The application of the maximum entropy function principle to fit the wind speed distribution

Transcription

1 Revue des Energes Renouvelables SMEE 0 Tpaza (200) 07 4 The applcaton of the maxmum entropy functon prncple to ft the wnd speed dstrbuton F. Chellal,3*, A. Khellaf 2, A. Belouchran 3, B. Batoun et S. Boualt Unté de Recherche Applquée en Energes Renouvelables URAER Route de Ouargla, B.P. 88, Gart Taam, Ghardaïa, Algére 2 Centre de Développement des Energes Renouvelables CDER B.P. 62, Route de l Observatore, Bouzaréah, Alger, Algére 3 Ecole atonale Polytechnque EP Avenue Hassen Bad, El-Harrach, Alger, Algére Abstract - In the feld of wnd energy converson, an accurate determnaton of the probablty dstrbuton of wnd speed guarantee an effcent use of wnd energy, thus enhance the poston of wnd energy aganst other forms of energy.in the followng paper we propose to use the maxmum entropy prncple to derve a famly of pre-exponental dstrbutons n order to descrbe wnd speed probablty dstrbutons. The statstcal performances of the developed dstrbutons are compared wth those of the conventonal Webull dstrbutons. As a result t has been found that the proposed dstrbutons provde better statstcal performances and they ft the wnd speed dstrbutons better than the Webull dstrbutons. Keywords: Maxmum Entropy Prncple - Webull dstrbuton - Wnd speed dstrbuton - Wnd energy.. ITRODUCTIO Wnd plays a prmordal role n many applcatons such as wnd energy exploratons, fght aganst desertfcaton and pollutants transport and dffuson. Wnd s consdered to be hghly varable from both of temporal and geographcal pont of vews. Thus the knowledge of the characterstcs of wnd speed s of great mportance. When the probablty dstrbuton of wnd speed s known, then ts characterstcs can be easly obtaned. From a lterature overvew on the probablty densty functons proposed to descrbe the wnd speed dstrbuton, we can fnd that the Webull s wdely used to ft the wnd speed dstrbuton. Accordng to Carta et al. [], the Webull dstrbuton s the most used n the specalsed lterature of wnd energy [2-9] and t s practcally the only dstrbuton whch s recommended n books related to wnd energy [0-3]. Recently, the maxmum entropy prncple (MEP) used n the nformaton theory has been proposed to ft the wnd speed dstrbutons [4]. The MEP has been ntroduced for the frst tme n the wnd energy feld by L and L [4-6]. The authors have proposed a set of MEP-type exponental famly of dstrbuton functons to ft the wnd speed dstrbuton. 2. WEIBULL DISTRIBUTIO The Webull dstrbuton wth two parameters can be wrtten as: * Chellal_faroul@yahoo.fr 07

2 08 F. Chellal et al. k k k f ( ) =..exp () c c c Where s the wnd s speed, c and k are the scale and the shape parameters respectvely. The scale factor has the dmenson of wnd speed and t should be greater than zero. The shape factor k has no dmenson and t ranges generally from.5 to 3 for most wnd condtons [8]. If k s set to be equal to 2, then the resultng dstrbuton s often called Raylegh dstrbuton. Several technques to estmate the parameters of Webull dstrbuton have been proposed. In the followng we use the maxmum lkelhood method (ML) and the moment method (MM). The ML estmator of the shape factor L and the scale factor ĉ L are gven by [7]: ln ( ) = ML = ln ( ) (2) = ML = / ĉ ML = (3) = Where s the total number of samples. The estmaton of the Webull parameters va the ML s very laborous because equaton (2) needs to be solved numercally va an teratve process. However one can use the moment method (MM) that requres less calculaton. The Webull s parameters are estmated va the MM as []: MM =.09 std ( ) mean ( ) (4) ĉmm = mean ( ) Γ.( + / MM ) (5) Where Γ ( ) s the Gamma functon. 3. MAXIMUM ETROPY BASED DISTRIBUTIO Accordng to Carta et al. [], the probablty densty functon f ( ) could be obtaned by mnmsng the Shannon s entropy under the followng restrctons: a- The sum of all of the probabltes wthn the defnton nterval must be equal to one. b- The M-low statstcal orders m ( =, 2, K, M ) wth respect to the theoretcal dstrbuton must be equal to the M-low statstcal orders wth respect to the emprcal dstrbuton. Mathematcally, the two restrctons can be expressed as:

3 SMEE 200: The applcaton of the maxmum entropy functon prncple 09 b f ( ) d ( ) = (6) a b f ( ) d ( ) = m (7) a Where [ a, b ] s the nterval wthn whch the probablty densty functon s defned. In our case, we take a = 0 and b = max ( ). The M-low statstcal orders m are obtaned emprcally as [5]: m = (8) = A general soluton of equatons (6) and (7) can be expressed as: M f ( ) = exp a j (9) j= Where a j are Lagrangan multplers and M s the number of the used low order moments. Thus, the probablty densty functon f ( ) s determned once the a j parameters are obtaned. For ths, one can solve numercally the followng set of equatons: M max exp a j 0 j= = M max exp a = j m 0 j= (0) M M max M exp a j 0 j= = m M In order to obtan better accuracy, L and L [5] have ntroduced a pre-exponental term to the MEP dstrbutons. The resultng dstrbutons are known as the MEP type dstrbuton. MEP type dstrbuton of order r s obtaned by addng the wnd speed to the power of r as a pre-exponental term. Thus, the MEP-type dstrbuton can be expressed as: M f ( ) r exp a = j j = r 0 ()

4 0 F. Chellal et al. In many applcatons such as n the wnd energy feld and n the structure s constructon feld, the wnd speed dstrbuton s used to estmate the wnd energy. For a surface A, the wnd power densty E s gven by [5]: P 3 E = =. ρ.. f ( ) d ( ) (2) A 2 0 Where P s the wnd power and ρ s the ar densty ( ρ =.225 kg/m 3 n the standard condtons). Thus, the wnd power dstrbuton can be expressed as [5]: 3 e ( ) =. ρ.. f ( ) (3) 2 For the regons of Algers (Algera), we propose n the followng study to nvestgate the possblty of developng wnd speed probablty dstrbutons that may have better performances than the conventonal Webull dstrbutons. MEP type dstrbutons constraned by two, three and four low order moments are obtaned and compared wth Webull dstrbutons. The comparson s based on the coeffcent of determnatons (COD) crteron and the root mean square error (RMSE) crterons. 4. STATISTICAL AALYSIS 4. Coeffcent of determnatons crterons The COD s used to ndcate how much the total varaton n the dependent varable can be accounted for by the theoretcal and the emprcal dstrbutons [5]. The COD s defned as: COD = R 2 = σ 2 2 x, y σ y (4) Where R s the correlaton coeffcent, σ 2 y s the varance of the measured data from ts own mean value y m and t s gven by: / 2 2 σy = = ( y y ) m Where y s the actual probablty value, y c s the predcted probablty values and s the number of samples. Smlarly σ y, x s gven by: / 2 ( y ) 2 yc = σ y,x = The greater the value of COS ndcates the better fttng of the wnd speed. (5) (6)

5 SMEE 200: The applcaton of the maxmum entropy functon prncple 4.2 Root mean square error crteron The second crteron s the RMSE and t s gven by: / 2 ( y ) 2 RMSE = y c (7) The smaller the RMSE ndcates the better fttng. 5. RESULTS AD DISCUSSIOS To llustrate the sutablty of the MEP-type dstrbutons, we propose n the followng secton to compare ther statstcal performances (RMSE and COD) wth those of the Webull dstrbuton. Data consdered n ths study are 0 mnutes wnd speed values measured at Algers, Algera (36.43, 3.5 E). Data are measured durng tme perod of two years (from January 2003-december 2004). Table, present some descrptve numercal measurements of the wnd speed n the analysed statons. As ndcated n Table, the ste s charactersed by low wnd potental. Table : Descrptve numercal measurements of the statons analysed Ste m m 2 m 3 m 4 std () max (m/s) (m 2 /s 2 ) (m 3 /s 3 ) (m 4 /s 4 ) (m/s) (m/s) Algers Fg. : A comparson of the MEP type dstrbutons wth the measured data and the Webull dstrbuton for the data measured at Algers a) Wnd probablty dstrbutons. a) Wnd densty dstrbutons.

6 2 F. Chellal et al. Table 2: The MEP and the Webull s parameters and ther statstcal analyss for the regon of Algers

7 SMEE 200: The applcaton of the maxmum entropy functon prncple 3 Table 2 shows the values of the estmated parameters of the two proposed dstrbutons n addton to ther statstcal performances. As mentoned prevously [5], one can see that the estmated parameters of the MEP type dstrbutons can not be expressed n closed form (closed form means that they can not be rounded and they should be expressed by 6 or 7 dgts). Table 2 ndcates that the Webull dstrbuton estmated usng the moment method fts slghtly better the wnd speed than the one estmated usng the maxmum lkelhood method. For the MEP dstrbuton ( r = 0 ), t has been found that t can not ft the wnd dstrbuton better than the Webull dstrbuton even when M s ncreased. However, for the MEP type dstrbutons ( r ), Table 2 ndcates that they can ft wnd speed dstrbuton n many cases. For M = 2 t has been found that the MEP type wth r = and r = 2 provde better performances than the Webull dstrbuton. But for the wnd power dstrbuton, both of the Webull based dstrbutons provde better fttng that the MEP type dstrbutons. For M 3, results ndcate that the RMSE of the MEP type dstrbuton s sgnfcantly lower than those obtaned usng the Webull dstrbutons. The lowest RMSE = 0.06 and RMSE = have been obtaned usng r = 2 for M = 3 and M = 4 respectvely. Ths means that the hgher the pre-exponental factor r do not mean necessarly the better fttng. Ths result s also vald for the wnd power densty where the lowest RMSE has been obtaned usng r = and r = 3 for M = 3 and M = 4 respectvely. One can see from Table that a good fttng of the wnd speed do not mply necessarly a good fttng of the wnd power densty. Smlar results have also been obtaned prevously [5]. Such results can be explaned by the fact that the wnd power densty s heavly weghted toward hgh wnd speeds. For the regon of Algers, we present n Fgure 2 the hstogram of the measured data, the ML and the MM Webull dstrbutons and the MEP type dstrbutons. For low wnd speed ( = m / s ), the MEP type fts better the wnd speed than the Webull dstrbuton (Fg. a-). For the wnd power densty dstrbuton, Fg. b- shows clearly that the MEP wth M = 4 and r = 3 has the best statstcal performances especally at wnd speed between ( 4 6 m / s ). 6. COCLUSIO In ths work, t has been found that the dstrbutons derved from the maxmum entropy prncple present a better alternatve to descrbe the wnd speed dstrbuton than the Webull dstrbuton. For the MEP dstrbutons, the results also ndcate that the ncrease of the statstcal constrants s benefcal n term of performances. However, for the MEP type dstrbutons, t has been found that the ncrease n the power of pre-exponental term do not necessarly yeld better performances. Fnally, as t has been already found n the prevous studes, obtaned results ndcate that a good fttng of the wnd speed dstrbuton do not mply necessarly good fttng of the wnd power densty dstrbuton

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