We considered a DR model where customers with unknown and heterogeneous marginal utilities respond to real time prices announced by the SO ahead of each time slot. The pricing mechanism is such that the SO announces a pricing function that linearly increases with total consumption per capita and decreases with increas- ing renewable energy generation in that time slot. The pricing provides the SO with the versatility to charge hourly prices that incentivizes users to behave according to
its goals. However, the users’ consumption preferences are random to the SO and it may be that the users behave in a manner that trumps the SO’s intentions in order to achieve their selfish goals. Our analysis shows that this won’t happen if agents, selfish as they may be, act rationally.
In particular, from the perspective of the SO, the peak-to-average ratio of con- sumption is reduced when the SO implements a PAR minimizing real time price, that is, users shift their consumption to time slots in which it is cheaper for SO to produce. The variance in demand caused by randomness in user preferences at each time slot reduces, increasing the demand forecast accuracy of the SO. From the perspective of a regulator invested in the well-being of the system, the proposed tariff by the SO is fair to the users [140] and the welfare is expected to be close to the efficient welfare. Furthermore, the renewable penetration is likely to increase given accurate forecasts of renewable generation due to deferrable loads serving as a buffer that absorbs the fluctuations of renewable generation. From the perspective of the users, the proposed tariff is expected to increase user utility, that is, users will consume the same amounts but at a cheaper price.
It has to be observed that the aforementioned implications depend on specific modeling choices, namely, the assumption of rational user behavior, the consideration of perfect knowledge of the preference distribution gh, and the use of a quadratic
form for the SO’s cost. These choices may be simplistic or unrealistic, but the results outlined here still provide meaningful guidelines if these restrictions are lifted. Consider, e.g., the case in which users are sub-rational and recall that we considered two models of rational behavior: price taking and price anticipating. If the users respond to announced price values, they would be price takers and the price is in expectation equal to the complete information competitive equilibrium price. If the users are selfish and anticipate their contribution to price function, then the price is
shown to approach the competitive price as N grows under certain conditions, and otherwise numerical results indicate that welfare reduction is tolerable. These models of behavior capture the two extremes of user behavior, and therefore, sub-rational behavior is likely to exhibit a behavior that falls in between these two extremes.
Regarding the assumption of perfect knowledge of the user preference distribu- tion Pgh it is likely that the SO will have some uncertain estimates, and that the
difference between the two is a random noise term. When the SO utilizes such noisy predictions of the mean preference ¯gh, the rational users will discount the weight on
the public information based on the uncertainty of the SO in their responses. While the overall performance of the system will degrade, the generalization will not affect the overall implications of the analysis. As for the use of quadratic energy costs, it is better to consider a model in which the cost for each device can be modeled as a linear function of the power dispatched from each device. In this case the cost model is an increasing piecewise linear function of total consumption as power is dis- patched from more costly generators with increasing total consumption [123]. The quadratic cost function is an approximation for the piecewise linear cost function which is tractable and captures the fundamental property that higher energy pro- duction requires bringing more costly sources online. The quantitative specifics may change for piecewise linear functions but the qualitative conclusions will be similar.
Chapter 6
Demand Response Management in
Smart Grids with Cooperating
Rational Consumers
6.1
Introduction
The specifics of a consumer behavior model and the information provided to the users impact the welfare of the overall system and is critical in assessing the benefits
or disadvantages of a pricing scheme in the electricity markets [129] 1. Based on
this observation, adopting the electricity market model in Chapter 5, we explore the effects of consumer behavior models where consumers respond rationally regarding selfish utility, the population’s aggregate utility or the welfare on the real-time pric- ing (RTP) scheme (Section 6.2.3). As time progresses, the consumption behavior of individuals reveal information about the preferences of others which individuals can
use to make better estimates of total consumption. For this, we provide three in- formation exchange models, namely, private, action sharing and broadcast (Section 6.2.4). In the private model, users do not receive any information besides the initial public signal by the SO. In action sharing there exists a communication network on which users exchange their latest consumption decisions with their immediate neigh- bors. In broadcasting, the SO broadcasts the total consumption after each time step. We assume that the customer’s power control scheduler can adjust the load consumption between time slots according to his preferences and information. That is, we are interested in modeling consumption behavior for shiftable appliances, e.g., electric vehicles, electronic devices, air conditioners, etc. [142].
We formulate each consumer behavior model and information exchange model pair as a repeated game of incomplete information and characterize equilibrium behavior (Section 6.4). Because the user payoff is quadratic (6.3), we can explicitly derive the BNE by solving a set of linear equations at each stage and updating beliefs depending on the information exchange model as is done in the QNG filter in Chapter 2. We use the QNG filter to rigorously analyze the effects of each pair of behavior and information exchange model on total consumption, aggregate consumer utility, SO’s net revenue (Section 6.6).
Our findings can be summarized as follows. Providing more information to the consumers do not hurt the expected net revenue of the SO and increases the ex- pected aggregate consumption utility. In addition, additional information to the users reduce the uncertainty in total demand. Furthermore, action sharing informa- tion exchange model eventually achieves the expected utility under full information when the communication network is connected. The positive effects of additional information are reduced with growing correlation among preferences. Furthermore, increasing correlation among consumption preferences has a decreasing effect on the
expected aggregate utility for all behavior models. Finally, the inefficiency due to selfish behavior diminishes with the growing number of customers.