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Problem StatementThe formal problem statement is given as follows:
The problem may be visualized as a superstructure, as shown in Figure 1, or alternatively in more compact form as an allocation matrix, as in Table 1. The objective is to determine the choice of cofiring technique to be implemented in each power plant (as denoted by variable O_{jh}) and to determine the amount of coal replaced with biomass (C_{;}). the allocation of biomass (denoted by w) and bioclrar (as denoted by v_{jk}), so as to minimize the total amount of carbon dioxide (CO,) generated by the system. Figure 1. Representation of superstructure in schematic. Table 1. Matrix form of the superstructure.
Model NomenclatureSets I Set of biomass sources J Set of power plants К Set of biochar sinks H Set of available technologies for cofiring Indices i Index for biomass source in set I j Index for power plant in set J к Index for biochar sinks in set К h Index for cofiring technologies in set H Parameters a CO, footprint of coal combustion in Mt CO,/Mt coal /? CO, footprint of transport in Mt CO,/Mtkm b_{jh} in Mt/y, represents the amount of biomass needed to replace the coal in power plant j using technology h d_{tj} Distance of biomass source i to power plant j in km F,_ Sequestration factor of biocliar sink к r_{Jk} Distance of power plant j to biocliar sink к in km 5 Available biomass from soiuce i
SEO_{k} Maximum amount of biocliar which can be sequestered by sink к z_{h} Biocliar yield when using technology h Variables C in Mt/y, refers to amount of coal replaced by biomass in power plant j O_{jh} Binary variable which indicates the activation of technology h for power plant j Xy in Mt/y, refers to the amount of biomass from soiu ce i which is used in power plant j y_{jk} in Mt/y, amount of biocliar generated from plant j and sequestered to sink к Model FormulationThe MILP model formulation is as follows: The objective function given by equation (1) is to minimize function Z that denotes the incremental amount of CO, generated by the network; it will assume a negative value if a reduction is adher ed. The first term in equation (1) corresponds to the amount of CO, reduction resulting from reduced coal consumption (C) in power plant j due to displacement by biomass using technology h. Parameter a is the CO, footprint per unit of coal (including both direct emissions from combustion and upstream contributions from the coal supply chain), and O_{jh} is a binary variable which indicates the use {O_{jh} = 1) or nonuse (O_{jh} = 0) of a cofiring technology. The second term corresponds to the CO, emission associated with the transport of biomass from source i to power plant j, which is proportional to the distance travelled, d_{jt} and the amount of biomass transported, x . The third term corresponds to the CO, generated in transporting the biocliar from power plant j to biochar sink к which is also proportional to the distance travelled. r_{jk}. and the amount of biocliar transported, y_{jk}. For the second and third terms of Equation (1), parameter/? refers to the emission factor or CO, footprint associated with transporting biomass and biocliar. Finally, the last term in Equation (1) corresponds to the amount of CO, sequestered from the biocliar which is proportional to the amount of biocliar, y_{Jk}. and the sequestration factor of the sink, F_{k}. The latter factor accounts for direct sequestration of the recalcitrant carbon in the biochar, as well as secondary effects due to changes in GFIG emissions from soil biota when biocliar is applied. Equation (2) ensures that the allocation of biomass source i to the different power plants will not exceed the total amount of biomass available, S_{r} The amount of biomass needed to replace the coal will depend on the type of cofiring technology selected and is given by b_{]h}. Equation (3) ensures that enough biomass is obtained fr om the different sources to satisfy the requirement of each power plant j. Equation (4) indicates that each power plant can only implement a maximum of one type of cofiring technology. Equation (5) on the other hand is the biochar balance which indicates that the total amount of biochar generated by a power plant as indicated by the biochar yield, z_{h}. should be sequestered and properly allocated to the available biochar sinks. Equation (6) ensures that the amount of biocliar sequestered in sink к should not exceed the sink capacity, SEO_{k}. Equation (7) indicates that the variable Q is binary. All other variables are nonnegative. Note that this MILP model can be readily solved to global optimality using the conventional branchandbound algorithm found in many commercial optimization software. For any given application, the model size is given by the formula in Table 2. Table 2. Model size as function or problem scale.
Case StudyThis representative case study considers a system with eight biomass sources, fir e power plants and four biochar sinks. This system size gives a cluster for which typical transportation distances for both biomass and biochar are reasonable. The representative case study gives rise to 77 continuous variables, 10 binary variables, and 27 functional constraints. The case study is implemented using the commercial software LINGO 17.0 which was nm using Intel® Core™ i76500U processor and 8.00 GB RAM with negligible CPU time. The biomass sources are assumed to be sites for biomass collection, consolidation, and storage. The limiting data for the biomass sources is shown in Table 3. The amount of coal that must be replaced by biomass for each power plant, if cofiring is implemented, is indicated in Table 4, along with other relevant technical characteristics. It is assumed here that the cofiring rate is 10%, based on thermal energy input. The biochar sinks are tracts of agricultural or setaside land to which biochar can be applied. The biochar sink characteristics are shown in Table 5. This includes the maximum amount of biocliar that a sink can hold and the sequestration factor of the sink which corresponds to the amount of CO, sequestered per unit of biocliar. This factor can also account for positive or negative changes in emissions of other GHGs from soil. The amount of biocliar generated from each power plant will depend on the type of cofiring technique selected, and becomes zero in the case of direct cofiring. Table 6 shows the biochar yield of each cofiring technology considered. In addition, each technology will require a different amount of biomass Table 3. Limiting data for biomass sources.
Table 4. Power plant characteristics.
Table 5. Biochar sink characteristics.
Table 6. Cofiring technology characteristics.
in order to supply the equivalent thermal energy of the replaced coal. The amounts of biomass required to generate the needed thermal energy are indicated in Table 7. Note that the total biomass requirement for indirect cofiring is greater than that of direct cofiring, because part of the biomass (i.e.. the biochar) remains unutilized as fuel. The distances between biomass sources and power plants are shown in Table 8, while the distances between the power plants and the potential biochar sinks are shown in Table 9. It is assumed that the biomass and biocliar are transported by truck, with an emission factor of 0.0001 Mt of CO,/Mt/km (Tan, 2016). The CO, footprint of coal is 3.16 Mt CO,/Mt of coal, including emissions from both the power plant and the upstream coal supply chain. The MILP model corresponding to this case study is coded in LINGO, as shown in the Appendix. Solving the model results in an optimal CO, emission increment of1.9619 Mt of CO,/y. This result Table 7. Biomass requirement in Mt/y.
Table 8. Distance between biomass source and power plant (d,) in km.
Table 9. Distance between power plant and biochar smk (r_{jk}) in km.
indicates a net reduction in CO, emissions, of which 77% is due to the replacement of coal with biomass and 23% to biochar sequestration. By comparison, the increment in CO, emission achieved when only direct cofiring is considered is 1.6967 Mt of СО,/у, which is 13.5% less than the reduction achieved with the optimal solution. Table 10 shows the flow of biomass from source to the power plant and the flow of biochar from the power plant to the sink (shown in the shaded region). Note that direct cofiring is used in power plant P3, due to the lack of biocliar sink capacity in the system. In addition to the identification of the optimum, Voll et al. (2015) argue that the analysis of near optimal solutions can pror ide valuable insights on the characteristics of good solutions to a particular problem. In addition, the actual differences in objective function values of optimal and nearoptimal solutions may be insignificant in practical situations; in such cases, the nearoptimal solutions may have advantages with respect to considerations that are not explicitly reflected in the optimization model formulation. Thus, an additional nine nearoptimal solutions were generated to evaluate which network connections occurred most frequently in the top ten solutions of the case. These solutions were generated automatically using the MILP solver in LINGO 17.0; in the absence of such a solver feature, these solutions can be generated sequentially using additional integercut constraints that eliminate previously Table 10. Optimal allocation of biomass and biocliar in Mt/y.
* Direct cofiiing option is selected. Table 11. Summary of CO, emissions in top ten CMNs.
determined network topologies (Voll et al., 2015). This approach can also lead to the identification of degenerate solutions (i.e., alternative topologies with equivalent objective function values). A summary of the amount of the incremental CO, emissions for these different solutions are summarized in Table 11. The worst solution in this set of networks is only 5.4% worse than the optimal network in terms of CO, emissions reduction. Examples of nearoptimal networks which correspond to the second and fifth best solutions are also shown in Tables 12 and 13, respectively. These networks give a systemwide CO, incremental change of 1.9546 Mt/y and 1.9346 Mt/y, respectively. These results are just 0.74% and 1.75% worse than the optimum solution. In real life applications, such small differences may not have practical significance, so that these solutions may be interpreted as having virtually equivalent performance. The decisionmaker may then select to implement a network based on other criteria not explicitly reflected in the optimization model. Two trends are also apparent in the optimal and nearoptimal solutions presented here. First, due to biocliar sink limitations, not all of the power plants use indirect cofiring in any given solution; some plants opt for either direct cofiring or no cofiring at all. Secondly, even if biocliar sink C3 has the largest capacity, as shown in Table 5, it is utilized only sparingly due to its low sequestration factor. Table 12. Nearoptimal allocation of biomass and biochar in Mt/y (Solution 2).
* Direct cofiring option is selected. Table 13. Nearoptimal allocation of biomass and biochar in Mt/y (Solution 5).
* Direct cofiring option is selected. The frequency of occurrence of network links in the top ten solutions is summarized in Table 14. and is indicated by the intensity of the shading of the cells. White indicates 0% occurrence, black indicates 100% occurrence, and intermediate shades of gray show partial occurrence in the set of solutions. Thus, it can be clearly seen which links in the network are critical, particularly for the connections between biomass sources and power plants. For example, the biomass sources B2, B4 and B5 are consistently linked to power plant P3, while B6, B7 and B8 are consistently linked to P4. These frequently occurring links represent robust features that will be relatively insensitive to deviations from modelling assumptions, such as changes in parameter values. By comparison, it can also be seen in the bottom four rows of Table 14 that there are more variations in the biocliar allocation schemes in the network. This result can be partly attributed to the selection of direct cofiring (which does not produce biocliar) in many of the solutions. For example, it can be seen that in each of the solutions in Tables 10, 12 and 13, a different power plant (i.e., PI, P5 and P2, Table 14. Frequency of connections m the top ten CMNs.
Legend: White  0% occurrence; Black  100% occurrence; Gray  199% occurrence. respectively) elects not to implement cofiring at all. The presence of such alternatives can potentially allow for more flexible decisionmaking in practical situations. These features also represent system components that are more sensitive to model assumptions; a decisionmaker may seek to acquire more data before making a final selection. 
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