Multi-level, Multi-stage and Stochastic Optimization Models for Energy Conservation in Buildings for Federal, State and Local Agencies

Energy Conservation Measure (ECM) project selection is made difficult given real-world constraints, limited resources to implement savings retrofits, various suppliers in the market and project financing alternatives. Many of these energy efficient retrofit projects should be viewed as a series of investments with annual returns for these traditionally risk-averse agencies. Given a list of ECMs available, federal, state and local agencies must determine how to implement projects at lowest costs. The most common methods of implementation planning are suboptimal relative to cost. Federal, state and local agencies can obtain greater returns on their energy conservation investment over traditional methods, regardless of the implementing organization. This dissertation outlines several approaches to improve the traditional energy conservations models. Any public buildings in regions with similar energy conservation goals in the United States or internationally can also benefit greatly from this research. Additionally, many private owners of buildings are under mandates to conserve energy e.g., Local Law 85 of the New York City Energy Conservation Code requires any building, public or private, to meet the most current energy code for any alteration or renovation. Thus, both public and private stakeholders can benefit from this research. The research in this dissertation advances and presents models that decision-makers can use to optimize the selection of ECM projects with respect to the total cost of implementation. A practical application of a two-level mathematical program with equilibrium constraints (MPEC) improves the current best practice for agencies concerned with making the most cost-effective selection leveraging energy services companies or utilities. The two-level model maximizes savings to the agency and profit to the energy services companies (Chapter 2). An additional model presented leverages a single congressional appropriation to implement ECM projects (Chapter 3). Returns from implemented ECM projects are used to fund additional ECM projects. In these cases, fluctuations in energy costs and uncertainty in the estimated savings severely influence ECM project selection and the amount of the appropriation requested. A risk aversion method proposed imposes a minimum on the number of “of projects completed in each stage. A comparative method using Conditional Value at Risk is analyzed. Time consistency was addressed in this chapter. This work demonstrates how a risk-based, stochastic, multi-stage model with binary decision variables at each stage provides a much more accurate estimate for planning than the agency’s traditional approach and deterministic models. Finally, in Chapter 4, a rolling-horizon model allows for subadditivity and superadditivity of the energy savings to simulate interactive effects between ECM projects. The approach makes use of inequalities (McCormick, 1976) to re-express constraints that involve the product of binary variables with an exact linearization (related to the convex hull of those constraints). This model additionally shows the benefits of learning between stages while remaining consistent with the single congressional appropriations framework.

Improved stochastic optimization of railway timetables

We present a general model to find the best allocation of a limited amount of supplements (extra minutes added to a timetable in order to reduce delays) on a set of interfering railway lines. By the best allocation, we mean the solution under which the weighted sum of expected delays is minimal. Our aim is to finely adjust an already existing and well-functioning timetable. We model this inherently stochastic optimization problem by using two-stage recourse models from stochastic programming, building upon earlier research from the literature. We present an improved formulation, allowing for an efficient solution using a standard algorithm for recourse models. We show that our model may be solved using any of the following theoretical frameworks: linear programming, stochastic programming and convex non-linear programming, and present a comparison of these approaches based on a real-life case study. Finally, we introduce stochastic dependency into the model, and present a statistical technique to estimate the model parameters from empirical data.

Two-Stage Stochastic Optimization. An Application in the Third Sector

[en] It is known that most of the problems applied in the real life present uncertainty. In the rst part of the dissertation, basic concepts and properties of the Stochastic Programming have been introduced to the reader, also known as Optimization under Uncertainty. Moreover, since stochastic programs are complex to compute, we have presented some other models such as wait-and-wee, expected value and the expected result of using expected value. The expected value of perfect information and the value of stochastic solution measures quantify how worthy the Stochastic Programming is, with respect to the other models. In the second part, it has been designed and implemented with the modeller GAMS and the optimizer CPLEX an application that optimizes the distribution of non-perishable products, guaranteeing some nutritional requirements with minimum cost. It has been developed within Hazia project, managed by Sortarazi association and associated with Food Bank of Biscay and Basic Social Services of several districts of Biscay.

Model specification tests in nonparametric stochastic regression models

In this paper, we consider testing for additivity in a class of nonparametric stochastic regression models. Two test statistics are constructed and their asymptotic distributions are established. We also conduct a small sample study for one of the test statistics through a simulated example. (C) 2002 Elsevier Science (USA).

Spinning reserve and emission unit commitment through stochastic optimization

This paper proposes a stochastic mixed-integer linear approach to deal with a short-term unit commitment problem with uncertainty on a deregulated electricity market that includes day-ahead bidding and bilateral contracts. The proposed approach considers the typically operation constraints on the thermal units and a spinning reserve. The uncertainty is due to the electricity prices, which are modeled by a scenario set, allowing an acceptable computation. Moreover, emission allowances are considered in a manner to allow for the consideration of environmental constraints. A case study to illustrate the usefulness of the proposed approach is presented and an assessment of the cost for the spinning reserve is obtained by a comparison between the situation with and without spinning reserve.

Spinning reserve and emission unit commitment through stochastic optimization

This paper proposes a stochastic mixed-integer linear approach to deal with a short-term unit commitment problem with uncertainty on a deregulated electricity market that includes day-ahead bidding and bilateral contracts. The proposed approach considers the typically operation constraints on the thermal units and a spinning reserve. The uncertainty is due to the electricity prices, which are modeled by a scenario set, allowing an acceptable computation. Moreover, emission allowances are considered in a manner to allow for the consideration of environmental constraints. A case study to illustrate the usefulness of the proposed approach is presented and an assessment of the cost for the spinning reserve is obtained by a comparison between the situation with and without spinning reserve.

Extremum estimators and stochastic optimization methods

Submitted in partial fulfillment for the Requirements for the Degree of PhD in Mathematics, in the Speciality of Statistics in the Faculdade de Ciências e Tecnologia

Calibration of stochastic volatility models via second order approximation: the Heston model case

Using a suitable Hull and White type formula we develop a methodology to obtain asecond order approximation to the implied volatility for very short maturities. Using thisapproximation we accurately calibrate the full set of parameters of the Heston model. Oneof the reasons that makes our calibration for short maturities so accurate is that we alsotake into account the term-structure for large maturities. We may say that calibration isnot "memoryless", in the sense that the option's behavior far away from maturity doesinfluence calibration when the option gets close to expiration. Our results provide a wayto perform a quick calibration of a closed-form approximation to vanilla options that canthen be used to price exotic derivatives. The methodology is simple, accurate, fast, andit requires a minimal computational cost.

A general decomposition formula for derivative prices in stochastic volatility models

We see that the price of an european call option in a stochastic volatilityframework can be decomposed in the sum of four terms, which identifythe main features of the market that affect to option prices: the expectedfuture volatility, the correlation between the volatility and the noisedriving the stock prices, the market price of volatility risk and thedifference of the expected future volatility at different times. We alsostudy some applications of this decomposition.

On the problematic of reserving and stochastic loss models

Abstract Traditionally, the common reserving methods used by the non-life actuaries are based on the assumption that future claims are going to behave in the same way as they did in the past. There are two main sources of variability in the processus of development of the claims: the variability of the speed with which the claims are settled and the variability between the severity of the claims from different accident years. High changes in these processes will generate distortions in the estimation of the claims reserves. The main objective of this thesis is to provide an indicator which firstly identifies and quantifies these two influences and secondly to determine which model is adequate for a specific situation. Two stochastic models were analysed and the predictive distributions of the future claims were obtained. The main advantage of the stochastic models is that they provide measures of variability of the reserves estimates. The first model (PDM) combines one conjugate family Dirichlet - Multinomial with the Poisson distribution. The second model (NBDM) improves the first one by combining two conjugate families Poisson -Gamma (for distribution of the ultimate amounts) and Dirichlet Multinomial (for distribution of the incremental claims payments). It was found that the second model allows to find the speed variability in the reporting process and development of the claims severity as function of two above mentioned distributions' parameters. These are the shape parameter of the Gamma distribution and the Dirichlet parameter. Depending on the relation between them we can decide on the adequacy of the claims reserve estimation method. The parameters have been estimated by the Methods of Moments and Maximum Likelihood. The results were tested using chosen simulation data and then using real data originating from the three lines of business: Property/Casualty, General Liability, and Accident Insurance. These data include different developments and specificities. The outcome of the thesis shows that when the Dirichlet parameter is greater than the shape parameter of the Gamma, resulting in a model with positive correlation between the past and future claims payments, suggests the Chain-Ladder method as appropriate for the claims reserve estimation. In terms of claims reserves, if the cumulated payments are high the positive correlation will imply high expectations for the future payments resulting in high claims reserves estimates. The negative correlation appears when the Dirichlet parameter is lower than the shape parameter of the Gamma, meaning low expected future payments for the same high observed cumulated payments. This corresponds to the situation when claims are reported rapidly and fewer claims remain expected subsequently. The extreme case appears in the situation when all claims are reported at the same time leading to expectations for the future payments of zero or equal to the aggregated amount of the ultimate paid claims. For this latter case, the Chain-Ladder is not recommended.

Stochastic approximation methods in stochastic optimization

Stochastic approximation methods for stochastic optimization are considered. Reviewed the main methods of stochastic approximation: stochastic quasi-gradient algorithm, Kiefer-Wolfowitz algorithm and adaptive rules for them, simultaneous perturbation stochastic approximation (SPSA) algorithm. Suggested the model and the solution of the retailer's profit optimization problem and considered an application of the SQG-algorithm for the optimization problems with objective functions given in the form of ordinary differential equation.

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from fluid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heaviside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic differential equations are also studied. Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic differential equations describing the evolution of the prices of N traders to a deterministic partial differential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is confirmed by the ability of the model to reproduce price spikes when their effects occur in a sufficiently long period of time.

Aggregations and Marginalization of GARCH and Stochastic Volatility Models

The GARCH and Stochastic Volatility paradigms are often brought into conflict as two competitive views of the appropriate conditional variance concept : conditional variance given past values of the same series or conditional variance given a larger past information (including possibly unobservable state variables). The main thesis of this paper is that, since in general the econometrician has no idea about something like a structural level of disaggregation, a well-written volatility model should be specified in such a way that one is always allowed to reduce the information set without invalidating the model. To this respect, the debate between observable past information (in the GARCH spirit) versus unobservable conditioning information (in the state-space spirit) is irrelevant. In this paper, we stress a square-root autoregressive stochastic volatility (SR-SARV) model which remains true to the GARCH paradigm of ARMA dynamics for squared innovations but weakens the GARCH structure in order to obtain required robustness properties with respect to various kinds of aggregation. It is shown that the lack of robustness of the usual GARCH setting is due to two very restrictive assumptions : perfect linear correlation between squared innovations and conditional variance on the one hand and linear relationship between the conditional variance of the future conditional variance and the squared conditional variance on the other hand. By relaxing these assumptions, thanks to a state-space setting, we obtain aggregation results without renouncing to the conditional variance concept (and related leverage effects), as it is the case for the recently suggested weak GARCH model which gets aggregation results by replacing conditional expectations by linear projections on symmetric past innovations. Moreover, unlike the weak GARCH literature, we are able to define multivariate models, including higher order dynamics and risk premiums (in the spirit of GARCH (p,p) and GARCH in mean) and to derive conditional moment restrictions well suited for statistical inference. Finally, we are able to characterize the exact relationships between our SR-SARV models (including higher order dynamics, leverage effect and in-mean effect), usual GARCH models and continuous time stochastic volatility models, so that previous results about aggregation of weak GARCH and continuous time GARCH modeling can be recovered in our framework.

Analysis of Some Stochastic Inventory Models with Pooling Retrial of Customers

The thesis deals with analysis of some Stochastic Inventory Models with Pooling/Retrial of Customers.. In the first model we analyze an (s,S) production Inventory system with retrial of customers. Arrival of customers from outside the system form a Poisson process. The inter production times are exponentially distributed with parameter µ. When inventory level reaches zero further arriving demands are sent to the orbit which has capacity M(<∞). Customers, who find the orbit full and inventory level at zero are lost to the system. Demands arising from the orbital customers are exponentially distributed with parameter γ. In the model-II we extend these results to perishable inventory system assuming that the life-time of each item follows exponential with parameter θ. The study deals with an (s,S) production inventory with service times and retrial of unsatisfied customers. Primary demands occur according to a Markovian Arrival Process(MAP). Consider an (s,S)-retrial inventory with service time in which primary demands occur according to a Batch Markovian Arrival Process (BMAP). The inventory is controlled by the (s,S) policy and (s,S) inventory system with service time. Primary demands occur according to Poissson process with parameter λ. The study concentrates two models. In the first model we analyze an (s,S) Inventory system with postponed demands where arrivals of demands form a Poisson process. In the second model, we extend our results to perishable inventory system assuming that the life-time of each item follows exponential distribution with parameter θ. Also it is assumed that when inventory level is zero the arriving demands choose to enter the pool with probability β and with complementary probability (1- β) it is lost for ever. Finally it analyze an (s,S) production inventory system with switching time. A lot of work is reported under the assumption that the switching time is negligible but this is not the case for several real life situation.