  One of the widely used peak reduction methods in smart grids is demand
response, where one analyzes the shift in customers' (agents') usage patterns
in response to the signal from the distribution company. Often, these signals
are in the form of incentives offered to agents. This work studies the effect
of incentives on the probabilities of accepting such offers in a real-world
smart grid simulator, PowerTAC. We first show that there exists a function that
depicts the probability of an agent reducing its load as a function of the
discounts offered to them. We call it reduction probability (RP). RP function
is further parametrized by the rate of reduction (RR), which can differ for
each agent. We provide an optimal algorithm, MJS--ExpResponse, that outputs the
discounts to each agent by maximizing the expected reduction under a budget
constraint. When RRs are unknown, we propose a Multi-Armed Bandit (MAB) based
online algorithm, namely MJSUCB--ExpResponse, to learn RRs. Experimentally we
show that it exhibits sublinear regret. Finally, we showcase the efficacy of
the proposed algorithm in mitigating demand peaks in a real-world smart grid
system using the PowerTAC simulator as a test bed.
