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Stan Zachary

Publications and source records attributed to Stan Zachary.

At least 19 recordsLinked to original sources
arXiv2022

Minimax decision rules for planning under uncertainty

It is common to use minimax rules to make decisions for planning when there is great uncertainty on what will happen in the future. Minimax regret is one popular version of this. We give an analysis of the behaviour of minimax rules in the case with a finite set of possible future scenarios. The use of minimax rules avoids the need to determine probabilities for each scenario, which is an attractive feature in many public sector settings. However, minimax rules will have sensitivity to the choice of scenarios. In many cases using a minimax approach will mean the requirement for what may be regarded as arbitrary probabilities on scenarios is replaced by a similarly arbitrary choice of a very small number of specific scenarios. We investigate this phenomenon. When regret-based rules are used there are also problems arising since the independence of irrelevant alternatives property fails, which can lead to opportunities to game the process. Our analysis of these phenomena considers cases where the decision variables are chosen from a convex set in $R^n$, as well as cases with a finite set of decision choices.

math.OC↗
arXiv2021

Scheduling and dimensioning of heterogeneous energy stores, with applications to future GB storage needs

Future ``net-zero'' electricity systems in which all or most generation is renewable may require very high volumes of storage, provided jointly by a number of heterogeneous technologies, in order to manage the associated variability in the generation-demand balance. We consider the problems of scheduling and dimensioning such storage. We develop a value-function based approach to optimal scheduling, and show that, to a good approximation, the problem to be solved at each successive point in time reduces to a linear programme with a particularly simple solution. We show that approximately optimal scheduling may be achieved without the need for a running forecast of the future generation-demand balance. We examine the applicability of the above theory to future GB storage needs, and discuss how it may be used to enable the most economic dimensioning of such storage, with possible savings of tens of billions of pounds, relative to the use of a single technology.

math.OC↗
arXiv2021

The integration of variable generation and storage into electricity capacity markets

We show how to value both variable generation and energy storage to enable them to be integrated fairly and optimally into electricity capacity markets. We develop theory based on balancing expected energy unserved against costs of capacity procurement, and in which the optimal procurement is that necessary to meet an appropriate reliability standard. For conventional generation the theory reduces to that already in common use. Further the valuation of both variable generation and storage coincides with the traditional risk-based approach based on equivalent firm capacity. The determination of the equivalent firm capacity of storage requires particular care; this is due both to the flexibility with which storage added to an existing system may be scheduled, and also because, when any resource is added to an existing system, storage already within that system may be flexibly rescheduled. We illustrate the theory with an example based on the GB system.

math.OC↗
arXiv2020

Scheduling of energy storage

The increasing reliance on renewable energy generation means that storage may well play a much greater role in the balancing of future electricity systems. We show how heterogeneous stores, differing in capacity and rate constraints, may be optimally, or nearly optimally, scheduled to assist in such balancing, with the aim of minimising the total imbalance (unserved energy) over any given period of time. It further turns out that in many cases the optimal policies are such that the optimal decision at each point in time is independent of the future evolution of the supply-demand balance in the system, so that these policies remain optimal in a stochastic environment.

math.OC↗
arXiv2019

The Scenario Culture

Scenario Analysis is a risk assessment tool that aims to evaluate the impact of a small number of distinct plausible future scenarios. In this paper, we provide an overview of important aspects of Scenario Analysis including when it is appropriate, the design of scenarios, uncertainty and encouraging creativity. Each of these issues is discussed in the context of climate, energy and legal scenarios.

stat.OT↗
arXiv2019

Using extreme value theory for the estimation of risk metrics for capacity adequacy assessment

This paper investigates the use of extreme value theory for modelling the distribution of demand-net-of-wind for capacity adequacy assessment. Extreme value theory approaches are well-established and mathematically justified methods for estimating the tails of a distribution and so are ideally suited for problems in capacity adequacy, where normally only the tails of the relevant distributions are significant. The extreme value theory peaks over threshold approach is applied directly to observations of demand-net-of-wind, meaning that no assumption is needed about the nature of any dependence between demand and wind. The methodology is tested on data from Great Britain and compared to two alternative approaches: use of the empirical distribution of demand-net-of-wind and use of a model which assumes independence between demand and wind. Extreme value theory is shown to produce broadly similar estimates of risk metrics to the use of the above empirical distribution but with smaller sampling uncertainty. Estimates of risk metrics differ when the approach assuming independence is used, especially when data across different historical years are pooled, suggesting that assuming independence might result in the over- or under-estimation of risk metrics.

stat.AP↗
arXiv2019

Optimal scheduling of energy storage resources

It is likely that electricity storage will play a significant role in the balancing of future energy systems. A major challenge is then that of how to assess the contribution of storage to capacity adequacy, i.e. to the ability of such systems to meet demand. This requires an understanding of how to optimally schedule multiple storage facilities. The present paper studies this problem in the cases where the objective is the minimisation of expected energy unserved (EEU) and also a form of weighted EEU in which the unit cost of unserved energy is higher at higher levels of unmet demand. We also study how the contributions of individual stores may be identified for the purposes of their inclusion in electricity capacity markets.

math.OC↗
arXiv2016

Least worst regret analysis for decision making under uncertainty, with applications to future energy scenarios

Least worst regret (and sometimes minimax) analysis are often used for decision making whenever it is difficult, or inappropriate, to attach probabilities to possible future scenarios. We show that, for each of these two approaches and subject only to the convexity of the cost functions involved, it is always the case that there exist two "extreme" scenarios whose costs determine the outcome of the analysis in the sense we make clear. The results of either analysis are therefore particularly sensitive to the cost functions associated with these two scenarios, while being largely unaffected by those associated with the remainder. Great care is therefore required in applications to identify these scenarios and to consider their reasonableness. We also consider the relationship between the outcome of a least worst regret and a Bayesian analysis, particularly in the case where the regret functions associated with the scenarios largely differ from each other by shifts in their arguments, as is the case in many applications. We study in detail the problem of determining an appropriate level of electricity capacity procurement in Great Britain, where decisions must be made several years in advance, in spite of considerable uncertainty as to which of a number of future scenarios may occur, and where least worst regret analysis is currently used as the basis of decision making.

math.OC↗
arXiv2016

Impact of storage competition on energy markets

We study how storage, operating as a price maker within a market environment, may be optimally operated over an extended period of time. The optimality criterion may be the maximisation of the profit of the storage itself, where this profit results from the exploitation of the differences in market clearing prices at different times. Alternatively it may be the minimisation of the cost of generation, or the maximisation of consumer surplus or social welfare. In all cases there is calculated for each successive time-step the cost function measuring the total impact of whatever action is taken by the storage. The succession of such cost functions provides the information for the storage to determine how to behave over time, forming the basis of the appropriate optimisation problem. Further, optimal decision making, even over a very long or indefinite time period, usually depends on a knowledge of costs over a relatively short running time horizon -- for storage of electrical energy typically of the order of a day or so. We study particularly competition between multiple stores, where the objective of each store is to maximise its own income given the activities of the remainder. We show that, at the Cournot Nash equilibrium, multiple large stores collectively erode their own abilities to make profits: essentially each store attempts to increase its own profit over time by overcompeting at the expense of the remainder. We quantify this for linear price functions We give examples throughout based on Great Britain spot-price market data.

math.OC↗
arXiv2015

The optimal control of storage for arbitrage and buffering, with energy applications

We study the optimal control of storage which is used for both arbitrage and buffering against unexpected events, with particular applications to the control of energy systems in a stochastic and typically time-heterogeneous environment. Our philosophy is that of viewing the problem as being formally one of stochastic dynamic programming, but of using coupling arguments to provide good estimates of the costs of failing to provide necessary levels of buffering. The problem of control then reduces to that of the solution, dynamically in time, of a deterministic optimisation problem which must be periodically re-solved. We show that the optimal control then proceeds locally in time, in the sense that the optimal decision at each time $t$ depends only on a knowledge of the future costs and stochastic evolution of the system for a time horizon which typically extends only a little way beyond $t$. The approach is thus both computationally tractable and suitable for the management of systems over indefinitely extended periods of time. We develop also the associated strong Lagrangian theory (which may be used to assist in the optimal dimensioning of storage), and we provide characterisations of optimal control policies. We give examples based on Great Britain electricity price data.

math.OC↗
arXiv2015

Optimal control of storage incorporating market impact and with energy applications

Large scale electricity storage is set to play an increasingly important role in the management of future energy networks. A major aspect of the economics of such projects is captured in arbitrage, i.e. buying electricity when it is cheap and selling it when it is expensive. We consider a mathematical model which may account for nonlinear---and possibly stochastically evolving---cost functions, market impact, input and output rate constraints and both time-dependent and time-independent inefficiencies or losses in the storage process. We develop an algorithm which is maximally efficient in the sense that it incorporates the result that, at each point in time, the optimal management decision depends only a finite, and typically short, time horizon. We give examples related to the management of a real-world system. Finally we consider a model in which the associated costs evolve stochastically in time. Our results are formulated in a perfectly general setting which permits their application to other commodity storage problems.

math.OC↗
arXiv2014

Estimation of Joint Distribution of Demand and Available Renewables for Generation Adequacy Assessment

In recent years there has been a resurgence of interest in generation adequacy risk assessment, due to the need to include variable generation renewables within such calculations. This paper will describe new statistical approaches to estimating the joint distribution of demand and available VG capacity; this is required for the LOLE calculations used in many statutory adequacy studies, for example those of GB and PJM. The most popular estimation technique in the VG-integration literature is `hindcast', in which the historic joint distribution of demand and available VG is used as a predictive distribution. Through the use of bootstrap statistical analysis, this paper will show that due to extreme sparsity of data on times of high demand and low VG, hindcast results can suffer from sampling uncertainty to the extent that they have little practical meaning. An alternative estimation approach, in which a marginal distribution of available VG is rescaled according to demand level, is thus proposed. This reduces sampling uncertainty at the expense of the additional model structure assumption, and further provides a means of assessing the sensitivity of model outputs to the VG-demand relationship by varying the function of demand by which the marginal VG distribution is rescaled.

stat.AP↗
arXiv2014

A note on Veraverbeke's theorem

We give an elementary probabilistic proof of Veraverbeke's Theorem for the asymptotic distribution of the maximum of a random walk with negative drift and heavy-tailed increments. The proof gives insight into the principle that the maximum is in general attained through a single large jump.

math.PR↗
arXiv2014

Optimal control of storage for arbitrage, with applications to energy systems

We study the optimal control of storage which is used for arbitrage, i.e. for buying a commodity when it is cheap and selling it when it is expensive. Our particular concern is with the management of energy systems, although the results are generally applicable. We consider a model which may account for nonlinear cost functions, market impact, input and output rate constraints and inefficiencies or losses in the storage process. We develop an algorithm which is maximally efficient in then sense that it incorporates the result that, at each point in time, the optimal management decision depends only a finite, and typically short, time horizon. We give examples related to the management of a real-world system.

math.OC↗
arXiv2013

Capacity Value of Additional Generation: Probability Theory and Sampling Uncertainty

The concept of capacity value is widely used to quantify the contribution of additional generation (most notably renewables) within generation adequacy assessments. This paper surveys the existing probability theory of assessment of the capacity value of additional generation, and discusses the available statistical estimation methods for risk measures which depend on the joint distribution of demand and available additional capacity (with particular reference to wind). Preliminary results are presented on assessment of sampling uncertainty in hindcast LOLE and capacity value calculations, using bootstrap resampling. These results indicate strongly that, if the hindcast calculation is dominated by extremes of demand minus wind, there is very large sampling uncertainty in the results due to very limited historic experience of high demands coincident with poor wind resource. For meaningful calculations, some form of statistical smoothing will therefore be required in distribution estimation.

math.PR↗
arXiv2013

Stochastic sequences with a regenerative structure that may depend both on the future and on the past

Many regenerative arguments in stochastic processes use random times which are akin to stopping times, but which are determined by the future as well as the past behaviour of the process of interest. Such arguments based on "conditioning on the future" are usually developed in an ad-hoc way in the context of the application under consideration, thereby obscuring underlying structure. In this paper we give a simple, unified and more general treatment of such conditioning theory. We further give a number of novel applications to various particle system models, in particular to various flavours of contact processes and to infinite-bin models. We give a number of new results for existing and new models. We further make connections with the theory of Harris ergodicity.

math.PR↗