Modelling a 2.5 MW direct driven wind turbine with permanent magnet generator

Transcription

1 Modelling a.5 MW direct driven wind turbine with permanent magnet generator Report for the final examination project of the Nordic PhD coure on Wind Power by Thoma P. Fugleth Department of Electrical Power Engineering Norwegian Univerity of Science and Technology NO-749 TRONDHEIM, NORWAY

2 Introduction Thi report i the concluion of the pot-coure work following the Nordic PhD-coure on Wind Power, which wa held on Smøla 5 th th June 5. The goal of the project wa to contruct a Simulink model of a.5 MW wind turbine with direct-driven permanent magnet generator connected to the grid via a back-to-back three-phae converter. The model wa then to be teted in imulation run.

3 Content Introduction... Content... 3 Sytem decription and modelling Mechanical dynamic Permanent magnet generator Generator-ide converter Grid ide converter and inductance DC-link dynamic Sytem parameter Wind model... 9 Control trategie.... Generator ide control.... Grid ide converter control... 3 Simulation reult... 3 Concluion and further work... 9 Reference... 3

4 Sytem decription and modelling Thi chapter um up the equation which were implemented in the imulink model. The ytem in it entirety conit of the following component. A wind turbine generate torque from wind preure. The torque i tranferred via the generator haft to the rotor of the generator. The generator produce an electrical torque, and the difference between the mechanical torque from the wind turbine and the electrical torque from the generator determine whether the mechanical ytem accelerate, decelerate or remain at contant peed. The generator i connected to a three-phae converter which rectifie the current from the generator to charge a DC-link capacitor. The DC-link feed a econd three-phae converter which i connected to the grid through a tranformer. Tranformer inductance Back-to-back converter ~ = + V _ dc = ~ PM generator V3 ϕ Grid figure Sytem overview. Mechanical dynamic It i not a part of thi exercie to model the wind turbine itelf and the pitching action of the blade. Intead it i aumed that the turbine i run at variable peed to enure that the tippeed ratio remain at a contant optimal point? opt =7.5 with a power coefficient C p =.474. The mechanical torque delivered by the wind turbine i given by: 3 Mt = π ρair Rwt Cp( λ) v (.) Here,? air i the denity of air, in kg/m 3. R wt i the radiu of the wind turbine rotor in meter and v 8 i the wind peed far away from the turbine. That i, the wind peed before it i lowed down by the wind turbine. C p (?) i the coefficient of power a a function of the tip-peed ratio?. Normally thi will alo be a function of the rotor pitch, but pitching dynamic are not modelled. The tip peed ratio? i calculated a: ωmek Rwt λ = v Thi value i fed to a look-up table to find value of C p (?). The turbine i connected to the rotor of the generator via a haft. The turbine, haft and generator are modelled a a ingle rotating ma: dωmek = ( Me + Mt) (.) dt J total 4

5 where M e i the electromechanical torque developed by the electrical machine (when the electrical machine i operated a a generator M e will be negative) and J total i the moment of inertia of the entire mechanical ytem. In p.u. value equation (.) become: dn = ( me + mt ) (.3) dt Tm For more accurate reult, the mechanical dynamic could be modelled a a two-ma-ytem with a pring and damper between them, but thi wa not done due to contraint in time and in available information.. Permanent magnet generator The generator model i implemented entirely in dq-coordinate. That i to ay, there are no AC-tate in the model. The generator i modelled with dc voltage and current in a rotorfixed rotating coordinate ytem with the d-axi being in the direction of the flux from the permanent magnet. Thi model i a caled per-unit model, and eentially the ame a that hown in [Nilen, 5]. The equation for the d- and q-axi voltage are a follow: dψ d ud = r id + n ψ q (.4) ω dt n dψ u = r i + + n (.5) q q q ψ d ωn dt Here, i d, i q, v d and v q are the p.u. d- and q-axi current and voltage repectively, r i the p.u. tator reitance.? n i the baic electrical frequency of the generator, n i the p.u. frequency og the generator and? d and? q are the d- and q-axi fluxe repectively. The d- and q-axi fluxe are given a follow: ψ = x i + ψ (.6) d d d m ψ = x i (.7) q q q We ee that by keeping the q-axi current equal to zero, we can orient the flux entirely in the d-axi direction. More on thi will follow later. Next, (.6) and (.7) are inerted into (.4) and (.5) to get the following: xd did ud = r id + n xq iq (.8) ω dt n x q diq q q d d ψ m ωn dt u = r i + + n ( x i + ) (.9) Tranforming thee equation a bit we get: did ωn = ( ud r id + xq ni q ) (.) dt x d q ωn di = ( uq r iq n ( xd id + ψ m) ) dt xq (.) The p.u. torque developed i given by: τ = ψ i ψ i (.) e d q q d 5

6 Thee equation are implemented in the imulink diagram hown below. r id id ud wn/xd Wn/Xd I_d -Cpim xd Gain pid pid*iq Me n*pid 3 n n piq*id n*piq piq xq Gain4 uq wn/xq Gain3 I_q iq 3 iq r figure Simulink model of permanent magnet generator.3 Generator-ide converter When working with a implified dq-model it i common to neglect witching dynamic, ripple current and other fat dynamic in the electrical ytem. In thi cae, we follow the example of [Nilen, 5] and model the generator ide converter a a imple time delay. That i, the voltage on the generator clamp are given a: u () t = u ( t t) u () t (.3) d contd, v dc q contq, v dc u () t = u ( t t) u () t (.4) where u cont,d and u cont,q are the control ignal given a output from the controller..4 Grid ide converter and inductance For the grid/tranformer inductance, the model given in [Molina et al. 5] i ued: g g g did g c ud = rg id + lg ng lg iq + ud (.5) dt g di g g q g c uq = rg iq + lg + ng lg iq + uq (.6) dt c Here, the upercript g tand for grid, to differ from the generator variable. v d and v c q are the voltage on the clamp of the front-end converter. 6

7 By tranforming thee equation we get: di dt g d g c g g = ( ud ud rg id + ng iq ) (.7) l g g diq g c g g = ( uq uq rg iq ng id ) (.8) dt lg Which i the model implemented in the imulink diagram hown in figure 3. ud_grid uq_grid /L i_dq u_dq_grid P i_dq Q Power calculation id 3 P 4 Q 3 ud_converter Add Gain Integrator R Gain [x] iq 4 uq_converter Product 5 n_grid Gain figure 3 grid inductance model The grid ide converter i modelled a a time delay jut like the generator ide converter..5 DC-link dynamic The equation for the DC-link dynamic are taken from the lecture note given at the Smøla coure [Norheim, 5], and are a follow: dudc c c c c = ( ( ud id + uq iq ) ( ud id + uq iq ) ( Prlo, + Pdlo, )) (.9) dt C udc Thi follow eaily from the principle of conervation of energy. The total power flowing into the capacitor bank mut equal the um of the power flowing out of the capacitor and the loe in the conductor. (.9) can alo be written a du dt d = ( Iv Id ) (.) C where ( Iv = ud id + uq iq Prlo, ) i the current flowing out of the generator ide udc converter and ( c c c c Id = ud id + uq iq Pdlo, ) i the current flowing into the grid ide udc converter. Thi mean that in when the PM machine i operating a a generator thee will both be negative. 7

8 The loe are modelled a imple reitance, IE Prlo, = rv ( id + iq ) and c c Pdlo, = rd ( id + iq ) Thi lo model give a imple approximation to the real loe in the ytem, but for more accurate imulation a better lo model that include witching loe and nonlinear loe in the generator would be preferable..6 Sytem parameter The p.u. value ued in the imulation are baed on a.5 MW wind turbine. The turbine itelf 6 i 8 meter in diameter, and with a moment of inertia Jwt = 9. kgm [DAWE 4]. A mentioned earlier, the pitching dynamic are not modelled. It i aumed that the blade are pitched to maintain optimum energy capture. The objective of the generator control i to keep the turbine running at a rotational peed correponding to a tip-peed ratio which give u the optimal power coefficient C p,opt =.474. In reality thi i imilar to controlling a fixed-pitch turbine. The C p -table data are taken from [DAWE, 4]. The generator i a permanent magnet generator with urface mounted magnet and ymmetrical rotor. In other word, the reactance i the ame in both the d- and q-axi. The p.u. value that have been ued in thi imulation are: r =. x d = x =.5 Thee are approximate value that are reaonably repreentative of a ynchronou PM machine in the.5 MW range. If we aume that p.u. peed and no load torque require p.u. tator voltage, we get a p.u. flux of ψ m =.865 Phyically, the rotor of the generator i aumed to conit of three cylinder ection: a center ection with a radiu of.5 meter and a length of.5 meter, a dic ection with an outer radiu of. meter and a length of.4 meter and an outer cylinder with an outer radiu of.5 meter and a length of meter. For the purpoe of calculating moment of inertia, the rotor i aumed to conit of homogenou teel. Uing the formula for moment of inertia for a cylinder 4 4 J = π ρ ( ) teel l router rinner with a denity? teel of 785 kg/m 3 6 we get a total moment of inertia J total = 9.93 kgm. The per unit mechanical time contant i Jtotal ωmekn, Tm = Sn where? mek,n i the nominal rotational frequency in rad/ec, which in thi cae equal π rad ω mek, n= 6rpm = and S n i the nominal power, in thi cae,5 MW. Thi give u a time contant of Tm = = q 8

9 The DC-link capacitor i dimenioned uing an empirical rule that i commonly ued in literature: CU dc, rated S n = τ dc (.) Here, C i the capacitance of the DC-link capacitor, U dc i the nominal DC-link voltage and S n i the nominal power of the generator. The time contant t dc i generally elected to be a proportion of a full cycle for the grid voltage. A full cycle for a 5 Hz grid i m, and a time contant of 6-8 m eem common. (.) give u: Sn τ dc C = U dcrated, Next, we know that the per unit capacitive reactance i: Idcrated, x = c ω CU (.) dcrated, If we inert the expreion for C into (.) and take into account that Idcrated, = Sn / Vdcrated, we get: Sn Udc, rated x = c Sn τ = dc ω U ωτ dc dcrated, Udc, rated We ee that the per unit capacitive reactance i independent of the phyical rated voltage and current. If we choe the DC-link time contant to be 7. m (.36 cycle) we get an x c value of: x c = =. 3 ( π 5 Hz)7. In teting, thi proved to be a bit too mall. In the end, x c wa et to be.5, which yielded better reult..7 Wind model The wind model ued i one developed at the Riø National Laboratory in Denmark. The imulink implementation i deigned at the Univerity of Aalborg, and i available from a part of the Wind Turbine Blocket beta, a toolbox for wind turbine imulation in Simulink. 9

10 Control trategie Thi chapter detail the trategie ued in the control ytem of the generator and the grid ide converter.. Generator ide control The generator ide control i baed on the tator voltage equation (.) and (.). To implify the control deign we divide the control ytem into two part: a decoupling controller and a PI controller. In other word, we give the input voltage the form: u = u + u (.) d di dii uq = uqi + uqii (.) We ee that if we give u dii and u qii the following form: u = n x i (.3) dii q q uqii = n xd id + n ψ m (.4) then we get the following firt order ytem to control: did ωn r ωn = i d + u di (.5) dt xd xd diq ωn r ωn = i q + u qi (.6) dt xq xq Equation (.5) and (.6) repreent two firt order ytem with one control variable each. In hort, we ue the control ytem to decouple the d- and q-axi equation from each other, a well a removing the peed-dependent term. The d- and q-axi current can now be controlled with two ordinary PI controller. The decoupling controller i dependent on meaurement of the current a well a knowledge of the impedance x d and x q and the flux generated by the permanent magnet. However, by uing PI controller, any error introduced by minor inaccuracie in the aumed value of? m, x d and x q are compenated for by the integrating controller. A hown in [Nilen, 5], the ytem defined by (.5) and (.6) can be repreented by the following implified tranfer function: id() ωn Td hd () = = (.7) u () x ( + T )( + T ) h q di d d um iq() ωn Tq () = = u () x ( + T )( + T ) qi q q um x x d q where Td = and Tq = ωn r ωn r T um i the um of all the maller time contant in the ytem uch a filtering of current meaurement and linearized witching delay. The current controller are PI controller with limited output and anti- wind-up feature. Antiwind-up i implemented by diabling the integrator while the output from the controller i equal to the upper or lower limit. The general tranfer function of the PI controller i a follow: (.8)

11 + Ti hpi() = Kp (.9) Ti Controller deign follow the guideline given in [Nilen, 5]. Since the generator ha a ymmetrical rotor with x d =x q the controller deign for both axe will be the ame. We tart out with what i termed the numerical optimum deign, which give u: xd K pd, = Tid, = Td ωn Tum When number are inerted, we get: K pd, = 5.65 T id, =.85 During teting, it wa revealed that thee controller value gave a controller that wa too aggreive, leading to rapid aturation and overhoot. After ome teting and tuning, the following current controller parameter were elected: K pd, =. T id, =.5 which yielded better performance. The peed controller i quite intereting compared to mot regular motor drive, a what we are really intereted in controlling i the tip peed ratio. While thi work quite well in imulation, tip peed ratio can be difficult to control in a real turbine. The rotational peed of the turbine can be meaured with a high degree of accuracy, but wind peed meaurement (which are neceary to calculate the tip peed ratio) are eaily affected by turbulence and other factor. In thi imulation we aume perfect wind peed meaurement, but in reality ome form of etimation baed on meaurement and modelling will probably have to be ued. Thi will again lead to ome lo of production, a the controller i unable to perfectly optimize the production. For peed control, the ytem wa teted with a PI-type controller deigned uing the ymmetrical optimum trategy, again taken from [Nilen, 5]. Thi method give the following controller parameter: Tm K = pn, T T in = 4 T ψ, umn, umn, m T m i the mechanical time contant of the generator and turbine, a detailed in ection.. T um,n i the um of the maller time contant: Tumn, = Teqi, + Tfin,. T fi,n i the peed filter time contant, while T eq,i i the equivalent time contant for the electrical ytem. In other word, the time contant we find if we repreent the electrical dynamic a a ingle firt-order ytem: T = ( T + T ). Thi method gave the following controller parameter: eqi, v f, i K pn, = 3978 T in, =, Thi illutrate that while method for obtaining controller parameter are a ueful tool for initial controller deign, they are not perfect. In thi cae the high proportional amplification meant that the controller for all practical purpoe worked like a hyterei controller. The reult wa a lot of overhoot and a high degree of ocillatory behaviour of the ytem, and the controller had to be tuned manually. Unfortunately none of the PI controller deign gave atifactory behaviour. Generally the ytem dynamic were either too low, or untable.

12 In the end, after a lot of teting and trial and error, it wa decided to go for a limited PID-type controller. The tranfer function of a PID controller i + Ti + Ti Td hpid() = K p (.) Ti The tructure of a PID controller with limited derivative control i illutrated in figure 4. The limitation in derivative i implemented with a rate limitation imulink block. The purpoe of thi limitation i to avoid controller aturation caued by high frequency ignal. K p T d d dt T i Controller parameter were elected motly through manual tuning to be: K pn, = T in, =.8 T dn, =. Thee were found to give atifactory performance.. Grid ide converter control The controller tructure for the grid ide converter i taken from [Norheim, 5]. The original deign conit of two PI controller, one for the amplitude and one for the phae angle of the converter voltage. That i: j u = m u e α (.) c The amplitude m control the reactive power to the grid, while the phae angle a control the DC-link voltage. In thi cae, the ytem wa difficult to tabilize with only PI controller. In particular, m wa prone to ocillatory behaviour. A relatively common occurrence wa that m would ocillate, while a teadily dropped. Thi would continue up to a point where a would drop harply, cauing high grid current, and making the DC-link voltage to drop almot immediately to zero. Naturally, thi would not happen in a phyical ytem. Thi behaviour wa only poible becaue of the implification done in modelling the ytem. However, it hould till be avoided. Eventually, a olution with a PI controller for a and a PID controller for m wa ettled upon, and wa found to give atifactory performance. The controller parameter were: K pm, =.5 T im, =.5 T dm, =.5 K α = T, α =.5 p, figure 4 PID controller i dc

13 3 Simulation reult The reult in thi chapter are baed on a 5 econd imulation with a mean wind peed of 6 m/. Simulation over a longer time frame would have been better, but imulation time wa limited by the amount of memory on the PC ued to run the imulation. The firt illutration, figure 5, how the wind peed time erie, and the correponding generator peed: 7.5 v w ind 7 [m/] n t [] figure 5 wind peed and correponding rotational peed A we can ee, the peed controller doe a good job of controlling the peed depending on the wind peed. The next illutration, figure 6, how the correponding tip peed ratio: 8 tip peed ratio t [] figure 6 tip peed ratio 3

14 A we can ee, the controller managed to keep the tip peed ratio within an acceptable region of the deired tip peed ratio of 7.5. Next, we look at the generator current and voltage. Figure 7 how the generator dq-axi current, while figure 8 how the correponding voltage..4 i d i q iq -.5 iq ref t [] figure 7 d- and q-axi generator current u d u q t [] figure 8 d- and q-axi generator voltage 4

15 In figure 7, the d-axi reference i contantly equal to zero. While the d-axi current tay reaonably cloe to zero the entire time, the controller performance probably could have been a bit better. The q-axi reference depend on the commanded torque from the peed controller, and i hown in red in the lower plot of figure 7. The obtained q-axi current i hown in blue. In general, the obtained current follow the commanded current quite well, but we ee that it overhoot the reference mot of the time, o perhap a lightly le aggreive controller would have been better. On the grid ide, we get the current hown in figure 9. i d,grid i q,grid t [] figure 9 grid d- and q-axi current The q-axi current tay quite well around zero, while the d-axi current varie with the power produced by the wind turbine. Note that the d-axi current i negative becaue the turbine produce power, in other word we are uing a load reference for the converter ytem. The grid ide converter voltage are hown in figure. Thee how pretty much what we would expect, with the d-axi voltage taying nearly contant around p.u. while the q-axi voltage varie with the reactive power required to control the dc-link voltage. Figure how the active and reactive power while figure how the commanded voltage vector given by m and a. 5

16 .5 u d,grid u q,grid t [] figure grid ide converter voltage Active power P Reactive Power Q t [] figure active and reactive power 6

17 -.4 m alpha -.5 [rad] t [] figure m and a 7

18 Finally, figure 3 how the DC-link voltage, a well a the current I v and I d. U dc I v I d t [] figure 3 DC-link voltage and current flowing into and out of the DC-link capacitor While the controller doe a reaonable job of controlling the DC-link voltage around the et point of.75, performance could be better. A imple way of tabilizing the DC-link voltage i by electing a bigger capacitor, but there probably i room for ome controller improvement a well. 8

19 Concluion and further work The imulation how that the model and controller are functioning well. The control trategy work well, although the DC-link voltage ideally hould be controlled more accurately. While the controller worked very well during initial teting with imple wind condition (contant or lowly varying wind peed), imulation with more complex wind condition how that the controller are not quite perfect yet. While the wind turbine model doe a reaonable job of imulating a real turbine, accuracy can be improved in ome repect. Implementation of a better lo model would improve the quality of the model greatly. Alo, it would be intereting to have data from a real wind turbine to bae the model parameter on intead of approximate value. The controller tructure hould be improved by adding a power limiting controller for when the wind peed increae above nominal wind peed. Thi hould be coupled with a model including pitching dynamic and better modelling of the aerodynamic to improve the controllability of the model. Now that a working wind turbine model i in place, it would be intereting to tet it with different controller tructure. One area that could be explored i in the ue of nonlinear control of the DC-link voltage, a thi i a ytem that i inherently nonlinear in nature. Alo, it could be intereting to compare the current olution with a pair of PI-controller and a decoupling grid for generator control with a multivariable nonlinear controller that include the cro-coupling term. 9

20 Reference [Nilen, 5] Roy Nilen, Elektrike Motordrifter, Department of Electrical Power Engineering, Norwegian Univerity of Science and Technology 5 [Norheim, 5] Ian Norheim, Dynamic modelling of turbine (), Generator, converter and control, Preentation from the Nordic Wind Coure, Smøla 7 th June 5 [DAWE, 4] Variou author, DAWE PhD Coure/Advanced chool 4, Compendium and coure material, Intitute of Energy Technology, Aalborg Univerity 4.