The control solution is based on the following steps: 1. A Markov Decision Process (MDP) model is introduced for an individual pool pump. 2. Using the Todorov's formulation, a randomized control architecture is proposed, motivated by the need for decentralized decision making, and the need to avoid synchronization that can lead to large and detrimental spikes in demand. 3. An aggregate model for a large number of pools is obtained from a mean field limit. 4. A linearization of the eigenvector problem in (2) provides an LTI-system approximation of the aggregate nonlinear model, with a scalar signal as the input and a measure of the aggregate demand as the output.
The final approximation is convenient for control design at the grid level. Simulations are provided to illustrate the accuracy of the models and effectiveness of the proposed control approach.
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