In this manuscript, we present a high-fidelity physics-based truth model of a Single Machine Infinite Bus (SMIB) system. W e also present reduced-order control-oriented nonlinear and linear models of a synchronous generator-turbine system connected to a power grid. The reduced-order control-oriented models are next used to design various control strategies such as: proportional-integral-derivative (PID), linear-quadratic regulator (LQR), pole placement-based state feedback, observer-based output feedback, loop transfer recovery (L TR)-based linear-quadratic-Gaussian (LQG), and nonlinear feedback-linearizing control for the SMIB system. The controllers developed are then validated on the high-fidelity physics- based truth model of the SMIB system. Finally , a comparison is made of the performance of the controllers at different operating points of the SMIB system. The material presented in this manuscript is part of a course on “Control and Optimization for the Smart Grid” that was developed in the Electrical and Computer Engineering Department at the Ohio State University in 2011-2012. This project was funded by the U.S. Department of Energy . 1 Contents 1 A Single Generation Unit 4 2 T ruth Model of the Synchronous Generator 5 2.1 Electrical Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 V oltage Equation in the Static F rame . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 V oltage Equation in the Synchronously Rotating F rame . . . . . . . . . . . . . . . . . 8 2.1.3 V oltage Equation of the Synchronous Generator in Per Unit System . . . . . . . . . . 10 2.2 A Synchronous Generator Connected to an Infinite Bus . . . . . . . . . . . . . . . . . . . . . 10 2.3 Mechanical Dynamics: Swing Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 T ruth Model of the Synchronous Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Model of the T urbine-Governor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 T ruth model of the combined Synchronous Generator and T urbine-Governor System con- nected to an infinite bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.7 Derivation of the Output Generator T erminal V oltage for the T ruth Model . . . . . . . . . . . 18 3 The Reduced Order Simplified Model 20 3.1 Reduced order model of the combined Synchronous Generator and T urbine-Governor system connected to an infinite bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Derivation of the Output Generator T erminal V oltage for the Reduced Order Model . . . . . 23 3.3 Linearization of the Reduced Order Model by T aylor series approximation . . . . . . . . . . . 25 3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Open Loop Input-Output Behavior of the Synchronous Generator and T urbine-Governor System 34 5 The Decoupled Reduced Order Model 41 5.1 LFC Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.1 PID controller design for the LFC loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 A VR Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.1 PID controller design for the A VR loop . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.3 LFC and A VR including coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6 PID Controller Design 53 6.1 PID Controller Design based on linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.2 Simulation results for the PID Controllers applied to the Reduced Order Nonlinear Model . . 56 6.3 Simulation results for the PID Controllers applied to the T ruth Model . . . . . . . . . . . . . 59 7 Linear State-Space Controller Design 62 7.1 State F eedback Controller design using LQR methodology . . . . . . . . . . . . . . . . . . . . 62 7.1.1 LQR Design based on linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 7.1.2 Simulation results for the F ull-State F eedback LQR applied to the Reduced Order Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.1.3 Simulation results for the F ull-State F eedback LQR applied to the T ruth Model . . . . 69 7.2 State F eedback Controller design using Pole Placement T echnique . . . . . . . . . . . . . . . 72 7.2.1 Pole Placement Design based on linear model . . . . . . . . . . . . . . . . . . . . . . . 72 7.2.2 Simulation Results for the F ull-State F eedback Pole Placement controller applied to the Reduced Order Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2.3 Simulation Results for the F ull-State F eedback Pole Placement controller applied to the T ruth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7.3 Output F eedback Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2 7.3.1 Observer-based LQR Design based on linear model . . . . . . . . . . . . . . . . . . . . 81 7.3.2 Observer-Based Pole Placement Controller Design based on linear model . . . . . . . . 88 7.3.3 Simulation Results for the Observer-based LQR applied to the Reduced Order Non- linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.3.4 L TR-based LQG Controller applied to the Reduced Order Nonlinear Model . . . . . . 93 7.3.5 L TR-based LQG Controller applied to the T ruth Model . . . . . . . . . . . . . . . . . 97 8 Nonlinear F eedback Linearizing Controller Design 100 8.1 Nonlinear F eedback Linearizing Controller Design for the Reduced Order Model . . . . . . . . 102 8.2 Simulation Results for the Nonlinear F eedback Linearizing Controller applied to the T ruth Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 9 Simulation results for the Controllers at different Operating Points 114 9.1 Simulation results for the Controllers at Operating Point II . . . . . . . . . . . . . . . . . . . 115 9.2 Simulation results for the Controllers at Operating Point III . . . . . . . . . . . . . . . . . . . 119 10 Appendix 124 10.1 Derivation of Park’s T ransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.1.1 Stator and Rotor Inductances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 10.1.2 Park’s T ransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 10.1.3 V oltage Equation in the Rotating F rame . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10.2 Per Unit Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 10.2.1 Choosing a base for stator quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10.2.2 Choosing a base for rotor quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10.2.3 The correspondence of per unit stator EMF to rotor quantities . . . . . . . . . . . . . 133 10.3 Sub-transient and T ransient Inductances and Time Constants . . . . . . . . . . . . . . . . . . 133 10.3.1 Sub-transient and T ransient Inductances . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.3.2 Time constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.4 Simplified Model of the Synchronous Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 136 10.4.1 The two-axis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 10.4.2 The one-axis or the third-order simplified model . . . . . . . . . . . . . . . . . . . . . 143 3 1 A Single Generation Unit F ossil fuels such as coal, oil, and natural gas have been the main resources of electrical energy for many years. However in recent years, there has been a gradual increase in the use of renewable energy resources for electricity generation, such as hydro, biogas, solar, wind, and geothermal energy . Electricity generation is basically the process of generating electric energy from other forms of energy . An electromechanical device called synchronous generator driven by a prime mover, usually a turbine or a diesel engine, converts the mechanical energy into alternating current (AC) electrical energy . Synchronous generator Measuring element T o grid through breakers and transformers Governor P f Pref ω T urbineW orking fluid inV alves W orking fluid out Shaft Exciter Automatic voltage regulator Vt It Vref Figure 1.1: Structure of a generation unit. The system shown in Figure 1.1 is a general structure of a single generation unit [ 7]. The turbine extracts the energy from the working fluid flowing into the turbine through valves . Typical working fluids are gas, steam, and water. The shaft is the rotary part of the turbine on which the synchronous generator is mounted. The opening and closing of the turbine valves or the frequency at which the turbine valves operate is regulated to a reference frequency of fref, by a turbine governor . The frequency of the grid fwhich is measured by the measuring element is directly related to the output power P. Thus, the output of the synchronous generator P and the angular frequency of the shaft ωare measured and fed back to the governor by the measuring element. Meanwhile, the measuring element also provides information about the output terminal voltage Vt and output current Itof the synchronous generator to the automatic voltage regulator (A VR), which is able to control the terminal voltage of the synchronous generator to a reference voltage Vref through the exciter . The excitation current generated by the exciter produces the magnetic field inside the generator. Thus, from the above figure we can see that in an interconnected power system, where a synchronous generator is connected to a grid, load frequency control (LFC) and automatic voltage regulator (A VR) equipment is installed for each generator. Figure 1.1 shows two control loops, namely the load frequency control (LFC) loop and the automatic voltage regulator (A VR) loop. The controllers are set for a particular operating condition and accommodate small changes in load demand to maintain the frequency and voltage magnitude within the specified limits. Small changes in real power are mainly dependent on changes in rotor angle δ, and thus the frequency ω. The reactive power is mainly dependent on the voltage magnitude (i.e., on the generator excitation). The excitation system time constant which is an indication of how fast the transients of the A VR loop decay exponential to zero, is much smaller than the prime mover time constant. Thus the transients of the excitation system and thus the A VR loop decay much faster than the transients of the LFC loop, hence it does not affect the LFC dynamics. Thus, the cross-coupling between the LFC loop and the A VR loop is negligible. Hence, load frequency control and excitation voltage control are usually analyzed independently [ 8]. 4 The operation objectives of the LFC are to maintain reasonably uniform frequency , and to divide the load between generators [ 8]. The change in frequency is sensed, which is a measure of the change in rotor angle δ, i.e., the error ∆δto be corrected. The error signal i.e., ∆f=fref−fis amplified, mixed, and transformed into a real power command signal ∆PV=Pref−P, which is sent to the prime mover to call for an increment in the torque. The prime mover, therefore, brings about a change in the generator output which will change the value of ∆fwithin the specified tolerance. The generator excitation system maintains the generator terminal voltage and controls the reactive power flow. The generator excitation of older systems may be provided through slip rings and brushes by means of DC generators mounted on the same shaft as the rotor of the synchronous machine. However, modern excitation systems also known as brush-less excitation systems, usually use AC generators with rotating rectifiers. The sources of reactive power are generators, capacitors, and reactors. The generator reactive power is controlled by field excitation using the A VR. The role of an A VR is to hold the terminal voltage magnitude of a synchronous generator at a specified level. An increase in the reactive power load of the generator is accompanied by a drop in the terminal voltage magnitude. The voltage magnitude is sensed through a potential transformer on one phase. This voltage is rectified and compared to a DC set point signal. The amplified error signal controls the exciter field and increases the exciter terminal voltage. Thus, the generator field current is increased, which results in an increase in the generated electromotiveforce (emf). The reactive power generation is increased to a new equilibrium, raising the terminal voltage to the desired value. 2 T ruth Model of the Synchronous Generator In the previous section we saw the basic working of a single generation unit and the respective roles of the load frequency control and the automatic voltage regulator. In this section we will derive the truth model of a synchronous generator. Before we proceed to the derivation of the truth model, we present some preliminaries about the synchronous generator. The two main parts of a synchronous generator can be described in either electrical or mechanical terms: • Electrical: – Armature: The power-producing component of an electrical machine. In a synchronous generator, the armature windings generate the electric current. The armature can be on either the rotor or the stator. – Field: The magnetic field component of an electrical machine. The magnetic field of the syn- chronous generator can be provided by either electromagnets or permanent magnets mounted on either the rotor or the stator. • Mechanical: – Rotor: The rotating part of the synchronous generator. – Stator: The stationary part of the synchronous generator. Because power transferred into the field circuit is much smaller than in the armature circuit, AC generators always have the field winding on the rotor and the stator has the armature winding. Thus, a classical synchronous generator has two main magnetic parts: the stator and the rotor, as shown in Figure 2.1 . The windings are represented by one-turn coils, specifically , the small circles “ ⃝” in the figure. The black dots and the crosses inside the small circles indicate the directions of the currents flowing in the windings, i.e., “×” means the current flowing in the direction from the outside of the paper vertically into the paper and “•” means the current flowing from the inside of the paper to the outside. 5 ×× × × ×× a′c b′ a c′bD F D′F′ QSQ′ N RotorStator Figure 2.1: Schematic structure of the synchronous generator. The armature winding , which carries the load current Itand supplies power to the grid, is placed in equidis- tant slots on the inner surface of the stator and consists of three identical phase windings, namely , aa′, bb′andcc′. The rotor is mounted on the shaft through which the synchronous generator is driven by the prime mover, for instance, a hydro turbine. The rotor consists of two poles, N pole and Spole, as seen in Figure 2.1 . The direct current (DC) excitation winding represented by FF′is wrapped around the rotor. F rom basic physics, we know that the DC flowing in the excitation winding generates a magnetic flux. Mag- netic flux is a measure of the amount of magnetic field (also called magnetic flux density ) passing through a given surface (such as a conducting coil). The SI unit of magnetic flux is the weber (in derived units: volt-seconds). The strength of the magnetic flux generated is proportional to the excitation current and its direction is known by using the right-hand rule. As the rotor rotates, the magnetic flux generated by the excitation winding wrapped on the rotor changes spatially . Thus, there are magnetic flux changes in the armature windings as a result of which an emf is induced in each phase of the three-phase stator armature winding. By connecting armature windings to the grid, a closed-loop circuit is formed which allows the AC to flow from the synchronous generator to the grid. The AC armature currents produce their own armature reaction magnetic flux which is of constant magnitude but rotates at the same speed as the rotor. The excitation flux and the armature reaction flux then produce a resultant flux that is stationary with respect to the rotor. T wo other windings represented by DD′andQQ′are the two short-circuit damper (or, amor- tisseur )windings which help to damp the mechanical oscillations of the rotor [ 2]. Hence, two dynamics will characterize the generator, i.e., electrical dynamics and mechanical dynamics. 2.1 Electrical Dynamics In this section we present the equations governing the electrical dynamics of a synchronous generator which are described in [ 2]. W e first present the voltage equations of a synchronous generator in the static frame, and then use Park’s transformation to convert these to the rotating frame. 2.1.1 V oltage Equation in the Static F rame In this subsection we present the voltage equations in the static frame. The static frame contains three reference axes a,b, and cwhich correspond to the three armature windings on the stator. Before presenting the details of the voltage equation of a synchronous generator, we start by considering the general case of a set of coupled coils in which one or more of the coils is mounted on a shaft and can rotate. The situation is shown schematically in Figure 2.2 . 6 L1 R1 i1 L2 R2 i2 L3 R3 i3 L4 R4 i4 Shaft+ −v1 + −v2 + −v3 + −v4 Figure 2.2: Coupled coils. Assume that for any fixed shaft angle θthere is a linear relationship between the flux linkage λand current i. Flux linkage is defined as the total flux passing through a surface (i.e. normal to that surface) formed by a closed conducting loop. Thus we get the relationship λ=L(θ)i, where, in the case of Figure 2.2 ,iand λare4×1vectors, and L is a4×4matrix. By applying Kirchhoff’s voltage law (KVL) to the circuit in Figure 2.2 , we have v=Ri+dλ dt(2.1) where R is a4×4matrix. Equation (2.1) indicates that the terminal voltage of each coil equals the sum of the voltage drop on the resistance and the derivative of the flux linkage. 7 ×× × × ×× θReference axis Direct axisQuadrature axis a′iacic ibb′ ia a c′ ibbD FiD iF D′F′ iQQQ′ ×d d′id ×q′ qiq Figure 2.3: Machine schematic. By applying Equation (2.1) and using the circuits convention on the associated reference directions in Figure 2.3 , we get the relationship between voltages, currents, and flux linkages [ 2]. 2 6666664va′a vb′b vc′c vFF′ vDD′ vQQ′3 7777775=2 6666664r r 0 r rF 0 rD rQ3 77777752 6666664ia ib ic iF iD iQ3 7777775+d dt2 6666664λaa′ λbb′ λcc′ λFF′ λDD′ λQQ′3 7777775 =Ri+dλ dt(2.2) W e simplify the equation above by using a single-subscript notation, i.e., va,vaa′=−va′a,vb,vbb′=−vb′b, vc,vcc′,vF,vFF′,vD,vDD′, and vQ,vQQ′. Here, we define v,[va, vb, vc,−vF,−vD,−vQ]Tto be the voltage vector consisting of the three phase terminal voltages ( va,vb,vc), and the voltage of the field winding ( vF) and two damper windings ( vD,vQ). The corresponding current vector is defined as i,[ia, ib, ic, iF, iD, iQ]T. Then Equation (2.2) can be written as follows: v=−Ri−dλ dt(2.3) 2.1.2 V oltage Equation in the Synchronously Rotating F rame The electrical dynamics as given in Equation (2.2) are derived in the static abc frame. The flux linkage in Equation (2.2) is dependent on the self and mutual inductances which are not constant, but are time varying. In the voltage equation as given in Equation (2.3) the˙λterm must be computed as ˙λ=L˙i+˙Li. Thus, to simplify the equations we make a coordinate transformation which transforms variables from the abc static frame to a synchronously rotating frame (which is also called dq frame, see Figure 2.3 ). As a result 8 of this transformation we introduce two fictitious windings dd′andqq′, as shown in Figure 2.3 . Thus we get vd=−rid−ωλq−dλd dt vq=−riq+ωλd−dλq dt vF=rFiF+dλF dt vD=rDiD+dλD dt vQ=rQiQ+dλQ dt(2.4) The details of the derivation of this transformation are given in the Appendix . The extra terms −ωλqand ωλdare introduced by the transformation. W e can rearrange Equation (2.4) to put the quantities on the direct axis together and the quantities on the quadrature axis together. Hence, Equation (2.4) is rewritten as follows: vd=−rid−ωλq−dλd dt vF=rFiF+dλF dt vD=rDiD+dλD dt vq=−riq+ωλd−dλq dt vQ=rQiQ+dλQ dt(2.5) As the damper windings are short-circuited, the terminal voltages are both zero. As shown in Figure 2.3 the direct axis is perpendicular to the windings dd′,FF′, and DD′; the quadrature axis is perpendicular to the windings qq′andQQ′. Using the right hand thumb rule we can see that the flux linkage due to the currents id,iF, and iD is along the direct axis and the flux linkage due to the currents iqandiQis along the quadrature axis. Thus, the flux linkage λdalong the dd′winding depends on the currents id,iF, and iD and is given by λd=Ldid+kMFiF+kMDiD, where Ldis the self inductance of dd′winding, MFis the mutual inductance between dd′andFF′windings, and MD is the mutual inductance between dd′and DD′windings, respectively . W e can derive equations for λF andλD in a similar fashion. Also, the flux linkage λqalong the qq′winding depends on the currents iq, and iQand is given by λq=Lqiq+kMQiQ, where Lqis the self inductance of the qq′winding, and MQis the mutual inductance between qq′andQQ′ windings, respectively . W e can derive λQusing the same approach. Also note that in Equation (2.6) and Equation (2.7) which gives a relationship between the flux and the current in each winding, the mutual inductance between the windings FF′andDD′is denoted by MR, and self-inductances of the windings are denoted by Ld,LF,LD,Lq, and LQ, respectively . Thus, the connection between the flux and the current is given by2 4λd λF λD3 5=2 4LdkMFkMD kMFLF MR kMDMR LD3 52 4id iF iD3 5 (2.6) and λq λQ =LqkMQ kMQLQiq iQ (2.7) where k=p 3/2. If we substitute Equation (2.6) and Equation (2.7) into Equation (2.5) and put it in 9 matrix form, we obtain 2 66664vd vF 0 vq 03 77775=2 66664−r 0 0−ωLq−ωkM Q 0 rF 0 0 0 0 0 rD 0 0 ωLdωkM FωkM D−r 0 0 0 0 0 rQ3 777752 66664id iF iD iq iQ3 77775+2 66664−Ld−kMF−kMD 0 0 kMF LF MR 0 0 kMD MR LD 0 0 0 0 0 −Lq−kMQ 0 0 0 kMQ LQ3 777752 66664˙id ˙iF ˙iD ˙iq ˙iQ3 77775 (2.8) Moving the derivative of the current to the left-hand side, we obtain 2 66664Ld kMFkMD 0 0 −kMF−LF−MR 0 0 −kMD−MR−LD 0 0 0 0 0 Lq kMQ 0 0 0 −kMQ−LQ3 777752 66664˙id ˙iF ˙iD ˙iq ˙iQ3 77775=2 66664−r 0 0−ωLq−ωkM Q 0 rF 0 0 0 0 0 rD 0 0 ωLdωkM FωkM D−r 0 0 0 0 0 rQ3 777752 66664id iF iD iq iQ3 77775−2 66664vd vF 0 vq 03 77775 (2.9) 2.1.3 V oltage Equation of the Synchronous Generator in Per Unit System A normalization of variables called the per unit normalization is always desirable. The idea is to pick base values for quantities such as voltages, currents, impedances, power, and so on, and to define the quantity in per unit as quantity in per unit =actual quantity base value of quantity(2.10) By carefully choosing the base quantities for both stator and rotor variables, the electrical dynamics expressed by Equation (2.9) can be expressed in the p.u. system as 2 66664Ld kMFkMD 0 0 −kMF−LF−MR 0 0 −kMD−MR−LD 0 0 0 0 0 Lq kMQ 0 0 0 −kMQ−LQ3 777752 66664˙id ˙iF ˙iD ˙iq ˙iQ3 77775=2 66664−r 0 0−ωLq−ωkM Q 0 rF 0 0 0 0 0 rD 0 0 ωLdωkM FωkM D−r 0 0 0 0 0 rQ3 777752 66664id iF iD iq iQ3 77775−2 66664vd vF 0 vq 03 77775p.u. (2.11) It is obvious that Equation (2.9) and Equation (2.11) are identical. This is always possible if base quantities are carefully chosen. The derivation of Equation (2.11) can be found in [ 1]. 2.2 A Synchronous Generator Connected to an Infinite Bus A typical configuration of a generation system model is a synchronous generator connected to an infinite bus as shown in Figure 2.4 . The figure shows a synchronous generator connected to an infinite bus through a transmission line having resistance Reand inductance Le. Only the voltages and currents for phase aare shown, where vais the phase voltage, iais the phase current, and V∞ is the infinite bus voltage. An infinite bus is an approximation of a large interconnected power system, where the action of a single generator will not affect the operation of the power grid. In an infinite bus, the system frequency is constant, independent of power flow, and the system voltage is constant, independent of reactive power consumed or supplied. 10 Le Re ia + − va + − V∞, α Figure 2.4: Synchronous generator loaded by an infinite bus. The constraints of the infinite bus are given by vd vq =Reid iq +Le˙id ˙iq −ωLe−iq id +√ 3V∞−sin (δ−α) cos (δ−α) (2.12) By including Equation (2.12) , we can rewrite Equation (2.11) as 2 66664Ld+LekMFkMD 0 0 −kMF−LF−MR 0 0 −kMD−MR−LD 0 0 0 0 0 Lq+LekMQ 0 0 0 −kMQ−LQ3 777752 66664˙id ˙iF ˙iD ˙iq ˙iQ3 77775=2 66664−r−Re 0 0−ω(Lq+Le)−ωkM Q 0 rF 0 0 0 0 0 rD 0 0 ω(Ld+Le)ωkM FωkM D−r−Re 0 0 0 0 0 rQ3 777752 66664id iF iD iq iQ3 77775 −2 666640 vF 0 0 03 77775−√ 3V∞2 66664−sin (δ−α) 0 0 cos (δ−α) 03 77775 (2.13) Thus, Equation (2.13) describes the electrical dynamics of a synchronous generator connected to an infinite bus. 2.3 Mechanical Dynamics: Swing Equation In this subsection, we present the mechanical dynamics of the synchronous generator. Under normal oper- ating conditions, the relative position of the rotor axis and the resultant magnetic field axis is fixed. The angle between the two is known as the power angle or torque angle . During any disturbance, the rotor will decelerate or accelerate with respect to the synchronously rotating air gap magneto-motive force (mmf), which is any physical driving (motive) force that produces magnetic flux, and a relative motion begins. In this context, the expression ’driving force’ is used in a general sense of work potential, and is analogous, but distinct, from force measured in Newton’s. In magnetic circuits the magneto-motive force (mmf) plays a role analogous to the role emf (voltage) plays in electric circuits. The equation describing this relative motion is known as the swing equation [1]. If, after this oscillatory period, the rotor locks back into synchronous speed, the generator will maintain its stability . If the disturbance does not involve any net change in power, the rotor returns to its original position. If the disturbance is created by a change in generation, load, or in network conditions, the rotor comes to a new operating power angle relative to the synchronously revolving field. The swing equation thus governs the motion of the machine rotor relating the moment of inertia (also referred to as the rotational inertia of the rotor) to the resultant of the mechanical and electrical torques on the rotor, i.e., J¨θ=TaN·m, where Jis the moment of inertia of all rotating masses attached to the 11 shaft, θis the mechanical angle of the shaft with respect to a fixed reference, and Tais the accelerating torque acting on the shaft. The torque is given by Ta=Tm−Te−Td, where Tm,Te, and Tdare mechanical, electrical, and damping torques, respectively . The mechanical torque Tm is the driving torque provided by the prime mover. The electrical torque Teis generated by the load currents of the armature windings on the stator. The damping torque Tdis produced by the damper windings on the rotor. The angular reference may be chosen relative to a synchronously rotating reference frame moving with constant angular velocity ωmR . The rotor angle in the static frame is given by θ(t) = (ωmRt+β) +δm, where βis a constant and δmis the rotor position also referred to as the mechanical torque angle , measured from the synchronously rotating reference frame. Let us denote the shaft angular velocity in the static frame as ωm in rad/sec, thus we have ωm=˙θ=ωmR+˙δm. By taking the derivative of ωm and second derivative of θwe obtain ˙ωm=¨θ=¨δm, if we substitute this in J¨θ=Tawe have J¨θ=J¨δm=J˙ωm=Ta=Tm−Te−Td[N·m] (2.14) The product of torque and angular velocity is the shaft power in watts, thus we have J¨δmωm=Pm−Pe−Pd[W] (2.15) The quantity Jωm is called the inertia constant and is denoted by M . It is related to the kinetic energy of the rotating masses Wk, where Wk=1 2Jω2 m.M is computed as M=Jωm=2Wk ωm[J·s] (2.16) Although M is called an inertia constant, it is not really constant when the rotor speed deviates from the synchronous speed ωmR . However, since ωm does not change by a large amount before stability is lost, M is evaluated at the synchronous speed ωmR and is considered to remain constant, i.e., M=Jωm∼=2Wk ωmR[J·s] (2.17) The swing equation in terms of the inertia constant becomes M¨δm=M˙ωm=Pm−Pe−Pd (2.18) In relating the machine inertial performance to the network, it would be more useful to write Equation (2.18) in terms of an electrical angle that can be conveniently related to the position of the rotor. Such an angle is the torque angle δ, which is the angle between the magneto-motive force (mmf) and the resultant magneto- motive force (mmf) in the air gap, both rotating at synchronous speed. It is also the electrical angle between the generated emf and the resultant stator voltage phasors. The torque angle δ, which is the same as the electrical angle, is related to the rotor mechanical angle δm, (measured from a synchronously rotating frame) by δ=p 2δm (2.19) where pis the number of poles of the synchronous generator. Figure 2.1 shows a schematic of a synchronous generator with two poles. Also, the synchronous speed ωmR used in the previous equations is actually the mechanical synchronous speed or the mechanical angular velocity at the synchronous reference value. It is related to the electrical synchronous speed ωRby ωR=p 2ωmR (2.20) By taking the derivative of Equation (2.19) on both sides, we get ˙δ=p 2˙δm (2.21) 12 Adding Equation (2.20) and Equation (2.21) we get ˙δ+ωR=p 2˙δm+p 2ωmR (2.22) Thus, the electrical angular velocity ωis related to the mechanical angular velocity ωm by ω=˙δ+ωR=p 2(˙δm+ωmR) =p 2ωm (2.23) Combining Equation (2.18) and Equation (2.19) we get 2M p¨δ=2M p˙ω=Pm−Pe−Pd (2.24) Thus, we can rewrite Equation (2.24) as follows: ¨δ= ˙ω=−p 2MPd+p 2M(Pm−Pe) (2.25) Since power system analysis is done in p.u. system, the swing equation is usually expressed in per unit. Dividing Equation (2.24) by the base power SB, and substituting for M results in 2 p2Wk ωmRSB¨δ=2 p2Wk ωmRSB˙ω=Pm SB−Pe SB−Pd SB(2.26) W e now define an important quantity known as the p.u. inertia constant H[8]. H=Wk SB(2.27) The unit of H is seconds. The value of H ranges from 1 to 10 seconds, depending on the size and type of machine. The per unit accelerating power is related to the per unit accelerating torque by Pa(p.u.)= Ta(p.u.)ω ωR. Recognizing that the electrical angular speed ωis nearly constant, and equal to ωR, we have the p.u. accelerating power Pato be numerically nearly equal to the p.u. accelerating torque Ta, i.e. Pa(p.u.)∼= Ta(p.u.). Substituting for H, and Pa(p.u.)∼=Ta(p.u.)in Equation (2.26) , we get 2 p2H ωmR¨δ=2 p2H ωmR˙ω=Pa(p.u.)=Pm(p.u.)−Pe(p.u.)−Pd(p.u.)∼=Ta(p.u.)=Tm(p.u.)−Te(p.u.)−Td(p.u.) (2.28) where Pm(p.u.),Pe(p.u.), and Pd(p.u.)are the per unit mechanical power, electrical power, and damping power respectively . Substituting ωR=p 2ωmR in Equation (2.28) we get 2H ωR¨δ=2H ωR˙ω=Pm(p.u.)−Pe(p.u.)−Pd(p.u.)∼=Tm(p.u.)−Te(p.u.)−Td(p.u.) (2.29) In Equation (2.29) , while the torque is normalized, the angular speed ωand the time tare not in per unit. Thus the equation is not completely in per unit. W e know that the angular speed ωand time tin per unit are given by ω(p.u.)=ω ωR t(p.u.)=ωRt(2.30) where the base angular velocity ωB=ωR. Substituting Equation (2.30) in Equation (2.29) the normalized swing equation can be written as τjdω(p.u.) dt(p.u.)=Tm(p.u.)−Te(p.u.)−Td(p.u.) (2.31) 13 where τj= 2HωR. The damping torque is calculated as Td(p.u.)=Dω , where D is the damping constant. The electrical torque Teϕ is calculated as Teϕ=iqλd−idλq (2.32) Also Te=Teϕ 3, where Teis the per unit electromagnetic torque defined on a three phase (3ϕ)V A base, and Teϕ is the per unit electromagnetic torque defined on a per phase V A base. Substituting Equation (2.6) and Equation (2.7) into Equation (2.32) and writing Tein the p.u. system, we obtain Teϕ(p.u.)= 3Te(p.u.)=Ldidiq+kMFiFiq+kMDiDiq−Lqidiq−kMQidiQ (2.33) F rom Equation (2.23) we have ω=˙δ+ωR. If we choose ωRas the frequency base and divide both sides of this equation by ωRwe have ω ωR=˙δ ωR+ωR ωR(2.34) Since, ω(p.u.)=ω ωRand ˙δ(p.u.)=˙δ ωRwe can write Equation (2.34) as˙δ(pu)=ω(pu)−1. Thus, from Equation (2.29) ,Equation (2.33) , and Equation (2.34) we can write the mechanical dynamics in the p.u. system as ˙ω=−1 τj(Ldidiq+kMFiFiq+kMDiDiq−Lqidiq−kMQidiQ) 3−1 τjDω+1 τjTm ˙δ=ω−1(2.35) 2.4 T ruth Model of the Synchronous Generator By combining the electrical dynamics and mechanical dynamics, we obtain the truth model of the syn- chronous generator which is highly nonlinear. Let us define L=2 66664Ld+LekMFkMD 0 0 −kMF−LF−MR 0 0 −kMD−MR−LD 0 0 0 0 0 Lq+LekMQ 0 0 0 −kMQ−LQ3 77775 Also denote µ= (Ld+Le)M2 R−LDLF(Ld+Le) +k2(LDM2 F+LFM2 D−2MDMFMR)andν=−k2M2 Q+ LQ(Le+Lq), we can derive the inverse matrix of Las L−1=2 6666641 µ(M2 R−LDLF)k µ(MDMR−LDMF)k µ(MFMR−LFMD) 0 0 −k µ(MDMR−LDMF)−1 µ(M2 Dk2−LD(Ld+Le))−1 µ((Ld+Le)MR−MDMFk2) 0 0 −k µ(MFMR−LFMD)−1 µ((Ld+Le)MR−MDMFk2)−1 µ(M2 Fk2−LF(Ld+Le)) 0 0 0 0 0LQ νkMQ ν 0 0 0 −kMQ ν−Le+Lq ν3 777775 =2 66664Ld1 kMF1kMD1 0 0 −kMF1−LF1−MR1 0 0 −kMD1−MR1−LD1 0 0 0 0 0 Lq1 kMQ1 0 0 0−kMQ1−LQ13 77775 (2.36) where Ld1=1 µ(M2 R−LDLF),LF1=1 µ(M2 Dk2−LD(Ld+Le)),LD1=1 µ(M2 Fk2−LF(Ld+Le)),MF1= 1 µ(MDMR−LDMF),MD1=1 µ(MFMR−LFMD),MR1=1 µ((Ld+Le)MR−MDMFk2),Lq1=LQ ν,LQ1= 14 Le+Lq ν, and MQ1=MQ ν. Using Equation (2.36) and Equation (2.13) we can write ˙id=−Ld1(r+Re)id+kMF1rFiF+kMD1rDiD−(Lq+Le)Ld1iqω−kMQLd1iQω +√ 3V∞Ld1sin(δ−α)−kMF1vF ˙iF=kMF1(r+Re)id−LF1rFiF−MR1rDiD+kMF1(Lq+Le)iqω+k2MF1MQiQω −√ 3V∞kMF1sin(δ−α) +LF1vF ˙iD=kMD1(r+Re)id−MR1rFiF−LD1rDiD+kMD1(Lq+Le)iqω+k2MD1MQiQω −√ 3V∞kMD1sin(δ−α) +MR1vF ˙iq=Lq1(Ld+Le)idω+kMFLq1iFω+kMDLq1iDω−Lq1(r+Re)iq+kMQ1rQiQ −√ 3V∞Lq1cos(δ−α) ˙iQ=−kMQ1(Ld+Le)idω−k2MQ1MFiFω−k2MQ1MDiDω+kMQ1(r+Re)iq−LQ1rQiQ +√ 3V∞kMQ1cos(δ−α)(2.37) Dividing both LHS and RHS of Equation (2.37) by√ 3we get ˙id√ 3=−Ld1(r+Re)id√ 3+kMF1rFiF√ 3+kMD1rDiD√ 3−(Lq+Le)Ld1iq√ 3ω−kMQLd1iQ√ 3ω +V∞Ld1sin(δ−α)−kMF1vF√ 3 ˙iF√ 3=kMF1(r+Re)id√ 3−LF1rFiF√ 3−MR1rDiD√ 3+kMF1(Lq+Le)iq√ 3ω+k2MF1MQiQ√ 3ω V∞kMF1sin(δ−α) +LF1vF√ 3 ˙iD√ 3=kMD1(r+Re)id√ 3−MR1rFiF√ 3−LD1rDiD√ 3+kMD1(Lq+Le)iq√ 3ω+k2MD1MQiQ√ 3ω −V∞kMD1sin(δ−α) +MR1vF√ 3 ˙iq√ 3=Lq1(Ld+Le)id√ 3ω+kMFLq1iF√ 3ω+kMDLq1iD√ 3ω−Lq1(r+Re)iq√ 3+kMQ1rQiQ√ 3 −V∞Lq1cos(δ−α) ˙iQ√ 3=−kMQ1(Ld+Le)id√ 3ω−k2MQ1MFiF√ 3ω−k2MQ1MDiD√ 3ω+kMQ1(r+Re)iq√ 3−LQ1rQiQ√ 3 +V∞kMQ1cos(δ−α) (2.38) Converting the state variables id,iF,iD,iq,iQ, and control input vFto their corresponding RMS quantities Id,IF,ID,Iq,IQ, and VFby substitutingid√ 3=Id,iF√ 3=IF,iD√ 3=ID,iq√ 3=Iq,iQ√ 3=IQ, andvF√ 3=VFin 15 Equation (2.38) we get ˙Id=−Ld1(r+Re)Id+kMF1rFIF+kMD1rDID−(Lq+Le)Ld1Iqω−kMQLd1IQω+V∞Ld1sin(δ−α) −kMF1VF ˙IF=kMF1(r+Re)Id−LF1rFIF−MR1rDID+kMF1(Lq+Le)Iqω+k2MF1MQIQω−V∞kMF1sin(δ−α) +LF1VF ˙ID=kMD1(r+Re)Id−MR1rFIF−LD1rDID+kMD1(Lq+Le)Iqω+k2MD1MQIQω−V∞kMD1sin(δ−α) +MR1VF ˙Iq=Lq1(Ld+Le)Idω+kMFLq1IFω+kMDLq1IDω−Lq1(r+Re)Iq+kMQ1rQIQ−V∞Lq1cos(δ−α) ˙IQ=−kMQ1(Ld+Le)Idω−k2MQ1MFIFω−k2MQ1MDIDω+kMQ1(r+Re)Iq−LQ1rQIQ +V∞kMQ1cos(δ−α) (2.39) Equation (2.35) can be written as ˙ω=−1 τj Ldid√ 3iq√ 3+kMFiF√ 3iq√ 3+kMDiD√ 3iq√ 3−Lqid√ 3iq√ 3−kMQid√ 3iQ√ 3 −1 τjDω+1 τjTm ˙δ=ω−1(2.40) Substitutingid√ 3=Id,iF√ 3=IF,iD√ 3=ID,iq√ 3=Iq,iQ√ 3=IQin Equation (2.40) ˙ω=−1 τj (Ld−Lq)IdIq+kMFIFIq+kMDIDIq−kMQIdIQ −1 τjDω+1 τjTm ˙δ=ω−1(2.41) Equation (2.39) and Equation (2.41) can be combined to get the truth model of the synchronous generator ˙Id=−Ld1(r+Re)Id+kMF1rFIF+kMD1rDID−(Lq+Le)Ld1Iqω−kMQLd1IQω+V∞Ld1sin(δ−α) −kMF1VF ˙IF=kMF1(r+Re)Id−LF1rFIF−MR1rDID+kMF1(Lq+Le)Iqω+k2MF1MQIQω−V∞kMF1sin(δ−α) +LF1VF ˙ID=kMD1(r+Re)Id−MR1rFIF−LD1rDID+kMD1(Lq+Le)Iqω+k2MD1MQIQω−V∞kMD1sin(δ−α) +MR1VF ˙Iq=Lq1(Ld+Le)Idω+kMFLq1IFω+kMDLq1IDω−Lq1(r+Re)Iq+kMQ1rQIQ−V∞Lq1cos(δ−α) ˙IQ=−kMQ1(Ld+Le)Idω−k2MQ1MFIFω−k2MQ1MDIDω+kMQ1(r+Re)Iq−LQ1rQIQ +V∞kMQ1cos(δ−α) ˙ω=−1 τj (Ld−Lq)IdIq+kMFIFIq+kMDIDIq−kMQIdIQ −1 τjDω+1 τjTm ˙δ=ω−1 (2.42) 16 F or simplification of the above expression let us denote: F11=−Ld1(r+Re), F 12=kMF1rF, F 13=kMD1rD, F 14=−(Lq+Le)Ld1, F 15=−kMQLd1, F16=V∞Ld1, G 11=−kMF1, F 21=kMF1(r+Re), F 22=−LF1rF, F 23=−MR1rD, F24=kMF1(Lq+Le), F 25=k2MF1MQ, F 26=−V∞kMF1, G 21=LF1, F 31=kMD1(r+Re), F32=−MR1rF, F 33=−LD1rD, F 34=kMD1(Lq+Le), F 35=k2MD1MQ, F 36=−V∞kMD1, G31=MR1, F 41=Lq1(Ld+Le), F 42=kMFLq1, F 43=kMDLq1, F 44=−Lq1(r+Re), F45=kMQ1rQ, F 46=−V∞Lq1, F 51=−kMQ1(Ld+Le), F 52=−k2MQ1MF, F 53=−k2MQ1MD, F54=kMQ1(r+Re), F 55=−LQ1rQ, F 56=V∞kMQ1, F 61=−1 τj(Ld−Lq), F62=−1 τjkMF, F 63=−1 τjkMD, F 64=1 τjkMQ, F 65=−1 τjD. (2.43) Thus, the 7thorder truth model of the synchronous generator connected to an infinite bus in per unit can be written in the nonlinear state-space form ˙Id=F11Id+F12IF+F13ID+F14Iqω+F15IQω+F16sin(δ−α) +G11VF ˙IF=F21Id+F22IF+F23ID+F24Iqω+F25IQω+F26sin(δ−α) +G21VF ˙ID=F31Id+F32IF+F33ID+F34Iqω+F35IQω+F36sin(δ−α) +G31VF ˙Iq=F41Idω+F42IFω+F43IDω+F44Iq+F45IQ+F46cos(δ−α) ˙IQ=F51Idω+F52IFω+F53IDω+F54Iq+F55IQ+F56cos(δ−α) ˙ω=F61IdIq+F62IFIq+F63IDIq+F64IdIQ+F65ω+F66Tm ˙δ=ω−1(2.44) 2.5 Model of the T urbine-Governor System In this section, we present the dynamics of the turbine-governor system. F or the sake of simplicity we assume a linear model of the turbine-governor system [ 9]. • T urbine dynamics: The dynamics of the turbine are modeled by ˙Pm=−1 τTPm+KT τTGV (2.45) where Pm is the mechanical power output of the turbine, GVis the gate opening of the turbine, τTis the time constant of the turbine, and KTis the gain of the turbine. As done in Equation (2.28) we have the per unit mechanical power numerically equal to the per unit mechanical torque, i.e. Pm(p.u.)=Tm(p.u.). Therefore, the per unit turbine dynamics are ˙Tm(p.u.)=−1 τTTm(p.u.)+KT τTGV(p.u.) (2.46) • Governor dynamics: The dynamics of the governor in per unit are ˙GV(p.u.)=−1 τGGV(p.u.)+KG τG uT−ω(p.u.) RT (2.47) where uTis the turbine valve control, τGis the time constant of the speed governor, KGis the gain of the speed governor, and RTis the regulation constant in per unit. Parameters of the turbine-governor system are KT= 1, K G= 1, τT= 0.5, τG= 0.2, R T= 20 (2.48) 17 2.6 T ruth model of the combined Synchronous Generator and T urbine-Governor Sys- tem connected to an infinite bus In this section, we present the truth model of the combined synchronous generator and turbine-governor sys- tem connected to an infinite bus. This model consists of 7nonlinear differential equations of the synchronous generator and 2linear differential equations of the turbine-governor system. Thus, the combined system consists of 9differential equations. Combining Equation (2.44) ,Equation (2.46) , and Equation (2.47) the truth model of the combined synchronous generator and turbine-governor system connected to an infinite bus, can be written as ˙Id=F11Id+F12IF+F13ID+F14Iqω+F15IQω+F16sin(δ−α) +G11VF ˙IF=F21Id+F22IF+F23ID+F24Iqω+F25IQω+F26sin(δ−α) +G21VF ˙ID=F31Id+F32IF+F33ID+F34Iqω+F35IQω+F36sin(δ−α) +G31VF ˙Iq=F41Idω+F42IFω+F43IDω+F44Iq+F45IQ+F46cos(δ−α) ˙IQ=F51Idω+F52IFω+F53IDω+F54Iq+F55IQ+F56cos(δ−α) ˙ω=F61IdIq+F62IFIq+F63IDIq+F64IdIQ+F65ω+F66Tm ˙δ=ω−1 ˙Tm=F81Tm+F82GV ˙GV=F91ω+F92GV+G92uT(2.49) where F81=−1 τT,F82=KT τT,F91=−KG τGRT,F92=−1 τG,G92=KG τG. Let x= [Id, IF, ID, Iq, IQ, ω, δ, T m, GV]T be the vector of state variables, u= [VF, uT]Tthe vector of control inputs, and y= [Vt, ω]Tthe vector of outputs, then Equation (2.49) can be written in the usual state-space form ˙x=F(x) +G(x)u y=h(x)(2.50) where F(x) =0 BBBBBBBBBBBB@F11x1+F12x2+F13x3+F14x4x6+F15x5x6+F16sin(x7−α) F21x1+F22x2+F23x3+F24x4x6+F25x5x6+F26sin(x7−α) F31x1+F32x2+F33x3+F34x4x6+F35x5x6+F36sin(x7−α) F41x1x6+F42x2x6+F43x3x6+F44x4+F45x5+F46cos(x7−α) F51x1x6+F52x2x6+F53x3x6+F54x4+F55x5+F56cos(x7−α) F61x1x4+F62x2x4+F63x3x4+F64x1x5+F65x6+F66x8 x6−1 F81x8+F82x9 F91x6+F92x91 CCCCCCCCCCCCA G(x) =0 BBBBBBBB@G11 0 G21 0 G31 0 0 0 0 0 0 0 0G921 CCCCCCCCA(2.51) 2.7 Derivation of the Output Generator T erminal V oltage for the T ruth Model The synchronous generator and turbine-governor system connected to an infinite bus is a MIMO system with two inputs: excitation field voltage VFand turbine valve control uT, i.e. u= [u1, u2]T= [VF, uT]T, and two regulated outputs: generator terminal voltage Vtand rotor angle δ, i.e. y= [y1, y2]T= [Vt, δ]T. Since 18 the rotor angle is diﬀicult to measure in a practical system, we use the angular frequency ωinstead of the rotor angle as the second output in all our work, as the frequency ωcan be easily measured. In this section we derive an expression for the generator terminal voltage Vt. F rom Equation (2.12) we have vd=Reid+Le˙id+ωLeiq−√ 3V∞sin(δ−α) vq=Reiq+Le˙iq−ωLeid+√ 3V∞cos(δ−α)(2.52) Dividing Equation (2.52) by√ 3and substitutingvd√ 3=Vd,vq√ 3=Vq,id√ 3=Id, andiq√ 3=Iq, where Vd,Vq, Id, and Iqare the corresponding RMS values, we get Vd=ReId+Le˙Id+ωLeIq−V∞sin(δ−α) Vq=ReIq+Le˙Iq−ωLeId+V∞cos(δ−α)(2.53) Substituting ˙Idand ˙Iqfrom Equation (2.49) in Equation (2.53) Vd=ReId+Le(F11Id+F12IF+F13ID+F14Iqω+F15IQω+F16sin(δ−α) +G11vF) +ωLeIq −V∞sin(δ−α) Vq=ReIq+Le(F41Idω+F42IFω+F43IDω+F44Iq+F45IQ+F46cos(δ−α))−ωLeId+V∞cos(δ−α) (2.54) Simplifying and rearranging Equation (2.54) we get Vd= (Re+LeF11)Id+LeF12IF+LeF13ID+ (LeF14+Le)Iqω+LeF15IQω+ (LeF16−V∞) sin( δ−α) +LeG11VF Vq= (LeF41−Le)Idω+LeF42IFω+LeF43IDω+ (Re+LeF44)Iq+LeF45IQ+ (LeF46+V∞) cos( δ−α) (2.55) F or simplification of Equation (2.55) let us denote Re+LeF11=y11,LeF12=y12,LeF13=y13,LeF14+ Le=y14,LeF15=y15,LeF16−V∞=y16,LeG11=i11,LeF41−Le=y21,LeF42=y22,LeF43=y23, Re+LeF44=y24,LeF45=y25, and LeF46+V∞=y26. Thus, Equation (2.55) can be simplified to Vd=y11Id+y12IF+y13ID+y14Iqω+y15IQω+y16sin(δ−α) +i11VF Vq=y21Idω+y22IFω+y23IDω+y24Iq+y25IQ+y26cos(δ−α)(2.56) The generator terminal voltage Vtis computed as Vt=q V2 d+V2q (2.57) where VdandVqare as given in Equation (2.55) . The output generator terminal voltage, y1=Vt, as a function of the states x, and control inputs u, is Vd=y11x1+y12x2+y13x3+y14x4x6+y15x5x6+y16sin(x7−α) +i11u1 Vq=y21x1x6+y22x2x6+y23x3x6+y24x4+y25x5+y26cos(x7−α) y1=Vt=q V2 d+V2q(2.58) Therefore, the output equation is given by y=h(x) =Vt x6 (2.59) where Vtis as given in Equation (2.57) . 19 3 The Reduced Order Simplified Model A detailed derivation of the reduced order simplified model of a synchronous generator connected to an infinite bus is given in the Appendix . 3.1 Reduced order model of the combined Synchronous Generator and T urbine-Governor system connected to an infinite bus The final system equations for the reduced order simplified model of the synchronous generator are summa- rized below ˙E′ q=1 τ′ d0(EFD−E′ q+ (Ld−L′ d)Id) ˙ω=1 τj[Tm−Dω−(E′ qIq−(Lq−L′ d)IdIq)] ˙δ=ω−1(3.1) where E′ dis the daxis voltage behind the transient reactance L′ q, and E′ qis the qaxis voltage behind the transient reactance L′ d, where L′ d=Ld−(kMF)2 LF.τ′ d0is the daxis transient open circuit time constant and is given by the relation τ′ d0=LF rF.EFD is the excitation field emf, and τj= 2HωR. Also IdandIqare the direct axis and quadrature axis currents respectively . E′ dis given by an algebraic constraint E′ d=−(Lq−L′ q)Iq (3.2) Note that all the variables in the third axis model of Equation (3.1) are RMS quantities. By applying Kirchhoff’s voltage law (KVL) to the daxis and qaxis stator circuits, the daxis and qaxis stator voltage equations of a synchronous generator in per unit are Vd=−rId−L′ qIq+E′ d Vq=−rIq+L′ dId+E′ q(3.3) On substituting E′ das given in Equation (3.2) in Equation (3.3) we get Vd=−rId−LqIq Vq=−rIq+L′ dId+E′ q(3.4) By applying KVL to a synchronous generator connected to an infinite bus the stator voltage equations can be written as V∞d=Vd−ReId−LeIq =−(r+Re)Id−(Lq+Le)Iq V∞q=Vq−ReIq+LeId =−(r+Re)Iq+ (L′ d+Le)Id+E′ q(3.5) where V∞dandV∞dare direct axis and quadrature axis infinite bus voltages respectively , Reis the resis- tance and Leis the inductance of the infinite bus. W e now solve the two simultaneous equations given in Equation (3.5) to determine the two unknowns IdandIq. On dividing V∞din Equation (3.5) by−(Lq+Le) V∞d −(Lq+Le)=−(r+Re) −(Lq+Le)Id+Iq (3.6) Similarly dividing V∞qin Equation (3.5) byr+Re V∞q (r+Re)=−Iq+(L′ d+Le) (r+Re)Id+E′ q (r+Re)(3.7) 20 Now we add Equation (3.6) and Equation (3.7) and compute Id Id=−(E′ q−V∞q)(Lq+Le)−V∞d(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)(3.8) Iqis determined in a similar fashion. Dividing V∞din Equation (3.5) by(r+Re) V∞d (r+Re)=−Id−(Lq+Le) (r+Re)Iq (3.9) Also dividing V∞qin Equation (3.5) byL′ d+Le V∞q (L′ d+Le)=−(r+Re) (L′ d+Le)Iq+Id+E′ q (L′ d+Le)(3.10) Now we add Equation (3.9) and Equation (3.10) and compute Iq Iq=(E′ q−V∞q)(r+Re)−V∞d(L′ d+Le) (r+Re)2+ (L′ d+Le)(Lq+Le)(3.11) W e substitute IdandIqas given in Equation (3.8) and Equation (3.11) in the reduced order simplified model of the synchronous generator as given in Equation (3.1) to get ˙E′ q=1 τ′ d0 EFD−E′ q+ (Ld−L′ d)−(E′ q−V∞q)(Lq+Le)−V∞d(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le) (3.12) F or simplification of the above equation we make the following substitutions: Lq+Le=L1, r+Re=R1, Ld−L′ d=L2and (r+Re)2+ (L′ d+Le)(Lq+Le) =M1 Thus, Equation (3.12) can be rewritten as ˙E′ q=1 τ′ d0 EFD−E′ q−L2L1 M1E′ q+L2L1 M1V∞q−L2R1 M1V∞d (3.13) Substituting V∞d=−V∞sin(δ−α)andV∞q=V∞cos(δ−α)in Equation (3.13) and rearranging it we get ˙E′ q=−(1 +L2L1 M1) τ′ d0E′ q+L2L1V∞ M1τ′ d0cos(δ−α) +L2R1V∞ M1τ′ d0sin(δ−α) +1 τ′ d0EFD (3.14) Let us denote−(1+L2L1 M1) τ′ d0=f11,L2L1V∞ M1τ′ d0=f12,L2R1V∞ M1τ′ d0=f13 and1 τ′ d0=g11. Thus, Equation (3.14) can be rewritten as ˙E′ q=f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD (3.15) Substituting IdandIqin the equation for ˙ω in the reduced order simplified model of the synchronous generator as given in Equation (3.1) we get ˙ω=1 τj Tm−Dω− E′ q(E′ q−V∞q)(r+Re)−V∞d(L′ d+Le) (r+Re)2+ (L′ d+Le)(Lq+Le) −(Lq−L′ d)−(E′ q−V∞q)(Lq+Le)−V∞d(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)(E′ q−V∞q)(r+Re)−V∞d(L′ d+Le) (r+Re)2+ (L′ d+Le)(Lq+Le) (3.16) 21 Let us denote L′ d+Le=L3andLq−L′ d=L4. Substituting V∞d=−V∞sin(δ−α)andV∞q=V∞cos(δ−α) in Equation (3.16) and on simplifying we get ˙ω=−R1 M1τj+L4L1R1 M2 1τj E′2 q+R1 M1τj+2L4L1R1 M2 1τj V∞E′ qcos(δ−α) −L3 M1τj+L4L1L3 M2 1τj−L4R2 1 M2 1τj V∞E′ qsin(δ−α) −L4R2 1 M2 1τj−L4L1L3 M2 1τj V2 ∞sin(δ−α) cos( δ−α)−L4L1R1V2 ∞ M2 1τj cos2(δ−α) +L4L3R1V2 ∞ M2 1τj sin2(δ−α)−D τjω+1 τjTm(3.17) Let us denote −R1 M1τj+L4L1R1 M2 1τj =f21,R1 M1τj+2L4L1R1 M2 1τj V∞=f22, −L3 M1τj+L4L1L3 M2 1τj−L4R2 1 M2 1τj V∞=f23,−L4R2 1 M2 1τj−L4L1L3 M2 1τj V2 ∞=f24, −L4L1R1V2 ∞ M2 1τj =f25,L4L3R1V2 ∞ M2 1τj =f26,−D τj=f27,and1 τj=f28(3.18) Thus, Equation (3.17) can be rewritten as ˙ω=f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm(3.19) W e can express the reduced order nonlinear model of the synchronous generator connected to an infinite bus in the usual state space form ˙E′ q=f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD ˙ω=f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm ˙δ=ω−1(3.20) F or the reduced order generator-turbine system we use the same turbine-governor model that we used for the truth model. The model of the turbine-governor system as given in Equation (2.46) and Equation (2.47) in per unit is ˙Tm=−1 τTTm+KT τTGV ˙GV=−KG τGRTω−1 τGGV+KG τGuT(3.21) The reduced order model of the generator-turbine system connected to an infinite bus consists of 3nonlinear differential equations of the synchronous generator and 2linear differential equations of the turbine-governor system. Thus, the combined system consists of 5differential equations. Combining Equation (3.20) and Equation (3.21) the reduced order model of the synchronous generator and turbine connected to an infinite bus can be written as ˙E′ q=f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD ˙ω=f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm ˙δ=ω−1 ˙Tm=f41Tm+f42GV ˙GV=f51ω+f52GV+g55uT(3.22) 22 where, −1 τT=f41,KT τT=f42,−KG τGRT=f51,−1 τG=f52, andKG τG=g55. Let us define the state variables as x= [E′ q, ω, δ, T m, GV]T, and the two control inputs as u= [u1, u2]T= [EFD, uT]T. W e can then express the simplified fifth order nonlinear model of the synchronous generator and turbine connected to an infinite bus as ˙x1=f11x1+f12cos(x3−α) +f13sin(x3−α) +g11u1 ˙x2=f21x2 1+f22x1cos(x3−α) +f23x1sin(x3−α) +f24sin(x3−α) cos( x3−α) +f25cos2(x3−α) +f26sin2(x3−α) +f27x2+f28x4 ˙x3=x2−1 ˙x4=f41x4+f42x5 ˙x5=f51x2+f52x5+g55u2(3.23) Thus, we can put the simplified fifth order nonlinear model of the synchronous generator and turbine connected to an infinite bus in the usual state-space form ˙x=f(x) +g(x)u y=h(x)(3.24) where f(x) =0 BBBBBB@f11x1+f12cos(x3−α) +f13sin(x3−α) f21x2 1+f22x1cos(x3−α) +f23x1sin(x3−α) +f24sin(x3−α) cos( x3−α)· ·· · · ·+f25cos2(x3−α) +f26sin2(x3−α) +f27x2+f28x4 x2−1 f41x4+f42x5 f51x2+f52x51 CCCCCCA g(x) =0 BBBB@g11 0 0 0 0 0 0 0 0g551 CCCCA(3.25) In Equation (3.24) and Equation (3.25) , the state variables are x= [E′ q, ω, δ, T m, GV]T, the two control inputs are u= [EFD, uT]T, and the two regulated outputs are y= [Vt, ω]T. The expression for the generator terminal voltage Vtwill be derived in the next subsection. 3.2 Derivation of the Output Generator T erminal V oltage for the Reduced Order Model The reduced order model of the synchronous generator and turbine connected to an infinite bus is a MIMO system with two inputs: excitation field EMF EFD , and turbine valve control uT, and two outputs: generator terminal voltage Vtand rotor angle δ. In this section we derive an expression for the generator terminal voltage Vtwhich is the first output of the MIMO system. F rom Equation (3.4) the direct axis and quadrature axis stator voltage equations of a synchronous generator are given by Vd=−rId−LqIq Vq=−rIq+L′ dId+E′ q(3.26) Since the stator resistance r≈0we can write Vd=−LqIq Vq=L′ dId+E′ q(3.27) 23 Substituting IdandIqas given in Equation (3.8) and Equation (3.11) in Equation (3.27) we get Vd=−LqIq =−Lq(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)E′ q +V∞Lq(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)cos(δ−α) −V∞Lq(L′ d+Le) (r+Re)2+ (L′ d+Le)(Lq+Le)sin(δ−α)(3.28) Let us denote −Lq(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)=Vd1 V∞Lq(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)=Vd2 −V∞Lq(L′ d+Le) (r+Re)2+ (L′ d+Le)(Lq+Le)=Vd3(3.29) Thus, Equation (3.28) can be rewritten as Vd=Vd1E′ q+Vd2cos(δ−α) +Vd3sin(δ−α) (3.30) Now let us derive the expression for Vq Vq=L′ dId+E′ q =−L′ d(Lq+Le) (r+Re)2+ (L′ d+Le)(Lq+Le)E′ q +V∞L′ d(Lq+Le) (r+Re)2+ (L′ d+Le)(Lq+Le)cos(δ−α) +V∞L′ d(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)sin(δ−α) +E′ q(3.31) Let us denote −L′ d(Lq+Le) (r+Re)2+ (L′ d+Le)(Lq+Le)=Vq1 V∞L′ d(Lq+Le) (r+Re)2+ (L′ d+Le)(Lq+Le)=Vq2 V∞L′ d(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le)=Vq3(3.32) Thus, Equation (3.31) can be rewritten as Vq=Vq1E′ q+Vq2cos(δ−α) +Vq3sin(δ−α) +E′ q (3.33) The generator terminal voltage Vtis given by Vt=q V2 d+V2q (3.34) The output generator terminal voltage, y1=Vt, as a function of the states x, is Vd=Vd1x1+Vd2cos(x3−α) +Vd3sin(x3−α) Vq=Vq1x1+Vq2cos(x3−α) +Vq3sin(x3−α) +x1 Vt=q V2 d+V2q(3.35) 24 Thus, the output equation consists of a nonlinear equation for Vtas given in Equation (3.34) and Equa- tion (3.35) , and a simple linear equation for ω y=h(x) =Vt x2 (3.36) 3.3 Linearization of the Reduced Order Model by T aylor series approximation In this section we linearize the fifth-order nonlinear model of the synchronous generator and turbine con- nected to an infinite bus by using the T aylor series approximation about a nominal operating point ( x0,u0). The operating condition is a steady state equilibrium of the system. The steady state equilibrium condition is attained by the system after all the transients die out or decay to zero. The equilibrium point ( x0,u0) is computed by solving the differential equation, ˙x=f(x0) +g(x0)u0= 0 . F rom Equation (3.25) we can write f1(x) =f11x1+f12cos(x3−α) +f13sin(x3−α) (3.37) Therefore, ∂f1 ∂x1 x0=f11 ∂f1 ∂x2 x0= 0 ∂f1 ∂x3 x0=−f12sin(x30−α) +f13cos(x30−α) ∂f1 ∂x4 x0= 0 ∂f1 ∂x5 x0= 0(3.38) f2(x) =f21x2 1+f22x1cos(x3−α) +f23x1sin(x3−α) +f24sin(x3−α) cos( x3−α) +f25cos2(x3−α) +f26sin2(x3−α) +f27x2+f28x4(3.39) Therefore, ∂f2 ∂x1 x0= 2f21x10+f22cos(x30−α) +f23sin(x30−α) ∂f2 ∂x2 x0=f27 ∂f2 ∂x3 x0=−f22x10sin(x30−α) +f23x10cos(x30−α) +f24cos 2( x30−α)−f25sin 2( x30−α) +f26sin 2( x30−α) ∂f2 ∂x40 x0=f28 ∂f2 ∂x5 x0= 0 (3.40) f3(x) =x2−1 (3.41) 25 Therefore, ∂f3 ∂x1 x0= 0 ∂f3 ∂x2 x0= 1 ∂f3 ∂x3 x0= 0 ∂f3 ∂x4 x0= 0 ∂f3 ∂x5 x0= 0(3.42) f4(x) =f41x4+f42x5 (3.43) Therefore, ∂f4 ∂x1 x0= 0 ∂f4 ∂x2 x0= 0 ∂f4 ∂x3 x0= 0 ∂f4 ∂x4 x0=f41 ∂f4 ∂x5 x0=f42(3.44) f5(x) =f51x2+f52x5 (3.45) Therefore, ∂f5 ∂x1 x0= 0 ∂f5 ∂x2 x0=f51 ∂f5 ∂x3 x0= 0 ∂f5 ∂x4 x0= 0 ∂f5 ∂x5 x0=f52(3.46) g1(x) =g11u1 (3.47) Therefore, ∂g1 ∂u1 u0=g11 ∂g1 ∂u2 u0= 0(3.48) 26 g2(x) = 0 (3.49) Therefore, ∂g2 ∂u1 u0= 0 ∂g2 ∂u2 u0= 0(3.50) g3(x) = 0 (3.51) Therefore, ∂g3 ∂u1 u0= 0 ∂g3 ∂u2 u0= 0(3.52) g4(x) = 0 (3.53) Therefore, ∂g4 ∂u1 u0= 0 ∂g4 ∂u2 u0= 0(3.54) g5(x) =g55u2 (3.55) Therefore, ∂g5 ∂u1 u0= 0 ∂g5 ∂u2 u0=g55(3.56) Therefore, the linear reduced order model of the synchronous generator and turbine-governor system con- nected to an infinite bus is ˙x=Ax+Bu (3.57) where xT= ∆E′ q∆ω∆δ∆Tm∆GV u=∆EFD ∆uT(3.58) In the above equation ∆ is the deviation from the nominal operating condition, i.e. E′ q−E′ q0= ∆E′ q, ω−ω0= ∆ω,δ−δ0= ∆δ,Tm−Tm0= ∆Tm,GV−GV0= ∆GV,EFD−EFD0= ∆EFD , and uT−uT0= ∆uT, 27 and A=2 66666666666666664∂f1 ∂x1∂f1 ∂x2∂f1 ∂x3∂f1 ∂x4∂f1 ∂x5 ∂f2 ∂x1∂f2 ∂x2∂f2 ∂x3∂f2 ∂x4∂f2 ∂x5 ∂f3 ∂x1∂f3 ∂x2∂f3 ∂x3∂f3 ∂x4∂f3 ∂x5 ∂f4 ∂x1∂f4 ∂x2∂f4 ∂x3∂f4 ∂x4∂f4 ∂x5 ∂f5 ∂x1∂f5 ∂x2∂f5 ∂x3∂f5 ∂x4∂f5 ∂x53 77777777777777775 x0 =2 66666666666664f11 0∂f1 ∂x30 0 ∂f2 ∂x1f27∂f2 ∂x3f28 0 0 1 0 0 0 0 0 0 f41f42 0f51 0 0 f523 77777777777775 x0(3.59) B=2 66666666666666664∂g1 ∂u1∂g1 ∂u2 ∂g2 ∂u1∂g2 ∂u2 ∂g3 ∂u1∂g3 ∂u2 ∂g4 ∂u1∂g4 ∂u2 ∂g5 ∂u1∂g5 ∂u23 77777777777777775 u0 =2 66664g11 0 0 0 0 0 0 0 0g553 77775 u0(3.60) In order to design a linear controller for the linearized model obtained by using T aylor series approximation, we need to find an expression for the first output ∆Vt. By using T aylor series approximation, we have Vt(x) =Vt(x0) +dVt dx x0(x−x0) +..... (3.61) Where, x= [E′ q, ω, δ, T m, GV]T, are the states of the reduced order nonlinear model. Let us denote Vt(x)− Vt(x0) = ∆ Vt, and Vt(x0) =Vt0. By neglecting the second and higher order derivatives in Equation (3.61) we have ∆Vt=dVt dx x0(x−x0) (3.62) F rom Equation (3.34) and Equation (3.35) the generator terminal voltage Vtis given by Vt=q V2 d+V2q (3.63) 28 where Vd=Vd1E′ q+Vd2cos(δ−α) +Vd3sin(δ−α) Vq=Vq1E′ q+Vq2cos(δ−α) +Vq3sin(δ−α) +E′ q(3.64) W e can write ∆Vt=dVt dx x0(x−x0) =∂Vt ∂VddVd dx x0(x−x0) +∂Vt ∂VqdVq dx x0(x−x0) (3.65) Differentiating Equation (3.63) with respect to VdandVqrespectively ∂Vt ∂Vd x0=Vd0q V2 d0+V2 q0=Vd0 Vt0 ∂Vt ∂Vq x0=Vq0q V2 d0+V2 q0=Vq0 Vt0(3.66) Also dVd dx x0(x−x0) = (E′ q−E′ q0)∂Vd ∂E′q x0+ (ω−ω0)∂Vd ∂ω x0+ (δ−δ0)∂Vd ∂δ x0+ (Tm−Tm0)∂Vd ∂Tm x0+ (GV−GV0)∂Vd ∂GV x0 dVq dx x0(x−x0) = (E′ q−E′ q0)∂Vq ∂E′q x0+ (ω−ω0)∂Vq ∂ω x0+ (δ−δ0)∂Vq ∂δ x0+ (Tm−Tm0)∂Vq ∂Tm x0+ (GV−GV0)∂Vq ∂GV x0 (3.67) Let us recall that the deviations of the state variables form their nominal operating condition are given by:E′ q−E′ q0= ∆E′ q,ω−ω0= ∆ω,δ−δ0= ∆δ,Tm−Tm0= ∆Tm,GV−GV0= ∆GV. Simplifying Equation (3.67) we get dVd dx x0(x−x0) =Vd1∆E′ q−Vd2sin(δ◦−α)∆δ+Vd3cos(δ◦−α)∆δ dVq dx x0(x−x0) =Vq1∆E′ q−Vq2sin(δ◦−α)∆δ+Vq3cos(δ◦−α)∆δ+△E′ q(3.68) Substituting Equation (3.66) and Equation (3.68) in Equation (3.65) and rearranging we get ∆Vt=Vd0 Vt0Vd1+Vq0 Vt0Vq1+Vq0 Vt0 ∆E′ q + −Vd0 Vt0Vd2sin(δ◦−α) +Vd0 Vt0Vd3cos(δ◦−α)−Vq0 Vt0Vq2sin(δ◦−α) +Vq0 Vt0Vq3cos(δ◦−α) ∆δ(3.69) Let us denote Vd0 Vt0Vd1+Vq0 Vt0Vq1+Vq0 Vt0 =T1  −Vd0 Vt0Vd2sin(δ◦−α) +Vd0 Vt0Vd3cos(δ◦−α)−Vq0 Vt0Vq2sin(δ◦−α) +Vq0 Vt0Vq3cos(δ◦−α) =T2(3.70) Thus, Equation (3.69) can be simplified to get ∆Vt=T1∆E′ q+T2∆δ (3.71) Thus, the linearized output equation of the reduced order model is given by y=Cx (3.72) 29 where y=∆Vt ∆ω (3.73) and the output matrix C is C=T10T20 0 0 1 0 0 0 (3.74) 3.4 Example In this section we calculate the (A, B, C )system matrices of the truth model and the reduced order model linearized about a given nominal operating point. The power, voltage, and current ratings as well as the parameters of the synchronous generator are given in T able 3.1 , which contains values for an actual synchronous generator with some quantities, denoted by an asterisk, being estimated for our study [ 1]. The power rating of a synchronous generator is equal to the product of the voltage per phase, the current per phase, and the number of phases. It is normally stated in megavolt-amperes (MV A) for large generators. The parameters given in T able 3.1 are not in the per unit system. The quantities in T able 3.1 are converted to the per unit system, which are given in T able 3.2 [1].T able 3.3 gives the parameters of the reduced order nonlinear model in the per unit system. Before we proceed to the calculation of the system matrices, we briefly explain a basic power system terminology , that is used in determining a suitable operating condition. Real power (also known as active power) ( P), measured in watts (W); apparent power ( S), measured in volt-amperes (V A); and reactive power (Q), measured in reactive volt-amperes (var) are related by the expression, S=P+jQ in vector form. If ϕis the phase angle between the current and voltage, then the power factor (PF) is equal to the cosine of the angle, |cosϕ|, and|P|=|S|cosϕand|Q|=|S|sinϕ. Power factors are usually stated as ”leading” or ”lagging” to show the sign of the phase angle ϕ. If a purely resistive load is connected to a power supply , current and voltage will change polarity in step, the power factor will be unity , and the electrical energy flows in a single direction across the network in each cycle. Inductive loads such as transformers and motors (any type of wound coil) consume reactive power with current waveform lagging the voltage. Capacitive loads such as capacitor banks or buried cable generate reactive power with current phase leading the voltage. Both types of loads will absorb energy during part of the AC cycle, which is stored in the device’s magnetic or electric field, only to return this energy back to the source during the rest of the cycle. Since a majority of the loads in a power grid are inductive, we assume power factor lagging conditions, where the generator armature current lags the generator terminal voltage. Also a high power factor is generally desirable in a transmission system to reduce transmission losses and improve voltage regulation at the load. A power factor of 0.85 is assumed, which is within the stable operating limits of a synchronous generator. The synchronous generator is connected to an infinite bus through a transmission line having Re= 0.02 p.u., and Le= 0.4p.u. The infinite bus voltage is 1.0p.u. The machine loading is given by , real power P= 1.0p.u. at 0.85 PF lagging conditions. The steady state operating conditions of a synchronous generator turbine system connected to an infinite bus depend on the synchronous generator turbine system parameters, transmission line parameters, and the machine loading. F or the synchronous generator with parameters and loading conditions given in T able 3.1 and T able 3.2 , a steady state operating point for the 30 truth model is evaluated which is as given in [ 1]: Id0=−0.9185 IF0= 1.6315 ID0=−4.6204×10−6 Iq0= 0.4047 IQ0= 5.9539×10−5 ω0= 1 δ0= 1 Tm0= 1.0012 GV0= 1.0012 Vq0= 0.9670 Vd0=−0.6628 Vt0= 1.172 δ0−α= 53.736◦(3.75) T able 3.1: Ratings and Parameters of the Synchronous Generator V ariables Rated MV A Rated voltage Excitation voltage Stator current V alues 160 MV A 15 k V, Y connected 375 V 6158.40 A V ariables Field current Power factor Ld LF V alues 926 A 0.85 6.341×10−3H 2.189 H V ariables LD Lq LQ kMF V alues 5.989×10−3H∗6.118×10−3H 1.423×10−3H∗109.01×10−3H∗ V ariables kMD MR kMQ r(125◦C) V alues 5.782×10−3H∗109.01×10−3H∗2.779×10−3H∗1.542×10−3Ω V ariables rF(125◦C) rD rQ Inertia constant H V alues 0.371 Ω 18.421×10−3Ω∗18.969×10−3Ω∗1.765kW·s hp V ariables Re Le D ω0(in rads) V alues 0.02 p.u. 0.4 p.u. 0 376.99 T able 3.2: Parameters of the T ruth model of the Synchronous Generator-T urbine System in p.u. V ariables (in p.u.) Ld LF LD Lq LQ kMF V alues 1.70 1.65 1.605 1.64 1.526 1.55 V ariables (in p.u.) kMD MR kMQ r rF rD V alues 1.55 1.55 1.49 0.001096 0.000742 0.0131 V ariables (in p.u.) rQ H (in s) Re Le D ω0(in rads) V alues 0.0540 2.37 0.02 0.4 0 376.99 V ariables (in p.u.) KT KG τT τG RT k V alues 1 1 0.5 0.2 20p 3/2 V ariables (in p.u.) L′ dτ′ d0τj V∞ α V alues 0.245 5.9 4.74 1.00 3.5598◦ 31 T able 3.3: Parameters of the Reduced Order Nonlinear Model of the SMIB in p.u. V ariables (in p.u.) Vd1 Vd2 Vd3 Vq1 Vq2 Vq3 V alues -0.0249 0.0249 -0.8037 -0.3797 0.3797 0.0037 V ariables (in p.u.) f11 f12 f13 f21 f22 f23 V alues -0.5517 0.3822 0.0037 -0.0101 0.0171 -0.3269 V ariables (in p.u.) f24 f25 f26 f27 f28 f41 V alues 0.2235 -0.0069 0.0022 0 0.2110 -2 V ariables (in p.u.) f42 f51 f52 g11 g55 EFDmax V alues 2 -0.2500 -5 0.1695 5 5 V ariables (in p.u.) EFDmin GV max GV min V alues -5 1.2 0 TheA matrix of the truth model linearized about the operating point is A=2 6666666666664−0.0361 0 .000437 0 .0142 −3.4883 −2.5478 −1.4119 1 .0115 0 0 0.0124 −0.0050 0 .0772 1 .2022 0 .8781 0 .4866 −0.3486 0 0 0.0228 0 .0044 −0.0964 2 .2077 1 .6125 0 .8936 −0.6401 0 0 3.5888 2 .6489 2 .6489 −0.0361 0 .0901 1 .0254 1 .3779 0 0 −3.5042 −2.5864 −2.5864 0 .0352 −0.1234 −1.0012 −1.3454 0 0 −0.000013 −0.00035 −0.00035 −0.0014 −0.00076 0 0 0 .00056 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 −2.0000 2 .0000 0 0 0 0 0 −0.2500 0 0 −5.00003 7777777777775 (3.76) TheB matrix of the truth model linearized about the operating point is B=2 6666666666664−0.5893 0 6.6918 0 −5.8933 0 0 0 0 0 0 0 0 0 0 0 0 5 .00003 7777777777775(3.77) theC matrix of the truth model linearized about the operating point is C=0.8510 0 .8739 0 .8708 0 .5673 0 .6059 0 .8691 −0.1048 0 0 0 0 0 0 0 1 .0000 0 0 0 (3.78) and the D matrix of the truth model linearized about the operating point is D=0.1333 0 0 0 (3.79) 32 The open loop eigenvalues of the A matrix of the truth model are −5.0000 −0.0359 + 0 .9983i −0.0359−0.9983i −2.0000 −0.0016 + 0 .0289i −0.0016−0.0289i −0.0007 −0.0995 −0.1217(3.80) Also a steady state operating point for the reduced order nonlinear model is x10=E′ q0= 1.1925 x20=ω0= 1 x30=δ0= 1 x40=Tm0= 1.0012 x50=GV0= 1.0012(3.81) TheA matrix of the reduced order model linearized about the operating point is A=2 66664−0.5517 0 −0.3060 0 0 −0.2776 0 −0.3054 0 .2110 0 0 1 0 0 0 0 0 0 −2 2 0 −0.25 0 0−53 77775(3.82) TheB matrix of the reduced order model linearized about the operating point is B=2 666640.1695 0 0 0 0 0 0 0 0 53 77775(3.83) theC matrix of the reduced order model linearized about the operating point is C=0.5258 0 0 .0294 0 0 0 1 0 0 0 (3.84) and the D matrix of the reduced order model linearized about the operating point is D=0 0 0 0 (3.85) The open loop eigenvalues of the A matrix of the reduced order model are −5.0069 −0.1048 + 0 .4778i −0.1048−0.4778i −0.3514 −1.9839(3.86) F rom Equation (3.80) and Equation (3.86) we can see that the eigenvalues of the open loop system lie on the left half plane. 33 0 200 400 600 800 100000.010.020.030.040.050.060.070.080.090.1Γ1 (Test Signal 1) time (sec)Figure 4.1: Plot of Γ1(T est Signal 1) vs time 0 200 400 600 800 100000.0050.010.0150.020.0250.030.0350.040.0450.05Γ2 (Test Signal 2) time (sec) Figure 4.2: Plot of Γ2(T est Signal 2) vs time 4 Open Loop Input-Output Behavior of the Synchronous Generator and T urbine-Governor System In this section we observe the open loop input-output behavior of the truth model and the reduced order nonlinear model of the SMIB for test signals given in Figure 4.1 and Figure 4.2 . The first control input VF of the truth model and the first control input EFD of the reduced order model are different. As given in Equation (10.41) the field voltage, VF, is related to excitation field emf, EFD , by the following expression VF=rF ωRkMF EFD =e15EFD(4.1) In the above expression, ωR= 1 p.u., and e15= (rF ωRkMF). W e first test the truth model and the reduced order nonlinear model of the SMIB for Γ1(T est signal 1) given in Figure 4.1 . In this case the two inputs for the truth model are, u= [VF, uT]T= [e15Γ1,Γ1]T, and the two inputs for the reduced order nonlinear model are, u= [EFD, uT]T= [Γ 1,Γ1]T. 34 Figure 4.3 ,Figure 4.4 , and Figure 4.5 show generator terminal voltage Vt, rotor angle δ, and frequency ωvs time plots for Γ1(T est Signal 1) applied to the reduced order nonlinear model. F rom Figure 4.3 we can see that after the addition of the fourth step of Γ1(T est Signal 1) at 600 seconds, the generator terminal voltage Vtsettles to a new steady state value of 0.83 p.u.. F rom Figure 4.4 we can see that the rotor angle δ first undergoes a large undershoot, followed by oscillations about 0.1 p.u. after the application of the second incremental step of Γ1at 200 seconds, which is further followed by reduced oscillations about 0.3 p.u. after the application of the third step of Γ1at 400 seconds, and finally after the application of the fourth and final step at 600 seconds the oscillations almost die out and the rotor angle δsettles to a value of 0.725 p.u.. Figure 4.5 shows that the frequency ωsettles to its steady state value of 1 p.u. after the application of the fourth step at 600 seconds. Like the rotor angle δ, the oscillations of the frequency ωalso decrease with the application of each incremental step. Figure 4.6 ,Figure 4.7 , and Figure 4.8 show generator terminal voltage Vt, rotor angle δ, and frequency ω vs time plots for Γ1(T est Signal 1) applied to the truth model. F rom Figure 4.6 we can see that the generator terminal voltage Vtconverges very slowly to the new steady state value of 0.83 p.u.. F rom Figure 4.7 we can see that the rotor angle δfirst undergoes a large undershoot after which it oscillates about 0.1 p.u., where the oscillations decay with time. Figure 4.8 shows that the frequency ω oscillates about its steady state value of 1 p.u.. W e observe a peculiar difference between the performance of the reduced order nonlinear model and the truth model, for Γ1(T est Signal 1). The addition of each incremental step of Γ1after a fixed interval has a distinct effect on the amplitude and damping of the output oscillations for the reduced order nonlinear model, where a rapid decrease in the output oscillations is observed with the application of each step. Thus, we observe transitions in the output response at time instants when incremental steps ofΓ1are applied to the reduced order nonlinear model. However, for the truth model, the effect of adding incremental steps at regular intervals is negligible on the output response, compared to the reduced order nonlinear model. W e do not see transitions in the output behavior at time instants when incremental steps ofΓ1are applied to the truth model, and the oscillations decay more uniformly in this case. 35 0 200 400 600 800 10000.750.80.850.90.9511.051.11.151.2Generator terminal voltage vt (p.u.) time (sec)Figure 4.3: Plot of the generator terminal voltage Vtvs time for Γ1(T est Signal 1) applied to the reduced order nonlinear model 0 200 400 600 800 1000−1.5−1−0.500.51delta (radians) time (sec) Figure 4.4: Plot of the rotor angle δvs time for Γ1(T est Signal 1) applied to the reduced order nonlinear model 0 200 400 600 800 10000.750.80.850.90.9511.051.1omega (radians/sec) time (sec) Figure 4.5: Plot of the frequency ωvs time for Γ1(T est Signal 1) applied to the reduced order nonlinear model 36 0 200 400 600 800 10000.850.90.9511.051.11.151.21.25Generator terminal voltage vt (p.u.) time (sec)Figure 4.6: Plot of the generator terminal voltage Vtvs time for Γ1(T est Signal 1) applied to the truth model 0 200 400 600 800 1000−0.8−0.6−0.4−0.200.20.40.60.81delta (radians) time (sec) Figure 4.7: Plot of the rotor angle δvs time for Γ1(T est Signal 1) applied to the truth model 0 200 400 600 800 10000.980.9850.990.99511.0051.011.015omega (radians/sec) time (sec) Figure 4.8: Plot of the angular velocity ωvs time for Γ1(T est Signal 1) applied to the truth model 37 W e now test the truth model and the reduced order nonlinear model of the SMIB for Γ2(T est signal 2) given in Figure 4.2 . In this case the two inputs for the truth model are, u= [VF, uT]T= [e15Γ2,Γ2]T, and the two inputs for the reduced order nonlinear model are, u= [EFD, uT]T= [Γ 2,Γ2]T. Figure 4.9 ,Figure 4.10 , and Figure 4.11 show generator terminal voltage Vt, rotor angle δ, and frequency ωvs time plots for Γ2(T est Signal 2) applied to the reduced order nonlinear model. F rom Figure 4.9 we can see that the generator terminal voltage Vtsettles to a new steady value of 0.82 p.u.. F rom Figure 4.10 we can see that the rotor angle δfirst undergoes a large undershoot after which it oscillates about 0.1 p.u.. Figure 4.11 shows that the frequency ω oscillates about its steady state value of 1 p.u.. The oscillations decay with time. F or Γ2(T est Signal 2) where the magnitudes of the incremental steps are reduced to half ofΓ1(T est Signal 1), the effect of adding incremental steps of Γ2after regular intervals to the reduced order nonlinear model, is not as significant as compared to the effect of Γ1(T est Signal 1). The oscillations decay uniformly with time, and we do not see the sharp transitions that we saw when Γ1was applied to the reduced order nonlinear model. Figure 4.12 ,Figure 4.13 , and Figure 4.14 show generator terminal voltage Vt, rotor angle δ, and frequency ωvs time plots for Γ2(T est Signal 2) applied to the truth model. These results are similar to the results obtained when Γ1(T est Signal 1) is applied to the truth model. 38 0 200 400 600 800 10000.750.80.850.90.9511.051.11.151.2Generator terminal voltage vt (p.u.) time (sec)Figure 4.9: Plot of the generator terminal voltage Vtvs time for Γ2(T est Signal 2) applied to the reduced order nonlinear model 0 200 400 600 800 1000−1.5−1−0.500.51delta (radians) time (sec) Figure 4.10: Plot of the rotor angle δvs time for Γ2(T est Signal 2) applied to the reduced order nonlinear model 0 200 400 600 800 10000.750.80.850.90.9511.051.1omega (radians/sec) time (sec) Figure 4.11: Plot of the frequency ωvs time for Γ2(T est Signal 2) applied to the reduced order nonlinear model 39 0 200 400 600 800 10000.850.90.9511.051.11.151.21.25Generator terminal voltage vt (p.u.) time (sec)Figure 4.12: Plot of the generator terminal voltage Vtvs time for Γ2(T est Signal 2) applied to the truth model 0 200 400 600 800 1000−0.8−0.6−0.4−0.200.20.40.60.81delta (radians) time (sec) Figure 4.13: Plot of the rotor angle δvs time for Γ2(T est Signal 2) applied to the truth model 0 200 400 600 800 10000.980.9850.990.99511.0051.011.015omega (radians/sec) time (sec) Figure 4.14: Plot of the angular velocity ωvs time for Γ2(T est Signal 2) applied to the truth model 40 5 The Decoupled Reduced Order Model Let us recall from the first section that small changes in real power are mainly dependent on changes in rotor angle δ, and thus the frequency (i.e. on the prime mover valve control), whereas the reactive power is mainly dependent on the voltage magnitude (i.e. on the generator excitation). The excitation system time constant is much smaller than the prime mover time constant and its transient decay much faster and does not affect the load frequency control (LFC) dynamics. Thus, the cross-coupling between the LFC loop and the automatic voltage regulator (A VR) loop is negligible. Hence, load frequency control and excitation voltage control can be analyzed independently . Therefore, we decouple the two-input two-output (MIMO) reduced order system into two single input single output (SISO) subsystems. The first SISO subsystem consists of the LFC loop, where the turbine valve control uTcontrols the rotor angle δand thus the frequency ω of the synchronous generator. The second SISO subsystem consists of the A VR loop, where the generator excitation EFD controls the terminal voltage Vtof the synchronous generator. F rom Equation (3.59) and Equation (3.60) the linear model of the synchronous generator and turbine connected to an infinite bus is ∆˙E′ q=f11∆E′ q+∂f1 ∂x3 x0∆δ+g11∆EFD ∆ ˙ω=∂f2 ∂x1 x0∆E′ q+f27∆ω+∂f2 ∂x3 x0∆δ+f28∆Tm ∆˙δ= ∆ω ∆˙Tm=f41∆Tm+f42∆GV ∆˙GV=f51∆ω+f52∆GV+g55∆uT(5.1) Note that all the coeﬀicients in the above expression are evaluated at the nominal operating point given in the previous section. Let us denote ∂f1 ∂x3 x0=A13 ∂f2 ∂x1 x0=A21 ∂f2 ∂x3 x0=A23(5.2) The MIMO system with control inputs ∆EFD and ∆uTcan be decoupled into the LFC loop with control input ∆uTas ∆ ˙ω=A21∆E′ q+f27∆ω+A23∆δ+f28∆Tm ∆˙δ= ∆ω ∆˙Tm=f41∆Tm+f42∆Gv ∆˙GV=f51∆ω+f52∆GV+g55∆uT(5.3) and the A VR loop with control input ∆EFD as ∆˙E′ q=f11∆E′ q+A13∆δ+g11∆EFD (5.4) The weak coupling A21∆E′ qbetween the LFC and the A VR loop can be neglected in the LFC loop in Equation (5.3) , and the weak coupling A13∆δbetween the LFC and the A VR loop can be neglected in the A VR loop in Equation (5.4) . Neglecting theses terms, the LFC loop with control input ∆uTis approximated as ∆ ˙ω=f27∆ω+A23∆δ+f28∆Tm ∆˙δ= ∆ω ∆˙Tm=f41∆Tm+f42∆Gv ∆˙GV=f51∆ω+f52∆GV+g55∆uT(5.5) 41 and the A VR loop with control input ∆EFD is approximated as ∆˙E′ q=f11∆E′ q+g11∆EFD (5.6) 5.1 LFC Dynamics In this section we present the dynamics of the (LFC) SISO system with control input ∆uT. W e then perform root locus analysis of the uncompensated (LFC) SISO system. A PID controller is designed next to stabilize the (LFC) SISO system which is originally unstable. F rom Equation (5.5) the LFC dynamics are ∆ ˙ω=f27∆ω+A23∆δ+f28∆Tm ∆˙δ= ∆ω ∆˙Tm=f41∆Tm+f42∆Gv ∆˙GV=f51∆ω+f52∆GV+g55∆uT(5.7) Differentiating ∆ ˙ωand substituting ∆˙δ= ∆ωwe have ∆¨ω=f27∆ ˙ω+A23∆ω+f28∆˙Tm (5.8) T aking the Laplace transform of Equation (5.8) with zero initial conditions we get s2∆ω(s) =f27s∆ω(s) +A23∆ω(s) +f28s∆Tm(s) (5.9) On rearranging Equation (5.9) and taking the Laplace transform of ∆˙δ= ∆ω ∆ω(s) ∆Tm(s)=f28s s2−f27s−A23 ∆δ(s) ∆ω(s)=1 s(5.10) Equation (5.10) evaluated at the nominal operating point gives ∆ω(s) ∆Tm(s)=0.211s s2+ 0.3054 ∆δ(s) ∆ω(s)=1 s(5.11) The dynamics of the turbine from Equation (5.7) can be written as ∆˙Tm=f41∆Tm+f42∆Gv (5.12) T aking the Laplace transform of Equation (5.12) with zero initial conditions s∆Tm(s) =f41∆Tm(s) +f42∆GV(s) (5.13) Rearranging and expressing Equation (5.13) as a transfer function ∆Tm(s) ∆GV(s)=f42 s−f41(5.14) Equation (5.14) evaluated at the nominal operating point gives ∆Tm(s) ∆GV(s)=2 s+ 2(5.15) 42 u_T(s) delta(s)g55 s−f52 −f51 s−f52f42 s−f41f28.s s −f27s−A232Step1 s Tm(s) omega(s)omega(s) Gv(s) delta(s)Figure 5.1: Simulink model of the (LFC) SISO system for a step input u_T(s) delta(s)g55 s−f52 −f51g55f42 s−f41f28.s s −f27s−A232Step1 s Tm(s) omega(s)omega(s) delta(s) Gv(s) Figure 5.2: Simulink model of the (LFC) SISO system for a step input F rom Equation (5.7) the governor dynamics can be written as ∆˙GV=f51∆ω+f52∆GV+g55∆uT (5.16) T aking the Laplace transform of Equation (5.16) with zero initial conditions s∆GV(s) =f51∆ω(s) +f52∆GV(s) +g55∆uT(s) (5.17) Rearranging and expressing Equation (5.17) as a transfer function ∆GV(s) =f51 s−f52∆ω(s) +g55 s−f52∆uT(s) (5.18) Equation (5.18) evaluated at the nominal operating point gives ∆GV(s) =−0.25 s+ 5∆ω(s) +5 s+ 5∆uT(s) (5.19) Figure 5.1 shows the simulink block diagram of the LFC loop for a step input. In all simulink models the subscript ∆ is omitted for convenience. By moving the summing point ahead of the blockg55 s−f52, the simulink model in Figure 5.1 can be simplified to get the simulink model shown in Figure 5.2 . In Figure 5.2 let G(s) =g55 s−f52f42 s−f41f28s s2−f27s−A23 =g55f42f28s (s−f52)(s−f41)(s2−f27s−A23)(5.20) and H(s) =−f51 g55(5.21) Using Equation (5.20) and Equation (5.21) , the simulink model in Figure 5.2 can be simplified to the simulink model shown in Figure 5.3 . The open loop transfer function of the SISO system shown in Figure 5.3 with output ω(s)and step input uT(s)is G(s)H(s) =−f42f28f51s (s−f52)(s−f41)(s2−f27s−A23)(5.22) 43 Figure 5.3: Simulink model of the (LFC) SISO system for a step input The closed loop transfer function of the SISO system between the output ∆δ(s)and control input ∆uT(s) is ∆δ(s) ∆uT(s)=G(s) 1 +G(s)H(s)1 s =g55f42f28 (s−f52)(s−f41)(s2−f27s−A23) 1−f42f28f51s (s−f52)(s−f41)(s2−f27s−A23) =g55f42f28 (s−f52)(s−f41)(s2−f27s−A23)−f51f42f28s(5.23) F or a step input, ∆uT(s) =1 s. F rom the final value theorem, the steady state value of ∆δ(s)is ∆δss= lim s→0s∆δ(s) = lim s→0s∆uT(s)g55f42f28 (s−f52)(s−f41)(s2−f27s−A23)−f51f42f28s =−g55f42f28 f52f41A23(5.24) Substituting numerical values for the constant parameters in Equation (5.24) we get ∆δss= 0.6909 (5.25) Also ∆ω(s) ∆uT(s)=G(s) 1 +G(s)H(s)=g55f42f28s (s−f52)(s−f41)(s2−f27s−A23) 1−f42f28f51s (s−f52)(s−f41)(s2−f27s−A23) =g55f42f28s (s−f52)(s−f41)(s2−f27s−A23)−f51f42f28s(5.26) F rom the final value theorem, the steady state value of ∆ω(s)for a step input is ∆ωss= lim s→0s∆ω(s) = lim s→0s∆uT(s)g55f42f28s (s−f52)(s−f41)(s2−f27s−A23)−f51f42f28s =−lim s→0g55f42f28 f52f41A23s = 0(5.27) Figure 5.4 and Figure 5.5 show the plots of ∆δ(s) and ∆ω(s) vs time for the uncompensated (LFC) SISO system for a step input, respectively . F rom these plots we can see that ∆δ(s) settles to a steady state value of 0.6909 and ∆ω(s) settles to a steady state value of 0 which are in agreement with the respective steady state values calculated in Equation (5.25) and Equation (5.27) . Also the root locus plot in Figure 5.6 clearly shows that the uncompensated (LFC) SISO system with step input uT(s)is unstable. 44 01002003004005006007008009001000−0.200.20.40.60.811.21.4∆ δ (radians) time (sec)Figure 5.4: Plot of ∆δ(s) vs time for the uncompensated (LFC) SISO system for a step input 01002003004005006007008009001000−0.4−0.3−0.2−0.100.10.20.30.4∆ ω (radians/sec) time (sec) Figure 5.5: Plot of ∆ω(s) vs time for the uncompensated (LFC) SISO system for a step input −20 −15 −10 −5 0 5−15−10−5051015Root Locus Real AxisImaginary Axis Figure 5.6: Root locus plot for the uncompensated (LFC) SISO system 45 u_T(s) delta(s)g55 s−f52 −f51g55f42 s−f41f28.s s −f27s−A232 PID ControllerPID(s) 1 s0 Tm(s) omega(s)omega(s) delta(s) Gv(s)e_T(s)Figure 5.7: Simulink model of the PID compensated (LFC) SISO system 5.1.1 PID controller design for the LFC loop In order to stabilize this system we design a proportional-integral-derivative (PID) controller. The simulink model of the PID compensated (LFC) SISO system is given in Figure 5.7 . The output of the PID controller ∆uT(s)in the frequency domain is given by ∆uT(s) =Kp1∆eT(s) +Ki1 s∆eT(s) +Kd1s∆eT(s) (5.28) where Kp1is the proportional gain, Ki1is the integral gain, Kd1is the derivative gain, and ∆eT(s)is the error signal. Expressing Equation (5.28) as a transfer function ∆uT(s) ∆eT(s)=Kp1+Ki1 s+Kd1s =Kd1s2+Kp1s+Ki1 s(5.29) F rom Figure 5.7 the open loop transfer function of the PID compensated (LFC) SISO system can be written as GPID 1(s) =Kd1s2+Kp1s+Ki1 sG(s) 1 +G(s)H(s) =Kd1s2+Kp1s+Ki1 sg55f42f28s (s−f52)(s−f41)(s2−f27s−A23)−f51f42f28s HPID 1(s) = 1 )GPID 1(s)HPID 1(s) =Kd1s2+Kp1s+Ki1 sg55f42f28s (s−f52)(s−f41)(s2−f27s−A23)−f51f42f28s (5.30) The PID gains are tuned to Kp1= 200 ,Ki1= 150 , and Kd1= 100 . Substituting these PID gains and numerical values for the constant coeﬀicients in Equation (5.30) we get GPID 1(s)HPID 1(s) =52.74s3+ 105 .5s2+ 79.11s 5s5+ 35s4+ 51.53s3+ 10.69s2+ 15.27s(5.31) Figure 5.8 and Figure 5.9 show the plots for ∆δ(s) and ∆ω(s) vs time for the PID compensated (LFC) SISO system, respectively . Also, Figure 5.10 shows the root locus plot for the PID compensated (LFC) SISO system. F rom this plot it is evident that the PID compensated (LFC) SISO system is stable since the root locus lies entirely in the left half s-plane. 46 0510152025303540455001234567x 10−5∆ δ (radians) time (sec)Figure 5.8: Plot of ∆δ(s) vs time for the PID compensated (LFC) SISO system 05101520253035404550−0.500.511.522.533.54x 10−5∆ ω (radians/sec) time (sec) Figure 5.9: Plot of ∆ω(s) vs time for the PID compensated (LFC) SISO system −6 −5 −4 −3 −2 −1 0 1−10−8−6−4−20246810Root Locus Real AxisImaginary Axis Figure 5.10: Root locus plot for the PID compensated (LFC) SISO system 47 5.2 A VR Dynamics In this section we present the dynamics of the (A VR) SISO system with control input ∆EFD . In addition we also design a PID controller for the (A VR) SISO system. F rom Equation (5.6) the dynamics of the (A VR) SISO system are ∆˙E′ q=f11∆E′ q+g11∆EFD (5.32) T aking the Laplace transform of Equation (5.32) assuming zero initial conditions we get s∆E′ q(s) =f11∆E′ q(s) +g11∆EFD(s) (5.33) Rearranging and expressing Equation (5.33) as a transfer function ∆E′ q(s) ∆EFD(s)=g11 s−f11(5.34) F rom Equation (3.71) the output terminal voltage ∆Vtis ∆Vt=T1∆E′ q+T2∆δ (5.35) F or the decoupled (A VR) SISO system with ∆EFD as the input and ∆Vtas the output the coupling T2∆δ between the (LFC) SISO system and the (A VR) SISO system is neglected. Therefore, ∆Vt=T1∆E′ q (5.36) T aking the Laplace transform of Equation (5.36) ∆Vt(s) =T1∆E′ q(s) (5.37) Combining Equation (5.34) and Equation (5.37) we get ∆Vt(s) ∆EFD(s)=T1g11 s−f11(5.38) Substituting numerical values for the coeﬀicients in Equation (5.38) evaluated at the nominal operating point we get ∆Vt(s) ∆EFD(s)=0.08913 s+ 0.5517(5.39) Figure 5.11 shows the simulink model of the uncompensated (A VR) SISO system for a step input. In Figure 5.11 let G1(s) =T1g11 s−f11(5.40) and H1(s) = 1 (5.41) Therefore, the open loop transfer function of the uncompensated (A VR) SISO system shown in Figure 5.11 is G1(s)H1(s) =T1g11 s−f11(5.42) The closed loop transfer function of the uncompensated (A VR) SISO system shown in Figure 5.11 is ∆Vt(s) ∆Eref(s)=G1(s) 1 +G1(s)H1(s)=T1g11 s−f11 1 +T1g11 s−f11 =T1g11 s−f11+T1g11(5.43) 48 V_tT1*g11 s−f11 StepV_t(s)E_ref E_FD(s)Figure 5.11: Simulink model of the uncompensated (A VR) SISO system for a step input F or a step input, ∆Eref(s) =1 s. F rom the final value theorem, the steady state value of ∆Vt(s)is ∆Vt(ss)= lim s→0s∆Vt(s) = lim s→0s∆Eref(s)T1g11 s−f11+T1g11 =T1g11 T1g11−f11(5.44) Substituting numerical values for the coeﬀicients in Equation (5.44) ∆Vt(ss)= 0.1391 (5.45) Figure 5.12 shows the terminal voltage ∆Vt(s) vs time plot for the uncompensated (A VR) SISO system for a step input. F rom this plot we can see that ∆Vt(s) settles to a steady state value of 0.1391 which is in agreement with the steady state value calculated in Equation (5.45) .Figure 5.13 shows the root locus plot for the uncompensated (A VR) SISO system. F rom this plot it is evident that the uncompensated (A VR) SISO system is stable. 49 0510152025303540455000.020.040.060.080.10.120.140.16Generator terminal voltage ∆ Vt (p.u.) time (sec)Figure 5.12: Plot of the terminal voltage ∆Vt(s) vs time for the uncompensated (A VR) SISO system for a step input −2 −1.5 −1 −0.5 0 0.5−0.08−0.06−0.04−0.0200.020.040.060.08Root Locus Real AxisImaginary Axis Figure 5.13: Root locus plot for the uncompensated (A VR) SISO system 5.2.1 PID controller design for the A VR loop W e design a PID controller to improve the transient and steady state response of the (A VR) SISO system. The output of this PID controller ∆EFD(s)in the frequency domain is given by ∆EFD(s) =Kp2∆eFD(s) +Ki2 s∆eFD(s) +Kd2s∆eFD(s) (5.46) where Kp2is the proportional gain, Ki2is the integral gain, Kd2is the derivative gain, and ∆eFD(s)is the error signal. Expressing Equation (5.46) as a transfer function ∆EFD(s) ∆eFD(s)=Kp2+Ki2 s+Kd2s =Kd2s2+Kp2s+Ki2 s(5.47) 50 V_T(s)T1*g11 s−f11 PID ControllerPID(s)0 V_T(s) e_FD(s) E_FD(s)Figure 5.14: Simulink model of the PID compensated (A VR) SISO system 05101520253035404550−0.500.511.522.533.54x 10−4Generator terminal voltage ∆ Vt (p.u.) time (sec) Figure 5.15: Plot of ∆Vt(s) vs time for the PID compensated (A VR) SISO system −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2−1.5−1−0.500.511.5Root Locus Real AxisImaginary Axis Figure 5.16: Root locus plot for the PID compensated (A VR) SISO system F rom Figure 5.14 the open loop transfer function of the PID compensated (A VR) SISO system can be written as GPID 2(s) =Kd2s2+Kp2s+Ki2 sT1g11 s−f11 HPID 2(s) = 1 )GPID 2(s)HPID 2(s) =Kd2s2+Kp2s+Ki2 sT1g11 s−f11(5.48) The PID gains Kp2= 10 ,Ki2= 10 , and Kd2= 4 are tuned so as to get the best system response. Substituting these PID gains and numerical values for the constant coeﬀicients in Equation (5.48) we get GPID 2(s)HPID 2(s) =0.3565s2+ 0.8913s+ 0.8913 s2+ 0.5517s(5.49) Figure 5.15 shows the generator terminal voltage ∆Vt(s) vs time plot for the PID compensated (A VR) SISO system. F rom this plot we can see that ∆Vt(s) attains its steady state value of 0 in approximately 10 seconds. Figure 5.16 shows the root locus plot for the PID compensated (A VR) SISO system. The PID compensated (A VR) SISO system is stable since the root locus lies in the left half s-plane. 51 5.3 LFC and A VR including coupling As there is weak coupling between the LFC and A VR subsystems, the rotor angle δand thus the frequency ω, and the terminal voltage Vtwere controlled separately . In this section we study the effect of coupling between the LFC and the A VR system. In the previous section we saw that the weak coupling A21∆E′ q between the LFC and the A VR loop which appeared in the coupled system in Equation (5.1) was neglected in the LFC loop in Equation (5.5) . Also, the weak coupling A13∆δbetween the LFC and the A VR loop which appeared in the coupled system in Equation (5.1) was neglected in the A VR loop in Equation (5.6) . In the coupled system including the LFC and A VR dynamics we do not neglect these terms. Thus, from Equation (5.1) we can write ∆˙E′ q=f11∆E′ q+A13∆δ+g11∆EFD ∆ ˙ω=A21∆E′ q+f27∆ω+A23∆δ+f28∆Tm ∆˙δ= ∆ω(5.50) T aking the Laplace transform of the expression for ∆˙E′ qin Equation (5.50) s∆E′ q(s) =f11∆E′ q(s) +A13∆δ(s) +g11∆EFD(s) (5.51) Rearranging Equation (5.51) we get ∆E′ q(s) =A13 s−f11∆δ(s) +g11 s−f11∆EFD(s) (5.52) Also from Equation (3.71) the first output which is the terminal voltage Vtof the synchronous generator for the coupled system including LFC and A VR dynamics is ∆Vt=T1∆E′ q+T2∆δ (5.53) The Laplace transform of Equation (5.53) gives ∆Vt(s) =T1∆E′ q(s) +T2∆δ(s) (5.54) Next, taking the derivative of ∆ ˙ωwe get ∆¨ω=A21∆˙E′ q+f27∆ ˙ω+A23∆ω+f28∆˙Tm (5.55) T aking the Laplace transform of Equation (5.55) s2∆ω(s) =A21s∆E′ q(s) +f27s∆ω(s) +A23∆ω(s) +f28s∆Tm(s) (5.56) Rearranging Equation (5.56) we get ∆ω(s) =A21s s2−f27s−A23∆E′ q(s) +f28s s2−f27s−A23∆Tm(s) (5.57) Also the Laplace transform of ∆˙δ= ∆ωiss∆δ(s) = ∆ ω(s). Thus, the second output which is the rotor angle ∆δ(s)of the synchronous generator for the coupled system including the LFC and A VR dynamics can be written as ∆δ(s) =A21 s2−f27s−A23∆E′ q(s) +f28 s2−f27s−A23∆Tm(s) (5.58) 52 omega(s)delta(s)V_T(s) A13 s−f11 A21.s s −f27s−A232 g55 s−f52g11 s−f11 −f51g55f42 s−f41f28.s s −f27s−A232T2T2T1T1 PID Controller 2PID(s) PID Controller 1PID(s) 1 s00 E_FD(s) Tm(s)u_T(s)omega(s)Eq_p(s) V_T(s) Gv(s)Figure 5.17: Simulink model of the PID compensated (LFC+A VR) MIMO system including coupling 6 PID Controller Design 6.1 PID Controller Design based on linear model The two PID controllers that were designed for the LFC and the A VR SISO systems in the previous section, are now tested on the combined LFC and A VR system including coupling which is the same as the linear model. A simulink block diagram which consists of the two PID controllers and the combined LFC and A VR system including coupling is constructed in Figure 5.17 .Figure 6.1 to Figure 6.5 show ∆δ(s),∆ω(s), ∆Vt(s),∆Efd(s), and ∆uT(s) vs time plots for the PID compensated combined (LFC+A VR) MIMO system including coupling. The plots obtained when the coupling coeﬀicients were set to zero i.e. in the decoupled LFC and A VR SISO systems, are identical to the plots obtained for the combined (LFC+A VR) MIMO system including coupling. Thus, separate frequency and voltage control of the synchronous generator and turbine-governor system connected to an infinite bus is justified. 53 05101520253035404550−101234567x 10−5∆ δ (radians) time (sec)Figure 6.1: Plot of ∆δ(s) vs time for the PID compensated (LFC+A VR) MIMO system including coupling 05101520253035404550−0.500.511.522.533.5x 10−5∆ ω (radians/sec) time (sec) Figure 6.2: Plot of ∆ω(s) vs time for the PID compensated (LFC+A VR) MIMO system including coupling 05101520253035404550−0.500.511.522.533.54x 10−4Generator terminal voltage ∆ Vt (p.u.) time (sec) Figure 6.3: Plot of ∆Vt(s) vs time for the PID compensated (LFC+A VR) MIMO system including coupling 54 05101520253035404550−20246810x 10−3Excitation field voltage ∆ Efd (p.u.) time (sec)Figure 6.4: Plot of ∆Efd(s) vs time for the PID compensated (LFC+A VR) MIMO system including coupling 05101520253035404550−6−4−2024681012x 10−3Turbine valve control ∆ uT(p.u.) time (sec) Figure 6.5: Plot of ∆uT(s) vs time for the PID compensated (LFC+A VR) MIMO system including coupling 55 6.2 Simulation results for the PID Controllers applied to the Reduced Order Nonlinear Model The two PID controllers that were designed for the decoupled LFC and A VR SISO systems and then tested on the (LFC+A VR) MIMO system including coupling are now tested on the reduced order nonlinear model. Figure 6.6 to Figure 6.11 show simulation results for the two PID controllers that are tested on the reduced order nonlinear model. F rom these results we can see that all the state variables, outputs, and the control inputs attain their respective steady state values in approximately 10 seconds. 56 0510152025303540455000.511.5State variables x (p.u.) time (sec) E'q omega delta Tm GVFigure 6.6: Plot of the state variables E′ q,ω,δ,Tm, and GVvs time for the PID controller applied to the reduced order nonlinear model 051015202530354045501.171.17051.1711.17151.1721.17251.1731.17351.1741.17451.175Generator terminal voltage Vt (p.u.) time (sec) Figure 6.7: Plot of the generator terminal voltage Vtvs time for the PID controller applied to the reduced order nonlinear model 051015202530354045500.950.960.970.980.9911.01delta (radians) time (sec) Figure 6.8: Plot of the rotor angle δvs time for the PID controller applied to the reduced order nonlinear model 57 051015202530354045500.950.960.970.980.9911.011.021.03omega (radians/sec) time (sec)Figure 6.9: Plot of the frequency ωvs time for the PID controller applied to the reduced order nonlinear model 051015202530354045501.71.81.922.12.22.32.42.52.6Excitation field voltage EFD (p.u.) time (sec) Figure 6.10: Plot of the generator excitation field control EFD vs time for the PID controller applied to the reduced order nonlinear model 05101520253035404550−50510152025Turbine valve control uT(p.u.) time (sec) Figure 6.11: Plot of the turbine valve control uTvs time for the PID controller applied to the reduced order nonlinear model 58 6.3 Simulation results for the PID Controllers applied to the T ruth Model The two PID controllers that were designed for the decoupled reduced order LFC and A VR SISO systems, and then tested on the (LFC+A VR) MIMO system including coupling and the reduced order nonlinear model, are now tested on the truth model. The PID controller which controls the LFC loop is re-tuned to Kp1= 2000 ,Ki1= 1500 , and Kd1= 1000 . Whereas, the PID controller which controls the A VR loop is re-tuned to Kp2= 2000 ,Ki2= 15000 , and Kd2= 4 .Figure 6.12 to Figure 6.16 show simulation results for the two PID controllers that are tested on the truth model. F rom these simulation results we can see that the generator terminal voltage Vt, angular velocity ω, rotor angle δ, and the two control inputs VFanduT, settle to their respective steady state values. 59 0 10 20 30 40 501.1721.17211.17211.17211.17221.17231.17231.17231.17241.17251.1725Generator terminal voltage vt (p.u.) time (sec)Figure 6.12: Plot of the generator terminal voltage Vtvs time for the PID controllers applied to the truth model 0 10 20 30 40 500.99970.99980.99980.99990.9999111.00011.00011.0002omega (radians/sec) time (sec) Figure 6.13: Plot of the angular velocity ωvs time for the PID controllers applied to the truth model 0 10 20 30 40 500.99970.99970.99980.99980.99990.9999111.00011.0001delta (radians) time (sec) Figure 6.14: Plot of the rotor angle δvs time for the PID controllers applied to the truth model 60 0 10 20 30 40 500.20.40.60.811.21.41.61.82x 10−3Excitation field voltage vF (p.u.) time (sec)Figure 6.15: Plot of the control input VFvs time for the PID controllers applied to the truth model 0 10 20 30 40 500.80.911.11.21.31.41.51.61.71.8Turbine valve control uT(p.u.) time (sec) Figure 6.16: Plot of the control input uTvs time for the PID controllers applied to the truth model 61 7 Linear State-Space Controller Design In this section we design linear state-space controllers for the synchronous generator and turbine connected to an infinite bus. 7.1 State F eedback Controller design using LQR methodology 7.1.1 LQR Design based on linear model W e first design a linear-quadratic regulator (LQR) for the linearized model of the synchronous generator and turbine connected to an infinite bus. Let us assume that we have sensors to measure all the states and we use a full-state feedback controller (regulator) of the form u=−Kx (7.1) that seeks to drive the states to zero. Since, the system is MIMO with 2inputs and 5states, LQR controller design entails finding the 2×5gain vector K. This can be done by directly using the lqr command in MA TLAB. F or this, letting J=Z∞ 0(xTQx+uTRu)dt (7.2) we seek to find the gain vector K to minimize the cost function J. Minimization of J results in driving x(t)to zero with as little control energy and state deviations as possible, with the balance between control energy and state deviations specified via the Q andR matrices. Here, assume the 5×5Q matrix is diagonal with diagonal elements qi≥0(each providing a weight for a different element of the deviation of the state) and the 2×2R matrix is diagonal with diagonal elements ri>0(each providing a weight for the deviation of the two control inputs. The values for Q andR are used as design parameters. A methodology to tune the Q andR matrices is given as follows: If all the qi= 0 , then the excursions of the states are high while the control input tries to force the state to zero. High values of qirelative to ri mean that you are willing to use lots of control energy to keep state excursions small while driving it to zero. Clearly , you cannot pick ri= 0 as this results in allowing infinite control energy to force the state to zero, typically then very fast. Finding the gain K to minimize Jinvolves solving the Riccati equation ATP+PA−PBR−1BTP+Q= 0 , where K=R−1BTP. W e use the Matlab lqr command to directly solve for the gain vector K given A,B,Q, and R. Thus, by using the weighting matrices Q=2 66664300 0 0 0 0 0 250 0 0 0 0 0 200 0 0 0 0 0 200 0 0 0 0 0 2503 77775(7.3) and R=0.5 0 0 0 .5 (7.4) and the state space matrices (A, B )as given in Equation (3.76) and Equation (3.77) the control gain K is found to be K=23.7240 −36.3457 −5.5938 −2.5612 −0.0454 −1.3381 21 .0340 1 .5703 9 .0242 21 .5437 (7.5) Figure 7.1 shows time history of the state variables ( ∆E′ q,∆ω,∆δ,∆Tm, and ∆GV) for the full-state feedback LQR controller applied to the reduced order linear model. F rom this plot we can see that the LQR drives all the states to 0in approximately 20 to 25 seconds. Similarly , from Figure 7.2 to Figure 7.6 we can see that ∆Vt,∆ω,∆δ,∆Efd, and ∆uTconverge to zero in approximately 25 seconds. 62 0 10 20 30 40 50−0.01−0.00500.0050.010.0150.02State variables x (p.u.) time (sec) ∆E’q ∆ω ∆δ ∆Tm ∆GVFigure 7.1: Plot of the state variables ∆E′ q,∆ω,∆δ,∆Tm, and ∆GV vs time for the full-state feedback LQR applied to the reduced order linear model 05101520253035404550−6−4−202468x 10−3Generator terminal voltage ∆ Vt (p.u.) time (sec) Figure 7.2: Plot of the generator terminal voltage ∆Vtvs time for the full-state feedback LQR applied to the reduced order linear model 05101520253035404550−8−6−4−20246810x 10−3∆ ω (radians/sec) time (sec) Figure 7.3: Plot of ∆ωvs time for the full-state feedback LQR applied to the reduced order linear model 63 05101520253035404550−0.01−0.00500.0050.010.0150.02∆ δ (radians) time (sec)Figure 7.4: Plot of ∆δvs time for the full-state feedback LQR applied to the reduced order linear model 05101520253035404550−0.04−0.0200.020.040.060.080.10.120.14Excitation field voltage ∆ Efd (p.u.) time (sec) Figure 7.5: Plot of the control input ∆Efd vs time for the full-state feedback LQR applied to the reduced order linear model 0 5 10 15 20 25 30−0.25−0.2−0.15−0.1−0.0500.050.1Turbine valve control ∆ uT(p.u.) time (sec) Figure 7.6: Plot of the control input ∆uTvs time for the full-state feedback LQR applied to the reduced order linear model 64 7.1.2 Simulation results for the F ull-State F eedback LQR applied to the Reduced Order Nonlinear Model The LQR based full-state feedback controller that was earlier designed for the reduced order linear model is now tested on the reduced order nonlinear model. Figure 7.7 and Figure 7.8 show plots for the generator terminal voltage Vt, and rotor angle δ, respectively . The generator terminal voltage Vtsettles to a new steady state value of 1.13 p.u. which deviates from the desired steady state value of 1.1723 p.u. by an amount of 0.0423 p.u. Similarly , the rotor angle δsettles to a new steady state value of 1.056 p.u. which deviates from the desired steady state value of 1 p.u. by an amount of 0.056 p.u. W e observe a steady state error when the LQR with original gains of subsection 7.1.1 is applied to the reduced order nonlinear model. Therefore, the gains of the controller are tuned once again so that the controller works eﬀiciently on the reduced order nonlinear model. W e use the Matlab lqr command to directly solve for the gain vector K given A, B, Q, and R. The system matrices (A, B )are evaluated at the nominal operating point, and Q andR matrices are appropriately tuned as per the procedure explained in the LQR design section. Thus, by using the feedback law, u=−Kx (7.6) the weighting matrices Q=2 6666440000 0 0 0 0 0 10000 0 0 0 0 0 250000 0 0 0 0 0 500 0 0 0 0 0 5003 77775(7.7) and R=0.07 0 0 0 .07 (7.8) and the state space matrices (A, B )as given in Equation (3.76) and Equation (3.77) the control gain K is found to be K=753.9172 −575.5829 −610.0649 −27.6436 −0.1301 −3.8375 1782 .567 1474 .307 128 .4938 84 .1272 (7.9) Figure 7.9 to Figure 7.14 show simulation results for the LQR-based full-state feedback controller applied to the reduced order nonlinear model with the gains re-tuned. F rom these results we can see that all the state variables, and outputs, attain their respective steady state values in approximately 8-10 seconds. Figure 7.13 and Figure 7.14 show plots of the two control inputs, Efd anduT, respectively . 65 0 5 10 15 20 25 301.081.11.121.141.161.181.2Generator terminal voltage Vt (p.u.) time (sec)Figure 7.7: Plot of the generator terminal voltage Vtvs time for the LQR-based full-state feedback controller with the original gains of subsection 7.1.1 applied to the reduced order nonlinear model 0 5 10 15 20 25 300.940.960.9811.021.041.061.08delta (radians) time (sec) Figure 7.8: Plot of the rotor angle δvs time for the LQR-based full-state feedback controller with the original gains of subsection 7.1.1 applied to the reduced order nonlinear model 0 5 10 15 20 2500.20.40.60.811.21.41.6State variables x (p.u.) time (sec) E'q omega delta Tm GV Figure 7.9: Plot of the state variables E′ q,ω,δ,Tm, and GVvs time for the re-tuned LQR-based full-state feedback controller applied to the reduced order nonlinear model 66 0 5 10 15 20 251.1451.151.1551.161.1651.171.1751.18Generator terminal voltage Vt (p.u.) time (sec)Figure 7.10: Plot of the generator terminal voltage Vtvs time for the re-tuned LQR-based full-state feedback controller applied to the reduced order nonlinear model 0 5 10 15 20 250.9650.970.9750.980.9850.990.99511.0051.01delta (radians) time (sec) Figure 7.11: Plot of the rotor angle δvs time for the re-tuned LQR-based full-state feedback controller applied to the reduced order nonlinear model 0 5 10 15 20 250.950.960.970.980.9911.011.021.03omega (radians/sec) time (sec) Figure 7.12: Plot of the frequency ωvs time for the re-tuned LQR-based full-state feedback controller applied to the reduced order nonlinear model 67 0 5 10 15 20−3−2−1012345Excitation field voltage EFD (p.u.) time (sec)Figure 7.13: Plot of the control input Efd vs time for the full-state feedback LQR applied to the reduced order nonlinear model 0 5 10 15 20 25 30−15−10−5051015Turbine valve control uT(p.u.) time (sec) Figure 7.14: Plot of the control input uTvs time for the full-state feedback LQR applied to the reduced order nonlinear model 68 7.1.3 Simulation results for the F ull-State F eedback LQR applied to the T ruth Model The LQR-based full-state feedback controller that was designed for the reduced order linear model and then re-tuned and tested on the reduced order nonlinear model, is now tested on the truth model. There is a non physical state E′ qin the reduced order model that needs to be reconstructed from the states of the truth model. The LQR-based full-state feedback controller can be implemented on the truth model by using either Equation (7.10) or Equation (7.11) to express E′ qas a function of state variables of the truth model. The gains of this controller are the same as the LQR controller that was re-tuned and tested on the reduced order nonlinear model. Figure 7.15 to Figure 7.19 show simulation results for the LQR based full state feedback controller applied to the truth model. F rom these simulation results we can see that the generator terminal voltage Vtoscillates about a steady state value of 1.1705 p.u. which deviates from the desired steady state value of 1.1723 p.u. by an amount of 0.0018 p.u. Angular velocity ω, oscillates about its desired steady state value of 1 p.u., and the rotor angle δ, oscillates about its desired state value of 1 p.u. These oscillations decay with time. Also, the generator excitation voltage VFsettles to its steady state value of 0.00121 p.u. and the turbine valve control settles to its steady state value of 1.0512 p.u. E′ q=e14 e11IF+e12 e11cos(δ−α) +e13 e11sin(δ−α) (7.10) E′ q=e14IF+L2Id (7.11) 69 0 200 400 600 800 10001.17021.17041.17061.17081.1711.17121.17141.17161.17181.1721.1722Generator terminal voltage vt (p.u.) time (sec)Figure 7.15: Plot of the generator terminal voltage Vtvs time for the LQR-based full-state feedback controller applied to the truth model 0 200 400 600 800 10000.99980.99980.99990.9999111.00011.00011.0002omega (radians/sec) time (sec) Figure 7.16: Plot of the angular velocity ωvs time for the LQR-based full-state feedback controller applied to the truth model 0 200 400 600 800 10000.99750.9980.99850.9990.999511.00051.0011.00151.0021.0025delta (radians) time (sec) Figure 7.17: Plot of the rotor angle δvs time for the LQR-based full-state feedback controller applied to the truth model 70 0 50 100 150 200−4−3−2−1012x 10−3Excitation field voltage vF (p.u.) time (sec)Figure 7.18: Plot of the control input VFvs time for the full-state feedback LQR applied to the truth model 0 0.2 0.4 0.6 0.8 100.511.522.533.544.55Turbine valve control uT(p.u.) time (sec) Figure 7.19: Plot of the control input uTvs time for the full-state feedback LQR applied to the truth model 71 7.2 State F eedback Controller design using Pole Placement T echnique 7.2.1 Pole Placement Design based on linear model The characteristic equation of a compensated linear system with controller u=−Kxis|sI−A+BK|= 0 . If the system is in the phase-variable canonical form then the characteristic equation of the compensated linear system is |sI−A+BK|=sn+ (an−1+kn) +· · · · +(a1+k2)s+ (a0+k1) = 0 (7.12) where an−1,an−2· · ·a1, and a0are the coeﬀicients of the characteristic equation |sI−A|= 0 and kn· · · k1, are gains of the control matrix K. F or the specified closed-loop pole locations −λ1,· · · − λnthe desired characteristic equation is αc(s) = (s+λ1)(s+λ2)· · ·(s+λn) =sn+αn−1sn−1+· · · · +α1s+α0= 0 (7.13) The design objective is to find the gain matrix K such that the characteristic equation for the controlled system is identical to the desired characteristic equation. Thus, the gain vector K is obtained by equating coeﬀicients of Equation (7.12) and Equation (7.13) and for the ithcoeﬀicient we get ki=αi−ai (7.14) If the state model is not in the phase-variable canonical form, we can use the transformation technique to transform the given state model to the phase-variable canonical form which results in the following formula, known as Ackermann’s formula [ 8]. K= [0 0 · · · · 0 1]S−1αc(A) (7.15) where the matrix Sis given by S= [B AB A2B· · · · An−1B] (7.16) andαc(A)is given by αc(A) =An+αn−1An−1+· · · · +α1A+α0I (7.17) The MA TLAB function K=place (A, B, p )can be used to design the controller gain matrix K, where pis a row vector containing the desired closed-loop poles. The closed-loop poles are selected such that all the state variables remain within a specific limit and converge to zero in minimum time. The closed-loop poles cannot be placed too close to the imaginary axis as the relative stability of the system decreases and the system oscillations increase. Also if they are placed far away from the imaginary axis the settling time of the state variables and hence the output decreases, but there is no guarantee that the state variables and the control inputs will remain within a reasonable physical limit. After some trial and error the desired closed-loop poles of the generator-turbine system were selected as p= [−0.8,−0.9,−0.7,−1.1,−1] (7.18) The controller gain matrix K corresponding to these closed loop poles is K=12.6444 −44.1610 0 .9926 −3.1472 −8.9082 0.0535 −0.3867 −0.0861 0 .0788 −1.0390 (7.19) Figure 7.20 to Figure 7.23 show that the state variables ∆E′ q,∆ω,∆δ,∆Tm,∆GV, and the outputs ∆Vt,∆ω and ∆δsettle to their steady state value of zero in approximately 10 seconds. Figure 7.24 and Figure 7.25 show plots for the two control inputs ∆Efd, and ∆uTrespectively . 72 05101520253035404550−0.015−0.01−0.00500.0050.010.0150.02State variables x (p.u.) time (sec) ∆E’q ∆ω ∆δ ∆Tm ∆GVFigure 7.20: Plot of the state variables ∆E′ q,∆ω,∆δ,∆Tm, and ∆GVvs time for the pole placement based full-state feedback controller applied to the reduced order linear model 05101520253035404550−8−6−4−2024681012x 10−3Generator terminal voltage ∆ Vt (p.u.) time (sec) Figure 7.21: Plot of the generator terminal voltage ∆Vtvs time for the pole placement based full-state feedback controller applied to the reduced order linear model 05101520253035404550−6−4−20246810x 10−3∆ ω (radians/sec) time (sec) Figure 7.22: Plot of ∆ωvs time for the pole placement based full-state feedback controller applied to the reduced order linear model 73 05101520253035404550−20246810121416x 10−3∆ δ (radians) time (sec)Figure 7.23: Plot of ∆δvs time for the pole placement based full-state feedback controller applied to the reduced order linear model 05101520253035404550−0.1−0.0500.050.10.150.20.250.3Excitation field voltage ∆ Efd (p.u.) time (sec) Figure 7.24: Plot of the control input ∆Efd vs time for the pole placement based full-state feedback controller applied to the reduced order linear model 05101520253035404550−2024681012x 10−3Turbine valve control ∆ uT(p.u.) time (sec) Figure 7.25: Plot of the control input ∆uTvs time for the pole placement based full-state feedback controller applied to the reduced order linear model 74 7.2.2 Simulation Results for the F ull-State F eedback Pole Placement controller applied to the Reduced Order Nonlinear Model W e now test the pole placement based full-state feedback controller that was designed earlier for the reduced order linear model, on the reduced order nonlinear model. Figure 7.26 and Figure 7.27 show plots for the generator terminal voltage Vt, and rotor angle δ, respectively . The generator terminal voltage Vtsettles to a new steady state value of 0.8116 p.u. which deviates from the desired steady state value of 1.1723 p.u. by an amount of 0.3607 p.u. Similarly the rotor angle δsettles to a new steady state value of 1.324 p.u. which deviates from the desired steady state value of 1 p.u. by an amount of 0.324 p.u. W e observe a large steady state error when the pole placement controller with original gains of subsection 7.2.1 is applied to the reduced order nonlinear model. Therefore, the gains of the controller are once again tuned by appropriately choosing the desired pole locations, to get satisfactory performance. The MA TLAB function K=place (A, B, p )can be used to design the controller gain matrix K, where pis a row vector containing the desired closed-loop poles. After some trial and error the desired closed-loop poles of the generator-turbine system were selected as p= [−300,−0.9,−280,−5,−70] (7.20) W e see here that poles p(1), p(3), and p(5) are placed far away from the imaginary axis so that the state variables E′ q,δ, and GVattain their desired steady state values in less time. At the same time care is taken that these state variables stay within a safe physical limit. The controller gain matrix K corresponding to these closed loop poles is K=2020 −376860 −304950 −18250 −130 −10 4840 3770 330 60 (7.21) Figure 7.28 -Figure 7.31 show that the state variables E′ q,ω,δ,Tm,GV, and the outputs Vtandδsettle to their respective steady state values in approximately 5 to 8 seconds. 75 051015202530354045500.70.80.911.11.2Generator terminal voltage Vt (p.u.) time (sec)Figure 7.26: Plot of the generator terminal voltage Vtvs time for the pole placement based full-state feedback controller with the original gains of subsection 7.2.1 applied to the reduced order nonlinear model 051015202530354045500.90.9511.051.11.151.21.251.31.351.4delta (radians) time (sec) Figure 7.27: Plot of the rotor angle δvs time for the pole placement based full-state feedback controller with the original gains of subsection 7.2.1 applied to the reduced order nonlinear model 0 5 10 15 20 2500.511.5State variables x (p.u.) time (sec) E'q omega delta Tm GV Figure 7.28: Plot of the state variables E′ q,ω,δ,Tm, and GVvs time for the pole placement based full-state feedback controller applied to the reduced order nonlinear model, with the gains re-tuned 76 0 5 10 15 20 250.850.90.9511.051.11.151.21.25Generator terminal voltage Vt (p.u.) time (sec)Figure 7.29: Plot of the generator terminal voltage Vtvs time for the pole placement based full-state feedback controller applied to the reduced order nonlinear model, with the gains re-tuned 0 5 10 15 20 250.990.99511.0051.011.0151.021.0251.031.035delta (radians) time (sec) Figure 7.30: Plot of the rotor angle δvs time for the pole placement based full-state feedback controller applied to the reduced order nonlinear model, with the gains re-tuned 0 5 10 15 20 250.960.9811.021.041.061.081.1omega (radians/sec) time (sec) Figure 7.31: Plot of the frequency ω vs time for the pole placement based full-state feedback controller applied to the reduced order nonlinear model, with the gains re-tuned 77 7.2.3 Simulation Results for the F ull-State F eedback Pole Placement controller applied to the T ruth Model W e now test the pole placement based full-state feedback controller on the truth model. The gains of the controller are re-tuned by appropriately choosing the desired pole locations, to get satisfactory performance. The MA TLAB function K=place (A, B, p )is used to design the controller gain matrix K, where pis a row vector containing the desired closed-loop poles. After some trial and error the desired closed-loop poles of the generator-turbine system were selected as p= [−8.00 + j0.05,−8.00−j0.05,−200,−250,−0.1] (7.22) The controller gain matrix K corresponding to these closed loop poles is K=1526.38−48186 .24−3062.64−1166.75−32.7 −0.9566 1321 .51 49 .346 103 .52 39 .965 (7.23) Figure 7.32 to Figure 7.36 show simulation results for the pole placement based full-state feedback controller applied to the truth model. F rom these results we can see that Vtoscillates about a steady state value of 1.17 p.u. which deviates from the desired steady state value of 1.1723 p.u. by an amount of 0.0023 p.u., ω oscillates about its desired steady state value of 1 p.u., and δoscillates about a steady state value of 1.0005 p.u. which deviates from the desired steady state value of 1 p.u. by an amount of 0.0005 p.u.. Small oscillations which decay with time are seen for ω, and δ.Figure 7.35 and Figure 7.36 show plots for the two control inputs VFanduTrespectively . As seen from these plots, the generator excitation voltage VFsettles to its steady state value of 0.00121 p.u. and the turbine valve control settles to its steady state value of 1.0512 p.u. 78 0 200 400 600 800 10001.11.111.121.131.141.151.161.171.181.19Generator terminal voltage vt (p.u.) time (sec)Figure 7.32: Plot of the generator terminal voltage Vtvs time for the pole placement based full-state feedback controller applied to the truth model 0 200 400 600 800 10000.99990.9999111.00011.0001omega (radians/sec) time (sec) Figure 7.33: Plot of ω vs time for the pole placement based full-state feedback controller applied to the truth model 0 200 400 600 800 10000.9970.9980.99911.0011.0021.0031.004delta (radians) time (sec) Figure 7.34: Plot of δvs time for the pole placement based full-state feedback controller applied to the truth model 79 0 2 4 6 8 10−0.6−0.5−0.4−0.3−0.2−0.100.1Excitation field voltage vF (p.u.) time (sec)Figure 7.35: Plot of the control input VFvs time for the pole placement based full-state feedback controller applied to the truth model 0 1 2 3 4 500.511.522.533.544.55Turbine valve control uT(p.u.) time (sec) Figure 7.36: Plot of the control input uTvs time for the pole placement based full-state feedback controller applied to the truth model 80 7.3 Output F eedback Controller Design 7.3.1 Observer-based LQR Design based on linear model In a practical generator-turbine system connected to an infinite bus not all the states are available for measurement. F or example the second output of the system which is the rotor angle cannot be measured using a sensor. Thus, we need to design an observer to estimate the rotor angle and the remaining unmeasured states. Once the estimator is designed, using the separation principle state estimates are used as the input to the state feedback controller to obtain a regulator for the system. W e first use only one sensor to measure only the first output ∆Vt. Later we use two sensors to measure both the outputs ∆Vt, and ∆ω. Using the A andB matrices as found earlier and y=C1x= T10T20 0 x i.e.∆Vt= 0.5258 0 0 .0294 0 0 x(7.24) where we use only the first output i.e. the generator terminal voltage ∆Vtmeasurement for the observer design. Using MA TLAB we can verify that the system is observable using the generator terminal voltage ∆Vti.e.C1as given in Equation (7.24) . The observer that produces an estimate of the state, ˆx, is of the form˙ˆx=Aˆx+Bu+L(y−ˆy) ˆy=C1ˆx(7.25) The observer gain matrix Lin the above equation is designed such that the error dynamics of the estimator decay faster than the remaining dynamics. Let us denote e=x−ˆxas the estimation error. Then, the error dynamics of the estimator are given by ˙e= (A−LC1)e (7.26) The procedure to design the observer gain matrix Lis as follows: W e first calculate the eigenvalues of the closed loop system, i.e. eigenvalues of the closed loop Acp matrix where Acp =A−BK . The real parts of the estimator poles are selected 10-15 times to the left of the closed loop poles of the system Acp matrix, so that the error dynamics of the estimator decay to zero faster. This can be achieved by appropriately selecting the scaling factor ρwhich places the estimator poles to the left of the closed loop poles of the system. The following MA TLAB command can be used to design the observer gain matrix poles =eig(Acp); p=rho∗[poles (1), poles (2), poles (3), poles (4), poles (5)]; Ltilde =place (A′, C′ 1, p); L=Ltilde′; The estimator gain matrix Lforρ= 12 is L=2 666641510.4 −65508 .0 −24107 .6 −1004232 .6 276190 .33 77775(7.27) Next, the estimated states are fed back to the feedback controller to obtain a regulator for the system u=−Kˆx (7.28) Choosing the weighting matrices Q=2 666641 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 13 77775(7.29) 81 and R=20 0 0 20 (7.30) and the state space matrices (A, B )as given in Equation (3.76) and Equation (3.77) the control gain K for the observer-based LQR with the output Vtmeasured is found to be K=0.0384 −0.0635 0 .0126 −0.0070 −0.0027 −0.0811 0 .1973 −0.0236 0 .0382 0 .0395 (7.31) The Q andR matrices are designed such that all the estimated state variables and the control signals stay within a specific limit. F rom the choice of the Q andR matrices for the estimator-based LQR with output ∆Vtmeasured, we can see that the weights of the Q matrix are reduced and weights of the R matrix are increased compared to the full state feedback LQR. In other words the amount of control energy that can be provided such that all the state variables and control inputs remain within a limit is significantly reduced. This is allowed by the separation principle where the controller and the observer gains are designed independently . Thus, the time taken by the estimated states and hence the outputs to reach the steady state value of 0 is larger than the state feedback case. This is evident from Figure 7.37 to Figure 7.40 where the estimated states ∆ˆE′q,∆ˆω,∆ˆδ,∆ˆTm,∆ˆGV, and the outputs ∆Vt,∆ω, and ∆δtake more time to converge to zero as compared to the LQR with full state feedback. Figure 7.41 and Figure 7.42 show plots for the two control inputs ∆Efd, and ∆uTrespectively . A better estimate of the states can be obtained by also using the angular velocity or frequency ∆ωof the synchronous generator which can be measured using a sensor, in addition to the generator terminal voltage ∆Vt. In this case we have y=C2x=T10T20 0 0 1 0 0 0 x i.e.∆Vt ∆ω =0.5258 0 0 .0294 0 0 0 1 0 0 0 x(7.32) Using MA TLAB we can verify that the system is observable using the generator terminal voltage ∆Vtand the frequency ∆ωi.e.C2as given in Equation (7.32) . The estimator gain matrix Lfor the output matrix C2as given in Equation (7.32) andρ= 12 is L=2 66664256.459 71 .2024 20.6146 34 .7633 −1846.26−922.386 −5032.595−1309.90 −531.7120 −135.42563 77775(7.33) Choosing the weighting matrices Q=2 666641 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 13 77775(7.34) and R=1 0 0 1 (7.35) and the state space matrices (A, B )as given in Equation (3.76) and Equation (3.77) the control gain K for the observer-based LQR with the outputs ∆Vtand ∆ωmeasured is found to be K=0.4722 −0.8024 0 .0599 −0.0726 −0.0195 −0.5758 1 .6563 −0.0271 0 .3948 0 .5217 (7.36) 82 00.511.522.533.544.55−30−25−20−15−10−505101520Estimated state variables xhat (p.u.) time (sec) ∆E’q hat ∆ω hat ∆δ hat ∆Tm hat ∆GV hatFigure 7.37: Plot of the estimated state variables ∆ˆE′q,∆ˆω,∆ˆδ,∆ˆTm, and ∆ˆGVvs time for the observer- based LQR applied to the reduced order linear model with output ∆Vtmeasured 010 20 30 40 50 60 70 80−3−2−101234567x 10−3Generator terminal voltage ∆ Vt (p.u.) time (sec) Figure 7.38: Plot of the generator terminal voltage ∆Vtvs time for the observer-based LQR applied to the reduced order linear model with output ∆Vtmeasured 010 20 30 40 50 60 70 80−8−6−4−20246810x 10−3∆ ω (radians/sec) time (sec) Figure 7.39: Plot of ∆ωvs time for the observer-based LQR applied to the reduced order linear model with output ∆Vtmeasured 83 010 20 30 40 50 60 70 80−0.015−0.01−0.00500.0050.010.0150.02∆ δ (radians) time (sec)Figure 7.40: Plot of the rotor angle ∆δvs time for the observer-based LQR applied to the reduced order linear model with output ∆Vtmeasured 02468101214161820−0.6−0.5−0.4−0.3−0.2−0.100.10.2Excitation field voltage ∆ Efd (p.u.) time (sec) Figure 7.41: Plot of the control input ∆Efd vs time for the observer-based LQR applied to the reduced order linear model with output ∆Vtmeasured 02468101214161820−1−0.500.511.522.5Turbine valve control ∆ uT(p.u.) time (sec) Figure 7.42: Plot of the control input ∆uTvs time for the observer-based LQR applied to the reduced order linear model with output ∆Vtmeasured 84 012345678910−0.6−0.5−0.4−0.3−0.2−0.100.10.20.30.4Estimated state variables xhat (p.u.) time (sec) ∆E’q hat ∆ω hat ∆δ hat ∆Tm hat ∆GV hatFigure 7.43: Plot of the estimated state variables ∆ˆE′q,∆ˆω,∆ˆδ,∆ˆTm, and ∆ˆGVvs time for the observer- based LQR applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 010 20 30 40 50 60 70 80−3−2−101234567x 10−3Generator terminal voltage ∆ Vt (p.u.) time (sec) Figure 7.44: Plot of the generator terminal voltage ∆Vtvs time for the observer-based LQR applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 010 20 30 40 50 60 70 80−8−6−4−20246810x 10−3∆ ω (radians/sec) time (sec) Figure 7.45: Plot of ∆ωvs time for the observer-based LQR applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 85 010 20 30 40 50 60 70 80−0.015−0.01−0.00500.0050.010.0150.02∆ δ (radians) time (sec)Figure 7.46: Plot of ∆δvs time for the observer-based LQR applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 010 20 30 40 50 60 70 80−0.03−0.025−0.02−0.015−0.01−0.00500.0050.010.0150.02Excitation field voltage ∆ Efd (p.u.) time (sec) Figure 7.47: Plot of the control input ∆Efd vs time for the observer-based LQR applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 0 510 15 20 25 30 35 40−0.1−0.0500.050.10.150.20.25Turbine valve control ∆ uT(p.u.) time (sec) Figure 7.48: Plot of the control input ∆uTvs time for the observer-based LQR applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 86 F rom the choice of the Q andR matrices for the second observer when the frequency ∆ωis measured in addition to the output ∆Vt, we can see that the weights of the R matrix are reduced compared to the case when only ∆Vtis measured. Thus, the amount of control effort that can be applied to the system such that all the states and the control inputs remain within a specific bound is more, and hence the state variables and the outputs converge to zero faster compared to the first observer with only ∆Vtmeasured, but not as fast as the full state feedback LQR controller. This is evident from Figure 7.43 to Figure 7.46 where the estimated states ∆ˆE′q,∆ˆω,∆ˆδ,∆ˆTm,∆ˆGV, and the outputs ∆Vt,∆ω, and ∆δof the observer based LQR with ∆Vtand∆ωmeasured take less time to converge to zero as compared to the observer-based LQR with only ∆Vtmeasured. However, the time taken is larger compared to the LQR with full state feedback, which gives the best results. Figure 7.47 and Figure 7.48 show plots for the two control inputs ∆Efd, and ∆uT respectively . 87 7.3.2 Observer-Based Pole Placement Controller Design based on linear model In this section we design an observer-based pole placement controller for the reduced order linear model of the generator-turbine system connected to an infinite bus. The procedure used to design an observer for the pole placement controller is similar to the one used earlier for the observer-based LQR. The two outputs, generator terminal voltage ∆Vtand frequency ∆ω, that can be measured using sensors are used for the observer design. Thus we have y=C2x=T10T20 0 0 1 0 0 0 x i.e.∆Vt ∆ω =0.5258 0 0 .0294 0 0 0 1 0 0 0 x(7.37) The estimator gain matrix Lfor the output matrix C2as given in Equation (7.32) andρ= 12 is L=2 6666468.7022 −2.8308 0.2306 18 .6252 −608.8626 34 .4711 −661.5803 387 .1455 −61.4420 58 .67003 77775(7.38) The desired closed-loop poles of the generator-turbine system for the observer-based pole placement controller were selected as p= [−0.7,−0.8,−0.5,−0.9,−0.8] (7.39) The controller gain matrix K corresponding to these closed loop poles is K=9.7459 −26.9495 4 .8337 −0.8674 −9.0425 0.0299 −0.2030 −0.0207 0 .1337 −1.1007 (7.40) F rom Figure 7.49 to Figure 7.52 we can see that for the observer-based pole placement controller the estimated state variables ∆ˆE′q,∆ˆω,∆ˆδ,∆ˆTm,∆ˆGV, and the outputs ∆Vt,∆ω, and ∆δconverge to zero in approximately 12 seconds. Figure 7.53 and Figure 7.54 show plots for the two control inputs ∆Efd, and ∆uTrespectively . The performance of the observer-based pole placement controller is comparable to the performance of the full-state feedback pole-placement controller. 88 0 5 10 15 20 25−0.16−0.14−0.12−0.1−0.08−0.06−0.04−0.0200.020.04Estimated state variables xhat (p.u.) time (sec) ∆E’q hat ∆ω hat ∆δ hat ∆Tm hat ∆GV hatFigure 7.49: Plot of the estimated state variables ∆ˆE′q,∆ˆω,∆ˆδ,∆ˆTm, and ∆ˆGVvs time for the observer- based pole placement controller applied to the reduced order linear model with outputs ∆Vtand∆ωmeasured 05101520253035404550−0.01−0.00500.0050.010.0150.02Generator terminal voltage ∆ Vt (p.u.) time (sec) Figure 7.50: Plot of the generator terminal voltage ∆Vtvs time for the observer-based pole placement controller applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 05101520253035404550−4−20246810x 10−3∆ ω (radians/sec) time (sec) Figure 7.51: Plot of ∆ω vs time for the observer-based pole placement controller applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 89 05101520253035404550−20246810121416x 10−3∆ δ (radians) time (sec)Figure 7.52: Plot of ∆δvs time for the observer-based pole placement controller applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 05101520253035404550−0.2−0.100.10.20.30.40.50.60.70.8Excitation field voltage ∆ Efd (p.u.) time (sec) Figure 7.53: Plot of the control input ∆Efd vs time for the observer-based pole placement controller applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 05101520253035404550−20246810x 10−3Turbine valve control ∆ uT(p.u.) time (sec) Figure 7.54: Plot of the control input ∆uTvs time for the observer-based pole placement controller applied to the reduced order linear model with outputs ∆Vtand ∆ωmeasured 90 7.3.3 Simulation Results for the Observer-based LQR applied to the Reduced Order Nonlin- ear Model W e now test the observer-based LQR on the reduced order nonlinear model. The generator terminal voltage Vtand the angular velocity ω are measured using sensors. The results obtained by directly applying the observer-based LQR that was designed for the reduced order linear model in subsection 7.3.1 are poor. Therefore, the gains of the controller and observer poles are adjusted so that the controller works eﬀiciently on the reduced order nonlinear model. Using the angular velocity or frequency ωof the synchronous generator which can be measured using a sensor, in addition to the generator terminal voltage Vtthe output equation can be written as y=C2x=T10T20 0 0 1 0 0 0 x i.e.∆Vt ∆ω =0.5258 0 0 .0294 0 0 0 1 0 0 0 x(7.41) The estimator gain matrix Lfor the output matrix C2as given in Equation (7.41) andρ= 12 is L=2 66664145.5739 29 .0262 −9.1126 22 .2527 −528.9546 −442.3042 −2676.907−678.8487 834.9962 219 .78403 77775(7.42) Choosing the weighting matrices Q=2 666645 0 0 0 0 0 5 0 0 0 0 0 0 .5 0 0 0 0 0 0 .05 0 0 0 0 0 53 77775(7.43) and R=1000 0 0 1000 (7.44) and the state space matrices (A, B )as given in Equation (3.76) and Equation (3.77) the control gain K for the observer-based LQR with the outputs Vtandωmeasured is found to be K=0.0021 −0.0027 0 .0005 −0.0003 −0.0001 −0.0036 0 .0079 −0.0014 0 .0009 0 .0028 (7.45) Figure 7.55 ,Figure 7.56 , and Figure 7.57 show simulation results for the re-tuned observer-based LQR applied to the reduced order nonlinear model. F rom these simulation results we can see that the generator terminal voltage Vtoscillates between 0.8 and 0.82 p.u. which is different from the desired steady state value of 1.172 p.u. by an amount of 0.372 p.u., the rotor angle δoscillates between 0 and 0.4 p.u. with the amplitude of oscillations decreasing with time, which is also different from the desired steady state value of 1 p.u., and the frequency ωoscillates about the desired steady state value of 1 p.u. Even after re-tuning the gains of the controller and the observer poles it is not possible to improve the results too much. A significant amount of steady state error is observed. The linear observer is not able to accurately estimate the states of the reduced order nonlinear model, i.e. it is not robust to uncertainties or state deviations between the reduced order nonlinear and the reduced order linear model. T o overcome this diﬀiculty i.e. to increase the robustness of the observer we design a linear quadratic gaussian (LQG) controller, in which the gains of the Kalman filter are tuned using loop transfer recover (L TR) procedure, in the next subsection. 91 0501001502002503003504004505000.750.80.850.90.9511.051.11.151.2Generator terminal voltage Vt (p.u.) time (sec)Figure 7.55: Plot of the generator terminal voltage Vtvs time for the observer-based LQR applied to the reduced order nonlinear model 01002003004005006007008009001000−1.5−1−0.500.51delta (radians) time (sec) Figure 7.56: Plot of the rotor angle δvs time for the observer-based LQR applied to the reduced order nonlinear model 0501001502002503003504004505000.750.80.850.90.9511.051.1omega (radians/sec) time (sec) Figure 7.57: Plot of the frequency ω vs time for the observer-based LQR applied to the reduced order nonlinear model 92 7.3.4 L TR-based LQG Controller applied to the Reduced Order Nonlinear Model In this section we present the design of a linear quadratic Gaussian (LQG) controller, where a design or gain adjustment procedure in the time domain is used which is analogous to loop shaping in the frequency domain, to adjust the gains of the Kalman filter. This gain adjustment procedure not only improves the robustness of the observer but also asymptotically achieves the same loop transfer function as a full-state feedback controller [ 3]. A basic requirement for every point of an adjustment trajectory is stability of the observer error dynamics. If this requirement is not satisfied then closed-loop stability of the system is also lost. One way to assure stable error dynamics is to restrict the observer to be a Kalman filter for some set of noise parameters or covariance matrices V1andV2, where V1is a5×5matrix of intensities related to the plant disturbances or process noise, and V2is a 2×2matrix of intensities related to the measurement or sensor noise. Thus let the observer gain H(q)be given by the Kalman filter expression H(q) = Σ( q)CTV−1 2 (7.46) with Σ(q)defined by the Riccati equation AΣ(q) + Σ( q)AT+V1(q)−Σ(q)CTV−1 2CΣ(q) = 0 (7.47) W e select V1=VT 1>0andV2=VT 2>0with (A, V1 2 1)and (C, A)stabilizable and observable, respectively . V1(q)which is a function of qis designed as V1(q) =V10+q2BV BT V2(q) =V20(7.48) where V10 andV20 are noise intensities appropriate for the nominal plant, and V is any positive definite symmetric matrix, which are chosen as V10=2 666641 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 13 77775 V20=1 0 0 1 V=1 0 0 1(7.49) With these selections, the observer gain for q= 0 corresponds to the nominal Kalman filter gain. However asqapproaches infinity , the gains are seen from Equation (7.47) to satisfy HV 2HT q2→BV BT(7.50) By substituting Equation (7.48) in Equation (7.47) and dividing Equation (7.47) byq2we get AΣ(q) q2 +Σ(q) q2 AT+V10 q2+BV BT−q2Σ(q) q2 CTV−1 2CΣ(q) q2 = 0 (7.51) F rom [ 3] it can be seen that Σ(q) q2→0 as q→ ∞ (7.52) whenever the transfer function C(sI−A)−1B has no right half plane zeros. Consequently , q2Σ(q) q2 CTV−1 2CΣ(q) q2 →BV BT(7.53) 93 and thus Equation (7.50) is established. Solutions of Equation (7.50) must necessarily be of the form H(q) q→BV1 2(V1 2 2)−1(7.54) where V1 2denotes some square root of V, i.e. (V1 2)TV1 2=V, and similarly V1 2 2 is some square root of V2. The design adjustment procedure defined by Equation (7.46) to Equation (7.48) will achieve the desired robustness improvement objective. W e can see that higher the value of q, higher will be the observer gain H and thus higher will be the robustness of the system which is maximum at q=∞. Our objective is to apply the L TR-based LQG controller on the reduced order nonlinear model. F rom simulations it is verified that for very large values of qthe system goes unstable. Thus the choice of qis very important and it cannot be arbitrarily selected to a large value, when we are applying this design adjustment procedure on a nonlinear model. The design parameter qis suitable selected to be equal to 9.0005 to get satisfactory closed loop performance of the system. F rom the separation principle we know that the gains of the Kalman filter H(q)and the controller gains K can be deigned independently . W e use the LQR algorithm to design the controller gains. Thus by using the feedback law, u=−Kˆ x (7.55) where ˆ xare the states estimated by the Kalman filter with loop shaping, the weighting matrices Q=2 666641254.75 0 0 0 0 0 1500 0 0 0 0 0 544 .5 0 0 0 0 0 142 .5 0 0 0 0 0 15003 77775(7.56) and R=1 0 0 1 (7.57) and the state space matrices (A, B )as given in Equation (3.82) and Equation (3.83) the control gain K is found to be K=34.5065 −49.5197 −5.6995 −4.1955 −0.0432 −1.2745 28 .0922 −0.0281 4 .3245 37 .7873 (7.58) Figure 7.58 to Figure 7.63 show simulation results for the L TR-based LQG controller applied to the reduced order nonlinear model. F rom these results we can see that all the state variables, outputs, and control inputs, attain their desired steady state values respectively . Thus a significant improvement in the robustness is observed when the Kalman filter gains are tuned using the loop-shaping design adjustment procedure explained above. The performance of the L TR-based LQG controller is comparable to that of the LQR- based full-state feedback controller. 94 010203040506070809010000.20.40.60.811.21.4State variables x (p.u.) time (sec) E'q omega delta Tm GVFigure 7.58: Plot of the state variables E′ q,ω,δ,Tm, and GV vs time for the L TR-based LQG controller applied to the reduced order nonlinear model 01020304050607080901000.9511.051.11.151.2Generator terminal voltage Vt (p.u.) time (sec) Figure 7.59: Plot of the generator terminal voltage Vtvs time for the L TR-based LQG controller applied to the reduced order nonlinear model 01020304050607080901000.90.9511.051.11.151.21.25delta (radians) time (sec) Figure 7.60: Plot of the rotor angle δvs time for the L TR-based LQG controller applied to the reduced order nonlinear model 95 01020304050607080901000.920.940.960.9811.021.041.061.081.11.12omega (radians/sec) time (sec)Figure 7.61: Plot of the frequency ωvs time for the L TR-based LQG controller applied to the reduced order nonlinear model 0102030405060708090100−5−4−3−2−1012345Excitation field voltage EFD (p.u.) time (sec) Figure 7.62: Plot of the generator excitation field control EFD vs time for the L TR-based LQG controller applied to the reduced order nonlinear model 0 5 10 15 20 25 3000.511.522.533.54Turbine valve control uT(p.u.) time (sec) Figure 7.63: Plot of the turbine valve control uTvs time for the L TR-based LQG controller applied to the reduced order nonlinear model 96 7.3.5 L TR-based LQG Controller applied to the T ruth Model It is not always possible to measure the rotor angle δof the synchronous generator. Thus in cases where the rotor angle cannot be measured, state feedback based control techniques are not applicable to the truth model. Hence it is necessary to design an observer-based output feedback controller which can be applied to the truth model. The loop transfer recovery (L TR)-based linear quadratic gaussian (LQG) controller that was tested on the reduced order nonlinear model, is now tested on the truth model. The covariance matrices V1,V2, that are used in the design of the Kalman filter, the loop shaping parameter q, and the controller gains are tuned once again so that the L TR-based LQG controller works eﬀiciently on the truth model. W e select V1=VT 1>0andV2=VT 2>0with (A, V1 2 1)and (C, A)stabilizable and observable, respectively . V1(q) which is a function of qis designed as V1(q) =V10+q2BV BT V2(q) =V20(7.59) V10 andV20 are noise intensities appropriate for the nominal plant, and V is any positive definite symmetric matrix, which are chosen as V10=2 666641 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 13 77775 V20=0.65 0 0 0 .65 V=1 0 0 1(7.60) The loop shaping parameter qis tuned to 5.25. W e use the LQR algorithm to design the controller gains. Thus by using the feedback law, u=−Kˆ x (7.61) where ˆ xare the states estimated by the Kalman filter with loop shaping, the weighting matrices Q=2 666647500 0 0 0 0 0 15000 0 0 0 0 0 16500 0 0 0 0 0 7500 0 0 0 0 0 75003 77775(7.62) and R=1 0 0 1 (7.63) and the state space matrices (A, B )as given in Equation (3.82) and Equation (3.83) the control gain K is found to be K=87.3944 −216.7677 −60.7947 −13.4353 −0.0618 −1.8244 98 .0650 17 .7303 42 .1399 85 .8027 (7.64) Figure 7.64 to Figure 7.68 show simulation results for the L TR-based LQG controller applied to the truth model. F rom Figure 7.64 ,Figure 7.65 , and Figure 7.66 we can see that Vt,ω, and δsettle to their respective steady state values. Small oscillations which decay with time are seen for ω, and δ. Figure 7.67 and Figure 7.68 show the plots for the two control inputs VF anduTrespectively . The generator excitation voltage VFsettles to its steady state value of 0.00121 p.u. and the turbine valve control settles to its steady state value of 1.0512 p.u. 97 051015202530354045500.80.850.90.9511.051.11.151.21.25Generator terminal voltage vt (p.u.) time (sec)Figure 7.64: Plot of the generator terminal voltage Vtvs time for the L TR-based LQG controller applied to the truth model 010020030040050060070080090010000.99980.999911.00011.0002omega (radians/sec) time (sec) Figure 7.65: Plot of the angular velocity ωvs time for the L TR-based LQG controller applied to the truth model 010020030040050060070080090010000.9960.9970.9980.99911.0011.0021.0031.0041.0051.006delta (radians) time (sec) Figure 7.66: Plot of the rotor angle δvs time for the L TR-based LQG controller applied to the truth model 98 0102030405060708090100−0.1−0.0500.050.10.150.20.250.30.350.4Excitation field voltage vF (p.u.) time (sec)Figure 7.67: Plot of the generator excitation voltage VFvs time for the L TR-based LQG controller applied to the truth model 01234567891000.20.40.60.811.21.41.61.82Turbine valve control uT(p.u.) time (sec) Figure 7.68: Plot of the turbine valve control uTvs time for the L TR-based LQG controller applied to the truth model 99 8 Nonlinear F eedback Linearizing Controller Design W e present the design of a nonlinear input-state feedback linearizing controller for the nonlinear model of the synchronous generator and turbine connected to an infinite bus. Before we proceed to the nonlinear controller design, we give the fundamentals of full-state feedback linearization for single-input, single-output (SISO) nonlinear control systems [ 5,6], which are aﬀine in the control input, that is, systems of the form ˙x=f(x) +g(x)u y=h(x)(8.1) where x∈Rn,u∈R,y∈R. W e assume that the vector fields f:D→Rn,g:D→Rnand the readout map h:D→Rare smooth in the domain D⊂Rn, that is their partial derivatives with respect to xof any order exist and are continuous in D. Our objective is to find a smooth full state feedback control law of the form u=α(x) +β(x)v (8.2) and a change of variables z=T(x)that transform the nonlinear system into an equivalent linear system, i.e. the closed loop system ˙x=f(x) +g(x)α(x) +g(x)β(x)v y=h(x)(8.3) is completely linearized. The ability to use state feedback control to convert a nonlinear state equation into a controllable linear state equation by canceling nonlinearities requires the nonlinear state equation to have the structure ˙x=Ax+Bγ(x)[u−α(x)] (8.4) where Aisn×n,B isn×p, the pair (A, B )is controllable, the functions α:Rn→Rpandγ:Rn→Rp×pare defined in a domain D⊂Rnthat contains the origin, and the matrix γ(x)is nonsingular for every x∈D. If the state equation takes the form Equation (8.4) , then we can linearize it via the state feedback u=α(x) +β(x)v (8.5) where β(x) =γ−1(x), to obtain the linear state equation ˙x=Ax+Bv (8.6) F or stabilization, we design v=−Kx such that A−BK is Hurwitz. The overall nonlinear stabilizing state feedback control is u=α(x)−β(x)Kx (8.7) Example 1 [ 6] T o introduce the idea of exact feedback linearization, let us consider the pendulum stabilization problem. Inspection of the state equation ˙x1=x2 ˙x2=−a[sin(x1+δ)−sinδ]−bx2+cu(8.8) shows that we can choose uas u=a c [sin(x1+δ)−sinδ] +v c(8.9) to cancel the nonlinear term a[sin(x1+δ)−sinδ]. This cancellation results in the linear system ˙x1=x2 ˙x2=−bx2+v(8.10) 100 Thus, the stabilization problem for the nonlinear system has been reduced to a stabilization problem for a controllable linear system. W e can proceed to design a stabilizing linear state feedback control v=k1x1+k2x2 (8.11) to locate the eigenvalues of the closed-loop system ˙x1=x2 ˙x2=k1x1+ (k2−b)x2(8.12) in the open left-half plane. The overall state feedback control law is given by u=a c [sin(x1+δ)−sinδ] +1 c(k1x1+k2x2) (8.13) However, if the nonlinear state equation does not have the structure of Equation (8.4) for one choice of state variables, it does not mean that we cannot linearize the system via feedback. A co-ordinate transfor- mation which transforms the old states into new state variables, can be used to convert the nonlinear state equation in the original states into a structure of Equation (8.4) in the new state variables. Example 2 [ 6] Consider the system ˙x1=asinx2 ˙x2=−x2 1+u(8.14) W e cannot simply choose uto cancel the nonlinear term asinx2. However, if we first change the variables by the transformation z1=x1 z2=asinx2= ˙x1(8.15) then z1andz2satisfy ˙z1=z2 ˙z2=acosx2(−x2 1+u)(8.16) and the nonlinearities can be canceled by the control u=x2 1+1 acosx2v (8.17) which is well defined for −π 2< x 2<π 2. The state equation in the new coordinates ( z1,z2) can be found by inverting the transformation to express ( x1,x2) in terms of ( z1,z2); that is, x1=z1 x2= sin−1z2 a(8.18) which is well defined for −a < z 2< a . The transformed state equation is given by ˙z1=z2 ˙z2=acos sin−1z2 a (−z2 1+u)(8.19) When a change of variables z=T(x)is used to transform the state equation from the x-coordinates to the z-coordinates, the map T must be invertible; that is it must have an inverse map T−1(·)such that x=T−1(z)for all z∈T(D), where D is the domain of T. Moreover, because the derivatives of zandx 101 should be continuous, we require both T(·)andT−1(·)to be continuously differentiable. A continuously differentiable map with a continuously differentiable inverse is known as diffeomorphism [ 5,6]. Thus, we can define a feedback linearizable system as Definition [ 6]: A nonlinear system ˙x=f(x) +g(x)u y=h(x)(8.20) where f:D→Rnandg:D→Rn×pare suﬀiciently smooth on a domain D⊂Rn, is said to be feedback linearizable (or input-state linearizable) if there exists a diffeomorphism T:D→Rnsuch that Dz=T(D) contains the origin and the change of variables z=T(x)transforms the system into the form ˙z=Az+Bγ(x)[u−α(x)] (8.21) with (A, B )controllable and γ(x)nonsingular for all x∈D 8.1 Nonlinear F eedback Linearizing Controller Design for the Reduced Order Model W e now present the design of an input-state feedback linearizing controller for the reduced order model of the synchronous generator and turbine-governor system connected to an infinite bus. The fifth order nonlinear model of the system in the original coordinates is ˙E′ q=f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD ˙ω=f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm ˙δ=ω−1 ˙Tm=f41Tm+f42GV ˙GV=f51ω+f52GV+g55uT(8.22) where x= [E′ q, ω, δ, T m, GV]Tare the original state variables and u= [u1, u2]T= [EFD, uT]Tare the two control inputs. W e now define a transformation z=T(x)that transforms the nonlinear state equations into a structure of Equation (8.4) . The new state variables are chosen as z1=δ z2= ˙z1=˙δ=ω−1 z3= ˙z2=¨δ= ˙ω=f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm z4=Tm z5= ˙z4=˙Tm=f41Tm+f42GV(8.23) Thus, we have ˙z1=˙δ=ω−1 =z2 ˙z2= ˙ω=¨δ=z3 ˙z3= ¨ω=... δ=σ1(x) +γ1(x)EFD ˙z4=˙Tm=z5 ˙z5=¨Tm=σ2(x) +γ2(x)uT(8.24) 102 The state transformations given above are invertible and exist throughout the domain of stable operation 0< δ < 180◦. F rom Equation (8.22) and Equation (8.24) , i.e. by taking the derivative of ˙ωwe have ˙z3= ¨ω= 2f21E′ q˙E′ q+f22˙E′ qcos(δ−α)−f22E′ qsin(δ−α)˙δ+f23˙E′ qsin(δ−α) +f23E′ qcos(δ−α)˙δ+f24cos2(δ−α)˙δ−f24sin2(δ−α)˙δ−2f25cos(δ−α) sin( δ−α)˙δ + 2f26cos(δ−α) sin( δ−α)˙δ+f27˙ω+f28˙Tm(8.25) Substituting ˙δ,˙E′ q,˙ω, and ˙Tm from Equation (8.22) in Equation (8.25) we get ˙z3= 2f21E′ q(f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD) +f22(f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD) cos( δ−α) −f22E′ qsin(δ−α)(ω−1) +f23(f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD) sin( δ−α) +f23E′ qcos(δ−α)(ω−1) +f24cos2(δ−α)(ω−1)−f24sin2(δ−α)(ω−1) −2f25cos(δ−α) sin( δ−α)(ω−1) + 2 f26cos(δ−α) sin( δ−α)(ω−1) +f27(f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm) +f28(f41Tm+f42GV)(8.26) On rearranging and simplifying Equation (8.26) we get ˙z3= (2f11f21+f27f21)E′ q2 + (2f21f12+f22f11−f23+f27f22)E′ qcos(δ−α) + (2f21f13+f22+f23f11+f27f23)E′ qsin(δ−α) + (f22f12−f24+f27f25) cos2(δ−α) + (f23f13+f24+f27f26) sin2(δ−α) + (f22f13+f23f12+ 2f25−2f26+f27f24) sin( δ−α) cos( δ−α) +f2 27ω+ (f27f28+f28f41)Tm+f28f42Gv + (f23)E′ qωcos(δ−α) + (−f22)E′ qωsin(δ−α) + (f24)ωcos2(δ−α) + (−f24)ωsin2(δ−α) + (−2f25+ 2f26)ωsin(δ−α) cos( δ−α) + (2f21g11E′ q+f22g11cos(δ−α) +f23g11sin(δ−α))EFD(8.27) F or simplification let us denote 2f11f21+f27f21=p31, 2f21f12+f22f11−f23+f27f22=p32, 2f21f13+f22+f23f11+f27f23=p33, f22f12−f24+f27f25=p34, f23f13+f24+f27f26=p35, f22f13+f23f12+ 2f25−2f26+f27f24=p36, f2 27=p37, f 27f28+f28f41=p38, f28f42=p39, f 23=q31,−f22=q32, f24=q33,−f24=q34,−2f25+ 2f26=q35, 2f21g11=r31, f 22g11=r32, f 23g11=r33(8.28) 103 On substituting Equation (8.28) in Equation (8.27) we get ˙z3=p31E′ q2+p32E′ qcos(δ−α) +p33E′ qsin(δ−α) +p34cos2(δ−α) +p35sin2(δ−α) +p36sin(δ−α) cos( δ−α) +p37ω+p38Tm+p39Gv +q31E′ qωcos(δ−α) +q32E′ qωsin(δ−α) +q33ωcos2(δ−α) +q34ωsin2(δ−α) +q35ωsin(δ−α) cos( δ−α) + (r31E′ q+r32cos(δ−α) +r33sin(δ−α))EFD(8.29) F rom Equation (8.24) we have ˙z3=σ1(x) +γ1(x)EFD=v1 (8.30) Thus, equating Equation (8.29) and Equation (8.30) we get σ1(x) =p31E′ q2+p32E′ qcos(δ−α) +p33E′ qsin(δ−α) +p34cos2(δ−α) +p35sin2(δ−α) +p36sin(δ−α) cos( δ−α) +p37ω+p38Tm+p39Gv +q31E′ qωcos(δ−α) +q32E′ qωsin(δ−α) +q33ωcos2(δ−α) +q34ωsin2(δ−α) +q35ωsin(δ−α) cos( δ−α) i.eσ1(x) =p31x12+p32x1cos(x3−α) +p33x1sin(x3−α) +p34cos2(x3−α) +p35sin2(x3−α) +p36sin(x3−α) cos( x3−α) +p37x2+p38x4+p39x5 +q31x1x2cos(x3−α) +q32x1x2sin(x3−α) +q33x2cos2(x3−α) +q34x2sin2(x3−α) +q35x2sin(x3−α) cos( x3−α) γ1(x) =r31E′ q+r32cos(δ−α) +r33sin(δ−α) i.eγ1(x) =r31x1+r32cos(x3−α) +r33sin(x3−α)(8.31) Thus, using Equation (8.30) and Equation (8.31) we can compute the excitation field EMF EFD as EFD=γ−1 1(x)(v1−σ1(x)) EFD=α1(x) +β1(x)v1(8.32) where α1(x) =−γ−1 1(x)σ1(x)andβ1(x) =γ−1 1(x). F rom Equation (8.22) and Equation (8.24) , i.e. by taking the derivative of ˙Tm we have ˙z5=¨Tm=f41˙Tm+f42˙Gv =f41(f41Tm+f42Gv) +f42(f51ω+f52Gv+g55uT) =f42f51ω+f2 41Tm+ (f41f42+f42f52)Gv+f42g55uT(8.33) Let us denote f42f51=p51,f2 41=p52,f41f42+f42f52=p53, and f42g55=r51. Thus, Equation (8.33) can be simplified as ˙z5=¨Tm=p51ω+p52Tm+p53Gv+r51uT (8.34) F rom Equation (8.24) we have ˙z5=σ2(x) +γ2(x)uT=v2 (8.35) Thus, equating Equation (8.34) and Equation (8.35) we get σ2(x) =p51ω+p52Tm+p53Gv i.eσ2(x) =p51x2+p52x4+p53x5 γ2(x) =r51(8.36) 104 Thus, using Equation (8.35) and Equation (8.36) we can compute the turbine valve control uTas uT=γ−1 2(x)(v2−σ2(x)) uT=α2(x) +β2(x)v2(8.37) where α2(x) =−γ−1 2(x)σ2(x)andβ2(x) =γ−1 2(x). W e next design the linear controllers v1andv2. The purpose of the linear controllers v1andv2is to regulate the state variables δandTmto their set points. The set points for the remaining state variables are found through the equilibrium condition δ(t) = 1 radian, and Tm(t) = 1 .0012 p.u.. Application of the state feedback control u1=EFD=α1(x) +β1(x)v1,u2=uT=α2(x) +β2(x)v2, and the state transformation z=T(x)to the nonlinear model of the system, results in a linear model of the system in the new coordinates as ˙z=Az+Bv (8.38) where zT= z1z2z3z4z5 v=v1 v2(8.39) A=2 666640 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 03 77775(8.40) B=2 666640 0 0 0 1 0 0 0 0 13 77775(8.41) The linear controller v=−Kzcan be designed either by pole placement or LQR technique such that A−BK is Hurwitz. W e use the LQR technique to design the controller gain matrix K. Thus, by using the feedback law, v=−Kz (8.42) the weighting matrices Q=2 66664300 0 0 0 0 0 250 0 0 0 0 0 200 0 0 0 0 0 200 0 0 0 0 0 2503 77775(8.43) and R=0.07 0 0 0 .07 (8.44) and the state space matrices (A, B )as given in Equation (8.40) and Equation (8.41) the control gain K is found to be K=65.4654 104 .0206 55 .3641 0 0 0 0 0 53 .4522 60 .6493 (8.45) Thus, we have v1=−K11(z1−z1d)−K12(z2−z2d)−K13(z3−z3d) =−K11(δ−δd)−K12(˙δ−˙δd)−K13(¨δ−¨δd)(8.46) 105 and v2=−K24(z4−z4d)−K25(z5−z5d) =−K24(Tm−Tmd)−K25(˙Tm−˙Tmd)(8.47) where z1d,z2d,z3d,z4d,z5dare the desired values of the new state variables. Since we want the set point to be an equilibrium, all the derivatives of z1andz4have to be zero. Thus z2d=z3d=z5d= 0 . Also z1d= 1 radian, and z4d= 1.0012 p.u. are appropriately chosen within a reasonable physical limit of the system. The limits of the excitation field voltage of the generator are given by EFD(max )= 5 p.u. and EFD(min)= -5 p.u. The limits of the turbine gate opening are given by GV(max )= 1.5 p.u. and GV(min)= 0 p.u. The operating conditions of the system are as given in example 3.4 with IF0= 1.6315 ,Iq0= 0.4047 , Id0=−0.9185 ,Vq0= 0.9670 ,Vd0=−0.6628 ,V∞= 1.00,Vt0= 1.172 ,δ0−α= 53.736◦,E′ q0= 1.1925 , and τ′ d0= 5.90 F rom Figure 8.1 we can see that the state variables E′ q,ω,δ,Tm, and GV attain their respective steady state values in approximately 5 to 7 seconds. Figure 8.2 shows that the generator terminal voltage Vtreaches a steady state value of 1.172 p.u. which is equal to the desired steady state operating point Vt0= 1.172 , in approximately 5 seconds. Figure 8.3 shows that the rotor angle δsettles to a steady state value of approximately 1 radians, which is equal to the steady state operating condition δ0= 1 radian, in approximately 7 seconds. The angular velocity or the frequency ωof the synchronous generator settles to a steady state value of 1 rad/sec which is equal to the desired operating condition ω0= 1 rad/sec which can be verified from Figure 8.4 . Also, from Figure 8.5 and Figure 8.6 we can see that the control inputs, EFD and uT, settle to their steady state values of 2.529 p.u. and 1.0512 p.u. respectively . T o verify our simulation results we analytically compute the steady state equilibrium values of the original state variables and the control inputs. The steady state equilibrium in the original co-ordinates can be computed by solving the differential equation, ˙x=f(x) +g(x)u= 0 . Thus using Equation (8.22) we can write ˙E′ q=f11E′ q+f12cos(δ−α) +f13sin(δ−α) +g11EFD= 0 ˙ω=f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm= 0 ˙δ=ω−1 = 0 ˙Tm=f41Tm+f42GV= 0 ˙GV=f51ω+f52GV+g55uT= 0(8.48) The steady state equilibrium in the original co-ordinates E′ q,ω,δ,Tm, and GV can be found by using z1d=δ= 1 radian, and z4d=Tm= 1.0012 p.u., and solving the differential equation ˙ω=f21E′2 q+f22E′ qcos(δ−α) +f23E′ qsin(δ−α) +f24sin(δ−α) cos( δ−α) +f25cos2(δ−α) +f26sin2(δ−α) +f27ω+f28Tm= 0 ˙δ=ω−1 = 0 ˙Tm=f41Tm+f42GV= 0(8.49) Also the steady state equilibrium of the control inputs EFD anduTcan be found by using ˙E′ q= 0 , and ˙GV= 0 , from Equation (8.48) , and the steady state equilibrium values of the original state variables E′ q(ss), ω(ss),δ(ss),Tm(ss), and GV(ss)obtained by solving Equation (8.49) i.e. 0 =f11E′ q0+f12cos(δ0−α) +f13sin(δ0−α) +g11EFD(ss) =−0.5517×1.1925 + 0 .3822×0.5915 + 0 .0037×0.8063 + 0 .1695×EFD(ss) 0 =f51ω0+f52GV0+g55uT(ss) =−0.25×1−5×1.0012 + 5 ×uT(ss)(8.50) Solving Equation (8.50) we get EFD(ss)= 2.529 anduT(ss)= 1.0512 , which are in agreement with our simulation results. 106 0510152025303540455000.20.40.60.811.21.4State variables x (p.u.) time (sec) E'q omega delta Tm GVFigure 8.1: Plot of the state variables E′ q,ω,δ,Tm, and GVvs time for the input-state nonlinear feedback linearizing controller applied to the reduced order model 051015202530354045500.80.850.90.9511.051.11.151.21.25Generator terminal voltage Vt (p.u.) time (sec) Figure 8.2: Plot of the generator terminal voltage Vtvs time for the input-state nonlinear feedback linearizing controller applied to the reduced order model 051015202530354045500.9650.970.9750.980.9850.990.99511.005delta (radians) time (sec) Figure 8.3: Plot of δvs time for the input-state nonlinear feedback linearizing controller applied to the reduced order model 107 051015202530354045500.950.960.970.980.9911.011.02omega (radians/sec) time (sec)Figure 8.4: Plot of ω vs time for the input-state nonlinear feedback linearizing controller applied to the reduced order model 05101520253035404550−5−4−3−2−1012345Excitation field voltage EFD (p.u.) time (sec) Figure 8.5: Plot of the generator excitation field control EFD vs time for the input-state nonlinear feedback linearizing controller applied to the reduced order model 051015202530354045500123456Turbine valve control uT(p.u.) time (sec) Figure 8.6: Plot of the turbine valve control uTvs time for the input-state nonlinear feedback linearizing controller applied to the reduced order model 108 8.2 Simulation Results for the Nonlinear F eedback Linearizing Controller applied to the T ruth Model F or the reduced order nonlinear model the state variables are x= [E′ q, ω, δ, T m, GV]T, the two control inputs are u= [EFD, uT]T, and the two regulated outputs are y= [Vt, ω]T. Whereas, for the truth model x= [Id, IF, ID, Iq, IQ, ω, δ, T m, GV]Tis the vector of state variables, u= [VF, uT]Tis vector of control inputs, andy= [Vt, ω]Tis the vector of outputs. Thus we can see that the first control input VFof the truth model and the first control input EFD of the reduced order model are different. As given in Equation (10.41) the excitation field emf, EFD , which is the first control input for the reduced order nonlinear model is related to field voltage, VF, which is the first control input for the truth model by the following expression VF rF ωRkMF=EFD (8.51) Equation (8.51) can be rearranged to get the following expression VF=rF ωRkMF EFD =e15EFD(8.52) In the above expression, ωR= 1 p.u., and e15= (rF ωRkMF). Also there is a non physical state E′ qin the reduced order model that needs to be reconstructed from the states of the truth model. The nonlinear feedback linearizing controller that was designed for the reduced order nonlinear model is now tested on the truth model. F rom Equation (8.32) the excitation field EMF, EFD , is computed as EFD=γ−1 1(x)(v1−σ1(x)) EFD=α1(x) +β1(x)v1(8.53) where α1(x) =−γ−1 1(x)σ1(x);β1(x) =γ−1 1(x)and σ1(x) =p31E′ q2+p32E′ qcos(δ−α) +p33E′ qsin(δ−α) +p34cos2(δ−α) +p35sin2(δ−α) +p36sin(δ−α) cos( δ−α) +p37ω+p38Tm+p39Gv +q31E′ qωcos(δ−α) +q32E′ qωsin(δ−α) +q33ωcos2(δ−α) +q34ωsin2(δ−α) +q35ωsin(δ−α) cos( δ−α) i.eσ1(x) =p31x12+p32x1cos(x3−α) +p33x1sin(x3−α) +p34cos2(x3−α) +p35sin2(x3−α) +p36sin(x3−α) cos( x3−α) +p37x2+p38x4+p39x5 +q31x1x2cos(x3−α) +q32x1x2sin(x3−α) +q33x2cos2(x3−α) +q34x2sin2(x3−α) +q35x2sin(x3−α) cos( x3−α) γ1(x) =r31E′ q+r32cos(δ−α) +r33sin(δ−α) i.eγ1(x) =r31x1+r32cos(x3−α) +r33sin(x3−α)(8.54) F rom Equation (8.52) the field voltage VFcan be written as VF=rF ωRkMF EFD=rF ωRkMF γ−1 1(x)(v1−σ1(x)) =e15(α1(x) +β1(x)v1) (8.55) In Equation (8.55) ,VFdepends on σ1(x)andγ1(x)which depend on the fictitious state variable E′ q, which is not a physical quantity that can be measured or a state variable of the truth model. Since E′ qcannot be measured using sensors, we cannot apply the field voltage VFdirectly on the truth model without eliminating the state variable E′ qin the above expression. This problem can be solved by expressing E′ qas a function of 109 any of the measurable states of the truth model. F rom Equation (10.136) as given in the derivation of the reduced order model in the appendix we have E=E′ q−(Ld−L′ d)Id (8.56) Substituting Idfrom Equation (3.8) in the above equation E=E′ q−(Ld−L′ d)−(E′ q−V∞q)(Lq+Le)−V∞d(r+Re) (r+Re)2+ (L′ d+Le)(Lq+Le) (8.57) F or simplification of the above expression let us denote Ld−L′ d=L2,Lq+Le=L1,r+Re=R1, (r+Re)2+ (L′ d+Le)(Lq+Le) =M1. Thus Equation (8.57) can be written as E=E′ q−L2−(E′ q−V∞q)L1−V∞dR1 M1 (8.58) The above expression can be simplified to get E= 1 +L1L2 M1 E′ q−L1L2V∞ M1 cos(δ−α)−R1L2V∞ M1 sin(δ−α) (8.59) Let us denote  1 +L1L2 M1 =e11 L1L2V∞ M1 =e12 R1L2V∞ M1 =e13(8.60) Substituting Equation (8.60) in Equation (8.59) and rearranging, E′ qcan be written as E′ q=1 e11E+e12 e11cos(δ−α) +e13 e11sin(δ−α) (8.61) F rom Equation (10.39) the excitation field emf, E, is related to the field current IFas E=ωRkMFIF=e14IF (8.62) Substituting Equation (8.62) in Equation (8.61) E′ q=e14 e11IF+e12 e11cos(δ−α) +e13 e11sin(δ−α) (8.63) E′ qis now expressed as a function of the field current IF, and rotor angle δ, which are state variables of the truth model. While the field current IFcan be measured using a sensor, it is not always possible to measure the rotor angle δ, using a sensor. In this case we assume that δcan be measured. Using Equation (8.56) and Equation (8.62) ,E′ qcan also be written as E′ q=e14IF+L2Id (8.64) where IFandIdare both state variables of the truth model which can be measured using senors. Thus the first control input for the truth model which is the field voltage, VF, as given in Equation (8.55) can be implemented on the truth model by using either Equation (8.63) or Equation (8.64) to eliminate E′ qin the expression for VF. W e prefer Equation (8.63) instead of Equation (8.64) since the rotor angle δ, is a state variable in both the reduced order and the truth model, whereas the direct axis current Id, is not a state 110 variable in the reduced order model. F rom Equation (8.37) the second control input for the truth model which is the turbine valve control uT, is given by uT=γ−1 2(x)(v2−σ2(x)) uT=α2(x) +β2(x)v2(8.65) where α2(x) =−γ−1 2(x)σ2(x),β2(x) =γ−1 2(x)and σ2(x) =p51ω+p52Tm+p53Gv i.eσ2(x) =p51x2+p52x4+p53x5 γ2(x) =r51(8.66) As seen earlier, in the section for nonlinear feedback linearizing controller design for the reduced order model, application of the state feedback control, u1=EFD=α1(x) +β1(x)v1,u2=uT=α2(x) +β2(x)v2, and the state transformation z=T(x)to the reduced order nonlinear model of the system, results in a linear model of the system in the new coordinates. However, this is true only for the reduced order model, but not true for the truth model. The resulting closed loop system in the case of the truth model is not linear at all. The linear controllers v1andv2were designed by using the LQR technique for the reduced order nonlinear model. The gains of this linear LQR controller are tuned once again so that the nonlinear controller works eﬀiciently on the truth model. Thus, by using the feedback law, v=−Kz (8.67) the weighting matrices Q=2 66664250 0 0 0 0 0 250 0 0 0 0 0 250 0 0 0 0 0 250 0 0 0 0 0 2503 77775(8.68) and R=30000 0 0 30000 (8.69) and the state space matrices (A, B )as given in Equation (8.40) and Equation (8.41) the control gain K is found to be K=0.0913 0 .4201 0 .9212 0 0 0 0 0 0 .0913 0 .4369 (8.70) Thus, we have v1=−K11(z1−z1d)−K12(z2−z2d)−K13(z3−z3d) =−K11(δ−δd)−K12(˙δ−˙δd)−K13(¨δ−¨δd)(8.71) and v2=−K24(z4−z4d)−K25(z5−z5d) =−K24(Tm−Tmd)−K25(˙Tm−˙Tmd)(8.72) where z1d,z2d,z3d,z4d,z5dare the desired values of the new state variables. Figure 8.7 to Figure 8.11 show simulation results for the nonlinear feedback linearizing controller applied to the truth model. F rom Figure 8.7 ,Figure 8.8 , and Figure 8.9 , we can see that Vt,ω, and δoscillate about their respective steady state values, where the oscillations slowly decay with time. Figure 8.10 and Figure 8.11 show the plots for the two control inputs, VFanduT, respectively . The generator excitation voltage VFoscillates about its steady state value of 0.00121 p.u. and the turbine valve control settles to its steady state value of 1.0512 p.u. 111 010020030040050060070080090010001.1661.1671.1681.1691.171.1711.1721.1731.1741.1751.176Generator terminal voltage vt (p.u.) time (sec)Figure 8.7: Plot of the generator terminal voltage Vtvs time for the nonlinear feedback linearizing controller applied to the truth model 010020030040050060070080090010000.9960.9970.9980.99911.0011.0021.0031.004omega (radians/sec) time (sec) Figure 8.8: Plot of the angular velocity ωvs time for the nonlinear feedback linearizing controller applied to the truth model 010020030040050060070080090010000.850.90.9511.051.11.15delta (radians) time (sec) Figure 8.9: Plot of the rotor angle δvs time for the nonlinear feedback linearizing controller applied to the truth model 112 010020030040050060070080090010000.80.911.11.21.31.41.51.6x 10−3Excitation field voltage vF (p.u.) time (sec)Figure 8.10: Plot of the generator excitation voltage VF vs time for the nonlinear feedback linearizing controller applied to the truth model 010203040506070809010000.20.40.60.811.21.4Turbine valve control uT(p.u.) time (sec) Figure 8.11: Plot of the turbine valve control uTvs time for the nonlinear feedback linearizing controller applied to the truth model 113 9 Simulation results for the Controllers at different Operating Points In this section we test the L TR-based LQG, nonlinear feedback linearizing controller, and the LQR-based full-state feedback controllers, which were tested on the truth model in the previous section, at new operating points. The operating points at which the controllers are tested are given in T able 9.1 . Operating point I, is the original desired operating point about which the reduced order model was linearized, and all the controllers were designed and tested in the previous sections. In this section we will test the controllers that we tested on the truth model, in the previous section at Operating Point I, at new operating points II and III respectively , without making any change to the controller gains, i.e. the controller gains that were computed for Operating Point I, are unchanged. Only parameters in the controllers that change with a change in the operating condition are the desired reference values of the state variables and the outputs. In all three operating conditions the desired steady state operating value of the frequency ωis 1 p.u., which is evident from the differential equation, ˙δ=ω−1. Also the infinite bus voltage V∞ is 1 p.u. at all three operating conditions. At Operating Condition I, machine loading or the real power P generated by the synchronous generator is 1 p.u. at 0.85 lagging power factor conditions. At Operating Point II, the real power Pgenerated by the synchronous generator is 0.6368 p.u. at 0.9892 lagging power factor conditions. At Operating Point III, the real power P generated by the synchronous generator is 1.3466 p.u. at 0.652 lagging power factor conditions. Thus, from Operating Point II we can see that as the power factor is increased from 0.85 to 0.9892 the real power generated by the synchronous generator decreases from 1 p.u. to 0.6368 p.u. Similarly , from Operating Condition III we can see that as the power factor is reduced to 0.652 the real power generated by the synchronous generator or the machine loading increases to 1.3466 p.u. Also the stator current Iaof the synchronous generator which is equal to 1.0037 p.u. at operating condition I, is reduced to 0.6323 p.u., when the machine loading is reduced at Operating Point II, and the stator current increases to 1.4764 p.u., when the load on the synchronous generator is increased at Operating Condition III. Thus, by varying the machine loading i.e. by increasing or decreasing the load on the generator, the operating conditions are varied. T able 9.1: Operating Points of the SMIB Operating Point I Operating Point II Operating Point III V ariables (in p.u.) Id0 -0.9185 -0.4818 -1.4281 IF0 1.6315 1.0228 2.37786 ID0 −4.6204×10−60 0 Iq0 0.4047 0.4094 0.37472 IQ0 5.9539×10−50 0 ω0 1 1 1 δ0 1 1.0325 0.88676 Tm0 1.0012 0.6373 1.34899 GV0 1.0012 0.6373 1.34899 Vq0 0.9670 0.7659 1.2575 Vd0 -0.6628 -0.6710 -0.6130 Vt0 1.172 1.0182 1.3990 Stator current Ia0 1.0037 0.6323 1.4764 V∞ 1.00 1.00 1.00 α 3.5598◦3.5598◦3.5598◦ E′ q0 1.1925 0.8844 1.6078 τ′ d05.90 5.90 5.90 Real power ( P=VtIacosϕ) 1.00 0.6368 1.3466 power factor ( PF= cos ϕ) 0.85 0.9892 0.652 114 9.1 Simulation results for the Controllers at Operating Point II Figure 9.1 ,Figure 9.2 , and Figure 9.3 show the simulation results for the L TR-based LQG controller applied to the truth model at Operating Point II. F rom these results we can see that the generator terminal voltage Vtsettles to a steady state value of 0.9923 p.u. which is different from the desired steady state value of 1.0182 p.u., with a steady state error of 0.0259 p.u. The angular velocity ω oscillates about the desired steady state value of 1 p.u., where the oscillations decay with time. Also the rotor angle δoscillates about a new steady state value of 1.09 p.u., which deviates from the desired steady state value of 1.0325 p.u., with a steady state error of 0.0575 p.u. Thus, we observe a small steady state error, when the L TR-based LQG controller is applied to the truth model at Operating Point II. Figure 9.4 ,Figure 9.5 , and Figure 9.6 show the simulation results for the nonlinear feedback linearizing controller applied to the truth model at Operating Point II. The generator terminal voltage Vtoscillates about the desired steady state value of 1.0182 p.u., the angular velocity ωoscillates about the desired steady state value of 1 p.u., and the rotor angle δoscillates about the desired steady state value of 1.0325 p.u. The magnitude of these oscillations are more than that seen for the L TR-based LQG controller. The transient response of the nonlinear feedback linearizing controller is not as good as the L TR-based LQG controller, but the steady state error seen in the linear controller is significantly reduced in the nonlinear controller at Operating Point II. Figure 9.7 ,Figure 9.8 , and Figure 9.9 show the simulation results for the LQR-based full-state feedback controller applied to the truth model at Operating Point II. The generator terminal voltage Vtsettles to the desired steady state value of 1.0182 p.u. The angular velocity ω oscillates about the desired steady state value of 1 p.u., where the oscillations decay with time. Also the rotor angle δ, oscillates about the desired steady state value of 1.0325 p.u. The steady state response of the LQR-based full-state feedback controller is much better than the L TR-based LQG controller at Operating Point II. Also, the transient response of the LQR-based full-state feedback controller is better than the transient response of the nonlinear feedback linearizing controller at operating condition II. 115 01020304050607080901000.650.70.750.80.850.90.9511.05Generator terminal voltage vt (p.u.) time (sec)Figure 9.1: Plot of the generator terminal voltage Vtvs time for the L TR-based LQG controller applied to the truth model (Operating Point II) 010020030040050060070080090010000.99850.9990.999511.00051.0011.0015omega (radians/sec) time (sec) Figure 9.2: Plot of the angular velocity ωvs time for the L TR-based LQG controller applied to the truth model (Operating Point II) 010020030040050060070080090010001.021.041.061.081.11.121.141.16delta (radians) time (sec) Figure 9.3: Plot of the rotor angle δvs time for the L TR-based LQG controller applied to the truth model (Operating Point II) 116 010020030040050060070080090010001.011.0121.0141.0161.0181.021.0221.024Generator terminal voltage vt (p.u.) time (sec)Figure 9.4: Plot of the generator terminal voltage Vtvs time for the nonlinear feedback linearizing controller applied to the truth model (Operating Point II) 010020030040050060070080090010000.9970.9980.99911.0011.0021.003omega (radians/sec) time (sec) Figure 9.5: Plot of the angular velocity ωvs time for the nonlinear feedback linearizing controller applied to the truth model (Operating Point II) 010020030040050060070080090010000.940.960.9811.021.041.061.081.11.12delta (radians) time (sec) Figure 9.6: Plot of the rotor angle δvs time for the nonlinear feedback linearizing controller applied to the truth model (Operating Point II) 117 0 200 400 600 800 100011.051.11.15Generator terminal voltage vt (p.u.) time (sec)Figure 9.7: Plot of the generator terminal voltage Vtvs time for the LQR-based full-state feedback controller applied to the truth model (Operating Point II) 0 200 400 600 800 10000.9980.99850.9990.999511.00051.0011.00151.002omega (radians/sec) time (sec) Figure 9.8: Plot of the angular velocity ωvs time for the LQR-based full-state feedback controller applied to the truth model (Operating Point II) 0 200 400 600 800 10000.9911.011.021.031.041.051.06delta (radians) time (sec) Figure 9.9: Plot of the rotor angle δvs time for the LQR-based full-state feedback controller applied to the truth model (Operating Point II) 118 9.2 Simulation results for the Controllers at Operating Point III Figure 9.10 ,Figure 9.11 , and Figure 9.12 show the simulation results for the L TR-based LQG controller applied to the truth model at Operating Point III. F rom these results we can see that the generator terminal voltage Vtsettles to a steady state value of 1.403 p.u. which is different from the desired steady state value of 1.3990 p.u., with a steady state error of 0.004 p.u. The angular velocity ωoscillates about the desired steady state value of 1 p.u., where the oscillations decay with time. Also the rotor angle δ, oscillates about a new steady state value of 0.896 p.u. which deviates from the desired steady state value of 0.88676 p.u., with a steady state error of 0.00924 p.u. Thus, we observe a small steady state error, when the L TR-based LQG controller is applied to the truth model at Operating Point III. Figure 9.13 ,Figure 9.14 , and Figure 9.15 show the simulation results for the nonlinear feedback linearizing controller applied to the truth model at Operating Point III. The generator terminal voltage Vtoscillates about the desired steady state value of 1.3990 p.u., the angular velocity ωoscillates about the desired steady state value of 1 p.u., and the rotor angle δoscillates about the desired steady state value of 0.88676 p.u. The magnitude of these oscillations are more than that seen for the L TR-based LQG controller. The transient response of the L TR-based LQG controller is better than that of the nonlinear feedback linearizing controller, but the steady state error seen in the linear controller is significantly reduced in the nonlinear controller at operating point III. Figure 9.16 ,Figure 9.17 , and Figure 9.18 show the simulation results for the LQR-based full-state feedback controller applied to the truth model at Operating Point III. The generator terminal voltage Vt settles to a steady state value of 1.3964 p.u. which is different from the desired steady state value of 1.3990 p.u., with a steady state error of 0.0026 p.u. The angular velocity ωoscillates about the desired steady state value of 1 p.u., where the oscillations decay with time. Also the rotor angle δoscillates about a steady state value of 0.885 p.u. which deviates from the desired steady state value of 0.88676 p.u., with a steady state error of 0.00176 p.u. A negligible steady state error is observed, when the LQR-based full-state feedback controller is applied to the truth model at Operating Point III. The steady state error is slightly less than that of the L TR-based LQG controller. Also the transient response of the LQR-based full-state feedback controller is better than that of the nonlinear feedback linearizing controller at operating condition III. F rom these simulation results we can conclude that for small variations in the operating conditions, i.e. when the rotor angle δ, is varied by a small amount from δ0= 1 at operating point I, to δ0= 1.0325 at operating point II, and δ0= 0.88676 at operating point III, the transient response of the two linear controllers is better than that of the nonlinear controller, but the steady state error is the least for the nonlinear controller, followed by the LQR-based full-state feedback controller which also has a negligible steady state error compared to the L TR-based LQG controller. The controllers were also tested for large variations in the operating conditions, especially when the rotor angle is varied by a large amount from δ0= 1 at operating point I. It is observed that as the new operating point is moved further away from operating point I, the L TR-based LQG controller applied to the truth model goes unstable, whereas the nonlinear feedback linearizing controller and the LQR-based full-state feedback controller applied to the truth model remain stable. The LQR-based full-state feedback controller shows a small steady state error, whereas the performance of the nonlinear feedback linearizing controller is independent of the operating condition. 119 01020304050607080901001.11.151.21.251.31.351.41.451.5Generator terminal voltage vt (p.u.) time (sec)Figure 9.10: Plot of the generator terminal voltage Vtvs time for the L TR-based LQG controller applied to the truth model (Operating Point III) 010020030040050060070080090010000.99960.99970.99980.999911.00011.00021.00031.00041.0005omega (radians/sec) time (sec) Figure 9.11: Plot of the angular velocity ωvs time for the L TR-based LQG controller applied to the truth model (Operating Point III) 010020030040050060070080090010000.880.8850.890.8950.90.9050.91delta (radians) time (sec) Figure 9.12: Plot of the rotor angle δvs time for the L TR-based LQG controller applied to the truth model (Operating Point III) 120 010020030040050060070080090010001.3951.3961.3971.3981.3991.41.401Generator terminal voltage vt (p.u.) time (sec)Figure 9.13: Plot of the generator terminal voltage Vtvs time for the nonlinear feedback linearizing controller applied to the truth model (Operating Point III) 010020030040050060070080090010000.9950.9960.9970.9980.99911.0011.0021.0031.0041.005omega (radians/sec) time (sec) Figure 9.14: Plot of the angular velocity ωvs time for the nonlinear feedback linearizing controller applied to the truth model (Operating Point III) 010020030040050060070080090010000.750.80.850.90.9511.05delta (radians) time (sec) Figure 9.15: Plot of the rotor angle δvs time for the nonlinear feedback linearizing controller applied to the truth model (Operating Point III) 121 0 200 400 600 800 10001.151.21.251.31.351.41.45Generator terminal voltage vt (p.u.) time (sec)Figure 9.16: Plot of the generator terminal voltage Vtvs time for the LQR-based full-state feedback controller applied to the truth model (Operating Point III) 0 200 400 600 800 10000.9940.9960.99811.0021.0041.006omega (radians/sec) time (sec) Figure 9.17: Plot of the angular velocity ωvs time for the LQR-based full-state feedback controller applied to the truth model (Operating Point III) 0 200 400 600 800 10000.820.840.860.880.90.920.940.960.981delta (radians) time (sec) Figure 9.18: Plot of the rotor angle δvs time for the LQR-based full-state feedback controller applied to the truth model (Operating Point III) 122