The motivation for this work can be found in the collection of papers listed HERE, which concern distributed control techniques for regulating the power grid. Flexibility of energy consumption can be harnessed for the purposes of ancillary services to provide virtual storage. In examples it is shown that the control architecture can be designed so that control of the loads is easy at the grid level: Tracking of a balancing authority reference signal is possible, while ensuring that the quality of service (QoS) for each load is acceptable on average.

The analysis is based on a mean field limit (as the number of loads approaches infinity). This project investigates individual risk in mean field models:

- The average performance is not an adequate measure of success. It is found empirically that a histogram of QoS is approximately Gaussian, and each node in the network will eventually receive poor service.
- The variance of QoS is estimated through a second order Taylor series expansion.

The mean-field model is essentially an application of the Law of Large Numbers; the approximation (ii) can be interpreted as the variance in the Central Limit Theorem. It is remarkable that these second-order statistics are computable, and so accurate in the examples considered.

In the power systems application considered, the variance of QoS may be unacceptably large. This is resolved through an additional layer of local control:

- The histogram of QoS is
*truncated*through this local control, so that strict bounds on service quality are guaranteed. - This has insignificant impact on the grid-level performance, beyond a modest reduction in capacity of ancillary service.

#### References (more to come)

@conference{chebusmey14,

Title = {**Individual risk in mean-field control models for decentralized control, with application to automated demand response**},

Author = {Chen, Yue and Bu\v{s}i\'{c}, Ana and Meyn, Sean},

Booktitle = {{Proc. of the 53rd IEEE Conference on Decision and Control}},

Month = {Dec.},

Pages = {6425-6432},

Year = {2014}} LINK

**Estimation and Control of Quality of Service in Demand Dispatch**},

Author = {{Chen}, Y. and {Bu{\v s}i{\’c}}, A. and {Meyn}, S.},

Title = {

**Ergodic Theory for Controlled {Markov} Chains with Stationary Inputs**},

Author = {Chen, Yue and Bu\v{s}i\'{c}, Ana and Meyn, Sean},

Journal = {ArXiv e-prints and submitted for publication, {Annals of Applied Prob.}},

Month = {April},

Year = {2016}} LINK