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Randall Martyr

Publications and source records attributed to Randall Martyr.

10 recordsLinked to original sources
arXiv2022

Markov risk mappings and risk-sensitive optimal prediction

We formulate a probabilistic Markov property in discrete time under a dynamic risk framework with minimal assumptions. This is useful for recursive solutions to risk-sensitive versions of dynamic optimisation problems such as optimal prediction, where at each stage the recursion depends on the whole future. The property holds for standard measures of risk used in practice, and is formulated in several equivalent versions including a representation via acceptance sets, a strong version, and a dual representation.

math.OC↗
arXiv2021

Discrete-time risk-aware optimal switching with non-adapted costs

We solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision maker is risk aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

math.OC↗
arXiv2018

Benchmarking the performance of controllers for power grid transient stability

As the energy transition transforms power grids across the globe, it poses several challenges regarding grid design and control. In particular, high levels of intermittent renewable generation complicate the task of continuously balancing power supply and demand, requiring sufficient control actions. Although there exist several proposals to control the grid, most of them have not demonstrated to be cost efficient in terms of optimal control theory. Here, we mathematically formulate an optimal centralized (therefore non-local) control problem for stable operation of power grids and determine the minimal amount of active power necessary to guarantee a stable service within the operational constraints, minimizing a suitable cost function at the same time. This optimal control can be used to benchmark control proposals and we demonstrate this benchmarking process by investigating the performance of three distributed controllers, two of which are fully decentralized, that have been recently studied in the physics and power systems engineering literature. Our results show that cost efficient controllers distribute the controlled response amongst all nodes in the power grid. Additionally, superior performance can be achieved by incorporating sufficient information about the disturbance causing the instability. Overall, our results can help design and benchmark secure and cost-efficient controllers.

math.OC↗
arXiv2018

Optimal control of a commercial building's thermostatic load for off-peak demand response

This paper studies the optimal control of a commercial building's thermostatic load during off-peak hours as an ancillary service to the transmission system operator of a power grid. It provides an algorithmic framework which commercial buildings can implement to cost-effectively increase their electricity demand at night while they are unoccupied, instead of using standard inflexible setpoint control. Consequently, there is minimal or no impact on user comfort, while the building manager gains an additional income stream from providing the ancillary service, and can benefit further by pre-conditioning the building for later periods. The framework helps determine the amount of flexibility that should be offered for the service, and cost optimized profiles for electricity usage when delivering the service. Numerical results show that there can be an economic incentive to participate even if the payment rate for the ancillary service is less than the price of electricity.

math.OC↗
arXiv2017

Nonzero-sum games of optimal stopping and generalised Nash equilibrium

In the nonzero-sum setting, we establish a connection between Nash equilibria in games of optimal stopping (Dynkin games) and generalised Nash equilibrium problems (GNEP). In the Dynkin game this reveals novel equilibria of threshold type and of more complex types, and leads to novel uniqueness and stability results.

math.PR↗
arXiv2016

Optimal Entry to an Irreversible Investment Plan with Non Convex Costs

A problem of optimally purchasing electricity at a real-valued spot price (that is, with potentially negative cost) has been recently addressed in De Angelis, Ferrari and Moriarty (2015) [SIAM J. Control Optim. 53(3)]. This problem can be considered one of irreversible investment with a cost functional which is non convex with respect to the control variable. In this paper we study the optimal entry into this investment plan. The optimal entry policy can have an irregular boundary arising from this non convexity, with a kinked shape.

math.OC↗
arXiv2016

Solving finite time horizon Dynkin games by optimal switching

This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black-Scholes market.

math.OC↗
arXiv2015

Finite-horizon optimal multiple switching with signed switching costs

This paper is concerned with optimal switching over multiple modes in continuous time and on a finite horizon. The performance index includes a running reward, terminal reward and switching costs that can belong to a large class of stochastic processes. Particularly, the switching costs are modelled by right-continuous with left-limits processes that are quasi-left-continuous and can take both positive and negative values. We provide sufficient conditions leading to a well known probabilistic representation of the value function for the switching problem in terms of interconnected Snell envelopes. We also prove the existence of an optimal strategy within a suitable class of admissible controls, defined iteratively in terms of the Snell envelope processes.

math.OC↗
arXiv2015

Dynamic programming for discrete-time finite horizon optimal switching problems with negative switching costs

This paper studies a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward and signed (positive and negative) switching costs. Using the martingale approach to optimal stopping problems, we extend a well known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs.

math.OC↗