Modeling and Control of a Marine Diesel Engine driving a Synchronous machine and a Propeller

Transcription

1 Moeling an Control of a Marine Diesel Engine riving a Synchronous machine an a Propeller Mutaz Tuffaha an Jan Tommy Gravahl Abstract In some esigns of power systems for marine vessels, large-size or meium-size Diesel engine(s) is(are) use to rive one synchronous machine to generate electricity, an the main propeller, simultaneously, through a gear box Such systems are subject to isturbances that may affect performance an fuel consumption The most important isturbances occur ue to the propeller torque, an loa eman on the electric network In this work, a simplifie state-space moel is suggeste for such systems base on well known moels of each component The moel consiers the ynamics of the shaft, Diesel engine, an synchronous machine with the propeller in their simplest moels The output voltage an torque coefficient were moele as uncertain parameters Then, exploiting feeback linearization, two controllers were suggeste for the propose moel to regulate the rotational spee of the shaft Firstly, by pole placement The secon is a robust controller by mixe H /H synthesis The results of the simulations of the propose controller are presente an compare INTRODUCTION The power systems on marine vessels take many configurations an esigns accoring to their purposes, an size Companies in the fiel conten for esigning an manufacturing better systems regaring efficiency, reliability, fuel saving, an environmental frienliness The Diesel engine is most commonly use ue to its efficiency, an low cost The propellers are riven either electrically (by a motor of any type), or mechanically (eg a Diesel engine), 3 In both This work was sponsore by Norwegian Research Council an SINTEF Fisheries an Aquaculture through the ImproVEDO-project M Tuffaha is with the Department of Engineering Cybernetics, Faculty of Information Technology, Mathematics an Electrical Engineering, Norwegian University of Science an Technology, NO-749 Tronheim, Norway mutaztuffaha@itkntnuno J T Gravahl is with the Department of Engineering Cybernetics, Faculty of Information Technology, Mathematics an Electrical Engineering, Norwegian University of Science an Technology, NO-749 Tronheim, Norway jantommygravahl@itkntnuno Figure : Schematic of the main network of the power systems on boar subject of research cases, Diesel engines are neee aboar to rive synchronous machines to generate electricity Lately, the control of Diesel Generating Set (Genset) has become one of the most important an inispensable topics in the fiel The authors in 4 note that the air ynamics of the turbo-charger may lea to unesire oscillations in the spee of the prime mover in power plants riven by Diesel engines Hence, they suggeste an aaptive controller to remey these oscillatory ynamics 4 Spurre by environmental reasons an the urge to save fuel, some systems, that have been evelope for marine vessels recently, comprise propellers that can be riven mechanically an/or electrically Such esigns exploit large or meium-size iesel engine(s) that can rive both the main propeller, an synchronous machine as shown in Fig The synchronous machine in such esigns works either as a motor or as a generator When the machine is riven by the electric power from the bus as a motor, it is escribe as Power-Take-In (PTI), whereas when it is riven by the mechanical power prouce by the iesel engine to generate electric power, it is escribe as Power-Take-Out (PTO) Hence, the system operate in several moes, an can be consiere as a Hybri system To the authors best knowlege, systems like the one in Fig have not been treate in the control literature as one system ue to its complexity The complexity

2 of the system arises from the fact that the propeller can be riven electrically, or mechanically Most of the authors in this arena treat the two cases separately For example, the authors in 5, an 6 propose two ifferent approaches to tackle the problem of controlling the Genset, but both of the works consiere the propeller as part of the loa since it is riven electrically Our target, as a future work, is to put a moel for this hybri system in all moes of operation, then, esign a supervisory power generation controller However, as a first step, a simplifie moel is propose for the above system in this work Inasmuch as the system escribe is complicate, the following assumptions were mae: The synchronous machine is working in one moe, PTO The Diesel-engine is meium-size 3 Low-level controllers are use to regulate the propeller torque, an output voltage of the generator Then, well known simplifie moels of each component of the system escribe were combine The frequency converter is not inclue in the moel as it is presume that regulating the shaft spee of the synchronous machine guarantees that the fluctuations in frequency coul be hanle by the low level controllers of the converter The output voltage an torque coefficient were moele as uncertain parameters Then, ue to its ability to hanle isturbances, feeback linearization was exploite to esign two ifferent controllers for the propose moel to regulate the shaft spee A controller by using the regular pole placement technique was propose Then, a mixe H /H controller was esigne for its robustness against uncertainties, base on the strategy propose in 7 Simulations of the propose moel were performe with the two control techniques for comparison purposes The propose moel is evelope in the next section In the thir section, the feeback linearization is applie to get a control law by pole placement Then, the robust controller is presente Section V presents the results of the simulations performe Finally, some conclusions are rawn in the last section MATHEMATICAL MODEL The shafts of the propeller, synchronous machine, an iesel engine are geare to each other, so before escribing the mathematical moel, let us efine the notation use for the gearing effect First, let R Y enote the gear ratio between gear an gear Y, such that: R Y = Ω, () Ω Y where Ω is the rotational spee in ra/sec Note that R Y = R Y Since the shafts are geare to each other, the rotational spees can be relate, accoring to (), as: Ω P = R PM Ω M (a) Ω E = R EM Ω M, (b) where the subscripts M, P, an E enote the machine, propeller, an iesel engine, respectively The torque applie to (obtaine from) any shaft will affect the angular acceleration of the other two shafts Let Q be the torque applie or obtaine Let the sub subscript enote the sie, eg, Q EM enote the engine torque as seen from the machine sie, then: Q EM = R EM Q EE, (3) an for simplicity, let Q EE be enote by Q E Shaft Dynamics The main iesel engine rives both the synchronous machine an the propeller The torque applie by the engine on the shaft will accelerate all the shafts Besies, there will be a counter torque on the propeller shaft, an the synchronous machine Hence, the engine main shaft ynamics can be written as: Q E = R ME J M Ω M + R PE J P Ω P + Q ME +Q PE + Q f = R MEJ M Ω E + R PEJ P Ω E + R ME Q M +R PE Q P + Q f = (R MEJ M + R PEJ P ) Ω E + R ME Q M +R PE Q P + Q f, (4) where J M is the moment of inertia of the machine an its shaft, J P is the moment of inertia of the propeller, an its shaft Q f is the frictional torque Alternatively, the ynamics of the engine shaft can be transferre to the machine sie to get: (J M + R PMJ P ) Ω M = R EM Q E Q M R PM Q P Q f(5) Synchronous Machine Moel The synchronous generator moel, in per unit (pu) notation, an in q-frame, can be state as 8: ψ = i + F i F ψ q = q i q ψ F = F i F F i, (6a) (6b) (6c) an = ω M ψ q + u (7a) = ω M ψ + u q (7b) ψ F = ω ( R F i F + u F ), (7c) where ψ, ψ q are the, an q axis components of the stator flux linkages, respectively u, an u q are the, an q axis components of the terminal voltage i, an i q are the, an q axis components of the stator current i F, ψ F an u F are the fiel circuit current, flux linkage, an voltage, respectively, an q are the, an q axis components of the stator self inuctance F, an R F are the fiel circuit self inuctance, an resistance, respectively F is the mutual inuctance between the fiel circuit an stator winings

3 Moreover, ω M is the rotational spee of the rotor in pu, which is given by: ω M = Ω M, (8) Ω Mbase where Ω Mbase = ω (p/), with p the number of poles, an the nominal frequency ω = π f, as given in Table Note that from (8), the pu values of the rotational spees are equal since: Ω M ω M = = R MPΩ P = ω P Ω Mbase R MP Ω Pbase R ME Ω E = = ω E, R ME Ω Ebase where the base quantities are given in Table Hence, from here on ω M = ω P = ω E = ω It is worth noting, that amper winings, an stator resistance are neglecte in (6), an (7), an so is stator flux ynamics (ie, ψ = ψ q = ) After some algebraic simplifications, The equations (6), an (7) can be rearrange to give: ( ψ F = ω τ F ψ F + F ) u q + u F, (9) ω where τ F = F R F is a time constant an = F F The electromagnetic torque obtaine from the machine in pu is given by (see 8): Q M = ψ i q ψ q i, () which can be rewritten by using (6), an (7) as: ( Q M = ) uq u q ω + F u ψ F F ω () From here on Q enotes the torque in pu, instea of the previous notation of Q Let δ = ( p )Ω Mt ω t + δ be the angle between the rotor an a synchronously rotating reference as epicte in Fig, then: δ = ω (ω M ) () Let U be the terminal voltage of the stator of the synchronous machine, then the terminal voltages in q-frame woul be, u = U sinδ, an u q = U cosδ Thus,(9), an () can be rewritten as: ψ F = ω ( U cosδ τ F ψ F + F ω an ( Q M = ) U sin(δ) q ω 3 Propeller Moel + F F ) + ω u F, (3) Uψ F sinδ ω (4) The propeller torque epens on the size an type of the propeller, an the operational moe of the vessel The moels can be complicate if more than one operational moe is consiere The propeller torque of the fixe pitch propeller (FPP) is moele by, 3, an Figure : q-frame in the synchronous machine an the angles involve 9 as: Q P = ρd 5 K Q n P (5) where n P = Ω P π is the rotational spee in rev/s, ρ is the ensity of water in Kg/m 3, D is the propeller iameter in m, an K Q is the imensionless propeller torque coefficient The coefficient K Q epens on the avance ratio (J), which in turn epens on the avance spee (u a ) Many moels have been suggeste in literature to moel the torque coefficient K Q an escribe its epenence on J, an u a, see eg, an 3 In this work, we suggest consiering K Q as an uncertain parameter To justify, propeller torque is usually controlle by multilevel controllers that are beyon the scope of this work Our target is to investigate the ability to regulate the rotational spee of the escribe system regarless of the propeller torque variation Thus, accepting the aforementione argument, the moel above can be escribe, in pu, as: Q P = ρd5 K Q n P Q Pbase ρd 5 = 4π K Q Ω Q P (6) Pbase Inserting (), an () in the above, one can get: Q P = C QP K Q ω, (7) where, C QP = ρd5 ΩP base is a constant, an the base quantities are as given in Table 4π Q Pbase 4 Diesel Engine Moel Many mathematical moels have been suggeste for the iesel engines The moels vary in their complexity accoring to the size an properties of the engine, the purpose of the moel, an the kin of the controllers use Because they were intereste in the air ynamics of the turbo-charger, the authors in 4 suggeste

4 a sixth orer moel for the Diesel engine, turbo-charger an generator shaft, in which they use a first orer moel for the fuel actuator, an a time elay to represent the engine itself Similarly, the following moel was use in : Q E = ( Q E + K u u DE (t τ D )), (8) τ E where, τ E is a time constant, u DE is the input signal to control the fuel actuator, an τ D is the time elay In the interest of keeping the moel simple, an inspire by the work in 5, we neglect the combustion elay an assume that the ynamics of the fuel actuator constitutes the essential ynamics of the engine Thus, iviing by Q Ebase given in Table, the moel above can be escribe in pu system as 5: Q E = ( Q E + K τ uu DE ), (9) E where K u = K u Q Ebase 5 Frictional Torque The main contributions to the frictional torque on the system are experience in the sie of the synchronous machine an the sie of the propeller Hence, the frictional torque can be moele in its simplest form as a combination of two components Linear viscous torque on the propeller shaft (see 9 for further etails on frictional torque), an amping torque in the synchronous machine to compensate for neglecting the amper winings an their effect on the electrical torque prouce by the machine as suggeste in 8 Thus, the frictional torque in pu can be written as: Q f = K f ω + K D (ω ), () where K f is the propeller linear friction coefficient, an K D is the amping torque coefficient of the synchronous machine, taking in consieration the conversion to pu system 6 Final Moel Diviing the moel in (5) by Q Mbase, an by using the base quantities given in Table, the moels in (3), (7), (9), an () can be groupe with the moel in (5) in one complete state-space moel as follows: δ = ω (ω ) ( ) ω = QE Q M Q P Q f H T ψ F = ω τ F ψ F + ω F U cosδ τ F + ω u F ω Q E = ( QE + K ) τ uu DE, () E where H T is the pu inertia constant given by: H T = (J M + R PM J P)ΩM base = (J M + R PM J P)Ω Mbase S base Q Mbase () The moel as given above has two input controls (u f, an u DE ) Usually, the output voltage U, an the propeller torque coefficient K Q are consiere as outputs an controllers are esigne to regulate them In this work, U an K Q are moele as uncertain parameters groupe in the vector θ = θ,θ T = U,K Q T The output of interest is the ifference between the rotational spee an the nominal spee (y = ω ), an the main objective of the propose controller is to regulate the shaft spee, uner the isturbance impose by the output voltage an the propeller torque coefficient Actually, regulating the shaft spee of the system is important in orer to regulate the frequency of the electrical output of the machine, since the output frequency epens on the rotational spee Further, it was assume that low level controllers can take care of small eviations in output frequency Since we are moeling the output voltage as uncertain parameter, we consier the flux linkage as another output (y = ψ F ), because it is relate to the output voltage Further, this assumption makes the system a square x MIMO system Thus, the rotational spee ω, an flux linkage ψ F were assume measurable Let the vector x = x,x,x 3,x 4 T = δ,ω,ψ F,Q E T be the state vector Let, also, u = u,u T = u F,u DE T be the input vector Then, the moel in () can be rewritten as: ẋ = f(x, θ) + g (x, θ)u + g (x, θ)u, y = h (x) y = h (x) (3) where, f (x, θ) ( ω (x ) ) f (x, θ) H x4 T Q M Q P Q f f(x, θ) = f 3 (x, θ) f 4 (x, θ) = ω τ F x 3 + ω τ F F θ cosx x, τ E x 4 where Q M, Q P, an Q f are as given in (4), (7), an () respectively Moreover, g (x, θ) = ω an, g (x, θ) = h (x) = x, h (x) = x 3 K u τ E 3 FEEDBACK LINEARIZATION, The uncertain parameters, in general, inuce isturbances on the ynamical systems The author in iscusse in etail the so-calle isturbance ecoupling problem, in which he iscusse the conitions require to make the feeback control law rener an output y inepenent of some isturbance Let us assume, for this section, that the uncertain parameter θ is fixe The moel in (3) has a relative egree two with respect to output y, ie, r = It also has relative egree one with respect to output y, ie, r = Following the proceure in, efine the characteristic matrix β(x, θ)

5 as: Lg L f h L g L f h f ω K u f β = L g L f h L g L f h = x 3 τ E x 4 ω (4) Then, accoring to, since the matrix β is nonsingular, a iffeomorphism (η,z) = T (x, θ) exists (see Appenix A), an is given by: z η = z () z () z () = h (x) L f h h (x) φ(x, θ) (5) η This transformation is use to linearize the system in () The ynamics of the new states with this transformation can be state as: ż η = ż () ż () ż () η = z L f h + β (x, θ)u L f h + β (x, θ)u φ(x, θ) (6) where β (x, θ) is the first row of β(x, θ), an so on Moreover, the outputs are: y = z () y = z () (7) By this transformation, the new moel is split into two parts; external part (z), an internal part (η) (see, an ) The iea now is to choose a function φ(x, θ) such that the internal ynamics are inepenent of the control input, because this makes the stability of the system easier to be obtaine as will be explaine later, otherwise more complicate techniques woul be require So, letting the internal ynamics be: φ(x, θ) = sinx x, (8) makes the internal ynamics η inepenent of the control input Let the erivative of η be expresse as a function of the new variables, ie, η = φ(η,z) = χ(η,z) The zero ynamics is efine as the internal ynamics aroun the zero of the linearize system, that is to say: η = χ(η,z e ), (9) where z e is the equilibrium point of the external ynamics, which can be assume, without loss of generality, zero Then, accoring to, an the origin of the system in (6) is asymptotically stable if the origin of (9) is asymptotically stable (see lemma 3 in an the proof therein) Straight forwar calculations show that the zero ynamics is, actually, asymptotically stable Thus, the concern is irecte now to the external ynamics only, which can be rewritten as: ż () = z () ż r = α(z,η) + β(x, θ)u, (3) where z r = z () given by: r,z () r T = z () L r α(z,η) = f h L r f h,z() T, an α(z,η) is L = f h L f h (3) If the feeback control u is chosen such that: u = β (v α(z,η)), (3) where v = v,v T is an auxiliary stabilizing input control, then the external ynamics can be written as two subsystems like follows: ż () = z () ż () = v, ż () = v, (33) which is linear an controllable Then, one can fin a control law: v = K p z, (34) that stabilizes the system in (33) by pole placement, for example 4 ROBUST CONTROL Although feeback linearization provies a goo tool to eal with isturbances through isturbance ecoupling, the isturbance inuce by nonlinear uncertain parameters can be problematic Firstly, the iffeomorphism T (x, θ) oes not provie exact linearization Further, the equilibrium point varies ue to its epenence on the parameters (see x e in Appenix B) The authors in 7 suggeste an elegant way to tackle these problems by mixe H /H synthesis The authors in 7 prove that if the uncertain parameter has a nominal value θ, an the function f(x, θ) can be linearize about θ as: f(x, θ) = f (x, θ ) + f (x, θ), (35) where f (x, θ) is calculate at θ, then the external ynamics in (6) can be rewritten, by the control law in (3) (calculate at the nominal values), as 7: ż () = z () + L f h ż () = L f L f h + v ż () = L f h + v (36) Now, let z = z (),z(),z() T, v = v,v, an y = y,y T Then, by using Taylor expansion for the nonlinear isturbances in (36) an by collecting all the nonlinear terms resulting from the expansion in the matrix Ã(θ,z), the authors in 7 prove that if the nonlinear perturbations are boune such that: Ãi W i i, i, i {,,3}, (37) where Ãi is the ith row of the matrix, i L, ), an W i are linear weights Then, the moel in (36) can be rewritten as 7: ż = A(θ)z + W + B(θ)v, (38) where W = iag(w,w,w 3 ), an =,, 3 T Mimicking the proceure above, one can obtain

6 f (x, θ) to be: f (x, θ) = an A = B(θ) = H T Q M θ θ =θ θ + Q P θ θ =θ θ ω F cosx τ F x θ D θ + C QP θ D θ D θ + C QP θ D θ D 3 θ, C =,, (39), (4) where θ = θ θ an so on, the parameters D, D, D 3, D, D, an C QP are as given in Appenix B Finally, following the proceure to esign a mixe H /H controller (see eg, 7), efine: ż = A(θ)z + W + B(θ)v Z = E z + F + F v Z = E z + F v (4) Now a feeback law v = K m z, (4) that stabilizes the system in (4), can be obtaine by using the Linear Matrix Inequality (LMI) toolbox in MATLAB This control law aims to minimize the combine objective of, an of the transfer functions of the isturbance to the efine output signals Z, an Z Straight-forwar calculations from (3) show that the control law u can be written as: ω v L u = h H T τ E τ f E F θ sinx (43) v L f h K u K u F 5 SIMULATIONS AND RESULTS The parameters use in the simulations are liste in Tables an By using the values liste in the tables, one can obtain C QP = 64, an H T = 735s The system in () was simulate in MATLAB/SIMULINK Both control gains were foun, K p in (34) was foun by pole placement, an K m in (4) was foun by the mixe H /H synthesis technique in (4) by using the LMI toolbox Actually, high gains are require to stabilize the slow ynamics To eluciate, the control law u in (43) cancels the nonlinear terms an stabilizes the states, but the inertia constant H t is large compare to the iesel engine time constant τ E, so high gains are require Hence, the weighting matrices W,E, E, F, F, an F in (4) were chosen to be: W =,E = E = 5 5, 5 85 F =,F = F =, 5 x Table : The Parameters Electrical Mechanical Quantity Value Quantity Value (pu) ρ (Kg/m 3 ) (pu) 5 D (m) 3 q (pu) τ E (s) F (pu) 8 K u (pu) F (pu) K f (pu) τ F (s) 3 K D (pu) 55 ω (ra/s) 377 J M (Kgm ) 7 p 6 J P (Kgm ) 3 Table : The Base Quantities Quantity Expression Value Ω Mbase (ra/s) ω /( p ) 566 Ω Ebase (ra/s) 8 Ω Pbase (ra/s) U base (V) 36 I base (A) 5555 S base (kva) 3 U base I base 3 Q Mbase (knm) S base Ω Mbase 39 Q Ebase (knm) S base Ω Ebase 375 Q Pbase (knm) S base Ω Pbase 5 an the gain K m was obtaine to be: K m = , For the gain K p, if the gain is not high enough the control law can not stabilize the system On the other han, raising the control gain coul lea to excessive control action The best results were foun when the poles were place at -8, -6, an -3 for the three states of the linearize moel in (33) The gain K p was obtaine to be: K p = 3 The control law was applie to the moel with arbitrary initial values twice, one with the gain K p, an one with the gain K m for comparison The nominal values of the uncertain parameters were assume θ =, an θ = 8 The simulations were run in two cases assume for the uncertainties of the parameters θ, an θ as shown in Table3 The trajectories of the output signals ψ F, an ω are shown in Fig 3 an Fig 4 for cases one an two, respectively We can see from Fig 3 that both output trajectories reach the equilibrium by both control gains, because the uncertainties assume in this case are not large On the other han, Fig 4 shows that control gain obtaine from H /H synthesis (K m ) rives the output to the steay state more quickly than

7 Table 3: The Uncertain parameters θ θ Case I % 5% Case II % 5% (a) (a) (b) (b) Figure 3: Comparison of the output trajectories: (a) The fiel circuit flux (ψ F ), an (b) Rotational spee (ω), obtaine by the two strategies: H /H synthesis (soli), an pole placement (ashe) when the the uncertainties of the two parameters θ = %, an θ = 5% the other gain K p, for the same chosen poles One can avocate that changing the gain K p may give better results The problem with the former argument is that the gain K p shoul be varie for every range of uncertainties Whereas, the gain K m coul stabilize the system for ifferent uncertainty ranges Because the propeller torque coefficient K Q, which was moele as uncertain parameter, varies significantly in reality ue to several factors such as weather conition, ship spee an sea conition, H /H robust controller is recommene for this moel Figure 4: Comparison of the output trajectories: (a) The fiel circuit flux (ψ F ), an (b) Rotational spee (ω), obtaine by the two strategies: H /H synthesis (soli), an pole placement (ashe) when the the uncertainties of the two parameters θ = %, an θ = 5% 6 FUTURE WORK In this work, we concentrate on the operation of the escribe system in one moe, PTO It is of interest to apply the propose controllers on the system in the other moe, ie, PTI Then, a hybri control strategy can be applie to smoothen the transition between the moes, an/or ecie the optimal moe of operation regaring fuel consumption Also, work on making less restrictive assumptions in the moeling is ongoing 7 CONCLUSIONS In this paper, a moel was propose for Gensets, that comprise a meium-size iesel engine riving a synchronous machine to generate electric power, an a propeller, simultaneously The output voltage of the synchronous machine, an the propeller torque coefficient were moele as uncertain parameters A feeback linearization was performe Then, two nonlinear

8 control laws were propose to regulate the shaft rotational spee, by pole placement, an by mixe H /H synthesis The simulations showe that the propose controllers coul stabilize the shaft spee In aition, we showe that the mixe H /H controller coul stabilize the shaft spee more quickly than the pole placement controller, regarless of how much the output voltage an/or the propeller torque coefficient may vary References L Guzzella, an A Amstutz, Control of Diesel Engines, IEEE Control Systems, vol 8, issue 5, pp 53-7, 998 M Blanke, K P Linegaar, an T I Fossen, Dynamic Moel for Thrust Generation of Marine Propellers, IFAC Conference of Maneuvering an Control of Marine craft (MCMC), pp , 3 A J Sørensen, an Ø Smogeli, Torque an Power Control of Electrically riven Marine Propellers, Control Engineering Practice, vol 7, no 9, pp 5364, 9 4 S Roy, O P Malik, an G S Hope, Aaptive Control Of Spee An Equivalence Ratio Dynamics of a Diesel Driven Power-Plant, IEEE Transactions on Energy Conversion, vol 8, no, pp 3-9, J F Hansen, A K Ånanes, an T I Fossen, Mathematical Moeling of Diesel-Electric Propulsion Systems for Marine Vessels, Mathematical an Computer Moeling of Dynamical Systems, vol 7, no, pp -33, 6 M L Huang, Robust Control Research of Chaos Phenomenon for Diesel-Generator Set on Parallel Connection, Applications of Nonlinear Control, Intec, 7 S N Kolavennu, S Palanki, an J C Cockburn, Robust controller esign for multivariable nonlinear systems via multi-moel H /H synthesis, Chemical Engineering Science, vol 56, pp , 8 P Kunur, Power System Stability An Control, McGraw-Hill, L Pivano, T A Johansen, Ø N Smogeli, an T I Fossen, Nonlinear Thrust Controller for Marine Propellers in Four-Quarant Operations, Proceeings of the American Control Conference, pp 9-95, 7 M Torres, an L A C Lopes, Inverter- Base Diesel Generator Emulator for the Stuy of Frequency Variations in a Laboratory-Scale Autonomous Power System, Energy an Power Engineering (EPE), vol 5, pp 74-83, 3 A Isiori, Nonlinear Control Systems, Springer- Verlag, Berlin, n eition, 989 H K Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 3r eition, A Transformation The transformation T (z,η) was foun to be: arcsin(η(z () + )) z () + z () T (z,η) = K f + ( q )θ η B Parameters + η + +) C QP θ (z () + ) + (K f + K D )z () (z () H T z () + F F θ z () η The parameters D, D, D 3, D, D, an C QP are as follows: θ η D = ( H T η q ) D = Fη H T F ω F D 3 = τ F η D = (K f + K D )D H T D = (K f + K D )D H T C QP = (K f + K D )C QP H T η = sin(x e ), x e where x e is the equilibrium point given by: ( arcsin x e = (K f +C QP θ ) θ ( q )+( F ) F F θ cos(x e ) ) ± nπ