Section: New Results

Applications to Renewable Resources and Energy

Participants : Sara Alouf, Alain Jean-Marie, Dimitra Politaki.

Stochastic models for solar power

In [31], D. Politaki and S. Alouf develop a stochastic model for the solar power at the surface of the earth. They combine a deterministic model of the clear sky irradiance with a stochastic model for the so-called clear sky index to obtain a stochastic model for the actual irradiance hitting the surface of the earth. Their clear sky index model is a 4-state semi-Markov process where state durations and clear sky index values in each state have phase-type distributions. They use per-minute solar irradiance data to tune the model, hence they are able to capture small time scales fluctuations. They compare this model with the on-off power source model developed by Miozzo et al. (2014) for the power generated by photovoltaic panels, and to a modified version that they propose. Computing the autocorrelation functions for all proposed models, they find that the irradiance model surpasses the on-off models and it is able to capture the multiscale correlations that are inherently present in the solar irradiance. The power spectrum density of generated trajectories matches closely that of measurements. This new irradiance model can be used not only in the mathematical analysis of energy harvesting systems but also in their simulation.

In [45], D. Politaki, S. Alouf and A. Jean-Marie in collaboration with F. Hermenier (Nutanix) aim at the performance analysis of a data center feb by renewable energy recources. They describe the data center system, proposing a new queuing model BMAP/PH/c which represents the queue length in a system having c servers, where arrivals are determined by a Batch Markov Arrival process and service times have a phase-type distribution. They validate this model using real traces. Next, they characterize the data center google workload traces which are available in the web and they validate that the jobs arrive to the system in groups (batches) and wait at the queue. The waiting time is diverse according to the available resources, job size etc. The authors then compute the empirical CDF of the service time and try to fit it with well-known distributions like exponential, Pareto etc. However, the Kolmogorov-Smirnov test rejects the null hypothesis at the 1% significance level which shows that service time doesn't fit with any well-known distribution.

Sustainable management of water consumption

Alain Jean-Marie, Mabel Tidball (INRA, Montpellier, France), Fernando Ordóñez and Victor Bucarey López (Univ. de Chile, Chile), consider in [36] a discrete time, infinite horizon dynamic game of groundwater extraction. A Water Agency charges an extraction cost to water users, and controls the marginal extraction cost so that it depends linearly on total water extraction (through a parameter $n$) and on rainfall (through parameter $m$). The water users are selfish and myopic, and the goal of the agency is to give them incentives them so as to, at the same time, improve their total welfare and improve the long-term level of the resource.

This problem is studied in two situations for a linear-quadratic model. In the first situation, the parameters $n$ and $m$ are considered to be fixed over time, and the Agency selects the value that maximizes the total discounted welfare of agents. A first result shows that when the Water Agency is patient (discount rate close to one), the optimal marginal extraction cost asks for strategic interactions between agents.

In the second situation, the authors look at the dynamic Stackelberg game where the Agency decides at each time what cost parameter they must announce in order to maximize the welfare function. This becomes a highly non-linear optimal control problem. Some preliminary results are presented.