Subsidize-free cost allocation method for infrastructure ... - Ozyrys

More documents

Recommendations

Info

problem (1)-(5) to nd the joint infrastructure constraints cost. Usually, in many practical cases the simplistic arbitrary cost-allocation rules are used. For instance, in a majority of the European electric energy national markets the cost of system infrastructure utilization is covered by arbitrary at ”postage stamp” fees – identical for all players or group of players, irrespectively of how much each player contributes to the costs of utilizing the system. In this approach any player who contributes to higher costs of resource utilization in fact achieves extra benets at the cost of other players – that's it, he is subsidized by other players. Schweppe et al. [11] introduced and Hogan [5] extended a concept of locational marginal nodal pricing (LMP) for electric energy transmitted over the network. The LMP method allows one to calculate marginal prices of energy at individual nodes of the infrastructure network. Price diversication between two adjacent nodes occurs when the transmission capacity between two nodes is limited and exhausted. Recently, the marginal nodal pricing became more and more popular method for designing the electrical real-time energy balancing markets. However, nodal pricing suffers from high sensitivity – in each node marginal player with local market power appears [12]. Finally, this is not a complete market solution – it is designed for systems where only transmission constraints are dominant. Recently, above drawbacks made researchers moved towards approaches based on game theory, which were successfully applied for cost allocation for other problems [2], [4], [6], [12], [13]. In this case it is assumed, that the contribution of each entity in every resource constraint (3) is known. For instance one can compute power ow in the transmission system using ow tracking methods and obtain share of each entity in each constrained transmission line [1]. Thus we only need to distribute infrastructure cost among the infrastructure resources constraints. Cooperative game theory approach is a natural framework for problems of cost allocation among collaborating agents [16]. However, most of so far researches assumed that game characteristic function C(b) is subadditive. In a typical problem this assumption is a consequence of problem nature – voluntariness of players participation in a common business. However, the problem of cost allocation for infrastructure market concerns the system constraints which usually are results of technical or physical conditions and must absolutely be satised. Thus, we are searching for fair solution for agents who must participate in a common business. Under this assumption characteristic function C(b) may not be subadditive, as it happens e.g. on the electrical energy balancing markets, but also in other industries. Classical game theory cost allocation methods, including Shapley value, Aumann-Shapley pricing, SCRB (Separable Cost Remaining Benets) are based on subadditivity assumption of C(b) [16]. More over, even for some subadditive functions C(b) these methods may produce solution which are not free from so called subsidizing phenomena. Subsidizing means that player entering into common business causes cost increase higher then the value he will be charged – thus, other entities must cover some portion of this cost. It is obvious that subsidizing phenomena open the doors for speculative, parasitical strategies which limit effective market functioning and is clearly undesired. Classical game theory approaches distribute exactly cost C(˜b), where ˜b represents actual resources availability. This is known as an efciency or break-even properties (or axiom). In the paper we show that subsidize-free allocation cannot meet efciency (break-even) condition. The main achievement of our investigation is a new method for allocation costs of market infrastructure which assures that allocation is free from subsidies. Obviously, it may produce an allocation that breaks the efciency axiom. II. GAME-THEORY MODEL FOR ALLOCATION PROBLEM We assume that during market balancing the resource constraints with right hand side equal ˜b i , i ∈ I must be satised. Total costs related to the resource constraints should be assigned to each constraint on the basis of constraint impact on market balance. Then assigned costs can be distributed among market entities according to given resource utilization degree. Charges should discourage market entities to take advantage of system constraints and their local market power as a tool for unfounded extra benets. The problem can be described as a cooperative game of I players related to i ∈ I system constraints. It is articial game in the sense that players decisions are already known and game theory is only to logically justify admitted allocation. The aim is to nd game rules (allocation rule) under which players decisions, which are known apriori, form equilibrium point. Formally, let I = {1, 2, ..., I} be a set of resource constraints indices, I = |I|, cost function is a function C : R I → R, with C((±∞)) = 0. An infrastructure market game, in short IMG, is a pair (C, ˜b), where C is a joint cost function described by a linear programme C(b) = min p,d ( ˆQ ±∞ − ˆQ(b)) subject to (2)-(5) and ˜b ∈ R I is a vector of actual available levels of resources. Classically, cost allocation method is a function ϕ(C, ˜b) = (x 1 , ..., x I ) ∈ R I and ∑ i∈I x i = C(˜b) (efciency condition). In this paper we will omit the efciency condition and we will refer to a cost allocation method just as a function ϕ(C, ˜b). Most of known cost allocation methods, including Shapley value, Aumann-Shapley pricing, SCRB method, core of the game, assume that an allocation must satisfy break-even property, which means the following condition is satised: ∑ x i = C(˜b). (6) i∈I Game players' decisions are known, however, cost allocation is based on hypothetical players' behaviors which potentially could mitigate or eliminate system constraints related to them. i − th resource can be considered in a continues range of 〈−∞; ˜b i 〉 or 〈˜b i ; +∞〉, however two marginal cases are of special importance: resource constraint is completely relaxed
and resource constraint is limited by ˜b i . Game players form group S ∈ I of players called coalition if only constraints related to them are xed with their values from vector ˜b and all other constraints are completely relaxed. We denote infrastructure cost for coalition S as a C(S). Further we use notation C(S) for infrastructure cost of subset S of players and C(b) for infrastructure cost for resources availability at level b, where C(S) = C(b ′ ), b ′ i = ˜b i for i ∈ S, b ′ i = ∞ or −∞ respectively for i ∉ S. Denition 1. Allocation is free from subsidizing if it satises incremental cost test a(S) ≥ C(N) − C(N \ S), (7) where a(S) = ∑ i∈S x i. Denition 2. Allocation rule ϕ(C, ˜b) is free from subsidizing for class of cost functions C if it produces allocation free from subsidizing for every admissible ˜b and C ∈ C. Coalition S is encouraged to dissolve if cost allocated on coalition S is lower then marginal benet related to coalition S dissolving, thus equations (7) are satised. Only in this case cost allocated on system constraints are sufciently incentive to motivate players to remove their constraints (at least not to make them more restrictive). Let's assume, that condition (7) is not satised for subset S. Eliminating constraints (7) reveals costs C(N)−C(N \S). Nevertheless, if allocation a(S) is lower then joint costs C(N)−C(N \S) then coalition S must be subsidized by other players. Notice, that efciency property (6) together with (7) constitute denition of game core. Core of the game is an nonempty set for many problems from important wellknown class of games pNAD. Set of games pNAD is a set of non-atomic, additive games of bounded variation with polynomial, continuously differentiable cost function [10]. Games class pNAD is important due to unique allocation scheme (Aumann-Shapley pricing) which meets widely acknowledged set of axioms for allocation rule. Unfortunately, IMG does not belong to pNAD, because C is not continuously differentiable and game is not additive. Lemma 1. Break-even and subsidize-free allocation rule exists only if characteristic function C satises m∑ C(I) ≤ [ i=1 C(I \ S i )]/(m − 1) (8) Proof. Let's ⋃split set I into m disjoint subsets S 1 , S 2 , . . . , S m , m i=1 S i = I, S j ∩ S k = ∅, ∀j, k ∈ {1, . . . , m}, j ≠ k. According to (7) an allocation is free from subsidizing if the following conditions are satised C(I) − C(I \ S i ) ≤ a(S i ) i = 1, . . . , m (9) By summing up the inequalities (9) we obtain m∑ m∑ mC(I) − C(I \ S i ) ≤ ∑ i=1 i=1 a(S i ) i = 1, . . . , m (10) An allocation satises break-even property if the equalities i∈I x i = ∑ m i=1 a(S i) = C(I) are met. By substituting this equation in the inequality (10) we obtain m∑ C(I) ≤ [ i=1 C(I \ S i )]/(m − 1) (11) Thus, the function C must satisfy conditions (8) to make the free from subsidizing property preserving possible. □ From the lemma 1 it comes that there is no universe free from subsidizing and break-even allocation rule for all possible functions C. For m = 2 this condition transforms into subadditivity denition. For m > 2 this condition is even more stronger. Theorem 1. Any break-even allocation rule for IMG is not subsidize-free. Proof comes straightly from previous lemma when we observe that characteristic function of IMG is not subadditive and does not meet condition (8). From theorem 1 it comes that only allocation rules that break the efciency axiom can be subsidize-free. Nevertheless, when core of game is empty, many researches relax the conditions (7) and use formulation of ɛ-core (e.g. for bin packing game see [8]) keeping the efciency axiom. ɛ-core is dened by the following conditions a(S) ≥ (1 − ɛ)(C(N) − C(N \ S)), ɛ ∈ 〈0; 1〉 (12) Sentence 1. Any allocation in ɛ-core, ɛ > 0, but not in 0-core is not subsidize-free. Proof of the sentence is evident, because inequalities (7) are in fact 0-core denition (12). Since for ɛ > 0 this obviously violates the conditions for subsidize-free allocation in denition 1, we are interested in holding 0-core (original, not relaxed core denition) property. Thus, the only way to achieve subsidize-free allocation is to relax efciency condition. Because majority of methods known in the literature assume that allocation must be break-even, they cannot allow to nd free from subsidizing allocation. Theoretically, subsidize free allocation can be easily obtained – it is easy to show that it is sufcient to allocate high enough cost on each player. In practice we are interested in the possible small charges which would assure freedom from subsidizing. III. MASIT – A NEW COST ALLOCATION METHOD Under the assumption, that incremental cost test (7) must be satised for each coalition, we obtain the following system assuring subsidize-free allocation: ∑ x i ≥ C(I) − C(I \ S) ∀S ⊆ I (13) i∈S Each market entity responsible for system constraints demands the lowest possible charges. Thus, the following multi-criteria problem MASIT (Minimal Allocation Satisfying Incremental cost Test) can be formulated. Problem MASIT: min (x 1 , x 2 , . . . , x n ) x 1 ,x 2 ,...,x n (14)
Page 1: Subsidize-free cost allocation meth
Page 5: l TABLE I OFFER DATA p max l s l 1

Subsidize-free cost allocation method for infrastructure ... - Ozyrys

Create successful ePaper yourself

Delete template?

Save as template?