Optimal Source–Sink Matching in Carbon Capture and Storage

Article pubs.acs.org/IECR

Optimal Source−Sink Matching in Carbon Capture and Storage Systems under Uncertainty Yi-Jun He,*,†,‡ Yan Zhang,‡ Zi-Feng Ma,† Nikolaos V. Sahinidis,‡ Raymond R. Tan,§ and Dominic C. Y. Foo∥ †

Institute of Electrochemical and Energy Technology, Department of Chemical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China ‡ Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, United States § Chemical Engineering Department/Center for Engineering and Sustainable Development Research, De La Salle University, 2401 Taft Avenue, 1004 Manila, Philippines ∥ Department of Chemical & Environmental Engineering/Centre of Excellence for Green Technologies, University of Nottingham, Malaysia, 43500 Semenyih, Selangor, Malaysia ABSTRACT: This study addresses the robust optimal source−sink matching in carbon capture and storage (CCS) supply chains under uncertainty. A continuous-time uncertain mixed-integer linear programming (MILP) model with physical and temporal constraints is developed, where uncertainties are described as interval and uniform distributed stochastic parameters. A worst-case MILP formulation and a robust stochastic two-stage MILP formation are proposed to handle interval and stochastic uncertainties, respectively. Then, two illustrative case studies are solved to demonstrate the effectiveness of the proposed models for planning CCS deployment under uncertainty.

1. INTRODUCTION Carbon capture and storage (CCS), which is the process of capturing carbon dioxide (CO2) from large industrial sources (such as oil refineries or fossil fuel-fired power plants), transporting it through pipelines, and storing it in various sinks (such as geological reservoirs and depleted oil or gas wells) is widely accepted as a potential technology for mitigating climate change by reducing industrial CO 2 emissions. CCS has attracted plenty of attention from both academia and industry. Various carbon capture techniques such as postcombustion,1 chemical looping combustion,2 oxy-fuel combustion,3 and integrated gasification combined cycle (IGCC)4 are capable of capturing relatively pure CO2 from industrial sources. Recently, pinch analysis techniques5−9 and mathematical programming methods10−20 have been developed to aid in planning large-scale CCS deployment. One of the core tasks of CCS deployment planning involves finding the optimal source−sink matching with physical and temporal constraints. Pinch analysis tools such as grid-wide carbon composite curve5 or CO2 capture and storage pinch diagram (CCSPD)6 provide good insights for energy planners to set various targets for the CCS problems. These targets may include the minimum extent of CCS retrofit5 or the minimum amount of unutilized CO2 storage capacity.6 Then network design methods such as nearest neighbor algorithm (NNA) are utilized to synthesize the allocation network to meet the targets established by pinch analysis.6 Another important method for planning of CCS deployment is to construct an integer linear programming (ILP) model,10 nonlinear programming (NLP) model,11 or mixed-integer linear/nonlinear programming (MILP/MINLP) model12−20 that is then solved by either deterministic © 2013 American Chemical Society

algorithms such as branch-and-bound method or stochastic algorithms such as a genetic algorithm.20 In addition, a recursive rule-based algorithm has been proposed to heuristically determine near-optimal low-cost pipeline networks.21 It has been demonstrated that inherent uncertainties confronted in CCS systems may affect both the target and network topology as well as costs.7,22 Accounting for such uncertainties is necessary to manage risks in implementing CCS on a large scale. The typical sources of uncertainty in CCS systems can be broadly divided into two categories:7,22−24 (1) CO2 source-related uncertainty, i.e., operating life of sources; and (2) CO2 sink-related uncertainty, i.e., injection rate and storage capacity limits. The optimal source−sink matching network without considering these uncertainties may be not optimal and may even be infeasible in practice. The effect of geological reservoir uncertainty on CO2 transportation and storage infrastructure has been investigated by Middleton et al.,22 who obtained different solutions by solving the optimization problem for different values of uncertain parameters. However, to the best of our knowledge, to date there have been no studies that explicitly account for robustness issues in optimal planning of CCS deployment under uncertainty. In this paper, the continuous time MILP model proposed by Tan et al.15 is reformulated to account for uncertainties. Note that the proposed model is a CO2 allocation model and not an infrastructure model.11−13,22 The effect of uncertainties on the final optimal network configuration of a Received: Revised: Accepted: Published: 778

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3. MILP MODELS WITH UNCERTAINTIES In this section, a deterministic continuous-time MILP model is first presented, followed by a worst-case MILP model and a robust MILP model. 3.1. Deterministic Continuous-Time MILP Model. As in the work of Tan et al.,15,18 the objective is to maximize the amount of CO2 that can be captured and stored in the CCS system given the specified temporal and physical constraints. Taking into account the additional CO2 emissions penalty arising from the need to generate more electricity to compensate for CCS power losses, the objective function is

CCS system are then thoroughly investigated through two illustrative case studies. The remainder of this paper is structured as follows. The source−sink matching problem in CCS systems is first stated and the sources of uncertainty in CCS systems are introduced. Then we propose an MILP model under uncertainty, along with a worst-case formulation as well as a robust stochastic programming formulation. The proposed models are then demonstrated through two illustrative case studies. Finally, conclusions and future research directions are provided.

2. PROBLEM STATEMENT The CCS deployment planning problem addressed in this work is as follows: (1) The CCS system consists of m CO2 sources and n CO2 sinks. (2) Each CO2 source i (i = 1, 2, ..., m) is characterized by four parameters: potential captured CO2 flow rate that corresponds to the available removal from the point source’s flue gas, the starting and ending times of the operating life of each source i, and the power losses resulting from carbon capture retrofit. (3) Each CO2 sink j (j = 1, 2, ..., n) is characterized by two parameters: an upper limit of CO2 storage capacity and the earliest time of availability for storing CO2 in the geological reservoir. (4) The source−sink matching problem requires satisfaction of the following physical constraints: (i) any given CO2 source i may be connected to only one CO2 sink j; (ii) the amount of CO2 stored in each CO2 sink j should not exceed the upper limit of storage capacity of sink j. (5) The following temporal constraints should be also satisfied: (i) the starting time of the connection between a source i and a sink j must be later than the earliest allowable time to begin CO2 capture in source i and the earliest time of availability for storing CO2 in sink j; (ii) a duration of connection between a source and a sink should be longer than a minimum duration from an economic standpoint. (6) A set of the problem parameters, i.e., ending time of the operating life of each source i, upper limit of CO2 storage capacity of each sink j and carbon footprint of compensatory power to make up for CCS energy losses, have uncertain values, and may be represented as interval uncertainties and stochastic uncertainties. The minimum and maximum values are defined for each uncertain parameter. All uncertain parameters are assumed to be uniformly distributed. (7) The objective is to determine the maximum amount of CO2 captured and stored in sinks given the above specified physical and temporal constraints. The optimization task should determine the source−sink matching network and starting time at which a source links to a sink. Both the objective function and constraints should account for these inherent uncertainties. The above problem statement extends to the uncertain case the problem stated by Tan et al.,15 who did not consider any uncertainties (items 6 and 7 above.)

max ∑ ∑ (Tiend − Tijstart)Bij Si − i

j

∑ ∑ (Tiend − Tijstart)BijPC i i

j

(1)

where Tstart is the time at which source i starts providing CO2 to ij sink j; Tend is the time at which source i ceases to operate; Si is i the CO2 capture rate from source i; Bij ∈ {0,1} is a binary variable, denoting the existence of a connection from source i to sink j; Pi is the power loss from source i if CO2 is captured; and C is assumed to be the carbon footprint of compensatory power to make up for CCS energy losses. The power loss incurred from retrofit includes requirements for CO2 capture and compression, and eq 1 thus implicitly accounts for the carbon footprint of pumping the CO2 via pipeline to the sinks. In practice, sources and sinks will be grouped according to geographical clusters with similar distances between source− sink pairs and hence similar specific pumping energy requirements.25 In this work, it is further assumed that once a connection between a source i and a sink j is established, it remains in place for the remainder of the planning horizon. The physical constraint that any given CO2 source i can be connected to only one CO2 sink j is represented as

∑ Bij ≤ 1

∀i (2)

j

Each CO2 sink j is also subjected to the following physical constraint

∑ (Tiend − Tijstart)BijSi ≤ Dj

∀j

i

(3)

where Dj denotes the limits of CO2 storage capacity of sink j. The temporal constraints shown in eqs 4 and 5 enforce the requirement that the starting time Tstart must be later than the ij earliest allowable time Ti to begin CO2 capture in source i and the earliest time of availability Tj for storing CO2 in sink j

Tijstart ≥ Ti ∀ i , j

(4)

Tijstart

(5)

≥ Tj ∀ i , j

Another temporal constraint on duration of connection between a source and a sink is expressed as Tiend − Tijstart ≥ T minBij ∀ i , j

(6)

min

where T is the minimum duration of viable connectivity. It is noted that the bilinear product of variables Tstart and Bij ij makes eqs 1 and 3 nonlinear. By accounting for parameter uncertainties, the reformulation approach through adding extra constraints proposed by Lee and Chen16 is not applicable. Hence, an exact linearization method26 by introducing a new continuous variable Zij is used to eliminate the bilinear terms with the following constraints: 779

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Zij ≤ Tijstart ∀ i , j

(7)

Zij ≤ MBij ∀ i , j

(8)

Zij ≥ Tijstart − (1 − Bij )M ∀ i , j

(9)

Zij ≥ 0 ∀ i , j

fmax − fmin ≤ εmax

where εmax is a prespecified allowable maximum deviation of the objective function. Setting εmax equal to +∞ implies that the decision-maker does not wish to add any restriction on the objective function deviation. It should be noted that, although it is easy to determine the minimum and maximum values of the objective function and constraints in this work, the optimization problem formulation is not so straightforward to solve because it involves nonlinear expressions that involve the uncertain parameters. A multilevel programming model needs to be established, and nesting optimization techniques need to be developed for the general case. 3.3. Robust MILP Formulation. For the case of uncertainties described by probability distributions, a robust stochastic two-stage MILP model is formulated. The optimization variables in the CCS deployment model could be broadly partitioned into design and operating variables, with the design variables consisting of the binary variables Bij that denote the existence of a connection from source i to sink j and operating variables that consist of continuous variables Tstart ij denoting the time at which source i links to sink j as well as the above-introduced auxiliary continuous variable Zij. The design and operating variables are often treated as first-stage and second-stage variables, respectively.27 In the two-stage modeling framework, the first-stage variables have to be decided before the actual realization of the uncertain parameters. However, the traditional two-stage stochastic approach often accounts only for the expected cost without any variability of these recourse costs. In this study, a robustness measure, namely upper partial mean (UPM), is introduced into the objective function for balancing the trade-off between expected value and variability.28 UPM is an asymmetric risk measure that would penalize only costs inferior to the expected or average value. UPM is defined as

(10)

where M is an arbitrary large number and is set equal to the duration of the planning horizon. Therefore, by incorporating the above exact linearization, eqs 1 and 3 are reformulated into eqs 11 and 12, respectively. max ∑ ∑ (TiendBij − Zij)(Si − PC i ) i

(11)

j

∑ (TiendBij − Zij)Si ≤ Dj

∀j (12)

i

3.2. Worst-Case MILP Formulation. If the uncertain parameters are described as intervals, the ending time of the operating life of each source i, upper limit of CO2 storage capacity of each sink j, and carbon footprint of compensatory power to make up for CCS energy losses could be represented end as Tend ∈ [Tend i i,min, Ti,max], Dj ∈ [Dj,min, Dj,max], and C ∈ [Cmin, Cmax], respectively. Given any values of optimization variables, the minimum and maximum of the objective function in eq 11 can be computed under the uncertain parameter ranges as

∑ ∑ (Tiend ,minBij − Zij)(Si − PC i max )

fmin =

i

fmax =

(13)

j

∑ ∑ (Tiend ,maxBij − Zij)(Si − PC i min) i

(14)

j

The uncertain objective function shown in eq 11 can be reformulated in a deterministic form as max λ = αfmax + (1 − α)fmin

(15)

where α denotes the risk factor that is defined as α=

Δ̅ =

λ − fmin fmax − fmin

(16)

∑ (Tiend ,maxBij − Zij)Si ≤ Dj ,min i

1 K

K

∑ Δk (20)

k=1

with

and λ denotes the deterministic objective function value. It should be noted that α and λ are within the ranges of [0, 1] and [f min, f max], respectively. If α is set equal to 1, the objective λ becomes equal to f max, denoting that the decision-maker wants to obtain an objective value equal to f max but would face maximum risk factor; if α is set equal to 0, then λ becomes equal to f min, denoting that the decision-maker is content with an objective value equal to f min with zero risk factor. The larger the value of α, the higher the risk assumed by the investor. To ensure that all temporal and physical constraints are exactly satisfied for any values of the uncertain parameters within their interval ranges, eqs 6 and 12 are reformulated into eqs 17 and 18, respectively, as follows: start Tiend ≥ T minBij ∀ i , j ,min − Tij

(19)

⎧ ⎪ 1 Δk = max⎨0, ⎪ K ⎩ −

k′ ) ∑ ∑ ∑ (Tiend,k′Bij − Zijk′)(Si − PC i k′

i

j

⎫ ⎪

k ∑ ∑ (Tiend,kBij − Zijk)(Si − PC i )⎬ ⎪ i

j

⎭

(21)

where K is the number of samples, Δk the positive deviation of the kth sample’s cost from the expected recourse cost, and Δ̅ the expectation of the positive deviations over all samples. Equation 21 indicates that for the maximizing optimization problem shown in eq 11, if the objective value of the kth sample is less than the expected value, Δk would be greater than 0. Note that UPM provides an intuitive measure of variability and avoids the need for a nonlinear formulation compared to using variance as the risk measure. Hence, in the robust optimization framework, the objective function shown in eq 11 is modified by introducing a weighted contribution of UPM and additional constraints that form a robust two-stage stochastic programming MILP formulation as follows:

(17)

∀j (18)

Note that eqs 17 and 18 represent conservative or pessimistic constraints. In addition, an extra constraint is introduced to restrict the deviation of objective function in a specified range: 780

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1 K

4. CASE STUDIES Two illustrative case studies adapted from Tan et al.15 are solved under uncertainty. For all case studies, the intervals of all uncertain parameters are set to a range from 90% to 110% of their nominal values; all stochastic uncertain parameters are assumed to be uniformly distributed with a range from 90% to 110% of their nominal values. Both worst-case and robust MILP models are implemented and solved in GAMS with CPLEX directly as the size of the formulation is moderate in our case studies. However, in the case of very large robust MILP problems, a heuristic solution strategy or efficient exact algorithms such as combination of Lagrangian relaxation and Benders decomposition should be applied.28 All computations were carried out on a PC with a 2.60 GHz processor and 8 GB of RAM. Before performing robust optimization, stochastic convergence analysis was conducted to determine the number of samples. Figures 1 and 2 show the convergence of the

k ∑ ∑ ∑ (Tiend,kBij − Zijk)(Si − PC i ) k

1 K

−ρ

Article

i

j

K

∑ Δk (22)

k=1

subject to

∑ Bij ≤ 1

∀i (23)

j

Δk ≥ −

1 K

k′ ) ∑ ∑ ∑ (Tiend,k′Bij − Zijk′)(Si − PC i k′

i

j

k ∑ ∑ (Tiend,kBij − Zijk)(Si − PC i ) ∀ k i

(24)

j

∑ (Tiend,kBij − Zijk)Si ≤ Djk

∀ j, k

i

(25)

Tijstart, k ≥ Ti ∀ i , j , k

(26)

Tijstart, k ≥ Tj ∀ i , j , k

(27)

Tiend, k − Tijstart, k ≥ T minBij ∀ i , j , k

(28)

Zijk ≤ Tijstart, k ∀ i , j , k

(29)

Zijk ≤ MBij ∀ i , j , k

(30)

Zijk ≥ Tijstart, k − (1 − Bij )M ∀ i , j , k

(31)

Tijstart, k ≥ 0 ∀ i , j , k

(32)

Zijk ≥ 0 ∀ i , j , k

(33)

Δk ≥ 0 ∀ k

(34)

Bij ∈ {0, 1} ∀ i , j

(35)

Figure 1. Convergence of expected value of objective function with the number of samples for case study 1 using robust stochastic MILP model with ρ = 0.

where ρ is prescribed goal programming weight. In general, with increasing ρ, the values of expectation and UPM will decrease. The first and second terms in eq 22 denote the expected amount of captured and stored CO2 and its variability, respectively. By investigating different values of ρ, the relationship between expectation and UPM can be extracted, which provides useful trade-off information for final decisionmaking. Note that the expectation is approximated by the sample average according to the law of large numbers. It is wellknown that the number of samples is directly linked to the fidelity of the expectation approximation. Theoretically, an infinite number of samples is needed for an exact calculation of the expectation. However, fast computation calls for minimizing the number of samples. This work implements a Monte Carlo (MC) method in the GAMS modeling system. Samples of three stochastic parameters are generated, and a stochastic convergence analysis is performed to determine the minimum number of samples required for convergence. Quasi-Monte Carlo methods such as Latin hypercube sampling and Hammersley sequence sampling could also be adopted to further reduce the number of samples.

Figure 2. Convergence of expected value of objective function with the number of samples for case study 2 using robust stochastic MILP model with ρ=0. 781

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expected value of objective function with the number of samples for case studies 1 and 2, respectively. Clearly, K = 400 is sufficient to accurately approximate the expectation for our two case studies. 4.1. Case Study 1. In this case study, there are five CO2 sources and two CO2 sinks with relevant nominal data as shown in Table 1. The minimum duration of viable

should be carefully selected for decision-making. The deterministic MILP model’s optimal objective value is 570 Mt,15 which implies the optimal configuration of the CCS network without considering uncertainties would suffer high risk and might violate some physical and temporal constraints. Figure 4 shows the expected value as a function of UPM for Case Study 1 using the robust MILP formulation. It is clearly

Table 1. CO2 Source and Sink Data for Case Study 1 source 1 2 3 4 5 total sink A B total

flow rate, Si (Mt/y) 3 4 2 7 4 20

time of operation, Ti and Tend (y) i

power loss, Pi (MW)

maximum capture (Mt)

0−30 0−40 5−30 10−40 0−40 n/a starting time, Tj (y)

100 90 100 160 70 50 200 210 220 160 690 670 maximum storage, Dj (Mt)

0 10 n/a

200 400 600

connectivity is assumed to be 20 years, and the nominal average carbon footprint of compensatory power generation is assumed to be C = 0.001 Mt y−1 MW−1.15 The planning horizon is 40 years, which corresponds to the end life of source 5. Figure 3 shows the objective function value as a function of

Figure 4. Expected value vs UPM for Case Study 1 using robust stochastic MILP model.

seen that as the expected value increases, the robustness measure UPM will also increase. In other words, expectation and UPM are, by nature, conflicting. Improvement of one objective will lead to degradation of the other objective. Thus, increasing the amount of CO2 emissions reduction incurs more risk. It can also be observed that, as the expectation varies from 483 to 521 Mt, the UPM increases from 0.14 to 9.58 Mt. Both expectation and UPM remain the same when the value of ρ is set below 2. The relationship between expectation and UPM shows an approximately piecewise linear characteristic: (1) At low expected value, i.e., [483, 493] Mt, the relationship shows a high slope of 11.4, indicating that a small increase of UPM would lead to a sharp increase of expectation. (2) At medium expected value, i.e., [493, 504] Mt, a moderate slope of 5.0 is calculated, indicating a transition zone in which a medium increase of UPM would lead to an intermediate increase of expectation. (3) At high expected value, i.e., [504, 521] Mt, a low slope of 2.6 is observed, indicating that a large increase of UPM would lead to a small increase of expectation. In general, this trend indicates that as risk level increases, there are diminishing returns to the optimal amount of CO2 captured and stored; or, conversely, seeking more ambitious levels of emissions reduction incurs rapidly increasing levels of risk. Although the solutions belonging to the transition zone provide potential trade-off between expectation and UPM, more preference information should be introduced to final decision-making. Moreover, a multicriteria decision-making method would be helpful to select a solution. Table 2 shows the optimal allocation network configuration (i.e., first-stage design variables) using the robust MILP formulation, in which sources 1 and 3 are linked to sink A, while sources 2, 4, and 5 are linked to sink B. It is found that (1) the optimal allocation network configuration stays the same for all setting of ρ and (2) the time at which allocation of CO2

Figure 3. Objective function value vs risk factor under different εmax for Case Study 1 using worst-case MILP model.

risk factor α under different allowable maximum deviation εmax for Case Study 1 using the worst-case MILP formulation. It is observed that when εmax is set above 120 Mt, the objective function values remain the same for different risk factors and the maximum and minimum of objective function are 401 and 520 Mt, respectively, indicating a quite large performance gap of 119 Mt. It means that the decision-maker could obtain an objective value equal to 401 Mt with zero risk factor, while one could obtain an objective value equal to 520 Mt with a high risk factor. If the value of εmax decreases below 120 Mt, the attainable maximum and minimum objective value decreases sharply as well. Note that εmax represents a robustness measure to some extent; the larger the value of the performance gap, the bigger the performance variability. Hence, both α and εmax 782

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Table 2. Optimal Allocation Network Structure of Robust MILP Model for Case Study 1 sources

sink A

sink B

1 2 3 4 5

1 0 1 0 0

0 1 0 1 1

Table 4. CO2 Source and Sink Data for Case Study 2 source 1 2 3 4 5 6 total sink

stream from source i to sink j begins (i.e., second-stage operating variables) are scenario-dependent and decided upon the realizations of uncertain parameters and would also vary with values of ρ. For the worst-case MILP formulation, it is found that the obtained optimal allocation network configuration strongly depends on the value of εmax and is almost independent of the risk factor α. When the value of εmax is approximately greater than or equal to 120 Mt, the obtained optimal allocation network configuration remains the same and is shown in Table 3. It is observed that source 2 is linked to sink

A B C total

flow rate, Si (Mt/y) 3 4 2 7 4 2 22

time of operation, Ti and Tend (y) i

power loss, Pi (MW)

maximum capture (Mt)

0−30 0−40 5−30 10−40 0−40 10−50 n/a starting time, Tj (y)

100 90 100 160 70 50 200 210 220 160 160 80 850 750 maximum storage, Dj (Mt)

0 10 15 n/a

200 400 250 850

Table 3. Optimal Allocation Network Structure of WorstCase MILP Model for Case Study 1 (εmax ≥ 120 Mt) sources

sink A

sink B

1 2 3 4 5

0 1 0 0 0

0 0 0 1 1

A, while sources 4 and 5 are linked to sink B. When the value of εmax is less than 20 Mt, there is no feasible solution. When the value of εmax is within the range of [20, 120] Mt, different optimal allocation network configurations are obtained in which only a subset of connections between a source and a sink exist. Compared with the results of the robust MILP, only some connections between sources and sinks exist in the worst-case MILP solution, which implies that the robust MILP formulation would provide a more practical feasible solution to this problem. Although it is difficult to compare the objective functions between the worst-case and robust MILP formulations directly, the optimal configuration obtained by the robust MILP formulation outperforms that found by the worst-case MILP formulation to some extent. In the optimal network configuration of the deterministic MILP model, sources 2 and 3 are linked to sink A, while sources 1, 4, and 5 are linked to sink B.15 Comparing the results of these three formulations, it is clear that the uncertain parameters posed in physical and temporal constraints would significantly affect the allocation network configuration as well as operating conditions. 4.2. Case Study 2. In this case study, there are six CO2 sources and three CO2 sinks with relevant nominal data as shown in Table 4. The minimum duration of viable connectivity is assumed to be 20 years and the nominal average carbon footprint of compensatory power generation is assumed to be C = 0.0025 Mt y−1 MW−1.15 The planning horizon is 50 years, which corresponds to the end life of source 6. Figure 5 shows the objective function value as a function of the risk factor α under different allowable maximum deviations εmax for Case Study 2 using the worst-case MILP formulation. As in Case Study 1, it is observed that when εmax is set above

Figure 5. Objective function value vs risk factor under different εmax for Case Study 2 using worst-case MILP model.

170 Mt, the objective function values stay the same for different risk factor values and the maximum and minimum of the objective function are 487 and 653 Mt, respectively, showing a quite large performance gap of 166 Mt. Hence, the decisionmaker could obtain an objective value equal to 487 Mt with zero risk factor, while one could obtain an objective value of 653 Mt with a risk factor of 1. If the value of εmax decreases below 170 Mt, the attainable maximum and minimum objective value would also show a decreasing trend. The deterministic MILP model’s optimal objective value is 596 Mt.15 Unlike Case Study 1, the attainable maximum objective value of the worstcase model is significantly greater than that of the deterministic model, indicating that the decision-maker can obtain an objective value equal to 596 Mt with a risk factor of 0.66 and the corresponding performance gap is 166 Mt. The expected value as a function of UPM for Case Study 2 using the robust MILP formulation is shown in Figure 6. It is observed that, as the expectation varies from 545 to 579 Mt, the UPM increases from 0.30 to 8.28 Mt. Similar to Case Study 1, both the expectation and the UPM remain the same when the value of ρ is set below 2. Moreover, the relationship between expectation and UPM also shows an approximately piecewise linear characteristic, as in the previous example: (1) At low expected value, i.e., [545, 550] Mt, a high slope of 18.2 is calculated, indicating that a small increase of UPM would lead to a sharp increase of expectation. (2) At medium expected 783

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Table 6. Optimal Allocation Network Structure of WorstCase MILP Model for Case Study 2 (εmax ≥ 170 Mt)

value, i.e., [550, 560] Mt, a moderate slope of 7.6 is calculated, indicating a transition zone in which a medium increase of UPM would lead to an intermediate increase of expectation. (3) At high expected value, i.e., [560, 579] Mt, a low slope of 2.8 is calculated, indicating that a large increase of UPM would lead to a small increase of expectation. As stated in Case Study 1, Figure 6 reveals only the relationship between expectation and UPM, where CO2 reduction levels decline at high risk levels; selection of an optimal solution requires additional decision-maker preference information as well as the use of advanced multicriteria decision-making methods. Table 5 shows the optimal allocation network configuration using the robust MILP formulation. In this solution, sources 1

sink A

sink B

sink C

1 0 1 0 0 0

0 1 0 1 0 0

0 0 0 0 1 1

sink B

sink C

1 0 1 0 0 0

0 1 0 0 1 1

0 0 0 1 0 0

5. CONCLUSIONS In this work, we have developed two MILP formulations for optimal planning of CCS deployment under uncertainty. Carbon allocation networks can be obtained by explicitly considering parameter uncertainties. Two illustrative case studies have been solved to demonstrate the applicability of the proposed models. It is found that uncertain parameters significantly affect not only the CCS allocation network configuration but also the operating conditions. Moreover, the optimal allocation networks obtained by a deterministic MILP model face high risk and violate both physical and temporal constraints for some extreme realizations of the uncertain parameters. Although both the worst-case and robust MILP formulation provide feasible network configurations, it is worth noting that the robust MILP formulation resulted in a more practical solution alternative. These results show that the modeling of uncertainty is critical in the optimal planning of source−sink matches in CCS systems. Future work should focus on general superstructure formulation for optimal source−sink matching design problem by explicitly considering economic objectives as well as network reliability and flexibility objectives. The distribution of uncertainties needs to be accurately characterized based on geological survey and power plant operating data. Moreover, for efficient and effective handling of large-scale optimal planning problems of CCS deployment under uncertainty, advanced derivative-based optimization algorithms or stochastic optimization methods may need to be explored for dealing with future large-scale CCS planning formulations. In addition, for problems involving multiple overlapping geographic clusters of sources and sinks, the model formulation may be modified to account for unique energy losses for pumping for each potential source−sink pair.

Table 5. Optimal Allocation Network Structure of Robust MILP Model for Case Study 2 1 2 3 4 5 6

sink A

1 2 3 4 5 6

value of εmax is less than 20 Mt, there is no feasible solution. When the value of εmax is within the range of [20, 170] Mt, different optimal allocation network configurations are obtained, in which only a subset of connections between a source and a sink exist. The optimal allocation network configuration using deterministic MILP model is that sources 2 and 3 are linked to sink A, sources 1, 4, and 5 are linked to sink B, and source 6 is linked to sink C.15 Comparing the results of these three formulations once again indicates that the uncertain parameters posed in physical and temporal constraints significantly affect the allocation network configuration as well as operating conditions.

Figure 6. Expected value vs UPM for Case Study 2 using robust stochastic MILP model.

sources

sources

and 3 are linked to sink A, sources 2 and 4 are linked to sink B, and sources 5 and 6 are linked to sink C. As with Case Study 1, it is also found for Case Study 2 that (1) the optimal allocation network configuration stays the same for all settings of ρ and (2) the times at which allocation of the CO2 stream from source i to sink j begins are scenario-dependent and would vary with values of ρ. For the worst-case MILP formulation, it is found that the obtained optimal allocation network configuration is sensitive to the value of εmax and not to the risk factor α. When the value of εmax is approximately set greater than or equal to 170 Mt, the obtained optimal allocation network configuration stays the same and is shown in Table 6. It is observed that sources 1 and 3 are linked to sink A, sources 2, 5, and 6 are linked to sink B, and source 4 is linked to sink C. When compared with Case Study 1, it is found that any one source is linked to one sink, again indicating that optimal solutions using worst-case MILP formulation is very sensitive to case-specific details. When the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (Y. J. He); yanzhang.zju@gmail. com (Y. Zhang); [email protected] (Z. F. Ma); sahinidis@cmu. 784

dx.doi.org/10.1021/ie402866d | Ind. Eng. Chem. Res. 2014, 53, 778−785

Industrial & Engineering Chemistry Research

Article

(9) Ooi, R. E. H.; Foo, D. C. Y.; Ng, D. K. S.; Tan, R. R. Planning of Carbon Capture and Storage with Pinch Analysis Techniques. Chem. Eng. Res. Des. 2013, 91 (12), 2721−2731. (10) Turk, G. A.; Cobb, T. B.; Jankowski, D. J.; Wolsky, A. M.; Sparrow, F. T. CO2 transport: A new Application of the Assignment Problem. Energy 1987, 12, 123−130. (11) Benson, H. Y.; Ogden, J. M. Mathematical Programming Techniques for Designing Minimum Cost Pipeline Networks for CO2 Sequestration. Proceedings of the 6th International Conference on Greeenhouse Gas Control Technologies, Kyoto, Japan, October 1−4, 2002; pp 627−632. (12) Middleton, R. S.; Bielicki, J. M. A Comprehensive Carbon Capture and Storage Infrastructure Model. Energy Procedia 2009, 1, 1611−1636. (13) Middleton, R. S.; Bielicki, J. M. A Scalable Infrastructure Model for Carbon Capture and Storage: SimCCS. Energy Policy 2009, 37, 1052−1060. (14) Pękala, Ł. M.; Tan, R. R.; Foo, D. C. Y.; Jeżowski, J. M. Optimal Energy Planning Models with Carbon Footprint Constraints. Appl. Energy 2010, 87, 1903−1910. (15) 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, 10015−10020. (16) 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. (17) Middleton, R. S.; Kuby, M. J.; Wei, R.; Keating, G. N.; Pawar, R. J. A Dynamic Model for Optimally Phasing in CO2 Capture and Storage Infrastructure. Environ. Modell. Software 2012, 37, 193−205. (18) Tan, R. R.; Aviso, K. B.; Bandyopadhyay, S.; Ng, D. K. S. Optimal Source−Sink Matching in Carbon Capture and Storage Systems with Time, Injection Rate, and Capacity Constraints. Environ. Prog. Sustainable Energy 2013, 32, 411−416. (19) Fimbres Weihs, G. A.; Wiley, D. E.; Ho, M. Steady-State Optimisation of CCS Pipeline Networks for Cases with Multiple Emission Sources and Injection Sites: South-East Queensland Case Study. Energy Procedia 2011, 4, 2748−2755. (20) Fimbres Weihs, G. A.; Wiley, D. E. Steady-state design of CO2 pipeline networks for minimal cost per tonne of CO2 avoided. Int. J. Greenhouse Gas Control 2012, 8, 150−168. (21) Kazmierczak, T.; Brandsma, R.; Neele, F.; Hendriks, C. Algorithm to Create a CCS Low-cost Pipeline Network. Energy Procedia 2009, 1, 1617−1623. (22) Middleton, R. S.; Keating, G. N.; Viswanathan, H. S.; Stauffer, P. H.; Pawar, R. J. Effects of Geologic Reservoir Uncertainty on CO2 Transport and Storage Infrastructure. Int. J. Greenhouse Gas Control 2012, 8, 132−142. (23) Holloway, S. Carbon Dioxide Capture and Geological Storage. Philos. Trans. R. Soc., A 2007, 365, 1095. (24) Bachu, S.; Bonijoly, D.; Bradshaw, J.; Burruss, R.; Holloway, S.; Christensen, N. P.; Mathiassen, O. M. CO2 Storage Capacity Estimation: Methodology and Gaps. Int. J. Greenhouse Gas Control 2007, 1, 430−443. (25) Nugent, J. A.; Nai, C. C. Prospects for Carbon Capture and Storage in Southeast Asia. Asian Development Bank: Mandaluyong City, Philippines, 2013. (26) Glover, F. Improved Linear Integer Programming Formulations of Nonlinear Integer Problems. Manage. Sci. 1975, 22, 455−460. (27) Sahinidis, N. V. Optimization under Uncertainty: State-of-theart and Opportunities. Comput. Chem. Eng. 2004, 28, 971−983. (28) Ahmed, S.; Sahinidis, N. V. Robust Process Planning under Uncertainty. Ind. Eng. Chem. Res. 1998, 37, 1883−1892.

edu (N. V. Sahinidis); [email protected] (R. R. Tan); [email protected] (D. C. Y. Foo). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (21006084). NOMENCLATURE

Uncertain Parameters

Dj CO2 storage limit of sink j C Carbon footprint of compensatory power generation Tend Ending time at which source i ceases to operate i Nominal Parameters

C K M Pi Si Tmin Ti Tj α ρ

Carbon footprint of compensatory power generation Number of samples Large positive number Power loss due to CCS in source i CO2 flow rate from source i Minimum duration of viable connectivity Starting time at which source i begins to operate Starting time at which sink j begins to operate Risk factor for worst-case MILP formulation Goal programming weight for robust MILP formulation

Optimization Variables

Bij Tstart ij Zij Δk

■

Binary variable denoting the existence of a connection from source i to sink j Time at which allocation of CO2 stream from source i to sink j begins Time at which allocation of CO2 stream from source i to sink j begins only if the connection between them exists positive deviation of the kth sample’s cost from the expected recourse cost for robust MILP formulation

REFERENCES

(1) Wang, M.; Lawal, A.; Stephenson, P.; Sidders, J.; Ramshaw, C. Post-combustion CO2 Capture with Chemical Absorption: A State-ofthe-Art Review. Chem. Eng. Res. Des. 2011, 89, 1609−1624. (2) Wall, T.; Liu, Y.; Spero, C.; Elliott, L.; Khare, S.; Rathnam, R.; Zeenathal, F.; Moghtaderi, B.; Buhre, B.; Sheng, C.; Gupta, R.; Yamada, T.; Makino, K.; Yu, J. An Overview on Oxyfuel Coal CombustionState of the Art Research and Technology development. Chem. Eng. Res. Des. 2011, 87, 1003−1016. (3) Fan, L. S.; Zeng, L.; Wang, W.; Luo, S. Chemical Looping Processes for CO2 Capture and Carbonaceous Fuel Conversion − Prospect and Opportunity. Energy Environ. Sci. 2012, 5, 7254−7280. (4) Falcke, T. J.; Hoadley, A. F. A.; Brennan, D. J.; Sinclair, S. E. The Sustainability of Clean Coal Technology: IGCC with/without CCS. Process Saf. Environ. Prot. 2011, 89, 41−52. (5) Tan, R. R.; Ng, D. K. S.; Foo, D. C. Y. Pinch Analysis Approach to Carbon-Constrained Planning for Sustainable Power Generation. J. Cleaner Prod. 2009, 17, 940−944. (6) Shenoy, A. U.; Shenoy, U. V. Targeting and Design of Energy Allocation Networks with Carbon Capture and Storage. Chem. Eng. Sci. 2012, 68, 313−327. (7) Diamante, J. A. R.; Tan, R. R.; Foo, D. C. Y.; Ng, D. K. S.; Aviso, K. B.; Bandyopadhyay, S. A Graphical Approach for Pinch-Based Source−Sink Matching and Sensitivity Analysis in Carbon Capture and Storage Systems. Ind. Eng. Chem. Res. 2012, 52, 7211−7222. (8) Foo, D. C. Y., Ng, D. K. S., Tan, R. R.; Smith R. Automated Targeting Technique to Carbon-Constrained Planning for Carbon Sequestration Retrofit in the Power Sector, International Conference on Applied Energy (ICAE2010), Singapore, 2010. 785

dx.doi.org/10.1021/ie402866d | Ind. Eng. Chem. Res. 2014, 53, 778−785