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Development of a Multiperiod Model for Planning CO2 Disposal and Utilization Infrastructure
Abstract
Reducing CO2 emissions economically and efficiently necessitates the construction of carbon capture and storage (CCS) infrastructure as a comprehensive network that is capable of disposing of, and of utilizing CO2. Most of the early attempts to design and model the future CCS infrastructure were either limited to examining an individual component of the CCS infrastructure or focused on design and operation of a deterministic, steady-state network in which CO2 emissions are constant over time. In this paper, a multiperiod model for planning CCS infrastructure is developed to consider variation of the emission reduction target, as well as the variation of CO2 emissions over a long-term planning interval, thus leading to phased infrastructure development. The proposed model can help determine where and how much CO2 to capture, store, transport, utilize or sequester for the purpose of maximizing the total profit of the CCS infrastructure while meeting the CO2 mitigation target during each time period of the planning interval. The features and capabilities of the model are illustrated by application of the future CCS infrastructure on the east coast of Korea. The results will be helpful to multiperiod planning of the development of a CCS infrastructure.
... There is little research about the model of the whole process of CCUS. The methods for different types of the model of the whole process of CCUS are compared in Table 4. Considering variation of the emissions reduction target, a multiperiod model for planning CCS infrastructure is developed by maximising the average annual profit (TAP) as the single-objective function (Han et al., 2012b). On the basis of Han et al. (2012b), a multi-period stochastic optimisation model for CCS infrastructure is established under uncertainty in CO 2 emissions, product prices and operating costs (Han et al., 2012c). ...
... The methods for different types of the model of the whole process of CCUS are compared in Table 4. Considering variation of the emissions reduction target, a multiperiod model for planning CCS infrastructure is developed by maximising the average annual profit (TAP) as the single-objective function (Han et al., 2012b). On the basis of Han et al. (2012b), a multi-period stochastic optimisation model for CCS infrastructure is established under uncertainty in CO 2 emissions, product prices and operating costs (Han et al., 2012c). The planning task is formulated as a MILP problem that finds minimised cost and environmental impact, and then, a multi-objective optimisation approach is provided for CCS infrastructure (Lee et al., 2012). ...
... All above studies are optimisation models about infrastructure for decision-making. Han et al., (2012bHan et al., ( , 2012cHan et al., ( , 2013 and Lee et al., (2012) all consider the uncertainty. ...
Carbon Capture, Utilisation and Storage (CCUS) technology is an effective and economic method to reduce incredibly increasing greenhouse gas emissions. Mathematical modelling and evaluating are the key technologies for CCUS development. In this paper, we discuss the model and evaluation of CO2 capture from power plant, CO2 transport by pipeline, CO2-Enhanced Oil Recovery (EOR) and storage, respectively. Then, we investigate the whole process model and evaluation of CCUS based on the aforementioned sub-models. The purpose of this paper is to review the state-of-the-art of such research and development, and point out the unsolved issues and the future progression work.
... 4,5 Carbon capture and storage (CCS) is considered to be an effective method to prevent the CO 2 from industrial processes into the atmosphere because of its economic, safety, and site-specific advantages. 6 In the CCS process, it is unavoidable to capture impurities, such as N 2 , O 2 , Ar, NO x , SO 2 , H 2 S, CO, H 2 , and CH 4 . 7 The specific impurities and their concentrations can vary significantly depending on the type of fuel, capture process (postcombustion, pre-combustion, and oxy-combustion), and separation technology (absorption, membranes, cryogenic distillation, gas hydrates, and chemical looping) used. ...
Anthropogenic greenhouse gases are mainly CO2, CH4, N2O, and fluorinated gases. However, due to the toxicity of the CO2–N2O mixture, there are few experimental pressure–volume–temperature–composition (PVTx) data available in the literature, which do not meet the temperature–pressure conditions for the applications of carbon capture and storage (CCS). In this work, the gaseous PVTx data of the CO2–N2O mixture with molar ratios of 0.6:0.4 and 0.8:0.2 were measured by using a constant-volume vapor phase density apparatus, over a temperature–pressure range of 298.15–473.15 K and 1.0–100.0 bar. The relative standard uncertainty of density ranged from 0.15 to 0.35%. Based on experimental data, a Helmholtz free energy equation of state (EOS) for the CO2–N2O mixture was developed, which can accurately reproduce the PVTx properties of the CO2–N2O fluid mixture. The EOS reveals that as the N2O content increases, the density of the CO2–N2O fluid mixture initially decreases and then increases. The overall trend shows that the density decreases with increasing temperature and increases with increasing pressure. The EOS can be used to quantitatively estimate the effect of impurity N2O on the CO2 storage capacity in the CCS process.
... The main objective of this work is to present an optimization framework for the CCUS supply chain design, with specific focus on the multi-sink CO2-EOR tactical planning for the long-term decisions. References for the supply chain design model are the work of Han et al. (2012), Montagna & Cafaro (2019) towards the design of efficient networks with no predetermined echelons in the superstructure, and the recent work of Duarte et al. (2022). Surrogate models proposed by Hasan et al. (2015) and commercial data is considered for the capture technologies alternatives. ...
This work describes a novel MILP formulation for the advanced optimization of CO2 Enhanced Oil Recovery (EOR) tactical planning. The goal is to address the long-term design of supply chain network (SCN) for the carbon capture, utilization and storage (CCUS) management, developing a decision support system for the CO2 management considering strategic and tactical decisions such as (1) the selection of carbon capture technologies, (2) the development of multi-modal transport infrastructure to match CO2 sources with potential storage and carbon utilization nodes, and (3) the tactical planning of EOR operations with CO2 injection. Even though many works in the literature address the CCUS problem, there is a lack of integration between all the decisions, both from the capture and the utilization side. The novelty of this paper relies on the integration of the CCUS supply chain design decisions considering a long-term planning horizon, with a particular focus in a comprehensive modelling of the CO2-EOR operations, which are one of the most important utilization options in order to mitigate and reduce CO2 emissions. The model maximizes the net present value considering capital and operational costs, as well as the revenues from the additional oil produced by CO2-EOR operations and the carbon credits associated with the CO2 sequestrated. The formulation is tested in a large case study, obtaining near optimal results in reasonable CPU times, showing the integration of capture facilities, pipelines network and the optimal allocation of captured CO2 in several fields over a large planning horizon.
... The main reduction policies include supplying renewable energy, expanding clean fuel generation, improving energy efficiency, implementing emissions trading, to demonstrate the proposed model. In Han et al.[10], a multi-period model was proposed for maximizing the average annual profit of CCS infrastructure (including utilization, capture, storage, and sequestration facilities) over a long-term planning interval considering the disposal and utilization of CO 2. In addition, the author referred to the concept of intermediate storage technologies that exist only to collect CO 2 captured from emission sources within a particular region and load the collected CO 2 for delivery by different transport modes. Other existing studies on CCS transport networks in different regions around the world have assumed that CO 2 flowing through a network is static throughout the life of the network. ...
Korea’s national carbon capture and storage (CCS) master plan aims to commercialize CCS projects by 2030. Furthermore, the Korean government is forced to reduce emissions from various sectors, including industries and power generation, by 219 million tons by 2030. This study analyzes a few scenarios of Korean CCS projects with a CO2 pipeline transportation network optimization model for minimizing the total facility cost and pipeline cost. Our scenarios are based on the “2030 basic roadmap for reducing greenhouse gases” established by the government. The results for each scenario demonstrate that the effective design and implementation of CO2 pipeline network enables the lowering of CO2 units cost. These suggest that CO2 transportation networks, which connect the capture and sequestration parts, will be more important in the future and can be used to substitute and supplement the emission reduction target in case the execution of other reduction options faces uncertainty. Our mathematical model and scenario designs will be helpful for various countries which plan to introduce CCS technology.
Colombia is formulating policies to accelerate the energy transition, decarbonize its industries and decrease its dependency on fossil fuels. In this work we present a novel optimization framework for the carbon capture, utilization, and storage (CCUS) design applied to the Colombian case. The model maximizes the net present value considering technologies for carbon capture, multimodal transportation, CO2 utilization for enhanced oil recovery (CO2-EOR) and geological storage to meet a given target on greenhouse gases (GHG). The novelty of this work is the integration of the CCUS design with the oil field development for CO2-EOR. Thus, the optimal petroleum production campaign (i.e., production wells for CO2 injection) is computed based on the oil production profiles, petrophysical properties and economic data. Moreover, this approach provides insights regarding investments on carbon capture technologies such as timing, sizing, location, and type of technology. The model also determines the optimal transportation mode considering on-shore (pipelines, railcars, and trucks) and off-shore (pipelines and ships). We tested our approach on a nationwide long-term planning horizon (20 years), considering 30 CO2 emissions sources distributed across 12 regions in the country, 16 carbon capture technologies, 3 different plant sizes (small, medium, and large), 6 potential oil fields for CO2-EOR, 2 sinks for geological storage, and 4 multimodal transportation modes. Results shown a carbon capture cost of 99.8 USD/t, with a cost breakdown of 45%, 30% and 25% allocated into the carbon capture processes, CO2 transportation and CO2-EOR utilization, respectively.
The present work aims to establish a methodology for the optimization of CO2 transportation networks by developing transportation levelized-cost functions for the mixture CO2:N2 (96:4%) within the ranges {25–2000 kg/s} and {50–2000 km}, locating the optimal solar cities (locations-sinks) to receive CO2 for further transformation into fine chemicals, and mapping the optimal routes which present the minimum transportation and utilization cost. Potential economies of scale of merging same-route sinks into a joint pipeline are scrutinized by selecting 17 CO2 sources from thermal power and combustion stations in the central EU region aggregating almost 3000 kg/s of CO2; these amounts are optimally allocated to 16 available sinks, which combine maximum solar capacity with the least distance to major ports. This study makes use of transport cost correlations, while the solar potential of the most promising solar cities is calculated in detail based on solar intensity databases and solar power plants cost assessment. Mixed-integer nonlinear programming (MINLP) techniques are incorporated in the optimization procedure under sinks capacity constraints. A major conclusion of this work is that for the mass flowrates considered and the sources locations chosen in the current study it is economically beneficial to transfer CO2 directly to the final sinks from the original sources rather than via intermediate sources. The MINLP formulation derives values of solar utilization cost of over €7B and transportation cost of over €2B per year covering transportation pipelines distances of 34,000 km: precise economic estimates of the proposed methodology regarding CO2 transportation and utilization might have significant implications on energy and environmental policy issues on European or global scale.
This paper describes a design and pre-feasibility study of a multi-user intermediate CO2 storage facility in the Grenland region of Norway, considering upstream and downstream issues. The study focuses on the principles for designing and installing a generic hub facility so that the results can be considered at other sites. The pre-feasibility study found that design pressures of 7 and 15 bar are feasible transport conditions; moreover, it showed that economies of scale might reduce the total cost for a CO2 network. It is recognised that cooperation across the chain is crucial in managing impurities due to the likely diverse sources of CO2. An intermediate storage facility can support the continuous supply of CO2 via a pipeline system for reservoir injection, improving the integrity of the injection well and equipment and the reservoir performance. A mixed-integer linear programming optimisation model has been developed for sizing and costing intermediate storage hubs of CO2 in Grenland and shipping connections between them and stationary emitters. The model considers an aggregated flow of 2 million tonnes per annum of CO2 for 30 years. The 7 bar design pressure has shown lower total costs when compared with the 15 bar scenario. This is due to the higher costs for shipping and intermediate storage when operating at higher pressure, which is larger than the cost reduction from liquefying CO2 to a higher pressure and higher temperature than those required for a 7 bar scenario. The estimated levelised costs were 21,6 € tonne⁻¹ at 7 bar pressure and 27,8 € tonne⁻¹ at 15 bar pressure.
In this study, a multi-period model of an organic waste-to-biodiesel phased supply chain network is developed to strategically consider variations in the biodiesel demand and the organic waste usage over a long-term planning interval. The proposed model can facilitate planning to determine the biorefinery location and the amount of biodiesel to produce, transport, or utilize to reduce the total cost of the organic waste-to-biodiesel supply chain network design while meeting the biodiesel demand during each period of the planning interval. The features and capabilities of the model are validated by application of a future organic waste-to-biodiesel supply chain network design in South Korea. The optimization results illustrate that the average total annual cost of the organic waste of biodiesel based on the multi-period model will be US 3.154–3.156/gallon of biodiesel containing 3% butanol derived from organic waste combined with diesel. A small portion (4.9–6.6%) of the total biodiesel demand was produced from organic waste because of low organic waste availability in the existing anaerobic digestion facilities, leading to increasing the outsourcing costs of butanol. The proposed approach will facilitate the strategic development of a sustainable supply chain network for organic waste recycling for material and energy valorization.
In this study, a stochastic model for strategic planning of the butyric acid-to-butanol supply chain network (Ba-to-Bu SCN) is developed to consider variations in the butanol (Bu) demand and butyric acid (Ba) supply derived from industrial/municipal waste. The proposed stochastic model can help determine where and how much Ba to process, Bu to produce, and Ba/Bu to transport to minimize the total cost of the Ba-to-Bu SCN design under Ba processing and Bu demand uncertainties. The features and capabilities of the stochastic model are validated and compared to those of the deterministic model by application of the future Ba-to-Bu SCN design for South Korea in 2030. The optimization results illustrate that the expected total cost of Ba-derived Bu by the stochastic model (US 4889.72 thousand per year). The goal of this study is to develop a decision making tool for a stochastic strategic problem to improve bio-economy caused by uncertainties. The proposed approach will help balance cost efficiency with stability in the uncertain future biorefinery infrastructure.
A multiperiod stochastic programming model is developed for planning a carbon capture and storage (CCS) infrastructure including CO2 utilization and disposal in an uncertain environment and with a time-varying investment environment. An inexact two-stage stochastic programming approach is used to analyze the effect of possible uncertainties in product prices, operating costs, and CO2 emissions. The proposed model determines where and how much CO2 to capture, store, transport, utilize, or sequester for the purpose of maximizing the total profit of handling the uncertainty while meeting the CO2 mitigation target during each time period of a given planning interval. The capability of the proposed model to provide correct decisions despite a changing uncertain environment is tested by applying it to designing and operating the future CCS infrastructure on the eastern coast of Korea over a 20-year planning interval (2011–2030).
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.
In this study, we address the design of a carbon capture and storage (CCS) infrastructure with economic and environmental concerns. Given a set of available technologies to capture, sequestrate, and transport CO2, the problem consists of determining the optimal planning of the CCS infrastructure capable of satisfying a predefined CO2 reduction target. The planning task is formulated as a multiobjective mixed-integer linear programming (moMILP) problem, which simultaneously accounts for the minimization of cost and environmental impact. The environmental impact is measured through all contributions made by operation and installation of the CCS infrastructure. The emissions considered in the environmental impact analysis are quantified according to the principles of Life Cycle Assessment (LCA), specifically the Eco-indicator 99 method. The multiobjective optimization problem was solved by using the ε-constraint method. 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.
Most industries consume a large amount of energy to produce goods and to reduce the resultant CO2 emissions. We develop a systematic process integration framework for the optimal design and techno-economic performance analysis of energy supply and CO2 mitigation strategies. We first generate an Integrated Energy Supply and CO2 Mitigation Network (IESCMN), which includes three processes, each of which consists of a large number of technologies. This model allows identification of energy requirements (sinks) and unused energy sources of the technologies. Developing an optimization model for the IESCMN allows identification of a promising strategy to minimize energy cost, because sources and sinks can be connected to each other to transfer unused energy. Our techno-economic analysis results show that the IESCMN results in a considerable decrease in energy cost, compared to the individual operation of each technology.
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.
Through combining insights from engineering, natural sciences, economics, and political science, one consistent, transparent, and comprehensive analytical framework for assessing and evaluating various CCS chains is developed. The presented methodology aims at improving knowledge on the design of efficient CCS chains by developing methods for assessing and comparing different CCS chains, their sensitivity with regard to internal factors and to external conditions, and what the most efficient policy tools and measures are for promoting CCS development.
To mitigate the climate change and global warming, various technologies have been internationally proposed for reducing greenhouse gas emissions. Especially, in recent, carbon dioxide capture and storage (CCS) technology is regarded as one of the most promising emission reduction options that be captured from major point sources (eg., power plant) and transported for storage into the marine geological structure such as deep sea saline aquifer. The purpose of this paper is to review the latest progress on the development of technologies for storage in marine geological structure and its perspective in republic of Korea. To develop the technologies for storage in marine geological structure, we carried out relevant R&D project, which cover the initial survey of potentially suitable marine geological structure fur storage site and monitoring of the stored behavior, basic design for transport and storage process including onshore/offshore plant and assessment of potential environmental risk related to storage in geological structure in republic of Korea. By using the results of the present researches, we can contribute to understanding not only how commercial scale (about 1 ) deployment of storage in the marine geological structure of East Sea, Korea, is realized but also how more reliable and safe CCS is achieved. The present study also suggests that it is possible to reduce environmental cost (about 2 trillion Won per year) with developed technology for storage in marine geological structure until 2050.
Hydrogen draws increasing attention as an alternative energy source. In order to provide hydrogen to various sectors such as industry, transportation on a global scale, how to produce and distribute it economically is an essential issue not to be missed. This study thereby addresses mathematical modeling of hydrogen supply networks. The proposed model is concerned with how much H2 can be produced, where can be stored with the aim of maximizing the total net profit. Particularly the physical form of the hydrogen in the network is explicitly taken into account in terms of whether it is stored as a gas or a liquid. The applicability of the proposed model will be demonstrated by a case study of the Korean H2 supply network with some remarks.
Numerous research works have been undertaken to plan carbon capture and storage (CCS) infrastructures for CO2 utilization and disposal. CO2 emissions are difficult to estimate precisely, because CO2 is emitted from various sources at varying rates. In this study, a two-stage stochastic programming model is developed for planning CCS infrastructure including CO2 utilization and disposal under stochastic CO2 emissions. The proposed model considers uncertainties in the variation of CO2 emissions. It can help determine where and how much CO2 to capture, store, transport, utilize, or sequester for the purpose of maximizing the total profit of handling uncertain CO2 emissions while meeting the CO2 mitigation target. The capability of the proposed model to provide correct decisions despite changing CO2 emissions is tested by applying it to an industrial complex on the eastern coast of Korea in 2020. The results will help to determine planning of, and budgeting for, development of a CCS infrastructure.
Much of the previous research on carbon capture and storage (CCS) has focused on individual technologies for disposing of CO2, such as capture, storage, sequestration, or transport. Moreover, recent research work considers utilization of CO2 as fuels, chemicals, or nutrients for bioreactors. To efficiently manage CO2 and the economic benefits achieved by this process, the CO2 transport and processing infrastructure supporting CCS will have to be constructed at a macro-scale. This paper introduces a scalable and comprehensive infrastructure model for CO2 utilization and disposal that generates an integrated, profit-maximizing CCS system. The proposed model determines where and how much CO2 to capture, store, transport, utilize or sequester to maximize total annual profit while meeting the CO2 mitigation target. The applicability of the proposed model is demonstrated using a case study for treating CO2 emitted by an industrial complex on the eastern coast of Korea in 2020. The results may be important in systematic planning of a CCS infrastructure and in assisting national and international policy makers to determine investment strategies for developing CCS infrastructures.
Introducing hydrogen as the fuel of the future necessitates a comprehensive, widespread supply chain network that is capable of producing, distributing, storing, and dispensing hydrogen to end users. Most of the early attempts to design and model the future hydrogen supply chain (HSC) were either limited to examining an individual component of the supply chain or focused on a predetermined hydrogen pathway. In these studies, a simulation-based approach has commonly been adopted rather than using a mathematical programming-based approach. The work presented here is an extension of an early attempt to design and operate a deterministic, steady-state HSC network using a mathematical modelling approach. In this paper, however, the model is developed to consider the availability of energy sources (i.e. raw materials) and their logistics, as well as the variation of hydrogen demand over a long-term planning horizon leading to phased infrastructure development. The proposed model is formulated as a mixed-integer linear programming (MILP) and solved via a commercial software tool, GAMS. The results show that the optimal design of the future HSC network of Great Britain (GB) starts with small-size plant together with using the hydrogen currently produced by chemical processing plants. As demand grows, more plants of different sizes should be built to meet the demand. The hydrogen produced will be transported using liquid hydrogen trucks and stored in different sizes of storage facilities.
In a power-generation system, power plants as major CO2 sources may be widely separated, so they must be connected into a comprehensive network to manage both electricity and CO2 simultaneously and efficiently. In this study, a scalable infrastructure model is developed for planning electricity generation and CO2 mitigation (EGCM) strategies under the mandated reduction of GHG emission. The EGCM infrastructure model is applied to case studies of Korean energy and CO2 scenarios in 2020; these cases consider combinations of prices of carbon credit and total electricity demand fulfilled by combustion power plants. The results highlight the importance of systematic planning for a scalable infrastructure by examining the sensitivity of the EGCM infrastructure. The results will be useful both to help decision makers establish a power-generation plan, and to identify appropriate strategies to respond to climate change.Highlights► We model an infrastructure for planning electricity generation and CO2 mitigation. ► We examine changes in prices of carbon credit and electricity demands, respectively. ► Results propose that the electricity demand fulfilled by combustion plants be low. ► The policy managers can make an optimal decision considering energy and environment.
Commercialization of carbon capture and storage from fossil fuelled power plants requires an infrastructure for transportation of the captured carbon dioxide (CO2) from the sources of emission to the storage sites. This paper identifies and analyses different transportation scenarios with respect to costs, capacity, distance, means of transportation and type of storage. The scenario analysis shows that feasible transportation alternatives are pipelines (on and off shore), water carriers (off shore) and combinations of these. Transportation scenarios are given for different transportation capacities ranging from a demonstration plant with an assumed capacity of 200 MWe (1Mt/y of CO2) up to a system of several large 1000 MWe power plants in a coordinated network (40 Mt/y up to 300 Mt/y of CO2). The transportation costs for the demonstration plant scenario range from 1 to 6€/ton of CO2 depending on storage type and means of transportation. The corresponding figure for the coordinated network scenario with off shore storage is around 2€/ton of CO2.
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.
Two-Stage Stochastic Programming Model for Planning CO2 Utilization and Disposal Infrastructure Considering the Uncertainty in the CO2 Emission Techno Economic Model for Carbon Dioxide Compression, Transport and Storage and Correlations for Estimating Carbon Dioxide Density and Viscosity
- J.-H. ; Han
- I.-B Lee
- D L Mccollum
Han, J.-H.; Lee, I.-B. Two-Stage Stochastic Programming Model for Planning CO2 Utilization and Disposal Infrastructure Considering the Uncertainty in the CO2 Emission. Ind. Eng. Chem. Res. 2011, 50 (23), 13435−13443. (14) McCollum, D. L. Techno Economic Model for Carbon Dioxide Compression, Transport and Storage and Correlations for Estimating Carbon Dioxide Density and Viscosity; Institute of Transportation Studies University of California: Davis, CA, 2006.
A Blueprint for the National Greenhouse Gas Emissions Reduction
- Presidental Committeeon Green
- Growth
Presidental Committeeon Green Growth. A Blueprint for the National Greenhouse Gas Emissions Reduction; 2011.