CHAPTER 2 MATHEMATICAL MODELING OF WIND ENERGY SYSTEMS

17 CHAPTER 2 MATHEMATICAL MODELING OF WIND ENERGY SYSTEMS 2.1 DESCRIPTION The development of wind enegy ytem and advance in powe electonic have enabled an efficient futue fo wind enegy. Ou imulation tudy compae thee contol cheme ued in wind enegy ytem. Thee widely ued contol cheme fo wind enegy ytem ae Pitch contol, Roto eitance contol and Vecto contol of double fed induction geneato and thei effectivene in contolling the fluctuation in the output powe occuing due to wind peed vaiation. The mathematical model built in SIMULINK to imulate the ytem ae decibed hee. The contol ytem deign fo the powe contol mathematical model i dicued and the chapte conclude with an analyi of the imulation eult compaing the thee contol technique. A taditional wind enegy ytem conit of a tall-egulated o pitch contol tubine connected to a ynchonou geneato though geabox. The ynchonou geneato opeate at fixed peed and one of ealiet oto contol cheme wa the oto eitance contol. The peed of an induction machine i contolled by the extenal eitance in the oto cicuit.

18 The dawback of the above two method i the inability of wind powe to captue at low wind peed. The double fed induction machine i an extenion of the lip powe ecovey cheme, wheein the machine can be made to act like a geneato at both ub ynchonou and upe ynchonou peed. Powe facto contol at the gid ide can by obtained by contolling the gid ide convete. The thee method ued in wind enegy ytem, the output vaiation fo the diffeent contol technique fo a change in the input wind velocity and a contant deied output powe efeence ae compaed and the method ae evaluated baed on the epone time and the magnitude of change in the output powe compaed to the deied output powe and alo compaed by imulation in SIMULINK. 2.2 TYPICAL WIND TURBINE GENERATOR The baic component involved in the epeentation of a typical wind tubine geneato ae hown in Figue 2.1. Figue 2.1 Component of a typical wind ytem The convete and the ectifie on the oto ide ae eplaced by bidiectional convete.

19 component: An entie wind enegy ytem can be ub divided into following 1. Model of the wind, 2. Tubine model, 3. Shaft and geabox model, 4. Geneato model and 5. Contol ytem model. The fit thee component fom the mechanical pat of the wind tubine geneato. The geneato fom the electo-mechanical link between the tubine and the powe ytem and the contol ytem contol the output of the geneato. The contol ytem model include the actuato dynamic involved, be it the hydaulic contolling the pitch of the blade, o the convete contolling the induction geneato. Thi chapte decibe the mathematical modeling of the vaiou component of the wind ytem and thei implementation in SIMULINK Manual (1997). 2.3 MODEL OF THE WIND The model of the wind hould be able to imulate the tempoal vaiation of the wind velocity, which conit of gut and apid wind peed change. Anjan Boe 1983), The wind velocity (V w ) can be witten a (P. M. Andeon and V w = V wb + V wg + V wr (2.1)

20 whee, V w = Tital wind velocity, V wb = Bae wind velocity, V wg = Gut wind component and V wr = Ramp wind component. The bae wind peed i a contant and i given by, V wb = C 1 ; C 1 = contant (2.2) by, The gut component i epeented a a (1-coine) tem and i given 0 t < T1 t T VwG = C 1 co π T t T 0 t T2 1 2 1 2 T2 T1 (2.3) whee C 2 i the maximum value of the gut component and T 1 and T 2 ae the tat and top time of the gut, epectively. which i given by, The apid wind peed change ae epeented by a amp function, 0 t < T3 t T VwR = C T t T 0 t T4 3 3 3 4 T4 T3 (2.4) whee C 3 i the maximum change in wind peed caued by the amp and T 3 and T 4 ae the tat and top time of the amp, epectively.

21 The noie component of the wind peed i not modeled, a the lage tubine inetia doe not epond to thee high fequency wind peed vaiation. The wind peed pofile ued to compae the thee powe contol method i hown in Figue 2.2. The S-function ue to model the wind velocity in SIMULINK i given in Appendix 1. Figue 2.2 Wind peed pofile ued fo imulation 2.4 TURBINE MODEL Aiteam aound the tubine i hown in Figue 2.3. Figue 2.3 Aiteam aound the tubine

22 whee, V a V 1 V 2 V 3 A 1 A 2 A 3 A R = Ai team volume element, = Unditibuted fa-upteam wind peed, = Wind peed at tubine, = Deceleated wind fa-downteam tubine, = Fa-upteam co ection of flow, = Co ection of flow at tubine, = Boading downteam co ection of flow and = Roto hift aea. The wind powe i dw Pw = dt w 1 2 ρ 2 2 The enegy dawn by wind tubine i, W w = V a ( V1 -V3 ) whee, W w ρ = Enegy dawn by wind tubine and = Ai denity. The Wind Powe ( ) P w 1 d V ρ V -V 2 = dt 2 2 ( ) a 1 3 An ai volume flow in the oto aea, dv ( A = A ) of =A V dt a 2 R R 2 yield in the quai-teady tate, Pw 1 = ρ A V -V 2 2 2 ( ) V R 1 3 2

23 Accoding to Betz (2006), the maximum wind tubine powe output (Siegfied et al 2006), 16 ρ 3 Pm = AR V1 i obtained when 27 2 V 2 = 2/3 V 1 and V 3 = 1/3 V 1 The tubine model epeent the powe captue by the tubine. The powe in the wind (P w ) in an aea i given by, 1 3 Pw = ρ Avw, AR = A (2.5) 2 whee vw i the wind velocity. Howeve, the tubine captue only a faction of thi powe. The powe captued by the tubine (P m ) can be expeed a (Andeon and Boe 1983), P = P m w Cp (2.6) whee C p i a faction called the powe coefficient. The powe coefficient epeent a faction of the powe in the wind captued by the tubine and ha a theoetical maximum of 0.55 (David Richad et al 1993). The powe coefficient can be expeed by a typical empiical fomula a Cp 1 2 ( γ 0.022β 5.6) 2 e 0.17 γ = (2.7) whee β i the pitch angle of the blade in degee and γ i the tip peed atio of the tubine, defined a

24 γ v ( mph) w = (ω b = Tubine angula peed) (2.8) ω b 1 ( ad ) Equation (2.5) - (2.8) decibe the powe captued by the tubine and contitute the tubine model. The SIMULINK implementation of the tubine model i hown in Figue 2.4. Figue 2.4 SIMULINK implementation of the wind tubine model 2.5 SHAFT AND GEARBOX MODEL The tubine i connected to the oto of the geneato though a geabox. The geabox i ued to tep up the low angula peed of the tubine (nomally about 25-30 pm) to the high otational peed of the geneato (nomally aound 1800 pm). Figue 2.5 how of the haft and geabox model, with all the toque acting on the ytem and the angula velocitie of the vaiou mae (Tony et al 2001). The tubine toque T m (poduced by the wind), acceleate the tubine inetia and i countebalanced by the haft toque T 1 (poduced by the toional action of the low peed haft). Thu,

25 T T = J m 1 m dωb dt (2.9) whee, ω b J m = Angula velocity of the tubine and = Moment of inetia of the tubine. Figue 2.5 Shaft and geabox model Similaly, the haft toque poduced by the high-peed haft (T 2 ) acceleate the oto and i countebalanced by the electomagnetic toque (T e ) poduced by the geneato. Thu, T T = J 2 e dω dt (2.10) whee, ω = J = Angula velocity of the oto and Moment of inetia of the oto. Auming that the geabox i ideal, with no backlah o loe and auming that the haft ae igid,

26 T T ω n = = ω n 1 1 2 b 2 (2.11) whee, n 1 / n 2 = Geabox atio. (2.11), Eliminating the haft toque fom equation (2.9) and (2.10) uing 2 n 2 n 2 dω Tm Te = Jm + J n1 n1 dt 14 424 43 Jeq (2.12) 2.6 INDUCTION GENERATOR MODEL The thee contol cheme pitch contol, oto eitance contol and vecto contolled doubly fed induction machine ue diffeent induction machine and diffeent excitation fo opeation. The pitch contol cheme ue a quiel cage induction machine, in which the oto cicuit i hotcicuited. The oto eitance contol method utilize a wound oto induction machine and the doubly fed induction geneato ue a oto cuent contolled wound oto machine. Since SIMULINK i an equation olve, diffeent machine model mut be ued depending on the input to the ytem. Howeve, only mino modification ae equied in the tandad induction machine model to obtain the thee diffeent model.

27 2.6.1 Standad Induction Machine Model The induction machine model i modeled on the ynchonouly otating efeence fame, otating at the ynchonou peed ω (Pete Va 1998). The block diagam epeentation of the machine model i hown in Figue 2.6. Figue 2.6 Block diagam epeentation of the induction machine model 2.6.2 Conveion fom abc to dq Quantitie Fo the tato, the quantitie (tato voltage i ued fo futhe deivation) in the tationay efeence fame fixed to the tato (α-β fame) can be expeed a a pace vecto given a (Segey 2005), 2 2 V ( ) = Va() t + avb() t + a Vc() t αβ 3 (2.13) V = V + jv αβ α β (2.14) ( )

28 whee, V( αβ ) = Time-vaying tato voltage pace vecto, ( ), ( ), ( ) V t V t V t = Thee phae voltage applied to the tato, a b c 2π j ( ) a = e 3, j = 1 and Vα, Vβ = Real and imaginay pat of V ( ), αβ epectively. The pace vecto howing the tato voltage pace vecto and the efeence fame i hown in Figue 2. 7. Figue 2.7 Space vecto fo tato quantitie The d-q axe hown coepond to the ynchonouly otating efeence fame, which i the common fame of efeence fo the tato and oto quantitie. Since the d-q axe otate at ynchonou peed, the angle θ can be expeed a, () t 0 t θ = θ + ω dτ (2.15) 0 whee, θ 0 = initial angle of the d-q fame w..t the α -β fame.

29 Expeed in the d-q efeence fame, the tato voltage pace vecto can be witten a, j( θαβ θ ) V = V e (2.16) αβ ( dq) ( ) V = V e ( dq) ( ) jθ (2.17) αβ Rewiting equation (2.17) in a matix fom, uing V = V + jv (2.18) ( dq) d q The following tanfomation i obtained. Vd coθ inθ Vα V = q in co V θ θ β (2.19) The oto quantitie (oto voltage i ued in futhe deivation) can imilaly be expeed in the tationay fame fixed to the oto a, 2 2 V ( ) = V () t + av () t + a V () t αβ 3 a b c (2.20) whee, V α β = Time vaying oto flux vecto in the oto α-β fame and ( ) ( ), ( ), ( ) V t V t V t = Thee phae voltage applied to the oto. a b c Figue 2.8 how the oto voltage pace vecto along with the oto efeence fame, the d-q efeence fame and the tato α-axi (ued a the efeence fo meauing the angle).

30 Figue 2.8 Space vecto fo the oto quantitie tato, Since the oto otate at an angula peed of ω with epect to the () t 0 t θ = θ + ω dτ (2.21) 0 whee, θ 0 = initial angle of the oto w..t the tato α-axi. fame a, The oto voltage pace vecto can be expeed in the d-q efeence ( dq) ( ) jθ dq V = V e (2.22) αβ = V ( αβ ) j( θαβ + θ θ) e ( dq) ( ) ( θ θ ) V = V e j (2.23) αβ Rewiting in matix notation ( θ θ) in ( θ θ) ( θ θ ) co( θ θ ) Vd co Vα V = q in V β (2.24)

31 Figue 2.9. The abc to dq conveion implemented in SIMULINK i hown in Figue 2.9 abc to dq conveion 2.6.3 Stato and Roto Flux Computation The tandad tato and oto equation fo an induction machine, expeed in an abitay efeence fame otating at a peed (ω k ) ae (Andeon 1983): d λ = + + k V R i jω λ Stato equation (2.25) dt d λ = + + ( k ) V R i j ω ω λ Roto equation (2.26) dt

32 whee, V, V = Stato and oto voltage pace vecto, λ, λ = Stato and voltage flux linkage pace vecto,, i i = Stato and oto cuent pace vecto and ω = Roto angula peed. The tato and oto flux linkage can be expeed in tem of the tato and oto cuent and the elf and mutual inductance a, λ = Li+ L i (2.27) m λ = L i + L i (2.28) m whee, L L L m = Stato inductance, = Roto inductance and = Mutual inductance. Solving equation (2.27) and (2.28) fo the tato and oto cuent and ubtituting in equation (2.25) and (2.26), we get and dλ R = V Lλ Lmλ jωkλ (2.29) dt K dλ R = V Lλ Lmλ j( ωk ω) λ (2.30) dt K whee, K = L L L m 2.

33 Splitting equation (2.29) and (2.30) into thei d and q component and ewiting in tate pace fom, RL RL m ωk 0 K K λd d V RL d RL λ m k 0 d λ ω q K K λq Vq = + dt λ d RL m RL d V d 0 ( k ) λ ω ω K K λq λq Vq RL m RL 0 ( ωk ω) K K (2.31) The above equation cannot be implemented diectly in SIMULINK a a tate pace fomulation, ince the matix vaie with time (due to the tem involving ω ). The flux linkage obtained a the olution to the tate pace poblem can be ued to compute the cuent uing equation (2.27) and (2.28). 2.6.4 Electomagnetic Toque Computation The electomagnetic toque (Te) poduced by the induction machine can be expeed a, 3 p L m * Te = i 22L λ (2.32) whee, p = Numbe of pole of the machine. The complete implementation of the induction machine in imlink i given in Appendix A2.

34 2.6.5 Squiel Cage Induction Machine Model In the Squiel cage induction machine, the oto i hot - cicuited. Thu, by etting the oto voltage to zeo, the quiel cage machine model i obtained. 2.6.6 Wound Roto Induction Machine Model fo Roto Reitance Contol In the cae of the wound oto machine model, an additional input i equied; thi coepond to the extenal eitance added to the oto cicuit. Since the extenal eitance added i going to be the ame in all the phae, the equied model can be obtained by uing (R ext + R ) in place of R in the equation and etting the oto voltage to zeo. The pinciple i illutated in Figue 2.10. Figue 2.10 Wound oto induction machine model ued fo oto eitance contol

35 2.6.7 Doubly Fed Induction Machine Model The doubly fed induction machine ha cuent contolled oto excitation. Thu, the oto voltage equation i upefluou, a the input ae the tato voltage and oto cuent (Pena et al 1996). The tato cuent can be expeed a (Aguglia et al 2007), i L i L m = λ (2.33) give, Uing equation (2.27) and ubtituting equation (2.33) into (2.25) dλ R = V λ L i jω λ (2.34) dt m k L The above equation can be expeed in tate pace fom a, R RLm ωk Vd + id d λd L λd L = + dt λq R λq RLm ωk Vq + iq L L (2.35) Thu, by computing the tato flux linkage pace vecto, all the othe cuent and flux linkage can be computed. 2.7 RESULTS OF THE SIMULATION The doubly fed induction geneato i found to have leat output powe vaiation at high velocitie, the output powe vaiation i 20 % and at low wind velocitie, the output powe vaiation i 10 % and the fatet epone of the thee contol technique imulated.

36 Moeove, the additional complexity of the ytem enable ove all ytem powe facto contol, to help opeate the ytem, at cloe to unity P.F. We can achieve almot unity powe facto futhe eeach wok i being caied out fo deiving analytically wind powe output in tem of RPM of oto haft. Thi will enable a to detemine the oto RPM at which wind powe output i highet. 2.8 SUMMARY The mathematical modeling of the vaiou component of the wind ytem (which ae to be ued to compae the contol technique) and thei SIMULINK implementation ae decibed.