Risk management for sulfur dioxide abatement under multiple uncertainties
C. DAI1,W. SUN2,Q. TAN3,2,*(),Y. LIU1,W.T. LU1,H.C. GUO1
1. College of Environmental Science and Engineering, Peking University, Beijing 100871, China 2. Institute for Energy, Environment and Sustainable Communities, University of Regina, Regina, Saskatchewan S4S 7H9, Canada 3. College of Water Resources & Civil Engineering, China Agricultural University, Beijing 100083, China
In this study, interval-parameter programming, two-stage stochastic programming (TSP), and conditional value-at-risk (CVaR) were incorporated into a general optimization framework, leading to an interval-parameter CVaR-based two-stage programming (ICTP) method. The ICTP method had several advantages: (i) its objective function simultaneously took expected cost and risk cost into consideration, and also used discrete random variables and discrete intervals to reflect uncertain properties; (ii) it quantitatively evaluated the right tail of distributions of random variables which could better calculate the risk of violated environmental standards; (iii) it was useful for helping decision makers to analyze the trade-offs between cost and risk; and (iv) it was effective to penalize the second-stage costs, as well as to capture the notion of risk in stochastic programming. The developed model was applied to sulfur dioxide abatement in an air quality management system. The results indicated that the ICTP method could be used for generating a series of air quality management schemes under different risk-aversion levels, for identifying desired air quality management strategies for decision makers, and for considering a proper balance between system economy and environmental quality.
Treated amount of SO2 emission, Xij,opt±/(tonne·day?1)
Treated amount of excess SO2 emission, Yijs,opt±/(tonne·day?1)
Treated amount of SO2 emission, Xij,opt±/(tonne·day?1)
Treated amount of excess SO2 emission, Yijs,opt±/(tonne·day?1)
Treated amount of SO2 emission, Xij,opt±/(tonne·day?1)
Treated amount of excess SO2 emission, Yijs,opt±/(tonne·day?1)
Power plant 1(i = 1)
s = 1
30
0
30
0
24
0
s = 2
30
0
30
0
24
0
s = 3
30
0
30
0
24
0
s = 4
30
0
30
0
24
0
s = 5
30
0
30
0
24
0
s = 6
30
0
30
0
24
[0, 2.3]
s = 7
30
0
30
0
24
[2.3, 5.9]
s = 8
30
0
30
0
24
[5.9, 9.5]
s = 9
30
0
30
0
24
[9.5, 13]
s = 10
30
0
30
0
24
[13, 16.6]
s = 11
30
0
30
0
24
[16.6, 20.2]
s = 12
30
0
30
0
24
[20.2, 23.8]
s = 13
30
[0, 3.4]
30
0
24
[23.8, 24]
s = 14
30
[5.8, 6.9]
30
0
24
[21.6, 24]
s = 15
30
12.6
30
0
24
[18.3, 21.9]
s = 16
30
[13.6, 14.1]
30
0
24
[20.9, 24]
s = 17
30
[17.4, 17.7]
30
0
24
[20.7, 24]
s = 18
30
16.1
30
5.2
24
[20.4, 24]
s = 19
30
13.1
30
12.1
24
[20.1, 23.6]
s = 20
30
10.3
30
18.7
24
[19.8, 23.4]
Power plant 2(i = 2)
s = 1
30
0
18
0
12
0
s = 2
30
0
18
0
12
0
s = 3
30
0
18
0
12
0
s = 4
30
0
18
0
12
0
s = 5
30
0
18
0
12
0
s = 6
30
0
18
0
12
0
s = 7
30
0
18
0
12
0
s = 8
30
0
18
0
12
[0, 0.1]
s = 9
30
0
18
0
12
[0.1, 2.5]
s = 10
30
0
18
0
12
[2.5, 4.8]
s = 11
30
0
18
0
12
[4.8, 7.2]
s = 12
30
0
18
0
12
[7.2, 9.5]
s = 13
30
0
18
0
12
[9.5, 11.9]
s = 14
30
[0, 2.2]
18
0
12
[11.9, 12]
s = 15
30
[2.2, 4.6]
18
0
12
12
s = 16
30
[4.6, 7]
18
0
12
12
s = 17
30
[7, 9.3]
18
0
12
12
s = 18
30
[9.3, 11.7]
18
0
12
12
s = 19
30
[11.7, 14]
18
0
12
12
s = 20
30
[14, 16.4]
18
0
12
12
Chemical industry plant (i = 3)
s = 1
0
0
0
0
[7.5, 7.7]
0
s = 2
0
0
0
0
[7.5, 7.7]
0
s = 3
0
0
0
0
[7.5, 7.7]
0
s = 4
0
0
0
0
[7.5, 7.7]
0
s = 5
0
0
0
0
[7.5, 7.7]
0
s = 6
0
0
0
0
[7.5, 7.7]
0
s = 7
0
0
0
0
[7.5, 7.7]
0
s = 8
0
0
0
0
[7.5, 7.7]
0
s = 9
0
0
0
0
[7.5, 7.7]
0
s = 10
0
0
0
0
[7.5, 7.7]
0
s = 11
0
0
0
0
[7.5, 7.7]
0
s = 12
0
0
0
0
[7.5, 7.7]
0
s = 13
0
0
0
0
[7.5, 7.7]
[0.2, 0.3]
s = 14
0
0
0
0
[7.5, 7.7]
[0.5, 0.6]
s = 15
0
0
0
0
[7.5, 7.7]
[0.8, 0.9]
s = 16
0
0
0
0
[7.5, 7.7]
1.1
s = 17
0
0
0
0
[7.5, 7.7]
[1.3, 1.4]
s = 18
0
0
0
0
[7.5, 7.7]
[1.6, 1.7]
s = 19
0
0
0
0
[7.5, 7.7]
[1.9, 2]
s = 20
0
0
0
0
[7.5, 7.7]
2.2
Petroleum refinery (i = 4)
s = 1
8
0
0
0
7.5
0
s = 2
8
0
0
0
7.5
0
s = 3
8
0
0
0
7.5
0
s = 4
8
0
0
0
7.5
0
s = 5
8
0
0
0
7.5
0
s = 6
8
0
0
0
7.5
0
s = 7
8
0
0
0
7.5
0
s = 8
8
0
0
0
7.5
0
s = 9
8
0
0
0
7.5
0
s = 10
8
0
0
0
7.5
[0, 0.6]
s = 11
8
0
0
0
7.5
[0.6, 1.1]
s = 12
8
0
0
0
7.5
[1.1, 1.7]
s = 13
8
1.2
0
0
7.5
[0.5, 1.1]
s = 14
8
2.3
0
0
7.5
[0, 0.6]
s = 15
8
0
0
0
7.5
[2.9, 3.4]
s = 16
8
3.4
0
0
7.5
[0, 0.6]
s = 17
8
4
0
0
7.5
[0, 0.6]
s = 18
8
4.6
0
0
7.5
[0, 0.6]
s = 19
8
5.2
0
0
7.5
[0, 0.5]
s = 20
8
5.7
0
0
7.5
[0, 0.6]
Steel mill (i = 5)
s = 1
2
0
2.7
0
[0, 0.2]
0
s = 2
2
0
2.7
0
[0, 0.2]
0
s = 3
2
0
2.7
0
[0, 0.2]
0
s = 4
2
0
2.7
0
[0, 0.2]
0
s = 5
2
0
2.7
0
[0, 0.2]
0
s = 6
2
0
2.7
0
[0, 0.2]
0
s = 7
2
0
2.7
0
[0, 0.2]
0
s = 8
2
0
2.7
0
[0, 0.2]
0
s = 9
2
0
2.7
0
[0, 0.2]
0
s = 10
2
0.2
2.7
0
[0, 0.2]
0
s = 11
2
0.4
2.7
0
[0, 0.2]
0
s = 12
2
0.6
2.7
0
[0, 0.2]
0
s = 13
2
0.8
2.7
0
[0, 0.2]
0
s = 14
2
1
2.7
0
[0, 0.2]
0
s = 15
2
1.1
2.7
0.1
[0, 0.2]
0
s = 16
2
0.9
2.7
0.5
[0, 0.2]
0
s = 17
2
0.7
2.7
0.9
[0, 0.2]
0
s = 18
2
0
2.7
1.8
[0, 0.2]
0
s = 19
2
0
2.7
2
[0, 0.2]
[0, 0.1]
s = 20
2
0
2.7
2.3
[0, 0.2]
0
Tab.3
Confidence level
Source
Technology
Risk parameters/(tonne·day?1)
λ = 0.5
λ = 1
λ = 2
λ = 5
λ = 10
β = 0.6
i = 1
j = 1
[34.2, 34.7]
[32.3, 32.5]
[31.5, 31.7]
[31, 31.7]
[30.9, 31.7]
j = 2
22
27.3
30.1
30.1
30
j = 3
[40.5, 43.4]
[37.6, 40.6]
[35.9, 38.8]
[36.4, 38.9]
[36.5, 38.9]
i = 2
j = 1
[31.4, 32.3]
[31.1, 31.9]
[30.6, 31.1]
[30.3, 30.6]
[30.3, 30.6]
j = 2
12.3
13.4
15.8
18
18
j = 3
[19.1, 20.3]
[18.5, 19.7]
[17.2, 18.4]
[16, 17.2]
[16, 17.2]
i = 3
j = 1
0
0
0
3.3
3.3
j = 2
0
0
0
0
0
j = 3
[7.3, 7.5]
[7.6, 7.8]
[8, 8.2]
[5, 5.2]
[5, 5.2]
i = 4
j = 1
7.9
8.5
8.5
8.3
8.1
j = 2
0
0
0
0
2.2
j = 3
[8, 8.4]
[7.8, 8.1]
[7.8, 8.1]
[7.9, 8.3]
[7.5, 7.6]
i = 5
j = 1
2.4
2.4
2.3
2.1
2.1
j = 2
2.6
2.6
2.8
3.2
3.3
j = 3
[0, 0.2]
[0, 0.2]
[0, 0.2]
[0, 0.2]
[0, 0.2]
β = 0.8
i = 1
j = 1
[33, 33.3]
[31.8, 32.1]
[31.4, 31.7]
[31, 31.7]
[30.9, 31.7]
j = 2
25.4
28.9
30.1
30.1
30
j = 3
[38.5, 41.6]
[36.6, 39.6]
[36, 38.8]
[36.5, 38.9]
[36.5, 38.9]
i = 2
j = 1
[31.1, 31.9]
[31.1, 31.9]
[30.5, 30.8]
[30.3, 30.6]
[30.3, 30.6]
j = 2
13.4
13.4
17.1
18
18
j = 3
[18.5, 19.7]
[18.5, 19.7]
[16.5, 17.7]
[16, 17.2]
[16, 17.2]
i = 3
j = 1
0
0
0.2
3.3
3.3
j = 2
0
0
0
0
0
j = 3
[7.3, 7.5]
[7.6, 7.8]
8.1
[5.3, 5.5]
[5.3, 5.5]
i = 4
j = 1
7.9
8.5
8.5
8.3
8.1
j = 2
0
0
0
1
3.1
j = 3
[8, 8.4]
[7.8, 8.1]
[7.8, 8.1]
[7.6, 7.8]
7.5
i = 5
j = 1
1.7
2.3
2.1
2.1
2.1
j = 2
2.8
2.7
3.1
3.3
3.3
j = 3
[0.5, 0.7]
[0, 0.2]
[0, 0.2]
[0, 0.2]
[0, 0.2]
β = 0.99
i = 1
j = 1
[33, 33.3]
[31.4, 31.7]
[31.3, 31.7]
[30.9, 31.7]
[31, 31.7]
j = 2
25.4
30.1
30.1
30
30
j = 3
[38.5, 41.6]
[36.1, 38.8]
[36.1, 38.9]
[36.5, 38.9]
[36.5, 38.9]
i = 2
j = 1
[31.1, 31.9]
[30.3, 30.6]
[30.3, 30.6]
[30.3, 30.6]
[30.3, 30.6]
j = 2
13.4
18
18
18
18
j = 3
[18.5, 19.7]
[16, 17.2]
[16, 17.2]
[16, 17.2]
[16, 17.2]
i = 3
j = 1
0
0
0.7
3.3
0
j = 2
0
0
0
0
3.3
j = 3
[7.3, 7.5]
[7.6, 7.8]
8.1
[6.4, 6.6]
[6.4, 6.6]
i = 4
j = 1
8
8.4
8.3
8.1
7.4
j = 2
0
0
1
6.5
7.2
j = 3
[8, 8.4]
[7.8, 8.1]
[7.6, 7.8]
[6.6, 7.2]
[6.6, 7.2]
i = 5
j = 1
1.7
2.3
2.1
2.1
2.1
j = 2
2.8
2.7
3.1
3.3
3.3
j = 3
[0.5, 0.7]
[0, 0.2]
[0, 0.2]
[0, 0.2]
[0, 0.2]
Tab.4
Fig.4
Fig.5
Fig.6
Fig.7
From power plant 1(i = 1)
From power plant 2(i = 2)
From chemical industry plant(i = 3)
From petroleum refinery(i = 4)
From steel mill(i = 5)
j = 1
j = 2
j = 3
j = 1
j = 2
j = 3
j = 1
j = 2
j = 3
j = 1
j = 2
j = 3
j = 1
j = 2
j = 3
Allowable SO2 emission ( Xij,opt±)/(tonne·day?1)
30
21.6
24
30
11.5
12
0
0
[6.6, 6.8]
6.3
0
7.5
1
2.75
[0.5, 0.68]
Excess SO2 emission ( Yijs,opt±)/(tonne·day?1)
s = 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
s = 2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
s = 3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
s = 4
0
0
[0, 3.6]
0
0
0
0
0
0
0
0
0
0
0
0
s = 5
0
0
[3.6, 7.1]
0
0
0
0
0
0
0
0
0
0
0
0
s = 6
0
0
[7.1, 10.7]
0
0
[0, 1.9]
0
0
0
0
0
0
0
0
0
s = 7
0
0
[10.7, 14.3]
0
0
[1.9, 4.3]
0
0
0
0
0
[0, 0.5]
0
0
[0.05, 0.07]
s = 8
0
0
[14.3, 17.9]
0
0
[4.3, 6.6]
0
0
0
0
0
[0.5, 1.1]
0
0
[0.25, 0.27]
s = 9
0
0
[17.9, 21.4]
0
0
[6.6, 9]
0
0
[0, 0.1]
0
0
[1.1, 1.7]
0
0
[0.4, 0.5]
s = 10
[0, 1]
0
[21.4, 24]
0
0
[9, 11.3]
0
0
[0.3, 0.4]
0
0
[1.7, 2.3]
0.15
0
[0.5, 0.52]
s = 11
[5.3, 5.6]
0
[19.7, 23]
[0, 1.7]
0
[11.3, 12]
0
0
[0.6, 0.7]
0
0
[2.3, 2.8]
0.35
0
[0.5, 0.52]
s = 12
9.95
0
[18.6, 22.3]
[1.7, 4]
0
12
0
0
0.9
0
0
[2.8, 3.4]
1
0
[0.05, 0.07]
s = 13
[11.1, 11.8]
0
[21.05, 24]
[4, 6.4]
0
12
0
0
[1.1, 1.2]
3.4
0
[0, 0.6]
1
0
[0.25, 0.27]
s = 14
[14.8, 15.3]
0
[21, 24]
[6.4, 8.7]
0
12
0
0
[1.4, 1.5]
4
0
[0, 0.6]
1
0.45
[0, 0.02]
s = 15
23.2
0
[16.1, 19.7]
[8.7, 11.1]
0
12
0
0
[1.7, 1.8]
0
0
[4.6, 5.1]
0.8
0.85
[0, 0.02]
s = 16
16.5
6
[20.4, 24]
[11.1, 13.5]
0
12
0
0
2
5.1
0
[0, 0.6]
0
1.85
[0, 0.02]
s = 17
13.5
12.8
[20.2, 23.8]
[13.5, 15.8]
0
12
0
0
[2.2, 2.3]
5.7
0
[0, 0.6]
0
2.05
[0, 0.02]
s = 18
10.6
19.6
[19.9, 23.5]
[15.8, 18.2]
0
12
0
0
[2.5, 2.6]
6.3
0
[0, 0.6]
0
2.25
[0, 0.02]
s = 19
19.4
21.6
[12.7, 16.2]
[13.3, 15.6]
4.9
12
0
0
[2.8, 2.9]
0
0
[6.9, 7.4]
0
2.45
[0, 0.12]
s = 20
17.4
21.6
[18.2, 21.8]
[9, 11.4]
11.5
12
0
0
3.1
6.3
0
[1.1, 1.7]
[0, 0.1]
2.75
0
Tab.5
1
Ahmed S (2004). Mean-risk objectives in stochastic programming. Technical report, Georgia Institute of Technology, available at www. optimization-online.org
2
An H, Eheart J W (2007). A screening technique for joint chance-constrained programming for air-quality management. Oper Res, 55(4): 792–798
https://doi.org/10.1287/opre.1060.0377
3
Athanasiadis I N, Mitkas P A (2007). Knowledge discovery for operational decision support in air quality management. Journal of Environmental Informatics, 9(2): 100–107
https://doi.org/10.3808/jei.200700091
4
Birbil S I, Frenk J, Kaynar B, Noyan N (2009). Risk measures and their applications in asset management. In: Gregoriou G N, ed. The VaR Implementation Handbook. New York: The McGraw-Hill Companies, 311–337
Byun D W, Kim S T, Chen F Y, Kim S B, Cuclis A, Moon N K (2003). Information infrastructure for air quality modeling and analysis: application to the houston-galveston ozone non-attainment area. Journal of Environmental Informatics, 2(2): 38–57
https://doi.org/10.3808/jei.200300021
7
Cai Y P, Huang G H, Tan Q, Yang Z F (2011a). An integrated approach for climate-change impact analysis and adaptation planning under multi-level uncertainties. Part I: methodology. Renewable & Sustainable Energy Reviews, 15: 2779–2790
8
Cai Y P, Huang G H, Tan Q, Liu L (2011b). An integrated approach for climate-change impact analysis and adaptation planning under multi-level uncertainties. Part II. case study. Renewable & Sustainable Energy Reviews, 15: 3051–3073
9
Chen C, Huang G H, Li Y, Li M (2013a). Model of risk analysis on site selection of biomass power plant based on stochastic robust interval method. Transactions of the Chinese Society of Agricultural Engineering, 29(20): 206–213
10
Chen C, Huang G H, Li Y P, Zhou Y (2013b). A robust risk analysis method for water resources allocation under uncertainty. Stochastic Environ Res Risk Assess, 27(3): 713–723
https://doi.org/10.1007/s00477-012-0634-5
11
Dai C, Li Y, Huang G (2012). An interval-parameter chance-constrained dynamic programming approach for capacity planning under uncertainty. Resour Conserv Recycling, 62: 37–50
https://doi.org/10.1016/j.resconrec.2012.02.010
Dong C, Huang G H, Cai Y P, Liu Y (2012). An inexact optimization modeling approach for supporting energy systems planning and air pollution mitigation in Beijing City. Energy, 37(1): 673–688
https://doi.org/10.1016/j.energy.2011.10.030.
Haurie A, Kübler J, Clappier A, Van Den Bergh H (2004). A metamodeling approach for integrated assessment of air quality policies. Environ Model Assess, 9(1): 1–12
https://doi.org/10.1023/B:ENMO.0000020886.39231.52
16
Hsu C P, Huang C W, Paul Chiou W J P (2012). Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets. Rev Quant Finance Account, 39(4): 447–468
https://doi.org/10.1007/s11156-011-0261-0
17
Huang G, Loucks D (2000). An inexact two-stage stochastic programming model for water resources management under uncertainty. Civ Eng Environ Syst, 17(2): 95–118
https://doi.org/10.1080/02630250008970277
Huang G H, Chang N B (2003). Perspectives of environmental informatics and systems analysis. J Environ Inform, 1(1): 1–6
https://doi.org/10.3808/jei.200300001
20
Inuiguchi M, Sakawa M (1998). Robust optimization under softness in a fuzzy linear programming problem. Int J Approx Reason, 18(1‒2): 21–34
https://doi.org/10.1016/S0888-613X(97)10002-0
21
Lejano R P, Ayala P M, Gonzales E A (1997). RESEARCH: optimizing the mix of strategies for control of vehicular emissions. Environ Manage, 21(1): 79–87
https://doi.org/10.1007/s002679900007
22
Li Y, Huang G, Nie X, Nie S (2008). An inexact fuzzy-robust two-stage programming model for managing sulfur dioxide abatement under uncertainty. Environ Model Assess, 13(1): 77–91
https://doi.org/10.1007/s10666-006-9077-z
23
Li Y, Huang G H, Veawab A, Nie X, Liu L (2006). Two-stage fuzzy-stochastic robust programming: a hybrid model for regional air quality management. J Air Waste Manag Assoc, 56(8): 1070–1082
https://doi.org/10.1080/10473289.2006.10464536
24
Lim C, Sherali H D, Uryasev S (2010). Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization. Comput Optim Appl, 46(3): 391–415
https://doi.org/10.1007/s10589-008-9196-3
25
Liu L, Huang G, Liu Y, Fuller G, Zeng G (2003). A fuzzy-stochastic robust programming model for regional air quality management under uncertainty. Eng Optim, 35(2): 177–199
https://doi.org/10.1080/0305215031000097068
26
Liu Y, Zou R, Guo H (2011). Risk explicit interval linear programming model for uncertainty-based nutrient-reduction optimization for the Lake Qionghai Watershed. J Water Resour Plann Manage, 137(1), 83–91
https://doi.org/10.1061/(ASCE)WR.1943-5452.0000099
27
Lv Y, Huang G H, Li Y P, Yang Z F, Sun W (2011). A two-stage inexact joint-probabilistic programming method for air quality management under uncertainty. J Environ Manage, 92(3): 813–826
https://doi.org/10.1016/j.jenvman.2010.10.027
28
Ma X, Zhang F (2002). A genetic algorithm based stochastic programming model for air quality management. J Environ Sci (China), 14(3): 367–374
29
Marti K, Ploöchinger E (1990). Optimal step sizes in semi-stochastic approximation procedures. I. Optimization, 21(1): 123–153
https://doi.org/10.1080/02331939008843527
30
Miller N, Ruszczynski A (2008). Risk-adjusted probability measures in portfolio optimization with coherent measures of risk. Eur J Oper Res, 191(1): 193–206
https://doi.org/10.1016/j.ejor.2007.06.052
31
Noyan N (2012). Risk-averse two-stage stochastic programming with an application to disaster management. Comput Oper Res, 39(3): 541–559
https://doi.org/10.1016/j.cor.2011.03.017
32
Rockafellar R T, Uryasev S (2000). Optimization of conditional value-at-risk. Journal of Risk, 2: 21–42
33
Ruszczyński A (1993). Parallel decomposition of multistage stochastic programming problems. Math Program, 58(1–3): 201–228
https://doi.org/10.1007/BF01581267
34
Schultz R (2011). Risk aversion in two-stage stochastic integer programming. Stochastic Programming: 165–187
35
Schultz R, Tiedemann S (2006). Conditional value-at-risk in stochastic programs with mixed-integer recourse. Math Program, 105(2‒3): 365–386
https://doi.org/10.1007/s10107-005-0658-4
Shao L G, Qin X S, Xu Y (2011). A conditional value-at-risk based inexact water allocation model. Water Resour Manage, 25(9): 2125–2145
https://doi.org/10.1007/s11269-011-9799-9
38
Tan Q, Huang G H, Wu C Z, Cai Y P (2011). IF-EM: an interval-parameter fuzzy linear programming model for environment-oriented evacuation planning under uncertainty. J Adv Transport, 45(4): 286–303
https://doi.org/10.1016/j.amc.2009.12.031
38
Tong X J, Qi L Q, Wu F, Zhou H (2010). A smoothing method for solving portfolio optimization with CVaR and applications in allocation of generation asset. Appl Math Comput, 216(6): 1723–1740
https://doi.org/10.1016/j.amc.2009.12.031
39
Turner D B (1970). Workbook of atmospheric dispersion estimates. Washington D C: Office of Air Programs, Publication No. AP-26, U.S. Environmental Protection Agency
40
Wang L, Milford J B (2001). Reliability of optimal control strategies for photochemical air pollution. Environ Sci Technol, 35(6): 1173–1180
https://doi.org/10.1021/es001358y
