Anticipated Cost Results

The next step is to calculate the anticipated cost errors for the inconsistent agents. We only calculate anticipated costs for an agent in horizon h if it has been identified that it was inconsistent (not notionally inconsistent or notionally consistent) between horizons h and h+ 1. The results for the greedy and uncertain agents are shown in figures 5.7 and 5.8 for the two load types respectively.

These results show the anticipated cost errors obtained for each scale value. These are split up into separate means depending on how far in ad- vance the anticipated cost error occurred (see section 5.10.1). Also provided are the 10th and 90th percentiles over all anticipated costs aggregated to- gether for all instances. The mean of the uncertain agent remains around zero, which indicates that on average the agent underestimates its costs in advance just as often as it overestimates them.

The overestimation of costs in earlier horizons is exactly how the greedy agents manipulate the other agents in this setting to gain an advantage. This is clearly seen as a non-zero mean for the greedy agents. This is more pronounced for the deferrable greedy agent because it gains a greater ad-

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Figure 5.7: Anticipated cost errors for fixed power greedy and uncertain agents. Means for each instance are taken separately based on how far in advance the cost was anticipated (given byd). The shaded sections provide the 10th and 90th percentiles over all instances for the greedy and uncertain agents.

vantage.

A threshold can be used to separate anticipated cost error means of the truthful uncertain agents from the greedy agents. This threshold can increase as the contracted agent power limits (scale parameter) varies. After a number of measurements, if the agent crosses over the threshold then they can either be targeted for further auditing or fined for manipulative behaviour. This will provide a strong deterrent.

It might still be possible for agents to gain some benefit by staying within the threshold we select, but the benefit is reduced. The computational burden is increased for the calculation of strategies that attempt to remain within the threshold. For example, it introduces bilinear terms for the fixed greedy agent, converting it from a MILP to a MINLP.

Using the anticipated cost error identifier with a threshold will limit the type of uncertainty that agents can exhibit. Those with more complex forms of uncertainty may be penalised if they produce results that look similar to manipulative actions. If too many false positives are made in a particu- lar setting then new identifiers can be developed. The great advantage of having privacy-preserving identifiers is that they can always be modified or more can always be added without affecting the operation of the underlying distributed algorithm.

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Figure 5.8: Anticipated cost errors for deferrable greedy and uncertain agents. Means for each instance are taken separately based on how far in advance the cost was anticipated (given byd). The shaded sections provide the 10th and 90th percentiles over all instances for the greedy and uncertain agents.

5.12

Related Work

Our development and calculation of greedy agent strategies is related to other work that has looked at equilibria in electricity markets. Hu and Ralph [2006] study equilibria in electricity markets with locational marginal prices where each agent solves a bilevel problem to obtain their strategy. They find sufficient conditions for the existence of pure Nash equilibria. Weber and Overbye [1999] similarly develop a method for finding Nash equilib- ria for producers and consumers that have linear price functions. Li and Shahidehpour [2005] develop a method for the case where agents have in- complete information about other agents.

Instead of searching for Nash equilibria, Kozanidis et al. [2013] develop optimal bidding strategies for a strategic producer in a single time period market with no network constraints. This is closer to what we developed in section 5.6, but we look at multiple time steps which overlap between horizons, and with different types of agents.

Compared to these works our problem is more complicated in certain areas but simpler in others. For example, instead of a single time horizon, we consider strategies over multiple overlapping horizons given by the receding horizon structure. We also focus more on a prosumer oriented setting where each agent can have a diverse set of preferences and constraints instead of a market dominated by large generator units. However, as a first step we

ignore network constraints.

Mechanism design and game theory have been used in demand response [Mohsenian-Rad et al., 2010, Chen et al., 2011, Tanaka et al., 2012, Akasi- adis and Chalkiadakis, 2013, Samadi et al., 2012], as well as other network problems including electricity markets and storage adoption [Andrianesis et al., 2010, Tanaka et al., 2012, Vytelingum et al., 2011].

VCG mechanisms have been utilised by Samadi et al. [2012] and Tanaka et al. [2012]. As discussed in section 5.2, VCG quickly becomes intractable for realistically sized problems, and requires agents to fully disclose their preferences. Tanaka et al. [2012] acknowledges these problems and proposes that future work looks at the development of approximate methods.

Chapman et al. [2013] provide a detailed review into the practicality of these existing game-theoretic approaches. They develop four important con- siderations which any realistic mechanism should take into account, which paraphrased are:

1. The power consumption in houses can take on both discrete and con- tinuous values.

2. Each house has a private state that represents the state of the occu- pants’ goals.

3. Houses have private preferences which are state-based, combinatorial and non-convex.

4. Household behaviour is strategic.

In section 4.10 we showed how the most common source of household discrete decisions can be effectively dealt with, and furthered this argument in section 5.4.3 by discussing other practical methods for dealing with non-convexities. To this list we add a fifth consideration which acknowledges the uncertain nature of the problem:

5. The system is inherently uncertain as a result of household occupant behaviour and weather patterns.

A realistic mechanism should at least work in the presence of uncertainty and ideally reduce its impact on the system performance.

Mhanna et al. [2015] use a scoring rule to charge consumer agents based on both their actual consumption and their deviation from day-ahead al- locations. By requiring agents to provide information on their uncertainty, they can reduce the incentive for agents to lie about their requirements over the day ahead. They find it to be “asymptotically” incentive compatible as system size or reported precision increases; however, the approach does not enable agents to share information they gain throughout the day and re-optimise their allocation in an online fashion.

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