Pareto efficiency

Pareto efficiency, sometimes called Pareto optimality, is an important concept in welfare economics with applications in engineering (for instance, resource scheduling problems) and computer science (i.e., file sharing tasks). By definition, scheduling a resource such as DR, for a number of agents, is Pareto efficient if no change from this schedule can increase benefit for one agent without reducing benefits for some other agents [75]. This apparently simple outcome plays a central role in the development of the mainstream microeconomic theory to date.

Pareto efficiency has significant implications for real-world markets. It guarantees that the market outcome associated with product scheduling cannot be improved further such that no agent would be disadvantaged from the revised schedule. As there is no further improvement across all market agents, the Pareto schedule is said to be globally optimal. This implication explains the reason why economic optimizations often rely on the Pareto efficient concept.

This important concept is illustrated by Fig. 4.3 where, for simplicity, there are only two agents—a buyer and a seller—who trade a product (i.e., DR) in the market. Trading benefits for the agents is given by U1and U2 which vary according to the trading outcome.

Following microeconomic theory, values of U1 and U2 are bounded within afeasible region,

R, defined by two axes and the bold curve having a concave form in general [75]. These boundaries could be explained by certain technical and economic factors associated with the resource production and scheduling. As can be seen from the graph, Pareto efficient outcome is obtained at the point P where the boundary curve “just touches” its tangent line U1+ U2 = b (i.e., the middle line in Fig. 4.3). If the pair (U1,U2) moves away from

this critical point but still within the feasible region (R), either U1 or U2 will be reduced.

This property which can be easily proven by geometry demonstrates the notion of Pareto efficiency given above.

Figure 4.3: Pareto efficiency for a two–agent market

Although Pareto efficient schedule has been well-understood, finding it from a large population of available alternatives, for instance, over the feasible regionR, is a challenging task. In fact, it is impractical to use the graphical representation given by Fig. 4.3 to identify the Pareto point, because the boundary curve cannot be determined accurately by the market operator due to lack of benefit data that must be collected from all local agents [5]. Additionally in a market with multiple agents (i.e., more than 3) to be represented in a multi-dimensional Euclidean space, finding Pareto efficient point using graphical tools is impossible. Thus, algebraic methods based on symbolic manipulations must be applied.

To derive an algebraic approach for solving the Pareto scheduling problem, we consider again the graph in Fig. 4.3. Supposed that we shift a line [originally given by U1+ U2= 0]

as far as possible from left to right until it “just touches” the boundary curve at only one point, this point will be P which represents Pareto efficiency and is given by U1+ U2= b.

If we continue shifting, all corresponding values of U1 and U2 [on the line] will go beyond

the feasible regionRand consequently are not accepted. In order words, b is the maximum horizontal (or vertical) distance for the line to be shifted from the origin 0, within the region

R. This observation results in the following optimization that algebraically represents the process of finding Pareto efficient point.

Formulation of the market clearing problem

max (U1+ U2), subject to U1,U2 ∈R (4.1)

This statement can be extended to the generic case with an economic system comprised of K agents, by using a similar illustration with Fig. 4.3.

max

K

X

k=1

Uk subject to Uk ∈R ∀k (4.2)

Now we determine how the problem (4.2) can be applied to the particular context of a DRX market comprised of multiple buying agents (i.e., Transcos, Discos, Recos) and selling agents (EScos on behalf of electricity customers). Here we recall all mathematical notations introduced in previous chapter. For each buyerj ∈ J—the set of all buyers in the market, its trading net benefit (Uj) is given by the difference between a gross benefit

(Bj) derived from using a DR and the payment Pj for this DR. In the case of a seller

i∈ I—the set of all sellers in the market, its trading net benefit (Ui) is represented by

a difference between the received DR payment (Pi) and the actual cost (Ci) of producing

DR by curtailing some of the electric loads. With these net benefits of buying and selling agents, the first part of (4.2) can be re-written as follows:

max    X j∈J (Bj−Pj) + X i∈I (Pi−Ci)    (4.3)

By assuming that all payments collected from the buyers are given to the sellers [5,8,9], we haveP jPj = P iPi. Then (4.3) becomes max    X j∈J Bj − X i∈I Ci    (4.4)

Equation (4.4) states that a DR schedule is Pareto efficient if the total market “surplus” derived from this schedule for all agents together is maximal. This surplus is measured as the difference between total gross benefit (P

jBj) for DR buyers and total cost (

P

iCi)

of producing DR by sellers. The surplus optimization here is consistent with our demand– supply analysis in previous chapter. It is considered the objective of an agent–based market clearing mechanism.

The constraint of this optimization problem has been illustrated by the feasible region

R in Fig. 4.3. In a DRX market, it is the matching between DR quantities demanded by buyers and those supplied by sellers. This constraint is called demand-supply balance given by

yj,n = X i∈I X l∈Li uj,n_i,lxi,l ∀j∈ J; n∈ Nj (4.5)

The left hand side of (4.5) is an aggregated quantity requested by a buyer j from a consumer groupn∈ N_j. Groups are composed from consumers having a common attribute of interest to the buyer. For example, in the case of a TSO, each group includes customers connected to a common load point at the transmission level of a power system. In the case of a distributor, one group contains customers connected to a common load point of a feeder at the distribution level. For a retailer, a group is comprised of customers holding the same type of supply contracts. Consequently, each buyerjinvolves a corresponding set of customer groups (i.e., Nj). Buyers, in general, require aggregated DR quantities from

these groups and do not need to know exactly which customers are the providers.

On the right hand side of (4.5), all individual quantitiesxi,lof those customers included

in groupnassociated with buyer j are added together to form an aggregated supply cor- responding to the demandyj,n. Note also thatl∈ Li is the index of customers represented

by the seller/aggregator i. Binary coefficient uj,n_i,l is a relational status of each customer

l to group n.uj,n_i,l is 1 if the customer is included in the group, and 0 otherwise. A more detailed description of these variables and their notations is included in previous chapter. In summary, Pareto efficiency in a DRX market can be obtained by solving the surplus optimization problem given by (4.4)–(4.5). Next subsection discusses how this problem can be solved.