Global optimization of a MINLP process synthesis model for thermochemical based conversion of hybrid coal, biomass, and natural gas to liquid fuels

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Abstract

A global optimization framework is proposed for a thermochemical based process superstructure to produce a novel hybrid energy refinery which will convert carbon-based feedstocks (i.e., coal, biomass, and natural gas) to liquid transportation fuels. The mathematical model for process synthesis includes simultaneous heat, power, and water integration and is formulated as a mixed-integer nonlinear optimization (MINLP) problem with nonconvex functions. The MINLP model is large-scale and includes 15,439 continuous variables, 30 binary variables, 15,406 equality constraints, 230 inequality constraints, and 335 nonconvex terms. The nonconvex terms arise from 274 bilinear terms, 1 quadrilinear term, and 60 concave cost functions. The proposed framework utilizes piecewise linear underestimators for the nonconvex terms to provide tight relaxations when calculating the lower bound. The bilinear terms are relaxed using a partitioning scheme that depends logarithmically on the number of binary variables, while the concave functions are relaxed using a linear partitioning scheme. The framework was tested on twelve case studies featuring three different plant capacities and four different feedstock-carbon conversion percentages and is able to solve each study to within a 3.22–8.56% optimality gap after 100 CPU hours. For 50% feedstock carbon conversion, the proposed global optimization framework shows that the break-even oil prices for liquid fuels production are $61.36/bbl for the small case study, $60.45/bbl for the medium case study, and $55.43/bbl for the large case study, while the corresponding efficiencies are 73.9%, 70.5%, and 70.1%, respectively.

Introduction

For the past several decades, the United States transportation sector has heavily relied on petroleum-based fuels. Current challenges including the volatility of the global oil market, an increased dependence on crude oil imports and the impact on domestic energy security, and high contributions of greenhouse gas emissions from the transportation sector have motivated efforts toward alternative sources of liquid transportation fuels. A diverse array of domestic feedstocks including coal, biomass, and natural gas are available for use in novel energy refineries that can produce gasoline, diesel, and kerosene that are compatible with the current transportation infrastructure. To properly incorporate these alternative feedstocks, it is imperative to investigate and develop optimal process designs that can be economically competitive to petroleum-based fuels.

Several studies have highlighted process design alternatives that can produce the three major liquid fuels along with electricity and other fuels such as methanol, ethanol, butanol, and hydrogen (Floudas, Elia, & Baliban, 2012). Generally, these processes have focused on the production of Fischer–Tropsch (FT) liquids from either coal, biomass, or natural gas (Adams and Jin, 2010, Adams and Bertucco, 2011, Agrawal et al., 2007, Baliban et al., 2010, Baliban et al., 2011, Bechtel Corp., 2003, Cao et al., 2008, Chen et al., 2011a, Chen et al., 2011b, Chiesa et al., 2005, Elia et al., 2010, Kreutz et al., 2008, Kreutz et al., 2005, Larson and Jin, 1999, Liu et al., 2009, Liu et al., 2010a, Liu et al., 2010b, Sudiro and Bertucco, 2007, Sudiro and Bertucco, 2009, Vliet et al., 2009). Hybrid processes involve the selection of multiple feedstocks to combine the benefits from each of the raw materials. Coal is an advantageous feedstock because the delivered cost is generally cheaper ($2.20/MM Btu (Energy Information Administration, 2011)) than natural gas ($5.39/MM Btu (Energy Information Administration, 2011)) or biomass ($4.0–$9.0/MM Btu (Kreutz et al., 2008, Larson et al., 2009, National Academy of Sciences, 2009)). The prices of all feedstocks are converted to 2009 $ using the GDP deflator index (US Government Printing Office, 2009). However, the high carbon content of coal will require that a significant portion of the feedstock carbon to be converted to CO2 where it can either be vented, sequestered, or converted back to CO using a non-carbon based source of hydrogen (Agrawal et al., 2007, Baliban et al., 2012, Baliban et al., 2010, Baliban et al., 2011, Elia et al., 2010). Natural gas provides a high hydrogen to carbon ratio that can help to increase the conversion rates of feedstock carbon to final liquid fuels. Recent prospects for shale gas production have helped reduce the delivered cost of natural gas and made this feedstock a more attractive choice for liquid fuels production (Energy Information Administration, 2011). Biomass is highly beneficial since it can provide an overall reduction in well-to-wheel greenhouse gas emissions and it is the only feedstock that is renewable if measures are taken to utilize crop and forest residues that will not interfere with the national food demand (Department of Energy, 2005).

Current hybrid feedstock processes in the literature focus on processes involving coal and biomass to liquids (Agrawal et al., 2007, Chen et al., 2011a, Chen et al., 2011b, Kreutz et al., 2008), coal and natural gas to liquids (Adams and Jin, 2010, Cao et al., 2008, Sudiro and Bertucco, 2007, Sudiro and Bertucco, 2009), biomass and natural gas to liquids, and coal, biomass, and natural gas to liquids (Baliban et al., 2010, Baliban et al., 2011, Baliban et al., 2012, Elia et al., 2010). Polygeneration systems have also been designed which are able to co-produce electricity and other liquid fuels along with gasoline, diesel, and kerosene (Adams and Bertucco, 2011, Baliban et al., 2011, Baliban et al., 2012, Chen et al., 2011a, Chen et al., 2011b, Chiesa et al., 2005, Kreutz et al., 2005, Liu et al., 2009, Liu et al., 2010a, Liu et al., 2010b, Sudiro et al., 2008). For a comprehensive account of the thermochemical based coal, biomass, and natural gas to liquid processes, the readers are directed to the recent review (Floudas et al., 2012). A majority of the studies conducted within the literature focus on process designs where the topology of the process is fixed. A process simulation is then conducted to determine the heat and mass balances for the process and a detailed economic analysis is performed to determine the viability of the plant (Adams and Jin, 2010, Adams and Bertucco, 2011, Baliban et al., 2010, Bechtel Corp., 2003, Cao et al., 2008, Chen et al., 2011a, Chen et al., 2011b, Chiesa et al., 2005, Elia et al., 2010, Kreutz et al., 2005, Kreutz et al., 2008, Larson and Jin, 1999, Liu et al., 2009, Liu et al., 2010a, Liu et al., 2010b, Sudiro and Bertucco, 2007, Sudiro and Bertucco, 2009, Vliet et al., 2009). Alternatively, studies focused on process synthesis of thermochemical hybrid energy plants have been introduced which initially postulate a process superstructure with several possible topologies and then use an optimization framework to examine the economic trade-offs between each topology. The solution with the best economic value is then selected as the optimal design (Baliban et al., 2011, Baliban et al., 2012). These processes are targeted to use a hybrid feedstock of coal, biomass, and natural gas to produce the desired liquid fuels (CBGTL). The process synthesis framework was enhanced by including a simultaneous heat and power integration (Baliban et al., 2011) using an optimization-based heat-integration approach (Duran & Grossmann, 1986) and a series of heat engines that can convert waste heat into electricity (Baliban et al., 2010, Baliban et al., 2011, Elia et al., 2010). More recently, the process synthesis model integrated a comprehensive wastewater treatment network (Baliban et al., 2012) that utilized a superstructure approach (Ahmetovic and Grossmann, 2010a, Ahmetovic and Grossmann, 2010b, Grossmann and Martín, 2010, Karuppiah and Grossmann, 2006) to determine the appropriate topology and operating conditions of process units that are needed to minimize wastewater contaminants and freshwater intake.

The optimization model for process synthesis with simultaneous heat, power, and water integration defines a mixed-integer non-linear and non-convex landscape where high-quality solutions are obtained by finding several locally optimal points from different initial conditions (Baliban et al., 2011, Baliban et al., 2012). However, no guarantee of global optimality was determined and it was not known how close the objective value of each solution was to the best possible value that can be obtained. In this paper, a global optimization framework is proposed for the CBGTL process superstructure that will determine a valid lower bound on the objective function using piecewise-linear underestimation of bilinear terms and concave cost functions and a branch-and-bound algorithm. The framework is tested on twelve case studies featuring three different plant capacities and four different feedstock-carbon conversion percentages. The overall greenhouse gas emission target for each case study is set to have a 50% reduction from petroleum based processes (Baliban et al., 2011, Baliban et al., 2012).

Section snippets

Conceptual design of process superstructure

The CBGTL superstructure is designed to co-feed biomass, coal, and natural gas to produce gasoline, diesel, and kerosene (Baliban et al., 2011, Baliban et al., 2012). Synthesis gas (syngas) is generated via gasification from biomass (Supp. Fig. S1) or coal (Supp. Fig. S2) or auto-thermal reaction of natural gas (Supp. Fig. S7) and is converted into hydrocarbon products in the Fischer–Tropsch (FT) reactors (Supp. Fig. S5) which are subsequently upgraded to the final liquid fuels (Supp. Fig. S6).

Mathematical model nonlinearities

This section will focus on the nonlinearities that are present within the mathematical model for process synthesis with simultaneous heat, power, and water integration. Specifically, each portion of the CBGTL process topology that gives rise to a nonlinear series of equations will be discussed along with the number of nonlinear terms introduced and the anticipated bounds of the variables present in these terms. The complete mathematical model is included for reference as Supplementary

Deterministic global optimization strategies

To solve the process synthesis with simultaneous heat, power, and water integration problem, a branch-and-bound global optimization algorithm (Misener et al., 2010, Misener et al., 2011, Misener and Floudas, 2010) is introduced as described below. At each node in the branch-and-bound tree, a mixed-integer linear relaxation of the mathematical model is solved using CPLEX 12.3 (CPLEX, 2009) and then the node is branched to create two children nodes. The solution pool feature of CPLEX is utilized

Computational results of twelve case studies

The proposed global optimization routine was used to analyze twelve distinct case studies using perennial biomass (switchgrass), low-volatile bituminous coal (Illinois #6), and natural gas as feedstocks. The ultimate and proximate analysis of the biomass and coal feedstocks and the molar composition of the natural gas feedstock (Elia, Baliban, & Floudas, 2011) are presented as Supplementary information. To examine the effects of potential economies of size on the final liquid fuels price, three

Conclusions

A novel global optimization framework has been proposed to address the large-scale coal, biomass, and natural gas to liquids (CBGTL) process synthesis mathematical model with simultaneous heat, power, and water integration. Using piecewise linear underestimators with a logarithmic partitioning scheme for the bilinear terms and piecewise linear underestimators with a linear partitioning scheme for the concave cost functions, twelve case studies for the CBGTL model have been optimized to within a

Acknowledgement

The authors acknowledge partial financial support from the National Science Foundation (NSF EFRI-0937706).

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