Multi-objective artificial bee colony algorithm for long-term scheduling of hydropower system: A case study of China

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1 Water Utlty Joural 7: 3-23, E.W. Publcatos Mult-objectve artfcal bee coloy algorth for log-ter schedulg of hydroower syste: A case study of Cha X. Lao *, J. Zhou, S. Ouyag, R. Zhag ad Y. Zhag School of Hydroower ad Iforato Egeerg, Huazhog Uversty of Scece ad Techology, Cha * e- al: laoxag@hust.edu.c Abstract: The ult-objectve log-ter ecooc dsatch hydroower syste s a colcated olear otzato roble wth a grou of colex costrats whch akes the otzato of coflct objectves through tradtoal ethods a hard task. Ths aer s a to reset a ovel ult-objectve evolutoary algorth aed ultobjectve artfcal bee coloy (MOABC) algorth ad coares the effcecy of MOABC ad establshed algorths log-ter cascaded hydroower syste dsatch. The troduced odfed eloyed hase roves the global otal caablty of MOABC algorth, ad a ovel robablty calculato ethod s eloyed to rove the search ablty of olooker bee hase. Moreover, a odfed eloyed bees hase cotrbutes to escae local extree value. Addtoally, a local search ethod based o chaos theory has bee troduced. The udate strategy of exteral archve set has bee troduced. The erforace of roosed MOABC has bee deostrated through a set of stadard test fuctos. I order to verfy the effectveess of roosed algorth further, the case of the world bggest hydroower syste, Three Gorges Project (TGP), has bee studed ths aer. Nuercal results ad coarsos deostrate the effectveess ad effcecy of roosed algorths whch alyg the log-ter schedulg of TGP hydroower systes Cha. The results showed the roosed ethod have a better covergece ablty ad dstrbuto of the Pareto frot. Key Words: hydroower schedulg; swar tellgece; ult-objectve; artfcal bee coloy. INTRODUCTION The hydroelectrcty cossts of early 20% of the global ower roducto the ew cetury (WCD 2000). Great otetal eergy of The Three Gorges Project (TGP), the bggest hydroower syste the world located the Yagtze Rver, could be develoed through rovg the syste oerato strateges. I ractce, rule curve whch deterg the acto real reservor oerato s extesvely utlzed. However, the schedulg of the hydroower syste s a colcated otzato roble assocated wth a set of colex costrats cotag hydraulc relatosh, varato of reservor storage, ower outut lt, water dscharges lt, head varato ad so o. Esecally ult-objectve otzato of the cascaded hydroower syste, the a hardsh s to fd the otal soluto fro a holstc ersectve by cosderg all objectves whch are geerally coflctg. Deterg the oerato olces of these colex characterstcs above through tradtoal rule curves ay acheve a alteratve oerato strategy but there s o guaratee that the schedulg result s otal. Deterstc otzato ethods used log-ter schedulg of the hydroower syste ca be classfed as atheatcal rograg ad coutatoal tellgece ethods. I the ast decades, ay atheatcal rograg ethods were aled schedulg of hydroower syste. Lear rograg (LP) (Jacovks et al. 989; Poabala et al. 989), o-lear rograg (NLP) (Barros et al. 2003), Lagrage relaxato (LR) (Gua et al. 997), dyac rograg (DP) (Ferrero et al. 998), are aled otzato of reservor oerato olces. However, the tradtoal atheatcal rograg ethod faced varous obstacles ractcal leetato. The a drawback of LP s that all the objectve fuctos ad costrats ust be lear. That eas these atheatcal odels are ot accurate because of the aroxato. For

2 4 X. Lao et al. NLP, the objectve fuctos eed to be cotuous ad dfferetal, ad the global otalty s also a roble egeerg alcato of NLP. The effects of otzato results of LR are flueced by the Lagrage ultlers udatg strategy ad tal solutos. Whe DP s aled schedulg of a hydroower syste, the decso varables eed to be dscretzed. Curse of desoalty occurred whe the ot of dscretzato s creasg. I recet decades, ore ad ore scholars focus o the alcato of coutatoal tellget ethods. The otzato ethods based o coutatoal tellget such as geetc algorth (Hcal et al. 20), sulated aealg(teegavarau ad Soovc 2002), at coloy otzato (Jalal et al. 2007), artcle swar otzato (Afshar 202; Kuar ad Reddy 2007), hoey bees atg otzato (Haddad et al. 2006), artfcal bee coloy (Lao et al. 202) are aled deterg the reservor oerato rules. However, the ethods etoed above are desged to solve sgle objectve robles. I order to hadle the ult-objectve robles, the coo ethod s weghtg each objectve fucto ad covertg to oe objectve fucto (Madal ad Chakraborty 20). Moreover, soe objectves are treated as costrats to decrease the uber of objectve fuctos. However, the redetered weghtg factor could ot reveal the relatosh betwee the each objectve fuctos. Actually, these ethods etoed above are sgle objectve otzato roble. Thus, ay ult-objectve evolutoary algorths (MOEA) such as o-doated sortg geetc algorth II (NSGA-II) (Deb et al. 2002), ehaced stregth Pareto evolutoary algorth (SPEA2) (Eckart et al. 200), ult-objectve artcle swar otzato algorth (Trath et al. 2007) (MOPSO) ad ult-objectve dfferetal evoluto algorth (MODE) (Xue et al. 2003) had bee develoed to hadle the ultobjectve otzato robles. MOEA have soe advatages of hadlg coflctg objectve fuctos because of the structure of algorth. The goal of MOEA s to fd a o-doated soluto set uforly dstrbutg o the true Pareto frot. However, these ethods etoed above are stll suffered fro the reature covergece because of the evolutoary echas. Therefore, the develoet of ew aroach ad roveet of exstg ethod are ecessary order to solve the colcated costrat ult-objectve robles. Sce the artfcal bee coloy algorth (Karaboga 2005) had bee develoed the ast few years, the swar tellget techque drawg fro the foragg behavors of bee coloy showed the great otetal of solvg the varous colex otzato roble (Karaboga ad Ozturk 2009; Ozturk et al. 202). A Pareto-based dscrete artfcal bee coloy (L et al. 20) was develoed for solvg the flexble job sho schedulg roble. However, t was desged for dscrete otzato roble. Hece, we roosed a ew ult-objectve otzato techque aed ult-objectve artfcal bee coloy algorth (MOABC) ad verfed the effectveess of MOABC by a set of test fuctos. Fally, we aled the roosed ethod the log-ter schedulg of TGP ad deostrate the effectveess ad effcecy of MOABC. The rest of aer s orgazed as follows: the secto 2 troduces the log-ter schedulg roble of the hydroower syste. MOABC s descrbed detal ad verfed by test fuctos secto 3. MOABC s alyg the ult-objectve log-ter schedulg of TGP secto 4. Secto 5 s the cocluso of our work. 2. PROBLEM FORMULATION The ult-objectve log-ter ecooc dsatch the hydroower syste s aed to utlze the otetal eergy of water TGP ad covert to the electrcal eergy. The aager of TGP always wats to ze the geerato beeft of the hydroower syste. O the other had, the ower dead of grd ust be satsfed through the oerato of the hydroower syste eve the dry seaso. Hece, ze the fr outut s aother ortat task the oerato of the hydroower syste. The fr outut s defed as the u average othly outut of the hydroower syste. I ths aer, we forulate the ult-objectve log-ter schedulg roble as follows:

3 Water Utlty Joural 7 (204) 5 2. Objectve fuctos Whe cosderg both geerato beefts ad fr outut sultaeously, the otzato becoes ult-objectve robles. Therefore, the roble ca be forulated as follows: I T F = C A Q H Δt, (), t, t = t= F = { A Q H }, (2) 2, t, t Ω I t= T where F s to ze the geerato beefts over the whole schedulg erod. F 2 s to ze the u total outut durg the schedulg erod. I reresets the uber of hydro lats; T s the legth of schedulg erod; C s the electrcty rce of hydro lat ; A s the ower geerato coeffcet of hydro lat ; Ω s the set whch cludes all the hydroower I ut; Q, H are the water dscharge ad et head of hydro lat at te t, resectvely Costrats The objectve fuctos etoed above subject to the followg costrats: Hydraulc coecto: I = Q + S + R t T (3), t, t Water dyac balace equlbru: V = V + ( I Q S ) Δt t T (4) Fal water level lts: Z = Z, start, ed (5) Reservor water level costrats: Z Z Z t T (6) Water release costrats: Q ( Q + S ) Q t T (7) Outut lts: P A Q H ) P t T (8) where I, R are atural flow ad local flow of hydro lat at te t, resectvely; S s, t abadoed flow of hydro lat at te t ; V s reservor storage volue at the ed of the

4 6 X. Lao et al. erod t. Q, Z, Z are u ad u urver water levels of hydro lat at te t ; Q are u ad u water dscharges of hydro lat at te t ; P, P are u ad u oututs of hydro lat at te t ; Z,, start Z are tal water, ed level ad fal water level of the schedulg erod. 3. MULTI-OBJECTIVE ARTIFICIAL BEE COLONY ALGORITHM The ult-objectve ssue are dfferet fro the sgle objectve otzato because the dfferet elte soluto keeg strateges ad the coarso echass of two solutos. I order to odfy the artfcal bee coloy for adatg ult-objectve robles, the followg odfcatos are establshed. 3.. Strategy of exteral archve set udatg Dfferet fro keeg oe otal soluto each cycle, exteral archves set Ω s used to kee the o-doated solutos U each coutg cycle. The relatosh betwee two solutos caot be descrbed sly as bgger or saller ad the fal otal result s ot a sgle soluto but a set of o-doated solutos. Thus, the exteral archve set udatg strategy ca be cocluded to three crteros: ) If Ω s ety, solutos U wll be added to Ω drectly. ) If a soluto u U has ot bee doated by ay solutos Ω, the soluto u wll be added to Ω, the solutos Ω whch doated by u wll be deleted. Otherwse, the soluto u wll be abadoed. ) If the uber of solutos Ω s larger tha the u sze, the redudat solutos wll be deleted accordg to evaluate the crowdg dstace etrc (Deb et al. 2002) of each soluto Modfcato of artfcal bee coloy algorth Eloyed bees hase I roosed ABC algorth, a araeter aed odfcato rate (MR) s used to roduce the ew food sources. I order to cotrol the robablty of roducg ew food source, the followg equato s used to detere the roducg rate. v v = x + φ ( x x ), f R < MR = x, otherwse k (9) where [0, D] reresets the deso of dvdual soluto sace (food resources), s a seral uber of a food source corresodg to a soluto of roble. v ad x rereset the ew ad old food source, resectvely. R s a rado real uber betwee [0,]. φ s a rado real uber dstrbuted uforly betwee [,] whch cotrols the effectveess of dstace betwee x ad x k. Obvously, ew food source s affected by the dstrbuto of bee coloy. MR s a rado uber betwee [0,] whch cotrols the utato robablty of solutos. x ad x k are chose fro the old oulato orgal ABC algorth, but the roosed ethod, both the

5 Water Utlty Joural 7 (204) 7 solutos are chose fro the exteral archves set. Because the solutos exteral archves set have a better ftess coare to the old oulato, the covergece seed wll ore quckly Probablty calculato I order to calculate the robablty araeter of olooker bees hase, the orgal ABC used ftess value. The better ftess value, the hgher ossbltes of solutos to be chose olooker bees hase, but ult-objectve roble, the ftess value of oe objectve s hard to descrbe the status of soluto sce the coflctg objectves. Hece, we use the crowdg dstace etrc to easure the solutos teratve coutato. The robablty s forulated as follow: volato 0.2, f volato > 0 volato = cd , cd otherwse (0) where s the robablty value of soluto. cd reresets the crowdg dstace etrc of the soluto, cd s the crowdg dstace etrc of the curret oulato. volato s the volato value of soluto, of the soluto oulato. volato s the volato value Select echas Dfferet fro sgle objectve roble, doace relatosh had bee used to detere whch oe s referred betwee two of solutos ult-objectve otzato robles. I the roosed ethod, we chose the soluto accordg to the followg logc: Coare two solutos, the soluto whch doate the other wll be added to the odoated set U. Both the two solutos wll be added to U whe they are o-doated wth each other Chaos local search strategy I order to rove the covergece effectveess, the Chaos local search had bee adoted MOABC. The detal rocedure of chaos local search s descrbed as follow: STEP. Logstc a has bee used to geerate the chaos sequece. The logstc a s forulated as follow: C + = µ C ( C ) () sk sk sk where µ s a cotrol araeter, C s chaos varable of k th o-doated soluto Ω after sk tes terato. C (0,). Oce the terato uber s creasg, the sk 0 chaotc ad dyac characterstc whe µ = 4 ad C {0.25, 0.5, 0.75}. STEP 2. We geerate the ew soluto by the followg equatos. sk sk C reflect the sk x = x + C ( x x ) (2) k

6 8 X. Lao et al. where xk s ew soluto geerated by C sk. STEP 3. Geerate the ew soluto as follow: ew best x = x M + ( M ) x (3) k k c c k where M c s chaos utato rate betwee [0,] whch cotrol the utato of best x. k STEP 4. Coare the doace relatosh betwee Ω. ew x k ad best x k, ad kee the better oe STEP 5. Go to STEP 2 utl the chaos terato uber acheved the u uber. 3.4 Fraework of MOABC The fraework of MOABC s show as Fg.. Start Italze food sources ad coutato codto Cycle= Eloyed bees hase Olooker bees hase Scout bee hase Udate the exteral archve set Chaos local search Cycle=Cycle+ Cycle>cycle? N Y Outut the result Fgure. The fraework of MOABC.

7 Water Utlty Joural 7 (204) Nuercal test of MOABC I order to verfy the effectveess of MOABC, a set of faous test fuctos had bee tested ths aer. The araeters settg of ZDT test fuctos are lsted as follow: MR=0.02, geerato uber s 500, the artfcal bee coloy sze s 00, exteral archve sets s set to True Pareto 0.8 True Pareto F F F F (a) ZDT (b) ZDT true Pareto 0.8 True Pareto frot F F F F (c) ZDT 3 (d) ZDT True Pareto frot 0.6 F F (e) ZDT 6 Fgure 2. Otzato results of MOABC ZDT test fuctos. The ZDT test fuctos deostrate the effectveess of MOABC algorth Fg.2. We ca clearly see fro the Fg.2 that all the solutos calculated by MOABC are dstrbuted aroxate

8 20 X. Lao et al. uforly o the true Pareto frot. The algorth has a accetable erforace of dstrbuto ad covergece ablty accordg the Fg CASE STUDY: MULTI-OBJECTIVE LONG-TERM SCHEDULING OF THREE GORGES PROJECT 4.. Paraeters settg of TGP hydroower syste The MOABC algorth had bee roosed to solve the ult-objectve log-ter ecooc dsatch of TGP cascaded reservors whch cotag Three Gorges Da ad Gezhou Da. I order to test the effectveess of roosed ethod real egeerg scee, the flow sequeces of flood ad dry seasos had bee used as the coutato codtos. For coveece, the local flow had ot bee cosdered the atheatcal odel. The schedulg erod s a water year ad the terval s a oth. The two dfferet water flow sequeces of TGP hydroower syste are lsted Table ad the boudary codtos are lsted Table 2. Table. Iflows of TGP hydroower syste. Water flow ( 3 /s) Water flow ( 3 /s) oth oth flood dry flood dry Table 2. Boudary codtos for TGP hydroower systes. araeter Three Gorges Gezhou Da Hydro lat dscharge rage ( 3 /s) [4500,98800] [4500,86000] Power geerato coeffcet Urver water level rage () [45,75] 65 Power geerato rage (MW) [4990,822.6] [946,288.6] Water head rage () [6,3] [6,27.8] Ital water level () The ult-objectve dfferetal evolutoary (Zhou et al. 20) (MODE) algorth had bee aled to solve the schedulg roble of TGP for deostratg the effectveess of MOABC. For both ethods, the oulato sze s set to 50, the exteral archve set sze s set to 30 ad the terato uber s set to I roosed ethod, the lt uber s set to 50. Ad MR=0.0. I MODE, the araeter F s set to 0.25 ad cross rate (CR) s set to Results dscusso We aled the roosed ethod to solve the schedulg roble of TGP ad coared wth the establshed ethod. The results are show Fg. 3. All the o-doated solutos whch geerated by flows dry seaso had bee lsted the Table 3. Through the results, t ca be clearly see that the o-doated frot geerated by MOABC has a better covergece effectveess coared to MODE the flood ad dry water flow codtos. Furtherore, the

9 Water Utlty Joural 7 (204) 2 results geerated by MOABC wth the flows dry seaso are obvously better tha by MODE. That shows the otetal of MOABC ethod whe the flows are suffcet. I addto, the solutos dstrbuted aroxate uforly o the o-doated frot. That eas the dstrbuto strategy of exteral archve set had bee verfed. Fro the Table 3 we ca see that all the solutos are o-doated. The results testfy the effectveess of exteral archve set udatg strategy aother asect. Fro the results we dscussed above, we ca coclude that the roosed MOABC s a effectve techology to solve the ult-objectve roble. The erforace of roosed ethod had bee satsfed by alyg log-ter schedulg of TGP MOABC MODE MOABC MODE fr outut 740 fr outut ower geerato ower geerato (a) flood (b) dry Fgure 3. Coarso results betwee MOABC ad MODE. Table 3. No-doated solutos of dry seaso Idex Power geerato Fr outut Idex Power geerato Fr outut

10 22 X. Lao et al. 5. CONCLUSION Ths aer resets a ovel ult-objectve evolutoary algorth aed ult-objectve artfcal bee coloy (MOABC) algorth ad coares the effcecy of MOABC ad establshed algorths log-ter cascaded hydroower syste dsatch. A set of test fuctos ad the logter schedulg roble of Three Gorges Project have bee used to verfy the effectveess of roosed ethod. The results showed that the roosed ethod has a better covergece ablty ad dstrbuto of the Pareto frot coarg to the establshed ethod. ACKNOWLEDGEMENTS Ths work was suorted by the Natoal Natural Scece Foudato of Cha (No ), ad the Secalzed Research Fud for the Doctoral Progra of Hgher Educato of Cha (No ). The authors also thak for the sghtful coets ad suggestos of aoyous revewers. REFERENCES Afshar, M. H. (202). "Large scale reservor oerato by Costraed Partcle Swar Otzato algorths." Joural of Hydro- Evroet Research, 6(), Barros, M. T. L., Tsa, F. T. C., Yag, S. L., Loes, J. E. G., ad Yeh, W. W. G. (2003). "Otzato of large-scale hydroower syste oeratos." Joural of Water Resources Plag ad Maageet-Asce, 29(3), Deb, K., Prata, A., Agarwal, S., ad Meyarva, T. (2002). "A fast ad eltst ultobjectve geetc algorth: NSGA-II." Ieee Trasactos o Evolutoary Coutato, 6(2), Eckart, Z., Marco, L., ad Lothar, T. (200). "SPEA2: Irovg the Stregth Pareto Evolutoary Algorth." Ferrero, R. W., Rvera, J. F., ad Shahdehour, S. M. (998). "Dyac rograg two-stage algorth for log-ter hydrotheral schedulg of ultreservor systes." Ieee Trasactos o Power Systes, 3(4), Gua, X. H., N, E. N., L, R. H., ad Luh, P. B. (997). "A otzato-based algorth for schedulg hydrotheral ower systes wth cascaded reservors ad dscrete hydro costrats." Ieee Trasactos o Power Systes, 2(4), Haddad, O. B., Afshar, A., ad Maro, M. A. (2006). "Hoey-bees atg otzato (HBMO) algorth: A ew heurstc aroach for water resources otzato." Water Resources Maageet, 20(5), Hcal, O., Alta-Sakarya, A. B., ad Ger, A. M. (20). "Otzato of Multreservor Systes by Geetc Algorth." Water Resources Maageet, 25(5), Jacovks, P. M., Gradowczyk, H., Fresztav, A. M., ad Tabak, E. G. (989). "A lear rograg aroach to water-resources otzato." Zetschrft für Oeratos Research, 33(5), Jalal, M. R., Afshar, A., ad Maro, M. A. (2007). "Mult-coloy at algorth for cotuous ult-reservor oerato otzato roble." Water Resources Maageet, 2(9), Karaboga, D. (2005). "A dea based o hoey bee swar for uercal otzato." Techcal Reort-TR06, Ercyes Uversty, Egeerg Faculty, Couter Egeerg Deartet. Karaboga, D., ad Ozturk, C. (2009). "Neural Networks Trag by Artfcal Bee Coloy Algorth o Patter Classfcato." Neural Network World, 9(3), Kuar, D. N., ad Reddy, M. J. (2007). "Multurose reservor oerato usg artcle swar otzato." Joural of Water Resources Plag ad Maageet-Asce, 33(3), L, J. Q., Pa, Q. K., ad Gao, K. Z. (20). "Pareto-based dscrete artfcal bee coloy algorth for ult-objectve flexble job sho schedulg robles." Iteratoal Joural of Advaced Maufacturg Techology, 55(9-2), Lao, X., Zhou, J., Zhag, R., ad Zhag, Y. (202). "A adatve artfcal bee coloy algorth for log-ter ecooc dsatch cascaded hydroower systes." Iteratoal Joural of Electrcal Power & Eergy Systes, 43(), Madal, K. K., ad Chakraborty, N. (20). "Short-ter cobed ecooc esso schedulg of hydrotheral systes wth cascaded reservors usg artcle swar otzato techque." Aled Soft Coutg, (), Ozturk, C., Karaboga, D., ad Gorkel, B. (202). "Artfcal bee coloy algorth for dyac deloyet of wreless sesor etworks." Turksh Joural of Electrcal Egeerg ad Couter Sceces, 20(2), Poabala, K., Vaell, A., ad Uy, T. E. (989). "A alcato of Kararkar's teror-ot lear rograg algorth for ult-reservor oeratos otzato." Stochastc Hydrology ad Hydraulcs, 3(), Teegavarau, R. S. V., ad Soovc, S. P. (2002). "Otal oerato of reservor systes usg sulated aealg." Water Resources Maageet, 6(5), Trath, P. K., Badyoadhyay, S., ad Pal, S. K. (2007). "Mult-Objectve Partcle Swar Otzato wth te varat erta ad accelerato coeffcets." Iforato Sceces, 77(22), WCD. (2000). "Das ad develoet: a ew fraework for decso-akg." Avalable va DIALOG, htt://

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