Investigation of the Relationship between Diesel Fuel Properties and Emissions from Engines with Fuzzy Linear Regression

Transcription

1 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl 2013 Investgaton of the Relatonshp between Desel Fuel Propertes and Emssons from Engnes wth Fuzz Lnear Regresson Yuanwang Deng College of mechancal and vehcle engneerng, Hunan Unverst, Changsha, , People s Republc of Chna denguanwang610@126.com Abstract Selectng the fuel propertes as the ndependent varables, and the emssons from engnes as the dependent varables, the fuzz lnear regresson equatons are constructed n ths paper wth the fuzz lnear regresson; as well as the smulaton and predcton results calculated wth the equatons are analzed. Satsfactor results are acheved b selectng an approprate degree of membershp. he effect of the degree of the membershp on the fuzz regresson coeffcents s also analzed. here was no effect on the fuzz centers of the fuzz regresson coeffcents of the equatons for total hdrocarbons (HC), carbon monoxde (CO), ntrogen oxdes (NOx). here was a lttle effect on those of the equaton for partculates (PM). he fuzz wdths of the fuzz regresson coeffcents ncrease when the degree of the membershp ncreases. When the bases of the center values of the smulaton results to the actual values are bg or the ntervals of the smulaton results are too bg to have meanng, addtonal restrctons should be used to mprove the smulaton accurac of the equatons. Kewords Emssons from Engnes; Fuel Propertes; Fuzz Lnear Regresson; Partculates Introducton he ar polluton of the ctes n Chna has become more and more serous due to automotve emssons n the last few ears, so new natonal standards for automotve emssons were enacted b the natonal envronmental protectve agenc on March 10 th, 1999 and put nto effect enacted on Januar 1 st, he standards requre that the levels of the automotve emssons must reach European EURO Ⅰemssons standards, thus the manufacturers must ntroduce new technques to control the emssons. Snce there are no specal vehcle desel fuels n Chna at present, ther standards mechancall appl the standards of the lght desel fuels, therefore ther qualt cannot reach the demand of the legslaton. Investgaton of the effect of the fuel propertes on the emssons from engnes s mportant for controllng the emssons from engnes and thus decreasng the ar polluton n our countr. Based on the test, selectng the fuel propertes as the ndependent varables and the emssons from engnes as the dependent varables, lnear regresson equatons can be constructed wth statstcal method. he relatonshp between the fuel propertes and the emssons can be analzed and the emssons can be predcted wth the equatons. Snce the fuel propertes are correlatve wth each other, when studng the effect of one of the propertes on the emssons, the propert must be separated from the other propertes. he relatonshp between the fuel propert and the emssons s fuzz. hs paper constructs the fuzz lnear regresson equatons on the relatonshp between the fuel propertes and the emssons wth fuzz mathematcs theor. he results obtaned wth the equatons are analzed and dscussed. Fuzz Lnear Regresson Assumng that x, x, 1 2, xn are the ndependent varables and s the dependent varable, the samples x, x, 1 2 xn,( 1,2,, m) are known. Accordng to the classcal lnear regresson, the multvarate lnear regresson model s denoted (Hu G et al., 1990): c c x c x c x n n, (1) Where, c 0, c1, c2,, cn are parameters to be estmated, and ( 1, 2,, m ) s an unobserved random 2 error, satsfng ε N(0, ). he classcal lnear regresson consders that the error ε between the actual and the calculated * of the 106

2 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl dependent varable s caused b the error n the test. he sum of squares of the errors should be mnmzed when constructng the lnear regresson equaton wth the known samples. hus the multvarate lnear regresson equaton s obtaned: c c x c c x n n, (2) Here, c 0, c1, c2,, c n are the estmated regresson coeffcents. he fuzz lnear regresson was founded n 1980s (Hdeo anaka et al., 1982) and has been appled n engne engneerng (L B et al., 2000; Wu J et al., 1999). It consders that the error ε between the actual and the calculated * of the dependent varable s caused b the fuzzness of the sstem. he followng fuzz lnear regresson equaton s obtaned wth the fuzz numbers nstead of the coeffcents n the classcal lnear regresson equaton: A Where, 0 1 A 2 x A 1 x 2 A n x n, (3) A, 0,1,2,,n are the estmated fuzz regresson coeffcents, usuall adoptng the smmetrcal trangular fuzz number, whose general * form s: A (, ), 0,1,2, n. Where, s the, fuzz center, ( 0 )s the fuzz wdth, and ts membershp functon s defned: a 1 a * ( a ) A. (4) 0 otherwse Assumng x 1, x, x,, x ), x ( 1 2 n 1, x, x,, x ),,,, ), ( 1 2 n ( 0 1 n ( 0, 1,, n ), accordng to the arthmetc operatons of the fuzz number (L H et al., 1994), ( x, x ) s obtaned. Its membershp functon s: 1 ( ) 1, 0, x, x x 0 x 0, 0 x 0, 0. (5) We am to obtan the fuzz numbers: A, 0,1,2,,n,now let s see the calculaton method. Accordng to the above mentoned, () must be decded b the decson maker, subectng to () H, where H ( H 0,1 ) s the degree of the membershp of the fuzz lnear regresson model selected b the decson maker. he fuzz wdth of * must be mnmzed to reduce the vagueness of the result. So the above fuzz lnear regresson problem s transformed nto the soluton of the followng lnear programmng problem: J mn( 0 1 n ), 0, 0,1,, n (6) Subectng to: x 1 H, 1,2,, m; 0,1,, n. (7) x hat s to sa: J mn( 0 1 n ), 0, 0,1,, n s. t.. (8) (1 H ) x x (1 H ) x x 1,2,, m; 0,1,, n he fuzz regresson coeffcents A (, ), 0,1,2,,n are obtaned b solvng the above lnear programmng problem. When usng the fuzz lnear regresson to solve the actual problem, we ma also meet another tpe of the samples: x, x, 1 2 xn (,e ), 1,2,,m,that s to sa, the outputs of the samples are fuzz numbers. Here, we also consder them as the smmetrcal trangular fuzz numbers. he degree of the membershp of the calculated actual h s denoted b h. Here: (1,2,,m) to the h, h max( h ),h satsfes 107

3 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl 2013 h ( ) μ 1.0 h h e and k ( h (9) h ) Ση x t x 1 H, 1,2,,m; 0,1,,n (12) x e hat s to sa: J mn( 0 1 n ), 0, 0,1,,n s.t. (13) (1H ) x x (1H )e (1H ) x x (1H )e 1,2,,m; 0,1,,n FIG.1 HE DEGREE OF HE MEMBERSHIP OF O Fgure 1 gves the degree of the membershp of to the actual. he followng formula s obtaned wth the smlart of the rght trangles: 1:(1 h ) ( x ):k Where, Hence, k x e(1 h ). h 1 t x t x e. (10) As mentoned above, the fuzz lnear regresson problem n ths condton can also be transformed nto the followng lnear programmng problem: J mn( 0 1 n ), 0, 0,1,, n Subectng to: x t ξ (11) he fuzz regresson coeffcents are also obtaned b the soluton of the above lnear programmng problem. Comparng the membershp functons () of the two tpes of the samples, t s found that, when e 0, the membershp functon () of the latter s the same as that of the former, so the former s the specal example of the latter. Applcaton and Analss he samples data used n ths paper come from the paper (M.Hubln et al., 1996). Eleven fuels wth dfferent propertes were used n that paper to do the tests wth several lght dut desel engnes. able 1 gves the propertes of the fuels. able 2 gves the emssons from engnes HC, CO, NOx, PM. B analzng the propertes of the fuels, fuels 1, 3, 4, 5, 7, 8, 9, 11, along wth the correspondng results of the emssons, are selected as the smulaton samples 1 8. he are used to construct the fuzz lnear regresson equatons. Fuels 2, 6, 10, along wth the correspondng results of the emssons, are selected as the predcton samples 1 3. he are used to analze the predcton ablt of the fuzz lnear regresson equatons. ABLE 1 HE PROPERIES OF HE FUELS No ρ/ kg/m p /% CN /r / No. fuel code, ρ denst, p polaromatcs, CN cetane number, 95 back end dstllaton 108

4 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl ABLE 2 HE EMISSIONS FROM ENGINES /g/km No HC CO NOx PM Selectng denst, polaromatcs, cetane number, and back end dstllaton as the ndependent varables, and one of the emssons HC, CO, NOx, or PM as the dependent varable, accordng to (3), the followng fuzz lnear regresson equaton s obtaned: A0 A1 x1 A2 x2 A3 A4 x4. (14) Accordng to (8), the soluton of (14) s transformed nto the soluton of the followng lnear programmng problem wth the smulaton samples1 8: J mn( ), 0, 0,1,,4 s.t.. (15) (1H ) x x (1H ) x x 1,2,,8; 0,1,,4 Because the szes of dfferent ndependent varables var wdel, n order to remove the effect of the szes of them on the regresson coeffcents, the data s transformed nto devatons from the average. Selectng H=0.8, the coeffcents are calculated wth the smplex method (He J et al., 1985), and shown n table 3. For example, for HC, usng the coeffcents n table 3, the fuzz lnear regresson equaton s denoted wth the followng formula: 4 5 ( , ) ( , ) x 4 ( ,0) x 5 4 ( , ) x 2 ( ,0) x 4 3 ABLE 3 HE FUZZY REGRESSION COEFFICIENS OF H=0.8 1 he equatons of CO, NOx, and PM can also be denoted wth a smlar formula usng the coeffcents n table 3. Comparng wth table 2, because the data of each ndependent varable has been transformed nto devatons from the average, the average of the dependent varable can be explaned b the sze of the fuzz center of the coeffcent for the constant x0. he sze of the fuzz center of each coeffcent s greatl dfferent from ts fuzz wdth, except for the sze of the fuzz centers of the coeffcents of the ndependent varable x4 of the equatons of HC and CO whch are an order of magntude smaller than ther fuzz wdths. he sze of the fuzz center of the coeffcent of the ndependent varable x3 of the equaton of PM s the same order of magntude as ts fuzz wdth. he szes of the fuzz centers of other coeffcents of the equatons are an order of magntude bgger than ther fuzz wdths. hs shows that the calculated coeffcents are ratonal. It s also found from table 3 that the fuzz wdths of some coeffcents are equal to zero, n ths case the coeffcents essentall become accurate numbers. here are some coeffcents whose fuzz wdths are not equal to zero, so the results obtaned wth these equatons are stll fuzz numbers. When all the coeffcents become accurate numbers, the equatons wll become the classcal lnear regresson equatons, and the results wll also become accurate numbers. x0 x1 x2 x3 x4 HC ξ η CO ξ η NOx ξ η PM ξ η

5 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl 2013 ABLE 4 HE SIMULAION RESULS OF H=0.8 CALCULAED WIH HE FUZZY LINEAR REGRESSION EQUAIONS he smulaton samples HC Upper lmts Centers Lower lmts CO Upper lmts Centers Lower lmts NOx Upper lmts Centers Lower lmts PM Upper lmts Centers Lower lmts able 4 gves the center values, upper lmts and lower lmts of the smulaton results. It s found from the table that, except for the actual value of PM of sample 4 whch s out of the nterval of the smulaton results, the other actual values le n the ntervals of the smulaton results, because the ntervals of the smulaton results are much smaller than the center values of the smulaton results whch are ratonal. hs shows that all of the fuzz lnear regresson equatons can present correctl the functonal relatonshp between the ndependent varables and the dependent varables. We can use the equatons to predct the emssons from engnes. able 5 gves the predcton results. Comparng wth the actual values n table 2, t s found that, except for the actual values of NOx of sample 3 (fuel 10) and PM of samples 1,2 (fuel 2,6) whch are out of the ntervals of the predcton results, the other actual values le n the ntervals of the predcton results. When the actual values do not le n the ntervals of the predcton results, the approach the upper lmts or the lower lmts of the predcton results. Comparng the upper lmts wth the lower lmts of the predcton results, t s also found that, when the actual values are not n the ntervals of the predcton results, the ntervals of the predcton results are small. hs s caused b the small fuzz wdth of the sstem. Hence, n order to mprove the predcton ablt of the equatons, the fuzz wdth of the coeffcents must be ncreased. ABLE 5 HE PREDICION RESULS OF H=0.8 ALCULAED WIH HE FUZZY LINEAR REGRESSION EQUAIONS he predcton samples HC Upper lmts Centers Lower lmts CO Upper lmts Centers Lower lmts NOx Upper lmts Centers Lower lmts PM Upper lmts Centers Lower lmts Further Analss and Dscusson of the Degree of the Membershp he degree of the membershp s selected b the decde maker n terms of the demand of the actual problem. Accordng to (15), H nfluences the soluton of the coeffcents of the equatons, so t s necessar to further analze H. Here, H=0.5 and H=0.9 are selected to analze the effect of them on the coeffcents of the equatons. 110

6 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl ABLE 6 HE FUZZY REGRESSION COEFFICIENS OF H=0.5 x0 x1 x2 x3 x4 HC ξ η CO ξ η NOx ξ η PM ξ η able 6 gves the coeffcents of the equatons for H=0.5. Comparng wth table 3, t s found that the fuzz centers of the coeffcents of the equatons of HC, CO and NOx are the same as those of H=0.8. However, the fuzz centers of the coeffcents of the equatons of PM are dfferent from those of H=0.8, but both fuzz centers are close. Except for the fuzz wdths whch are equal to zero, the other fuzz wdths of the coeffcents of the equatons are smaller than those of H=0.8. So when H=0.5, smulaton HC, CO and NOx wth the equatons, the center values of the smulaton results are not changed. However, smulatng PM wth the equaton, the center values of the smulaton results are changed. he ntervals of the smulaton results are decreased. able 7 gves the center values, upper lmts and lower lmts of the smulaton results. It s found from the table that the actual values of HC depart from the center values and le n the upper or lower lmts of the smulaton results. For CO, except for sample 3 (fuel 4) where the actual values approach the center value of the smulaton result, the other actual values present the same law as those of HC. For NOx, except for 3, 6 (fuels 4, 8) where the actual values approach the center values of the smulaton results, the other actual values also present the same law as those of HC. Comparng the actual values wth the smulaton results of PM, t s found that the actual values of samples 2, 3, 4, 8(fuels 3, 4, 5, 11)le n the nterval of the smulaton results. he other actual values do not le n the ntervals of the smulaton results. Hence, when H=0.5, the smulaton accurac of the equatons s worse than that of H=0.8. ABLE 7 HE SIMULAION RESULS OF H=0.5 CALCULAED WIH HE FUZZY LINEAR REGRESSION EQUAIONS he smulaton samples HC Upper lmts Centers Lower lmts CO Upper lmts Centers Lower lmts NOx Upper lmts Centers Lower lmts PM Upper lmts Centers Lower lmts

7 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl 2013 ABLE 8 HE PREDICION RESULS OF H=0.5 CALCULAED WIH HE FUZZY LINEAR REGRESSION EQUAIONS he predcton samples HC Upper lmts Centers Lower lmts CO Upper lmts Centers Lower lmts NOx Upper lmts Centers Lower lmts PM Upper lmts Centers Lower lmts able 8 gves the predcton results. Comparng wth the actual values of table 2, t s found that the actual values of HC le n the ntervals of the predcton results. For CO, the actual value of sample 2 (fuel 6) les n the nterval of the predcton result, otherwse the other two actual values do not le n the ntervals of the predcton results. For NOx, the actual value of sample 1 (fuel 2) les n the nterval of the predcton result, et the other two actual values exclude to do so. However, for PM, none of the actual values le n the ntervals of the predcton results. Hence, the predcton results are not good. able 9 gves the coeffcents of the equatons for H=0.9. Comparng wth table 3, t s found that the fuzz centers of the coeffcents of the equatons for HC, CO and NOx are the same as those of H=0.8. he fuzz centers of the coeffcents of the equaton for PM are dfferent from those of H=0.8, except that when the fuzz wdths are equal to zero, the other fuzz wdths are bgger than those of H=0.8. Smlarl comparng wth table 6, the same concluson s also reached. Hence, the ntervals of the smulaton and predcton results of the equatons are bgger than those of H=0.8 and H=0.5. able 10 gves the center values, upper lmts and lower lmts of the smulaton results. Comparng wth the actual values of table 2, t s found that all of the actual values le n the ntervals of the smulaton results. Comparng wth the smulaton results of table 4, the center values of the smulaton results of HC, CO and NOx are same, and the center values of the smulaton results of PM are dfferent, but ther bases to the actual values are all small. However, the ntervals of the smulaton results ncrease. Smlarl, comparng wth table 7, the same concluson s also found. ABLE 9 HE FUZZY REGRESSION COEFFICIENS OF H=0.9 x0 x1 x2 x3 x4 HC ξ η CO ξ η NOx ξ η PM ξ η

8 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl ABLE 10 HE SIMULAION RESULS OF H=0.9 CALCULAED WIH HE FUZZY LINEAR REGRESSION EQUAIONS he smulaton samples HC Upper lmts Centers Lower lmts CO Upper lmts Centers Lower lmts NOx Upper lmts Centers Lower lmts PM Upper lmts Centers Lower lmts able 11 gves the predcton results. Comparng wth the actual values n table 2, t s found that, except for the actual values of NOx of sample 3 (fuel 10) and PM of the samples 1,2 (fuel 2,6) whch are out of the ntervals of the predcton results, the other actual values le n the ntervals of the predcton results. When the actual values do not le n the ntervals of the ABLE 11 HE PREDICION RESULS OF CALCULAED WIH HE FUZZY LINEAR REGRESSION EQUAIONS he predcton samples H=0.9 HC Upper lmts Centers Lower lmts CO Upper lmts Centers Lower lmts NOx Upper lmts Centers Lower lmts PM Upper lmts Centers Lower lmts In concluson, the predcton results of the equatons for H = 0.8 are the best of the selected degrees of the membershp, and those of H = 0.9 take second place, predcton results, the approach the upper lmts or the lower lmts of the predcton results, whch s smlar to that of H=0.8. Snce the ntervals of the predcton results are bgger than those of H=0.8, the predcton accurac s worse than that of H=0.8. Comparng wth table 8, t s also found that ts predcton accurac s better than that of H=0.5. whle those of H = 0.5 are the worst. When H s changed, the fuzz centers of the coeffcents of the equatons for HC, CO and NO x are not changed, but the fuzz centers of the coeffcents of the equatons for PM are changed. Except when the fuzz wdths are equal to zero, the other fuzz wdths ncrease wth H. Hence the ntervals of the smulaton and predcton results also ncrease. Further Analss and Dscusson on the Restrctons he smulaton accurac of the equatons s decded b the fuzz centers and fuzz wdths of the coeffcents. he fuzz centers decde the bases of the center values of the smulaton results to the actual values. he fuzz wdths decde the szes of the ntervals of the smulaton results. Hence, n order to make the bases small, an addtonal condton should be used to restrct the fuzz center of the coeffcents. In order to make the actual values le n the ntervals of the smulaton results and the ntervals not be too bg to have meanng, an addtonal condton should also be used to restrct the fuzz wdths of the coeffcents (Behrooz Heshmat et al., 1985). 113

9 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl 2013 In terms of (7), the bas can be controlled b x. It s denoted that wth the rato of x to, the rato s less than a percent selected b the decson maker. It can also be consdered as the form of the degree of the membershp of the trangular membershp functon,.e., ( x ) L, where L s a value selected b the decson maker n the L set. hat s to sa: x ( x ) 1 L (16) A smlar dea can be appled to the fuzz wdth ( x M ), here, M s a value also selected b the decson maker n the M set. he degree of the membershp s also smlarl defned wth the trangular membershp functon, and the dea s shown wth the followng formula: x ( x ) 1 M (17) Addng (16) and (17) nto the restrcton of the lnear programmng problem (8), a new lnear programmng problem s obtaned. he fuzz regresson coeffcents can be also obtaned b the soluton of the new lnear programmng problem. However, n terms of (7), snce the sze of H s related to and x, a bgger value of H ma be x correspondng to a less value of bgger value of and s bg and x and a x. Onl when the bas of x x closes or exceeds, can satsfactor results be acheved b addng (16) and (17) nto the restrcton of (8). In order to test the effect of the restrctons of (16) and (17) on the soluton of the coeffcents, H=0.8, L=0.95 and M=0.8 are selected and the coeffcents of the equatons are calculated. he coeffcents of the equatons of HC, CO and NOx are the same as those of H=0.8, the coeffcents of the equatons of PM are dfferent from those of H=0.8 and are shown n table 12. It shows that when H=0.8, the smulaton accurac of the equatons of HC, CO and NOx acheves or exceeds that of H=0.8, L=0.95 and M=0.8, so the restrctons do not operate. However, when calculatng the coeffcents of the equatons of PM, the restrctons operate and dfferent coeffcents are obtaned. Comparng the smulaton results n table 13 wth those n table 9, t s found that the bases of the center values of the smulaton results to the actual values of PM for H=0.8 are bgger than those of H=0.8, L=0.95 and M=0.8, as well as the ntervals of the smulaton results are also bgger than those of H=0.8, L=0.95 and M=0.8, so the smulaton accurac of the equaton for H=0.8, L=0.95 and M=0.8 s hgher than that of H=0.8. able 12 HE FUZZY REGRESSION COEFFICIENS OF H=0.8, L=0.95 AND M=0.8 x0 x1 x2 x3 x4 PM ξ η ABLE 13 HE SIMULAION RESULS OF H=0.8, L=0.95 AND M=0.8 CALCULAED WIH HE FUZZY LINEAR REGRESSION EQUAIONS he smulaton samples PM Upper lmts Centers Lower lmts

10 Internatonal Journal of Energ Scence (IJES) Volume 3 Issue 2, Aprl hus, an approprate H must be selected to solve the coeffcents. If the bases of the center values of the smulaton results to the actual values are small and the ntervals of the smulaton results are much smaller than the actual values, the restrctons are not used; otherwse, the restrctons should be used to obtan the satsfactor results. Conclusons he coeffcents of the ndependent varables obtaned wth the fuzz lnear regresson are fuzz numbers, so the results obtaned wth the equatons are also fuzz numbers. For the soluton of the coeffcents s related to H, whose sze s decded b the decson maker. When the sze of H s changed, the coeffcents wll be changed, and the smulaton and predcton accurac of the equatons wll also be changed. hus, the approprate H must be selected. In ths paper, when H=0.8 s selected, the results are satsfactor. When the bases of the center values of the smulaton results to the actual values are bg, or the ntervals of the smulaton results approach the actual values or are bgger than the actual values, the restrctons should be used to obtan the satsfactor results. REFERENCES Regresson and Its Applcatons to Forecastng n Uncertan Envronment. Fuzz Sets and Sstems 1985; 15: Hu G, Zhu R. Multple data analss method, anng: Nanka Unverst Press, Hdeo anaka, Satoru Uema, and Ko Asa. Lnear Regresson Analss wth Fuzz Model. IEEE ransactons on Sstem, Man, and Cbernetcs 1982; 6: He J, Jang D, Chen S. Practcal lnear programmng and computer program, Beng: snghua Unverst Press; L B, Zhu M, and Xu K. A practcal engneerng method for fuzz relablt analss of mechancal structures. Relablt Engneerng and Sstem Safet 2000; 67: L H, Wang P. Fuzz mathematcs, Beng: Natonal defense ndustr press, M.Hubln, P.G.Gadd, D.E.Hall, etc. European Programmes on Emssons, Fuels and Engne echnologes (EPEFE) Lght Dut Desel Stud, SAE Journal of Fuels & Lubrcants 1996; 4: Wu J. Devce fault dagnoss usng possblstc theor. Fuzz Sstems and Mathematcs 1999; 12: Behrooz Heshmat and Abraham Kandel, Fuzz Lnear 115