How Can Smart Buildings Be Price-Responsive?

Ho w Can Smart Bu ildings Be Price-Responsi v e? Ricardo Fern ´ andez-Blan c o Gr oup O ASYS University of Mala ga Malaga, Spain Ricardo.FCarram olino@uma.es Juan Migue l Morale s Dept. Applied Mathematics University of Mala ga Malaga, Spain Juan.Mor a le s@um a.es Salvador Pineda Dept. Electrical Engineering University of Mala ga Malaga, Spain spinedamo rente@gmail.co m Abstract —The pr ospectiv e participation of smart buildings in the electricity system i s strongly related to the increasing active role of demand-side r esources in the electrical grid. In addition, the gro win g penetration of smart meters and rece nt advances on home automation technologies will spur the dev elopment of new mathematical tools to help opti mize th e l ocal re sources of these buildings. Within this context, this paper first p ro vid es a comprehensiv e model to determine the electrical consumpti on of a si n gle-zone household based on economic model predictiv e control. The goal of this problem is to minimize the electricity consumption cost while acco unting for the heating d ynamics of the building, smart home appliances, and comfort constraints. This paper then id entifies and analyzes the key p arameters re- sponsible fo r the price-responsive behavio ur of smart households. Index T erms —Model predictive control, price-responsiv e loads, smart appl iances, smart build ings. N O M E N C L AT U R E A bar above ( x ) or belo w ( x ) any variable denotes its maximum and m inimum b o unds, respectively . A. Sets and In dices C i Set of power p h ases o f load i , indexed by c . C Set of user-defined co mfort co nstraints. I Set of uninterrup tible load s, indexed by i . T Set of time perio ds, ind exed by t . T i Set of desirable oper ation times of load i . T occ Set of time period s du ring which the ho usehold is occupied , indexed by t . T Set of te c h nical constrain ts pertaining to co ntrol actions. B. P ar a meters C T i Cycle duty of uninterru ptible load i [# time perio d s]. N T Number of time perio ds. P ul ci Power of phase c o f un interruptib le load i [kW]. P sb t Standby p ower consump tion in period t [kW]. T R it Binary-valued p a rameter specifying the perio ds of operation of th e un interrup ta b le load i . U A r,a Heat transfer coe fficient between the room air and the ambient [W/ ◦ C]. This proje ct has recei ved fund ing from the European Research Counci l (ERC) unde r the European Union’ s Horizon 2020 research and innov ation pro- gramme (grant a greement No 755705) . This projec t has also been support ed by Fundaci ´ on Iberdrola Espa ˜ na 2018. The authors thankful ly ackno wledge the computer resourc es, technical exp ertise and assistance provi ded by t he SCBI (Supercomput ing and Bioinformat ics) center of the Uni versit y of Malaga. z amb t Outdoor ambient temp erature in per io d t [ ◦ C]. z hw d t Hot water deman d in period t [l] . z occ t Occupancy schedu le in period t . φ t Luminan ce from n atural light [lum en]. η le Indoo r lu minous efficac y [lumen/kW]. λ t Electricity price in per iod t [ e /k Wh]. ρ l Penalty term for the light le vel boun ds [ e /lumen]. ρ temp Penalty term for temper a ture bou nds [ e / ◦ C]. C. V ariables l t Light le vel in period t [lume n ]. u b t Power consu mption of the building in p eriod t [kW]. u bl t W indow b lind p osition in period t [ p.u.]. u ı t Power c o nsumption in perio d t . Superscr ipt ı = { heat , cool , hp , w h , r f , al } and th ey denote heating an d co o ling sy stem , heat pu mp f or the floor heating system, water heater, ref rigerator, and artificial lighting, re sp ecti vely [kW]. u ul it Power consu mption of uninterru ptible load i in period t [kW]. v l t Slack variable for the light level bound s in per iod t [lumen ] . v r /wh/r f t Slack variable for the temper ature bounds of the room/water hea ter/refriger ator in perio d t [ ◦ C]. x β t T empe rature in per iod t . Sup erscrit β = { r, f , w , w h, rf } and they denote temper ature of room, flo or , water in floor heating p ipes, water heater, refrigeratio n c h amber, respectively [ ◦ C]. δ a cit Auxiliary variable to relate p ower phase c with th e scheduling of un interrup tible lo ad i in period t . δ ul it Binary variable indicating whethe r uninter ruptible load i is sch eduled on (1) or off (0) in perio d t . D. V ector s u V ector of con trol variables. v V ector of slack variables. x V ec to r of state variables. y V ector o f measured sign a ls. z V ector of external disturban c es. E. Matrices A Matrix o f h eat transf e r coefficients (d ynamics matrix). B Matrix of coefficients relating the state and con trol variables ( control m atrix). C Matrix of coe ffi cients relating the state variables and the m e asured signals (sensor matrix). D Matrix of coefficients relating the measured signals directly with the con tr ol variables (dire ct term). E Matrix of co efficients relating the state v ariables and external disturb a nces (disturb a n ces matrix). I . I N T RO D U C T I O N Demand-sid e resources are expe c ted to play an im p ortant role in future electricity systems [1]. Their acti ve par ticip ation in pr ovid ing flexibility to the gr id will also help integrate more and more renewable energy , thus paving the way towards the decarbo nization o f the electricity sector . Prospectiv e demand - side resou rces are storag e devices and smart buildings. Large- scale integration of th e former is expected to increase pr ovided capital cost reduc tions an d increased charging/discharging efficiencies [2], wher e as th e latter is gainin g mo re attention due to recent advances on home automation and the growing penetration of sm a rt me te r s in the d istribution network [3]. This pap er is focu sed on the smart buildings’ ability to provide flexibility , which ma y b e of utm o st interest to incentivize the activ e particip ation of demand -side resources, thu s leading to new ways of electricity trading [4]. There is a vast literature on building simulation b y : (i) individually modelling thermal loa ds such as refriger ators [5], water heaters [6], o r smart s olar tan ks [7]; o r (ii) mo delling the building h eating dynamics [8]. Halvgaard et al. [8] analyzed a single-zon e building with a water - based floor heater ( FH) via econom ic model pred icti ve control (MPC) in o rder to shift power to periods with lower p rices. Note that an econ omic MPC is a model that represents co m plex system dyn amics while minimizing an econ omic objec tive function. The active participation of smart buildings in the electricity system has b e en widely investigated in the literature [9]–[15]. Most of the works consider ther mostatically-con trolled loads (TCLs) only [9]–[12]. Li et al. [9] devised a market-based coordin ation o f a po ol of TCLs which were mo d elled by a second-o rder equiv alent ther mal para m eter (ET P) m odel. Authors of [10] in vestigated the effects of dynamic-price retail contracts on integrated retail an d power market ope r ations wherein the househ olds ar e represented by a simplified ETP model of th eir air con ditioning systems. Autho rs o f [11] simulated a pool of price-respo nsiv e h ousehold s eq uipped with heating pump s to ap p ly in verse optimization in order to forecast their co nsumption . Zhao an d Zhang [12] developed a fra m ew o r k to aggregate price-r esponsive loa d s repr esented by a second- o rder ETP mo del. Ho wev er , little attention has been paid to an integrated formulation of the smart b uilding with s mart appliances [13]– [15], whic h is cr ucial to accurately capture th e potential flexibility o f their buildings. Anjos et al. [14] proposed a Dan tz ig -W o lfe decom p osition approa c h to dea l with numer ous and heterogeneo us building s managed by a central ag gregator . Althou gh, they m o del on e o f each load type that a building m a y have, the building heatin g dynamics wer e disregard ed, thus ign oring the effect of the comfor t constrain ts. Contr eras-Oca ˜ na et al. [13] presented one of the mo st com pleted building mo dels a nd an a ly zed po ssible interactions between commercial buildings and an aggregator of electric vehicles. Gon zalez et al. [ 15] presented a r e sidential load simulator with appliance s. Howe ver , their focu s was o n providing load profiles rather th an on how the househo ld could provide flexibility . Recently , Junker et al. [16] de v ised a flexibility metric to characterize g e neric buildings (o r districts) that can m e a sure their reaction s to different penalty sign a ls. The major co ntributions o f this p a per are thus twofold: 1) W e p ropose a compact formula tio n for mo delling smart buildings w ith smart ap pliances, and a c o mprehen si ve formu latio n for a single- z o ne h ousehold . It includes a state-space mod el with five states in ord er to capture the building h e at dynamics, comfor t constrain ts with user-defined parameters, an d tech nical constrain ts. As major novelties, the m odel includ es the im p lementation of two d ifferent space heaters (water-based floor heater or H V A C systems) a long with other thermal load s (e.g. refriger a to r and /or water heater); and the uninterr uptible loads are m o delled with d iscrete variables to accurately account f or th eir variable cycle power . 2) W e analyz e the key drivers leading to price-respon si ve househo lds based on th eir comfor t settings and structural characteristics. I I . C O M PAC T P RO B L E M F O R M U L A T I O N Managing buildings with smart appliances c a ll for an inte- grated o p erational mod e l that takes in to acco unt the heating dynamics o f the building. Econom ic MPC becom e s th us a solid option to con trol the b uilding and it has b e e n a p plied to building dynamic simulation in the last decad e (see [17] an d referenc e s therein). This section provides a compact mathe- matical formulatio n for mod elling a smart building by using econom ic MPC. Th e model can be mathem atically expressed as th e following op timization pr o blem: min u , v , x , y h ( u , v ) (1a) subject to : ˙ x = Ax + B u + E z (1b) y = C x + D u (1c) u , v , y ∈ C (1d) u ∈ T , (1e) where h ( · ) comprises the electricity cost incurr ed by the smart building and the pena lty cost due to discomfo rt, which is minimized in co nstraint (1a). Constraints (1b)–(1c) d efine the state-sp ace mod e l of the building heat dynam ics ( in cluding therm al loads), which ar e represented by a linear system o f first-o rder differential eq u a- tions. Exter nal disturba n ces c a n be accounted fo r in this system. Specifically , con stra in ts ( 1 b) mod e l the heat b alance in each building zone and therma l load (e.g. refrigerator or water heater). Constraints (1c) accou nt for th e relationship amo n g the measured signals with the state and co ntrol variables. Eq uation (1d) models the set of user-defined comfo r t co nstraints, which include contr o l over th e indoo r air tempe r ature and lighting, among others. Fin ally , equation ( 1e) represents the set of tech- nical constraints including the scheduling of uninter ruptible loads such as washing machine, tumble d ryer, dishwasher, smart oven, and so on. The sy stem o f differential equatio n s can be discretized using Euler’ s m ethod an d thu s th e prob lem can be recast as: min u , v , x , y h ( u , v ) (2a) subject to: x t = A d x t − 1 + B d u t − 1 + E d z t − 1 ; ∀ t ∈ T (2b) y t = C d x t − 1 + D d u t − 1 ; ∀ t ∈ T (2c) u , v , y ∈ C (2d) u ∈ T , (2e) where expression s ( 2 a), (2d)–(2 e) ar e iden tical to ( 1a), (1d)– (1e), in that or der , b ut expressed in discrete time. Similarly , constraints (2b)–(2 c) co mprise the discrete state-space model described in (1 b)–(1c). Subscript d deno tes that the matrices are in discr e te time. I I I . P R I C E - R E S P O N S I V E H O U S E H O L D F O R M U L AT I O N Giv e n a sing le-zone household , we explore two option s for space heating: 1) a water-based FH, an d 2 ) an HV A C 1 system. Apa r t f rom these thermal loads, we mode l a resi- dential refriger ator and a water heater , as well as the indoor lighting. Uninterru ptible loads are also rep resented in th e model. Assuming that th e state variables are directly observed (i.e., C d and D d are, respectively , equal to the iden tity and null matr ices of appr opriate dimensions), a price-respo nsiv e househo ld c a n be fo r mulated as: min Ξ X t ∈T λ t u b t + X t ∈T h ρ temp  v r t + v wh t + v r f t  + ρ l v l t i (3) subject to:       x r t x f t x w t x r f t x wh t       | {z } x t = A d       x r t − 1 x f t − 1 x w t − 1 x r f t − 1 x wh t − 1       | {z } x t − 1 + B d           u hp t − 1 u heat t − 1 u cool t − 1 u al t − 1 u bl t − 1 u r f t − 1 u wh t − 1           | {z } u t − 1 + E d   z amb t − 1 z occ t − 1 z hw d t − 1   | {z } z t − 1 ; ∀ t ∈ T (4) x r t − v r t ≤ x r t ≤ x r t + v r t ; ∀ t ∈ T (5) x wh t − v wh t ≤ x wh t ≤ x wh t + v wh t ; ∀ t ∈ T (6) x r f t − v r f t ≤ x r f t ≤ x r f t + v r f t ; ∀ t ∈ T (7) l − v l t ≤ l t ≤ l + v l t ; ∀ t ∈ T occ (8) l t = φ t u bl t + η le u al t ; ∀ t ∈ T occ (9) u bl t ≥ u bl ; ∀ t ∈ T occ (10) u b t = X ı u ı t + X i ∈I u ul it + P sb t ; ∀ t ∈ T (11) 1 FH stands for Floor H eater , while HV AC m eans Heating, V entila tion, and Air Condi tioning system. 0 ≤ u ı t ≤ u ı ; ∀ ı = { heat, cool , hp, wh, r f , al } , t ∈ T (12) δ ul it ≤ T R it ; ∀ i ∈ I , t ∈ T (13) X t ∈T δ ul it = C T i ; ∀ i ∈ I (14) t + C T i − 1 X k = t δ ul ik ≥ C T i ( δ ul it − δ ul i,t − 1 ); ∀ t ≤ N T − C T i + 1 (1 5 ) δ a 1 it ≥ δ ul it − δ ul i,t − 1 ; ∀ i ∈ I , t ∈ T (16) δ a cit ≥ δ a c − 1 ,i,t − 1 ; ∀ c > 1 , i ∈ I , t ∈ T (17) u ul it = X c ∈C δ a cit P ul ci ; ∀ i ∈ I , t ∈ T (18) δ a cit ∈ [0 , 1]; ∀ c ∈ C , i ∈ I , t ∈ T (19) δ ul it ∈ { 0 , 1 } ; ∀ i ∈ I , t ∈ T (20) u bl t ∈ [0 , 1]; ∀ t ∈ T (21)  v r t , v r f t , v wh t , v l t  T | {z } v t ≥ 0 ; ∀ t ∈ T , (22) where Ξ =  l t , x t , u t , u b t , v t , δ a cit , δ ul it  is the set o f variables. The g oal of this optimizatio n prob lem, giv e n in ( 3), is the minimization o f the electricity costs and penalty costs on violations of u ser-defined com fort con straints. W e assume that a set of p rices is k nown a priori b y the hou sehold’ s energy managem ent system. The state-space mod el of the househo ld heat dynamics is g i ven by constraints (4), whose matrices are d e scr ibed in [1 8]. This mode l includes a three- state model for the water-based FH, which is represen ted by the inside air temperatur e, the floor temperature, and the w ater temperatur e in the floor heating pipes, as similarly done in [8]. In add ition, the state-space model (4) takes into accoun t two additional states, i.e., the therm al dynam ics of a residen tial refriger a to r and a water heater located outside th e ho usehold. The interested reader is referred to [5 ] , [6] f or comp lex models on these two appliances. Mo reover , the HV A C system is also represented in the matrix equations gi ven in (4) by a sing le- state model ( only th e indoor air temperatur e). The external disturbanc e s a r e the ambien t temper ature, room o ccupancy , and hot water demand . Comfort co nstraints ar e set in (5)–(10), e xpressions (1 1)– (18) represent te c h nical constraints, wh ereas (19) –(22) declare the character of variables δ a cit , δ ul it , u bl t , and the slack v ariables v r t , v r f t , v wh t , v l t . Regarding the co mfort con straints, (5)–(7) set the user- defined min imum a nd maximu m boun d s on the indoo r air temperatur e, water tempera tu re in the water heater, and air temperatur e in the r efrigerato r c h amber, in th at ord er , for each time pe r iod t . Th e comfo rt boun ds f or the indoo r air temperatur e can be cho sen as: x r t = x r,se t t + αw t and x r t = x r,se t t − αw t , where x r,se t t is the set-po int indoor air temperatur e, α is the m aximum tem perature d ifference with respect to the set-point that the u ser is able to withstand, and w t is a vecto r o f con tin uous parameters varying between 0 and 1. Thu s, com fort bou nds were set u p in two ways: • Price-in depende n t comfo rt bound s (PI-CB): w t is a vector of o nes. • Price-d ependen t comfort bound s (PD- CB): w t is a vector of n ormalized prices over a given d a y . Constraints (8)– (10) mo d el the household light lev e ls when occupan ts are in the h ousehold , as d escribed in [13]. Con- straints ( 8) enfor ce recommen ded min im um and max imum light levels. Th e hou sehold ligh t level g iv en in (9) can be computed as the natu ral ligh t comin g thr o ugh th e windows, which dep ends on th e blind po sitions (i.e., φ t u bl t ), an d the indoor artificial lighting (i.e., η le u al t ). Constraints ( 10) m ay enforce a lower b ound f or the position of the blin d s. Regarding th e technical constrain ts, expression (11) sets the building electricity consumption equal to the con tributions from the heat pump of the FH, HV A C system, indoor artificial lighting, water h eater , ref rigerator, uninter ruptible lo a ds, and stand-by p ower . Constraints ( 1 2) impose the bou nds on th e power for th e h eating pum p of the FH, HV A C system, artificial lighting, water heater, an d refrigerato r . The set of constraints (13)–(18) de fin es the op eration of the uninterr uptible loads. W e assume that each uninterru p tible load is characterized by its cycle time C T i and the cycle power at each phase c , i.e., P ul ci . In additio n , the user may pre-define the time interv al T i at which the load could be sched uled o n so that the b inary-valued parameter T R it = 1 if the u ninterru ptible load i could be on in period t , and T R it = 0 o th erwise. Expressions (13) enforce the unin terruptible loads to b e scheduled off outside the pre- defined time interval, wh ereas constrain ts (14) enfor ce b inary variables δ ul it to b e 1 durin g the cycle time within th e p re- defined time interval. Constra in ts ( 1 5) im pose th at th e schedul- ing dur ing the cycle time must be consecu ti ve. Constraints (16)–(17) set the relatio nship be tween the schedu ling variable δ ul it and the activ ation variable δ a cit . Finally , expressions (18) define the p ower of the uninterrup tible load i in period t as the power in the corr esponding cycle p hase. I V . C A S E S T U DY The single-zone hou sehold presented in [8] is u sed to analyze what comf o rt settings an d structu ral attributes will help make it pr ice-responsive. Th e floor and wind ow areas are respectively 3 0 and 1 m 2 . For the water -based FH, the data are b ased on [8]. The mass of w ater in th e FH is 400 kg. Th e compressor has a coefficient of p erform ance (COP) of 3 and a nomin al power o f 1 kW . When the househ o ld is equippe d with an HV A C system, the COP for the heating and cooling system is resp ecti vely 1.67 an d 3.67 and its electric a l capacity is 1 kW . The household is eq uipped with a 30-l w ater heater located outside and a residential refrigerato r . Data can be fo und in T able I. In let water temper ature and the hot water daily consumption pr ofile is given in [1 8]. Artificial lighting cap acity is 60 W with an indoor lum inous efficac y eq ual to 90 lumen/W . W e assume that the lu minous efficac y of d aylight is 10 5 lumen /W in order to compute the outdoo r illuminance in lux and the comfo r t light levels are set to 10 0 and 10000 lu x. No te that the outdoor illum in ance T ABLE I D ATA F O R T H E R E F R I G E R A T O R [ 5 ] , [ 1 5 ] A N D W A T E R H E A T E R [ 6 ] Refrige rator W ater heater Nominal po wer capacit y [kW] 0.35 1.26 COP 0.76 0.92 Thermal ca pacit y [Wh/ ◦ C] 6.65 34.85 Heat t ransfer coef ficient [W/ ◦ C] 0.678 0.5 T ABLE II D ATA F O R T H E C O M F O RT C O N S T R A I N T S noflex flex extraflex x r,se t t [ ◦ C] 20 20 20 α [ ◦ C] 0 2 5 { x wh , x wh } [ ◦ C] { 54, 56 } { 50, 60 } { 45, 65 } { x r f , x r f } [ ◦ C] { 4.9, 5.1 } { 4, 5 } { 3, 6 } T i W ashing machine 06:00-14: 00 Dishwashe r (first) 06:00-14: 00 Dishwashe r (second) 16:00-00: 00 T umble dryer 15:00-00: 00 Oven 10:00-15: 00 and th e comfor t light le vels are respectively multiplied by the window and flo or ar ea to convert th em to lumen. W e conside r 4 u ninterrup tible load s: oven, washing ma- chine, tu m ble d ryer, and dishwasher . Their schedules are described in [1 8], their desirab le op erating h ours are g iven in T able II, and their p ower o n each cycle phase can b e foun d in [1 9] f or y ear 2 020. The discretization step ∆ t is assumed to b e 1 5 min to proper ly captu re th e building dynamics and the p e nalty terms ρ temp and ρ l are set to 1 000. W e ru n daily simulations with 15-min time steps for on e year . For ea c h simulation, we use a look-ahead window of one d a y to acc o unt for the future impacts of thermal dynamics when control action s are taken in a given period of time. Ambient temperature, solar radiation, and electricity p rices are gi ven in [18] for the sake of reprod ucibility . This refe r ence also in cludes the occup ancy schedules. No te th at the standb y power is neglected. Three different cases regarding th e d egree of h ousehold flexibility ar e analyzed : noflex , fl ex , and extrafle x cases. T able II also provides the data on setting comfor t b ounds for th e indoor air , refrigera tor , and water he ater temp erature. The simulations h av e been perfo rmed on a Linux- based server with one CPU clockin g at 2.6 GHz and 2 GB of RAM using CPLEX 12 .6.3 [20] und e r Pyomo 5.2 [2 1]. Optim ality gap is set to 0%. A. Effect of Comfort Settings T able III provides results for a single-zon e h ousehold under two types of space heatin g (FH and HV A C), three c a ses for the degree of flexibility ( nofl e x , flex , and e x traflex ), and the strategy followed to set the c o mfort b ounds (PI - CB and PD-CB). This table sh ows the annu al e le c tr icity cost, the total violations ( i. e., nu mber of Celsius degrees out of the correspo n ding comfort bounds over the 3504 0 time period s of the yea r ), the percen tage of 15- minute time intervals in which the in d oor air te m perature lies with in thr e e different ranges, T ABLE III R E S U L T S F O R A S I N G L E - Z O N E H O U S E H O L D T ype Comf ort Case Annual Vi olations Freq. at Fre q . [18,20)– Freq. [15,18)– Building u b u hp/heat bounds cost [ e ] [ ◦ C · h] 20 ◦ C [%] a (20,22] ◦ C [%] a (22,25] ◦ C [%] a cons. [kWh] [%] b [%] b FH PI-CB nofle x 103.5 862.5 37.5 62.5 0.0 1944.0 54.5 51.3 flex 78.9 0.0 1.5 98.5 0.0 1569.1 62.7 86.6 ext raflex 68.4 0.0 0.0 0.3 99.7 1372.7 63.5 85.8 PD-CB nofle x 103.5 862.5 37.5 62.5 0.0 1944.0 54.5 51.3 flex 93.2 55.9 22.2 77.8 0.0 1843.9 61.9 70.0 ext raflex 87.0 35.2 21.7 78.3 0.0 1751.3 64.1 73.1 HV A C PI-CB nofle x 107.0 0.0 100.0 0.0 0.0 2044.1 57.2 60.2 flex 89.9 0.0 1.1 98.9 0.0 1793.0 63.5 77.2 ext raflex 76.1 0.0 0.5 19.6 79.9 1538.8 64.7 80.1 PD-CB nofle x 107.0 0.0 100.0 0.0 0.0 2044.1 57.2 60.2 flex 94.5 0.0 11.1 88.9 0.0 1867.7 62.3 70.6 ext raflex 84.2 0.0 8.0 61.6 30.4 1691.2 64.4 73.9 a Columns 6–8 r epresent the pe rcentage of 15-minute time int ervals lying withi n the gi ven temperature inte rv als throughout the year . b Columns 10–11 rep resent the share of power lying within lo w-price periods. the building energy con su mption, and the share of building and heating p ower ly ing in low-price periods (i.e., those per iods where the annual normalized prices ar e lo wer than 0 .5). Note that we show the frequ e ncy of time in tervals in which th e indoor air temperature lies within: (i) 20.0 ◦ C (co lumn 6), ( ii) [18, 20)–(20, 2 2] ◦ C (co lumn 7), and (iii) [15, 18)– (22, 25] ◦ C (column 8). For all cases, computin g times for running annual simulations were in the rang e of 960–36 0 0 s. The HV A C system leads to more expensive solution s than the FH, but p rovides in gener al less discom fort assum ing that the referen c e set-poin t of 20 ◦ C is the tempera ture at which the occupan ts exper ience th e highest comfort. Note that for the n oflex case, the HV A C is able to keep the air tem perature within comfor t bounds (i.e. , at the refer e nce set-point), u nlike a household with FH (this is also due to the absence of air condition ing wh en considerin g FH). W e can no tice that the annual cost decreases when increasing the de gree o f flexibility at the exp ense of a higher degree of discom fort regardle ss o f the comf ort settings. Also, the co st reduction is higher und e r the PI-CB stra tegy when increasing the degree of flexibility (33.9% vs. 15 .9% for the extr afl ex case with respect to the noflex case with the FH). Similar conclusions can be drawn with the HV A C system. Howe ver the PD-CB strategy leads to higher co sts than under th e PI-CB when the occupants ar e flexible regardle ss o f the type of spa ce heater . The PD-CB stra tegy results in ti me-varying com fort b ounds throug hout the day , which is the reason wh y it leads to highe r costs than th e PI-CB. Howe ver, we can clearly observe that the PD-CB lead s to less discom fort (assum in g that the referen ce set-point of 20 ◦ C is the temperature at which the occu p ants experience the highest comfort). Note that the percen tage o f 15-min u te time intervals lying within [1 5, 18)–(2 2, 25] ◦ C is significantly reduced unde r the PD-CB comp ared to that percentag e under the PI-CB. Columns 10–11 in T ab le III provid e the sha r e of power (total in the building and that of the space h eating system) lying in low-price perio ds. This w o uld be an indicator wheth er a hou sehold is more or less price-re sponsive. Perc e ntage of total power (column 1 0) in low-price period s incr eases as th e T ABLE IV E F F E C T O F U A r,a O N A N N U A L C O S T A N D D I S C O M F O RT M E T R I C S T ype Cost and metrics Fac tor 0.5 1 2 4 FH Cost [ e ] 76.6 86 105.8 149.8 V iolatio ns [ ◦ C · h] 0 0 1.1 93.4 Freq. at 20 ◦ C [%] 0 0 2.9 4.2 Freq. [18 ,20)–(20,22] ◦ C [%] 100 100 97 95.4 Freq. [15 ,18)–(22,25] ◦ C [%] 0 0 0.1 0.2 HV A C Cost [ e ] 87 106.5 146 225 V iolatio ns [ ◦ C · h] 0 0 0 0 Freq. at 20 ◦ C [%] 0.9 0.6 0.3 0.2 Freq. [18 ,20)–(20,22] ◦ C [%] 99.1 99.4 99.7 99.8 Freq. [15 ,18)–(22,25] ◦ C [%] 0 0 0 0 occupan ts are more fle xible. This trend becomes even more noticeable when it comes to the h eating power since space heaters h av e a higher d egre e of control than other loads such as refr igerators, dishwashers, etc. Both spac e heating systems allow fo r sh if ting energy to low-price per iods when increasing the flexibility (around 7.2–7.5 % of d ifference b etween the share of building power for the extr afl ex and nofl ex cases with HV AC versus 9.0– 9.6% with FH). Although th ere are som e differences, bo th systems can be price-r esponsive when u sing a look-ahead win dow . B. Effect of Structural P arameters Structural parameters such as the heat transfer coef fi- cients 2 could b e also responsib le fo r the househo ld price - responsiveness. T able IV shows the effect of th e heat transfer coefficient between the roo m air and the ambient, U A r,a , fo r the flex case o n the ann ual cost, total n umber of viola tio ns of comfor t b ounds, and the per c entage of 15-minute time inter- vals in which the room tempera tu re lies within three different ranges, wherea s T able V shows th e share of b uilding power 3 lying within low-price per iods fo r different flexibility cases. In order to isolate the effect of the heat transfer coe fficient, we 2 The heat transfer coef ficient U A is the product of the heat conduct i vity U and the sur face area A where the heat tra nsfer tak es place. 3 Buildi ng power refer s to the total consumption of the smart household. T ABLE V E F F E C T O F U A r,a O N T H E S H A R E O F B U I L D I N G P OW E R C O N S U M P T I O N L YI N G W I T H I N L OW - P R I C E P E R I O D S [ % ] FH HV A C Fac tor noflex flex extraflex noflex flex extraflex 0.5 54.9 64.2 65.5 56.1 63.0 6 5.2 1.0 54.0 67.2 68.1 55.9 62.3 6 4.6 2.0 53.1 71.0 71.7 55.7 60.2 6 2.2 4.0 51.7 72.5 73.4 55.6 57.8 5 8.6 assume that the household does not ha ve any windo w and that the lighting constraints a r e ignored. The v a lu e of U A r,a has been multiplied b y f ac tors of 0.5, 1, 2, and 4 , respecti vely . The greater the v alu e of U A r,a , the less insulated the household is. It can b e ob served than when the h ousehold is less in sulated, the annu al costs increase because there are mo re thermal ex- changes through the walls at the exp ense of slightly incr easing the occup ants’ discomfo rt (see T able IV). Since this table shows the r esults fro m the flex case, most of the time periods are ab ove 1 8 ◦ C and below 22 ◦ C. The household with an FH is mo r e price-r e sp onsiv e when increasing the value o f U A r,a (see T able V). W e can observe that the d ifferences b etween the extrafle x and nofl ex cases vary between 10.6–21 .7% for the factors 0.5–4, r espectiv ely . Con versely , the hou seh old with an HV AC system is mor e price-respo nsiv e when decre a sing th e value of U A r,a . Th is is due to the fact that the HV AC system is an appliance with fast dynamics whereas the water-based FH is characterized by slow dynam ics. V . C O N C L U S I O N S W e p ropose a compact and integrated for mulation for a smart building with smart applian ces via econ omic mod el predictive contro l. W e mod el fiv e states to ca p ture therm al dynamics of space heaters, refrigerato r and water h eater; an d we accur ately model variable power cycles for uninterru ptible loads. Substantial cost sa v ings can be achie ved when increasing the co mfort boun ds under HV AC systems (21– 29%) and water- based floor h eating systems (1 6–34%) . Price-responsiveness of smart h ouseholds is quite similar betwe e n space heaters of slow (water - b ased FH) o r fast im p act (HV AC system) if a lo ok- ahead win d ow is used. The a m ount o f building consump tion lying within low-price perio ds increa ses from 54.5 –57.2% to 64 .1–64. 7% wh en using wid e r comfor t bou nds fo r the smart ap pliances. In g eneral, the mor e price-resp o nsiv e the househo ld is, the high er discom f ort the occu pants exper ien ce. Howe ver, price-d ependen t comfor t bound s co uld help reduce the o ccupants’ discom f ort, thus red u cing the nu m ber of time intervals at temp eratures far from the ref erence set-point. 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