Robust Optimization for Process Synthesis and Design of

Article pubs.acs.org/IECR

Robust Optimization for Process Synthesis and Design of Multifunctional Energy Systems with Uncertainties Harresh Kasivisvanathan,*,† Aristotle T. Ubando,*,‡ Denny K. S. Ng,*,† and Raymond R. Tan*,§ †

Department of Chemical and Environmental Engineering / Centre of Excellence for Green Technologies, The University of Nottingham, Malaysia Campus, Broga Road, 43500 Semenyih, Selangor, Malaysia ‡ Mechanical Engineering Department and §Chemical Engineering Department, Center for Engineering and Sustainable Development Research, De La Salle University, 2401 Taft Avenue, Manila, Philippines S Supporting Information *

ABSTRACT: Generally, synthesis and design of an optimal process is a challenging task. The procedure includes specifying and optimizing system configurations in order to achieve a certain aspect, such as maximizing economic performance, minimizing environmental impact, etc. However, uncertainties of the design parameters (e.g., volatility of raw materials and products price, variability of feedstock supply and product demand, etc.) may undermine the effectiveness of such systematic design approaches. In response to this issue, various optimization works have been presented to address such problems. In this paper, a robust mixed integer linear programming (MILP) with input−output model is presented to aid decision-makers in addressing process synthesis problems due to uncertainties that arise from variation in feedstock supply and product demand. This work primarily encompasses multifunctional energy systems that can be described by a system of linear equations, which entails “black box” modeling. The robust model helps to determine the design capacity of each process unit in a flexible network which involves sizing of equipment. This network is assumed to be able to operate in all uncertain scenarios considered, with a minimum cost of the plant. In addition, the intended model also helps to determine the requirement to operate additional equipment in a plant in the presence of uncertainties. This is especially true with designs of plants that are incapable of meeting any increase in demand. These aspects of the work are novel and an important contribution as such analysis is not available in the literature, to date. Three case studies that include a polygeneration plant and palm oil based integrated biorefinery are presented to demonstrate the proposed novel approach in a more descriptive manner.

1. INTRODUCTION Early design stages in process systems engineering are characterized by limited knowledge for decision-making in the presence of potential uncertainties. Assumptions and justifiable experience-based engineering judgment become necessary to aid decision-makers in determining strategies to synthesize the involved chemical processes. Uncertainties may arise in various forms, concerning either the process model itself or the environmental factors that ultimately restrict the operational flexibility of a design. For example, volatility of raw materials and products price, variability of feedstock supply and product demand, kinetic and transfer coefficients, etc.1 will influence the process design. Conventionally, these uncertainties are dealt with via the use of heuristics, expertise from experienced design engineers and consultants who recommend empirical arbitrary design factors from nominal limits. This is normally done with the definition and sizing of equipment based on computer aided steady state simulation models, and followed by determination of the choice and design of the control system.2 However, solutions of this nature do not guarantee optimality, and they may lead to unreasonably conservative designs.3 Violation of key assumptions and design parameters used during design often results in process bottlenecks and further design infeasibility, which are unfavorable. This may also require changes in design after the system becomes operational.4 Uncertainties involve a number of internal and external key factors at various design stages of a © 2014 American Chemical Society

process. This essentially requires explicit and systematic approaches in order to make the best decision for the process. In the past few decades, researchers have extensively explored this area and proposed several approaches to quantitatively and qualitatively address the decision-making under uncertainty.3 Various formulations have been developed addressing operational concerns such as process flexibility, 5−8 process controllability,9,10 and process robustness,2,11,12 as well as other issues in relation to the value of information.13−15 Grossmann and Sargent5 proposed a strategy for the optimum design of chemical plants that is formulated as a nonlinear program. Based on the proposed strategy,5 the uncertain parameters are first expressed as bounded variables. Next, an efficient method is derived to solve the constrains which do not change with parameters. On the other hand, Halemane and Grossmann6 presented a two-stage nonlinear infinite program in which the infinite number of inequality constraints is satisfied. The major objectives of their work6 were to ensure the optimality and feasibility of operation in a range of parameter values. Later, Straub and Grossmann7 presented a nonlinear model that aids in evaluating and optimizing the probability of operating a feasible design. It is represented by stochastic Received: Revised: Accepted: Published: 3196

June 9, 2013 December 22, 2013 January 3, 2014 January 3, 2014 dx.doi.org/10.1021/ie401824j | Ind. Eng. Chem. Res. 2014, 53, 3196−3209

Industrial & Engineering Chemistry Research

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flexibility in which the feasible region with uncertain parameters is shown in the form of a cumulative distribution. A stochastic programming problem was also presented by Pistikopoulos and Ierapetritou8 as a two-stage problem which determines the design with maximum revenue and ensuring design feasibility in parallel. In comparison to previous approaches, discretizing uncertainty is not required to simultaneously obtain a feasible design with an optimal economic performance. The variety of process synthesis and design problems that face optimization issues due to uncertainties have received the attention of many solution seekers who have incorporated various methods to address this concern. They mainly include, but are not limited to, fuzzy optimization methods,16,17 chance constrained programming,18−20 fuzzy chance constrained programming, stochastic programming,1,21−23 stochastic dynamic programming, and robust programming. Fuzzy optimization approach is a mathematical programming that solves for multiobjective functions and trades-off between uncertain parameters which are often contradictory in nature. In recent years, many researchers presented fuzzy optimization methods in their works. For example, Tay et al.24 and Kasivisvanathan et al.25 presented a fuzzy optimization approach to synthesize and design a sustainable integrated biorefinery with maximum economic potential and minimum environmental impact. Chance constrained programming, on the other hand, describes constraints in the form of probability levels, and this nature allows for multiobjective analysis. The decision-making here can take a managerial level, as it requires a trade-off between minimization of risk and cost or profit as part of a multiobjective analysis.19 Some earlier works that were done adapting chance constrained programming include capital budgeting,26,27 production planning,28 and process optimization.29 The fuzzy chance constrained programming problem is an extension from chance constrained programming in the presence of ambiguous and vague information.30 Problems of this nature have earned the attention of many researchers in the field such as Liu and Iwamura31 and Luhandjula,32,33 who have derived different methods to solve them. These techniques account for different scenarios involving fuzziness and randomness of parameters. Stochastic programming is solved with the same methodology of chance constrained programming29 for decision-making under uncertainties. The uncertanties and risks associated with decisions are modeled in a form suitable for optimization. Applicability of stochastic programming is restricted to the analysis of real-life applications due to the limited information on probability distributions of uncertain parameters.34−37 In the stochastic dynamic programming sense, it is a multistage decision procedure where the global optimum is achieved by optimizing a sequence of small subproblems. The problems are solved iteratively from the last stage to the first stage, where the decision made at each stage is dependent on future stage consequences. The stages here refer to the point in time just before the decision changes, as most dynamic programming analyzes long-term operation planning. Generally, this technique is used to solve sequential optimization problems by enumerating states that are likely to happen in each stage.38 Some earlier works involving stochastic and dynamic programming include optimization of hydroelectric systems,39,40 reservoir networks,41,42 and hydrothermal power systems.38,43 As shown in the literature,44−48 robust programming is an attractive and computationally efficient approach. It ultimately gives attention to models that ensure feasible solutions given

the possible scenario-based outcomes of uncertain parameters. As the data change in this approach, the suboptimal solution obtained with the nominal values is essentially acceptable by decision-makers to ensure a solution that remains feasible and near optimal.47 In addition to this, robust programming is a single-stage optimization problem entailing minimal computational cost. Soyster48 initiated robust optimization based on the worst-case scenario which can be equated to all random parameters.47 Recently, robust optimization was further extended by including a nonlinear term in the objective function with the intention of presenting a less conservative robust programming.44−46 This results in the capability of providing solutions that are not overly conservative. In consideration of its various advantages, robust programming has been widely applied to address concerns of uncertainties in investment portfolio selection,35 and also on engineering grounds such as supply chain planning, 49 production scheduling, capacity expansion,50−52 and resource allocation,53,54 as well as synthesis of an integrated biorefinery.55,56 Tay et al.55 extended the use of robust optimization to the synthesis of integrated biorefineries which consider uncertainties in raw material supply and product demand. Meanwhile, Tang et al.56 synthesized an integrated biorefinery with consideration of cost fluctuation. On the other hand, the mixed integer linear programming (MILP) method was also used for resolving cases with process synthesis problems. MILP was first proposed by Grossmann and Santibanez57 using simplified representations of process units as “black boxes” governed by fixed material and energy balance ratios. Their original formulation assumes a multiperiod, profit maximization problem, where product demand and raw material supply limits (if any) are deterministic. MILP modeling has been widely used, and there are numerous papers written on it in scientific journals. The most common solution technique for MILP problems is the LP-based branch and bound algorithm,58,59 which has been implemented in most computer codes. These methods have later been improved using preprocessing techniques that reduce the dimensionality and integrality gap of MILP.60 MILP has been used in many different sectors such as in building refurbishment to reduce the life cycle cost (LCC) for the building.61 Papageorgiou and Lakhdar62 have used MILP modeling in a biopharmaceutical manufacturing facility to determine optimal production plans and allow for uncertain fermentation titers. MILP has also been used in flood diversion plans with a minimized system cost and a maximized safety level under multiple uncertainties.63 Tarau et al.64 have presented MILP modeling in their work for automated baggage handling systems for a predictive route control. In a recent work, Kasivisvanathan et al.65 presented MILP modeling to determine optimal operational adjustment in multifunctional energy systems as a result of partial inoperability of any process units within the system. In this paper, a robust MILP model is presented to aid decision-makers in addressing process synthesis problems due to uncertainties that arise for variety of reasons in multifunctional energy systems (e.g., polygeneration plants and biorefineries). The MILP approach is embedded within the robust programming framework,66 which allows multiple supply and demand scenarios to be considered. Multifunctional energy systems, while being highly efficient due to opportunities for process integration, are also to some extent by the interdependencies among individual process units.65 Fundamentally, this approach assumes that multifunctional 3197

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process units at partial capacity when necessary. Likewise, safety factors allow the processes to operate above the design conditions if necessary. The different distribution and reallocation of streams in the plant define the different and unique scenarios k (k = 1, 2, 3, ..., K). At the baseline state which makes up for the operation for scenario 1, the material (or energy) balance magnitude of each stream i (i = 1, 2, 3, ..., I) from each process unit j (j = 1, 2, 3, ..., J) is given by the coefficient aij. This coefficient does not change and is common to all scenarios K. In this convention, process inputs are denoted by negative values whereas process outputs are positive. The net baseline output of each stream from the plant is then ∑j aij. In every scenario k, the utilized operating capacity of each process unit j is written as xj,k which must fall within the feasible operating range limits q·xj as the lower limit and xj as the upper limit. Hence, xj is the maximum capacity of process unit j that is purchased to be able to operate a plant in all scenarios K, while q is the lowest fraction of maximum capacity at which the process unit can feasibly operate. Each net output of the plant is also defined by limits (yiL and yiU). The objective of this robust optimization is to minimize the total cost associated with operating a plant, Ctotal, that adapts to a number of scenarios K encountered due to uncertainties.

energy systems can be described by a system of linear equations, which entails “black box” modeling. In MILP formulations, the handling of discrete and continuous variables enhances the capability for selecting optimum configurations of chemical processes. Thus, the use of several linear approximations in the model to obtain solutions is justifiable in the preliminary stage of synthesizing a design.57 Besides, MILP techniques also can potentially tackle synthesis problems that are usually very large in size with thousands of variables and constraints. Even though most synthesis problems possess nonlinear structures, in many cases these problems can be formulated to incorporate MILP. Reasonable linear approximations are made in these instances, and where required binary variables can be used for approximating nonlinearities.57 In this work, each process unit in the system is characterized by a fixed set of material and energy balance coefficients. In addition, each process unit also has a viable operating range, ranging from a minimum partial load value below which the process becomes unstable or infeasible, up to a ceiling corresponding to the rated capacity. Essentially, multifunctional energy systems efficiently produce product portfolios via production of energy, fuels, utilities, and commodity chemicals. Besides, such systems also provide valuable solutions toward efficiency, sustainability, and economics by taking advantage of optimal integrations among process units. As mentioned previously, uncertainties can manifest in the volatility of the purchase price of both raw materials and products, or as variability in or the supply of process inputs, and demand for the product portfolio. Therefore, it becomes necessary to have unit operations that are capable of accommodating changes in parameters. In line with this, the robust MILP model aids in making operational and capital decisions for multifunctional energy systems with uncertainties. Making an operational decision involves equipment sizing that directly determines the design capacity of selected process units in the system network. A capital decision, on the other hand, is made when selecting a more favorable process unit on top of the other, which caters to multiple scenarios while having the same function. This work adapts the input−output model67 built within the robust MILP model for multifunctional energy systems. The optimization model allows determining the required maximum equipment sizes within the multifunctional energy systems while minimizing the plant’s total associated cost. These aspects of the work are novel and important contributions as such analysis is not available in the literature, to date. The rest of the paper is organized as follows. A formal problem statement is given in section 2 followed by a description for the robust optimization model itself in section 3. Industrial case studies involving a polygeneration plant (section 4) and a palm oil based integrated biorefinery (sections 5 and 6) are then given to demonstrate the proposed approach. Finally, conclusions and prospects for future work are given in section 7.

3. ROBUST OPTIMIZATION MODELING The material and energy balance for a plant which entails linear correlations can be expressed in the form of an input−output matrix as shown in eq 1:

∑ aijxj ,k = yi ,k

∀i

∀k

j

(1)

where aij represents the matrix of the input or output flow rate of stream i to or from process unit j (common in all scenarios K), xj,k is the feasible operating capacity of process unit j in scenario k, and yi,k is the net output of stream i from the plant in scenario k. A nonnegative value for yi,k indicates a stream that is purely a product, a negative value indicates a stream that is purely an input, and a value of 0 indicates that it is purely an intermediate stream. A scenario k simply defines the unique scheduling of a plant that compromises for any uncertainties that manifest in the plant. A single plant can operate with different schedules corresponding to the number of scenarios that the plant may face during operations. Let us consider a system comprised of a single process unit with operating capacity of x1,k to better illustrate the idea in eq 1. The process unit requires 4 kg of material A (y1,k) and 2.5 kg of material B (y2,k) to produce 2 kg of material C (y3,k). Figure 1 shows a sample flow sheet of this process. Expressing all component streams, i.e., A, B, and C in terms of eq 1 gives the following: (i) A: −4x1,k = y1,k (ii) B: −2.5x1,k = y2,k (iii) C: 2x1,k = y3,k

2. PROBLEM STATEMENT The problem addressed in this work may be formally stated as follows. We are given a plant that can be described by a system of linear equations, and hence represented with an input− output process matrix. Material and energy balances for each process unit are assumed to be a scale invariant “black box” model. It is assumed that the plant that is designed to operate at a baseline state (normal operation) can be adjusted in operation to meet other new demands or where an uncertainty occurs; such adjustments are made possible by operating

Figure 1. Sample flow sheet for illustration. 3198



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Figure 2. Schematic diagram of a simple polygeneration plant.

where in this scenario k, matrix aij for streams y1,k, y2,k, and y3,k flow rates are 4, 2.5, and 2, respectively, considering A and B are completely consumed in x1,k to produce C. The operating capacity, x1,k, is defined in fractions of percentage. For example, assuming a process unit operates at 100% under normal operation, x becomes 1 (=100/100). Therefore, if the process unit in Figure 1 is operating at the baseline state, x1,k is equal to 1. This then returns −4, −2.5, and 2 kg for A, B, and C respectively. Note that the negative value for streams A and B are only to denote that they are inputs and hence being consumed in the process. During an operation of the plant for a given scenario k, xj,k is adjusted within predefined limits as qxj ≤ xj , k ≤ xj

∀j

∀k

where bj is the binary variable that indicates whether a process j operates (bj = 1) or not (bj = 0) in all scenarios K. It is worth noting that the philosophy of robust programming is to make two types of decisions, which are operational and capital decisions. Making an operational decision involves making a choice to include a process unit in the design. In this case, the binary variable bj is 1 for all process units that are chosen as part of the plant. Therefore, it is assumed that all chosen process units have to operate in all scenarios K. The most important element here is to decide on the design capacity, xj, which involves sizing of each equipment. On the other hand, the binary variable bj may take on a greater significance if there is a choice of different equipment that competes for the same function. This involves a capital decision where the model favorably selects the equipment which can cater to multiple scenarios by virtue of greater operating range. In this particular case, bj appears to be 1 for equipment that is selected over other equipment performing the same function for which bj becomes 0. The limits for the net output flow rates leaving the plant are defined by eq 5:

(2)

where xj is the maximum capacity of process unit j that is purchased to be able to operate a plant in all scenarios K, and q is the partial load fraction to which the plant still operates feasibly below its maximum capacity and defines its lower bound in all scenarios K. Since xj represents the selected capacity or size of a unit j, it can be bounded by any sensible positive values as its upper, xjU, and lower, xjL, limits: L

U

xj bj ≤ xj ≤ xj bj

bj ∈ {0, 1}

∀j

∀j

yi , k L ≤ yi , k ≤ yi , k U

(3)

∀i

∀k

(5)

where yi,kL and yi,kU are the lower and upper limits of the stream yi,k flow rate in scenario k, respectively. Note that there may be cases where a given stream may be sold as product, or

(4) 3199



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Both the GT-HRSG and the boiler consume fuel oil to convert into useful energy. The gas turbine converts the fuel oil to power, while the heat recovery steam generator salvages some of the heat rejected from the gas turbine to useful heat energy. Some of the power produced from the GT-HRSG is then used by the electric chiller to produce cooling energy, while a part of electricity is utilized in the reverse osmosis module to treat fresh water. The excess power from the polygeneration is then fed into the grid. Meanwhile, the boiler converts fuel oil to heat energy. Table 1 shows the process

purchased as an input, depending on supply and demand conditions. In cases where the raw material or production capacity of a particular stream is known and a fixed value, yi,kL becomes equal to yi,kU, which is yi,k itself. As for intermediate products, a complete consumption returns a yi,k of 0 during normal or baseline state operation. However, the intermediate products need not always remain zero across different scenarios. This can arise due to a network disruption in an individual process unit or when a process unit is completely shut down. The primary objective behind this robust modeling relies on minimizing the total cost to run a plant that adapts to a number of scenarios K encountered during operation. The cost of equipment, Cequipment, that is applicable to all scenarios K is given by eq 6: C equipment =

Table 1. Process Matrix of a Polygeneration System: Case Study 170

∑ Cj fixedbj + (∑ Cjcapxj)/top j

(6)

j

At the outset, the cost of a plant in a given scenario k, Ck can be represented by eq 7: Ck operating = C equipment −

∑ Ci streamyi ,k

operating

,

∀k (7)

i

where Cj is the fixed cost of process unit j, is the annualized capital cost of process unit j at the baseline state (i.e., as when xj = 1), Cistream is the unit cost of stream i, and top is plant operational hours in a year. Note that Ckoperating may assume negative values for cases where there is profit from operation of the plant. Likewise, some streams may also have negative cost coefficients, as in the case of economic penalties for generation of output of wastes such as CO2. The total cost of the plant, Ctotal, can be represented by the correlation fixed

C total =

Cjcap

∑ PkCk operating k

boiler

CHP module

chiller

RO module

heat (kW) power (kW) cooling (kW) treated water (l/h) fuel oil (l/h) fresh water (l/h) rejected water (l/h)

1 −0.01 0 −2.16 −0.108 0 0

1.5 1 0 −9.72 −0.54 0 0

0 −0.2 1 0 0 0 0

0 −3 0 3600 0 −9000 5400

matrix of the polygeneration plant. Note that the energy and material balances of the polygeneration plant (shown in Table 1) are based on the case study presented by Tan et al.4 Note also that a negative value in the process matrix signifies that the stream is an input of a process while a positive value indicates that the stream is an output of a process. The feasible partial load operating range of each process unit in the polygeneration plant shows the optimum range capacity that each process can be operated in as seen in Table 2.

∀k

Table 2. Feasible Operating Ranges of Process Units in Polygeneration: Case Study 1

(8)

where Pk is the probability that scenario k occurs. Ultimately, the objective function to minimize C total taking into consideration all scenarios K encountered, can be written as min C total

material and energy streams

(9)

The resulting model is a mixed integer linear program (MILP), for which any optimal solution found is guaranteed to be globally optimal. Solutions of such models of realistic size present no significant computational difficulties. In sections 4−6, the model is demonstrated with polygeneration and palm oil based biorefinery case studies in a more descriptive manner to thoroughly guide throughout the proposed robust optimization modeling. All case studies are implemented using the commercial optimization software Lingo 13.0.68

process unit

feasible operating range (%)

boiler GT-HRSG chiller reverse osmosis

75−100 80−100 75−100 10−100

For this case study, two demand scenarios are considered. Each of the scenarios corresponds to a different set of demands. It is assumed that the operations of the polygeneration plant run 70% of the time using the scenario 1 setting, while in the remaining time it runs using the scenario 2 setting. In order to fulfill the demands encountered in the scenarios, the polygeneration plant needs to produce different amounts of energy and steam (Table 3). To simplify the cost estimation, this case study assumes linear cost functions with fixed costs of zero. Meanwhile, the annualizing factor of the capital cost is assumed as 0.067 year−1. The annualized capital cost coefficients for each of the process

4. CASE STUDY 1 This case study considers a polygeneration plant which supplies heat, power, cooling, and treated water, based on the example of Tan et al.4 The simple polygeneration plant consists of four components: a utility boiler, a gas turbine with heat recovery steam generator (GT-HRSG) system, an electric-driven chiller, and a reverse osmosis module. It is assumed that the polygeneration plant is to be installed in a remote location without existing gas distribution infrastructure. In such cases, small gas turbines for stationary applications may use fuel oil.69 The process flow sheet of the polygeneration plant is shown in Figure 2.

Table 3. Final Demands for Scenarios 1 and 2: Case Study 1

3200

output stream

scenario 1

scenario 2

heat (kW) power (kW) cooling (kW) water (l/h)

25 000 10 000 9000 360 000

23 000 9000 11 000 288 000



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more profitable (by 133%) compared to scenario 2. The optimal operating states of the polygeneration plant for scenarios 1 and 2 are shown in Figures 3 and 4. On the basis of the results of both scenarios, the maximum capacities of the polygeneration product for heat, power, cooling, and treated water are 25 000 kW, 10 000 kW, 11 000 kW, and 360 000 l/h, respectively. The utilized capacities for both scenarios 1 and 2 including the nominal capacities for all process units are shown in Table 7. As shown, the nominal capacity for each process unit is the maximum value between scenarios 1 and 2. The maximum value for each process unit now becomes the design capacity for the polygeneration plant satisfying both scenarios in the future.

units of the polygeneration differ from each other based on the actual rating given in the references as shown in Table 4. The Table 4. Annualized Capital Cost Coefficients for Each Process Unit: Case Study 1 capital cost coeff

value

ref

GT-HRSG (US$/kW) boiler (US$/kW) chiller (US$/kW) reverse osmosis (US$/l h−1)

1,653 305 467 13.31

Carvalho Carvalho Carvalho Seider et

et al.71 et al.71 et al.71 al.72

Table 5. Price of Material and Energy Streams: Case Study 1 material and energy stream cost

price

heat (US$/kWh) power (US$/kWh) cooling (US$/kWh) treated water (US$/L) fuel oil (US$/L) fresh water (US$/L)

0.050 0.090 0.060 0.025 0.900 0.0001

5. CASE STUDY 2 This case study presents a palm oil based integrated biorefinery involving a multiple process system, which is also energy selfsufficient.25 Figure 5 illustrates the process flow diagram of the network of an integrated biorefinery retrofitted from a palm oil mill. The network is simply defined by a system of linear equations, where the mass and energy balances are scaleinvariant. In a baseline state operation, i.e., scenario 1, the whole plant is designed for a feedstock of 50 tons/h fresh fruit bunches (FFBs) and 24.31 tons/h external fresh water supply. The whole plant is entirely energy self-sufficient while also allocating for an export of middle-pressure (MP) steam for sale to other industrial facilities in the vicinity. The process units in the plant that include pyrolysis, transesterification, fermentation, etc. are designed to produce a net total of seven valuable end products, namely, biochar, gasoline, bio-oil, acetone, butanol, ethanol, and animal feed. All unit operations in the baseline state are optimized to operate at 100% capacity. Table 8 shows the annualized capital cost of each process unit and its individual required capacity at baseline operation together with its feasible operating range. The partial load fraction of each process unit j is taken as 80% of its maximum capacity. In order to simplify the cost estimation, this case study too assumes linear cost functions with fixed costs of zero. In any one calendar year of operation after commissioning, the plant is foreseen to mostly operate at its baseline state, which would cover approximately 60% of the time frame (scenario 1). It is assumed that, for another 20% of the analyzed time frame in the same year (scenario 2), only 80% of the initial feedstock of palm press fiber (PPF), i.e., 5 tons/h (=0.8 × 6.25 tons/h) would be made available for processing in the pyrolysis unit. In the remaining 20% toward the end of the year (scenario 3), it is foreseen that there will be an incremental in demand for bio-oil by 10%, i.e., a requirement to produce a total of 0.594 ton/h (=1.1 × 0.54 ton/h) of bio-oil. In scenarios 2 and 3, the plant would operate away from its baseline state which requires optimal reallocation of streams in the plant. In line with these different operational schedules of the plant throughout the year, it is necessary to adapt an operational strategy in order to minimize the cost to construct and install a plant that accommodates these uncertainties. A robust representation as discussed under Robust Optimization Modeling is adapted for these scenarios, and the net output flow rates of each stream in the network as shown in Figure 5 is defined with the aid of eq 5. In scenario 1, the lower limit becomes equal to the upper limit to incorporate the material and energy balance values into the model (available in the Supporting Information). In scenario 2, eq 5 can be rewritten as follows:

ref Tan Tan Tan Tan Tan Tan

et et et et et et

al.70 al.70 al.70 al.70 al.70 al.70

prices of the product streams are shown in Table 5. Since the total cost is expressed annually, the price of the materials and energy streams must also be expressed on an annual basis by using an annualizing factor of 8000 h year−1. Thus, all of the terms in eq 7 are expressed annually. Table 5 shows the material and energy cost per unit value of each stream. It is further assumed that the binary variables bj are all set to a value of 1, indicating that the process units are all selected in the final design. Therefore, on the basis of the proposed model, a linear program results. The model consists of a total of 23 variables. Solving the proposed model via Lingo 13.0, the optimized results show that the minimum total cost (Ctotal) for polygeneration is determined as −US$47,106/h. Note that a negative value of cost indicates the profit for the plant. The annual net profits of the plant from the production of the energy streams for scenarios 1 and 2 (less fuel oil and fresh water consumption) are determined as −US$57,169/h and −US$24,577/h, respectively. The optimal sales and costs of the material and energy streams are shown in Table 6. As shown, Table 6. Resulting Sales and Costs of the Material and Energy Streams sales and costs

scenario 1

scenario 2

heat power cooling treated water fuel oil fresh water total

1,250 900 540 216,000 −158,560 −2,961 57,169

1,150 810 660 172,800 −148,366 −2,477 24,577

treated water contributed to about 98.77% (= 216,000/(1,250 + 900 + 540 + 216,000)) of the total sales of the plant while fueloil accounted for 98.17% (= 158,560/(158,560 + 2,961)) of the total cost of the plant. The resulting annualized costs of the plant CkOperating for scenarios 1 and 2 are found to be −US$56,883/h and −US $24,292/h, respectively. It is noted that scenario 1 is much 3201



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Figure 3. Operating state of polygeneration plant in scenario 1.

y20,2 = y20,2 L = y20,2 U = (1 − 0.8)(6.25x1,1)

It is noted that, in this case study, the biomass feedstocks required to operate this biorefinery are purchased from the preexisting palm oil mill, which is currently retrofitted with the integrated biorefinery. Therefore, the cost of feedstocks is made variable due to the total consumption of biomass types, i.e., palm kernel shell (PKS), empty fruit bunches (EFBs), palm press fiber (PPF), palm oil mill effluent (POME), and palm kernel cake (PKC), and also the price for the external water treatment, for the plant. The cost computation for this biorefinery also includes the monetary carbon credits of US $0.4 obtained from burning every kilogram of methane. The costs and prices of the inputs and products respectively are summarized in the second data column of Table 9. It is further assumed that the binary variables bj are all set to a value of 1, indicating that the process units are all selected in the final design. In consideration that the plant operates at its baseline capacity for 60% (P1) of the duration of a calendar year, 20% (P2) with decreased PPF input in scenario 2, and a remaining 20% (P3) with an incremental demand for bio-oil in scenario 3, the proposed MILP model (eqs 1−8, 10, and 12) is solved to minimize Ctotal as defined by the objective function in eq 9 via Lingo 13.0. The model consists of a total of 93 and 10 linear and binary variables, respectively. Based on the optimized result, the resulting total cost of the plant for the period of analysis is found to be −US$9,888/h. Note that the nonpositive value is to indicate that the plant operates with an economic performance of the mentioned number in a time frame of 1 year. Table 8 summarizes the inclusion of the plant’s optimal response in scenarios 2 and 3 along with their operating

(10)

As in scenario 1, eq 10 states that the lower and upper limits for the net output flow rate of PPF are equal. Hence, the net output is defined as the product of the remainder fraction of the unused PPF and the total amount that is otherwise processed in the pyrolysis unit in the baseline state (scenario 1). Similarly, scenario 3 reformulates eq 5 as y6,3 = y6,3 L = y6,3U = (1 + 0.1)y6,1

(11)

Equation 11 limits the net output flow rate of bio-oil in scenario 2 to the product of the additional fraction required from the baseline state and the total amount of bio-oil extracted as one of the products in the baseline state. As an additional constraint to these, the net energy stream flow rates are bounded by nonnegative values as the lower limits in scenarios 2 (yi,2L) and 3 (yi,3L) to essentially meet the energy self-sufficient criterion of the integrated biorefinery. Equation 12 represents the additional process constraint to be included in the model for all energy streams only. yi ,2 L = yi ,3 L = 0

(12)

With this constraint in place, the model ensures that any potential internal energy source, namely electricity (power), high-pressure (HP) steam, middle-pressure (MP) steam and and low-pressure (LP) steam are first utilized to satisfy the requirements of the integrated biorefinery before allocating for potential export. 3202



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Figure 4. Operating state of polygeneration plant in scenario 2.

6. CASE STUDY 3 Case study 2 is revisited in this case study by narrowing down the analysis to the boiler unit that is shown within a red dashed square in Figure 5. The more detailed configuration of the boiler is shown in Figure 6, where the plant is designed to consist of three different boiler units in total, namely boilers 1, 2, and 3. Boilers 1 and 2 are biomass combustion boilers ideally possessing the same functionality in burning biomass as fuel to produce high-pressure (HP) steam. However, they are relatively dissimilar with the former being more efficient and having a rather narrow feasible operating range as compared to the latter. These technically suggest a lower efficiency and a wider feasible operating range for boiler 2. However, the partial load fractions of both boilers 1 and 2 still remain 80% of their respective maximum capacities. On the other hand, boiler 3 is an ordinary water tube boiler which burns only methane produced from pyrolysis and anaerobic digestion units as fuel to produce HP steam. All boilers in the network are assumed to be available at the same capital cost per unit of capacity despite their process and mechanical differences. This case study essentially analyzes three different operating conditions throughout one calendar year, where in each operating case the feedstock, intermediate, and product stream allocations differ quantitatively based on the different production demands. It is assumed that the plant is expected to operate at its baseline state (normal operating condition) only for 60% of a year’s period (scenario 1). The baseline state operation in this case study is similar to that of case study 2 and

Table 7. Comparison of Utilized Capacities in Scenarios 1 and 2: Case Study 1 process units

nominal capacity

utilized capacity (scenario 1)

utilized capacity (scenario 2)

boiler GT-HRSG chiller reverse osmosis

6584 12 277 11 000 493 557

6584 12 277 9 000 493 557

5600 11 600 11 000 412 849

capacities in each scenario. Table 9 shows the corresponding net flow rates of all products and inputs as opposed to the baseline state. Table 10 shows the maximum capacities of all process units that can be purchased to be able to operate the plant in all scenarios with a minimum total cost, or rather a maximum economic performance of the plant. Each process unit is also provided with an additional margin for safety in addition to its maximum operating capacity. As a result of the robust optimization, most process units are found to be operating at either a capacity similar to the baseline state or lower. Therefore, it is sufficient that these unit operations be purchased for a capacity at the baseline operation, i.e., 100% as portrayed in Table 10. The robust optimization shows that only the anaerobic digestion unit requires a capacity that is slightly greater than the baseline operation. This is due to a relatively higher distribution of POME to this unit operation in scenario 2 for a higher production of boiler fuel (methane) in order to ensure of an energy self-sustained integrated biorefinery. 3203



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Figure 5. Schematic diagram of palm oil based integrated biorefinery.

Table 8. Capacities and Operating Ranges of Process Units: Case Study 2 no.

process unit

baseline annual capital cost (US$)


baseline state (scenario 1) 3.13 tons/h (100%) 1.255 tons/h (100%) 18.78 tons/h (100%) 1.14 tons/h (100%) 419.77 tons/h (100%) 0.704 ton/h (100%) 0.94 ton/h (100%) 425 kW (100%) 564 kW (100%) 1.65 tons/h (100%)

1 2

palm oil mill pyrolysis

0 (in existence) 3,810,822

80−100 80 −150

3

anaerobic digestion

23,904,509

80−150

4 5

fermentation boiler

26,463,183 6,212,652

80−150 80−150

transesterification Fischer−Tropsch steam turbine engine export as animal feed

3,036,236 17,503,328 366,052.3 1,940,146 0 (no equipment)

80−150 80−150 80−150 80−150 80−100

6 7 8 9 10

differs only where the boiler configuration is concerned. The biomass and methane in this case study (case study 3) are channelled to different boilers (boilers 1 and 3) to produce HP steam, instead of a single boiler as in case study 2. The need for operation of an additional boiler, i.e., boiler 2, is only considered if boiler 1 alone is insufficient to accommodate any demand variation from normal operating condition. However, the possible operation of more than one boiler 1 is first analyzed before choosing to operate boiler 2. This decision will be made based on an economic standpoint, i.e., a more economically feasible option. It is noted that when boiler 2 is in operation, the fraction to which PKS and EFB mixes in scenario 1 (feed to boiler 1) has to be retained. This is to keep the desirable condition in burning the biomasses as fuel. Besides

optimal response (scenario 2)


3.13 tons/h (100%) 1.00 ton/h (80%)

3.13 tons/h (100%) 1.255 tons/h (100%)

18.85 tons/h (100.4%)

18.78 tons/h (100%)

1.12 tons/h (98.7%) 419.54 tons/h (99.9%)

1.00 ton/h (87.6%) 419.77 tons/h (100%)

0.68 ton/h (96.9%) 0.75 ton/h (80%) 424.77 kW (99.9%) 546.62 kW (96.9%) 1.65 tons/h (100%)

0.65 ton/h (92.4%) 0.94 ton/h (100%) 425 kW (100%) 521.40 kW (92.4%) 1.65 tons/h (100%)

the baseline operation, it is also assumed that in another 20% of the analyzed time frame in the same year (scenario 2), the plant is operated with an incremental demand for MP steam export by 10% from the baseline state, i.e., 128.7 tons/h (=1.1 × 117 tons/h). Subsequently, the plant is operated with a further incremental demand for the MP steam export by 50% in total from the baseline state, i.e., 175.5 tons/h (=1.5 × 117 tons/h) in the remaining 20% of the same calendar year (scenario 3). The robust model discussed under Robust Optimization Modeling is adapted to determine the stream allocation in the process network for all three scenarios of this case study. In scenario 1, the net output flow rates of each stream is defined with the aid of eq 5 by equating the lower and upper limits and 3204



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Table 9. Baseline and Rescheduled Flow rates of Net Inputs and Outputs: Case Study 2 no.

stream

cost

baseline state (scenario 1)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

acetone animal feed biochar biodiesel biofuel bio-oil butanol carbon dioxide ethanol methane fresh fruit bunches syngas fresh water exported electricity exported HP steam exported MP steam exported LP steam palm kernel shell empty fruit bunches palm press fiber palm oil mill effluent palm kernel cake

US$1.44/kg US$0.22/kg US$0.50/kg N/A US$2.80/kg US$0.103/kg US$1.93/kg N/A US$0.661/kg N/A N/A N/A US$0.00078/kg US$0.0714/kW US$0.018/kg US$0.012/kg US$0.009/kg US$0.057/kg US$0.007/kg US$0.007/kg US$0.013/kg US$0.143/kg

1.14 tons/h 1.65 tons/h 0.64 tons/h 0 ton/h 0.94 ton/h 0.54 ton/h 3.33 tons/h 11.42 tons/h 1.7 tons/h 0 ton/h −50 tons/h 0 ton/h −24.31 tons/h 0 kW 0 ton/h 117 tons/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h

1.12 tons/h 1.65 tons/h 0.512 ton/h 0 ton/h 0.75 ton/h 0.31 ton/h 3.29 tons/h 11.209 tons/h 1.68 tons/h 0 ton/h −50 tons/h 0 ton/h −21.448 tons/h 0 kW 0 ton/h 125.5 tons/h 0 ton/h 0.002 ton/h 0.01 ton/h 1.25 tons/h 0 ton/h 0 ton/h

1.00 ton/h 1.65 tons/h 0.64 ton/h 0 ton/h 0.94 ton/h 0.594 ton/h 2.92 tons/h 11.42 tons/h 1.49 tons/h 0 ton/h −50 tons/h 0 ton/h −24.31 tons/h 0 kW 0 ton/h 138.37 tons/h 0.10 ton/h 0 ton/h 0 ton/h 0 ton/h 1.01 tons/h 0 ton/h

1−8 and 12−14) is solved with an aim to minimize the cost as defined by the objective function in eq 9 in Lingo 13.0. The model consists of a total of 100 and 13 linear and binary variables, respectively. The robust model developed results in a Ctotal of −US$9,089/h, which essentially indicates the economic performance of the plant during the analyzed one calendar year. The optimal response of the plant in scenarios 2 and 3 along with their operating capacities in each scenario are summarized in Table 11, whereas the corresponding net flow rates of all inputs and products are tabulated in Table 12. In this case study, the maximum capacities of process units that can be purchased to accommodate the operation of the plant in all scenarios with minimal cost are shown in Table 13. Each process unit is also provided with an additional margin for safety on top of its maximum operating capacity as in case study 2. On the basis of Table 13, it can be seen that most process units are found to be operating either at a capacity similar to the baseline state or higher to accommodate the increase in demand of MP steam export. Unit operations that are invovled in a higher capacity purchase include anearobic digestion, boiler 3, and the steam turbine. This clearly reflects the need for higher distribution of POME to the anaerobic digestion unit for production of more methane to be burned in boiler 3 and hence resulting in the desired higher production rate of MP steam. On a more important note, it can be seen that the operation of a second biomass combustion boiler, i.e., boiler 2, is not required to meet the variation in demand in the process network design. This is denoted by the binary variable, bj, that returned a value of 0 or the maximum operating capacity required that is estimated as 0% by the model.

Table 10. Installed Capacities of Process Units: Case Study 2 no.

process unit

bj

max operating capacity (%)

margin for safety (%)

1 2 3 4 5 6 7 8 9 10

palm oil mill pyrolysis anaerobic digestion fermentation boiler transesterification Fischer−Tropsch steam turbine engine export as animal feed

1 1 1 1 1 1 1 1 1 1

100.0 100.0 100.4 100.0 100.0 100.0 100.0 100.0 100.0 100.0

20 20 20 20 20 20 20 20 20 20

incorporating these values into the robust model. Equation 5 is rewritten in scenario 2 as y16,2 = y16,2 L = y16,2 U = (1 + 0.1)y16,1

(13)

Equation 13 suggests that the total production of MP steam in scenario 2 is the product of an additional fraction required from the baseline state and the total amount of MP steam produced in the baseline state. Scenario 3, which has an almost similar MP steam stream reallocation but by an additional 50% from the baseline state, reformulates eq 5 as follows: y16,3 = y16,3 L = y16,3U = (1 + 0.5)y16,1

(14)

Equation 12 from case study 2 is again adapted in this case study to ensure that the energy requirements of the integrated biorefinery are self-supported before considering for external export. Considering that the plant operates at baseline capacity for 60% (P1) of a calendar year’s time frame, 20% (P2) with incremental demand for MP steam export in scenario 2, and a remaining 20% (P3) with a further increase in demand for MP steam export in scenario 3, the proposed MILP model (eqs

7. CONCLUSION A robust optimization model has been developed in this work in order to address process synthesis problems due to the various uncertainties that can manifest in a multifunctional energy system that produces multiple products. The method3205



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Figure 6. Detailed configuration of boiler.

Table 11. Capacities and Operating Ranges of Process Units: Case Study 3 no.

process unit

baseline annual capital cost (US$)




3.13 tons/h (100%) 1.255 tons/h (100%) 18.78 tons/h (100%) 1.14 tons/h (100%) 55.35 tons/h (100%) 0 ton/h (0%) 364.42 tons/h (100%) 0.704 ton/h (100%) 0.94 ton/h (100%) 425 kW (100%) 564 kW (100%) 1.65 tons/h (100%)

3.13 tons/h (100%) 1.255 tons/h (100%)

3.13 tons/h (100%) 1.255 tons/h (100%)

18.97 tons/h (101%)

19.74 tons/h (105.1%)

1.10 tons/h (96.3%) 55.35 tons/h (100%)

0.93 ton/h (81.7%) 55.35 tons/h (100%)

0 ton/h (0%) 368.08 tons/h (101%)

0 ton/h (0%) 382.72 tons/h (105%)



1 2

palm oil mill pyrolysis

0 (in existence) 3,810,822

80−100 80−150

3

anaerobic digestion

23,904,509

80−150

4 5

fermentation boiler 1

26,463,183 6,212,652

80−150 80−150

6 7

boiler 2 boiler 3

6,212,652 6,212,652

80−200 80−150

transesterification Fischer−Tropsch steam turbine engine export as animal feed

3,036,236 17,503,328 366,052.3 1,940,146 0 (no equipment)

80−150 80−150 80−150 80−150 80−100

8 9 10 11 12


Future work may focus on extensions of robust program-

ology assumes that a plant is designed to be able to operate at multiple states to adapt to different scenarios due to the uncertainties encountered during the life of the plant. In line with this, the model developed allows searching for an optimal operational response for each process unit of the plant. In addition, it also minimizes the average cost associated with building and operating a plant that is able to adapt to all potential scenarios occurring over a time frame considered for analysis. This approach has been demonstrated using case studies involving a polygeneration plant and a palm oil based integrated biorefinery.

ming, taking into account degrees of risk associated with different alternative scenarios. In such cases, hybrid approaches may be developed that combine robust optimization with strategies such as fuzzy or stochastic techniques. In addition, the approach developed here may be extended further to apply to the case where interactions exist among multiple plants in industrial symbiosis (IS) systems. 3206



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Table 12. Baseline and Rescheduled Flow Rates of Net Inputs and Outputs: Case Study 3 no.

stream




1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

acetone animal feed biochar biodiesel biofuel bio-oil butanol carbon dioxide ethanol methane fresh fruit bunches syngas fresh water exported electricity exported HP steam exported MP steam exported LP steam palm kernel shell empty fruit bunches palm press fiber palm oil mill effluent palm kernel cake

1.14 tons/h 1.65 tons/h 0.64 ton/h 0 ton/h 0.94 ton/h 0.54 ton/h 3.33 tons/h 11.42 tons/h 1.7 tons/h 0 ton/h −50 tons/h 0 ton/h −24.31 tons/h 0 kW 0 ton/h 117 tons/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h

1.10 tons/h 1.65 tons/h 0.64 ton/h 0 ton/h 0.94 ton/h 0.56 ton/h 3.21 tons/h 11.52 tons/h 1.64 tons/h 0 ton/h −50 tons/h 0 ton/h −24.31 tons/h 0 kW 0 ton/h 128.7 tons/h 0.2 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h

0.93 ton/h 1.65 tons/h 0.64 ton/h 0 ton/h 0.94 ton/h 0.64 ton/h 2.72 tons/h 11.94 tons/h 1.39 tons/h 0 ton/h −50 tons/h 0 ton/h −24.31 tons/h 0 kW 0 ton/h 175.5 tons/h 1.0 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h 0 ton/h

authors would also like to acknowledge financial support from the Minister of Higher Education, Malaysia, through an LRGS Grant (Project Code 5526100) and the Philippine Commission of Higher Education via the PHERNet Program on Sustainability Studies.

Table 13. Installed Capacities of Process Units: Case Study 3 no.

process unit

bj

max operating capacity (%)

margin for safety (%)

1 2 3 4 5b1 5b2 5b3 6 7 8 9 10

palm oil mill pyrolysis anaerobic digestion fermentation boiler 1 boiler 2 boiler 3 transesterification Fischer−Tropsch steam turbine engine export as animal feed

1 1 1 1 1 0 1 1 1 1 1 1

100.0 100.0 105.1 100.0 100.0 0.0 105.0 100.0 100.0 104.4 100.0 100.0

20 20 20 20 20 0 20 20 20 20 20 20

■

■

Indices

i = stream index (i = 1, 2, ..., I) j = process unit index (j = 1, 2, ..., J) k = scenario (k = 1, 2, ..., K) Parameters

aij = output of stream i from process unit j at the baseline state Cequipment = cost of equipment in plant applicable to all scenarios K Cjfixed = fixed cost of process unit j Cjcap = annualized capital cost of process unit j at the baseline state Cistream = unit cost of stream i Pk = probability that scenario k occurs in any one calendar year period of analysis xjL = lower limit of operating capacity of process unit j xjU = upper limit of operating capacity of process unit j yi,kL = lower limit of net output of stream i from plant in scenario k yi,kU = upper limit of net output of stream i from plant in scenario k

ASSOCIATED CONTENT

S Supporting Information *

Tables to be read in conjunction with case studies 2 and 3, summarizing overall material and energy balances. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: *E-mail: *E-mail: *E-mail:

NOMENCLATURE

[email protected] [email protected] [email protected]. [email protected]

Variables

Notes

bj = binary variable indicating operation or nonoperation of process unit j Ckoperating = cost of plant in scenario k Ctotal = total cost of plant in consideration of all scenarios K q = fraction to which the plant still operates feasibly from its maximum capacity/lower bound top = plant operational hours in a year

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the University of Nottingham Research Committee through the New Researcher Fund (NRF 5021/A2RL32). In addition, the 3207



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xj = purchased maximum operating capacity of process unit j xj,k = feasible operating capacity of process unit j in scenario k yi,k = net output of stream i from plant in scenario k

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