Constraints

Constraints need to be defined in the proposed market clearing optimization model. The most important constraint is demand–supply balance given by:

yj,n = X i∈I X l∈Li uj,n_i,lxi,l ∀j∈ J;n∈ Nj (3.9)

The left hand side of (3.9) represents aggregated quantity yj,n which is the demand

of buyerj over customer group n. On the right hand side, all individual quantities xi,l of

customers included in the group are added together to form an aggregated supply matching the demand. Binary coefficient uj,n_i,l representsrelational status of each customerl to the groupn.uj,n_i,l is 1 if the customer is included in the group, and 0 otherwise.

To illustrate this balancing equation, we consider again the power system in Fig. 3.5. For simplicity we assume that each load point at a distribution feeder level represents a single customer, therefore there are totally 14 customers in the system. We also assume that the Reco offers two different types of electricity retail contracts—type A for customers 1, 2, ..., 10 and type B for customers 11, ..., 14. Consequently we haveNReco={A,B},

N_Transco ={I,II},N_Disco ={1,2, ...,14}, as well as J ={Reco,Transco,Disco}. Note also thatI ={ESCo} and LESCo ={1,2, ...,14}. According to (3.9):

yReco,A=xESCo,1+xESCo,2+...+xESCo,10

yReco,B=xESCo,11+...+xESCo,14

yTransco,I=xESCo,1+xESCo,2+...+xESCo,6

Market clearing optimization model

yDisco,1=xESCo,1

yDisco,2=xESCo,2

...

yDisco,14=xESCo,14

Since customerl supplies a common DR to one Reco, one Disco, and the Transco, the customer is included in three different groups associated with these three DR buyers, re- spectively. Therefore, its quantityxi,l appears in three corresponding balancing equations,

as illustrated by the above example. This repetition shows that DR from a customer can be consideredpublic good, which is a special type of resource with each single quantity jointly used by multiple players. Indeed, treating DR as a public good is central to the analysis of a DRX, not only in this chapter but also others within this thesis.

The next constraint is related to the contribution of each player in overall payment for the public good. These contributions can be specified in an assurance contract that is signed between involved players [77–80]. This contract refers to a financial mechanism for guaranteeing an efficient provision of the public good in the face of the free riding

problem. In general, such a problem occurs whenever there is an action (i.e., scheduling DR as a public good) that would benefit several players (i.e., a Transco, a Disco, a Reco), but once the action is taken, there is no way to exclude those who did not pay for the action from the benefits. This non–excludability leads to free riding opportunity—some self-interested players may make a decision to let other players pay for the action, then to enjoy the benefits for free. This situation is unfair to the voluntary payers and reduces the overall benefit of taking the action. In the worst case where no player pays, the action would not be taken.

Assurance contract is considered a powerful mechanism to avoid the free-riding prob- lem, that is, to encourage every beneficiary contribute to the overall payment of a public good. The core idea is to let the beneficiaries voluntarilypledgeto contribute paying for the good. If the total payment is enough, the good will be supplied; otherwise, the pledges are refunded in such a way that benefits the contributors more than others. Such a refunding policy was proven to motivate all beneficiaries making pledges for payments [77, 78].

Here we formulate the use of assurance contract which imposes the second constraint of the DRX market clearing optimization model. Our analysis focuses on practical issues particularly relevant for the DRX, with numerical study given in Section 3.4. It isnot our purpose to re-examnie the theoretical aspects already described elsewhere in the economic literature.

Let us consider three arbitrary DR buyers:j (e.g., Reco),j0 (e.g., Disco), andj00 (e.g., Transco), who together intend to purchase a common quantityxi,l at a marginal cost ci,l

from customerlas a DR seller. As per assurance contract, the payment allocation among these buyers for that quantity must satisfy [77]:

P_ki,l≥δ_ki,l.(ci,lxi,l) ∀k∈ {j, j0, j00} (3.10)

The left hand side of (3.10) represents an actual payment, denoted by P_ki,l, made by only one buyer for the common DR justafter the market is cleared. This payment must be at least equal to a threshold amount according to the assurance contract. This amount is referred to as an “obligatory contribution”. It is determined by multiplying the DR revenue

ci,lxi,lof the customerlby the so-called contribution rateδki,l—a fixed parameter specified

in the assurance contract.

Without obligatory contributions from buyers, one buyer may pay less than the others, regardless of how much benefit it gains from DR. Some buyers may avoid paying anything at all but at the same time enjoy the benefits of DR. Non-paying beneficiaries are referred to as “free riders”, and they can cause substantial distortions of a market [78]. Consequently, an assurance contract specifying the contribution rate of each buyer in the market is necessary to avoid this free-rider problem, and thus ensure market efficiency.

For a given DR quantityxi,l to be supplied, the total obligatory contributions Pkδ i,l k

.(ci,lxi,l) must match the customer revenue [77]:

δi,l_j +δ_ji,l0 +δi,l_j00 = 1 (3.11)

For this constraint, we can easily deduce that:P

kP i,l

k ≥ci,lxi,l. This means all buyers

collectively may have to pay an amount that is greater than the payment made to the customer. In economic theory, this discrepancy is commonly known as “payment excess”, which represents an imbalance between the buyer’s payment and the seller’s revenue. Fig. 3.6 shows that as the quantity x deviates from the equilibrium point x∗ payment excess occurs (as depicted by the area (B +C)). There are two common ways to deal with this excess [6]. First, if the excess is small, it may be kept by the DRXO to recover the cost of running the market. Second, if the excess is relatively high, part of it will be

refunded back to the buyers using certain refunding policies (i.e., proportional to their own contributions). These two payment excess scenarios are also considered part of the assurance contract arrangement [79].

Now, without the loss of generality, we consider only the buyerj. It buys DR not only from the customerl, but also many others—each included in a group n ∈ Nj associated

with buyer j. Following the condition in (3.10), the buyer total payment to all these customers must satisfy:

P_jtotal =X i,l P_ji,l≥X i∈I X l∈Li

Numerical example HoweverP_jtotal=P n∈Njpj,nyj,n, then we imply: X n∈Nj pj,nyj,n≥ X i∈I X l∈Li

uj,n_i,l.δi,l_j .(ci,lxi,l) (3.13)

A practical issue stemming from this condition is that the buyer j purchases the DR from a large number of customers. This results in a large number of corresponding pa- rametersδi,l_j being considered in the contract. In this paper, for illustrative purposes, we assume that these parameters are all equal to a common valueδj. This assumption does

not affect the comparison between the DRX and partial DR approaches. The assumption implies the following variation on the previous constraint:

X n∈Nj pj,nyj,n ≥δj   X i∈I X l∈Li

uj,n_i,l.(ci,lxi,l)



 ∀j∈ J (3.14)

For further simplification, we assume that the value ofδj is: 1) commonlyδRfor every

buyer j who is a Reco; 2) commonly δD for every buyer j who is a Disco; 3) δT for the

Transco only. Following the condition in (3.11), we imply:

δR+δT +δD = 1 (3.15)

We call the vector δ = [δR δT δD] the “contribution rates” vector, which is a core,

pre-determined parameter of the DRX model. Since the value ofδ is decided by agreement between the buyers via an assurance contract prior to market clearing, it plays the role of beingpledges mentioned above. Specifically if no buyer pledges to contribute then no DR as a public good will be scheduled. The more amount of pledges is made in the contract, the more DR quantity is to be supplied. Such an impact of the pre-determined parameter

δ on DRX market clearing outcome will be analysed via numerical study in Section 3.4. Overall, the developed DRX optimization model has xi,l and yj,n as the decision vari-

ables, (3.6) as the objective function, (3.9) and (3.14) as the constraints, as well as (3.8) and (3.15) as the supporting calculations.