A COMPARATIVE ANALYSIS OF WIND SPEED DISTRIBUTION EVALUATION

Transcription

1 BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Uiversitatea Tehică Gheorghe Asachi di Iaşi Tomul LVII (LXI), Fasc. 2, 20 Secţia ELECTROTEHNICĂ. ENERGETICĂ. ELECTRONICĂ A COMPARATIVE ANALYSIS OF WIND SPEED DISTRIBUTION EVALUATION BY CIPRIAN-MIRCEA NEMEŞ * Gheorghe Asachi Techical Uiversity of Iaşi Faculty of Electrical Egieerig, Eergetics ad Applied Iformatics Received: March 2, 20 Accepted for publicatio: Jue, 20 Abstract. A comparative aalysis of some methods for estimatig Weibull parameters is performed. The Weibull distributio is a widely used distributio, especially for modelig the radom variable of wid speed. The techiques require historical wid speed data, collected over a certai time iterval, to establish the parameters of the wid speed distributio for a particular locatio. Key words: wid speed data; parameter estimatio; Weibull distributio.. Itroductio World-wide utilizatio of reewable eergy i electric power systems is growig rapidly due to evirometal cocers ad a costat icrease of eergy prices due to reducig the fossil fuels amout used to covetioal eergy sources. Wid power is oe of the fastest growig electric geeratio techologies i the whole world. Wid geeratio brigs a great amout of beefits to power systems, such as: a cheaper eergy comparig with the thermal geeratio, emissio reductio; wid eergy is available for large areas, developmet of a wid power farm ca be implemeted much easier ad faster tha buildig a thermal or hydro plat, etc. Meatime, wid geeratio brigs a * cemes@ee.tuiasi.ro

2 46 Cipria-Mircea Nemeş series of difficulties to the traditioal power systems, as: ucotrollability of power geeratio, the wid geeratio depedig o wid availability, irregularly fluctuatig ad itermittece of power geeratio, respectively a poor predictability of the wid geeratio. The wid property of iterest whe the power geeratio is aalysed, is the wid speed. This is because the wid power output obtaied from the wid is directly proportioal to the cube of the wid speed. Oe of the mai characteristics of wid is that this is highly variable ad its properties vary from oe locatio to aother. 2. The Wid Speed Probability Desity Fuctio Wid speed probability desity fuctio plays a importat role i electric power geeratio applicatios of wid turbie. A large umber of studies have bee published i scietific literature related to wid eergy, which propose to use a variety of probability desity fuctios (e.g. ormal, logormal, gamma, Rayleigh, Weibull, etc.) to describe wid speed distributios (Villaueva & Feijoo, 200; Carta & Ramirez, 2009; Lu & Lam, 2000). The commo coclusio of these studies is that the Weibull distributio with two parameters may be successfully utilized to describe the priciple wid speed variatio. For accout the variability of wid speed, it is assumed to be characterized by a Weibull distributio with a scale parameter, α, [m/s], ad a shape parameter, β, (dimesioless). The Weibull probability desity ad cumulative distributio fuctios are give, respectively, by v β β v β v β fw() v = exp, Fw() v = exp. α α α α () The most importat requiremets for effective wid power plaig ad operatio i power systems is a accurate estimatio of wid speed distributio. Ivestigatio of wid geeratio itegratio must be carefully performed i accordace with the wid speed probabilistic character. 3. Methods of Parameters Estimatio The wid speed distributio is determied whe its parameters are established. The parameters estimate of the Weibull distributio ca be foud usig differet estimatio methods. Some methods are graphical ad others are aalytical, each method havig a criterio which yields estimates that are best i some situatios. The most commoly aalytical methods are: maximum likelihood estimator (MLE), method of momets (MOM) ad least squares method (LSM) (Razali et al., 2009; Bai & Egelhardt, 992). Because the estimate parameters play a major role i developig a

3 Bul. Ist. Polit. Iaşi, t. LVII (LVXI), f. 2, model of electric power wid geerator, it is importat that differet estimatio methods to be compared to fit parameters of Weibull distributio from wid speed database. I what follows we try to fid a aswer to the questio: what method gives the best Weibull parameters estimatio? For that, will be aalysed the performace of these methods with the same wid speed database. 3.. The Maximum Likelihood Estimator The MLE is a aalytical method, widely applied i egieerig ad mathematics problems. For our case, for Weibull distributio of wid speed, i accordace with MLE theory, the likelihood fuctio is built as the joit desity of the radom variables be a fuctio of the ukow two parameters β β vi vi L( αβ, ) = f( vi) = exp α α α i= i=, (2) where α ad β values ca be achieved by usig iterative methods or limits method. Last method of parameters evaluatio ivolves takig the partial derivatives of the likelihood fuctio with respect to the parameters, settig the resultig equatios equal to zero β i= i= ll β = + lvi vi lvi = 0, β β α ll β = v 0. 2 i = α α α i= (3) The values of α ad β result as solutios of these equatios The Method of Momets The MOM is aother aalytical method to establish the distributio parameters. If it is kow the set of wid data, the momets of ukow parameters that deped o the two-parameter Weibull distributio will be equalized with the empirical momets, as i the followig equatios: vi, σ ( ) 2 i. (4) v= = v v i= i= The aalytical expressio of mea ad the variace of Weibull distributios are give by the followig expressios:

4 48 Cipria-Mircea Nemeş M() v = αγ + ad D () v = α Γ + + Γ +, (5) β β β where Γ( ) is the gamma fuctio. We ca obtai the β parameter from the coefficiet of variatio (by dividig the variace with the square mea) ad, after that, the α parameter ca be established based o first eq. (5) Least Squares Method For the estimatio of Weibull parameters, the LSM is extesively used i egieerig problems. The method provides a liear relatio betwee the two parameters havig as start poit the twice logarithms of Weibull cumulative distributio fuctio amely ll l( ) l( ) w () = β v β α. (6) F v This relatioship represets a straight lie, whose eq. is Y = ax + b, where Y = ll, l, ad l Fw () v X = v a = β b = β α. (7) Usig the simple liear regressio, the α ad β parameters result from coefficiets of polyomial liear fittig. 4. Case Study I order to compare the methods above described, a Matlab program has bee developed to evaluate the Weibull parameters, based o previous methods ad same wid speed database. To evaluate the performace of these methods, the mea squared errors (MSE) has bee used to evaluate the accuracy of estimated probability desity fuctio to real distributio. The MSE is a method to evaluate the differece betwee values provided by a estimated probability desity fuctio ad the true values of the database distributio (Lage, 2005). The wid data, used for aalyses, was collected from the North-East area of Romaia, over oe year period, for the year The data collectio

5 Bul. Ist. Polit. Iaşi, t. LVII (LVXI), f. 2, was made at oe hour iterval, the hourly average values beig recorded. Fig. presets the wid speed values collected at aemometer height (0 m above the groud). Fig. The hourly wid speed database. Based o these measuremets, the wid speed distributio ad it parameters have bee estimated, the Weibull distributio beig a good approximatio of the real database of wid speed frequecy. The Table shows the Weibull parameters for the aalysed database, scale ad shape parameters beig determied usig above methods. Table Weibull Parameters for Whole Year Parameters/error Method MLE MOM LSM Scale parameter, α Shape parameter, β MSE As ca bee see, the parameters of Weibull distributio are very close, the scale parameter lies betwee ad m/s, ad the shape parameter betwee.7898 ad.866. Likewise, the last lie of Table shows the values of MSE for each method. It is foud that MOM is superior i accuracy ad has a smaller error compared with the MLE ad LSM methods. The order of the methods based o their accuracy is as follows: MOM, MLE ad LSM.

6 50 Cipria-Mircea Nemeş I order to compare the results of these estimatio methods usig differet sample sizes of database, the above methods have bee applied for the wid speed values from the moths from each of the four seasos. The average seasoal Weibull parameters are preseted i Table 2. Seaso Sprig Summer Autum Witer Table 2 Weibull Parameters for Seasos Parameters/error Method MLE MOM LSM Scale parameter, α Shape parameter, β MSE Scale parameter, α Shape parameter, β MSE Scale parameter, α Shape parameter, β MSE Scale parameter, α Shape parameter, β MSE As ca bee observed, the scale ad shape parameters, from the whole database ad from seasoal values, have the best estimatio i case of MOM. Thus, we ca say that MOM is the best method used to estimate the parameters for the two-parameter Weibull distributios takig ito cosideratio the MSE as a measuremet for compariso. 5. Coclusios I practice, it is very importat to describe the variatio of wid speeds for optimal desig of the wid geeratio systems. The wid variatio for a typical site is usually described usig the Weibull distributio. Therefore it is very importat to kow the best method for parameters evaluatio, with miimal errors. This study has bee performed to compare the results of three methods of parameters estimatio, for the same database. It has bee deduced, from computatioal results, that method which gives the lowest values of MSE, is the MOM, i both cases, for whole year database ad for seasoal values. However, from accuracy viewpoit, the LSM ad the MLE of fittig Weibull fuctio were also good methods because these oes give close values of parameters as MOM ad, more over, Matlab package cotais fuctios ad tools that estimate the parameters ad cofidece itervals for Weibull data.

7 Bul. Ist. Polit. Iaşi, t. LVII (LVXI), f. 2, 20 5 Ackowledgmets. This paper was supported by the project PERFORM-ERA Postdoctoral Performace for Itegratio i the Europea Research Area (ID ), fiaced by the Europea Social Fud ad the Romaia Govermet. REFERENCES Bai L., Egelhardt M., Itroductio to Probability ad Mathematical Statistics. Duxbury Press, Califoria, USA, 992. Carta J.A., Ramırez P., A Review of Wid Speed Probability Distributios Used i Wid Eergy Aalysis. Case Studies i the Caary Islads. Reew. a. Sustai. Eergy Rev., 3, 5, (2009). Lage P.M., O the Ucertaity of Wid Power Predictios Aalysis of the Forecast Accuracy ad Statistical Distributios of Errors. J. of Solar Eergy Egg., 27, (2005). Lu I.Y.F., Lam J.C., A Study of Weibull Parameters Usig Log-Term Wid Observatios. Reew. Eergy J., 20, 2, (2000). Razali A.M., Salih A.A., Mahdi A.A., Estimatio Accuracy of Weibull Distributio Parameters. J. of Appl. Sci. Res., 5, 7, (2009). Villaueva D., Feijoo A., Wid Power Distributios: A Review of their Applicatios. Reew. a. Sustai. Eergy Rev., 4, (200). ANALIZA COMPARATIVĂ A METODELOR DE ESTIMARE A DISTRIBUŢIILOR PROBABILISTICE ALE VITEZEI VÂNTULUI (Rezumat) Estimarea distribuţiilor costituie tehica pri care se determiă valorile parametrilor distribuţiilor statistice, plecâd de la u set de date obţiut pe cale experimetală, di statistici sau simulări. Sut aalizate trei metode de estimare a parametrilor fucţiilor de distribuţie (MLE, MOM şi LSM), particularizate petru cazul distribuţiei Weibull, î vederea stabilirii metodei care furizează cel mai ridicat grad de îcredere asociat parametrilor distribuţiei. Cele trei metode au fost ierarhizate, di puct de vedere al preciziei, cu ajutorul metodei celor mai mici pătrate. Metodele au fost evaluate pe aceeaşi bază de date, reprezetâd valorile vitezei vâtului, îregistrate î zoa de ord-est a Româiei, petru aul Parametrii distribuţiei Weibull asociate bazei de date au fost evaluaţi cu cele trei metode, stabilid că metoda MOM coduce la cele mai bue rezultate.

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