Probabilistic computation of wind farm power generation based on wind turbine dynamic modeling

1 Abstract —This paper ad dresses the pro blem of pre dicting a w ind farm ’s power g eneratio n w hen no or few statistical data is available. The s tudy is based on a time- series w ind speed model and on a simp le dynam ic mod el of a doubly-fed indu ction generator w ind turbine i nclud ing cut-off and cut-in behaviours . The w ind t urbin e is mode led as a stochast ic hybrid system w ith three op eration m ode s. Num erical res ults, obtai ned usin g M onte- Carlo sim ulations, provide the ann ual distribution of a wi nd farm ’s active power generation . For differen t num bers of w ind turbines , we com pa re the nu mer ical resu lts ob tained using the dynam ic mo del w ith those ob tained conside ring the w ind turbine’ s steady-state power curve. Simul ations show that consider ing the wind turb ine’s dyna mics does n ot significan tly enhan ce the accurac y of the annual distribution of a w ind farm generation . Index Ter m s -- w ind farm, stochast i c hybrid system, cut-off behav iour I. I NTRODUCTI ON UMEROUS report s em phasize that operatio n and planning in modern po w er s yst ems are highly impacted by wind power generatio n [1 ]-[3]. T he signifi cant integration of wind turbines req uires thus utilities to develop new tools for po wer sy stem managem ent. Actually , though statistical data can be collected b y system o perators, proba bilistic tools are needed in o rder to achieve a more efficient us e o f the sy stem capabilities and better planning f or new investm ents. Fo r example, several re cent works have focused on proba bilistic loa d-flow computation [4 ]-[5] in order to ease operational tasks for transm ission sy stem operat ors. A s a load-flow computation r equires the knowledge of every i nj ection and demand, each wind t urbine o utput power, w hich is naturally volatile, should be account ed f or through its p robability density function (pdf) in those applicatio ns [6]-[7]. There exist several appro aches for assessing the pro bability density fun ctio n of w ind turbine power generation. One could take advantage of the progress i n w ind forecasting [8] and eventually assess the scheduled pdf of a w ind turbine’ s power injection. System o perator s could then schedule power system H. Bayem, Y. Phu lpin an d P. De ssante are with the Department of Powe r and Energy S ystems in S UPELEC, Paris, France (e-mail: {herman. baye m, yannick.ph ulpin, philipp e.dessante}@supelec.fr) . J. Bect is w ith the Department of Signal Processing and El ectronic Systems in SUPELEC, Paris, France (e -mail: julien.bect@ supelec.fr). operation depending on t he f orec asted hou r ly average wind speed. However, t his app roach seems difficult to implement since wind speed fo recasts are highly uncertain. On the other hand, one co uld consider, as in [6] or [9 ], the annual distribution of a wind turbine’s pow er generation based o n its steady- state power curve and using annual wind speed distributions. Ho wever, this ap proach does n ot consider the w ind t urbine’s dynami cs, which may impact i ts act ive po wer injection [10] -[11]. Our appro ach aims to analyze w hether the consideratio n of a w ind turbine’s dynam ics impacts the annual d istribution of its power generation. T he study is bas ed on a time-s er ies model for th e wind speed . While an appr opriate model should be designed for every type of generator , we w ill main ly focus in this paper on a Doubly -Fed Induction Gen er ator (DFIG) wi nd turbine which will be modelled as a stochastic hybrid syst em w ith three oper ation modes. T his dynamic modeling includes cut-off and cut-in behaviours, w hich c an have a great influence on the wind turbine’s generation availability in certain wind speed ranges [12]. Finally, numerical results, obtained using Monte-Carlo simu lations, provid e the annual distribution of the active power generation fo r a single wind turbine a nd for a w ind farm with several w ind turbines. T hose numerical results are compared w ith the annual distrib ution ob tained using t he steady- state power curve of the wi nd turbine. This paper is organised as follows: i n Section I I, the wind model is detailed. Section III describes th e w ind turbine while Section IV defines the three mode control schem e used in our simu lations. Finally, the simulation proce ss, the numerical data and the simu latio n results are presented in Section V. II. W I ND MODEL As introduced in [1 3], w e consider a “w ide ba nd” model for the horizontal wind speed v(t) of the form: ) ( ) ( ) ( t w t v t v + = (1) w here ) ( t v is a slowly varying signal, modeling the hourl y w ind speed varia tions, and w( t) a high ly f luctuating si gnal, modeling th e high frequency t urbulent phenomena. Moreover, conditionally to v t v = ) ( , the fast component w (t) is modeled as a sta tionary Gaussian process [ 13]-[14 ] with mean v and standard de viation v ⋅ = κ σ (with κ a positive co nstant). We use the same model as i n [11] and [14] for t he dynamics of the fast component: Herman B ayem, Yannick Phulpin, Philippe Dess ante and Jul ien Bect Probabili stic c omputation of wind farm power generat ion based on wind t urbine dyna m ic modeling N 2 ) ( / 2 ) ( ) ( ) ( t T t v T t w t dt dw ξ κ + − = (2) where ) ( t ξ is a Gaussian white noise ( formal derivative of standard Brownian motion) and v L T / = , with L t he turbu- lence length scale. Experimental values o f both κ and L c an be found in [14 ], for example. The slow component ) ( t v is inspired by the ARMA(3,2) proces s used in [ 15] . T he modifications are a “reflection” at 0 = v , introd uced to keep the time series positive at a ll times, and a linear interpolatio n to convert th e time series in to a continuous-tim e process. For numerical p urposes, we will use the ARMA parameters given in [ 15] for the “S wift Current” location. The time- series model of the horizontal wind speed v(t) is thus com pletely specif ied . Although e xtremely s implified in several respects (long-tim e corre lations are obviously underestimated b y the ARMA model, for instance), it should be s ufficient for the p urpose of this p aper. W e refer to [11] for a more detailed d iscussion concerni ng the model for the fast component. III. W IND TURBIN E MODEL A wind turbine m od el is generally composed of three subsystem s repres enting it s main components, namely the rotor, the gener ator, and the gearbo x, which c onnects the roto r and generato r shafts. In t his paper, we use a variab le speed win d t urbine, w hose generator i s co nnected to th e grid thro ugh a p ower electr onic converter. T his type of win d turbine is generally equipped w ith control lers which allow the control of the wind t urbine’s pitch angle and active/reactive p ower output. While an exhaustive descript ion o f such a wind turbine can b e found in [13], we pre sent hereafter a simple dynami c model. A. Gen eral representatio n of a variable spee d DFIG wind turbine We co nsider a vari able speed pitch controlled wind turbine with a doubly fed induction generato r. As d escribed in [13], the stator of the DFIG is directly co nnected to the grid whi le its rotor is coupled to the gr id through a power c onverter, whi ch allow s controlling the a ctive and reactive pow er ou tp ut. This scheme is presented in Figure 1. The grid is repre sented by a sl ack bus, into wh ich the wind turbine’s active p ower generation is injected at any time. In this co ntext, the dynam ics of the wind turbi ne relate d to the variations of the w ind speed are controlled trough three controller s, namely t he r otor sp eed controller, the pitch angle controller and the state c ontroller. Figure 2 shows how those elements (roto r, generator and co ntrollers) interact. Section III-B d etails t he d ifferent elements o f the wind turbine while the control scheme is presented in sectio n IV. B. Wind t urbine model 1) Tu rbine mod el The mechanical power cap tured by the turbine is given by the following equation: Figure 1: Connection of a DFI G wind turbine to th e ele c trical networ k [16 ]. Figure 2: interactions between wind turbine’s elements. 3 2 ) , ( 2 ) , , ( v C R v P P m ⋅ ⋅ ⋅ ⋅ = θ λ ρ π ω θ (3) w here v i s the wind speed , ω the wind turbine r otational spee d, ρ is the ai r d ensity, R t he turbine radius, θ the pitch a ngle, λ the t ip-speed r atio ( λ=R.ω/v ) and C p is the power c oefficient, w hich de scribes the turbine’s ability t o convert wind kinetic energy in to mech anical energy. 2) Gen erator drive train model The rel ation between d riving and braking torques and t he acceleration of the turbi ne can be expressed as follo w s: ω ω g m P P J − = ⋅ & (4) w here J is the total inertia of the s ystem, P m is the mechanical power that resulting from the force recei ved by the roto r, and P g is the electro -m echanical power exercised by the D FIG. For the sake o f sim p licity, we do not consider friction tor que i n this study . 3) DFI G model We c onsider a generator efficiency η such that the active power inject ed in the grid P g,out is equal t o 90 % of the electro- mechani cal p ower P g . 4) Roto r Speed controller The r otor speed controller sets the a ctive power set po int P g according to the rotor spee d. P g also depends on the functioning modes of the wind turbine, wh ich are detailed in Section IV. 5) Pitch controller As represented in Figure 3, the power co efficient C p is dependent on the tip-speed ratio λ and o n the pitch angle θ . Rotor subsystem Ge nerator / Inv erter Pitch co ntrol Rotor spee d con trol State control ler Rotor spee d Mechan ical pow er P m Pitch se t poi nt Powe r s et poin t Speed set poi nt Pitch angle Rotor spee d Powe r Wind speed + - Wind speed 3 Figure 3: Power coefficient as a fun ction of the tip speed ratio for d iffere nt pitc h angle. When the t urbine is generating, the pitch angle controller limits the mechanical power cap tured by the turbine in orde r to maintain the rotor spe ed with in accep table limits. As in [1 6], it is modeled as a pro portio nal co ntroller with saturation limits on the pitch rate:      − ⋅ ≥ = ≤ = = otherwise ) ) ( ( and if 0 and 0 if 0 2 max nom nom nom K h ω ω ω ω θ θ ω ω θ θ & (5) with : ( ) ( ) ( ) θ θ θ θ & & & & , max , min min max = h . (6) 6) S tate controller In ord er to operate the DFIG wind turbine in an acceptable speed range, t he state controller enables the pitch control and the rot or speed contro l to operat e depending on the wind speed and the roto r speed as descr ibed in section IV. IV. C ONTRO L SCHEME FOR THE W IND T URBINE In order to ease the un de rstanding of the w ind turbine’s dynam ics with a time-varyi ng wind s peed, w e use a simple control scheme with three modes. All ty pe s of o peration are covered, includin g cut-off and cut-in behaviours. Figu re 4 summ arizes the co ntrol scheme of this hybrid system , w hich is detailed in this section. Figure 4: Control scheme of the varia ble speed DF IG wind tu rbine. In the above figure, the condition v>v c o mean s: off cut off cut v v or v v − − > > 60 60 5 5 (7) w here 5 v and 0 6 v are the average wind speed over the last 5 and 60 seconds, respec tively , in cut v − 5 i s the cut-in wind speed and off cut v − 5 and off cut v − 60 the short term and long te rm cut-off wind speeds, respectively. The conditions given in equation (7) were inspired by [17]. A. Mode 0: no load “Mode 0” correspo nds to situations where the turbine do es not generate any electrical po w er. T his situation occurs when the turbine is completely sto pped due to low or high wind speeds or d ue to a manual stop. B y extension, w e also co nsider that t he w ind turbine is in this mode during starting p hases, w hen the generat or is not yet coupled t o t he grid. Since we ar e concerned about the annual distribution of the power generation, this ex tension s hould have a low impact on the simu lation results. Transition from “m ode 0” is activated w hen the average w ind speed meets the “norm al fun ctio ning” requirement s, w hich are deduced from [17] and defined as follows :      ≤ ≤ ≥ − − − off cut off cut in cut v v v v v v 60 60 5 5 60 60 (8) In this mode, the power demand is set to zero ( P g =0 ), and neither ω nor θ a re explicitly m odeled (since th ey are not needed). When the transition is activated, the wind turbine is sw itched to mode 1 with ω = ω min and θ = 0. B. Mode 1: partial load For medium wind speed s, the r otor speed is co ntrolled by setting the generator outp ut power according t o the following equation inspired by [18]: ( ) ( ) ( ) 3 1 1 2 0 , 2 ω ω ω ρ π ω v v R C R P p g ⋅         ⋅ ⋅ = , (9) w ith: ( ) ( ) ci ci nom nom v v v v + − − − = min min 1 ω ω ω ω ω (10) w here ω nom is the nominal value for the r otor sp eed and v nom the wind spee d for which this r otor speed is achieved under steady- state c onditions. Two transitions from “ mode 1 ” can be activated: - when the rotor speed ω becomes lower t han 0.95ω min or the average wind speed b ecomes higher than the cut-off win d sp eed ( v>v c- o ) , t hen the wind turbine is s w itched to “mode 0”. - when the rotat ional speed ω b ecomes h igher than ω nom, defined as the rotation speed for whi ch P g reaches its nominal value P nom , then t he wind turbine is switched to “mode 2”. C. Mode 2: f ull load “Mode 2” cor responds to higher w ind speeds for which the rotor speed is close to ω nom and t he pitch controller is activated in ord er to limit the m echanical power and t o m aintain t he Mode 1: Partial loa d Mode 2: Full load Mode 0: No load v>v ci,min and v= ω no m ω<= 0.95ω no m ω<= 0.95ω min or v>v co v>v co 4 output power ar ound its nominal value. T he generator po w er reference is: ( ) nom g nom g P P , ω ω ω = (11) where P g,nom is the rated power of the wind turbine. Two transitions can be activated from mode 2: - wh en the average wind speed becomes higher than the cut-out w ind speed ( v>v c o ) , then the w ind turbine is swi tched to “mode 0”. - wh en the rotational speed ω b ecomes low er than 0,95. ω nom , the wi nd turbine is sw itched to “mo de 1”. V. S IMUL AT ION RESUL TS A. Si mulation pro cess The wind and wind turbine models under co nsideration are relevant for simulatin g the power productio n of a wind farm in a large time-scale (months or years). F or t his study, the tim e- scale is one year. The first par t o f the simu lation pr ocess is the Monte-Carlo sampling of hourly mean wind speed values accord ing to the probab ilistic wind model descr ibed in Section II . T hen, the power production of the wind turbine is co mputed accordi ng to its dyn amic model and t o its steady-state po wer curve, respectively. When the wind farm is composed o f several wi nd turbines, each turbine i i s suppo sed to receive a wind with the same slowly varying component ) ( t v and an independent fluctuating signal w i (t). B. Numerica l data Numerical d ata for th e w ind turbine and the win d speed model are presented in T able I and Ta ble I I, r espectively. Two values ( 5,4 6 m /s and 1 0,00 m /s) have been chosen for the a nnual mean w ind speed value. TABLE I W IND TURBINE FEATURES . Name Value R Rotor rad ius 37.5m ω min Minimal rotor speed 9 RPM ω nom Nominal rotor speed 18 RPM Pg,nom Nominal power 2,03 MW v nom Nominal w ind speed 14 m/s v ci,min Cut-in w ind speed 3.5 m/s v ci,max Restart win d speed (after cut-off ) 19 m/s off cut v − 5 “Fast” cut-out speed 25 m/s off cut v − 60 “Slow” cu t-out speed 20 m/s J Turbine inertia 1,4 10 6 kg.m 2 TABLE II WIND MODEL FEATURES . Name Value ρ Air density 1.134 kg/m 3 L Turbulence length scale 300 m κ Std / mean ratio for v 0.15 C. Single wind turbine generation d istribution The simul atio n proce ss was first app lied to a single wind turbine with two different w ind speed time series (w ith an annual mean wind speed of 5.46 m/s and 10 m/s resp ectively). The joint probability function of the w ind turbine po w er production and obta ined w ind speed is shown in Figure 5 and Figure 6. One can notice that t he distributions of t he wind turbine power generation as a fun ctio n o f the i nstantan eous wind speed that is re presented in Figure 5 is similar to t he experimental dis tributio n reported in [1 9]. It ca n also be observed on Figures 5 and 6 that, for certai n instantaneous values of the wind speed , there may be a considerabl e uncertainty in the power production in com pa rison to the steady- state power. This uncertainty is e ven more significant w hen co nsidering sites with high er annual mean win d spe ed since the cut-off behaviour of the wi nd turbine is poor ly modeled by the steady- state power curve. Figure 5: Po w er output obtain ed f rom the dynamic model (grey ) for a wind time-series of 5.46 m /s an nual mean. The steady-state power curve is repres ented in black. Figure 6: Po w er output obtain ed f rom the dynamic model (grey ) for a wind time se ries of 1 0.00 m/s a nnua l mean. The steady-state p owe r curve is repres ented in black. 5 Figure 7: Cumulative d istributi on funct ion of the wind turbine ge neration obtain ed with d ynamic modeling and st eady-s t ate power curve with an ann ual mean wind speed of 5.46 m/s. Figure 8: Cumulative d istributi on funct ion o f the w ind turbin e generation obtain ed with d ynamic modeling and st eady-s t ate power curve with an ann ual mean wind speed of 10 m/s. For a low annual mean wind spee d, similar results are observed w hen computing the cumulative distribution function of the wind turbine power pro duction with dyn amic m od eling and with the stea dy-s tate power curve of the wind turbine (Figure 7 ). T his observation may be explained by a lower uncertainty wh en it is computed over a long period of time (i.e. one year instead of one second). A slight difference is remarkable when considering a site with higher annual mean win d sp eed (Figure 8). In t his case, the dynamic modeling of the wind turbi ne with high winds and the cu t-off consideration may indeed be significant. On F igure 8, it can be observed that the prob ability to have no po w er o utput is close to zero. Although this may be surprising with regard to the three mode scheme previously designed, this result is mainly caused by the wind m odel (mainly b ased on the average wind spe ed), which rarely generates w ind speed values low er than 2 or 3 m /s f or an average sp eed o f 10 m/s. Nevertheless, this o bservation emphasizes the necessit y of developing a model more adequate to the simulation of long- term win d var iations. Figure 9 : C umulative distrib ution f unction of a 5 turbin e wind farm generation o bt ained with dynamic mo deling and steady- stat e po wer cur ve with an annu al mean wind speed of 10 m/s. Figure 10: Cumulat ive d istribu tion function of a 10 turbine wind fa rm generation o bt ained with dynamic mo deling and steady- stat e po wer cur ve with an annu al mean wind speed of 10 m/s. D. Wind farm genera tion distribut ion Simulations were a lso p erformed for wind farms with 5 and 10 turbines, respecti vely . As emphasized for Figure 7, the annual distribution of a w ind farm on a site w ith 5.46 m /s annual mean wind spe ed are similar wh en co nsidering the steady- state power curve or the dynami c model for the wind turbines. Figures 9 an d 1 0 represent the p ower generation of t he wind farm ( P g,out ) normalized b y t he farm size ( 5 and 10 turb ines, respectively) for a site with 1 0 m/s annual mean wind speed . Differences be tw een the two wind turbine models are particularly small. As introduced for Figure 8, small wind values are p articularly unexpected with such a n annual mean w ind speed . Consequently , the probab ility of no production is close to zer o. 6 VI. C ONCL USION In this paper , Monte-Carlo sampling is performed o n a probab ilistic wind model to obtain a wind speed time series for one year. The resulting distribution is combined with a dynam ic model of a wind t urbine in order to compute its generation distributio n. Fo r comparison, this d istribution is also ap plied to the steady-s tate po w er curve of the wind turbine. Simu latio ns lead to sm all differences b etween those two models, w hich are mai nly observed for high wind speeds, close to the cut-off values. Moreo ver, the pro babilistic computation of a mu lti-turbine w ind farm using a correlation betw een w ind speeds for eac h turbine shows t hat this difference may decre ase wh en considering sites with s everal wind turbines. This pap er t hus emphasizes that the dynam ics of the win d turbines are not signi ficant w ith respec t to the annu al distribution of wind farm power generation. 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