  This paper studies the static economic optimization problem of a system with
a single aggregator and multiple prosumers in a Real-Time Balancing Market
(RTBM). The aggregator, as the agent responsible for portfolio balancing, needs
to minimize the cost for imbalance satisfaction in real-time by proposing a set
of optimal personalized prices to the prosumers. On the other hand, the
prosumers, as price taker and self-interested agents, want to maximize their
profit by changing their supplies or demands and providing flexibility based on
the proposed personalized prices. We model this problem as a bilevel
optimization problem. We first show that the optimal solution of this bilevel
optimization problem can be found by solving an equivalent convex problem. In
contrast to the state-of-the-art Mixed-Integer Programming (MIP)-based approach
to solve bilevel problems, this convex equivalent has very low computation time
and is appropriate for real-time applications. Next, we compare the optimal
solutions of the proposed personalized scheme and a uniform pricing scheme. We
prove that, under the personalized pricing scheme, more prosumers contribute to
the RTBM and the aggregator's cost is less. Finally, we verify the analytical
results of this work by means of numerical case studies and simulations.
