Risk-Limiting Dynamic Contracts for Direct Load Control

Transcription

1 1 Risk-Limiing Dynamic Conracs for Direc Load Conrol Insoon Yang, Duncan S. Callaway, and Claire J. omlin arxiv: v3 [mah.oc] 9 Oc 214 Absrac his paper proposes a novel coninuous-ime dynamic conrac framework ha has a risk-limiing capabiliy. If a principal and an agen ener ino such a conrac, he principal can opimally manage is performance and risk wih a guaranee ha he agen s risk is less han or equal o a pre-specified level and ha he agen s expeced payoff is greaer han or equal o anoher pre-specified hreshold. We achieve such risk-managemen capabiliies by formulaing he conrac design problem as mean-variance consrained risk-sensiive conrol. A dynamic programming-based mehod is developed o solve he problem. he key idea of our proposed soluion mehod is o reformulae he inequaliy consrains on he mean and he variance of he agen s payoff as dynamical sysem consrains by inroducing new sae and conrol variables. he reformulaions use he maringale represenaion heorem. he proposed conrac mehod enables us o develop a new direc load conrol mehod ha provides he load-serving eniy wih financial risk managemen soluions in real-ime elecriciy markes. We also propose an approximae decomposiion of he opimal conrac design problem for muliple cusomers ino muliple low-dimensional conrac problems for one cusomer. his allows he direc load conrol program o work wih a large number of cusomers wihou any scalabiliy issues. Furhermore, he conrac design procedure can be compleely parallelized. he performance and usefulness of he proposed conrac mehod and is applicaion o direc load conrol are demonsraed using daa on he elecric energy consumpion of cusomers in Ausin, exas as well as he Elecriciy Reliabiliy Council of exas locaional marginal price daa. I. INRODUCION o reduce he greenhouse gases caused by elecriciy generaion, here has been growing ineres in and effors oward inegraing renewable energy sources ino he elecric power grid. California, for example, plans o use renewable resources o serve 33% of he elecriciy load by 22 [1]. In paricular, he peneraion of solar and wind energy resources is expeced o significanly increase. However, he uilizaion of hese resources is challenging because hey are uncerain and inermien. o absorb he uncerainy in solar and wind power, for example, he reserve capaciy mus be sufficienly large. In California, he increase in he reserve coss needs o be compensaed his work was suppored by he NSF CPS projec AcionWebs under gran number , NSF CPS projec FORCES under gran number , Rober Bosch LLC hrough is Bosch Energy Research Nework funding program and NSF CPS Award number I. Yang and C. J. omlin are wih he Deparmen of Elecrical Engineering and Compuer Sciences, Universiy of California, Berkeley, CA 9472, USA {iyang, omlin}@eecs.berkeley.edu D. S. Callaway is wih he Energy and Resources Group, Universiy of California, Berkeley, CA 9472, USA dcal@berkeley.edu

2 2 by all cusomers and he amoun of he compensaion per cusomer may no be negligible when California achieves is goal of 33% renewable energy peneraion [2]. Supply-side approaches have been proposed o address he uncerainy of renewable resources in economic dispach and uni commimen using sochasic dynamic programming [3], mixed-ineger sochasic programming [4] and robus opimizaion [5], among ohers. However, hese aforemenioned mehods do no examine he poenial of demand-side resources in managing uncerain renewable generaion or loads. o invesigae his poenial, his paper proposes a demand-side soluion ha manages he uncerainy of cusomers solar and wind generaion and loads. he load-serving eniy or he aggregaor for he cusomers procures power or generaion reserves in a dayahead marke and he amoun of procuremen is deermined based on a day-ahead load forecas. Because he acual oal load deviaes from he procured power, he load-serving eniy mus purchase he deviaed amoun of power in a real-ime marke o balance supply and demand. I is desirable for he load-serving eniy o minimize he realime purchase of energy because he energy price and he reserve cos in he real-ime marke are normally higher and more volaile han hose in he day-ahead marke. As he peneraion of cusomers solar and wind generaion increases, however, he elecriciy price in he real-ime marke is highly volaile and he cusomers demand is very difficul o predic. herefore, he risk of spending a subsanial budge in he real-ime marke increases. If he load-serving eniy bears his financial risk, he energy price in he cusomers ariff would ineviably increase. o reduce his risk ha he load-serving eniy mus face, we propose a conrac approach for direc load conrol in which he cusomer ransfers he auhoriy o conrol his or her load o he load-serving eniy. Once he loadserving eniy and is cusomer ener ino he conrac, he load-serving eniy can allocae a porion of he risk o he cusomer hrough he compensaion scheme and he conrol sraegy for he cusomer s load specified in he conrac. A cusomer migh reasonably worry ha such compensaion and conrol could increase he risk for large energy coss and a disrupion of comfor. he proposed conrac addresses his concern by guaraneeing ha he risk in he cusomer s payoff, a weighed sum of energy coss and discomfor level, is limied by a pre-specified hreshold and ha he mean of he cusomer s payoff is greaer han anoher pre-specified level. he former is called he risk-limiing condiion, and he laer is called he paricipaion payoff condiion. he compensaion scheme and he conrol sraegy wrien in he conrac mus be designed such ha heir combinaion miigaes he load-serving eniy s financial risk in he real-ime marke while saisfying he risk-limiing and paricipaion payoff condiions for he cusomer. he key elemen of he proposed conrac for such a demand-side managemen is direc load conrol ha allows he load-serving eniy o acively use he cusomer s load o manage is financial risk. A number of direc conrol mehods have been suggesed for various ypes of elecric loads such as hermosaically conrolled loads [6], [7], elecric vehicles [8], [9] and deferrable loads [1], [11]. he objecives of exising direc load conrol mehods include shifing demand (e.g., valley-filling ), providing ancillary services (e.g., frequency regulaion) and energy arbirage [12], [13], [14]. o he bes of he auhors knowledge, however, he poenial of direc load conrol for financial risk managemen in real-ime elecriciy markes has no ye been sudied. We bridge he gap beween direc load conrol and risk managemen by proposing a conrac-based approach.

3 3 A variey of conrac mehods have been suggesed for demand-side managemen in elecric power sysems. he use of conracs for reducing energy price risk in spo markes has been invesigaed [15]. Inerrupible service conracs have been exensively sudied, in which a cusomer akes risk of service inerrupion in reurn for a discoun in he energy price [16], [17], [18]. More recenly, deadline-differeniaed deferrable energy conracs have been proposed o preven he risk of a cusomer no receiving energy delivery by a pre-specified deadline [19]. A differen conrac approach associaed wih duraions-differeniaed loads is sudied in [2]. None of he aforemenioned conrac mehods, however, akes ino accoun deailed elecric load dynamics, which are essenial in direc load conrol o guaranee he cusomer s comfor and he load s sysem consrains. o incorporae dynamics of elecric loads in conracs, we adop a dynamic conrac framework, also called a coninuous-ime principal-agen problem [21], [22]. In such a problem, a principal (e.g., a company) and an agen (e.g., a worker) make a conrac ha specifies a compensaion scheme and a conrol sraegy in an uncerain environmen. he seing we consider in his work is called he firs-bes, in which he principal and he agen have he same informaion and share he risk in he principal s revenue sream. he principal can monior he agen s conrol or effor and, herefore, can enforce he conrol sraegy wrien in he conrac. In he elecriciy seing, regarding a load-serving eniy as he principal and is cusomer as he agen, his firs-bes case is appropriae for direc load conrol because he load-serving eniy has he auhoriy o monior and conrol is cusomer s elecric loads. A dynamic conrac problem for he firs-bes case was firs considered in [23]. I uses a simple principalagen model wih exponenial uiliy funcions inroduced by Holmsrom and Milgrom for moral hazard [21]. A more general class of he firs-bes case dynamic conrac problems is addressed in [24] by using he maringale and convex dualiy mehods [25], [26], [27], [28]. However, he proposed soluion approach requires ha he payoff funcions be differeniable, sricly increasing and sricly concave and ha he dynamical sysem be a sochasic inegral equaion. hese resricions are accepable in many applicaions in economics and finance, bu hey may exclude some imporan engineering problems, including direc load conrol, because hey ofen require dynamical sysems and payoff funcions be complicaed. Furhermore, he aforemenioned mehods assume ha he uiliy funcion of he agen wih respec o his or her payoff is given. However, in pracice, i is difficul o have complee knowledge of he agen s uiliy funcion. In paricular, if he agen s uiliy funcion used in designing he conrac deviaes from his or her acual uiliy, he agen may no wan o ener ino he conrac again. In his paper, we propose a novel dynamic conrac mehod ha overcomes he limiaions of exising mehods for he firs-bes case. he proposed mehod uses he variance of he agen s payoff as he risk measure for he agen. By imposing a consrain on he variance, we can limi he risk he agen needs o bear. We call his consrain he agen s risk-limiing condiion. In addiion, he proposed mehod guaranees ha he mean of he agen s payoff exceeds some pre-specified hreshold. From he principal s poin of view, by execuing an appropriaely designed compensaion scheme and conrol sraegy specified in he conrac, he principal can ransfer some porion of is financial risk o he agen, respecing he agen s risk-limiing condiion. his variance approach does no require complee knowledge of he agen s uiliy funcion, which is difficul o obain in pracice. Such a risk-limiing capabiliy disinguishes our mehod from exising conrac mehods. o ake ino accoun he principal s risk aversion,

4 4 we formulae he conrac design problem as risk-sensiive conrol [29], [3]. Due o he consrains on he mean and he variance of he agen s payoff, however, dynamic programming is no direcly applicable. One may be able o handle he consrains using he sochasic maximum principle or he dualiy mehod [31], [28]. However, hese approaches do no, in general, allow a globally opimal soluion. he heoreic conribuion of he paper is o develop a mehod ha gives a globally opimal soluion of such meanvariance consrained sochasic opimal conrol problems. More specifically, using he maringale represenaion heorem, we reformulae he consrains on he mean and he variance of he agen s payoff funcion, which are difficul o handle, as wo new dynamical sysems conrolled by new conrol variables. he firs new sysem sae represens he agen s fuure expeced payoff wih a modified diffusion erm. he second new sysem sae can be inerpreed as he remaining amoun of risk he agen can bear. he former and laer sysems are used o reformulae he consrain on he mean and he variance of he agen s payoff, respecively. I urns ou ha he reformulaed problem is risk-sensiive conrol wih a sochasic arge consrain in he augmened sae space of he original sysem and he new dynamical sysems. A globally opimal soluion o he reformulaed problem can be obained by using he dynamic programming principle in he augmened sae space. he value funcion of he problem is compued by numerically solving an associaed Hamilon-Jacobi-Bellman equaion and is hen used o synhesize opimal compensaion and conrol sraegy. he proposed soluion mehod allows more general sysem models for loads and payoff funcions for he principal and he agen han exising dynamic conrac mehods for he firs-bes case. his flexibiliy and he risk-limiing capabiliy of he proposed conrac mehod make i appropriae for direc load conrol policy which explicily reas financial risk in real-ime elecriciy markes. We also propose an approximae decomposiion mehod for he conrac design problem: he problem for n agens can be decomposed ino n opimal conrac design problems each for a single agen. his decomposiion allows an approximae conrac wih a provable subopimaliy bound. Due o he decomposiion, he compuaional complexiy of he proposed mehod increases linearly wih he number of agens. Furhermore, he decomposed conrac design problem for an agen is independen of ha for anoher agen. herefore, he conrac design procedures for muliple cusomers can be compleely parallelized. he res of his paper is organized as follows. he problem seing for direc load conrol and he definiion of he risk-limiing dynamic conrac are presened in Secion II. We reformulae he consrain on he variance of he agen s payoff, which is called he risk-limiing condiion, as a consrain on he compensaion provided o he agen by inroducing a new conrol variable in Secion III. Using he reformulaed consrain, we propose he mehod for designing a globally opimal risk-limiing dynamic conrac and discuss is decenralized implemenaion in Secion IV. Finally, he performance of he proposed conrac mehod and is applicaion o direc load conrol are demonsraed wih daa on he elecric energy consumpion of Ausin cusomers as well as he Elecriciy Reliabiliy Council of exas (ERCO s) locaional marginal price (LMP) daa in Secion V.

5 5 II. HE SEING We consider a siuaion in which he load-serving eniy wans o make a conrac wih n heerogeneous cusomers o direcly conrol each cusomer s personal elecric load, such as an air condiioner or a waer heaer. For simpliciy, we assume ha each cusomer allows he load-serving eniy conrol over only one of his or her loads, alhough he proposed mehod is also applicable o he case of muliple loads per cusomer. he load-serving eniy s goal is o manage he risk of spending a subsanial budge in a real-ime energy marke by conrolling he cusomers loads in he direc load conrol program. We consider a finie ime horizon conrac: le [, ] be he period in which he conrac is effecive. A. oal Power Consumpion Le η i R be he energy consumpion (in kwh) up o ime [, ] by cusomer i and u i R be he power consumpion (in kw) by cusomer i s elecric load in he direc load conrol program. Noe ha even when u i =, he oal power consumpion by cusomer i is no, in general, zero a ime due o he exisence of he cusomer s oher loads and possibly solar or wind generaion (which can be considered as negaive loads). If all he cusomers ener ino he conrac, he load-serving eniy has he auhoriy o deermine u i := {u i } for i = 1,, n. he number, n, of cusomers is ypically in he order of Le u := (u 1,, u n ). he uncerainy in he cusomers loads and solar and wind power generaion causes he energy consumpion process {η } o be sochasic. o describe he energy consumpion process, we use a sochasic differenial equaion (SDE) model of he form dη i = ( l i () + u i ) d + σi ()dw i, (1) where l i () R,, is he forecas of cusomer i s loads (in kw) oher han hose in he direc load conrol program. he effec of he load forecas error is modeled by he diffusion erm, σ i ()dw i, where W i := {W i } is a one-dimensional sandard Brownian moion on a probabiliy space (Ω, F, P) and he diffusion coefficien σ i : [, ] R is a bounded funcion. We assume ha W i and W j are independen for any i, j {1,, n} such ha i j. he funcions l i and σ i can be esimaed from daa on he elecric energy consumpion of cusomers in Ausin as explained in Secion V. Furhermore, he validiy of he sandard Brownian moion in he model for he proposed conrac framework is esed using he daa in Secion V. B. Energy Price and load-serving Eniy s Revenue in Real-ime Markes Le p() R,, be he amoun of power procured by he load-serving eniy in he day-ahead marke. We assume ha p() is given. he energy price in he real-ime marke is chosen as he locaional marginal price (LMP). Le λ be he LMP a ime. he dynamics of he LMP can be modeled as he following SDE [32], [18]: dλ = r (ν() ln λ )λ d + σ ()λ dw, (2) where W := {W } is a one-dimensional sandard Brownian moion on (Ω, F, P) and he price volailiy σ : [, ] R is a bounded funcion. For simpliciy, we assume ha W is independen of W i for i = 1,, n,

6 6 bu our conrac mehod can easily be exended o he case in which hey are dependen. his model is suiable o capure he mean-revering behavior of energy prices in he real-ime (spo) marke: when he energy price is high (resp. low), he supply ends o increase (resp. decrease) and, herefore, causes he price o decrease (resp. increase) [33]. Le w := ln λ, hen w := {w } saisfies dw = r (ν() w )d + σ ()dw. (3) We esimae r, ν() and σ () using he ERCO LMP daa in Secion V. In principle, he LMP is no compleely exogenous because i is influenced by he power consumpion of he cusomers loads. In his work, however, we assume ha his effec is negligible and, herefore, ha he LMP is exogenous. he load-serving eniy s revenue in he real-ime marke up o ime, denoed as z R, is given by ) n z = λ s (p(s)ds. Noe ha we assume ha excess power is sold as easily as deficis are procured. his sochasic inegral can be rewrien as he following SDE: dz = λ (p() i=1 ) n (l i () + u i ) d i=1 dη i s n λ σ i ()dw i. (4) he load-serving eniy s revenue is affeced by he conrol u of he cusomers loads in he direc load conrol program. he se of feasible conrols is chosen as U i := {u i : [, ] U i u i progressively measurable wih respec o F (i) }, where U i is a compac se in R and {F (i) } is he filraion generaed by he wo dimensional Brownian moion W (i) := (W, W i ). We also le U := U 1 U n. i=1 C. Cusomers Loads Consider cusomer i s load in he direc load conrol program, and le x i R be he sysem sae a ime [, ]. If he load is an air condiioner uni, hen x i would represen he indoor emperaure; if he load is a waer heaer, i would represen he waer emperaure. hen, he sysem dynamics can be modeled as he following sochasic differenial equaion: dx i = f i (x i, u i )d (5) wih he iniial condiion x i = x i for i = 1,, n, where he conrol u i is a sochasic process. Alhough our conrac mehod can handle sochasic sysem models wih a diffusion erm, we use he model (5) for simpliciy. Here, we assume ha f i : R U i R is coninuous and ha f i (x, u) is differeniable in x for any u U i. We furher assume ha here exiss a consan K such ha for all (x, u) R U i f i (x, u) x K, f(x, u) K(1 + x + u ).

7 7 hen, here exiss a unique soluion x i := {x i } L 2 (, ) for i = 1, n, where L 2 (, ) denoes he [ ] space of all real-valued, progressively measurable sochasic processes x such ha E x2 d <. See [34] for he proof. Example 1. Le x i denoe he indoor emperaure of cusomer i a ime, and le Θ i () represen he corresponding oudoor emperaure. hen, he dynamics of he indoor emperaure can be described as he following equivalen hermal parameer (EP) model [35]: dx i = [α i (Θ i () x i ) κ i u i ]d (6) for i = 1,, n. Here, α i = R 1,i /R 2,i, where R 1,i denoes he hermal conducance beween he oudoor air and indoor air and R 2,i is he hermal conducance beween he indoor air and he hermal mass for cusomer i s room. he posiive consan κ i convers an increase in energy (kwh) o a reducion in emperaure ( C) for cusomer i s air condiioner. D. Payoff Funcions 1) load-serving eniy s payoff: he load-serving eniy s payoff funcion is chosen as is profi in he direc load conrol program. Le C i R be he end-ime compensaion paid o cusomer i in he direc load conrol program and µ i () be he energy price per uni kwh a ime specified in cusomer i s elecriciy ariff. We assume ha µ i : [, ] R is bounded. he load-serving eniy s oal payoff in real ime, i.e., neglecing he cos of power procured in he day-ahead marke, which is is revenue obained from he cusomers, is hen given by n n dz + µ i ()dη i C i = n i=1 i=1 i=1 [ (µi () λ ) ( l i () + u i ) + λ p i () ] d + n i=1 (µ i () λ ) σ i ()dw i n C i, where {p 1 (),, p n ()} is a se saisfying n i=1 p i() = p() and he se of feasible compensaion values is chosen as C i := {C i R C i is F (i) -measurable}. Le C := C1 C n. We define he payoff funcion of he load-serving eniy as J P [C, u] := ( n i=1 where r P i : [, ] R R R R and σ P i : [, ] R R are such ha ) ri P (, w, x i, u i )d + σi P (, w )dw i C i, (7) r P i (, w, x i, u i ) := (µ i () e w )(u i + l i ()) + e w p i (), σ P i (, w ) := (µ i () e w ) σ i () for i = 1,, n. he superscrip P represens he fac ha he load-serving eniy plays he role of he principal in he conrac. Noe ha in he direc load conrol applicaion r P i i=1 (8) is independen of x i. We use he models (8) for direc load conrol bu he proposed conrac design mehod is applicable o more general models of running payoff and volailiy. For noaional simpliciy, we will suppress he dependency of he funcions on ime.

8 8 2) Cusomer s payoff: Each cusomer s oal payoff depends on (i) his or her economic profi and (ii) his or her comfor level. Cusomer i s profi can be compued as he compensaion received minus he energy coss, i.e., C i = µ i ()dη i µ i ()(l i () + u i )d µ i () σ i ()dw i + C i. Cusomer i s payoff funcion can be represened as Ji A [C i, u i ] := ri A (x i, u i )d + σi A ()dw i + C i, (9) where r A i : [, ] R R R and σ A i : [, ] R are such ha r A i (, x i, u i ) := µ i ()(l i () + u i ) + r i (x i, u i ), σ A i () := µ i () σ i () (1) for i = 1,, n. Here, r i (x i, u i ) represens cusomer i s comfor level given he sysem sae x i and conrol u i. he superscrip A represens he fac ha he cusomer is he agen in he conrac. We assume ha here exis consans B and B 1 such ha ri A(x, u) B + B 1 x for all x R given any u U i. Example 2. Cusomer i s discomfor level is zero if he indoor emperaure, x i, is wihin a desirable emperaure range, [Θ, Θ]. he discomfor level increases as he indoor emperaure increases above Θ or drops below Θ. If we se he comfor level as he negaive value of he discomfor level, hen we can model cusomer i s comfor level as r i (x i, u i [ ) = ω i (x i Θ) + + (Θ x i ] ) +, (11) where he consan parameer ω i represens he cusomer i s valuaion of comfor and (a) + := a if a > and (a) + := oherwise for any a R. E. Risk-Limiing Dynamic Conracs he load-serving eniy (principal) offers cusomer i a conrac ha specifies he compensaion scheme, C i, and is conrol sraegy, u i := {u i } for i = 1,, n. he conrac is dynamic in he sense ha he load-serving eniy uses he sae feedback conrol sraegy wrien in he conrac o dynamically choose he conrol acion each cusomer mus follow. Cusomer i (agen i) acceps he conrac only if 1) (paricipaion-payoff condiion) he mean of cusomer i s payoff is greaer han or equal o some hreshold, b i R, i.e., E[Ji A [C i, u i ]] b i, (12) and 2) (risk-limiing condiion) he variance of cusomer i s payoff is less han or equal o some hreshold, S i R, i.e., Var[Ji A [C i, u i ]] S i. (13)

9 9 Noe ha variance is used as he risk measure of he cusomer s payoff. We call b i and S i he paricipaion payoff and he risk share of cusomer i, respecively. Le Λ := {(b 1, S 1 ),, (b M, S M )} be a se of given pairs of paricipaion payoffs and risk shares. hese pairs are designed by he load-serving eniy and provided o he cusomers. Cusomer i selecs a pair (b i, S i ) Λ and he conrac deermined from his pair. Once each cusomer eners ino a conrac, he load-serving eniy direcly conrols each cusomer s load following he conrol sraegy specified in he conrac. A he end of he conrac period, he load-serving eniy compensaes each cusomer according o he compensaion scheme specified in he conrac. More specifically, once cusomer i agrees o ener ino he conrac, he load-serving eniy company has he auhoriy o conrol cusomer i s load for maximizing he load-serving eniy s expeced payoff under he consrains 1) and 2). aking ino accoun he risk of he load-serving eniy s payoff being small as well, we formulae he problem of designing such a dynamic conrac (C, u) as he following consrained risk-sensiive conrol problem: max C C,u U 1 θ log E [ exp( θj P [C, u]) ] (14a) subjec o dw = r (ν() w )d + σ ()dw (14b) dx i = f i (x i, u i )d E[J A i [C i, u i ]] b i Var[J A i [C i, u i ]] S i, (14c) (14d) (14e) where θ R \ {} is a consan, called he coefficien of load-serving eniy s risk-aversion. When θ is posiive, he risk-sensiive objecive funcion penalizes he risk of he load-serving eniy s payoff being small because exp( θj P ) is concave increasing in J P. herefore, he load-serving eniy can make a risk-averse decision by solving (14). If θ <, he load-serving eniy is risk-seeking. For inuiion, noe ha he risk-sensiive objecive funcion is well approximaed by a weighed sum of he mean and he variance of he payoff when θ is small because he aylor expansion of he risk-sensiive objecive funcion is given by 1 θ log E [ exp( θj P ) ] = E[J P ] θ 2 Var[J P ] + O(θ 2 ) (15) as θ. Noe ha he variance of he payoff is penalized when θ >. he soluion, (C OP, u OP ), o his problem is said o be he opimal risk-limiing dynamic conrac for direc load conrol. his problem of risk-limiing dynamic conrac design is a mean-variance consrained-sochasic opimal conrol problem ha is no direcly solvable via dynamic programming. In he following secions, we carefully characerize he necessary and sufficien condiions for he consrains on he mean and he variance of he cusomers payoff funcions. he characerizaions allow us o show ha he conrac design problem can be reformulaed as a risk-sensiive conrol problem wih a sochasic arge consrain ha can be solved by dynamic programming. he informaion availabiliies o he load-serving eniy and he cusomers in he direc load conrol program are symmeric, as opposed o he case of indirec load conrol, in which he load-serving eniy has limied observaion

10 1 capabiliy (e.g., [36]). More specifically, in he proposed framework, he load-serving eniy can monior he conrol and sae of he cusomers loads in he direc load conrol program as well as he energy price in he real-ime marke. Furhermore, he load-serving eniy has all he parameers and funcions needed o design an opimal conrac. ha is, i has he informaion of p, l i, σ i, µ i, r, σ, ν, which can be esimaed from daa as shown in Secion V, and he cusomers comfor funcions r i and load models for i = 1,, n. In pracice, he comfor funcions and load models can be idenified using a raining period. Each cusomer, in principle, can have he same informaion. However, cusomer i needs o know only p, l i, σ i, µ i, r, σ, ν, his or her own comfor funcion and load model. he proposed framework assumes ha each cusomer can monior he conrol and sae of his or her load in he direc load conrol program and he energy price in he real-ime marke. Anoher imporan feaure of he proposed conrac mehod is ha he ineracions beween he load-serving eniy and is cusomers can be decoupled from each oher because one cusomer s load does no affec hose of oher cusomers and he paricipaion payoff and risk-limiing condiions are personalized. his feaure will allow us o decenralize he conrol of loads as shown in Secion IV-B. We assume ha he conrac period [, ] is a ime inerval wihin 24 hours, bu he proposed mehod can handle arbirary finie ime horizons. herefore, he cusomers and he load-serving eniy, in principle, can renew he conracs every day. However, i may no be convenien for each cusomer o choose a conrac or, equivalenly, a paricipaion payoff and a risk share pair every day. his issue can be resolved by auomaically choosing he conrac for he curren day as ha for he previous day unless he cusomer explicily wans o change i. Daily conracs have a pracical advanage: he day-ahead forecass of he LMP model parameers and demand uncerainy (and oudoor emperaure in he case of air condiioners) can be incorporaed ino he conracs. herefore, he conracs can be designed using accurae models. We also assume ha each cusomer does no sraegically conrol oher loads o modify he forecased σ i by he load-serving eniy. his assumpion can be jusified in wo ways. Firs, he load-serving eniy can make a conrac o conrol muliple loads of a cusomer so ha he cusomer has lile flexibiliy o change σ i. Second, even if he cusomer sraegically affecs σ i in one day, he cusomer s gain in he nex day is marginal because he conrac is renewed wih a new esimae σ i ha incorporaes any sraegic behavior. Formally, his problem can be formulaed as a Sackelberg differenial game, in which he he load-serving eniy chooses he esimaes of l i and σ i for he conrac period [, ] assuming ha he cusomer has no incenive o deviae from σ i in he conrac period. A similar problem is considered in our previous work [36]. his Sackelberg differenial game problem is ou of he scope of his paper and will be addressed in our fuure work on risk-limiing dynamic conracs for indirec load conrol. III. RISK-LIMIING COMPENSAION he risk-limiing condiion (13) is an inequaliy consrain on he variance of each agen s payoff. his consrain hinders us from using he dynamic programming principle o solve he conrac design problem (14). In his secion, we characerize a condiion on he end-ime compensaion, which is equivalen o he risk-limiing condiion. I urns ou ha he new equivalen condiion allows us o formulae he conrac design problem as a risk-sensiive

11 11 conrol problem ha can be solved by dynamic programming. We begin by defining he following se of sochasic processes: le Γ i be he se of processes ξ i := {ξ i }, ξ i = (ξ i,1, ξ i,2 ) R 1 2 such ha (i) ξ i is F (i) -progressively measurable; [ ] (ii) E ξi 2 d < for i = 1,, n. We also le Γ := Γ 1 Γ n. In he nex lemma, we show ha here exiss a unique process in his se such ha is inegral over he Brownian moion W (i) := (W, W i ) corresponds o he difference beween he agen s payoff and is mean value. Lemma 1. Fix i {1,, n}. Given C i C i and u i U i, here exiss a unique (up o a se of measure zero) sochasic process ξ i = {ξ i } Γ i such ha J A i [C i, u i ] E[J A i [C i, u i ]] = Proof: Fix C i C i and u i U i. We inroduce a new process [ ] q i := E ri A (x i s, u i s)ds + C i F (i). ξ i dw (i). (16) Here, he expecaion is condiioned over he filraion {F (i) } generaed by he Brownian moion W (i) = (W, W i ). We noice ha he process q i + r A i (x i s, u i s)ds = E [ is maringale. Recall ha here exis consans B and B 1 such ha r A i (x, u) B + B 1 x for all x R and u U i. herefore, we have ( 2 ri A (x i s, us)ds) i which implies ha for [, ] E [ ( because x i L 2 (, ). From he definiion of q i, we deduce ha Due o he inequaliies (17) and (18), we obain [ ( E q i + r A i (x i s, u i s)ds + C i F (i) (B + B 1 x i s ) 2 ds, ) 2 ] ri A (x i s, u i s)ds < (17) E [ (q i ) 2] <. (18) ) 2 ] ri A (x i s, u i s)ds <. ]

12 12 he Maringale represenaion heorem (e.g., [37], [38]) suggess ha here exiss a unique (up o se of measure zero) process ξ i = { ξ i } Γ i such ha We also noe ha q i + ri A (x i, u i )d = q i + ξ dw i (i). q i = C i, q i = E[J A i [C i, u i ]]. herefore, we obain Ji A [C i, u i ] = E[Ji A [C i, u i ]] + σi A ()dw i + ξ dw i (i). Se ξ 1,i 1,i := ξ and ξ 2,i 2,i := ξ + σi A(), hen ξi is in Γ i and saisfies (16). his lemma represens he agen s payoff as he sum of is mean value and he Iô inegral of he new process ξ i along he Brownian moion W (i). he following heorem suggess ha his represenaion allows reformulaion of he risk-limiing condiion (13). heorem 1. Fix i {1,, n} and u i U i. he risk-limiing condiion holds, i.e., Var[J A i [C i, u i ]] S i (19) if and only if here exiss a unique (up o se of measure zero) γ i Γ i such ha and C i = E[Ji A [C i, u i ]] ri A (x i, u i )d σi A ()dw i + γdw i (i) (2) [ ] E γ i 2 d S i. (21) Proof: Suppose ha here exiss γ i Γ i such ha (2) and (21) hold. hen, Due o he Iô s isomery, we have J A i [C i, u i ] E[J A i [C i, u i ]] = γ i dw (i). [ ] Var[Ji A [C i, u i ]] = E γ i 2 d. Combining his equaliy and (21), we obain he risk-limiing condiion (19). Suppose now ha he risk-limiing condiion (19) holds. Lemma 1 suggess ha here exiss ξ i Γ i such ha J A i [C i, u i ] E[J A i [C i, u i ]] = ξ i dw (i). he variance of agen i s payoff is given by [ ] Var[Ji A [C i, u i ]] = E ξ i 2 d.

13 13 Due o he risk-limiing condiion (19), we have [ ] E ξ i 2 d S i. herefore, ξ i Γ i saisfies boh (2) and (21). Such a process mus be unique up o a se of measure zero due o Lemma 1. he nex corollary suggess a way o consruc he end-ime compensaion given u U and γ Γ. Corollary 1. Fix u U and γ Γ such ha [ ] E γ i 2 d S i (22) for i = 1,, n. he risk-limiing condiion (13) holds if and only if he end-ime compensaion, C C, saisfies for i = 1,, n. C i = E[Ji A [C i, u i ]] ri A (x i, u i )d σi A ()dw i + γdw i (i) heorem 1 and Corollary 1 imply ha, given u i U i, deermining C i C i is equivalen o choosing γ i Γ i such ha i saisfies (22). In he nex secion, we consider γ i Γ i as a wo-dimensional decision variable and hen consruc he end-ime compensaion using an opimal γ i. We also show ha he mean of agen i s payoff is given by he paricipaion payoff b i if an opimal conrac is chosen. However, even if we reformulae he conrac design problem (14) as a sochasic opimal conrol in which he decision variables are u i and γ i, i = 1,, n, he inegral consrain (22) prohibis us from using dynamic programming o solve he reformulaed problem. We resolve his issue in he following secion by inroducing new sae and conrol variables. IV. RISK-LIMIING DYNAMIC CONRAC DESIGN We now propose he soluion mehod for risk-limiing dynamic conrac design problem (14) given (b i, S i ) Λ for i = 1,, n. Recall ha he principal s objecive is o maximize is risk-sensiive payoff wih a guaranee ha he paricipaion payoff and risk-limiing condiions for all he agens are saisfied. he risk of each agen s payoff being oo small is limied by he variance consrain (14e), which is he risk-limiing condiion. On he oher hand, small values of he principal s payoff are penalized by maximizing he risk-sensiive objecive funcion (14a) when he risk-aversion coefficien θ is posiive. In oher words, if he principal is risk-averse, she ransfers her financial risk o he agens as long as he agens risk-limiing condiions are respeced. he conrac design problem (14) is a consrained sochasic opimal conrol problem ha canno be direcly solved by dynamic programming [39]. he sochasic maximum principle approach may be used o handle he consrains [31], [4]. However, i is no capable of finding a globally opimal soluion unless he problem is concave in boh C and u. o obain a globally opimal soluion, we reformulae he problem as a risk-sensiive conrol problem ha can be solved by dynamic programming. he key idea is o inroduce wo new sae variables. he firs sae variable s value a he erminal ime allows us o consruc he end-ime compensaion using heorem

14 14 1. he second variable is used o reformulae he consrain (21) on he new decision variable, γ, as an SDE. However, he dynamic programming approach, in general, has an inheren scalabiliy issue: he compuaional complexiy exponenially increases as he sysem dimension increases (e.g., [41]). We overcome his scalabiliy issue by proposing an approximae decomposiion of he conrac design problem for all agens ino n lowerdimensional conrac design problems, each for a single agen, where n is he number of agens. A. Reformulaion We show ha he soluion of he conrac design problem (14) can be obained by solving he following risksensiive conrol problem: max u U,γ Γ,ζ Γ 1 θ log E [ exp ( θ J P [u, γ, ζ] )] (23a) subjec o dw = r (ν() w )d + σ ()dw (23b) dx i = f i (x i, u i )d (23c) dv i = r A i (x i, u i )d + γ i,1 dw + (γ i,2 σ A i ())dw i (23d) v i = b i dy i = γ i 2 d + ζ i dw (i) y i = S i (23e) (23f) (23g) where J P is he reformulaed principal s payoff given by ( n J P [u, γ, ζ] := ri P (w, x i, u i ) + y i a.s., (23h) i=1 ) σi P (w )dw v i. Noe ha we now view γ Γ as a decision variable insead of C C. his is feasible due o heorem 1 and Corollary 1. he new sae y i and new decision variable ζ i handle he inegral consrain (21) in heorem 1. Inuiively speaking, he firs new sae variable v i represens agen i s expeced fuure payoff wih a modified diffusion erm. he second new sae y i can be inerpreed as he remaining amoun of risk ha agen i can bear from ime. Having hese inerpreaions, we can show ha he erminal value of he firs new sae variable can be used o consruc an opimal end-ime compensaion and ha of he second new sae variable mus be greaer han or equal o zero o saisfy agen i s risk-limiing condiion. hese wo claims are shown in heorem 2. Anoher imporan observaion is ha he problem is now defined in he augmened sae space of w, x := (x 1,, x n ) R n, v := (v 1,, v n ) R n and y := (y 1,, y n ) R n. herefore, he oal sysem dimension is 3n + 1. his reformulaed problem can be decenralized ino n hree dimensional risk-sensiive conrol problems, as shown in Secion IV-B.

15 15 heorem 2. Le (u OP, γ OP, ζ OP ) be he soluion o (23). We also le x OP, v OP and y OP denoe he processes driven by (23c), (23d) and (23f) wih (u OP, γ OP, ζ OP ), respecively. Define C OP,i := v OP,i (24) for i = 1,, n. hen, (C OP, u OP ) is an opimal risk-limiing dynamic conrac, i.e., i solves (14). Proof: We firs observe ha J A i [C OP,i, u OP,i ] = = b i + ri A (x OP,i, u OP,i )d + γ OP,i dw (i) due o he SDE (23d) wih he iniial condiion (23e). herefore, we have σ A i ()dw i + v i (25) E[J A i [C OP,i, u OP,i ]] = b i, (26) which implies ha he paricipaion payoff condiion (14d) holds. Furhermore, he variance of agen i s payoff wih (C OP,i, u OP,i ) is given by due o he Iô s isomery. We also noice ha which suggess ha Hence, if y OP,i conrac. Var[J A i [C OP,i, u OP,i ]] = E y OP,i = S i Var[J A i [C OP,i, u OP,i ]] = E [ γ OP,i 2 d + [ S i y OP,i + γ OP,i 2 d ] ζ OP,i dw (i), ζ OP,i dw (i) a.s., he risk-limiing condiion (14e) also holds. herefore, (C OP, u OP ) is a feasible dynamic Suppose ha (C OP, u OP ) is no a soluion of (14) and selec a soluion, (Ĉ, û), of (14). heorem 1 suggess ha here exiss a unique (up o se of measure zero) ˆγ Γ such ha Ĉ i = E[Ji A [Ĉi, û i ]] ri A (ˆx i, û i )d σi A ()dw i + ˆγ dw i (i) and [ ] E ˆγ i 2 d S i. (27) We claim ha (û, ˆγ) saisfies all he consrains of (23) wih he sochasic process ˆv := {ˆv } defined as ˆv i := ˆv i for some iniial value ˆv i R such ha r A i (ˆx i s, û i s)ds + (ˆγ i s σ A i (s))dw i s ˆv i = Ĉi. (28) ].

16 16 I is clear ha he process ˆv i saisfies he SDE (23d) for i = 1,, n by definiion. Addiionally, noe ha E[J A i [Ĉi, û i ]] = ˆv i. Suppose ha ˆv does no saisfy he iniial condiion (23e). We firs assume ha here exiss j {1,, n} such ha ˆv j > b j. Define a new end-ime compensaion C as C i Ĉ j (ˆv j := b j) We hen have Ĉ i if i = j oherwise. E[J A j [C j, û j ]] C j = E[J A j [Ĉj, û j ]] Ĉj = ˆv j Ĉj, which implies ha E[J A j [C j, û j ]] = b j. herefore, (C, û) saisfies he paricipaion payoff condiion (14d) for all i = 1,, n. he risk-limiing condiion also holds wih (C, û) because he difference beween C and Ĉ is deerminisic. On he oher hand, we noice ha J P [C P, û] > J [Ĉ, û] because C j < Ĉj and C i = Ĉi for i j. he conrac (C, û) saisfies all he consrains of (14) and is sricly beer han (Ĉ, û). his is a conradicion because (Ĉ, û) solves he conrac design problem (14). herefore, ˆvi mus be equal o he paricipaion payoff b i for i = 1,, n. Hence, (û, ˆγ) saisfies all he consrains of (23) wih he processes ŷ and ˆx, where ˆx solves (23c) wih he conrol û. We define a sochasic process ỹ i := {ỹ} i as [ ] ỹ i := E ˆγ i 2 d F (i). Noe ha ỹ i + [ ] ˆγ s i 2 ds = E ˆγ i 2 d F (i) [ ( is maringale. Furhermore, E ỹ i + ) ] 2 ˆγi s 2 ds < because ˆγ i Γ i. herefore, he maringale represenaion heorem suggess ha here exiss a unique (up o a se of measure zero) ˆζ i Γ i such ha ỹ i + ˆγ i 2 d = ỹ i + ˆζ i dw (i). herefore, he process ỹ i solves (23f) wih (ˆγ, ˆζ). Due o (27), we also have ỹ i S i and hence he consrain (23g) is saisfied. We define anoher sochasic process ŷ i := {ŷ i } as ŷ i := ỹ i + S i ỹ i

17 17 for [, ]. herefore, we have ŷ i = S i and ŷ i because ỹi = by definiion. Hence, ŷi saisfies he consrains (23g) and (23h). Since (Ĉ, û) solves (14), while (C OP, u OP ) does no, he following inequaliy holds: his inequaliy can be rewrien as [ E E [ E[ exp( θj P [Ĉ, û])] > E[ exp( θj P [C OP, u OP ])]. exp exp ( ( θ θ ( n ))] ri P (w, ˆx i, û i )d ˆv i > i=1 ( n i=1 r P i (w, x OP,i, u OP,i )d v OP,i due o (28) and (24). his is conradicory o he fac ha (u OP, γ OP, ζ OP ) is a soluion o (23). herefore, (C OP, u OP ) should solve (14) and hence an opimal risk-limiing dynamic conrac. We observe ha he agens expeced payoff mus be equal o heir paricipaion payoffs from (26), i.e., he inequaliies (14d) for he paricipaion payoff are always binding a an opimal conrac. Inuiively speaking, if he agen s expeced payoff is sricly greaer han his or her paricipaion payoff, he principal has an incenive o decrease he end-ime compensaion for he agen. he equaliy (25) also suggess ha each agen s payoff can be compleely characerized by his or her paricipaion payoff and he new conrol variable γ if an opimal conrac is execued. ))] Corollary 2. Agen i s payoff wih an opimal risk-limiing dynamic conrac (C OP, u OP ) is given by for i = 1,, n. herefore, J A i [C OP,i, u OP,i ] = b i + E [ J A i [C OP,i, u OP,i ] ] = b i. γ OP,i dw (i) B. Decoupled Conrac Design and Decenralized Conrol We propose an approximae decomposiion of he conrac design problem (14) ino n low dimensional problems using he fac ha he sysem dynamics (14c), he paricipaion payoff condiion (14d) and he risk-limiing condiion (14e) for one agen are decoupled from hose for oher agens and ha W 1,, W n are muually independen. he approximae soluion obained using his decomposiion has a guaraneed subopimaliy bound. his decomposiion enables he direc load conrol program wih he proposed dynamic conracs o handle a large populaion of agens wihou scalabiliy issues. More specifically, for each i {1,, n}, he approximae risk-limiing dynamic conrac for agen i can be

18 18 obained by solving he following risk-sensiive conrol problem: max u i U i, γ i Γ i,ζ i Γ i 1 θ log E [ exp ( θ J i P [u i, γ i, ζ i ] )] subjec o dw = r (ν() w )d + σ ()dw dx i = f i (x i, u i )d dy i = γ i 2 d + ζ i dw (i) (29) where y i = S i y i a.s., J P i [u i, γ i, ζ i ] := b i + (r P i (w, x i, u i ) + r A i (x i, u i ))d + Noe ha he sysem for v i is absorbed ino he modified payoff funcion J P i (σi P (w ) + σi A ())dw i γdw i (i).. I is clear ha his decomposiion is exac when σ due o he muual independence of {W 1,, W n }. In addiion, he following proposiion suggess ha he approximae conrac obained using he proposed decomposiion has a provable subopimaliy bound, which can be compued a poseriori. Proposiion 1. Le (u OP, γ OP, ζ OP ) and (u i, γ i, ζ i ) be he soluions o (14) and (29), respecively. We also le ū i be he soluion o max E[ J P u i U i i [u i,, ]] subjec o dw = r (ν() w )d + σ ()dw (3) dx i = f i (x i, u i )d. Suppose ha E[ J P [u,, ]] >, where ū := (ū 1,, ū n ). Se ρ := 1 θ log E [ exp ( θ J P [u, γ, ζ ] )] E[ J. P [ū,, ]] hen, he following subopimaliy bound holds: ( ρ 1 θ log E [ exp ( θ J P [u OP, γ OP, ζ OP ] )]) 1 θ log E [ exp ( θ J P [u, γ, ζ ] )] (31) for θ >. Proof: Using Jensen s inequaliy, we have 1 θ log E [ exp ( θ J P [u OP, γ OP, ζ OP ] )] 1 θ log exp ( θe[ J P [u OP, γ OP, ζ OP ]] ) = E[ J P [u OP, γ OP, ζ OP ]]. When he principal is risk-neural, he principal s payoff is independen of (γ, ζ) and seing (γ, ζ) = (, ) always saisfies he risk-limiing condiion. From his observaion, we claim ha (ū,, ) solves he problem (14) when

19 19 θ =, i.e., he objecive funcion is replaced wih E[ J P [u, γ, ζ]]. We firs noe ha (ū,, ) saisfies all he consrains in (14). Suppose ha (ū,, ) does no solve (14) and choose a soluion, (û, ˆγ, ˆζ), of (14). Because E[ J i P [ûi, ˆγ, ˆζ]] = E[ J i P [ûi,, ]] and û saisfies he consrain of (3), he following inequaliy holds: E[ J i P [û i, ˆγ i, ˆζ i ]] = E[ J i P [û i,, ]] E[ J i P [ū i,, ]]. herefore, we have E[ J P [û, ˆγ, ˆζ]] = n E[ J i P [û i, ˆγ i, ˆζ i ]] i=1 n E[ J i P [ū i,, ]] = E[ J P [ū,, ]]. i=1 his inequaliy is conradicory o he fac ha (ū,, ) does no solve (14). herefore, E[ J P [u OP, γ OP, ζ OP ]] E[ J P [ū,, ]]. As a resul, he subopimaliy bound (31) holds. Noe ha he subopimaliy bound can be compued by solving (29) and (3) for each i while i is no feasible o direcly solve (14) for n > 1. his proposiion implies ha he proposed decomposiion ends o be exac as he coefficien θ of he principal s risk aversion goes o zero because ρ 1 as θ. Furhermore, he approximae conrac (C, u ) saisfies he paricipaion-payoff condiion (12) and he risk-limiing condiion (13). Due o his decomposiion, he conrac design for agen i only requires he sae space, R 3, of (w, x i, y) i raher han he full join sae space, R 3n+1, of (w, x, v, y ). herefore, he compuaional complexiy of designing a risk-limiing conrac for an agen is independen of he oal number of agens. he decomposed problem for agen i is solved via dynamic programming over he reduced sae space, R 3, of (w, x i, y) i as follows. We se he feasible se of conrol as Ω i := {(u i, γ i, ζ i ) U i Γ i Γ i y i a.s.}. o synhesize a risk-limiing dynamic conrac for agen i, we firs define he value funcion of (29) associaed wih agen i as φ i (w, x i, y i, ) := max 1 (u i,γ i,ζ i ) Ω i θ log E w,x i,y i, [ ( ( ))] exp θ R i (w s, x i s, u i s)ds + G i (w s, γs)dw i s (i) b i, (32) where E w,xi,y i,[a] denoes he expecaion of A condiioned on (w, x i, y i ) = (w, x i, y i ), and o handle he consrain y i R i (w, x i, u) := ri P (w, x i, u) + ri A (x i, u), ] G i (, w, γ) := [ γ 1 γ 2 + σ Pi (w) + σai ()). a.s., which is ofen called he sochasic arge consrain, we use he Hamilon- Jacobi-Bellman (HJB) characerizaion proposed in [42]. his characerizaion convers he arge consrain ino a classical sae consrain using he geomeric dynamic programming principle [43]. he reformulaed consrain is

20 2 embedded in an auxiliary value funcion. his auxiliary value funcion is a viscosiy soluion of an HJB equaion. Applying he dynamic programming principle on he original value funcion (32), one can derive a consrained-hjb equaion in which he sochasic arge consrain is reformulaed as he consrains on he auxiliary value funcion and he conrol. In our case, he auxiliary value funcion is a zero funcion. Le U i (y i ) := {(u, γ, ζ) U i R 2 R 2 γ = ζ = if y i }. hen, he sochasic arge consrain is simply incorporaed ino he following consrained-hjb equaion: φ i + max { (Fi (w, x i, u, γ) θσ(ζ)g i (w, γ) ) Dφ i (u,γ,ζ) U i(y i) +R i (w, x i, u) θ 2 G i(w, γ) 2 θ 2 Σ(ζ) Dφ i } 2 r(σ(ζ)σ(ζ) D 2 φ i ) =, φ i (w, x i, y i, ) = b i, whose viscosiy soluion corresponds o he value funcion (32) [44], [34], [42], where r (ν() w) σ () F i (, w, x i, u, γ) := f i (x i, u), Σ(, ζ) :=. γ 2 ζ 1 ζ 2 In general, an analyic soluion of he HJB equaion is difficul o find. herefore, we grid up he sae space, which corresponds o he domain of he PDE, and we numerically evaluae he soluion a he grid poins using convergen schemes, e.g., [45], [46]. Afer solving he HJB equaion, we can use he value funcion o compue an opimal compensaion scheme, C, and an opimal conrol sraegy, u. Se (w, x i, y i ) = (ln λ, x i, S i ) for i = 1,, n. Given he process x i s sraegy as (u i, γ i, ζ i ) = arg max (u,γ,ζ) U i(y i) for i = 1,, n and for [, ]. Noe ha v i := (w s, x i s, y i s ) R 3 for s [, ], we can deermine an opimal conrol { (Fi (w, x i, u, γ) θσ(ζ)g i (w, γ) ) Dφ i (x i, ) + R i (w, x i, u) θ 2 G i(w, γ) 2 θ 2 Σ(ζ) Dφ i (x i, ) r ( Σ(ζ)Σ(ζ) D 2 φ i (x i, ) )} and y i can be compued by inegraing he SDEs (23d) and (23f) over ime wih he conrol (u i s, γ i s, ζ i s ) for s [, ), respecively. hen, an opimal compensaion scheme can be obained as C = v which is proposed in heorem 2. A more deailed discussion regarding how o synhesize an opimal conrol using a viscosiy soluion of an associaed HJB equaion can be found in [47] even when he viscosiy soluion is no differeniable. he synhesized opimal conrol is wrien ino he conrac and he cusomer mus follow he conrol sraegy if he or she eners ino he conrac. Noe ha he conrol for one load is independen of ha for anoher load. Furhermore, he opimal conrol for a load is given as sae-feedback, where he sae variables only require he energy price in he real-ime marke and he local informaion of he load. herefore, he proposed conrol can

21 21 Load serving eniy (LSE) C 1 C n W 1 Cusomer 1. Cusomer n W n u 1 local conroller u n Load 1 Load n x 1,u 1 x n,u n Fig. 1: Implemenaion of he proposed conracs: he conrols of loads can be decenralized wih a broadcas of price (LMP) informaion, while a cenralized monioring is required. he compensaions are provided a he end of he conrac period. be decenralized wih a broadcas of he price informaion, i.e., he local conroller in which he opimal conrol sraegy is programmed is sufficien for he implemenaion of he conrac as depiced in Fig. 1. On he oher hand, he load-serving eniy sill needs o monior he sae, he conrol and he forecas error for each cusomer o ensure ha each cusomer follows he opimal conrol sraegy wrien in he conrac. he oal power consumpion of each cusomer moniored by a smar meer can be used o compue he forecas error. In addiion, he sensors for loads such as hermosas provide he sae and conrol informaion. he moniored informaion could be ransferred o he load-serving eniy hrough a one-way daa connecion such as he Inerne. he informaion gahered by he monioring is also used o compue he opimal compensaion provided o each cusomer a he end of he conrac period. V. APPLICAION O DIREC LOAD CONROL FOR FINANCIAL RISK MANAGEMEN In his secion, we apply he proposed risk-limiing dynamic conracs o direc load conrol. he performance and usefulness of he novel direc load conrol program for financial risk managemen are demonsraed using he daa of LMPs in he ERCO and he elecric energy consumpion of cusomers in Ausin, exas. A. Daa and Seing We consider a scenario in which each cusomer provides one air condiioner for he proposed direc load conrol program. We use he EP model (6) for he cusomer s indoor emperaure dynamics given in Example 1. he se of feasible conrol values is chosen as U i := {, 2}, assuming ha cusomer i s air condiioner consumes kw in is OFF sae and 2kW in is ON sae. he model parameers are chosen as α i =.1 and κ i = 1.5, which are calculaed based on he Residenial module user s guide from GridLAB-D and are physically reasonable [48]. Cusomer i s comfor level is chosen as (11) in Example 2, wih ω i =.15. We se he cusomer s desirable indoor emperaure range, [Θ, Θ], as [2 C, 22 C]. We choose he conrac period as [1h, 18h]. We use cusomers elecric energy consumpion daa in Ausin, exas [49] o esimae he load profile l i () and he diffusion coefficien σ i () in (1). he load l i () is chosen as he mean value of he cusomer i s power consumpion