A Cooperative Demand Response Scheme Using ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS, VOL. 11, NO. 6, DECEMBER 2015

A Cooperative Demand Response Scheme Using Punishment Mechanism and Application to Industrial Refrigerated Warehouses Kai Ma, Guoqiang Hu, Member, IEEE, and Costas J. Spanos, Fellow, IEEE

Abstract—This paper proposes a cooperative demand response (CDR) scheme for load management in smart grid. The CDR scheme is formulated as a constrained optimization problem that generates a Pareto-optimal response strategy profile for consumers. Comparing with the noncooperative response strategy (i.e., Nash equilibrium) obtained from the one-shot demand management game, the Pareto-optimal response strategy reduces the electricity costs to the consumers. We further develop an incentivecompatible trigger-and-punishment mechanism to avoid the noncooperative behaviors of the selfish consumers. Furthermore, the CDR scheme is applied to achieve load management of industrial refrigerated warehouses. To implement the CDR scheme in largescale systems, we group the refrigerated warehouses into clusters and utilize the CDR scheme within each cluster. Numerical results demonstrate that the CDR scheme can reduce the electricity costs, drop the electricity prices, and curtail the total energy consumption in comparison with the noncooperative demand response scheme. Index Terms—Cooperation, demand response (DR), industrial refrigerated warehouses, Pareto optimality, punishment mechanism, smart grid.

N OMENCLATURE N , N, i M, M, m Nd Nc Nm k l ld

Set, number, and index of consumers. Set, number, and index of clusters. Set of noncooperative consumers. Set of cooperative consumers. Set of consumers within cluster m. Index of time slots. Strategy profile of consumers. Strategy profile of consumers with noncooperative behaviors.

Manuscript received September 11, 2014; revised March 25, 2015; accepted April 23, 2015. Date of publication May 07, 2015; date of current version December 02, 2015. This work was supported in part by the Republic of Singapore’s National Research Foundation through the Berkeley Education Alliance for Research in Singapore (BEARS) for the Singapore–Berkeley Building Efficiency and Sustainability in the Tropics Program (SinBerBEST), and in part by the National Natural Science Foundation of China under Grant 61203104. BEARS has been established by the University of California, Berkeley, as a Center for Intellectual Excellence in Research and Education in Singapore. Paper no. TII-14-1399. K. Ma and G. Hu are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]; [email protected]). C. J. Spanos is with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TII.2015.2431219

lc Si li ˆli limin limax lmin lmax lid lic p(l) pd (ld ) pt Vi Vit Vim Viq Vip V¯i Vir Ui Uit Uim UiNE Uic ΔU cn ΔU cd ΔU e ¯e ΔU ¯ Ui β

Socially optimal strategy profile of consumers. Set of possible strategies of consumer i. Actual energy consumption of consumer i (kWh). Normal energy consumption of consumer i (kWh). Minimal energy consumption of consumer i (kWh). Maximal energy consumption of consumer i (kWh). Minimal energy consumption of the consumers (kWh). Maximal energy consumption of the consumers (kWh). Energy consumption of noncooperative consumer i (kWh). Socially optimal energy consumption of consumer i (kWh). Electricity price ($/kWh). Electricity price with noncooperative behaviors ($/kWh). Electricity price for price-taking consumers ($/kWh). Total costs to consumer i ($). Total costs to price-taking consumer i ($). Total costs to consumer i within cluster m ($). Discomfort costs to consumer i ($). Electricity payments of consumer i ($). Average costs to consumer i over multiple time slots ($). Costs to refrigerated warehouse i ($). Payoff of consumer i ($). Payoff of price-taking consumer i ($). Payoff of consumer i within cluster m ($). Payoff of consumer i obtained from one-shot demand management game ($). Payoff of consumer i obtained from cooperative demand response scheme ($). Loss of social optimality due to false alarm ($). Loss of social optimality due to false detection ($). Loss of social optimality ($). Average loss of social optimality ($). Average payoff of consumer i over multiple time slots ($). Probability of detecting noncooperative behavior.

1551-3203 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

MA et al.: CDR SCHEME USING PUNISHMENT MECHANISM AND APPLICATION TO INDUSTRIAL REFRIGERATED WAREHOUSES

β min , β max Minimal and maximal probability of detecting noncooperative behavior. δ Discount factor. Lower bound of discount factor. δ min L Forecast demand (kWh). Forecast demand of cluster m (kWh). Lm ΔL Change of total energy consumption (kWh). ¯ ΔL Average change of total energy consumption with noncooperative behaviors (kWh). η Detection threshold (kWh). η max , η min Maximal and minimal detection thresholds (kWh). Φ(η) False alarm probability. Ψ(η) False detection probability. α Probability of noncooperative behaviors. Critical probability of noncooperative behaviors. αc qˆ Indicator of detection results. q Indicator of the existence of noncooperative behaviors. T Number of time slots for punishment. Minimal number of time slots for punishment. T min Time slot at which the punishment starts. T0 Actual indoor temperature set point (◦ F). Qin i in ˆ Q Desired indoor temperature set point (◦ F). i out Outdoor temperature (◦ F). Qi Cost coefficient. γi Thermal parameters. ωi , θ i Pricing parameter and base price (cents/kWh). λ, p0 I. I NTRODUCTION

D

EMAND response (DR) is defined as the changes in electricity usage by end-use consumers in response to the power grid needs from electricity markets [1]. In general, there are two categories of DR schemes: 1) incentive-based scheme; and 2) price-based scheme [2]. The incentive-based scheme includes direct load control, interruptible programs, demand bidding, emergency DR, capacity markets, and ancillary services markets. For the direct load control programs, the energy provider manages the loads of the participating consumers directly [3]–[5]. In the interruptible programs, the consumers receive the incentive payments or rate discounts so as to reduce their predefined values [6]. In the demand bidding programs, the consumers bid on specific load reductions and curtail their loads according to the amount specified in the bid [7], whereas in the emergency DR programs, the participating consumers are paid for measured load reductions under the emergency conditions [8]. Capacity market programs require the consumers to provide predefined load reductions as needed [9], and ancillary services programs are designed to provide regulation or load following services [10]. For the price-based DR scheme, the energy provider adjusts the loads by flexible pricing, such as critical peak pricing [11], real-time pricing [12], and regulation pricing [13]. To support the DR, an advanced metering infrastructure (AMI) is developed to collect the energy consumption and announce the electricity price [14], [15]. Typically, there are two types of consumers for pricebased DR: 1) price-taking (PT) consumers [16]–[18]; and 2) price-anticipating (PA) consumers [19]–[25]. The PT

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consumers assume that their energy consumption cannot affect the electricity price, whereas the PA consumers believe that their energy consumption can change the electricity price. In general, the PA consumers refer to large energy consumers such as industrial facilities, commercial buildings, plug-in hybrid electric vehicles (PHEVs), and data centers. It was proved that both the industrial facilities and the commercial buildings have large potential in DR [26]–[28], the PHEVs can provide DR by charging or discharging [29], [30], and the data centers can be used to achieve DR via pricing approaches [31]–[33]. Recently, game theory has been used for studying the DR of PA consumers. For example, noncooperative games were utilized to study the cost minimization of interactive consumers [19]–[21] and the charging control of plug-in electric vehicles [22], [23]. Stackelberg games were employed to model the interactions between the consumers and the utility companies [24], [25], [34]. In fact, neither the Nash equilibrium nor the Stackelberg equilibrium are Pareto optimal in the two game models. Generally, the Pareto optimality is used for evaluating the efficiency of resource allocation in economic systems. If the resource allocation in any economic system is not Pareto optimal, there is potential to improve the Pareto efficiency. Repeated game was utilized to improve the Pareto efficiency for the DR of PA consumers [35]. However, two critical problems have not been addressed: How to detect the noncooperative behavior and what is the punishment strength to stop the noncooperative behavior? In [36], the cooperative demand response (CDR) was achieved by using the repeated game, and a punishment mechanism was designed to avoid the noncooperative behaviors of the buildings. Furthermore, the noncooperative behavior detection method was developed, and the punishment strength was given to stop the noncooperative behavior under accurate measurable data. In this study, the CDR scheme is formulated as a social optimization problem. Specifically, the punishment mechanism is developed under inaccurate measurable data, and the DR scheme is applied to achieve load management of industrial refrigerated warehouses. The novelty of this work is twofold. 1) We study the noncooperative behavior detection with missing measurable data and give the optimal detection threshold to minimize the loss of social optimality. 2) We propose a cluster-based method to transform the social optimization problem to several suboptimization problems and obtain a suboptimal solution of the social optimization problem. This paper is organized as follows. Some preliminaries are given in Section II. In Section III, the CDR is formulated as a social optimization problem, and the socially optimal response strategy is obtained. In Section IV, an incentivecompatible trigger-and-punishment mechanism is developed to avoid the noncooperative behaviors of the selfish consumers. In Section V, the CDR scheme is applied to achieve load management of industrial refrigerated warehouses with heating, ventilation, and air conditioning (HVAC) systems, and a heuristic method is developed to obtain the suboptimal response strategy by grouping the refrigerated warehouses into different clusters. Numerical results are given in Section VII, and conclusions are summarized in Section VII.

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II. P RELIMINARIES A. Noncooperative Game and Nash Equilibrium Definition 1: [37] A noncooperative game is defined as a triple G = {N , (Si )i∈N , (Ui (l))i∈N }, where N = {1, 2, . . . , N } is the set of active players participating in the game (1) Si = li li ∈ limin , limax is the set of possible strategies that player i can take, and Ui (l) is the payoff function. Definition 2: [37] For a noncooperative game G = {N , (Si )i∈N , (Ui (l))i∈N }, a vector of strategies ∗ ) is a Nash equilibrium if and only if l∗ = (l1∗ , l2∗ , . . . , lN ∗ ∗ ∗ ) for all i ∈ N and any other li ∈ Si , Ui (li , l−i ) ≥ Ui (li , l−i where l−i = (l1 , l2 , . . . , li−1 , li+1 , . . . , lN ) denotes the set of strategies selected by all the players except for player i, (li , l−i ) = (l1 , l2 , . . . , li−1 , li , li+1 , . . . , lN ) denotes the strategy profile, and Ui (li , l−i ) is the resulting payoff of the player i given the strategies of the other players. B. Taguchi Loss Function Definition 3: [38] The Taguchi loss function captures the cost to society due to the manufacture of imperfect products. The loss function is defined as V = τ (y − yˆ)2

A0 . Δ20

the power function [43], and the weighted linear function [44], [45]. The discomfort cost function is defined as Viq (li , ˆli ), and the electricity payments of consumer i are denoted as Vip = p(l)li ,

(3)

III. P ROBLEM F ORMULATION We consider a demand management system composed of an energy provider and several consumers, as shown in Fig. 1. The energy provider can adjust the loads by periodically announcing the pricing curve to the consumers. We assume that the price is affected by the energy consumption of the consumers, i.e., PA consumers. According to the updated electricity price, the consumers can adjust their energy consumption to reduce the electricity costs. The electricity costs are composed of two parts: 1) the discomfort costs; and 2) the payments. Generally, the discomfort costs are increased with the change from normal energy consumption and can be denoted as a continuous, increasing, and convex function,1 such as the quadratic function [16], [21], [39], [40], the logarithmic function [41], [42],

Vi = Viq (li , ˆli ) + p(l)li .

(4)

(5)

The discomfort costs and the electricity payments usually conflict with each other, and the consumers need to make a tradeoff between them. It is shown from (5) that the energy consumption of one consumer can change the electricity price and further affect the electricity costs to the other consumers. Thus, the DR can be formulated as the following noncooperative game. Definition 4: (One-shot demand management game) A oneshot demand management game is defined as a triple G = {N , (Si )i∈N , (Ui )i∈N }, where N = {1, 2, . . . , N } is the set of active consumers participating in the game, Si is the set of possible strategies that consumer i can take, and Ui = −Vi = −Viq (li , ˆli ) − p(l)li is the payoff function. The stable solution of the one-shot demand management game is the Nash equilibrium, which can be obtained from ∂Ui /∂li = 0, i ∈ N , i.e., −dViq (li , ˆli )/dli − ∂p(l)/∂li · li − p(l) = 0, i ∈ N .

(6)

Generally, the Nash equilibrium is not a Pareto-optimal solution, and thus it is possible to improve the payoffs of all the consumers simultaneously.2 Next, we develop a CDR scheme to improve the Pareto efficiency of Nash equilibrium. The CDR scheme is formulated as a social optimization problem (P1) maximize Ui subject to Ui ≥

1 The

PA consumers generally refer to industrial or commercial consumers that have continuous aggregate loads.

i∈N

where N = {1, 2, . . . , N } denotes the set of consumers, i denotes the index of consumer, ˆli is the normal energy consumption, li is the actual energy consumption, l = {l1 , l2 , . . . , lN } is the strategy profile, and p(l) is the announced electricity price, which is assumed to be an increasing function of the total energy consumption. Then, the electricity costs to consumer i can be defined as

(2)

where y is the value of quality characteristic, yˆ is the desired value of y, V is the loss in dollars, and τ is a constant coefficient. The Taguchi loss function defines the relationship between the economic loss and the deviation of the quality characteristic from the desired value. For a product with desired value yˆ, yˆ ± Δ0 denotes the deviation limit at which some countermeasures must be undertaken. Assuming the cost of countermeasure is A0 at yˆ + Δ0 or yˆ − Δ0 , we define the constant τ as τ=

Fig. 1. Demand management system with PA consumers.

2 Improving

i∈N UiNE ,

i∈N

the payoff is equivalent to reducing the electricity costs.


where UiNE denotes the payoff of consumer i obtained from c } the one-shot demand management game. Let lc = {l1c , . . . , lN denote the socially optimal energy consumption obtained from (P1) and Uic denote the corresponding payoff of consumer i. It is easy to see that lc is a Pareto-optimal solution and Uic is not smaller than UiNE for all i ∈ N . Since the price is related to the energy consumption of all the consumers, the objective function of (P1) is a multivariable function. The condition for the concavity of the objective function is dependent on the concavity of the Hessian matrix ⎤ ⎡ 2 ∂ U/∂l12 ∂ 2 U/∂l1 ∂l2 . . . ∂ 2 U/∂l1 ∂lN ⎢ ∂ 2 U/∂l2 ∂l1 ∂ 2 U/∂l22 . . . ∂ 2 U/∂l2 ∂lN ⎥ ⎥ ⎢ H=⎢ ⎥ (7) .. .. .. .. ⎦ ⎣ . . . . 2 2 2 2 ∂ U/∂lN ∂l1 ∂ U/∂lN ∂l2 . . . ∂ U/∂lN where U = i∈N Ui . In general, H is not negative definite, and the constraints cannot contribute to a convex set. Therefore, the optimization problem (P1) is nonconvex, and the optimal solution lc is hard to obtain. In Section V, we will develop a heuristic method to obtain a suboptimal solution that meets the constraints in (P1). In the CDR scheme, some of the consumers are possible to improve their payoffs by taking the noncooperative strategies when the other consumers keep cooperative. We assume that some of the consumers (i ∈ N d ) take the noncooperative strategies while the other consumers (j ∈ N c ) keep cooperative, where N d is the set of noncooperative consumers and N c is the set of cooperative consumers. Then, the energy consumption of the noncooperative consumers can be obtained from −∂Viq (li , ˆli )/∂li − ∂pd (ld )/∂li · li − pd (ld ) = 0,

i ∈ Nd (8)

where pd (ld ) is the price when some of the consumers take the d c c noncooperative strategies and ld = {l1d , . . . , lN d , l1 , . . . , lN c }, d d where li (i ∈ N ) denotes the energy consumption of the noncooperative consumers and ljc (j ∈ N c ) denotes the energy consumption of the cooperative consumers. The corresponding payoffs of the noncooperative consumers are denoted as Uid = −Viq (lid , ˆli ) − pd (ld )lid ,

i ∈ Nd

(9)

and the payoffs of the cooperative consumers are denoted as Ujd = −Vjq (ljc , ˆlj ) − pd (ld )ljc , j ∈ N c .

(10)

For example, the noncooperative consumer can increase its payoff (9) by taking the noncooperative strategy when all of the other consumers keep cooperative, Therefore, each consumer has the motivation to take the noncooperative strategy, and the socially optimal energy consumption is not a stable solution in one-shot DR. Remark 1: To compare the CDR scheme for PA consumers with the DR scheme for PT consumers, we give the formulation of PT consumers. The electricity cost to consumer i is denoted as Vit = Viq (li , ˆli ) + pt li

(11)

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where pt is a constant price for PT consumers. Minimizing the electricity costs to consumer i, we obtain the optimal energy consumption li∗ from dViq (li , ˆli )/dli + pt = 0. Uit

(12) −Vit

= for the PT We also define the utility function consumers. Remark 2: In practice, the discomfort cost function may be a combination of multiple step functions. In that case, we can employ the continuous convex function to approximate it and obtain the optimal energy consumption for the consumers. Then, the suboptimal strategy can be obtained by approximating the optimal energy consumption to the step value. Remark 3: In the problem formulation, we assume the consumers are all adaptable. In practice, the demand management system may include nonadaptable consumers, which do not adapt to the price, i.e., the energy consumption of the nonadaptable consumers are ˆli , independent of the price. The nonadaptable consumers will increase the total loads and thus the price, because the price is an increasing function of the total loads. The increased price will further reduce the energy consumption of the other adaptable consumers. IV. T RIGGER - AND -P UNISHMENT M ECHANISM To make the socially optimal energy consumption stable, we consider giving punishments to the consumers if they adopt the noncooperative strategies. In that case, the consumers will care more about the long-term electricity costs. The average electricity costs to consumer i over multiple time slots are defined as ∞ V¯i = δ k−1 Vi (k) (13) k=1

where k is the index of the time slot and δ ∈ (0, 1) is the discount factor, which denotes how the consumers discount their future costs. In that case, the consumers not only value the current electricity costs but also the future electricity costs. Therefore, each consumer needs to keep a good reputation to avoid the increased costs in the future. Similarly, we define ¯i = −V¯i = theaverage payoff function of consumer i as U ∞ k−1 Vi (k). − k=1 δ Next, we will develop a trigger-and-punishment mechanism to avoid the noncooperative behaviors. All of the consumers are assumed to adopt the cooperative strategies in the first time slot. In the subsequent time slots (i.e., k ≥ 2), the cooperation will be maintained if all of the consumers adopt the cooperative strategies in the previous time slot. If the energy provider observes noncooperative behaviors in the previous time slot, it will keep the consumers noncooperative during the subsequent T time slots and restart the cooperation at the (T + 1)th time slot. There are two questions to be answered in designing the trigger-and-punishment mechanism: How the energy provider detects the noncooperative behaviors and what is the punishment strength that can stop the noncooperative behaviors? In the subsequent sections, we will answer these two questions and omit the time slot index k without causing confusions.

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have large influences on the accuracy of the noncooperative behaviors detection and thus the social optimality of the CDR scheme. Next, we will give the optimal detection threshold to minimize the loss of social optimality. Proposition 1: Assuming that the probability, with which one consumer performs noncooperative behavior, is α at time slot T0 , the average loss of social optimality due to the false alarm and false detection is minimal if η max , α > αc η= (16) η min , α ≤ αc where η max and η min are the maximal and minimal detection thresholds, and αc is denoted as Fig. 2. Distribution of the change of total energy consumption (i.e., ΔL) with and without noncooperative behaviors.

A. Noncooperative Behaviors Detection The noncooperative behaviors of the consumers will change the electricity price and the total energy consumption. In this section, we utilize the change of the total energy consumption3 as the indicator of the noncooperative behaviors that exist in the demand management system. The change of the total energy consumption is defined as li − lic . (14) ΔL = i∈N

i∈N

In practice, the total energy consumption is measured by the energy provider based on the AMI. It is shown that the missing meter data cause errors to the measurement of the total energy consumption [46], [47]. As shown in [46]–[48], the change (i.e., the errors in the measurement) of the total energy consumption can be assumed to follow a normal distribution N (μ, σ 2 ), where σ is the standard variance, μ = 0 if there does ¯ if there not exist any noncooperative behavior, and μ = ΔL exist noncooperative behaviors. To detect the noncooperative behaviors of the consumers, we define the detection rule as 1, if |ΔL| ≥ η qˆ = (15) 0, if |ΔL| < η where η is the detection threshold and qˆ is the detection result. Specifically, qˆ = 1 denotes that the energy provider detects the noncooperative behaviors, and qˆ = 0 denotes that the energy provider does not detect any noncooperative behavior. As shown in Fig. 2, the detection rule (15) causes false alarm and false detection. The false alarm occurs when the noncooperative behaviors are detected in the demand management system that does not have any noncooperative behavior, and the false detection occurs when the noncooperative behaviors are not detected in the demand management system that has noncooperative behaviors. Both the false alarm and false detection 3 In practice, the change of the total energy consumption is affected by the scale of the demand management system (e.g., the number of consumers). In the simulations, we will discuss it in detail.

α =1− c

ΔU cn ΔU cn + ΔU cd

N1 (17)

T0 +T ( i∈N Uic (k) − i∈N UiNE (k)) and with ΔU cn = k=T 0 +1 ΔU cd = i∈N Uic (T0 + 1) − i∈N Uid (T0 + 1). Proof: Given that the probability with which one consumer performs noncooperative behavior is α, the probability of the noncooperative behaviors that occurs in the demand management system can be denoted as 1 − (1 − α)N . We set the indicator q = 1 when there exist noncooperative behaviors and q = 0 when there does not exist any noncooperative behavior. Then, the false alarm probability can be defined as Pr[ˆ q = 1|q = 0] = Pr[|ΔL| ≥ η, ΔL ∼ N (0, σ 2 )] = Φ(η)

(18)

and the false detection probability can be defined as Pr[ˆ q = 0|q = 1] ¯ σ 2 )] = Ψ(η) = Pr[|ΔL| < η, ΔL ∼ N (ΔL,

(19)

where Φ(η) is a decreasing function of η and Ψ(η) is an increasing function of η. Under the detection rule (15), the loss of social optimality due to the false alarm (i.e., qˆ = 1, q = 0) or the false detection (i.e., qˆ = 0, q = 1) is denoted as T 0 +T e c NE Ui (k) − Ui (k) ΔU = qˆ(1 − q) k=T0 +1

+ (1 − qˆ)q

i∈N

i∈N

Uic (T0

i∈N

+ 1) −

Uid (T0

+ 1) .

i∈N

(20) In (20), the first part is the loss of social optimality due to false alarm and is defined as the sum of the loss of social optimality in each time slot with punishment, because the punishment strategy is to make all the consumers adopt the noncooperative strategies (i.e., Nash equilibrium) in the subsequent T time slots. The second part is the loss of social optimality in the next time slot due to the false detection of noncooperative behaviors in the current time slot. Given the false alarm probability


and the false detection probability, the average loss of social optimality is denoted as ¯ e = E[ΔU e ] ΔU

T 0 +T

= β(1 − (1 − α) ) N

k=T0 +1

+ (1−β)(1−α)

N

Uic (k)

i∈N

Uic (T0

−

Uid (T0

To make the socially optimal energy consumption stable and achieve the incentive compatibility of the trigger-and¯ d for all i ∈ ¯c > U punishment mechanism, there should be U i i N , i.e., T 0 +T

UiNE (k)

i∈N

+ 1) −

i∈N

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k=T0

δ k−1 Uic (k) > δ T0 −1 Uid (T0 ) +

T 0 +T

δ k−1 UiNE (k)

k=T0 +1

(26)

+ 1) .

i∈N

(21) In (21), β = E[ˆ q ] represents the probability, estimated by the energy provider, that there exist noncooperative behaviors in the demand management system and it can be calculated by β = Pr[ˆ q = 1|q = 0]Pr[q = 0] + Pr[ˆ q = 1|q = 1]Pr[q = 1] = (1 − α)N Pr[ˆ q = 1|q = 0] q = 0|q = 1]) + (1 − (1 − α)N )(1 − Pr[ˆ = (1 − α)N Φ(η) + (1 − (1 − α)N )(1 − Ψ(η))

from which, we can obtain the lower bound of the discount factor δ min and the minimal duration of punishment T min . The period of the punishment can affect the implementation of the CDR mechanism. If the period is too short, the punishment mechanism cannot stop the noncooperative behavior. However, if the period is too long, the cooperative consumers will suffer a great deal from the punishment. Therefore, there exists an optimal period that can stop the noncooperative behavior without imposing too much cost on the cooperative consumers. The optimal period can be obtained by rounding up the minimal duration of punishment, i.e., [T min ]+1.

= f (η). (22) Since Φ(η) is decreasing and Ψ(η) is increasing with η, we conclude that β is decreasing with η. Assuming η min ≤ η ≤ η max , we have β min = f (η max ) and β max = f (η min ). To minimize the average loss of social optimality (21), we obtain the optimal detection threshold: η max , (1 − (1 − α)N )ΔU cn > (1 − α)N ΔU cd η= η min , (1 − (1 − α)N )ΔU cn ≤ (1 − α)N ΔU cd . (23) The critical condition (23) indicates that there exists a critical probability of the noncooperative behaviors (i.e., αc ) such that η = η max if α > αc and η = η min if α ≤ αc . The critical probability αc is obtained from (1 − (1 − α)N )ΔU cn = (1 − α)N ΔU cd . It is shown in (17) that αc is decreased with the number of consumers. In practical demand management system, the number of consumers is very large such that αc is extremely small. Therefore, η = η max is always the optimal threshold. B. Punishment Strength Suppose all of the consumers adopt the cooperative strategies, the average payoff of one consumer is denoted as ¯ic = U

∞

δ k−1 Uic (k).

(24)

k=1

The average payoff of the consumer when adopting the noncooperative strategy at time slot T0 is denoted as ¯d = U i

T 0 −1

δ k−1 Uic (k) + δ T0 −1 Uid (T0 )

k=1

+

T 0 +T k=T0 +1

δ k−1 UiNE (k) +

∞ k=T0 +T +1

δ k−1 Uic (k).

(25)

V. A PPLICATION TO L OAD M ANAGEMENT OF I NDUSTRIAL R EFRIGERATED WAREHOUSES In this section, we consider the application to load management of industrial refrigerated warehouses with HVAC systems. Changing the cold storage temperature set points of the refrigerated warehouses will cause the reduction of product quality and further increase economic costs to the industrial consumers. Taguchi loss function is a method that captures economic costs due to the manufacture of imperfect products [38]. According to Definition 3, the economic costs can be defined as in 2 ˆ in Viq (Qin i (k)) = γi (Qi (k) − Qi (k)) ,

i∈N

(27)

ˆ in where γi is the cost coefficient, Qin i (k) and Qi (k) denote the actual temperature set point and the desired temperature set point in time slot k, respectively. The indoor temperature of refrigerated warehouse i evolves according to the following linear dynamics [18]: in out in Qin i (k) = Qi (k − 1) + ωi (Qi (k) − Qi (k − 1)) + θi li (k) (28)

where ωi and θi specify the thermal characteristics of the operating environment and the HVAC system, Qout i (k) denotes in (k) − Q (k − 1)) models the the outdoor temperature, ωi (Qout i i heat transfer, θi li (k) (θi > 0) models the energy-heat transformation of the HVAC system. Assuming that the refrigerated warehouse i requires ˆli (k) kWh energy to maintain the desired indoor temperature, we have in out in ˆ ˆ in Q i (k) = Qi (k − 1) + ωi (Qi (k) − Qi (k − 1)) + θi li (k) (29)

where ˆli (k) is different for the refrigerated warehouses with different desired temperature set points. For example, the recommended storage and transit temperatures for food products are from 32 ◦ F to 64 ◦ F for vegetables and fruits, from 32 ◦ F to

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39 ◦ F for milk and meat, and from −22 ◦ F to 0 ◦ F for seafood and ice cream [49]. Substituting (28) and (29) into (27) and omitting the time slot index k, we transform the cost function to Viq = γi θi2 (li − ˆli )2 ,

i ∈ N.

The electricity price is defined as li − L + p 0 p(l) = λ

(30)

(31)

i∈N

where λ is a pricing parameter to implement elastic pricing, p0 is the base price, and L is the forecast demand. Then, the costs to refrigerated warehouse i can be denoted as r 2 2 ˆ V = γi θ (li − li ) + λ li − L + p 0 li (32) i

i

i∈N

with which, (P1) is a nonconvex optimization problem, and the global optimal solution is hard to obtain. Next, we propose a heuristic method to search for the suboptimal solution of (P1) and group the refrigerated warehouses into M clusters. The number of refrigerated warehouses in cluster m (m ∈ M = {1, 2, . . . , M }) is Nm , and the set of refrigerated warehouses in cluster m is denoted as Nm = {1, 2, . . . , Nm }. The details of the cluster-based CDR scheme are given as follows. The refrigerated warehouses are first grouped into M clusters according to their normal energy consumption ˆli . Specifically, assuming that the highest and lowest normal energy consumption are lmax and lmin , respectively, the refrigerated warehouses belong to cluster m if their normal energy consumption lies within [lmin + (m − 1)(lmax − lmin )/M, lmin + m(lmax − lmin )/M ], m ∈ {1, 2, . . . , M }. The forecast demand is allocated to each cluster according to the ratio of the total normal energy consumption in one cluster to the total normal energy consumption in the demand management system, i.e., ˆli (33) Lm = i∈Nm L, m ∈ M ˆli i∈N

where Lm is the forecast demand of cluster m. Similarly, the CDR scheme in cluster m can be formulated as Uim (P2) maximize i∈Nm

subject to Uim ≥ UiNE ,

i ∈ Nm

where Uim is the payoff function of refrigerated warehouse i in cluster m m m 2 2 U = −V = − γi θ (li − ˆli ) − λ li − Lm +p0 li . i

i

i

i∈Nm

(34) To solve (P2), we first consider the following unconstrained optimization problem: (P3) maximize Uim . i∈Nm

Next, we give the condition to guarantee a unique global optimal solution of (P3).

Proposition 2: Given the payoff function Uim defined by (34), the optimization problem (P3) has a unique global optimal solution if λ≤

γi θi2 , Nm − 2

i ∈ Nm .

(35)

Proof: Given Uim defined by (34), the Hessian matrix of (P3) is denoted as ⎡ ⎤ −2γ1 θ12 − 2λ −2λ ... −2λ ⎢ ⎥ −2λ −2γ2 θ22 − 2λ . . . −2λ ⎢ ⎥ H=⎢ ⎥. .. .. .. .. ⎣ ⎦ . . . . 2 −2λ −2λ . . . −2γNm θNm − 2λ (36) Given (35), it is sufficient to show that H is strictly diagonally dominant, i.e., |Hi,i | ≥ |Hi,j |, j=i,j∈Nm

|Hi,i | ≥

|Hj,i |

∀i ∈ Nm .

(37)

j=i,j∈Nm

Following Gershgorin’s theorem [50], all the eigenvalues are negative, and H is a negative definite matrix. Therefore, the optimization problem (P3) is convex and has a unique global optimal solution. Supposing the condition (35) is satisfied, we can obtain the optimal solution of (P3), i.e., lc = H −1 C

(38)

where C is defined by ⎡

p0 − λLm − 2γ1 θ12 ˆl1 ⎢ p0 − λLm − 2γ2 θ2 ˆl2 2 ⎢ C=⎢ .. ⎣ . p0 − λLm − 2γN θ2 ˆlN m

Nm

⎤ ⎥ ⎥ ⎥. ⎦

(39)

m

Next, we will check the feasibility of the constraints of (P2). If any constraint of (P2) is not satisfied, the energy provider will increase the number of clusters from M to M + 1 and reallocate the energy consumption to the refrigerated warehouses according to (38) until all the constraints of (P2) are satisfied. The clustering algorithm is shown in Algorithm 1. The number of clusters obtained by Algorithm 1 is a minimal value to guarantee that the constraints of (P2) are satisfied. In practice, the number of clusters can be larger than this minimal value. The impact of the number of clusters on the performance of the CDR scheme will be studied in the simulations. Algorithm 1. Clustering algorithm Input: Refrigerated warehouses set: N = {1, 2, . . . , N }; Parameters: γi , θi , λ; Normal energy consumption: ˆli ; Forecast demand: L; Number of clusters: M = 1. Output: Number of clusters: M ; Clusters set: M = {1, 2, . . . , M }; Refrigerated warehouses set in cluster m: Nm .


g = 0; for all i ∈ N do Calculate the energy consumption lc according to (38); if Ui ≤ UiNE then g = 1; end if end for while g = 1 do M ← M + 1; for m ∈ M do for i ∈ N if ˆli ∈[lmin +(m−1)(lmax −lmin )/M, lmin + m(lmax − lmin )/M ] then Add refrigerated warehouse i to cluster m; end if end for Allocate the forecast demand L to cluster m according to (33); Calculate the energy consumption lc for the refrigerated warehouses in cluster m according to (38); end for g = 0; for i ∈ N do if Uim ≤ UiNE then g = 1; end if end for if M = N then The clustering algorithm is terminated; end if end while

In each cluster, we introduce a cluster head that is responsible for setting the electricity price within the cluster and allocating the energy consumption to the refrigerated warehouses using the CDR scheme, as shown in Fig. 3. When the cluster head observes the change in the total energy consumption from the normal value, it will investigate the reasons for the change, such as the noncooperative behaviors or the detection errors. If the change is caused by the noncooperative behaviors of the refrigerated warehouses, the cluster head will announce the start of the punishment to all the refrigerated warehouses in the next time slot and restart the cooperation after at least T min time slots. In practice, to obtain the socially optimal energy consumption, the energy provider needs to periodically measure the energy consumption and collect the parameters (e.g., θi ) from the refrigerated warehouses. However, the information update is not frequent because of large communication overhead. The infrequent communications will further make the energy provider harder to distinguish the noncooperative behaviors from the errors in the total energy consumption. In the simulations, it is shown that the clustering method can benefit the detection of the noncooperative behaviors and reduce the motivations of the refrigerated warehouses to adopt the noncooperative strategies.

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Fig. 3. Cluster-based DR scheme with industrial refrigerated warehouses.

VI. N UMERICAL R ESULTS In this section, the performance of the CDR scheme is evaluated. We assume that the normal energy consumption of the refrigerated warehouses is uniformly distributed in [100 kWh, 150 kWh], the cost coefficients γi θi2 are uniformly distributed in [2, 4] or [3, 5], the base price p0is 0.05$/kWh, the forecast demand is estimated by L = μ i∈N ˆli , and the pricing parameter λ is calculated by λ = 2/N or λ = 1/N .

A. Comparison With Related Works Assuming that the number of refrigerated warehouses is 100, the cost coefficients γi θi2 are uniformly distributed in [2, 4], and the pricing parameter λ is calculated by λ = 2/N , we compare the proposed CDR scheme with the DR scheme for PT consumers and the noncooperative demand response (NDR) scheme for PA consumers in Table I. It is shown that the proposed DR scheme reduces the electricity price,4 the total costs,5 and the total energy consumption effectively. To evaluate the total cost reduction (TCR) of the refrigerated warehouses obtained from the proposed CDR scheme, we define the TCR as TCR =

c i∈N (Ui

i∈N

− UiN ) × 100% UiN

(40)

where UiN is replaced by Uit when comparing with the DR scheme for PT consumers, and UiN is replaced by UiNE when compared with the NDR scheme for PA consumers. As shown in Fig. 4, the TCR obtained from the proposed DR scheme increases with the number of refrigerated warehouses and starts to saturate when the number of refrigerated warehouses is larger than 60. 4 To evaluate the total costs and the total energy consumption, we assume the electricity price in the DR scheme for PT consumers is the same as the price in the proposed scheme. 5 The total costs are composed of the discomfort costs and the payments, and the payments are equal to the product of the electricity price and the total energy consumption.

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TABLE I C OMPARISONS W ITH THE DR S CHEME FOR PT C ONSUMERS AND THE NDR S CHEME FOR PA C ONSUMERS

TABLE II CDR S CHEME W ITH AND W ITHOUT N ONCOOPERATIVE B EHAVIORS (WNB AND WTNB)

Fig. 4. Total cost reduction of the refrigerated warehouses obtained from the CDR scheme versus number of refrigerated warehouses.

B. CDR With and Without Noncooperative Behavior To evaluate the cost reduction of the refrigerated warehouse when adopting the noncooperative strategy, we define the cost reduction due to the noncooperative behavior (CRN) as CRN =

Uid − Uic × 100%, Uic

i ∈ N d.

(41)

To evaluate the increase of total energy consumption when a refrigerated warehouse adopts the noncooperative strategy, we define the total energy consumption increase due to the noncooperative behavior (EIN) as d c d (l − l ) i c i × 100%. (42) EIN = i∈N i∈N li Assuming that one refrigerated warehouse has the noncooperative behavior and the other refrigerated warehouses keep cooperative, the electricity price, the average costs, and the total energy consumption are all increased, as shown in Table II. Furthermore, we study the impact of the number of refrigerated warehouses on the performance of the CDR scheme. The cost reduction of the refrigerated warehouse that has the noncooperative behavior increases with the number of refrigerated warehouses, as shown in Fig. 5, and the increase in the total energy consumption decreases with the number of refrigerated warehouses, as shown in Fig. 6. Both of them saturate when the number of refrigerated warehouses becomes large. From Figs. 5 and 6, we can also observe that a larger pricing parameter (i.e., λ) gives higher CRN and EIN, while a larger cost weight (i.e., γi θi2 ) gives lower CRN and EIN. Furthermore, it

Fig. 5. Cost reduction of the refrigerated warehouse that has the noncooperative behavior versus number of refrigerated warehouses.

is also shown that the noncooperative refrigerated warehouse has relatively large cost reduction and thus strong motivation to adopt the noncooperative strategy, and the increase in the total energy consumption in the demand management system is relatively small when the number of refrigerated warehouses is large. Thus, it is hard to distinguish the noncooperative behavior from the errors in the total energy consumption. To solve this problem, we divide the refrigerated warehouses into different clusters. C. Cluster-Based CDR Assuming that the number of refrigerated warehouses is 100, we study the impact of the number of clusters on the performance of the CDR scheme. The total costs to all the refrigerated warehouses with clustering are given in Fig. 7. It is shown that the total costs increase with the number of clusters, which indicates that the clustering reduces the social optimality (i.e., the negative total costs) of the CDR scheme. The loss of social optimality is evaluated in Fig. 8. It is shown that the loss of social optimality is bounded by 1.4% of the optimal costs when the number of clusters is less than 10, and thus the clustering


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TABLE III P ERFORMANCE OF CDR S CHEME W ITH D IFFERENT N UMBERS OF C LUSTERS

Fig. 6. Increase in the total energy consumption when a refrigerated warehouse has the noncooperative behavior versus number of refrigerated warehouses.

Fig. 7. Total costs to all the refrigerated warehouses in the CDR scheme versus number of clusters.

Fig. 9. Critical probability of the noncooperative behavior (αc ) versus number of refrigerated warehouses.

cooperative, the cost reduction of the noncooperative refrigerated warehouse decreases with the number of clusters, and the increase of the total energy consumption increases with the number in clusters. It is shown that clustering can be helpful for detecting the noncooperative behavior and reducing the motivation of the refrigerated warehouses to adopt noncooperative strategies. If there is no clustering, i.e., fully decentralized management, the problem becomes NDR. In Table I and Fig. 4, we have compared the proposed scheme with the noncooperative scheme in terms of the total costs, the total energy consumption, and the price. D. Noncooperative Behavior Detection and Punishment

Fig. 8. Loss of social optimality versus number of clusters.

gives a suboptimal solution. As shown in Table III, the TCR obtained from cooperation also decreases with the number of clusters. Assuming that a refrigerated warehouse has the noncooperative behavior and the other refrigerated warehouses keep

As shown in Fig. 9, the critical probability decreases with the number of refrigerated warehouses. Specifically, even when the number of refrigerated warehouses is 10, the critical probability is smaller than 0.06%, which indicates that η max is always the optimal detection threshold,6 because the number of 6 The optimal threshold η max indicates that the false alarm can cause more loss of social optimality than the false detection. Thus, the punishment mechanism should not be triggered more often.

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that the clustering method can help with the detection of noncooperative behaviors and reduce the motivation of the consumers to adopt noncooperative strategies.

R EFERENCES

Fig. 10. Minimal duration of punishment versus discount factor. TABLE IV M INIMAL D URATION OF P UNISHMENT AND L OWER B OUND OF D ISCOUNT FACTOR

refrigerated warehouses in the demand management system is larger than 10 and thus α > αc is satisfied almost everywhere. Furthermore, it is also shown in Fig. 9 that a larger λ gives a lower critical probability and a larger γi θi2 gives a higher critical probability. Assuming that one refrigerated warehouse has the noncooperative behavior, we study the minimal duration of punishment (i.e., T min ) and the lower bound of the discount factor (i.e., δ min ) under different parameter settings. As shown in Fig. 10, the minimal duration of punishment decreases with the discount factor because the future costs play a more significant role in the average costs and thus less duration of punishment is needed to stop the noncooperative behavior. It is shown in Table IV that a larger λ gives a higher T min and δ min , and a larger γi θi2 gives a lower T min and δ min .

VII. C ONCLUSION In this study, we formulate the CDR scheme as a constrained social optimization problem. It is shown that the CDR scheme reduces the electricity price, the total costs, and the total energy consumption comparing with the NDR scheme. We design the trigger-and-punishment mechanism to keep cooperation and avoid the noncooperative behaviors of the PA consumers. We develop the method to detect the noncooperative behavior and establish the condition on the duration of punishment to stop the noncooperative behavior. The CDR scheme is further applied to achieve load management of industrial refrigerated warehouses with HVAC systems. We propose a clustering algorithm to obtain a suboptimal solution of the nonconvex optimization problem. Specifically, the refrigerated warehouses are grouped into different clusters that are managed by the cluster heads. The CDR scheme is executed within each cluster. It is shown

[1] U. S. Dept. of Energy. (2006, Feb.) Benefits of demand response in electricity markets and recommendations for achieving them [Online]. Available: http://web.mit.edu/cron/project/urban-sustainability/Old%20 files%20from%20summer%202009/Ingrid/Urban%20Sustainability%20 Initiative.Data/doe_demand_response_rpt_t---ss_feb_17_06_final.pdf [2] P. Palensky and D. Dietrich, “Demand side management: Demand response, intelligent energy systems, and smart loads,” IEEE Trans. Ind. Informat., vol. 7, no. 3, pp. 381–388, Aug. 2011. [3] N. Lu, “An evaluation of the HVAC load potential for providing load balancing service,” IEEE Trans. Smart Grid, vol. 3, no. 3, pp. 1263–1270, Sep. 2012. [4] D. Guo, W. Zhang, G. Yan, Z. Lin, and M. Fu, “Decentralized control of aggregated loads for demand response,” in Proc. Amer. Control Conf. (ACC), 2013, pp. 6601–6606. [5] J. L. Mathieu, S. Koch, and D. S. Callaway, “State estimation and control of electric loads to manage real-time energy imbalance,” IEEE Trans. Control Syst. Technol., vol. 28, no. 1, pp. 430–440, Feb. 2013. [6] R. Argiento, R. Faranda, A. Pievatolo, and E. Tironi, “Distributed interruptible load shedding and micro-generator dispatching to benefit system operations,” IEEE Trans. Power Syst., vol. 27, no. 2, pp. 840–848, May 2012. [7] Y. Liu and X. Guan, “Purchase allocation and demand bidding in electric power markets,” IEEE Trans. Power Syst., vol. 18, no. 1, pp. 106–112, Feb. 2003. [8] M. H. Albadi and E. El-Saadany, “A summary of demand response in electricity markets,” Elect. Power Syst. Res., vol. 78, no. 11, pp. 1989– 1996, 2008. [9] H. Aalami, M. P. Moghaddam, and G. Yousefi, “Demand response modeling considering interruptible/curtailable loads and capacity market programs,” Appl. Energy, vol. 87, no. 1, pp. 243–250, 2010. [10] B. Kirby, “Ancillary services: Technical and commercial insights,” 2007 [Online]. Available: http://www.science.smith. edu/~jcardell/Courses/EGR325/Readings/Ancillary_Services_Kirby.pdf [11] K. Herter, P. McAuliffe, and A. Rosenfeld, “An exploratory analysis of California residential customer response to critical peak pricing of electricity,” Energy, vol. 32, no. 1, pp. 25–34, 2007. [12] G. Barbose, C. Goldman, and B. Neenan, “A survey of utility experience with real time pricing,” Lawrence Berkeley Nat. Lab., Tech. Rep. LBNL54238, 2004. [13] J. Taylor, A. Nayyar, D. Callaway, and K. Poolla, “Consolidated dynamic pricing of power system regulation,” IEEE Trans. Power Syst., vol. 28, no. 4, pp. 4692–4700, Nov. 2013. [14] V. C. Gungor et al., “A survey on smart grid potential applications and communication requirements,” IEEE Trans. Ind. Informat., vol. 9, no. 1, pp. 28–42, Feb. 2013. [15] H. Tung et al., “The generic design of a high traffic advance metering infrastructure using zigbee,” IEEE Trans. Ind. Informat., vol. 10, no. 1, pp. 836–844, Feb. 2014. [16] P. Samadi, A. Mohsenian-Rad, R. Schober, V. W. Wong, and J. Jatskevich, “Optimal real-time pricing algorithm based on utility maximization for smart grid,” in Proc. IEEE Int. Conf. Smart Grid Commun., 2010, pp. 415–420. [17] N. Gatsis and G. B. Giannakis, “Residential load control: Distributed scheduling and convergence with lost AMI messages,” IEEE Trans. Smart Grid, vol. 3, no. 2, pp. 770–786, Jun. 2012. [18] L. Chen, N. Li, L. Jiang, and S. H. Low, “Optimal demand response: Problem formulation and deterministic case,” in Proc. Control Optim. Methods Elect. Smart Grids, 2012, pp. 63–85. [19] A. Mohsenian-Rad, V. W. Wong, J. Jatskevich, R. Schober, and A. LeonGarcia, “Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid,” IEEE Trans. Smart Grid, vol. 1, no. 3, pp. 320–331, Dec. 2010. [20] R. Deng, Z. Yang, J. Chen, N. Asr, and M.-Y. Chow, “Residential energy consumption scheduling: A coupled-constraint game approach,” IEEE Trans. Smart Grid, vol. 5, no. 3, pp. 1340–1350, May 2014. [21] K. Ma, G. Hu, and C. J. Spanos, “Distributed energy consumption control via real-time pricing feedback in smart grid,” IEEE Trans. Control Syst. Technol., vol. 22, no. 5, pp. 1907–1914, Sep. 2014.


[22] Z. Ma, D. S. Callaway, and I. A. Hiskens, “Decentralized charging control of large populations of plug-in electric vehicles,” IEEE Trans. Control Syst. Technol., vol. 21, no. 1, pp. 67–78, Jan. 2013. [23] W. Cao, B. Yang, C. Chen, and X. Guan, “PHEV charging strategy with asymmetric information based on contract design,” in Proc. IFAC Symp. Large Scale Complex Syst. Theory Appl., 2013, pp. 520–525. [24] J. Chen, B. Yang, and X. Guan, “Optimal demand response scheduling with stackelberg game approach under load uncertainty for smart grid,” in Proc. IEEE Int. Conf. Smart Grid Commun., 2012, pp. 546–551. [25] B. Chai, J. Chen, Z. Yang, and Y. Zhang, “Demand response management with multiple utility companies: A two-level game approach,” IEEE Trans. Smart Grid, vol. 5, no. 2, pp. 722–731, Mar. 2014. [26] J. L. Mathieu, P. N. Price, S. Kiliccote, and M. A. Piette, “Quantifying changes in building electricity use, with application to demand response,” IEEE Trans. Smart Grid, vol. 2, no. 3, pp. 507–518, Sep. 2011. [27] J. L. Mathieu, D. S. Callaway, and S. Kiliccote, “Variability in automated responses of commercial buildings and industrial facilities to dynamic electricity prices,” Energy Build., vol. 43, no. 12, pp. 3322–3330, 2011. [28] Y. Ding, S. Hong, and X. Li, “A demand response energy management scheme for industrial facilities in smart grid,” IEEE Trans. Ind. Informat., vol. 10, no. 4, pp. 2257–2269, Nov. 2014. [29] Z. Darabi and M. Ferdowsi, “An event-based simulation framework to examine the response of power grid to the charging demand of plugin hybrid electric vehicles,” IEEE Trans. Ind. Informat., vol. 10, no. 1, pp. 313–322, Feb. 2014. [30] Q. Dong, D. Niyato, P. Wang, and Z. Han, “The PHEV charging scheduling and power supply optimization for charging stations,” IEEE Trans. Veh. Technol., 2015, to be published. [31] Z. Liu, I. Liu, S. Low, and A. Wierman, “Pricing data center demand response,” in Proc. ACM Int. Conf. Meas. Model. Comput. Syst., 2014, pp. 111–123. [32] Z. Zhou, F. Liu, Z. Li, and H. Jin, “When smart grid meets geo-distributed cloud: An auction approach to datacenter demand response,” in Proc. IEEE Int. Conf. Comput. Commun. (INFOCOM), 2015, pp. 1–9. [33] Z. Zhou, F. Liu, and Z. Li, “Pricing bilateral electricity trade between smart grids and hybrid green datacenters,” in Proc. ACM Int. Conf. Meas. Model. Comput. Syst. (SIGMETRICS), 2015, pp. 1–2. [34] O. Kilkki, A. Alahaivala, and I. Seilonen, “Optimized control of pricebased demand response with electric storage space heating,” IEEE Trans. Ind. Informat., vol. 11, no. 1, pp. 281–288, Feb. 2015. [35] L. Song, Y. Xiao, and M. van der Schaar, “Demand side management in smart grids using a repeated game framework,” IEEE J. Sel. Areas Commun., vol. 32, no. 7, pp. 1412–1424, Jul. 2014. [36] K. Ma, G. Hu, and C. J. Spanos, “Cooperative demand response using repeated game for price-anticipating buildings in smart grid,” in Proc. Int. Conf. Control Autom. Robot. Vis. (ICARCV), 2014, pp. 565–570. [37] J. Nash, “Non-cooperative games,” Ann. Math., vol. 54, no. 2, pp. 286– 295, 1951. [38] G. Taguchi et al., Taguchi’s Quality Engineering Handbook. Hoboken, NJ, USA: Wiley, 2005. [39] R. Deng, Z. Yang, and J. Chen, “Load scheduling with price uncertainty and coupling constraints,” in Proc. IEEE Power Energy Soc. Gen. Meeting (PES), 2013, pp. 1–5. [40] S. Coogan, L. J. Ratliff, D. Calderone, C. Tomlin, and S. S. Sastry, “Energy management via pricing in LQ dynamic games,” in Proc. Amer. Control Conf. (ACC), 2013, pp. 443–448. [41] Z. Fan, “A distributed demand response algorithm and its application to PHEV charging in smart grids,” IEEE Trans. Smart Grid, vol. 3, no. 3, pp. 1280–1290, Sep. 2012. [42] S. Maharjan, Q. Zhu, Y. Zhang, S. Gjessing, and T. Basar, “Dependable demand response management in the smart grid: A stackelberg game approach,” IEEE Trans. Smart Grid, vol. 4, no. 1, pp. 120–132, Mar. 2013. [43] P. Yang, G. Tang, and A. Nehorai, “A game-theoretic approach for optimal time-of-use electricity pricing,” IEEE Trans. Power Syst., vol. 28, no. 2, pp. 884–892, May 2013. [44] A.-H. Mohsenian-Rad and A. Leon-Garcia, “Optimal residential load control with price prediction in real-time electricity pricing environments,” IEEE Trans. Smart Grid, vol. 1, no. 2, pp. 120–133, Sep. 2010. [45] N. Rahbari-Asr and M.-Y. Chow, “Cooperative distributed demand management for community charging of PHEV/PEVs based on KKT conditions and consensus networks,” IEEE Trans. Ind. Informat., vol. 10, no. 3, pp. 1907–1916, Aug. 2014. [46] D. Niyato, P. Wang, Z. Han, and E. Hossain, “Impact of packet loss on power demand estimation and power supply cost in smart grid,” in Proc. IEEE Wireless Commun. Netw. Conf. (WCNC), 2011, pp. 2024–2029.

1531

[47] G. Mateos and G. B. Giannakis, “Load curve data cleansing and imputation via sparsity and low rank,” IEEE Trans. Smart Grid, vol. 4, no. 4, pp. 2347–2355, Dec. 2013. [48] Y. Hu, A. Kuh, T. Yang, and A. Kavcic, “A belief propagation based power distribution system state estimator,” IEEE Comput. Intell. Mag., vol. 6, no. 3, pp. 36–46, Aug. 2011. [49] A. Lekov, “Opportunities for energy efficiency and automated demand response in industrial refrigerated warehouses in california,” Lawrence Berkeley Nat. Lab., Tech. Rep. LBNL-1991E, 2009. [50] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 2012.

Kai Ma received the B.Eng. degree in automation, and the Ph.D. degree in control science and engineering from Yanshan University, Qinhuangdao, China, in 2005 and 2011, respectively. In 2011, he joined Yanshan University. From 2013 to 2014, he was a Postdoctoral Research Fellow with Nanyang Technological University, Singapore. He is currently an Associate Professor with the Department of Automation, School of Electrical Engineering, Yanshan University. He has authored more than 30 refereed journal and conference papers. His research interests include demand response in smart grid and resource allocation in communication networks.

Guoqiang Hu (M’06) received the B.Eng. degree in automation, the M.Phil. degree in automation and computer-aided engineering, and the Ph.D. degree in mechanical engineering from the University of Science and Technology of China, Hefei, China; the Chinese University of Hong Kong, Hong Kong; and the University of Florida, Gainesville, FL, USA, in 2002, 2004, and 2007, respectively. He is currently with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. Prior to his current position, he was a Postdoctoral Research Associate with the University of Florida, in 2008, and an Assistant Professor with Kansas State University, Manhattan, KS, USA, from 2008 to 2011. His research interests include analysis, control, and design of distributed intelligent systems, with applications to smart grids, smart buildings, and networked robots. Dr. Hu serves as a Subject Editor for the International Journal of Robust and Nonlinear Control, and an Associate Editor for Unmanned Systems, Asian Journal of Control, and the Conference Editorial Board (CEB) of IEEE Control Systems Society.

Costas J. Spanos (M’77–SM’92–F’98) received the Diploma in electrical engineering from the National Technical University of Athens, Athens, Greece, in 1980, and the M.S. and Ph.D. degrees in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, PA, USA, in 1981 and 1985, respectively. In 1988, he joined as the Faculty with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley, Berkeley, CA, USA. He has served as the Director of the Berkeley Microlab, the Associate Dean for Research with the College of Engineering, and the Chair of the Department of Electrical Engineering and Computer Sciences (EECS). He works in statistical analysis in the design and fabrication of integrated circuits, and on novel sensors and computer-aided techniques in semiconductor manufacturing. He also works on statistical data mining techniques for energy efficiency applications. He has participated in two successful startup companies, Timbre Tech (acquired by Tokyo Electron), Tokyo, Japan, and OnWafer Technologies (acquired by KLA-Tencor), CA. He is currently the Director of the Center of Information Technology Research in the Interest of Society (CITRIS), Berkeley, and the Chief Technical Officer for the Berkeley Educational Alliance for Research in Singapore (BEARS), Singapore.