19 research outputs found
A product form for the general stochastic matching model
We consider a stochastic matching model with a general compatibility graph,
as introduced in \cite{MaiMoy16}. We show that the natural necessary condition
of stability of the system is also sufficient for the natural matching policy
'First Come, First Matched' (FCFM). For doing so, we derive the stationary
distribution under a remarkable product form, by using an original dynamic
reversibility property related to that of \cite{ABMW17} for the bipartite
matching model
Probabilistic cellular automata, invariant measures, and perfect sampling
A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The
cells are updated synchronously and independently, according to a distribution
depending on a finite neighborhood. We investigate the ergodicity of this
Markov chain. A classical cellular automaton is a particular case of PCA. For a
1-dimensional cellular automaton, we prove that ergodicity is equivalent to
nilpotency, and is therefore undecidable. We then propose an efficient perfect
sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm
does not assume any monotonicity property of the local rule. It is based on a
bounding process which is shown to be also a PCA. Last, we focus on the PCA
Majority, whose asymptotic behavior is unknown, and perform numerical
experiments using the perfect sampling procedure
Reversibility and further properties of FCFS infinite bipartite matching
The model of FCFS infinite bipartite matching was introduced in
caldentey-kaplan-weiss 2009. In this model there is a sequence of items that
are chosen i.i.d. from and an independent
sequence of items that are chosen i.i.d. from ,
and a bipartite compatibility graph between and
. Items of the two sequences are matched according to the
compatibility graph, and the matching is FCFS, each item in the one sequence is
matched to the earliest compatible unmatched item in the other sequence. In
adan-weiss 2011 a Markov chain associated with the matching was analyzed, a
condition for stability was verified, a product form stationary distribution
was derived and the rates of matches between compatible types
and were calculated.
In the current paper, we present several new results that unveil the
fundamental structure of the model. First, we provide a pathwise Loynes' type
construction which enables to prove the existence of a unique matching for the
model defined over all the integers. Second, we prove that the model is
dynamically reversible: we define an exchange transformation in which we
interchange the positions of each matched pair, and show that the items in the
resulting permuted sequences are again independent and i.i.d., and the matching
between them is FCFS in reversed time. Third, we obtain product form stationary
distributions of several new Markov chains associated with the model. As a by
product, we compute useful performance measures, for instance the link lengths
between matched items.Comment: 33 pages, 12 figure
Demand Dispatch with Heterogeneous Intelligent Loads
A distributed control architecture is presented that is intended to make a collection of heterogeneous loads appear to the grid operator as a nearly perfect battery. Local control is based on randomized decision rules advocated in prior research, and extended in this paper to any load with a discrete number of power states. Additional linear filtering at the load ensures that the input-output dynamics of the aggregate has a nearly flat input-output response: the behavior of an ideal, multi-GW battery system. \
Feature Projection for Optimal Transport
Optimal transport is now a standard tool for solving many problems in
statistics and machine learning. The optimal "transport of probability
measures" is also a recurring theme in stochastic control and distributed
control, where in the latter application the probability measure corresponds to
an empirical distribution associated with a large collection of distributed
agents, subject to local and global control. The goal of this paper is to make
precise these connections, which inspires new relaxations of optimal transport
for application in new and traditional domains. The proposed relaxation
replaces a target measure with a "moment class": a set of probability measures
defined by generalized moment constraints. This is motivated by applications to
control, outlier detection, and to address computational complexity. The main
conclusions are (i) A characterization of the solution is obtained, similar to
Kantorovich duality, in which one of the dual functions in the classical theory
is replaced by a linear combination of the features defining the generalized
moments. Hence the dimension of the optimization problem coincides with the
number of constraints, even with an uncountable state space; (ii) By
introducing regularization in the form of relative entropy, the solution can be
interpreted as replacing a maximum with a softmax in the dual; (iii) In
applications such as control for which it is not known a-priori if the moment
class is non-empty, a relaxation is proposed whose solution admits a similar
characterization; (iv) The gradient of the dual function can be expressed in
terms of the expectation of the features under a tilted probability measure,
which motivates Monte-Carlo techniques for computation
Risk-Averse Equilibrium Analysis and Computation
We consider two market designs for a network of prosumers, trading energy:
(i) a centralized design which acts as a benchmark, and (ii) a peer-to-peer
market design. High renewable energy penetration requires that the energy
market design properly handles uncertainty. To that purpose, we consider risk
neutral models for market designs (i), (ii), and their risk-averse
interpretations in which prosumers are endowed with coherent risk measures
reflecting heterogeneity in their risk attitudes. We characterize analytically
risk-neutral and risk-averse equilibrium in terms of existence and uniqueness ,
relying on Generalized Nash Equilibrium and Variational Equilibrium as solution
concepts. To hedge their risk towards uncertainty and complete the market,
prosumers can trade financial contracts. We provide closed form
characterisations of the risk-adjusted probabilities under different market
regimes and a distributed algorithm for risk trading mechanism relying on the
Generalized potential game structure of the problem. The impact of risk
heterogeneity and financial contracts on the prosumers' expected costs are
analysed numerically in a three node network and the IEEE 14-bus network
Arbitrage with Power Factor Correction using Energy Storage
The importance of reactive power compensation for power factor (PF)
correction will significantly increase with the large-scale integration of
distributed generation interfaced via inverters producing only active power. In
this work, we focus on co-optimizing energy storage for performing energy
arbitrage as well as local power factor correction. The joint optimization
problem is non-convex, but can be solved efficiently using a McCormick
relaxation along with penalty-based schemes. Using numerical simulations on
real data and realistic storage profiles, we show that energy storage can
correct PF locally without reducing arbitrage profit. It is observed that
active and reactive power control is largely decoupled in nature for performing
arbitrage and PF correction (PFC). Furthermore, we consider a real-time
implementation of the problem with uncertain load, renewable and pricing
profiles. We develop a model predictive control based storage control policy
using auto-regressive forecast for the uncertainty. We observe that PFC is
primarily governed by the size of the converter and therefore, look-ahead in
time in the online setting does not affect PFC noticeably. However, arbitrage
profit are more sensitive to uncertainty for batteries with faster ramp rates
compared to slow ramping batteries.Comment: 10 pages, 8 figure
Can we use perfect simulation for non-monotonic Markovian systems ?
International audienceSimulation approaches are alternative methods to estimate the stationary be- havior of stochastic systems by providing samples distributed according to the stationary distribution, even when it is impossible to compute this distribution numerically. Propp and Wilson used a backward coupling to derive a simu- lation algorithm providing perfect sampling (i.e. which distribution is exactly stationary) of the state of discrete time finite Markov chains. Here, we adapt their algorithm by showing that, under mild assumptions, backward coupling can be used over two simulation trajectories only