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

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