Equivalent conditions

From a practical point of view, the Pareto optimization problem (4.4)–(4.5) could be centrally solved by the DRXO, using data reflecting benefitBj and costCi collected from

buyers and sellers, respectively. This is the case of a pool-based DRX developed in previous chapter. Such a centralized optimization, however, faces a major difficulty—participating agents may not be willing to provide the market operator with their own true information. For example, while a seller may “lie” by declaring an over-estimated cost for supplying DR, a buyer could declare an under-estimated benefit gained from DR trading. The aim here is to “game” the DRX market in a way that benefits a particular agent more than others. Such gaming behaviour would make market clearing difficult because the calculation of total surplus using (4.4) is no longer realistic due to incorrect cost and benefit data collected from the agents [98].

In a pool-based DRX market, the above information gaming issue can be mitigated using an assurance contract signed between buyers and sellers prior to the point of market clearing. Under this contract, agents will hurt themself [in that their benefits are automat- ically reduced] if they untruthfully declare cost and benefit data [78–80]. This interesting

Formulation of the market clearing problem

point signifies that the agents are better off not lie on purpose when sending information for market clearing. This “truth-revealing” advantage of an assurance contract, however, is only applicable to the case of pool-based market clearing where all relevant information are to be aggregated for a centralized profit optimization. In an agent-based DRX market where most private data is kept confidential by each agent and only used for their local optimizations, assurance contract approach is difficult to apply.

For this reason, rather than using an assurance contract and then directly solving the market clearing problem (4.4)–(4.5), we should convert it into an equivalent condition, and then solve this condition using a realistic mechanism considering the agent’s gaming behaviour. This conversion is based on the following proposition.

Proposition 4.3.1 Assuming that the cost functionCi is convex overxi,l and the benefit function Bj is concave over yj,n, the problem (4.4)–(4.5) is equivalent to the following conditions: ∂Ci ∂xi,l =X j∈J X n∈Nj uj,n_i,l ∂Bj ∂yj,n ∀i∈ I; l∈ Li (4.6) yj,n = X i∈I X l∈Li uj,n_i,lxi,l ∀j∈ J; n∈ Nj (4.7)

Proof With the convexity ofCi and the concavity ofBj, the problem given by (4.4)–(4.5)

belongs to a well-known class of convex optimization with affine (linear) equality con- straints [99]. By Karush-Kuhn-Tucker (KKT) theorem, necessary and sufficient conditions for this optimality are

(a) ∂L/∂yj,n = 0 ∀j ∈ J; n∈ Nj (b) ∂L/∂xi,l= 0 ∀i∈ I; l∈ Li (c) ∂L/λj,n = 0 ∀j∈ J; n∈ Nj where L , (P jBj − P iCi) + P j P n[λj,n(yj,n− P i P lu j,n

i,l xi,l)] is called a Lagrange

function and λj,n are called KKT multipliers. By taking partial derivatives of L in (a),

(b), and (c), we imply the following: (d) −λj,n=∂Bj/∂yj,n (e) ∂Ci/∂xi,l=− P j P nu j,n i,lλj,n (f) yj,n=P_iP_luj,n_i,lxi,l

Figure 4.4:Assumptions on cost and benefit functions

Remark Equation (4.6) entails the marginal cost (measured by ∂Ci/∂xi,l) associated

with producing an individual quantity xi,l by consumer l, must be equal to the sum of

marginal benefits (measured by ∂Bj/∂yj,n) gained by those buyers who jointly use this

individual quantity. This equation is consistent with the well known Samuelson rule which is an alternative formalization of Pareto efficiency for optimal resource scheduling [100].

The mathematical derivation of (4.6)–(4.7) relies on two major assumptions—the con- vexity ofCi overxi,l and the concavity of Bj overyj,n. That is, for any t1, t2 ∈[0,1]

(a)Ci(t1x0_i,l+ (1−t1)x00_i,l)≤t1Ci(x0_i,l) + (1−t1)Ci(x00_i,l) ∀x0_i,l, x00_i,l

(b)Bj(t2y0_j,n+ (1−t2)y00_j,n)≥t2Bj(y_j,n0 ) + (1−t2)Bj(y_j,n00 ) ∀y_j,n0 , y00_j,n

These mathematical assumptions are illustrated by Fig. 4.4 where both Ci and Bj

generally are increasing functions, but their exact trends are different. The former raises exponentially, meaning that the cost associated with producing an additional DR quantity increases at a higher rate than before. BenefitBj, on the other hand, tends to be “saturate”

with the increasing DR quantity.

In fact, the assumptions of cost convexity and benefit concavity are common in eco- nomic literature. For example, the generation cost in an electricity market is often modelled as a convex and quandratic curve to be used for the market design and analysis [101]. These assumptions will be discussed further in Section 4.5, for the DRX context.