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Efficient Coordinations of Large-scale Elastic Loads with MCP and PSP Auction Mechanisms ZOU Suli, MA Zhongjing, LIU Xiangdong School of Automation, Beijing Institute of Technology, Beijing 100081, P. R. China E-mail:
[email protected],
[email protected],
[email protected] Abstract: In this paper, we propose a distributed method to coordinate large-scale elastic load units with auction mechanisms which have been widely applied in resource allocation problems. Distinct from the progressive second price (PSP) auction mechanism, under the market clearing price (MCP) auction mechanism, the incentive compatibility does not hold in general but holds with respect to the efficient bid profiles of other agents. We observe that the difference between the payments under PSP and MCP mechanisms vanishes as the population size goes to infinity and consequently show that the efficient bid strategy is an epsilon-Nash equilibrium in MCP auction games with epsilon converges to zero as the population size goes to infinity. As another key result developed in the paper we further show that the efficient bid strategy is an epsilon-Nash equilibrium of auction game systems with epsilon is on the second order of the second derivative of the generation cost which goes to zero as the population size goes to infinity in scalable systems. Key Words: Elastic loads, Distributed control, Uniform market clearing price mechanism, Progressive second price mechanism, Nash equilibrium.
1 Introduction Elastic demands, e.g. air-conditioners, heaters and emerging plug-in electric vehicles, could be properly coordinated in time or shape to flatten the load curve, reduce peak demand in the grids, decrease generation costs, and reduce the emission of green gas and so on, e.g. [1], [2], [3] etc. and references therein. With the restructuring and deregulation of electricity market, and the development of advanced information technology applied in smart grids, distributed coordinations for large-scale individual users become feasible. In the literature, enormous studies have been dedicated to the management and coordination of elastic demands following certain pricing mechanisms, see [4], such as predetermined fixed retail price, dynamic price dependent upon system conditions, e.g. [4], [5], etc. In [6, 7], the authors proposed decentralized valleyfilling strategies for large-population plug-in electric vehicles where the electricity charging price at any time is determined by the aggregated demand at that time. [8] presents an economic model of price/incentive response loads based on the concept of flexible price elasticity of demand and customer benefit function. [2] assesses an extended responsive load economic model to improve load profile characteristics and achieve customers’ satisfaction. A method of multi attribute decision making is used for handling the optimal solution which reflects the perspectives of each DR stakeholder. In [9], an autonomous and distributed demand-side energy management system based on game theory is proposed taking advantage of a two-way digital communication infrastructure. Besides the above distributed control methods, auction mechanisms [10]. More specifically uniform market clearing price (MCP) [11] and pay as bid (PAD) [12] have been widely applied to coordinate the dispatch of generations in day-ahead electricity market. Moreover parallel to the aucThis work is supported by National Natural Science Foundation (NNSF) of China under Grant 61174091.
tion mechanisms applied in electricity market, auction-based resource sharing problems have been studied in other fields, e.g. [13, 14] where the so called progressive second price auction mechanism was proposed to allocate the network resources among users. In the auction mechanisms listed above, each agent only reports a two-dimensional bid profile which is composed of a maximum demand and an associated buying price, and is used to replace his complete (private) utility function. In [14], the authors verified that the efficient bid profile is a Nash equilibrium for network resources, and in [15], the authors of the paper showed that efficient charging bidding profiles of electric vehicles over a multi-time interval is a Nash equilibrium over dynamic progressive second price auction mechanism. J.Peng and P.Caines, in [16], applied and extended the quantized-PSP auction mechanism firstly proposed in [13], to power electricity system such that the social optimal allocation is implemented. In this paper, we proposed a distributed method to coordinate large-scale elastic load units with auction mechanisms which have been widely applied in resource allocation problems. Distinct from the progressive second price (PSP) auction mechanism, under the market clearing price (MCP) auction mechanism, the incentive compatibility does not hold in general but holds with respect to the efficient bid profiles of other agents. We observe that the difference between the payments under PSP and MCP mechanisms vanishes as the population size goes to infinity and consequently show that the efficient bid strategy is an epsilon-Nash equilibrium in MCP auction games with epsilon converges to zero as the population size goes to infinity. As another key result developed in the paper we further show that the efficient bid strategy is an epsilon-Nash equilibrium of auction game systems with epsilon is on the second order of the second derivative of the generation cost which goes to zero as the population size goes to infinity in a series of scalable systems. The rest of the paper is organized as follows. In Section 2, we formulated a class of coordination problems for elastic loads in a scalable power system. In Section 3 auction-
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based distributed coordination problems under PSP and MCP mechanisms are established and the comparison of payments under distinct auction mechanisms are analyzed. In Section 4, we show that under MCP mechanism the efficient bid strategy is an epsilon-Nash equilibrium with epsilon vanished in the population limit. Conclusions of the paper and some ongoing research are listed in Section 6.
Subject to a collection of admissible demand allocation x, the total cost of the system, denoted by JN , is specified as below.
2 Formulations of optimal coordination problems of large-scale elastic loads
We formally specify a class of optimal coordination problems for elastic loads below:
Consider a typical electricity power system with generators and loads (including inelastic loads and elastic loads). Suppose the demand of the entire inelastic loads is a parameter that is pre-known and the population of elastic loads in the system is N ; we would like to analyze the economic operation of (elastic) loads at a single instant. The population of loads is denoted N = {1, 2, · · · , N }. For each individual load n (n ∈ N ), denote admissible demand allocation by xn with the constraints follows:
Problem 1
0 ≤ xn ≤ Γn ,
(1)
where Γn , with Γn < Γ, respects the maximum demand resources of load n. The set of admissible demand allocations for the entire load is denoted by x ≡ (xn , n ∈ N ). Subject to an admissible demand allocation xn , each load n has a valuation function denoted as vn (xn ). We suppose all the resources suppliers are in the same type and integrated as one generator with a series of cost functions cN (y) subject to the populations N of elastic loads, where y denotes the total generation (total system demand); then N by the power conservation law, we have y = DN + xn , n=1
where DN is the integrated inelastic load demand in the system which is assumed to be dependent on the populations N . Suppose DN takes the following linear form, DN = N Q,
(2)
where Q is a constant value denoting the normalized inelastic demand over the population size in the grid. We consider the following two assumptions in the paper: (A1) cN (y), is monotonic increasing, strictly convex and differentiable on y; (A2) vn (xn ), for all n ∈ N , is monotonic increasing, strictly concave and differentiable on xn . In this paper, we consider the following specification for generation cost cN : In order to reflect the real feature of cost in the market, we assume that the generation cost cN satisfies the following property: cN (N Qmax ) = cM (M Qmax ),
(3)
for any pair of population sizes N and M , where Qmax represents a normalized generation capacity over the population N 1 size N , such that Qmax ≥ Q + Γ Q + Γn . N n=1 Interpretation of (3): It implies that the generation of grid is scalable with respect to the size of load populations which need to be coordinated.
JN (x) = cN (DN +
N
N
xn ) −
n=1
vn (xn ).
(4)
n=1
min JN (x)
(5)
x
such that x satisfies constraints (1). That is to say, the objective of the system is to implement an optimal allocation x∗∗ for each load to minimize the system cost (4). We also call the optimal allocation x∗∗ is efficient. The optimal allocation x∗∗ of Problem 1 can be characterized by the KKT conditions as follows. Firstly, the Lagrangian for Problem 1 with population size N is specified as below: LN (x, λ) = JN (x) +
N
λn (xn − Γn ),
n=1
where λn is the Lagrange multiplier associated with the constraint xn ≤ Γn in (1). The KKT conditions to Problem 1 are specified in (6). ⎧ ∂LN ∂LN ⎨ ≥ 0, xn ≥ 0, xn = 0 ∂xn ∂xn ⎩ x −Γ ≤ 0, λ ≥ 0, (x − Γ )λ =0 n
n
n
n
n
n
(6) for all n ∈ N , where
N ∂LN = cN (DN + xn )−vn (xn )+ ∂xn n=1
λn . Remark: By the strict convexity of cost function JN (x) and the inequality constraints (1) under Assumptions (A1,A2), Problem 1 is a constrained convex optimization problem. Thus under Assumptions (A1,A2), the optimal allocation x∗∗ of Problem 1 is unique. The centralized optimal dispatch strategy can be effectively implemented in the case that the system has complete information and can directly schedule the behaviors of all the agents. However, in practice the individuals do not want to share their private information with others. Requiring complete information may create heavy communication and the centralized control method might be computationally infeasible. Thus we will adapt an auction-based distributed coordination method to control the elastic loads.
3 Auction-based distributed coordination mechanism We present a distributed coordination method in which each load decides his own bid autonomously and bids a two-dimensional variable to replace his completely valuation function, to ease the disadvantages of centralized control.
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3.1 Bid profiles of individual agents Each load n (n ∈ N ), submits a (two-dimensional) bid profile bn , such that bn = (βn , dn ), with 0 ≤ dn ≤ Γn , which specifies the maximum per unit price βn that load n is willing to pay and demands up to dn units of the electricity resources. The set of the bid profiles of agent n is denoted Bn . The corresponding feasible allocation xn with respect to bn satisfies: 0 ≤ xn ≤ dn ,
(7)
vn (xn ; bn ) = βn min(xn , dn ). Hence following (7), we have vn = βn xn . The system cost with respect to a collection of bid profiles b ≡ (bn , n ∈ N ) is as follows: N
xn ) −
n=1
N
vn (xn ; bn ). (8)
n=1
for all n ∈ N .
Considering a collection of bid profiles b, the payoff function of individual agent n, denoted fn (b), is specified below:
Problem 2 ∗ JN (b) = min JN (x(b); b) x
(9)
such that x satisfies constraint (7). That is, the auctioneer will assign an optimal allocation x∗ (b) = argminx JN (x(b); b) to agents to minimize the total cost JN . The optimal solution x∗ to Problem 2 can be characterized by the KKT conditions as follows. Firstly, the Lagrangian for Problem 2 is specified as below: N
(12)
where x∗n (b) is the optimal allocation for agent n of Problem 2 with respect to b, and τn (b) represents the payment of agent n with respect to b. In Section 3.2 and Section 3.3 below, we will specify the payment τn of individual agent under the so-called progressive second price (PSP) mechanism and uniform market clearing price (MCP) mechanism respectively. Moreover in Section 3.4, we will study the difference of the individual payoff functions under PSP and MCP auction mechanisms. 3.2 Progressive second price auction mechanism
The auction based distributed coordination problem can be specified as below:
N (x, σ; b) = JN (x(b); b) + L
n=1
fn (b) = vn (x∗n (b)) − τn (b)
for all n ∈ N . Its revealed valuation function is:
JN (x(b); b) = cN (DN +
Lemma 3.2 Suppose x∗ is the efficient allocation with re∗∗ spect to (βn∗ , d∗n ) = (vn (x∗∗ n ), xn ) for all n ∈ N . Hence the following holds: ⎧ N ⎪ ⎪ ⎪ = c (D + d∗n ), in case x∗n > 0 ⎪ N N ⎨ n=1 βn∗ , (11) N ⎪ ⎪ ∗ ∗ ⎪ ≤ c (DN + ⎪ dn ), in case xn = 0 N ⎩
σn (xn − dn ),
n=1
where σn is the Lagrange multiplier associated with the constraint (7). The KKT conditions of Problem 2 are specified in (10). ⎧ N N ⎨ ∂L ∂L ≥ 0, xn ≥ 0, xn = 0 ∂xn ∂xn ⎩ xn − dn ≤ 0, σn ≥ 0, (xn − dn )σn = 0 (10) N N ∂L = cN (DN + xn ) − β n + σ n . for all n ∈ N , where ∂xn n=1 ∗∗ Lemma 3.1 Consider bid profiles b∗n = (vn (x∗∗ n ), xn ), for ∗ ∗ ∗∗ all n ∈ N ; then x (b ) = x , i.e. the associated allocation x∗ with respect to b∗ is efficient.
The progressive second price mechanism is introduced in [13], which is an extension of VCG-style auction mechanism where each agent only reports a two-dimensional bid instead of reporting his type or complete utility function. In PSP mechanism, each agent will calculate its payment expressed as follows: the payment of an agent n is that, the summation of all agents’ utility when agent n does not join the auction games, minus the summation of the other agents’ utility when agent n joined the auction process. That is, the payment of each agent is exactly the externality they impose on others through their participation. Then the payment of agent n under PSP mechanism with a collection of bid profile b, denoted τnpsp (b), is given as below: ∗ ∗ τnpsp (b) −JN (b−n ) − [−JN (b) − βn x∗n (b)],
where for any bid profile b, b−n denotes the bid profile without agent n’s participation, that is, with the bid dn substituted with dn = 0, and x∗ (b) represents the optimal allocation with respect to b. The specified form of τnpsp (b) is given below: τnpsp (b) =cN (DN + +
N
x∗n ) − cN (DN +
x−n m )
m=n
n=1
βm (x−n m
−
x∗m ),
(13)
m=n
where x−n denotes the optimal allocation with respect to b−n . 3.3 Uniform market clearing price auction mechanism A market clearing price is the price of goods or a service at which quantity supplied is equal to quantity demanded. The market clearing price [17] is the bid price of the most
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expensive supplier that is needed to completely meet the demand, and is used as the basis for the settlement of market commitments. Suppose x∗ denotes the optimal allocation with respect to a bid profile b. Following the MCP mechanism, we define q(x∗ ) ≡ (qn (x∗n ), n ∈ N ) such that βn , in case x∗n > 0 qn (x∗n ) = , (14) p, in case x∗n = 0 where p represents a high regulated price, i.e. q(x∗ ) represents the collection of the bid price of the agents whose allocation is larger than zero. The uniform market clearing price, denoted p(b), with respect to b, is given in (15). ⎧ ⎪ = min qn (x∗n ), ⎪ ⎪ n∈N ⎪ ⎪ ⎪ ∗ ⎨ in case min qn (x∗n ) = cN (DN + N n=1 xn ) n∈N p(b) N ⎪ ∈ [cN (DN + n=1 x∗n ), min qn (x∗n )], ⎪ ⎪ n∈N ⎪ ⎪ ⎪ ∗ ⎩ in case min qn (x∗n ) > cN (DN + N n=1 xn ) n∈N
(15) Interpretation of (15): N ∗ ∗ • In case min qn (xn ) = cN (DN + n=1 xn ): n∈N
•
p(b) is determined by the the minimal bid price of the agent whose price is larger than zero, as illustrated in Fig. 1(a); N In case min qn (x∗n ) > cN (DN + n=1 x∗n ): n∈N
p(b) is in a region determined by the generation price and the minimal bid price with allocation larger than zero, as illustrated in Fig. 1(b). βn
βn
bids of agents
cN (DN +
bids of agents
N
n=1 xn )
cN (DN +
N
n=1 xn )
p(b) p(b)
N n=1
N n=1
Fig. 1: Specification of p(b) w.r.t. bid profile b Thus, we can define a parameter η with 0 ≤ η ≤ 1 and can further specify the market clearing price given in (15), as p(b; η) with η as follows: p(b; η) cN (DN +
x∗n )
n=1
+ η(min {qn (x∗n )} − cN (DN + n∈N
sup {cN (DN + y)}.
y∈[0,N Γ]
(18)
Remark: By the scalability property specified in (3), sup{cN } converges to zero asymptotically as N goes to infinity. Theorem 3.1 Considering any bid profile b, |τnmcp (b; η) − τnpsp (b)| ≤ σN ,
(19)
such that σN = O(sup{cN }), in case η = 0.
4 Nash equilibrium property of efficient bid profiles under PSP and MCP auction mechanisms Suppose b∗ is the bid profile specified in Lemma 3.1, such that the corresponding optimal allocation is efficient. In this section, we will study the ε–Nash Equilibrium of b∗ for auction games with finite populations and Nash equilibrium of b∗ for auction games in the population limit under the two auction mechanism. Before that we first define the ε–Nash Equilibrium in Definition 4.1 below. Definition 4.1 (ε–Nash equilibrium) A collection of bid profiles b0 is an ε–Nash equilibrium (ε– NE for short), with ε ≥ 0, for auction game system 2 if the following holds: fn (b0n , b0−n ) ≥ fn (bn , b0−n ) − ε,
(20)
for all bn ∈ Bn . That is to say an auction game system is at an ε–NE with b0 , if any individual agent n can benefit himself with a gain of at most ε by unilaterally deviating from his bid profile b0n .
4.1 Best bid strategy of individual agents
x∗n
(b)
N
sup{cN } ≡
Remark: An ε–Nash equilibrium degenerates into a Nash equilibrium in case ε equal to zero.
x∗n
(a)
3.4 Comparison of agent payments under PSP and MCP auction mechanisms For notational simplicity, in the rest of the paper, we consider
N
x∗n )).
(16)
n=1
Following the MCP mechanism, the payment of agent n, denoted τnmcp (b), is given as below: τnmcp (b; η) p(b; η)x∗n (b),
Define the set of best replies of agent n subject to a bid profile b−n of opponents’ bids, denoted Bn∗ (b−n ), as follows:
Bn∗ (b−n ) bn ∈ Bn ;
s.t. fn (bn , b−n ) = max fn (bn , b−n ) ; (21) bn ∈Bn
and we call Bn∗ (b−n ) the set of best bid strategy of agent n with respect to b−n the collection of bid profiles of other agents except agent n. The optimal bidding set of agent n with respect to the efficient opponents’ bid profiles can be written as Bn∗ (b∗−n ). We define Bnt as the set of truth-telling bid profiles of agent n, such that Bnt {bn ≡ (βn , dn ) ∈ Bn ; s.t. βn = vn (dn )}.
(17)
where p(b), specified in (16), denotes the market price with respect to b, and a fixed parameter η ∈ [0, 1].
(22)
Lemma 4.1 (Incentive compatibility of PSP auction) [13] Under the PSP auction mechanism and for any collection of
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βn
bid profiles of agents except n, b−n , Bn∗ (b−n )
k = maxdn {vn (dn )}
k = mindn {cN (DN +
(23)
However, unlike PSP mechanism, the truth telling property usually does not hold under the MCP mechanism. We show the truth telling at efficient bid profile under MCP mechanism in Lemma 4.2.
Bnt = ∅, for any η ∈ [0, 1],
x n
(26)
we have Δfn ≤ 0. Similarly, for any agent n, we have: Δfn ≥ −(d∗n − x n )[μmax n + νnmax (d∗n − x n )], N x = supy∈[0,Γn ] {cN (DN + m=n d∗m +y)} where μmax N and νnmax = 1/2 supy∈[0,Γn ] {vn (y)}. Then we have
Corollary 4.1 Under MCP mechanism in case η = 0, the efficient bid profile b∗ is an σN –NE, such that σN = O(sup{cN }).
S2
dn
−νnmin ∗ ∗ d ,d , x n ∈ min n n μmin N − νn
Theorem 4.1 [13, 14] The efficient bid profile b is a Nash equilibrium in PSP auction game systems.
S1
dn
∗
Δfn ≤ − μmin (d∗n − x n ) xn − νnmin (d∗n − x n )2 , N
d∗n
min with νnmin = 1/2 inf y∈[0,Γ n ] {v∗n (y)} < 0, μN = inf y∈[0,Γn ] {cN (DN + m=n dm + y)} ≥ 0, and x n ∈ [0, d∗n ], as illustrated in Fig. 2. Thus, when x n satisfies the follows:
4.2 ε-Nash equilibrium property of efficient bid profiles in scalable systems under MCP auction mechanism
with respect to b∗ and we have where p∗ represents the MCP ∗ ∗ p = cN (DN + m=n dm + d∗n ) by (15), and p represents the MCP with respect to (bn , b∗−n ). In the following, we will and p in case dn ≥ d∗n and dn < d∗n analyze the value of x respectively. m = d∗m for all (I) In case dn ≥ d∗n . We can show that x m = n, and x n ≤ d∗n . ∗ , we have the MCP in this case By (15) and βn ≤ βm n ). p = βn = cN (DN + m=n d∗m + x By (25), we can have the following:
d∗m + dn )}
Fig. 2: Illustration of linearized “payoff” with respect to b∗ and (bn , b∗−n ) with dn ≥ d∗n
i.e. there exists a bid btn ∈ Bnt , such that btn is a best response bid profile of agent n with respect to the efficient bid profiles of other agents b∗−n .
x n
m=n
S1
p = βn
(24)
Theorem 4.2 Under MCP mechanism, the efficient bid profile b∗ is an εN –NE for auction game system with population size of N , such that εN = O([sup{cN }]2 ) for all η ∈ [0, 1]. the optimal allocationSketch of Proof. Denote x∗ and x s with respect to b∗ and (bn , b∗−n ) respectively. Also, we denote p∗ and p the MCP with respect to b∗ and (bn , b∗−n ) respectively. By Lemma 3.1, we have x∗ = d∗ . By Lemma 4.2, we can set that bn ∈ Bnt . By (12) and (17), we can obtain that the difference of fn (b∗ ) and fn (bn , b∗−n ), denoted Δfn , is specified as follows:
d∗n vn (x)dx − p∗ (d∗n − x n ) − (p∗ − p) xn . (25) Δfn =
S2
p∗ = βn∗
Lemma 4.2 (Incentive compatibility under MCP mechanism w.r.t. efficient bid profiles of other agents) Under the MCP mechanism and the efficient collection of bid profiles of other agents except n, denoted b∗−n ,
Payoff of agent n w.r.t. (bn , b∗ −n )
Bnt = ∅,
i.e. there exists a truth-telling bid btn ∈ Bnt , such that btn is a best bid strategy of agent n with respect to b−n .
Bn∗ (b∗−n )
∗ Payoff of agent n w.r.t. (b∗ n , b−n )
[μmax ]2 εn,N = maxN max d∗n 2 . 4 μN − ν n
(27)
(II) In case dn < d∗n . We can show (27) holds in this case, following the same technique as in (I). In summary, by (20) holds for any n ∈ N , we can take εN as the following, [μmax ]2 Γ2 , εN = maxN 4 μN − νmax
(28)
where ν max = max νnmax . n∈N
5 Numerical simulations Under Assumption (A1) and by (3), we suppose the form of cN as follows: cN = aN (DN +
N
x n )b ,
(29)
n=1
where aN is a coefficient depending on the population size N , and b is a constant larger than 1 denoting the convexity of generation cost. For the purpose of demonstration, we select b = 1.3 and cN (N dmax ) = 0.25. Suppose that the inelastic load demand DN = N kW, and all the elastic loads are in the same type with a capacity of Γn = 10 kW for all n ∈ N . Then we can calculate that aN = 0.0937N −0.3 . By Assumption (A2), we assume the load valuation function is vn = 0.3x0.9 n .
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5.1 Non Nash equilibrium property of efficient bid profiles under MCP mechanism We obtain by Theorem 4.2 that for any auction game with a finite population of units, the efficient bid b∗ can not be an NE. i.e. there must exist a bid bn ∈ Bn such that fn (b∗ ) < fn (bn , b∗−n ). Set N = 10 as an example and then aN = 0.0469. The efficient bid profile of agent n is b∗n = (βn∗ , d∗n ) = (0.2236, 6.6020) for all n ∈ N . Consider an individual agent n, whose bid strategy changes to bn = (βn , dn ) = (0.2223, 7), while the others keep fixed as the efficient bid profiles, denoted b∗−n . We can calculate that the allocation of agent n with respect to (bn , b∗−n ) is x n = 5.0118 kW. Thus, the difference of payoffs subject to b∗ and (bn , b∗−n ), denoted Δfn , is specified as below: xn ) + βn x n Δfn = vn (d∗n ) − βn∗ d∗n − vn ( = 0.1640 − 0.1658 = −0.0018 < 0,
References
which implies that there exists a bid bn specified above such that fn (b∗ ) < fn (bn , b∗−n ). 5.2 εN -NE property of efficient bid profiles under MCP mechanism 0.9 4
0.8 3.5
0.7 0.6 The value of ε
The value of σ
3 2.5 2 1.5
0.4
0.2 0.1
0.5 0
0.5
0.3
1
0
0.5
1
1.5 lgN
2
2.5
3
(a)
0
0
0.5
1
1.5 lgN
2
2.5
3
(b)
Fig. 3: Variations of σN and εN with respect to the population N By Theorem 3.1 and Theorem 4.2, following the parameters above, we can calculate the value of σN and εN with respect to N , as illustrated in Table 1. As we can observe from Fig. 3, εN decreases much quickly than σN as the population size N increases. Table 1: The values of σN and εN w.r.t. N N 2 5 10 100 1000
σN 1.1247 0.2369 0.0729 0.0015 2.9027 × 10−5
the payments under PSP and MCP mechanisms vanishes as the population size goes to infinity and consequently show that the efficient bid strategy is an epsilon-Nash equilibrium of auction games with epsilon converges to zero as the population size goes to infinity. As another main result developed in the paper we further show that the efficient bid strategy is an epsilon-Nash equilibrium of auction systems with epsilon is on the second order of the second derivative of the generation cost which goes to zero as the population size goes to infinity in a series of scalable systems. As an ongoing research, we are extending the work established in the paper to multi-time coordination problems, e.g. the charging coordinations of large-population plug-in electric vehicles over a finite charging interval. In such coordination problems, the loads are in the context of cross-elasticity loads, see [18].
εN 0.2632 0.0408 0.0074 4.8681 × 10−6 1.9638 × 10−9
6 Conclusions and ongoing research In this paper, we proposed a distributed method to coordinate large populations of elastic loads in electricity market with sealed auction mechanisms. Distinct from the progressive second price (PSP) auction mechanism, under the MCP mechanism, the incentive compatibility does not hold in general but holds with respect to the efficient bid profiles of other agents. We also observe that the difference between
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