Deriving Reservoir Operating Rules via Fuzzy Regression and ANFIS

Transcription

1 Dervng Reservor Operang Rules va Fuzzy Regresson and ANFIS S. J. Mousav K. Ponnambalam and F. Karray Deparmen of Cvl Engneerng Deparmen of Sysems Desgn Engneerng Unversy of Scence and Technology Unversy of Waerloo, Canada Tehran, Iran, Absrac The mehods of ordnary leas-squares regresson (OLSR, fuzzy regresson (FR, and adapve nework fuzzy nference sysem (ANFIS are compared n nferrng operang rules for a reservor operaons problem. Dynamc programmng (DP s used o provde he npu-oupu daa se o be used by OLSR, FR, and ANFIS models. The coeffcens of an FR model are found by solvng a lnear programmng (LP problem. A raned fuzzy nference sysem (ANFIS s also used o derve he reservor operang rules as fuzzy f-hen rules. The OLSR, FR, and ANFIS based rules are hen smulaed and compared. The mehods are appled o a long-erm plannng problem and o a medum-erm mplc sochasc opmzaon model. FR s useful o derve operang rules for he long-erm model, where paral nformaon s avalable. ANFIS s benefcal n he medum erm mplc sochasc model as s able o exrac mporan feaures of he sysem from he generaed npu-oupu se. Keywords: operang rules, fuzzy regresson, fuzzy nference sysem. 1. Inroducon Fuzzy regresson (FR and adapve-nework-based FIS (ANFIS are used n nferrng operang rules for reservor operaons. A summary of dfferen mehods used for surface reservor managemen can be found n Yeh [1] and Ponnambalam [2]. Inferrng operang rules s o ge general rules by whch reservor operaons can be conrolled. Young [3], Bhaskar and Whlach [4] and Karamouz and Houck [5] used mulple lnear regresson o derve he operang rules. Oher mehods ncluded usng he Arfcal Neural Neworks (ANN and fuzzy rulebased echnque. In hs sudy fuzzy regresson, frs nroduced by Tanaka e al. [6], s examned n dervng operang rules for reservor operaons. Two problems are addressed here, a long-erm opmzaon problem and an mplc sochasc opmzaon model. The OLSR, FR, and ANFIS are used o derve operang rules and hey are hen smulaed and compared. 2. Mehods of nferrng operang rules Le us assume ha Dynamc Programmng (DP s used o derve opmal releases for a sequence of nflows, as here. Once he opmal releases are avalable, he problem s o derve an operang polcy for any feasble sorage and nflow from hese releases. Gven he varous nflow scenaros and he correspondng opmal releases and sorage volumes, a mulple regresson analyss can be used n he followng manner. 2 2 r = a + bs + c + ds (1 where r s he release and s he nflow durng perod. s s he begnnng sorage volume Fuzzy Regresson (FR Suppose a smple lnear relaon as: ^ y 1 = a + a x (2

2 h where ^y s he model predced value and x s a crsp observed value. The FR problem s o fnd fuzzy coeffcens a c( a, w( ] and a c( a, w( ] so [ a [ a ha he wdhs of he observed fuzzy values y (s s whn he wdhs of he predced fuzzy values ^ y (s a a confdence level h [,1 ] (see Fgure 1. Ths mples ha he nerval DE n Fgure 1 has o be whn he nerval BC. Noe ha c( a and w( a are he cener pon and half wdh of fuzzy number a for a rangular symmercal fuzzy number. m ( B y y A Mnmum-fuzzness-based FR Models Therefore, he oal fuzzness of FR model relaes o he wdh of fuzzy coeffcens used n he model as hey are mulpled by he observed values. Hence, he problem of fndng he parameers of Equaon (2 leads o he soluon of an LP wh an obecve funcon mnmzng he oal fuzzness of he model. Ths LP model s as follows: E Fgure 1: Inerpreaon of fuzzy regresson y Mnmzez = nw. + w. s. : = = c ( a. x + (1 h. c ( a. x (1 h. = 1 = 1 n = x w( a. x y w( a. x y + (1 h. w ( y = 1,... n (1 h. w ( y = 1,... n c ( a, c1 ( a1 (3 Where s he number of ndependen varables and n s he number of daa pars and h s called he credbly level. In he consrans of he above LP model, c ( a. x s he cener pon of he predced values (pon A n Fgure 1, a predcon whch would be reached o f a non-fuzzy regresson equaon was used. The second erm ± (1 h. w ( a. x represens he fuzzness = of he predced values shown by pons C and B n Fgure 1, when hey are added o he cener pon A. The rgh hand sdes of he consrans are he maxmum and mnmum values of a fuzzy observed y (pons E and D n Fgure 1, respecvely. Bardossy e al. [7] used anoher formulaon. They consdered more general nonlnear, nonsymmercal LR ype fuzzy coeffcens (Fgure 2 as follows: L( x = R( x = (1 x m a ( c( a x ( x = L wl ( a p for x w ( a ( x c( a ma ( x = R for x wr ( a wr ( a (4 where p akes neger values and m( a ( x s he membershp degree of fuzzy number a n x. Also wl ( a and wr ( a are he lef and rgh wdhs of a fuzzy number havng cener value of c( a. l

3 m( a L( a R( a c ( a w r ( a Fgure 2: An LR Fuzzy number 2.2 Fuzzy nference sysem (FIS a Operang rules can be represened by fuzzy IF- THEN rules n a Takag-Sugeno FIS as follows: If s s v 1 and s v 2 hen r = b1. + b2. s + b3 where, v k s he value of he k h explanaory varable and b are he parameers of he consequence par of rule. Each value of an explanaory varable s represened by a fuzzy se. Therefore, a k s a fuzzy se. The parameers b are esmaed usng avalable daa or operaor experence. There s no sysemac way o know wha ype of membershp funcons of premse varables s he bes n a defned FIS. An effcen way for dong hs s usng arfcal neural nes raned by npu-oupu daa lke ANFIS [8]. ANFIS uses an nal FIS and unes wh a backpropagaon algorhm. 3. Implemenaons In hs sudy, Dez reservor locaed n Iran has been seleced o whch he mehods are appled. For he mplc sochasc model used n hs sudy, a me seres model was fed o he hsorcal nflows. Then was used o generae a large number (1 of scenaros of monhly nflows. Usng eher he hsorcal or he generaed nflows n DP, he opmal ses of reservor sorage and release volumes are obaned correspondng o he long-erm deermnsc model n he frs case and he medum erm mplc sochasc model. These opmal ses are hen used n nferrng operang rules usng OLSR, FR, and ANFIS. The nferred operang rules are hen smulaed o see how hey perform. 3.1 Long-erm plannng opmzaon Implemenaon of he FR model The opmal sorages and releases along wh hsorcal nflows were used n he FR model o nfer he general operang rules. For each monh an FR model was bul separaely. The regresson equaon used s as follows: r = a + a1 + a2s (5 Takng fory years of opmal sorage and release ses obaned by DP along wh he hsorcal nflows, he FR was appled o he problem. Implemenaon of ANFIS ANFIS s used o exrac he relaon of sorage, nflow, and release varables from he daa pars obaned by DP and represen hem as fuzzy f-hen rules. The premse par of fuzzy f-hen rules s reservor nflow and sorage volumes. The consequen par s he release volume. For each monh, an ANFIS model was developed separaely. To make smulaon resuls ndependen of nal sorage volume, reservor operaon was frs smulaed by OLSR wh dfferen nal reservor sorages. These smulaed pahs and he orgnal DP opmal pah were used for ranng ANFIS. Thus, when comparng dfferen mehods, ANFIS resuls are he resuls of hs reraned ANFIS. Fgure 3 shows he mean square error (MSE of predced values by he prelmnary FIS, ANFIS, OLSR, and FR for dfferen monhs. In erms of fness capably, ANFIS s performed superor o oher mehods. However, he mporan hng s o analyse her performance n smulaon. mean square error of fness 12 x Comparson of MSE of FR, OLSR, FIS, and ANFIS FR OLSR FIS ANFIS monh Fgure 3: Comparson of mean square error of dfferen models o f opmal daa (long erm problem

4 3.2 Medum-erm mplc sochasc model In he second problem, we derve operang rules from npu-oupu daa obaned from a Mone-Carlo DP model. DP model was solved for 3 synhec scenaros. Each scenaro has one year of horzon wh 12 monhly me seps. The fnal sorage a he end of each year was forced o be more han half of he reservor capacy. Inferrng operang rules As was done n he long erm plannng model, OLSR, FR, and ANFIS parameers are deermned based on he npu-oupu pars obaned by Mone- Carlo DP. Dfferen opmal pahs of DP, for dfferen nal sorage were used n ANFIS ranng; hs s o overcome he dffculy of sorage fallng ou of he ranng ranges n smulaon. Table 1 presens he resuls for comparng he overall performance of OLSR, FR, FIS, and ANFIS rules n erms of her fng capably. Also he monhly dsrbuon of MSE of predced values s llusraed n Fgure 4. These resuls mply ha ANFIS performs que superor o oher mehods n erms of fng. However, he smulaon wll show he real success of he mehods. mean square error of fness Comparson of MSE of FR, OLSR, FIS, and ANFIS FR OLSR FIS ANFIS monh Fgure 4: Monhly dsrbuon of MSE of predced values by dfferen mehods (medum erm problem 4. Resuls and comparsons The OLSR, FR, and ANFIS are compared based on smulang he reservor operaons usng correspondng polces. The smulaon performance ndces are compued. The ndces seleced are: mean value of he smulaed obecve funcon (LOSS, coeffcen of varaon of LOSS (CV, and he percenage of he me n whch he release s greaer han.9 of demand (REL. 4.1 Long erm operaons For comparson purposes, he resuls of he so-called SOP polcy ha smply ses release volume equal o demand f possble and he orgnal DP releases (for long erm plannng only are ncluded. Table 2 presens he resuls. As s clear n Table 2, mean values of he smulaed obecve funcon of he OLSR, FR, and ANFIS mehods are more or less close whle FR has he smalles (bes LOSS value. In erms of relably of meeng he waer demand, he SOP polcy s he bes. SOP produces severe shorages or splls over he smulaed horzon and hence he larger LOSS value. FR has he smalles coeffcen of varaon (CV of he obecve funcon. Overall, he FR s he bes and s resuls are he closes o he DP resuls, alhough s he wors n erms of he MSE. Several cases are found when, for example, us 2 years of sreamflow record s avalable and s hard or naccurae o exend he record usng hydrologcal mehods. On he oher hand, plannng and desgn of he proecs need o be done for 4 or 5 years of her proeced lfe. To examne wheher FR s useful n such a suaon, he resuls were repeaed wh eher half or one-hrd of he sreamflow record for esmang parameers, bu carryng ou he smulaon for he whole fory years. ANFIS was no able o ackle hs case because of he lack of daa for ranng. Therefore, only OLSR and FR are compared n hs case. The resuls are n Table 3. Agan, he FR performs que well especally n erms of he value of smulaed obecve funcon. 4.2 Medum erm operaons For hs problem, smulaon was carred ou usng 3 synhecally generaed scenaros no used n ranng sage. The resuls are presened n Table 4. Accordng o Table 4, unlke he plannng model, ANFIS s superor o oher mehods and performs well boh n fng and smulaon modes. I s obvous ha SOP s he bes n erms of he relably, bu s smulaed loss as he mos mporan ndex s much hgher han oher polces. The resuls of he mplc sochasc model show ha he performance of he FR s beer han OLSR model bu no as good as ANFIS. As n long-erm plannng, we are neresed o he effec of a reduced number of he scenaros used n he opmzaon model and he parameer esmaon of he nferrng

5 mehods. The resuls are n Table 5 for 5 and 3 scenaros, respecvely. As we see, here s no a sgnfcan dfference beween he cases wh 3 and 5 or even 3 scenaros. 4.3 Sensvy Analyss The mos mporan facor n an FR s he degree of fuzzness consdered for observed values of dependen varable. Take he FR defned n Equaon (3; f we smply se h =, he fuzzness of y wll be defned by y * y = w ( y. The varaon of smulaed obecve funcon (LOSS, wh he varaon of y showed ha for y = 1%, he LOSS s he lowes bu he oher LOSS values are no oo far off. As he degree of fuzzness ncreases, he degree of fuzzness of he dependen varable y consdered n he FR model should be ncreased oo. Earler we consdered he case n whch he uncerany of he model s ncreased due o lack of daa. In hs case, he combned effec of randomness of nflows and mprecson of dscrezaon and nal sorage effec wll provdes he possbly for smulaed releases beng far from he releases o be suggesed by regresson models. Therefore, s raonal o use an FR model assumng a hgher degree of fuzzness of dependen varable y refleced n y. Examnaons showed ha where half pars of daa are used n he parameer esmaon mode, he bes degree of fuzzness for y s around 4 percen and where only one hrd of daa s used; s around 7 percen usng LOSS as he creron. We close hs secon by rasng hs queson: when can a complcaed mehod lke ANFIS be useful? We realzed ha he capably of ANFIS n erms of fness o daa s much beer han OLSR and FR. However, ANFIS dd no perform well n he longerm plannng problem whle performed que well n he mplc sochasc problem. ANFIS s powerful n erms of fng and exracng nonlnear and ll-defned relaons ha may exs n he daa. Therefore, f he knowledge and nformaon requred are refleced n he npu-oupu daa, ANFIS wll be able o exrac and conver hem o s knowledge-based core fuzzy rule base. Thus, f he npu-oupu daa nclude he nformaon on uncerany and mprecson of he real world sysem o be conrolled by ANFIS, ANFIS s expeced o be successful. However, f hose aspecs of he sysem behavour dd no exs n he npu-oupu daa or even hey exsed bu are purely whe nose, hen lnear rules are enough o represen hem. 5. Concluson The problem of nferrng operang rules usng an ordnary regresson, a fuzzy regresson, and an adapve neuro-fuzzy model explored. These mehods esed n wo ypes of models, a long-erm plannng and a medum-erm mplc sochasc opmzaon. For he long-erm plannng model, he fuzzy regresson showed promsng resuls, especally n he suaons where he lengh of sreamflow record s lmed. For he mplc sochasc opmzaon model snce he man nformaon by whch he reservor should be operaed s whn he daa se, ANFIS performed superor o oher mehods. Resuls ndcae ha he fness error of he models used n dervng operang rules does no necessarly show how well hey wll perform whle smulang her polces. Acknowledgemens Graeful hanks o Mr. Shahab Aragh-Nead of Cvl Engneerng Deparmen of Amrkabr Unversy of Technology for hs help wh he daa and s smulaon. References [1] Yeh, W., W.-G Reservor managemen and operaon models: A sae of he ar revew. Waer Resources Research, 21 (12, pp [2] Ponnambalam, K. 22 Opmzaon n waer reservor sysems. Handbook of Appled Opmzaon, Eded by Padalos, P. M., and Maurco, G. C. R., pp [3] Young, G. K Fndng reservor operang rules. Journal of Hydrology, 93 (6, pp [4] Bhaskar, N. R., and Whlach, Jr. E. E. 198 Dervng of monhly reservor release polces. Waer Resources Research, 16 (6, pp

6 [5] Karamouz, M., and Houck, M. H Annual and monhly reservor operang rules. Waer Resources Research, 18 (5, pp [6] Tanaka, H., Uema, S., and Asa, K Lnear regresson analyss wh fuzzy models. IEEE Trans. Sys. Man., Cybern., vol. 12, pp [7] Bardossy, A., bogard, I., and Ducksen, L Fuzzy regresson n hydrology. Waer Resources Research, 26 (7, pp [8] Jang, J.-S. R. (1993 ANFIS: Adapve - Nework-Based Fuzzy Inference Sysems. IEEE Transacons on Sysems, Man, and Cybernecs, 23 (3, pp Table 1: Comparson of nferred operang rules n erms of fed npu-oupu daa, mplc sochasc model MehodsOLSRFRFISANFIS MSE Mehods Table 2: Comparson of smulaon performance of OLSR, FR and ANFIS, long erm plannng model Smulaed obecve funcon LOSSCV MSEREL OLSR2.98e e588.6 FR1.9821e e584.6 ANFIS2.121e e49.6 SOP3.4946e DP1.4393e Mehod Table 3: Comparson of OLSR and FR where paral daa are used, long erm plannng model Scenaro wh half pars of daascenaro wh one-hrd pars of daa LOSSCVRELLOSSCVREL OLSR2.679e FR2.1398e e SOP3.4946e e DP1.4393e e Table 4: Comparson of smulaon performance of OLSR, FR and ANFIS, mplc sochasc model Mehods Smulaed obecve funcon LOSSCV OLSR FR ANFIS SOP REL Table 5: Comparson of OLSR and FR where paral daa are used n parameer esmaon, mplc sochasc model Mehod Wh 5 generaed scenaroswh 3 generaed scenaros LossCVRELLossCVREL OLSR FR ANFIS SOP