. Since there are a variety of generator models employed in any power system simulation, it is an obvious advantage to treat them all in terms of some bounding function. The potential drawback to this approach would seem either to be a degradation of control performance or equivalently or to introduce some conservativeness into control design. However, as the simulation results show, this is not a problem. In addition, gains for better performance may be generated automatically by combining the proposed robust control design with an auto-tuning algorithm [4]. 3) When for all , then by definition we have , . In the other words, if , for all , then , . Based on the above development, we can conclude that 1) is always bounded and 2) will converge to zero as converges to zero for all . Thus, we can write as (12)
where
and
are vectors consisting of and , , respectively. Recall that . are some scalar functions of both and . Coefficients Expression (12) is an equality which is rewritten by a form that reveals the second property of . Therefore, functions may take positive or negative values. By the first property of , we know that there exist constants such that, for all
Constant can be viewed as Lipschitzian constants of or their bounds. In the preliminary version of this paper [3], explicit expressions of and are found for generators described by two-axis models. Similar derivations can be done for any other generator models. We are now in a position to state the main result of this paper. It should be noted that the theorem presented here is related to many existing results on decentralized control of composite systems. For example, it has been shown in [1], [10], and [11] using state transformation and M-matrix approaches that linear interconnected systems in the form of (9) are globally stabilizable. The theorem given here extends these ideas to power systems whose interconnections are nonlinear. That this extension is possible is not unexpected since nonlinearities in terms of sinusoidal functions can be bounded in norm by the state. However, the approach taken in this paper differs from existing results in [1], [10], and [11] because it provides explicit choices for the sub-Lyapunov
functions. Through direct computation of time derivatives, these choices of Lyapunov functions provide explicit solutions for the control gains, which is a major advantage in control design. As will be shown later, this approach allows one to consider easily any nonlinear couplings. Theorem 1: The error system (11) is guaranteed to be asymptotically stable with an exponential rate of convergence if the local feedback gains , , , and , are chosen such that, for all
(13) and
(14) , , , and are the entries of matrix , which where is defined in the Appendix. Proof: The proof uses Lyapunov’s direct method. Choose the Lyapunov function candidate to be , which is defined in the Appendenote the Euclidean norm or its induced matrix dix. Let is symmetric norm. Then, since
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diag
and the nonzero elements of
are defined by
for all . By the Gershgorin theorem in [12], we know that the matrix is positive definite if we choose the gains such that the inequality (14) holds for all . Thus, we have , and
where for all
. Let and that
. It follows that
which shows that decreases to zero exponentially. Therefore, system (11) is exponentially stable in the large under control (6) with finite local feedback gains. Several remarks are worth making at this point. 1) When is large enough, we have
Hence, (14) holds for large . 2) The control guarantees that the system approaches the postdisturbance steady-state at an exponential rate, even though the steady-state is not explicitly known. 3) Although the lower bounds of the gains that guarantee the stability are given by (14), it is not necessary to know the parameters , , and , only the range of values of these parameters. In the case that the time constants in (10) are unknown as well, gains cannot be calculated through ( ) but can simply be made large enough to satisfy the polynomial orders dictated by (13). 4) The lower bounds given by (14) provide only a very conservative estimate for the magnitude of the gains. It is unavoidable that this inequality is not a necessary condition, since Lyapunov’s second method always yields very conservative results. The magnitude of the gains really needed to control the power system can and should be determined by computer simulation or by an autotuning algorithm [4]. We have run a large number of simulations at various gain levels and the results of those simulations can be briefly summarized as follows. The damping improves as the individual gains are increased from very small values to values in the rage of 50 to 100. Further increases in the gains does not perceptively change the damping or increase a measure of stability such as fault clearing time. One can conclude from this that in all probability the region of attraction for gains in the range 50 to 100 is only marginally different from the region of attraction for the conservative gains obtained from the proof of stability. The analysis is, of course, less than completely satisfying. However, it is better than the following alternative: 1) reduce the model in size (number of generators) by some aggregation technique; 2) linearize the model about an operating point; 3) design a control based on the linearized model; and 4) apply the control to the actual, nonlinear power system. This latter approach is the one that has dominated power system control for many years, and still does. However, as deregulation proceeds this latter approach becomes less and less appealing. First, with the topology of the grid in constant flux as various IPP’s bid for the power generation, it becomes very difficult to do the initial model reduction that is always done. Second, the steadystate operating point is not going to be as predictable as it has been in the past. We now consider the second case where the system is subject to a Type 2 disturbance where the system can become unstable if the control is inadequate. In this case (12) is not as easy to justify. From a practical point of view it seems likely that we can still find satisfactory ’s. However, for stability analysis, a cleaner solution is to find ’s that depend only on relative angles and internal voltage magnitudes, and not on the other error states. Then, as long as the relative angles stay within assumed limits, the stability analysis can proceed.
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First expand
about
,
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, as
(15) ’s, are the same as the ’s in such a way that in (12), but depend on internal voltage magnitude and relative machine angles. The other ’s, , , depend only on machine angles. Hence, for all , the ’s are finite, for all and can either be finite or infinite. Since , , , for . Define , then
(a)
(16) for
. Recalling that diag
and defining
, we obtain (17)
Then, using the same control and Lyapunov function used earlier, we have
(18) where (b) Fig. 2. System response to Disturbance 1 without control: (a) rotor angles of Machines 6 and 7 with respect to the system synchronous angle, and (b) rotor angles of Machines 6 and 7 with respect to the angle of Machine 10.
Rewriting (18), we have (19) Since whenever
It follows that:
, it follows from (19) that for some . Now, suppose , where
It is obvious that . There are two possibilities: 1) , , then (i.e., ) as , 2) such that but . It is the second possibility that is the source of trouble. However, we have , that is, , or
Consequently, contradicts . So, there is no such that and . Thus, case 1 is the only valid case, or equivalently, as . From the above discussion, we can conclude that under a Type 2 disturbance, the system (11) is locally asymptotically stable and that the region depends on (15). The system is guaranteed to be stabilized under the proposed control, if the initial states are in the region of attraction, or equivalently, the fault (or other contingency) is cleared before some (possibly unknown) maximum critical clearing time. A control can certainly lengthen the maximum clearing time, but obviously there is a limit on the increase, dictated by the amount of control effort available.
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(a)
at s and cleared after six cycles by dropping the line from bus 26 to 25. Fig. 2 shows the angle trajectories of machine 6 and 7 under disturbance 1, the others are similar, with the simulation carried out with no control. Fig. 2(a) uses the system synchronous angle as the angle reference. From this diagram, we can see that the machine angles move away from the prefault angles, and never come back, but the sustained oscillation can not be seen clearly. Fig. 2(b) uses machine 10 as the reference. The sustained oscillation in this case is obvious. Fig. 3 is the simulation of the system under the same disturbance as Fig. 2 with the system synchronous angle as reference, but in this case the proposed control is included with . Fig. 3(a) again shows the angle trajectories of machine 6 and 7. The oscillations are damped in less than 10 s, and the trajectories converge to values very close to the prefault angles. Fig. 3(b) shows the corresponding mechanical power. The flat areas are the result of limiting the maximum mechanical power to 1.05 of the initial mechanical power. To further test the proposed controller, we used Disturbance 2. Fig. 4(a) shows that the system without control is unstable. Fig. 4(b) shows the same case with the proposed control. The system is stable and reaches steady state after about 5 s. The rotor angles shown are for machines 8 and 9, the two machines closest to the fault. The rotor angle deviations of the other machines are smaller and are omitted. Fig. 4(c) shows the changes in mechanical power. The results shown in Fig. 4 indicate that the robust controller is very effective.
V. SUMMARY
(b) Fig. 3. System response to Disturbance 1 with the proposed control: (a) rotor angles of Machines 6 and 7, and (b) mechanical power of Machines 6 and 7.
IV. VALIDATION OF THE CONTROL ALGORITHMS The proposed control algorithms were tested using the loading of the 39 Bus New England System given in [14]. This test system is commonly used for testing stability since the topology of the system and the load data are readily available. A single axis machine model is used for all generators, and every generator has excitation control with the exception of machine 10 which is used as the reference. This machine has an inertia constant ten times greater than that of any other machine in the system and thus is a good candidate for the reference. The mechanical power of all machines is in per unit on a 100 MVA base. To test the control we chose two scenarios. Disturbance 1 is s, which is cleared at a fault at bus 2 at time s by dropping the line from bus 2 to bus 1. This is a six cycle fault, quite long by most standards, but our goal was to produce a sustained oscillation using the loading given in [14]. Disturbance 2 is a three phase fault at bus 26 initiated
AND
CONCLUSION
In this paper we have introduced a robust controller based on a nonlinear model of the power system. We have shown that the control is robust against essentially arbitrary disturbances that cause the system to either exhibit a sustained oscillation or become unstable. The simulation results presented show that the control is very effective and the transient response is quite acceptable. The control is also very feasible since it is simply an augmentation of the control already present on every generator. The steady state at which the control stabilizes the system, will not be, in general, the steady state desired by the system operators. However, once the system has been stabilized, the operators can easily adjust the steady-state operating point. In other words, once the disturbance has been controlled, the system operators can move the system to the exact operating point they desire, just as they routinely move this operating point many times per day. The control is decentralized since it depends solely upon measurements made and fed back at each generator. Further, the control scheme can easily be integrated with the existing controls that already exist on all major generators. An important contribution of the paper is the introduction of bounding functions that allows any generator model to be used in the proof of stability without introducing undue complexity into the Lyapunov function and hence into the proof. As discussed in detail in the body of the paper, the modeling process accounts for the presence of excitation control indi-
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(a)
(b)
(c) Fig. 4. System response to Disturbance 2: (a) rotor angles of all machines without control, (b) rotor angles of Machines 8 and 9 using robust control, and (c) mechanical power of Machines 8 and 9 under robust control.
rectly. That is, we do not require that the internal voltages of the machines remain constant, only that the internal voltages are bounded, which is true in a real power system. In modeling the mechanical side of the system, we have used a secondorder model of the turbine/governor. The real turbine/governor systems are higher order, but the second-order model, in our estimation, captures the essential dynamic behavior as far as the stability analysis is concerned. A procedure similar to that used to account for the generator dynamics could, of course, be used to account for turbine/governor models of higher order. We plan to address this issue in a future paper. The control scheme presented here looks to the needs of the utilities in the next decade during which time the stability of the interconnected grid will most probably continue to decline. At present, most utilities protect themselves on the basis of “worst case” scenarios. This is a workable approach at present, but in the near future, it may be necessary to protect power systems against a much wider class of disturbances. In that case the control scheme presented here may be very attractive,
particularly since it is completely decentralized and can be implemented without exorbitant cost. APPENDIX In this section, we derive a Lyapunov function for the linear part of the system (11), that is the Lyapunov function for (20) , for or equivalently Lyapunov function candidate with diag
We want diag
. Consider the , where
, where . The matrix is positive
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definite if
, depending on the choices of , , , , , , and . The point is to find one set of the these choices such that (21) and (33) hold for all the solutions of (22)–(33). One set of choices is as follows: (21)
The equation equations in the entries of the matrix
yields the following : (22) (23) (24)
(34)
(25) (26) (27)
It is easy to see that for these choices (33) holds for larger than a positive finite number. Under these choices, the are as follows: elements of
and (28) (29) (30) and . where In order to apply Lyapunov’s second method, we must show that the matrix is positive definite for all . To this end, we first determine the conditions under which is stable
Using the Routh–Hurwitz stability criterion, it is easy to verify that is stable if where (31)
(32) In the normal parameter range, , and . Hence, in order for inequalities (31) and (32) to hold, it is sufficient that
(33) is positive definite if the matrix is stable and The matrix the matrix is positive definite. In other words, is positive definite if the inequalities (33) and (21) are satisfied. Once and the gains are determined, can be uniquely determined by solving (22) to (30). There are many possible solutions for
We now check the positive definiteness of matrix . If is large enough, the signs of , , and are determined by the coefficients of the corresponding leading terms in the , , power of . Hence, for a large ,
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and
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consequently, expression of
. Substituting , we have
into the
Clearly, into the matrix
for large . Substituting the above results , we have that, for large ,
Hence, matrix is positive definite, consequently matrix is positive definite. We can conclude that is a Lyapunov function of (20).
Haibo Jiang was born in 1957 in the People’s Republic of China. He received the B.S. degree from Zhejiang University, Hangzhou, China, in 1982, and the M.S. degree from Tianjin University, Tianjin, China, in 1987. He received the Ph.D. degree in electrical engineering at the Georgia Institute of Technology, Atlanta, in 1994. He is currently a Financial Analyst for the Robinson Humphries Company in Atlanta, GA.
Hongzhi Cai received the B.S. and M.S. degrees from Zhejiang University, Hangzhou, China, in 1984 and 1987, respectively. He received the Ph.D. degree from the Department of Electrical and Computer Engineering, University of Central Florida, Orlando, in 1997. He was with the Electrical Power Planning and Engineering Institute, Beijing, China, from 1987 to 1993. Currently he is a Senior Application Engineer with the Foxboro Co., Foxboro, MA. His research interests include power system analysis, control, and automation.
REFERENCES
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-matrices to the stability problems of [1] M. Araki, “Applications of composite dynamical systems,” J. Math. Anal. Applicat., vol. 52, pp. 309–321, 1975. [2] A. R. Bergen, Power Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1986. [3] H. Jiang, J. F. Dorsey, Z. Qu, and J. Bond, “Toward a global decentralized control for large-scale power systems,” in Proc. 32nd IEEE Conf. Decision Contr., San Antonio, TX, Dec. 1993, pp. 3716–3721. [4] H. Jiang, J. F. Dorsey, Z. Qu, and T. Habetler, “Global robust adaptive control of a power system: Primary control on the electrical side,” in Proc. IFAC World Congr., Sydney, Australia, vol. 7, July 1993, pp. 525–528. [5] Q. Lu and Y. Z. Sun, “Nonlinear stabilizing control of multimachine systems,” IEEE Trans. Power Syst., vol. 31, pp. 1159–1163, 1986. [6] Y. Mansour, “Application of eigenanalysis to the western North American power system,” Eigenanalysis and Frequency Domain Methods for Syst. Dynamic Performance, Power Eng. Soc. publication 90TH02923-PWR, 1990. [7] M. A. Pai, Power System Stability. New York: North Holland, 1981. [8] Z. Qu, J. F. Dorsey, J. Bond, and J. McCalley, “Robust transient control of power systems,” IEEE Trans. Circuits Syst., vol. 39, pp. 470–476, June 1992. [9] Z. Qu, “Robust control of a class of nonlinear uncertain system,” IEEE Trans. Automat. Contr., vol. 37, pp. 1437–1442, Sept. 1992. [10] D. D. Siljak and M. Vukcevic, “Decentralized stabilizable linear and bilinear large-scale systems,” Int. J. Contr., vol. 26, pp. 289–305, 1977. [11] D. D. Siljak, Decentralized Control of Complex Systems. New York: Academic, 1991. [12] G. W. Steward, Introduction to Matrix Computations. New York: Academic, 1973. [13] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1978. [14] Western Electric Corp., “Phase II: Frequency domain analysis of lowfrequency oscillations in large electric power system,” Electric Power Res. Inst., Palo Alto, CA, Final Rep., EPRI EL-2348, Apr. 1982.
John F. Dorsey (S’77–M’79–SM’90) received the Ph.D. degree in systems science from Michigan State University, East Lansing, in 1980. Since then, he has been a member of faculty of the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta.
Zhihua Qu (M’90–SM’93) was born in Shanghai, China in 1963. He received the B.S. and M.S. degrees in electrical engineering from the Changsha Railway Institute in 1983 and 1986, respectively. He received the Ph.D. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1990. From 1986 to 1988, he worked as a Faculty Member at the Changsha Railway Institute. Since 1990, he has been with the Department of Electrical and Computer Engineering at the University of Central Florida. Currently, he holds the rank of Associate Professor and is the Assistant Chair of the Department. He has been elected to hold the CAE/LINK Professorship in the College of Engineering from 1997 to 2000. His main research interests are nonlinear control techniques, robotics, and power systems. He has authored or coauthored more than 60 refereed papers in these areas and is the author of the book, Robust Tracking Control of Robotic Manipulators (New York: IEEE Press, 1996).
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Toward a Globally Robust Decentralized Control for Large-Scale Power Systems Haibo Jiang, Hongzhi Cai, John F. Dorsey, Senior Member, IEEE, and Zhihua Qu, Senior Member, IEEE Abstract—A robust control scheme is presented that stabilizes a nonlinear model of a power system to a very large class of disturbances that includes any disturbances causing the system to exhibit sustained oscillation. The disturbance can be anywhere in the power system. The fact that the improvement in stability is significant and system wide leads to the name globally robust control. The control is local or decentralized in the sense that the control of each generator depends only on information available at that generator, and is derived using Lyapunov’s direct method. The derivation is quite general, permitting a second-order representation of the turbine/governor and any generator model. Simulation results are presented which show the effectiveness of the proposed control against instabilities of current importance including sustained oscillations following a major system disturbance such as a fault or major line outage. The control is also effective for steady-state operation. Index Terms—Control, Lyapunov direct method, power system, robust control, stability, steady state.
I. INTRODUCTION
O
VER the last decade, the interconnected power system in the United States has become less stable. This is a trend that will continue as deregulation of the energy industries continues. As a result, the large interconnected subsystems that coordinate their activities have begun to detect sustained oscillations in their simulation analysis. Real oscillations have also been observed. These oscillations usually follow a major disturbance, for instance removal of a fault or following the loss of a major transmission line. One of the more prominent examples of this phenomenon is the 0.7-Hz oscillation that arises in the Western System Coordinating Council (WSCC) following the loss of one of the ac or dc interties between the Pacific Northwest and California [6]. Preventing this sustained oscillation is of great interest since the potential for an oscillation causes lower operational limits to be set on transfer levels on the interties. Oscillations can also arise during “normal” steady-state operation. This is a phenomenon that will only increase in frequency as deregulation proceeds. Thus, there is a need to formulate a control strategy that will work effectively both during transient periods following major interruptions and at steady state. Further, if deregulation proceeds as proposed, Manuscript received September 19, 1995. Recommended by Associate Editor, S. Kumagai. H. Jiang is with the Robinson Humphries Co., Atlanta, GA 30332 USA. H. Cai is with the Foxboro Co., Foxboro, MA 02035 USA. J. F. Dorsey is with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. Z. Qu is with the Department of Electrical Engineering, University of Central Florida, Orlando, FL 32816-0450 USA. Publisher Item Identifier S 1063-6536(97)03275-2.
and the suppliers of power are determined on the basis of a regular “auction,” then the configuration of the interconnected grid will routinely be in a state of change. In this situation it will be increasingly difficult to predict the strength or location of disturbances. Thus, the distinction between transient and steady-state stability will be moot. This paper considers the damping of oscillations because they are forerunners to more complex disturbances that will be an inevitable byproduct of deregulation. The proposed control, however, is very general and is formulated to be effective against the expected control problems of the foreseeable future. We have given the proposed control the name globally robust decentralized control. The term globally robust is one that requires careful explanation. No power system control strategy can claim to be globally stable. There is always a disturbance waiting in the wings that will defeat any reasonable control strategy. On the other hand, control strategies can be distinguished by the class of disturbances against which they are effective. Some control strategies are designed to work against specific disturbances. Indeed, this was the normal procedure before deregulation since in the regulated environment, the topology of the interconnected grid was reasonably well fixed, and certain contingencies (disturbances) could be identified as the most destabilizing. Under deregulation what is required is a control strategy that will counteract a wide variety of disturbances that may occur anywhere is the system. The proposed control is globally robust in this sense. The damping of sustained oscillations has been studied by a number of investigators. However, the approach has been to use a linearized model of the power system as the basis for designing a power system stabilizer (PSS) which augments existing excitation systems to improve the damping of the electric power system during low-frequency oscillations. The basic idea is to add PSS to specific machines to damp oscillations due to a specific disturbance, or set of disturbances, at particular points in the system. As we have already indicated, this is not a strategy that will be effective in the future. The existing literature does not, with a few exceptions, address the problem of finding a globally robust control that will protect the power system against a large class of disturbances, anywhere in the system. Further, since the existing analysis is based almost exclusively on a linearized model of the power system, it is not applicable to cases where the system suffers a major disturbance, such as a fault. In such cases a nonlinear model of the power system is required. What is really needed under deregulation is a control, based on a nonlinear model of the power system that is effective
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against system wide disturbances and can be used for control both during the transient period following a major disturbance and at steady state. There is some work on nonlinear stabilizing control of power systems [5], [8], [9]. In [5], the major drawback is that the control requires global feedback information and exact knowledge about the power system. In our earlier papers [8] and [9], we first began the development of a decentralized robust control that stabilized a power system for disturbances anywhere in the power system. However, in [8] and [9], we used the classical generator model and a first-order model of the turbine/governor. The excitation system was not modeled and consequently not included in the derivation of the control or the proof of stability by Lyapunov’s direct method. Also the effect of the simplified modeling on local stability was not considered. The motivation of this paper is to present a scheme for designing power system control (PSC), which uses a nonlinear model of the generator and a second-order model of the turbine/governor, making it distinctly different from conventional designs of PSS’s. The excitation system is included implicitly. PSC is more general than PSS. It is designed not only to damp out sustained oscillations, but also to stabilize a power system against almost all reasonable disturbances. It cannot, of course, be effective against unforseen catastrophic contingencies, such as earthquakes, where no amount of control can hold the system together. However, even in these extreme cases, the proposed control strategy will have a strong mitigating effect that can help the graceful islanding of the system. As with our previous efforts, the control is based on local linear feedback at each generator, and thus is completely decentralized. We note that currently every major generator in a power system has integral feedback through the turbine governor to maintain frequency, and excitation control fed back through the electrical side to prop up the internal voltage of the generators in the hope of improving stability. What we are proposing is a straightforward modification or augmentation of the existing control. The payoff is high, because we obtain a control that is very robust, in the sense that it stabilizes the power system to a much wider class of disturbances. In establishing PSC, we combine the swing equations with a second-order model of the turbine/governor. In addition, we make the following observations. 1) For all kinds of generator models, the swing equations are essentially the same with the electrical power being a function of the model chosen. 2) For any excitation model, the internal voltage is bounded both by saturation effects within the generator and by stability limitations in the magnitude and angle differences between buses in the power system. Based on the above observations, we can design a PSC which is independent of the internal voltages, and then apply the PSC to any kind of generator model. This makes the PSC more realistic than our earlier results. The control presented in this paper is derived from a proof of stability based on Lyapunov’s direct method. The proposed control is linear and local, avoiding the shortcomings of [5]. A
judicious choice of the Lyapunov function allows us to show that the control is robust to parameter and load variations, and to changes in the power system topology. In the next section we give a detailed formulation of the problem discussed in this paper. In Section III a transient control and corresponding stability result are investigated under different type of disturbances. In this same section we also discuss the general form of electrical power functions under different disturbances. Section IV provides simulation results that validate the control scheme. The derivation of the Lyapunov function is in the Appendix. II. PROBLEM FORMULATION In formulating the power system model for constructing and analyzing a robust control scheme, one would obviously want as detailed a machine model as possible. In particular, for 20-s response times following a major disturbance, or for steady-state analysis, we would like to consider the effects of excitation systems and turbine/governor control. Choosing a more detailed generator model and including the dynamics of the excitation system and turbine/governor leads to extremely complex algebraic expressions that obscure the design of the control. In the next section we will introduce general forms of electrical power functions that depend on the type of disturbance, and which implicitly include all generator and excitation models in the Lyapunov analysis that leads to the control and the proof of stability. With the help of the new electrical power functions, we can avoid the complex algebraic expressions of the generator models, and use only the swing equations and the turbine/governor equations. The electrical dynamics of the generator is included through the electrical power functions. A final observation is the following. Although any kind of turbine/governor model can, at least theoretically, be used in the formulation, the use of a model of order more than two leads to algebraic expressions that will be too complicated to make it easy to find a suitable Lyapunov function. This causes difficulty in proving stability. It should be pointed out that “the difficulty in proof of the stability” does not mean the proposed control will fail for the high-order turbine/governor model. Simulations show the control is effective when used with higher order turbine/governor models. Our feeling is that a second-order turbine/governor model combined with the swing equations captures all the essential dynamics of the mechanical side of the turbine/generator system. We plan to address the issue of higher order turbine/governor models in a paper in the very near future. For the present analysis, we chose the second-order turbine/governor model given in [2]. In this paper we consider a power system of generators, given by with the swing equation of machine , (1) where is the angle of machine relative to the synchronous angle of the system, and , , , , and are the inertia constant, damping coefficient, electrical output power, prefault steady-state mechanical input power, and mechanical power deviation, respectively, of machine .
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Fig. 1. Block diagram of a second-order turbine/governor.
The block diagram of a second-order turbine/governor model is shown in Fig. 1, and yields the following differential equation:
(2) , , , , and are the turbine time where constant, governor time constant, governor gain, frequency regulation, and control signal, respectively, of machine . The frequency feedback included in the proposed control, effectively replaces the standard frequency feedback. Hence, in order to keep the system equations simple, we remove the standard frequency feedback in Fig. 1. Equation (2) then becomes (3) Rewriting (1) and (3) as first-order differential equations yields
(4) and are the frequency deviation and mechanwhere ical power derivative of the machine , respectively. Let the equilibrium points of (4) be characterized by , , , , , and , which are the solutions of the following equations:
The problem now becomes how to design the control such that the equilibrium points of (4) are the steady-state operating points of the power system. Since the system is operating under excitation control and the system parameters are not known exactly, the equilibrium points cannot be found explicitly. In the next section, we design a controller which controls the power system smoothly to a steady-state operating point. If the postfault system configuration remains the same as that of the prefault system, the steady-state operating point will be close, if not identical, to the prefault steady-state operating point. Otherwise, the post fault steady-state operating point will be different, depending on the change in power system topology. We emphasize that the actual steady-state operating point does not need to be known to implement the proposed control. It is introduced as a mathematical convenience to facilitate the proof of robustness. It is worth noting that not knowing the precise steady-state operating point is not a practical limitation. In practice, once the disturbance has been controlled, the system operators can move the system to the exact operating point they desire, just as they routinely move this operating point many times per day. III. A ROBUST CONTROL The control developed in this paper will stabilize the power system to almost any major disturbance anywhere in the power system. As discussed earlier, contingencies can occur that call for more control effort than any reasonable control strategy can supply. However, the proposed control represents a significant improvement in stability over that presently available. One of the system disturbances that we correct in the simulation is a sustained oscillation that arises after the disturbance is removed. However, we emphasize that this is not the only potential postdisturbance system instability that the control can operate against. To that end we also consider a case where the system is unstable without control. The overall control consists of local controllers, one at each machine, with each local controller using only information available locally, that is at the machine. We emphasize that the proposed control represents a straightforward augmentation of the control already in place on most machines. The proposed robust controller for the th machine is given by
(6) or
(7) (5)
, , , and The differences are the errors between the states of the power system
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and the desired steady states. The variable is the transient control input into the th machine. It should be noted that the desired steady states are arbitrary, local references, and that , , , and are local measurements of the machine . The control does not require the knowledge of the states of the other machines. The choices for the desired states do not change the stability of the overall power system, and therefore can be made arbitrarily; but they will affect the actual post fault steady state. Specifically, post fault steady state is the solution of (5) after replacing the variables in (7) by their corresponding ones with superscript and then substituting (7) into (5). To study the stability of power system (4) under control (6), we need to derive the error system. We first define the error states as
and
(10) (8) , and are the steady-state solutions where , , of (4). Combining (4)–(6), we have
The term is the electrical power deviation from the steady-state value of the th bus. Defining , we can write the error system as follows: (11) where
diag diag
Writing the above equations in matrix form yields (9) where
The function is the electrical power deviation at bus . It is a function of the angles and the internal voltages , . is tailored to the generator model chosen. Before proceeding, we first define two different types of disturbances. Type 1 disturbance: After the fault is removed, the system exhibits a sustained oscillation under conventional turbine/governor and excitation control but without the proposed PSC. Type 2 disturbance: After the fault is removed, the system will collapse under conventional control. We define these disturbances in terms of faults, since faults are usually the most pronounced insults the system receives. There are certainly other sources of both types of disturbances. The analysis presented here would go through in the same way for all but the most pathological of contingencies. In the following discussion, we assume that the system suffers a Type 1 Disturbance. We will discuss the case of a Type 2 Disturbance later. We note here that for a Type 2 disturbance the best that the proposed PSC can do is increase the fault clearing time significantly. If the fault duration is long enough,
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no control can keep the machine angles from separating too far with a consequent loss of synchronism. In a power system under sustained oscillation, the following facts are worth noting. 1) is a combination of sinusoidal functions of . 2) The internal voltage is always bounded. Its exact value, however, depends on the generator model chosen. Since there are a variety of generator models employed in any power system simulation, it is an obvious advantage to treat them all in terms of some bounding function. The potential drawback to this approach would seem either to be a degradation of control performance or equivalently or to introduce some conservativeness into control design. However, as the simulation results show, this is not a problem. In addition, gains for better performance may be generated automatically by combining the proposed robust control design with an auto-tuning algorithm [4]. 3) When for all , then by definition we have , . In the other words, if , for all , then , . Based on the above development, we can conclude that 1) is always bounded and 2) will converge to zero as converges to zero for all . Thus, we can write as (12)
where
and
are vectors consisting of and , , respectively. Recall that . are some scalar functions of both and . Coefficients Expression (12) is an equality which is rewritten by a form that reveals the second property of . Therefore, functions may take positive or negative values. By the first property of , we know that there exist constants such that, for all
Constant can be viewed as Lipschitzian constants of or their bounds. In the preliminary version of this paper [3], explicit expressions of and are found for generators described by two-axis models. Similar derivations can be done for any other generator models. We are now in a position to state the main result of this paper. It should be noted that the theorem presented here is related to many existing results on decentralized control of composite systems. For example, it has been shown in [1], [10], and [11] using state transformation and M-matrix approaches that linear interconnected systems in the form of (9) are globally stabilizable. The theorem given here extends these ideas to power systems whose interconnections are nonlinear. That this extension is possible is not unexpected since nonlinearities in terms of sinusoidal functions can be bounded in norm by the state. However, the approach taken in this paper differs from existing results in [1], [10], and [11] because it provides explicit choices for the sub-Lyapunov
functions. Through direct computation of time derivatives, these choices of Lyapunov functions provide explicit solutions for the control gains, which is a major advantage in control design. As will be shown later, this approach allows one to consider easily any nonlinear couplings. Theorem 1: The error system (11) is guaranteed to be asymptotically stable with an exponential rate of convergence if the local feedback gains , , , and , are chosen such that, for all
(13) and
(14) , , , and are the entries of matrix , which where is defined in the Appendix. Proof: The proof uses Lyapunov’s direct method. Choose the Lyapunov function candidate to be , which is defined in the Appendenote the Euclidean norm or its induced matrix dix. Let is symmetric norm. Then, since
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diag
and the nonzero elements of
are defined by
for all . By the Gershgorin theorem in [12], we know that the matrix is positive definite if we choose the gains such that the inequality (14) holds for all . Thus, we have , and
where for all
. Let and that
. It follows that
which shows that decreases to zero exponentially. Therefore, system (11) is exponentially stable in the large under control (6) with finite local feedback gains. Several remarks are worth making at this point. 1) When is large enough, we have
Hence, (14) holds for large . 2) The control guarantees that the system approaches the postdisturbance steady-state at an exponential rate, even though the steady-state is not explicitly known. 3) Although the lower bounds of the gains that guarantee the stability are given by (14), it is not necessary to know the parameters , , and , only the range of values of these parameters. In the case that the time constants in (10) are unknown as well, gains cannot be calculated through ( ) but can simply be made large enough to satisfy the polynomial orders dictated by (13). 4) The lower bounds given by (14) provide only a very conservative estimate for the magnitude of the gains. It is unavoidable that this inequality is not a necessary condition, since Lyapunov’s second method always yields very conservative results. The magnitude of the gains really needed to control the power system can and should be determined by computer simulation or by an autotuning algorithm [4]. We have run a large number of simulations at various gain levels and the results of those simulations can be briefly summarized as follows. The damping improves as the individual gains are increased from very small values to values in the rage of 50 to 100. Further increases in the gains does not perceptively change the damping or increase a measure of stability such as fault clearing time. One can conclude from this that in all probability the region of attraction for gains in the range 50 to 100 is only marginally different from the region of attraction for the conservative gains obtained from the proof of stability. The analysis is, of course, less than completely satisfying. However, it is better than the following alternative: 1) reduce the model in size (number of generators) by some aggregation technique; 2) linearize the model about an operating point; 3) design a control based on the linearized model; and 4) apply the control to the actual, nonlinear power system. This latter approach is the one that has dominated power system control for many years, and still does. However, as deregulation proceeds this latter approach becomes less and less appealing. First, with the topology of the grid in constant flux as various IPP’s bid for the power generation, it becomes very difficult to do the initial model reduction that is always done. Second, the steadystate operating point is not going to be as predictable as it has been in the past. We now consider the second case where the system is subject to a Type 2 disturbance where the system can become unstable if the control is inadequate. In this case (12) is not as easy to justify. From a practical point of view it seems likely that we can still find satisfactory ’s. However, for stability analysis, a cleaner solution is to find ’s that depend only on relative angles and internal voltage magnitudes, and not on the other error states. Then, as long as the relative angles stay within assumed limits, the stability analysis can proceed.
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First expand
about
,
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, as
(15) ’s, are the same as the ’s in such a way that in (12), but depend on internal voltage magnitude and relative machine angles. The other ’s, , , depend only on machine angles. Hence, for all , the ’s are finite, for all and can either be finite or infinite. Since , , , for . Define , then
(a)
(16) for
. Recalling that diag
and defining
, we obtain (17)
Then, using the same control and Lyapunov function used earlier, we have
(18) where (b) Fig. 2. System response to Disturbance 1 without control: (a) rotor angles of Machines 6 and 7 with respect to the system synchronous angle, and (b) rotor angles of Machines 6 and 7 with respect to the angle of Machine 10.
Rewriting (18), we have (19) Since whenever
It follows that:
, it follows from (19) that for some . Now, suppose , where
It is obvious that . There are two possibilities: 1) , , then (i.e., ) as , 2) such that but . It is the second possibility that is the source of trouble. However, we have , that is, , or
Consequently, contradicts . So, there is no such that and . Thus, case 1 is the only valid case, or equivalently, as . From the above discussion, we can conclude that under a Type 2 disturbance, the system (11) is locally asymptotically stable and that the region depends on (15). The system is guaranteed to be stabilized under the proposed control, if the initial states are in the region of attraction, or equivalently, the fault (or other contingency) is cleared before some (possibly unknown) maximum critical clearing time. A control can certainly lengthen the maximum clearing time, but obviously there is a limit on the increase, dictated by the amount of control effort available.
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(a)
at s and cleared after six cycles by dropping the line from bus 26 to 25. Fig. 2 shows the angle trajectories of machine 6 and 7 under disturbance 1, the others are similar, with the simulation carried out with no control. Fig. 2(a) uses the system synchronous angle as the angle reference. From this diagram, we can see that the machine angles move away from the prefault angles, and never come back, but the sustained oscillation can not be seen clearly. Fig. 2(b) uses machine 10 as the reference. The sustained oscillation in this case is obvious. Fig. 3 is the simulation of the system under the same disturbance as Fig. 2 with the system synchronous angle as reference, but in this case the proposed control is included with . Fig. 3(a) again shows the angle trajectories of machine 6 and 7. The oscillations are damped in less than 10 s, and the trajectories converge to values very close to the prefault angles. Fig. 3(b) shows the corresponding mechanical power. The flat areas are the result of limiting the maximum mechanical power to 1.05 of the initial mechanical power. To further test the proposed controller, we used Disturbance 2. Fig. 4(a) shows that the system without control is unstable. Fig. 4(b) shows the same case with the proposed control. The system is stable and reaches steady state after about 5 s. The rotor angles shown are for machines 8 and 9, the two machines closest to the fault. The rotor angle deviations of the other machines are smaller and are omitted. Fig. 4(c) shows the changes in mechanical power. The results shown in Fig. 4 indicate that the robust controller is very effective.
V. SUMMARY
(b) Fig. 3. System response to Disturbance 1 with the proposed control: (a) rotor angles of Machines 6 and 7, and (b) mechanical power of Machines 6 and 7.
IV. VALIDATION OF THE CONTROL ALGORITHMS The proposed control algorithms were tested using the loading of the 39 Bus New England System given in [14]. This test system is commonly used for testing stability since the topology of the system and the load data are readily available. A single axis machine model is used for all generators, and every generator has excitation control with the exception of machine 10 which is used as the reference. This machine has an inertia constant ten times greater than that of any other machine in the system and thus is a good candidate for the reference. The mechanical power of all machines is in per unit on a 100 MVA base. To test the control we chose two scenarios. Disturbance 1 is s, which is cleared at a fault at bus 2 at time s by dropping the line from bus 2 to bus 1. This is a six cycle fault, quite long by most standards, but our goal was to produce a sustained oscillation using the loading given in [14]. Disturbance 2 is a three phase fault at bus 26 initiated
AND
CONCLUSION
In this paper we have introduced a robust controller based on a nonlinear model of the power system. We have shown that the control is robust against essentially arbitrary disturbances that cause the system to either exhibit a sustained oscillation or become unstable. The simulation results presented show that the control is very effective and the transient response is quite acceptable. The control is also very feasible since it is simply an augmentation of the control already present on every generator. The steady state at which the control stabilizes the system, will not be, in general, the steady state desired by the system operators. However, once the system has been stabilized, the operators can easily adjust the steady-state operating point. In other words, once the disturbance has been controlled, the system operators can move the system to the exact operating point they desire, just as they routinely move this operating point many times per day. The control is decentralized since it depends solely upon measurements made and fed back at each generator. Further, the control scheme can easily be integrated with the existing controls that already exist on all major generators. An important contribution of the paper is the introduction of bounding functions that allows any generator model to be used in the proof of stability without introducing undue complexity into the Lyapunov function and hence into the proof. As discussed in detail in the body of the paper, the modeling process accounts for the presence of excitation control indi-
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(a)
(b)
(c) Fig. 4. System response to Disturbance 2: (a) rotor angles of all machines without control, (b) rotor angles of Machines 8 and 9 using robust control, and (c) mechanical power of Machines 8 and 9 under robust control.
rectly. That is, we do not require that the internal voltages of the machines remain constant, only that the internal voltages are bounded, which is true in a real power system. In modeling the mechanical side of the system, we have used a secondorder model of the turbine/governor. The real turbine/governor systems are higher order, but the second-order model, in our estimation, captures the essential dynamic behavior as far as the stability analysis is concerned. A procedure similar to that used to account for the generator dynamics could, of course, be used to account for turbine/governor models of higher order. We plan to address this issue in a future paper. The control scheme presented here looks to the needs of the utilities in the next decade during which time the stability of the interconnected grid will most probably continue to decline. At present, most utilities protect themselves on the basis of “worst case” scenarios. This is a workable approach at present, but in the near future, it may be necessary to protect power systems against a much wider class of disturbances. In that case the control scheme presented here may be very attractive,
particularly since it is completely decentralized and can be implemented without exorbitant cost. APPENDIX In this section, we derive a Lyapunov function for the linear part of the system (11), that is the Lyapunov function for (20) , for or equivalently Lyapunov function candidate with diag
We want diag
. Consider the , where
, where . The matrix is positive
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definite if
, depending on the choices of , , , , , , and . The point is to find one set of the these choices such that (21) and (33) hold for all the solutions of (22)–(33). One set of choices is as follows: (21)
The equation equations in the entries of the matrix
yields the following : (22) (23) (24)
(34)
(25) (26) (27)
It is easy to see that for these choices (33) holds for larger than a positive finite number. Under these choices, the are as follows: elements of
and (28) (29) (30) and . where In order to apply Lyapunov’s second method, we must show that the matrix is positive definite for all . To this end, we first determine the conditions under which is stable
Using the Routh–Hurwitz stability criterion, it is easy to verify that is stable if where (31)
(32) In the normal parameter range, , and . Hence, in order for inequalities (31) and (32) to hold, it is sufficient that
(33) is positive definite if the matrix is stable and The matrix the matrix is positive definite. In other words, is positive definite if the inequalities (33) and (21) are satisfied. Once and the gains are determined, can be uniquely determined by solving (22) to (30). There are many possible solutions for
We now check the positive definiteness of matrix . If is large enough, the signs of , , and are determined by the coefficients of the corresponding leading terms in the , , power of . Hence, for a large ,
JIANG et al.: GLOBALLY ROBUST DECENTRALIZED CONTROL
and
319
consequently, expression of
. Substituting , we have
into the
Clearly, into the matrix
for large . Substituting the above results , we have that, for large ,
Hence, matrix is positive definite, consequently matrix is positive definite. We can conclude that is a Lyapunov function of (20).
Haibo Jiang was born in 1957 in the People’s Republic of China. He received the B.S. degree from Zhejiang University, Hangzhou, China, in 1982, and the M.S. degree from Tianjin University, Tianjin, China, in 1987. He received the Ph.D. degree in electrical engineering at the Georgia Institute of Technology, Atlanta, in 1994. He is currently a Financial Analyst for the Robinson Humphries Company in Atlanta, GA.
Hongzhi Cai received the B.S. and M.S. degrees from Zhejiang University, Hangzhou, China, in 1984 and 1987, respectively. He received the Ph.D. degree from the Department of Electrical and Computer Engineering, University of Central Florida, Orlando, in 1997. He was with the Electrical Power Planning and Engineering Institute, Beijing, China, from 1987 to 1993. Currently he is a Senior Application Engineer with the Foxboro Co., Foxboro, MA. His research interests include power system analysis, control, and automation.
REFERENCES
M
-matrices to the stability problems of [1] M. Araki, “Applications of composite dynamical systems,” J. Math. Anal. Applicat., vol. 52, pp. 309–321, 1975. [2] A. R. Bergen, Power Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1986. [3] H. Jiang, J. F. Dorsey, Z. Qu, and J. Bond, “Toward a global decentralized control for large-scale power systems,” in Proc. 32nd IEEE Conf. Decision Contr., San Antonio, TX, Dec. 1993, pp. 3716–3721. [4] H. Jiang, J. F. Dorsey, Z. Qu, and T. Habetler, “Global robust adaptive control of a power system: Primary control on the electrical side,” in Proc. IFAC World Congr., Sydney, Australia, vol. 7, July 1993, pp. 525–528. [5] Q. Lu and Y. Z. Sun, “Nonlinear stabilizing control of multimachine systems,” IEEE Trans. Power Syst., vol. 31, pp. 1159–1163, 1986. [6] Y. Mansour, “Application of eigenanalysis to the western North American power system,” Eigenanalysis and Frequency Domain Methods for Syst. Dynamic Performance, Power Eng. Soc. publication 90TH02923-PWR, 1990. [7] M. A. Pai, Power System Stability. New York: North Holland, 1981. [8] Z. Qu, J. F. Dorsey, J. Bond, and J. McCalley, “Robust transient control of power systems,” IEEE Trans. Circuits Syst., vol. 39, pp. 470–476, June 1992. [9] Z. Qu, “Robust control of a class of nonlinear uncertain system,” IEEE Trans. Automat. Contr., vol. 37, pp. 1437–1442, Sept. 1992. [10] D. D. Siljak and M. Vukcevic, “Decentralized stabilizable linear and bilinear large-scale systems,” Int. J. Contr., vol. 26, pp. 289–305, 1977. [11] D. D. Siljak, Decentralized Control of Complex Systems. New York: Academic, 1991. [12] G. W. Steward, Introduction to Matrix Computations. New York: Academic, 1973. [13] M. Vidyasagar, Nonlinear Systems Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1978. [14] Western Electric Corp., “Phase II: Frequency domain analysis of lowfrequency oscillations in large electric power system,” Electric Power Res. Inst., Palo Alto, CA, Final Rep., EPRI EL-2348, Apr. 1982.
John F. Dorsey (S’77–M’79–SM’90) received the Ph.D. degree in systems science from Michigan State University, East Lansing, in 1980. Since then, he has been a member of faculty of the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta.
Zhihua Qu (M’90–SM’93) was born in Shanghai, China in 1963. He received the B.S. and M.S. degrees in electrical engineering from the Changsha Railway Institute in 1983 and 1986, respectively. He received the Ph.D. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1990. From 1986 to 1988, he worked as a Faculty Member at the Changsha Railway Institute. Since 1990, he has been with the Department of Electrical and Computer Engineering at the University of Central Florida. Currently, he holds the rank of Associate Professor and is the Assistant Chair of the Department. He has been elected to hold the CAE/LINK Professorship in the College of Engineering from 1997 to 2000. His main research interests are nonlinear control techniques, robotics, and power systems. He has authored or coauthored more than 60 refereed papers in these areas and is the author of the book, Robust Tracking Control of Robotic Manipulators (New York: IEEE Press, 1996).
