Research Article Dynamic Environmental/Economic Scheduling for Microgrid Using Improved MOEA/D-M2M

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1 Mathematcal Problems n Engneerng Volume 26, Artcle ID 26753, 4 pages Research Artcle Dynamc Envronmental/Economc Schedulng for Mcrogrd Usng Improved MOEA/D-M2M Xn L and Yanjun Fang Department of Automaton, Wuhan Unversty, Wuhan 4372, Chna Correspondence should be addressed to Xn L; xnl53@hotmal.com Receved 6 February 26; Accepted 24 Aprl 26 Academc Edtor: Roman Aubry Copyrght 26 X. L and Y. Fang. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. The envronmental/economc dynamc schedulng for mcrogrds (MGs) s a complex multobjectve optmzaton problem, whch usually has dynamc system parameters and constrants. In ths paper, a bobjectve optmzaton model of MG schedulng s establshed. And varous types of mcrosources (lke the conventonal sources, varous types of renewable sources, etc.), electrcty markets, and dynamc constrants are consdered. A recently proposed MOEA/D-M2M framework s mproved (I-MOEA/D-M2M) to solve the real-world MG schedulng problems. In order to deal wth the constrants, the processes of solutons sortng and selectng n the orgnal MOEA/D-M2M are revsed. In addton, a self-adaptve decomposton strategy and a modfed allocaton method of ndvduals are ntroduced to enhance the capablty of dealng wth uncertantes, as well as reduce unnecessary computatonal work n practce and meet the tme requrements for the dynamc optmzaton tasks. Thereafter, the proposed I-MOEA/D-M2M s appled to the ndependent MG schedulng problems, takng nto account the load demand varaton and the electrcty prce changes. The smulaton results by MATLAB show that the proposed method can acheve better dstrbuted fronts n much less runnng tme than the typcal multobjectve evolutonary algorthms (MOEAs) lke the mproved strength Pareto evolutonary algorthm (SPEA2) and the nondomnated sortng genetc algorthm II (NSGAII). Fnally, I-MOEA/D-M2M s used to solve a 24-hour MG dynamc operaton schedulng problem and obtans satsfactory results.. Introducton The energy shortages and envronmental polluton have gven an mpetus to the development of the mcrogrds (MGs), whch are more flexble, energy savng, and envronmentally frendly [ 4]. A mcrogrd system usually contans not only the conventonal sources but also varous types of renewable sources, such as photovoltacs, wnd power, gas turbnes, and mcroturbnes [5, 6]. Usually, battery banks are also needed to balance the effects of the volatlty of renewable energy. To reduce the cost and emsson smultaneously, one of the key strateges s to optmze the output power of the dstrbuted generators (DGs) and the dstrbuted storage (DS) unts by usng multobjectve optmzaton approaches, known as MG envronmental/economc schedulng. For the engneerng applcaton, there are manly two knds of MG operaton schedulng modes, namely, statc mode and dynamc mode. And the dynamc mode can adapt to the complcated changes n the real-world MG systems better than the statc one. However, the multobjectve optmzaton problems (MOPs) of MG dynamc schedulng always have dynamc system parameters, tme-varyng constrants, and more uncertantes to deal wth. Several studes have been made on the mprovement of the nonlnear optmzaton methods, such as dual and quadratc programmng [7], and ther applcaton on MOPs of schedulng for all knds of power systems. However, these methods cannot handle nonconvex objectve functons [8]. In recent years, many efforts have been made on the development of multobjectve evolutonary algorthms (MOEAs) n dealng wth the MG schedulng problems. In [9], an adaptve modfed partcle swarm optmzaton (AMPSO) framework was proposed. The applcaton on a day-ahead envronmental/economc schedulng problem for MG system showed that the approach performed better n convergence and populaton dversty than the standard PSO dd. In [], a heurstcsbased ant colony optmzaton (ACO) method was proposed and appled on a multobjectve schedulng problem for MG.

2 2 Mathematcal Problems n Engneerng The development of the basc ACO had enhanced the capablty of handlng complex constrants. Besdes, MOEAs arewdelyusednotherpowersystemsschedulngproblems, whch have smlartes wth MG systems. In [, 2], the strength Pareto evolutonary algorthm (SPEA) based approaches were presented and used to solve the envronmental/economc power dspatch problem for tradtonal power system and hybrd power system, whch ncludes wnd and solar thermal energes. The results show that the methods couldsolvethemopseffcently,obtanwelldstrbuted results,andhavegoodpopulatondverstes.in[3],the nondomnated sortng genetc algorthm II (NSGAII) was mproved to solve a bobjectve dynamc schedulng problem for a hydrothermal power generaton system, whch could adequately track Pareto-optmal fronters onlne. However, some necessary practcal constrants consderng tme change have not been nvolved. It s evdent from a growng number of researches on MG schedulng MOPs that these problems, especally the dynamc problems, offer serous challenges to the MOEAs for the followng reasons: ()AnMGsystemusuallycontansvaroustypesofcomponents, whch have dverse model functons of fuel cost, mantenance cost, emsson, and so forth. Thus, the search space of the MOP s usually nonlnear, nonconvex, and nondfferentable [4]. (2) In a real-world MG dynamc schedulng problem, there are always dynamc changes n system parameters (lke the electrcty prces) and constrants (such as the load demand), as a result of whch the search space vares wth tme and the optmzaton algorthm may not always work well. (3) In a dynamc procedure, the operators and the decson-makers usually cannot evaluate the qualty of the results n a very short perod of tme wthout professonal optmzaton knowledge. (4) The dspatch sequences are dependent on ther varaton n prevous tme steps. That s to say, the selected soluton wll certanly be affected by the former results, and hence the errors between dfferent optmzaton methods would be accumulated. (5) In engneerng problems, the ntermedate trade-off solutons n the Pareto front are what the decsonmakers really care about. However, n a dynamc schedulng MOP, snce the search spaces usng dfferentmoeasmaynotbethesameeverytmeandthe trueparetofrontsalwaysunknown,tsdffcultto compare the selected ntermedate solutons drectly. Consderng the above challenges, the optmzaton approachesnhandlngthemopsofmgdynamcschedulng should obtan the set of solutons as close to the true Pareto front as possble every tme, whch means the robustness of the algorthm performance s hghly requred under uncertantes. However, the algorthm robustness and the causes of falure n solvng these MOPs have not been further studed. Meanwhle, the optmzaton methods need to be smple and self-adaptve to meet the frequent changes of the system parameters. In the remander of ths paper, the system models and objectve functons are ntroduced n Secton 2. The MOEA/ D-M2MalgorthmsdescrbedandmprovednSecton3. Thereafter, the mproved algorthm s appled to several practcal scenaros to study the effects of dfferent parameters and the algorthm performance by comparng the results wth the other two typcal MOEAs n Secton 4. Fnally, the conclusons of the study are presented. The Appendx contans the parameters used n ths paper. 2. Problem Statement 2.. System Descrpton. MG systems can have varous structures. In ths paper, the MG conssts of three photovoltac (PV) arrays, three wnd turbnes (WTs), two mcroturbnes (MTs), two fuel cells (FCs), and a battery (BAT) bank. Assume that the output power of PVs and WTs s uncontrollable and has nether cost nor emsson. On the other hand, the MTs and FCs generate power wth natural gas as fuel and exhaust emssons, whle the output power can be controlled by operators. Battery banks are electrochemcal devces that store energy when the DGs generate more power than the load demand and supply the load when the power from DGs and themangrdfalstomeettheload System Components Models Uncontrollable DGs. In ths MG system, three kw PVs and three kw WTs are consdered, of whch the output power at tme t can be calculated as follows: P PVt =I STC G AC ( + k (T c T τ )),, V t <V c, { a w V 3 t +b w V 2 t +c w V t +d w, V c <V t <V r, P WTt = P r, V r <V t <V co, { {, V t >V co, where I STC s the standard test condton (STC) coeffcent; G AC s the ncdent rradance; k s the temperature coeffcent of power; T c and T τ are the cell temperature and the reference temperature, respectvely; P r s the rated power; V c and V co are the cut-n and cut-out wnd speed, respectvely; V r and V t are the rated and actual wnd speed at tme t,respectvely; a w, b w, c w,andd w are the parameters dependng on the wnd turbne types. The values of all the parameters above can be found elsewhere [5] Controllable DGs. In ths paper, two MTs (65 kw Capstone C65) and two FCs (4 kw IFC PC-29) are used as controllable DGs n the MG system. Ther fuel cost models at tme t can be expressed below, and the parameter settngs of each DG are taken from [5]. Hence, C G,t (P t ) =C ng () LHV ng P t η t, (2)

3 Mathematcal Problems n Engneerng 3 where P t s the output power of th DG at tme t; C ng s the natural gas prce to supply the DGs; LHV ng s the fuel lower heatng rate; η t s the effcency of the DGs at tme t. The effcency curves of MT and FC can be found from manufacturers. Thus, the effcency functons for the controllable DGs are obtaned usng curve-fttng methods, whch are shown below: η t =a ( P t 65 ) 3 +b ( P t 65 ) 2 +c ( P t 65 )+d, η t =a P t +b, ( =3,4), ( =,2), where η t s the effcency at tme t. If = or 2, η t s the effcency of the MTs; otherwse, t s the effcency of the FCs. The coeffcent values n the models are dfferent dependng on the types of DGs, whch are shown n Table Dstrbuted Storage Unts. Thestateofcharge(SOC)of the battery bank can be descrbed by calculatng the dscharge/charge power of the battery, as expressed below: (3) SOC = SOC max P +P +, (4) where SOC max sthemaxmumstateofchargeandp and P + are the dscharge and charge power, respectvely Optmzaton Problem. MG dynamc optmal schedulng s to optmze the output power of every dstrbuted energy resource (DER) to meet the load demand as well as satsfy a seres of dynamc constrants through the whole optmzaton process, whle takng nto account the economc and envronmental effect of the MG system. In ths secton, the proposed objectve functons and constrants are dscussed The Proposed Objectve Functons. One of the man goals of MG dynamc optmal schedulng s to mnmze the economc cost at tme t, such as fuel cost, mantenance cost, deprecaton cost, and the electrcty exchange cost wth the man grd, whch can be descrbed below: mn C t (P) = mn N = (C G,t (P t )+OM t (P t )+C DP,t (P t )) +C Grd,t (P Grd,t ), where N s the amount of all the DERs n the MG; P s the decson varable vector; C G,t (P t ) s the fuel cost of th DER at tme t;om t (P t ) s the mantenance cost of th DER at tme t; C DP,t (P t ) s the deprecaton cost of th DER at tme t; P Grd,t s the power exchanged wth the grd at tme t; C Grd,t (P Grd,t ) sthecostofpurchasedpowerattmet f P Grd,t >or the ncome of sold power at tme t f P Grd,t <. (5) The mantenance cost and deprecaton cost at tme t are descrbed as follows, respectvely. And all the parameter settngs n the functons can be found n Table 3. Hence, OM t (P t )=K OM P t, C DP,t (P t )= ADCC P r, 876 cf P t, where K OM,ADCC, P r,,andcf are the mantenance cost factor, the average deprecaton cost, the maxmum output power, and the capacty factor of the th DER n the MG, respectvely. In addton to the economc cost mnmzaton, ths paper manly takes nto account the carbon emsson mnmzaton as another optmzaton objectve, as expressed below: mn E t (P) = M = (6) (α P 2 t +β P t +γ ), (7) where M s the amount of all the controllable DGs; the values of emsson factors α, β,andγ canbefoundntable Constrants Descrpton Power Balance Constrant. All the output power, ncludng the power exchanged wth the man grd, should meet the load demand (P L (t))attmet, whch can be expressed as M = P t +(P PVt +P WTt +P BTt ) P L (t) =. (8) Rated Power Constrants. All the DGs have power generaton lmts, whch can be descrbed as P mn P t P max, (9) where P mn s the mnmum power output of th DG, whch s settobezeronthspaper,andp max s the maxmum power output of the th DG. StateofChargeConstrants.Thebatterybankcannotbe overchargedoroverused,sothelmtsofthestateofcharge (SOC)ofthebatterybankareasfollows: SOC mn SOC t SOC max, () where SOC mn and SOC max are the mnmum and maxmum stateofcharge,whcharesettobe3kwandkw, respectvely. Power Exchange Constrant. In ths paper, the power exchanged between the MG system and the man grd has constrants as descrbed below: P Grd, P Grd,t P Grd,+, () where P Grd, and P Grd,+ are the two bounds of exchange power. In ths paper, P Grd, and P Grd,+ are set to be 5 kw and 5 kw, respectvely.

4 4 Mathematcal Problems n Engneerng Ramp Rate Constrant. The ncrease/decrease of output power of controllable DGs n unt tme s called ramp rate, whch reflects the performance of the DGs. Ths constrant can be expressed as R down, Δt P t P,t R up, Δt, (2) where R down, and R up, are ramp-down and ramp-up rate of the output power of the th DG, respectvely. The values of them are shown n Table 3. Δt s the tme of the process. Charge/Dscharge Rate Constrant. In ths paper, the maxmum allowable charge/dscharge current s 2% of the battery bank capacty, as shown below: P BAT,+ (.2 V sys U BAT ), Δt P BAT, (.2 V sys U BAT ), Δt (3) where V sys s the system voltage at the DC bus and U BAT s the batterybankcapactynah.thevaluesettngscanbefound n [5]. 3. The I-MOEA/D-M2M Algorthm In ths secton, the orgnal MOEA/D-M2M s brefly descrbed, whch s appled n ths paper as the basc optmzaton framework. Then, the algorthm s modfed and extended to mprove the performance n dealng wth realworld MOPs. 3.. Descrpton of MOEA/D-M2M. MOEA/D-M2M, ntroduced by Lu et al. [6] n 24, s a varant of MOEA/D (MOEA based on decomposton). However, unlke MOEA/ D usng aggregaton methods, MOEA/D-M2M decomposes amopntoasetofsmplesub-mopsandsolvesthemnone sngle run, whch can be called multple to multple (M2M). Ths algorthm framework s desgned for some MOPs whch render t dffcult for other MOEAs to acheve a good dversty of populaton. The procedure of the algorthm can be found n [6] and s reproduced n Fgure. The major advantages of MOEA/D-M2M nclude the followng [6, 7]: () MOEA/D-M2M transforms a MOP nto several sub- MOPs, whch s stll an equvalent optmzaton process, whereas the orgnal MOEA/D makes an approxmate transformaton. (2) Compared wth SPEA2, NSGAII, MOEA/D, and so forth, MOEA/D-M2M protects every subpopulaton by decomposton and solves every subproblem usng multobjectve optmzaton approaches, whch balances the dversty and convergence smultaneously at each generaton. (3) The optmzaton process requres much less manual operaton. The operators only need to set the drecton vectors at the very begnnng nstead of adjustng varous complex parameters, whch means the algorthm s more adaptve for the MG dynamc schedulng problem wth dynamc system parameters and constrants. (4) The algorthm can be customzed wth elements from other MOEAs, accordng to dfferent real-world MOPs.ThePareto-basedMOEAscanbeappledn the subproblems to handle specfc dffcultes wth some pror knowledge, whch can mprove the accuracy and effcency n the whole optmzaton procedure Improvement of MOEA/D-M2M. The orgnal MOEA/D- M2M solved the modfed ZDT and DTLZ test nstances [6, 8] well, n whch there are no constrants and both the search space and the objectve space are [, ] n.however, areal-worldmop,suchasthemgdynamcschedulng problem, wth some equalty and nequalty constrants can be generally formulated as follows: mn F t (x) =(f,t (x),...,f m,t (x)) subject to g k,t (x) =, k=,...,k h l,t (x), l=,...,l, (4) where f,t (x),...,f m,t (x) are objectve functons at tme t; x s the decson vector at tme t; g k,t (x) and h l,t (x) are equalty and nequalty constrants at tme t, respectvely.the realworld MOPs as descrbed n (4) often have varous constrants, dynamc varable bounds wth the change of tme, and uncertan objectve spaces. Thus, the algorthm needs to be modfed to meet the practcal requrements. Also, some other changes are needed to mprove the performance of MOEA/D-M2M Constrants Handlng. The constrant-handlng method, presented by Deb et al. [9], s ntroduced n ths paper durng the process of sortng and selecton. Accordng to ths method, a soluton has a better nondomnaton rank than soluton j,f () soluton s feasble whle soluton j s not; (2) both of them are feasble and soluton domnates soluton j; (3) both of them are nfeasble and soluton volates the overall constrant less. Ths constraned-domnaton prncple guarantees that all thefeasblesolutonshaveprortytobeselected,andthe computatonal complexty of the orgnal algorthm does not change [9] Self-Adaptve Decomposton Strategy. In [6], the objectve space and the Pareto fronts of the test MOPs are always [, ] n ; hence, the startng pont of the drecton vectors can be fxed as the orgnal pont. However, n the real-world MOPs wth constrants, usually the bounds of the Pareto fronts are unknown and may vary wth tme. Therefore, to acheve a more precse result and reduce unnecessary

5 Mathematcal Problems n Engneerng 5 Start Set the parameters ncludng the followng: () The number of subproblems K. () The drecton vectors. () The sze of the subpopulaton S. (v) The maxmum generaton MaxGen. (v) The genetc operators. Combne R and K k= P k to create a new populaton Q Intalze K subpopulatons (P,...,P K ) and set gen = Set k= gen = gen + k=k+; ntalze P k Set R=Φ, k= Yes Is P k >S? k=k+; fnd the sze (G) of the subpopulaton P k Set g= Rank the solutons n P k usng the nondomnated sortng method Remove the P k S lowest ranked solutons from P k No Is P k <S? Yes Select S P k solutons from Q randomly and add them to P k No g=g+ For the soluton x g n P k, randomly choose y from P k and apply genetc operators on x g and y to obtan a new soluton z Is k=k? Yes No Compute the objectve functon values (F-values) of z; R=R {z} Fnd all the nondomnated solutons n K k= P k and output them No Is g=g? Yes No Is gen = MaxGen? No Is k=k? Yes Yes Fnd all the feasble solutons from the results End Fgure : Flowchart of MOEA/D-M2M. computatonal work n practce, t s mportant to revse the decomposton strategy. In ths paper, the self-adaptve decomposton strategy s formulated based on the change of startng pont n each generaton. In the decomposton procedure at the frst generaton, the mnmum values for each objectve f,,mn,...,f m,,mn, whch are calculated usng all the feasble solutons n the current populaton, form the frst startng pont O (f,,mn,...,f m,,mn ) for the decson varables. For the jth generaton, the selected mnmum values f,j,mn,...,f m,j,mn

6 6 Mathematcal Problems n Engneerng Table : The h-values, extreme solutons, and CPU tme usng the three algorthms under dfferent load demands. Load demand (kw) I-MOEA/D-M2M SPEA2 NSGAII 5 h-value Best Mean Extreme solutons For obj (2.37,.746) (2.45,.745) (2.44,.2894) For obj2 (4.724,.2) (4.62,.3) (7.5578,.5) h-value Best Mean Extreme solutons For obj (3.868,.743) (3.8788,.7358) (4.293,.358) For obj2 (6.967,.3) (6.9822,.5) (8.2485,.8) 5 h-value Best Mean Extreme solutons For obj (9.87,.948) (9.32,.8863) (9.2865,.8563) For obj2 (3.4595,.249) (3.259,.386) (3.2488,.236) Average CPU tme (s) The terms obj and obj2 represent to the two optmzaton objectves, respectvely. for each objectve n the current populaton are compared wth the assocated values of O j at the (j )th generaton, respectvely, and the smaller ones are pcked to form O j (f,j,mn,...,f m,j,mn ).Inthsway,thealgorthmcanfocus on the effectve search space and avod beng trapped nto the useless or nfeasble regons, whch wll mprove the effcency n dealng wth the constraned real-world MOPs wth uncertan Pareto fronts Allocaton of Indvduals. When allocatng ndvduals after decomposton, the remanng S P k solutons are randomly chosen from the combned populaton Q f P k < S, accordng to Fgure, whereas t s better to select the solutons whch have small acute angles wth the drecton vector n the objectve space, snce they can conduct to a better front for the subregon Ω k. Thus, the method to allocate the remanng ndvduals n ths stuaton can be modfed as the followng steps. Step. Choose two subregons nearest to Ω k.iftheamountof feasble ndvduals n the two subregons s suffcent, select the S P k ones and add them to P k ; otherwse, select all the feasble ndvduals and then move to Step 2. Step 2. Choose two more subregons nearest to Ω k,andselect the rest of the needed ndvduals from them. If the quantty of feasble ndvduals s stll nsuffcent, repeat ths step. If all the feasble ndvduals out of Ω k are selected and the total number s stll less than S,movetoStep3. Step 3. Select the rest of the ndvduals from the nfeasble ones by comparng ther overall constrant and add them to P k. 4. Smulaton Results and Dscusson In ths secton, the MG optmzaton model descrbed n the prevous secton s appled n three scenaros, consderng the load demand varaton, the electrcty prce change, and a 24-hour dynamc optmzaton process. In each scenaro, three MOEAs ncludng I-MOEA/D-M2M, NSGAII, and SPEA2 [2] are used and ther optmzaton results are compared. The populaton sze s and the maxmum generaton number s 5 for each algorthm. For SPEA2, the archve sze s. And, for I-MOEA/D-M2M, K = S =.The smulated bnary crossover [2, 22] and polynomal mutaton methods are appled n the three algorthms wth the same control parameters n [9]. And both NSGAII and SPEA2 use the same constrant-handlng strategy as I-MOEA/D- M2M does. The reference pont of the hypervolume values s (3, ) based on plenty of expermental results. All the experments are accomplshed usng MATLAB 23, on a PC wth an Intel Core 3 (3.3 GHz) processor under Wndows 7 usng 6 GB of RAM. 4.. Scenaro One. In Scenaro One, the load demand varaton s consdered as the prmary factor to affect the optmzaton results. In ths case, the other system parameters, lke the envronmental parameters, the electrcty prce, and so forth, are fxed, whch are shown n Table 4. The ramp rate constrant s not appled here snce the optmzaton procedures are ndependent. Fgures 2(a) 2() show the optmzaton results under dfferent load demands usng I-MOEA/D-M2M, SPEA2, and NSGAII, respectvely. The red ponts represent the populatonnthelastgeneraton,andtheblackonesarethefeasble nondomnated solutons. The comparsons of hypervolume values (h-values), feasble extreme solutons, and average CPU tme are presented n Table. It s clear from Fgures 2(a), 2(d), and 2(g) that the results acheved by I-MOEA/D-M2M are always well dstrbuted. Accordng to Table, the best and mean h-values have very small dfference when usng I-MOEA/D-M2M, whch means the algorthm can converge to the Pareto fronts n less teraton. The extreme solutons for each objectve are the best compared wth those usng the other two algorthms. Therefore, ths algorthm s more robust and can provde more dverse trade-off solutons for the decson-makers under

7 Mathematcal Problems n Engneerng (a) (b) (c) (d) (e) (f) (g) (h) () Fgure 2: Columns from left to rght: () the optmzaton results usng I-MOEA/D-M2M; (2) the optmzaton results usng SPEA2; (3) the optmzaton results usng NSGAII. Rows: () the optmzaton results under 5 kw load demand; (2) the optmzaton results under kw load demand; (3) the optmzaton results under 5 kw load demand. varous load demands. And snce t needs much less CPU tme, I-MOEA/D-M2M s more sutable for MG dynamc schedulng problems. For SPEA2, when the load s 5 kw, the populaton s dstrbuted unformly. The best h-value s smlar to that obtaned by I-MOEA/D-M2M, but the mean value s much smaller. The mnmum cost and mnmum emsson do not have much dfference compared wth those obtaned by I- MOEA/D-M2M. For nstance, the mnmum cost s 2.47$, whch s only.38% more than the I-MOEA/D-M2M optmzed soluton. Wth the load ncreasng, SPEA2 loses the populaton dversty gradually (shown n Fgures 2(b), 2(e), and 2(h)). It can be seen from Fgure 2(h) that part of the populaton gets trapped nto the nfeasble regon usng SPEA2

8 8 Mathematcal Problems n Engneerng Power output (kw) Varable number Solutons n Regon A Solutons out of Regon A Fgure 3: The feasble solutons obtaned by NSGAII under kw load demand. when theloads5kw.ontheotherhand,theh-values and the mnmum cost and mnmum emsson are smaller than those usng I-MOEA/D-M2M, and the dfference between them ncreases wth the load. Also, t should be noted that SPEA2 costs much more tme ( s) than the other two methods do. Although t performs better than NSGAII wthn 5 generatons, SPEA2 s not a good choce n dealng wth ths MG dynamc MOP. For NSGAII, the change s smlar to that usng SPEA2, whle the h-values and the obtaned extreme solutons are worse. It can be seen from Fgures 2(c), 2(f), and 2() that the amount of the feasble nondomnated solutons reduces wth the load ncreasng, and some of the solutons always aggregate n a small regon. For example, when the load demandskw,45%ofthepopulatonhavesmlarcostsn the objectve space, whch are between 4.624$ and $. In ths paper, ths regon s called Regon A. To analyze ths, all the feasble solutons n Fgure 2(f) are shown n Fgure 3 by parallel coordnates. It s obvous that only the thrd and the fourth varables (.e., output of FC and FC2) of the red solutons have dfferent values, whle others reman unchanged as ther boundary values. Hence, t s easer to fnd nondomnated solutons n Regon A snce only two varables need to be changed. As a result, NSGAII may get trapped nto the regons lke ths, as convergence takes precedence over populaton dversty n ts search strategy. The teratve processes of NSGAII and I-MOEA/D-M2M are shown n Fgure 4, whch descrbe how the populaton vares wth generatons. ItcanbeseenfromFgure4(a)that,atthe6thgeneraton, more than half of the populaton stays n Regon A by usng NSGAII, and only a few solutons have the cost values hgher than $ n the objectve space. In ths paper, the regon s called Regon B, where the cost value s hgher than $. After 4 generatons, the majorty of the populaton extends to Regon B and s dstrbuted unformly, and there are only a few solutons whch have the cost values lower than 4.624$ n the objectve space. The regon s called Regon C, where the cost value s lower than 4.624$. When the generaton reaches 2, there are more solutons n Regon C and the whole front s smlar to that obtaned by I-MOEA/D-M2M. Although NSGAII can also acheve a good result fnally, t costs more runnng tme. Furthermore, when the load changes, t s dffcult to decde how many generatons the algorthm needs to get satsfyng results, lke the case n Fgure 2(). Wth the load demand constrants changng, the feasble search space vares frequently and uncertanly, and NSGAII cannot guarantee to fnd the Pareto front every tme, snce t may get trapped nto some regons lke Regon A. SPEA2 cannot deal wth these problems ether (shown n Fgure 2(h)) because they both adopt the convergence frst and dversty second selecton strategy [6], although SPEA2 performs better than NSGAII. In contrast, I-MOEA/D-M2M can get a well dstrbuted front wthn just generatons due to ts decomposton strategy, and the results at 3th generaton are almost the same as those at 5th generaton, whch means the algorthm s much faster n handlng these MOPs. By usng the self-adaptve decomposton strategy, I-MOEA/D-M2M can avod searchng the useless regons and pay more attenton to populaton dversty. So, whatever the load demand s, I- MOEA/D-M2M can always quckly obtan the feasble tradeoff solutons Scenaro Two. In Scenaro Two, the effect of electrcty prce change s studed. The envronmental parameters are held constant as shown n Table 4, and the load demand s 5 kw. The ramp rate constrant s stll not appled here as Scenaro One. The electrcty prces settngs are shown n Table 5. Fgure 5 shows the optmzaton results n three cases usng I-MOEA/D-M2M, SPEA2, and NSGAII, respectvely. It s evdent that I-MOEA/D-M2M performs well n all the three cases, the solutons obtaned are dstrbuted unformlyntheobjectvespace,andtcanbeseenfrom Fgures 5(a), 5(d), and 5(g) that the fronts are qute dfferent n the three MOPs. For SPEA2, when the electrcty purchase prce drops to.2$/kwh, the qualty of the results s better than that n Case One snce all the nondomnated solutons are feasble, and the front s smlar to that usng I-MOEA/D- M2M, whereas the amount of the feasble nondomnated solutons reduces when the electrcty sale prce ncreases to.3$/kwh, and nearly half of the solutons are nfeasble. For NSGAII, no matter how the electrcty prces change, t only fnds part of the Pareto front wthn 5 generatons, accordng to Fgures 5(c), 5(f), and 5(). And there are a large number of nfeasble or domnated solutons left. In concluson, I-MOEA/D-M2M can adapt to the MG MOP wth electrctyprcesvaryng,whlethequaltyofresultsusng SPEA2 s based on the combnaton of electrcty purchase andsaleprces,andnsgaiimayneedfarmorethan5 generatonstogetsmlarparetofrontstoi-moea/d-m2m Scenaro Three. Wth the above ndependent optmzaton runs and analyses of the algorthms performance, t s tme to demonstrate the use of the proposed optmzaton methodologytoacomplex24-hourmgdynamcoperaton

9 Mathematcal Problems n Engneerng (a) (b) (c) (d) (e) (f) Fgure 4: The teratve processes of NSGAII and I-MOEA/D-M2M under kw load demand. Rows: () the optmzaton results after 6,, and 2 generatons usng NSGAII; (2) the optmzaton results after, 3, and 5 generatons usng I-MOEA/D-M2M. schedulng problem nvolvng economc and envronmental consderatons. The load demand n a typcal day s shown n Fgure 6, and the envronmental data for wnd speed, rradance, and ar temperature can be found n [5]. Besdes, the tme-varyng electrcty prces are consdered, shown n Table 6. The pseudoweght vector method [23, 24] s ntroduced n ths paper to select the solutons based on the decsonmakers preference. After the optmzaton process n each hour, the pseudoweght ω for the th objectve for every soluton s calculated usng the followng equaton: ω = (f max M j= ((fmax j f )/(f max f j )/(f max j f mn ) f mn j )), (5) where f max and f mn are the maxmum and mnmum values for the th objectve n the obtaned results, respectvely; f s the value of the th objectve functon for every soluton; M s the number of the optmzaton objectves. Thereafter, the followng decson functon can be appled: Mn d n = ω n ω, (6) where d n and ω n are the decson functon value and the pseudoweght vector for the nth soluton n the fnal populaton and ω stheweghtvectorsetbythedecson-maker. The soluton wth the mnmum decson functon value wll be selected, whch s satsfed by the decson-maker. In ths paper,threecasesarestuded,ofwhchtheω values are (, ), (, ), and(.5,.5), respectvely. The optmzaton results by I-MOEA/D-M2M are shown n Fgure 7. Fgure 7(a) shows the optmzaton results for the 24- hour MG dynamc operaton schedulng problem when ω = (, ), whch means the soluton wth the mnmum cost n theobjectvespacesselectedneachhour.thepoweroutput curves of PVs and WTs are not shown here snce they are used frst n each case and can be calculated by the forecast envronmentaldata.itcanbeseenthat,forthetwomts, MT s used frst and MT2 starts to work only f MT outputs full power under the relatve constrants. For the two FCs, smlarly, FC always generates more power than FC2, whle sometmes FC2 starts runnng even though FC does not reach the full power under ts constrants. The dfferences can be explaned by computng the average prces of power generated by the DGs usng (2). For the BAT, to avod frequent chargng and dschargng, t wll be charged when

10 Mathematcal Problems n Engneerng (a) (b) (c) (d) (e) (f) (g) (h) () Fgure 5: Columns from left to rght: () the optmzaton results usng I-MOEA/D-M2M; (2) the optmzaton results usng SPEA2; (3) the optmzaton results usng NSGAII. Rows: () the optmzaton results n Case One; (2) the optmzaton results n Case Two; (3) the optmzaton results n Case Three. the state s 3 kw (SOC mn ) and wll not dscharge untl t reaches kw (SOC max ). Snce there s no need to provde more power than the load demand by the other DGs n the MG n ths paper, the BAT s manly charged usng the free powerobtanedbyuncontrollabledgs.asaresult,theoutput curveofbatdoesnotchangemuchnthethreecases.forthe power exchanged wth the man grd, the value depends on theelectrctyprceandtheaverageprcesofoutputpowerof the controllable DGs. For nstance, t can be calculated by (5) that no matter how much power the DGs output the average prces are always hgher than the electrcty purchase prce n valley perod (: 7:). Thus, t s more economcal to buy power from the grd before usng the components n the MG. Fgure 7(b) shows the optmzaton results when ω = (, ), whch means mnmzng the emsson level s the most

11 Mathematcal Problems n Engneerng 3 25 Load demand (kw) Tme (h) Fgure 6: Varaton of load demand n 24 hours Power (kw) Power (kw) Tme (h) Tme (h) MT MT2 FC FC2 BAT Power exchanged wth grd MT MT2 FC FC2 BAT Power exchanged wth grd (a) 6 (b) 5 4 Power (kw) X: 9 Y: Tme (h) MT MT2 FC (c) FC2 BAT Power exchanged wth grd Fgure 7: Optmzaton results usng I-MOEA/D-M2M n the three cases of the 24-hour MG dynamc operaton schedulng problem.

12 2 Mathematcal Problems n Engneerng Table 2: The total cost and emsson obtaned by the three algorthms n dfferent cases. Results Case One Case Two Case Three I-MOEA/D-M2M SPEA NSGAII Table 3: Parameter settngs of the MG system components. Parameters MT FC BAT PV WT K OM ($/kwh) ADCC ($) cf (%) a b c d.68.9 α (kg/kw) β (kg/kw) γ (kg/kw) Ramp rate (kw/mn) Table 4: Parameter settngs of the envronmental factors and the electrcty prce. Wnd speed (m/s) Irradance (W/m 2 ) Ar temperature ( C) Electrcty sale prce ($/kwh) Electrcty purchase prce ($/kwh) concerned objectve. Interestngly, the output curves of the two MTs have the same varyng tendency under dfferent load demands, whle MT2 supples a lttle more power (less than 5 kw) than MT does every tme. For the FCs, the stuaton s smlar. Ths s qute dfferent from Case One but can also be explaned by (7). For the power exchanged wth the man grd,thevaluesalwaysthemaxmumsncetsassumedto produce no harmful emsson n ths paper. Fgure 7(c) shows the optmzaton results when ω = (.5,.5), whch consder trade-offs between cost and emsson. It can be seen that the curves of the power output by the DGs and the power exchanged wth the man grd change rregularly wth tme, whch cannot be derved mathematcally by the MG models n Secton 2. However, n practcal engneerng problems, the ntermedate solutons lke those ncasethreearetheonesthedecson-makersreallycare about. Therefore, the extreme solutons obtaned n Case One and Case Two are of mportant reference sgnfcance for evaluatng the performance of the optmzaton method. Wth the above smulatons, t s clear that I-MOEA/D-M2M does better n searchng the extreme solutons than the other two algorthms. Table 2 shows the total cost and emsson obtaned by the three algorthms wth dfferent ω values. It canbeseenthattheresultsusngi-moea/d-m2mhavegot themnmumtotalcostncaseoneandtotalemssonn Case Two compared wth those usng the other algorthms. Partcularly, n Case Two, the mnmum emsson soluton Table 5: Parameter settngs of the electrcty prces. Electrcty sale prce ($/kwh) Electrcty purchase prce ($/kwh) Case One Case Two Case Three usng I-MOEA/D-M2M ncurs a total emsson of kg, whch s 3.47% and 27.57% better than the SPEA2 and NSGAII optmzed solutons, respectvely. In Case Three, the cost and emsson levels by I-MOEA/D-M2M are lower than those by NSGAII. But the emsson by I-MOEA/D-M2M s more than that by SPEA2. Accordng to Case One and Case Two,theformerresultsstllthefrstchocecomparedwth the latter, snce I-MOEA/D-M2M can approxmate the true Pareto front better than SPEA2 and reflect the decsonmakers preference more precsely. 5. Concluson In ths paper, an mproved MOEA/D-M2M algorthm s proposed for solvng the dynamc MG envronmental/economc schedulng problem. In ths I-MOEA/D-M2M framework,

13 Mathematcal Problems n Engneerng 3 Perod Peak perod Normal perod Valley perod Table 6: The electrcty prce settngs n 24 hours. Tme (h) : 5: 8: 2: 7: : 5: 8: 2: 23: : 7: 23:-24: Prce ($/kw) Sale Purchase a constrant-handlng method s suggested durng the process of sortng and selecton. Meanwhle, the self-adaptve decomposton and the ndvduals allocaton strateges are employed to enhance the capablty of dealng wth uncertantes and make the algorthm more effcent. And then I-MOEA/D-M2M s appled to a seres of ndependent statc MG envronmental/economc schedulng problems. The results are compared wth those obtaned by SPEA2 and NSGAII n terms of hypervolume values, extreme solutons, and CPU tme. It s evdent that I-MOEA/D-M2M can acheve a well dstrbuted front every tme wth much less tme, whle the other two algorthms cannot guarantee the qualty of the solutons set and take more tme. The optmzaton processes n solvng the two problems are studed and the shortcomngs of the two typcal MOEAs are ponted out, whch also llustrates that the I-MOEA/D- M2Mframeworkcanadapttothesearchspacechangeswell. Fnally, the feasblty of the I-MOEA/D-M2M s verfed by the smulatons of a complex 24-hour MG dynamc operaton schedulng problem. Further studes are stll necessary on more realstc MG models and the effects of other dynamc parameters. Appendx Parameters Settngs See Tables 3 6. Competng Interests The authors declare that there are no competng nterests regardng the publcaton of ths paper. Acknowledgments Ths work was supported by the Natonal Natural Scence Foundaton of Chna (Grants nos and ), by the Natonal Key Technology Support Program of Chna (Grant no. 23BAAB), by the Fundamental Research Funds for the Central Unverstes (Grant no ), and by the Chna Scholarshp Councl (Grant no ). References [] A.K.Basu,S.P.Chowdhury,S.Chowdhury,andS.Paul, Mcrogrds: energy management by strategc deployment of DERs a comprehensve survey, Renewable and Sustanable Energy Revews,vol.5,no.9,pp ,2. [2] C. Pan and Y. Long, Evolutonary game analyss of cooperaton between mcrogrd and conventonal grd, Mathematcal Problems n Engneerng, vol.25,artcleid326,pages, 25. [3] H. Lang and W. Zhuang, Stochastc modelng and optmzaton n a mcrogrd: a survey, Energes, vol.7,no.4,pp , 24. [4]J.La,H.Zhou,X.Lu,X.Yu,andW.Hu, Droop-baseddstrbuted cooperatve control for mcrogrds wth tme-varyng delays, IEEE Transactons on Smart Grd,26. [5] F. A. Mohamed and H. N. Kovo, System modellng and onlne optmal management of McroGrd usng mesh adaptve drect search, Internatonal Electrcal Power & Energy Systems,vol.32,no.5,pp ,2. [6] H.Lan,S.Wen,Q.Fu,D.C.Yu,andL.Zhang, Modelnganalyss and mprovement of power loss n mcrogrd, Mathematcal Problems n Engneerng, vol.25,artcleid49356,8pages, 25. [7] F. J. Roojers and R. A. M. van Amerongen, Statc economc dspatch for co-generaton systems, IEEE Transactons on Power Systems,vol.9,no.3,pp ,994. [8] M. Basu, Combned heat and power economc emsson dspatch usng nondomnated sortng genetc algorthm-ii, Internatonal Electrcal Power and Energy Systems,vol.53, no., pp. 35 4, 23. [9]A.A.Moghaddam,A.Sef,T.Nknam,andM.R.Alzadeh Pahlavan, Mult-objectve operaton management of a renewable MG (mcro-grd) wth back-up mcro-turbne/fuel cell/ battery hybrd power source, Energy, vol. 36, no., pp , 2. []C.M.Colson,M.H.Nehrr,andC.Wang, Antcolonyoptmzaton for mcrogrd mult-objectve power management, n Proceedngs of the IEEE/PES Power Systems Conference and Exposton (PSCE 9),pp. 7,Seattle,Wash,USA,March29. [] M. A. Abdo, Envronmental/economc power dspatch usng multobjectve evolutonary algorthms, IEEE Transactons on Power Systems,vol.8,no.4,pp ,23. [2] S. Brn, H. Abdallah, and A. Oual, Economc dspatch for power system ncluded wnd and solar thermal energy, Leonardo Scences, vol. 4, pp , 29. [3] K. Deb and S. Karthk, Dynamc mult-objectve optmzaton and decson-makng usng modfed NSGA-II: a case study on hydro-thermal power schedulng, n Evolutonary Mult- Crteron Optmzaton, vol.443oflecture Notes n Computer Scence, pp , Sprnger, Berln, Germany, 27. [4] L. Guo, W. Lu, J. Ca, B. Hong, and C. Wang, A two-stage optmal plannng and desgn method for combned coolng, heat and power mcrogrd system, Energy Converson and Management,vol.74,pp ,23. [5] L. Le, Study of Economc Operaton n Mcrogrd, North Chna Electrc Power Unversty, Bejng, Chna, 2. [6]H.-L.Lu,F.Gu,andQ.Zhang, Decompostonofamultobjectve optmzaton problem nto a number of smple multobjectve subproblems, IEEE Transactons on Evolutonary Computaton,vol.8,no.3,pp ,24.

14 4 Mathematcal Problems n Engneerng [7] S. Jang and S. Yang, An mproved multobjectve optmzaton evolutonary algorthm based on decomposton for complex pareto fronts, IEEE Transactons on Cybernetcs, vol.46,no.2, pp , 25. [8] S. Huband, P. Hngston, L. Barone, and L. Whle, A revew of multobjectve test problems and a scalable test problem toolkt, IEEE Transactons on Evolutonary Computaton, vol., no. 5, pp , 26. [9] K. Deb, A. Pratap, S. Agarwal, and T. Meyarvan, A fast and eltst multobjectve genetc algorthm: NSGA-II, IEEE Transactons on Evolutonary Computaton, vol.6,no.2,pp.82 97, 22. [2] E. Ztzler, M. Laumanns, and L. Thele, SPEA2: mprovng the strength Pareto evolutonary algorthm, Tech. Rep. 3, Swss Federal Insttute of Technology, 2. [2] K. Deb and H. G. Beyer, Self-adaptve genetc algorthms wth smulated bnary crossover, Evolutonary Computaton, vol.9, no. 2, pp , 2. [22] K. Deb and R. B. Agrawal, Smulated bnary crossover for contnuous search space, Complex Systems, vol. 9, no. 3, pp. 5, 994. [23] K. Deb, Mult-Objectve Optmzaton Usng Evolutonary Algorthms, John Wley & Sons, Chchester, UK, 2. [24] K. Deb and S. Chaudhur, I-MODE: an nteractve multobjectve optmzaton and decson-makng usng evolutonary methods, n Evolutonary Mult-Crteron Optmzaton, vol. 443 of Lecture Notes n Computer Scence, pp , Sprnger, Berln, Germany, 27.

15 Advances n Operatons Research Volume 24 Advances n Decson Scences Volume 24 Appled Mathematcs Algebra Volume 24 Probablty and Statstcs Volume 24 The Scentfc World Journal Volume 24 Internatonal Dfferental Equatons Volume 24 Volume 24 Submt your manuscrpts at Internatonal Advances n Combnatorcs Mathematcal Physcs Volume 24 Complex Analyss Volume 24 Internatonal Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Mathematcs Volume 24 Volume 24 Volume 24 Volume 24 Dscrete Mathematcs Volume 24 Dscrete Dynamcs n Nature and Socety Functon Spaces Abstract and Appled Analyss Volume 24 Volume 24 Volume 24 Internatonal Stochastc Analyss Optmzaton Volume 24 Volume 24