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Guaranteed-cost reliable control with regional pole placement of a power system
Journal of the Franklin Institute 348 (2011) 884–898 www.elsevier.com/locate/jfranklin
Guaranteed-cost reliable control with regional pole placement of a power system H.M. Solimana, A. Dabroumb, M.S. Mahmoudc,n, M. Solimand,1 a
Electrical Engineering Department, Faculty of Engineering, Cairo University, Egypt b Yanbu Industrial College, Yanbu Industrial city, Saudi Arabia c Systems Engineering Department, KFUPM, PO Box 5067, Dhahran 31261, Saudi Arabia d Electrical Engineering Department, Faculty of Engineering, Cairo University, Egypt Received 24 August 2010; received in revised form 2 September 2010; accepted 27 February 2011 Available online 17 March 2011
Abstract This paper deals with the simultaneous coordinated design of power system stabilizer (PSS) and the flexible ac transmission systems (FACTS) controller. The problem of guaranteed cost reliable control with regional pole constraint against actuator failures is investigated. The state feedback controllers are designed to guarantee the closed loop system satisfying the desired pole region, thus achieving satisfactory oscillation damping and settling time, and having the guaranteed cost performance simultaneously. The proposed controllers satisfy desired dynamic characteristics even in faults cases. The controller’s parameters are obtained using the linear matrix inequalities (LMI) optimization. Simulation results validate the effectiveness of this approach. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Power system stabilizer (PSS); FACTS; Fault-tolerant control; Reliable control; Actuator failures; Pole constraint; Guaranteed cost control; LMI
1. Introduction Small-signal stability is an important requirement for power system operation. Power systems are experiencing low frequency oscillation due to small load disturbances. The oscillations may sustain and grow to cause system blackouts if no adequate damping is n
Corresponding author. E-mail addresses: [email protected] (H.M. Soliman), [email protected], [email protected] (M.S. Mahmoud), [email protected] (M. Soliman). 1 Currently on leave to the University of Calgary, Canada. 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.02.013
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Nomenclature Pm Pe Xd0 Xd, Xq Xe 0 Td0 M d o Efd E0 q u1 u2 kA,TA KC,TC V XCref P, Q s
mechanical input power of the generator electrical output power of the generator generator direct-axis transient reactance direct and quadrature-axis synchronous reactances, respectively transmission line reactance d-axis open circuit field time constant inertia constant torque angle angular velocity field voltage q-axis voltage behind transient reactance stabilizing signal of the PSS output stabilizing signal of the TCSC stabilizer exciter gain and time constant TCSC gain and time constant infinite bus voltage TCSC reference reactance machine loading the Laplace operator
available. To enhance system damping, the generators are equipped with power system stabilizers (PSSs) that provide supplementary feedback stabilizing signals in the excitation systems. PSSs extend the power system stability limit by enhancing the system damping of low frequency oscillations associated with the electro-mechanical modes. Design techniques for the lead-type conventional PSS are found in [1]. Several design approaches have been proposed in the past, from modern control theory to evolutionary optimization, to PSS design problem. These include robust control [2–5], adaptive control [6,7], and intelligent control [8]. Besides PSSs, flexible ac transmission (FACTS) controllers are also applied to enhance system stability [3,9–11]. Particularly, in multi-machine systems, using only conventional PSS may not provide sufficient damping for inter-area oscillations. In these cases, FACTS power oscillation damping controllers are effective solutions. Furthermore, in recent years, with the deregulation of the electricity of market, the traditional concepts and practices of power systems have changed. Better utilization of the existing power system to capacities by installing FACTS devices becomes imperative. FACTS devices are playing an increasing and major role in the operation and control of competitive power systems. FACTS devices can be: (1) series connected, for example thyristor controlled series capacitors (TCSC), or thyristor controlled phase angle regulators (TCPAR); or (2) shunt connected device such as static Var compensators (SVC). TCSC devices are the key devices of the FACTS family and they are recognized as effective and economical means to damp power system oscillation.Therefore, in this research, a series FACTS device, the thyristorcontrolled series capacitor (TCSC) is employed for damping system oscillations.
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However, uncoordinated design of TCSC controller and PSS may cause destabilizing interactions. To improve system performance, researches were made on the coordination between PSSs and TCSC controllers [11,12]. Ref. [23] investigates the application of a genetic algorithm for optimal, minimal effort, FACTS-based controller design. The optimal location of a static synchronous compensator (STATCOM) and its coordinated design with power system stabilizers (PSSs) for power system stability improvement are presented [24]. It is well known that controlling a plant using two controllers is better than using only one. Reliable control, also termed fault-tolerent control, is ability to stabilize the system when both controllers are sound or one of them fails. It should be pointed out that redundancy is the key attribute in reliable control systems. The fundamental difference between the robustness and reliability of control systems could be explained as follows. A robust controller can function acceptably with small and medium size parameter variations and/or model uncertainties, plausibly due to different loading conditions. Meanwhile, a reliable controller accommodates more drastic changes in system configurations, probably caused by component failures and/or outages. Reliable control systems are extremely vital in practice (for example. in avionics [15], networked control systems [16,28,29], fuzzy systems [30], adaptive systems [31], stochastic systems [32,33], to name a few). Although there are at least one hundred research papers dealing with PSS for excitation channel, it seems that very little effort is done to use additional redundant controller, e.g. governor channel, for reliable stabilization of power systems [13,14]. Fault-tolerant power system control with large governor delay is considered in [27]. Loss of one control signal is equivalent to an actuator failure. Such faults may be attribute to a loss of signal, a communication channel, a controller malfunction, or a combination of these. It should be noticed that in [27] the case of sudden complete actuator failure is considered and it is tackled by state feedback control. Gradual controller’s deterioration and failure using lead controllers via Kharitonov theorem is treated in [34]. In control literature, the problem of fault-tolerant control has received considerable attention in recent years (see for example [15] and the references therein). In actual implementation, systems are often required to have good transient characteristics, which can be described by regional closed-loop poles constraints using state feedback, [17,25] or output feedback PID control [26]. However, most existing fault tolerant control results(see, for example, [18]) did seldom take such performance requiements into account. Therefore, it is our motivation to investigate the guaranteed-cost fault-tolerant control with pole region constaints for power systems subject to actuator failures (either in PSS, or TCSC-control). Then by LMI method, a convex optimization problem is formulated to find the corresponding controller. System configuration, mathematical model and problem formulation are given in Section 2 of the manuscript. The problem solution is presented in Section 3, while Section 4 handles the problem simulation results and analytical testing. 2. Notation and facts In the sequel, Rn ; Rnm denote, respectively, the n-dimensional Euclidean space and the set of n m real matrices. In the sequel, WT and W1 denote, respectively, the transpose, and the inverse of any square matrix W. The notation W40, (Wo0) is used to denote
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a symmetric positive (negative) definite matrix W; I denotes the identity matrix of appropriate dimension. The symbol d is as an ellipsis for terms in matrix expressions that are induced by symmetry e.g. " # L þ ðW þ N þ dÞ N L þ ðW þ N þ W T þ N T Þ N ¼ d R NT R Fact#1: The congruence transformation zTWz does not change the definiteness of W. Fact#2: (Schur complement) This fact is used to transform a non linear matrix inequality to a linear one. Given constant matrices W1,W2,W3 where W1=WT1 and 0oW2=WT2 . Then " # W1 T 1 W1 þ W3 W2 W3 o03 o0 W3 W2 3. Problem formulation The present work involves the design procedures for robust controllers that tolerate the fault of control-signal. The research considers a single-machine infinite-bus power system equipped with TCSC, Fig. 1. The system includes a PSS, F1, acting on the generator exciter and another controller, F2, setting the thyristor firing angle, which determines the reactance of the TCSC. The infinite bus and transmission line reactance could be considered the Thevenin equivalent of a multi-machine interconnected power system. In practice, not all of the machines in a power system are equipped with FACTS devices. Therefore, if more than one generator is equipped with TCSC in a power system, the proposed routine could be separately applied to all of these generators one at a time and considering the rest of the system as an infinite bus. The controllers are designed in such an optimal way that the system would maintain 10–15 s of settling time, and a desired damping ratio following any small disturbance as recommended by power system operators. Low frequency oscillations are very plausible in single-machine infinite bus systems due to small disturbances such as load changes. A TCSC is placed in series with the
G
Exciter u1
TCSC -jXc
x
u2 failure#2
failure#1
F1
Transmission line jXe
YL
F2
Fig. 1. Single machine infinite bus system with TCSC.
Inf. bus V
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L
C Fig. 2. Thyristor controlled series capacitor (TCSC) topology.
transmission line to change the line flow. Therefore, a TCSC can extend the power transfer capability and provide additional damping for low frequency oscillation. A TCSC is a capacitive reactance compensator, which consists of a series capacitor bank shunted by a thyristor controlled reactor in order to provide a smooth variation in series capacitive reactance, Fig. 2. Controlling the firing angle of the thyristor regulates the TCSC reactance and its degree of compensation. The fourth order model of the system [19] represents the machine dynamics around a certain operating point as given below. x_ ¼ Ax þ Bu n
ð1Þ m
where x(t)AR is the state vector and u(t)AR is the control vector given as h iT 0 x ¼ Dd Do DEq DEf u ¼ ½u1 ; u2 T
ð2Þ ð3Þ
In our case, n=4, m=2, and u2=DXC. It is important to notice that the dynamic characteristic of the TCSC is very fast; its time constant is 0.02 s, as compared with that of the system under study, hence neglected. The synchronous machine under study is provided with a thyristor-based excitation system of transfer function T(s), KE TðsÞ ¼ 1 þ TE s The system data are in p.u. as follows: Synchronous machine: Exciter: Transmission line: Machine loading (nominal): Local load: Machine terminal voltage:
Xd=0.973, Xd0=0.19, Xq=0.55, 0 o0=377 rad/s, Td0 =7.76 s, M=9.26 s KE ¼ 50; TE ¼ 0:05s Xe=0.997 S=PþjQ=1.0þj0.015pu YL=0.249þj0.262 1.05þj0
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The resulting matrices of the system model are given below. 2 3 2 3 0 377 0 0 0 0 6 0:0588 6 0 0 0:1303 0 7 0:0704 7 6 7 6 7 A¼6 7; B ¼ 6 7 4 0:09 5 4 0 0:1957 0:1289 0 0:0177 5 95:532 0 815:93 20 1000 93:846
889
ð4Þ
The open loop system represented in Eq. (4) is unstable with eigenvalues: þ0:37j4:96;
10:397j3:28
The model is standard, based on Heffron–Phillips approach with FACTS device taken into consideration, and provides fair enough accuracy [1]. It is to be noted that the states are measurable or can be easily calculated. Consider the problem of determining the state feedback, given as: u ¼ Fx
ð5Þ T
for excitation and TCSC controllers u=[u1,u2] that ensure stability when either one fails or when both are active. Therefore, the following decomposition is used: " # F1 B ¼ b1 b2 ; F ¼ ð6Þ F2 To cope with the possible fault scenarios: (1) only controller#1 is active (fault in controller#2), (2) only controller #2 is active (fault in controller#1), and (3) no faults; the following matrices are respectively defined B1 ¼ b1 0 ; B2 ¼ 0 b2 ; and B3 ¼ b1 b2 ð7Þ It is worth mentioning that the possibility of simultaneous failure of both controllers is very remote, so it is excluded. Thus, the system (1) under all possible faults becomes x_ ¼ Ax þ Bi u
; i ¼ 1; 2; 3
ð8Þ
It is assumed that (A, Bi),8i, are controllable. The objectives of the present work could be briefly introduced as follows. For system (8), find a state-feedback controller F, such that the faulty closed-loop system x_ ¼ ðA þ Bi F Þx;
8i
ð9Þ
will meet either of the following constraints (selected by the designer): (i) Reliable stability: The closed loop poles must lie to the left hand side of the complex plane for all possible faults. In practice this might not be enough to provide satisfactory dynamic performance. The desired performance indicates a closed loop minimum damping ratio (zmin) of 0.1–0.25, while the system oscillations should settle within ts=10–15 s as typically followed by power system utilities [20]. (ii) Controlled settling timeþ (i): The speed of response given by the settling time, ts=4/a, can be controlled if the closed loop poles lie to the left of a vertical line a, a40 in the complex plane (Fig. 3a). Where a is called the degree of stability, also termed relative stability. (iii) Controlled overshootþ(ii): To avoid generator shaft fatigue and possible breakdown, the overshoot or equivelently the damping ratio must not be less than a minimum value.
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ζmin -α
α q relative stability
Circular region D
Fig. 3. (a)Shifted region. (b) Circular region D(q,r).
If the closed-loop poles lie within the circular region D(q,r) with center at –q and radius roq (Fig.3 b), both damping ratio and settling time can be achieved. Selecting (zmin=0.25) and (a=0.36), we find the desired circular region D(q,r) as q ¼ 11; and r ¼ 10:64
ð10Þ
(iv) Guaranteed cost controlþ(iii): Although pole placement in a circular region puts interesting practical constraints on the transient response of power systems, in practice it might be desirable that the controller be chosen to minimize a cost function as well. The cost function associated with the faulty system (8) is Z 1 J¼ ðxT Qx þ uT RuÞdt ð11Þ 0
where Q ¼ QT 40; R ¼ RT 40 are given weighting matrices. With the state feedback (5), the cost function of the closed loop is Z 1 J¼ xT ðQ þ F T RF Þxdt ð12Þ 0
The guaranteed cost control problem is to find F such that cost function J exists and to have an upper bound J*, i.e. satisfying JrJ*.
4. Problem solution Design case 1: Theorem1. The closed loop faulty system (9) is reliably stabilized by state feedback u=Fx if there exist matrices Y=YT40, SARm n, such that the following LMIs AY þ Bi S þ o0;
8i
ð13Þ
have a feasible solution. The controller satisfying reliable stabilization is given by F ¼ SY 1
ð14Þ
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Proof. Define a positive definite Lyapunov function V ¼ xT Px; P ¼ PT 40. Evaluating the derivative of V along with (9) results in V_ o0, and thus stability, if PðA þ Bi F Þ þ o0; 8i
ð15Þ
Pre- and post-multiply (15) byP1, Fact #1, and letting Y=P1, S=FY; controller (14) is obtained. & Design case 2: Theorem 2. The closed loop faulty system (9) is reliably stabilized with a degree of stability a, by state feedback u=Fx if there exist matricesY=YT40, S such that the following LMIs A:Y þ aY þ Bi :S þ o0;
8i
ð16Þ
have a feasible solution. The controller achieving reliable stabilizationþdegree of stability is given by F ¼ SY 1 Proof. Following the same lines as above, replacing A with AþaI, the proof is straightforward. & Design case 3: Theorem 3. The closed loop faulty system (9) is reliably stabilized with poles lie in a disk D (q,r), by state feedback u=Fx if there exist matricesY=YT40, S such that the following LMIs " # d r2 Y o0; 8i ð17Þ AY þ qY þ Bi S Y have a feasible solution. The controller fulfilling reliable stabilization with regional pole constraint D(q,r) is given by F ¼ SY 1 Proof. The poles of a matrix A lie in the disk D (q,r) if and only if there exists a matrix P=PT40, such that [21] " # r2 P d o0 ð18Þ ðAY þ qIÞ P1 It follows that the stablility of the closed loop faulty system (9) with poles lying in the disk D(q,r) is achieved by replacing A with the closed loop matrix Aci=AþBiF, 8i. Pre- and post-multiplying by diag[P1,I], we get " # r2 P1 o0 ðAci þ qIÞP1 P1 This is a nonlinear matrix inequality. To convexify the result, we let Y=P1, S=FY and obtain the LMI (17) as desired. &
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Design case 4: Theorem 4. Consider the faulty system (9) and the cost function (12), if there exists a feedback matrix F and Y=YT40, S such that the following LMIs 2 3 r2 Y 6 AY þ qY þ B S Y 7 i 6 7 ð19Þ 6 7o0; 8i 4 YT 0 Q1 5 S
0
0
R1
hold for all possible faults, then the controller providing reliable stabilization with regional pole constraint D(q,r) and guaranteed cost is given by F ¼ SY 1 Moreover, the cost function has an upper bound J ¼ qxT0 Px0 ;
x0 ¼ xð0Þ
ð20Þ
Proof. The proof consists of two parts. In the first part, we show reliable stabilization with regional pole constraint D(q,r). In the second part, we provide the guaranteed cost. For the first part, we build on Theorem 3 and conclude that the controller achieving reliable stabilization with regional pole constraint is given by LMI (18). By Fact #2, this LMI is equivalent to the nonlinear matrix inequality r2 P þ ðAci þ qIÞT PðAci þ qIÞo0;
8i
ð21Þ
For the second part, we apply the results of [17] to reach that the faulty system (9) with cost function (12) is stable provided that r2 P þ ðAci þ qIÞT PðAci þ qIÞ þ Q þ F T RF o0;
8i
ð22Þ
hold, where (Q40, R40). It is clearly evident that if (22) is satisfied, it implies that (21) is fulfilled as well, Pre- and post-multiplying (22) by P1 and using the Schur complements operation, the nonlinear matrix inequalities (22) can be linearized and expanded to get LMI (19). This establishes that controller (19) achieves reliable stabilization with regional pole placement and guaranteed cost. To show that controller (19) provides an upper bound of the cost function, consider a Lyapunov function V ¼ xT Px; P ¼ PT 40 Notice that (22) is equivalent to ATci P
1 T q2 r2 Q þ F T RF Q þ F T RF T o0; P þ o Aci PAci o q q q q
Differentiating V(x(t)) with respect to time and using (23), we obtain Q þ F T RF V_ ¼ xT fPAci þ gxrxT x q
8i
ð23Þ
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Therefore, integrating both sides of the above inequality from 0 to N gives Z 1 Q þ F T RF xdtrV ðx0 ÞV ðxð1ÞÞ xT q 0 Since the stability of the system has already been established, x(t)-0 as t-N, it can be concluded that V(x(t))-0 as t-N. This completes the proof. &
5. Simulation results The proposed feedback control schemes by Eqs. (13), (16), (17), and (19) are used for the stabilization of the power system model described by the linear model (9). The linear matrix inequalities (13), (16), (17), and (19) are solved using the matlab LMI control toolbox [22] to get the feedback matrix for the design cases mentioned above. The results are summarized in Table 1. The closed loop poles for every design case representing all possible actuator failure are summarized in Table 2.
Table 1 Proposed controllers. Design case
Feedback matrix
1
F¼
0:1869
42:134
0:6186 0:00395
2:2198 441:94 7:742 0:0812 N/A, the desired relative stability is already achieved by F, above. 0:0293 110:7953 1:8771 0:0098 F¼ 0:4942 103:6476 1:9155 0:0101 For Q=I,R=I, 0:0074 120:8538 2:1108 0:0113 F¼ 0:4988 103:2079 2:0051 0:0111
2 3 4
Table 2 Closed loop eigenvalues for faulty and sound controllers (different designs) Design case
Comment
Closed loop poles
1
Only u1 is active(u2 fails) Only u2 is active(u1 fails) u1 and u2 are active (no failure)
0.8247j6.7015, 11.257j1.5856 1.1054, 8.094, 17.1757j10.042 2.2201, 8.64037j4.9658, 28.002
3
Only u1 is active(u2 fails) Only u2 is active(u1 fails) u1 and u2 are active (no failure)
4.2357j7.08, 10.777j2.94 1.6971, 3.3342, 10.73927j3.51 1.518, 10.12127j5.2864, 14.5624
4
Only u1 is active(u2 fails) Only u2 is active(u1 fails) u1 and u2 are active (no failure)
4.647j6.77, 11.0867j3.07 1.422, 3.71, 10.637j3.496 1.56, 10.147j4.67, 15.81
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It is clear from Table 2, design case#1 that the controller achieves reliable stabilizationþrelative stability for any possible actuator faults. Figs. 4 and 5 show respectively that using design 3 and 4, the closed loop poles locations satisfy the required performance, reliable stabilizationþregional pole placement, for all possible actuator failures. Next, the simulation results when using the proposed controllers are presented. For an impulse load increase causing Dd=0.1 rad, the initial condition of the SMIB system is taken as x0 ¼ ½0:1; 0; 0; 0T . Design case#3:closed loop poles 12
10
only u1 on only u2 on both u1 & u2 on
zeta=.25
pole region
imaj
8
6
4
2
0 −25
−20
−15
−10
−5
0
real
Fig. 4. Closed loop pole locations for all possible faults. Design 3: reliable stab.þpole region.
Design case#4:closed loop poles 12
10
zeta=.25
only u1 on only u2 on both u1 & u2 on pole region
imaj
8
6
4
2
0 −25
−20
−15
−10
−5
0
real
Fig. 5. Closed loop pole locations for all possible faults. Design 4: reliable stab.þpole regionþ guaranteed cost.
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angle dev,rad
only u1 on 0.1 0 −0.1 0
2
4
6
8
10
6
8
10
6
8
10
angle dev,rad
time,s only u2 on 0.1 0.05 0
0
2
4 time,s
angle dev,rad
u1 & u2 on 0.1 0.05 0
0
2
4 time,s
angle dev,rad
Fig. 6. Angle deviation response. Design case 1.
only u1 on
0.1 0 −0.1 0
2
4
6
8
10
6
8
10
6
8
10
angle dev,rad
time,s only u2 on
0.1 0.05 0 0
2
4
angle dev,rad
time,s u1 & u2 on
0.1 0.05 0 0
2
4 time,s
Fig. 7. Angle deviation response. Design case 4.
The simulation results when using the first controller, design case #1, is shown in Fig. 6. From the figures it can be seen that the rotor angle deviation converges to 0 in about 5 s. While, Fig. 7 shows the response when the fourth controller, design case 4, is used. It is evident that reliable stabilizationþregional pole placementþguaranteed cost are satisfied, the angle deviation decays to 0 within 3 s, for any possible failure. This manifest the superiority of design 4 over design 1. To show the effectiveness of the design case #4, the conventional linear quadratic regulator (LQR) is used. For the same values used before, the matrices Q and R are taken
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angle dev,rad
896
only u1 on
0.1 0 −0.1
0
2
4
6
8
10
6
8
10
6
8
10
angle dev,rad
time,s only u2 on
0.1 0.05 0
0
2
4
angle dev,rad
time,s u1 & u2 on
0.1 0.05 0
0
2
4 time,s
Fig.8. Angle deviation response. Conventional LQR.
as I; the conventional optimal feedback regulator is obtained as 0:0620 8:9587 0:5248 0:9761 F¼ 0:4697 69:8057 1:2027 0:0910 The simulations for different actuator failures using the conventional LQR are shown in Fig. 8. It is clear that the standard LQR is very vulnerable in face of actuators failure. 6. Conclusions This article presents a new design approach for fault-tolerant stabilization of a power system. The two-channel control encloses two stabilizers acting on the excitation and TCSC of a synchronous alternator operating in an interconnected power system. Four different control schemes to satisfy desired dynamic performance constraints are derived. Taking the guaranteed cost constraint into account, the problem of fault-tolerant controller design with regional pole constraint is tackled by LMI approach for a power system subject to actuator failures. Simulation results show that using the proposed controllers, the system performs fairly well. It provides satisfactory stability, transient property, and quadratic cost performance despite of possible actuator faults. The proposed controller outperforms the standard LQR. Since power systems usually operate on the nominal loading condition, the present manuscript develops four theorems to achieve reliable stabilization of the study system under nominal loading. Development of new theorems is underway to achieve reliable stability at different loading points. Acknowledgment Dr M. S. Mahmoud would like to thank the Deanship of Scientific Research (DSR) at KFUPM for research support through project IN100018.
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References [1] W. Sauer, M.A. Pai., Power System Dynamics and Stability, Pearson Education, New Delhi, 2003. [2] Z.Z. Zhong, J.C. Wang, A comparison and simulation study of robust excitation control strategies for singlemachine infinite bus power system, International Journal of Modeling, Identification and Control 7 (4) (2009) 351–356. [3] B. Pal, B. Chaudhuri, in: Robust Control in Power Systems, Springer, 2005. [4] P.S. Rao, I. Sen, Robust pole placement stabilizer design using linear matrix inequalities, IEEE Trans. Power Systems 15 (1) (2000) 313–319. [5] K.A. El-Metwally, A. El-Shafei, H.M. Soliman, A robust power system stabilizer design using swarm optimization, International Journal of Modeling, Identification and Control 1 (4) (2006) 263–271. [6] O.P. Malik, Adaptive and intelligent control applications to power system stabilizer, International Journal of Modeling, Identification and Control 6 (1) (2009) 51–61. [7] M.A. Awadallah, H.M. Soliman, An adaptive power system stabilizer based on fuzzy and swarm intelligence, International Journal of Modeling, Identification and Control 1 (4) (2006). [8] H.M. Soliman, E.H.E. Bayoumi, M.F. Hassan, Power system stabilizer design for minimal overshoot and control constraint using swarm optimization, Electric Power Components and Systems 37 (2009) 111–126. [9] Y.H. Song, A.T. Johns, Flexible AC Transmission Systems (FACTS), IEE Press, London, U.K, 1999. [10] X. Lei, D. Jiang, D. Retzmann, Stability improvement in power systems with nonlinear TCSC control strategies, European Transactions on Electric Power 10 (6) (2000) 339–345. [11] X. Lei, E.N. Lerch, D. Povh, Optimization and coordination of damping controls for improving system dynamic performance, IEEE Transactions on Power Systems 16 (2001) 473–480. [12] L. Cai, I. Erlich, Simultaneous coordinated tuning of PSS and FACTS damping controllers in large power systems, IEEE Transactions on Power Systems 20 (1) (2005) 294–299. [13] H.M. Soliman, A.L. El_Shafei, K.A. El_Metwally, E.M.K. Makkawy, Fault-tolerant wide-range stabilization of a power system, International Journal of Modeling, Identification and Control 3 (2) (2008) 173–179. [14] H.M. Soliman, M.F. Morsi, M.F. Hassan, M.A. Awadallah, Power system reliable stabilization with actuator failure, Electric Power Components and Systems 37 (2009) 61–77. [15] D. Ye, G. Yang, Adaptive fault-tolerant tracking control against actuator faults with application to flight control, IEEE Transactions on Control Systems Technology 14 (6) (2006) 1088–1096. [16] D. Zhang, Z. Wang, S. Hu, Robust satisfactory fault-tolerant control of uncertain linear discrete-time systems: an LMI approach, International Journal of Systems Science 38 (2) (2007) 151–165. [17] S.O. Reza Moheimani, I.R. Petersen, Quadratic guaranteed cost control with robust pole placement in a disk, IEE Proceedings on Control Theory Applications 143 (1) (1996) 37–43. [18] X. Nian, J. Feng, Guaranteed–cost control of a linear uncertain system with multiple time-varying delays: an LMI approach, IEE Proceedings: Control Theory and Applications 150 (1) (2003) 17–22. [19] M.A. Abido, Pole placement technique for PSS and TCSC-based stabilizer design using simulated annealing, Electrical power and energy systems 22 (2000) 543–554. [20] J. Passrba, Analysis and control of power system oscillations CIGRE, Special Brochure111, 1996. [21] W.M. Haddad, D.S. Bernstein, Controller design with regional pole constraints, IEEE Transactions on Automatic Control AC 37 (1992) 54–69. [22] P. Gahinet, A. Nemirovski, A.J. Laub and M. Chilali, LMI-Control Toolbox The Math Works, Boston, Mass., 1995. [23] S. Panda, Application of non-dominated sorting genetic algorithm-II technique for optimal FACTS-based controller design, Journal of The Franklin Institute 347 (7) (2010) 1047–1064. [24] S. Panda, N.P. Padhy, Optimal location and controller design of STATCOM for power system stability improvement using PSO, Journal of The Franklin Institute 345 (2) (2008) 166–181. [25] J.H. Kim, S.J. Ahn, S.J. Ahn, Guaranteed cost and HN filtering for discrete-time polytopic uncertain systems with time delay, Journal of The Franklin Institute 342 (4) (2005) 365–378. [26] T.K. Liua, S.H. Chen, J.H. Chou, C.Y. Chen, Regional eigenvalue-clustering robustness of linear uncertain multivariable output feedback PID control systems, Journal of The Franklin Institute 346 (3) (2009) 253–266. [27] M.T. Al-Rifai, M.F. Hassan, H.M. Soliman, Fault-tolerant stabilization for time-delayed power systems, International Journal of Innovative Computing, Information, and Control 6 (7) (2010) 3247–3263. [28] Z. Mao, B. Jiang, P. Shi, Observer based fault-tolerant control for a class of nonlinear networked control systems, Journal of The Franklin Institute 347 (6) (2010) 940–956.
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[29] C.X. Yang, Z.H. Guan, J. Huang, Stochastic fault tolerant control of networked control systems, Journal of The Franklin Institute 346 (10) (2009) 1006–1020. [30] J. Wu, S.K. Nguang, J. Shen, G.J. Liu, Y.G. Li, Fuzzy guaranteed cost tracking control for boiler-turbines via TS fuzzy model, International Journal of Innovative Computing, Information, and Control 6 (12) (2010) 5575–5586. [31] X. Zhao, J. Han, Yaw control of helicopter: an adaptive guaranteed cost control approach, International Journal of Innovative Computing, Information, and Control 5 (8) (2009) 2267–2276. [32] K.C. Yao, Reliable robust output feedback stabilizing computer control of decentralized stochastic singularly-perturbed systems, ICIC Express Letters 1 (1) (2007) 1–7. [33] Z. Gu, C. Peng, E. Tian, Reliable Control for a class of discrete-time state-delayed nonlinear systems with stochastic actuators failures, ICIC Express Letters 4 (6(B)) (2010) 2475–2480. [34] H.M. Soliman, E.H.E. Bayoumi, and M.A. Awadallah Reconfigurable fault-tolerant PSS and FACTS controllers Electric Power Components and Systems,38, 1446–1468, 2010.
Guaranteed-cost reliable control with regional pole placement of a power system H.M. Solimana, A. Dabroumb, M.S. Mahmoudc,n, M. Solimand,1 a
Electrical Engineering Department, Faculty of Engineering, Cairo University, Egypt b Yanbu Industrial College, Yanbu Industrial city, Saudi Arabia c Systems Engineering Department, KFUPM, PO Box 5067, Dhahran 31261, Saudi Arabia d Electrical Engineering Department, Faculty of Engineering, Cairo University, Egypt Received 24 August 2010; received in revised form 2 September 2010; accepted 27 February 2011 Available online 17 March 2011
Abstract This paper deals with the simultaneous coordinated design of power system stabilizer (PSS) and the flexible ac transmission systems (FACTS) controller. The problem of guaranteed cost reliable control with regional pole constraint against actuator failures is investigated. The state feedback controllers are designed to guarantee the closed loop system satisfying the desired pole region, thus achieving satisfactory oscillation damping and settling time, and having the guaranteed cost performance simultaneously. The proposed controllers satisfy desired dynamic characteristics even in faults cases. The controller’s parameters are obtained using the linear matrix inequalities (LMI) optimization. Simulation results validate the effectiveness of this approach. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Power system stabilizer (PSS); FACTS; Fault-tolerant control; Reliable control; Actuator failures; Pole constraint; Guaranteed cost control; LMI
1. Introduction Small-signal stability is an important requirement for power system operation. Power systems are experiencing low frequency oscillation due to small load disturbances. The oscillations may sustain and grow to cause system blackouts if no adequate damping is n
Corresponding author. E-mail addresses: [email protected] (H.M. Soliman), [email protected], [email protected] (M.S. Mahmoud), [email protected] (M. Soliman). 1 Currently on leave to the University of Calgary, Canada. 0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.02.013
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Nomenclature Pm Pe Xd0 Xd, Xq Xe 0 Td0 M d o Efd E0 q u1 u2 kA,TA KC,TC V XCref P, Q s
mechanical input power of the generator electrical output power of the generator generator direct-axis transient reactance direct and quadrature-axis synchronous reactances, respectively transmission line reactance d-axis open circuit field time constant inertia constant torque angle angular velocity field voltage q-axis voltage behind transient reactance stabilizing signal of the PSS output stabilizing signal of the TCSC stabilizer exciter gain and time constant TCSC gain and time constant infinite bus voltage TCSC reference reactance machine loading the Laplace operator
available. To enhance system damping, the generators are equipped with power system stabilizers (PSSs) that provide supplementary feedback stabilizing signals in the excitation systems. PSSs extend the power system stability limit by enhancing the system damping of low frequency oscillations associated with the electro-mechanical modes. Design techniques for the lead-type conventional PSS are found in [1]. Several design approaches have been proposed in the past, from modern control theory to evolutionary optimization, to PSS design problem. These include robust control [2–5], adaptive control [6,7], and intelligent control [8]. Besides PSSs, flexible ac transmission (FACTS) controllers are also applied to enhance system stability [3,9–11]. Particularly, in multi-machine systems, using only conventional PSS may not provide sufficient damping for inter-area oscillations. In these cases, FACTS power oscillation damping controllers are effective solutions. Furthermore, in recent years, with the deregulation of the electricity of market, the traditional concepts and practices of power systems have changed. Better utilization of the existing power system to capacities by installing FACTS devices becomes imperative. FACTS devices are playing an increasing and major role in the operation and control of competitive power systems. FACTS devices can be: (1) series connected, for example thyristor controlled series capacitors (TCSC), or thyristor controlled phase angle regulators (TCPAR); or (2) shunt connected device such as static Var compensators (SVC). TCSC devices are the key devices of the FACTS family and they are recognized as effective and economical means to damp power system oscillation.Therefore, in this research, a series FACTS device, the thyristorcontrolled series capacitor (TCSC) is employed for damping system oscillations.
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However, uncoordinated design of TCSC controller and PSS may cause destabilizing interactions. To improve system performance, researches were made on the coordination between PSSs and TCSC controllers [11,12]. Ref. [23] investigates the application of a genetic algorithm for optimal, minimal effort, FACTS-based controller design. The optimal location of a static synchronous compensator (STATCOM) and its coordinated design with power system stabilizers (PSSs) for power system stability improvement are presented [24]. It is well known that controlling a plant using two controllers is better than using only one. Reliable control, also termed fault-tolerent control, is ability to stabilize the system when both controllers are sound or one of them fails. It should be pointed out that redundancy is the key attribute in reliable control systems. The fundamental difference between the robustness and reliability of control systems could be explained as follows. A robust controller can function acceptably with small and medium size parameter variations and/or model uncertainties, plausibly due to different loading conditions. Meanwhile, a reliable controller accommodates more drastic changes in system configurations, probably caused by component failures and/or outages. Reliable control systems are extremely vital in practice (for example. in avionics [15], networked control systems [16,28,29], fuzzy systems [30], adaptive systems [31], stochastic systems [32,33], to name a few). Although there are at least one hundred research papers dealing with PSS for excitation channel, it seems that very little effort is done to use additional redundant controller, e.g. governor channel, for reliable stabilization of power systems [13,14]. Fault-tolerant power system control with large governor delay is considered in [27]. Loss of one control signal is equivalent to an actuator failure. Such faults may be attribute to a loss of signal, a communication channel, a controller malfunction, or a combination of these. It should be noticed that in [27] the case of sudden complete actuator failure is considered and it is tackled by state feedback control. Gradual controller’s deterioration and failure using lead controllers via Kharitonov theorem is treated in [34]. In control literature, the problem of fault-tolerant control has received considerable attention in recent years (see for example [15] and the references therein). In actual implementation, systems are often required to have good transient characteristics, which can be described by regional closed-loop poles constraints using state feedback, [17,25] or output feedback PID control [26]. However, most existing fault tolerant control results(see, for example, [18]) did seldom take such performance requiements into account. Therefore, it is our motivation to investigate the guaranteed-cost fault-tolerant control with pole region constaints for power systems subject to actuator failures (either in PSS, or TCSC-control). Then by LMI method, a convex optimization problem is formulated to find the corresponding controller. System configuration, mathematical model and problem formulation are given in Section 2 of the manuscript. The problem solution is presented in Section 3, while Section 4 handles the problem simulation results and analytical testing. 2. Notation and facts In the sequel, Rn ; Rnm denote, respectively, the n-dimensional Euclidean space and the set of n m real matrices. In the sequel, WT and W1 denote, respectively, the transpose, and the inverse of any square matrix W. The notation W40, (Wo0) is used to denote
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a symmetric positive (negative) definite matrix W; I denotes the identity matrix of appropriate dimension. The symbol d is as an ellipsis for terms in matrix expressions that are induced by symmetry e.g. " # L þ ðW þ N þ dÞ N L þ ðW þ N þ W T þ N T Þ N ¼ d R NT R Fact#1: The congruence transformation zTWz does not change the definiteness of W. Fact#2: (Schur complement) This fact is used to transform a non linear matrix inequality to a linear one. Given constant matrices W1,W2,W3 where W1=WT1 and 0oW2=WT2 . Then " # W1 T 1 W1 þ W3 W2 W3 o03 o0 W3 W2 3. Problem formulation The present work involves the design procedures for robust controllers that tolerate the fault of control-signal. The research considers a single-machine infinite-bus power system equipped with TCSC, Fig. 1. The system includes a PSS, F1, acting on the generator exciter and another controller, F2, setting the thyristor firing angle, which determines the reactance of the TCSC. The infinite bus and transmission line reactance could be considered the Thevenin equivalent of a multi-machine interconnected power system. In practice, not all of the machines in a power system are equipped with FACTS devices. Therefore, if more than one generator is equipped with TCSC in a power system, the proposed routine could be separately applied to all of these generators one at a time and considering the rest of the system as an infinite bus. The controllers are designed in such an optimal way that the system would maintain 10–15 s of settling time, and a desired damping ratio following any small disturbance as recommended by power system operators. Low frequency oscillations are very plausible in single-machine infinite bus systems due to small disturbances such as load changes. A TCSC is placed in series with the
G
Exciter u1
TCSC -jXc
x
u2 failure#2
failure#1
F1
Transmission line jXe
YL
F2
Fig. 1. Single machine infinite bus system with TCSC.
Inf. bus V
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L
C Fig. 2. Thyristor controlled series capacitor (TCSC) topology.
transmission line to change the line flow. Therefore, a TCSC can extend the power transfer capability and provide additional damping for low frequency oscillation. A TCSC is a capacitive reactance compensator, which consists of a series capacitor bank shunted by a thyristor controlled reactor in order to provide a smooth variation in series capacitive reactance, Fig. 2. Controlling the firing angle of the thyristor regulates the TCSC reactance and its degree of compensation. The fourth order model of the system [19] represents the machine dynamics around a certain operating point as given below. x_ ¼ Ax þ Bu n
ð1Þ m
where x(t)AR is the state vector and u(t)AR is the control vector given as h iT 0 x ¼ Dd Do DEq DEf u ¼ ½u1 ; u2 T
ð2Þ ð3Þ
In our case, n=4, m=2, and u2=DXC. It is important to notice that the dynamic characteristic of the TCSC is very fast; its time constant is 0.02 s, as compared with that of the system under study, hence neglected. The synchronous machine under study is provided with a thyristor-based excitation system of transfer function T(s), KE TðsÞ ¼ 1 þ TE s The system data are in p.u. as follows: Synchronous machine: Exciter: Transmission line: Machine loading (nominal): Local load: Machine terminal voltage:
Xd=0.973, Xd0=0.19, Xq=0.55, 0 o0=377 rad/s, Td0 =7.76 s, M=9.26 s KE ¼ 50; TE ¼ 0:05s Xe=0.997 S=PþjQ=1.0þj0.015pu YL=0.249þj0.262 1.05þj0
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The resulting matrices of the system model are given below. 2 3 2 3 0 377 0 0 0 0 6 0:0588 6 0 0 0:1303 0 7 0:0704 7 6 7 6 7 A¼6 7; B ¼ 6 7 4 0:09 5 4 0 0:1957 0:1289 0 0:0177 5 95:532 0 815:93 20 1000 93:846
889
ð4Þ
The open loop system represented in Eq. (4) is unstable with eigenvalues: þ0:37j4:96;
10:397j3:28
The model is standard, based on Heffron–Phillips approach with FACTS device taken into consideration, and provides fair enough accuracy [1]. It is to be noted that the states are measurable or can be easily calculated. Consider the problem of determining the state feedback, given as: u ¼ Fx
ð5Þ T
for excitation and TCSC controllers u=[u1,u2] that ensure stability when either one fails or when both are active. Therefore, the following decomposition is used: " # F1 B ¼ b1 b2 ; F ¼ ð6Þ F2 To cope with the possible fault scenarios: (1) only controller#1 is active (fault in controller#2), (2) only controller #2 is active (fault in controller#1), and (3) no faults; the following matrices are respectively defined B1 ¼ b1 0 ; B2 ¼ 0 b2 ; and B3 ¼ b1 b2 ð7Þ It is worth mentioning that the possibility of simultaneous failure of both controllers is very remote, so it is excluded. Thus, the system (1) under all possible faults becomes x_ ¼ Ax þ Bi u
; i ¼ 1; 2; 3
ð8Þ
It is assumed that (A, Bi),8i, are controllable. The objectives of the present work could be briefly introduced as follows. For system (8), find a state-feedback controller F, such that the faulty closed-loop system x_ ¼ ðA þ Bi F Þx;
8i
ð9Þ
will meet either of the following constraints (selected by the designer): (i) Reliable stability: The closed loop poles must lie to the left hand side of the complex plane for all possible faults. In practice this might not be enough to provide satisfactory dynamic performance. The desired performance indicates a closed loop minimum damping ratio (zmin) of 0.1–0.25, while the system oscillations should settle within ts=10–15 s as typically followed by power system utilities [20]. (ii) Controlled settling timeþ (i): The speed of response given by the settling time, ts=4/a, can be controlled if the closed loop poles lie to the left of a vertical line a, a40 in the complex plane (Fig. 3a). Where a is called the degree of stability, also termed relative stability. (iii) Controlled overshootþ(ii): To avoid generator shaft fatigue and possible breakdown, the overshoot or equivelently the damping ratio must not be less than a minimum value.
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ζmin -α
α q relative stability
Circular region D
Fig. 3. (a)Shifted region. (b) Circular region D(q,r).
If the closed-loop poles lie within the circular region D(q,r) with center at –q and radius roq (Fig.3 b), both damping ratio and settling time can be achieved. Selecting (zmin=0.25) and (a=0.36), we find the desired circular region D(q,r) as q ¼ 11; and r ¼ 10:64
ð10Þ
(iv) Guaranteed cost controlþ(iii): Although pole placement in a circular region puts interesting practical constraints on the transient response of power systems, in practice it might be desirable that the controller be chosen to minimize a cost function as well. The cost function associated with the faulty system (8) is Z 1 J¼ ðxT Qx þ uT RuÞdt ð11Þ 0
where Q ¼ QT 40; R ¼ RT 40 are given weighting matrices. With the state feedback (5), the cost function of the closed loop is Z 1 J¼ xT ðQ þ F T RF Þxdt ð12Þ 0
The guaranteed cost control problem is to find F such that cost function J exists and to have an upper bound J*, i.e. satisfying JrJ*.
4. Problem solution Design case 1: Theorem1. The closed loop faulty system (9) is reliably stabilized by state feedback u=Fx if there exist matrices Y=YT40, SARm n, such that the following LMIs AY þ Bi S þ o0;
8i
ð13Þ
have a feasible solution. The controller satisfying reliable stabilization is given by F ¼ SY 1
ð14Þ
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Proof. Define a positive definite Lyapunov function V ¼ xT Px; P ¼ PT 40. Evaluating the derivative of V along with (9) results in V_ o0, and thus stability, if PðA þ Bi F Þ þ o0; 8i
ð15Þ
Pre- and post-multiply (15) byP1, Fact #1, and letting Y=P1, S=FY; controller (14) is obtained. & Design case 2: Theorem 2. The closed loop faulty system (9) is reliably stabilized with a degree of stability a, by state feedback u=Fx if there exist matricesY=YT40, S such that the following LMIs A:Y þ aY þ Bi :S þ o0;
8i
ð16Þ
have a feasible solution. The controller achieving reliable stabilizationþdegree of stability is given by F ¼ SY 1 Proof. Following the same lines as above, replacing A with AþaI, the proof is straightforward. & Design case 3: Theorem 3. The closed loop faulty system (9) is reliably stabilized with poles lie in a disk D (q,r), by state feedback u=Fx if there exist matricesY=YT40, S such that the following LMIs " # d r2 Y o0; 8i ð17Þ AY þ qY þ Bi S Y have a feasible solution. The controller fulfilling reliable stabilization with regional pole constraint D(q,r) is given by F ¼ SY 1 Proof. The poles of a matrix A lie in the disk D (q,r) if and only if there exists a matrix P=PT40, such that [21] " # r2 P d o0 ð18Þ ðAY þ qIÞ P1 It follows that the stablility of the closed loop faulty system (9) with poles lying in the disk D(q,r) is achieved by replacing A with the closed loop matrix Aci=AþBiF, 8i. Pre- and post-multiplying by diag[P1,I], we get " # r2 P1 o0 ðAci þ qIÞP1 P1 This is a nonlinear matrix inequality. To convexify the result, we let Y=P1, S=FY and obtain the LMI (17) as desired. &
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Design case 4: Theorem 4. Consider the faulty system (9) and the cost function (12), if there exists a feedback matrix F and Y=YT40, S such that the following LMIs 2 3 r2 Y 6 AY þ qY þ B S Y 7 i 6 7 ð19Þ 6 7o0; 8i 4 YT 0 Q1 5 S
0
0
R1
hold for all possible faults, then the controller providing reliable stabilization with regional pole constraint D(q,r) and guaranteed cost is given by F ¼ SY 1 Moreover, the cost function has an upper bound J ¼ qxT0 Px0 ;
x0 ¼ xð0Þ
ð20Þ
Proof. The proof consists of two parts. In the first part, we show reliable stabilization with regional pole constraint D(q,r). In the second part, we provide the guaranteed cost. For the first part, we build on Theorem 3 and conclude that the controller achieving reliable stabilization with regional pole constraint is given by LMI (18). By Fact #2, this LMI is equivalent to the nonlinear matrix inequality r2 P þ ðAci þ qIÞT PðAci þ qIÞo0;
8i
ð21Þ
For the second part, we apply the results of [17] to reach that the faulty system (9) with cost function (12) is stable provided that r2 P þ ðAci þ qIÞT PðAci þ qIÞ þ Q þ F T RF o0;
8i
ð22Þ
hold, where (Q40, R40). It is clearly evident that if (22) is satisfied, it implies that (21) is fulfilled as well, Pre- and post-multiplying (22) by P1 and using the Schur complements operation, the nonlinear matrix inequalities (22) can be linearized and expanded to get LMI (19). This establishes that controller (19) achieves reliable stabilization with regional pole placement and guaranteed cost. To show that controller (19) provides an upper bound of the cost function, consider a Lyapunov function V ¼ xT Px; P ¼ PT 40 Notice that (22) is equivalent to ATci P
1 T q2 r2 Q þ F T RF Q þ F T RF T o0; P þ o Aci PAci o q q q q
Differentiating V(x(t)) with respect to time and using (23), we obtain Q þ F T RF V_ ¼ xT fPAci þ gxrxT x q
8i
ð23Þ
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Therefore, integrating both sides of the above inequality from 0 to N gives Z 1 Q þ F T RF xdtrV ðx0 ÞV ðxð1ÞÞ xT q 0 Since the stability of the system has already been established, x(t)-0 as t-N, it can be concluded that V(x(t))-0 as t-N. This completes the proof. &
5. Simulation results The proposed feedback control schemes by Eqs. (13), (16), (17), and (19) are used for the stabilization of the power system model described by the linear model (9). The linear matrix inequalities (13), (16), (17), and (19) are solved using the matlab LMI control toolbox [22] to get the feedback matrix for the design cases mentioned above. The results are summarized in Table 1. The closed loop poles for every design case representing all possible actuator failure are summarized in Table 2.
Table 1 Proposed controllers. Design case
Feedback matrix
1
F¼
0:1869
42:134
0:6186 0:00395
2:2198 441:94 7:742 0:0812 N/A, the desired relative stability is already achieved by F, above. 0:0293 110:7953 1:8771 0:0098 F¼ 0:4942 103:6476 1:9155 0:0101 For Q=I,R=I, 0:0074 120:8538 2:1108 0:0113 F¼ 0:4988 103:2079 2:0051 0:0111
2 3 4
Table 2 Closed loop eigenvalues for faulty and sound controllers (different designs) Design case
Comment
Closed loop poles
1
Only u1 is active(u2 fails) Only u2 is active(u1 fails) u1 and u2 are active (no failure)
0.8247j6.7015, 11.257j1.5856 1.1054, 8.094, 17.1757j10.042 2.2201, 8.64037j4.9658, 28.002
3
Only u1 is active(u2 fails) Only u2 is active(u1 fails) u1 and u2 are active (no failure)
4.2357j7.08, 10.777j2.94 1.6971, 3.3342, 10.73927j3.51 1.518, 10.12127j5.2864, 14.5624
4
Only u1 is active(u2 fails) Only u2 is active(u1 fails) u1 and u2 are active (no failure)
4.647j6.77, 11.0867j3.07 1.422, 3.71, 10.637j3.496 1.56, 10.147j4.67, 15.81
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It is clear from Table 2, design case#1 that the controller achieves reliable stabilizationþrelative stability for any possible actuator faults. Figs. 4 and 5 show respectively that using design 3 and 4, the closed loop poles locations satisfy the required performance, reliable stabilizationþregional pole placement, for all possible actuator failures. Next, the simulation results when using the proposed controllers are presented. For an impulse load increase causing Dd=0.1 rad, the initial condition of the SMIB system is taken as x0 ¼ ½0:1; 0; 0; 0T . Design case#3:closed loop poles 12
10
only u1 on only u2 on both u1 & u2 on
zeta=.25
pole region
imaj
8
6
4
2
0 −25
−20
−15
−10
−5
0
real
Fig. 4. Closed loop pole locations for all possible faults. Design 3: reliable stab.þpole region.
Design case#4:closed loop poles 12
10
zeta=.25
only u1 on only u2 on both u1 & u2 on pole region
imaj
8
6
4
2
0 −25
−20
−15
−10
−5
0
real
Fig. 5. Closed loop pole locations for all possible faults. Design 4: reliable stab.þpole regionþ guaranteed cost.
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angle dev,rad
only u1 on 0.1 0 −0.1 0
2
4
6
8
10
6
8
10
6
8
10
angle dev,rad
time,s only u2 on 0.1 0.05 0
0
2
4 time,s
angle dev,rad
u1 & u2 on 0.1 0.05 0
0
2
4 time,s
angle dev,rad
Fig. 6. Angle deviation response. Design case 1.
only u1 on
0.1 0 −0.1 0
2
4
6
8
10
6
8
10
6
8
10
angle dev,rad
time,s only u2 on
0.1 0.05 0 0
2
4
angle dev,rad
time,s u1 & u2 on
0.1 0.05 0 0
2
4 time,s
Fig. 7. Angle deviation response. Design case 4.
The simulation results when using the first controller, design case #1, is shown in Fig. 6. From the figures it can be seen that the rotor angle deviation converges to 0 in about 5 s. While, Fig. 7 shows the response when the fourth controller, design case 4, is used. It is evident that reliable stabilizationþregional pole placementþguaranteed cost are satisfied, the angle deviation decays to 0 within 3 s, for any possible failure. This manifest the superiority of design 4 over design 1. To show the effectiveness of the design case #4, the conventional linear quadratic regulator (LQR) is used. For the same values used before, the matrices Q and R are taken
H.M. Soliman et al. / Journal of the Franklin Institute 348 (2011) 884–898
angle dev,rad
896
only u1 on
0.1 0 −0.1
0
2
4
6
8
10
6
8
10
6
8
10
angle dev,rad
time,s only u2 on
0.1 0.05 0
0
2
4
angle dev,rad
time,s u1 & u2 on
0.1 0.05 0
0
2
4 time,s
Fig.8. Angle deviation response. Conventional LQR.
as I; the conventional optimal feedback regulator is obtained as 0:0620 8:9587 0:5248 0:9761 F¼ 0:4697 69:8057 1:2027 0:0910 The simulations for different actuator failures using the conventional LQR are shown in Fig. 8. It is clear that the standard LQR is very vulnerable in face of actuators failure. 6. Conclusions This article presents a new design approach for fault-tolerant stabilization of a power system. The two-channel control encloses two stabilizers acting on the excitation and TCSC of a synchronous alternator operating in an interconnected power system. Four different control schemes to satisfy desired dynamic performance constraints are derived. Taking the guaranteed cost constraint into account, the problem of fault-tolerant controller design with regional pole constraint is tackled by LMI approach for a power system subject to actuator failures. Simulation results show that using the proposed controllers, the system performs fairly well. It provides satisfactory stability, transient property, and quadratic cost performance despite of possible actuator faults. The proposed controller outperforms the standard LQR. Since power systems usually operate on the nominal loading condition, the present manuscript develops four theorems to achieve reliable stabilization of the study system under nominal loading. Development of new theorems is underway to achieve reliable stability at different loading points. Acknowledgment Dr M. S. Mahmoud would like to thank the Deanship of Scientific Research (DSR) at KFUPM for research support through project IN100018.
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