Optimization Days 2019
HEC Montréal, May 1315, 2019
JOPT2019
HEC Montréal, 13 — 15 May 2019
TB9 Methods and Models for Large Systems Optimization
May 14, 2019 10:30 AM – 12:10 PM
Location: Dutailier International
4 Presentations

10:30 AM  10:55 AM
Optimal and approximate solutions to linear quadratic regulation of a class of graphon dynamical systems
In this paper, we study the linear quadratic regulation (LQR) problem for dynamical systems coupled over large scale networks and obtain locally computable lowcomplexity solutions. The underlying large or even infinite networks are represented by graphons and the couplings appear in both the dynamics and the quadratic cost. The optimal solution is obtained for graphon dynamical systems where the graphons are exactly characterized by finite spectral summations. Based on this, we provide a suboptimal solution for problems with general graphon couplings via spectral approximations. The complexity of generating these control solutions involves solving d + 1 scalar Riccati equations where d is the number of non zero eigenvalues in the graphon spectral representation. A numerical example is given to illustrate the explicit solution and demonstrate the simplicity of the solution.

10:55 AM  11:20 AM
Optimal Control of LargeScale Networks of Stochastic Linear Systems
Owing to the extremely large number of nodes and dynamic elements in largescale complex networks, it is effectively impossible to achieve desired control objectives .
In this work, we develop an approximate control theory for largescale network stochastic linear systems by the use of stochastic control theory of limiting (infinite dimensional) stochastic systems. First, in order to represent arbitrarysize networks of linear systems with additive noise, dynamical stochastic system models are formulated in an appropriate infinite dimensional space. Second, control of the infinite dimensional system is analysed. Finally, from the stochastic control for the limit infinite dimensional system we obtain feedback control laws for the network system, with guaranteed approximation errors. In particular, this is carried out within the framework of the linear quadratic regulator problem for largescale network stochastic systems. 
11:20 AM  11:45 AM
An affine decision rule approximation for modeling uncertainty of demand response in smart grids
In this research, we design a linear program to model expansion of smart energy systems. In this modeling, consumers are able to contribute to the network by means of flexible loads (demand response). We have considered this demand response uncertain. An affine adjustable robust optimization technique is used to cope with uncertainty in this problem.
Keywords: Demand response management, Robust optimization, affine decision rule 
11:45 AM  12:10 PM
A quantilized mean field game approach to energy pricing with application to fleets of plugin electric vehicles.
We consider the problem of designing the price of electricity by an energy provider to a pool of homogeneous loads.
The energy provider is risk sensitive and considers that its energy production cost at any particular time is related to the instantaneous maximum excursion of the random aggregate demand of the loads. A statistical measure of this excursion is the $\alpha$quantile of the distribution of the individual electricity demands of the loads, or equivalently the value da at risk $\alpha$, of the electricity demand per vehicle. The price is assumed to be a known and possibly time varying function of $d_\alpha$. The loads are associated with individual price sensitive costs. For a very large number of loads, in particular a large fleet of electric vehicles, this results in a mean field game (MFG). The existence of an MFG equilibrium associated with a price trajectory, and the epsilon Nash property of the resulting limiting control
laws, are established in this work.