Correspondence pubs.acs.org/IECR
Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Simplified Model for Source−Sink Matching in Carbon Capture and Storage Systems Munawar A. Shaik* and Amit Kumar Department of Chemical Engineering, IIT Delhi, Hauz Khas, New Delhi 110 016, India
T
an et al.1 proposed a continuous-time optimization model for CO2 source−sink matching in carbon capture and storage (CCS) systems. Their model has different constraints related to source−sink allocation, start times of source and sink connection, capacity of CO2 storage sink, and penalty on CO2 emissions from compensatory power generation. Later, Lee and Chen2 suggested improvement over the model of Tan et al.1 leading to reduced problem size in terms of number of continuous variables and constraints. Our correspondence emphasizes reformulation and simplification of the models proposed by Tan et al.1 and Lee and Chen2 leading to further reduction in problem size.
penalty on CO 2 emissions from compensatory power generation. max ∑ ∑ (TiendBij − Zij)Si − i
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MODIFIED MILP MODEL OF LEE AND CHEN2 Lee and Chen2 eliminated the variable Zij and the relevant constraints in eqs 5−7. They proposed eq 4 differently as given end in eq 10, which enforces Tstart when Bij = 0. ij =Ti
∀i
Tiend(1 − Bij ) ≤ Tijstart ≤ T iend ̀ − T minBij
(1)
The starting time at which the connection is established between source i and sink j should be greater than the starting time of the source i as well as sink j as given in eqs 2 and 3. Tijstart ≥ Tj
∀ i, j
(2)
Tijstart ≥ Ti
∀ i, j
(3)
∀ i, j
∑ (Tiend − Tijstart)Si ≤ Dj
∀ i, j
max ∑ ∑ (Tiend − Tijstart)Si − i
Zij ≤ MBij
∀ i, j
∀ i, j
i
j
Remark 2. The model of Lee and Chen leads to reduction in number of continuous variables and constraints due to elimination of Zij. However, there is no direct correspondence between Bij and Tstart in their model due to the absence of Zij. ij When Bij = 0, Tstart is not enforced to zero in both the models ij of Tan et al.1 and Lee and Chen.2 Moreover, there is no need to arbitrarily enforce Tstart = Tend when Bij = 0. The model can be ij i reformulated without this requirement and still linearity can be maintained as described in the following modification.
(6)
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SIMPLIFIED MILP MODEL In the proposed modification, allocation constraint (1) remains the same. Equations 2 and 3 are replaced by a single constraint in eq 13 where the start time, at which the connection is established between source i and sink j, would be greater than both the start times of source i and sink j.
The amount of CO2 injected into sink j should be less than its maximum storage capacity as given in eq 8. ∀j
i
j
∑ ∑ (Tiend − Tijstart)PC i 2
(7)
∑ (TiendBij − Zij)Si ≤ Dj
(11)
(12) (4)
(5)
Zij ≥ Tijstart − (1 − Bij )M
(10)
∀j
i
The connectivity between source and sink will be viable only if there exists a suitable sink j which will provide connectivity to source i. Continuous variable Zij was defined to linearize the bilinear term Zij = Tstart ij Bij as given in eqs 5−7.
Zij ≤ Tijstart
∀ i, j
Equations 8 and 9 were modified and replaced with eqs 11 and 12. CO2 injected by source i to sink j is accounted in eqs 11 and 12 only if Bij = 1, when Tstart ≤ Tend ij i ; else if Bij = 0 then start (Tend − T ) would be zero. i ij
From an economic prospective the connectivity between source i and sink j will be profitable only when it is more than the minimum viable connectivity duration as given in eq 4. Tiend − Tijstart ≥ T minBij
j
(9)
LINEARIZED MILP MODEL OF TAN ET AL.1 The model proposed by Tan et al.1 is summarized below. The allocation constraint for CO2 source−sink matching is given in eq 1. j
i
Remark 1. When binary variable Bij = 0, then from eq 7 continuous variable Zij is forced to be zero. But when Bij = 0, eq start 4 reduces to Tstart ≤ Tend is not enforced to zero. ij i , and Tij 1 Hence, Tan et al. defined Zij separately in eqs 5−7 and used Zij instead of Tstart in the objective function and other constraints. ij However, the model can be reformulated without using Zij and the corresponding linearization equations can be eliminated.
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∑ Bij ≤ 1
j
∑ ∑ (TiendBij − Zij)PC i
(8)
Objective Function. The objective is to maximize the reduction in CO2 emissions as defined in eq 9 subject to © XXXX American Chemical Society
A
DOI: 10.1021/acs.iecr.8b00704 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Industrial & Engineering Chemistry Research Tijstart ≥ max(Ti , Tj)Bij
∀ i, j
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(13)
∀ i, j
i = Set of CO2 sources j = Set of CO2 sinks Parameters
(14)
C = CO2 emission intensity of compensatory power generation Dj = CO2 storage limit of sink j M = Large number in big-M constraints Si = CO2 flow rate from source i Pi = Power loss due to CCS in source i Tmin = Minimum viable duration of connectivity Ti = Time at which source i begins to operate Tend = Time at which source i ceases to operate i Tj = Time at which sink j begins to operate
Here Tstart is enforced to zero when Bij = 0, unlike in the ij previous two models: that is, eq 4 of Tan et al.1 or eq 10 of Lee and Chen,2 where Tstart can take some arbitrary value. ij As Tijstart now exactly replaces Zij, we eliminated the redundant variable Zij and the relevant linearization constraints in eqs 5−7, while still maintaining linearity in other constraints. The CO2 sink storage constraint, eq 8, and the objective function, eq 9, are modified by replacing Zij directly with Tstart ij as shown in eqs 15 and 16.
∑ (TiendBij − Tijstart)Si ≤ Dj
Variables
∀j (15)
i
max ∑ ∑ (TiendBij − Tijstart)Si − i
∑ ∑ (TiendBij − Tijstart)PC i
j
i
j
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(16)
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RESULTS AND DISCUSSION The proposed simplified MILP model (eqs 1, 13−16) was applied to solve the two case studies presented by Tan et al.1 and the results are compared with Tan et al.1 and Lee and Chen2 in Table 1.
case study 1
discrete variables continuous variables constraints total CO2 captured (Mt)
Tan Lee and et al.1 Chen2
case study 2
simplified model
Tan et al.1
Lee and Chen2
simplified model
10
10
10
18
18
18
21
11
11
37
19
19
68 590
48 590
28 590
118 660
82 660
46 660
Bij = Binary variable, denoting existence of CO2 connection from source i to sink j Tijstart = Positive variable, denoting start time of CO2 connection from source i to sink j
REFERENCES
(1) Tan, R. R.; Aviso, K. B.; Bandyopadhyay, S.; Ng, D. K. S. Continuous-Time Optimization Model for Source-Sink Matching in Carbon Capture and Storage Systems. Ind. Eng. Chem. Res. 2012, 51 (30), 10015−10020. (2) Lee, J.-Y.; Chen, C.-L. Comments on ‘Continuous-Time Optimization Model for Source-Sink Matching in Carbon Capture and Storage Systems’. Ind. Eng. Chem. Res. 2012, 51, 11590−11591. (3) Rosenthal, R. E. GAMS−A User’s Guide; GAMS Development Corporation: Washington, DC, 2012.
Table 1. Comparison of Different Models model statistics
NOMENCLATURE
Sets
If Bij = 1, then the minimum viable connectivity duration constraint, eq 4, is modified as given in eq 14. Tijstart ≤ (Tiend − T min)Bij
Correspondence
The case studies are solved using GAMS3 14.0.1 software using CPLEX solver on a DELL XPS L15 laptop with CORE i5, 2.50 GHz processor, and 6 GB RAM. The objective function value and the CO2 source−sink allocation network obtained are exactly the same, as expected, in all three models. Because of small problem size of the case studies considered, the CPU time is less than 1 s for all models. Compared to the other two models, the proposed simplified model gives a further reduction in the number of constraints, which could be potentially useful when solving larger problems.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Munawar A. Shaik: 0000-0002-4364-483X Notes
The authors declare no competing financial interest. B
DOI: 10.1021/acs.iecr.8b00704 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
