Pareto Optimal Demand Response Based on Energy Costs and Load Factor in Smart Grid

1 P areto Optimal Demand Response Based on Ener gy Costs and Load F actor in Smart Grid W ei-Y u Chiu, Member , IEEE , Jui-Ti ng Hsieh, and Chia-Ming Chen Abstract —Demand response f or residential users is essential to the realization of modern smart grids. This paper proposes a multiobjective appro ach to designing a demand re sponse program that consid ers the energy costs of residential users and the load factor of the und erlying grid. A multiobjectiv e optimization problem (MOP) is formulated and Pare to optimality is adopted. Stochastic sear ch methods of generating feasible values f or decision variables are p roposed. Theore tical analysis is perf ormed to show that the p roposed methods can effectiv ely generate and p reser ve f easible points during the solution p rocess, which comparable methods can hardly achiev e. A multiobjective ev olutionary algorithm is developed to solve the M OP , producing a Par eto optimal demand r esponse (PODR) progra m. Simulations re veal th at the proposed method outperfo rms th e comparable methods in terms of energy costs while pro ducing a satisfying load factor . The propose d PODR program is abl e to systematically balance the needs of the grid and residential users. Index T erms —Cost minimization, day-ahead pricing, demand response, energy consumption schedu ling, EV charging, load factor maximization, Pareto optimality , Pareto optimal demand response. I . I N T RO D U C T I O N I N e xisting power grids, power plants usually deliver a unidirectio nal power fl ow to customers, converting only one-thir d of the total energy in th e ir fuel into electr icity and wasting the heat p rodu c e d. T wenty percent of a power grid’ s gen e ration cap a city is o f ten u sed only to cover peak loads that acco u nt for approx imately five percent of the time [1], [2]. When n atural disasters occur , conv entional power grids are unab le to r esist or self-h eal. Next-g eneration p ower grids, k nown as smart g rids, h ave been proposed an d d e sig ned to replace traditio nal power grid s with the aim of redu c- ing transmission loss, generating electricity mo r e e fficiently , and resisting or self-healing after natural d isasters. A few governments hav e ad opted p roactive policies to popula r ize smart grids and co nstruct related in frastructur e. Smar t grids are expected to have higher electricity transmission stability , detect faults and self-h e al, allow bid ir ectional energy an d inf ormation This work wa s supported by the Ministry of Scien ce and T e chnology of T ai wan under Grant MOST 108-2221-E-007-10 0. (Correspondi ng author: W e i-Y u Chiu.) W .-Y . Chiu and C.-M. Chen are with the Department of Electrica l En- gineeri ng, Nationa l T sing Hua Univ ersity , Hs inchu 30013, T aiw an (e-mail: chiuwei yu@gmail.com). J.-T . H s ieh is with the Depa rtment of Electrical Enginee ring, Y uan Ze Uni versit y , T ao yuan 32003, T aiwa n. c  2019 IEEE. Personal use of this materi al is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, includi ng reprinting/re publishing this materi al for adverti sing or promotional purposes, creating new collect iv e works, for resale or redistri butio n to servers or lists, or reuse of any copy righted component of this work in other works. Digital Objec t Ide ntifier 10.1109/TII.2019.2928 520 flow between customers and suppliers, and reliab ly defend against attacks o r natur al disasters [3], [4]. In a smart grid, the real-time energy co nsumptio n profiles of residential users are crucial to power supplier s [5]. W ith this informa tio n, suppliers or local tradin g cen ters can design a dynamic pricing scheme that stren gthens m arket mec h anisms and encou rages users to shif t their pe a k loads [6] , [7]. Power scheduling can then be achieved through the direc t m inimiza- tion of energy consumption costs [8]–[15]. The change of demand cu rves in respo nse to pricing signals is term ed deman d response. Common pricing models include real-time pricing, day- ahead pricing, time-of- use pr icing, and critical-peak pricin g models [1 6]. Regardless of wh ich dy namic pricing sche m e is employed, suppliers can lower the cost of power gener ation if users’ peak lo ads are shifted. This ca n be achieved throu gh the use of demand response prog rams. Genera lly , shifting pea k loads is r elated to incre a sing t he load factor, which is the reciproca l o f the peak -to-average r atio [17], [18]. Howe ver , to fully utilize the grid capacity , the load factor is a m ore approp riate metric. Sev eral studies have incorp o rated the co ncept of load factor or peak-to- av erage ratio in to p ower schedu ling. Gam e theory approa c h es h ave been widely u sed for design ing dem and re- sponse p rogra m s [19]–[22]. Alth ough u nder cer tain co n ditions Nash eq uilibrium strategies can be Pareto optimal [23], it is no t always the case: payo ffs o f gam e players m ay b e improved simultaneously af te r a Nash equilib rium is attained [24]. Stochastic op timization su ch as gen etic a lgorithms has been app lied to attain a satisfactory load factor as well [25], yielding sing le-objective o ptimization p roblem s. So me draw- backs may inherit from such single-o bjective formulatio ns, such as the n eed of p rior decision making for the trad eoff between objectives [26]. Because demand response is closely related to pr icing signals, a d e mand response pro gram can be derived from a pricing scheme [27]. Kunwar et al. [28] pro posed an area- load b ased p ricing schem e fo r de mand side managem ent. The load factor and energy cost were jo intly optimized. Given pricing signals, the propo sed metho dolog y cou ld produ ce a prom ising deman d response program addr e ssing the two objectives. In contrast with single - objective optimizatio n, such a mu ltiobjective appr o ach could avoid, for example, the need of heur istic assignm ent for weightin g coe fficients and the need of pr io r decision making for tradeoffs ind uced by conflict- ing objectiv es. T here are, howe ver , a fe w importan t issues that were not covered by [28]. First, althou gh the approach considered shiftab le load s, they were addressed statistically . 2 In this c ase, n o integer or discrete decision variables wer e in volved in optimization , but they ar e importan t f or explicit control of shiftable loads. Second , owing to th e employed statistical mo delling, the impact of electric vehicles (EVs) was not explicitly examined. In gene r al, EVs pose different physical constraints and shou ld be distinguished from ordin ary home app liances. Third, renewable energy sourc es (RESs) th at have b een wid ely ad opted in residen tial comm unities wer e no t considered . Motiv ated by [ 28], we pro pose a m ultiobjective appr o ach that addresses the load factor and energy con sumption costs of residential users in separate dimen sions, leadin g to a mul- tiobjective optimiz a tio n pro b lem (MOP). Our system models include EVs and RESs. Regarding the MOP , power sch eduling profiles serve as the decision variables, consisting of c on- tinuous a n d discrete ones. For th ese two types of decision variables, feasible search me chanisms are developed by ex- ploiting constraint a n d variable structures. Relevant analysis is provided to sho w their effecti veness. Solving the MOP yields an ap p roxima te Pareto set and a Pareto front. A Pareto op timal solution is then selected to indicate how power loads should be adjusted over time, yielding a Pareto optimal de m and response (PODR) prog ram. This prog ram can be offered by utilities to residential u sers. The associated algo rithms or technologies are implemented in residen tial homes. The loa d f actor is re late d to gr id reliability fo r utilities while the e n ergy co st pertain s to the p articipation of re sid e ntial users. The main co n tributions of th is study are summarized as follows: First, we examine the grid load factor and en ergy costs of r esidential users using a multiobjective framework, and consider a combin a tio n of various typ e s of applian ces, an EV , and RESs. Th is com bination h as n ot been fully in vestigated in the literature when multiobjective op timization is applied for residential p ower scheduling . Second, we p ropose a few algorithm s that gen erate f e a sible values for decision variables, leading to a solution method for our MOP . Owing to physical constraints inv olving discrete and continu ous variables, this MOP can hardly b e solved by con ventional multiobjective ev olutionary algorithm s. Third, num erical simulations d emon- strate that o u r approach can be b eneficial to residential user s while achieving a satisfy in g load factor as com p ared to co m- parable methods. Finally , in addition to numerical evaluations, the p ropo sed alg orithms ar e the oretically analyz e d, provid ing a rig orous fou ndation for the prop osed PODR pr ogram. The rem ainder of th is paper is organized as follows : Sec- tion II discusses related work. Section III descr ibes mathe- matical mo d els of various h ome app liances, the en ergy co n- sumption costs of re sidential users, and the load factor of the power grid . The PODR program is proposed in Sectio n III. Section IV presents our simulatio n results inv olving compar- isons b etween existing deman d resp onse strategies. Fin ally , the paper conc ludes in Sectio n V . I I . R E L A T E D W O R K This section examines various meth ods employed to design demand respo nse p rogra m s for re sid e ntial users. T o facilitate discussions, the section is divided into two sub sections. The first subsection p r esents existing residential dema n d response methods inv olving the con c e pt o f th e load factor but withou t explicitly addressing it as an o bjective. This in cludes m ethods that have the load factor a s an o p timization co nstraint or hav e the load factor num erically analyzed in the simu lations. The second subsectio n investigates meth o ds that do n ot in volve the concept of the lo ad factor but perta in to ou r theme. A. Method s W ith Loa d F actor Game theo ry , co ntrol methods, mixed integer nonline a r progr amming, conve x optimization , and machine learning methods have been the m o st pro mising techniqu es emp loyed to r ealize residential dem and response inv olving the con c ept of the load factor . When gam e theory is used to m odel deman d re- sponse pro grams [29]–[3 5], two games ar e typically inv olved: one for suppliers such a s utilities and a g gregators, an d the other fo r cu sto m ers. A deman d respon se progr am is realized by attaining the Nash equilibrium of the gam es. Game theo ry approa c h es often inv olve b idding, but a bidding scheme for residential dem a nd respo nse can also be realized u sing other technique s, see, for example, [3 6]. Game theory approaches can be further comb ined with block chain tec h nolog ies to improve network security [37]. Control strategies h ave been applied for demand r esponse services as well [38]. In this case, state-spa c e represen tations are mostly employed to model system dynam ic s. For instance, Luo et al. [39] prop o sed a mu ltistage ho me e n ergy manag e- ment system co nsisting of for ecasting, d a y -ahead schedulin g, and actual oper ation. At the day-ahe a d sched uling stage, a co- ordinated ho me en ergy resourc e sch eduling m odel con strained by the peak-to- av erage ratio was construc te d to m inimize a one-da y home operation c o st. At the actual ope ration stage, a mod el predictive con trol based op erational strategy was propo sed to co rrect ho me en ergy resource op erations with the update of real-time inf ormation . Models for r esidential energy deman d have drawn muc h attention [40], [41]. When mod els are certain o r can be reliably estimated, mixed integer n onlinear pro gramm ing that in volves discrete decision variables can be applied [ 42]–[45]. In this case, discrete or integer decision variables often represent shiftable lo ads that can be adjusted to change demand curves. If a ssociated pro blem form ulations do n ot inv olve discrete or integer decision variables, then c o n ventional line a r pro gram- ming is applicable to residential demand man agemen t [46]. Con vex optimization methods have been applied to solve a subpr o blem fo r demand r esponse pro grams [47], [48]. In [47], fo r example, an optimization pr oblem that addressed a trade-off between paym e nts and the discom fort of u sers was formu late d , yield in g bo th integer an d co ntinuou s variables. The pro blem was further separated into two subpro b lems, an d one of them was solved using co nvex optimizatio n methods. T o relax assump tions on entities in the system or pro bably to ad dress system uncertainties, r einforc ement lear n ing fo r demand respon se has been extensively studied recently [4 9]. Throu g h the learn ing pro cess, an age n t or agents can lear n how to optimize users’ consumption patterns. Con sid e ring the peak-to- av erage ratio, Bahrami et al. [50] p roposed an actor- critic structure th at reduced the expected cost o f users in 3 the a g gregate lo ad. Deh ghanp our et a l. [5 1] co nsidered air condition ing loads as agen ts and optimized their consumption patterns throu gh modify ing the temperature set- p oints of the devices. Both consum ption costs an d users’ comfor t prefer- ences were addr e ssed . Giv en the aforemention e d state- o f-the-a rt me th odolo gies, a few points sho uld be no ted. While shifting loads thr ough demand response pr ogram s proves promising , it is worth mentionin g that load shifting may ha ve impac t on d istribution systems [52], [53]. In additio n to routine oper ations, demand response programs can be utilized for emergen cy ope r ations [54]. For a c ompreh ensive r evie w , the read e r can refer to [55] and [ 5 6] o n resid e n tial dema nd and to [57] on a case study illustrating th e bene fits of demand respon se in c o nsideration of residen tial users and utilities’ load factor . B. Method s W ithout Loa d F actor This subsection examines recent studies on dema nd re- sponse programs withou t explicitly in volving the con c ept of the load factor; an energy manage m ent system has been a dominan t way o f realizing a dem and respon se prog r am in response to utility pr icing sign als in the literature . Here is a list of examples. Hansen et al. [5 8] inv estigated a home energy manag e m ent system th at au tomated the energy usage. Observable Markov d ecision process appro aches were pro- posed to min imize the hou sehold electricity bill. Shafie-Kh ah et al. [59] prop osed a stoch astic energy mana g ement system that considered un c ertainties o f the distributed ren ewable re- sources and th e EV av ailab ility fo r charging and discha rgin g. Adika and W ang [36] examined a day-ahead de m and-side bidding appro ach that m aximized residential users’ benefits. Rastegar et al. [60] constructed a two-lev el framework for residential energy manag ement. Customer s minim ized their paymen t costs and sent out the desired power scheduling of ap p liances at the first lev el; a m ultiobjective op timization framework was employed to improve tech nical ch aracteristics of the distribution system at the seco nd level. Some auth ors fo c uses on the redu ction o f peak lo ads in residential demand respon se. V ivekananthan et al. [61] in- vestigated a reward based demand respo nse alg orithm that could shave network pe aks. Zho u et al. [62] established a multiobjective mod el of time-of-u se and stepwise power tariff for residen tial users, yielding load shifting from p e ak to off- peak pe riods. Most recently , learnin g b ased appro aches to residen tial demand response have em e rged, see, f or example, [ 6 3], [6 4] and [65]. In addition to direct in vestigations on meth o dolog ies, tools have been developed to facilitate existing dem and re- sponse p rocesses. For instance, Paterakis et al. [66] employed artificial neural networks and the wa velet tran sform to predict the r esponse of r e sidential load s to price sign a ls. W ang et a l. [67] developed a method for estima tin g the residen tial demand response b aseline. Finally , it is worth mention ing th at residen tial u ser beh aviors may b e studied in an aggregate manner, thereb y intro ducing the concept of residential demand aggregation [68]. T wo emerging topics a r e dema n d response method s for r e sid en- tial hea tin g, ventilation , and air cond itioning [69]–[7 1] an d /ϴϬ Ϯ  ϯ /ϴϬϮϭϭ / ϭϵ Ϭϭ W Žǁ Ğ ƌƉ ůĂŶ ƚƐ ^Ƶ ď Ɛ ƚ Ăƚ ŝŽŶ Ğ ŵĂŶ Ě ƌĞƐƉ Ž Ŷ ƐĞ Ɖ ƌŽŐƌĂŵ ůĞĐ ƚƌ ŝĐŝƚLJ  ƚƌĂ ŶƐ ŵ ŝƐ Ɛ ŝŽŶ ůĞ Đƚƌ ŝĐŝƚLJ  ƚƌĂ ŶƐ ŵŝƐ Ɛ ŝŽŶ ůĞ Đƚƌ ŝĐŝƚLJ  ƚƌĂ ŶƐ ŵŝƐ Ɛ ŝŽŶ ůĞ Đƚƌ ŝĐŝ ƚLJ ƚƌĂ ŶƐ ŵŝ Ɛ Ɛ ŝŽŶ Ă ƚĂ  ƚƌĂ ŶƐ ŵŝ Ɛ Ɛ ŝŽŶ Ă ƚĂ  ƚƌĂ ŶƐ ŵŝ Ɛ Ɛ ŝŽŶ ĞŶƚƌĂů ĐŽŶƚƌŽů ůĞƌ ŝƌ ĞĐƚ ĐŽŶƚƌŽů ^Ƶ ď Ɛ ƚ Ăƚ ŝŽŶ Ă ƚĂ  ƚƌĂ ŶƐ ŵŝƐ Ɛ ŝŽŶ Fig. 1. System diagr am of power suppliers and users in an Internet of T hings based env ironment. demand response methods for residential plug-in EVs [72]– [74]. I I I . S Y S T E M M O D E L S This section discusses math e matical m odels describing the power consump tio n of home appliances, en ergy costs of residential users, and load factor of th e grid. T he day -ahead pricing scheme is con sidered. T o realize an automatic d emand response prog ram, we consider an Internet of Th ings based en viron m ent with home appliances ab le to transmit an d receiv e signals [75], [76]. Th ese home ap pliances can be con trolled by a centra l controller using IEEE standar ds, such as IEEE 802.3, IEEE 8 0 2.11 , or I EEE 19 01. Fig. 1 presen ts the system diagram and T ab le I summarizes n otations and acronym s used throug hout this paper . A. Home A ppliance s Home ap pliances can be classified in to thre e types [77]: unshiftable an d in flexible applian ces (the corr e sp onding set is denoted by A ), shif ta b le and inflexible app lian ces (the correspo n ding set is denoted by A S ), and shiftable and flexible appliances (th e cor respond ing set is denoted by A S F ). Th e loads in duced by u sing those ap pliances can be classified accordin g ly . Let h ∈ H d enote the time slot and ∆ h de note the slot duration . 1) Unshifta ble a nd Inflexible Ap pliances: So me h ome ap- pliances are u sed du ring specific time p eriods. For example, lights must be turned on in the ev ening, and it may not be possible to shif t the time of u se or adju st their switching time. Cooking appliance s, such as elec tric pots, roasters, and microwa ve ovens, a re also used in specific time slots. Residential users may u se en tertainmen t electronics such as 4 T ABLE I N O TA T I O N S h Time slot h where h ∈ H H Observ ation horizon (the size of H ) A Set of unshiftable and infle xible appliance s a Unshiftabl e and inflexible applia nce p h a Amount of power for applianc e a to operate in time slot h A S Set of shiftable and infle xible applianc es b Shiftabl e and inflexibl e appl iance h b W orking time slot of appl iance b s b and e b Start and end time slots of appliance b w b T otal working hours of appli ance b p h b ( h b ) Amount of po wer for applianc e b to operat e in time slot h A S F Set of shiftable and flexi ble appliances c Shiftabl e and flexible appli ance s c and e c Start and end time slots of appliance c p h c Amount of po wer for appli ance c to operat e in time slot h p max c Maximum opera ting power of applianc e c p min c Minimum opera ting power of applianc e c p min c, agr Aggreg ate minimum po wer for using appliance c s ev and e ev Start and end time slots of EV battery char ging p h ev EV char ging po wer in time slot h p max ev Maximum char ging rate of an EV B 0 ev Initia l ca pacity of the EV battery B min ev Minimum capa city of the EV battery B max ev Maximum capa city of the EV batte ry B 0 Initia l ca pacity of the reside ntial energy storage system B h Energy lev el of the residential energ y storage system in time slot h B min Minimum capacity of the residential energy storage system B max Maximum capac ity of the residenti al energy storage system u h Char ging and dischargi ng control for the residenti al energ y storage s ystem in time slot h r h Amount of powe r provided by rene wable energy sources in time slot h televisions and comp uters only after work or scho ol. These appliances are thus regarded as unshiftable and inflexible. For a ∈ A , we deno te p h a as the amount of power for ap p liance a to o perate in time slot h . 2) Sh ifta ble an d Infl exible A p pliances: Shiftable and in- flexible appliances are used at potentially any time, but their power consu mption when perf orming a spe c ified job is fixed (and thus inflexible). V acuum cleaners and w ashing mach ines belong to this type . The time at which a n au tomatic vacuum machine o r a washing m achine is operated may no t be c r ucial (assuming th a t they do not make too m uch no ise; other wise, they should not be o perated at nigh t) . A user co uld prescribe a possible time window , and then the demand response prog ram could d ecide when the device op erates accor d ing to these instructions. The time window mu st be sufficient for the appliance to comp le te the job. Let w b denote the total working hours required b y ap- pliance b ∈ A S . Suppose th at a residential user sets the acceptable star t an d end time slots of applianc e b ’ s opera tion as s b and e b , r espectively ( s b ≤ e b ). Let C s b → e b w b denote the set of all w b -combin ations of workin g time slots within the time wind ow [ s b , e b ] . A d emand respo n se progr am thu s selects certain working time slots represented by h b from C s b → e b w b for appliance b , i.e., h b ∈ C s b → e b w b ∀ b ∈ A S . (1) For instan ce, if s b = 15 , e b = 17 , and w b = 2 ( i.e., a p pliance b n eeds 2 h ours to finish its work), the n C s b → e b w b = {{ 15 , 16 } , { 15 , 17 } , { 16 , 17 }} and h b could be equal to { 15 , 16 } , { 15 , 17 } or { 16 , 17 } . If h b = { 15 , 16 } , then appliance b oper ates f rom 14:00 to 16:0 0 . The associa te d po wer f o r ap pliance b to operate in time slot h is denoted p h b ( h b ) where p h b ( h b ) = 0 if h b ∩ { h } = ∅ . 3) Sh ifta ble and Flexible App liances: The u se o f shiftable and flexible appliances can be adjusted in two d imensions: when the appliances are used an d how much power th ey should consume. Heaters, air con d itioners, and EV charging stations can yield shif table and flexible loads. For example, residential users can lower an air co ndition e r’ s temp erature setting (flexibility) and de lay its u se (shiftability) as they wish [9]. Let p h c denote th e amount of power for appliance c to operate in time slot h , where c ∈ A S F , and s c and e c denote the start and end time slots of ap pliance c , resp ectiv ely . The following constraints are im posed on p h c [13]: p min c ≤ p h c ≤ p max c and e c X h = s c p h c ≥ p min c, agr (2) where p min c and p max c represent the m inimum and maximu m operating power , respectively , and p min c, agr represents the aggre- gate minimum p ower for using appliance c , wh ich pertains to user c o mfort. T o ensur e th at feasible p h c , h = s c , ..., e c , exist, we assume ( e c − s c + 1) p max c > p min c, agr . (3) Although charging an EV can b e consider e d as a shiftable and flexible load, we exclude d EVs fr o m A S F because the charing power of an EV depend s on the rem aining capa c- ity o f the vehicle’ s b attery , which imposes addition al co n- straints [ 7 8]. Let p h ev denote the EV ch arging power in time slot h , and s ev and e ev be the start and e n d tim e slots o f th e battery charging, r espectively . The following constraints on the charging rate and EV ba tter y capac ity m u st be satisfied: 0 ≤ p h ev ≤ p max ev and B min ev ≤ B 0 ev + e ev X h = s ev p h ev ∆ h ≤ B max ev (4) where p max ev represents the maximu m charging rate, B min ev is the minim u m ca p acity of the EV battery , B 0 ev is the initial capacity , an d B max ev is the max imum cap acity . T o ensure that feasible p h ev , h = s ev , ..., e ev , exist, we assume B 0 ev + ( e ev − s ev + 1) p max ev ∆ h > B min ev . (5) B. Energy Cost and Loa d F a ctor A r esidential h ome can be integrated with ren ew able en ergy sources (RESs) and equip ped with a storage system for energy managem ent. Let B h be the en ergy le vel of the sto rage system ( B 0 represents the initial energy level), r h be the expected charging po wer from RESs, and u h be the control law that dictates the amou nt of power bein g charged to o r discharged 5 from the energy storage system. The storage dy n amics can be described as [79] B h = B h − 1 + ( r h − u h )∆ h. (6) The con straints on th e en ergy storage system are 0 ≤ B h ≤ B max ∀ h ∈ H (7) where B max is the maximu m storage capacity . The total energy extracted fr om th e grid in time slot h , denoted by E h total , can be expressed as E h total = max { X a ∈A p h a + X b ∈A S p h b ( h b ) + X c ∈A S F p h c + p h ev − u h , 0 } ∆ h. (8) If u h is greater than the total p ower deman d, then E h total = 0 and the excess energy is discarded. This situation can happen when r h is too large to b e stored in th e energy storage system and consumed by residential appliances. If u h < 0 , then − u h is the amo unt of power delivered to the e n ergy stor age system from the power grid. The total en ergy cost can be obtained by multip ly ing the to tal energy con sumption by the electricity price λ h : X h ∈H E h total λ h . (9) Unlike custom e r s, utilities are principally co ncerne d with the load factor . The load factor is critical because ge n eration cost an d grid quality have a direct connection with th e load factor . A higher load factor implies a m ore stable power grid and a lower cost o f p ower gene r ation [80]. T he loa d factor can be defined as the ratio o f the average demand to the maximum demand [17], [1 8]. The fo llowing p erform ance index can b e used in a dem and r esponse p rogram offered by utilities to improve their load factor: P h ∈H E h total /H max h ∈H E h total (10) where H is the size of H and rep resents the observation horizon . I V . P A R E T O O P T I M A L D E M A N D R E S P O N S E P RO G R A M This section p ropo ses the PODR p rogr a m that can be offered by utilities to residential user s. An MOP pertaining to power scheduling is form ulated. The objectives of the optimization prob lem are to m inimize the energy cost of a residential u ser in (9) and m aximize the load factor in (10). T o construct a stochastic sear c h scheme for explor ation and exploitation of the decision spac e , we analyze the associated decision variables and prop o se a few methods of feasible value gener a tions and mutatio n and cro ssover op erations tha t preserve feasibility . A mu ltiobjective ev olution ary algorithm for constructing the PODR program is de veloped accord ingly . Finally , the algorithm com plexity is discussed. The MOP is fo r mulated as min x f cost ( x ) = X h ∈H E h total λ h max x f LF ( x ) = P h ∈H E h total /H max h ∈H E h total subject to (2) , (4) , and (7) ∀ c ∈ A S F (11) where x denotes the decision variable vector that contains decision variables h b , p h c , p h ev , and u h for all b, c , and h . The objective functions are conflictin g , so the glob al optimal solution that optimizes both ob jectiv e function s simultane o usly does no t exist. T o solve th e M O P in ( 11), Pareto o ptimality is adopted [81]. A feasib le p oint x ′ , i.e., an x ′ that satisfies the co nstraints, d o minates an o ther fe asible p oint x ′′ if the condition s f cost ( x ′ ) ≤ f cost ( x ′′ ) and f LF ( x ′ ) ≥ f LF ( x ′′ ) hold true with at least one stric t inequality . A solutio n is nondo minated (a lso called Pareto optimal or Pareto efficient) if improving one ob je c tive value must y ield a degradation in the other ob jectiv e value [82]. A set of Pareto op timal solutio ns or nondo minated solution s is desired. A nondomin ated solution should b e selected from the set on the basis of its associated perfor mance represented by th e Pareto fr o nt. In the following discussions, we ad o pt a f ew terminolo- gies u sed in genetic alg o rithms [83]. Genetic algorithm s are stochastic search tech n iques that have roots in g enetics and employ a po pulation based method to find solutions to o p- timization p r oblems conventionally with a single o bjective. These algo r ithms begin with a set of points rand o mly in i- tialized. The set is termed the initial population. Poin ts in the populatio n are ev alu ated on the basis o f the o bjective fun ction, yielding function values called the fitne ss. With the h elp of the fitness, a set of new points are gener ated using mutation and crossover oper ations. Th ese opera tio ns togeth er with a selection operation based o n th e fitness ar e applied to improve an a verag e fitn ess value from population to p opulatio n . The aforemen tioned p rocedu re repeats iterati vely to produ ce new populatio ns until a stopping criter io n is satisfied. In genetic algor ithms, the mutation is performed on a candidate poin t and deri ves a new point term ed a mutant. The probab ility of having a mutan t is dictated b y a mutation rate. If the cand idate point is r epresented by a bin ary string using an encodin g scheme, then a typical mutation can be designe d as stochastically complemen ting bits from 0 to 1 or vice versa. While the m utation is applied to one candid ate p o int, the crossover is perfo rmed on a pair o f can didate points called the paren ts and pro d uces a correspond ing pair of poin ts called the o ffspring. T he p robab ility o f performin g the crossover operation is d ictated by a cro ssover rate. In the case of using the binary representation, a typical cr ossover op eration can be realized thr ough an exchan ge of sub strin gs of th e paren ts. T o add r ess con straints o f optimization prob le m s in genetic algorithm s, penalty method s can be u sed. A penalty functio n is added to the objective fu nction for p oint ev aluation. The fitness of a po in t beco mes a sum of the obje c tive functio n value an d a p enalty functio n value. For an infeasible po int, i.e., v iolating th e constraints, its penalty f unction v alue is 6 nonzer o (n egati ve for max imization p roblems an d p o siti ve for minimization problem s) a nd th us p enalizes the o bjective value. Because p oints with better fitness ar e p rone to be kept in populatio ns during the algo rithm iteration s, in feasible p oints can be grad ually removed, leading to a feasible set. Genetic alg orithms pr ovid e a gen eric framework for solvin g optimization problem s. Su itable modifications can r ender them more efficient and powerful. For example, we may use ce rtain schemes dedicated to th e proble m of interest that rand o mly generate feasible points and provide mutation and crossover operation s that preser ve the f easibility o f those p oints. As such, algor ith m effi ciency can be imp roved beca use more computatio ns are spent on impr oving f easible points instead of finding feasible ones. Further more, by inc o rpor a ting the concept of Pareto optimality into the fitness ev aluation, we may design algo rithms that can a d dress multiple objectives, which are termed m ultiobjective ev olution a ry algorithms. Although a few m ultiobjective e volutionary a lg orithms ar e av ailab le for findin g solutions of MOPs, most of the m consider either pure continu ous de cision variables or pure discrete decision v ariables. Th e situation in which both co ntinuou s an d discrete decision variables are in volved, as in our scenario, has n o t b een well add ressed [84]. In addition, the constra in ts in (11) can be difficult to address using con ventional con straint handling techniq ues. Metho ds o f feasible value gene rations and m utation an d crossover o peration s tha t preserve fea sibility are requ ired. T o solve (11), we first co nsider an enco ding scheme for discrete variables and present the associated mutatio n and crossover operation s. Continu ous variables are then addressed. For d iscrete variable h b , a feasib le value can b e readily g en- erated according to (1). Th e fo llowing en coding schem e and associated m utation and crossover opera tions are ado pted. Let φ ( h b ) b e a binary repre sen tation of h b with leng th e b − s b + 1 and [ φ ( h b )] j denote the j th b it of φ ( h b ) . The encod ing sch eme φ ( · ) is design ed as [ φ ( h b )] j =  1 , if j + s b − 1 ∈ h b 0 , otherwise (12) for j = 1 , 2 , ..., e b − s b + 1 . For instance, if s b = 15 , e b = 17 , w b = 2 , and h b = { 15 , 16 } , the n φ ( h b ) = 110; if h b = { 15 , 17 } , then φ ( h b ) = 10 1 . For the mutation oper ation, we rando mly find a pair ([ φ ( h b )] j , [ φ ( h b )] j ′ ) = (0 , 1) and switch their values. For the crossover oper ation, we apply logic operation “OR” to two b inary r epresentatio n s, and then random ly ch oose w b bits with value 1 from the offspring and conv ert the other b its with value 1 to value 0. T he co ndition in (1) is thus satisfied throu gh the use o f the m utation and crossover operations. For con tin uous d ecision variables p h c , p h ev , and u h , we note that th e f o llowing mu tation an d crossover op erations can produ ce feasible points in the decision space if points in volved are all feasible. Theorem 1: Let x h new denote a mutan t or offspring. The mutation or crossover is perf ormed accord ing to x h new = δ x h + (1 − δ ) x ′ h (13) where δ ∈ [0 , 1] is a rand om n umber f or all h . For the mutation operation , x h is a point in the p o pulation and x ′ h is a po int that is randomly generated . For the cr ossover operatio n, x h and x ′ h are d istinc t points in the population . If x h and x ′ h in (1 3) are feasible, then the mutan t or o ffspring x h new is feasible. Proof: Consid e r x h = p h c . If p h c and p ′ h c are f easible, then p h c ∈ [ p min c , p max c ] , p ′ h c ∈ [ p min c , p max c ] , e c X h = s c p h c ≥ p min c, agr , and e c X h = s c p ′ h c ≥ p min c, agr . W e have δ p h c + (1 − δ ) p ′ h c ∈ [ p min c , p max c ] and e c X h = s c δ p h c + (1 − δ ) p ′ h c ≥ p min c, agr for δ ∈ [0 , 1] . Therefo r e, p h new = δ p h c + (1 − δ ) p ′ h c is f easible. A similar argum ent can be per f ormed to show the feasibility proper ty when x h = p h ev . Consider x h = u h . If u h and u ′ h are feasible, then we have B h − 1 + ( r h − u h )∆ h ∈ [0 , B max ] and B ′ h − 1 + ( r h − u ′ h )∆ h ∈ [0 , B max ] . (14) Define B h new = δ B h + (1 − δ ) B ′ h and u h new = δ u h + (1 − δ ) u ′ h . By (1 4), we have B h − 1 new + ( r h − u h new )∆ h ∈ [0 , B max ] . Note that B h new = δ ( B h − 1 + ( r h − u h )∆ h ) + (1 − δ )( B ′ h − 1 + ( r h − u ′ h )∆ h ) = B h − 1 new + ( r h − u h new )∆ h. Therefo re, B h new ∈ [0 , B max ] , which implies that u h new is feasible.  If the r e are metho ds of random ly generating f easible p o ints, then the feasibility can be preserved during the solution process using the mutation and crossover o peration s in Th eo- rem 1. Algorithms 1, 2, and 3 presen t th ose meth ods, and th eir effecti vene ss are further con firmed by the f ollowing theorem. Algorithm 1 Gen eration of Feasible p h c Input: s c , e c , p min c , p max c , and p min c, agr . Output: Feasible p h c , h = s c , ..., e c , that satisfy (2). Generate p h c random ly from [ p min c , p max c ] . while P e c h = s c p h c < p min c, agr do p h c := p h c + ( p max c − p h c ) δ h (15) where δ h ∈ [0 , 1 ] is a r andom numb er . end while 7 Algorithm 2 Generatio n of Feasible p h ev Input: s ev , e ev , p max ev , B 0 ev , B min ev , an d B max ev . Output: Feasible p h ev , h = s ev , ..., e ev , that satisfy (4). Generate p h ev random ly from [0 , p max ev ] . while P e ev h = s ev p h ev ∆ h < B min ev − B 0 ev do p h ev := p h ev + ( p max ev − p h ev ) δ h (16) where δ h ∈ [0 , 1 ] is a r andom nu mber . end while if P e ev h = s ev p h ev ∆ h > B max ev − B 0 ev then p h ev := B max ev − B 0 ev P e ev h = s ev p h ev ∆ h p h ev (17) end if Algorithm 3 Generatio n of Feasible u h Input: B 0 and r h . Output: Feasible u h , h ∈ H (i.e., the constrain ts in (7) are satisfied). for h = 1 : H do Randomly gen erate u h such that u h ∈ [ B h − 1 + r h ∆ h − B max ∆ h , B h − 1 + r h ∆ h ∆ h ] . (18) Evaluate B h = B h − 1 + ( r h − u h )∆ h. end for Theorem 2: Under the assump tions described in (3 ) and (5), Algorithms 1 , 2, and 3 pr oduce f e a sible values for p h c , p h ev , an d u h . Proof: Accor ding to (15) an d th e fact that p h c , h = s c , ..., e c , are initially g enerated fr om [ p min c , p max c ] , variables p h c , h = s c , ..., e c , with p h c ≥ p min c approa c h p max c from th e lef t as the number o f iteratio ns in creases. Owin g to the assum ption in (3), the cond itio ns in (2) ho ld tr ue eventually when the values of p h c , h = s c , ..., e c , increase. By a similar argument and ac cord- ing to (5) and (16), we have B min ev ≤ B 0 ev + P e ev h = s ev p h ev ∆ h ev entually when the values of p h ev , h = s ev , ..., e ev , increase; howe ver , it is possible that an increm ent is too large to h av e B 0 ev + P e ev h = s ev p h ev ∆ h ≤ B max ev . T o remedy this, we use (1 7) to r e d uce the values of p h ev , h = s ev , ..., e ev . When (17) is executed, we have the following two resu lts: p h ev on th e left- hand side of (17) satisfies p h ev < p max ev because the term ( B max ev − B 0 ev ) / P e ev h = s ev p h ev ∆ h on the right-h and side of (17) is less than 1; an d p h ev on the lef t-hand side o f ( 17) satisfies P e ev h = s ev p h ev = ( B max ev − B 0 ev ) / ∆ h. The conditions in (4) are then satisfied. Finally , we no te that (18) imp lies (7), illustrating the f easibility of u h .  W ith the help of Alg o rithms 1 – 3, Algorithm 4 presents the PODR progr am that can be offered by utilities to residential users. In Algorith m 4, in formatio n about ap pliances and acce p table times of use is set first. Mu ta tio n an d crossover operation s are then performed over X ( t c ) in Step 2.1. I n Step 2 .2, the Algorithm 4 Pareto O p timal Power Scheduling Input: Electricity pr ice λ h ; MOP in (11) with parameters p h a , p h b , w b , s b , e b , p max c , p min c , p min c, agr , s c , e c , p max ev , s ev , and e ev ; mutation r a te µ ; n ominal popu la tio n size N nom ; m aximum populatio n size N max ; an d maximum iteratio n nu mber t max . Output: PODR progra m. Step 1 ) In itialize X (0) : rand omly gener ate working hou rs h b for shiftab le but inflexible app liances; apply Algo- rithms 1, 2, an d 3 to generate feasible values fo r p h c , p h ev , and u h . Remove domina te d p oints fro m X (0) . Step 2) Let t c = 0 . while t c ≤ t max do Step 2.1) Clon e po ints in X ( t c ) with c lone rate N max / N nom . Apply mu tation operatio n with rate µ and crossover op eration with rate 1 − µ d efined in Theorem 1 to cloned points. Store the m utants and offspring in X ( t c ) . Step 2.2) Remove d ominated p oints from X ( t c ) . If | X ( t c ) | > N nom , then use an archive update method to redu c e its size. Step 2.3) Let X ( t c + 1) = X ( t c ) and t c = t c + 1 . end while Step 3) Select the kne e of th e ap prox im ate Pareto fr ont associated with the app roxima te P areto set X ( t max ) . archive is u p dated by removing some nondo minated points if the po pulation size | X ( t c ) | is too large. After a numb er of iterations, an appro x imate Pareto set an d fr ont are obtained in Step 3. In p r actice, the knee of the app roxim a te Pareto fro nt is often prefe r red for several reasons: it can achiev e excellent overall system perfo rmance if th e fro n t is b e nt; it represen ts the solution closest to the ideal one that is not re a c hable; and it has rich geometrical and physical meanings [85], [86]. In Step 3, the kn ee solution is selected accord ing to [8 7]: x ∗ = arg min x ∈ X ( t max ) f cost ( x ) − min x ′ ∈ X ( t max ) f cost ( x ′ ) L cost + max x ′ ∈ X ( t max ) f LF ( x ′ ) − f LF ( x ) L LF (19) where L cost = max x ∈ X ( t max ) f cost ( x ) − min x ∈ X ( t max ) f cost ( x ) and L LF = max x ∈ X ( t max ) f LF ( x ) − min x ∈ X ( t max ) f LF ( x ) (20) are the maximum spread s of th e appro ximate Pareto front in the first and second dimensio n s, respectively , and  min x ∈ X ( t max ) f cost ( x ) L cost max x ∈ X ( t max ) f LF ( x ) L LF  T is the ideal vector . As sho wn in (19), the final Pareto o ptimal solu tion is se- lected usin g the sy stem’ s knowledge of the maxim um spread s, critical inform a tion in multicriteria decision makin g. Such informa tio n is not obtained usin g conventional single- objective optimization approach es. The knee so lu tion x ∗ correspo n ds 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time slot 2 4 6 8 10 12 14 16 18 20 22 Price (cent/kWh) 27 July 2017 (maximum price range) 6 July 2017 (25th percentile) 23 July 2017 (75th percentile) 29 July 2017 (minimum price range) Fig. 2. Pricing schemes used in our s imulati ons. to a P areto optimal power consumption pro file, yielding the PODR p rogra m . The complexity of Algorithm 4 mainly dep ends on the use o f Pareto con cepts that are the most compu tationally expensiv e. The comp lexity can be r ough ly expressed as [84] O (2 t max N nom ( t com + N nom − 1)) where t com represents the average compu tational time for objective evaluation. V . S I M U L A T I O N R E S U LT S This section presents an examina tio n of the power schedu l- ing of 400 residential users. 1 Each residence was equipped with at most thirteen unshiftab le and in flexible applian ces, three shiftab le and inflexible app liances, two shiftable and flexible appliances, o ne EV , and po ssibly one energy stor age system integrated with solar energy sou r ces. The exact n u mber of app liances and associated typ es for a residential user wer e random ly chosen. Solar e n ergy data in [88] were used. Let H = { 1 , 2 , ..., 24 } and ∆ h = 1 . T ab le II presents the a sso- ciated settings, an d the notation “un if ( h 1 , h 2 , t )” therein ind i- cates that an appliance require d t hours of working time start- ing f r om a nu m ber ran domly ch osen f rom { h 1 , h 1 + 1 , ..., h 2 } . For examp le , applian ce a 4 needed 5 h ours to comp lete the task, and th e start time slot could hav e b een 1 6, 1 7, or 18. Most parameter values were ch osen acco rding to [ 89] an d [90]. The day-a head prices illustrated in Fig . 2 were used [9 1]. After analyzin g total energy consump tion o f each mo nth in 2017, we discovered that energy consumptio n peaked in July . Four representative day s in July were selected and listed in a decreasing ord er in terms of their price r ange as follows: 27 July 201 7 (maximum price range), 6 July 201 7 (25th percentile), 23 July 2017 (75 th percen tile), and 29 July 2017 (minimum p rice r ange). Amo ng the m, 27 July 2 017 also had the h ighest price. 1 In [18], the total ava ilable capacit y of a region was set at 2296 kW . In our scenari o, each residentia l user consumes 5.5 kW at most, which thus account s for approxi mately 400 residentia l users in the region. T ABLE II S I M U L AT I O N P A R A M E T E R S Notatio ns Po wer W orking Schedule p h a 1 0.02 kW h ∈ { 17 , 18 , ..., 24 } p h a 2 0.22 kW unif(18, 22, 3) p h a 3 0.2 kW unif(11, 13, 3) p h a 4 0.2 kW unif(16, 18, 5) p h a 5 0.7 kW unif(18, 22, 1) p h a 6 1.3 kW unif(14, 16, 1) p h a 7 0.2 kW unif(18, 22, 1) p h a 8 0.08 kW unif(18, 20, 3) p h a 9 0.05 kW h ∈ { 1 , 2 , ... , 24 } p h a 10 1.5 kW h = 8 p h a 11 1.6 kW h ∈ { 17 , 18 } p h a 12 0.2 kW h ∈ { 1 , 2 , ... , 24 } p h a 13 0.8 kW h = 17 Notatio ns V alues s b 1 , e b 1 Random number from { 10 , 11 , 12 , 13 } , e b 1 = s b 1 + 7 p h b 1 , w b 1 1 kW if h b 1 ∩ { h } 6 = ∅ , w b 1 = 1 s b 2 , e b 2 Random number from { 12 , 13 , 14 , 15 } , e b 2 = s b 2 + 4 p h b 2 , w b 2 1 kW if h b 2 ∩ { h } 6 = ∅ , w b 2 = 2 s b 3 , e b 3 Random number from { 13 , 14 , 15 , 16 } , e b 3 = s b 3 + 7 p h b 3 , w b 3 2 kW if h b 3 ∩ { h } 6 = ∅ , w b 3 = 2 s c 1 12 e c 1 24 s c 2 Random number from { 20 , 21 , 22 , 23 } e c 2 s c 2 + 9 s ev Random number from { 18 , 19 , .. ., 22 } e ev s ev + 11 p max c 1 , p max c 2 3 kW p min c 1 , p min c 2 0.5 kW p min c 1 , agr 29 kW p min c 2 , agr 12 kW p max ev 3 kW B max , B 0 4 kWh, 1 kWh B max ev , B min ev 24 kWh, B min ev = 0 . 8 B max ev B 0 ev Random number from [0 . 3 B max ev , 0 . 6 B max ev ] 175 180 185 190 195 200 205 Cost (cents) 0.6 0.65 0.7 0.75 0.8 0.85 Load factor (%) Approximate Pareto front Knee Fig. 3. Sampled approximate Pareto front obtained by solving (11 ). 9 The PODR pr o gram fo r users was obtain ed using Alg o- rithm 4 with µ = 0 . 8 , N nom = 40 , N max = 40 0 and t max = 400 . Ap prox imate Pareto fronts associated with residential users were produced . Fig . 3 plots a sampled ap prox imate Pareto front. Each p o int on the front c orrespo n ded to two values: energy cost a n d lo ad factor . The sh a pe o f the fr o nt confirmed that minimizin g the co st and m aximizing the load factor were conflictin g objectives. The k nee was selected accordin g to (19). Our multiobjective approach was compared with an area- load method m odified from [28], a paym e n t minimizatio n method mod ified fr o m [18], and load factor maximizatio n and load variance minimization metho ds m odified fr om [92] ( see Append ix ). Ow in g to constra in t structures and different ty p es of variables th at were in volved, conv ention a l stoch astic search methods could hard ly prod uce feasible points. Alg orithms 1 – 3 were thus ap plied to prod uce a portion of initial points that were used throu gh the so lution processes associated with the compara b le meth ods. Fig. 4 presents indi vidual per f orman ce on the representativ e days. T able I II summarizes th e perfor- mance (expressed as a p ercentage ) of ea c h method. For th e energy cost, the p ercentag es were e valuated with respect to our metho d . A sm a ller percentage indicated a lower cost. For the load factor, the pe rcentages were ev alu ated with respect to the load variance min imization method. A larger pe rcentage indicated a higher load factor . Among tho se meth ods, the PODR prog ram balanced th e two conflicting objectives in an advantageo us manner . This should be generally the case because our app roach consider ed join optimization of the two objectives, had robust constrain t handling techniq ues p resented in Algo rithms 1-3, and em- ployed d edicate mutation and crossover oper a tions to p reserve solution f e asibility , a s shown in Th e orem 2 . Th e m odified area-load method also adopted a multiob jective appro ach; as compare d to the PODR, it yielded a small improvement in the load factor by appro ximately 2.8 % (- 7.8%+10 .6%) but a large d egradation in th e energy costs by 14.5%. F or single- objective op timization methods, load variance m inimization and load factor max imization yielded larger load factors than other methods because performa n ce m etrics related to the load factor were optimized ; howe ver, the resulting energy costs for residen tial users, which were n o t considere d dur ing the solution process, were the worst amo ng other m ethods. Finally , the mo dified pay m ent minimization metho d yielded satisfactory costs for r esidential user s b ut the worst load factor; the associated energy cost was h igher th an that o f th e PODR because that metho d lacked suitable constraint handling technique s. V I . C O N C L U S I O N This paper inv estigated two critical aspects of power scheduling : r esidential users’ energy costs an d the power grid ’ s load factor . Th ese two aspects shou ld b e con sidered simu lta- neously when utilities are to o ffer demand response programs to r esidential users. In practice, min imizing the cost and maximizing th e load factor a re two conflicting objectiv es—one cannot b e o ptimized with out worsening the oth e r . T o a ddress 27 July 2017 6 July 2017 23 July 2017 29 July 2017 0 50 100 150 200 250 300 350 400 Cost (cents) PODR Load Variance Minimization Load Factor Maximization Area-load Method Payment Minimization (a) 27 July 2017 6 July 2017 23 July 2017 29 July 2017 0 0.5 1 1.5 Load factor (%) PODR Load Variance Minimization Load Factor Maximization Area-load Method Payment Minimization (b) Fig. 4. (a) Energy costs and (b) load fac tors of 400 resident ial users. T ABLE III C O M PA R I S O N O F V A R I O U S P O W E R S C H E D U L I N G M E T H O D S Cost Increase Wi th Respect to PODR Program Date Load V ariance L oad Factor Area-load Payment Minimization Minimization Maximization Me thod July 27, 2017 26.4% 30.6% 14.4% 8.5% July 6, 2017 18.6% 26% 15.5% 2.9% July 23, 2017 15% 23.1% 14.1% 7.3% July 29, 2017 12.4% 19.6% 14% 6.7% average 18.1% 24.8% 14.5% 6.4% Load Fac tor Reductio n W ith Respect to Modified Load V ariance Minimizat ion Met hod Date PODR L oad Factor Area-load Payment Minimization Program Maximization Method July 27, 2017 -13.1% -4.1% -10.6% -24.5% July 6, 2017 -8.2% -3.8% -6.5% -23.6% July 23, 2017 -12.7% -4.4% -9.3% -31.4% July 29, 2017 -8.4% -4.4% -4.7% -18.8% average -10.6% -4.2% -7.8% -24.6% 10 this challen g e, we prop osed a Pareto optima l demand respon se progr am that d etermined the Pareto op timal power schedu lin g. This p r ogram was constructed by solv ing a multiobjective optimization problem. T o explore and exploit the d ecision space a ssocia ted with the p roblem, a f ew algor ith ms were developed to gen erate feasible v a lu es f or decision v ariables. Rele vant analysis w as p erform ed to show the effecti veness of the algor ithms. Com pared with existing methods for power scheduling , the Pareto optimal deman d response p rogr a m found a satisfying balance between the two objectives by sacrificing one objective slightly while substantially imp roving the other . This stud y was mainly focu sed on deman d response for r esidential users. Our fu ture work includ es the in vestigation on multiob jectiv e app roach e s to the d esign of d emand respon se progr ams f or co mmercial o r industrial custome rs and for vehicle-to-gr id systems. A P P E N D I X D E S C R I P T I O N S O F C O M PA R A B L E M E T H O D S For mo dified area- load and pay ment minimiz a tion metho ds, the co nstraints in (11) w e r e addr essed using the following constraint f unction : U ( x ) = X c ∈A S F max { p min c, agr − e c X h = s c p h c , 0 } + max { B min ev − B 0 ev − e ev X h = s ev p h ev ∆ h, 0 } + max { B 0 ev + e ev X h = s ev p h ev ∆ h − B max ev , 0 } + X h ∈H max { B h − B max , − B h , 0 } . If U ( x ) > 0 , th e n the power scheduling profile x violates the constraints an d x is infe a sib le. For the ar ea-load meth od modified fr om [28], we solved min x f cost ( x ) + σ 1 U ( x ) + σ 2 f Penalty ( x ) max x f LF ( x ) − σ 1 U ( x ) (21) where σ 1 and σ 2 represent th e weightin g factors o f the co n- straint violation an d pena lty , respectively . Con straint handling was not discussed in [28], but it is essential because ph ysical systems always induce constra in ts. W e m odified the objec tives by including U ( x ) . The penalty of ene rgy load deviating from the average load was evaluated as f Penalty ( x ) = X h ∈H | E h total − E av | (22) where E av = X h ∈H E h total /H . (23) The n ondo minated sorting g enetic a lg orithm- II was used to solve (21). For the paym ent minimization metho d mod ified from [18], we solved min x f cost ( x ) + σ 1 U ( x ) (24) using par ticle swarm optimiza tio n. The load factor m aximization me th od from [92] was mo d - ified as max x f LF ( x ) subject to (2) , (4) , and (7) ∀ c ∈ A S F . (25) For th e load variance minimization meth od modified fr o m [92], we solved min x X h ∈H ( E h total − E av ) 2 /H subject to (2) , (4) , and (7) ∀ c ∈ A S F (26) where E av is defined in (2 3). R E F E R E N C E S [1] H. Farha ngi, “The pat h of the smart grid, ” IEEE P ower Energy Mag. , vol. 8, no. 1, pp. 18–28, Jan. 2010. [2] A. Ipakchi and F . 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