Incentive Design and Utility Learning via Energy Disaggregation

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1 Preprints of the 19th Word Congress The Internationa ederation of Automatic Contro Incentive Design and Utiity Learning via Energy Disaggregation Liian J. Ratiff Roy Dong Henrik Ohsson S. Shankar Sastry University of Caifornia, Berkeey, Berkeey, CA USA {ratiff, roydong, ohsson, Abstract: The utiity company has many motivations for modifying energy consumption patterns of consumers such as revenue decouping and demand response programs. We mode the utiity company consumer interaction as a reverse Stackeberg game and present an iterative agorithm to design incentives for consumers whie estimating their utiity functions. Incentives are designed using the aggregated as we as the disaggregated (device eve) consumption data. We simuate the iterative contro (incentive design) and estimation (utiity earning and disaggregation) process for exampes. Keywords: Game Theory, Economic Design, Energy Management Systems 1. INTRODUCTION Currenty, most eectricity distribution systems ony provide aggregate power consumption feedback to consumers, in the form of a energy bi. Studies have shown that providing device-eve feedback on power consumption patterns to energy users can modify behavior and improve energy efficiency (Gardner and Stern, 2008; Laitner et a., 2009). The current infrastructure ony has sensors to measure the aggregated power consumption signa for a househod. Even advanced metering infrastructures currenty being depoyed have the same restriction, abeit at high resoution and frequency (Arme et a., 2013). Additionay, depoying pug-eve sensors woud require entering househods to insta these devices. Methods requiring pug-eve sensors are often referred to as intrusive oad monitoring, and the network infrastructure required to transmit high resoution, high frequency data for severa devices per househod woud be very costy. A ow cost aternative to the depoyment of a arge number of sensors is non intrusive oad monitoring. We consider the probem of nonintrusive oad monitoring, which, in the scope of this paper, refers to recovering the power consumption signas of individua devices from the aggregate power consumption signa avaiabe to our sensors. This is aso sometimes referred to as energy disaggregation, and we wi use the two terms interchangeaby. This probem has been an active topic of research atey. Some works incude The work presented is supported by the NS Graduate Research eowship under grant DGE , NS CPS:Large:ActionWebs award number , TRUST (Team for Research in Ubiquitous Secure Technoogy) which receives support from NS (award number CC ), and ORCES (oundations Of Resiient CybErphysica Systems), the European Research Counci under the advanced grant LEARN, contract , a postdoctora grant from the Sweden-America oundation, donated by ASEA s eowship und, and by a postdoctora grant from the Swedish Research Counci. Dong et a. (2013a); roehich et a. (2011); Johnson and Wisky (2012). We propose that the utiity company shoud use incentives to motivate a change in the energy consumption of consumers. We assume the utiity company cares about the satisfaction of its consumers as we as atering consumption patterns, but it may not be abe to directy observe the consumption patterns of individua devices or a consumer s satisfaction function. In brief, the probem of behavior modification in energy consumption can be understood as foows. The utiity company provides incentives to myopic energy consumers, who seek to maximize their own utiity by seecting energy consumption patterns. This can be thought of as a contro probem for the utiity company. Additionay, the utiity company does not directy observe the energy consumption patterns of individua devices, and seeks to recover it from an aggregate signa using energy disaggregation. This can be thought of as an estimation probem. urther, the consumer does not report any measure of its satisfaction directy to the utiity. Thus, it must be estimated as we. There are many motivations for changing energy consumption patterns of users. Many regions are beginning to impement revenue decouping poicies, whereby utiity companies are economicay motivated to decrease energy consumption (Eom, 2008). Additionay, the cost of producing energy depends on many variabes, and being abe to contro demand can hep aeviate the costs of inaccurate oad forecasting. Demand response programs achieve this by controing a portion of the demand at both peak and off-peak hours (Mathieu et a., 2012). We propose a mode for how utiity companies woud design incentives to induce the desired consumer behavior. In this paper, we consider three cases of utiity earning and incentive design. In the first, the utiity company designs an incentive based entirey on the aggregate power consumption signa. We propose an agorithm to estimate Copyright 2014 IAC 3158

2 19th IAC Word Congress the satisfaction function of the consumer based on the consumer s aggregated power consumption signas in Section 3. Then, in Section 4.1, we consider the case where the utiity company knows the power consumption signa of individua devices and an unknown satisfaction function. inay, in Section 4.2, we consider the case when the utiity company ony has access to the aggregated power consumption signa, and uses an energy disaggregation agorithm to recover the power consumption of individua devices. This disaggregated signa is used to aocate incentives, but the resuts wi depend on the accuracy of our estimator, the energy disaggregation agorithm. We concude the paper by showing the resuts from simuations of two exampes of designing incentives whie estimating the consumer s satisfaction function in Section 5. inay, in Section 6 we make concuding remarks and discuss future research directions. {γi }D i=1 y 3. INCENTIVE DESIGN USING AGGREGATE POWER SIGNAL We cast the utiity consumer interaction in a reversed Stackeberg game framework in which the utiity company is the eader and the consumer is the foower (see igure 1). The eader s true utiity is assumed to be given by JL (v, y) = g(y) v + βf (y) (4) where g( ) is a concave function of the consumer s energy usage y over a biing period, v is the vaue of the incentive paid to the consumer, f : Y R is the consumer s satisfaction function for energy consumption which we assume is concave and β is a mutipying factor capturing the degree of benevoence of the utiity company. We assume that v V = [0, v ] since the utiity company shoud not take additiona money away from the consumer on top of the cost of their usage and v shoud be ess than some maxima amount the eader is wiing to pay to the consumer v. Simiary, et y Y = [0, y ] where y is the upper bound on the aowed energy usage and et Y = (0, y ). Utiity Company {y i }D i=1 Incentive Utiity Learning/ NILM ig. 1. Cosing the Loop: Behavior modification via incentives γi is a contro probem. The consumer decides when to use devices resuting in device eve consumption yi. Non intrusive oad monitoring (NILM) is used to estimate probem device eve usage y i. Simiary, utiity earning is an estimation probem 2. INCENTIVE DESIGN PRELIMINARIES A reverse Stackeberg game is a hierarchica contro probem in which sequentia decision making occurs; in particuar, there is a eader that announces a mapping of the foower s decision space into the eader s decision space, after which the foower determines his optima decision variabes (Groot et a., 2012). Both the eader and the foower wish to maximize their pay off determined by the functions JL (v, y) and J (v, y) respectivey. The eader s decision is denoted v; the foower s decision, y; and the incentive, γ : y 7 v. The basic approach to soving the reversed Stackeberg game is as foows. Let v and y take vaues in V R and Y R, respectivey; JL : R R R; J : R R R. We define the desired choice for the eader as (v d, y d ) = arg max JL (v, y) v V, y Y }. (1) v,y The incentive probem can be stated as foows: Probem 1. ind γ : Y V, γ Γ such that arg max J (γ(y), y) = y d y Y (2) γ(y d ) = v d (3) where Γ is the set of admissibe incentive mechanisms. In a reguated market with revenue decouping in pace, a simpified mode may consider g(y) = y (5) representing the fact that the utiity wants the consumer to use ess energy. Simiary, if the utiity company has aspirations to institute a demand response program, a simpified mode may consider 2 (6) g(y) = y y ref where y ref is the reference signa prescribed by the demand response program. The consumer s true utiity is assumed to be J (γ(y), y) = py + γ(y) + f (y) (7) where p is the price of energy set and known to a and γ : Y R is the incentive mechanism. Thus, the consumer soves the optimization probem max{j (γ(y), y) y Y }. (8) y We assume that the consumer is a househod who is not strategic in the sense that they take the incentive γ and the price p and optimize their utiity function without strategicay choosing y. In particuar, we assume that the consumer is myopic in that he does not consider past or future incentives in his optimization probem. Incentives are designed by soving Probem 1 where we assume Γ to be the set of quadratic poynomias from Y to R. The eader does not know the foower s satisfaction function f ( ), and hence, must estimate it as he soves the incentive design probem. We wi use the notation fˆ for the estimate of the satisfaction function and JˆL and Jˆ for the payer s cost functions using the estimate of f. We propose an agorithm for iterativey estimating the agent s satisfaction function and choosing the incentive γ( ). We do so by using a poynomia estimate of the agent s satisfaction function at each iteration and appying 3159

3 19th IAC Word Congress first-order optimaity conditions. The use of more genera sets of basis functions is eft for future research. Suppose that γ (0) and γ (1) are given a priori. At each iteration the eader issues an incentive and observes the foower s reaction. The eader then uses the observations up to the current time aong with his knowedge of the incentives he issued to estimate the foower s utiity function. ormay, at the k-th iterate the eader wi observe the foower s reaction y (k) to a deivered incentive γ (k) where we suppress the dependence of the incentive on y. The foower s reaction y (k) is optima with respect to J (γ (k) (y), y) subject to y (k) Y. We use the observations y (0),..., y (k) to estimate the parameters in the foower s satisfaction function given by j ˆf (k) (y) = α i y i+1 (9) i=0 where j is the order of the poynomia estimate to be determined in the agorithm and we restrict α = (α 1,..., α j ) A a convex set, e.g. A = R j+1 +. As in Keshavarz et a. (2011), we assume that an appropriate constraint quaification hods and we use Kharush- Khun-Tucker (KKT) conditions to define a notion of approximate optimaity. Thus, we can aow for some error in the estimation probem either from measurement noise or suboptima consumer choice. In particuar, for each i = 0,..., k et γ (i) = γ (i) (y (i) ) and define y (i) r (i) ineq = (g (γ (i), y (i) )) y (i) +, = 1, 2 (10) r (i) stat(α, λ (i) ) = J (γ (i) y (i), y (i) ) + 2 =1 λ (i) g i(γ (i), y (i) ) y (i) (11) r comp(λ (i) (i) ) = λ (i) g (γ (i), y (i) ), = 1, 2 (12) y (i) where ( ) + = max{, 0}, g 1 (γ, y) = y 0 and g 2 (γ, y) = y ȳ 0 (13) with Lagrange mutipiers λ (i) = (λ (i) 1, λ(i) 2 ). Then, for {(γ (i), y (i) )} k i=0, we can sove { k min α,λ i=0 } φ(r stat, (i) r comp) (i) α A, λ (i) 0, i = 0,..., k (P-2) where the inequaity for λ (i) is eement-wise and φ : R R 2 R + is a nonnegative convex penaty function (e.g. any norm on R R 2 ) satisfying φ(x 1, x 2 ) = 0 {x 1 = 0, x 2 = 0}. (14) The optimization probem (P-2) is convex since r (i) r (i) stat and comp are inear in α and λ (i) and the constraints are convex. If we sove (P-2) and φ is zero at the optima soution, r (i) stat and r comp (i) are zero at the optima soution for each i. If, in addition, r (i) ineq is zero at the optima soution for each i, then the estimate ˆf (k) at iteration k is exacty consistent with the data. If y (i) Y, the probem simpifies to a checking a inear agebra condition. Indeed, consider Ĵ (γ(y), y) = py + γ(y) + ˆf(y). (15) In the case that ˆf is concave and under our assumption that the foower is rationa and hence pays optimay, the observation y (i) is a goba optimum at iteration i. Otherwise, the observation y (i) is a oca optimum; the foower pays myopicay. In both cases, we use the necessary condition (Bertsekas, 1999) Ĵ (i) (γ(y(i) ), y (i) ) = 0 (16) for each of the past iterates i {0,..., k} to determine estimates of the coefficients in ˆf (k). At the k-th iteration, we have data {(γ (i), y (i) )} k i=0. Since we require γ (i) Γ, we can express each γ (i) as γ (i) (y) = ξ (i) 1 y + ξ(i) 2 y2. (17) Then, using Equation (16), we define b (i) = p (ξ (i) 1 + 2ξ (i) 2 y(i),d ) (18) and ỹ (i) j = [ 1 2y (i) (j + 1)(y (i) ) j] (19) for i {0,..., k}. We want to find the owest order poynomia estimate of f given the data. We do so by checking if b (k) range(y (k) ) where ỹ (0) j b (0) Y (k) =., b(k) =. (20) ỹ (k) j b (k) starting with j = 2 and increasing it unti (20) is satisfied or we reach j = k. Suppose that it is satisfied at j = N, 2 N k. Then, we estimate ˆf (k) to be an (N + 1)-th order poynomia. We determine α i for i {0,..., N} by soving b (k) Y (k) α = 0, where α = [α 0... α N ] T. (21) If b (k) / range(y (k) ) for any j [2, k], we terminate. Our agorithm prescribes that the eader check if y (i) Y for each i. If this is the case, then he sha find the minimum order poynomia given the data as described above. On the other hand, he sha sove the convex probem (P-2). Using {α i } j i=0, Ĵ (k) (k) L, and Ĵ, the eader soves the incentive design probem. That is, the eader first soves (v (k+1),d, y (k+1),d ) = arg min v V,y Y = arg min v V,y Y (v, y) (22) { g(y) v + β ˆf } (k) (y) Ĵ (k) L Then, the eader finds γ (k+1) Γ such that (23) arg max Ĵ (k) (γ(k+1) (y), y) = y (k+1),d (24) y Y γ (k+1) (y (k+1),d ) = v (k+1),d (25) If y (k+1),d Y, then since we restrict γ (k+1) to be of the form (17) the above probem reduces to soving A (k+1) ξ (k+1) = b (k+1) where 3160

4 19th IAC Word Congress [ ] [ ] A (k+1) 1 2y = (k+1),d y (k+1),d (y (k+1),d ) 2, ξ (k+1) ξ (k+1) = 1 ξ (k+1), (26) 2 and [ ] b (k+1) p α0 2α = 1 y (k+1),d. (27) v (k+1),d If b (k+1) range(a (k+1) ) then a soution ξ (k+1) exists and if A (k+1) is fu rank, the soution is unique. Otherwise, if y (k+1),d / Y, we terminate the agorithm. Remark 1. The agorithm is motivated by the case when the consumer s satisfaction function is a poynomia of order k and the utiity company does not know k, by foowing the agorithm past even k + 1 iterations, the utiity company wi be paying optimay. Aternativey, if incentives γ (i) were chosen randomy, the utiity company woud not know when to stop choosing random γ (i) s; thus, after k + 1 iterations woud begin paying suboptimay. Proposition 1. Let f be poynomia of order k+1, y (0) Y and γ (0), γ (1) be given a priori. Suppose that at each iteration of the agorithm b () range(y () ), rank(y () ) = + 1, y (i) Y and b (+1) range(a (+1) ). Then, after k iterations the satisfaction function is known exacty and the incentive γ (k+1) induces the consumer to use the desired contro. Proposition 2. Suppose that f is poynomia up to order k+1 and that the eader has k+1 historica measurements γ ( k),..., γ (1), y ( k),..., y (1) (28) such that Y (k) is fu rank where y (i) Y for i = 0,..., k, then the eader can estimate the foower s satisfaction function exacty and if there exists an incentive γ (k+1), then it induces the desired equiibrium. We concude this section by providing an exampe of the iterative process when f is a concave function. Exampe 1. irst, we suppose that γ (0), γ (1) Γ are chosen a priori and are parameterized as foows: γ (i) (y) = ξ (i) 1 y + ξ(i) 2 y2 (29) for i {0, 1}. Then, the procedure goes as foows. The eader issues γ (0) and observes y (0). Subsequenty, he issues γ (1) and observes y (1). Suppose y (0), y (1) Y. The eader determines α 1, α 0 in the estimation of ˆf(y) = α1 y 2 + α 0 y by computing the derivative of Ĵ (0) (γ(0) (y), y) and (γ(1) (y), y) with respect to y, evauating at y (0) and Ĵ (1) y (1) and equating to zero, i.e. he soves p 2y (0) + 2(α 1 + ξ (0) 2 )y(0) + α 0 + ξ (0) 1 = 0 (30) p 2y (1) + 2(α 1 + ξ (1) 2 )y(1) + α 0 + ξ (1) 1 = 0 (31) for α 0 and α 1. If either y (0) or y (1) are on the boundary of Y, then the eader soves (P-2) for α = (α 1, α 0 ). Using α 0, α 1, the eader soves the foowing incentive design probem for γ (2). irst, find (v (2),d, y (2),d ) V Y such that Ĵ (2) L (v, y) = y v + α 1y 2 + α 0 y (32) is maximized. Since we restrict to quadratic incentives, we parameterize γ (2) as in Equation (29) with i = 2. Now, given the utiity Ĵ (2) (γ(2) (y), y), we find ξ (2) 1, ξ(2) 2 such that arg max y Y Ĵ (2) (y; ξ(2) 1, ξ(2) 2 ) = y(2),d (33) ξ (2) 1 y(2),d + ξ (2) 2 (y(2),d ) 2 = v (2),d (34) Assuming that y (2),d Y, it wi be an induced oca maxima under the incentive γ (2). Hence, Equation (33) can be reformuated using the necessary condition y Ĵ (2) (y(2),d ; ξ (2) 1, ξ(2) 2 ) = 0. (35) Now, Equations (34) and (35) give us two equations in the two unknowns ξ (2) 1, ξ(2) 2 that can be soved; indeed, p + ξ (2) 1 + α 0 + 2(ξ (2) 2 + α 1 )y (2),d = 0 (36) ξ (2) 1 y(2),d + ξ (2) 2 (y(2),d ) 2 = v (2),d (37) Soving these equations gives us the parameters for γ (2). Now, the eader can issue γ (2) to the foower and observe his reaction y (2). The eader can then continue in the iterative process as described above. 4. DEVICE LEVEL INCENTIVE DESIGN USING DISAGGREGATION ALGORITHM In a manner simiar to the previous section, we consider that the consumer s satisfaction function is unknown. However, we now consider that the utiity company desires to design device eve incentives. We remark that the utiity company may not want to incentivize every device; the process we present can be used to target devices with the highest consumption or potentia to offset inaccuracies in oad forecasting. 4.1 Exact Disaggregation Agorithm We first describe the process of designing device eve incentives assuming the utiity company has a disaggregation agorithm in pace which produces no error. That is, they observe the aggregate signa and then appies their disaggregation agorithm to get exact estimates of the device eve usage y for {1,..., D} where D is the number of devices. The utiity company has the true utiity function D J L (v, y) = g (y ) v + β f (y ) (38) =1 and the consumer has the true utiity function D J (γ(y), y) = py + γ (y ) + f (y ). (39) =1 The utiity company coud choose ony to incentivize specific devices such as high consumption devices. This fits easiy into our framework; however, for simpicity we just present the mode in which incentives are designed for each device. The impicit assumption that the payer utiities are separabe in the devices aows us to generaize the agorithm presented in the previous section. Let us be more precise. We again assume that γ (0), γ (1) for {1,..., D} are given a priori. At the k-th iteration the utiity company issues an incentive γ (k) for each device {1,..., D} and observes the 3161

5 19th IAC Word Congress aggregate signa y (k). Then they appy a disaggregation agorithm to determine the device eve usage y (k) for {1,..., D}. The utiity company forms an estimate of the consumer s device eve satisfaction function j ˆf (k) (y ) = α i, y i+1 (40) i=0 and then soves the probem of finding the α i, s by soving for α = (α 0,,..., α j, ) as in the previous section for each device {1,..., D}. Proposition 3. or {1,..., D}, et f be poynomia up to order k + 1, γ (0), γ (1) be given a priori, and y (0), y (1) Y. Suppose that at each iteration of the agorithm b (m) range(y (m) ), rank(y (m) ) = m + 1, y (m) Y (m+1), and b A (m+1) for each {1,..., D}. Then, after k = max k (41) {1,...,D} iterations, the satisfaction function is known exacty and the incentives γ (k +1) induce the desired equiibrium. Note that the notation ( ) indicates the object defined in Section 3 for the -th device. 4.2 Disaggregation Agorithm with Some Error Now, we consider that the eader has some error in his estimate of the device eve usage due to inaccuracies in the disaggregation agorithm, i.e. the eader determines ŷ such that y ŷ ε where ε > 0 is the resuting error from the estimation in the disaggregation agorithm. Bounds on ε can be determined by examining the fundamenta imits of non intrusive oad monitoring agorithms (Dong et a., 2013b). We again assume that γ (0), γ (1) for {1,..., D} are given a priori. oowing the same procedure as before, at the k-th iterate the eader wi issue γ (k) for {1,..., D} and observe y (k). Then appy a disaggregation agorithm to determine ŷ (0) where y ŷ ε (42) for {1,..., D}. The incentive design probem foows the same steps as provided in the previous section with the exception that the y s are repaced with the estimated ŷ s and we toerate an error in soving for the minima poynomia estimate of f. 5. NUMERICAL EXAMPLES We simuate two exampes of designing incentives whie estimating the consumer s satisfaction function. In both exampes we assume a unit price per unit of energy, i.e. p = Aggregate Signa and Log Satisfaction unction We simuate a system in which the consumer has the true utiity given by J (γ(y), y) = py + γ(y) + f(y) (43) Satisfaction ˆf (2) f y d y y ig. 2. Estimated satisfaction function ˆf (2) and true satisfaction function f. The true response y = 6.56 and the desired response y d = 6.5. Notice that the sope of the estimated satisfaction function and the sope of the true satisfaction function are roughy equa at y d and y. where the satisfaction function is f(y) = 10 og(y + 1). (44) We assume the utiity company is in a reguated market; hence wants the consumer to consume ess. Thus, the utiity company has utiity function J L (v, y) = y v + βf(y) (45) where the benevoence factor is β = We et ȳ = v = 100. We choose two concave incentive function γ (0) (y) and γ (1) (y) defined as foows: γ (0) (y) = y y and γ (1) (y) = y y. We use the agorithm presented in Section 3 to design incentives whie estimating α 0 and α 1. We simuate the utiity company issuing γ (0) and then γ (1) where the consumer chooses his optima response to each of the incentives. The responses are y (0) = 5.29 and y (1) = After two iterations, we get a reasonabe approximation of the true f and a quadratic incentive γ (2) ; ˆf (2) (y) = 2.57y 0.093y 2, γ (2) (y) = 0.33y 0.05y 2. (46) The optima power usage under the incentive γ (2) is y = 6.56 and the desired power usage is y d = 6.5. It is cear that the utiity company coud do better if he new the true satisfaction function. igure 2 shows ˆf (2) (y) and f(y). It is important to notice that y is neary equa to y d and at these two points the sope of ˆf (2) is approximatey equa to that of the true f. This indicates that ˆf (2) is a good estimate of f. 5.2 Disaggregated Signa We simuate a system in which the consumer s the true utiity 10 J (γ(y), y) = py + γ (y ) + f (y ) (47) =1 where the satisfaction functions f (y ) are exacty quadratic for each device {1,..., 10}; f (y ) = α 1, y 2 + α 0, y (48) The utiity company s utiity is given by 3162

6 19th IAC Word Congress αi,1 α i,1 / α i, α 1,1 α 2, Iteration ig. 3. Reative error in estimate of α i,1 s for device 1 with disaggregation error bound ε = αi is the true vaue. The reative error eventuay decreases beow the noise bound ε = J L (v, y) = y v + β f (y ) (49) =1 where the benevoence factor (i.e. a representation of how much the utiity company cares about the satisfaction of the consumer) is β = 1 for each. The utiity company must disaggregate the aggregated energy signa y giving rise to estimates ŷ. If ŷ = y, i.e. there is no error in the disaggregation agorithm, then after two iterations the utiity company woud know the satisfaction function of each device exacty. Let s expore the case when the disaggregation agorithm has ε error. In our exampes we randomy generate noise within a given ε bound and add that to the true y i s to simuate the error in the disaggregation step resuting from the disaggregation agorithm. igure 3 shows the reative error on the estimates of α i,1 for i {1, 2} as a function of the iteration. The reative error for other devices are simiar. We used the error bound ε = 0.15 for the disaggregation error. The reative error decreases as the number of iterations increase. As we iterate the noise introduced via disaggregation has minima impact on the estimate of α i, for i = {1, 2} and {1,..., D}. We note that the designed incentive for this probem converges to zero as we increase the iterations and the impact of the noise is minimized. It becomes zero since the benevoence factor is β = 1 and the price p = 1; hence, the agent and the eader have the same utiity functions after the eader earns the agent s satisfaction function. As we increase the noise threshod ε, the estimation of α i, degrades. 6. DISCUSSION AND UTURE WORK We modeed the utiity company consumer interaction as a reversed Stackeberg game. We defined a process by which the utiity company can jointy estimate the consumer s utiity function and design incentives for behavior modification. Whether the utiity company is interested in inducing energy efficient behavior or creating an incentive compatibe demand response program, the procedure we present appies. We are studying fundamenta imits of non intrusive oad monitoring in order to determine precise bounds on the payoff to the utiity company when a disaggregation agorithm is in pace and incentives are being designed. We seek to understand how these fundamenta imits impact the quaity of the incentive design probem as we as how they can be integrated into a stochastic contor framework for incentive design when faced with non-strategic agents with unknown preferences. The eectrica grid is a socia cyber-physica system (S- CPS) with human actors infuencing the trajectory of the system. Inherent to the study of S-CPS s are privacy and security considerations. We remark that consumers may consider their satisfaction function to be private information. We are currenty exporing the design of privacy aware mechanisms for ε incentive compatibe probems (Nissim et a., 2012). REERENCES Arme, K.C., Gupta, A., Shrimai, G., and Abert, A. (2013). Is disaggregation the hoy grai of energy efficiency? The case of eectricity. Energy Poicy, 52, Bertsekas, D.P. (1999). Noninear programming. Athena Scientific. Dong, R., Ratiff, L., Ohsson, H., and Sastry, S.S. (2013a). Energy disaggregation via adaptive fiter. In Proceedings of the 50th Aerton Conference on Communication, Contro, and Computing. Dong, R., Ratiff, L., Ohsson, H., and Sastry, S.S. (2013b). undamenta imits of non intrusive oad monitoring. arxiv: v1. Eom, J. (2008). 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