Slip Control of a Squirrel Cage Induction Generator Driven by an Electromagnetic Frequency Regulator to Achieve the Maximum Power Point Tracking

energies

Article Slip Control of a Squirrel Cage Induction Generator Driven by an Electromagnetic Frequency Regulator to Achieve the Maximum Power Point Tracking

Thales Ramos 1,*, Manoel F. Medeiros Júnior 2, Ricardo Pinheiro 2 and Arthur Medeiros 1 1 Federal Institute of Education, Science and Technology of Rio Grande do Norte, Natal 59015-000, Brazil; [email protected] 2 Federal University of Rio Grande do Norte, Natal 59078-970, Brazil; ﬁ[email protected] (M.F.M.J.); [email protected] (R.P.) * Correspondence: [email protected]

Received: 26 February 2019; Accepted: 17 May 2019; Published: 1 June 2019

Abstract: A new topology was recently developed to drive generators, aiming to avoid power electronic devices directly connected to the grid, and making possible the hybridization of the wind power with other sources. The system is composed by an induction machine with rotor in squirrel cage, and a rotating armature endowed with a three-phase winding that may be fed by a secondary source. The previous purpose was to convert a variable velocity imposed by the wind turbine to the armature in a constant velocity to be developed by the cage rotor, driving a shaft of synchronous generator. This article proposes the use of an induction generator instead of a synchronous one in order to explore the maximum available wind energy (MPPT). The simulation results show that the proposed topology is viable and supports both variations in wind speed and disturbances in power grid.

Keywords: Electromagnetic Frequency Regulator (EFR); induction generator; maximum power point tracking (MPPT); hybrid generation; low voltage ride-through; wind turbine

1. Introduction Human necessity for electric power grows continually. This way, the use of renewable resources which requires minimal impacts to environment is strongly desirable. Among renewables mentioned by [1] study wind and hydraulic resources are predominant. Data obtained from [2] shows that the growth of wind energy in the last years is worldwide the most significant. Nowadays, there are several topologies of wind generators, each one presenting advantages and disadvantages. In [3], most used topologies in wind farms are presented. They are based in synchronous machines or in induction machines, either with wounded rotor or with squirrel cage. Except rare cases, they are coupled to the wind turbine by means of a gearbox and are connected to the grid by means of electronic converters. New topologies, however, have been proposed in recent years, looking for reducing effects relative to power quality or for optimizing generated power, among others. For example, articles [4,5] present a generation system based on two double fed induction machines in cascade. One of them achieves the whole control whereas the other is responsible for power generation. Another concept is a new generator using permanent magnet addressing a large number of poles as proposed in [6], with the main objective of eliminating necessity of gearbox. More topologies based on permanent magnet are presented in [7]. Considering the increasing use of renewables as well the search for more efficient generation topologies, some efforts have also been carried out with the aim of integrating different resources in a hybrid generation scheme. In this sense, the proposal presented in [8] consists in a hybrid

Energies 2019, 12, 2100; doi:10.3390/en12112100 www.mdpi.com/journal/energies Energies 2019, 12, 2100 2 of 19 system composed by a wind turbine, and a micro turbine driven by vapor produced by solar h eaters. The results of a study showing the effects of variation in solar intensity and wind velocity in the performance of a hybrid isolated generation system is presented in [9]. Some other studies present alternatives to hybridization of solar and wind generations [10–12]. Following this trend, a new topology was recently developed to drive generators, aiming to avoid power electronic devices directly connected to the grid, and making possible the hybridization of the wind power source with another one. This system is named Electromagnetic Frequency Regulator (EFR) and consists of an induction machine with rotor in squirrel cage and a rotating armature endowed with a three-phase winding that may be fed by a secondary source [13,14]. Although the term “stator” is proper for induction machines, it will not be used here because this term suggests something stationary (from Latin). The authors of references [13,14] preferred to designate it as an asynchronous rotor because the other rotor would drive a synchronous machine and, therefore, would rotate at a synchronous speed. Since in this work it is not necessary to classify the parts in relation to a synchronous speed, since the adopted generator is an induction machine, the authors preferred to designate the parts of the EFR simply as armature (connected to the gearbox) and rotor (connected to the generator shaft). The main function of the EFR is to convert a variable speed imposed by the wind turbine to the armature in a desirable speed to be developed by the cage rotor. Another generation topology similar to that here proposed is presented in [15]. It also consists of a squirrel cage induction machine in which there is no stationary part, i.e., both parts are rotating. The device presented in reference [15] is named Electromagnetic Coupler. Articles [13–15] still have the similarity of the use of a synchronous generator connected to high speed shaft of the induction machine. There are two main differences between the devices. Whereas the EFR uses the cage connected to the synchronous generator, the Electromagnetic Coupler has the cage connected to the shaft of the gearbox. Furthermore, the EFR’s inverter is DC/AC and needs to be connected to an external DC power source, whereas the structure with Electromagnetic Coupler uses a back-to-back converter that receives power from the grid. The advantage of both topologies is the mechanical decoupling between the wind turbine and the generator. However, the EFR like presented in [13,14] is not as efficient as the Electromagnetic Coupler of [15], for driving a synchronous generator, because the EFR is not associated with control devices necessary to allow a reasonable range of active and reactive power generation. This way, the topology present in this article uses an EFR, but not driving a synchronous generator. This article presents a new application for the EFR: instead of using a generator with constant speed (synchronous), coupled to its shaft, the use of an induction generator will be explored, thus enabling the tracking of the point of maximum power extraction of the wind (MPPT). This may be achieved by imposing to the generator the torque corresponding to the maximum wind power extraction, and so defining the reference velocity to be controlled by the EFR. An analysis in steady state was carried out in order to determine the contributions to the generated power from the wind source and from the other source. Dynamic simulations are introduced to evaluate the robustness of the new topology in the face of events such as wind disturbances like steps, ramp, turbulence and gust as well as the ability to not disconnect after short circuits, according the ride-through curves of some countries.

2. Mathematical Modeling of the System

2.1. Horizontal Axis Wind Turbine The extraction of mechanical power from the winds for a horizontal axis wind turbine is determined by [16–18]: 1 P = ρπR2C (λ, β)V3 , (1) t 2 p w EnergiesEnergies2019 2018, 12,, 11 2100, x FOR PEER REVIEW 3 of3 of 19 19

where 𝜌 is air density, 𝑅 is the radius of the blade, 𝐶 is the power coefficient which depends whereuponρ theis air tip-speed density, ratioR is ( the𝜆) and radius blade of thepitch-angle blade, C (p𝛽is), theand power𝑉 is the coe windfficient speed. which The depends tip-speed upon ratio theis tip-speedcalculated ratio by (λ) and blade pitch-angle (β), and Vw is the wind speed. The tip-speed ratio is calculated by 𝜆ω tR = , (2) λ = , (2) Vw where 𝜔 is the turbine angular speed. The power coefficient (𝐶 ) describes the aerodynamic where ω is the turbine angular speed. The power coefficient (C ) describes the aerodynamic performancet of the turbine whose theoretical maximum value is knownp as the Betz limit and is equal performance of the turbine whose theoretical maximum value is known as the Betz limit and is equal to 0.593. Each wind turbine has its own power coefficient curve. For the purposes of this work the to 0.593. Each wind turbine has its own power coefficient curve. For the purposes of this work the coefficient defined in (3) was adopted [19]: coefficient defined in (3) was adopted [19]: . . 𝐶(𝜆, 𝛽) = 0.73 −0.58𝛽−0.02𝛽 − 13.2! 𝑒 , (3) 18.4 151 2.14 −λ Cp(λ, β) = 0.73 0.58β 0.02β 13.2 e i , (3) where λi − − − where 1 0.003 𝜆 = + ! 1 (4) 𝜆1 − 0.02𝛽0.003𝛽 +1− λi = + (4) λ 0.02β β3 + 1 and 𝛽 is the blade pitch angle in degrees. − and β isFigure the blade 1 shows pitch the angle curves in degrees. of the power coefficient as a function of tip-speed ratio (𝜆) and for

someFigure blade1 shows pitch the angle curves ( 𝛽 of) thevalues. power The coe maximumﬃcient as avalue function of ofthe tip-speed power ratiocoefficient (λ) and ( for𝐶_ some) is bladeapproximately pitch angle (0.441,β) values. corresponding The maximum to 𝜆=𝜆 value of= the 7.206 power, when coeﬃ cientthe blade (Cp_max pitch) is approximately angle is at its 0.441,minimum corresponding (𝛽=0°). to λ = λopt = 7.206, when the blade pitch angle is at its minimum (β = 0◦).

FigureFigure 1. 1.Power Power coe coefficientﬃcient curves curves as as a functiona function of of tip-speed tip-speed ratio ratio and and blade blade pitch pitch angle. angle.

TheThe mechanical mechanical torque torque produced produced by the by wind the turbinewind turbine can be obtained can be fromobtained the following from the equation: following equation: ( ) Pt 1 3 Cp λ, β 2 Tt = =𝑃 ρπ1R 𝐶 (𝜆,Vw 𝛽). (5) ωt 2 λ 𝑇 = = 𝜌𝜋𝑅 𝑉 . (5) 𝜔 2 𝜆 2.2. Induction Generator 2.2.Considering Induction Generator the simplifying assumptions presented in [20], the voltage equations of a squirrel cage induction generator can be expressed as [21]: Considering the simplifying assumptions presented in [20], the voltage equations of a squirrel cage induction generator can be expressed as [21]: d v = r i ωeλ + λ , (6) d1g 1g d1g q1g d1g 𝑣 = 𝑟 𝑖− − 𝜔 𝜆 dt+ 𝜆 , (6) d v = r i + ω λ + λ , (7) q1g𝑣 1=g q𝑟1g 𝑖 +e d𝜔1g𝜆 dt+ q1g𝜆 , (7)

Energies 2019, 12, 2100 4 of 19

! Pg d v = 0 = r gi ωe ωm λq g + λ , (8) d2g 2 d2g − − 2 2 dt d2g ! Pg d vq g = 0 = r giq g + ωe ωm λ + λq g, (9) 2 2 2 − 2 d2g dt 2 where vd1g, vq1g are the terminal voltages in d and q axes at the stator, r1g is the stator resistance, id1g, iq1g are the electrical currents in the d and q axes at the stator, ωe is the synchronous angular frequency, i.e., the grid voltage pulsation, λd1g, λq1g are the flux linkages in the d and q axes at the stator, vd2g, vq2g are the rotor voltages in d and q axes, r2g is the rotor resistance, id2g, iq2g are the rotor currents in the d and q axes, Pg is the number of poles of the induction generator, ωm is the angular speed of rotor shaft and λd2g, λq2g are the flux linkages in the d and q axes of the rotor. The relationships between the electric currents and the flux linkages are expressed by:

λd1g = L11gid1g + L12gid2g, (10)

λq1g = L11giq1g + L12giq2g, (11)

λd2g = L22gid2g + L12gid1g, (12)

λq2g = L22giq2g + L12giq1g, (13) where: L = L L , (14) 11g s1g − m1g 3 L = L , (15) 12g 2 mg

L g = Ls g Lm g, (16) 22 2 − 2 where Ls1g is the self-inductance of any stator winding, Ls2g is the self-inductance of any rotor winding, Lm1g is the mutual inductance between any two windings in the stator, Lm2g is the mutual inductance between any two windings in the rotor and Lmg is the maximum mutual inductance between stator and rotor windings. The electromagnetic torque of the induction generator Teg can be obtained from:

3 Pg Teg = λ i λ i . (17) 2 2 d1g q1g − q1g d1g Using the convention of electric currents adopted, the numerical evaluation of Equation (17) results as a negative value.

2.3. Electromagnetic Frequency Regulator The Electromagnetic Frequency Regulator (EFR) is a system consisting of an induction machine with a squirrel cage rotor and rotating armature, as well as a frequency inverter that adjusts the electric current injected into the armature. The purpose of this system is to receive a variable angular speed (such as that of a wind turbine) and deliver a desired angular speed to the EFR’s rotor [13,14]. The conventional induction machine has as stationary element the stator (armature), housing a set of coils connected to a three-phase power grid. The innovation of the EFR is to allow the armature to rotate integrally with the wind turbine. Thus, in this system there is not a stationary part but two components that rotate with different speeds. In order to have electrical access to the rotating armature it is necessary to insert collector rings that do not normally exist in an induction machine with a rotor in a squirrel cage. Considering that the speed of wind turbines is significantly lower than the rotating field speed required for generating voltage at the grid frequency, several topologies of wind turbines adopt the use of a gearbox. Its effect is to multiply speeds and as well to reduce torque. This component is Energies 2019, 12, 2100 5 of 19

Energies 2018, 11, x FOR PEER REVIEW 5 of 19 also considered to compose the EFR-driven generation system as shown in Figure2. The relationship consideredbetween the to turbine compose and the EFR EFR-driven armature speed generation is shown system in the as following shown in equation: Figure 2. The relationship between the turbine and EFR armature speed is shown in the following equation: ωa Tt gb = 𝜔= 𝑇, (18) 𝑔𝑏ω = t =Ta , (18) 𝜔 𝑇 where ωa is the armature speed and Ta is the armature torque. where 𝜔 is the armature speed and 𝑇 is the armature torque. Figure2 shows a schematic 3D visualization, highlighting some constructive aspects of the Figure 2 shows a schematic 3D visualization, highlighting some constructive aspects of the proposed system. The shaft attached to the gearbox, which supports the collector rings which receive proposed system. The shaft attached to the gearbox, which supports the collector rings which receive the power from the inverter, is also mechanically connected to the EFR armature, although this coupling the power from the inverter, is also mechanically connected to the EFR armature, although this is not visible in the ﬁgure. coupling is not visible in the figure.

Figure 2. SchematicSchematic 3D 3D visualization visualization of of EFR.

Figure 33 presentspresents thethe proposedproposed generationgeneration systemsystem adaptedadapted fromfrom [[13],13], replacing replacing the synchronous synchronous generator by an induction one. Since the objective of the EFR is to keep the rotor speed at the desired reference, it is necessary to control the frequency of the electric current injected in the armature. The armature speed added to the rotating field speed of the armature causes a rotor speed corresponding to the desired slip. This is achieved by the frequency inverter connected to the armature. The frequency inverter needs to be supplied by a direct current source, which could be provided by a solar panel, a hydrogen cell, or even a set of smaller wind turbines, for example. The whole development presented in this article considers that the direction of the mechanical angular speed of the armature is the same as the rotational magnetic field caused by the armature currents. This way, the resulting magnetic field speed is the sum of the mechanical speed of the armature and the speed of the rotating magnetic field caused by the electric frequency of the inverter output currents. As sketched in Figures2 and3, the EFR rotor and the induction generator rotor are mechanically coupled and therefore have the same mechanical speed ωm. Although the rotor speeds of both machines are the same, their slips present opposite signs. The EFR operates with positive slip, i.e., in the motor region once the speed of the rotating field is greater than the speed of the rotor, whereas the induction generator operates with negative slip, in order to generate electrical energy.

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Figure 3. Schematic diagram of EFR. Figure 3. Schematic diagram of EFR. The speed of the rotating field relative to the armature is given by the difference between the Since the objective of the EFR is to keep the rotor speed at the desired reference,PR it isP Rnecessary absolute speed of the rotating field and the speed of the turbine, that is: ωeR + ωa ωa. Thus, to control the frequency of the electric current injected in the armature. The armature2 speed− 2 added to the voltage equations for the armature are identical to those of conventional induction machine: the rotating field speed of the armature causes a rotor speed corresponding to the desired slip. This is achieved by the frequency inverter connected to the armature.d v = r i ωeRλ + λ , (19) The frequency inverter needsd 1toR be supplied1R d1R − by a directq1R currentdt d1R source, which could be provided by a solar panel, a hydrogen cell, or even a set of smaller wind turbines, for example. d The whole development presentedv = rin thisi +articleω λconsiders+ λ that, the direction of the mechanical (20) q1R 1R q1R eR d1R dt q1R angular speed of the armature is the same as the rotational magnetic field caused by the armature currents.where vd 1ThisR, vq 1way,R are the the resulting terminal magnetic voltages infield d and speed q axes is the at thesum EFR’s of the armature, mechanicalid1R speed, iq1R areof the armatureelectrical currentsand the speed in the of d andthe rotating q axes at magnetic the EFR’s fiel armature,d causedω eRbyis the the electric inverter frequency voltage pulsation,of the inverterλd1R, outputλq1R are currents. the flux linkages in the d and q axes at the EFR’s armature. AsRegarding sketched to in rotor Figures voltages, 2 and 3, the the rotation EFR rotor imposed and the to induction the armature generator causes rotor an additional are mechanically rotating voltage superposing the one caused by the armature currents, so that the resulting equations become: coupled and therefore have the same mechanical speed 𝜔 . Although the rotor speeds of both machines are the same, their slips present opposite signs. The EFR operates with positive slip, i.e., in PR PR d v = r Ri ωeR + ωa ωm λq R + λ = 0, (21) the motor region onced2R the speed2 d2 Rof− the rotating2 field− is 2greater than2 thedt speedd2R of the rotor, whereas the induction generator operates with negative slip, in order to generate electrical energy. P P d The speed of thev rotating= r ifield+ relativeω + to Rtheω armatureR ω isλ given+ byλ the difference= 0, between (22)the q2R 2R q2R eR a m d2R q2R absolute speed of the rotating field and the speed2 of the− turbine,2 that is: dt𝜔 + 𝜔 − 𝜔 . Thus, thewhere voltagevd2R, equationsvq2R are the for EFR’s the armature rotor voltages are identica in d andl to qthose axes, ofid conventional2R, iq2R are the induction EFR’s rotor machine: currents in the d and q axes and λd2R, λq2R are the flux linkages in the d and q axes of the EFR’s rotor. 𝑣 = 𝑟 𝑖 − 𝜔 𝜆 + 𝜆 , The relationship between the electric currents and flux linkages are the same as for a conventional(19) induction machine. Unlike a conventional induction𝑣 machine= 𝑟 𝑖 in+ which𝜔 𝜆 only+ one𝜆 part, rotates in the EFR both the (20) armature and the rotor are movable. Thus, two dynamic angular acceleration equations are required, 𝑣 𝑣 𝑖 𝑖 whereone for each, part. are the terminal voltages in d and q axes at the EFR’s armature, , are the 𝜔 electricalIn order currents to arrive in the at d the and balance q axes equationsat the EFR’s of thearmature, EFR it is necessary is the inverter to keep voltage in mind pulsation, that the 𝜆 𝜆 electromagnetic, are the torque flux linkages is responsible in the for d acceleratingand q axes at the the rotor EFR’s and armature. at the same time for the deceleration of theRegarding armature. to Thenrotor voltages, the armature the rotation is accelerated imposed by to the the wind armature turbine causes torque an additional and braked rotating by the voltage superposing the one caused by the armature currents, so that the resulting equations become:

Energies 2019, 12, 2100 7 of 19 electromagnetic torque. While the EFR rotor shaft is accelerated by its electromagnetic torque, it is braked by the electromagnetic torque of the induction generator connected to the shaft. Equations (23) and (24) describe this relationship.

dωm J = TeR + Teg KDωm, (23) Σ dt −

dωa Ja = Ta TeR KDtωt, (24) dt − − where JΣ is the sum of the inertia of the EFR rotor with the inertia of the induction generator rotor, Teg is the electromagnetic torque of the induction generator (a negative value), Ja is the inertia of the turbine added to the inertia of the rotating armature (referred to the high speed side of the gearbox), KDt is the turbine friction and TeR is the electromagnetic torque developed in the EFR that may be calculated by:

3 PR TeR = λ i λ i . (25) 2 2 d1R q1R − q1R d1R In addition to dynamic equations of both machines, the power conservation principle must be considered according to: Pt + Pi = Pger + Plosses, (26) where Pi is the power contribution delivered by the inverter, Plosses are the electrical and mechanical power losses, and Pger is the generated power. It must be pointed out that Equation (26) is used only to prove validity of results obtained by simulations.

3. Control Strategy The generation topology of the Electromagnetic Frequency Regulator has been designed so that the EFR receives a variable speed in the armature of the machine and delivers a desired speed in the rotor shaft. Then, when connecting an induction generator to the rotor shaft it is possible to control the torque of the generator through the slip. The electromagnetic torque of an induction generator as a function of the slip is given by the Equation (27) [22]:   n V2 (r /s) 1  ph 1,eq 2  T =  , (27) eg  2 2  ωs   r1,eq + (r2/s) + X1,eq + X2 where ωs is the synchronous angular mechanical speed, nph is the number of phases, s is the slip, X2 is the per-phase reactance of the rotor windings and V1,eq, r1,eq and X1,eq can be obtained by:

! jXm V1,eq = V1 , (28) r1 + j(X1 + Xm)

jXm(r1 + jX1) Z1,eq = , (29) r1 + j(X1 + Xm) where V1 is the grid phase voltage, Xm is the magnetizing reactance and X1 is the per-phase reactance of the stator windings. According to Equations (23) and (24), by a steady state the turbine torque, the electromagnetic torque of the EFR and the electromagnetic torque of the induction generator are equal. This allows the torque throughout the system to be controlled by the slip of the induction generator. According to Equation (5) there is only one optimal torque for each wind speed. This is achieved when the power coeﬃcient is at its maximum value. This way it is possible to achieve the Maximum Power Point Tracking (MPPT). Thus, the desired torque for each wind speed can be calculated from Equation (5), using Cp_max = 0.441 and λopt = 7.206, resulting in: Energies 2018, 11, x FOR PEER REVIEW 8 of 19

𝑗𝑋 𝑉, =𝑉 , (28) 𝑟 + 𝑗(𝑋 +𝑋)

𝑗𝑋(𝑟 + 𝑗𝑋) 𝑍, = , (29) 𝑟 + 𝑗(𝑋 +𝑋)

where 𝑉 is the grid phase voltage, 𝑋 is the magnetizing reactance and 𝑋 is the per-phase reactance of the stator windings. According to Equations (23) and (24), by a steady state the turbine torque, the electromagnetic torque of the EFR and the electromagnetic torque of the induction generator are equal. This allows the torque throughout the system to be controlled by the slip of the induction generator. EnergiesAccording2019, 12, 2100 to Equation (5) there is only one optimal torque for each wind speed. This is achieved8 of 19 when the power coefficient is at its maximum value. This way it is possible to achieve the Maximum

Power Point Tracking (MPPT). Thus, the desired torque for each2 wind speed can be calculated from 1 3 0.441 Vw Equation (5), using 𝐶_ = 0.441 andT 𝜆∗ = =ρπ 7.206R , resulting, in: (30) 2 7.206 gb 1 0.441 𝑉 𝑇∗ = 𝜌𝜋𝑅 , (30) where T∗ is the desired torque for the system. 2 7.206 𝑔𝑏 whereEntering 𝑇∗ is the the desired desired torque torque for in the Equation system. (27), it leads to Equation (31), having the desired slip as unknown:Entering the desired torque in Equation (27), it leads to Equation (31), having the desired slip as 2 2 2 2 2 T∗ωs r + X + X s∗ + r 2T∗ωsr n V s∗ + T∗ωsr = 0, (31) unknown: 1,eq 1,eq 2 2 1,eq − ph 1,eq 2 ∗ ∗ ∗ ∗ ∗ where s∗ is the desired𝑇 𝜔 𝑟 slip., +𝑋 Solving, +𝑋 Equation 𝑠 +𝑟 (31)2𝑇 results𝜔𝑟, in two−𝑛 real𝑉, roots.𝑠 +𝑇 The𝜔 choice𝑟 =0, between the(31) two slips should be the one with the lowest absolute value. This choice results in lower electrical current 𝑠∗ andwhere lower reactive is the desired power, slip. corresponding Solving Equation to improved (31) results system in two performance. real roots. The choice between the two slips should be the one with the lowest absolute value. This choice results in lower electrical The rotational speed of the rotor shaft may be calculated from the desired slip according to [23]: current and lower reactive power, corresponding to improved system performance. The rotational speed of the rotor shaft may be calculated from the desired slip according to [23]: ωm∗ = (1 s∗)ωs. (32) ∗ − ∗ 𝜔 = (1−𝑠 )𝜔. (32) Thus, if the EFR controls the rotation speed of the shaft at the desired value, the electromagnetic Thus, if the EFR controls the rotation speed of the shaft at the desired value, the electromagnetic torque required by the induction generator will be equal to the turbine torque that delivers the torque required by the induction generator will be equal to the turbine torque that delivers the maximum power for each wind speed. maximum power for each wind speed. Figure4 shows the block diagram of the control used in this article whose objective is to control Figure 4 shows the block diagram of the control used in this article whose objective is to control the frequency and the voltage supplied by the inverter to feed the EFR armature windings. the frequency and the voltage supplied by the inverter to feed the EFR armature windings.

Figure 4. 4. BlockBlock diagram diagram of of EFR EFR control. control.

4.4. Results Results AA simulation simulation programprogram was developed developed using using Scilab Scilab platform platform to tosolve solve the the above above differential diﬀerential and and algebraicalgebraic equations.equations. A A 4th 4th order order Runge-Kutta Runge-Kutta algorithm algorithm was was adopted adopted for for solving solving time-varying time-varying equations.equations. Although Although power power electronic electronic devices devices can ca providen provide signiﬁcantly significantly faster faster responses, responses, all simulated all disturbancessimulated disturbances could be controlled could be by controlled slower actuation, by slower thus actuation, in order to thus accelerate in order simulations to accelerate the step sizesimulations of 1 ms was the step preferred. size of 1 ms was preferred. TheThe induction induction machine machine parameters parameters used used for thefor EFRthe simulationEFR simulation were takenwere fromtaken [24 from] considering [24] considering the same rated power of 2 MW, which is compatible to the informed rated power of the same rated power of 2 MW, which is compatible to the informed rated power of turbine. The values turbine. The values are shown in Table 1. are shown in Table1.

Table 1. Parameters of of the the Electromagnetic Electromagnetic Frequency Frequency Regulator. Regulator.

Parameter Value Unity Number of poles 4 - Armature resistance 0.01 p.u. Rotor resistance 0.01 p.u. Armature leakage inductance 0.10 p.u. Rotor leakage inductance 0.08 p.u. Mutual inductance 3.0 p.u. Inertia constant 0.5 s

The parameters used for the simulation of the induction generator are same as adopted for EFR, including the number of poles. Finally, the simulated wind turbine parameters are presented in Table2 below. Energies 2018, 11, x FOR PEER REVIEW 9 of 19

The parameters used for the simulation of the induction generator are same as adopted for EFR, includingEnergies 2019 the, 12, number 2100 of poles. 9 of 19 Finally, the simulated wind turbine parameters are presented in Table 2 below. Table 2. Parameters of the wind turbine. Table 2. Parameters of the wind turbine. Parameter Value Unity Parameter Value Unity DiameterDiameter 9090 m m Gearbox ratio 100 - GearboxNominal ratio turbine speed 18 100 RPM - NominalNominal turbine wind speed speed 11 18 m/s RPM NominalInertia wind constantspeed 2.5 11 s m/s Inertia constant 2.5 s In relation to the data reported in [24], the turbine diameter and the nominal wind speed were In relation to the data reported in [24], the turbine diameter and the nominal wind speed were adjusted to match the nominal turbine and machines power ratings. adjusted to match the nominal turbine and machines power ratings. The moment of inertia J can be obtained in terms of inertia constant H by [18]: The moment of inertia 𝐽 can be obtained in terms of inertia constant 𝐻 by [18]: 2H2𝐻 J = Sb, (33) 𝐽=ω 2 𝑆, (33) 𝜔0 where ω𝜔0is is rated rated angular angular speed speed in in mechanical mechanical radians radians per per second second and andS 𝑆bis is the the volt-ampere volt-ampere rating. rating.

4.1. Steady Steady State State Analysis Analysis The proposed system using the Electromagnetic Frequency Regulator was simulated in steady state for several wind speeds. The The contribution contribution of of the wind wind turbine and the inverter to the generated power by the induction generator is shown in Figure5 5..

Figure 5. ActiveActive power as a function of wind speed.

When the wind speed reaches 11 m/s the active power in the induction generator is equal to the rated power of the machines. For wind speeds above this value, the pitch control of the wind turbine would act in order to maintain the power values at the same level. The wind turbine design was developed so that the rated wind speed was 11 m/s, in order to better take advantage of the winds of the Brazilian Northeast since most of the time the winds have speeds between 5 m/s and 7 m/s. Another relevant point in the graph analysis of Figure5 is that, the maximum active power provided by the inverter is less than 18% of the rated power of the induction generator. Nevertheless, dimensioning of inverter must consider also possible stresses due to disturbances. The slip of the induction generator for each wind speed are sketched in Figure6. Energies 2018, 11, x FOR PEER REVIEW 10 of 19

When the wind speed reaches 11 m/s the active power in the induction generator is equal to the rated power of the machines. For wind speeds above this value, the pitch control of the wind turbine would act in order to maintain the power values at the same level. The wind turbine design was developed so that the rated wind speed was 11 m/s, in order to better take advantage of the winds of the Brazilian Northeast since most of the time the winds have speeds between 5 m/s and 7 m/s. Another relevant point in the graph analysis of Figure 5 is that, the maximum active power provided by the inverter is less than 18% of the rated power of the induction generator. Nevertheless, Energiesdimensioning2019, 12, 2100 of inverter must consider also possible stresses due to disturbances. 10 of 19 The slip of the induction generator for each wind speed are sketched in Figure 6.

FigureFigure 6. 6.Induction Induction generator generator slipslip in steady st stateate as as a a function function of of wind wind speed. speed.

TheThe effi efficiencyciency of of the the electrical electrical generation generation system system can can be be defined defined as as the the active active power power delivereddelivered by theby generator the generator to the to power the power grid dividedgrid divided by the by sum the ofsum the of power the power supplied supplied by the by wind the wind turbine turbine and the powerand the delivered power delivered by the inverter: by the inverter: P𝑃ger η =𝜂= .. (34) Pt + Pi (34) 𝑃 +𝑃 ConsideringConsidering just just cooper cooper losses, losses, thethe eefficiencyfficiency is greater than than 98% 98% for for any any wind wind speed speed from from 3 m/s 3 m /s upup to to the the nominal nominal speed. speed. Here Here must must be be pointed pointed out out that that this this result resultwas was obtainedobtained usingusing thethe machinesmachines of referenceof reference [24], [24], whose whose data data set isset limited, is limited, omitting omitting information information like, like, e.g., e.g., iron iron losses. losses.

4.2.4.2. Dynamic Dynamic Analysis Analysis TheThe objective objective of of this this subsection subsection isis toto presentpresent the dynamic performance performance of of the the proposed proposed system system duringduring the the main main disturbances disturbances in in a a wind wind farm.farm. Disturbances in in the the wind wind and and in in the the grid grid connected connected to to thethe generator generator are are considered considered for for analysis. analysis. It It must must be be emphasized emphasized that that because because of of very very high high inertiasinertias all graphicsall graphics sketched sketched in this in sectionthis section are thoughtare thought to giveto give an ideaan idea of theof the responses responses immediately immediately after after the disturbances,the disturbances, but not but comprehending not comprehending the wholethe whol periode period until until steady steady state state is reached.is reached.

4.2.1.4.2.1. Step Step Response Response TheThe proposed proposed system system was was submitted submitted toto threethree steps of of 1 1 m/s m/s in in wind wind speed, speed, starting starting from from a steady a steady stateEnergiesstate of of 2018 8 8 m ,m/s. /11s., x Thus, Thus,FOR PEER at the theREVIEW end end of of the the simulation, simulation, the theinduction induction genera generatortor will be will at rated be at power. rated11 of power.The 19 Thesimulation simulation result result for forthe the rotor rotor speed speed andand the desired the desired speed speed is shown is shown in Figure in Figure 7. 7.

Figure 7. 7. AngularAngular speed speed of of the the rotor rotor shaft. shaft.

As described in the control strategy, the control variable that guides the speed to the desired value is the frequency of the current at the inverter output. The frequency variation due to the wind speed steps is sketched in Figure 8.

Figure 8. Frequency at the inverter output.

The power generated by the induction generator and its turbine and inverter contributions are presented in Figure 9.

Energies 2018, 11, x FOR PEER REVIEW 11 of 19

Energies 2019, 12, 2100 11 of 19 Figure 7. Angular speed of the rotor shaft.

AsAs described described in in the controlcontrol strategy,strategy, the the control control variable variable that that guides guides the the speed speed to the to desiredthe desired value valueis the is frequency the frequency of the of current the current at the at inverter the invert output.er output. The frequency The frequency variation variation due to due the to wind the wind speed speedsteps steps is sketched is sketched in Figure in Figure8. 8.

FigureFigure 8. 8. FrequencyFrequency at at the the inverter inverter output. output.

TheThe power power generated generated by by the the induction induction generator generator and and its its turbine turbine and and inverter inverter contributions contributions are are presentedpresented in in Figure Figure 9.9.

Figure 9. Generated power.

4.2.2. Wind Speed Variations The simulated wind speed model is similar to that proposed in [24]. It is assumed that the wind speed is composed of the sum of four components:

The initial average value of the wind speed • A ramp component • A gust component • Turbulence • The simulation run was carried out in order to illustrate the response of EFR operating under normal conditions for 60 s. The applied wind speed signal is composed by all four components. The simulation starts with wind velocity equal to 8 m/s and there is a ramp with amplitude of 2.5 m/s − up to 10 s. Between the instants of 10 s to 20 s there is a gust amplitude equal 4 m/s. After the instant EnergiesEnergies 20182018,, 1111,, xx FORFOR PEERPEER REVIEWREVIEW 1212 ofof 1919

FigureFigure 9.9. GeneratedGenerated power.power.

4.2.2.4.2.2. WindWind SpeedSpeed VariationsVariations TheThe simulatedsimulated windwind speedspeed modelmodel isis similarsimilar toto thatthat proposedproposed inin [24].[24]. ItIt isis assumedassumed thatthat thethe windwind speedspeed isis composedcomposed ofof thethe sumsum ofof fourfour components:components: •• TheThe initialinitial averageaverage valuevalue ofof thethe windwind speedspeed •• AA rampramp componentcomponent •• AA gustgust componentcomponent •• TurbulenceTurbulence TheThe simulationsimulation runrun waswas carriedcarried outout inin orderorder toto ilillustratelustrate thethe responseresponse ofof EFREFR operatingoperating underunder normalnormalEnergies conditions2019conditions, 12, 2100 forfor 6060 s.s. TheThe appliedapplied windwind spspeedeed signalsignal isis composedcomposed byby allall fourfour components.components.12 TheThe of 19 simulationsimulation startsstarts withwith windwind velocityvelocity equalequal toto 88 m/sm/s andand therethere isis aa rampramp withwith amplitudeamplitude ofof −−2.52.5 m/sm/s upup toto 1010 s.s. BetweenBetween thethe instantsinstants ofof 1010 ss toto 2020 ss ththereere isis aa gustgust amplitudeamplitude equaequall 44 m/s.m/s. AfterAfter thethe instantinstant ofofof timetime time equalequal equal toto to 3030 30 s,s, s, therethere there isis is aa a newnew new rampramp ramp withwith with amplitudeamplitude amplitude ofof of 55 5 m/sm/s m/ s duringduring during 2525 25 s.s. s. TheThe The averageaverage average valuevalue value componentcomponentcomponent isis is equalequal equal toto to 5.55.5 5.5 m/s,m/s, m/s, andand and turbulenceturbulence turbulence isis is presentpresent present throughoutthroughout throughout thethe the simulation.simulation. simulation. TheThe The simulatedsimulated simulated windwindwind speedspeed speed isis is shownshown shown inin in FigureFigure Figure 10.10. 10 .

FigureFigureFigure 10.10. 10. SimulatedSimulatedSimulated windwind wind speed.speed. speed.

TheTheThe generationgeneration generation topologytopology topology usingusing using EFREFR waswas simulated simulated consideringconsidering the thethe wind windwind speed speedspeed shown shownshown in inin Figure FigureFigure 10 . 10.10.The TheThe rotational rotationalrotational speed speedspeed of ofof the thethe rotor rotorrotor and andand its itsits desired desireddesired speed speedspeed are areare shown shownshown in Figureinin FigureFigure 11 . 11.11.

FigureFigureFigure 11.11. 11. AngularAngularAngular speedspeed speed ofof of thethe the rotorrotor rotor byby by windwind wind speedspeed speed variations.variations. variations.

The frequency variation in time is presented in Figure 12, and the powers of generator, turbine

and inverter are shown together in Figure 13, in order to make possible a comparison. Energies 2018, 11, x FOR PEER REVIEW 13 of 19 Energies 2018, 11, x FOR PEER REVIEW 13 of 19 The frequency variation in time is presented in Figure 12, and the powers of generator, turbine Energiesand inverterThe2019 frequency, 12, are 2100 shown variation together in time in Figure is presented 13, in order in Figure to make 12, andpossible the powersa comparison. of generator, turbine13 of 19 and inverter are shown together in Figure 13, in order to make possible a comparison.

Figure 12. Frequency at the inverter output by wind speed variations. FigureFigure 12. Frequency at at the the inverter inverter output output by by wind wind speed speed variations. variations.

FigureFigure 13. GeneratedGenerated power power by by wind wind speed speed variations. variations. Figure 13. Generated power by wind speed variations. 4.2.3.4.2.3. The The Fault-Ride-Through Fault-Ride-Through Capability 4.2.3. The Fault-Ride-Through Capability TheThe fault-ride-through fault-ride-through requirements are are represen representedted through through a voltage a voltage against against time time profile proﬁle as as shownshownThe in in Figurefault-ride-through Figure 14 14[ [25].25]. According requirements to to this this figure,are ﬁgure, represen the the German Germanted through regulations regulations a voltage impose impose against the the mosttime most restrictiveprofile restrictive as limitsshownlimits for for in fault-ride-through fault-ride-throughFigure 14 [25]. According capabilitycapability to this of of wind windfigure, turbines, turbines, the German requiring requiring regulations that that wind wind impose farms farms the must most must have restrictive have voltage voltage Energies 2018, 11, x FOR PEER REVIEW 14 of 19 withstandlimitswithstand for fault-ride-through capability capability upup toto zerozero capability during of 150 wind ms. ms. turbines, requiring that wind farms must have voltage withstand capability up to zero during 150 ms.

FigureFigure 14. Low-Voltage Ride Ride Through Through (LVRT) (LVRT) re requirementsquirements of ofsome some grid grid codes. codes.

In order to simulate the most severe case of voltage deep, the proposed system was subjected to a stator voltage of the induction generator equal to zero for 150 ms. The disturbance occurs at the instant 100 ms and returns to the rated voltage at the instant 250 ms. The line-to-line voltage (p.u.) at the induction generator terminals representing the disturbance is presented in Figure 15.

Figure 15. Line-to-line voltage (p.u.) at the induction generator terminals.

The desired speed and the speed of the rotor shaft for the admitted voltage disturbance are shown in Figure 16.

Energies 2018, 11, x FOR PEER REVIEW 14 of 19

Energies 2019, 12, 2100 14 of 19 Figure 14. Low-Voltage Ride Through (LVRT) requirements of some grid codes.

InIn order order to to simulate simulate the the most most severe severe case case of ofvoltage voltage deep, deep, the the proposed proposed system system was was subjected subjected to a tostator a stator voltage voltage of the of theinduction induction generator generator equal equal to tozero zero for for 150 150 ms. ms. The The disturbance disturbance occurs occurs at at the the instantinstant 100 100 ms ms and and returns returns to to the the rated rated voltage voltage at at the the instant instant 250 250 ms. ms. The The line-to-line line-to-line voltage voltage (p.u.) (p.u.) at at thethe induction induction generator generator terminals terminals representing representing the the disturbance disturbance is is presented presented in in Figure Figure 15. 15 .

FigureFigure 15. 15. Line-to-lineLine-to-line voltage voltage (p.u.) (p.u.) at at the the induction induction generator generator terminals. terminals.

EnergiesTheThe 2018 desired desired, 11, x FOR speed PEER and andREVIEW thethe speed speed of of the the rotor rotor shaft shaft for for the admittedthe admitted voltage voltage disturbance disturbance are15 shown ofare 19 shownin Figure in Figure 16. 16.

FigureFigure 16. 16. RotorRotor speed speed for for ride ride-through-through analysis. analysis.

ThisThis is is the the disturbance disturbance with with the the major major speed speed variat variationion in in generator generator rotor rotor shaft. shaft. Nevertheless, Nevertheless, the the systemsystem is is able able to to re-establish re-establish the the speed speed close close to to the the desired desired one one in in less less than than 3 3 s safter after the the return return of of the the voltagevoltage to to the the rated rated value. value. As As can can be be seen seen in in Figu Figurere 17, 17 ,the the frequency frequency of of the the current current at at the the inverter inverter outputoutput returns returns smoothly smoothly to the steady state,state, afterafter aa briefbrief undershoot. undershoot. This This occurs occurs because because of of inertia inertiaJa ,

𝐽as, as can can be be observed observed in in the the curve curve of of Figure Figure 18 18,, describingdescribing the behavior of of speed speed 𝜔ωa. The The voltage voltage dip dip occurredoccurred at at the the generator generator stator stator produces produces a major decreasedecrease in thethe electromagneticelectromagnetic torque,torque, leadingleadingto to a afast fast gain gain in in armature armature speed. speed. However, However, a slowa slow return return to to the the its its steady steady state state value value occurs. occurs. Note Note that that the thefrequency frequency assumes assumes a similar a similar behavior, behavior, in order in order to assure to assure a fast a fast return return of rotor of rotor speed speed to the to steadythe steady state state(see Figure(see Figure 16).Although 16). Although the armature the armature speed speed reaches reaches its maximum its maximum value in value a time in closea time to close 2 s, the to rotor2 s, thespeed rotor and speed the currentsand the currents at the same at the time same are time already are already close to close their to steady their statesteady value. state value. This occurs This occursbecause because of the compensationof the compensation performed performed by the controlby the control action onaction the frequencyon the frequency of the inverter of the inverter currents. currents.

Figure 17. Frequency behavior at the inverter output for ride-through analysis.

Energies 2018, 11, x FOR PEER REVIEW 15 of 19

Figure 16. Rotor speed for ride-through analysis.

This is the disturbance with the major speed variation in generator rotor shaft. Nevertheless, the system is able to re-establish the speed close to the desired one in less than 3 s after the return of the voltage to the rated value. As can be seen in Figure 17, the frequency of the current at the inverter output returns smoothly to the steady state, after a brief undershoot. This occurs because of inertia

𝐽, as can be observed in the curve of Figure 18, describing the behavior of speed 𝜔. The voltage dip occurred at the generator stator produces a major decrease in the electromagnetic torque, leading to a fast gain in armature speed. However, a slow return to the its steady state value occurs. Note that the frequency assumes a similar behavior, in order to assure a fast return of rotor speed to the steady state (see Figure 16). Although the armature speed reaches its maximum value in a time close to 2 s, the rotor speed and the currents at the same time are already close to their steady state value. This Energies occurs2019, 12 because, 2100 of the compensation performed by the control action on the frequency of the inverter 15 of 19 currents.

Energies 2018Figure, 11,Figure x FOR 17. PEERFrequency17. Frequency REVIEW behavior behavior at the the inverter inverter output output for ride-through for ride-through analysis. analysis. 16 of 19

Energies 2018, 11, x FOR PEER REVIEW 16 of 19

FigureFigure 18. 18.Armature Armature speed speed for ride-through ride-through analysis analysis..

The currentsThe currents of the of inductionthe Figureinduction 18. generator Armature generator speed andand for at ride-through th thee output output analysisof ofthe the.inverter inverter are both are shown both shownin in Figure 19Figure. As 19. can As be can observed be observed in thisin this ﬁgure figure the the resultingresulting stator stator current current of the of generator the generator during during the the simulatedThe currents short-circuit of the isinduction very high, generator if compared and withat th ecorresponding output of the steady-state inverter are value, both whereasshown in at simulatedFigurethe short-circuitinverter, 19. As itcan does be is notobserved very happen. high, in Anthis ifother comparedfigure relevant the re with sultingaspect corresponding isstator that thcurrente maximum of steady-state the short-circuitgenerator value,during current the whereas of at the inverter,simulatedthe generator it does short-circuit notis almost happen. is 13 very times Anotherhigh, the if steady-state compared relevant with cu aspectrrent, corresponding iswhereas that the the steady-state maximumcurrent in thevalue, short-circuit inverter whereas is only at current of the generatorthe20% inverter, higher is almost it than does its not 13 steady-state timeshappen. the An value. steady-stateother Thus,relevant it isaspect current, unnecessary is that whereas th toe maximum switch the off current short-circuitthe inverter in the currentduring inverter ofthe is only 20% higherthedisturbance, generator than its isas steady-state almostthis overcurrent 13 times value. the is compatible steady-state Thus, itwith iscurrent, unnecessarythe levels whereas supported the to switchcurrent by actual in oﬀ the devices.the inverter inverter Therefore, is only during the disturbance,20%extra higher protection as this than overcurrent its for steady-state the inverter is compatiblevalue. is not Thus,required, withit is asunnecessarythe by Rotor levels Side to supported switch Converter off bythe (RSC) actualinverter in a devices. duringDoubly-Fed the Therefore, disturbance,Induction Generator as this overcurrent (DFIG). is compatible with the levels supported by actual devices. Therefore, extra protection for the inverter is not required, as by Rotor Side Converter (RSC) in a Doubly-Fed extra protection for the inverter is not required, as by Rotor Side Converter (RSC) in a Doubly-Fed InductionInduction Generator Generator (DFIG). (DFIG).

Figure 19. Currents of the induction generator and inverter output for ride-through analysis.

5.Figure DiscussionFigure 19. Currents 19. Currents of theof the induction induction generatorgenerator and and inverter inverter output output for ride-through for ride-through analysis. analysis. The results show that the topology using the Electromagnetic Frequency Regulator coupled with 5. Discussion an induction generator is feasible in sense of exploring the maximal wind power available for the wholeThe wind results speed show range, that the starting topology from using the theusual Electromagnetic cut-in speed untilFrequency that one Regulator corresponding coupled withto the anrated induction power. generator In this way, is feasibleit is possible in sense to have of exploring a hybrid generationthe maximal source wind using power a singleavailable topology. for the wholeThe wind generation speed range, topology starting proposed from the in usualthis article cut-in presentsspeed until some that similarities one corresponding with the alreadyto the ratedknown power. Doubly-Fed In this way, Induction it is possible Generator to have (DFIG), a hybrid as: use generation of gearbox, source converter using a not single dimensioned topology. for full Thepower, generation and use topologyof an induction proposed machine in this for article generation. presents Therefore, some similarities the comparison with the between already the knowntwo topologies Doubly-Fed is inevitable. Induction Generator (DFIG), as: use of gearbox, converter not dimensioned for full power, and use of an induction machine for generation. Therefore, the comparison between the two topologies is inevitable.

Energies 2019, 12, 2100 16 of 19

5. Discussion The results show that the topology using the Electromagnetic Frequency Regulator coupled with an induction generator is feasible in sense of exploring the maximal wind power available for the whole wind speed range, starting from the usual cut-in speed until that one corresponding to the rated power. In this way, it is possible to have a hybrid generation source using a single topology. The generation topology proposed in this article presents some similarities with the already known Doubly-Fed Induction Generator (DFIG), as: use of gearbox, converter not dimensioned for full power, and use of an induction machine for generation. Therefore, the comparison between the two topologies is inevitable. By the DFIG topology a back-to-back converter is required while the EFR uses a unidirectional inverter, fed by another source, and not connected to the grid. This aspect allows the use of a supplementary primary source to feed the rotating armature windings. Furthermore, no harmonic sources interact galvanically to the grid. An apparent disadvantage of the proposed topology is the use of two induction machines instead of only one, like by DFIG, but avoiding the back-to-back converter.

6. Conclusions Besides the advantages highlighted in previous section, the generation topology using the Electromagnetic Frequency Regulator driving an induction generator has proved to be able to withstand large variations in wind speed, like a gust, without necessity to disconnect. This topology proved also to be effective to ride-through a severe short-circuit at the induction generator terminals, even without being endowed with a crowbar, as is usual in DFIG topologies. It happens because the current at the EFR inverter output during a disturbance is not high as the current in the DFIG converter, once the inverter has no connection to the grid. These statements are consistent to the results of dynamic simulation presented in Section4. In addition, high values of e fficiency for the set of electrical machines have resulted within the wind speed operational range. According to the steady state simulations the EFR inverter may be dimensioned for about 20% of the generator rated power. The same percentage may be applied to define the EFR armature rated power, although the rotor must be dimensioned for the total generated power. This should be considered during EFR design, resulting in economy of manufacture costs. Concluding, all presented simulations results encourage to use the proposed system for practical applications, in sites where a supplementary source to the wind power is available.

Author Contributions: Conceptualization, M.F.M.J. and R.P.; Data curation, T.R.; Formal analysis, T.R. and M.F.M.J.; Investigation, T.R., M.F.M.J., R.P. and A.M.; Methodology, T.R., M.F.M.J., R.P. and A.M.; Project administration, M.F.M.J.; Resources, M.F.M.J. and R.P.; Software, T.R. and A.M.; Supervision, M.F.M.J. and R.P.; Visualization, T.R. and A.M.; Writing—original draft, T.R. and A.M.; Writing—review & editing, M.F.M.J. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest. Energies 2019, 12, 2100 17 of 19

Nomenclature

Pt Turbine mechanical power ρ Air density R Radius of the blade Cp Turbine Power Coeﬃcient Vw Wind speed λ Tip-Speed Ratio ωt Turbine angular speed β Blade pitch angle Tt Turbine mechanical torque vd1, vq1 Voltage in d and q axes of the stator r1 Stator resistance id1, iq1 Electrical current in the d and q axes of the stator ωe Synchronous angular frequency λd1, λq1 Flux linkage in the d and q axes of the stator vd2, vq2 Voltage in d and q axes of the rotor r2 Rotor resistance id2, iq2 Electrical current in the d and q axes of the rotor P Number of poles of the machine ωm Angular speed of rotor shaft λd2, λq2 Flux linkage in the d and q axes of the rotor Ls1 Self-inductance of any stator winding Ls2 Self-inductance of any rotor winding Lm1 Mutual inductance between any two windings in the stator Lm2 Mutual inductance between any two windings in the rotor Lm Maximum mutual inductance between stator and rotor windings J Inertia of the rotor Te Electromagnetic torque Tm Mechanical torque KD Rotor friction constant gb Gearbox ratio ωa Angular speed of the EFR’s armature Ta Mechanical torque at the EFR’s armature vd1R, vq1R Voltage in d and q axes of the EFR’s armature id1R, iq1R Electrical current in the d and q axes of the EFR’s armature λd1R, λq1R Flux linkage in the d and q axes of the EFR’s armature ωeR Inverter voltage pulsation vd2R, vq2R Voltage in d and q axes of the EFR’s rotor id2R, iq2R Electrical current in the d and q axes of the EFR’s rotor λd2R, λq2R Flux linkage in the d and q axes of the EFR’s rotor JΣ Sum of inertia of the EFR’s rotor with the inertia of induction generator rotor TeR Electromagnetic torque developed in the EFR Teg Electromagnetic torque in the induction generator Jt Inertia of the turbine added to the inertia of the rotating armature KDt Turbine friction constant ωs Synchronous angular mechanical speed nph number of phases of the induction generator s Slip of the induction generator X2 Reactance of the rotor windings V1 Grid phase voltage Xm Magnetizing reactance X1 Reactance of the stator windings Energies 2019, 12, 2100 18 of 19

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