Environ. Sci. Technol. 2001, 35, 1173-1180
Reliability of Optimal Control Strategies for Photochemical Air Pollution LIHUA WANG AND JANA B. MILFORD* Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309-0427
This study illustrates how consideration of modeling uncertainties can affect optimal control strategies for urban ozone. Control strategies are investigated for illustrative cases of air parcel trajectories ending at Azusa, CA, and Riverside, CA, on August 28, 1987. The control strategies are designed to achieve a specified air quality target with a given reliability, considering uncertainties in the California Institute of Technology’s trajectory model and its inputs, including uncertainties in emissions and in the SAPRC-97 chemical mechanism. A decoupled stochastic optimization scheme is used to solve the chance-constrained programming problem. Least-cost control strategies derived using nominal model inputs and parameter values have low reliability for some target O3 concentrations when uncertainties are taken into account. For the case considered, reducing volatile organic compound (VOC) emissions from motor vehicles is identified as the leastcost approach to meeting O3 targets at Azusa. However, the optimal control strategies for Riverside depend on the target O3 concentrations and the level of reliability required. Consideration of model uncertainty is found to shift the focus from VOC controls to nitrogen oxide controls for the Riverside trajectory.
Introduction Control strategies for urban ozone that are developed using photochemical air quality models may prove to be more or less effective than anticipated because of meteorological variability and uncertainties in models and their inputs. Considering the expense of implementing controls and the harmful effects of ozone, cost-effective control strategies that have the desired effectiveness with high reliability need to be identified. One of the most widely recognized methods for designing cost-effective control strategies under uncertainty is chance-constrained programming (1-3). This approach finds the emission reductions that have minimum cost and satisfy a predefined probability of meeting an air quality standard, considering the effects of uncertainties and/ or variability in emissions, meteorology, and other model assumptions. However, chance-constrained programming has not been widely used for secondary air pollutants such as ozone, because it results in highly nonlinear systems that are hard to solve using conventional optimization algorithms or that require high computational expense (4). The objective of this study is to demonstrate the applicability of chanceconstrained programming to urban ozone. * Corresponding author phone: (303) 492-5542; fax: (303) 4922863; e-mail:
[email protected]. 10.1021/es001358y CCC: $20.00 Published on Web 02/06/2001
2001 American Chemical Society
FIGURE 1. Pictorial representation of the stochastic optimization framework (adapted from ref 13). The formal problem of finding optimal control strategies for urban ozone was first explored in the 1970s and early 1980s (5-8). These early studies relied on empirical relationships between ozone and its precursor emissions or used graphical optimization methods that superimpose control cost contours onto ozone isopleth diagrams generated using box models with simplified chemical mechanisms. They did not explicitly consider modeling uncertainty. More recent studies have investigated a variety of mathematical programming techniques and used updated, but still simplified, models of ozone-emissions relationships (4, 9-12). The optimization method used in this study builds on the approach of Heyes et al. (12), who used a reduced-form model to describe the formation of ozone from its precursors. The reduced-form model was found by applying regression analysis to the results of several hundred runs of a complex photochemical air quality model. One year’s meteorological data were used to account for meteorological variability, but no modeling uncertainties were considered. Optimal control strategies were then found by solving an ambient air qualityconstrained least-cost optimization problem with the reduced-form relationship. The resulting nonlinear optimization problem was easily solved using conventional algorithms. In another relevant study, Loughlin (9) applied chance-constrained programming using a genetic algorithm to minimize the costs of meeting an ambient air quality standard for ozone with a specified reliability. A box model was used to simulate the relationship between emissions and ozone concentrations. Hypothetical uncertainties in emissions and control costs were considered. Although Loughlin demonstrated how genetic algorithms could be used for this problem, he found them to be extremely computationally intensive, even for a single-cell photochemical model. This study solves a chance-constrained programming problem for ozone by using a decoupled stochastic optimization scheme (Figure 1) (13) with a reduced-form model for the probability of violating the ozone standard. Optimal control strategies are investigated for Azusa, CA, and Riverside, CA, for a high-ozone case ending on August 28, 1987. Emission reductions for stationary and mobile sources of volatile organic compounds (VOCs) and nitrogen oxides VOL. 35, NO. 6, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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(NOx) are considered as separate control variables. The reduced-form model is developed through regression analysis of the results of more than a thousand runs of the California Institute of Technology (CIT) trajectory model (14), considering the uncertainties in the SAPRC-97 chemical mechanism (15), emissions, and trajectory paths, among other model parameters.
Methods CIT Photochemical Trajectory Model. The basis for the CIT trajectory model is the atmospheric diffusion equation in the Lagrangian coordinate system, with associated initial and boundary conditions
(
)
∂〈ci〉 ∂〈ci〉 ∂ K + Ri(〈c1〉, ... 〈ci〉, ... 〈cN〉; t) ) ∂t ∂t zz ∂z
(1)
〈ci(z,0)〉 ) c0i (z) ∂〈ci〉 νig〈ci〉 - Kzz ) Ei ∂z ∂〈ci〉 Kzz )0 ∂z
at the ground surface
at the top of the column
where ci(z, t) is the concentration of compound i at time t and height z along the trajectory and νig and Ei are the surface deposition velocity and surface emission rate for compound i, respectively. The version of the CIT trajectory model used here subdivides the trajectory column into five vertical cells, with spatial derivatives calculated using finite difference approximations. Then the complete set of coupled species conservation equations is integrated numerically using the hybrid algorithm described by Young and Boris (16). The chemical mechanism employed in the CIT trajectory model in this study is SAPRC-97, which is composed of 214 reactions involving 102 species. Peterson’s (17) actinic flux estimates are adopted to calculate the 20 photolysis rates in the SAPRC97 mechanism under clear-sky conditions. Then UV scaling factors are applied to correct the photolysis rates for the conditions encountered during the August 1987 episode. The deposition rates are calculated as the product of the deposition affinity, which accounts for removal at less than the transport-limited rate, and the upper limit of the dry deposition velocity based on local surface roughness and meteorological conditions. Emissions are aggregated into three categories: motor vehicle emissions, biogenic emissions, and other anthropogenic emissions. All emissions are input to the model at the surface. Trajectory models allow users to investigate the transport and chemical transformation of pollutants, as well as the effect of emissions controls, with relatively low computational cost. Considering the high computational requirements for uncertainty analysis and optimization algorithms, the CIT trajectory model is used in this study to explore the optimal ozone control strategies for Azusa and Riverside. The limitations of the CIT trajectory model include the neglect of vertical advective transport, horizontal diffusion, and wind shear. Due to these assumptions, this study should be viewed as only an illustration of the stochastic optimization approach. Simulation Conditions. The simulation conditions used in this analysis are those of the Southern California Air Quality Study (SCAQS) episode of August 27-28, 1987, and are the same as those used by Bergin et al. (18). The August 1987 case was used because of the availability of comprehensive meteorological, emissions, and air quality data and the 1174
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occurrence of high ozone concentrations (19). Peak ozone concentrations of ∼240 ppb were observed at both Azusa and Riverside on August 28. Two-day trajectories ending at the time of the observed peak O3 concentrations, 1:30 p.m. local time at Azusa and 2:30 p.m. at Riverside, were modeled for both locations. Spatially and temporally distributed emissions for the airshed were estimated using the official inventory for stationary sources and the fuel and remotesensing based inventory described by Harley et al. (20) for motor vehicles. The total emissions of CO, VOCs and NOx in the South Coast Air Basin (SoCAB) were estimated to be about 8900 × 103, 2950 × 103, and 1150 × 103 kg/day (20) with about 91, 61, and 62%, respectively, contributed by on-road motor vehicle emissions. For the trajectory corresponding to the peak O3 concentration at Riverside, ∼77% of the anthropogenic VOC and ∼76% of the NOx emissions come from motor vehicles. About 67 and 56%, respectively, of the anthropogenic VOC and NOx emissions along the Azusa trajectory are from motor vehicles. Additional information on the model inputs is provided by Bergin et al. (18). Uncertainty in Model Parameters. The uncertainties considered in this study include those in trajectory paths due to windfield uncertainties and those in motor vehicle and other anthropogenic NOx and VOC emissions, mixing heights, O3 and NO2 deposition affinities, and key parameters of the SAPRC-97 mechanism. The 23 parameters treated as uncertain were found previously to be influential for ozone concentrations in the base case and/or for the reductions in ozone estimated to result from reducing VOC or NOx emissions from motor vehicles by 25% (18). Bergin et al. (18) provide a complete list of the uncertain parameters and explain how they are treated in the model. To summarize the uncertainties for the most important categories, those for the chemical parameters range from 10% (one standard deviation, σ, relative to the nominal value) for the rate constant of O3 plus NO to 66% for the rate constant of PPN decomposition; uncertainties for emissions range from 15% for stationary anthropogenic NOx emissions to 30% for stationary anthropogenic VOC emissions; and uncertainties in trajectory paths result in ∼15% uncertainties in the sum of the emissions along the trajectories. One change was made in this study from the input uncertainties used by Bergin et al. (18). The treatment of uncertainty in parameters of the mechanism for oxidation of aromatic compounds was revised on the basis of recent work (21). Here, the product yields for AFG2 (highly photoreactive uncharacterized ring fragmentation products formed from the alkylbenzenes) and methylglyoxal from the reaction of the lumped aromatics class ARO2 with OH are treated as having (1σ uncertainties of 23 and 20%, respectively, which are smaller uncertainty estimates than those used by Bergin et al. (18). Ambient Least-Cost Model under Uncertainty. One of the objectives of this study is to investigate the relative effectiveness of NOx and VOC controls for mobile sources and stationary anthropogenic sources. Consequently, values of four decision variables, the percentage emission reductions or emission reduction factors (ERFs) for mobile NOx emissions (MVNOx), mobile VOC emissions (MVVOC), stationary anthropogenic NOx emissions (OTHNOx), and stationary anthropogenic VOC emissions (OTHVOC), are to be determined to optimize the ozone control strategies. ERFs are defined as the ratio of the mass emissions reductions to the mass emissions in the base case for a given source category and pollutant. The ERFs are assumed to be applied uniformly over all of the emissions in the SoCAB, for the specified pollutant and source category. According to the stochastic optimization scheme (Figure 1), the ambient least-cost,
chance-constrained model used in this study is defined as
min cost ) CSERFMVNOx + CSERFMVVOC + CSERFOTHNOx + CSERFOTHVOC (2) s.t. P[O3] g [O3]standard e R
tions to estimate the corresponding violation probability. This process is applied in turn to all 60 ERF combinations, requiring a total of 1200 model runs. Regression analysis is then applied to the 60 violation probabilities to find the reduced-form relationship:
Pviolation ) f(ERFMVNOx, ERFMVVOC, ERFOTHNOx, ERFOTHVOC) (4)
BLERFMVNOx e ERFMVNOx e BUERFMVNOx BLERFMVVOC e ERFMVVOC e BUERFMVVOC BLERFOTHNOx e ERFOTHNOx e BUERFOTHNOx BLERFOTHVOC e ERFOTHVOC e BUERFOTHVOC where CSERFj is the cost of ERF × 100% reductions for the jth emissions category and BUERFj and BLERFj represent the upper and lower bounds for the ERF for source j. The expression P([O3] g [O3]standard) e R represents the constraint that the modeled probability of violating the air quality standard is less than the predefined level R, which is termed the violation probability. In other words, the reliability of the control strategy can be expressed as P([O3] e [O3]standard) ) (1 - R). [O3] is a function of uncertain variables (δ) considered in the air quality models, as well as the four ERFs:
[O3] ) f(δ,ERFMVNOx,ERFMVVOC,ERFOTHNOx,ERFOTHVOC) (3) Therefore, the constraint P([O3] g [O3]standard) e R is a complex nonlinear constraint that must be evaluated using an air quality model. Decoupled Stochastic Optimization. In theory, the chance-constrained cost minimization problem (eq 2) can be solved through stochastic optimization (Figure 1). Given one set of values for the four ERFs, the corresponding probability distribution function of ozone concentrations is estimated in the uncertainty analysis loop, by considering the uncertainties in the input variables in the model. The violation probability for the given ERFs can be estimated from the probability distribution function. The optimization loop will then find the search directions and step sizes for the four decision variables for the next trial. This process continues until the optimal solution is found. However, the computational load for stochastic optimization is extremely high, considering the parametric method used to evaluate gradients for search directions in common optimization algorithms and the Monte Carlo calculations required in the uncertainty loop. Therefore, we decoupled the uncertainty analysis and optimization loops in this study. First, a reduced-form model for the relationship between the violation probability and the ERFs is found through regression analysis of the results of more than a thousand Monte Carlo simulations. Two different kinds of samples are used in the simulations. The first are samples for the 23 uncertain input variables of the air quality model, described under Uncertainty in Model Parameters. For computational efficiency, samples of uncertain input variables are drawn using Latin hypercube sampling (LHS) (22), with a sample size of 200. The second kind of sample used is for the ERFs. Each of these samples, which are also drawn using LHS, represents one combination of control levels for the four source categories. For this study, 60 combinations of ERFs were used, with values drawn from uniform distributions spanning the ranges from 0 to 100% for ERFMVNOx and ERFOTHNOx, from 0 to 80% for ERFMVVOC, and from 0 to 50% for ERFOTHVOC. Given 1 of the 60 combinations of ERFs, the uncertainty analysis loop performs 200 Monte Carlo simula-
Using the reduced-form relationship for the violation probabilities, the control cost minimization problem (eq 2) becomes a general nonlinear optimization problem with smooth functions, which can be easily solved using traditional optimization algorithms. Successive quadratic programming (SQP) (23, 24) is used here. The accuracy of the results obtained from this approach will depend on the accuracy of the reduced-form model for the violation probabilities. The difference between this approach and that of Heyes et al. (12) is that the reduced-form model here is for the violation probabilities under uncertainty, instead of for deterministic ozone concentrations. Control Measures for Mobile and Stationary Sources. The least-cost model in eq 2 requires cost information for the emission reductions in each category. In this study, control measures and associated marginal costs are taken from McBride et al. (11) for VOC controls, from SCAQMD (25) for stationary NOx controls, and from Cackette (26) and SCAQMD (27, 28) for mobile NOx controls. The full data set is available as Supporting Information. The data include the detailed source category (e.g., aerosol consumer products), its uncontrolled emissions rate in 1987 (e.g., 45.3 tons/day), and proposed control measures with their marginal costs and control factors. For each source, at most four control measures were identified. We assume that different control measures for one emission source will be applied successively according to the sequence of their availability. We also assume that the fractional control factor (i.e., 1 - ERF) for each control measure is known, irrespective of uncertainty in the base case emissions estimates. Thus, for example, although the official motor vehicle emissions inventory is thought to be underestimated by a factor of ∼2 (20), the ratio of the emissions reduction from each control measure to the total emissions is assumed not to be affected by that underestimation. The reductions are further assumed to apply uniformly in time and to be immediately available, although planning documents may have listed them as available only in future years. On the basis of the above information and assumptions, cumulative cost functions (in 1991 dollars per ton) are derived for the four controllable source categories. We note that the information about the control measures is estimated for the total summertime 1987 emission inventory in the SoCAB, not for the specific episodes and trajectories in this study. The cumulative emission reductions are expressed as ERFs (Figure 2) and are assumed to be applied uniformly throughout the whole air basin. The total cost for a particular reduction level of the emission source can be calculated from the cumulative percent reduction cost and the total emissions of that source in the SoCAB. Thus, the control problem is implemented by applying uniform percentage emissions reductions throughout the airshed to meet violation probability constraints at either Azusa or Riverside. Figure 2 shows the cumulative control cost functions for total VOC and NOx controls, which are derived on the basis of the costeffectiveness and prioritization of all of the control measures for both motor vehicle and stationary emissions. Ordered from least to most expensive, the first 40% of NOx emission reductions come mainly from mobile sources. The first 50% of VOC emission reductions come from mobile sources, aerosol consumer products, and surface coatings. Separate VOL. 35, NO. 6, 2001 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. Cumulative control costs for total (a) VOC and (b) NOx emissions. cumulative control cost functions for mobile and stationary sources are given by Wang (29). Overall, the control costs increase with increased emission reductions as quadratic or higher order functions. As a result, beyond some emissions reduction level, the costs increase sharply when emissions are further reduced. On the basis of control technologies identified in the air quality management plans, the upper limits for mobile VOC and NOx emissions reductions are about 81 and 48%, respectively, and those for stationary VOC and NOx emissions reductions about 49 and 86%, respectively. The upper bounds on the total VOC and NOx emission reductions are about 66 and 56%, respectively.
Results Nominal Simulation Results. Using nominal parameter values, the simulated maximum ozone concentrations are about 212 and 256 ppb for Azusa and Riverside, respectively, whereas the point observed concentrations are 240 ppb for both sites. The nominal simulated peak ozone concentrations are thus about 12% lower for Azusa and 6% higher for Riverside than the corresponding observed values. Ozone isopleth graphs calculated using nominal input and parameter values are shown in Figure 3. The nominal results indicate that VOC and NOx controls are almost equally effective for reducing the maximum 1-h average ozone concentration at Riverside, whereas VOC control is more effective than NOx control for Azusa. For Riverside, the relative effectiveness of VOC and NOx controls depends on the target ozone level. NOx control is more effective than VOC control for reducing the maximum ozone concentration to levels >200 ppb, whereas VOC control appears more to be effective for target levels of 0.98. Due to the collinearity of the terms in the quadratic functions, there exist many different forms of the violation probability equations for the four ERFs with similar R2 values. However, the optimization results were not very sensitive to the choice of equations. Regression equations for the other target levels are given by Wang (29). In using the violation functions for optimal control strategy analysis, two different cases are considered. In the “ideal case”, it is assumed that the anthropogenic VOC and NOx emissions can be reduced to zero. Of course, this will not happen in practice. In the second case considered, called the “constrained case”, the ranges of the emission reductions from mobile and stationary sources are limited by the measures identified in the air quality maintenance plans. Graphical Analysis for Optimal Control Strategies. Figure 5 shows the violation probability functions for total VOC and NOx emission reductions for the 120 and 80 ppb levels for Azusa and Riverside. [Graphs for the 150 ppbv level are presented by Wang (29).] Figure 5 shows that when one starts from the base case, VOC controls decrease the violation probabilities, whereas initially NOx control alone would increase violation probabilities for the 120 and 80 ppb levels. Of note, with NOx emissions reduced by >60%, decreasing VOC emissions can increase violation probabilities. Overall, Figure 5 suggests that VOC controls are more effective than NOx controls for Azusa, whereas the relative control effectiveness of VOC and NOx controls for Riverside depends on the target level and the required violation probability. For Riverside, NOx controls are more effective for reducing ozone to the 150 ppb level (not shown) and to the 120 ppb level if violation probabilities
