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IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 3, SEPTEMBER 2013 1341

Decentralized Coordination of Energy Utilization forResidential Households in the Smart Grid

Yuanxiong Guo, Student Member, IEEE, Miao Pan, Member, IEEE, Yuguang Fang, Fellow, IEEE, andPramod P. Khargonekar, Fellow, IEEE

Abstract—In this paper, we investigate the minimization of thetotal energy cost of multiple residential households in a smartgrid neighborhood sharing a load serving entity. Specifically,each household may have renewable generation, energy storageas well as inelastic and elastic energy loads, and the load servingentity attempts to coordinate the energy consumption of thesehouseholds in order to minimize the total energy cost within thisneighborhood. The renewable generation, the energy demand ar-rival, and the energy cost function are all stochastic processes andevolve according to some, possibly unknown, probabilistic laws.We develop an online control algorithm, called Lyapunov-basedcost minimization algorithm (LCMA), which jointly considers theenergy management and demand management decisions. LCMAonly needs to keep track of the current values of the underlyingstochastic processes without requiring any knowledge of their sta-tistics. Moreover, a decentralized algorithm to implement LCMAis also developed, which can preserve the privacy of individualhousehold owners. Numerical results based on real-world tracedata show that our control algorithm can effectively reduce thetotal energy cost in the neighborhood.

Index Terms—Demand response, energy management, energystorage, inelastic and elastic energy loads, Lyapunov optimization,renewable generation, smart grid.

I. INTRODUCTION

T HE GROWING demands of electricity and concernsover global climate change and carbon emission have

motivated the grid modernization, which transforms the currentpower grids to the future “smart grid.” As stated in [1], thesmart grid will enable deep penetration of renewable genera-tion, customer driven demand response, widespread adoption ofelectric vehicles, and electric energy storage. Sensing, commu-nication, computation, and control technologies in conjunctionwith advances in renewable generation, energy storage, powerelectronics, etc., are critical to realizing the vision and promiseof the smart grid.

Manuscript received June 13, 2012; revised December 14, 2012 and March15, 2013; accepted May 12, 2013. Date of publication August 07, 2013; date ofcurrent version August 21, 2013. This work was partially supported by the U.S.National Science Foundation under Grants CNS-1147813, ECCS-1129061,ECCS-1129062, and Eckis Professor endowment at the University of Florida.Paper no. TSG-00366-2012.Y. Guo, Y. Fang, and P. P. Khargonekar are with the Department of Electrical

and Computer Engineering, University of Florida, Gainesville, FL 32611 USA(e-mail: [email protected]; [email protected]; [email protected]).M. Pan is with the Department of Computer Science, Texas Southern Univer-

sity, Houston, TX 77004 USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSG.2013.2268581

Within the smart grid, demand side management (DSM) isa key component, which can help reduce peak load, increasegrid reliability, and lower generation cost [2]. There are mainlytwo types of demand side management techniques: direct loadcontrol (DLC) and demand response based on time-varyingpricing [3]. In DLC, the load serving entity, usually a utilitycompany, enters into a contract with the consumers beforehand,so that certain amount of energy load can be curtailed duringthe peak hours in order to release the congestion on the powergrid or to avoid the operation of high cost peak generators.Currently, it is mainly employed by large industrial and com-mercial customers. On the other hand, the demand responsebased on time-varying pricing encourages the customers toeither reduce or shift their normal energy consumption basedon the pricing signal issued by the load serving entity (LSE)in return for some benefits, such as electricity bill reduction.Several popular schemes already exist in this regard, such ascritical-peak pricing (CPP), time-of-use (TOU) pricing, andreal-time pricing (RTP). With the introduction of the advancedmetering infrastructure (AMI), which can provide two-waycommunication between utility companies and smart meters,it is expected that there will be a widespread deployment ofsuch demand response programs for residential and businesscustomers in the smart grid [4].Meanwhile, nearly 7% of electricity is lost during trans-

mission and distribution (T&D) from remote power plants todistant homes [5]. Distributed generation (DG) from manysmall on-site energy sources deployed at individual homes andbusinesses can be used to decrease both T&D losses and carbonemissions. Typical examples of these small on-site energysources include rooftop solar panels, fuel cells, microturbines,and micro-wind generators. Distributed energy storage devicesare usually used in combination with these renewable sourcesto better utilize them. We envision residential households inthe smart grid which use on-site renewable generation, modestenergy storage, and the electric grid to meet their energy de-mands, within which some are elastic and can be served in aflexible manner. How to simultaneously manage these compo-nents for households within a neighborhood in order to reducethe total energy cost as well as the impact on the distributionnetwork of the power grids is a challenging problem, especiallyconsidering the random dynamics in the system.There have been many previous studies on energy consump-

tion scheduling in households, renewable energy integration,and demand response schemes. On the residential energy con-sumption scheduling side, Mohsenian-Rad et al. [6] formulatethe optimal control of multiple flexible appliances as a linear

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1342 IEEE TRANSACTIONS ON SMART GRID, VOL. 4, NO. 3, SEPTEMBER 2013

program to achieve a desired trade-off between the electricitypayment and the waiting time for the operation of each appli-ance in a household, where customers are subject to a real-time pricing tariff combined with inclining block rates. A gametheory based approach is proposed in [7] to handle the caseof multiple households. Kim et al. [8] use dynamic program-ming to solve the problem of scheduling power consumption ofa single appliance in order to minimize the expected cost. On theside of renewable energy integration, the problem of supplyingrenewable energy to demand-flexible customers is investigatedin [9], [10]. Specifically, Papavasiliou et al. [10] address the op-timal allocation for renewable sources to demand-flexible cus-tomers in a real-time pricing environment using dynamic pro-gramming, while in [9], the authors develop a Lyapunov opti-mization based method. Bitar et al. [11] consider optimal sellingstrategies for uncertain and variable wind production into thecurrent electricity market. On the demand response side, Li etal. [12] consider the problem of optimal demand response asa convex optimization problem and study the role of dynamicpricing. Jiang et al. [13] propose a model that integrates two-pe-riod electricity markets, uncertainty in renewable generation,and real-time dynamic demand response, and derive the optimalcontrol decisions to optimize the social welfare. However, mostof the previous studies either only consider optimization for onehousehold, or assume perfect future information, or do not con-sider on-site distributed generation and energy storage.This paper extends the single household case in our previous

work [14] to that of multiple households within a smart gridneighborhood. In our work, not only does the energy cost of anindividual household matter, but also the total energy cost in aneighborhood is of equal importance.We show that, through ourcollaborative and decentralized energy consumption schedulingalgorithm in multiple households, the impact of the householdenergy consumption on the power system and the total energycost can be greatly reduced. Due to the decentralized and on-line properties of our proposed algorithm, it can be easily im-plemented in the smart grid. In summary, our paper makes thefollowing contributions:• In the setting of multiple households within a neighbor-hood, we propose a new system architecture to incorporatethe following essential components in the smart grid: dis-tributed renewable generation, energy storage, demand re-sponse, and smart appliances.

• We develop a decentralized online algorithm, called Lya-punov-based cost minimizing algorithm (LCMA), to ap-proximately minimize time-average total energy cost forhouseholds within a neighborhood without the knowledgeof the statistics of related stochastic models.

• Through theoretical analysis, we show that our algorithmcan obtain an explicit trade-off between cost saving andenergy storage capacity. Moreover, through extensive sim-ulations based on real-world data sets, we demonstrate theeffectiveness of our proposed algorithms.

The paper is organized as follows. In Section II, we describeour systemmodel and formulate the problem as a stochastic pro-gramming problem.We describe the design principle behind ouralgorithm and present an online algorithm in Section III. We

Fig. 1. A schematic diagram of household energy management in a smart gridneighborhood.

then analyze our algorithm in Section IV. We present numer-ical results based on real-world data in Section V. Finally, someconcluding remarks are presented in Section VI.

II. SYSTEM MODEL AND PROBLEM FORMULATION

In this section, we provide mathematical descriptions for theload serving entity (LSE), energy load, energy storage, and dis-tributed renewable generation in residential households. Basedon these definitions, we will formulate our control problem as astochastic program.Consider a set of households/customers that are served

by the LSE in a smart grid neighborhood setting as depictedin Fig. 1. The LSE may be a utility company and the smartgrid neighborhood may cover all households connected to astep-down transformer in the distribution network. The LSEmay participate into wholesale electricity markets (day-ahead,hour-ahead, real-time balancing, ancillary service) to purchaseelectricity from power generators and then sell it to the cus-tomers in the retail market. The electricity price in the retailmarket is typically set at a fixed level that reflects the broadaverage of the hourly costs to serve customers over a year orseason. However, it does not encourage efficient usage of elec-tricity, causing high peak demand and low load factor. We con-sider a time-slotted model with an infinite horizon. Each slotrepresents a suitable period for control decisions (e.g., 1 hour or15 min) and is indexed by .1

A. Load Serving Entity

The LSE serves as an agent that is responsible for purchasingenough electricity from wholesale electricity markets to servethe energy demand of the households in its service area. The re-tail price is set in order to at least recover the running cost ofthe LSE. In the future smart grid, a field area network (FAN)would be deployed, which can provide convenient communi-cations between utility companies and smart meters of residen-tial households. For simplicity, we make the assumption that thecost of the LSE can be represented by a cost function thatspecifies the cost of providing amount of electricity to thecustomers during one period. We assume that the cost function

is increasing, continuously differentiable, and convex in

1In this paper, all power quantities such asare in the unit of energy per slot, so the energy produced/consumed in timeperiod is , respectively.

GUO et al.: DECENTRALIZED COORDINATION OF ENERGY UTILIZATION FOR RESIDENTIAL HOUSEHOLDS IN THE SMART GRID 1343

with a bounded first derivative. We use and to de-note the minimum and the maximum first derivatives of ,respectively.

B. Energy Load

In general, the energy loads in a household can be roughly di-vided into two categories: inelastic and elastic loads. Examplesof inelastic energy loads include lights, TVs, microwaves, andcomputers. For this type of energy loads, the energy requestsmust be met exactly at the time when needed. In contrast,there are some energy loads in households that are elastic in thesense that they can be controlled (using smart appliances, forexample,) to adjust the times of their operations and the amountof their energy usage without impacting the satisfaction of cus-tomers. Examples include refrigerators, dehumidifiers, air con-ditioners, and electric vehicles. As observed in [15], while theelastic energy loads comprise less than 7.5% of the total loadsin a household, they account for 59% of the average energyconsumption. Therefore, there is great hidden potential in ex-ploiting the inherent flexibility of such elastic loads for variousimportant individual and system level objectives.Inside a household, electric loads can communicate with the

smart meter via the home area network (HAN), which may beWi-Fi or ZigBee. For each household , denote bythe inelastic energy loads (in unit of kWh) and by theelastic energy loads (in unit of kWh) at time . As in [9], we as-sume that the elastic energy loads are “buffered” (i.e., the energyrequests are held or delayed) first in a queue before beingserved. Denote by the amount of energy that is used forserving the queued energy loads at time . Then the dynamicsof are as follows:

(1)

For each , we assume that

(2)

where so that the queue can always be stabi-lized. For any feasible control decision, we need to ensure thatthe average delay of the elastic loads in the queue is finite. Inother words, the service of elastic energy loads cannot be de-layed for an arbitrarily long time. This can be stated as follows:

(3)

C. Energy Storage

In addition to energy loads, each household may have somekind of energy storage device, possibly in the form of the bat-tery in the PHEV. For each household , we denote bythe battery capacity, by the energy level of the battery attime , and by the power charged to (when )or discharged from (when ) the battery during slot .Assume that the battery energy leakage is negligible and bat-teries at households operate independently of each other. Thenwe model the dynamics of the battery energy level by

(4)

For each household , the battery usually has an upper boundon the charge rate, denoted by , and an upper bound onthe discharge rate, denoted by , where andare positive constants depending on the physical properties ofthe battery as well as the charging infrastructure. Therefore, wehave the following constraint on :

(5)

The battery energy level should always be nonnegative andcannot exceed the battery capacity. So in each time slot , weneed to ensure that for each household ,

(6)

However, the cost of battery use cannot be ignored. In prac-tice, batteries can only be charged a finite number of times. Be-sides, conversion loss occurs both in charging and dischargingprocesses. Stored energy is also subject to leakage with time. Allthese factors depend on how fast/much/often it is charged anddischarged. Instead of modeling these factors exactly, we use anamortized time-invariant cost function (in unit of dollars)to model the impact of charging or discharging operation onthe battery during one slot for household . Each battery costfunction is assumed to be continuously differentiable inwith a bounded first derivative and . We use

and to denote the minimum and the maximum first deriva-tives of for each household , respectively.

D. Renewable Distributed Generation (DG)

Each household may possess a distributed renewable gener-ator installed on its site, such as rooftop PV panel or small windturbine. Since renewable sources such as wind and solar, areusually intermittent, uncertain, and uncontrollable, we modelthe renewable energy generated by the renewable DG by a dis-crete time random process , which has the maximum valuegiven by its nameplate capacity . Therefore, we have

(7)

Note that the power generation from a renewable generator isusually lower than the normal power consumption of residen-tial households. Residential households need to connect to theutility electric grid for backup power and, therefore, are mostlygrid-tied systems. In this paper, we assume that the marginalcost of renewable energy is zero and should be utilized as muchas possible.

E. Problem Formulation

With the above models for the battery and the distributed re-newable generator, at each time , the total power demand ofhousehold needed from the utility electric grid is

(8)

Note that in the formula above, we have assumed that powercannot be fed from the household into the utility electric gridthrough, for example, net metering. Since we assume that eachhousehold has an energy storage device, excess renewable en-ergy generation can be stored into it without spillage as long as


the storage has enough capacity. We plan to incorporate the op-tion of two-way energy flow in our future investigation.In this paper, we are interested in minimizing the LSE’s total

cost of providing the electricity to the whole smart grid neigh-borhood in a sufficiently long horizon. Note that reducing thecost of supplying electricity for the LSE is both beneficial to theLSE as well as individual customers since the cost will be finallytransferred to the customer’s electricity bill. Therefore, the con-trol problem can be stated as follows: for the dynamic systemdefined by (1) and (4), design a control strategy which, giventhe past and present random renewable supplies, the battery en-ergy levels, the energy demands, and the energy cost function,chooses the battery charge/discharge vector and the elasticload serving rate vector such that the time-average total en-ergy cost of the whole smart grid neighborhood is minimized.It can be formulated as the following stochastic programmingproblem, called :

(9)subject to constraints (1), (2), (3), (4), (5), and (6).Here the expectation in the objective is w.r.t. the random

renewable generation , the random inelastic energy loads, and the random elastic energy loads for each

household. Define as the infimum time average cost asso-ciated with , considering all feasible control actions subjectto the queue stability and the finite battery energy level. We willdesign a control algorithm, parameterized by a constant ,that satisfies the constraints above and achieves the average costwithin of the optimal value , while guaranteeingthat the worst-case delay is within .

III. ONLINE DECENTRALIZED ALGORITHM

In this section, we design algorithms to solve . Onechallenge of solving the stochastic optimization problem aboveis the uncertainty of future renewable generation, time-varyingcost function, inelastic or elastic energy loads. Moreover, theconstraints on bring the “time-coupling” property tothe stochastic optimization problem above. That is to say, thecurrent control action may impact the future control actions,making it more challenging to solve. Our solution is basedon the technique of Lyapunov optimization [16] and requiresminimum information on the random dynamics in the system.

A. Delay-Aware Virtual Queue

Since the constraint only ensures finite averagedelay for the elastic energy loads in household , worst-casedelay guarantee is usually desired in practice. For this purpose,we leverage the technique of “virtual queue” in the Lyapunovoptimization framework. Specifically, the following virtualqueues , are defined to provide theworst-case delay guarantee on any buffered elastic energy loadsin :

(10)

where is an indicator function that is 1 ifor 0 otherwise; is a fixed positive parameter to be specifiedlater. The intuition behind this virtual queue is that sincehas the same service process as , but has an arrival processthat adds whenever the actual backlog is nonempty, this en-sures that grows if there are energy loads in the queue

that have not been serviced for a long time. The followinglemma shows that if we can control the system to ensure thatthe queues and have finite upper bounds, then anybuffered energy load is served within a worst-case delay as fol-lows:Lemma 1: Suppose we can control the system to ensure that

and for all slots , whereand are some positive constants. Then, the worst-casedelay for all buffered energy loads in household is upperbounded by slots where

(11)

Proof: See Appendix A.We will show that there indeed exist such constants

and for all households later.

B. The Lyapunov-Based Approach

The idea of our algorithm is to construct a Lyapunov-basedscheduling algorithm with perturbed weights for determiningthe optimal power usage. By carefully perturbing the weights,we can ensure that whenever we charge or discharge the battery,the energy level in the battery always lies in the feasible region.First, we choose a perturbation vector (to be

specified later). We define a perturbed Lyapunov function asfollows:

(12)

Now define , and define a one-slotconditional Lyapunov drift as follows:

(13)

Here the expectation is taken over the randomness of load ar-rivals, cost function, and renewable generation, as well as therandomness in choosing the control actions. Then, following theLyapunov optimization framework, we add a function of the ex-pected cost over one slot (i.e., the penalty function) to (13) toobtain the following drift-plus-penalty term:

(14)where is a positive control parameter to be specified later.Then, we have the following lemma regarding the drift-plus-penalty term:


Lemma 2: For any feasible action under constraints (2), (5),and (6) that can be implemented at slot , we have

(15)

where is a constant given by

(16)

Proof: See Appendix B.We now present the LCMA algorithm. The main design prin-

ciple of our algorithm is to choose control actions that approxi-mately minimize the R.H.S. of (15).Lyapunov-Based Cost Minimization Algorithm (LCMA):

Initialize and . At each slot , observe, and , and do:

• Choose control decisions and as the optimal solutionto the following optimization, called :

(17)

• Update according to the dynamics (1), (4), and (10),respectively.

The intuition behind our algorithm is trying to store excessrenewable energy for later use, recharge the battery during theperiod of low electricity price while discharging it during the pe-riod of high electricity price, and delay elastic energy loads tolater slots with lower electricity price. Note that we do not needto consider the time-coupling constraints (6) of the battery en-ergy level in the algorithm, since they can be automatically sat-isfied during our operation of the queues, as proven in Theorem1 below. Moreover, the algorithm only requires the knowledgeof the instantaneous values of system dynamics and does notrequire any knowledge of the statistics of these stochastic pro-cesses. However, the algorithm above should be able to run ina decentralized manner in order to be implemented in practice.In the ensuing subsection, we design a decentralized algorithmto solve the optimization problem .

C. Decentralized Algorithm to

First, we introduce the following slack variables: toupper bound individual grid power demand and to upperbound the total grid power demand. Then, we can transforminto the following formulation, called :

(18a)

(18b)

(18c)

(18d)

where the maximum grid power consumption is imposedbecause of security and reliability considerations for household, and is the transformer capacity. Since is a strictlyincreasing function, we can easily prove by contradiction thatthe formulation above and are equivalent and have exactlythe same optimal solutions in terms of and . Since isa convex optimization problem and has decomposability struc-tures, it motivates us to design the following distributed subgra-dient-based algorithm to iteratively solve it based on the ideaof consistency price in network utility maximization [17], [18].In each time slot , the algorithm implements the steps as indi-cated in Algorithm 1. The flow chart for Algorithm 1 is shownin Fig. 2. When the constant step-size is small enough, the al-gorithm above converges to the optimal solution [19]. Note thatother types of step-size can also be used with different conver-gence properties [20]. A desirable feature of our decentralizedalgorithm is that the LSE does not need to know the detailed in-formation about the energy usage in each individual householdand only requires the total grid energy usage for all house-holds. By operating in this manner, our algorithm can help pre-serve the privacy of homeowners, who are shown to be con-cerned with some privacy issues associated with the the smartgrid [21].

Algorithm 1: Decentralized Algorithm to

1 Initialization: Set equal to some nonnegative value,

2 foreach Iteration

3 while Not satisfying convergence criterion do

4 The home energy management system (HEMS) inhousehold ’s smart meter updates , , andafter receiving the Lagrangian multiplier according tothe solution to the following optimization problem:

(19a)


Fig. 2. Flow chart for the Algorithm 1.

(19b)

(19c)

(19d)

(19e)

5 The LSE collects the predications of total utility powerdemands from all households over the FAN.Then, its neighborhood energy management system (NEMS)obtains the optimal generation power and updates the Lagrangemultiplier as follows:

(20)

(21)

where is a constant step-size, and then, broadcaststo all households over the FAN

6 Set

Note that the optimal Lagrangian multiplier is similarto the dynamic electricity retail price charged by the LSE to eachhousehold at time .

IV. PERFORMANCE ANALYSIS

In this section, we analyze the performance of LCMA underthe case that the renewable energy generation , energyload arrival processes and are all i.i.d.Note that our results can also be extended to the more generalsetting where , , and all evolveaccording to some finite state irreducible and aperiodic Markovchains according to the Lyapunov optimization framework [16].

Theorem 1: If andfor all households , then under the LCMA al-

gorithm for any fixed parameters , and, where

(22)

we have the following properties:1) The queues and are deterministically upperbounded by and at every slot, where

(23)

(24)

Further, are upper bounded by where

(25)

2) The worst-case delay of any buffered elastic energy load isgiven by:

(26)

3) The energy queue satisfies the following for all timeslots :

(27)

4) All control decisions are feasible.5) If , , and are i.i.d. over slots,then the time-average expected operating cost under ouralgorithm is within bound of the optimal value, i.e.,

(28)

where is the constant specified in (16).Proof: The proof is also a straightforward extension of the

results in our previous work [14] into the case of multiple house-holds.We provide the sketch of our proof as follows. Details canbe found in our technical report [22].1) We prove the results by induction. First, if

, the maximum increase during one slot is .Therefore, we obtain the upper bound in this case. Second,if , LCMAwill choosethe maximum possible value for since the partialderivative of the objective function in w.r.t. isnegative. The arrival amount can not be larger than theserved amount by our assumption. Therefore, the queuelength cannot increase. This completes the proof. Theupper bound of and can be provedsimilarly.

2) This follows directly from Lemma 1.3) Once again, we prove the result by induction. If

, then LCMA willchoose the maximum value for . Therefore the bat-tery would charge as much as possible, i.e.,


. Second, assuming that,

then, where we have used the upper bound of .

Third, suppose ,then LCMA will choose the minimum value for .Therefore, the battery would discharge as much as pos-sible, i.e., .This completes the proof.

4) Since we choose our decisions to satisfy all constraints in, combining it with the results above together, all con-

straints of are satisfied. Therefore, our control deci-sions are feasible to .

5) As we have mentioned before, the LCMA is always tryingto greedily minimize the R.H.S. of the upper bound (15) ofthe drift-plus-penalty term at every slot over all possiblefeasible control policies. Therefore, by plugging this policyinto the R.H.S. of the inequality (15), and comparing itwith a stationary, randomized control policy to the problemwithout the time coupling constraint (6), we can obtainthe performance result following the performance resultderivation in Lyapunov optimization framework.

V. NUMERICAL EXPERIMENTS

In this section, we provide numerical results based on real-world data sets to complement the analysis in the previous sec-tions.

A. Experiment Setup

We consider a simple power system consisting of eight house-holds in one neighborhood that share the same load serving en-tity and have on-site renewable generation, energy storage, andelastic and inelastic energy loads. The households are dividedinto two categories. For the first type of households (indexedby , 2, 3, 4), both the elastic and inelastic energy load ar-rivals during one slot are i.i.d. and take value from [1, 5] kWhuniformly at random. For the second type of households (in-dexed by , 6, 7, 8), both the elastic and inelastic energyload arrivals during one slot are also i.i.d. and take value from[1.5, 7.5] kWh uniformly at random. For the renewable gener-ation, we use the hourly average solar irradiance data for LosAngeles area from the Measurement and Instrumentation DataCenter (MIDC) [23] at the National Renewable Energy Labo-ratory. The period we consider in this paper is half a year fromJanuary 1, 2011 to June 30, 2011. In total, this duration includes181 days or 4344 1-hour slots. The control interval is chosen tobe one hour. We use different scaling factors to characterize theheterogeneity of households. Specifically, we choose the scalingfactors such that the average solar energy production during oneslot is about 3 kWh for the first type of households and 4.5 kWhfor the second type of households. We fix the maximum chargeand discharge rates of batteries in households as follows: for

, , , and for, , . Also,

we choose for all . As [12], the battery cost isassumed to be a simple quadratic function as follows:

(29)

where is a constant coefficient. For the purpose of simpleillustration, we choose the same battery cost function for allhouseholds in the evaluations.For the LSE, we assume that the energy cost function is a

smooth quadratic function as follows:

(30)

where , , and are constant coefficients. We choose, , and in our evaluations.

B. Results and Analysis

In order to analyze the performance improvement due to ourLCMA, we compare it with the following two approaches: i) Nostorage, no demand response (B1): The household tries to usethe renewable energy as much as possible. When the renewableenergy is not sufficient, the household draws energy from theutility grid. Unused renewable energy is wasted; ii) Storage, nodemand response (B2): The household uses renewable energyonly as a supplement to the grid by consuming it whenever itis available. The household stores any extra renewable energyin its battery, but never charges the battery from the grid. Thestored energy would be used to serve the future demands. Notethat LCMA differs from the approaches above in the sense thatLCMA would actively charge the battery when the grid poweris cheap while discharging it when the grid power is expensive.Moreover, LCMA differentiates between inelastic and elasticenergy loads and delays the elastic energy loads to later timewhen the grid power cost is low.First, we start by considering the convergence of the distri-

bution algorithm. Fig. 3(a) illustrates the convergence of Algo-rithm 1. We use a step size and the algorithm shows nosign of lack of convergence.Then, we compare our algorithm with the two approaches

above using the real-world solar power generation. Note that theperformance of LCMA depends on the battery capacity, the bat-tery cost, and the control parameters and . We choose, , , and ,

. The initial battery energy level at each house-hold is chosen to be zero. Let and .As can be seen in Fig. 3(b), our proposed LCMA can reducethe total energy cost by approximately 20% compared with B1and 13% compared with B2 in the six-month period. Also, theslopes of the lines are different, meaning that the savings areunbounded as the time increases. Meanwhile, the LCMA hason average a much smaller delay than the worst-case guarantee(11), as shown in Fig. 3(c).In the following, we consider the impact of varying control

parameters on the performance of LCMA.• Impact of Battery Capacity: In this evaluation, wevary the battery capacities of households with otherparameters fixed. We setfor , for

, and . The result is illustratedin Fig. 4(a). From the figure, it is clear that the larger thecapacity is, the more cost saving LCMA can obtain, whichcoincides with the algorithmic performance results of ouralgorithm in Theorem 1. As we have mentioned before,


Fig. 3. (a) Convergence of Algorithm 1; (b) comparison of the total energy cost in three approaches; (c) histogram of delay for the elastic demands in the servicequeue for one household.

Fig. 4. (a) The impact of battery capacity on the cost saving; (b) the impact of battery cost on the cost saving; (c) the impact of on the cost saving.

the saving comes from the fact that our algorithm chargesthe battery when the marginal energy cost is low, whiledischarging it when the marginal energy cost is high.

• Impact of Battery Cost: Currently, batteries are still ex-pensive. The charging or discharging operation would re-duce the lifetime of the battery. However, it is expected thatthe cost of the battery would decrease greatly in the nextdecade. In this evaluation, we estimate the impact of bat-tery cost on the cost saving of our algorithm. We set

and keep , ,and , fixed. The result is shownin Fig. 4(b). Note that when the battery cost per usageduring one period is very large (e.g., 200 $), our algo-rithm would not charge or discharge the battery at all, soit is the same as the approach B1. As the battery cost in-creases, the total cost saving of LCMA compared with B1would decrease until they are the same since the opportu-nity to utilize the temporal variation of electricity prices issmaller.

• Impact of Worst-case Delay Requirement: In this set-ting, we adjust the parameters while fixing other param-eters to see the impact of the worst-case delay guaranteefor elastic energy loads on the performance of LCMA. Wechoose , respectively. As observedin Fig. 4(c), the increase of (i.e., the worst-case delay)gives more opportunity to optimize the energy cost, since

the elastic energy loads are more likely to be served in thelow energy cost period.

VI. CONCLUSIONS

In this paper, we present an algorithm (LCMA) for the de-centralized and coordinated stochastic optimization of flexibleenergy resources in a smart grid setting. The total system costcan be reduced if more energy loads are elastic and can tol-erate being served with some delay. Our algorithm is simpleand was shown to be able to operate without knowing the statis-tical properties of the underlying dynamics in the system. Withthe increase of energy storage capacities, the performance ofour algorithm is proved to be arbitrarily close to the optimalvalue. Moreover, our algorithm provides an explicit relation-ship between energy storage capacity, worst-case delay, and costsaving. Extensive numerical evaluations based on real-worlddata sets show the effectiveness of our approach.

APPENDIX APROOF OF LEMMA 1

Here we prove Lemma 1.Proof: For all households , consider any slot for which

. We will show that this energy load is servedon or before time slot by contradiction. Suppose not,then during slots , it must be that


. Otherwise, the energy load would have beenserved before . Therefore, , and from the update(10) of , we have for all :

(31)

Summing the above over yields:

(32)

Rearranging the terms and using the facts thatand yields:

(33)

Since the energy loads are queued in a FIFO mannerand , they would be served on or beforetime whenever there are at least units of en-ergy served during . Since we haveassumed that the energy loads are not served by time

, it must be that . Comparingthis inequality with (33) yields:

(34)

which implies that , contradictingthe definition of in (11).

APPENDIX BPROOF OF LEMMA 2

Proof: From (4), subtracting both sides by , and squaringboth sides, we have for each household ,

(35)

Moreover, we have the following inequality:

(36)

Taking expectations of both sides of (4) given , and sum-ming over all households , we can get the following upperbound for the Lyapunov drift for :

(37)

Also, from (1), squaring both sides, and using the followinginequality:

(38)

we obtain

(39)

Similarly, from (10), we have

(40)

Then, we obtain the following inequality:

(41)

Combining these three bounds together, summing overall households, taking the expectation w.r.t. on bothsides, and adding penalty term

to both sides of the above inequality, wearrive at the conclusion in the Lemma.

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Yuanxiong Guo (S’11) received his B.Eng. degreefrom the Department of Electronics and InformationEngineering, Huazhong University of Science andTechnology, Wuhan, China, in 2009. He has beenworking towards the Ph.D. degree in the Departmentof Electrical and Computer Engineering at Univer-sity of Florida, Gainesville, FL, USA, since August2010. His current research interests are in the areaof cyber-physical systems including smart grids,sustainable data centers, and cloud computing. Heis a recipient of the Best Paper Award from IEEE

GLOBECOM 2011, Houston, TX, USA.

Miao Pan (S’07–M’12) received his B.Sc. degreein electrical engineering from Dalian Universityof Technology, China, in 2004, M.A.Sc. degree inelectrical and computer engineering from BeijingUniversity of Posts and Telecommunications, China,in 2007 and Ph.D. degree in electrical and com-puter engineering from the University of Florida,Gainesville, FL, USA, in 2012, respectively.He is now an Assistant Professor in the Depart-

ment of Computer Science at Texas Southern Uni-versity, Houston, TX, USA. His research interests in-

clude cognitive radio communications and networking, wireless security andnetwork economics, and resource management in cyber-physical systems suchas smart grids, cloud computing, and wireless sensor networks.

Yuguang Fang (F’08) received a Ph.D. degree insystems engineering from Case Western ReserveUniversity, Cleveland, OH, USA, in January 1994and a Ph.D. degree in electrical engineering fromBoston University, Boston, MA, USA, in May 1997.He was an Assistant Professor in the Department

of Electrical and Computer Engineering at NewJersey Institute of Technology from July 1998to May 2000. He then joined the Department ofElectrical and Computer Engineering at Universityof Florida, Gainesville, FL, USA, in May 2000 as

an Assistant Professor, got an early promotion to an Associate Professor withtenure in August 2003, and to a full Professor in August 2005. He holds aUniversity of Florida Research Foundation (UFRF) Professorship from 2006 to2009, a Changjiang Scholar Chair Professorship with Xidian University, Xi’an,China, from 2008 to 2011, and a Guest Chair Professorship with TsinghuaUniversity, China, from 2009 to 2012. He has published over 300 papers inrefereed professional journals and conferences.Dr. Fang received the National Science Foundation Faculty Early Career

Award in 2001 and the Office of Naval Research Young Investigator Award in2002, and is the recipient of the Best Paper Award in IEEE Globecom (2011),IEEE International Conference on Network Protocols (ICNP, 2006) and the re-cipient of the IEEE TCGNBest Paper Award in the IEEE High-Speed NetworksSymposium, IEEE Globecom (2002). He has also received a 2010–2011 UFDoctoral Dissertation Advisor/Mentoring Award, 2011 Florida Blue Key/UFHomecoming Distinguished Faculty Award and the 2009 UF College ofEngineering Faculty Mentoring Award. Dr. Fang is also active in professionalactivities. He is a member of ACM. He served as the Editor-in-Chief for IEEEWireless Communications (2009–2012) and serves/served on several editorialboards of technical journals including IEEE TRANSACTIONS ON MOBILECOMPUTING (2003–2008, 2011–present), IEEE Network (2012–present), IEEETRANSACTIONS ON COMMUNICATIONS (2000–2011), IEEE TRANSACTIONSON WIRELESS COMMUNICATIONS (2002–2009), IEEE JOURNAL ON SELECTEDAREAS IN COMMUNICATIONS (1999–2001), IEEE Wireless CommunicationsMagazine (2003–2009) and ACMWireless Networks (2001–present). He servedon the Steering Committee for IEEE TRANSACTIONS ON MOBILE COMPUTING(2008–2010). He has been actively participating in professional conferenceorganizations such as serving as the Technical Program Co-Chair for IEEE IN-OFOCOM’2014, the Steering Committee Co-Chair for QShine (2004–2008),the Technical Program Vice-Chair for IEEE INFOCOM’2005, the TechnicalProgram Area Chair for IEEE INFOCOM (2009–2013), Technical ProgramSymposium Co-Chair for IEEE Globecom’2004, and a member of TechnicalProgram Committee for IEEE INFOCOM (1998, 2000, 2003–2008).

Pramod P. Khargonekar (F’93) received theB.Tech. degree in electrical engineering from theIndian Institute of Technology, Bombay, and theM.S. degree in mathematics and Ph.D. degree inelectrical engineering from the University of Florida,Gainesville, FL, USA.He has held faculty positions at the University

of Minnesota and the University of Michigan. Hewas Chairman of the Department of ElectricalEngineering and Computer Science at Michiganfrom 1997 to 2001, and also held the title Claude

E. Shannon Professor of Engineering Science there. From 2001 to 2009, hewas Dean of the College of Engineering and is now Eckis Professor Electricaland Computer Engineering at the University of Florida. He served as DeputyDirector for Technology at U. S. Department of Energys Advanced ResearchProjects Agency C Energy (ARPA-E). He is currently serving the U.S. NationalScience Foundation as Assistant Director for Engineering. His current researchinterests are focused on renewable energy and electric grid, neural engineering,and systems and control theory.Dr. Khargonekar is a recipient of the NSF Presidential Young Investigator

Award, the American Automatic Control Council’s Donald Eckman Award,the Japan Society for Promotion of Science Fellowships, and a DistinguishedAlumnus Award and Distinguished Service Award from the Indian Institute ofTechnology, Bombay. He is a co-recipient of the IEEE W. R. G. Baker PrizeAward, the IEEE CSS George S. Axelby Best Paper Award, and the AACCHugo Schuck Best Paper Award. He was a Springer Professor the University ofCalifornia, Berkeley in 2010. He is on the list of Web of Science Highly CitedResearchers. At the University of Michigan, he received a teaching excellenceaward from the EECS department, a research excellence award from the Col-lege of Engineering, and the Arthur F. Thurnau Professorship. At the Universityof Minnesota, he received the George Taylor Distinguished Research Award.

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