Problem Statement

The formal problem statement is given as follows:

  • • Given M available sources of biomass, with each source having a maximum availability Sr
  • • Given N power plants that may opt to implement biomass co-firing, each with the equivalent amount of coal displaced by biomass given by bjh (for each power plant j, a fixed percentage of its thermal energy requirement will be replaced with biomass if a decision to implement co-firing is made).
  • • Given V biomass co-firing technologies, each characterized by factors b h (the biomass requirement when technology h is implemented in power plant j) and zh (the biochar yield per unit of biomass).
  • • Given P biochar sinks, with each sink having a sequestration factor Fk and a maximum biochar capacity of SEOt.
  • • Given distance d.. between any given biomass source i and power plant j.
  • • Given distance t between any given power plant j and biocliar sink k.

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 co-firing technique to be implemented in each power plant (as denoted by variable Ojh) and to determine the amount of coal replaced with biomass (C;). the allocation of biomass (denoted by w) and bioclrar (as denoted by vjk), so as to minimize the total amount of carbon dioxide (CO,) generated by the system.

Representation of superstructure in schematic

Figure 1. Representation of superstructure in schematic.

Table 1. Matrix form of the superstructure.

Power Plant 1

Power Plant 2

Power Plant 3

Biomass Source 1

Biomass Source 2

Biomass Source 3

Biomass Source 4

Biochar Sulk 1

Biochar Sulk 2

Biochar Sulk 3

Biochar Sulk 4

Model Nomenclature

Sets

I Set of biomass sources

J Set of power plants

К Set of biochar sinks

H Set of available technologies for co-firing

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 co-firing technologies in set H

Parameters

a CO, footprint of coal combustion in Mt CO,/Mt coal

/? CO, footprint of transport in Mt CO,/Mt-km

bjh in Mt/y, represents the amount of biomass needed to replace the coal in power plant j using

technology h

dtj Distance of biomass source i to power plant j in km

F,_ Sequestration factor of biocliar sink к

rJk Distance of power plant j to biocliar sink к in km

5 Available biomass from soiuce i

SEOk Maximum amount of biocliar which can be sequestered by sink к zh Biocliar yield when using technology h

Variables

C in Mt/y, refers to amount of coal replaced by biomass in power plant j

Ojh 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

yjk in Mt/y, amount of biocliar generated from plant j and sequestered to sink к

Model Formulation

The 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 Ojh is a binary variable which indicates the use {Ojh = 1) or non-use (Ojh = 0) of a co-firing 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, djt 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. rjk. and the amount of biocliar transported, yjk. 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, yJk. and the sequestration factor of the sink, Fk. 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, Sr The amount of biomass needed to replace the coal will depend on the type of co-firing 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 co-firing 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, zh. 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, SEOk. Equation (7) indicates that the variable Q is binary. All other variables are non-negative. Note that this MILP model can be readily solved to global optimality using the conventional branch-and-bound 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.

Model feature

Number

Binary variables

Continuous variables

Constraints

Case Study

This 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™ i7-6500U 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 co-firing is implemented, is indicated in Table 4, along with other relevant technical characteristics. It is assumed here that the co-firing rate is 10%, based on thermal energy input. The biochar sinks are tracts of agricultural or set-aside 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 co-firing technique selected, and becomes zero in the case of direct co-firing. Table 6 shows the biochar yield of each co-firing technology considered. In addition, each technology will require a different amount of biomass

Table 3. Limiting data for biomass sources.

Biomass source

Available biomass (Mt/y)

B1

0.10

B2

0.15

B3

0.10

B4

0.25

B5

0.10

B6

0.20

B7

0.12

B8

0.10

Table 4. Power plant characteristics.

Power plant

Capacity (MW)

Baseline coal consumption (Mt/y)

Amount of displaced coal (Mt/y)

PI

200

0.60

0.060

P2

250

0.75

0.075

P3

600

ISO

0.180

P4

500

1.50

0.150

P5

250

0.75

0.075

Table 5. Biochar sink characteristics.

Power plant

Maximum amount of sequestered biochar (Mt/y)

Sequestration factor (Fk) (Mt CO;/Mt of biochar)

Cl

0.04

3.20

C2

0.05

3.00

C3

1.00

2.60

C4

0.06

3.00

Table 6. Co-firing technology characteristics.

Technology

Biochar yield

T1

Direct co-firing

0

T2

Induect co-firing

0.20

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 co-firing is greater than that of direct co-firing, 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 of-1.9619 Mt of CO,/y. This result

Table 7. Biomass requirement in Mt/y.

Co-firing technology

T1

T2

PI

0.1200

0.1500

P2

0.1500

0.1875

P3

0.3600

0,4500

P4

0.3000

0.3750

P5

0.1500

0.1875

Table 8. Distance between biomass source and power plant (d,) in km.

PI

P2

P3

P4

P5

B1

60

120

160

220

240

B2

40

120

140

200

220

B3

30

90

140

200

220

B4

70

30

140

210

200

B5

40

40

60

140

130

B6

120

70

90

120

60

B7

80

140

80

120

160

B8

100

150

60

100

140

Table 9. Distance between power plant and biochar smk (rjk) in km.

Cl

C2

C3

C4

PI

100

60

100

140

P2

140

100

30

60

P3

70

50

50

80

P4

100

100

110

120

P5

140

130

80

80

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 co-firing 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 co-firing 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 near-optimal solutions may be insignificant in practical situations; in such cases, the near-optimal solutions may have advantages with respect to considerations that are not explicitly reflected in the optimization model formulation. Thus, an additional nine near-optimal 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 integer-cut constraints that eliminate previously

Table 10. Optimal allocation of biomass and biocliar in Mt/y.

PI

P2

P3*

P4

P5

B1

0

0

0

00900

0

B2

0

0

0.0975

0.0525

0

B3

0

0

0.1000

0

0

B4

0

0.1875

0.0625

0

0

B5

0

0

0.1000

0

0

B6

0

0

0

0.0125

0.1875

B7

0

0

0

0.1200

0

B8

0

0

0

0.1000

0

Cl

0

0

0

00400

0

C2

0

00150

0

0.0350

0

C3

0

0

0

0

0

C4

0

0.0225

0

0

0.0375

* Direct co-fiiing option is selected.

Table 11. Summary of CO, emissions in top ten CMNs.

Solution rank

Incremental CO, emission (Mt/y)

1

-1,9691

2

-1,9546

3

-1,9530

4

-1,9365

5

-1,9346

6

-1.8958

7

-1.8953

8

-1,8943

9

-1,8922

10

-1.8628

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 near-optimal networks which correspond to the second and fifth best solutions are also shown in Tables 12 and 13, respectively. These networks give a system-wide 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 decision-maker 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 near-optimal solutions presented here. First, due to biocliar sink limitations, not all of the power plants use indirect co-firing in any given solution; some plants opt for either direct co-firing or no co-firing 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. Near-optimal allocation of biomass and biochar in Mt/y (Solution 2).

PI*

P2*

P3

P4

P5

B1

0

0

0.0750

0

0

B2

0.0200

0

0.1300

0

0

B3

0.1000

0

0

0

0

B4

0

0.1500

0.1000

0

0

B5

0

0

0.1000

0

0

B6

0

0

0

0.2000

0

B7

0

0

0.0450

0,0750

0

B8

0

0

0

0.1000

0

Cl

0

0

0

0,0400

0

C2

0

0

0.0500

0

0

C3

0

0

0 0150

0

0

C4

0

0

00250

0,0350

0

* Direct co-firing option is selected.

Table 13. Near-optimal allocation of biomass and biochar in Mt/y (Solution 5).

PI

P2

P3

P4*

P5

B1

0.0500

0

0

0,0175

0

B2

0

0

0.1000

0,0500

0

B3

0.1000

0

0

0

0

B4

0

0

0.2500

0

0

B5

0

0

0.1000

0

0

B6

0

0

0

0.0125

0,1875

B7

0

0

0

0.1200

0

B8

0

0

0

0.1000

0

Cl

0

0

0.0400

0

0

C2

0.0300

0

0.0200

0

0

C3

0

0

0.0075

0

0

C4

0

0

0.0225

0

0,0375

* Direct co-firing 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 co-firing (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.

Р1

Р2

РЗ

Р4

Р5

В1

В2

вз

В4

В5

В6

В7

В8

С1

С2

СЗ

С4

Legend: White - 0% occurrence; Black - 100% occurrence; Gray - 1-99% occurrence.

respectively) elects not to implement co-firing at all. The presence of such alternatives can potentially allow for more flexible decision-making in practical situations. These features also represent system components that are more sensitive to model assumptions; a decision-maker may seek to acquire more data before making a final selection.

 
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