4.5 Robustness evaluation
4.5.1 Agent local optimization
Myopic situation
In the proposed mechanism with a radial structure, all communications must go through the DRXO as a central node. Therefore, agents cannot collect private information directly from each other. The only information available to each agent and issued by the DRXO, is the price of DR quantities they intend to buy or sell. Under this market arrangement, it would be certainly very hard for any agent to anticipate actions taken by other agents to increase their local benefits [105]. Such a situation is referred in economic literature as “myopia”, in that a local agent cannot see what happens outside.
There are two common approaches [for an agent to deal with myopic situation when optimizing its local profit] including statistical and guidence-based [106]. The former aims at modeling the actions taken by other [competing] agents, using stochastic methods due to lack of relevant information. For example, the myopic trading in financial markets, under multiple sources of uncertainties (i.e., volatilities) over time and across all agents, often relies heavily on statistical modeling [5, 8, 9, 73, 105]. Despite its advantages, the statistical
approach cannot be completely realistic, consequently causing some risks incurred by those agents who use such an approach. For instance, many financial crises over the world can be explained by the underestimated market volatilities [9,73]. Due to these uncertainties, any agent is better off using an alternative [myopic] approach, that is, to follow the guidence of a designated market operator who has some information of other agents [105]. In the context of a DRX, both buying and selling agents can myopically follows the market price issued from the DRXO as an agent guider. In other words, each agent under the proposed mechanism is considered as a price-taker.
Optimization convexifying
Accepting prices issued by the DRXO, each buying agentj performs local profit opti- mization at every market clearing roundt, as follows:
max Bjt− X n∈Nj ptj,nyj,nt ∀j∈ J (4.12)
Similarly, each price-taking selling agentiperforms local optimization according to:
max X l∈Li pti,lxti,l−Cit ∀i∈ I (4.13) In (4.12) and (4.13), pt
i,l and ptj,n are constants, while xti,l and yj,nt are treated as
optimization variables under the corresponding agent. Cit and Bjt are the DR cost and benefit that are functions of those variables, respectively. Note that these functions must be kept private by agents. In a DRX market with radial communication structure, agents are not allowed to exchange information directly with each other.
Generally, the problems given by (4.12) and (4.13) may be non-convex, and thereby, having either multiple solutions, one solution, or even no solution at all [99]. These cases together unnecessarily complicate the agent local optimizations and then the whole market clearing mechanism. Hence, there is a need for proposing a “convexifying rule” as given by Fig. 4.8, where all agents are required to approximate their DR costs [or DR benefits] as strictly convex [or concave] functions to be embedded within the local optimizations. If such a rule can be successfully applied, it is not only the problems (4.13) and (4.12) that have unique solutions but also the whole DRX market that have a unique equilibrium point as was proven above.
Convexification is not a theoritical limitation of the proposed agent-based mechanism but a practical necessity for achieving a derised market outcome with unique equilibrium solution. From local agents perspective, the convexifying rule keeps their optimizing task simple. In fact, such rules have long been discussed in electricity markets as an important tool to deal with the non-convexity issues in generating unit commitment [107].
Robustness evaluation
Figure 4.8:Convexification as an approximation tool
Under convexification, rational agents choose the optimal DR quantities to buy or sell following (4.12) or (4.13), as given by:
ptj,n = ∂B t j ∂yj,nt ∀j∈ J; n∈ Nj (4.14) pti,l= ∂C t i ∂xt i,l ∀i∈ I;l∈ Li (4.15)
Equation (4.14) implies that rational buyers will increase their demand up to the point at which the marginal benefits they gain from DR are equal to the corresponding prices they have to pay. Equation (4.15), on the other hand, implies that rational sellers will increase their DR provisions up to the point at which the marginal costs of producing these DRs are equal to the corresponding prices they receive. These observed buying and selling behavious are consistent with microeconomic theory [75].
By substituting both (4.14) and (4.15) into (4.10) we obtain:
∂Cit ∂xt i,l =X j∈J X n∈Nj uj,ni,l ∂B t j ∂yt j,n ∀i∈ I; l∈ Li (4.16)
Equation (4.16) implies that market clearing at every round t satisfies (4.6)—one of the two conditions of Pareto efficiency. With this condition satisfied, the remaining issue of the mechanism evaluation is to demonstrate that market clearing always converges to a unique equilibrium point satisfying a demand supply balance given by (4.7)—the other condition of Pareto optimality.