How To Model A Powerhouse

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1 Opimal demand response: problem formulaion and deerminisic case Lijun Chen, Na Li, Libin Jiang, and Seven H. Low Absrac We consider a se of users served by a single load-serving eniy (LSE. The LSE procures capaciy a day ahead. When random renewable energy is realized a delivery ime, i manages user load hrough real-ime demand response and purchases balancing power on he spo marke o mee he aggregae demand. Hence opimal supply procuremen by he LSE and he consumpion decisions by he users mus be coordinaed over wo imescales, a day ahead and in real ime, in he presence of supply uncerainy. Moreover, hey mus be compued joinly by he LSE and he users since he necessary informaion is disribued among hem. In his paper we presen a simple ye versaile user model and formulae he problem as a dynamic program ha maximizes expeced social welfare. When random renewable generaion is absen, opimal demand response reduces o join scheduling of he procuremen and consumpion decisions. In his case, we show ha opimal prices exis ha coordinae individual user decisions o maximize social welfare, and presen a decenralized algorihm o opimally schedule a day in advance he LSE s procuremen and he users consumpions. The case wih uncerain supply is repored in a companion paper. 1 Inroducion 1.1 Moivaion There is a large lieraure on various forms of load side managemen from he classical direc load conrol o he more recen real-ime pricing [1, 2]. Direc load conrol in paricular has been pracised for a long ime and opimizaion mehods have been Lijun Chen, Na Li, Libin Jiang and Seven H. Low Engineering and Applied Science, California Insiue of Technology, USA {chenlj, nali, libinj, slow}@calech.edu 1

2 2 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low proposed o minimize generaion cos e.g. [3, 4, 5, 6], maximize uiliy s profi e.g. [7], or minimize deviaion from users desired consumpions e.g. [8, 9], someimes inegraed wih uni commimen and economic dispach e.g. [4, 10]. Almos all demand response programs oday arge large indusrial or commercial users, or, in he case of residenial users, a small number of hem, for wo, among oher, imporan reasons. Firs demand side managemen is invoked rarely o mosly cope wih a large correlaed demand spike due o weaher or a supply shorfall due o fauls, e.g., during a few hoes days in summer. Second he lack of ubiquious wo-way communicaion in he curren infrasrucure prevens he paricipaion of a large number of diverse users wih heerogeneous and ime-varying consumpion requiremens. Boh reasons favor a simple and saic mechanism involving a few large users ha is sufficien o deal wih he occasional need for load conrol, bu boh reasons are changing. Renewable sources can flucuae rapidly and by large amouns. As heir peneraion coninues o grow, he need for regulaion services and operaing reserves will increase, e.g., [11, 12]. This can be provided by addiional peaker unis, a a higher cos, or supplemened by real-ime demand response [13, 14, 15, 12, 16]. We believe ha demand response will no only be invoked o shave peaks and shif load for economic benefis, bu will increasingly be called upon o improve securiy and reduce reserves by adaping elasic loads o inermien and random renewable generaion [17]. Indeed, [12, 18, 19] advocaes he creaion of a disribuion/reail marke o encourage greaer load side paricipaion as an alernaive source for fas reserves. Such applicaion however will require a much faser and more dynamic demand response han pracised oday. This will be enabled in he coming decades by he large-scale deploymen of a sensing, conrol, and wo-way communicaion infrasrucure, including he flexible AC ransmission sysems, he GPS-synchronized phasor measuremen unis, and he advanced meering infrasrucure, ha is currenly underway around he world [20]. Demand response in such conex mus allow he paricipaion of a large number of users, and be dynamic and disribued. Dynamic adapaion by hundreds of millions of end users on a sub-second conrol imescale, each conribuing a iny fracion of he overall raffic, is being pracised everyday on he Inerne in he form of congesion conrol. Even hough boh he grid and he Inerne are massive disribued nonlinear feedback conrol sysems, here are imporan differences in heir engineering, economic, and regulaory srucures. Noneheless he precedence on he Inerne lends hope o a much bigger scale and more dynamic and disribued demand response archiecure and is benefi o grid operaion. Ulimaely i will be cheaper o use phoons han elecrons o deal wih a power shorage. Our goal is o design algorihms for such a sysem.

3 Opimal demand response: problem formulaion and deerminisic case Summary Specifically we consider a se of users ha are served by a single load-serving eniy (LSE. The LSE may represen a regulaed monopoly like mos uiliy companies in he Unied Saes oday, or a non-profi cooperaive ha serves a communiy of end users. Is purpose is (possibly regulaed o promoe he overall sysem welfare. The LSE purchases elecriciy on he wholesale elecriciy markes (e.g., day-ahead, real-ime balancing, and ancillary services and sells i on he reail marke o end users. I provides wo imporan values: i aggregaes loads so ha he wholesale markes can operae efficienly, and i hides he complexiy and uncerainy from he users, in erms of boh power reliabiliy and prices. Our model capures hree imporan feaures: Uncerainy. Par of he elecriciy supply is from renewable sources such as wind and solar, and hus uncerain. Supply and demand. LSE s supply decisions and he users consumpion decisions mus be joinly opimized. Two imescale. The LSE mus procure capaciy on he day-ahead wholesale marke while user consumpions should be adaped in real ime o miigae supply uncerainy. Hence he key is he coordinaion of day-ahead procuremen and real-ime demand response over wo imescales in he presence of supply uncerainy. Moreover, he opimal decisions mus be compued joinly by he LSE and he users as he necessary informaion is disribued among hem. The goal of his paper is o formulae his problem precisely. Due o space limiaion, we can only fully rea he case wihou supply uncerainy. Resuls for he case wih supply uncerainy are summarized here, bu fully developed in a companion paper [21]. Suppose each user has a se of appliances (elecric vehicle, air condiioner, lighing, baery, ec.. She (or her energy managemen sysem is o decide how much power she should consume in each period = 1,...,T of a day. The LSE needs o decide how much capaciy i should procure a day ahead and, when he random renewable energy is realized a real ime, how much balancing power o purchase on he spo marke o mee he aggregae demand. In Secion 2, we presen our user and supply models, and formulae he overall problem as an (1+T -period dynamic program o maximize expeced social welfare. The key idea is o regard he LSE s day-ahead decision as he conrol in period 0 and he users consumpion decisions as conrols in he subsequen periods = 1,...,T. By unifying several models in he lieraure, our user model incorporaes a large class of appliances. Ye, i is simple, hus analyically racable, where each appliance is characerized by a uiliy funcion and a se of linear consumpion consrains. In Secion 3, we consider he case wihou renewable generaion. In he absence of uncerainy i becomes unnecessary o adap user consumpions in real-ime and hence supply and consumpions can be opimally scheduled a once insead of over wo days. We show ha opimal prices exis ha coordinae individual users decisions in a disribued manner, i.e., when users selfishly maximize heir own surplus

4 4 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low under he opimal prices, heir consumpion decisions urn ou o also maximize he social welfare. We develop an offline disribued algorihm ha joinly schedules he LSE s procuremen decisions and he users consumpion decisions for each period in he following day. The algorihm is decenralized where he LSE only knows he aggregae demand bu no user uiliy funcions or consumpion consrains, and he users do no need o coordinae among hemselves bu only respond o common prices from he LSE. Wih renewable generaion, he uncerainy precludes pure scheduling and calls for real-ime consumpions decisions ha adap o he realizaion of he random renewable generaion. Moreover, his mus be coordinaed wih procuremen decisions over wo imescales o maximize he expeced welfare. Disribued algorihms for opimal demand response in his case and he impac of uncerainy on he opimal welfare are developed in he companion paper [21] We make wo remarks. Firs he effeciveness of real-ime pricing for demand response is sill in acive research. On he one hand, empirical sudies have shown consisenly ha price elasiciy is low and heerogeneous; see [22, 23, 24] and references herein. On he oher hand, here are srong economic argumens ha real-ime reail prices improve he efficiency of he overall sysem by allowing users o dynamically adap heir loads o shorages, wih poenial benefis far exceeding he cos of implemenaion [18]. Moreover, he long-run efficiency gain is likely o be significan even if demand elasiciy is small, bu unforunaely, he popular open-loop ime-of-use pricing may capure a very small share of he efficiency gain of realime pricing [25]. We neiher argue for nor agains real-ime pricing. Indeed we do no consider in his paper he economic issues associaed wih such a sysem, such as locaional marginal prices, revenue-adequacy, ec. Wha we refer o as prices are simply conrol signals ha provide he necessary informaion for users o adap heir consumpion in a disribued, ye opimal, manner. Wheher his conrol signal should be linked o moneary paymens o provide he righ incenive for demand response is beyond he scope of his paper, i.e., we do no address he imporan issue of how o incenivize users o respond o supply and demand flucuaions. 1 Second, unlike many curren sysems, he kind of large-scale disribued demand response sysem envisioned here mus be fully auomaed. Human users se parameers ha specify uiliy funcions and consumpion consrains and may change hem on a slow imescale, bu he algorihms proposed here will execue auomaically and ransparenly o opimize social welfare. The radiional direc load conrol approach assumes ha he conroller (e.g. a uiliy company knows he user consumpion requiremens, in he form of payback characerisics of he deferred load, and can opimally schedule deferred consumpions and heir paybacks cenrally. This is reasonable for he curren sysem where he paricipaing users are few and heir requiremens are relaively saic. We ake he view ha he uiliies and requiremens of user consumpions are diverse and privae. I is no pracical, nor necessary, o have direc access o such informaion in order o opimally coordinae heir consumpions in a large, disribued, and dynamic sysem of he fuure. The algorihm 1 See however [19] for a discussion on some implemenaion issues of real-ime pricing for reail markes and a proposal for he Ialian marke.

5 Opimal demand response: problem formulaion and deerminisic case 5 presened here is an example ha can achieve opimaliy wihou requiring users o disclose heir privae informaion. 1.3 Oher relaed work A large lieraure exiss on demand response. Besides hose cied above, more recen works include, e.g., [26, 27] on load conrol of hermal mass in buildings, [28, 29, 30] on residenial load conrol hrough coordinaed scheduling of differen appliances, [31, 32, 33] on he scheduling of plug-in elecric vehicle charging, and [34] on he opimal allocaion of a supply defici (raioning among users using heir supply funcions. Load side managemen in he presence of uncerain supply has also been considered in [16, 10, 35, 36, 12, 37]. Unlike he convenional approach ha compensaes for he uncerainy o creae reliable power, [16] advocaes selling inerrupible power and designs service conracs, based on [38], ha can achieve greaer efficiency han he convenional approach. In [10] various opimizaion problems are formulaed ha inegrae demand response wih economic dispach wih ramping consrains and forecass of renewable power and load. Boh cenralized dispach using model predicive conrol and decenralized dispach using prices, or supply and demand funcions, are considered. A wo-period sochasic dispach model is sudied in [35] and a selemen scheme is proposed ha is revenue-adequae even in he presence of uncerain supply and demand. A queueing model is analyzed in [36] where he queue holds deferrable loads ha arise from random supply and demand processes. Convenional generaion can be purchased o keep he queue small and sraegies are sudied o minimize he ime-average cos. The models ha are closes o ours, developed independenly, are [12, 37]. All our models include random renewable generaion, consider boh day-ahead and realime markes, and allow demand response, bu our objecives and sysem operaions are quie differen. [12] advocaes he esablishmen of a reail marke where users (e.g., PHEVs can buy power from or sell reserves, in he form of demand response capabiliy, o heir LSE. The paper formulaes he LSE s and users problems as dynamic programs ha minimize heir expeced coss over heir bids, which can be eiher simple, uncorrelaed (price, quaniy pairs for each period, or complex, (price, quaniy pairs wih emporal correlaions. The model in [37] includes nonelasic users ha are price non-responsive, and elasic users ha can eiher leave he sysem or defer heir consumpions when he elecriciy price is high. The goal is o maximize LSE s profi over day-ahead procuremen, day-ahead prices for nonelasic users, and real-ime prices for elasic users.

6 6 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low 1.4 Noaions Given quaniies such as demands q ia ( from appliance a of user i in period, q ia := (q ia (, T denoes he vecor of demands a differen imes, q i ( := (q ia (,a A i he vecor of demands of differen appliances, q i := (q ia,a A i he vecor of demands of i s appliances a differen imes, and q := (q i, i he vecor of all demands. Similarly for aggregae demands Q i ( = a Ai q ia (, Q ia := q ia (, Q i, Q, ec. Scrip leers denoe ses, e.g., N,A i,t. Small leers denoe individual quaniies, e.g., q ia (, q ia, q i (, q i, q, ec. Capial leers denoe aggregae quaniies, e.g., Q i (, Q ia, P d (,P r (,P o (,P b (, ec. We use q ia (,q ia,q i (, ec for loads and P d (,P r (, ec for supplies. We someimes wrie i a Ai q ia ( as i,a q ia (. For any real a,b,c, [a] + := max{a,0} and [a] c b := max{b,min{a,c}}. Finally, we wrie a vecor as x = (x i, i wihou specifying wheher i is a column or row vecor so we can ignore he ranspose sign o simplify he noaion; he meaning should be clear from he conex. 2 Model and problem formulaion Consider a se N of N users ha are served by a single load-serving eniy (LSE. We use a discree-ime model wih a finie horizon ha models a day. Each day is divided ino T periods of equal duraion, indexed by T := {1,2,,T }. The duraion of a period can be 5, 15, or 60 mins, corresponding o he ime resoluion a which energy dispach or demand response decisions are made. 2.1 User model Each user i N operaes a se A i of appliances such as HVAC (hea, venilaion, air condiioner, refrigeraor, and plug-in hybrid elecric vehicle. User i may also possess a baery which provides furher flexibiliy for opimizing is elecriciy consumpion across ime. Appliance model. For each appliance a A i of user i, q ia ( denoes is energy consumpion in period T, and q ia he vecor (q ia (, over he whole day. An appliance a is characerized by: a uiliy funcion U ia (q ia ha quanifies he uiliy user i obains from using appliance a; a K ia T marix A ia and a K ia -vecor η ia such ha he vecor of power q ia saisfies he linear inequaliy A ia q ia η ia. (1 In general U ia depends on he vecor q ia. In his paper, however, we consider four ypes of appliances whose uiliy funcions ake one of hree simple forms. These

7 Opimal demand response: problem formulaion and deerminisic case 7 models are summarized in Table 1 and jusified in deail in he Appendix. The uiliy of a ype 1 or ype 2 appliance is addiive in : 2 U ia (q ia := U ia (q ia (,. (2 The uiliy of a ype 3 appliance depends only on he aggregae consumpion: U ia (q ia := U ia ( q ia (. (3 The uiliy of a ype 4 appliance depends on he inernal emperaure and power consumpions in he pas. I is of he form: U iq (q ia := U ia (T ia ( + β (1 α τ q ia (τ (4 where T ia ( is a given sequence of emperaures defined in equaion (29 in he Appendix and α,β are given hermal consans. All uiliy funcions are assumed o be coninuously differeniable and concave funcions for each. For example, some of our simulaions in [39, 21] use he following ime independen and addiive uiliy funcion of form (2: le y ia ( be a desired energy consumpion by appliance a in period ; hen he funcion τ=1 U ia (q ia (, := U ia (q ia ( := (q ia ( y ia ( 2 (5 measures he uiliy of following he desired consumpion profile y ia (. Such uiliy funcions minimize user discomfor as advocaed in [8, 9]. Table 1: Srucure of uiliy funcions and consumpion consrains for appliances. Appliances Uiliy funcion Consumpion consrains Examples Type 1 (2 (6 Lighings Type 2 (2 (6, (7 TV, video game, compuer Type 3 (3 (6, (7 PHEV, washers Type 4 (4 (6, (8 HVAC, refrigeraor Baery D i (r i (6, (7 r i = q ia for baery a The consumpion consrains (1 for hese appliances ake hree paricular forms. Firs, for all appliances, he (real power consumpion mus lie beween a lower and an upper bound, possibly ime-dependen: q ia ( q ia ( q ia (. (6 2 We abuse noaion o use U ia o denoe boh a funcion of vecor q ia and ha of a scalar q ia (; he meaning should be clear from he conex.

8 8 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low An imporan characer of an appliance is is allowable ime of operaion; e.g., an EV can be charged only beween 9pm and 6am, TV may be on only beween 7 9am and 6 12pm. If an appliance operaes only in a subse T ia T of periods, we require ha q ia ( = q ia ( = 0 for T ia and U ia (0 = 0. We herefore do no specify T ia explicily in he descripion of uiliy funcions and always sum over all T. The second kind of consrain specifies he range in which he aggregae consumpion mus lie: Q ia q ia ( Q ia. (7 The las kind of consrain is slighly more general (see derivaion in he Appendix: η ia A ia q ia η ia. (8 Baery model. We denoe by B i he baery capaciy, by b i ( he sae of charge in period, and by r i ( he power (energy per period charged o (when r i ( 0 or discharged from (when r i ( < 0 he baery in period. We use a simplified model of baery ha ignores power leakage and oher inefficiencies, where he sae of charge is given by b i ( = r i (τ + b i (0. (9 τ=1 The baery has an upper bound on charge rae, denoed by r i, and an upper bound on discharge rae, denoed by r i. We hus have he following consrains on b i ( and r i (: 0 b i ( B i, r i r i ( r i. (10 We assume any baery discharge is consumed by oher appliances (zero leakage, and hence i canno be more han wha he appliances need: r i ( a A i q ia (. (11 Finally, we impose a minimum on he energy level a he end of he conrol horizon: b(t γ i B i where γ i [0,1]. The cos of operaing he baery is modeled by a funcion D i (r i ha depends on he vecor of charged/discharged power r i := (r i (,. This cos may correspond o he amorized purchase and mainenance cos of he baery over is lifeime, and depends on how fas/much/ofen i is charged and discharged; see an example D i (r i in [39]. The cos funcion D i is assumed o be a convex funcion of he vecor r i. Noe ha in his model, a baery is equivalen o an appliance: is uiliy funcion is D i (r i and is consumpion consrains (9, (10, and b(t γ i B i are of he same form as (6 (7 wih q ia = r i. Therefore a baery can be specified simply as anoher appliance, in which case he consrain (11 requires ha i s aggregae demand be nonnegaive, a Ai q ia ( + r i ( 0. This is summarized in Table 1. Henceforh,

9 Opimal demand response: problem formulaion and deerminisic case 9 we will ofen use appliances o also include baery and may no refer o baery explicily when his does no cause confusion. 2.2 Supply model We now describe a simple model of he elecriciy markes. The LSE procures power for delivery in each period, in wo seps. Firs i procures day-ahead capaciies P d ( for each period a day in advance and pays for he capaciy coss c d (P d (;. The renewable power in each period is a nonnegaive random variable P r ( and i coss c r (P r (;. I is desirable o use as much renewable power as possible; for noaional simpliciy only, we assume c r (P; 0 for all P 0 and all. Then a ime (real ime, he random variable P r ( is realized and used o saisfy demand. The LSE saisfies any excess demand by some or all of he day-ahead capaciy P d ( procured in advance and/or by purchasing balancing power on he real-ime marke. Le P o ( denoe he amoun of he day-ahead power ha he LSE acually uses and c o (P o (; is cos. Le P b ( be he real-ime balancing power and c b (P b (; is cos. These real-ime decisions (P o (,P b ( are made by he LSE so as o minimize is oal cos, as follows. Given he demand vecor q( := (q ia (,a A i, i, le Q( := i,a q ia ( be he oal demand and (Q( := Q( P r ( he excess demand, in excess of he renewable generaion P r (. Noe ha (Q( is a random variable in and before period 1, bu is realizaion is known o he LSE a ime. Given excess demand (Q( and day-ahead capaciy P d (, he LSE chooses (P o (,P b ( ha minimizes is oal real-ime cos, i.e., i chooses (P o (,P b ( ha solves he problem: c s ( (Q(,P d (; := min { c o(p o (; + c b (P b (; P b ( 0, P o (,P b ( P o ( + P b ( (Q(, P d ( P o ( 0}. (12 Clearly Po ( + Pb ( = (Q( unless (Q( < 0. The oal cos is c(q(,p d (;P r (, := c d (P d (; + c s ( (Q(,P d (;. (13 wih (Q( := Q( P r (. We assume ha, for each, c d ( ;, c o ( ; and c b ( ; are increasing, convex, and coninuously differeniable wih c d (0; = c o (0; = c b (0; = 0. Example: supply cos Suppose c b (0 > c o(p, P 0, i.e., he marginal cos of balancing power is sricly higher han he marginal cos of day-ahead power, he LSE will use he balancing power only afer he day-ahead power is exhaused, i.e., P b ( > 0 only if (Q( > P d (. The soluion c s ( (Q(,P d (; of (12 in his case is paricularly simple and (13 can be wrien explicily in erms of c b,c o,c b :

10 10 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low c(q(,p d (;P r (, = c d (P d (; + ( c o [ (Q(] P ( d( 0 ; + c b [ (Q( Pd (] + ;. (14 i.e., he oal cos consiss of he capaciy cos c d and he energy cos c o of day-ahead power, and he cos c b of he real-ime balancing power. 2.3 Problem formulaion: welfare maximizaion Recall ha q := (q(, T and Q( := i,a q ia (. The social welfare is he sandard user uiliy minus supply cos: W(q,P d ;P r := U ia (q ia i,a T =1 c(q(,p d (;P r (,. (15 As menioned above he LSE s objecive is no o maximize is profi hrough selling elecriciy, bu raher o maximize he expeced social welfare. Given he day-ahead decision P d, he real-ime procuremen (P o (,P b ( is deermined by he simple opimizaion (13. This is mos ransparen in (14 for he special case: he opimal decision is o use day-ahead power P o ( o saisfy any excess demand (Q( up o P d (, and hen purchase real-ime balancing power P b ( = [ (Q( P d(] + if necessary. Hence he maximizaion of (15 reduces o opimizing over day-ahead procuremen P d and real-ime consumpion q in he presence of random renewable generaion P r (. I is herefore criical ha, in he presence of uncerainy, q( should be decided afer P r ( have been realized a imes. P d however mus be decided a day ahead before P r ( are realized. The radiional dynamic programming model requires ha he objecive funcion be separable in ime. The welfare funcion in (15 is no as he firs erm U ia (q ia depends on he enire conrol sequence q ia = (q ia (,. So does he consumpion consrain (1. We now inroduce an equivalen sae space formulaion of ha will allow us o sae precisely he overall opimizaion problem as an (1 + T -period dynamic program. Consider a dynamical sysem over an exended ime horizon = 0, 1,..., T. The conrol inpus are he LSE s day-ahead decision P d := (P d (, in period 0 and he user s decisions q( in each subsequen period. Le v( denoe he inpus, i.e., v(0 = P d and v( = q(, = 1,...,T. Noe ha v(0 R T + whereas q( R M where M := N i=1 A i. The sysem sae x( := ( x 1 (,x 2 ia (,x3 (,x 4 ia (, a A i, i has four componens, defined as follows: Wihou loss of generaliy, x(0 sars from he origin. x 1 ( R T keeps rack of he day-ahead decisions P d : for each = 1,...,T, x 1 ( = P d = (P d (τ,τ = 1,...,T.

11 Opimal demand response: problem formulaion and deerminisic case 11 xia 2 ( Rk ia of appropriae dimension k ia for each (i,a pair keeps rack of he consumpion consrain (1. The sae definiion and is ransiion are problem specific; see a concree example in Secion 2.4. x 3 ( R + keeps rack of he random renewable power x 3 (0 = 0, x 3 ( = P r (, = 1,...,T. The purpose of his sae definiion is merely noaional, so ha he conrol policy can depend on he realizaion of he random renewable power P r ( hrough is dependence on sae x 3 (. xia 4 ( RT 1 for each (i,a pair racks he user decisions v ia ( 1 = q ia ( 1 in he previous period: xia 4 (1 = 0 T 1, he T 1 dimensional zero vecor; for each = 2,...,T, he ( 1h componen [xia 4 (] 1 of xia 4 ( is se o be he inpu v ia ( 1 and all he oher componens [xia 4 (] τ of xia 4 ( remain he same as hose of xia 4 ( 1, so ha he final sae x4 ia (T is he vecor (q ia(, = 1,...,T 1 of inpus up o period T 1. The firs erm in (15 is hen a funcion of he sae and inpu in period T, U ia (q ia = U ia (xia 4 (T,v ia(t. This allows us o rewrie he welfare funcion in (15 in a form ha is separable in ; see below. The above discussion is summarized by a ime-varying sae ransiion funcion f : x( + 1 = f (x(,v(,p r ( + 1, = 0,...,T i.e., he new sae x( + 1 depends on he curren sae x(, he inpu v(, and he new random variable P r (, and is herefore random. The consumpion consrains (1, which may include he baery consrains, generally ranslae ino consrains on he sae x 2 ( and inpu v( and we represen his by x( X ( and v( V ( R M, M := N i=1 A i. Someimes hese consrains also give rise o a erminal reward ha we denoe by W T +1 (x(t + 1. Consider he class of feedback conrol laws v( = φ (x(, where φ 0 : X (0 R T + specifies he day-ahead decision P d and φ : X ( V ( specifies he user decisions q( for each period = 1,...,T. Hence he conrol v( depends only on he curren sae x(. Under he conrol law φ := (φ, = 0,...,T, he sae evolves (sochasically according o x( + 1 = f (x(,φ (x(,p r ( + 1. (16 We emphasize ha x( is obained under policy φ even hough his may no be explici in he noaion. To make he welfare funcion in (15 separable in, use (13 o define he welfare in each period, under he conrol law φ, as a funcion of he curren sae x( and he curren inpu v( = φ (x(: W φ := W φ (x(,v( T τ=1 ( c d ([v(0] τ ;τ, = 0 := c s (Q φ (,[x 1 (] ;, 1 < T i,a U ia ((xia 4 (T,v ( ia(t c s (Q φ (T,[x 1 (T ] T ;T, = T (17

12 12 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low where Q φ ( = i,a [v(] ia is he aggregae demand in period under φ, and v ia (T = q ia (T are he real-ime consumpion decisions in he las conrol period T. Then he welfare funcion in (15 is equivalen o J φ := T =0 W φ (x(,v( +W φ T +1 (x(t + 1 where he definiion of he erminal reward W φ T +1 (x(t + 1 is problem specific. We can now sae precisely our objecive as he consrained maximizaion of he expeced welfare over he conrol law φ: max φ E J φ = E ( T W φ +W φ T +1 =0 where he expecaion is aken over P r (, = 1,...,T. s.. x φ ( X (. (18 Remark. An imporan assumpion in his formulaion is ha he consumpion consrains (1 can be modeled by an appropriae definiion of saes xia 2 (, heir ransiions f, he consrain ses X (,V (, and possibly a erminal reward W T +1 (x(t + 1. We now illusrae he problem formulaion using a concree example. 2.4 Example To simplify he noaion we make wo assumpions ha do no cause any loss of generaliy. Firs we use he oal cos funcion c in (14 in he definiion of he welfare funcion (15. Second we assume each user i has a single ype-2 appliance and no baery (so we drop he subscrip a. From Table 1, user uiliy funcions are addiive in ime, U i (q i = U i (q i (; and he consumpion consrains are q i ( q i ( q i (, i (19 Q i T =1 q i(. (20 Since he uiliy funcions are separable in, we do no need o define x 4 (. We now describe he (1 + T -period dynamic program by specifying he definiion of x 2 (, he sae ransiion funcion f, and he consrain ses X (,V (. The sysem sae x( := (x 1 (,x 2 (,x 3 ( consiss of hree componens of appropriae dimensions wih x( = (P d,x 2 (,P r (, = 1,...,T where x 2 ( is deermined by he consrain (20. Define x 2 i ( o be he remaining demand of user i a he beginning of each period : x 2 i (1 = Q i, and for each =

13 Opimal demand response: problem formulaion and deerminisic case 13 1,...,T, xi 2( +1 = x2 i ( v i( where v i ( = q i (. To enforce ha x 2 (T +1 0, we define he erminal cos c T +1 (x(t + 1 = 0 if x 2 (T N and c T +1 (x(t + 1 = oherwise, where 0 n is he n-dimensional zero vecor. Le he iniial sae be x(0 = 0 T +N+1. Denoe Q := (Q i, i. The sysem dynamics is hen linear imevarying: ( 0 I T x(1 = x(0 + T v(0 + Q 0 (N+1 T P r (1 ( IT x( + 1 = +N 0 T +N x( 0 T N ( 0T I 0 T +N 0 N v( + +N P 1 r ( + 1, 1 T 0 where I n is he n n idenify marix, 0 m n he m n zero marix, and P r (T +1 := 0. The welfare in each period, under inpu sequence v, is (using (14 and for = 1,...,T, W v 0 (x(0,v(0 := T τ=1c d (P d (τ;τ = W v (x(,v( ( := U i (q i (; c o [Q( P r (] P d( 0 ; i ( [1v( = U i (v i (; c o x 3 ( ] [x 1 (] ; 0 i T τ=1 c d ([v(0] τ ;τ c b ( [Q( Pr ( P d (] + ; c b ( [1v( x 3 ( [x 1 (] ] + ; where 1 is he (row vecor of 1 s. The consrain (19 yields he inpu consrain ses V (0 := R T + and, for = 1,...,T, V ( := {q( R N q( q( q(}. There is no consrain on he sae, i.e., X ( = R T +N+1. Le φ := {φ 0 : R T +N+1 R T +, φ : R T +N+1 V (, = 1,...,T } be he conrol policy so ha v( = φ (x(, 0 T. Then he welfare maximizaion problem (18 is ( max φ E W φ T 0 (x(0,v(0 + W φ (x(,v( c T +1 (x(t + 1 =1 where he sae x( and he inpu v( are obained under policy φ. In [21] we sudy he problem (21 in deail. We propose a disribued heurisic algorihm o solve he (1 + T -period dynamic program. We prove ha he algorihm is opimal when he welfare is quadraic and he LSEs procuremen decisions are sricly posiive. Oherwise, we bound he gap beween he welfare achieved by he heurisic algorihm and he maximum. Simulaion resuls sugges ha he performance of he heurisic algorihm is very close o opimal. As we scale up he size of a renewable generaion plan, boh is mean producion and is variance will likely (21

14 14 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low increase. As expeced, he maximum welfare increases wih he mean producion, when he variance is fixed, and decreases wih he variance, when he mean is fixed. More ineresing, we prove ha as we scale he size of he plan up, he maximum welfare increases. 3 Opimal scheduling wihou supply uncerainy In his paper we only fully rea he case where here is no supply uncerainy, i.e., P r ( 0. Our goal is o opimally coordinae supply and demand o maximize social welfare. In he absence of uncerainy (our model also ignores demand uncerainy, i becomes unnecessary o adap user consumpions in real-ime and hence supply and consumpions can be opimally scheduled a once insead of over wo days. Welfare maximizaion (18 hen akes a simpler form and we develop an offline disribued algorihm ha joinly opimizes he LSE s procuremens and he users consumpions for each period in he following day. 3.1 Opimal procuremens and consumpions We firs consider LSE s procuremen decisions. Recall ha Q i ( := a Ai q ia ( and i Q i ( is he aggregae demand in period. Wih supply uncerainy, while P d is decided a day ahead, he opimizaion (12 mus be carried ou in real ime afer P r ( has been realized o obain opimal P o (,P b (. Here, on he oher hand, all hree decisions (P d (,P o (,P b ( can be compued in advance in he absence of uncerainy. Hence, given an aggregae demand i Q i (, he LSE solves (insead of (12 (13: ( c Q i (; i := min P d (,P o (,P b ( c d(p d (; + c o (P o (; + c b (P b (; (22 s.. P o ( + P b ( Q i (, P d ( P o ( 0, P b ( 0 i o obain he oal cos. The soluion of (22 specifies he opimal decisions (P d (,P o (,P b ( o saisfy he aggregae demand i Q i ( for each period in he following day. I is no difficul o show ha c(, is an non-decreasing, convex, and coninuously differeniable funcion for each, so he problem (22 is convex. Since c d (P; > 0, he KKT condiion implies ha P d ( = P o ( a opimaliy, i.e., i is opimal o exhaus all he day-ahead capaciy. This is always possible because all procuremen decisions are compued joinly wihou uncerainy. If we furher assume ha he marginal cos of he balancing power is higher han ha of he dayahead power, i.e., c b (0; > c d (P; + c o(p; for all P 0, hen KKT implies ha

15 Opimal demand response: problem formulaion and deerminisic case 15 i will never pay o use balancing power, i.e., Pb ( = 0 a opimaliy. In his case, Pd ( = P o ( = i Q i (. Hence welfare maximizaion reduces o he compuaion of he user consumpions q ia (; he corresponding procuremen decisions are hen given by (22. The opimizaion of he social welfare in (15 hen becomes: ( max U ia (q ia c Q i (; (23 q i,a i s.. A ia q ia η ia, a A i, i, (24 0 Q i ( Q i, i (25 The inequaliies in (24 are he consumpion consrains (1 of user i s appliances and baery. The lower inequaliy in (25 is he same as (11; see he discussion a he end of Secion 2.1 on baery consrains. The upper inequaliy in (25 imposes a bound on he oal power drawn by user i. By assumpion, he objecive funcion is concave and he feasible se is convex. Hence an opimal poin can in principle be compued offline cenrally by he LSE. This however will require ha he LSE know all he users uiliy and baery cos funcions and all he consrains, which is impracical for echnical or privacy reasons. The objecive funcion in (23 and he consrains (24 (25 can be decomposed ino subproblems ha are solvable in a decenralized manner where he LSE only needs o know he aggregae demand bu no he individual privae informaion. The key idea is for he LSE o se prices π := (π(, o induce he users o individually choose socially opimal consumpions q i := (q ia (, in response. Indeed, given prices π, we assume ha each user i chooses is own demand q i so as o maximize is ne benefi, her oal uiliy minus he elecriciy cos, i.e., each user i solves: max q i U ia (q ia a A i π(q i ( s.. (24 (25. (26 Given prices π, we denoe an opimal soluion of (26 and he corresponding aggregae demand by ( q i (π := (q ia (;π,, a A i, Q i (π := (Q i (;π, := q i,a (;π,. a A i Recall q(π := (q i (π, i. I is a remarkable fac in economics ha here exis prices π ha align he users objecives and he LSE s objecive of maximizing welfare, i.e., here are prices π such ha if q i (π opimize i s objecives for all users i hen hey also opimize he social welfare. Definiion 1. A consumpion vecor q is called opimal if i solves (23 (25. A price vecor π is called opimal if q(π is opimal, i.e., any soluion q(π of (26 also solves (23 (25.

16 16 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low The following resul follows from he welfare heorem in economics. I implies ha seing he prices o he marginal coss of power is opimal. Theorem 1. The prices ha saisfy π ( := c ( i Q i (;π ; 0 are opimal. Proof. Wrie he welfare maximizaion problem as ( max U ia (q ia q i Q i,y i c Y i (; i i,a s.. Y i ( = a A i q ia (, i, where he feasible se Q i is defined by he consrains (24 (25. Clearly, an opimal soluion q exiss. Moreover, here exis Lagrange mulipliers πi (, i,, such ha (aking derivaive wih respec o Y i ( πi ( = c ( Yi (; = c ( q ia(; 0. i i a A i Since he righ-hand side is independen of i, he LSE can se he prices as π ( := πi ( 0 for all i. One can check ha he KKT condiion for he welfare maximizaion problem are idenical o he KKT condiions for he collecion of users problems. Since all hese problems are convex, he KKT condiions are boh necessary and sufficien for opimaliy. This proves he heorem. 3.2 Offline disribued scheduling algorihm Theorem 1 moivaes a disribued algorihm o compue he opimal prices π and user decisions q(π. The LSE ses prices o be he marginal coss of power and each user solves is own maximizaion problem (26 in response. The model is ha a he beginning of each day he LSE and (he energy managemen sysems of he users ieraively compue he elecriciy prices π( and consumpions q i ( for each period of he following day. These decisions are hen carried ou for ha day. This is an offline algorihm since all decisions are made a once before he day sars. I is decenralized where he LSE only knows he aggregae demand bu no user uiliy funcions or consumpion consrains and he users do no need o coordinae among hemselves bu only respond o common prices. Algorihm 1: Opimal scheduling wihou supply uncerainy For each ieraion k = 1,2,..., afer iniializaion: 1. The LSE collecs aggregae demand forecass, denoed by (Q k i (,, from all users i over a communicaion nework. I updaes he prices o he marginal coss π k+1 ( := c ( i Q k i (; and broadcass (π k+1 (, o all users. 2. Each user i updaes is demands q k+1 i afer receiving π k+1 according o

17 Opimal demand response: problem formulaion and deerminisic case 17 [ ( ( ] Uia q k i q k+1 ia ( = q k ia( + γ q k ia ( πk+1 ( where γ > 0 is a consan sepsize, and [ ] Qi denoes he projecion ono he feasible se Q i specified by consrains (24 (25. User i s aggregae demand forecas in period is updaed o Qi k+1 ( = a Ai q k+1 ia (. 3. Incremen ieraion index o k + 1 and goo Sep 1. Algorihm 1 converges asympoically o opimal prices π and opimal consumpions q(π, provided he sepsize γ > 0 is small enough. Theorem 2. Suppose he uiliy funcions U ia (q ia are sricly concave for all i,a. Suppose he Hessian marices 2 U ia and he second derivaive c ( ; are boh uniformly bounded. Then he sequence (π k,q k generaed by Algorihm 1 converges o he opimal price and consumpion vecor (π,q(π, provided γ > 0 is sufficienly small. Proof. Le he welfare funcion be h(q := i,a U ia (q ia c ( Q i (; i Then h(q is sricly concave since U ia (q ia are sricly concave. The gradien h(q has componens ( [ h(q] ia ( = U ia (q i q ia ( c Q i (; i Hence Algorihm 1 is a gradien projecion algorihm where in each ieraion k, he variable q k is updaed o q k+1 according o: [ ] q k+1 = q k + γ h(q k where Q := Q 1 Q N. Moreover he assumpion in he heorem on 2 U ia and c implies ha h(q is Lipschiz. Then, provided γ > 0 is small enough, any accumulaion poin q of he sequence q k generaed by Algorihm 1 is opimal, i.e., maximizes welfare h(q [40, p. 214]. The consrains (24 (25 imply ha he sequence q k lies in a compac se and hence mus have a convergen subsequence. Bu sric concaviy of h implies ha he opimal q is unique. Therefore all convergen subsequences, hence he original sequence q k, mus converge o q. By coninuiy of c, π k ( = c ( i Q k i (; converges o he unique price c ( i Q i (; which, by Theorem 1, is opimal. We simulae his algorihm in [39] wih realisic sysem parameers. The simulaion resuls show ha, as expeced, he prices are capable of coordinaing he Q Q i

18 18 Lijun Chen, Na Li, Libin Jiang, and Seven H. Low decisions of differen appliances in a decenralized manner, o reduce peak aggregae demand and flaen is profile, grealy increasing he load facor. Furhermore, baery amplifies he benefis of demand response. Appendix: Deailed appliance models We describe deailed models of common elecric appliances summarized in Secion 2.1. Type 1. This caegory of appliances includes lighing ha mus be on for a cerain period of ime. The consumpion consrain is (6, wih he undersanding ha q ia ( = q ia ( = 0 for periods ha are ouside is ime of operaion. User i aains a uiliy U ia (q ia (, from consuming power q ia ( independen of is consumpion in oher periods, and he overall uiliy (2 is herefore separable in. Type 2. This caegory includes TV, video games, and compuers. For hese appliances, a user s uiliy depends on her consumpion in each period she wishes o use i as well as he oal amoun of consumpion in a day. Hence he consumpion consrains are (6 and (7. For example, a user may have a favorie TV program ha she wishes o wach everyday. Wih DVR, she can wach he program a any ime. However he oal power demand of TV should a leas cover he program. Type 2 appliances have he same kind of uiliy funcions (2 as Type 1 appliances. The ime dependen uiliy funcion models he fac ha a user may ge differen benefis from consuming he same amoun of power a differen imes, e.g., she may enjoy a TV program o differen levels a differen imes. Type 3. This caegory includes PHEV, dish washer, clohes washer. For hese appliances, a user only cares abou wheher he ask is compleed by a cerain ime. This means ha he aggregae power consumpion by such an appliance mus exceed a hreshold wihin is ime of operaion [28, 29, 33]. Hence he consumpion consrains are (6 and (7. The uiliy depends only on he oal power consumed, hence (3. Type 4. This caegory includes HVAC (heaing, venilaion, air condiioning and refrigeraor ha conrol he emperaure of a user s environmen. Le Tia in ( and Tia ou ( denoe he emperaures a ime inside and ouside he place ha appliance (i,a is in charge of, and T ia denoes he se of imes when user i cares abou he emperaure. For insance, for air condiioner, Tia in ( is he emperaure inside he house, Tia ou( is he emperaure ouside he house, and T ia is he se of imes when she is a home. The inside emperaure evolves according o he following linear dynamics [27, 9, 26]: T in ia ( = T in ia ( 1 + α(tia ou ( T in ia ( 1 + βq ia ( (27

19 Opimal demand response: problem formulaion and deerminisic case 19 where α and β are parameers ha specify hermal characerisics of he appliance and he environmen in which i operaes. The second erm in equaion (27 models hea ransfer. The hird erm models he hermal efficiency of he sysem; β > 0 if appliance a is a heaer and β < 0 if i is a cooler. Here, we define Tia in (0 as he emperaure Tia in(t from he previous day. Le [T ia, T ia ] be a range of preferred emperaure, leading o he consrain: T ia T in ia ( T ia, T ia. (28 Using Equaion (27, we can wrie T in ia ( in erms of (q ia(τ,τ = 1,...,: Tia in ( = (1 α Tia in (0 + Define Then τ=1 T ia ( := (1 α Tia in (0 + T in ia ( = T ia ( + β (1 α τ αt ou (τ + β τ=1 τ=1 ia τ=1 (1 α τ q ia (τ. (1 α τ αtia ou (τ. (29 (1 α τ q ia (τ. (30 Wih (30, he consrain (28 becomes a linear consrain on he load vecor q ia : for any T ia, T ia T ia ( + β τ=1 (1 α τ q ia (τ T ia. This is he consrain (8, in addiion o (6. Assume user i aains a uiliy U ia (Tia in( when he emperaure is Ti,a in (. Then (30 gives he uiliy funcion (4. References 1. C. W. Gellings and J. H. Chamberlin. Demand-Side Managemen: Conceps and Mehods. The Fairmon Press, M. H. Albadi and E. F. El-Saadany. Demand response in elecriciy markes: An overview. In Proceedings of he IEEE Power Engineering Sociey General Meeing, June A. I. Cohen and C. C. Wang. An opimizaion mehod for load managemen scheduling. IEEE Transacions on Power Sysems, 3(2: , May Y. Y. Hsu and C. C. Su. Dispach of direc load conrol using dynamic programming. IEEE Transacions on Power Sysems, 6(3: , Augus D. C. Wei and N. Chen. Air condiioner direc load conrol by muli-pass dynamic programming. IEEE Transacions on Power Sysems, 10(1: , February J. Chen, F. N. Lee, A. M. Breipohl, and R. Adapa. Scheduling direc load conrol o minimize sysem operaion cos. IEEE Transacions on Power Sysems, 10(4: , November 1995.

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