Transient stability improvement through excitation control

Transcription

1 Transient stability improvement through excitation control Alexandre S. Bazanella, Cauê L. da Conceição Department of Electrical Engineering Universidade Federal do Rio Grande do Sul Porto Alegre RS BRAZIL Abstract In this paper we present, discuss, extend and compare different design methods for excitation control aiming at improving transient and steadystate stability. 1 Introduction Transient stability is essentially a nonlinear phenomenon. As such, nonlinear methods must be applied to its analysis and to control design aiming at its improvement. The application of nonlinear control methods to enhance transient stability has been given much attention in the literature since the late eighties. Both turbine control and the more challenging and widely applicable excitation control have been explored. These excitation controllers are proposed to replace the traditional Automatic Voltage Regulator and Power System Stabilizer (AVR+PSS) control structure, and questions about the benefits of this replacement have not yet been answered. On the other hand, with new policies for provision of ancillary services and the increasing number of small sized distributed generators competing for voltage regulation, it is not clear which role AVR s are to play in the future. Indeed, many small distributed generating units are currently operating without AVR s and this trend might spread to larger independent producers in the future. Feedback linearization was one of the early strategies to be explored for nonlinear excitation control design, with applications proposed to both single and multimachine systems, even with output feedback and state observers (see This work has been supported by CAPES and CEEE - Cia. Estadual de Energia Eltrica. 1

2 [9, 6, 14, 19] among others). However, robustness issues against parameter uncertainties and unmodelled dynamics have not been solved, neither stability analysis in the output feedback case. The potential fragility of these methods, which are based on nonlinearity cancellations, has motivated recent works on energy related design techniques, usually developed for the single machine infinite bus (SMIB) third-order flux-decaying model and its associated energy like Lyapunov function [16]. Most results are based on the application of L g V controllers to replace the AVR and PSS [12, 8, 4]. More recently, application of the more general interconnection and damping assignment (IDA) design methodology [15] has also been proposed [7]. In this paper we summarize the main concepts associated to the improvement of transient stability through control, review the above mentioned design methods and illustrate their performance in a benchmark. Furthermore, we tackle one issue that is not usually taken into account: the uncertainty of the system s operating point. All these controllers rely on the knowledge of the equilibrium point to be stabilized for their implementation, which is not a reasonable assumption for the problem at hand. We present a simple extension for these controllers which does not require knowledge of the equilibrium, preserves most properties of the original controllers and is shown to result in similar performance for the closed-loop system in the benchmark. This extension consists in a linear filter conceived from an adaptive control concept as first presented in [3, 5]. The paper is organized as follows. In Section 2 the SMIB model and its properties are presented and briefly discussed under the light of Lyapunov theory. The main theoretical concepts of transient stability relevant to control design are summarized in Section 3. These concepts are illustrated in the benchmark. The Lyapunov-based controllers are presented in Section 4, showing their main theoretical properties and their performance in improving the benchmark s transient stability. In Section 5 the controller s extensions are presented, with similar results. A summary of the results and some concluding remarks are given in Section 6. All simulations in this paper were performed using the power systems toolbox in MATLAB. All the simulations use the complete sixth-order model [10, 17] for the synchronous machine. 2 Model and Lyapunov Function Excitation control design is usually performed based on a single machine infinite bus (SMIB) model, in which only the dynamics of the generator being controlled is kept, all the remaining of the system being modelled as the Thèvenin equivalent of an active network composed of inductances (the lines and loads) and ideal voltage sources (the remaining generators). Controllers designed in this way have been successfully applied to real (multimachine) systems for decades. Recently the use of multimachine models for control design has been proposed both for linear control and for nonlinear control methods [12, 6]. Although the 2

3 use of a more complete model is certainly desirable and promising results have been reported, the present state of the theory has some drawbacks in dealing with such models. Nonlinear analysis and control design methods presented to date can only handle multimachine models which neglect the transfer conductances, and thus the effect of loads in the system, which raises doubts on what is really gained with this more complex modelling. Some nice properties and a great amount of the physical intuition present in the SMIB model are lost, or at least are not visible. This usually leads to black-box control design, and control laws that are either extremely complex or depend on distant measures or both. On the other hand, the use of the SMIB as a design model allows to derive design choices and parameter tuning procedures that can be expected to carry on to the multimachine case. The synchronous machine is modelled by the flux-decay E q model [16, 2, 10] for the purpose of analysis. Most stability studies performed to date are based on this model or - in the case of transient stability - on the so-called classical model, which is simpler. Model-based control design has also been mainly performed with this model. Then the model for the SMIB can be written as follows δ = ω 2H w w 0 = P m P e D ω T d0ė q = (X d X d ) (X d + X E) E B cos δ X d + X E (X d + X E) E q + E 0 fd + T d0u 1 with P e = (X d +X E) E BE q sin δ. The state variables of this system are the load angle (δ), the shaft speed deviation from the synchronous speed ( w), and the quadrature axis internal voltage (E q ); the remaining notation is given in the Appendix. The mechanical power P m is assumed to be constant and the field voltage has been decomposed into a constant term Efd 0 and a supplementary signal T d0 u, with u representing the control input. To simplify the notation we find convenient to rewrite the SMIB model in the compact form ẋ 1 = x 2 ẋ 2 = b 1 x 3 sin x 1 Dx 2 + P (1) ẋ 3 = b 3 cos x 1 b 4 x 3 + E + u. The coefficients b i, i = 1,..4 are positive (appendix A), and the inputs P and E, which represent the mechanical power and field voltage 1, respectively, are held constant. Let us study the uncontrolled (u = 0) behavior of the SMIB system. The equilibria of the uncontrolled SMIB system (1) are given by x = [ x 1, 0, x 3 ], 1 The correspondence between the new b parameters and the traditional parameters is obvious. Notice only that P and E are not the pu values of the mechanical power and field voltage, but are proportional to these values. 3

4 where x 1 and x 3 are the solutions of E = b 4 x 3 b 3 cos x 1, P = b 1 x 3 sin x 1. It is easy to see that a necessary condition for existence of equilibria is E > b4p b 1 b 3. If this inequality holds, there exists an asymptotically stable equilibrium, which we denote with the subindex ( ), i.e., x = [x 1, 0, x 3 ], to underscore the fact that it is the operating point. Besides x, the open loop system has an additional equilibrium (that we denote x u ) which may be shown to be unstable [16, 13]. The equilibrium x of the uncontrolled SMIB model (1) is locally stable with an energy-like Lyapunov function [16] H(x) = 1 2 x2 2 + b 1x 3(cos x 1 cos x 1) P x 1 + b1b4 2b 3 x 2 3 (2) where we have defined x i = xi x i, i = 1, 3. Indeed, H(x) has an isolated local minimum at x, hence it is a positive definite function in some neighborhood of x, and its time derivative is Ḣ(x) = { b 1 b 3 [b 3 (cos x 1 cos x 1 ) b 4 x 3 ] 2 + Dx 2 2} 0 (3) Then asymptotic stability of x is established invoking La Salle s invariance principle. The best estimate of the domain of attraction is the connected part of the level set x : H(x) < H(x u ) and the best estimate for the critical clearing time is given by the time that the faulted system s trajectory takes to reach the boundary of this region. The worst-case system damping can also be estimated through the Lyapunov function H(x). If we define γ = min Ḣ(x) H(x), then for any initial condition x(0): x(t) x(0) e γ t Thus the system damping can be enhanced by adding a negative definite term to the Lyapunov derivative. 3 Transient stability 3.1 General concepts The problem of transient stability is related to the system s trajectory leaving or not the domain of attraction of the stable post-fault equilibrium during a fault. A power system can be said to be transient stable if no admissible fault is capable of driving the systems trajectory outside the domain of attraction of the equilibrium. Hence, the larger the domain of attraction and the shorter the systems trajectory during any admissible fault, the more stable the system is. Any fault in the system must be cleared for the system to return to a normal operating state. The time a fault takes to be cleared is called the clearing time. 4

5 Figure 1: Estimate of the region of attraction for the benchmark; the stable and unstable equilibria are also marked. The larger the clearing time, the farther away from the equilibrium the system s trajectory will go. For a given fault, the maximum clearing time before the system s trajectory leaves the domain of attraction of the operating point is called the critical clearing time. Critical clearing times for the most relevant faults are important measures of power systems stability margins. To illustrate the transient stability concepts and the controller s performance, we take a benchmark from [10]. Data for the system are given in the Appendix. Figure 1 shows the estimate of the region of attraction of the operating point obtained with the Lyapunov function (2), whereas Figure 2 shows its projection onto the plane x 1 x 3. The behavior of the system after a shortcircuit with t cl = 80 ms is presented in Figure 3. By simulating the fault with different clearing times, it is possible to determine the exact critical clearing time for this fault. In this case it is determined to be t cr = 124 ms. Recall that the simulations are performed with a sixthorder model for the synchronous machine. An estimate for the critical clearing 5

6 c = x3 (V) x1 (rad) Figure 2: Projection of the estimate of the region of attraction onto the plane x 1 x 3 for the benchmark. 6

7 105 Base Delta (Degrees) Time (s) Figure 3: Load angle oscillations after a short-circuit at the machine s terminal with t cl = 80 ms. 7

8 time can be obtained with the aid of the Lyapunov function (2) by simulating the faulted system until it reaches the boundary of the estimate of the region of attraction. In so doing for the model (1), an estimate of ˆt cr = 83 ms is obtained, which is 33% smaller than the real one. The controllers should enhance these critical clearing times. One can think of two conceptually different forms of doing this: i) shortening the trajectory of the system during the fault, and ii) enlarging the domain of attraction. 3.2 Shortening the faulted trajectory During a severe fault the voltages in the system drop, and so does the electrical power provided by the synchronous generator to the network. Since the mechanical power does not vary substantially in such a short time, the generators tend to accelerate, driving the system state away from the operating point. It is possible do reduce this acceleration, and thus shorten the systems trajectory during the fault, by somehow increasing the electrical power delivered during the fault. There are currently applied power systems controllers that perform this task, the AVR being the most important in this regard. Modern AVR s raise very rapidly the generators field voltage to its maximum available value (also called the ceiling voltage ), which helps increasing the terminal voltage and thus the power delivered by the machine. This can reduce considerably the system s trajectory during the fault. The appearance of fast, high gain, high ceiling AVR s during the 1950 s has been regarded as an important means of improving the transient stability, which was the main stability problem found by power systems engineers until then. This contribution has been described and celebrated in numerous papers since, and is well known by power systems engineers that fast high gain and high ceiling voltage regulators improve transient stability. Yet, the theoretical reasons for this fact (which are given above) are rarely seen in the literature. 3.3 Enlarging the domain of attraction Enlargement of the domain of attraction can also be obtained by different control devices. Turbine control would appear to be the most natural and easier way to control a generator, as the turbine acts directly on the mechanical power provided to the generator [11]. However, most turbines are unable to yield the fast torque response required to act in such small time frames as the ones involved in transient stability. Network control through FACTS (Flexible AC Transmission Systems) devices constitute a new and very promising alternative. FACTS devices seem to provide strong control authority to improve both transient and steady-state stability. However, the availability of these devices is still limited, and their installation can not be economically justified only on the basis of improving stability, a situation which will probably not be changed in a short horizon. Hence, excitation control remains the main input through which one can improve power systems short-term (steady-state and transient) stability. 8

9 4 Passivity based controllers Early nonlinear excitation controllers proposed were based on feedback linearization [14, 6, 9]. These were very important results to establish limits and possibilities for nonlinear control in power systems, but direct practical application of these design methods by themselves seems unlikely. The cancellation of all systems nonlinear behavior usually results in complex control laws and high control effort. Besides that, this control might be wasted cancelling beneficial nonlinearities. Because of these features, such controllers tend to suffer from lack of robustness against parameter uncertainties and do not lend themselves easily to a physical interpretation of their action on the system. The application of these methods along with linear robust control design tools has been proposed to cope with these issues [18]. On the other hand, Lyapunov and passivity analysis provide, besides guarantees of stability, insight into the system s behavior. It is possible to partially assign the system s behavior and design choices can be made in order to obtain that. This has motivated the recent proposals of passivity based approaches to the design of excitation control. Some papers have presented L g V controllers [12, 4, 5], with results similar to each other. More general controllers can be designed by means of the interconnection and damping assignment (IDA) procedure [15]. This methodology, of which L g V controllers can be regarded as a particular case, has been applied to the design of excitation controllers in [7]. In the sequel we review two of the most up-to-date results and apply these controllers to a benchmark in order to illustrate their properties and compare their performance. Although L g V controllers can be regarded as a particular case of the IDA controllers, they are a very important class with strong properties - that do not generalize to all IDA controllers - and whose effect on the system performance can be better appreciated when applied by itself. 4.1 L g V controller If V (x) is a Lyapunov function for the open-loop system, then a control law in the form u = k(l g V (x)) T, k > 0 is called an L g V controller and V(x) is called an L g V control Lyapunov function. The closed-loop system with this control is described by ẋ = f(x) kg(x)(l g V (x)) T. (4) The L g V control can also be interpreted as a unit gain negative output feedback imposed on the passive system defined choosing for and the output map ẋ = f(x) + g(x)u (5) y = k(l g V (x)) T. (6) Then stability of the closed-loop system can be seen as a consequence of the passivity of the plant (5), (6) and the strictly positive real (SPR) property of the unit gain feedback. 9

10 A well-know property of L g V controllers is that they guarantee infinite gain margin. Also, an L g V controller does not shift the position of the equilibrium, since V (x) x x=x = 0 (7) which implies that the control vanishes at x. Furthermore, as shown in [5], L g V controllers enlarge the estimate of the region of attraction of the stable equilibrium obtained with the same Lyapunov function. Estimates of the region of attraction - and hence of the critical clearing times - are usually conservative, like the one presented above for the benchmark. This implies that enlarging the estimate of the region of attraction does not necessarily mean that the actual region of attraction is enlarged. In order to design - by any method - controllers that guarantee the enlargement of the domain of attraction, it will be important to develop less conservative estimates. The critical clearing time obtained with the L g V controller in our benchmark is t cr = 118 ms, a bit smaller than in open-loop. On the other hand, the system damping is strongly enhanced, as can be seen in Figure Interconnection and damping assignment controller An excitation controller based on the IDA procedure [15] has been presented in [7] and is reviewed in the following. To apply the IDA technique we rewrite the equations of the SMIB system (1) in the so-called port controlled Hamiltonian form: ẋ = (J R) H x + gu where the damping matrix (R = R 0), interconnection matrix (J = J ), and input matrix (g) matrices are given by R = 0 D b 3 b 1, J = , g = With the system described in this form, the Lyapunov derivative can be written as Ḣ(x) = H T (x)r(x) H (x) 0 (8) x x It is seen that the matrix R determines the system damping and that J determines the interconnections among the state variables, which justifies their names. Stabilization in IDA control is achieved assigning the closed loop dynamics ẋ = (J d R d ) H d (x) (9) x where H d (x) is the desired closed-loop energy function, which has a minimum at x, J d = J d and R d = R d 0 are some desired interconnection and damping matrices, respectively. 10

11 110 k= delta (Degrees) Time (s) Figure 4: Load angle oscillations with u = kl g V, k = 5, after a short-circuit at the machine s terminal with t cl = 80 ms. 11

12 The choice of the desired matrices J d and R d is based on physical considerations. Given the form of R, all the open-loop damping is in the third state variable. Injecting additional damping into the electrical variable (x 3 ) is easily achieved feeding back H x 3, which is exactly the action of an L g V controller. On the other hand, it can be seen that the damping in the mechanical coordinates (x 1, x 2 ) is weak, since D is usually very small. Furthermore, since the interconnection matrix J does not contain any coupling between the electrical and the mechanical dynamics, the propagation of the damping injected in the mechanical coordinate is far from obvious. This suggests the new interconnection and damping matrices J d = 1 0 α α , R d = 0 D 0 (10) k v where k v 0 and α 1 are constants to be defined. Then we have the following result, reproduced from [7]. Proposition 1 Consider the SMIB system (1) in closed-loop with the static state feedback controller u = k v b 1 (cos x 1 cos x 1 ) α 1 α 2 ( b3 b 1 + k v ) x 1 α 1 x 2 ( b (11) 3 b 1 α 2 b 4 + k v α 2 ) x 3 with the tuning parameters k v 0, α 2 > 0 and α 1 < 0 verifying α 2 b 1b 4 b 3, α 1 < b 1 α 2 (12) Then, the closed loop system takes the form (9), (10) and (i) x is the unique equilibrium such that x 1e π 2. (ii) x is asymptotically stable with Lyapunov function H d (x) = H(x) + H a (x), where H a = b 1 α 1 [ x 1 cos x 1 sin x 1 + b4α1 2b 3 ( x α 1 x 1 x 3 + x 2 1 )] (α 2 b 1b 4 b 3 )(α 1 x 1 + x 3 ) 2 (13) and H is given by (2). (iii) Estimates of its domain of attraction are the sublevel sets Ω d c = {x H d (x) c}. The tuning parameters have a clear physical significance: k v represents the additional damping injected into the electrical subsystem and α 1 determines the degree of interconnection, that is, how much of this damping is transferred to the mechanical variables. Also, notice that with k v = 0 the controller is linear. Moreover, taking α 2 = b 1b 4 b 3 b 3 b 1 yields the partial state feedback control u = α 1 (b 4 x 1 + x 2 ), with α 1 < b 3 b 4. To better illustrate the effect of the interconnection assignment alone, we set k v = 0, thus eliminating the damping assignment (the L g V part of the IDA 12

13 105 a1= a2= Delta (Degrees) Time (s) Figure 5: Load angle oscillations with the IDA controller, α 1 = 0.654, α 2 = , after a short-circuit at the machine s terminal with t cl = 80 ms. controller). Application of this design procedure to the benchmark requires, according to the result above, that α , α 1 < Different design choices have been made, with k v = 0, all of them resulting in increased critical clearing times. Better performance has been obtained in this case for values not very far from the above limits, so we present the results obtained for α 2 = , α 1 = These design choices result in t cr = 182 ms and the dynamic behavior presented in Figure 5. Notice that the critical clearing time has been dramatically increased (in 47%), but not the dynamic performance. An estimate of the region of attraction is shown in Figure 6, where it can be seen that it is indeed much larger than the open-loop estimate. This is the opposite to what happened with the L g V controller, which dramatically improves damping and reduces the critical clearing times. Similar results have been obtained for other examples. 13

14 x x Figure 6: Projection onto the x 1 x 3 plane of the estimate of the region of attraction the IDA controller (dashed line). The open-loop estimate is (dotted line) is also shown for comparison. 14

15 5 Dynamic extensions 5.1 An equilibrium estimation mechanism A practical problem in implementing excitation controllers in power systems is that assuming knowledge of the equilibrium to be stabilized is a dangerous assumption, for several reasons. First, the design model is of much lower order than the actual system and the unmodelled dynamics, although not altering significantly the dynamic behavior or steady-state input-output relationships, may have a strong effect on the equilibrium values of the state variables in the reduced model. On the other hand, even if the unmodelled dynamics were not so important and the model parameters are assumed to be known to a good precision, calculation of the equilibrium point based on these parameter values would require the solution of ill-conditioned equations [16, 13, 10], possibly leading to large errors in this calculation. The controllers discussed in this paper should not affect the steady-state operation of the system, changing its operating point. The operating point is defined by the output regulators, such speed governors and AVR s, which indirectly enforce the correct equilibrium to provide the desired output values. If the open-loop equilibrium - defined by these regulators - is not the one enforced in the control law, then the closed-loop equilibrium can be far from the actual operating condition desired for the system, implying loss of output regulation. In currently applied controllers in power systems and other applications this problem is managed by means of so called wash-out filters [10, 1]. These are high-pass filters connected in series with the controller in order to eliminate the DC component of the control. These filters are usually not taken into account in the control design and are added a posteriori. This strategy has been applied for a long time, although it is clearly not valid for any system 2, and only quite recently [3, 5] a theoretical background has been provided to its application. In [3] the local behavior of control systems with wash-out filters has been studied. A generalization of these filters has been proposed that allows the separate design of the controller and the filter under the single assumption that the equilibrium to be stabilized is hyperbolic. A similar filter has been proposed in [5], where its global behavior has been studied when applied along with L g V controllers. In this paper we apply these results to remove the assumption of equilibrium knowledge from the above passivity based controllers. We follow the presentation in [5]. Let the control be given by u = φ(x) φ(x ) (14) where φ( ) is a differentiable function of the state and x is the operating point. Most controllers proposed for power systems - and other applications - can be written in this form, including both controllers presented in the previous section. If the equilibrium is not known, then we can treat φ(x ) as an unknown 2 Think of the control of a first-order open-loop unstable system, for instance. 15

16 parameter in the control law and apply a certainty equivalence adaptive control in the form u = φ(x) ˆθ (15) ˆθ = A(φ(x) ˆθ) (16) with A R m m, A = A T > 0. This control structure will be referred to as the dynamic extension of the controller (14). Since we are considering a single input system, in the present case the matrix A in the dynamic extension control reduces to a scalar, which we shall denote β. In this case, we can rewrite the control as u = p p + β φ(x) where p is the derivative operator. The control law amounts to high-pass filtering of the signal φ(x) before inserting it into the control; this is usually called a wash-out filter. The knowledge of the equilibrium is required for the controller only at the point of implementation, not for design. Since the equilibrium does not appear in (15) and (16), the implementation of this control does not require its knowledge. Thus, the dynamic extension does not require the knowledge of the operating point x, which allows its direct implementation in systems with unknown operating point. Moreover, the operating point is invariant under this feedback. Proposition 2 To each equilibrium x e of the open-loop system (5) there corresponds one and only one equilibrium of the closed-loop system (5), (15), (16), and this equilibrium is [x T e ϕ(x e ) T ] T. The positions of the original equilibria are maintained in a robust way, in the sense that this is a structural property of the control scheme and therefore does not depend on the parameters of the controller. On the other hand, all the equilibria of the original system are maintained, not only the operating point. 5.2 L g V controller It has been proven that for L g V controllers, the dynamic extension preserves at least three important properties: stability, infinite gain margin and the enlargement of the estimate of the region of attraction. The issue of the size of the region of attraction deserves some explanation, since the dynamic controller increases the dimension of the state-space of the system. What we mean is that the size of the estimate of the closed-loop region of attraction in the x-directions is larger than the open-loop estimate. This has been proven by taking the following Lyapunov function candidate for the closed-loop system with the feedback (15), (16): 16

17 U(x, θ) = V (x) + 1 2k (θ θ) T A 1 (θ θ) (17) Then the Lyapunov derivative is U(x, θ) = L f V (x) + L g V (x)(ϕ(x) θ) + 1 k (θ θ) T A 1 ( A(ϕ(x) θ)) (18) = L f V (x) 1 k (θ θ) T (θ θ) < 0 which is zero only at the equilibrium [x T e ϕ(x 0 e) T ] T, therefore establishing its asymptotic stability. That this control has infinite gain margin is clear from (18), since U is negative definite for all k > 0. Proving the enlargement of the estimate of the region of attraction is more involved and for this reason is not presented here. We can summarize these properties as follows. Proposition 3 The control (15)(16) with φ(x) φ(x ) = k. H(x) x, k > 0 has the following properties: x is an asymptotically stable equilibrium of the closed-loop system (1)(15)(16); the control has infinite gain margin; the estimate of the domain of attraction obtained with the Lyapunov function U(x) for the closed-loop system (1)(15)(16) contains (in the sense defined above) the estimate of the region of attraction for the open-loop system (1). To apply the dynamic extension of the L g V controller, we set the filter in the controller to β = 0.1 and use the same gain k = 5 as in Section 4. The critical clearing time obtained with the dynamic L g V controller in this case is t cr = 118ms, the same as obtained with the L g V controller in the previous Section. It can be seen in Figure 7 that in this case the damping is also similar. 5.3 Damping assignment controller Only results on local stability properties can be directly applied to the dynamic extension of the IDA controller, as summarized in the following proposition. Proposition 4 Consider a nonlinear system whose equilibrium point x is asymptotically stabilized by the control law u = φ(x) φ(x ) and its jacobian matrix calculated in open-loop at the equilibrium J(x ) = f(x) x x. Then β > 0 such that x is an asymptotically stable equilibrium of the closed-loop system with the control (15)(16) if and only if the number of eigenvalues of J(x ) in the right-half plane of the complex plane is either 0 or even. 17

18 110 k=5 Dynamic Extension delta (Degrees) Time (s) Figure 7: Load angle oscillations with dynamic L g V, k = 5, after a short-circuit at the machine s terminal with t cl = 80 ms. 18

19 105 a1= a2= Dynamic Extension Delta (Degrees) Time (s) Figure 8: Load angle oscillations with dynamic IDA controller, after a shortcircuit at the machine s terminal with t cl = 80 ms. Since the linearization of the SMIB system around the operating point is stable, the hypothesis of the proposition is satisfied and hence the controller can be applied with a wash-out filter. We take β = 0.1 s and the same control parameters in the previous Section, again resulting in the same critical clearing time t cr = 182 ms and similar damping, as seen in Figure 8. 6 Discussion The interconnection and damping assignment control design presented in this paper provides means to suitably tune nonlinear excitation control to improve both dynamic performance and critical clearing times for major faults. Injecting interconnection between the mechanical and electrical subsystems enlarges the region of attraction and thus increases the critical clearing times. The critical clearing times obtained in each control scenario are given in Table 1. 19

20 Table 1: Summary of the critical clearing times in each control scenario. CONTROLLER t cl (ms) Open-loop 124 L g V 118 IDA 182 Dynamic L g V 118 Dynamic IDA 182 On the other hand, damping injection (L g V control), although important to improve the system s dynamic performance, does not seem to improve transient stability. Hence, the best performance from both criteria can be achieved by assigning damping and interconnection, as in [7]. The extension of these controllers for the unknown operating point case is another step towards the practical application of these results. It has been shown that very similar performance, regarding both criteria, can be achieved with this extensions. Additional results, obtained for different control parameters, are presented in the Appendix. Some drawbacks are still to be solved regarding passivity based control design for excitation control. The extension of these results to the output feedback case is one of under study, which is a necessary step in the direction of their practical application. Also, in order to guarantee enlargement of the region of attraction, it is necessary to start with good estimates for the open-loop system, which is not usually the case for the currently used Lyapunov function. Hence, work on the development of more accurate models and/or Lyapunov functions to improve these estimates is another relevant topic for future research. A System Data The data of the synchronous machine are taken from Kundur(1994), examples 13.2 and The table below shows the parameters: 20

21 Pb = 2220 MVA Pm = 1998 MVA Vb = 24 kv fn = 60 Hz Xd = 1.81 pu X d = 0.3 pu X d = 0.23 pu Xq = 1.76 pu X q = 0.65 pu X q = 0.25 pu Xl = 0.15 pu Td 0 = 8.0 s Td 0 = 0.3 s Tq 0 = 1.0 s Tq 0 = 0.07 s Ra = pu H = s D = 0 XL = 0.5 pu Base Power Mechanical Power Base Voltage Base Frequency d-axis synchronous reactance d-axis transient reactance d-axis subtransient reactance q-axis synchronous reactance q-axis transient reactance q-axis subtransient reactance Leakage reactance d-axis transient open-circuit d-axis subtransient open-circuit q-axis transient open-circuit q-axis subtransient open-circuit Stator resistance Inertia constant Damping coefficient Reactance line b 1 = b 3 = b 4 = ω s 2H(X d + X l) (X d + X d ) T d0 (X d + X l) (X d + X l ) T d0 (X d + X l) B Controllers Data B.1 IDA Controller u = {α 1 x 2 + ( b3 b1 α 2 b 4 )(x 3 x 3e ) + + b 3 b 1 α 1 α 2 (x 1 x 1e )} Case α1 α2 A B C

22 B.2 L g V Controller where: u = k.l g V = k[φ(x) φ(x )] φ(x) = b 3.cos(x 1 ) b 4.x 3 Case k A 1 B 2 C 5 C Critical Times The critical time in base case is 124ms. IDA Controller: L g V Controller: Case tc(ms) A 182 B 168 C 165 A* 182 B* 168 C* 165 Case tc(ms) A 118 B 115 C 98 A* 118 B* 115 C* 98 * Dynamic Extension. The dynamic has a filter that can be modelled by a transference function: with β = 0.1. H(s) = s s + β (19) 22

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