No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Developing a Two-Stage Stochastic Programming Model for CO2 Disposal Planning under Uncertainty
Abstract
It is difficult to clearly estimate CO2 emissions because CO2 is emitted from various sources according to changing environments. Any approach relating to CO2 disposal has to deal with the presence of such uncertainty in CO2 disposal demand. A two-stage stochastic programming model is developed for planning CO2 emissions disposal networks by taking into account this uncertainty. The proposed CO2 disposal network model allows us to determine where and how much captured CO2 can be held for storage, and where to sequester the given amount of CO2 among multiple potential candidates for the purpose of minimizing the total cost of handling in the face of uncertainty in CO2 sequestration demand. The proposed model is applied to a case study that examines the robustness of the model in terms of handling changing environments in the context of CO2 emissions and disposal targets in Korea. The results gained aid in determining policy to plan the budget for the disposal of CO2.
... Different factors influence reliability of carbon supply chains planning. Regarding the storage site, uncertainties are related to permeability, capacity and porosity of reservoirs [27][28][29]. In addition to these, uncertainties are present inside a supply chain due to the fluctuation of carbon dioxide sources, variability of construction and operation costs, capture technology and unpredictable events [29][30][31]. ...
... Regarding the storage site, uncertainties are related to permeability, capacity and porosity of reservoirs [27][28][29]. In addition to these, uncertainties are present inside a supply chain due to the fluctuation of carbon dioxide sources, variability of construction and operation costs, capture technology and unpredictable events [29][30][31]. In addition, external factors, such as carbon policy, technology, engineering performance and market forces can influence the design of these systems [32]. ...
This paper develops a two-stage stochastic mixed integer linear programming model to optimize Carbon Capture, Utilization and Storage (CCUS) supply chains in Italy, Germany and the UK. Few works are present in the literature about this topic, thus this paper overcomes this limitation considering carbon supply chains producing different products. The objective of the numerical models is to minimize expected total costs, under the uncertainties of the production costs of carbon-dioxide-based compounds. Once carbon dioxide emissions that should be avoided are fixed, according to environmental protection requirements for each country, the optimal design of these supply chains is obtained finding the distribution of carbon dioxide captured between utilization and storage sections, the amount of different carbon-based products and the best connection between each element inside the system. The expected total costs for the CCUS supply chain of Italy, Germany and the UK are, respectively, 77.3, 98.0 and 1.05 billion€/year (1004, 613 and 164 €/ton CO2 captured). A comparison with the respective deterministic model, analyzed elsewhere, is considered through the evaluation of the Expected Value of Perfect Information (EVPI) and the Value of Stochastic Solution (VSS). The former is 1.29 billion€/year, 0.18 million€/year and 8.31 billion€/year, respectively, for the CCUS of Italy, the UK and Germany. VSS on the other hand is equal to 1.56 billion€/year, 0 €/year and 0.1 billion€/year, respectively, for the frameworks of Italy, the UK and Germany. The results show that the uncertain production cost in the stochastic model does not have a significant effect on the results; thus, in this case, there are few advantages in solving a stochastic model instead of the deterministic one.
... Two-stage stochastic optimization models are used in a wide array of transportation planning problems, including disaster response Barbarosoǧlu and Arda (2004), supply chain management Marufuzzaman et al. (2014), pavement maintenance and rehabilitation Ameri et al. (2019), and CO 2 disposal planning Han et al. (2012). Capacity, supply, demand, technology advancement, and budget represent some of the widely used stochastic parameters in such studies. ...
This paper investigates inland port infrastructure investment planning under uncertain commodity (such as coal, petroleum, manufactured products, nonmetallic minerals) demand conditions. A two-stage stochastic optimization is developed to model the impact of demand uncertainty on infrastructure planning and transportation decisions. The model minimizes expected total costs, including capacity expansion costs, associated with handling equipment and storage infrastructure, and the expected transportation costs. To solve the problem, an accelerated Benders decomposition algorithm is implemented. The use of a stochastic approach is justified by comparing the value of stochastic solution with its corresponding deterministic solution. For demonstration, the model is applied to the Arkansas section of the McClellan-Kerr Arkansas River Navigation System (MKARNS). Given data availability, the model is generalizable to other regions. Results show that as investment in port capacities (handling equipment and storage infrastructure) increases by 21 M per year. The model serves as a decision-making tool for optimal, distributed allocation of monetary investments, that encourages mode shift to inland waterways.
... Two-stage stochastic optimization models are used in a wide array of transportation planning problems, including disaster response Barbarosoǧlu and Arda (2004), supply chain management Marufuzzaman et al. (2014), pavement maintenance and rehabilitation Ameri et al. (2019), and CO 2 disposal planning Han et al. (2012). Capacity, supply, demand, technology advancement, and budget represent some of the widely used stochastic parameters in such studies. ...
This paper investigates inland port infrastructure investment planning under uncertain commodity demand conditions. A two-stage stochastic optimization is developed to model the impact of demand uncertainty on infrastructure planning and transportation decisions. The two-stage stochastic model minimizes the total expected costs, including the capacity expansion investment costs associated with handling equipment and storage, and the expected transportation costs. To solve the problem, an accelerated Benders decomposition algorithm is implemented. The Arkansas section of the McCllean-Kerr Arkansas River Navigation System (MKARNS) is used as a testing ground for the model. Results show that commodity volume and, as expected, the percent of that volume that moves via waterways (in ton-miles) increases with increasing investment in port infrastructure. The model is able to identify a cluster of ports that should receive investment in port capacity under any investment scenario. The use of a stochastic approach is justified by calculating the value of the stochastic solution (VSS).
... To overcome this, a scenario-based stochastic approach (Birge and Louveaux, 2011;Han et al., 2012) was adopted to consider the uncertainty in energy requirements of the EAF process and electricity price. e developed model considers the uncertainty of the Köhle thermodynamic parameters, including the total charged scrap, alloys, and slag formers, tapping temperature, power on/o time, consumed natural gas, oxygen injection, and added oxygen for postcombustion. ...
A stochastic multi-objective mathematical model based on material and energy flows under the system uncertainty of specific electricity demand-related scraps characteristics and operation conditions was developed to simulate the electric arc furnace steelmaking process and optimize carbon dioxide emission and cost. Considering several energy-saving technologies based on stochastic thermodynamic energy efficiency and electricity price, this paper suggests an optimal cost-saving and carbon-dioxide-emission-reducing strategy. The suggested model provides a trade-off relationship between cost and CO2 emission. To minimize CO2 emissions, a tunnel preheater without additional fuel consumption was suggest In contrast, to minimize the cost of utilizing cost-effective fossil fuels instead of electricity as the system energy requirement, an oxy-fuel burner and shaft furnace type of preheater were proposed. This problem was formulated as a mixed-integer linear programming model.
... For model K, the error data of 37 years were clustered into 9 clusters and the cluster centers were used as scenario values. For model U, the maximum scenario was set to be 20% larger than the deterministic value and the minimum scenario was set to be 20% smaller than the deterministic value, which is frequently used in other uncertainty studies (Han et al., 2012). ...
To investigate the effect of uncertainty in biodiesel production from microalgae, a supply chain network (SCN) of the process from cultivation of microalgae to distribution of biodiesel is developed using mixed-integer linear programming. Biodiesel demandis the crucial factor in the design of SCN, but is quite uncertain, so a stochastic approach that considers uncertain scenarios is developed, and its recommendations are compared to those of a deterministic model. Also, three different scenario-generation methodologies are employed to illustrate the applicability of the approach. The proposed model determines: (1) the numbers, locations, and sizes of a carbon capture and storage system; (2) the numbers, locations, sizes, and types of bio-refineries; (3) the transportation paths of CO2 and water from feedstock fields to bio-refineries; and (4) the transportation paths of biodiesel from bio-refinery to demand cities, while minimizing the expected total cost considering several constraints such as locations of power plants as carbon sources, potential locations of bio-refineries, and demands for biodiesel at each site. The proposed model is validated by applying it to a case study based on the predicted biodiesel demand of Korea in the year 2030. A numerical example illustrates that the unit production cost of alga-derived biodiesel by the stochastic model (US 2.99 per liter). The proposed approach is able to respond to uncertain demand situations in SCN design.
... Remarkable contributions have been done surrounding the optimal resources planning and one of the achievements is twostage programming (TSP), which is capable of modifying the predetermined targets (called first-stage decision) based on the overall influence of uncertain events and generating corresponding decision (called second-stage decision) after the uncertain events happen. Resources planning optimization research on TSP covers all walks of life, ranging from agriculture (Fu et al., 2018;Zhang et al., 2017), hydrology Hu et al., 2016;Yu et al., 2016), environment (Han et al., 2012;Han et al., 2013;Han and LEE, 2011), energy (Yun et al., 2017), and transportation (Barbarosoglu and Arda, 2004) to medicine (Dillon et al., 2017), manufacture (Alfieri et al., 2012), and networks (Wu and Kucukyavuz, 2018). A highlighted part in TSP is penalty quantification caused by uncertain events, which is usually expressed as benefits loss from resources deficiency according to previous studies. ...
... An interval-fuzzy two-stage stochastic programming model for planning carbon dioxide trading under uncertainty . Energy and environmental systems planning under uncertainty-An inexact fuzzy-stochastic programming approach (Han et al. 2012). A two-stage inexact-stochastic programming model for planning carbon dioxide emission trading under uncertainty . ...
This book comes out from the materials I used to refer while doing my research on the optimization issues in logistics. I brought together some of these materials to form a guidance material on the fundamentals of the optimization concepts along with my own studies on the application of optimization methods. This book consists of two parts and six chapters. The first part of the book, which consists of three chapters, is about introduction to optimization with typical base problems and algorithms for solving problems. The second part of this book consists of three my own researches on the application of optimization methods. Each chapter of this book is independent of each other. I hope you will find this book useful, informative, beneficial and appropriate for your needs. Turkay Yildiz, M.A., Ph.D.
... ). Two-Stage Stochastic Programming Model for Planning CO2 Utilization and Disposal Infrastructure Considering the Uncertainty in the CO2 Emission (Han and Lee 2011). Developing a Two-Stage Stochastic Programming Model for CO2 Disposal Planning under Uncertainty(Han et al. 2012). A two-stage inexact-stochastic programming model for planning carbon dioxide emission trading under uncertainty. ...
Transportation and logistics systems are characterized by their highly dynamic structures along with numerous interconnected processes. The natures of these systems involve various levels of resource allocation decisions where usually it is not always possible to execute these decisions in the field on time at the best possible way because of the unpredictable factors in plans. By considering the uncertain operational environment, this study explores the uncertainty issue within operational systems and deals with the problem of allocating resources to maximize expected total profit and minimize inefficiencies under uncertainty. The aim is to design, develop, visualize, and effectively deal with a more realistic model to satisfy uncertain demand nodes by leaving minimal or none unsatisfied zones within an operational environment at seaports, transportation, logistics and supply chain systems. A representative optimization model, which is developed to address the uncertainty issue, has been solved by using an optimization algorithm. The results show that operational plans without the utilization of uncertainty models could have negative impacts, including increased emissions, negative environmental effects, along with higher costs to organizations.
We present a stochastic decision-making algorithm for the design and operation of a carbon capture and storage (CCS) network; the algorithm incorporates the decision-maker’s tolerance of risk caused by uncertainties. Given a set of available resources to capture, store, and transport CO2, the algorithm provides an optimal plan of the CCS infrastructure and a CCS assessment method, while minimizing annual cost, environmental impact, and risk under uncertainties. The model uses the concept of downside risk to explicitly incorporate the trade-off between risk and either economic or environmental objectives at the decision-making level. A two-phase-two-stage stochastic multi-objective optimization problem (2P2SSMOOP) solving approach is implemented to consider uncertainty, and the ε-constraint method is used to evaluate the interaction between total annual cost with financial risk and an Eco-indicator 99 score with environmental risk. The environmental impact is measured by Life Cycle Assessment (LCA) considering all contributions made by operation and installation of a CCS infrastructure. A case study of power-plant CO2 emission in Korea is presented to illustrate the application of the proposed modeling and solution method.
We develop an economic process design for separation of CO2 from the off-gas in ISMPs (iron and steel making plants). Based on the characteristics of the off-gas from which CO2 must be separated, we design two process configurations: PSA (pressure-swing adsorption) and MEA (monoethanolamine)-based chemical absorption. We also develop a simulation model of each process, and perform an economic evaluation of the configurations. Our technical performance analyses show that the CO2 recovery and purity are >90% in the both processes and that highly-concentrated combustible gas (CO and H2) can be obtained as a byproduct. Our economic performance analyses show that the designed processes lead to cost-effective and competitive options ($62/t CO2 separated), compared to the CO2 separation processes used in power plants. Using the combustible gas as a fuel for the boiler of power cycle greatly reduces the cost of CO2 separation in ISMPs.
In this study, a comprehensive infrastructure assessment model for carbon capture and storage (CiamCCS) is developed for (i) planning a carbon capture and storage (CCS) infrastructure that includes CO2 capture, utilization, sequestration and transportation technologies, and for (ii) integrating the major CCS assessment methods, i.e., techno-economic assessment (TEA), environmental assessment (EA), and technical risk assessment (TRA). The model also applies an inexact two-stage stochastic programming approach to consider the effect of every possible uncertainty in input data, including economic profit (i.e., CO2 emission inventories, product prices, operating costs), environmental impact (i.e., environment emission inventories) and technical loss (i.e., technical accident inventories). The proposed model determines where and how much CO2 to capture, transport, sequester, and utilize to achieve an acceptable compromise between profit and the combination of environmental impact and technical loss. To implement this concept, fuzzy multiple objective programming was used to attain a compromise solution among all objectives of the CiamCCS. The capability of CiamCSS is tested by applying it to design and operate a future CCS infrastructure for treating CO2 emitted by burning carbon-based fossil fuels in power plants throughout Korea in 2020. The result helps decision makers to establish an optimal strategy that balances economy, environment, and safety efficiency against stability in an uncertain future CCS infrastructure.
A carbon capture and storage (CCS) plays a very important role to reduce dramatically in emission sources which are distributed throughout various areas. Numerous research works have been undertaken to analyze the techno-economic feasibility of planning the CCS infrastructure. However, uncertainties such as emissions, reduction costs, and carbon taxes may exist in various impact factors of the CCS infrastructure. However, few research works have adopted these uncertainties in designing the CCS infrastructure. In this study, a two-stage stochastic programming model is developed for planning the CCS infrastructure under uncertain operating costs and carbon taxes. It can help determine where and how much to capture, store or transport for the purpose of minimizing the total annual reduction cost in handling the uncertainties while meeting the mitigation target. The capability of the proposed model to provide correct decisions despite changing the operating costs and carbon taxes is tested by applying it to a real case study based on Korea. The results will help to determine planning of a CCS infrastructure under uncertain environments.
To effectively reduce CO2, CO2 mitigation technologies should be employed tactically. This paper focuses on carbon capture and storage (CCS) as the most promising CO2 reduction technology and investigates how to establish CCS strategy suitably. We confirm a major part of the optimal strategy for CCS infrastructure planning through a literature review according to mathematical optimization criteria associated with facility location models. In particular, the feasibility of large scale CCS infrastructure is evaluated through economic, environmental, and technical assessment. The current state-of-the-art optimization techniques for CCS infrastructure planning are also addressed while taking numerous factors into account. Finally, a list of issues for future research is highlighted.
A large number of research works were undertaken for the planning of sustainable electricity generation and CO2 mitigation (EGCM) infrastructure design under uncertainty. The typical methodologies assessed the performance of the problem under the variability of the uncertain parameters by optimizing the expected value of the objective function. This approach can have large probabilities of the value optimized in unfavorable scenarios. In this paper, we present a mathematical programming model in planning sustainable electricity generation and CO2 mitigation (EGCM) infrastructure design, including financial risk management under uncertainty. The proposed model allows us to determine available technologies to produce electricity and treat CO2 on the purpose of maximizing the expected total profit and minimizing the financial risk of handling uncertain environments (i.e. CO2 mitigation operating costs, carbon credit prices and electricity prices, etc.), while fulfilling electricity demands and CO2 mitigation standards. The multi-objective optimization problem was solved by using the weighted-sum method that imposes a penalty for risk to the objective function. The capability of the proposed modeling framework is illustrated and applied to a real case study based on Korea, for which valuable insights are obtained.
This Intergovernmental Panel on Climate Change (IPCC) Special Report provides information for policymakers, scientists and engineers in the field of climate change and reduction of COâ emissions. It describes sources, capture, transport, and storage of COâ. It also discusses the costs, economic potential, and societal issues of the technology, including public perception and regulatory aspects. Storage options evaluated include geological storage, ocean storage, and mineral carbonation. Notably, the report places COâ capture and storage in the context of other climate change mitigation options, such as fuel switch, energy efficiency, renewables and nuclear energy. This report shows that the potential of COâ capture and storage is considerable, and the costs for mitigating climate change can be decreased compared to strategies where only other climate change mitigation options are considered. The importance of future capture and storage of COâ for mitigating climate change will depend on a number of factors, including financial incentives provided for deployment, and whether the risks of storage can be successfully managed. The volume includes a Summary for Policymakers approved by governments represented in the IPCC, and a Technical Summary. 5 annexes.
In the carbon capture and storage (CCS) process, CO2 sources and geologic reservoirs may be widely spatially dispersed and need to be connected through a dedicated CO2 pipeline network. We introduce a scalable infrastructure model for CCS (simCCS) that generates a fully integrated, cost-minimizing CCS system. SimCCS determines where and how much CO2 to capture and store, and where to build and connect pipelines of different sizes, in order to minimize the combined annualized costs of sequestering a given amount of CO2. SimCCS is able to aggregate CO2 flows between sources and reservoirs into trunk pipelines that take advantage of economies of scale. Pipeline construction costs take into account factors including topography and social impacts. SimCCS can be used to calculate the scale of CCS deployment (local, regional, national). SimCCS’ deployment of a realistic, capacitated pipeline network is a major advancement for planning CCS infrastructure. We demonstrate simCCS using a set of 37 CO2 sources and 14 reservoirs for California. The results highlight the importance of systematic planning for CCS infrastructure by examining the sensitivity of CCS infrastructure, as optimized by simCCS, to varying CO2 targets. We finish by identifying critical future research areas for CCS infrastructure.
Much of the early research in the hydrogen supply chain area was focused on individual technologies of the supply chain, such as production, storage, or distribution, rather than dealing with the supply chain as a whole. The motivation behind this paper is the need to: (1) design a hydrogen supply chain that integrates the previously mentioned components within a single framework, (2) understand the important trade-offs in such a supply chain, and (3) have a full understanding of the data requirements and uncertainties in such an exercise. Optimization techniques were implemented to develop the hydrogen supply chain for the transport sector, therefore, determining the optimum infrastructural and operational costs. Of course, cost is not likely to be the sole determinant of performance in practice. The network of interest was formulated as a mixed-integer linear programming (MILP) problem. Also, the network is presented as a steady state ‘snapshot’ problem using Great Britain as a backdrop. The model and assumptions presented in this paper reveal that the optimum future hydrogen supply chain might consist of medium-to-large, centralized methane steam reforming plants. The hydrogen produced from these plants will then be delivered as a liquid via tanker trucks and stored in centralized storage facilities.
In this study, a two-stage inexact-stochastic programming (TISP) method is developed for planning carbon dioxide (CO2) emission trading under uncertainty. The developed TISP incorporates techniques of interval-parameter programming (IPP) and two-stage stochastic programming (TSP) within a general optimization framework. The TISP can not only tackle uncertainties expressed as probabilistic distributions and discrete intervals, but also provide an effective linkage between the pre-regulated greenhouse gas (GHG) management policies and the associated economic implications. The developed method is applied to a case study of energy systems and CO2 emission trading planning under uncertainty. The results indicate that reasonable solutions have been generated. They can be used for generating decision alternatives and thus help decision makers identify desired GHG abatement policies under various economic and system-reliability constraints.
This study addresses the design problem of the hydrogen supply chain consisting of various activities such as production, storage and transportation. The purpose of this study is to (1) develop a stochastic model to take into account the effect of the uncertainty in the hydrogen activities and (2) examine the total network costs of various configurations of a hydrogen supply chain in an uncertain environment for hydrogen demand. A deterministic optimization model was first improved from previously existed model. A stochastic formulation based on the two-stage programming approach was then proposed to assure more realistic results. Uncertainty was introduced in the hydrogen demand of each region, which is estimated with an energy economy model and statistic data. This model was applied to evaluate the future hydrogen supply chain of Korea. Results include not only the investment strategy for the optimal supply chain configuration but also the effect of uncertain demands.
It has been observed by many people that a striking number of quite diverse mathematical problems can be formulated as problems in integer programming, that is, linear programming problems in which some or all of the variables are required to assume integral values. This fact is rendered quite interesting by recent research on such problems, notably by R. E. Gomory [2, 3], which gives promise of yielding efficient computational techniques for their solution. The present paper provides yet another example of the versatility of integer programming as a mathematical modeling device by representing a generalization of the well-known “Travelling Salesman Problem” in integer programming terms. The authors have developed several such models, of which the one presented here is the most efficient in terms of generality, number of variables, and number of constraints. This model is due to the second author [4] and was presented briefly at the Symposium on Combinatorial Problems held at Princeton University, April 1960, sponsored by SIAM and IBM. The problem treated is: (1) A salesman is required to visit each of n cities, indexed by 1, … , n . He leaves from a “base city” indexed by 0, visits each of the n other cities exactly once, and returns to city 0. During his travels he must return to 0 exactly t times, including his final return (here t may be allowed to vary), and he must visit no more than p cities in one tour. (By a tour we mean a succession of visits to cities without stopping at city 0.) It is required to find such an itinerary which minimizes the total distance traveled by the salesman.
Note that if t is fixed, then for the problem to have a solution we must have tp ≧ n . For t = 1, p ≧ n , we have the standard traveling salesman problem.
Let d ij ( i ≠ j = 0, 1, … , n ) be the distance covered in traveling from city i to city j . The following integer programming problem will be shown to be equivalent to (1): (2) Minimize the linear form ∑ 0≦ i ≠ j ≦ n ∑ d ij x ij over the set determined by the relations ∑ n i =0 i ≠ j x ij = 1 ( j = 1, … , n ) ∑ n j =0 j ≠ i x ij = 1 ( i = 1, … , n ) u i - u j + px ij ≦ p - 1 (1 ≦ i ≠ j ≦ n ) where the x ij are non-negative integers and the u i ( i = 1, …, n ) are arbitrary real numbers. (We shall see that it is permissible to restrict the u i to be non-negative integers as well.)
If t is fixed it is necessary to add the additional relation: ∑ n u =1 x i 0 = t Note that the constraints require that x ij = 0 or 1, so that a natural correspondence between these two problems exists if the x ij are interpreted as follows: The salesman proceeds from city i to city j if and only if x ij = 1. Under this correspondence the form to be minimized in (2) is the total distance to be traveled by the salesman in (1), so the burden of proof is to show that the two feasible sets correspond; i.e., a feasible solution to (2) has x ij which do define a legitimate itinerary in (1), and, conversely a legitimate itinerary in (1) defines x ij , which, together with appropriate u i , satisfy the constraints of (2).
Consider a feasible solution to (2).
The number of returns to city 0 is given by ∑ n i =1 x i 0 . The constraints of the form ∑ x ij = 1, all x ij non-negative integers, represent the conditions that each city (other than zero) is visited exactly once. The u i play a role similar to node potentials in a network and the inequalities involving them serve to eliminate tours that do not begin and end at city 0 and tours that visit more than p cities. Consider any x r 0 r 1 = 1 ( r 1 ≠ 0). There exists a unique r 2 such that x r 1 r 2 = 1. Unless r 2 = 0, there is a unique r 3 with x r 2 r 3 = 1. We proceed in this fashion until some r j = 0. This must happen since the alternative is that at some point we reach an r k = r j , j + 1 < k .
Since none of the r 's are zero we have u r i - u r i + 1 + px r i r i + 1 ≦ p - 1 or u r i - u r i + 1 ≦ - 1. Summing from i = j to k - 1, we have u r j - u r k = 0 ≦ j + 1 - k , which is a contradiction. Thus all tours include city 0. It remains to observe that no tours is of length greater than p . Suppose such a tour exists, x 0 r 1 , x r 1 r 2 , … , x r p r p +1 = 1 with all r i ≠ 0. Then, as before, u r 1 - u r p +1 ≦ - p or u r p +1 - u r 1 ≧ p .
But we have u r p +1 - u r 1 + px r p +1 r 1 ≦ p - 1 or u r p +1 - u r 1 ≦ p (1 - x r p +1 r 1 ) - 1 ≦ p - 1, which is a contradiction.
Conversely, if the x ij correspond to a legitimate itinerary, it is clear that the u i can be adjusted so that u i = j if city i is the j th city visited in the tour which includes city i , for we then have u i - u j = - 1 if x ij = 1, and always u i - u j ≦ p - 1.
The above integer program involves n ² + n constraints (if t is not fixed) in n ² + 2 n variables. Since the inequality form of constraint is fundamental for integer programming calculations, one may eliminate 2 n variables, say the x i 0 and x 0 j , by means of the equation constraints and produce an equivalent problem with n ² + n inequalities and n ² variables.
The currently known integer programming procedures are sufficiently regular in their behavior to cast doubt on the heuristic value of machine experiments with our model. However, it seems appropriate to report the results of the five machine experiments we have conducted so far. The solution procedure used was the all-integer algorithm of R. E. Gomory [3] without the ranking procedure he describes.
The first three experiments were simple model verification tests on a four-city standard traveling salesman problem with distance matrix [ 20 23 4 30 7 27 25 5 25 3 21 26 ]
The first experiment was with a model, now obsolete, using roughly twice as many constraints and variables as the current model (for this problem, 28 constraints in 21 variables). The machine was halted after 4000 pivot steps had failed to produce a solution.
The second experiment used the earlier model with the x i 0 and x 0 j eliminated, resulting in a 28-constraint, 15-variable problem. Here the machine produced the optimal solution in 41 pivot steps.
The third experiment used the current formulation with the x i 0 and x 0 j eliminated, yielding 13 constraints and 9 variables. The optimal solution was reached in 7 pivot steps.
The fourth and fifth experiments were used on a standard ten-city problem, due to Barachet, solved by Dantzig, Johnson and Fulkerson [1]. The current formulation was used, yielding 91 constraints in 81 variables. The fifth problem differed from the fourth only in that the ordering of the rows was altered to attempt to introduce more favorable pivot choices. In each case the machine was stopped after over 250 pivot steps had failed to produce the solution. In each case the last 100 pivot steps had failed to change the value of the objective function.
It seems hopeful that more efficient integer programming procedures now under development will yield a satisfactory algorithmic solution to the traveling salesman problem, when applied to this model. In any case, the model serves to illustrate how problems of this sort may be succinctly formulated in integer programming terms.
This paper presents linear models of the most common components in the value chain for capture and storage. The optimal investment planning of new gas power plants traditionally includes the cost of fuel versus sales of electricity and heat from the plant. If a new power plant also causes additional investments in gas infrastructure, these should be included in the optimization. With the increasing focus on global emissions, yet another aspect is introduced in the form of technology and infrastructure for capture, transport, and storage of . To be able to include all these aspects in the planning of new power plants, linear models for capture and storage are formulated consistent with current models for gas, electricity, and heat infrastructures. This paper presents models for the following infrastructure: source, combined cycle gas turbine producing electricity, heat and exhaust, capture plant, pipeline, liquefaction plant, storage, ship transport, injection pump, and demand/market.
Developing a deterministic mathematical model of carbon capture and storage networks
- J H Han
- J H Ryu
- I B Lee
Han, J. H. ; Ryu, J. H. ; Lee, I. B. Developing a deterministic mathematical model of carbon capture and storage networks. Unpublished work.