mathematics

Article Grid Frequency and Amplitude Control Using DFIG Wind Turbines in a Smart Grid

José Antonio Cortajarena 1,*, Oscar Barambones 2 , Patxi Alkorta 1 and Jon Cortajarena 3

1 Engineering School of Gipuzkoa, University of the Basque Country, Otaola Hirib. 29, 20600 Eibar, Spain; [email protected] 2 Engineering School of Vitoria, University of the Basque Country, Nieves Cano 12, 01006 Vitoria, Spain; [email protected] 3 Engineering School of Gipuzkoa, University of the Basque Country, Europa Plaza 1, 20018 Donostia, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-943-033-041

Abstract: Wind-generated energy is a fast-growing source of renewable energy use across the world. A dual-feed induction machine (DFIM) employed in wind generators provides active and reactive, dynamic and static energy support. In this document, the droop control system will be applied to adjust the amplitude and frequency of the grid following the guidelines established for the utility’s smart network supervisor. The wind generator will work with a maximum deloaded power curve, and depending on the reserved active power to compensate the frequency drift, the limit of the reactive power or the variation of the voltage amplitude will be explained. The aim of this paper is to show that the system presented theoretically works correctly on a real platform. The real-time experiments are presented on a test bench based on a 7.5 kW DFIG from Leroy Somer’s commercial machine that is typically used in industrial applications. A synchronous machine that emulates the wind profiles moves the shaft of the DFIG. The amplitude of the microgrid voltage at load variations


is improved by regulating the reactive power of the DFIG and this is experimentally proven. The
 contribution of the active power with the characteristic of the droop control to the load variation is Citation: Cortajarena, J.A.; made by means of simulations. Previously, the simulations have been tested with the real system to Barambones, O.; Alkorta, P.; ensure that the simulations performed faithfully reflect the real system. This is done using a platform Cortajarena, J. Grid Frequency and based on a real-time interface with the DS1103 from dSPACE. Amplitude Control Using DFIG Wind Turbines in a Smart Grid. Mathematics Keywords: double feed induction generator; grid frequency and amplitude support; smart grid 2021, 9, 143. https://doi.org/ 10.3390/math9020143

Received: 29 November 2020 1. Introduction Accepted: 8 January 2021 Published: 11 January 2021 Wind energy is progressively gaining importance in the world’s electricity production, with important engineering aspects to be addressed for its integration into conventional

Publisher’s Note: MDPI stays neu- electricity grids. In fact, it contributes about 7% of total energy production worldwide tral with regard to jurisdictional clai- and onshore and offshore wind energy together would generate more than a third of all ms in published maps and institutio- electricity needs, becoming the main source of generation by 2050 [1]. nal affiliations. Recently, some studies have been developed in order to analyze the installation of small wind turbines in urban areas. Installing wind turbines in all the possible extents can mitigate the rising energy demand. Built-up areas possess high potential for wind energy, including the rooftop of high-rise buildings, railway track, the region between or Copyright: © 2021 by the authors. Li- around multistoried buildings, and city roads. However, harnessing wind energy from censee MDPI, Basel, Switzerland. these areas is quite challenging due to dynamic environments and turbulence for higher This article is an open access article roughness on urban surfaces [2]. Some studies have been done in order to estimate the distributed under the terms and con- wind resource in an urban area [3,4]. These studies evaluate the urban wind resource by ditions of the Creative Commons At- employing a physically-based empirical model to link wind observations at a conventional tribution (CC BY) license (https:// creativecommons.org/licenses/by/ meteorological site to those acquired at urban sites. The approach is based on urban climate 4.0/). research that has examined the effects of varying surface roughness on the wind-field

Mathematics 2021, 9, 143. https://doi.org/10.3390/math9020143 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 143 2 of 18

between and above buildings. These papers aim to provide guidance for optimizing the placement of small wind turbines in urban areas by developing an improved method of estimating the wind resource across a wide urban area. There are different types of wind turbines like induction generator [5] or permanent magnet synchronous generators [6,7] based wind turbines, however the synchronous generators are usually used for a small-scale wind turbines. In addition, the DFIM is the most commonly installed type of generator in wind turbines to date. DFIG generators provide access to the rotor windings and regulating the rotor voltage, the generator active and reactive powers are fully controllable. Therefore, the design and implementation of a new control scheme for a DFIG based wind turbine system has attracted the attention of several authors in the last years [8–12]. The main reason for using the doubly fed induction generator is that the power converters have to manage only a fraction of the total system power, about 30%. For this reason, there are fewer losses in the power electronics unit than in a full-power converter topology. The reduction in costs due to the use of a smaller inverter is another significant factor [13]. The regulation of the active and reactive powers in decoupled form with the DFIG is regulated using the field-oriented control (FOC) technique [14–19]. A smart grid has among its goals one dedicated to providing a more robust, efficient, and flexible electric power system [20,21]. In the last decade, the model predictive control (MPC) has been applied to microgrid systems to optimally schedule and control the microgrid, owing to the advantages of MPC such as fast response and robustness against parameter uncertainties. In the work presented in [22], a stochastic model predictive control framework to optimally schedule and control the microgrid with large scale renewable energy sources is proposed. This microgrid consists of fuel cell-based, wind turbines, PV generators, battery/thermal energy storage system gas fired boilers, and various types of electrical and thermal loads scheduled according to the demand response policy. In the work presented in [23], the authors propose a MPC for regulating frequency in stand- alone microgrids. This work analyzes the impact of system parameters on the control performance of MPC for frequency regulation, using a typical stand-alone microgrid, which consists of a diesel engine generator, an energy storage system, a wind turbine generator, and a load. A novel sensorless model predictive control (MPC) strategy of a wind-driven doubly fed induction generator (DFIG) connected to a dc microgrid is proposed in [24]. In this work, the MPC strategy has been used as a current controller to overcome the weaknesses of the inner control loop and to consider the discrete-time operation of the voltage source converter that feeds the rotor. The extensive use of power generation using wind has forced countries to establish a set of rules for the operation and connection to the grid of wind generators. Amongst all the standards, those related to smart networks aim to guarantee a safe supply, and ensure the reliability and quality of the energy generated [25,26]. With the strong growth of grid-connected wind power plants, there is a need for grid-integrated wind farms with the ability to withstand grid voltage and frequency also during perturbations in the network [18]. To withstand the voltage and frequency, the wind generator must be able to change its operating point according to the needs. The most widely used and robust method is the well-known droop control. This control method is based on a concept known from the electrical networks and is based on the reduction of the frequency of the generator when its active power consumption increases [27]. This control strategy can be applied to different generators such as wind turbines in order to increase the reliability of the system. In wind turbines, when control is carried out by means of the power drivers, as is the case of the DFIM, the speed is adjusted according to the wind speed in order to optimize energy production. This allows the regulation of the generator at the point of maximum Mathematics 2021, 9, x FOR PEER REVIEW 3 of 18

Mathematics 2021, 9, 143 3 of 18 energy production. This allows the regulation of the generator at the point of maximum power in a wide range of power. When using the droop technique, the generator must be running at a different speed than the maximum power speed. Consequently, when droop control is demanded,power the control in a wide system range will of adjust power. the When active using and reactive the droop powers technique, to bal- the generator must ance out deviations bein runningfrequency at and a different amplitude speed of the than grid the voltage maximum respectfully. powerspeed. Consequently, when In the work presenteddroop control in [11] is the demanded, authors propose the control a Power system Delta will Control adjust thefor activea wind and reactive powers turbine in order to toparticipate balance out in the deviations primary in and frequency secondary and frequency amplitude regulation. of the grid voltageThis respectfully. work presents a readily Inindustrializable the work presented set of inalgorithms [11] the authors for torque propose and apitch Power control. Delta Control for a wind However, the proposedturbine control in order scheme to participate are only validated in the primary by means and of secondary simulation frequency using regulation. This the NREL’s FAST software.work presents a readily industrializable set of algorithms for torque and pitch control. The motivationHowever, of the present the proposed work is to control validate scheme the simulate are onlyd validated algorithms by in means a real of simulation using system based on windthe generators NREL’s FAST to contribute software. to the compensation of the frequency and amplitude variations of theThe voltage motivation of the of microgrids. the present work is to validate the simulated algorithms in a real This control schemesystem was based initially on wind proposed generators and simulated to contribute in the to the2018 compensation IEEE Interna- of the frequency and tional Conference onamplitude Industrial variations Electronics of for the Sustainable voltage of theEnergy microgrids. Systems [28]. However, in this previous work, theThis proposed control schemecontrol wasapproach initially was proposed only validated and simulated by means in theof 2018 IEEE Interna- some simulation resultstional using Conference a simple on wind Industrial turbine Electronics model. This for new Sustainable work goes Energy further, Systems [28]. However, providing more simulationsin this previous using a more work, detailed the proposed model, controlwhich accurately approach represents was only validatedthe by means of dynamics of the realsome system. simulation This will results allow simulations using a simple to be wind made turbine when model.real tests This cannot new work goes further, be performed. Moreover,providing some more real simulations time experiments using a more are presented detailed model, over a which test bench accurately represents the based on a commercialdynamics machine of, the typically real system. used in This industrial will allow applications. simulations Thus, to be the made wind when real tests cannot generator is a 7.5 kWbe Leroy performed. Somer DFIM Moreover, driven some by a realsynchronous time experiments machine that are presentedemulates over a test bench the desired wind profiles.based on Different a commercial experiments machine, were typically developed used inusing industrial this test applications. bench, Thus, the wind designed and built generatorad hoc, and is a these 7.5 kW real Leroy experiments Somer DFIM validated driven the by results a synchronous previously machine that emulates obtained in the simulations.the desired Thus, wind these profiles. experimental Different results experiments can be used were to developed demonstrate using this test bench, the applicability of designedthis control and scheme built adin industrial hoc, and theseapplications. real experiments It should be validated noted that the results previously the experimental validationobtained in of the the simulations. new controlThus, schemes these is experimental a considerable results research can be ad- used to demonstrate vance, since it facilitatesthe applicability its implementation of this control in real scheme industrial in industrial applications. applications. This article It should be noted that the experimental validation of the new control schemes is a considerable research advance, deals with the process of controlling the DFIG in the contribution of active and reactive since it facilitates its implementation in real industrial applications. This article deals with power for frequency and voltage regulation. The technique used is the well-known droop the process of controlling the DFIG in the contribution of active and reactive power for control. frequency and voltage regulation. The technique used is the well-known droop control. The work is organized as follows; Sections 2 and 3 present the equations for the con- The work is organized as follows; Sections2 and3 present the equations for the control trol of the DFIG. Sections 4, 5, and 6 cover the droop control, the deloading process for of the DFIG. Sections4–6 cover the droop control, the deloading process for frequency frequency control and explain the voltage compensation and the calculation of the maxi- control and explain the voltage compensation and the calculation of the maximum reactive mum reactive power based on the active working power. Section 7 introduces the labora- power based on the active working power. Section7 introduces the laboratory results and tory results and the simulations. Lastly, the conclusions are presented. the simulations. Lastly, the conclusions are presented.

2. DFIG Control Equations2. DFIG and Control Reference Equations System and Reference System ′ ′ Figure 1 shows theFigure stationary1 shows 훼훽 thestator stationary referenceαβ system,stator the reference rotor 훼 system,훽 reference the rotor α0β0 reference system and the dq referencesystem and system the linked dq reference to the doubly system fed linked induction to the machine doubly fedstator induction flux machine stator vector. flux vector.

  q Reference system linked to the d

 Rotor reference system

 Stator reference system Figure 1. DFIGFigure reference 1. DFIG systems. reference systems.

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The doubly fed induction generator is controlled in the revolving reference frame aligned with the stator flux as shown in Figure1. The next equations explain the operation of the DFIG in the dq frame [29,30].

→ → → dψs,dq → v = R i + + jω ψ (1) s,dq s s,dq dt e s,dq → → → dψr,dq → v = R i + + j(ω − ω )ψ (2) r,dq r r,dq dt e r r,dq → → → ψs,dq = Ls i s,dq + Lm i r,dq and Ls = Lm + Lls (3) → → → ψr,dq = Lr i r,dq + Lm i s,dq and Lr = Lm + Llr (4) 3 T = P ψ i − ψ i
(5) e 2 p sd sq sq sd dω T − T = J m + Bω (6) e L dt m 3 3 P = v i + v i
and Q = v i − v i
(7) s 2 sd sd sq sq s 2 sq sd sd sq → → → where v s,dq, i s,dq, and ψs,dq are the stator voltage, current and flux vectors in the syn- → → → chronous dq reference system. v r,dq, i r,dq, and ψr,dq are the rotor voltage, current, and flux vectors in the synchronous dq reference system. As can be seen in Figure1, the stator flux q component is zero, and when operating with Equation (3) the next two equations are obtained, → ψ s Lm isd = − ird (8) Ls Ls

Lm isq = − irq (9) Ls Equations (8) and (9) shown that the stator current is controllable with the rotor current. With the omission of the stator resistance because it is so small, the stator flux can be assumed to be constant and its value is,

→ → v s

ψs = (10) ωe The stator voltage d component is almost zero because the reference system is oriented along the stator flux, so it can be obtained that,

3 Lm Ps ≈ ωeψs irq (11) 2 Ls  →  v s 3 → Lm Qs ≈ v s  − ird (12) 2 ωe Ls Ls

Equations (11) and (12) show that the stator active power is controlled with the q component of the rotor current and the stator reactive power with the rotor current d component. Figure2 shows the block diagram for the control of the DFIG from the rotor side using ∗ ∗ the rotor side converter (RSC). The current references irq and ird are calculated with (11) and (12) and with the required active and reactive power references. These current references are then compared with the real currents and the differences are the inputs signals of two PI Mathematics 2021, 9, 143 5 of 18 Mathematics 2021, 9, x FOR PEER REVIEW 5 of 18 Mathematics 2021, 9, x FOR PEER REVIEW 5 of 18

regulators, obtaining in their outputs the references of the rotor voltage v∗ and v∗ . Lastly, 푣∗ and 푣∗ . Lastly, the DC/AC inverter pulses SA, SB, and SC are produced using the rq rd 푟푞 푟푑 the푣∗ DC/ACand 푣∗ inverter. Lastly, pulsesthe DC/AC SA, SB, inverter and SC pulses are produced SA, SB, usingand SC the are seven produced segments using space the seven segments space vector 푟푞pulse width푟푑 modulation (SVPWM). vectorseven pulsesegments width space modulation vector pulse (SVPWM). width modulation (SVPWM).

dq   dq      SVPWM  SVPWM

abc abc d dt d dt

d dt d dt

Figure 2. DFIG control structure. Figure 2. DFIG control structure. Figure 2. DFIG control structure. 3. Grid Side Converter Control3. Grid Equations Side Converter Control Equations 3. Grid Side Converter Control Equations For the control of the gridFor side the converter control of(GSC), the grid the revolving side converter reference (GSC), frame the revolving is reference frame is For the control of the grid side converter (GSC), the revolving reference frame is aligned with the grid voltagealigned vector with as illustrated the grid voltage in Figure vector 3. as illustrated in Figure3. aligned with the grid voltage vector as illustrated in Figure 3.

 Reference system linked to the  Reference system linked to the

 

FigureFigure 3. 3. GridGrid voltage voltage reference reference system. system. Figure 3. Grid voltage reference system. The following equations describeThe following the behavior equations of the describe grid side the converter behavior ofconnected the grid side converter connected to the gridThe trough following a line equations filter Lg anddescribeRg in the the behavior mentioned of rotatingthe grid referenceside converter system, connected to the grid trough a line filter 퐿푔 and 푅푔 in the mentioned rotating reference system, to the grid trough a line filter 퐿푔 and 푅푔 in the mentioned rotating reference system, 푑푖⃗ → 푔,푑푞 → 푣⃗ = 푅 푖⃗ + 퐿 → + 푣⃗ d 푑i 푖⃗g푔,dq,푑푞 (13)→ 푐표푛푣,푑푞 푔 푔,푑푞 푔 푑푡v 푣⃗ 푔,푑푞= R= 푅i 푖⃗ ++L퐿 ++ v푣⃗ (13)(13) conv푐표푛푣,dq ,푑푞 g 푔g,푔dq,푑푞 g 푔 dt푑푡 푔g,,푑푞dq 3 3 3   3   푃푔 = (푣푔푑푖푔푑 + 푣푔푞푖푔푞) and 푄푔 3= (푣푔푞푖푔푑 − 푣푔푑푖푔푞) 3 (14) 2 P푃g == (v푣2gdi푖gd ++v푣gqigq푖 ) and andQ 푄g == (v푣gqigd푖 −−v푣gdigq푖 ) (14)(14) 푔 22 푔푑 푔푑 푔푞 푔푞 푔 22 푔푞 푔푑 푔푑 푔푞 where 푣⃗푐표푛푣,푑푞, 푖⃗푔,푑푞, and 푣⃗푔,푑푞 are→ the GSC→ output voltage,→ the GSC current and grid wherewhere v푣⃗conv푐표푛푣,dq,푑푞,, i푖⃗푔g,푑푞dq,, and v푣⃗푔g,,푑푞dq areare the the GSC GSC output output voltage, voltage, the the GSC GSC current current and and grid grid voltage vectors. 푃푔 and 푄푔 are the active and reactive powers regulated by the GSC. voltagevoltage vectors.vectors. P푃푔and andQ 푄푔are are the the active active and and reactive reactive powers powers regulated regulated by by the the GSC. GSC. Thus, the grid voltage q component is zerog and gthe active power is regulated with Thus,Thus, thethe gridgrid voltagevoltage qqcomponent component isis zerozero andand thethe activeactive powerpower isis regulatedregulated withwith the grid current d component and the reactive power is regulated with the grid current q thethe gridgrid currentcurrent dd componentcomponent andand thethe reactivereactive powerpower isis regulatedregulated withwith thethe gridgrid currentcurrent qq component [31,32]. componentcomponent [[31,3231,32].]. 3 3 3 3 푃 = 푣 푖 and 푄 = − P푣g =푖3 v i and Qg = − 3v (15)igq (15) 푔 2 푔푑 푔푑 푔 2푃 푔푑= 푔푞2 푣gd 푖gd and 푄 = −2 gd푣 푖 (15) 푔 2 푔푑 푔푑 푔 2 푔푑 푔푞 Figure 4 illustrates a block Figurediagram4 illustrates of the implemented a block diagram grid side of the power implemented converter grid side power converter Figure 4 illustrates a block diagram of the implemented grid side power converter control structure. control structure. control structure.

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C 

dq  C      SVPWM dq         SVPWM 

abc

abc PLL

PLL

Figure 4. Grid side converter control structure. Figure 4. Grid side converterFigure control 4. Grid structure. side converter control structure. The actual DC bus voltage is compared with the reference of the DC bus voltage and the difference will be the inputThe actual of a PI DCregulator bus voltage to obtain is compared at its output with the the reference reference of of the the DC bus voltage and ∗ ∗ active current componentthe difference 푖푔푑. Then, will 푖푔푑 be is the compared input of with a PI regulatorthe real 푖 to푔푑 obtain current at itscompo- output the reference ofof thethe ∗ ∗ nent to obtain at the output of the controller 푣푐표푛푣∗∗ ,푑. The ∗controller∗ 푣푐표푛푣,푞 is obtained active current componentcomponent i푖gd푔푑.. Then, Then,i gd푖푔푑is is compared compared with with the the real realigd current푖푔푑 current component compo- from the reactive power controller section. Lastly, the DC/AC inverter∗ pulses∗ SA, SB, ∗and ∗ nentto obtain to obtain at the at output the output of the controllerof the controllervconv,d. The푣푐표푛푣 controller,푑. The controllervconv,q is 푣 obtained푐표푛푣,푞 is fromobtained the SC are obtained usingfromreactive the the powersevenreactive controllersegments power controller section. space vector Lastly,section. pulse the Lastly DC/AC width, the DC/AC inverter modulation inverter pulses SA,pulses SB, SA, and SB SC, and are (SVPWM). SCobtained are obtaine using thed usingseven segments the seven space segments vector pulsespace width vector modulation pulse width (SVPWM). modulation (SVPWM). 4. Droop Control 4. Droop Control One DFIM could 4.be Droop picturedOne Control DFIM as a couldvoltage be source pictured that as is a connected voltage source to the that principal is connected or to the principal or local network, throughlocal an impedanceOne network, DFIM through couldline Z beas an picturedillustrated impedance as in a line voltageFigure Z as 5 source illustrated[33,34 ].that inis Figureconnected5[33 ,to34 the]. principal or local network, through an impedance line Z as illustrated in Figure 5 [33,34].

Figure 5. DFIGFigure generator 5. DFIG connected generator to connectedthe main grid to the trough main line grid impedance. trough line impedance. Figure 5. DFIG generator connected to the main grid trough line impedance. In a system working inIn a asinusoidal system working steady instate, a sinusoidal the apparent steady power, state, S, the which apparent flows power, S, which flows between the DFIG and the grid of Figure5, could be defined as, between the DFIG and the Ingrid a system of Figure working 5, could in bea sinusoidal defined as, steady state, the apparent power, S, which flows between the DFIG and the grid∗ of2 Figure 5→, could→ be!∗ defined as, 푣⃗푠 − 푣⃗푔 →∗푉푠 푉푠푉푔 2 ∗ → 푗휃→ v s −푗(v휃g+훿) Vs j VsVg j( + ) 푆 = 푃 + 푗푄 = 푣⃗푠푖⃗푠 ==푣⃗푠 (+ =) = =푒 − 푒 = θ −(16) θ δ S P 푍jQ∠휃 v s i s푍 v s 푍 ∗ 2 e e (16) 푣⃗푠Z−∠θ푣⃗푔 푉푠 Z 푉푠푉푔Z 푆 = 푃 + 푗푄 = 푣⃗ 푖⃗∗ = 푣⃗ ( ) = 푒푗휃 − 푒푗(휃+훿) (16) 푠 푠 푠 푍∠휃 푍 푍 whereas, 푉푠 and 푉푔 respectively are the voltage module of the DFIG and the main grid whereas, V and V respectively are the voltage module of the DFIG and the main grid voltage vectors, 훿 is the angle betweens gthe stator voltage of the DFIG and the grid volt- voltagewhereas vectors,, 푉푠 andδ 푉is푔 therespectively angle between are the the voltage statorvoltage module of of the the DFIG DFIG and and the the grid main voltage, grid age, also known as thevoltage power vectors angle,, Z훿 andis the 휃 angleare the between magnitude the stator and phasevoltage of of the the line DFIG and the grid volt- ∗ also known as the power angle, Z and θ are the magnitude and phase of the line impedance impedance and 푖⃗푠 is theage conjugated, →also∗ known complex as the vectorpower ofangle, the statorZ and current 휃 are .the The magnitude active and and phase of the line reactive power of Equationand i (16)s is can the be conjugated decomposed∗ complex as, vector of the stator current. The active and reactive impedance and 푖⃗푠 is the conjugated complex vector of the stator current. The active and power of Equation2 (16) can be decomposed as, reactive power푉푠 of Equation푉푠푉 푔(16) can be decomposed as, 푃 = 푐표푠(휃) − 푐표푠(휃 + 훿) 푍 푍 V22 VsVg P = 푉푠s cos(θ) − 푉푠푉푔cos(θ + δ) Z ( ) Z ( ) (17) 2 푃 = 2 푐표푠 휃 −V V 푐표푠 휃 + 훿 (17) 푉푠 푉푠푉푔 = V푍s ( ) − s푍g ( + ) 푄 = 푠푖푛(휃) − 푠푖푛Q (휃 +Z sin훿) θ Z sin θ δ (17) 푍 푍 2 푉푠 푉푠푉푔 푄 = 푠푖푛(휃) − 푠푖푛(휃 + 훿) Supposing that the impedance of the line is mostly inductive푍 , |푍|푍= 휔푔퐿 and θ = 90º, and that δ is near to zero, sin(δ) ≈ δ and cos(δ) ≈ 1, therefore (17) can be simplified to, Supposing that the impedance of the line is mostly inductive, |푍| = 휔푔퐿 and θ = 90º, and that δ is near to zero, sin(δ) ≈ δ and cos(δ) ≈ 1, therefore (17) can be simplified to,

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◦ Supposing that the impedance of the line is mostly inductive, |Z| = ωg L and θ = 90 , and that δ is near to zero, sin(δ) ≈ δ and cos(δ) ≈ 1, therefore (17) can be simplified to,

푉푠푉푔 푃 = 훿 VsVg (18) 휔푔퐿 P = δ (18) ωg L 푉 − 푉 푠 푔 Vs − Vg 푄 = 푉푠 (19) 휔 퐿Q = Vs (19) 푔 ωg L Equations (18)Equations and (19) show (18) andthat (19)regulating show thatthe active regulating power the the active angle powerδ or the the fre- angle δ or the quency of the frequencynetwork can of thebe adjusted network and can the be adjustedvoltage difference and the voltage 푉푠 − 푉푔 difference is regulatedVs − byV g is regulated the reactive power.by the It reactive should power.be stressed, It should that bethis stressed, is just true that when this is the just line true impedance when the is line impedance mainly inductive,is mainly which inductive, is often the which case. is The often aim the of case. droop The regulation aim of droop is to regulation adapt the is to adapt the frequency andfrequency amplitude and of the amplitude grid voltage of the independently grid voltage independently by regulating bythe regulating active and the active and reactive power.reactive power. The frequencyThe and frequency voltage droop and voltagecharact drooperistics characteristics are shown in Figure are shown 6. in Figure6.

FigureFigure 6. Droop 6. Droop characteristics. characteristics. Left Left frequency, frequency, right right voltage. voltage.

where, Where, ω − ω = − 0 Pmax 휔0 − K휔p푃푚푎푥 (20) 퐾푝 = − Pmax − Pdel (20) 푃푚푎푥 − 푃푑푒푙 Vg0 − VgQmax Kq = − (21) 푉푔0 − 푉푔푄푚푎푥 QSmax − QS0 퐾푞 = − (21) and, ω0 is the rated frequency given푄푆푚푎푥 to− the푄푆0 power Pdel and Vg0 is the rated voltage amplitude given to the reactive power Q . Pmax is the maximum active power that the wind turbine and, 휔0 is the rated frequency given to theS power0 푃푑푒푙 and 푉푔0 is the rated voltage am- can obtain from the wind speed and QSmax is the maximum reactive power of the stator. plitude given to the reactive power 푄푆0. 푃푚푎푥 is the maximum active power that the wind KP and KQ are the frequency and voltage droop coefficients, respectively. ωP and turbine can obtain from the wind speed and 푄푆푚푎푥 is the maximum reactive power of the max Vg are the admitted minimum grid frequency and amplitude for the possible Pmax and stator. 퐾푃 and 퐾Qmax푄 are the frequency and voltage droop coefficients, respectively. 휔푃푚푎푥 QSmax respectively. and 푉푔푚푎푥 are the admitted minimum grid frequency and amplitude for the possible 푃푚푎푥 and 푄푆푚푎푥5. Gettingrespectively. Additional Active Power of the Wind Generator In order to make a frequency regulation when the frequency of the system falls, some 5. Getting Additional Active Power of the Wind Generator extra energy must be extracted from the wind turbine. Therefore, DFIG wind turbines In order tomust make run a frequency with a deloaded regulation power when curve the when frequency the plant of the is runningsystem falls, under some regular frequency extra energy mustrequirements be extracted [35,36 from]. The the power wind transferred turbine. Therefore, to the shaft DFIG of a wind generatorturbines is stated as must run with a deloaded power curve when the plant is running under regular frequency requirements [35,36]. The power transferred to the shaft1 of a wind generator is stated as P = C (λ, β)ρ πR2v3 (22) turb 2 P air w 1 2 3 푃푡푢푟푏 = 퐶푃(휆, 훽)휌푎푖푟휋푅 푣푤 (22) where ρair, is the mass density2 of the air, R is the radius of the propeller, CP(λ, β) is the power performance coefficient, v is the wind speed, β is the pitch angle and λ is the blade where 휌푎푖푟, is the mass density of the air, R isw the radius of the propeller, 퐶푃(휆, 훽) is the tip speed ratio and is defined as, power performance coefficient, 푣푤 is the wind speed, β is the pitch angle and λ is the Rωpr blade tip speed ratio and is defined as, λ = (23) vw 푅휔푝푟 Taking into account Equation휆 = (22), the wind turbine is deloaded by(23) influencing the 푣 performance of the power coefficient.푤 Cp depends on the tip speed ratio and pitch angle; Taking intothus, account by adjusting Equation one (22), or both the wind factors, turbine the wind is deloaded turbine could by influencing be deloaded. the performance of the power coefficient. Cp depends on the tip speed ratio and pitch angle; thus, by adjusting one or both factors, the wind turbine could be deloaded.

Mathematics 2021Mathematics, 9, x FOR PEER2021 REVIEW, 9, 143 8 of 18 8 of 18

The term appliedThe to term the applied Cp factor to is the describedCp factor in is Equation described (24), in Equation (24), 𝑛1𝜆𝑛6 + −𝑛4 − 𝑛3 2.5 + 𝛽 +𝑛2𝐴 −𝑛8𝐵 n1(λn6+(−n4−n3(2.5+β)+n2 A−n8B)) 𝐶 𝜆, 𝛽 = CP(λ, β) = 𝑒 e(n5 A−n8B) 1 1 (24) (24) A = and B = 3 1 λ+1n7(2.5+β) 1+(2.5+β) 𝐴 = and 𝐵 = 𝜆+𝑛72.5 + 𝛽 1+2.5 + 𝛽 −3 where, n1 = 0.645, n2 = 116, n3 = 0.4, n4 = 5, n5 = 21, n6 = 9.12 × 10 , n7 = 0.08 and where, n1 = 0.645, n2 = 116, n3 = 0.4, n4 = 5, n5 = 21, n6 = 9.12e-3, n7 = 0.08 and n8 = 0.035 [37]. n8 = 0.035 [37]. The power of Thethe turbine power ofproduced the turbine with produced the Cp coefficient, with the Cp forcoefficient, a zero pitch for angle a zero and pitch angle and for various windfor variousspeeds in wind relation speeds to the in relationspeed of to the the DFIG speed rotor of the is illustrated DFIG rotor in is Figure illustrated 7. in Figure7.

1.4 β= 0º 13m/s 1.2 Cp max

1 12m/s A 0.9Cp max B 0.8 11m/s

0.6 10m/s Turbinepower (p.u.) 0.4 9m/s Speed limit 8m/s 0.2 7m/s 0 ω 0.4 0.6 0.8 1 1.2 1.4 rmax 1.6 Rotor speed (p.u.) FigureFigure 7. Turbine 7. Turbine deloaded deloaded and maximum and maximum power power curves. curves.

With EquationsWith (24) Equations and (22), (24) various and (22),turbine various power turbine characteristics power characteristics are determined are determined as as illustrated inillustrated Figure 7. inThe Figure red one7. The belongs red one to the belongs maximum to the achievable maximum power, achievable in which power, in which the power coefficientthe power is the coefficient highest isavailable. the highest The available. blue and Thegreen blue curves and are green the curves deloaded are the deloaded power curves.power The blue curves. curve The represents blue curve a 10% represents deloaded a 10% power deloaded curve at power each curvegiven atwind each given wind speed, being ablespeed, to beingextract able when to extractnecessary when a 10% necessary of the maximum a 10% of the power maximum for that power wind for that wind speed. However,speed. the However, green curve the greenallows curve 10% allowsof the maximum 10% of the maximumpower to be power extracted to be extractedat at any any wind speed.wind The speed. deload The deloadprocedure procedure decreases decreases the efficiency the efficiency of a ofwind a wind turbine turbine by by raising the tip speed ratio. Thus, the deload curves shift to the right of the maximum power curve. raising the tip speed ratio. Thus, the deload curves shift to the right of the maximum Because of the increase in speed, kinetic energy is stored in the inertia of the wind turbine power curve. Because of the increase in speed, kinetic energy is stored in the inertia of the and is expressed as wind turbine and is expressed as 1   E = J ω2 − ω2 (25) 1 kinetic 2 T d1 opt 𝐸 = 𝐽 𝜔 −𝜔 (25) 2 where JT is the wind turbine inertia, ωd1 is the turbine deloaded speed and ωopt is the where 𝐽 is themaximum wind turbine power inertia, speed or𝜔 optimum is the turbine speed asdeloaded illustrated speed in Figure and 𝜔7. This is the energy could be maximum poweremployed speed or for optimum short-term speed frequency as illustr controlated in [38 Figure]. The 7. additional This energy active could power be that can be employed forachieved short-term for frequency the deloading control operation [38]. The can additional be derived active from power Equation that (22) can and be is dependent achieved for theon deloading the wind speed operation and thecan deloadingbe derived percentage from Equation used (22) as and is dependent on the wind speed and the deloading percentage used as 1 ∆P = ∆C (λ, β)ρ πR2v3 (26) 1 turb 2 P air w Δ𝑃 = Δ𝐶 𝜆, 𝛽𝜌 𝜋𝑅𝑣 (26) 2 ∆Pturb is the power reserve, ∆CP is the increase in the power coefficient when the Δ𝑃 isturbine the power moves reserve, from Δ𝐶 the deloadedis the increase working in the point power to acoefficient greater power when line. the This power turbine movesincrease from the is utilized deloaded for working long-term point frequency to a greater adjustment power [30 ].line. This power increase is utilizedThe for long-term reference calculation frequency ofadjustment the downloaded [30]. DFIG speed for every wind speed starts The referencewith thecalculation specification of the ofdownloaded the optimal DFIG tip speed speed relation. for every Thereafter, wind speed with starts the extra active with the specificationpower demanded of the optimal by the tip smart speed grid relation. supervisor, Thereafter, the new with tip speed the extra ratio active is computed based power demandedon Equations by the smart (23) grid and supervisor (24). , the new tip speed ratio is computed based on Equations (23) andObserving (24). Figure7, for 12 m/s of wind speed, the wind turbine can run 10% deloaded as in point B. When the grid frequency decreases, due to a sudden increase in the load, the

Mathematics 2021, 9, x FOR PEER REVIEW 9 of 18

Mathematics 2021, 9, 143 9 of 18 Observing Figure 7, for 12 m/s of wind speed, the wind turbine can run 10% deloaded as in point B. When the grid frequency decreases, due to a sudden increase in the load, theDFIG DFIG must must boost boost the the set set point point of of the the desired desired active active power. power. Consequently, Consequently, the the operating operat- ingpower pow coefficienter coefficient of the of the wind wind turbine turbine moves moves from from the pointthe point B of theB of deloaded the deloaded curve curve to the topoint the point A of theA of higher the higher power power curve. curve. InIn this this way, way, the the increase increase in in active power supportssupports thethe controlcontrol ofof thethe grid grid frequency. frequency If. Ifthe the highest highest deloaded deloaded powerpower isis achieved,achieved, (point B) due to to increased increased wind wind speed, speed, the the pitch pitch controllercontroller will will be be required required to to maintain maintain the the required required power power reserve, reserve, which which is is the the difference difference betweenbetween the the red red and and blue blue horizontal horizontal lines. lines. TheThe pitch pitch controller controller can can be be used used also also to to avoid avoid overs overstresstress or or to to limit limit the the speed speed of of the the windwind turbine turbine while while regulating regulating the the frequency frequency [20]. [20 ]. TheThe major major issue issue of of the the wind wind generator generator is is the the uncontrollability uncontrollability of of the the wind wind as as its its speed speed variationsvariations affect affect the the amount amount of of energy energy stored stored.. The The manner manner of of saving saving the the desired desired amount amount of of power,power, regardless regardless of of the the wind wind speed, speed, is is to to deload deload the the wind wind turbine turbine for for a a specified specified power power valuevalue as as shown shown in in Figure Figure 77.. For For instance, instance, the the green green curve curve shows shows that that the the reserved reserved power power is is 10%10% of of the the maximum maximum power power for for all all speeds. speeds. However,However, working working in in this this way way wastes wastes a large a large amount amount of wind of wind energy energy at low at low speeds. speeds. In FigureIn Figure 8, a8 diagram, a diagram is shown is shown for the for generation the generation of the of active the active power power reference reference to be intro- to be ducedintroduced in the inP* the inputP* inputof the of DFIG the DFIG control control scheme scheme presented presented in Figure in Figure 2. Fi2gure. Figure 8 also8 also in- cludesincludes droop droop control, control, inertial inertial energy energy,, and and the thepitch pitch controller. controller. The Thepitch pitch controller controller will willact whenact when the activethe active power power surpa surpassessses thethe maximum maximum deloaded deloaded power, power, the the peak peak mechanical mechanical powerpower or or when when the the wind wind turbine turbine speed speed is is greater greater than than the the higher higher lim limitit [39]. [39]. The The control control commandcommand Δ∆푃푠푢푝푒푟푣푖푠표푟Psupervisor isis applied by by the the network supervisor supervisor for for other other control control strategies strategies suchsuch as as automatic automatic control control of of generation generation and and energy energy flow flow or or to to limit limit the the energy energy level level to to a a specifiedspecified level level [20,40 [20,40]. ].

1.4 β = 0º 13m/s 1.2 Cp max

.) 1 12m/s p.u A 0.9Cp max B 0.8 11m/s

0.6 10m/s

Turbine power ( power Turbine 0.4 9m/s Speed limit 8m/s 0.2 7m/s 0 0.4 0.6 0.8 1 1.2 1.4 r max 1.6 Rotor speed (p.u.) Measured

FigureFigure 8. 8. DFIGDFIG reference reference generation generation of of active active power. power. D Drooproop and and smart grid supervisor defineddefined control.con- trol. 6. Reactive Power and Voltage Amplitude Control 6. ReactiveThe amplitude Power and of Voltage the stator Amplitude voltage is Control regulated by adjusting the reactive power as describedThe amplitude in the Section of the4 .stator voltage is regulated by adjusting the reactive power as de- scribedThe in the reactive Section energy 4. is controlled by the GSC and by the RSC [17]. Both converters are dimensionedThe reactive to handleenergy is about controlled 30% of by the the nominal GSC and power by the of RSC thegenerator. [17]. Both Convertersconverters are are dimensionedprincipally utilizedto handle to about supply 30% the of active the nominal energy frompower the of rotor the generator. to the stator Converters or vice versa. are The GSC can be employed to supply reactive energy together with the stator to the grid. principally utilized to supply the active energy from the rotor to the stator or vice versa. The It should be noted that the maximum grid-side converter current cannot be exceeded GSC can be employed to supply reactive energy together with the stator to the grid. and therefore the amount of reactive power that can be injected by the grid-side converter It should be noted that the maximum grid-side converter current cannot be exceeded will depend on the amount of active power flowing through the rotor. Thus, when the and therefore the amount of reactive power that can be injected by the grid-side converter speed of the rotor is synchronous, the active power through the rotor is approximately zero will depend on the amount of active power flowing through the rotor. Thus, when the and therefore the reactive power could be maximum. However, as the power through the speed of the rotor is synchronous, the active power through the rotor is approximately rotor increases due to an increase in slip, the reactive power must be reduced. As shown in Equation (12), the reactive power through the stator is regulated by the rotor current d-component. When the load increases suddenly, not only does the frequency

MathematicsMathematics 2021, 20219, x FOR, 9, x PEERFOR PEER REVIEW REVIEW 10 of 1810 of 18

Mathematicszerozero 2021and, 9thereforeand, 143 therefore the reactive the reactive power power could could be maximum. be maximum. However, However, as the as power the power through through 10 of 18 the rotorthe rotor increases increases due dueto an to increase an increase in slip, in slip,the reactive the reactive power power must must be reduced. be reduced. As shownAs shown in Equation in Equation (12), (12),the reactivethe reactive power power through through the statorthe stator is regulated is regulated by the by the rotor current d-component. When the load increases suddenly, not only does the frequency rotor current d-component.decrease, When the but load also increases the amplitude suddenly, of the not grid only voltage does the tends frequency to be reduced. Because of that, decrease, but also the amplitude of the grid voltage tends to be reduced. Because of that, the decrease, but also the amplitudethe active of the and grid reactive voltage power tends increases to be reduced. trying Because to correct of thethat, frequency the and the amplitude active and reactive power increases trying to correct the frequency and the amplitude of the active and reactive powerof increases the grid trying voltage. to correct the frequency and the amplitude of the grid voltage. grid voltage. The rotor current must be limited to its maximum value taking into account the active The rotorThe rotor current current mustand must be reactive limited be limited power to its to values.maxi its maximummum Equation value value (27),taking taking derived into intoaccount from account Equations the active the active (11) and (12), provides the and andreactive reactive power power values. values.maximum Equation Equation active (27), and(27),derived reactivederi vedfrom powersfrom Equations Equations permitted (11) (11)and considering and(12), (12),provides theprovides rotor maximum current, the maximumthe maximum active active and reactive and reactive powers powers permitted permitted considering considering the rotor the maximum rotor maximum cur- rent,current,  → 2 2 v s  2 2 3 3 → Lm   2|(2P|) + Q − 2  = v i 2 + i 2 (27) 3 |푣⃗푠3| 𝑣⃗ s 3 3퐿s 푚 𝐿 s rd rq 2  | |2 ωe L2s  2 2 Ls (27) (푃푠) 𝑃+ (푄+𝑄푠 − − ) =( =|푣⃗푠| 𝑣⃗) (푖푟푑 +𝑖푖푟푞+𝑖) (27) 2 휔푒퐿2푠𝜔𝐿 2 2퐿푠 𝐿 A graphicalA graphical representation representationA graphical of Equation of Equation representation (27) (27)in Figure in Figure of 9 Equation, illustrates 9, illustrates (27) the in full Figurethe rangefull9 ,range illustrates of of the full range of 𝑃 −𝑄 𝑄 푃푠 − 푄푠 generation generation in steady inPs steady− Qstate.s generation state. If 푄 푠If is greater in is steady greater than state. than zero,If zero, Qthes is DFIGthe greater DFIG absorbs than absorbs zero,reactive reactive the DFIG absorbs reactive energyenergy due dueto its to inductive its inductiveenergy feature duefeature [41 to] its ,[41], however, inductive however, 푄 feature푠 values𝑄 values [41 lower], however, lower than than zeroQs valueszero correspond correspond lower than zero correspond to to theto reactive the reactive power power thatthe thatthe reactive DFIG the DFIG is power able is ableto that supply to the supply DFIG to the to is grid. ablethe grid. to supply to the grid.

= =

𝑃 −𝑄 FigureFigure 9. StatorFigure 9. Stator 푃 9.푠 −Stator 푄푠 boundariesPs − boundariesQs boundaries for rated for rated current forrated current of currentthe of rotor the of rotor.. the rotor.

FigureFigure 10 illustrates 10 illustrates the red theFigure curvered 10curve of illustrates maximum of maximu the mechanical redm mechanical curve ofpower maximum power that thatcan mechanical becan obtained be obtained power that can be obtained for thefor specified the specified wind wind speeds.for speeds. the The specified blueThe blueand wind andgreen speeds. gr curveseen curves The show blue show the and reactive the green reactive curvespower power showthat thatthe the the reactive power that the DFIGDFIG can providecan provide to the to gridDFIG the gridwhen can when providethe generator the togene therator is grid operating is when operating the at rated generator at rated rotor rotor iscurrent. operating current. at rated rotor current.

1 1 Q β = 0º β= 0º Qs s P Pmechanic .) mechanic 12m/s 0.8 0.8 12m/s

p.u 11m/s 11m/s 0.6 0.6 10m/s Q Qs 10m/s 0.4 0.4s

0.2 0.2 Active and reactive power (p.u.) Active and reactive power ( power reactive and Active 8m/s 8m/s 6m/s 6m/s 0 0 0.4 0.6 0.8 1 1.2 1.4 1.6 0.4 0.6 0.8 1Rotor speed1.2 (p.u.) 1.4 1.6 Rotor speed (p.u.) FigureFigure 10. Maximum 10. Maximum mechanical mechanicalFigure power 10. powerMaximum and reactiveand mechanicalreactive powers powe obtainable powerrs obtainable and for reactive maximum for maximum powers rotor obtainable rotor current. current. for maximum rotor current. DFIGDFIG 7.5 kW, 7.5 greenkW, green curve. curve. DFIG DFIG 7.51.2 kW,MW, 1.2 greenMW, blue blue curve. curve. DFIG 1.2 MW, blue curve.

The blue Qs line corresponds to a 1.2 MW DFIG and the green line to a 7.5 kW DFIG. The blueThe blue 푄푠 line𝑄 corresponds line corresponds to a 1.2to a MW 1.2 MW DFIG DFIG and andthe green the green line toline a 7.5to a kW 7.5 DFIG.kW DFIG. WhenWhen the DFIG the DFIG is taking is takingWhen the maximum the the maximum DFIG power is taking power of the of wind, maximumthe wind, all the all power rotor the rotor ofcurrent the current wind, is fo allrmedis formed the rotor current is formed by theby qthe component q component andby andthe the valuethe q componentvalue of the of dthe component andd component the value will ofwillbe the zero. be d zero. componentThis Thisimplies implies will that be thatthe zero. the This implies that the reactive power is absorbed from the grid and its value will be defined by Equation (12). As reactivereactive power power is absorbed is absorbed from from the gridthe gridand andits value its value will willbe defined be defined by Equation by Equation (12). (12). shown in the figure, the lower the mechanical power, the more reactive power the DFIG As shownAs shown in the in figure,the figure, the lowerthe lower the mechanicalthe mechanical power, power, the morethe more reactive reactive power power the the can provide to the grid to compensate for the voltage drop in the grid. DFIGDFIG can providecan provide to the to grid the gridto compensate to compensate for the for voltage the voltage drop drop in the in grid. the grid. The selection of the maximum active power in the grid frequency compensation does not provide any choice to assist in the correction of the grid amplitude from the stator, but the GSC could be used if their current limit is maintained. This is why certain criteria must be detailed by the smart grid controller in the distribution of the maximum active and reactive powers while respecting the current limits of the stator, rotor and the GSC. Mathematics 2021, 9, x FOR PEER REVIEW 11 of 18

The selection of the maximum active power in the grid frequency compensation does not provide any choice to assist in the correction of the grid amplitude from the stator, but Mathematics 2021, 9, x FOR PEER REVIEWthe GSC could be used if their current limit is maintained. This is why certain criteria11 mustof 18

be detailed by the smart grid controller in the distribution of the maximum active and reactive powers while respecting the current limits of the stator, rotor and the GSC. Figure 11 providesThe selection a diagram of the formaximum generating active the power reactive in thepower grid references frequency ofcompensation the stator and does the notGSC provide converter. any choiceThe value to assist of 퐾 in푄 thedepends correction on the of capacitythe grid amplitudeof the DFIG from to supply the stato reactiver, but Mathematics 2021, 9, 143 11 of 18 thepower GSC andcould this be used value if changestheir current as the limit active is maintained. power varie Thiss as is why shown certain in Figure criteria 10. must The be∆ 푄detailed푠푢푝푒푟푣푖푠표푟 by control the smart signal grid is controller employed in by the the distribution grid controller of the for maximum additional active regulation and reactivestrategies, power likes forwhile example respecting power the factor current compensation. limits of the stator, rotor and the GSC. Figure 11 providesFigure a diagram 11 provides for generating a diagram the reactive for generating power thereferences reactive of power the stator references and the of the stator GSC converter.and The the GSCvalue converter. of 퐾푄 depends The value on the of KcapacityQ depends of the on DFIG the capacity to supply of thereactive DFIG to supply power and reactive this value power changes and this as the value active changes power as varie the actives as shown power in varies Figure as shown 10. The in Figure 10. ∆푄푠푢푝푒푟푣푖푠표푟 Thecontrol∆Qsupervisor signal iscontrol employed signal by is employedthe grid controller by the grid for controller additional for regulation additional regulation strategies, likestrategies, for example like for power example factor power compensation. factor compensation. Measured

Measured Figure 11. Diagram for generating the reactive power references of the stator and the GSC converter.

7. Experimental Results Figure 12 shows the picture of the real test bench for testing the suggested controlling solutions. The system consists of a 7.5 kW DFIM and a 10 kW synchronous machine Figure(PMSM) 11. Diagram thatFigure performs for 11. generatingDiagram the tasks for the generatingof reactive a household power the reactive references wind powerturbine of th referencese [42 stator–45]. and of The thethe probes statorGSC converter and for the mon- GSC. converter. itoring all the currents and voltages of the system are matched and wired to a DS1103 7.dSPACE Experimental [44]7. Experimentalco Resultsntrol system. Results FigureThe gate 12 shows signalsFigure the of12 picture theshows RSC of the theand picture real GSC test ofIGBTs thebench real are for test generated testing bench the for by suggested testing the DS1103 the controlling suggested and the controlling solutions. The system consists of a 7.5 kW DFIM and a 10 kW synchronous machine solutions.rotor position The and system speed consists are determined of a 7.5 kW by a DFIM 4096 encoder and a 10 wired kW to synchronous the DS1103. machineThe main (PMSM) that performs the tasks of a household wind turbine [42–45]. The probes for (PMSM)characteristic that performss of the doublethe tasks feed of a induction household machine wind turbine are given [42– 45].in Table The probes 1. The for RSC mon- and monitoring all the currents and voltages of the system are matched and wired to a DS1103 itoringGSC are all two the NFScurrents-200 convertersand voltages with of Mitsubithe systemshi IGBTsare matched fabricated and by wired Dutt to[46 a]. DS1103 Three 2 dSPACE [44] control system. dSPACEmH @ 15A [44] inductors control system. form the line filter between the GSC and the grid. The gate signals of the RSC and GSC IGBTs are generated by the DS1103 and the rotor position and speed are determined by a 4096 encoder wired to the DS1103. The main characteristics of the double feed induction machine are given in Table 1. The RSC and GSC are two NFS-200 converters with Mitsubishi IGBTs fabricated by Dutt [46]. Three 2 mH @ 15A inductors form the line filter between the GSC and the grid.

FigureFigure 12. 12.DFIG DFIG platform platform used used to testto test the the proposed proposed control control strategies. strategies.

The gateTable signals 1. Leroy of the Somer RSC DFIM and GSCmain IGBTsparameters are generated by the DS1103 and the rotor position and speed are determined by a 4096 encoder wired to the DS1103. The main Parameter Value characteristics of the double feed induction machine are given in Table1. The RSC and Stator voltage 380 V GSC are two NFS-200 converters with Mitsubishi IGBTs fabricated by Dutt [46]. Three FigureRotor 12. voltageDFIG platform used to test the proposed control190 strategies. V 2 mH @ 15A inductors form the line filter between the GSC and the grid. Rated stator current 18 A Rated rotorTable current 1. Leroy Somer DFIM main parameters 24 A

Parameter Value Stator voltage 380 V

Rotor voltage 190 V Rated stator current 18 A Rated rotor current 24 A

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Rated speed 1447 rpm @ 50 Hz Rated torque 50 Nm Stator resistance 0.325 Ω Rotor resistance 0.275 Ω Magnetizing inductance 0.0664 H Mathematics 2021, 9, 143 Stator leakage inductance 0.00264 H 12 of 18 Rotor leakage inductance 0.00372 H Inertia moment 0.07 Kg*m2 Table 1. Leroy Somer DFIM main parameters Before connecting the DFIM to the grid, the DC bus formed by the RSC and GSC ca- pacitors in a back-toParameter-back configuration must be charged. Once the Value DC Bus is charged and if the output voltageStator vector voltage of the GSC is aligned with the grid voltage 380 V vector, the GSC con- nects to the grid andRotor starts voltage the regulation of the DC Bus voltage to 190a set V value of 580 V. When theRated wind stator (emulated current by the PMSM) reaches the minimum 18 A speed of 5 m/s, the double fed inductionRated rotor machine current starts the process of hooking up to 24the A grid. This is done in Rated speed 1447 rpm @ 50 Hz two stages. In the first stage, the encoder offset with respect to the stator flux must be ob- Rated torque 50 Nm tained. In the secoStatornd stage, resistance the voltage vector generated by the stator 0.325 mustΩ be aligned with the grid voltage Rotorvector. resistance When the last process is complete, the DFIM 0.275’s statorΩ is connected to the grid andMagnetizing the actual regulation inductance process begins. 0.0664 H FigureStator 13 shows leakage the inductanceregulation of the DFIG when the wind 0.00264speed changes H from 7 to 12 and then toRotor 15 m/s. leakage This inductance means a rotor speed from 900 (sub-synchronous 0.00372 H speed) to 1500 Inertia moment 0.07 Kg*m2 (synchronous speed) and then to 1900 rpm (super-synchronous speed) respectively. In the DFIG the energy from the stator always flows into the grid. The flow of energy via theBefore rotor connectingchanges sense the depending DFIM to the on grid,the speed the DCof the bus DFIG. formed Thus, by when the RSC the andspeed GSC of thecapacitors DFIG is inlower a back-to-back than the synchronous configuration speed, must the be energy charged. of the Once rotor the is DCabsorbed Bus is from charged the gridand and if the goes output from voltage the rotor vector to the of thestator, GSC in isthe aligned figure withthe interval the grid from voltage 0.5 to vector, 1.5 s theat 7 GSCm/s ofconnects wind speed. to the Because grid and of starts this, the regulationtotal power of is the inferior DC Bus to the voltage power to of a setthe value stator. of When 580 V. the DFIGWhen speed the windis higher (emulated than the by synchronous the PMSM) speed, reaches the the rotor minimum energy speed changes of 5sense m/s, and the goesdouble from fed the induction rotor to the machine grid. starts the process of hooking up to the grid. This is done in twoAs stages.soon as In the the total first power stage, exceeds the encoder the nominal offset with power respect (7.5 tokW) the and stator the fluxwind must speed be increases,obtained. the In the pitch second regulator stage, forces the voltage a modification vector generated of the pitch by the angle stator to mustensure be that aligned the generatorwith the griddoes voltagenot exceed vector. the nominal When the power. last process As it can is be complete, observed the in DFIM’sthe figure, stator since is noconnected changes toare the made grid to and the the reactive actual power regulation reference, process the begins. d component of the rotor current is constant.Figure However,13 shows thedue regulation to changes of in the wind DFIG speed when and the therefore wind speed in mechanical changes from power, 7 to the12 andq component then to 15 of m/s. the Thisrotor means changes a rotor as does speed the from powe 900r of (sub-synchronous the stator. speed) to 1500 (synchronous speed) and then to 1900 rpm (super-synchronous speed) respectively.

Figure 13. DFIG signals for a wind speed from 7 to 15 m/s. Upper graph, wind speed reference. FigureSecond 13 graph,. DFIG stator, signals rotor, for turbinea wind andspeed total from electrical 7 to 15 power.m/s. Upper Third graph, graph, wind pitch speed angle. reference. Fourth graph, Secondreferences graph, and stator, actual rotor, values turbine of rotor and current total delectricaland q components. power. Third graph, pitch angle. Fourth graph, references and actual values of rotor current d and q components. In the DFIG the energy from the stator always flows into the grid. The flow of energy via the rotor changes sense depending on the speed of the DFIG. Thus, when the speed of the DFIG is lower than the synchronous speed, the energy of the rotor is absorbed from the grid and goes from the rotor to the stator, in the figure the interval from 0.5 to 1.5 s at 7 m/s of wind speed. Because of this, the total power is inferior to the power of the stator. When the DFIG speed is higher than the synchronous speed, the rotor energy changes sense and goes from the rotor to the grid. As soon as the total power exceeds the nominal power (7.5 kW) and the wind speed increases, the pitch regulator forces a modification of the pitch angle to ensure that the generator does not exceed the nominal power. As it can be observed in the figure, since no Mathematics 2021, 9, 143 13 of 18

Mathematics 2021, 9, x FOR PEER REVIEW 13 of 18 Mathematics 2021 , 9, x FOR PEER REVIEW 13 of 18 changes are made to the reactive power reference, the d component of the rotor current is constant. However, due to changes in wind speed and therefore in mechanical power, the q component of the rotor changes as does the power of the stator. Figure 14 shows the implementation made on the real platform to experimentally check Figure 14 shows the implementationFigure 14 shows made the implementationon the real platform made to experimentally on the real platform check to experimentally the contribution of the DFIG on the voltage amplitude when the load is connected to the the contribution of thecheck DFIG the on contribution the voltage of amplitude the DFIG onwhen the voltagethe load amplitude is connected when to thethe load is connected to grid. grid. the grid.

Figure 14. Single line diagram of the system implemented and used for testing. Figure 14. SingleFigure line 14. diagramSingle line of the diagram system of im theplemented system implemented and used for and testing. used for testing.

To observeTo the observe voltageTo the variation observe voltage the at variation the voltage PCC at variation(point the PCC of atcommon (point the PCC of coupling common (point of), coupling an common inductive), coupling), an inductive an inductive load is set where 푅퐿 = 0 and 퐿퐿 = 20 푚퐻. The wind speed is fixed to 6 m/s and Figure 15 load is set where 푅퐿load= 0 isand set 퐿 where퐿 = 20R 푚퐻L =. 0Theand windLL = speed20 mH is. fixed The windto 6 m/s speed and is Figure fixed to15 6 m/s and Figure 15 shows theshows grid phase theshows grid voltage phase the amplitude grid voltage phase amplitude reduction voltage amplitude reductionwhen the reductionloadwhen is the connected whenload is the connected at loadt = 0.5 is s. connected at t = 0.5 s. at t = 0.5 s. Due to theDue inductive to theDue inductivecharacteristic to the inductivecharacteristic of the characteristic line ofand the the line inductive of and the linethe inductiveload, and the there inductive load, is a 5there V load, reduc- is therea 5 V isreduc- a 5 V reduction tion in thetion PCC. in theAtin 0.75 PCC. the sPCC. Atthe 0.75 Atvoltage 0.75 s the s compensation the voltage voltage compensation compensation is activated, is is activated, supplying activated, supplying the supplying DFIG the the DFIG DFIG through through thethrough stator withthethe stator statora capacitive with with a capacitive areactive capacitive power reactive reactive of 7power KVAR power of (maximum 7 of KVAR 7 KVAR (maximum rotor (maximum current). rotor rotor current). current). This This resultsThis in aresults voltageresults in increasea voltage in a voltage on increase the PCC increase on of the 1.5 on PCC V. the Since of PCC 1.5 the V. of GSCSince 1.5 V. ( gridthe Since GSC side the (convertergrid GSC side (grid )converter side converter)) is is able to provideis able to reactive provideable to power, providereactive 14 reactive power,KVAR of14 power, capacitiveKVAR 14 of KVAR capacitive reactive of capacitive power reactive is injected reactivepower isat power injected 1.25 is injectedat 1.25 at 1.25 s s ordered sby ordered the grid orderedby s upervisor,the grid by the supervisor, which grid supervisor, causes which an whichcausesadditional causesan additional 3 anV increase additional 3 V bringing increase 3 V increase thebringing bringing the the PCC PCC voltagePCC closer voltage tovoltage the closer grid closer to voltage. the to grid the gridvoltage. voltage.

Figure 15. LoadFigure voltage 15. Load for inductive voltage for load inductive and voltage load compensation and voltage compensation with the controlled with the DFIG controlled re- DFIG Figure 15. Load voltage for inductive load and voltage compensation with the controlled DFIG re- active powerreactive for Figure power 14 forstructure. Figure 14 structure. active power for Figure 14 structure. Due to theDue impossibility to theDue impossibility of to varying the impossibility of the varying frequency of the varying frequencyon the the real frequencyon platform, the real on aplatform, simulation the real platform,a simulation is a simulation is is performedperformed to observeperformed to the observe frequency tothe observe frequencyvariation the on variation frequency the PCC. on variationThe the gridPCC. is on The configured the grid PCC. is configured Theso that grid its is so configured that its so that frequency frequencyvaries as a its variesfunction frequency as aof function the varies active of as powerthe a function active consumed, power of the consumed, as active occurs power in as a occurs generator. consumed, in a generator.Thus, as occurs Thus, in a generator. the grid decreasesthe grid bydecreasesThus, 1 Hz when the by grid 1 there Hz decreases when is an thereactive by is 1power an Hz active when consumption power there consumption is an of active10 kW.power Therefore,of 10 kW consumption. Therefore, of 10 kW. in our testin the our frequency testTherefore, the frequencyvalue indrops our value testfrom thedrops 50 frequency Hz from when 50 value thereHz when dropsis no there consumption from is 50 no Hz consumption when to 49 there Hz isto no49 consumptionHz when the consumptionwhen the consumptionto 49in Hzthe whengrid in is the the10 consumptionkW.grid is 10 kW. in the grid is 10 kW. If the powerIf theis negative powerIf theis (the negative power grid isabsorbs (the negative grid energy), absorbs (the grid the energy), absorbsfrequency the energy), frequencyincreases the frequencyinincreases the same in increases the same in the same ratio. The activeratio. Thepower activeratio. is consumed Thepower active is consumed when power the is loadwhen consumed is theconnected load when is connectedin the the load line inof is the connectedFigure line 1 of4 withFigure in the 1 line4 with of Figure 14 a resistivewith load awhere resistive 푅퐿 = load29 Ω where and 퐿R퐿L==029, thatΩ andis 5.5L kWL = are0, thatconsumed is 5.5 kW for the are load. consumed for the a resistive load where 푅퐿 = 29 Ω and 퐿퐿 = 0, that is 5.5 kW are consumed for the load. Figure 16 illustratesFigureload. 16 illustratesthe behavior the of behavior the system of the by systemshowing by the showing frequency the frequencyof the PCC, of the PCC, the wind speedthe wind and speed the activeFigure and thepowers 16 activeillustrates of powersthe load, the of behavior the the grid load, ofand the the the grid system DFIG. and by theThe showing DFIG. wind Thespeed the frequencywind speed of the PCC, has been sethas at been 7 m/s setthe up at windto 7 m/s1 s. speed Withup to andthis 1 s. thewindWith active thethis power powerswind the that of power thethe load,DFI thatG the deliversthe grid DFI andG is delivers1.5 the kW. DFIG. is 1.5 The kW. wind speed has been set at 7 m/s up to 1 s. With this wind the power that the DFIG delivers is 1.5 kW. Up to 0.5 sUp the to grid 0.5 absorbss the grid this absorbs power this and power the grid and frequency the grid frequencyat the PCC at point the PCC is 50.15 point Hz. is 50.15 Hz. Up to 0.5 s the grid absorbs this power and the grid frequency at the PCC point is 50.15 Hz. At 0.5 s, theAt load 0.5 sthat, the absorbs load that 5.5 absorbs kW is connected 5.5 kW is connectedproducing producing a decrease a in decrease frequency in frequencyof 0.55 of 0.55 At 0.5 s, the load that absorbs 5.5 kW is connected producing a decrease in frequency of Hz, until itHz, reaches until 49.6it reaches Hz. From 49.6 1Hz. s, theFrom win 1d s ,speed the win increasesd speed as increases shown inas theshown figure in theuntil figure until it reaches 10it reaches m/s, producing 10 m/s, producing an increase an in increase the contribution in the contribution of active power of active to thepower grid, to up the grid, up to 3.9 kW andto 3.9 reducing kW and the reducing contribution the contribution to be made to by be the made grid by to the 1.6 gridkW. to 1.6 kW.

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Mathematics 2021, 9, x FOR PEER REVIEW 14 of 18

0.55 Hz, until it reaches 49.6 Hz. From 1 s, the wind speed increases as shown in the figure This reductionuntil it of reaches the active 10 m/s, power producing contribution an increase by the grid in the means contribution a recovery of activeof the power to the frequency in thegrid, grid up at to 49.83 3.9 kW Hz. and reducing the contribution to be made by the grid to 1.6 kW.

Figure 16. PCCFigure frequency, 16. PCC wind frequency, speed and wind active speed power. and Frequencyactive power. drop Frequency compensation drop forcompensation the resistive for load the in the scheme of Figure 14. resistive load in the scheme of Figure 14. This reduction of the active power contribution by the grid means a recovery of the Figure 17 presents the behavior of the system by showing the frequency of the PCC, frequency in the grid at 49.83 Hz. the power coefficient CP, the DFIG mechanical speed and the active powers of the load, Figure 17 presents the behavior of the system by showing the frequency of the PCC, the grid and the DFIG. The wind speed has been set at 12 m/s for all the time. From the the power coefficient CP, the DFIG mechanical speed and the active powers of the load, beginning the DIFG is running to 2300 rpm, this speed is higher than the optimum speed, the grid and the DFIG. The wind speed has been set at 12 m/s for all the time. From the thus the DFIG is working 35% deloaded generating 3.5 kW when the obtainable maximum beginning the DIFG is running to 2300 rpm, this speed is higher than the optimum speed, power is 5.4 kW. The load is connected at 0.75 s, which means a drop in frequency from thus the DFIG is working 35% deloaded generating 3.5 kW when the obtainable maximum 50.16 to 49.6 Hz. In the instant of 1 s, the energy is extracted from the wind, going from power is 5.4 kW. The load is connected at 0.75 s, which means a drop in frequency from the deloaded working speed to the maximum power speed. This speed deceleration (from 50.16 to 49.6 Hz. In the instant of 1 s, the energy is extracted from the wind, going from 2300 to 1780 rpm) provides extra kinetic energy which produces an increase in frequency the deloaded working speed to the maximum power speed. This speed deceleration (from until it stabilizes2300 atto 49.8 1780 Hz rpm) when provides the maximum extra kinetic possible energy power which is produces extracted an from increase the in frequency wind. Obviously,until i itf it stabilizes is not possible at 49.8 to Hz extract when more the maximum energy from possible the wind power it is is not extracted possible from the wind. to reduce theObviously, frequency deviation if it is not further. possible The to extractimportant more currents energy that from reflect the wind the process it is not possible to shown in Figurereduce 17, thecan frequencybe seen in deviationFigure 18. further.In this figure, The important the grid current, currents the that current reflect the process consumed byshown the load, in the Figure stator 17 current, can be of seen the inDFIG, Figure the 18 current. In this of figure,the grid the side grid converter current, the current and finally theconsumed current through by the load, the rotor the stator of the current DFIG ofcan the be DFIG, seen. the current of the grid side converter and finally the current through the rotor of the DFIG can be seen. Until 0.75 s the load is disconnected and the DFIG works in deloaded mode. The generator works in super-synchronous speed and the power is delivered to the grid by the stator and the rotor or GSC converter. The generated power of 3.5 kW produces a stator peak current of 4.4 A and a GSC peak current of 1.7 A with the imposed reactive power of 0 VAR. When the load is connected at 0.75 s the power consumption is 5.54 kW and the load peak current is 11.2 A. Since the power delivered by the DFIG is less than the power absorbed by the load, the grid must supply the difference and therefore the grid current peak becomes 5.1 A. In the instant 1 s it passes from the state of deloaded to the point of maximum power. As the deceleration from 2300 to 1780 rpm occurs, the power delivered by the DFIG increases as the CP of the turbine increases and also recovers kinetic energy in deceleration. For this reason, the stator and grid side converter currents increase while the grid current decreases. Finally, when the DFIG reaches full power speed, the current through the stator is 8.4 A and through the GSC is 1.3 A. The rest of the current, until reach the 11.2 A that the load absorbs, is provided by the grid. The amplitude variation of the rotor current is small since the d component of the current predominates to impose a reactive power of 0 VAR. The Ird value is 16.5 A and the amplitude variations are due to Figure 17. PCC frequency, CP coefficient, DFIG speed, and active power. Frequency drop compen- sation for the resistive load in the scheme in Figure 14 with the DFIG working deloaded.

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This reduction of the active power contribution by the grid means a recovery of the frequency in the grid at 49.83 Hz.

Figure 16. PCC frequency, wind speed and active power. Frequency drop compensation for the resistive load in the scheme of Figure 14.

Figure 17 presents the behavior of the system by showing the frequency of the PCC, the power coefficient CP, the DFIG mechanical speed and the active powers of the load, the grid and the DFIG. The wind speed has been set at 12 m/s for all the time. From the beginning the DIFG is running to 2300 rpm, this speed is higher than the optimum speed, thus the DFIG is working 35% deloaded generating 3.5 kW when the obtainable maximum power is 5.4 kW. The load is connected at 0.75 s, which means a drop in frequency from 50.16 to 49.6 Hz. In the instant of 1 s, the energy is extracted from the wind, going from the deloaded working speed to the maximum power speed. This speed deceleration (from 2300 to 1780 rpm) provides extra kinetic energy which produces an increase in frequency

Mathematics 2021, 9,until 143 it stabilizes at 49.8 Hz when the maximum possible power is extracted from the 15 of 18 wind. Obviously, if it is not possible to extract more energy from the wind it is not possible to reduce the frequency deviation further. The important currents that reflect the process shown in Figure 17, can be seen in Figure 18. In this figure, the grid current, the current consumed by thethe load, variation the stator of the current Irq component of the DFIG, to manage the current the active of the power. grid side The converter frequency reduction of and finally thethe current current through is observed the rotor as it of approaches the DFIG can the be synchronism seen. speed.

Mathematics 2021, 9, x FOR PEER REVIEW 15 of 18 Figure 17. PCCFigure frequency, 17. PCC CP frequency, coefficient, CP DFIGcoefficient, speed, DFIG and activespeed, power. and active Frequency power. dropFrequency compensation drop compen- for the resistive load in the schemesation for in Figurethe resistive 14 with load the in DFIG the scheme working in deloaded. Figure 14 with the DFIG working deloaded.

Figure 18. Grid, load, stator, grid side converter, and rotor currents for the process shown in Figure 17. Figure 18. Grid, load, stator, grid side converter, and rotor currents for the process shown in Figure 17.

8. ConclusionsUntil 0.75 s the load is disconnected and the DFIG works in deloaded mode. The generator works in super-synchronous speed and the power is delivered to the grid by This article proposes the use of the double feed induction generator in wind power the stator and the rotor or GSC converter. The generated power of 3.5 kW produces a systems to control the frequency and the amplitude of the point of common coupling stator peak current of 4.4 A and a GSC peak current of 1.7 A with the imposed reactive voltage. It has been demonstrated that the DFIG must be deloaded to extract the active power fromof 0 VAR. the wind When that the is load required is connected to compensate at 0.75 fors the the power frequency consumption of the grid, is 5.54 and kW the smartand the grid load controller peak current must is establish11.2 A. Since the quantitythe power deloaded. delivered Inby addition,the DFIG theis less smart than grid the controllerpower absorbed has to by choose the load, the required the grid maximummust supply reactive the difference power, taking and therefore into account the grid the electricalcurrent peak limits becomes of the double 5.1 A. In fed the induction instant 1 generator, s it passes to from balance the outstate the of amplitude deloaded ofto the gridpoint voltage. of maximum The capacity power. given As the to adeceleration smart grid tofrom handle 2300 frequency to 1780 rpm and voltageoccurs, amplitudethe power enhancesdelivered theby the reliability DFIG increases of the installation. as the CP of the turbine increases and also recovers kinetic energyThe in efficiencydeceleration. of the For wind this reason, turbine the isreduced stator and when grid itside is desired converter to reservecurrents the increase wind energywhile the in agrid wind current turbine decreases. to compensate Finally, for thewhen frequency. the DFIG Since reaches the maximum full power current speed, value the current through the stator is 8.4 A and through the GSC is 1.3 A. The rest of the current, until reach the 11.2 A that the load absorbs, is provided by the grid. The amplitude varia- tion of the rotor current is small since the d component of the current predominates to impose a reactive power of 0 VAR. The Ird value is 16.5 A and the amplitude variations are due to the variation of the Irq component to manage the active power. The frequency reduction of the current is observed as it approaches the synchronism speed.

8. Conclusions This article proposes the use of the double feed induction generator in wind power systems to control the frequency and the amplitude of the point of common coupling volt- age. It has been demonstrated that the DFIG must be deloaded to extract the active power from the wind that is required to compensate for the frequency of the grid, and the smart grid controller must establish the quantity deloaded. In addition, the smart grid controller has to choose the required maximum reactive power, taking into account the electrical limits of the double fed induction generator, to balance out the amplitude of the grid volt- age. The capacity given to a smart grid to handle frequency and voltage amplitude en- hances the reliability of the installation. The efficiency of the wind turbine is reduced when it is desired to reserve the wind energy in a wind turbine to compensate for the frequency. Since the maximum current value of the DFIG limits the active and reactive power set, by using the converter on the grid side, the reactive power injection can be increased to compensate the grid voltage. Frequency compensation has been tested by performing simulations creating a weak grid.

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of the DFIG limits the active and reactive power set, by using the converter on the grid side, the reactive power injection can be increased to compensate the grid voltage. Frequency compensation has been tested by performing simulations creating a weak grid. However, the voltage compensation has been implemented in a real platform with a commercial DFIG that has successfully validated the algorithms presented, allowing the commercial implementation of such a system.

Author Contributions: Methodology, J.A.C. and O.B.; Software, O.B., P.A. and J.C.; Hardware and tests, J.A.C., P.A. and J.C. Writing—original draft, J.A.C.; Writing—review editing, O.B., P.A. and J.C. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Basque Government through the project SMAR3NAK (ELKARTEK KK-2019/00051), the Diputación Foral de Álava (DFA) through the project CONAVAUTIN 2 and the University of the Basque Country (UPV/EHU) through (PPGA20/06). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

RSC Rotor side converter GSC Grid side converter α, β Direct and quadrature axes expressed in the stationary reference frame α0, β0 Rotor direct and quadrature axes expressed in the rotor reference frame d, q Direct and quadrature axes expressed in the synchronous rotating reference frame → v s Stator voltage vector → i s Stator current vector → ψs Stator flux vector Lm DFIG mutual inductance Ls DFIG stator inductance Lr DFIG rotor inductance Lls DFIG stator leakage inductance Llr DFIG rotor leakage inductance Rs DFIG stator resistance Rr DFIG rotor resistance ωe Synchronous speed ωr Rotor electrical speed ωm Rotor mechanical speed Te DFIG electromagnetic torque TL DFIG load torque J DFIG inertia B DFIG friction coefficient RL Load resistance LL Load inductance Ps DFIG stator active power Qs DFIG stator reactive power Pp Pair of poles → v g Grid voltage vector → i g Grid current vector Rg Grid filter resistance Mathematics 2021, 9, 143 17 of 18

Lg Grid filter inductance ωg Grid frequency → v conv Grid side converter output voltage vector Pg Grid side converter active power Qg Grid side converter reactive power P DFIG total active power Q DFIG total reactive power S DFIG total apparent power Z Line impedance KP Voltage droop coefficient K Frequency droop coefficient Q → Vs v s

→ Vg v g δ Angle between the DFIG stator voltage and the grid voltage ωpr Propeller speed ωd1 Deloading propeller speed ωopt Propeller optimum speed L Line inductance IGBT Insulated gate bipolar transistor

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