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./ NONLINEAR STABILITY ANALYSIS AND ADAPTIVE CONTROL OF POWER SYSTEMS by Jamil Mahdi ALMajid B.S., University of Colorado, 1988 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering and Computer Science 1991 L\Y j d :i
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This thesis for the Master of Science degree by Jamil Mahdi ALMajid has been approved for the Department of Electrical Engineering and Computer Science by Marvin F. Anderson William R. Roemish '7219/ Date
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ALMajid, Jamil Mahdi (M.S., Electrical Engineering) Nonlinear Stability Analysis and Adaptive Control of Power Systems Thesis directed by Professor Edward T. Wall In this thesis a stability analysis of power systems, using the second method of Lyapunov, and a control design for power systems using adaptive modelfollowing control design are presented. The energy metric algorithm (EMA) is used to generate a Lyapunov function for the timevarying nonlinear system. The algorithm is based upon a generalization of the total energy of a system and is applied to the equivalent twomachine system as a case study. The second part of this thesis presents a method for stabilizing the equivalent twomachine system using an adaptive modelfollowing control (AMFC). The algorithm presented in this thesis is based qn the model reference adaptive control technique using the PopovLandau method. The design procedure is simple, effective, and does not require the accurate modeling of the dynamic system. The problem of perfect asymptotic adaptation is interpreted as a stability problem. Through the use of hyperstability, in conjunction with the properties of the dynamic system, the largest family of adaptation laws assuring the stability of the overall adaptive control is obtained and the most suitable adaptation law for a specific application may be chosen. The form and content of this abstract are approved. I recommend its publication. SignedEdward T. Wall iii
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CONTENTS CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . 1 2. EQUIVALENT TWOMACHINE SYSTEM DYNAMIC EQUATION . . . . . . . . . 3 2.1. Introduction . . . . . . . . . . . . . . . . 3 2.2. Derivation of the Swing Equation . . . . . . . 3 2.3. Synchronous Machines . . . . . . . . . . . 8 2.3.1. The Park Transformation . . . . . . . . 10 2.3.2. Revised Equations of a Synchronous Machine . . . . . . . . 13 2.3.3. Transient State . . . . . . . . . . . 14 2.3.4. PowerAngle Calculation of a SalientPole Machine . . . . . . . . . 15 2.4. Calculation of Change of Field Flux Linkage . . . 17 2.5. Calculation of Positive Sequence Damping . . . . 21 2.5.1. Accelerating Power . . . . . . . . . . 25 3. THE GENERATION OF LYAPUNOV FUNCTIONS AND STABILITY ANALYSIS . . . . . 26 3.1. Introduction . . . . . . . . . . . . . . . 26 3.2 Generation of Lyapunov Functions by the Energy Metric Algorithm . . . . . . . . 27
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3.3. Stability Study Using the Method of Lyapunov . . . . . . . . . 30 3.4. Equivalent TwoMachine System Study . . . . . 30 3.4.1. Special Cases . . . . . . . . . . . . 32 4. ADAPTIVE CONTROLLER DESIGN . . . . . . . 40 4.1. Introduction . . . . . . . . . . . . . . . 40 4.2. Dynamic Equation of the Plant in the StateSpace ...................... 41 4.3. The Reference Model . . . . . . . . . . . 44 4.4. Perfect ModelFollowing Control ............... 45 4.5. Adaptive ModelFollowing Control (AMFC) ...... 48 4.6. Adaptive Law and Stability .................... 51 4.6.1. Hyperstability of the AMFC . . . . . . 56 5. SIMULATION RESULTS . . . . . . . . . . . . 60 5.1. Introduction . . . . . . . . . . . . . . . 60 5.2. Numerical Equations of the Plants . . . . . . . 60 5.3. Numerical Equation of the Reference Models . . . . . . . . . . . 62 5.4. Results . . . . . . . . . . . . . . . . . 64 5.4.1. Without Adaptation . . . . . . . . . . 64 5.4.2. Step Input . . . . . . . . . . . . . 65 5.4.3. Square Wave Input . . . . . . . . . . 66 6. CONCLUSIONS . . . . . . . . . . . . . . . . 95 v
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APPENDIX A CALCULATION OF f(Xp1 (t)) AND P D (Xp1 (t)) . . . . 97 B. GAIN MATRICES Kp, Ku, AND D CALCULATIONS ............................ 100 C. GAIN MATRICES G, F, F, M, M, AND N SELECTION . . . . . . . . . . . . . . . . 109 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . 111 vi
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ACKNOWLEDGEMENTS This study would not have been possible without the cooperation and support of many people to whom I will always be grateful. I especially wish to express my sincere gratitude to my advisor Dr. Edward T. Wall for his support and guidance during the preparation of this thesis. I would also like to thank Professor Marvin F. Anderson and Dr. William R. Roemish for their encouragement and also for serving on my committee. Also, I would like to thank Mr. Yehia Dib, Mr. David Ince, and Mr. Mohammed Mashroom for their help. Finally, this thesis is dedicated to my family for their patience, support and encouragement. vii
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CHAPTER 1 INTRODUCTION In the early days of power system development, interruptions to electric service were more frequent than they are today. However, in the economy of today a power system interruption is very serious and costly. This greater dependence on a continuous supply of electrical energy has resulted in the need to build into modem power systems a high degree of reliability and performance. Power system stability is defined as the property of a power system which insures that it will remain in operating equilibrium through normal and abnormal conditions. The powersystem stability problem is concerned with the performance of the various synchronous machines and loads on the system during unexpected faults and switching conditions. In this research, the transient stability analysis of a power system, using the second method of Lyapunov together with an adaptive control scheme are developed and synthesized. The energy metric algorithm (EMA) [1, 2] for the generation of a Lyapunov function is applied to an equivalent twomachine system. The adaptive control scheme used for the stabilization of the equivalent two machine system is based on an adaptive modelfollowing control systems (AMFC) technique [8, 9], where the design of the adaptation mechanism is
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based on use of Popov hyperstability analysis. The objective of the adaptive control design is to cause the equivalent twomachine system to follow a given desired performance specified by a reference model. The control law is composed of two parts. The first part of the controller is linear in the system state and the reference input, which insures perfect modelfollowing control. The second part of the control is produced by an additional adaptive feedback loop, where the adaptive gains are determined using hyperstability theory. The adaptive feedback actively compensates for the parameter variations and uncertainties in the dynamical equations of the equivalent twomachine system. The novelty of this control strategy is that the adaptive controller requires neither accurate modeling nor precision dynamic coefficients, and the asymptotic stability of the overall system can be assured automatically using the hyperstability and positivity concepts. 2
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2.1 Introduction CHAPTER2 EQUIVALENT TWOMACHINE SYSTEM DYNAMIC EQUATION The first step in the design of the control law for a power system is the derivation of an analytical dynamic model for the power system. In this chapter, the structure of the dynamic equation of the equivalent two machine system is derived. In section 2.2, using the laws of rotating bodies, the swing equation is derived. In section 2.3, a threephase salientpole synchronous machine is introduced. The powerangle equation is derived using the Park transformation. Section 2.4, will discuss the calculation of change of field flux linkage. Finally, in section 2.5, the derivation of the positive sequence damping equations and the complete swing equation are given. 2.2 Derivation of the Swing Equation [3] Since a synchronous machine is a rotating body, the laws of mechanics applying to rotating bodies apply. Rotation An angle is define as the ratio of arc s to radius r, thus s 6=r (radian) 2.1
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Angular velocity is de (a)=2.2 dt and angular acceleration is de.> d2e ==2.3 dt dt2 The relations between the angular modes and linear modes of a particle of a rotating body at distance r from the axis of rotations are displacement s = re 2.4 velocity v=rc.> 2.5 acceleration a= rex 2.6 The torque on a body due to a tangential force F at a distance r from the axis of rotation is T=rF 2.7 The total torque is T= frdF 2.8 dF = adm = rcxdm where m is the mass of the particle. Since this force acts with lever arm r, the torque required for the particle is 4
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and that required for the whole body is where is the moment of inertia. The work done on a body through an angle de is Also and the power is Then in rotary motion, The angular momentum is W= Jrde T= dW de de P= T= Tcv dt M =lev 5 2.9 2.10 2.11 2.12 2.13 2.14
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Also M= fTdt= = 2.15 The kinetic energy is then 2.16 The Swing Equation From equation (2.9) Ia. = T 2.17 and 2.18 where T is the sum of all the torques acting on the machine. Next, let T m = the shaft torque, corrected for torque due to rotational losses, and Te = the electromagnetic torque. Both of these torques are taken as positive for generator action, that is, with mechanical input and electrical output. The net torque, which produces acceleration, is the algebraic difference of the accelerating shaft torque and the retarding electromagnetic torque 2.19 6
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In the steady state this difference is zero, and there is no acceleration. During the transient state, however, a difference exists, and there is a net acceleration or retardation. Define where 1 is the rated normal synchronous speed, then da de =(a) dt dt 1 and Substituting equation (2.22) in equation (2.18) d2a 1=T dt2 or Multiplying this equation by the speed , there results where M = I is the angular momentum. P m = Te is the mechanical power input. 7 2.20 2.21 2.22 2.23 2.24 2.25
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Pe = T6 is the electrical power output. Pa is the accelerating power. 2.3 Synchronous Machines [ 4] A threephase salientpole machine will be considered as shown in Figure 2.1. Such a machine has four windings: the field winding and three armature windings. The terminal voltage of any one of these windings is d v= n + "' dt 2.26 where r is the resistance, i is the current, and lJr is the flux linkage of the winding. By definition 2.27 where L is the inductance. Denoting the armature windings by subscripts a, b, c, and the field winding by f then dlJra v=n+a a dt 2.28a 2.28b dlJrc v = rz +c c dt 2.28c 2.28d 8
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Axis phase c Axis phase a Direct axis Quadrature axis Figure 2.1. A threephase salientpole machine. 9
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where 2.29a 2.29b ' = L i + L b ib + L i + L 1 i, ca a c cc c c J 2.29c 2.29d where the Li i 1 s are the selfinductances, and the Lij 1 s are the mutualinductances between two armature phases. 2.3.1 The Park Transformation [5] Let the actual armature phase currents ia, ib, and ic be replaced by new fictious currents id, iq, and i0 and related to them by the following equations 2.30a 2.30b 2.30c 2.30d Similarly define 10
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where Ld = directaxis synchronous inductance Lq = quadratureaxis synchronous inductance L0 = zerosequence inductance M1 mutual inductance Similarly, define dlfr v = ri + __ q + (J)lJr q q dt d Steady State Equations Let the field be excited with constant current and let the armature current be steady and of positive sequence 11 2.31a 2.3lb 2.31c 2.3ld 2.32 2.33 2.34 2.35
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id = I cos P = constant = Id iq = I sin p = constant= Iq where Then the flux components are 1Jr q = Lq Iq = constant $0 = 0 = constant The Park voltage equations (2.32) and (2.33) are where vd =rIdC&> Wq =rIdC&>Lqlq =ridxq Iq = vd = constant vq = riq + C&> 1Jrd = r Iq + C&>Ldld + = r Iq + xd Id + Eq = Vq = constant is the excitation voltage 12 2.36a 2.36b 2.36c 2.37a 2.37b 2.37c 2.37d 2.38 2.39 2.40
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2.3.2 Revised Equations of a Synchronous Machine The algebraic signs of the armature currents id and iq will be revised. Consequently, positive iq and id will now represent generator action with lagging current. Moreover, to make the equations applicable to a machine with solid, round rotor, a circuit denoted by subscript g will be assumed present on the quadrature axis of the rotor. This new circuit is analogous to the main field windings denoted by f on the direct axis of the rotor. The revised equations are 2.41 2.42 2.43 2.44 Similarly 2.45 dlJI v = ri + __ q + q q dt q 2.46 2.47 2.48 13
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2.49 2.50 2.3.3 Transient State Consider a fast change of armature current. During such a change, the flux linkage v1 remains substantially constant. A new fictious internal armature voltage will be defined I 6> Ml E = lJrl q L 'II 2.51 This voltage will remain constant during the change of armature current. The difference between EqE/ is 2.52 Given 2.53 where L/ = directaxis transient inductance, and by the use of equation (2.43), equation (2.52) becomes 2.54 14
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or During rapid changes, equations (2.47) and (2.48) are still valid. Using equation (2.55), equations (2.47) and (2.48) become 2.3.4 PowerAngle Calculation of a SalientPole Machine Steady State 2.55 2.56 2.57 The electrical power output Pe may be expressed in terms of the terminal voltage V, the excitation voltage Eq, and the angle a between V and Eq. From equations (2.47) and (2.48) with Ed= 0 Vdr + (EqVq)xq Idr2 + xdxq 2.58 I = (EqVq)r + Vdxd q r2 + xdxq 2.59 The power output is P=Vdld+VI e q q 2.60 Substituting for Id and Iq in equation (2.60) and rearranging 15
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p = Eq(Vq r + Vdxq)(vi+ v:)r + Vd Vq (xdxq) 2 .61 e r2 + xd xq Since vq = v cos a and Vd = Vsin a 2.62 and 2.63 then 2.64 Substitution of equations (2.62), (2.63), and (2.64) into equation (2.61) gives Eq V (r cos a + xq sin a)V2 r + V2 (xdxq) sin 25 pe = 72 + xdxq 2.65 Next, let 2.66 and 2 2 2 Z = r + x q q 2.67 then 2.68 and 16
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Z r xdx P =E V......!l..sin(a+y)V2+V2 qsin2a 2.69 8 q z2 z2 2z2 If the machine is connected to an infinite bus through a network having shunt branches, the network and the voltage of the infinite bus can be replaced by an equivalent e.m.f. and series impedance. If the series resistance is negligible, then equation (2.35) simplifies to E V x x p = _qsin a + V2 d q sin 2 a e xd 2xdxq and the transient term is E 1v x x 1 Pe = xq I sin a V2 q I d sin 2a a 2xaxq 2.70 2.71 If an external reactance xe is connected in series with the armature, then equation (2.71) becomes 2.72 2.73 2.4 Calculation of Change of Field Flux Linkage Applying the Kirchhoff voltage law to the field circuit gives I dtjll E = R1z'l+e% dt 2.74 17
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where E 1 = exciter armature e.m.f. u R1 = fieldcircuit resistance Multiplying equation (2.74) by CA>M1! R 1 gives Define I Lu T = do R I and by the use of equations (2.49) and (2.51), there results and 2.75 2.76 2.77 2.78 where Eex = exciter voltage referred to the armature circuit. Substitution of equations (2.49), (2.77), and (2.78) into equation (2.75) gives I 1 dEq E=E+Tdoex q dt 18 2.79
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or 2.80 Eq depends upon E/ and !d. From equation (2.54) 2.81 and from equation (2.58), neglecting the resistance 2.82 where 2.83 By the use of equations (2.82) and (2.83), equation (2.81) becomes I x 1 (x x ) E = E d d V cos 5 q xl q xl d d 2.84 given I T I= xd T I d X do d 2.85 where r/ is the transient shortcircuit time constant. Multiplying equation (2.84) by 1/ and using equation (2.85) there results 19
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E E I V(x x 1 ) q = __!!_ a a cos a Td/ T/ x/Td/ Using equation (2.86), equation (2.80) becomes dEl __ q dt I V(x x ) a a cos a IT I xd do 2.86 2.87 If an external reactance xe is connected in series with the annature, then equation (2.87) for the flux decay becomes 2.88 or 2.89 The governor action can be approximated by an equivalent firstorder system as dPm T +P =P Ka e dt m mo 2.90 and 2.91 20
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where Te = equivalent time constant of the governor system pm = mechanical power input pmo = steady state mechanical power input K governor amplification factor Yt 1/ Te Yz KITe 2.5 Calculation of Positive Sequence Damping For large slip Rrd Is is negligibly small and the impedance seen from the armature terminals is the subtransient reactance x/! Thus 2.92 and 2.93 The starter current is 2.94 which, at small slip, becomes 21
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The rotor current is I= E s j (xe + x/) I =I _JX...;;;_d_ s R,.d Is Next substitution of equation (2.95) into (2.97) gives and the damping power is Using equations (2.92) and (2.93) gives and the time constant Td/1 is 22 2.95 2.96 2.97 2.98 2.99 2.100
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2.101 from which R,.d = (a) (x I_ x I A T II d d J do 2.102 substituting (2.102) in (2.99) gives 2.103 If the rotor is unsymmetrical, the damping power fluctuates at twice the slip frequency and the resultant fluctuating damping power is found by substituting for E in equation (2.103), and in the corresponding quadratureaxis equation, the directand quadratureaxis components Ed= E sin a 2.104 Eq=Ecosa 2.105 Finally, the directaxis damping power is 2.106 and the quadratureaxis damping power is 23
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2.107 Therefore, the total damping power P D is P Dd + P Dq where PD = s (a) = E 2.108 damping power slip = 1/360/ dS /dt; I = frequency of infinite bus 21t/ v = voltage of infinite bus machine reactances external reactance open circuit subtransient time constant angle by which the machine leads the angle of the infinite bus 24
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2.5.1 Accelerating Power The damping power P D is taken into account as an additional component of the positive sequence electric output power. The accelerating power (replacing Pa in equation (2.25)) is 2.109 2.110 Rewriting equation (2.25), the swing equation becomes 2.111 In the next chapter, the generation of Lyapunov functions by the energy metric algorithm (EMA) using the swing equation as an example, and the study of the stability of the nonlinear timevarying swing equation will be shown. 25
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CHAPTER3 THE GENERATION OF LYAPUNOV FUNCTIONS AND STABILITY ANALYSIS 3.1 Introduction For a given power system, stability is usually the most important concept to be determined. If the system is linear and timeinvariant, many stability criteria are available. Among them are the Nyquist stability criterion, Routh stability criterion, etc. If the system is nonlinear and/or timevarying, however, then such stability criteria do not apply. A M. Lyapunov published a general theory of dynamic stability which is known as the second or direct method. This approach of Lyapunov has been widely recognized as the principal tool in analyzing linear and nonlinear stability criteria since it does not require a solution to the system equation. It is applicable to all control systems that can be represented by a mathematical model consisting of a system of differential equations. The second method of Lyapunov may be viewed as an extension of the principle of conservation of energy, which states that the total energy is constant in a stable conservative configuration, and is decreasing in a stable nonconservative one. The scalar function of the state variables and its total derivative, commonly called a Lyapunov function, can be treated as an extension of the energy concept. Since an energy concept is available, and
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since Lyapunov used energy definitions in his original investigations of stability, it appears quite reasonable to use the definitions of energy in the search for a fundamental approach to the generation of Lyapunov functions in stability analysis. This chapter is organized as follows. In section 3.2, the generation of Lyapunov functions by the use of the Energy Metric Algorithm (EMA) is presented. An outline of the procedure is listed. Section 3.3 presents a stability study using the second method of Lyapunov. It will be shown how to determine the stability region. Section 3.4 presents the equivalent two machine system study that includes the generation of Lyapunov functions and the stability conditions for three cases: Case 1: Neglecting transient saliency Case 2: Considering transient saliency, with constant damping power Case 3: Considering transient saliency, with variable damping power. At the end of the section, computer simulations for the stability regions for the three cases are presented. 3.2 Generation of Lyapunov Functions by the Energy Metric Algorithm [1, 2] The Energy Metric Algorithm A procedure for the generation of Lyapunov functions for time varying nonlinear systems is described. The procedure permits the generation of Lyapunov functions directly from the system equations. The method is based upon a generalization of the total energy of a system and is 27
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obtained from the line integral of a composite differential oneform derived directly from the system equations. Outline of the Procedure Step 1 Describe the system as a set of firstorder timevarying nonlinear differential equations. .i1 = F1 (x, t), 1 i n, t 0 where x is the vector (x1 x2 xn). Step 2 Form a set of differential equations of integral curves by taking quotients of the firstorder equations in (3.1) and eliminating dt in each case. dx1 F1 (x, t) = lj (x, t)' There will be n (n1) /2 such equations. Step 3 j>i The differential equations of integral curves are converted to the same number of differential oneforms by multiplying and clearing the denominator terms. lj (x, t) dx1 F1 (x, t) dxi = 0, j>i 28 3.1 3.2 3.3
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Step 4 By addition and subtraction these differential oneforms are reduced to a single oneform. cu = cu1 (x, t) dx1 + cu2 (x, t) dx2 3.4 + cu3 (x, t) dx3 + .. + n (x, t) dxn Step 5 A line integration of the resulting oneform is taken. The path chosen is along the state coordinates, with time t held constant. The result is a state function of the form v = f (a) = L'l (a)l c 'tl' o, ... o, t) d'tl + J:2 c..>2 (xl' '!2 0, ... 0, t) d't2 + + J:n (l)n (xl' x2, ... xn1' 'tn' t) d'tn With suitable bounds on t, this function is a candidate for a Lyapunov function for the timevarying system (3.1 ). Step 6 The total derivative of V with respect to t is taken to obtain V. The proper theorems must then be applied to V and V to determine the type of stability. 29 3.5
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3.3 Stability Study Using the Second Method of Lyapunov [6, 7] To apply the second method of Lyapunov, the system equations must first be in the state variable form. The Lyapunov function V(x) obtained by any method must satisfy the following conditions: 1. V(x) and its first partial derivatives are continuous. 2. V(x) is positive definite in a region around the equilibrium point. 3. V is negative definite in a region around the equilibrium point, or negative semidefinite in a region around the equilibrium point and not identically equal to zero on a solution of the system other than the equilibrium point. The region of asymptotic stability is defined by the largest closed curve V = in the domain where V s: 0. To determine the stability region, it is necessary to determine first the unstable equilibrium state closest to the stable equilibrium state. The surface v = passing through the unstable equilibrium state, together with the part of the plane perpendicular to the x1axis with negative x2 will give a stability region. By completing the quadratic terms in V(x), a variable p is introduced. The union of all stable regions for the admissible values of p is the required stability region. 3.4 Equivalent TwoMachine System Study The system under consideration consists of a synchronous generator connected to an infinite bus through a double transmission line. One of the transmission lines develops a ground fault and is isolated from the system in 30
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a time t by the circuit breakers at the two ends. The problem is to find a stable region of operation for the generator. All resistances are neglected. The system is described by equation (2.111) The positive sequence damping PD (a) is given by equation (2.108) The electrical power output Pe considering transientsaliency is given by equation (2.72). The equation for the flux decay is given by equation (2.88) 31 3.6 3.7 3.8 3.9
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The governor action is given by an equivalent firstorder system given by equation (2.91) 3.4.1 Special Cases Case 1 dPm T +P =P Ka e dt m mo dPm = _..., (P P ) ..., a dt 11 m mo 12 Neglecting Transient Saliency 3.10 Assuming constant damping, constant field flux linkage and constant input power, there results The system equations are 3.11 Using the energy metric algorithm, the Lyapunov function is 32
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Completing the quadratic terms To eliminate the term x1 x2 in V proceed as follows = DP 2M' V is nonnegative if p 2 /2M. where 0 p D 2 Pot P 313 V = (D(} )x2 xt[sin (x1 + 50)sin 50 ] Case 2 Considering Transient Saliency with Constant Damping Power Assuming constant field flux linkage and constant input power, there results 33
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The system equations are Mx2 = Dx2 P01 [sin(x1 + 50)sin 50 ] + P02 [sin(2x1 + 250)sin250 ] Using the energymetric algorithm, the Lyapunov function is V = Mxi + P01 [cos 50 cos (x1 + 50)x1 sin 50 ] Completing the quadratic terms To eliminate the term x1 x2 in V set a;= DP 2M' where 0 P D + P01 [cos 50 cos(x1 +50)x1 sin 50 ] P02[cos250 cos(2x1 + 250)2x1sin2C>0 ] Vis nonnegative if a P2/2M. 34 3.14 3.15
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Case 3 V= (DP)xi!xt{P01[sin(x1 + 30)sin30 ] P02[sin(2x1 + 230)sin230]} Considering Transient Saliency with Variable Damping Power 3.16 Assuming constant field flux linkage, and constant input power, there results The system equations are M:i2 = [Pdlsin2(x1 + 30 ) + Pd2cos2(x1 + 30)]x2 P0t(sin (x1 + 30 ) sin 30 ] + P02 [ sin(2x1 + 230)sin2 30 ] Using the energymetric algorithm, the Lyapunov function is 35 3.17
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V= {[Pdlsin2(xl + ao) + Pd2cos2(xt + ao)]P}x; p {P01 [sin(x1 + a0)sin a0J P02 [sin(2x1 + 2a0)sin2a0]} 3.18 3.19 The stability regions case 1, case 2, and case 3 are shown in Figures 3.1, 3.2, and 3.3 respectively. 36
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rn ell .... 't;j ell '"' .d ..... (.;.) l p = 0; A _! D t' 2 . . . P=D 2 1 0 1 2 1 0.5 0 0.5 1 1.5 X1 in radians Figure 3.1. Stability region neglecting transient saliency forM = 1; D = .31; P01 = 3.189; 50 = 27.5. 2 2.5 3
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{fl l=l as ..... "C as J.4 l=l ..... !.Jl 00 p 0; : p .! D 2 P=D 2 1 2 .... 3L_ ____ Figure 3.2. X1 in radians Stability region considering transient saliency with constant damping forM = 1; D = .31; P01 = 3.189; P02 = 0.685; a0 = 27.5.
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rn ctl ...... "C ctl '"' 1::1 ...... w \0 p = 0; . A _!_ D t' 2 . . .. P=D 2 1 0 1 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 3 X1 in radians Figure 3.3. Stability region considering transient saliency with variable damping forM = 1; Pd1 = 0.01161; P d2 = 0.00890; P01 = 3.189; P02 = 0.685; 50 = 27.5.
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CHAPTER4 ADAPTIVE CONTROLLER DESIGN 4.1 Introduction Model reference adaptive control (MRAC) is the most widely used adaptive control and is relatively easy to implement when compared to other adaptive methods. The concept of MRAC is based on selecting an appropriate reference model which incorporates within it the system design specifications together with an adaptation mechanism that is driven by the error between the reference model output and the plant output. The adaptation mechanism modifies the feedback gains of the controller in order for the closed loop performance characteristics to closely match the desired performance characteristics embodied in the behavior of the reference model. A continuoustime adaptive modelfollowing control algorithm [8, 9] is considered in this chapter for a class of nonlinear timevarying plants, where the hyperstability theorem of Popov [10] is used to determine the stability bound. The procedure presented is simple and effective. Moreover, it leads to a control system that assures asymptotic stability of the system. Accurate modelling and the solution of the complicated dynamical equations of the system is not required in this approach, which tends to make the control system design simple and attractive. The continuoustime
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adaptive controller considered is capable of compensating for the uncertainties and nonlinearities in the system dynamical equation. This chapter is organized as follows. In section 4.2, the nonlinear, timevarying plant of the equivalent twomachine system is described. In section 4.3, a linear timeinvariant reference model is given which allows the adaptive controller plant to follow the reference model as closely as possible. In section 4.4, perfect modelfollowing control is presented, which guarantees the existence of a solution to the linear modelfollowing control. In sections 4.5 and 4.6, an adaptive modelfollowing control and the adaptation laws are presented. 4.2 Dynamic Equation of the Plant in the StateSpace The dynamic equation of the process is given by equation (3.6) as or For a system with a constant voltage and therefore constant field flux linkages, and neglecting governor action, equation (4.1) becomes 41 4.1 4.2
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If the plant state space vector is defined as Xp = [XPl' xnt with XPl = a and Xn = a, then in normalized canonical phasevariable form, equation ( 4.2) becomes XPI = Xn(t) 1 1 Xn = Mf(Xp1(t))M PD (Xp1(t))Xn(t) where 4.3 4.4 Now, if the synchronous machine has a control Up(t) added to the constant input power P m' then the system of equations as given by equation ( 4.3) become Next define and XpJ = XP2(t) 1 1 Xn = Mf(Xp1(t))MPD(Xp1(t))Xn(t) + Up(t) 0 Ap(t) = !t(Xp1(t)) Bp(t) = [ l 42 4.5 4.6 4.7
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To illustrate this, two out of the three cases shown in section 3.5 will be discussed. Case 1 Transient Saliency with Constant Damping Power Taking the equilibrium state as the origin of the system, equation (3.8) becomes and Then f(Xp1(t)) = P01 [sin (Xp1(t) + 30)sin(B0)] + P02 [sin(2Xp1(t) + 230)sin (230)] and P D = constant Case 2 Transient Saliency with Variable Damping Power Taking the equilibrium state as the origin of the system, equation (3.7) becomes PD(Xp1(t)) = Pd1 sin2(Xp1(t) + 30 ) + Pd2 cos2(Xp1(t) + 30 ) and 43 4.8 4.9 4.10 4.11 4.12
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f(xp1(t)) =P01[sin(xp1(t) +50)sin(50)] 4.13 + P02[sin(2xp1(t) + 250)sin(250)] 4.3 The Reference Model A linear timeinvariant decoupled reference model is chosen in order to obtain the desired performance. The desired characteristics, such as rise time, overshoot, damping, and settling time are specified in the reference model system equation The state space representation of the reference model equation (4.11) is where xm is the model state vector r is the model input vector. Then A =[ 0 1 l m <..>2 The reference model is asymptotically stable provided that (Am, Bm) is a controllable pair, and Am is a Hurwitz matrix. 44 4.14 4.15 4.16
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The control design objective is to have the plant state Xp(t) follow the reference model state Xm (t). To do this let the state error vector e(t) be defined as 4.17 Then the objective is to constrain e(t) to asymptotically approach zero. That is lim e(t) = 0 4.18 t+ 00 4.4 Perfect ModelFollowing Control The design problem for linear modelfollowing control, consists of finding matrices Kp and K" so that the plant state follows the model state. In other words, the transfer matrices of the controlled plant and of the reference model are constrained. to be equal. Consider next, the linear modelfollowing control system shown in Figure 4.1. The reference model is 4.19 The plant to be controlled is 4.20 And the plant control input is 4.21 45
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r ..... Model Xm .... + e Ku ...._+. .... Plant Xp ,. ... Kp ..... ..... Figure 4.1. Linear control system. 46
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where Kp and K14 are the feedback and feedforward gains respectively, and Am, Bm, Ap, Bp, Km, and Ku, are constant matrices of appropriate dimensions. The pairs (Am, Bm) and (AP, Bp) are stabilizable, and furthermore, Am is a Hurwitz matrix. The error differential equation must satisfy the condition e(t) = im(t) Xp(t) =Am Xm(t)Bm r(t)AP Xp(t) + Bp KP Xp(t)Bp K14 r(t) e(t) =Am e(t) +(AmAp + Bp Kp) Xp(t) + (BmBp Ku) r(t) For the perfect modelfollowing (PMF) conditions to be satisfied, it is required that e(t) = e(t) = o therefore equation ( 4.22) becomes as t+ 00 (AmAp + Bp KP) Xp(t) + (BmBp Ku) r(t) = 0 which implies 4.22 4.23 4.24 4.25 4.26 Equations ( 4.25) and ( 4.26) will have solutions for K P and Ku if and only if rank Bp =rank [ Bp, (AmAp)] =rank [ Bp, Bm] The solutions for equations ( 4.25) and ( 4.26) will be obtained by the use of. the Penrose pseudo inverse of B P given by 47
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4.27 Then K P and K, are given by 4.28 4.29 The Erzberger Conditions for (P:MF) [11] Substituting the values of Kp and K, in equations (4.25) and (4.26) gives the Erzberger conditions (IBp B;) (Am Ap) = 0 (IBp B;) Bm = 0 where I is the identity matrix. Note that the Erzberger conditions are satisfied if the reference model and the plant are structurally related. 4.5 Adaptive ModelFollowing Control (A:MFC) The linear modelfollowing control presented in section 4.4 gives good results, but cannot overcome the nonlinearities of the plant 4.30 4.31 parameters, or the variation of parameters due to unknown faults or disturbances to the synchronous machine. For these situations, an AMFC system should be considered in order to force the plant to follow the reference model. An AMFC system can be used only if PMF conditions exist. 48
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Two possible implementations of the AMFC system can be used: (1) parameter adaptation, and (2) signal synthesis adaptation. Since the two are equivalent in principle, signal synthesis adaptation will be used. The parallel AMFC system with signal synthesis adaptation shown in Figure 4.2 is described as follows The reference model is The controlled plant is And the generalized state error is e(t) = Xm(t)Xp(t) The differential equation of the error e (t) may be expressed in the form e(t) =Am e(t) +(AmAp) Xp(t) + Bm r(t)Bp U(t) U(t) = u1 (t) + U2 (t) where U1 (t) is the linear modelfollowing control which guarantees the perfect modelfollowing given by equation ( 4.21) as U1 (t) = Kp Xp(t) + Ku r(t) 4.32 4.33 4.34 4.35 4.36 and U2 (t) is the control signal produced by the adaptive feedback loop which is determined by using hyperstability and positivity theory. It will be assumed that A P and B P are timeinvariant during the adaptation process. 49
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Model + ,,. rr ..... Ku ... + U1 .. u ... Plant ,..+ ... +J l U2 / Kp Adaptation .::Mechanism .::...Figure 4.2. Parallel AMFC system with signal synthesis. 50
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4.6 Adaptation Law and Stability The objective of the adaptation mechanism is to generate the adaptation laws Jl. Kp (e, t) and Jl. Ku ( e, t) to ensure that the generalized state error e (t) goes to zero under certain conditions. Landau [8, 9] used the hyperstability and positivity approach to obtain the largest possible family of adaptation laws which will assure asymptotic stability of the AMFC system, from which the most desired adaptation law may be selected. Theorem 4.1 (Popov Hyperstability) [10] Consider a feedback system which can be split into two blocks as shown in Figure 4.3. The linear timeinvariant feedforward path is described as e(t) =A e(t) + B W(t) 4.37 V(t) = De(t) 4.38 W1 ... Linear v ... timeinvariant w Nonlinear timevarying ..... Figure 4.3. Equivalent feedback system. 51
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where e is a n dimensional vector and v, w are m dimensional vectors. The pair (A, B) is completely controllable and the pair (A, D) is completely observable. The feedback path is described as follows W = f(V('t), t) 0< 't' < t The system will be asymptotically hyperstable if and only if 1. 11 (0, tt) = fot1 VT('t') W d't' a2 where a2 is a finite positive constant. 2. The transfer matrix of the linear timeinvariant block Z(s) = D(slAr1 B 4.39 4.40 4.41 4.42 is strictly positive real (SPR). Equivalently there exist a P and H such that p A + AT p = H 4.43 4.44 where P and H are symmetric positive definite matrices. In order to apply the Popov hyperstability theorem, the equivalent feedback representation of the AMFC system, which is the parallel AMFC system error and control given by equations ( 4.35) and ( 4.36), respectively need to be found. The error is 4.45 52
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The control law is U(t) = U1 (t) + U2 (t) U1 (t) = KpXp(t) + Kur(t) U2(t) = llKp(e, t)Xp(t) + llKu(e, t)r(t) KP = B; (AmAP) 4.46 4.47 4.48 4.49 4.50 where the two timevarying vector functions llKp(e, t) and llK"(e, t) are required to assure that the generalized state error e (t) goes to zero asymptotically. The general form for llKp(e, t) and llK"(e, t), which is developed by Landau [8, 9], that is llKp(e, t) = llKp(v, t) =fat cj>1 (v, 't, t) d't + 4>2 (v, t) 4.51 4.52 where the vector functions cl>l' 4>2 1Jr 1' and 1Jr2 are selected so that the hyperstability conditions are satisfied. gives Combining equations ( 4.45) through ( 4.50), with some rearrangement e(t) =Am e(t)Bp{[ llKp(e, t)B; (AmAp) KP] Xp (t) + [IlK" (e, t) + KuB; Bm]r(t)} 53 4.53
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Let W(t) = W1 (t) = ( aKP (e, t) B; (AmAP)Kp) Xp(t) + (AK,(e, t) + K,B;Bm)r(t) Then equation ( 4.53) becomes e(t) = Ame(t)Bp W(t) Let V(t), the output of the linear feedforward block, be expressed as V(t) = De(t) 4.54 4.55 4.56 The equivalent feedback representation of the adaptation mechanism is shown in Figure 4.4. This type of adaptation scheme is called proportional plus integral adaptation (PI). Equations (4.57 to 4.61) descnbe the system where e(t) = Ame(t) + Bp WI (t) v(t) = De(t) W(t) = W1(t) = (AKp(e, t)A0)Xp(t) + ( AK,(e, t)A1)r(t) 54 4.57 4.58 4.59 4.60 4.61
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I D r M F F G I I I Figure 4.4. Proportional plus integral adaption (PI). 55
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The vector functions Jl.Kp(e, t) and Jl.K,(e, t) given by equations (4.51) and ( 4.52) are ll.Kp(e, t) = ll.Kp(v, t) =Jot <1>1 (v, 't', t) d't' + 4>2(v, t) ll.K,(e, t) = ll.Ku(v, t) = fot 1Jr 1 (v, 't', t) d't' + 1Jr2(v, t) Note that the integral terms <1>1 and 1Jr1 constitute the memory of the adaptation mechanism. This integral relation assures zero steadystate error. The proportional terms <1>2 and v 2 are used to speed up the reduction of the error at the beginning of the process. <1>2 and v2 will vanish when e is equal to zero. The adaptation laws ll.Kp(e, t) and ll.K,(e, t) given by equations (4.51) and (4.52) can be chosen as ll.Kp(v, t) =Jot Fv(G XP)T dt + Fv(G Xp)T ll.Ku(v, t) =Jot Mv(Nr)T dt + Mv(Nr)T 4.62 4.63 where F, M, G and N are symmetric positive definite matrices and F, M are positive semidefinite matrices with appropriate dimensions. 4.6.1 Hyperstability of the AMFC The hyperstability conditions given by equations (4.41) and (4.42) are 1. 11 (0, tl) = fotl yT ('r) Wd't' :?: a;2 where a2 is a finite positive constant. 2. Z(s) = D(sl Ar1 B is strictly positive real. 56
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The expressions for AKp(e, t) and AKu (e, t) may be used to show that the resulting feedback block satisfies the Popov integral inequality. Substituting equation ( 4.59) into the Popov integral inequality, equation ( 4.41) gives and V(t) is given by equation ( 4.56). Combining equation ( 4.64) with equations ( 4.62) and ( 4.63) results in 11 (0, t1 ) = J:1 yT[ fat F V(G Xp)T d A0 ] Xpdt + fot1 yT[ fot M V(Nr)T d "t' A1 ] rdt + fotl ( yT i V) (x; G x p) d t 4.64 4.65 The last two integrals in equation ( 4.65) are greater than or equal to zero because F, M are positive semidefinite matrices and G and N are positive definite symmetric matrices. Therefore, for the Popov inequality to hold, it is sufficient that each of the first two integrals in ( 4.65) be greater than a negative finite constant. Using the properties of the positive definite symmetric matrix, the first integral of equation ( 4.65) can be written as 57 4.66
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where But the integral I 1 can also be described as The second integral in equation ( 4.65) also produces such an inequality and therefore it may be concluded that the feedback block satisfies the Popov integral inequality. The other requirement, for the closed loop to be hyperstable, is that the transfer matrix of the feedforward block must be strictly positive real (SPR). The feedforward transfer matrix shown in Figure 4.4, is 4.68 This transfer matrix can be made SPR if an appropriate linear compensator matrix D is chosen, assuming that B P has that same structure as B m. Then matrix D may be selected as 58
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where P is a positive definite matrix solution of the Lyapunov equation T A P+PA =H m m where H is a symmetric positive definite matrix. Equation ( 4.68) can be written as 4.69 4.70 4.71 which is strictly positive definite, and hence the second condition of the Popov hyperstability requirement is satisfied. Note that for the closedloop system to be hyperstable implies that the generalized state error approaches zero asymptotically. 59
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5.1 Introduction CHAPTERS SIMULATION RESULTS In this chapter several simulations are given to illustrate the application of the adaptive modelfollowing control algorithm. Two cases of the equivalent twomachine system, Case 1, considering transient saliency with constant damping power and Case 2, considering transient saliency with variable damping power, will be used to demonstrate the capability of the adaptive modelfollowing control algorithm using two different reference models. 5.2 Numerical Equation of the Plants As given earlier, the dynamic equation of the equivalent twomachine system is For a system with constant voltage and field flux linkages and neglecting governor actions, equation (5.1) becomes 5.1 5.2 The differential equation (5.2) is described by the state space representation given by equations ( 4.6) and ( 4.7).
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Ap = [!J(:p1(t)) ! P:(Xp1(t))l Bp(t) = [ l A complete description of the evaluation of f(Xp1(t)) and PD(Xp1(t)) is given in Appendix A Case 1 Transient Saliency with Constant Damping Power f(Xp1(t)) = P01 [sin (Xp1 (t) + 50)sin(50)] + P02[sin(2Xp1(t) + 25Jsin50(t)] PD =constant The linearized state space representation of Plant 1 is Ap = [ 0 1 l 2.04 0.31 The step response of Plant 1 is given in Figure 5.1. 61 5.3 5.4 5.5
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Case 2 Transient Saliency with Variable Damping Power f(Xp1(t)) = P01 [sin (XP1 (t) + 50)sin(50)] + P02[sin(2Xp1(t) + 25o)sin(2 5o)] The linearized state space representation of Plant 2 is [ 0 A p2.04 The step response of plant 2 is given in Figure 5.2. 5.3 Numerical Equation of the Reference Models 5.6 The stabilizing of the equivalent twomachine system can be achieved by a suitable choice of the reference model. The desired time domain performance characteristics, such as rise time, overshoot, settling time, and damping can be specified with a minimum number of parameters. To achieve two different time domain performance characteristics, two different reference models were used. The reference model system equation is 5.7 62
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The state space representation of equation (5.5) is 5.8 5.9 5.10 Model 1 The parameters of Model 1 are chosen for = 1 and c.> = 4. [ 0 1] A = m 16 8 5.11 The step response of Model 1 is given in Figure 5.3. Model 2 The parameters of Model 2 are chosen for = 1.584 and c.> = 31.56. A=[ 0 1 l m 996 100 5.12 The step response of Model 2 is given in Figure 5.4. 63
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The reference models given by equations (5.10) and (5.11) and the controlled plant given by equations (5.5) and (5.6) satisfy the Erzberger conditions. A complete discussion on how the gains Kp, Ku and D were calculated is shown in Appendix B. In the simulation of the AMFC system, two different reference inputs were used, which are the step input of unit amplitude and a square wave of unit amplitude. All the gain matrices of the control law U(t) are given in Appendix C. 5.4 Results A computer simulation study is performed to investigate the quality of performance of the AMFC algorithm. 5.4.1 Without Adaptation The step response of plant 1 and plant 2 before the use of adaptation are shown in Figures 5.1 and 5.2. For the square wave input, Figures 5.13, 5.17, 5.21, and 5.25 show both the plant output and the model output for the different cases. From these figures, the plants are very sluggish and unstable. The objective of adaptation is to have the output of the plants track the output of the reference models. In other words, the AMFC system objective is to force the generalized state error e to go to zero asymptotically. Next, the final results using the AMFC algorithm is shown. 64
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5.4.2 Step Input From Figures 5.5, 5.6, 5.7, and 5.8, the position errors for Plant 1 and Plant 2 using Model 1 have a maximum of 2.2 X 106 radians and 6.0 X 106 radians respectively. The adaptation time is 2.084 seconds for Plant 1 and 3.3 seconds for Plant 2. From Figures 5.9, 5.10, 5.11, and 5.12, the position errors for Plant 1 and Plant 2 using Model 2 have a maximum of 1.45 X 10s radians respectively. The adaptation time is 0.55 seconds for Plant 1 and 2.45 seconds for Plant 2. Comparison In both cases using Model 1 and Model 2 the position error is extremely small. The adaptation time for Plant 1 is faster than the adaptation time for Plant 2. It can be seen from Figures 5.6, 5.8, 5.10, and 5.12 that the output of the plants track the output of the models identically. The use of very large values for the gain matrix F, as shown in Appendix C, reduced the position error significantly. In the practical world of power systems analysis, when a fault or disturbance occurs, it is necessary to make a correction as fast as possible before the system falls out of synchronism. As seen for the position error Figures 5.5, 5.7, 5.9, and 5.11, the error is significantly small, which will keep the system from falling out of synchronism. 65
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5.4.3 Square Wave Input From the error Figures 5.15, 5.20, 5.24, and 5.28, the maximum error for Plant 1 Model 1 is 3 X 103 radians, for Plant 2 Modell; it is 15 X 103 radians, for Plant 1 Model 2 is 8 X 104 radians; and for Plant 2 Model 2 it is 17.5 X 10"4 radians. The adaptation time is very fast. It takes about two samples for the error to go to zero except for Plant 1 Model 2 where the error is about 5 X 106 radians at t = 6 seconds until t = 15 seconds. In Figures 5.14, 5.18, 5.22, and 5.26 it can be seen that the plants follow the models identically. The control signals U(t) applied to the controlled plants are shown in Figures 5.15, 5.19, 5.23, and 5.27. Comparison Using Modell large values of the gain matrices F and M and small values for F and M, were required as shown in Appendix C. Using Model 2 large values for the gain matrix F and small values for F, M, and M were needed. For both input signals, step input and square wave input, increasing the adaptive gain F resulted in smaller error and better tracking. The output of the controlled plant and the output of the reference are virtually indistinguishable. The selection of the gains F, M, and M did not appear to have any special pattern. The gain matrices are limited by the saturation existing in the adaptation loop. The simulation results show that the proposed control method has the potential for higher performance with a very simpler structure. 66
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0.9 0.8 0.7 0.6 :ag 0.5 8. 0.4 0\ .....} 1:'1) 0.3 0.2 0.1 0 0.0 10.0 20.0 30.0 40.0 Time (sec) Figure 5.1. Plant 1 before adaptation.
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0.9 0.8 0.7 0.6 g 0.5 c: 8. "' 0.4 0\ .....1 tfl 0.3 0.2 0.1 0 0.0 10.0 20.0 30.0 40.0 Time (sec) Figure 5.1. Plant 1 before adaptation.
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1 n 1\ 0.9 0.80.7 ........ g "' 0.6 .... .... 0 i; S!: 0.5 i 0.4 0\ 00 tl) 0.3 0.20.1 v v v 0 0.0 10.0 20.0 30.0 40.0 Time (sec) Figure 5.2. Plant 2 before adaptation.
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;;. 1.2 1.1 1 0.9 0.8 '8' g 0.7 = 0.6 8. 0.5 0\ 1,0 til 0.4 0.3 0.2 0.1 0 0.0 1.0 2.0 3.0 4.0 5.0 Time (sec) Figure 5.3. Modell step response.
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1.2 1.1 1 0.9 0.8 g 0.7 c: 8. 0.6 ....3 j' 0.5 0 Ill 0.4 0.3 0.2 0.1 0 0.5 LO 2.0 Time (sec) Figure 5.4. Model2 step response.
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4 3 J A t::2 \ .g. ...... E< .....) l f' ll< 1 0.000 2.084 Time (sec) Figure 5.5. Plant 1 Modell position error (unit step input).
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1.2 1.1 1 0.9 0.8 B' s 0.7 v "' c 0.6 & "' 0.5 tj Vl 0.4 0.3 0.2 0.1 0 0.0 1.0 2.0 3.0 4.0 5.0 Time (sec) Figure 5.6. Plant 1 Modell step response.
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10 9 8 7 6 ,, 5 ...... cu., s:::cu 4 .g. .... E< ......:. 3 (j.) 2 1 0 1 0.0 1.0 2.0 3.0 4.0 5.0 Time (sec) Figure 5.7. Plant 2 Modell position error (unit step input).
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1.2 1.1 1 0.9 0.8 :a g 0.7 c 0.6 8. 0.5 i til 0.4 0.3 0.2 0.1 0 0.0 1.0 2.0 3.0 4.0 5.0 Time (sec) Figure 5.8. Plant 2 Model 1 step response.
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2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 :a' 1.2 iS""' 1.1";; ggs 1 J::.. o.,., 0.9 c:O .g. 0.8"C 0.7 0.60.50.4 0.3 0.20.1 l 0 0.1 0.0 0.5 1.0 1.5 2.0 Time (sec) Figure 5.9. Plant 1 Model 2 position error (unit step input).
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1.2 1.1 1 0.9 0.8 :0 A 0.7 I 0.6 0.5 .....} fl) 0\ 0.4 0.3 0.2 0.1 0 2.0 Time (sec) Figure 5.10. Plant 1 Model 2 step response.
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0.0002 0.00019 0.00018 0.00017 0.00016 0.00015 0.00014 0.00013 0.00012 0.00011 0.0001 E = 0.00009 0 0.00008 .... til 0.()()007 0.00006 0.00005 0.00004 0.00003 0.00002 0.00001 0 0.00001 0.000 2. 50 Time (sec) Figure 5.11. Plant 2 Model 2 position error (unit step input).
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1.2 1.1 1 0.9 0.8 :0 g 0.7 8. 0.6 0.5 'l tl) 00 0.4 0.3 0.2 0.1 0 0.5 1.0 1.5 2.0 Time (sec) Figure 5.12. Plant 2 Model 2 step response.
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1.6 1.4 1.2 1 0.8 {\ (\ , Pr {t, {\ {t, 'r11 V1 ,, H V1 ,, 0.6 0.4 '" ..... = 0.2 t. l, "' L,. L, ", .... Cl) 0 ::t 0.2 ......:J 0.4 \0 rl) r ( ( ( r r r' 0.6 0.8 1 1.2 !', '' \}N \) \}N ' v 1.4 1.6 10 20 30 40 50 Time (sec) Figure 5.13. Plant 1 Modell, no adaptation (T = 6.5).
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1.1 1.0 0.9 0.8 I r ( (I 0.7 0.6 0.5 0.4 ,.... 0.3 '' 0.2 .... = c. 0.1 c:: .... Q) 0.0 0.1 0.2 00 0.3 0 c< [/) 0.4 0.5 0.6 0.7 0.8 0.9 1.0 L L L L L 1.1 10 20 30 Time (sec) Figure 5.14. Plant 1 Modell, square wave input response (T = 6.5).
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8 7 6 5 4 3 2 '' 1 .w r l ( r ::3 Q.. 0 .s g 1 c 0 00 u 2 !"" \ \ \ 3 4 5 6 7 8 0.0 10.0 20.0 30.0 Time (sec) Figure 5.15. Plant 1 Model 1, square wave control signal (T = 6.5).
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0.004 0.003 0.002 0.001 ... '"C Cl:l .._ 0 0 00 0.001 N 1\ .. / ...... 0.002 0.003 0.004 I 10 20 30 40 50 Time (sec) Figure 5.16. Plant 1 Model, square wave input error (T = 6.5).
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2 1.5 {\ {\ {\ {\ 1 0.5 .... ::I ...... a> 0 1; ( ,........, I( n I( ,...., (I I(" \ ( r ,J I I ,J ..., 00 0.5 w 0" V) ', r 1 ( I 1 1.5 'f' 1'f.' \ .___j LJ \ h 'v v v v 2 10 20 30 40 50 Time (sec) Figure 5.17. Plant 2 Modell, no adaptation (T = 6.5).
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1.2 1 0.8 0.6 "'0 ("\1 0.4 ..... .......... 0.2 .s0 0 St 0 0.2 00 la +:>;:j 0.4 0"' en 0.6 0.8 1 1.2 10 20 30 Time (sec) Figure 5.18. Plant 2 Model 1, square wave input response (T = 6.5).
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30 20 10 :a( v ..... = p. .s 0 g = 0 00 u Vl 10 , 20 30 0.0 10.0 20.0 30.0 Time (sec) Figure 5.19. Plant 2 Model 1, square wave control signal (T = 6.5).
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0.02 0.015 0.01 0.005 \ 0 00 0.005 0'1 t1/ 10.01 0.015 0.02 10 20 30 40 50 Time (sec) Figure 5.20. Plant 2 Modell, square wave input error (T = 6.5).
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1.6 1.4 1.2 1 0.8 0.6 0.4 ..... g. 0.2 ...... Cl) f; 0 0.2 00 i 0.4 ...l ell 0.6 0.8 1 1.2 1.4 1.6 0.0 5.0 10.0 1 .0 Time (sec) Figure 5.21. Plant 1 Model 2, no adaptation (T = 2).
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1.6 1.4 1.2 1 0.8 (I ,., rl 1 (I (I r I 0.6 . u 0 0.4 rn .._ "S 0.2 0.. .s 0 0 Ea 0.2 00 0.4 00 0 0.6 & (/) 0.8 1 \.._j L L L L L \__j 1.2 1.4 1.6 0.0 5.0 10.0 15.0 Time (sec) Figure 5.22. Plant 1 Model 2, square wave input response (T = 2).
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20 15 10 5 ....... .... :I Q. .s 0 g c 0 00 u 5 \0 10 15 20 Time (sec) Figure 5.23. Plant 1 Model 2, square wave control signal (T = 2).
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0.001 0.0009 0.0008 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 :;0.0001 g 0 "'" 0 Jl 0.0001 0.0002 \0 0.0003 0 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.001 0.0 5.0 10.0 15.0 Time (sec) Figure 5.24. Plant 1 Model 2, square wave input error (T = 2).
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1.2 1 0.8 0.6 0.4 ....... 0.2 5. .5 0 0 i3': 0 0.2 10 f" til 0.4 0.6 0.8 1 1.2 0.0 5.0 10.0 15.0 Time (sec) Figure 5.25. Plant 2 Model2, no adaptation (T = 2).
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1.6 1.4 1.2 1 0.8 n n n n n n n I 0.6 .. ] 0.4 0.2 ....... ;:I .s0 Q) 0.2 \0 0.4 N Q) 0.6 ;:I c::r 0.8 Cl) 1 l l l l L u 1.2 1.4 1.6 ) 5 10 15 Time (sec) Figure 5.26. Plant 2 Model 2, square wave input response (T = 2).
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40 30 20 ...... 10 '" \ k 1\ [\_ .... :I c. 0 = g c 0 \0 u 10 w I/ v 20 .30 '40 ) 5 1 0 Time (sec) Figure 5.27. Plant 2 Model 2, square wave control signal (T = 2).
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0.002 0.0015 rh 0.001 e 0.0005 ...._, 11 0 0 s:l 0 "+=1 ....... til 0.0005 \0 0 .j::>. Pot ij_ 0.001 0.0015 0.002 0.0 4.0 8.0 12.0 Time (sec) Figure 5.28. Plant 2 Model 2, square wave input error (T = 2).
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CHAPTER 6 CONCLUSIONS For a given power system design, the stability is usually the most important concept to be investigated. The stability for a generalized equivalent twomachine system using an adaptive control design together with Lyapunov stability has been investigated in this thesis. Lyapunov functions for the equivalent twomachine system are obtained using the energy metric algorithm [1, 2]. The Lyapunov function obtained is optimized by completing the quadratic form of the kinetic energy. This algorithm permits the generation of the Lyapunov functions directly from the system equations. The adaptive modelfollowing control approach uses Popov hyperstability and adaptive modelfollowing control and produces an adaptive controller capable of achieving the desired performance given by the reference model. The controller is continuous and consists of two parts, the first part is linear in the state and the reference input, where its linear gains are determined using the known linear portion of the system dynamics. The second part of the controller is produced using adaptive gains determined by the use of the hyperstability concept. The AMFC design chosen is. simpler than other existing methods, without loss in performance. The adaptation law implementation does not
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require the real time solution of the system equations. Simulation using a digital computer made it possible to verify the theoretical results. Despite its simplicity AMFC system is robust and performs very well. The stability of the equivalent twomachine system using the AMFC technique is guaranteed by the use of the hyperstability and positivity concepts. The equivalent twomachine system stability model is widely used to simulate larger scale power systems and in general gives good results for decision making. 96
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APPENDIX A CALCULATION OF f(Xp1(t)) and PD(Xp1(t))
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A 60cycle salientpole synchronous generator having the following constants: Reactances in per unit I xd = 0.37 x/1= 0.24 xq = 0.75 x/ = o.75 x/1= o.34 Time constants in seconds I xdo = 5.0 T dDII = 0.035 II Tqo = 0.035 is delivering a current of 1.00 per unit at a power factor of 0.91 (lagging) to an infinite bus having a voltage of 1.00 per unit. Using the above values, equations ( 4.8), ( 4.9), ( 4.10) and ( 4.12) become A1 P m = 3.18919 sin (50)0.68468 cos(50 ) A2 f(Xp1(t)) = P,Pe (Xp1(t)) = 3.18919 [sin (Xp1(t) + 50)sin0>0)] A3 + 0.68468[sin(2Xp1(t) + 25J2sin(250)] A4 98
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Case I Considering Transient Saliency with Constant Damping Power 0 Ap(t) = J_ f(X (t)) M Pl Linearizing equation (A5) Ap = [ 0 1 l Bp = [0 1 ] 2.04 0.31 Case II Transient Saliency with Variable Damping Power Linearizing equation (A7) Ap = [_:04 ]. BP = 99 A5 A6 A8
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APPENDIXB GAIN MATRICES KP, Ku, AND D CALCULATIONS
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The implementation of linear model following control (LMFC) requires the calculation of the constant gain matrices Kp, K11 And in order to achieve Popov hyperstability conditions the constant gain matrix D must be calculated. The Penrose pseudo inverse {B;) needed to calculate Kp and K11, B;, Kp, K11 and D are given by equations (4.25), (4.26), (4.27) and (4.42) Plant 1 Model 1 + ( T )1 T Bp = Bp Bp Bp T D=Bp P [ 0 A m 16 where P is a symmetric positive definite matrix solution of the Lyapunov equation 101
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T A P+PA = H m m and H is a symmetric positive definite matrix also fo1 16] [pll p12] + [pll p12] [ o 1] [1 o] l 8 pl2 p22 pl2 p22 16 8 = 0 1 p = [1.3125 0.0313] 0.0313 0.0664 This P matrix is used for both plants using Model 1, since B P is the same for both plants. 102
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Kp = [13.96 7.69] 1 [1.3125 0.0313] D = [O ] 0.0313 0.0664 D = [0.0313 0.0664] Plant 2 Model 1 103
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B; = (ro 1] [0 1] Kp = [13.96 7.993] T [1.3125 0.0313] D = B P = [0 1] p 0.0313 0.0664 D = [0.0313 0.0664] 104
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Plant 1 Model 2 As in the case of Modell, Pis chosen from T A P+PA = H m m [01 996] [pll p12] + [pll p12] [ 0 1 ] [1 0 l 100 pl2 p22 p12 p22 996 100 = 0 1 p = [ 5.030 0.0005] 0.0005 0.005 This P matrix will be used for both plant using Model 2 since B P is the same for both plants. + ( T )1 T Bp = Bp Bp Bp 105
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Plant 2 Model 2 K = [0 1] [ O l = 996 u 996 Kp = [993.96 99.96] T D=Bp P 1] [ 5.030 0.0005] D= [0 0.0005 0.005 D = [0.0005 0.005] 106
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+ ( T )1 T Bp = Bp Bp Bp B; = (ro 1] ro 11 K, = [0 1] [ O l = 996 996 KP = [993.96 99.993] 107
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D = [0 T D=BpP 1] [ 5.030 0.0005] 0.0005 0.005 D = [0.0005 0.005] To that the AMFC scheme will overcome the variation of the plant parameters due to unexpected disturbances, or faults or the uncertainty in the knowledge of plant parameters Plant 1 and Plant 2 were varied. Plant 1 Original Plant 1 Ap = [ 0 1 l Bp = [01]2.04 0.31 Varied Plant 1 [ 0 1] Ap = 4 1 Plant 2 Original Plant 2 A P = [ 0 1 l B P = [0 1 ] 2.04 0.007, Varied Plant 2 [ 0 1] Ap = 12 3' 108
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APPENDIX C GAIN MATRICES G, F, F, M, M, AND N SELECTION
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The G matrix is the same for all cases and F, F, M, M and N are shown below. I Step Input I F I I M I I N I F M Plant 1 Model 1 5 X 105 2 21 0.005 1 Plant 2 Modell 8 X 105 2 21 0.005 1 Plant 1 Model 2 15,000 1 0.8 0.05 4 Plant 2 Model 1 7,500 1 0.005 0.1 4 I I I I I I I I Square Input I F I I M I I N I F M Plant 1 Model 1 650 0.5 400 0.05 1 Plant 2 Modell 650 0.5 400 0.05 1 Plant 1 Model 2 195 0.05 1 1 1 Plant 2 Model 2 15,000 0.05 8 0.5 1 110
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BffiLIOGRAPHY [1] Wall, E.T. "The generation of Lyapunov functions in control theory by an energy metric algorithm." Preprints of JACC, 1968, pp. 172179. [2] Wall, E.T., and Moe, M.L. "Generation of Lyapunov functions for timevarying nonlinear system." IEEE Trans. Automatic Control, Apri11969, Vol. AC14, p. 211. [3] Kimbark, E.W. Power System Stability. Vol. I. New York: John Wiley and Sons, Inc., 1967. [4] Kimbark, E.W. Power System Stability. Vol. III. New York: John Wiley and Sons, Inc., 1956. [5] Park, R.H. "TwoReaction Theory of Synchronous MachinesPart I, Generalized Method of Analysis." AI.E.E. Trans., July 1929, Vol. 48, pp. 716730. [6] Vidyasagar, M., Nonlinear System Analysis. PrenticeHall, Inc., Englewood Cliffs, N.J., 1978. [7] Hsu, J.C. and Meyer, AU. Modern Control Principles and Applications. McGraw Hill Book Co., Inc., 1968. [8] Landau, I.D. Adaptive Control The Model Reference Approach. New York: Marcel Dekker, 1979. [9] Landau, I.D. "A Hyperstability Criterion for Model Reference Adaptive Control System." IEEE Trans. Aut. Control AC14, 1969, pp. 552555. [10] Popov, V.M. Hyperstability of Control Systems. SpringerVerlag, Berlin, 1973 . [11] Erzberger, H. "Analysis and Design of Model Following System by State Space Techniques." Proc. of JACC, 1968, pp. 572581. 111
