Methodological Models for Optimal Control of Marine Oil Spill

ABSTRACT

The frequency of accidental discharge of oil into aquatic environment has presented a significant threat to marine biota with related adverse effects on the supply of products and services of importance to human cultures. This threat of economic and environmental devastation led to the development of a number of monitoring and clean-up alternatives such as remote sensing and chemical dispersants, respective. This study proposed a new possible research direction in marine oil spill modeling where the monitoring and clean-up alternatives would be optimized to enhance locations selection for the deployment of containment and combating technique evaluation before actual usage. A novel optimal control theory has been developed through operational research formalism as a critical first step in mitigating the problem of oil spill look-alike phenomenon associated with remote sensing, and the conflicting priorities in the application of chemical dispersants, which may be toxic to marine biota, during marine oil spill clean-up. Markovian decision processes with sequential optimization techniques were utilized in formulating the control-theoretic methodological models that would aid environmental managers in minimizing the uncertainty in the remote sensing data to reduce the high number of false alarms (oil slick look-a-likes) phenomena, minimizing the apparent toxicological effect of clean-up technique like chemical dispersants, determining the control measure that would cause a process to satisfy the physical constraints of chemical dispersants applications, and at the same time optimizing some performance criteria for all future earnings from marine biota. A dynamic model for a new strategy based on a diffusion process, which makes the distinction between two types of optimization objectives: increasing awareness and changing predisposition to adopt coherent pluralistic technique has been developed and the optimality condition for minimization of false alarm in marine oil spill detection was obtained as: ̇ ( ( )) ( ( )) , where   1 n t denote the regions within which the spill originated in the system,   2 n t denote other regions beyond with probable spill threat due to the spill diffusion process and is captured via remote sensing application, 3 n is the potential regions within 2 n after verification for disparity classification, is the decay variable and is time. Furthermore, a penalty cost for taking, at each time period, any decision following each possible signal was obtained as a decision rule:

TABLE OF CONTENTS

Title page – – – – – – – – – – i
Certification – – – – – – – – – – ii
Dedication – – – – – – – – – – iii
Acknowledgement – – – – – – – – – iv
Abstract – – – – – – – – – – v
Table of contents – – – – – – – – – vii
Chapter One: Introduction – – – – – – – – 1
1.0 Preamble – – – – – – – – – 1
1.1 Background of the Study – – – – – – – 1
1.2 Statement of the Problem – – – – – – – 3
1.3 Aim and Objectives of the Study – – – – – – 3
1.4 Significance of the Study – – – – – – – 4
1.5 Scope of the Study – – – – – – – – 4
1.6 Organization of the Thesis – – – – – – – 4
1.7 Research Methodology – – – – – – – 5
Chapter Two: Review of Related Literature – – – – – 7
2.0 Oil Spill Control Modelling – – – – – – – 7
2.1 Remote Sensing in Oil Spill Control – – – – – – 9
2.2 Conceptual Framework of Operational Research (OR) Techniques in Marine
Oil Spill Control – – – – – – – – 10
2.2.1 The Coherent Pluralism Technique – – – – – – 10
2.2.2 Statistical Methods as Complimentary Tool – – – – – 11
Chapter Three: The Methodological Models – – – – – 15
3.0 Methodological Model for Combination Methodology – – – 15
3.1 The Conceptual Model – – – – – – – 15
3.1.1 The Computational Procedure – – – – – – 17
3.1.2 Formulation of Analytical Scheme – – – – – – 21
3.1.3 Formulation of Analytical Scheme for Penalty Cost and Decision Rule – 33
3.2 Methodological Model for Sequential Optimization Technique in Marine oil Spill
Control – – – – – – – – – 35
3.2.1 Mathematical Preliminaries and Definition of Terms – – – 35
3.2.2 The Conceptual Model Oil Spill Transport – – – – – 38
3.2.3 Formulation of the Optimal Control Problem – – – – 40
3.2.4 The Necessary Conditions for Optimality – – – – – 42
3.2.5 Development of Sequential Optimization Processes for Optimal Response –
Strategy – – – – – – – – – 45
3.2.6 A Decision Rule for Sequential Optimization Processes – – – 47
3.2.7 Formulation of Optimal Costs Model for Marine Oil Spill Response – 49
Chapter Four: Optimal Control Theory for Marine Oil Spill Problem – – 54
4.0 Problem Statement – – – – – – – – 54
4.1 Formulation of the Optimal Control Model – – – – – 55
4.2 The Solution Module – – – – – – – – 57
4.3 Statistical Analogy – – – – – – – – 60
Chapter Five: Concluding Remarks – – – – – – 64
5.0 Conclusions – – – – – – – – – 64
5.1 Contribution to Knowledge – – – – – – – 64
References – – – – – – – – – – 65
Appendix – – – – – – – – – – 70
List of Publications from the Work – – – – – – 70

CHAPTER ONE

INTRODUCTION

1.0 PREAMBLE

Control theory is an interdisciplinary branch of operations research that deals with the behaviour of dynamical systems. while the term „operations research‟, quite often, is associated almost exclusively with the use of mathematical techniques to model and analyse decision problems, optimal control theory, as a distinct discipline in operations research, concerns the choice of disposition of a set of control instruments ( )to achieve a given objective function ( ), taking into account the dynamic properties of the system under consideration. it can be stated in general terms as the optimization of an objective function of system behaviour that uses a given set of control instruments over a specified period of time (see also, wiegerinck et al., 2006).

This area of research has witnessed remarkable progress over the past decades towards the development of methods for the stabilization and control of a host of linear and nonlinear systems problems (shastriet al., 2008). it is finding increased application in both biological and economic disciplines (williams and nichols, 1984; behncke, 2000; sethi and thompson, 2000;joshi, 2003), as well as theoretical and applied ecology (houston et al., 1988). according to runge and johnson (2002), it is also central in applied sciences and natural resources management.

1.1 background of the study

Degradation of aquatic ecosystem is generally agreed to be undesirable. historically, most evaluations of ecological effects of petroleum contamination have related impacts to effects on the supply of products and services of importance to human cultures. according to xu and pang (1992), most pollution control and environmental laws were enacted in order to protect public health and ecological objectives. in these laws, a substance is considered a pollutant if it has been perceived to have an adverse effects on human health and wildlife. in recent years, a Number of substances appear to pose such threats. among them is crude oil spillage which first came to public attention with the torrey canyon disaster in 1967 (pierce et al., 1997).

According to redondo and platonov (2009), the risk of crude oil spillage to the sea presents a major threat to the marine ecology compared with other sources of pollution in the oceans. before now, it was earlier reported that oil spills affect wildlife and their habitats in many ways, which include, the modification of the environmental conditions of marine ecology, which can be translated into the transformations of the chemical composition of the environment; alterations in its physical properties; destruction of the marine biomass nutritional capita; changes in the environmental biological equilibrium; and danger to human health (usfws, 2004). the same can also be said about Nigeria, where oil spillage is a major environmental problem and its coastal zone rated one of the most polluted spots on the earth planet in the year 2006 (brown, 2006). according to nwilo and olusegun (2007), from 1976 to 2007, over 1,896,960 barrels of oil were sunk into the Nigerian coastal waters resulting in a serious pollution of drinkable water, destruction of resort centres, properties and lives along the coastal zone. this was seen to be a major contributor to the regional crisis in the Nigeria Niger-Delta region (alao, 2008).

As a case in point, after a spill in the ocean, oil in water body, regardless of whether it originated as surface or subsurface spill, spreads to form a thin film called oil slick. according to yapa (1996), the movement of this slick is governed by the advection and turbulent diffusion due to current and wind action. the slick always spreads over the water surface due to a balance between gravitational, inertia, viscous and interfacial tension forces, while the composition of the oil changes from the initial time of the spill. it follows that light (low molecular weight) fractions will evaporate, water soluble components dissolve in the water column, and immiscible components become emulsified and disperse in the water column as small droplets.

In essence, the frequency of accidental oil spills in aquatic environments has presented a growing global concern and awareness of the risks of oil spills and the damage they do to the environment. however, it is widely known that oil exploration is a necessity in our industrial society, and a major sustainer of our lifestyle as most of the energy used in Canada and the united states, for instance, is for transportation that runs on oil and petroleum products. thus, following the trends in energy usage, this is not likely to decrease much in the future because industry uses oil and petroleum derivatives to manufacture such vital products as plastics, fertilizers, and chemical feedstock, which will still be required in the future. in what follows, it is a global belief that the production and consumption of oil and petroleum products might continue to increase worldwide while the threat of oil pollution is also likely to increase accordingly (esa, 1998).

Consequently, a fundamental problem in environmental research in recent time has been identified in the literature to be how properly to assess and control the spatial structure of pollution fields at various scales, and several studies showed that mathematical models were the only available tools for rapid computations and determinations of spilled oil fate, and for the simulation of the various clean-up operations. this study was carried out to sustain this objective.

1.2 statement of the problem

Specifically, the problem of oil slick look-a-likes phenomena associated with optical detectors (remote sensing), and the environmental impacts of some cleanup techniques (conflicting priorities) that makes chemical dispersant a limiting factor in the decision-making process has been a major snag in marine oil spill management since early 21st century (congalton, 1991; marghany et al., 1996; woodcock, 2002; tkalich et al., 2003; keramitsoglou et al., 2006; topouzelis, 2008; mansor et al., 2010). radar satellites was proposed by mansor et al. (2010) to replace the commonly used remote sensors while tkalich et al. (2003)posited that combating techniques should be carefully evaluated before actual usage in order to enhance oil transport and fates models. nevertheless, studies on the optimization of the remote sensor and chemical dispersants being the most widely and commonly used oil spills monitoring and clean-up alternatives, to the best of our knowledge, was conspicuously missing in the modern marine oil spill literature. this study was therefore motivated by the need to bridge this gap as an innovation in the marine literature.

1.3 aim and objectives of the study

The aim of this study is to formulate mathematical abstractions of the control processes where decisions would be taken at several stages through an optimal control path to achieve the following specific objectives:

1. minimizing the uncertainty in the remote sensing data to reduce the high number of false alarms (oil slick look-a-likes) phenomena,

2. minimizing the apparent toxicological effect of clean-up technique like chemical dispersants,

3. determining the control measure that would cause a process to satisfy the physical constraints of chemical dispersants applications, and at the same time

4. optimizing some performance criteria for all future earnings from marine biota.

1.4 significance of the study

Important attributes of marine oil spill, such as the nature of the spill, scale, type, time, etc., which are usually uncertain and more complex (Grigaluras and Opaluch, 1988), have been studied and applied in formulating decision rules in order to aid contingency planning in case of oil spill disaster, especially with regard to the possibility of using chemical dispersant and other techniques, in minimizing or preventing shoreline oiling which reduces oil spill clean-up costs. the mathematical expressions have also been developed to assist in optimal computation in order to enhance the sustainability of ecological values of aquatic biota for future earnings.

1.5 Scope of the Study

Generally, the problem of marine oil spillage has a wider dimension and varies in terms of environmental contents. this study focuses specifically on formulation of optimization processes for remote sensing data processing and chemical dispersants application procedure using or formalism, theoretical illustration of optimal control theory in a specific marine oil spill problem and statistical relevance of the optimal control theory in marine oil spill management.

1.6 Organization of the thesis

The subsequent section of this thesis is organized as follows:

In section 1.7, the research methodology used in this study is presented. chapter 2 focused on the review of related literature with specific attention to oil spill control modelling, the relevance of remote sensing, operational research techniques, and statistical methods.

Chapter 3 is devoted to the development of methodological models used in the course of this research work with section 3.1 and 3.2 centring on the mathematical formulations of the combination methodology and the sequential optimization processes. in section 3.1.2, the computational procedure of the combination methodology is presented, while the analysis scheme is given in section 3.1.3. the optimal control theory is presented in section 3.2.1, and in section 3.2.3 the oil spill dynamical model is given, while the necessary conditions for optimality are established in section 3.2.5. the oil spill response cost problem is also treated in section3.2.8. chapter 4 considered the formulation of optimal control theory for a typical oil spill problem. the optimal model is presented in section 4.2 with the statistical analogy given in section 4.3. the thesis is concluded in chapter 5 with useful recommendation, contribution to knowledge in section 5.1.

1.7 Research Methodology

Several studies showed that mathematical models are tools for rapid computations and determination of spilled oil, and for the simulation of the various clean-up operations (chigbu and bassey, 2010).in this work, the basic problem of marine oil spill was specified and formulated mathematically using operational research formalism. the methodological model for the optimization of the monitoring strategy in order to minimize false alarm phenomena was derived using coherent pluralism (jackson, 1999 and mingers, 2001), while sequential optimization processes were formulated to objectify the evaluation of combating technique application before actual usage, as proposed by tkalich et al. (2003).

The pluralistic method involved the use of statistical technique for disparity classification of information from remote signals based on prior information on the characteristics of the oil density, water content, viscosity, etcetera, while the sequential processes involved a closed-loop optimization based on finding an optimal rule for selecting at each period t, a control for each possible state value in a dynamic system.

In other word, a combination methodology that suggested a move past an analysis of standalone data to one incorporating and synthesizing information from many associated sources was used to tackle the problem of false alarm. in developing the optimization technique for remote sensing data, the outcomes of remote sensing application were consolidated through reanalyzing possible disparate results within the context of their common endpoints, increasing the sensitivity of the analysis to detect the presence of environmental or human effects, and providing a quantitative scheme for the phenomenon of interest based on the combined information.

Furthermore, a fundamental principle upon which oil spill fate and transport models are based was considered in the development of a sequential optimization technique. we considered the law of conservation of mass, derived from navier-stockes equations (tkalich et al., 2003):
(1)
Where ,
H = oil slick thickness;
C= oil concentration;
V = oil slick drifting velocity;
D= oil fluid velocity;
E = dispersion-diffusion coefficient;
 = computational slick spreading function;
H r and r = physical chemical kinetic terms;
U = grid size;
 = cartesian coordinate; and
T = time,

And introduced a control system into equation (1), which described a change in the Concentration of the spilled oil in the marine environment, resulting from the random motion Of oil slick in the transporting medium, to obtain a new dynamical model for optimal control.

In what followed, we set out the necessary conditions for optimality using the fundamental Theorem of variational calculus and partial differential equation (pde). thus, a dynamical Model that could cause the control system to follow a path such that the objective functional Of the marine environment could be optimized was formulated, and a sequential optimization Technique that followed a bellman‟s optimality principle (ji and zhou, 2006; crespo and sun,
2003) was employed to minimize clean-up operational constraint and enhanced contingency Planning.

This philosophy was illustrated theoretical by formulating methodological models for optimal Control of a typical oil spill situation with specific objective function a specific situation Where the region was assumed to be exposed to a serious oil spill effect that resulted in series of conflicts between the host community in the exploration region and the operating oil firms.

A statistical analogy of the methodological models has also been presented to envisage their Statistical relevance.