A Node Formulation for Multistage Stochastic Programs with Endogenous Uncertainty

This paper introduces a node formulation for multistage stochastic programs with endogenous (i.e., decision-dependent) uncertainty. Problems with such structure arise when the choices of the decision maker determine a change in the likelihood of future random events. The node formulation avoids an explicit statement of non-anticipativity constraints, and as such keeps the dimension of the model sizeable. An exact solution algorithm for a special case is introduced and tested on a case study. Results show that the algorithm outperforms a commercial solver as the size of the instances increases

Generation Expansion Planning with Large Amounts of Wind Power via Decision-Dependent Stochastic Programming

Power generation expansion planning needs to deal with future uncertainties carefully, given that the invested generation assets will be in operation for a long time. Many stochastic programming models have been proposed to tackle this challenge. However, most previous works assume predetermined future uncertainties (i.e., fixed random outcomes with given probabilities). In several recent studies of generation assets\u27 planning (e.g., thermal versus renewable), new findings show that the investment decisions could affect the future uncertainties as well. To this end, this paper proposes a multistage decision-dependent stochastic optimization model for long-term large-scale generation expansion planning, where large amounts of wind power are involved. In the decision-dependent model, the future uncertainties are not only affecting but also affected by the current decisions. In particular, the probability distribution function is determined by not only input parameters but also decision variables. To deal with the nonlinear constraints in our model, a quasi-exact solution approach is then introduced to reformulate the multistage stochastic investment model to a mixed-integer linear programming model. The wind penetration, investment decisions, and the optimality of the decision-dependent model are evaluated in a series of multistage case studies. The results show that the proposed decision-dependent model provides effective optimization solutions for long-term generation expansion planning

Optimization Approaches for Electricity Generation Expansion Planning Under Uncertainty

In this dissertation, we study the long-term electricity infrastructure investment planning problems in the electrical power system. These long-term capacity expansion planning problems aim at making the most effective and efficient investment decisions on both thermal and wind power generation units. One of our research focuses are uncertainty modeling in these long-term decision-making problems in power systems, because power systems\u27 infrastructures require a large amount of investments, and need to stay in operation for a long time and accommodate many different scenarios in the future. The uncertainties we are addressing in this dissertation mainly include demands, electricity prices, investment and maintenance costs of power generation units. To address these future uncertainties in the decision-making process, this dissertation adopts two different optimization approaches: decision-dependent stochastic programming and adaptive robust optimization. In the decision-dependent stochastic programming approach, we consider the electricity prices and generation units\u27 investment and maintenance costs being endogenous uncertainties, and then design probability distribution functions of decision variables and input parameters based on well-established econometric theories, such as the discrete-choice theory and the economy-of-scale mechanism. In the adaptive robust optimization approach, we focus on finding the multistage adaptive robust solutions using affine policies while considering uncertain intervals of future demands. This dissertation mainly includes three research projects. The study of each project consists of two main parts, the formulation of its mathematical model and the development of solution algorithms for the model. This first problem concerns a large-scale investment problem on both thermal and wind power generation from an integrated angle without modeling all operational details. In this problem, we take a multistage decision-dependent stochastic programming approach while assuming uncertain electricity prices. We use a quasi-exact solution approach to solve this multistage stochastic nonlinear program. Numerical results show both computational efficient of the solutions approach and benefits of using our decision-dependent model over traditional stochastic programming models. The second problem concerns the long-term investment planning with detailed models of real-time operations. We also take a multistage decision-dependent stochastic programming approach to address endogenous uncertainties such as generation units\u27 investment and maintenance costs. However, the detailed modeling of operations makes the problem a bilevel optimization problem. We then transform it to a Mathematic Program with Equilibrium Constraints (MPEC) problem. We design an efficient algorithm based on Dantzig-Wolfe decomposition to solve this multistage stochastic MPEC problem. The last problem concerns a multistage adaptive investment planning problem while considering uncertain future demand at various locations. To solve this multi-level optimization problem, we take advantage of affine policies to transform it to a single-level optimization problem. Our numerical examples show the benefits of using this multistage adaptive robust planning model over both traditional stochastic programming and single-level robust optimization approaches. Based on numerical studies in the three projects, we conclude that our approaches provide effective and efficient modeling and computational tools for advanced power systems\u27 expansion planning

Tehokas ratkaisumenetelmä stokastisiin optimointiongelmiin endogeenisillä ja eksogeenisillä epävarmuuksilla

Despite multi-stage decision problems being common in production planning, there is a class of such problems for which a general solution framework does not exist, namely problems with endogenous uncertainty. Methods from decision analysis and stochastic programming can be used, but both require significantly constraining assumptions. In order to overcome the current challenges, Decision Programming combines approaches from these two fields, making it possible to acquire optimal strategies for different decision problems. Decision Programming is strictly limited to problems in which uncertainty events and decisions are taken from a finite discrete set, reducing its applicability to problems with continuous decision spaces. Discretizing a continuous decision space increases the problem size and can lead to computational intractability. This thesis presents a problem decomposition approach extending the Decision Programming framework. The decomposition approach allows for considering continuous decision and uncertainty spaces in problems with a suitable structure. The proposed framework was applied to three different problems, including a large-scale production planning problem from literature. The main example in this thesis is a novel approach on climate change mitigation cost-benefit analysis, where R&D is carried out simultaneously with the emissions abatement decisions. The R&D projects provide information on the climate damage severity and decrease the price of abatement. Problems with similar structure have not been discussed in the literature, and the extended Decision Programming framework is able to solve the problem to optimality.Vaikka monivaiheiset päätöksenteko-ongelmat ovat yleisiä tuotannon suunnittelussa, erääseen ryhmään näitä ongelmia ei ole yleistä ratkaisumenetelmää. Tämä johtuu niinsanotusta endogeenisestä epävarmuudesta. Näihin ongelmiin voidaan soveltaa stokastisen optimoinnin ja päätösanalyysin menetelmiä, mutta kummatkin vaativat merkittäviä rajoittavia oletuksia. Uusi menetelmä, Decision Programming, yhdistää stokastisen optimoinnin ja päätösanalyysin menetelmiä mahdollistaen optimistrategioden löytämisen erilaisissa päätösongelmissa. Decision Programming rajoittuu ongelmiin joissa satunnaistapahtumat ja päätökset valitaan äärellisistä diskreeteistä joukoista. Tämä rajoittaa sen soveltuvuutta ongelmiin joissa päätösjoukot ovat jatkuvia, sillä tällaisen päätösjoukon diskretointi kasvattaa ongelman kokoa ja saattaa johtaa laskennallisiin haasteisiin. Tässä työssä esitellään Decision Programming -viitekehystä laajentava hajotusmenetelmä jonka avulla voidaan ratkaista ongelmia, jotka sisältävät jatkuvia päätös- ja epävarmuusjoukkoja. Menetelmän soveltaminen vaatii kuitenkin ongelmalta sopivan rakenteen. Työssä esitettyä menetelmää sovellettiin kolmeen esimerkkiongelmaan, joista yksi on suuren mittakaavan tuotannonsuunnitteluongelma kirjallisuudesta. Työn pääesimerkki on uudenlainen lähestymistapa ilmastonmuutoksen hillinnän kustannus-hyötyanalyysiin, jossa tutkimustyötä tehdään samanaikaisesti päästövähennysten kanssa. Tutkimusprojekteilla saadaan lisätietoa ilmastovaikutusten vakavuudesta ja lasketaan päästövähennysten hintaa. Vastaavanlaisia ongelmia ei ole ennen käsitelty kirjallisuudessa ja laajennettu Decision Programming -viitekehys mahdollistaa optimiratkaisun löytämisen tässä esimerkissä

Adaptive Multistage Stochastic Programming

Multistage stochastic programming is a powerful tool allowing decision-makers to revise their decisions at each stage based on the realized uncertainty. However, in practice, organizations are not able to be fully flexible, as decisions cannot be revised too frequently due to their high organizational impact. Consequently, decision commitment becomes crucial to ensure that initially made decisions remain unchanged for a certain period. This paper introduces adaptive multistage stochastic programming, a new optimization paradigm that strikes an optimal balance between decision flexibility and commitment by determining the best stages to revise decisions depending on the allowed level of flexibility. We introduce a novel mathematical formulation and theoretical properties eliminating certain constraint sets. Furthermore, we develop a decomposition method that effectively handles mixed-integer adaptive multistage programs by adapting the integer L-shaped method and Benders decomposition. Computational experiments on stochastic lot-sizing and generation expansion planning problems show substantial advantages attained through optimal selections of revision times when flexibility is limited, while demonstrating computational efficiency of the proposed properties and solution methodology. Optimizing revision times in a less flexible case can outperform arbitrary selection in a more flexible case. By adhering to these optimal revision times, organizations can achieve performance levels comparable to fully flexible settings

Adaptive Multistage Stochastic Programming

Multistage stochastic programming is a powerful tool allowing decision-makers to revise their decisions at each stage based on the realized uncertainty. However, in practice, organizations are not able to be fully flexible, as decisions cannot be revised too frequently due to their high organizational impact. Consequently, decision commitment becomes crucial to ensure that initially made decisions remain unchanged for a certain period. This paper introduces adaptive multistage stochastic programming, a new optimization paradigm that strikes an optimal balance between decision flexibility and commitment by determining the best stages to revise decisions depending on the allowed level of flexibility. We introduce a novel mathematical formulation and theoretical properties eliminating certain constraint sets. Furthermore, we develop a decomposition method that effectively handles mixed-integer adaptive multistage programs by adapting the integer L-shaped method and Benders decomposition. Computational experiments on stochastic lot-sizing and generation expansion planning problems show substantial advantages attained through optimal selections of revision times when flexibility is limited, while demonstrating computational efficiency of the proposed properties and solution methodology. Optimizing revision times in a less flexible case can outperform arbitrary selection in a more flexible case. By adhering to these optimal revision times, organizations can achieve performance levels comparable to fully flexible settings

A decomposition strategy for decision problems with endogenous uncertainty using mixed-integer programming

Despite methodological advances for modeling decision problems under uncertainty, faithfully representing endogenous uncertainty still proves challenging, both in terms of modeling capabilities and computational requirements. A novel framework called Decision Programming provides an approach for solving such decision problems using off-the-shelf mathematical optimization solvers. This is made possible by using influence diagrams to represent a given decision problem, which is then formulated as a mixed-integer linear programming problem. In this paper, we focus on the type of endogenous uncertainty that received less attention in the introduction of Decision Programming: conditionally observed information. Multi-stage stochastic programming (MSSP) models use conditional non-anticipativity constraints (C-NACs) to represent such uncertainties, and we show how such constraints can be incorporated into Decision Programming models. This allows us to consider the two main types of endogenous uncertainty simultaneously, namely decision-dependent information structure and decision-dependent probability distribution. Additionally, we present a decomposition approach that provides significant computational savings and also enables considering continuous decision variables in certain parts of the problem, whereas the original formulation was restricted to discrete variables only. The extended framework is illustrated with two example problems. The first considers an illustrative multiperiod game and the second is a large-scale cost-benefit problem regarding climate change mitigation. Neither of these example problems could be solved with existing frameworks.Comment: 26 pages, 10 figure