## 3 Demand-Response Aggregator Frameworks in Future US Electricity Markets
### Scalability in Demand-Response Aggregators' Operations  
Decentralized solutions for the prosumer-aggregator Stackelberg game are limited in the literature. In [73; 74; 209], a decentralized algorithm was designed for the DAM DR problem to solve the bi-level program, sharing only optimal solutions between prosumers and the DR aggregator. Other decentralized solutions were mentioned in Section 3.4 from a few papers [237; 238; 239].  
The rest of this section will focus on decentralized solutions for general bi-level optimization programs (Stackelberg games) to facilitate the future decentralization of DR aggregator operations in the case of thousands/millions of prosumers with hundreds/thousands of constraints each. A proper game-theoretic framework setup and the proper utilization of one of the decentralized algorithms listed below may additionally allow for a solution to the prosumer privacy problem (constraints and other problem details remain hidden). The specifics of such research remain open questions, and if addressed, they may facilitate the large-scale deployment of DR aggregators in future US electricity markets, contributing significantly to the decarbonization and net-zero emission goals analyzed in Section 1.  
In the decentralized algorithm proposed in [243], the lower-level program is reformulated into an unconstrained one with the penalty method, and then, the transformed bi-level program is solved iteratively by applying the penalty function method to the lower-level problems. It was also proven that a sequence of approximated solutions converges to the optimal solution, but privacy concerns for the lower-level programs are not taken into consideration in the solution method, nor is it mentioned whether the same convergence result applies to bi-level programs with multiple followers. In [244], the authors proposed a decentralized bi-level program algorithm that is solved via the collaboration of the lower-level programs. Specifically, all lower-level variables are updated simultaneously based on a gradient descent algorithm, exchanging information about their current variable updates and their objective derivative values at the current variable value (not a completely private algorithm), while the upper-level variable is updated based on lower-level program's variable update information. After some iterations of lower-level program variable updates, the upper-level program's variable value is updated based on the current lower-level variable values. The drawback of this method is that the number of iterations is pre-defined and the same for all lower-level programs, and the method may converge to a local optimum. Last but not least, in paper [245], the authors proposed a similar approach to this dissertation, which is a bi-level evolutionary algorithm, but they restricted the solution's mapping to quadratic polynomial functions. The authors do not study the effect of the approximation and learning errors on the solution's quality, but they do provide a very insightful visualization of a generic solution map. Moreover, the authors propose a way to find the best solution map approximation model by choosing the model that minimizes the empirical risk error on a given dataset.  
To facilitate the large-scale deployment of DR aggregators in future US electricity markets and, hence, the DR benefits, a scalable and private algorithm that is able to solve deterministic or uncertain game-theoretic frameworks that take into consideration prosumer-aggregator competition is necessary. In this way, DR benefits can be extracted on a large scale in future US markets, the probability of electricity generation decarbonization is increased, and a net-zero-emission US economy can be realized.