2540 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 12, DECEMBER 2015
Demand Response in Smart Grids: A RandomizedAuction Approach
Ruiting Zhou, Zongpeng Li, Chuan Wu, and Minghua Chen
Abstract—The smart grid is a modern power grid that achieveshigh efficiency and robustness through sophisticated informa-tion and communications technology. Demand response has greatpotential in helping balance demand and supply in a smart grid,cutting generation cost and carbon footprint, and improving sys-tem stability. Auctions represent a natural and efficient approachfor carrying out demand response between the power grid andlarge electricity users, microgrids, and electricity storage devices.This work explores the modeling and design space of demandresponse auctions, targeting expressive power, truthful informa-tion revelation, computational efficiency, and economic efficiency.We present a randomized auction that explores the underlyingproblem structure of demand response, and prove that it is truth-ful, runs in polynomial time, and achieves (1 + ε)-optimal socialcost for an arbitrarily small constant ε. The key technique liesin the marriage of smoothed analysis and randomized reduction,which makes its debut in this work among literature on mecha-nism design, and can be applied to problems where social welfareoptimization is NP-hard but admits a smoothed polynomial-timealgorithm.
Index Terms—Smart grids, demand response, energy efficiency,smoothed analysis, randomized auctions.
I. INTRODUCTION
T HE smart grid, emerging as a convergence of ICTwith power system engineering, is a modern electric
power grid infrastructure for enhanced efficiency and relia-bility through automated control, communication, sensing andmetering, and the strategic optimization of demand, energy, andnetwork availability [1], [2]. As in a traditional power grid, thequintessential problem in a smart grid is the realtime balancebetween supply and demand. Imbalances are to be correctedwithin seconds, to avoid frequency deviations that threaten grid
Manuscript received on March 29, 2015; revised on July 11, 2015; acceptedon September 3, 2015. Date of publication September 23, 2015; date ofcurrent version November 16, 2015. This work was supported in part byHidaca Inc., in part by Hong Kong RGC under Grant HKU 718513 and Grant17204715, in part by the National Basic Research Program of China (ProjectNo. 2013CB336700), and in part by the University Grants Committee of HongKong (Theme-based Research Scheme Project no. T23-407/13-N and GeneralResearch Fund no. 14201014).
R. Zhou is with the University of Calgary, Calgary, AB, Canada (e-mail:[email protected]).
Z. Li is with the University of Calgary, Calgary, AB, Canada, andalso with the School of Computer and Collaborative Innovation Center ofGeospatial Technology, Wuhan University, Wuhan, China (e-mail: [email protected]).
C. Wu is with the University of Hong Kong, Kowloon, Hong Kong (e-mail:[email protected]).
M. Chen is with the Chinese University of Hong Kong, Kowloon, Hong Kong(e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JSAC.2015.2481208
stability [3]. Demand response facilitates cost savings by reduc-ing and temporally shifting peak loads, arbitraging betweenperiods of over- and under-generation. Essentially all powergrids dispatch generators in a merit order, and wholesale elec-tricity prices are in line with the highest marginal cost. Asa result, significant economic benefits can be gleaned from aseemingly small reduction in peak consumption, as illustratedin Fig. 1. It was observed that in a regional power grid withinthe USA Eastern Interconnection, a 10% peak demand shed-ding translates into $28 billion annual savings in electricitycost [4].
In its definition of demand response, the Federal EnergyRegulatory Commission (FERC) envisions both elastic andemergent versions: “changes in electric usage by end-use cus-tomers from their normal consumption patterns to incentivepayments designed to induce lower electricity use at timesof high wholesale market prices or when system reliabilityis jeopardized.” The first demand response model we study(Sec. III) assumes emergent demand response with a fixed tar-get of demand reduction, while the richer problem of elasticdemand response is considered later in Sec. VI.
A demand response target, measured as the reduction in netelectricity consumption, can be achieved through three types ofactions:
� Demand curtailing and temporal shifting. Large electric-ity users exemplified by data centres are ideal candidates forparticipating in demand response. In 2013, U.S. data centersconsumed an estimated 91 billion kilowatt-hours of electric-ity, equivalent to the annual output of 34 large (500-megawatt)coal-fired power plants. They incur 9 billion in electricity billsand emit 97 million metric tons of carbon pollution per year [5].At the same time, computing tasks are often elastic by nature[6], making temporal workload shifting feasible.
� Onsite generation. Another salient characteristic of amodern smart grid is the growing penetration of distributedgeneration [7], where microgrids and large users are in pro-cession of their own generation units that include renewablegeneration with unstable output (wind, solar) and stand-by gen-eration that can be started and tuned on-demand (diesel, fuelcell). Such quick-start generation can contribute to a demandresponse process by increasing the output level during peakdemand periods.
� Electricity release to the grid. Recent technology advancesare making active (e.g., batteries) and passive (e.g., Plug-inElectric Vehicles, or PEVs) electricity storage economicallyfeasible. Ideally, these devices are charged during periods oflow demand and low prices, and discharged at periods of highdemand and high prices. A demand response auction provides
Fig. 1. Benefit of demand response: a small reduction in peak consumption(and hence peak generation rate) leads to significant cost savings and carbonfootprint reduction. Electricity generation cost is non-linear, and the marginalcost per Watt increases as the generation rate increases.
the necessary catalyst that makes such electricity arbitragepractically possible.
While a vast literature from the past three decades focuson engineering challenges of demand response [8], [9], theyoften fail to address a fundamental question of practical impor-tance: why would individual loads, microgrids and storagedevices voluntarily respond to grid instability at their own cost?As admitted by the UK government [10]: “an effective mar-ket mechanism must be created to reward installation of the(demand response) technology fairly.”
The evolution of a traditional power grid to the smart gridmakes demand response auctions particularly suitable. First,the two-way information communication infrastructure enablesrealtime bid submission and auction result declaration [1].Second, agents in the grid now have their own intelligence mod-ule based on software algorithms, and are capable of submittingdemand response bids and executing auction algorithms [1].Large scale realtime auctions in a network environment havenow proven practically feasible. Thousands of ad impressionauctions are executed on the Internet by Google per second,or billions per day [11]; myThings, the personalized retarget-ing company in Europe, handles over 50 million realtime bidsper day [12]; Plethora Mobile receives up to 40, 000 bids persecond for audience targeting opportunities [13].
A main alternative to auction is pre-defined electricity priceoffers, e.g., a substantially high metering price for discouragingconsumption when supply is tight [4]. While straightforward,fixed price schemes have their limitations. First, what price tooffer is always a tricky question; the grid may end up resort-ing to heuristic guesses and trial-and-error. Second, it is hard topredict by how much the demand would decrease as a result,problematic for emergent demand response. Third, on-site gen-eration and electricity release need to be considered and pricedseparately. A well designed demand response auction automati-cally resolves all three problems. First, when properly designed,an auction discovers the market price of a demand responsebid automatically. Second, a demand response target can beexplicitly set and achieved in an auction. Third, it is naturalto implement a unified type of bid that models all three optionsof demand response: curtailing electricity consumption, on-sitegeneration, and electricity discharges by storage devices.
This work explores the modelling and design space ofdemand response auctions in a smart grid, and aims to testthe limits of the performance of such auction mechanisms,
in terms of expressive power, truthful information revelation,computational efficiency, and economic efficiency. First, weconsider both emergent demand response where a fixed targetin net demand reduction is to be achieved, and elastic demandresponse where the grid has a concave utility function over arange of flexible reduction ranges. We model demand responsebids in both forms of demand reduction and supply augmenta-tion. Quick-start generation at the power grid itself, with linearor non-linear generation cost, is further included. Second, werequire that the auction be truthful — all demand responseparticipants achieve their respective maximum utility by bid-ing truthfully, regardless of other bidders’ strategies. Third, forpractical feasibility, we assume that computational power at thegrid is not unlimited, and the auction algorithm must executein polynomial-time. Fourth, we aim at tuning the auction algo-rithm to elicit desirable behaviors from agents in the grid, suchthat a demand response goal is met with minimum possiblegrid-wide cost.
We present the design of demand response auctions that con-sider all three types of bids, execute in polynomial time, andachieve near-optimal social welfare. A key technique is thecombination of smoothed analysis with randomized MIDR auc-tion design. This is enabled by a pair of associated perturbationsthat facilitates the design of a smoothed polynomial time algo-rithm and turns it into a truthful auction. This new technique ofdesigning smoothed polynomial-time auctions is applicable toa broad range of auction design problems, where social welfareoptimization can be modelled into a linear integer program thatis NP-hard in general, but admits a smoothed polynomial timealgorithm.
In the rest of the paper, we review related literature inSec. II, and define the demand response problem in Sec. III.Sec. IV designs a smoothed polynomial-time demand responsealgorithm, and Sec. V converts the algorithm into an FPTASauction. Simulation studies are presented in Sec. VII, andSec. VIII concludes the paper.
II. RELATED WORK
Over the past decade, demand response has been exten-sively studied for various management objectives in powergrids. Logenthiran et al. [2] use a day-ahead load shifting tech-nique to help providers reshape the load profile and reducepeak demand. They formulate demand side management asminimizing the gap between objective consumption and actualconsumption, with a heuristic evolutionary algorithm adopted.Qian et al. [14] propose a real-time pricing scheme that helpsreduce the peak load and realize demand response manage-ment in smart grid systems. Shi et al. [15] consider residentialdemand response in a power distribution network with powerflow and system operational constraints, and propose a dis-tributed scheme can to compute an optimal demand schedule.Saber et al. [16] study two possible models to utilize PEVs: theload-leveling model and the smart grid model, and show that thelatter with renewable energy sources is a promising approach.Different from the above literature, this work focuses on theauction design that provides the necessary financial catalyst
2542 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 12, DECEMBER 2015
TABLE ICOMPARISON BETWEEN EXISTING DEMAND RESPONSE LITERATURE AND THIS WORK. DC = DEMAND CURTAILING AND TEMPORAL
SHIFTING; OG = ONSITE QUICK-START GENERATION; ER = ELECTRICITY RELEASE TO THE GRID
for realizing demand response. More importantly, these exist-ing demand response mechanisms sidestep the computationalchallenges by avoiding making win-lose decisions and assumemandatory participation of every agent, which compromisesoptimal social welfare.
A series of recent work start to examine the design of auc-tion mechanisms for realizing demand response in smart grids.Samadi et al. [17] propose a VCG mechanism that aims to max-imize the social welfare of a smart grid. Their design requiresusers to report their energy demand, and computes each user’selectricity bill payment. They verify that their mechanism guar-antees economic efficiency and user truthfulness. Zhou et al.[18] propose an truthful online auction to incentivize the par-ticipation of storage devices in power demand response. Theapproximation ratio of their primal-dual approach is not a con-stant, but is close to 2 in typical scenarios. Another recent work[19] study datacenter demand response where geo-distributedclouds participate in demand response activities at multiplepower grids. A decentralized mechanism is designed for eachdatacenter to elicit truthful bids and to determine the winningones. Again, mandatory participation in the demand response isassumed in the first work [17]. Although latest studies [18], [19]model voluntary participation, most of them provide no provenguarantee for approximation ratio in social welfare. Our workis among the first that applies smoothed analysis techniquesto design auction mechanisms, and guarantees (1 + ε)-optimalsocial cost for an arbitrarily small ε.
On the optic of integrating microgrids into a modern smartgrid, Lu et al. [20] propose an online algorithm for the micro-grid generation scheduling problem, which achieves a smallcompetitive ratio below 3. Moreover, a few studies have startedto investigate auction design for microgrids. An auction frame-work for electricity trading between a power grid and micro-grids is presented by Zhang et al. [7]. Both grid-to-microgridand microgrid-to-grid energy sales are studied, with truthfulbidding guaranteed for the latter case only. The above litera-ture models the voluntary participation of agents, but does notalways guarantee truthful bidding, and cannot provide a guar-antee of near-optimal social welfare. Table I summarizes thecomparison between existing literature and this work.
A polynomial-time approximation scheme (PTAS) [21] is atype of approximation algorithm for NP-hard problems. It takestwo parameters: ε > 0 and problem size n, and produces a solu-tion that is (1 + ε)-optimal for minimization problems, or (1 −ε)-optimal for maximization problems. The running time of aPTAS is required to be polynomial in n, but can be exponential
Fig. 2. An illustration of the demand response auction in a smart grid.
to 1ε. If we further require the complexity to be polynomial
in both n and 1ε, a PTAS become a fully polynomial-time
approximation scheme (FPTAS).Smoothed analysis [22] is a relatively new technique for
analyzing the expected running time of an algorithm with a ran-domly perturbed problem instance. It originates from attemptsto understand and analyze the behavior of algorithms thathave a bad worst-case performance but a good performancein practice, such as the simplex algorithm for linear program-ming. To our knowledge, this work is the first that adopts theidea of smoothed analysis to mechanism design. Dough andRoughgarden [23] studied mechanism design where social wel-fare maximization has a packing structure. They show that if anFPTAS exists when truthful bids are known, then such truthfulbids can be elicited through a truthful auction that retains theFPTAS property. While this work has been inspired in part bytheir randomization techniques, we do not require the existenceof an FTPAS in the first place. We resort to the art of smoothedpolynomial-time algorithm design instead.
III. SYSTEM MODEL AND PRELIMINARIES
We consider a smart grid system where a power grid is con-nected with agents that include microgrids, large electricityusers (e.g., data centers), and storage devices (e.g., batteries,PEVs), as illustrated in Fig. 2. When the power grid predicts atime period in which supply may fail to meet demand, it actsas the auctioneer and initiates a demand response process bycalling for bids from agents through a reverse auction, a.k.a. a
ZHOU et al.: DEMAND RESPONSE IN SMART GRIDS 2543
procurement auction. Each agent’s bid is a pair (em, bm). Hereem is the power it can supply to the power grid or the amountof power consumption (in W) it is willing to shed. bm is thecorresponding remuneration asked for. The power grid is in pos-session of its own stand-by generators (e.g., diesel), which canbe turned on to supply electricity as well.
Let [M] denote the integer set {1, 2, . . . , M}. Assume Magents participate in the auction. D is the demand response tar-get, i.e., power shortage (in W) in the upcoming time period.We define c and zmax as the unit cost (in $ per W) and max-imum available power (in W) of the grid’s diesel generators.We also assume that any (M − 1) agents’ bids can cover thedemand response target, i.e.,
∑m∈[M−n] em ≥ D, ∀n ∈ [M]. At
the end of the auction, the auctioneer announces: (i) A binarynumber xm corresponding to each agent m, where xm is 1 if thegrid accepts its bid, and 0 otherwise. (ii) A payment pm to eachwinning agent m. Finally, the power grid determines the totaloutput rate z of its stand-by generators.
Let vm be the true cost of em , which is private informationknown to agent m itself only. Let b−m be the set of all bidsexpect that of agent m. The utility of agent m is:
um(bm, b−m) ={
pm − vm if xm = 1
0 otherwise
In term of strategic behaviours, an agents is assumed to be self-ish and rational, with a natural goal of maximizing its ownutility. An agent may choose to misreport its cost (bm �= vm),if doing so leads to a higher utility. The auctioneer instead aimsto maximize the social welfare of the entire grid, for which it isimportant to elicit truthful bids from agents.
Definition (Truthful auction): An auction is truthful if forany agent m, its dominant strategy is to report the true cost vm
of em , regardless of other agents’ bids. In other words, for allbm �= vm and b−m , the following always holds: um(vm, b−m) ≥um(bm, b−m).
Definition (Social welfare, social cost): The social welfare ina demand response auction is the aggregate utility of the grid(−∑m∈[M] pm − cz) and the bidding agents (
∑m∈[M](pm −
vm xm)). Payments between agents and the grid cancel them-selves, and the social welfare is equal to −∑m∈[M] vm xm − cz.Maximizing the social welfare is equivalent to minimizing thesocial cost
∑m∈[M] vm xm + cz, which in turn is equivalent to
minimize∑
m∈[M] bm xm + cz under truthful bidding.Social Cost Optimization in A Demand Response
Auction. Under the assumption of truthful bidding, the socialcost minimization problem for demand response can be mod-elled by the following mixed integer linear program (MILP):
Minimize∑
m∈[M]
bm xm + cz (1)
Subject to:∑
m∈[M]
em xm + z ≥ D (1a)
xm ∈ {0, 1}, ∀m ∈ [M] (1b)
0 ≤ z ≤ zmax (1c)
Constraint (1a) guarantees that the successful bids and thediesel generation are together sufficient to cover the grid’sdemand response target D. Constraint (1b) models binary deci-sion making. Constraint (1c) limits the output of the dieselgenerators by their maximum capacity.
MILP (1) is a generalization of the NP-hard problem of min-imum knapsack [24], and is hence unlikely to have optimalpolynomial-time algorithms. We are interested in polynomial-time demand response auctions that are truthful and canapproach the optimal social cost as closely as possible. A tableof notations is provided below for ease of reference.
IV. THE SMOOTHED POLYNOMIAL-TIME ALGORITHM
In this section, we first formulate a complementary prob-lem to MILP (1) in Sec. IV-A, such that a feasible solution tothe complementary problem can be easily converted to a feasi-ble solution to MILP (1). Then we design an exact algorithmto solve the complementary problem in Sec. IV-B. Sec. IV-C applies the smoothed analysis technique to perturb biddingprices bm . With a carefully designed perturbation matrix, theexact algorithm to the complementary problem can be utilizedto return a (1 + ε)-optimal solution to MILP (1), with expectedpolynomial running time.
A. A Complementary Problem to MILP (1)
Let’s define Dc = ∑m∈[M] em − D. A complementary
MILP to MILP (1) is:
Maximize∑
m∈[M]
bm ym − cz (2)
Subject to:∑
m∈[M]
em ym − z ≤ Dc (2a)
ym ∈ {0, 1}, ∀m ∈ [M] (2b)
0 ≤ z ≤ zmax (2c)
2544 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 12, DECEMBER 2015
Let (x, z1) and (y, z2) be solutions to (1) and (2), respec-tively, where x = [x1 x2 . . . xM ]T and y = [y1 y2 . . . yM ]T .Let y = �1 − x and z2 = z1. Clearly, if (x, z1) is a feasible solu-tion to MILP (1), then (y, z2) is a feasible solution to MILP (2).Consequently, (x, z1) is an optimal solution to MILP (1) if andonly if (y, z2) is an optimal solution to MILP (2), where �1 is aM × 1 vector of 1’s.
B. An Exact Algorithm to the Complementary Problem
We next design an algorithm to solve the complementaryproblem in MILP (2). The algorithm is exact in that it alwaysreturns the optimal solution, but it may not terminate in poly-nomial time. The solution consists of two sets of values: y andz2. A naive approach is to enumerate all the possible combina-tions of y, and complement each with a z2: if
∑m∈[M] em ym
is already smaller than Dc, then z2 is set to zero; otherwise,z2 = ∑
m∈[M] em ym − Dc. The optimal solution is the one withthe maximum value in (
∑m∈[M] bm ym − cz2). However, this is
inefficient since the number of possible y’s grows exponentiallyas the size of the input increases. Let b = [b1 b2 . . . bM ]T ,e = [e1 e2 . . . eM ]T . Our first idea is to only enumerate the“good” y’s and ignore the “bad” ones, based on the followingobservation: a vector y cannot be optimal if it is dominated byanother vector y′, i.e., if bT y′ is larger than bT y and eT y′ issmaller than eT y. We formalize the concept of “good” vectorsusing Pareto optimal vectors:
Definition (Pareto optimal vector): A vector y is Paretooptimal if there does not exist a vector y′ dominating y, i.e.,� ∃y′ such that bT y′ ≥ bT y and eT y′ ≤ eT y, with at least oneinequality being strict.
Lemma 1: Let P(m) be the set of all Pareto optimal vec-tors when only the first m agents are considered. If y(m) ∈P(m), then the vector obtained from y(m) by removing itsm-th element is a Pareto optimal vector in P(m − 1), ∀m ∈[2, 3, . . . , M].
Proof: Consider a vector y(m) ∈ P(m). By the definitionof Pareto optimal vectors, y(m) is not dominated by anothervector. By way of contradiction, suppose that a vector y(m−1)
obtained by removing the last element y(m)m from y(m) is
not Pareto optimal. Then there exists a Pareto optimal vectory(m−1)′ dominating y(m−1). In addition, y(m−1)′ + y(m)
m (i.e.,the vector obtained by appending y(m)
m to the end of y(m−1)′ )dominates y(m), which leads to a contradiction. �
Lemma 1 suggests that the Pareto optimal set P(m) can becomputed from P(m − 1). Furthermore, it must be containedin the set P(m − 1) + 0 ∪ P(m − 1) + 1, where P(m − 1) + 0is obtained by appending 0 as the m-th element to each vectorin P(m − 1), similar for P(m − 1) + 1. An exact algorithm isshown in Algorithm 1, adopting the classic dynamic program-ming method for constructing the Pareto optimal set. First, itinitializes an empty set A and constructs the bottom set P(1) atline 1. By the definition of a Pareto optimal vector, both solution1 (accept the first agent’s bid) and 0 (reject the the first agent’sbid) are included in the set P(1). Then a for loop in lines 2-5computes P(2), . . . ,P(M). At each iteration, P(m) is derived
by eliminating all the dominated vectors (line 4) from the setP(m − 1) + 0 ∪ P(m − 1) + 1. Another for loop in lines 6-12computes the value of variable z2. For each y in Pareto optimalset P(M), if the total eT y is smaller than or equal to Dc, z isset to zero to maximize the cost. Otherwise, z2 is set to the gapbetween eT y and Dc. If z2 satisfies constraint (2c), the value ofy and z2 is stored in the set A at line 10. Line 13 returns thesolution with maximum objective value among all the feasiblesolutions in set A.
Algorithm 1. An Exact Algorithm for MILP (2)
Input: b, e, Dc
Output: optimal solution y∗ and z∗ to MILP (2)1: A = ∅; P(1) = {0, 1};2: for all m ∈ [2, 3, . . . , M] do3: Merge P(m − 1) + 0 and P(m − 1) + 1 into P(m)′ such
that P(m)′ is sorted in non-decreasing order of socialcost;
dominates y(m)};5: end for6: for all y ∈ P(M) do7: if eT y ≤ Dc then z2 = 0;8: else z2 = eT y − Dc;9: end if
10: if z2 ≤ zmax A = A ∪ (y, z2); then11: end if12: end for13: Return y∗, z∗ = arg max(y,z2)∈A bT y − cz2
Lemma 2: The number of Pareto optimal vectors |P(m)|does not decrease when m increases, i.e., P(1) ≤ · · · ≤ P(M).
Proof: In Algorithm 1, |P(m)| is computed from P(m)′by pruning the non-Pareto optimal vectors. When we mergeP(m − 1) + 0 and P(m − 1) + 1 into P(m)′, if all solutions inP(m − 1) + 0 are retained in P(m), then |P(m)| ≥ |P(m − 1)|.If some solutions in P(m − 1) + 0 are eliminated, then there areother solutions in P(m − 1) + 1 that dominate them. Therefore,we can always find a vector to replace the removed one. Thatfinishes the proof of |P(m)| ≥ |P(m − 1)|. �
Lemma 3: Upon termination, Algorithm 1 returns an optimalsolution (y∗, z∗) to MILP (2).
Proof: We first prove that the vector y∗ in the optimalsolution set (y∗, z∗) for MILP (2) must be a Pareto optimal vec-tor in P(M). Otherwise, there exists a Pareto optimal vector y′that dominates y∗, i.e., bT y′ ≥ bT y∗ and eT y′ ≤ eT y∗ with atleast one inequality being strict. Therefore, y∗ is not an optimalsolution to MILP (2), which is a contradiction. Hence, the opti-mal y∗ comes from Pareto optimal set P(M). In Algorithm 1,we subsequently calculate the corresponding z2 for each Paretooptimal vector in P(M) to satisfy constraints (2a) and (2c),and output the one with maximum objective value as the final(optimal) solution. �
Theorem 1: The running time of Algorithm 1 is poly-nomial to the number of Pareto optimal vectors, and isO(M |P(M)|).
ZHOU et al.: DEMAND RESPONSE IN SMART GRIDS 2545
Proof: Lines 1 in Algorithm 1 can be executed in O(1)
steps. The running time of line 13 is polynomial to |P(M)|.During each iteration of the first loop, line 3 constructs setP(m)′ by merging the two sets P(m − 1) + 0 and P(m − 1) +1. Both of the these sets are sorted in non-decreasing order ofcost due to the assumption of P(m − 1). Thus, we can computeP(m)′ in O(|P(m − 1)|) steps such that it is also sorted. Giventhis order of vectors in P(m)′, the set P(m) can be foundedin linear time at line 4. The running time of the second loop(lines 6-12) is polynomial to the number of vectors in the Paretooptimal set P(M). In summary, the overall time complex-ity of Algorithm 1 is upper bounded by
We next design a randomized algorithm with expected poly-nomial running time for MILP (1), based on randomizedperturbation. The basic idea is to first construct a set of fea-sible solutions to the complementary problem (2) throughthe exact Algorithm 1, then randomly output a solution fromthis set following a well-designed distribution, such that thegap between the expectation of the chosen solution and theoptimal solution is very small. Next, we utilize the comple-mentarity between MILP (2) and MILP (1) to compute acorresponding feasible solution to MILP (1), and prove thatthe expectation of our solution is away from optimum by atmost an additive factor ε maxm∈[M] bm , where ε is a smallparameter that can be arbitrarily close to zero. In order tocompute the feasible set efficiently, we use a perturbationmatrix to perturb the input bids based on smoothed analysistechniques.
Given an approximation parameter α ∈ (0, 1), we draw Mrandom variables uniformly from [0, α/M], forming a vectorβ = [β1 β2 . . . βM ]T . Define a perturbation matrix:
P = (1 − α)I + β�1T
M, (3)
where I is a M × M identity matrix. We utilize the perturbationmatrix to perturb the cost vector b into a new vector b = Pb,such that each new cost bm can be expressed as:
bm = (1 − α)bm + βm∑M
j=1 b j
M,∀m ∈ [M]. (4)
The perturbed complementary problem is:
Maximize∑
m∈[M]
bm ym − cz
Subject To: Constraints (2a)(2b)(2c).
(5)
Algorithm 1 can be executed to solve the above perturbed prob-lem (5), and it outputs an optimal solution (y p, z p). The valueof the objective function is P O B J = bT y p − cz p.
Let (y∗, z∗) and O B J2 = bT y∗ − cz∗ be the optimal solu-tion to the complementary problem in (2) and the value of thecorresponding objective function, respectively. We then have:
P O B J = bT y p − cz p = (Pb)T y p − cz p ≥ (Pb)T y∗ − cz∗
= bT
((1 − α) I+
�1βT
M
)y∗ − cz∗ ≥ (1−α)bT y∗−cz∗
= O B J2 − αbT y∗ (6)
The first inequality holds because (y p, z p) is the optimal solu-tion to the perturbed problem (5). We can observe that apossible solution to problem (2) is (y = PT y p, z p), whichhas only a small loss αbT y∗. However, (y, z p) may not bea feasible solution because PT y p may have fractional entriesdue to the setting of P . Hence, constraint (2b) is violated,and constant (2a) may not be satisfied either. Although wecan not use (y, z p) directly as the solution for problem (2),we can design a randomized algorithm that outputs a samplefollowing a well-designed distribution, such that the expecta-tion of the random sample equals (y, z p). As a result, ourapproach can solve problem (2) with a small loss αbT y∗ inexpectation.
Let y be a M-dimensional all-zero vector �0, then (�0, z p) isa feasible solution to (5). If we assume em ≤ Dc,∀m ∈ [M],let l1, l2, . . . , lM denote basis vectors, i.e., lm
m = 1 and lm′m = 0,
∀m′ �= m. Then for any m, (lm, z p) lies in the feasible set of(5). Note that (y p, z p) is also a feasible solution to (2) becausethe constrains in MILP (5) are identical to those in MILP (2).The final output for MILP (2) is (y f , z f ), where y f is a sam-ple randomly produced from the set {y p, l1, l2, . . . , lM , �0} andz f = z p. The final output follows the distribution D(y f , z f ):
D(
y f , z f)
=
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
Pr[y f = y p, z f = z p
] = 1 − α
Pr[y f =lm, z f =z p
]=∑Mj=1 β j y p
jM ,∀m ∈ [M]
Pr[
y f = �0, z f = z p]
= 1 − Pr[y f = y p
]−M∑
m=1Pr[y f = lm
].
(7)
We can verify that the expectation of y f is equal to y:
E[y f ] = (1 − α)y p +∑M
j=1 β j y pj
M
(M∑
m=1
lm
)= PT y p = y
(8)
Thus, the expected value of the objective function when(y f , z f ) follows the distribution D(y f , z f ) is
E[bT y f − cz f
]= bT y − cz p ≥ O B J2 − αbT y∗ (9)
We know that MILP (2) is a complementary problem to MILP(1). A solution (x f , z f ) to MILP (1) can be obtained by let-ting x f = �1 − y f and z f = z p. According to the assumption∑
m∈[M−n] em ≥ D,∀n ∈ [M] (Sec. III), (x f , z f ) must be a
2546 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 12, DECEMBER 2015
feasible solution to MILP (1). Algorithm 2 is our random-ized algorithm that utilizes this property to solve the originalMILP (1). We next analyze the approximation guarantee ofAlgorithm 2.
Theorem 2: The expected social cost of the solution (x f , z f )
returned by Algorithm 2 is at most an additive ε maxm∈[M] bm
more than the optimal social cost, where ε = αM .
Proof: Define O B J1 as the optimal social cost of problem(1). The expected objective value returned by Algorithm 2 is
E[bT x f + cz f
]= E
[bT(�1 − y f
)+ cz p
]=
∑m∈[M]
bm−E[bT y f −cz p
]≤∑
m∈[M]
bm−(1 − α)bT y∗ + cz∗
=⎛⎝ ∑
m∈[M]
bm − bT y∗ + cz∗⎞⎠+ αbT y∗
≤ O B J1 + ε/M∑
m∈[M]
bm ≤ O B J1 + ε maxm∈[M]
bm . (10)
�We next show in Lemma 4 and Theorem 3 that the expected
running time of Algorithm 2 is polynomial to the input size.Intuitively, the time complexity of Algorithm 2 depends on thenumber of Pareto optimal vectors, for which we establish anupper-bound that is a polynomial of M and 1
ε.
Lemma 4: For the perturbed complementary maximiza-tion problem (5) with perturbation matrix P produced fromequation (3), the expected number of Pareto optimal vectors
E[|P(M)|] is upper bounded by 1 + M4
α.
Proof: Let c(y) be the social cost under vector y and per-turbed cost b. Let e(y) = eT y. Each Pareto optimal vector hasa total cost in [0, Mbmax ] because each agent’s perturbed costis at most bmax . Assuming that no two vectors are identical, wecan partition [0, Mbmax ] into small intervals such that there isat most one Pareto optimal vector in each small interval. As aresult, the expected number of Pareto optimal vectors is:
E[|P(M)|] = 1 + limN→∞
N−1∑n=0
Pr
[∃y ∈ [P(M)] :
c(y) ∈(
Mbmax n
N,
Mbmax (n + 1)
N
]].
Where the additional 1 corresponds to the vector �0, which isPareto optimal by definition. To estimate the probability ineach interval, we first define some variables. Let yn∗ be thevector that has the largest e(y) and satisfies c(y) ≤ Mbmax n
N ,
i.e., yn∗ = arg max{e(y)|c(y) ≤ Mbmax nN }. For n ≥ 0, yn∗ must
always exist. Let yn = arg min{c(y)|e(y) > e(yn∗) ∩ c(y) >Mbmax n
N } be the vector that has the smallest cost such that
e(y) > e(yn∗) and c(y) > Mbmax nN .
If c(yn) exists, then we define a random variable�( Mbmax n
N ) = c(yn) − Mbmax nN , and claim that,
Claim 1: If and only if �( Mbmax nN ) ≤ Mbmax
N , there exists a
Pareto optimal vector y such that c(y) ∈ ( Mbmax nN ,
Mbmax (n+1)N ].
Proof: Assume there is a Pareto optimal vector with thecost in ( Mbmax n
N ,Mbmax (n+1)
N ], and let yn be the Pareto opti-
mal vector with the smallest cost in ( Mbmax nN ,
Mbmax (n+1)N ]. Then
according the definition, yn = yn and �( Mbmax nN ) = c(yn) −
Mbmax nN ∈ (0, Mbmax
N ]. Conversely, if �( Mbmax nN ) ≤ Mbmax
N , yn
must be a Pareto optimal vector whose cost lies in the rangeof ( Mbmax n
N ,Mbmax (n+1)
N ]. �Hence, we can rewrite the expected number of Pareto optimal
vectors as:E[|P(M)|] = 1 + limN→∞
∑N−1n=0 Pr [�( Mbmax n
N ) ≤Mbmax
N ].Furthermore, we define y(n,m−) as the vector that
rejects agent m’s bid and has a cost of at most Mbmax nN .
Let S(n,m−) = {y|c(y) ≤ Mbmax nN ∩ ym = 0} be the set
of all y(n,m−). Further define y(n∗,m+) as: y(n∗,m−) =arg max{e(y)|c(y) ≤ Mbmax n
N ∩ ym = 0}. We define another
variable y(n,m+) as y(n,m+) = arg min{c(y)|e(y) >
e(yn∗,m−) ∩ c(y) > Mbmax nN ∩ ym = 1}. Similarly, we define
a random variable �m( Mbmax nN ) = c(yn,m+) − Mbmax n
N when
yn,m+ exists, then we have the following claim:Claim 2: When �( Mbmax n
N ) is defined, there exists an index
m ∈ [M] such that �( Mbmax nN ) = �m( Mbmax n
N ).
Proof: When �( Mbmax nN ) is defined, there exist the corre-
sponding yn and yn∗. Because c(yn∗) < c(yn), there must existat least one agent’s bid, indexed by m ∈ [M], being acceptedby yn but rejected by yn∗ i.e., yn
m = 1, yn∗m = 0. We claim that
for this index m, �( Mbmax nN ) = �m( Mbmax n
N ). In order to proveit, we first observe that yn∗ = yn∗,m− This is due to the reasonthat yn∗ is the vector with the highest e(y) among all vectorswith cost at most Mbmax n
N . It is belong to the set Sn,m−, it isin particular the vector with the highest e(y) among all vec-tors that reject agent m’s bid and have the cost at most Mbmax n
N .
Similarly arguments can be applied to prove yn = yn,m+. Thisdirectly implies that �( Mbmax n
N ) = �m( Mbmax nN ). �
Claim 3: : ∀m ∈ [M], Pr [�m( Mbmax nN ) ≤ Mbmax
N ] ≤ M3
αN .
Proof: In the perturbed MILP (5), the value of bm lies
in [(1 − α)bm, (1 − α)bm + α∑M
j=1 b j
M2 ] since βm is drawn from
[0, α/M]. The length of the interval isα∑M
j=1 b j
M2 , which is no
smaller than αbmaxM2 as
∑Mj=1 b j ≥ bmax . The density of bm
is upper-bounded by M2
αbmaxeverywhere in the interval [(1 −
α)bm, (1 − α)bm + α∑M
j=1 b j
M2 ]. �In order to prove this lemma, we exploit the randomness
of cost bm for a given m, other agents’ cost b j , j �= m, canbe considered as arbitrarily fixed parameters. Then the vectorsfrom set Sn,m− are fixed and hence also the vector yn∗,m− isfixed. Let S = {y|e(y) > e(yn∗,m−) ∩ c(y) > Mbmax n
N ∩ ym =1}. If the vector yn∗,m− is fixed, then set S is also fixed.yn,m+ is the vector with the minimal cost in set S. To proveClaim 3, we only need to find the probability of c(yn,m+) ∈( Mbmax n
N ,Mbmax (n+1)
N ]. Because other agents’ costs can be con-
sidered as fixed parameters, the value of bm determines whetherc(yn,m+) lies in the interval ( Mbmax n
N ,Mbmax (n+1)
N ] or not.
ZHOU et al.: DEMAND RESPONSE IN SMART GRIDS 2547
We can rewrite this event as { Mbmax nN <
∑j �=m b j + bm ≤
Mbmax (n+1)N ]}. Let λ = Mbmax n
N −∑j �=m b j , then the above
event is the same as the event bm ∈ (λ, λ + MbmaxN ]. Hence,
the probability of this event is upper-bounded by MbmaxN ×
M2
αbmax= M3
αN . �Combining Claim 1, Claim 2 and Claim 3, we have:
Pr
[∃y ∈ [P(M)
]: c(y) ∈
(Mbmax n
N,
Mbmax (n + 1)
N
]]
= Pr
[�
(Mbmax n
N
)≤ Mbmax
N
]
≤ Pr
[∃m ∈ [M] : �m
(Mbmax n
N
)≤ Mbmax
N
]
=M⋃
m=1
Pr
[�m
(Mbmax n
N
)≤ Mbmax
N
]
≤M∑
m=1
Pr
[�m
(Mbmax n
N
)≤ Mbmax
N
]≤ M4
αN.
Therefore, we have: E[|P(M)|] ≤ 1 + limN→∞∑N−1
n=0M4
αN =1 + M4
α. �
Theorem 3: The expected running time of Algorithm 2 forsolving MILP (1) is polynomial.
Proof: In Algorithm 2, lines 1-2 generate the perturbationmatrix P , and their running time is polynomial to M . Lines4-5 output a random solution y f according to the distributionD(y p), and the running time is polynomial as well. Line 6takes one step to compute x f and returns the result. Now, weonly need to examine the running time of line 3. CombiningTheorem 1 and Lemma 4, the expected running to solve theperturbed MILP (5) is
O(M E
[|P(M)|]) ≤ O
(M
(1 + M4
α
))
≤ O
(M5 1
α
)= O
(M6 1
ε
).
Hence, the overall expected running time of Algorithm 2 isO(M6 1
ε). �
Note that Algorithm 2 is effectively a randomized additiveFPTAS, since it solves MILP (1) with expected polynomial-time, and outputs a solution that is at most an additiveε maxm∈[M] bm more than the optimal social cost.
Further define γ = maxi, j∈[M]{bi/b j }. We have thatmaxm∈[M] bm/O B J1 ≤ γ , as O B J1 includes at least oneagent’s bid. Thus, the expectation of social cost achieved byAlgorithm 2 is: E[bT x f + cz f ] ≤ (1 + γ ε)O B J1. In otherwords, Algorithm 2 can achieve (1 + γ ε)-optimality whereε can be arbitrarily close to zero. With γ being a constant,Algorithm 2 becomes a randomized FPTAS that can output a(1 + ε)-optimal solution.
V. AN FPTAS DEMAND-RESPONSE AUCTION
We now translate the smoothed polynomial-time algorithm(Algorithm 2) into a truthful auction, adding truthfulness to the
Algorithm 2. A Smoothed Polynomial-Time Algorithm forMILP (1)
Input: α ∈ (0, 1), b, e, Dc
Output: A solution (x f , z f ) for MILP (1)1: Generate β = [β1 β2 . . . βM ]T uniformly randomly from
the interval [0, α/M];
2: Compute the perturbation matrix: P = (1 − α)I + β�1T
M ;3: Run Algorithm 1 with the input (Pb, e, Dc), and obtain
output (y p, z p);4: Produce a distribution function D(y f , z f ) as shown in (7));5: Choose (y f , z f ) according to the distribution function
D(y f , z f );6: x f = 1 − y f ; Return (x f , z f ).
algorithm while retaining its FPTAS property. This is achievedthrough a recent technique of maximal-in-distributional range(MIDR) algorithms [25], the second main technique in thiswork besides smoothed analysis. An MIDR algorithm refers toa randomized algorithm that outputs a sample from a set of fea-sible solutions, according to a distribution that does not dependon agent bids, achieving the largest social welfare among allsuch distributions in the range. Combined with VCG-style pay-ments following a similar distribution, an MIDR algorithmyields an auction that is truthful in expectation. We will still uti-lize the complementarity between MILPs (1) and (2). First, wegenerate a distribution range for MILP (2), then covert it to onefor MILP (1). Then we prove that our randomized Algorithm 2is an MIDR algorithm. At the end, a randomized VCG-like pay-ment scheme that works in concert with the MIDR algorithm isdesigned to obtain a truthful demand response auction.
Theorem 4: Algorithm 2 is an MIDR algorithm for theoriginal cost minimization problem in MILP (1).
Proof: An MIDR algorithm pre-commits to a distributionrange (a set of probability distributions over feasible solutions)independent of agents’ bids, and returns a sample based ona distribution that is from the distribution range to maximizethe expected social welfare. In a procurement auction, socialwelfare maximization is equivalent to social cost minimization.Therefore, an MIDR algorithm for MILP (1) is to output a sam-ple form a distribution that minimizes the expected social costover the distribution range.
First, we construct a distribution range for MILP (2), andconvert it to a distribution range for the original problem inMILP (1). Let S denote the set of all feasible solutions (y′, z′)of MILP (5). For each feasible solution (y′, z′) ∈ S, we canconstruct a distribution for (y, z2) similar to the one in (7):
D(y, z2) =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
Pr[y = y′, z2 = z′] = 1 − α
Pr[y = lm, z2 = z′] =
∑Mj=1 β j y′
jM ,∀m ∈ [M]
Pr[
y = �0, z2 = z′]
= 1 − Pr[y = y′]−∑M
m=1 Pr[y = lm
].
(11)
Where (y, z2) is a feasible solution for MILP (5) sampledfollowing distribution D(y, z2). Constraints in MILP (5) is
2548 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 12, DECEMBER 2015
the same as those in MILP (2), hence (y, z2) is also a fea-sible solution to MILP (2) and (y, z2) is independent ofagents’ bids. Therefore, we can construct a compact set Ry ={D(y, z2),∀(y′, z′) ∈ S} including all the distributions indexedby feasible solution (y′, z′), and Ry is the distribution range forMILP (2).
Let D(x f , z f ) be the distribution that draws (y f , z f ) accord-ing to distribution D(y f , z f ) and outputs (x f = �1 − y f , z f =z f ). Similarly, let D(x, z1) be the distribution that draws (y, z2)
according to distribution D(y, z2) and computes a solution(x = �1 − y, z1 = z2). Then, the distribution range Rx for MILP(1) includes all the possible D(x, z1). We have
E(x f ,z f )∼D(x f ,z f )
[bT x f + cz f
]= E(y f ,z f )∼D(y f ,z f )
[bT(�1 − y f
)+ cz f
]=
∑m∈[M]
bm − bT(
PT y p)
+ cz p
=∑
m∈[M]
bm − max(y′,z′)∈S
(bT(
PT y′)− cz′)
=∑
m∈[M]
bm − max(y′,z′)∈S
E(y,z2)∼D(y,z2)
[bT y − cz2
]
= min(y′,z′)∈S
E(y,z2)∼D(y,z2)
[bT(�1 − y
)+ cz2
]= min
D(x,z1)∈RxE(x,z1)∼D(x,z1)
[bT x + cz1
](12)
The first two equalities above follow from the definition ofD(x f , z f ) and y f ’s expected value in equation (8). Thethird equality holds because (y p, z p) is the optimal solutionto the perturbed MILP (5). The fourth equality holds sinceE(y,z2)∼D(y,z2)[y] = PT y′, which can be derived according to(8). The last two equalities come from the definition of D(x, z1)
and Rx . In summary, Algorithm 2 is an MIDR algorithm thatachieves the smallest expected social cost among all the solu-tions produced following the distributions in the distributionrange Rx . �
Next, towards designing a truthful-in-expectation auction, wefirst describe an important property of an MIDR algorithm:analogous to the VCG mechanism, there is a deterministic pay-ment rule pvcg
m that can be coupled with an MIDR algorithm toyield a truthful-in-expectation mechanism, and
pvcgm = E[T (x f
−m, z f−m) − (T (x f , z f ) − bm x f
m)],∀m ∈ [M].(13)
Here pvcgm is the payment for each agent m. We do not need to
consider the payment to the power grid’s own quick-start gener-ators. (x f
−m, z f−m) is the output of Algorithm 2 by setting agent
m’s asking price to infinity. T (x f−m, z f
−m) is the total social costwith agent m excluded from the auction. T (x f , z f ) is the totalsocial cost when agent m participates and (x f , z f ) is the solu-tion returned by Algorithm 2. We define x f
m as the m-th elementof x f , then T (x f , z f ) − bm x f
m is the overall cost except agentm, when every agent participates in the auction.
It is not always possible to compute the expected valuein (13) efficiently. Nonetheless, if the expectation of a ran-domized payment scheme is equal to pvcg
m , then this paymentalso guarantees truthfulness in expectation [23]. Therefore, wecompute the payments as follows:
pm = T (x f−m, z f
−m) − (T (x f , z f ) − bm x fm),∀m ∈ [M] (14)
Lemma 5: The payment scheme in (14) yields a truthfulauction in expectation.
Proof: Intuitively, the randomized MIDR auction is truth-ful because both winner determination and payment compu-tation are bid independent, and it is known that an auctionis truthful if and only if it is bid independent. More specif-ically, It is easy to observe that E[pm] = E[T (x f
−m, z f−m) −
(T (x f , z f ) − bm x fm)] = pvcg
m . According to the properties ofMIDR algorithms [23], [25], such VCG-type payment rendersan MIDR algorithm truthful in expectation. �
Theorem 5: The randomized algorithm in Alg. 3 combinedwith the randomized VCG payment (14) is a truthful-in-expectation mechanism, parametrized by ε, that runs in polyno-mial time in expectation, and outputs a solution with expectedsocial cost at most an additive ε maxm∈[M] bm more than theoptimal value.
Proof: The theorem follows from Theorem 2, Theorem 3and Lemma 5. �Algorithm 3. A Randomized Auction Mechanism
Input: α ∈ (0, 1), b, e, Dc
Output: A solution (x f , z f ) to MILP (1) and payment pm
1: Run Algorithm 2 with the input (α, b, e, Dc), the output is(x f , z f );
2: Compute the payment for each winning agent, pm =T (x f
−m, z f−m) − (T (x f , z f ) − bm x f
m),∀m ∈ [M];3: Return a solution (x f , z f ); Return the payment pm for
each winning agent m.
VI. EXTENSIONS AND DISCUSSIONS
A. Non-linear Cost in Electricity Generation
In the real world, the unit cost for the power grid to gener-ate its own power is not linear to the quantity of generation z.The cost of stand-by generation is a function of the generationrate, �(z). It is determined by all the costs over the operationtime period t , and varies for different values of z. The objectivefunction of MILP (1) can be rewritten as:
Minimize∑
m∈[M]
bm xm + �(z)
For example, the total cost �(z) of a diesel generator to gen-erate zW power consists of the following components [26]:�(z) = It + Mt (z) + Ft (z) + Wt (z). During the generator’srunning time period t , It is investment cost, Mt (z) is operationand maintenance cost, Ft (z) is fuel cost, and Wt (z) is wastedisposal and emission control cost. It is a constant value, which
ZHOU et al.: DEMAND RESPONSE IN SMART GRIDS 2549
is the one-time purchase cost of the generator amortized to therunning period t . Mt (z), Ft (z) and Wt (z) are functions of thegeneration rate z.
The same strategy (Algorithm 3) can be adapted to sucha non-linear cost model for designing an FPTAS demand-response mechanism. The difference lies in the last step inAlgorithm 1. It returns a solution (y, z2) ∈ A with maximumvalue of bT y − �(z2).
B. Elastic Demand Response
While the power grid may have a most preferred value forthe demand response target, such a target is often not the onlyoption, and a small variation is tolerable though less preferred.In a real-world power grid, such deviation, up to a certainthreshold value, can be absorbed by the generating plantsthrough ex post primary frequency control [3]. Realtime imbal-ances in grid-wide electricity demand and supply are reflectedin measured deviations in power frequency from its nominalvalue (50 Hz in the majority of the world, 60 Hz in NorthAmerica and small parts of Asia). Generating units are thenregulated on a second-by-second basis to correct such devia-tion in a closed-loop control fashion. Such primary frequencycontrol comes at its own cost [9], and we capture that througha utility function of the power grid, which associates each fea-sible demand response target value with a difference level ofpreference:
Minimize∑
m∈[M]
bm xm + �(Z) − U (D) (15)
Subject To:∑
m∈[M]
em xm + z ≥ D (15a)
xm ∈ {0, 1}, ∀m ∈ [M] (15b)
0 ≤ z ≤ zmax (15c)
Dmin ≤ D ≤ Dmax (15d)
U (D) is the grid’s utility function that depends on the targetreduction D in power consumption. U (D) is often a concavefunction in practice [17]. Dmin and Dmax are the lower-boundand upper-bound of the demand response target of the powergrid, respectively, which bound the net reduction within anacceptable range. The power grid can accept an aggregated netreduction from agents that is smaller or larger than the actualdemand response target D′, but it has the highest preferencelevel at point D′. When D ∈ [Dmin, D′], U (D) is increas-ing and concave; when D ∈ [D′, Dmax ], U (D) is decreasingand concave. The corresponding complementary problem isformulated as:
Maximize∑
m∈[M]
bm ym − �(Z) + U (D) (16)
Subject To:∑
m∈[M]
em ym − z ≤∑
m∈[M]
em − D (16a)
ym ∈ {0, 1}, ∀m ∈ [M] (16b)
0 ≤ z ≤ zmax (16c)
Dmin ≤ D ≤ Dmax (16d)
Fig. 3. Power supply and demand in Ontario, Canada, October 27, 2014 toNovember 2, 2014.
Let (x, z1, D1) and (y, z2, D2) be a feasible solution for prob-lem (15) and problem (16), respectively. We have y = �1 − x ,z2 = z1 and D2 = D1. (x, z1, D1) is an optimal solution toproblem (15) if and only if (y, z2, D2) is an optimal solutionto problem (16).
A challenge in the auction design for such a new model lies incomputing the optimal solution (y, z2, D2) to the complemen-tary problem (16) (Sec. IV-B). The rest of the FPTAS auctiondesign is similar. We perturb the bids bm to obtain a randomizedalgorithm with excepted polynomial running time. The expec-tation of the solution is at most ε maxm∈[M] bm more than theoptimal social cost. A similar VCG-type randomized paymentscheme then completes the truthful-in-expectation auction.
In Algorithm 1, for every Pareto optimal vector y ∈ P(M),we need to compute values for z2 and D2 that maximizeU (D) − �(Z) and satisfy constraints (16c) and (16d):
Maximize U (D) − �(Z) (17)
Subject To: D − z ≤∑
m∈[M]
em − eT y (17a)
0 ≤ z ≤ zmax (17b)
Dmin ≤ D ≤ Dmax (17c)
After computing z2 and D2 for each Pareto optimal vec-tor y ∈ P(M), Algorithm 1 selects a solution that maximizes∑
m∈[M] bm ym − �(Z) + U (D).Assume that �(z) is an increasing convex function, as the
marginal cost of electricity generation grows with the outputrate z increases, then −�(z) is a concave function. Recall thatU (D) is a concave function. The sum of concave functionsis still a concave function [27]. Therefore, the objective func-tion U (D) − �(Z) is concave, and problem (17) becomes aclassic convex minimization problem, which can be solved inpolynomial-time using standard convex optimization methodssuch as the interior-point algorithm [28].
VII. PERFORMANCE EVALUATION
We evaluate our FPTAS demand response mechanismthrough trace-driven simulation studies, based on real-worlddemand data in Ontario, Canada in 2014. The left of Fig. 3shows hourly demand and capacity of Ontario’s grid fromOctober 27, 2014 to November 2, 2014 [29]. Available capac-ity represents the capacity of Ontario’s power market, includingboth local generation and imports. Ontario demand representsthe actual power demand within Ontario, and is calculated
2550 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 12, DECEMBER 2015
Fig. 4. Percentage of cost saving with different D and number of agents
by subtracting exports from the total power generation. Wecan obverse that the market capacity is always larger thanactual demand. Furthermore, power generation and demandare imbalanced at times. Consequently, the power grid needsto import/export energy from/to other provinces. The right ofFig. 3 illustrates hourly electricity trading data. In the follow-ing simulations, the demand response target D is set to 100MW,or one fifth of the average hourly import, under the assumptionthat one fifth of the shortage is supplied by the auction and four-fifths is purchased from other provinces directly. Following areport about cost of electricity by source [26], the value of bm
is generated uniformly randomly from the interval [200, 2000].The amount of supply/reduction em is a uniformly distributednumber between 0MW to 10MW. The unite cost of the dieselgenerator is set to $180 per MW, with a maximum output capac-ity of 10MW [30]. Each set of simulation is repeated ten times,and results are averaged.
Percentage of Cost Savings. Fig. 4 shows the percentageof social cost savings by our demand response auction withdifferent demand response target D and different number ofagents. Let Co be the cost when the power grid uses its dieselgenerators to cover the demand-supply gap instead of resort-ing to demand response. Let Cn be the social cost returnedby our randomized algorithm. The percentage of cost savingis computed as Co−Cn
Co. The cost data is taken from reports
of US Energy Information Administration (EIA) of the U.S.Department of Energy [26], [30]. From Fig. 4, we can see thatthe demand response approach can save more than 50% of thecost when 40 agents submit demand response bids. Even whenthere are less agents (20 agents) submitting bids, it can still savemore than 20% of the total cost. Moreover, the change of thedemand response target D doesn’t influence the percentage ofcost saving.
Approximation Ratio. A salient feature of our random-ized auction is its FPTAS property, i.e., it achieves (1 + ε)-optimal social cost. We first evaluate the approximation ratioof our Algorithm 2 under different system settings. Recall thatAlgorithm 2 solves the original minimization problem (1) byfirst solving the complementary problem (2), then convertingit to a solution for problem (1). Fig. 5 and Fig. 6 compare theapproximation ratio achieved by Algorithm 2 to solve MILP(1) and MILP (2), respectively. The ratios are computed bycomparing the social cost achieved by Algorithm 2 to theoptimal social cost.
Given α = 0.03, Fig. 5 shows the approximation ratio withdifferent number of agents. We can observe that the approxi-mation ratio (red bars on the right side) for MILP (2) remains
Fig. 5. Approximation ratio with different number of agents.
Fig. 6. Approximation ratio with different α.
around 0.96 with the growth of the number of agents. That isbecause equation (9) indicates the gap between the approxima-
tion ratio of MILP (2) and optimal ratio 1 is αbT y∗bT y∗−cz∗ , which is
determined by the value of α as αcz∗ is a very small number.The blue bars on the left side that represents the approximationratio to solve the original problem (1) increases when the num-ber of agents increases. This trend is in line with the theoreticalanalysis in Theorem 2. The difference between the objectivevalue return by Algorithm 2 and the optimum is αbT y∗. Recallthat y∗ is the optimal solution to maximization problem (2). Thevalue of bT y∗ increases when number of agents grows, since y∗includes larger bids to maximize bT y∗. We have proved that thegap between the blue bars (left side) and dotted line is boundedby γ ε = γ Mα. Our simulation results suggest that the gap issubstantially smaller than the theoretical bound. Fig. 6 illus-trates the approximation ratio under different α when 30 agentsparticipate in the demand response process. With the increaseof α, the approximation ratio of MILP (2) decreases and theapproximation ratio of MILP (1) increases. Similar to the expla-nation for Fig. 5, the difference between approximation ratioof MILP (2) and optimal ratio 1 is close to α, while the gapbetween approximation ratio of MILP (1) and optimal ratio 1 islarger than α.
At the end of Sec. IV, we mention that the approximationratio of our algorithm is upper bounded by (1 + γ ε). Fig. 7shows the approximation ratio achieved by Algorithm 2 tosolve MILP (1) with different γ , where γ = maxi, j∈[M]{bi/b j }.Although the theoretical analysis proves that (1 + γ ε) is theupper bound of the approximation ratio, our simulations reveala more rosy picture in practice. Furthermore, the value of γ
doesn’t affect the ratio, while α dominates the final ratio. Wecan observe that the approximation ratio fluctuates with thedecrease of γ , but decreases monotonically when α drops. Thiscan be intuitively explained as following: γ is used merely tobound the approximation ratio. It indicates the ratio in the worstcase scenario rather than determines the ratio. Theorem 2 showsthe real ratio still depends on the value of α. We also compare
ZHOU et al.: DEMAND RESPONSE IN SMART GRIDS 2551
Fig. 7. Approximation ratio with different γ .
Fig. 8. Comparison between our algorithm+ with Algorithm 2 from Zhanget al. [7] ∗.
Fig. 9. Percentage of winners.
our algorithm with Algorithm 2 from Zhang et al. [7] to exam-ine the approximation ratio when α = 0.01, as shown in Fig. 8.It can be observed that both algorithms perform relatively wellunder the given input, achieving small approximation ratios thatare between 1 and < 1.2. For all different number of agentstested, our algorithm outperforms that of Zhang et al. slightly.
Percentage of Winners. We next study the performance ofour randomized algorithm in terms of winner satisfaction, asmeasured by the percentage of agents whose bid is acceptedby the grid. Fig. 9 shows that more agents are selected bythe grid when the number of participating agents is small.This is because the grid needs a large fraction of the agents tocover its shortage when there is only small number of choice.The cost and maximum capacity of diesel generators alsoinfluence the percentage of winners. Compare the percentageof winners when c = $120 and c = $250, we can observethat the percentage of winners with high cost is always larger.This can be explain as follows: when diesel generation iseconomical, our algorithm will first utilize diesel generators,and then consider demand response bids. But the grid selectsall the bids from agents and avoids to use diesel generatorswhen their costs are high.
Social Cost. Fig. 10 illustrates social cost computed by ourrandomized algorithm with varying number of agents and α.The smallest value occurs at the left bottom of the surface where
Fig. 10. Social cost with different α and number of agents.
Fig. 11. Grid’s utility function
α takes the smallest value and the number of agents is large. Thesocial cost decreases when the number of agents increases. Thisis because a larger agent pool includes a larger number of agentswho submit low-cost bids. With a small number of agents, thegrid is forced to select expensive bids to meet the demandresponse target. Furthermore, given the same number of agents,a smaller α means the social cost is closer to the smallest socialcost. This is the reason why we can observe a downward trendfrom the right side to the left side of the surface.
In Sec. VI, we extended our studies to non-constant demandresponse targets that are captured by the grid’s utility function.Fig. 12 shows social cost achieved by 40 agents with differentutility functions. We consider four quadratic utility functions[17] shown in Fig. 11. They are concave functions of the targetD. The maximum and minimum target that grid can accept is150 MW and 50 MW, respectively. The actual target of grid isset to 100 MW. When D = 100MW, grid has the highest levelof preference. For example, the dotted line represents the util-ity function U (D) = 4(D − 50)(D − 150), the maximum levelof preference is equivalent to to $10000 when D = 100MW.When D is smaller or larger than 100 MW, the level of pref-erence decreases and reaches zero when D = 50MW or D =150MW. However, the trend in Fig. 12 is not the same as inFig. 11. We use circles to indicate the social cost when D isequal to the grid’s actual demand response target, and trian-gles to mark the optimal social cost. Although the grid prefers100 MW the most, it is clear that the lowest cost may not occurat such a preferred point. Our algorithm can quickly compute a(1 + ε)-optimal solution to these models in polynomial time.
Usage of Diesel Generators. Finally, we evaluate the usageof on-site quick-start generators, as exemplified by diesel gener-ators, under different cost and generation capacity. We assumesthat 40 agents participate in the auction. The Y axis in Fig. 13represents the ratio of the output rate to the maximum power
2552 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 12, DECEMBER 2015
Fig. 12. Social cost with utility function
Fig. 13. Usage of diesel generators under different c and zmax .
of the diesel generators, and the X axis is the maximum avail-able power zmax . Here c is the unit cost, which equals $120, or$180 or $250 per MW. Clearly, the higher the unit cost is, thelower the usage of diesel generators is. When the unit cost is low(light blue bars on the bottom layer), diesel generators operateat the maximum capacity. Red bars on the top layer show thatthe grid avoids to start its diesel generators when it is expensiveto operate diesel generators. Furthermore, it is apparent that thecost, rather than the generation capacity, determines the usageof diesel generators.
VIII. CONCLUSIONS
This work formulates general and expressive models fordemand response auctions. Through a new technique thatcombines smoothed polynomial-time algorithm design withrandomized reduction, we designed demand response mecha-nisms that are truthful, polynomial-time computable, and canapproach optimal social cost arbitrarily closely. The new tech-nique of designing randomized auction mechanisms throughsmoothed polynomial-time algorithms may be applied to abroad range of problems where social welfare maximizationis hard in the worst case, but the hard cases are relativelyrare and isolated. However, similar to many other FPTAStypes of algorithms and mechanisms in the future, our auc-tion algorithms can be sometimes slow in practice despiteits polynomial time running time, due to large exponents inthe running time. Designing demand response auctions thatare even more efficient for large scale practical applicationsremains an interesting and challenging problem.
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Ruiting Zhou received the B.E. degree in telecom-munication engineering from Nanjing University ofPost and Telecommunication, Nanjing, China, in2007, the M.S. degree in telecommunications fromHong Kong University of Science and Technology,Hong Kong, in 2008, and the M.S. degree in com-puter science from the University of Calgary, Calgary,AB, Canada, in 2012. Since 2014, she has been pursu-ing the Ph.D. degree at the Department of ComputerScience, University of Calgary. Her research inter-ests include smart grids, cloud computing, and mobile
network optimization.
Zongpeng Li received the B.E. degree in computerscience and technology from Tsinghua University,Beijing, China, in 1999, the M.S. degree in com-puter science, and the Ph.D. degree in electrical andcomputer engineering from the University of Toronto,Toronto, ON, Canada, in 2001 and 2005, respectively.Since August 2005, he has been with the Departmentof Computer Science, University of Calgary, Calgary,AB, Canada. From 2011 to 2012, he was a Visitor atthe Institute of Network Coding, Chinese Universityof Hong Kong, Hong Kong. His research interest
includes computer networks.
Chuan Wu received the B.Eng. and M.Eng. degreesin computer science and technology from TsinghuaUniversity, Beijing, China, in 2000 and 2002, respec-tively, and the Ph.D. degree in electrical and computerengineering from the University of Toronto, Toronto,AB, Canada, in 2008. Since September 2008, shehas been with the Department of Computer Science,University of Hong Kong, Hong Kong, where she iscurrently an Associate Professor. Her research inter-ests include cloud computing, online, and mobilesocial networks. She was the corecipient of the Best
Paper Award of HotPOST 2012.
Minghua Chen received the B.Eng. and M.S.degrees in EE from Tsinghua University, Beijing,China, in 1999 and 2001, respectively, and thePh.D. degree in EECS from the University ofCalifornia at Berkeley, Berkeley, CA, USA, in2006. He visited Microsoft Research Redmond as aPostdoc Researcher for one year. He joined with theDepartment of Information Engineering, the ChineseUniversity of Hong Kong, Hong Kong, in 2007,where he is currently an Associate Professor. He isalso an Adjunct Associate Professor with the Institute
of Interdisciplinary Information Sciences, Tsinghua University. He is currentlyan Associate Editor of the IEEE/ACM TRANSACTIONS ON NETWORKING.His research interests include energy systems, distributed optimization, multi-media networking, wireless networking, network coding, and delay-constrainedcommunication. He was the recipient of the Eli Jury Award from UC Berkeleyin 2007 and the Chinese University of Hong Kong Young Researcher Award in2013. He was also recipient of the IEEE ICME Best Paper Award in 2009, theIEEE Transactions on Multimedia Prize Paper Award in 2009, and the ACMMultimedia Best Paper Award in 2012.
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