Optimization under Variability and Uncertainty: A Case Study for

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Environ. Sci. Technol. 2004, 38, 6741-6747

Optimization under Variability and Uncertainty: A Case Study for NOx Emissions Control for a Gasification System JIANJUN CHEN AND H. CHRISTOPHER FREY* Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Campus Box 7908, Raleigh, North Carolina 27695-7908

Methods for optimization of process technologies considering the distinction between variability and uncertainty are developed and applied to case studies of NOx control for Integrated Gasification Combined Cycle systems. Existing methods of stochastic optimization (SO) and stochastic programming (SP) are demonstrated. A comparison of SO and SP results provides the value of collecting additional information to reduce uncertainty. For example, an expected annual benefit of $240 000 is estimated if uncertainty can be reduced before a final design is chosen. SO and SP are typically applied to uncertainty. However, when applied to variability, the benefit of dynamic process control is obtained. For example, an annual savings of $1 million could be achieved if the system is adjusted to changes in process conditions. When variability and uncertainty are treated distinctively, a coupled stochastic optimization and programming method and a two-dimensional stochastic programming method are demonstrated via a case study. For the case study, the mean annual benefit of dynamic process control is estimated to be $700 000, with a 95% confidence range of $500 000 to $940 000. These methods are expected to be of greatest utility for problems involving a large commitment of resources, for which small differences in designs can produce large cost savings.

Introduction Environmental regulations are one of the factors that drive the development of new coal-based electric power generation technologies. Typical emission control systems for a pulverized coal-fired power plant include a wet limestone flue gas desulfurization system for SO2 control, an electrostatic precipitator for PM removal, and combustion control for NOx reduction. Selective Catalytic Reduction (SCR) is also available for post combustion NOx control. Integrated Gasification Combined Cycle (IGCC) systems are an alternative to pulverized coal (PC) combustion system. IGCC systems have the potential to realize higher efficiency and better environmental performance for power generation (1). The U.S. Department of Energy and others are pursuing development of a new generation of gasification systems intended to offer an environmentally and economically viable alternative for power generation in the United States (2). * Corresponding author phone: (919)515-1155; fax: (919)515-7908; e-mail: [email protected]. 10.1021/es0351037 CCC: $27.50 Published on Web 10/23/2004

 2004 American Chemical Society

For IGCC systems, there are a wide variety of feedstocks, products, and technologies. Optimization is an important technique to achieve a balance between cost and environmental performance. For example, Diwekar et al. optimized SO2 control for an IGCC system (3). However, there also remain many uncertainties in the cost estimates, coal utilization, and environmental control in IGCC systems (47). Optimization methods combined with uncertainty analysis provide a powerful and rigorous tool for design of process technologies (e.g. IGCC system in this paper). Diwekar et al. summarized two methods for optimization of process models under uncertainty (8). Stochastic optimization (SO) involves selection of one optimal design based upon consideration of selected statistics, such as expected value, variance, and so on, for the objective function, constraints, or both. The numerical implementation of SO involves two iterative loops: (1) an inner Monte Carlo sampling loop, in which uncertainty is simulated conditional on point estimates selected for each design variable, and (2) an outer optimization loop in which the values of the design variables are manipulated. Stochastic programming (SP) involves estimation of optimal decision variable values for each sample of a Monte Carlo simulation for uncertainty, thereby resulting in a distribution of uncertainty for each decision variable. SP features: (1) an inner optimization loop, in which the system is optimized conditional on a given realization of uncertainty, and (2) an outer Monte Carlo sampling loop in which realizations of uncertainty are simulated. SO is used to make a decision now regarding a system for which uncertainty cannot be further reduced. SP is used to assess the probable range of optimal solutions if uncertainty is first realized before choosing an optimal design. SO is a “here-and-now” formulation, while SP is a “wait-and-see” formulation (9). Schematic diagrams for SO and SP are provided as Figures S-1 and S-2 of the Supporting Information (SI). As demonstrated later in this paper, a comparison between these two methods provides insight regarding (1) the robustness of the SO solution and (2) the value of reducing uncertainty. Diwekar et al. applied SO and SP to NOx emissions control for an IGCC system (8). Process technologies are subject not only to uncertainty but also to variability in feedstock composition, unit costs of consumables, and the performance of unit operations. Variability is a heterogeneity of values for a quantity over time, space, or among different members of a population, while uncertainty results from lack of information (10-13). The need for quantitative distinction between variability and uncertainty has gained wide acceptance in fields such as nuclear and human health risk assessment (14-19) and air pollutant emissions estimation (16, 20-24). However, distinction of variability and uncertainty during optimization are rarely made. Recently, an analytical method for optimal process design based upon consideration of uncertainty in some inputs, variability in other inputs, and adjustment of decision variables dynamically during operation to compensate for variability has been developed and demonstrated through case studies, for selected generic problems in chemical process design (25). However, the method does not account for simultaneous variability and uncertainty components for a given input. Thus, the objective of this paper is to demonstrate optimization of NOx emissions control for an IGCC system when both variability and uncertainty in inputs are considered. The paper is organized in this way: the methodology for optimization under variability and uncertainty is introduced. NOx emissions control for an IGCC system is used as the basis for case studies VOL. 38, NO. 24, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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to demonstrate the application and interpretation of the methodology. Results of case studies are presented and discussed.

Methodology The key methodological elements for optimization under both variability and uncertainty include the following: (1) specification of variability, uncertainty, or both, for model inputs; (2) specification of alternative optimization methods based upon the notions of SO and SP; and (3) selection of an optimization solver. Quantification of Variability and Uncertainty in Model Inputs. Frequency distributions are used to characterize variability in a quantity, and probability distributions are used to represent uncertainty of a quantity (13). For example, uncertainty regarding knowledge of the true but unknown population frequency distribution for variability can be depicted using probability ranges estimated for each percentile of the distribution for variability, resulting in a twodimensional or second-order distribution (10, 13-21, 23, 24). Monte Carlo simulation can be employed to generate random numbers for such distributions (16). A useful method for quantifying uncertainty regarding the true but unknown frequency distribution is bootstrap simulation, which was introduced by Efron (26) primarily to simulate confidence intervals for any statistic of a frequency distribution, including its moments (e.g., mean, variance), parameters, and percentiles. A best estimate of the true but unknown population frequency distribution is specified to represent variability. Bootstrap simulation is used to generate u alternative frequency distributions to simulate the uncertainty in the frequency distribution. For each alternative distribution, a total of v random samples are simulated to represent one possible realization of variability within a population. The output is a v × u two-dimensional random sample matrix, with one dimension representing variability (v samples) and the other one representing uncertainty (u realizations) (16, 20, 21, 23, 24). A software tool, AuvTool developed for the purpose of providing variable and uncertain inputs to a risk assessment model, is employed to generate the bootstrap estimates of uncertainty regarding frequency distributions for variability (27). Optimization Methods under Both Variability and Uncertainty. Two methods were employed for optimization of process models when both variability and uncertainty are considered. The first is referred to as coupled stochastic optimization and programming (CSOP), and the second is referred to as two-dimensional stochastic programming (TDSP). When applied to only uncertainty, SO provides a single optimal design taking into account uncertainty. When applied to only variability, SO provides a single optimal solution that would represent the best prior estimate of how to operate the system when the specific realizations of variability (e.g., such as for feedstock composition) are not known or cannot be measured in real time (e.g., due to cost or lack of appropriate methods). By comparison, CSOP involves repeated application of stochastic optimization to each alternative frequency distribution which represents variability in an input. Thus, CSOP can be used to assess the uncertainty with regard to how to optimize the design taking into account variability. TDSP differs from CSOP in that it involves development of optimal solutions for each iteration of the two-dimensional probabilistic analysis, thereby producing a two-dimensional distribution of variability and uncertainty in optimal values. This method enables one to evaluate the effect of both variability and uncertainty on optimal solutions, assuming in each case that the system can be optimized after the specific values of variability and uncertainty are realized. Therefore, TDSP should typically provide optimal solutions as good as 6742

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or better than that of CSOP, since TDSP assumes that more is known at the time that the optimal decision must be made. An example illustrating the algorithms of CSOP and TDSP is provided in the Supporting Information. Flow diagrams for implementation of CSOP and TDSP are given as Figures S-3 and S-4 in the Supporting Information. Selection of an Optimization Solver. The most popular mathematical methods for constrained nonlinear optimization problems are generalized reduced gradient (GRG) and successive quadratic programming (SQP) and their variants (8). However, these mathematical methods require extrapolation of the objective function from one point to its neighborhood based on the gradient. Thus, these methods cannot easily be applied to “black box” computer models for which an analytical objective function is not specified. Alternative approaches which do not require gradient information include genetic algorithm (GA), simulated annealing (SA), and tabu searches (28). Evolver, a GA based optimizer, was selected in this study (29). GA is a powerful stochastic search and optimization technique based on principles from evolution theory (30). It has been applied to a variety of problems, such as air quality management (31) and process design (28, 32). GA is further described elsewhere (30, 33).

Process Simulation Model and Probabilistic Inputs CSOP and TDSP were applied to optimize NOx emissions control under variability and uncertainty for a 730 MW IGCC system featuring the use of a bituminous coal-fueled airblown gasifier and hot gas cleanup. Figure 1 shows the schematic of the IGCC system studied in this paper (34). Basically, coal, steam, and oxygen enter a high pressure, hightemperature gasifier reactor vessel. A portion of the coal is combusted to release heat, while the remainder participates in endothermic gasification reactions with steam to produce a syngas containing CO and H2. The syngas that exits from the gasifier enters a high temperature gas cooling unit, where it is quenched by water. Subsequently it is cleaned of impurities such as particulate matter and sulfur compounds. The clean fuel gas is sent to a gas turbine, where recovered energy is used to rotate a generator for electricity. The hot exhaust gas from the gas turbine passes through a Heat Recovery Steam Generator (HRSG), where the exhaust gas is cooled and the transferred heat is used to generate superheated steam. Superheated steam is later used to produce electric power via a steam turbine. The hot gas cleanup system includes the following: in-bed desulfurization in the fluidized bed gasifier with limestone, subsequent sulfur removal with a zinc ferrite sorbent, and high efficiency cyclones and ceramic filters for particulate removal. Selective Catalytic Reduction (SCR) is located within the HRSG for post combustion NOx control. Performance, emissions, and cost models of this system have previously been developed using the ASPEN chemical process simulator and are reported in detail elsewhere (3, 5, 6, 8, 34, 35). Because the run time for the ASPEN process simulation model is prohibitively long with respect to the objective of performing v × u simulations, a response surface model (RSM) with substantially faster run time was developed (36, 37). The RSM includes 9 inputs and 60 outputs. The outputs of RSM serve as inputs to the cost model. The accuracy of the RSM is typically within one percent compared to the ASPEN model. The cost model was updated using the Chemical Engineering Plant Cost Index (CI) and Industrial Chemicals Producer Price Index (CICPPI) to represent January 2002 values (38). Seven decision variables were chosen. With regard to the postcombustion NOx emission control system, key decision variables include the SCR NOx removal efficiency, SCR ammonia slip, and SCR catalyst layer replacement interval.

FIGURE 1. A schematic of a fluidized bed gasifier-based integrated gasification combined cycle system with hot gas cleanup.

TABLE 1. Design Variables and Their Adjusting Ranges description gasifier oxygen-to-carbon molar ratio gasifier steam-to-carbon molar ratio sulfur retained in the gasifier bottom ash (mass ratio) SCR NOx removal efficiency SCR NH3 slip (ppm) SCR catalyst replacement interval (hours) capacity factor

upper and lower bounds 0.45-0.47 0.445-0.455 0.80-0.95 0.50-0.90 5.0-20.0 5000-25000 0.50-0.90

With regard to the gasifier and sulfur control, key decision variables include the gasifier oxygen to carbon molar ratio, gasifier steam to carbon molar ratio, and sulfur retained via reaction with limestone in the gasifier bottom ash. In addition, the capacity factor was included as a design variable. The upper and lower bounds of these variables are given in Table 1. Twenty-seven model inputs were identified as uncertain based on literature review and expert judgment (5, 6, 35, 36). An additional 26 model inputs were identified as subject to both variability and uncertainty. The distribution assumptions for uncertain variables and variables with variability are given in Tables S-1 and S-2 of the Supporting Information, respectively. Uncertainty in the mean values of the inputs with both variability and uncertainty was quantified and is given in Table S-5 of the Supporting Information.

Optimization Case Studies Optimization case studies were conducted for the following cases: (1) SO and SP applied to only variability; (2) SO and SP applied to only uncertainty; (3) CSOP applied to variability under multiple realizations of uncertainty; and (4) TDSP applied to all iterations of both variability and uncertainty. For all simulations, u ) 100 and v ) 100. Optimization Considering Only Variability in Model Inputs. SO and SP were conducted, respectively, when only variability in model inputs was considered using the same

sequence of random numbers for model inputs in each case to facilitate direct comparisons between the methods. Uncertain input variables were kept at the default values shown in Table S-1. For SO, the expected (mean) Cost of Electricity (COE) was minimized subject to each of four probabilistic constraints that the mean, 90th percentile, 95th percentile, or 99th percentile of NOx emissions be less than or equal to 0.2 lb/ 106 BTU as summarized in Table 2. Six of the decision variables are bound to either a lower or an upper limit, as given in Table S-6 of the Supporting Information. The SCR removal efficiency was the only decision variable that changed over the four case studies and is included in Table 2. As the probabilistic NOx constraint becomes more stringent, the optimal mean COE increases from 50.45 to 50.62 mills/kWh. Although this change may seem small on a relative basis, it is equivalent to approximately $1 million per year for a single plant. Thus, there is a cost premium associated with achieving the NOx constraint with increasing degrees of confidence in the face of variability. For SP, the COE was minimized when the NOx emissions were constrained to be less than or equal to 0.2 lb/106Btu. As shown in Table 2, the mean optimal COE is 50.45 mills/ kWh. The 95% range of the optimal COE is from 48.59 to 53.09 mills/kWh. The 95% probability ranges of optimal values for design variables are summarized in Table S-7 of the Supporting Information, and the mean value for the SCR removal efficiency is given in Table 2. Six of the design variables either did not vary, varied only slightly, or are insensitive with respect to the objective function. The SCR removal efficiency was the only sensitive design variable in this case and had a 95% probability range of 0.50 to 0.73. This range of uncertainty is large compared to the best estimate design obtained from stochastic optimization for the mean NOx constraint case, thereby indicating the potential risk of imposing a removal efficiency decision before variability in system inputs can be measured or assessed during actual operation. Comparison of SO and SP results enables insight into the benefits that would accrue from real time monitoring of variability and adjustment of operations. For this purpose, VOL. 38, NO. 24, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 2. Comparison of Stochastic Optimization and Stochastic Programming Results When Considering Only Variability in Model Inputs level 1 constraint (lb/106Btu) expected cost of electricity (mills/kWh) SCR removal efficiency

level 2

Optimal Value of Stochastic Optimization expected NOx probability (NOx emissions emissions e0.2 e0.2) g 0.9 50.45 50.55 0.60 0.68

Average Optimal Value of Stochastic Programming constraint (lb/106Btu) NOx emissions e0.2 cost of electricity (mills/kWh) 50.45 SCR removal efficiency 0.60 expected value of perfect information (EVPI) mills/kWh 0.00 0.10 106$/year 0.00 0.60

the concept of Expected Value of Perfect Information (EVPI) is applied to variability rather than uncertainty. Traditionally, EVPI is the difference between the expected loss (or cost) of optimal management decision based on the results of uncertainty analysis and the expected loss of the optimal management decision if all uncertainty were eliminated in one or all uncertainty quantities. The EVPI is an upper bound of the value of efforts to reduce uncertainty (39). When applied to variability, rather than uncertainty, EVPI indicates the benefit of dynamically adjusting decision variables for optimal control. It is calculated as the difference between the optimal expected COE from SO and the expected optimal COE from SP. Of course, these two results are not exactly comparable, since for SO a chance constraint is used for NOx, allowing for the possibility that the constraint is exceeded for some variability realizations, whereas SP requires that the NOx constraint never be violated for any realization of variability. The EVPI increases as the degree of confidence associated with the chance constraint increases. The benefit of dynamically adjusting to variability in model inputs is as high as $1 million per year when comparing the most stringent NOx constraint for SO with the SP results. The loss function here was simply the COE. If the cost of fines or shutdowns because of noncompliance with permits were also included in the analysis, the benefits of dynamic control would likely be higher. As shown in Table 2, the positive EVPI is because of a lower average optimal SCR removal efficiency, and hence lower average costs, from the stochastic programming results. Variability in the SCR removal efficiency is influenced by variability in the fraction of coal bound nitrogen converted to NH3 in the gasifier and the fraction of NH3 converted to NOx in gas turbine. These two variables are used to estimate the uncontrolled NOx emissions. An implication, therefore, is that cost can be reduced if better information is made available regarding the uncontrolled NOx emissions, such as based upon real time monitoring data. Thus, the results imply that there is a potential benefit from dynamic process control. The cost of monitoring and control system would have to be compared with the estimated maximum benefit. Optimization Considering only Uncertainty in Model Inputs. Optimization considering only uncertainty in model inputs was conducted in a similar manner as that for only variability in model inputs. Using SO, the expected COE was optimized subject to the chance constraint that the 90th percentile of NOx emissions be less than or equal to 0.2 lb/ 106Btu. The optimal expected COE was found to be 51.65 mills/kWh, and the required SCR removal efficiency was 0.65. Other optimal design values are consistent with those found when considering only variability and are given in Table S-8 6744

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level 3

level 4

probability (NOx emissions e 0.2) g 0.95 50.59 0.71

probability (NOx emissions e 0.2) g 0.99 50.62 0.74

0.14 0.84

0.17 1.02

of the Supporting Information. Using SP, the COE was optimized subject to the constraint that NOx emissions be less than or equal to 0.2 lb/106Btu. The average optimal COE is 51.61 mills/kWh, and the 95% range of the optimal COE is from 48.25 to 60.30 mills/kWh. Average optimal design values and their 95% uncertainty ranges are given in Table S-9 of the Supporting Information. Pearson correlation coefficients between the optimal COE and random samples for the uncertain inputs were calculated to identify the key contributors to uncertainty in the optimal COE (13). Other sensitivity analysis approaches for pinpointing key contributors to uncertainty can be found in refs 40-45. As summarized in Table S-10 of the Supporting Information, zinc ferrite absorbent attrition rate was the most important contributor, followed by the project contingency factor, the error term of the regression model for HRSG direct cost, and the unit cost of coal. The difference in the average optimal COE from SP is 0.04 mills/kWh compared to the SO solution, which is equivalent to an expected benefit of reducing uncertainty of $240 000 per year. The true EVPI would probably be higher, since the cost of violating an emission permit might include fines or a forced shutdown of the plant, which is not accounted for here. The insights from the results based upon uncertainty are similar to those for the results based upon variability in the previous section, but the implications are slightly different. The positive EVPI is because of lower average SCR removal efficiency, and hence lower cost, from SP versus SO. Although zinc ferrite absorbent attrition rate contributes most to the overall uncertainty in the optimal COE, it is the uncertainties in the factors affecting uncontrolled NOx emissions that lead to the uncertainty in the SCR removal efficiency. Thus, reducing uncertainty regarding estimates of uncontrolled NOx emissions, such as by obtaining better data, not the uncertainty in the zinc ferrite absorbent attrition rate, would enable a design to be selected with a lower average cost than if the design must be robust to a large range of uncertainty. Whether a decision maker is willing to reduce the uncertainties depends on the cost of additional research compared to the magnitude of the EVPI. Optimization Considering Both Variability and Uncertainty in Model Inputs. In applying CSOP, SO was conducted for each realization of uncertainty. The objective of each SO was to minimize the expected COE over variability when the 90th percentile of NOx emissions was constrained to be less than or equal to 0.2 lb/106 Btu. Figure 2 shows the cumulative probability distribution of the optimal expected COE. The average optimal expected COE is 51.73 mills/kWh, and 90% of the estimates are less than 55 mills/kWh. Correlation coefficients were used to identify the key contributors to uncertainty in the optimal expected COE, which included

FIGURE 2. Cumulative probability distribution for optimal expected cost of electricity from the coupled stochastic optimization and programming method subject to the constraint of 90th percentile of NOx emissions e0.2 lb/106 Btu.

FIGURE 4. Cumulative probability distribution of uncertainty in EVPI with respect to variability. from $500 000 to $940 000 per year. These results are comparable to those from the one-dimensional case but provide insight that the range of uncertainty is approximately a factor of 2 from the lowest to the highest end of the range. Thus, the benefits of dynamic control can be assessed with more certainty if uncertainty in key model inputs could first be reduced.

Discussion

FIGURE 3. Two-dimensional distributions for optimal cost of electricity from the two-dimensional stochastic programming method subject to the constraint that NOx emissions e0.2 lb/106 Btu. the same key factors as for the one-dimensional simulation of uncertainty only. Details of the optimal results are given in Table S-11, and the sensitivity analysis results are given in Table S-12 of the Supporting Information. For TDSP, the COE was minimized subject to NOx emissions less than or equal to 0.2 lb/106Btu, and the twodimensional results for the optimal COE are shown in Figure 3. The figure depicts uncertainty intervals with respect to the variability in the optimal cost. The wide range of the uncertainty confidence intervals suggests that uncertainty contributes as much or more to variation in the optimal solution as does variability. This implies that there is more variation in optimal designs because of uncertainty, rather than variability. Thus, it would be more beneficial in the long run to collect better information in order to reduce uncertainty so as to arrive at a robust optimal solution. The mean optimal COE is 51.62 mills/kWh. The 95% range of uncertainty for the 50th percentile of variability of the optimal COE extends from 47.61 to 57.88 mills/kWh. Using Pearson correlation coefficients, key contributors to the variability and uncertainty in the optimal COE were identified, and results are detailed in Table S-13 of the Supporting Information. Similar to the SO results, the zinc ferrite adsorbent attrition rate was the most important contributor to uncertainty in the optimal COE, while contingency in the gas turbine direct cost was the most important contributor to variability in the optimal cost. Average values and the 95% range of uncertainty for the 50th percentile of variability of optimal design values are summarized in Table S-14 of the Supporting Information. CSOP involves SO for each alternative distribution of variability (i.e. a realization of uncertainty), and TDSP involves SP for each alternative distribution of variability. A comparison of SO results and average results of SP enables one to calculate the benefit of dynamic control with respect to variability. Thus, using CSOP and TDSP results, the cumulative distribution for the benefit of dynamic control can be constructed. Figure 4 shows the cumulative distribution for the benefit of dynamic control based on the CSOP and TDSP case study results. The average benefit of dynamic control is 0.112 mills/kWh, or $700 000 per year, with a 95% probability range from 0.08 mills/kWh to 0.15 mills/kWh or

In conventional point-estimate approaches to optimization, a single design choice is obtained based upon assuming point estimates for all model inputs. However, if the true value of any of the inputs differs from the assumed point estimate, the decision maker may have no idea as to whether the selected design would still be optimal or close to optimal. This paper illustrates the use of several methods that can be used to assist decision makers with design choices taking into account variability, uncertainty, or both in one or more variables. Furthermore, comparisons of results can help the decision maker gain insight into whether it is better to wait until more information is available before making an optimal design choice. For situations in which variability in feedstock composition, operating conditions, or other inputs to a model is considered to be significant, but for which uncertainties may be negligible, the decision maker typically must consider whether it is possible to dynamically monitor the system and adjust the design solution in real time, or whether a single design choice must be made now that must be robust to future variability in the system. Stochastic programming mimics the former case, while stochastic optimization represents the latter case. In the example case study, the optimal design of the NOx control system was shown to vary depending upon individual realizations of variability, leading to a range of optimal solutions from SP. However, by specifying a chance constraint, the decision maker can obtain one optimal solution using SO that is robust or reliable with respect to variability. On average, the SP solution is better than the SO solution, which implies that there is an economic benefit to adapting the design to specific realizations of variability, rather than to develop only one design that must account for all possible realizations of variability. This benefit must be weighed against the additional cost of dynamically adjusting the system. In some cases, it may not be possible to change a design variable in real time (e.g., sizes of equipment). Thus, a comparison of SP and SO in these cases could provide insight into whether the range of variability in inputs should be constrained more narrowly (e.g., through purchasing specifications on feedstock compositions). However, there are situations where we have only limited information concerning the variability in the model inputs (e.g. feedstock composition). This is particularly true at the design phase. We have demonstrated a technique for quantifying both variability and uncertainty in inputs to a process simulation model using bootstrap simulation. The “two-dimensional” representation of both variability and uncertainty can be conceptualized as being alternative realizations of frequency distributions for variability, in which VOL. 38, NO. 24, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 3. Summary of Potential Benefits from Techniques Demonstrated in the Paper techniques

results

potential benefits

example

Stochastic Optimization (SO)

reliability-based optimal solution with Optimal solution is reliable at or point estimate of design values respect to either variability or greater than a specified level of (e.g., SCR removal efficiency) uncertainty, assuming the exact confidence with respect to either subject to chance constraint value of variable or uncertain variability or uncertainty in (e.g., 95% probability of inputs is not known design constraints. achieving emission standard) Stochastic multiple optimal solutions assuming The range of SP results indicates how There are a wide range of optimal Programming that the specific realization of much the optimal solution could designs for NOx control and on (SP) variability or uncertainty is known vary if variability or uncertainty average there is a benefit to before an optimal choice must could be resolved prior to reducing uncertainty or knowing be made making a design. the specific value of variability before a design choice is made. Coupled Stochastic A distribution of SO results. Each SO The distribution of SO solutions with The optimal NOx control efficiency Optimization and corresponding to a specific respect to uncertainty provides a that is reliable with respect to Programming realization of uncertainty provides measure of how much the optimal variability could change (CSOP) a design that is robust to variability. design should change depending depending upon future but upon uncertainty. currently unknown values of uncertain inputs. Two-Dimensional Many optimal solutions are estimated The variability and uncertainty in Optimal NOx designs vary Stochastic for each combination of optimal solutions indicate whether substantially with respect to Programming realizations for all inputs that have the optimal design would differ variability and uncertainty (TDSP) variability, uncertainty, or both. significantly depending upon specific realizations of variability, uncertainty, or both.

the range of such realizations accounts for uncertainty. The coupled stochastic optimization and programming technique involves stochastic optimization, with respect to variability, for each possible alternative frequency distribution. Using this method, the decision maker is able to evaluate the uncertainty in the design, assuming that the system cannot be adapted in real-time with respect to variability. The two-dimensional stochastic programming technique simulates dynamic control for each alternative frequency distribution. Thus, by comparing CSOP and TDSP results, one is able to know not only the point estimate but also the associated uncertainty in the benefit of dynamic control, as is demonstrated in the case study. Simultaneous applications of CSOP and TDSP are expected to be of greatest utility for problems involving large commitment of resources, for which even small differences in designs can produce large absolute cost savings. We summarize the main features and potent benefits of each method demonstrated in the paper in Table 3. A key limitation is the computational intensity of the techniques, which in practical applications may mandate the prior development of RSMs of systems of interest, such as was demonstrated here. In addition, although the concept of EVPI as applied provides a benchmark for evaluating the benefits of dynamic control or for reducing uncertainty, it is typically an upper bound approximation subject to compatibility between the chance constraints required for the SO methods versus the exact constraints used in the SP methods. The loss functions used for such analyses can be extended to include a variety of real world factors, such as fines, shutdowns, or other penalties for failing to comply with emission permits.

Acknowledgments This work was supported by the National Science Foundation via Grant No. BES-9701502. We appreciate the assistance of Kaishan Zhang with aspects of the two-dimensional simulation and Junyu Zheng with AuvTool.

Supporting Information Available Text, tables, and figures pertaining to SO and SP algorithms, bootstrap simulation, CSOP and TDSP algorithms, genetic algorithm, uncertainty and variability assumptions for model inputs, and summaries of optimal objective function and design values for each case study and a brief description as to how the uncertainty and variability distribution assump6746

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tions are developed. This material is available free of charge via the Internet at http://pubs.acs.org.

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Received for review October 6, 2003. Revised manuscript received August 30, 2004. Accepted September 7, 2004. ES0351037

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