- Email: [email protected]
A grey nonlinear programming approach for planning coastal wastewater treatment and disposal systems
e:>
Pergamon
OI273-1223(95)00565-X
Wat. ScL Tech. Vol. 32. No. 2, pp. 19-29. 1995. Copyright C I99S IAwQ Printed ill Oreal Britaio. All righll ruerved. 0273-1223I9S 59'SO + 0'00
A GREY NONLINEAR PROGRAMMING APPROACH FOR PLANNING COASTAL WASTEWATER TREATMENT AND DISPOSAL SYSTEMS Ni-Bin Chang and S. F. Wang Department ofEnvironmental Engineering, National Cheng-Kung University, Tainan. Taiwan
ABSTRACT Coastal w~tewater treatment and ocean outfall facilities are usually planned as two separate subsystems m the conventional design strategy. However. the technical efforts in planning these sUbsrs~m~ can be ~~bined by considering the minimization of system costs and the maxlml.za~on of utilization rate of environmental assimilative capacity. But the system unc~rtamties should be further addressed in the modelling framework as the variations of env~onmentalf~~ t.>ecome significant The grey systems theory is found to be an effective tool to m?dify the determlOlStic models. Therefore, a grey nonlinear programming model is developed in this paper to s~pport suc~ an uncertainty analysis. The case study of the Guishuic wastewater treatment and disposal project in Taiwan is used as a numerical demonstration. KEYWORDS Grey nonlinear programming, ocean outfall, system analysis, wastewater treatment INTRODUCTION Many coastal cities dispose of their sewage to the sea through wastewater treatment and ocean outfall facilities. The coastal wastewater treatment and ocean outfall facilities ~ usually planned as two separate subsystems in the conventional design strategy. In the wastewater treatment subsystem, the choice of a primary or a secondary treatment process depends on the fmancia!, land availability, and managerial factors. A set of ocean outfall pipes must then ensure compliance with the environmental quality standards in the designated sea area, based on the conditions of assimilative capacity in the nearby ocean environment and the effluent quality from the wastewater treatment plant From the aspects of system analysis, it is known that the optimal length of ocean outfall pipe is not only related to the required sea water quality and ambient diffusion environment but also the treatment efficiency simultaneously. As a result. the minimum system costs required for wastewater treatment and disposal subject to the required sea water quality in a user-specified region may be further integrated and explored by a mathematical programming model. A trade-off betweenl'lI;;incc rin ll invt:slments corresponding to tht: coasllll Md marint: facilities that exists in lh~pllulOing proct:ss. 19
N. -B. CHANG and S. F. WANG
20
Many previous studies put forward their views on the nearfield and farfield diffusion models as well as on the unit operations of the wastewater treatment processes, few of them are found to be directly related to this analysis. One pertinent piece of rese,'U'Ch is the work carried out by Bonazountas et al' (1988).
Adeterministic multi-objective optimization model was used for analyzing similar
issues in Greece. But the length of ocean outfall pipe and the performance of wastewater treatment plant were formulated in separate objectives such that the direct trade-off in engineering costs cannot be proceeded with. Besides. the deterministic optimization techniques are not sufficient to model such a complex problem because of the inherent uncertainty from environmental variations. This paper proposes a new approach: a grey nonlinear programming (GNLP) model to optimally integrate the planning work of ocean outfall pipe and wastewater treatment plant The practical implementation of this model is assessed by a case study of the Guishuic wastewater treatment and disposal project in Taiwan.
PRE·OPTIMAL ANALYSES In this analysis, the overall steps in planning an integrated wastewater treatment and disposal project consist of: (i) deriving construction cost functions corresponding to ocean outfall and wastewater treatment facilities separately; (ii) selecting mathematical models of mixing and dispersion as well as investigating site specific parameters; (iii) formulating optimization model and searching for the optimal solution (i.e., determining the best length of outfall pipe and the removal efficiency of treatment process simultaneously, based on the estimated flowrate and designed pipe diameter); and (iv) conducting sensitivity llIIalyses in relation to different environmental and engineering factors. The first two steps are usually regarded as the pre-optimal analyses.
Derivation of Construction Cost Function of Ocean Outfall Pipes The major affecting factors of the construction cost of an outfall pipe include both outfall diameter and length. Wallis (1979), based on the information of 38 outfalls in the U. S, has derived several unit cost functions in terms of outfall diameter or length. Since unit construction costs plotted as a separate function of outfall length and diameter are not appropriate for this analysis, a linear construction cost function is thus formulated in terms of both outfall length and diameter as follows:
where:
=
TC I the total construction cost of an ocean outfall pipe ($). a O' ai' ~ = the regression coefficients. L
=the total length of outfall pipe (pipe length plus diffuser length)
(m).
o =the diameter of the main outfall pipe (m).
el
=the error term in regression formulation.
Planning coastal wastewater trealment
Derivation of Construction Cost Function for Wastewater Treatment Plants Although Robert. et al. (1975) have investigated a series of linear construction and operating cost functions in terms of the flowrate associated with each different unit operation in tpe wastewater treatment plant. those cost functions are of no use in this analysis. The required cost formulation here needs to be considered in terms of overall removal efficiency of target pollutants in the treatment process instead of flowrate associated with each unit operation. Beside~. Graves et aI. (1970) derived a nonlinear construction cost function in terms of both flowrate and removal efficiency. However. the emphasis of highly nonlinear formulation in his work obviously increases the computational difficulties in the optimization procedure. A linear construction cost function is therefore formulated in terms of overall removal efficiency of a target pollutant and total sewage flowrate as below:
where:
TC2 = the total construction cost of a large scale wastewater treatment plant ($). boo b l • b2 = the regression coefficients. d = the removal efficiency of biological oxygen demand (BOD) (%). Q = the incoming flowrate for wastewater treatment (CMD). e2 =the error term in regression formulation.
Selection of Diffusion Models The dispersion of pollutants in coastal water has been described in vast literature. These topics cover ~ixing characteristics in nearfield and f3!field a.mbient environment, sea water quality simulation and assessment, etc. An outfall model is essentially a combination of three submodels. The first part calculates the initial dilution. the second part the horizontal dispersion. and the third part bacterial mortality.
Effluent dilution in the initial mixing region has been the subject of innumerable studies in the laboratory, in the field. and with mathematical models. The prediction of initial dilution can be expected and results for simple configurations. such as single-port or multiport discharges in either stagnant or flowing ambient, can be found in vast literature. It is noticed that a good estimate of initial dilution in calm. homogeneous receiving waters can be expressed by several mathematical models. But there is no universally accepted model for moving waters. The dilution submodel used for illustrating the effect of initial dilution in this study is (Metcalf & Eddy Inc.• 1979): 2
Dl = O.2~UZ where:
MlII-Z.e
u = the ambient current speed (mls). z = water depth (=sin t. L) (m). Q = the discharged sewage flowrate (CMS).
21
22
N. -B. CHANG and S. F. WANG
L = the length of main outfall pipe (m). t = the average angle between the surface of sea and the sea bed at the disposal site (degree). Furthermore, beyond the initial mixing region, the plume undergoes additional mixing by turbulent diffusion in the transition and farfield regions. Various similar models can be found in the literature. Brooks equation is finally chosen because of its simple and representative characteristics. But it should be remarked that several assumptions have been made in Brooks' model to simplify the analytical framework, as illustrated in the literature.(Brooks and Koh, 1986)When the power of four thirds is applied, this dilution submodel of horizontal (farfield) dispersion becomes (Metcalf & Eddy Inc" 1979 and Markham, 1983): D2 =--;:~:::;;::::. 1.5
where:
x = the distance along the current path to the site where dilutions (or concentrations) is required (m). b = the length of diffuser perpendicular to the ambient current (m). Eo = the initial transverse diffusion coefficient (=4.53xlO-4 b413) (mils). ~ = a dimensionless number (=12Eo!(ub». y
erf (y) = the standard error function = _ ~ "V1t
Jexp(-r )dr. 2
0
Finally, microbial mortality is usually measured by T90 which stands for the time in hours for the inactivation of 90% of the initial number of microbes. The component of microbial reduction can be expressed as a dilution factor "D3", and its definition is: D3=1O (tfr90) where:
t = the travel time(s) to where dilution is measured (hr). T90 = the time(s) for microbial density to decrease 90% (hr).
MODEL FORMULATION The objective function is comprised of the direct construction costs of ocean outfall pipe and wastewater treatment plant respectively, in which only two decision variables with uncertain characteristics, the BOD removal efficiency (B) and the length of outfall pipe (L), are defined. This analysis assumes that the discharge rate at outfall pumping station for sewage disposal is consistent with the incoming f10wrate of raw sewage at the entrance of wastewater treatment plant. The diameter of outfall pipe can then be externally decided according to the estimated sewage f10wrate and the engineering criterion that the design velocity of treated sewage is no more than an upper bound in the main outfall pipe. In the constraint formulation, more environmental factors have to be considered rather than economic requirements. The spectrum of the components in sewage that endanger human health and marine life includes pathogenic bacteria, metals, lower dissolved oxygen (DO) level, ere.
In such a profile, the mortality of E. coli is usually selected as the surrogate
Planning coastal wastewater lreabnent
index of pathogenic bacteria, while BOD or DO level is a significant environmental index for marine wildlife. This model tries to illustrate both impacts of BOD and E. coli in the constraint set The chlorination facilities must provide a specified efficiency of disinfection which is irrelevant to the levels of sewage treatment for BOD or SS removal. Hence. there is no trade-off between engineering costs in selecting the length of outfall pipe and the type of dis~~ction facility for E. coli control. But the ultimate dilution effect of E. coli is linked with the assimilative constraint for BOD control through the process of initial dilution. since the length of outfall pipe is an unknown in this model. Therefore. the triple product of the component dilutions (01 0 02.03) represents the total dilution of non-conservative substances. while 03 is equal to 1 for conservative substances specifically. In general. the assimilative constraint for BOD control may dominate the optimization process. But the removal efficiency would technologically encounter an upper bound. as formulated in the third constraint Moreover. many real-world problems contain unclear information which cannot be depicted by conventional probability theory. Both fuzzy and grey systems theories are designed to supplement the interpretation of such uncertainties for those real-world issues. It should be remarked in this analysis that the use of certain numbers to express uncertain parameters in the model is not adequate. and the optimal solutions in the traditional deterministic and fuzzy programming models are not flexible enough in this implementation. The grey systems theory developed by Dr. J. Deng in 1984 in China (Oeng. 1984 and 1986) can improve this drawback. In his theory. all systems are divided into three categories: white. grey. and black. A white system shows completely certain and clear messages. while the black system has totally unknown characteristics. The messages released in a grey system is in between. Thus. a grey number ~(a) in the grey system may be delineated by an open interval with upper and lower limits ~a). ~a)]. A whitening number is defmed as one of the exact number within this interval. Such an illustrative method supplements the expression of system uncertainties by the conventional probability theory and fuzzy sets theory whenever the probability density and membership functions cannot be fully obtained. The proposed grey nonlinear optimization model is then formulated as follow. in which several environmental variables are defined as grey variables.
The proposed nonlinear optimization model is then formulated as: Min
TCI + TC2 =f(~L.~B)
subject to: (1) assimilative constraint for BODS control:
(2) constraint of E. coli control:
(3) constraint of BOD removal efficiency:
23
N. -B. CHANG and S. F. WANG
24
OS I8S S MAX
(4) non-negativity constraint: in which
~L
2: 0
Dl=
O.29t.?u(sint*GlIL)2
QQ
1 D2 =----;~==::::. 1.5
where: R =the maximum allowable limit of a specific conservative pollutant in a specified sea area (ppm ormgll). BO = the background concentration of a specific conservative pollutant in a specified sea area (ppm or mg/l). p =the initial concentration of a specific conservative pollutant in the inflow of wastewater prior to treatment. such as BOD~. metal. and so on (ppm or mg/l). Q =the wastewater flowrate in the treatment plant (CMD) and in the outfall pipe (CMS). RD =the required total dilution for microbial control. MAX =the highest performance of BOD removal efficiency if an advanced treatment process is designed.
In model implementation. the first constraint can further be simplified as:
However. grey arithmelics below:
can be used to integrate the grey numbers GlIA and GlIB, as illustrated
~A + QB = [Q(A)~(B). ~A)+~B)l QA - QB = [Q(A)-~B). ~A)-Q(B)l ~A * ~B = [Q(A)*Q(B). ~A)*il5(B)]
CASE STUDY Ouishuic. the second largest stream in Tainan County. Taiwan. has been seriously polluted for a long time. One of the management strategies is to install an intercept system along the upper and middle sections of the stream in order to carry most of industrial and domestic wastewater toward a wastewater treatment plant near to the estuary area. While the level of treatment efficiency has not
Planning coastal wastewater treatment
25
been detel1I1ined yet, the ocean outfall pipe, more than five kilometres in length, has being built. The geographical location of this system is shown in Figure I. A lot of debate has been focused on whether the primary or the secondary treatment process has to be selected to satisfy the effluent quality requirements; but the system analysis using optimization techniques is never being proposed to achieve the purpose of cost minimization. This analysis is prepare~ to illustrate the methodology, using a grey nonlinear programming model in search of flexible planning strategies for coastal wastewater treatment and disposal system, as demonstrated by this case study.
Keelung City Taipei City
Taichung Ci ty
Taiwan Strait Guishuic Taman Count.y_
'.
/~.
'\J:
Taioan
Pacific Ocean
CitY~;..
/"'-)J
Kaohsiung City
ocean outfall pipe
\
Fig. I The geographical location of Guishuic wastewater treatment and disposal system.
TABLE 1 DATABASE OF OCEAN OUTFALL PROJECTS IN TAIWAN location of outfall Dine Pa-li (Taioei County) Chunl!-Chou (Kaohsiunl! City) Tzuoo-Ynl! (Kaohsiunl! City) Dah-Lin-Puu (Kaohsiunl! City) Keelunl! (Keelunl! City) Guishuic (Tainan County)
tola! length (m)
6,600 3,000 5,082 3300 200 5,500
• estllDated costs by deSIgn consulung ftrms •• the currency ratio is 26 NT$/l US$ in 1993
pipe diameter (m) 3.6 1.8 1.52 1.5 1.6 1.3
construction cost (millsNT$) 3200 325 500 243 80* 410*
year
of bid 1992 1983 1980 1980 1993 1993
nonnalized cost (millsNT$) 3.243 397 741 360 80 410
CPI index 104.5 86.8 71.5 71.5 105.9 105.9
26
N. -B. CHANG and S. F. WANG
TABLE 2 DATABASE OF COASTAL WASTEWATER TREATMENT PLANTS IN TAIWAN flowrate
location of wastewater treatment oIant Pa-Ii ITaioei Countv) Cbunl!-ehou (KaohsiuDl! CilV) Lin-Hae (Kaohsiun~ CiM De-Hwa CTainei Citv) Taichunl! ITaicbun~ City) Tainan ITainan CiM Keelun~ (Keelun~ CilV) Guishuic ITainan County)
trealment level
(CMD)
1320000 S60 000 S4 000 274000 87 SOO 110000 63500 78 000
nrimllrv orimarv
CODsttucUon cost (millsNTS) 7 000 167S
orimsv secon< I3rv secon I3rv
secon I3rv
orimarv
1200 94 2060 38S0"
3500" 440"
year of
bid
1992 1983 1992 1980 1993 1993 1993 1993
IIOI'IIl3hzed
cost
I (millsNTS) 7094 2043 1216 139 2060
CPI index 100.S
86.8 100.S 7l.S 105.9
38S0
IOS.9
3500 440
105.9 IOS.9
" estunated cosls by design consulting firms "" the currency ratio i. 26 NTSII US$ in 1993
To fulfil the pre-optimal analyses, two multiple linear regression analyses need to be conducted separately to identify the construction cost functions, based on a database of six ocean outfall projects and eight large-scale wastewater treatment projects in Taiwan, as listed in Tables 1 and 2. Since the cost data are adapted from different base of time, the consumer price index (CPI) is used to normalize these cost data by the year 1993 in this analysis before the regression analyses are performed. But the normalization for the locational difference is negligible because Taiwan is a spatially small area. SAS software package is used as a computer solver. The regression results appear well suited 10 the proposed models as shown below: Tel
=-1841.75 + 0.16 L + 1113.17 D (-2.082)
TC2
(3.389
(5.379)
=-2641.0338 + 60.4756 S + 0.0058 Q (-10.257)
(4.188)
(11.282)
R 2 =0.9888 R 2 =0.8531
The numbers shown in the above parentheses beneath those model estimators represent t-ratios which are all statistically significant under the 5% level of significance in these two regression practices. In addition, R2 values are also fairly good in both cases. Besides for the identification of the objective functions, other quantitative parameters in the constraint
set should also be assessed before pursuing the optimal solution. According to an investigation conducted by a consulting fum (SINO-Tech Inc., 1989), a dry weather flow between 1.5 and 1.7 CMS of raw sewage is to be collected in the local communities and the estimated BODs cpncentration in the inflow of raw sewage
ranges
from 250 to 350 ppm. The main outfall pipe and its diffuser
have already been designed with the length of 5,014 m and 360 m respectively. A primary treatment process was proposed in an earlier planning report. Traditional engineering criteria suggest that the maximum velocity of discharged sewage in the main outfall pipe should be limited by an upper bound of 1 mls and the velocity of surface current toward the beach is around 0.3 mls. It is assumed that a designated swimming beach is located at several kilometres from the diffuser which is subject to a protection by the stipulated class A standard of sea water quality. The maximum allowable limit
Planning coaslal wastewater lreabnent
27
of BOD, is 2 ppm within the class A sea area, but the background concentration of BOD, is ranged from 0.6 to 1.4 ppm (Yen etnl•• 1983 nod SINO-Tech. 1989). The low water tidal current runs northeast at 0.4-0.8 m/s and the wind rose shows a NE wind to be the most predominant during the winter months (SINO-Tech. 1989). A mild slope of sea bed exists in this coastal area. Investigation shows that the average angle between the sea bed and the surface of ~a is only 0.18 degree. In addition, the upper bound of the number of E. coli after the disinfection process is regulated by the level of 106/100ml, and the mandatory standard for microbial control in the class A sea area is 1,000 E. coli/IOOml in Taiwan. Therefore, a total required dilution, RD, greater than Or equal to 1,000 must be achieved in this case study. For clear coastal waters around Taiwan, a T90 of 2-6 hours is a reasonable value for this parameter (Brooks nod Koh, 1986). A lower bound of 1.8-2 hours is chosen here according to the repons of Yen et aI.• (1983) and SIND-Tech (1989). The solution procedure is handled via an interactive and screening approach to reach the approximately global optimal solution, as shown in Figure 2. GINO software package is chosen as a computer solver for optimization analysis in the case study. The solution procedure proposed by Huang, Baetz, and Patry (1993) for solving the grey or grey fuzzy linear programming model cannot be applied to solve such a nonlinear programming model. This analysis thus initializes a new solution procedure by an interactive fashion associated with the grey arithmetics. The grey optimal solutions, as shown in Table 3, provide a set of appealing and flexible solutions to the problem of planning a large scale wastewater treatment and disposal system. The grey design strategy suggests that an outfall with 2,634 m to 3,183 m in length along with an advanced treatment process is the best combination in this case. TABLE 3 RESULTS OF GREY OPIlMI1AnON ANALYSIS scenario QA QB e!)B e!)A
case 1 case 2
..J.
case 3 case 4
..J
II?L
..J
..J
..J ..J
..J
=rQ L, ~ L] =[2634 m, 3183 ml
..J
IiIL
~
IiITC
2,637
93.7
5,154.66
3,059
97.7
5,224.55
3.183
98.4
5.244.81
2.634
90.0
5,152.02
1I?5 ::: [& 5. e!) 5] ::: [90.0%. 98.4%]
IiITC::: fa TC. e!) Tel::: (5152.02 Mills 1993NT$. 5244.81 Mills 1993NT$] • represents the selection ot such a deslgnated condition
CONCLUDING REMARKS This analysis successfully elaborates an integrated approach between ocean outfall design and the selection of sewage treatment efficiency by a novel class of GNLP model. In a field-proven case study, the total system costs can be minimized and the stipulated environmental quality levels can also be satisfied through a spectrum of environmental and engineering factors. An interval predictions of decision variables may contribute a flexible insight in engineering decision making. It is believed that such a grey analytical framework invariably provides a concise and simple tool for environmental system engineers in planning a large-scale coastal wastewater treatment and disposal system. More sophisticated diffusion models and other environmental quality constraints could be
28
N. -8. CHANG and S. F. WANG
incorporated into the constraint set to improve the soundness of this model Incidentally, if budget is a significant issue in decision making, the cost expression tenns in the objective function can be funher illustrated by fuzzy membership functions easily according to the decision maker's preference. Thus, it allows grey messages related to the input parameter values and fuzzy goals pertaining to the decision maker's aspiration levels to be communicated into the nonlinear optimization processes, and therefore creates a set of more flexible optimal solution.
GNLP Model
Using grey arithmetic to merge grey coeffiCients
Scanning procedure
for solving GNLP
Is the solution global optimal?
no
yes
I GNLP Solullon I Fig. 2 The solution procedure of grey nonlinear programming model.
Planning coastal wastewater treabnent
29
REFERENCES Brooks, N. H. and Koh, R. C. Y. (1986). Outfall Djffuser System for Dischar~e of Effluent from TreaUDent Works on PsytaJja Island. Athens. Hydraulic Desj~n and Water Quality AnaJysis for Near and Intermedjate Fjeld, Final Report submitted by Woodward Clyde Consultants, California, San Francisco, USA. , Bonazountas, M. and Kallidromitou, D. and Dimou, N. (1988). OUTFALL: A Sea Outfall Model, Computer TechnjQues in Enyironmental Studies, Proceedings of the Second International Conference, Porto Carras, Greece, 95-109. Deng, J. (1984). The DeolY and Methods of Socio-economic Grey Systems, Science Press _Beijing, China (in Chinese).
Deng,-!. (1986). Grey Prediction and Decisjon, Huazbong Institute of Technology Wuhan. China (in Chinese).
.
Press
Graves, G. W., Whinston, A. B. and Hatfield, G. B. (1970). Mathematical Pro~rammin~ for Re~ional Water Ouality Mana~ment, Department of Interior, Washington D. C., USA Huang, G.H., Baetz, B. W., and Patry, G. G. (1992). A Grey Linear Programming Approach for Municipal Solid Waste Management Planning Under Uncertainty Civil Engineering Systems& 9.319-335.
Huang, G.H., Baetz, B. W., and Patry, G. G. (1993). A Grey Fuzzy Linear Programming Approach for Municipal Solid Waste Management Planning Under Uncertainty, Civil Engineering Systems& 10. 123-\46. Metcalf & Eddy, Inc. (1979). Wastewater: Engineering: Treatment. Disposal. Reuse. McGraw• Hill, Inc., New York, USA, 2nd edition, 852. Markham, A (1983). Modelling of Sewage Outfalls in the Marine Environment, In: An Introduction to Water OuaUty Model1jn~, 2nd OOn.. ed A. James, John Wiley & Sons, New York, USA, 293-308. Robert, H. V. N. et al. (1975). A Guide to the Selection of Cost-effective Wastewater Treatment Systems, U.S. EPA, July. SINO-.Tech, Inc. (1989). En~jneerin~ Plannin~ of Gujshuic Ocean Outfall System, Taipei, ~8J.wan, R.O.C. (in Chinese). Wallts, L. G. (1979). Ocean Outfall Construction Costs, Journal ofWpcF, May, 951-957, USA. Yen, P. H.! Wen, C. G., et al. (1983). Investi~ations of Gujshujc OutfaJl of Marine Enyjronment. Te~h01<:al Report N~. 61, Institute of Hydraulic Experiment, National Cheng-Kung Umverslty, Taman, T8J.wan, R.O.C. (in Chinese).
Pergamon
OI273-1223(95)00565-X
Wat. ScL Tech. Vol. 32. No. 2, pp. 19-29. 1995. Copyright C I99S IAwQ Printed ill Oreal Britaio. All righll ruerved. 0273-1223I9S 59'SO + 0'00
A GREY NONLINEAR PROGRAMMING APPROACH FOR PLANNING COASTAL WASTEWATER TREATMENT AND DISPOSAL SYSTEMS Ni-Bin Chang and S. F. Wang Department ofEnvironmental Engineering, National Cheng-Kung University, Tainan. Taiwan
ABSTRACT Coastal w~tewater treatment and ocean outfall facilities are usually planned as two separate subsystems m the conventional design strategy. However. the technical efforts in planning these sUbsrs~m~ can be ~~bined by considering the minimization of system costs and the maxlml.za~on of utilization rate of environmental assimilative capacity. But the system unc~rtamties should be further addressed in the modelling framework as the variations of env~onmentalf~~ t.>ecome significant The grey systems theory is found to be an effective tool to m?dify the determlOlStic models. Therefore, a grey nonlinear programming model is developed in this paper to s~pport suc~ an uncertainty analysis. The case study of the Guishuic wastewater treatment and disposal project in Taiwan is used as a numerical demonstration. KEYWORDS Grey nonlinear programming, ocean outfall, system analysis, wastewater treatment INTRODUCTION Many coastal cities dispose of their sewage to the sea through wastewater treatment and ocean outfall facilities. The coastal wastewater treatment and ocean outfall facilities ~ usually planned as two separate subsystems in the conventional design strategy. In the wastewater treatment subsystem, the choice of a primary or a secondary treatment process depends on the fmancia!, land availability, and managerial factors. A set of ocean outfall pipes must then ensure compliance with the environmental quality standards in the designated sea area, based on the conditions of assimilative capacity in the nearby ocean environment and the effluent quality from the wastewater treatment plant From the aspects of system analysis, it is known that the optimal length of ocean outfall pipe is not only related to the required sea water quality and ambient diffusion environment but also the treatment efficiency simultaneously. As a result. the minimum system costs required for wastewater treatment and disposal subject to the required sea water quality in a user-specified region may be further integrated and explored by a mathematical programming model. A trade-off betweenl'lI;;incc rin ll invt:slments corresponding to tht: coasllll Md marint: facilities that exists in lh~pllulOing proct:ss. 19
N. -B. CHANG and S. F. WANG
20
Many previous studies put forward their views on the nearfield and farfield diffusion models as well as on the unit operations of the wastewater treatment processes, few of them are found to be directly related to this analysis. One pertinent piece of rese,'U'Ch is the work carried out by Bonazountas et al' (1988).
Adeterministic multi-objective optimization model was used for analyzing similar
issues in Greece. But the length of ocean outfall pipe and the performance of wastewater treatment plant were formulated in separate objectives such that the direct trade-off in engineering costs cannot be proceeded with. Besides. the deterministic optimization techniques are not sufficient to model such a complex problem because of the inherent uncertainty from environmental variations. This paper proposes a new approach: a grey nonlinear programming (GNLP) model to optimally integrate the planning work of ocean outfall pipe and wastewater treatment plant The practical implementation of this model is assessed by a case study of the Guishuic wastewater treatment and disposal project in Taiwan.
PRE·OPTIMAL ANALYSES In this analysis, the overall steps in planning an integrated wastewater treatment and disposal project consist of: (i) deriving construction cost functions corresponding to ocean outfall and wastewater treatment facilities separately; (ii) selecting mathematical models of mixing and dispersion as well as investigating site specific parameters; (iii) formulating optimization model and searching for the optimal solution (i.e., determining the best length of outfall pipe and the removal efficiency of treatment process simultaneously, based on the estimated flowrate and designed pipe diameter); and (iv) conducting sensitivity llIIalyses in relation to different environmental and engineering factors. The first two steps are usually regarded as the pre-optimal analyses.
Derivation of Construction Cost Function of Ocean Outfall Pipes The major affecting factors of the construction cost of an outfall pipe include both outfall diameter and length. Wallis (1979), based on the information of 38 outfalls in the U. S, has derived several unit cost functions in terms of outfall diameter or length. Since unit construction costs plotted as a separate function of outfall length and diameter are not appropriate for this analysis, a linear construction cost function is thus formulated in terms of both outfall length and diameter as follows:
where:
=
TC I the total construction cost of an ocean outfall pipe ($). a O' ai' ~ = the regression coefficients. L
=the total length of outfall pipe (pipe length plus diffuser length)
(m).
o =the diameter of the main outfall pipe (m).
el
=the error term in regression formulation.
Planning coastal wastewater trealment
Derivation of Construction Cost Function for Wastewater Treatment Plants Although Robert. et al. (1975) have investigated a series of linear construction and operating cost functions in terms of the flowrate associated with each different unit operation in tpe wastewater treatment plant. those cost functions are of no use in this analysis. The required cost formulation here needs to be considered in terms of overall removal efficiency of target pollutants in the treatment process instead of flowrate associated with each unit operation. Beside~. Graves et aI. (1970) derived a nonlinear construction cost function in terms of both flowrate and removal efficiency. However. the emphasis of highly nonlinear formulation in his work obviously increases the computational difficulties in the optimization procedure. A linear construction cost function is therefore formulated in terms of overall removal efficiency of a target pollutant and total sewage flowrate as below:
where:
TC2 = the total construction cost of a large scale wastewater treatment plant ($). boo b l • b2 = the regression coefficients. d = the removal efficiency of biological oxygen demand (BOD) (%). Q = the incoming flowrate for wastewater treatment (CMD). e2 =the error term in regression formulation.
Selection of Diffusion Models The dispersion of pollutants in coastal water has been described in vast literature. These topics cover ~ixing characteristics in nearfield and f3!field a.mbient environment, sea water quality simulation and assessment, etc. An outfall model is essentially a combination of three submodels. The first part calculates the initial dilution. the second part the horizontal dispersion. and the third part bacterial mortality.
Effluent dilution in the initial mixing region has been the subject of innumerable studies in the laboratory, in the field. and with mathematical models. The prediction of initial dilution can be expected and results for simple configurations. such as single-port or multiport discharges in either stagnant or flowing ambient, can be found in vast literature. It is noticed that a good estimate of initial dilution in calm. homogeneous receiving waters can be expressed by several mathematical models. But there is no universally accepted model for moving waters. The dilution submodel used for illustrating the effect of initial dilution in this study is (Metcalf & Eddy Inc.• 1979): 2
Dl = O.2~UZ where:
MlII-Z.e
u = the ambient current speed (mls). z = water depth (=sin t. L) (m). Q = the discharged sewage flowrate (CMS).
21
22
N. -B. CHANG and S. F. WANG
L = the length of main outfall pipe (m). t = the average angle between the surface of sea and the sea bed at the disposal site (degree). Furthermore, beyond the initial mixing region, the plume undergoes additional mixing by turbulent diffusion in the transition and farfield regions. Various similar models can be found in the literature. Brooks equation is finally chosen because of its simple and representative characteristics. But it should be remarked that several assumptions have been made in Brooks' model to simplify the analytical framework, as illustrated in the literature.(Brooks and Koh, 1986)When the power of four thirds is applied, this dilution submodel of horizontal (farfield) dispersion becomes (Metcalf & Eddy Inc" 1979 and Markham, 1983): D2 =--;:~:::;;::::. 1.5
where:
x = the distance along the current path to the site where dilutions (or concentrations) is required (m). b = the length of diffuser perpendicular to the ambient current (m). Eo = the initial transverse diffusion coefficient (=4.53xlO-4 b413) (mils). ~ = a dimensionless number (=12Eo!(ub». y
erf (y) = the standard error function = _ ~ "V1t
Jexp(-r )dr. 2
0
Finally, microbial mortality is usually measured by T90 which stands for the time in hours for the inactivation of 90% of the initial number of microbes. The component of microbial reduction can be expressed as a dilution factor "D3", and its definition is: D3=1O (tfr90) where:
t = the travel time(s) to where dilution is measured (hr). T90 = the time(s) for microbial density to decrease 90% (hr).
MODEL FORMULATION The objective function is comprised of the direct construction costs of ocean outfall pipe and wastewater treatment plant respectively, in which only two decision variables with uncertain characteristics, the BOD removal efficiency (B) and the length of outfall pipe (L), are defined. This analysis assumes that the discharge rate at outfall pumping station for sewage disposal is consistent with the incoming f10wrate of raw sewage at the entrance of wastewater treatment plant. The diameter of outfall pipe can then be externally decided according to the estimated sewage f10wrate and the engineering criterion that the design velocity of treated sewage is no more than an upper bound in the main outfall pipe. In the constraint formulation, more environmental factors have to be considered rather than economic requirements. The spectrum of the components in sewage that endanger human health and marine life includes pathogenic bacteria, metals, lower dissolved oxygen (DO) level, ere.
In such a profile, the mortality of E. coli is usually selected as the surrogate
Planning coastal wastewater lreabnent
index of pathogenic bacteria, while BOD or DO level is a significant environmental index for marine wildlife. This model tries to illustrate both impacts of BOD and E. coli in the constraint set The chlorination facilities must provide a specified efficiency of disinfection which is irrelevant to the levels of sewage treatment for BOD or SS removal. Hence. there is no trade-off between engineering costs in selecting the length of outfall pipe and the type of dis~~ction facility for E. coli control. But the ultimate dilution effect of E. coli is linked with the assimilative constraint for BOD control through the process of initial dilution. since the length of outfall pipe is an unknown in this model. Therefore. the triple product of the component dilutions (01 0 02.03) represents the total dilution of non-conservative substances. while 03 is equal to 1 for conservative substances specifically. In general. the assimilative constraint for BOD control may dominate the optimization process. But the removal efficiency would technologically encounter an upper bound. as formulated in the third constraint Moreover. many real-world problems contain unclear information which cannot be depicted by conventional probability theory. Both fuzzy and grey systems theories are designed to supplement the interpretation of such uncertainties for those real-world issues. It should be remarked in this analysis that the use of certain numbers to express uncertain parameters in the model is not adequate. and the optimal solutions in the traditional deterministic and fuzzy programming models are not flexible enough in this implementation. The grey systems theory developed by Dr. J. Deng in 1984 in China (Oeng. 1984 and 1986) can improve this drawback. In his theory. all systems are divided into three categories: white. grey. and black. A white system shows completely certain and clear messages. while the black system has totally unknown characteristics. The messages released in a grey system is in between. Thus. a grey number ~(a) in the grey system may be delineated by an open interval with upper and lower limits ~a). ~a)]. A whitening number is defmed as one of the exact number within this interval. Such an illustrative method supplements the expression of system uncertainties by the conventional probability theory and fuzzy sets theory whenever the probability density and membership functions cannot be fully obtained. The proposed grey nonlinear optimization model is then formulated as follow. in which several environmental variables are defined as grey variables.
The proposed nonlinear optimization model is then formulated as: Min
TCI + TC2 =f(~L.~B)
subject to: (1) assimilative constraint for BODS control:
(2) constraint of E. coli control:
(3) constraint of BOD removal efficiency:
23
N. -B. CHANG and S. F. WANG
24
OS I8S S MAX
(4) non-negativity constraint: in which
~L
2: 0
Dl=
O.29t.?u(sint*GlIL)2
1 D2 =----;~==::::. 1.5
where: R =the maximum allowable limit of a specific conservative pollutant in a specified sea area (ppm ormgll). BO = the background concentration of a specific conservative pollutant in a specified sea area (ppm or mg/l). p =the initial concentration of a specific conservative pollutant in the inflow of wastewater prior to treatment. such as BOD~. metal. and so on (ppm or mg/l). Q =the wastewater flowrate in the treatment plant (CMD) and in the outfall pipe (CMS). RD =the required total dilution for microbial control. MAX =the highest performance of BOD removal efficiency if an advanced treatment process is designed.
In model implementation. the first constraint can further be simplified as:
However. grey arithmelics below:
can be used to integrate the grey numbers GlIA and GlIB, as illustrated
~A + QB = [Q(A)~(B). ~A)+~B)l QA - QB = [Q(A)-~B). ~A)-Q(B)l ~A * ~B = [Q(A)*Q(B). ~A)*il5(B)]
CASE STUDY Ouishuic. the second largest stream in Tainan County. Taiwan. has been seriously polluted for a long time. One of the management strategies is to install an intercept system along the upper and middle sections of the stream in order to carry most of industrial and domestic wastewater toward a wastewater treatment plant near to the estuary area. While the level of treatment efficiency has not
Planning coastal wastewater treatment
25
been detel1I1ined yet, the ocean outfall pipe, more than five kilometres in length, has being built. The geographical location of this system is shown in Figure I. A lot of debate has been focused on whether the primary or the secondary treatment process has to be selected to satisfy the effluent quality requirements; but the system analysis using optimization techniques is never being proposed to achieve the purpose of cost minimization. This analysis is prepare~ to illustrate the methodology, using a grey nonlinear programming model in search of flexible planning strategies for coastal wastewater treatment and disposal system, as demonstrated by this case study.
Keelung City Taipei City
Taichung Ci ty
Taiwan Strait Guishuic Taman Count.y_
'.
/~.
'\J:
Taioan
Pacific Ocean
CitY~;..
/"'-)J
Kaohsiung City
ocean outfall pipe
\
Fig. I The geographical location of Guishuic wastewater treatment and disposal system.
TABLE 1 DATABASE OF OCEAN OUTFALL PROJECTS IN TAIWAN location of outfall Dine Pa-li (Taioei County) Chunl!-Chou (Kaohsiunl! City) Tzuoo-Ynl! (Kaohsiunl! City) Dah-Lin-Puu (Kaohsiunl! City) Keelunl! (Keelunl! City) Guishuic (Tainan County)
tola! length (m)
6,600 3,000 5,082 3300 200 5,500
• estllDated costs by deSIgn consulung ftrms •• the currency ratio is 26 NT$/l US$ in 1993
pipe diameter (m) 3.6 1.8 1.52 1.5 1.6 1.3
construction cost (millsNT$) 3200 325 500 243 80* 410*
year
of bid 1992 1983 1980 1980 1993 1993
nonnalized cost (millsNT$) 3.243 397 741 360 80 410
CPI index 104.5 86.8 71.5 71.5 105.9 105.9
26
N. -B. CHANG and S. F. WANG
TABLE 2 DATABASE OF COASTAL WASTEWATER TREATMENT PLANTS IN TAIWAN flowrate
location of wastewater treatment oIant Pa-Ii ITaioei Countv) Cbunl!-ehou (KaohsiuDl! CilV) Lin-Hae (Kaohsiun~ CiM De-Hwa CTainei Citv) Taichunl! ITaicbun~ City) Tainan ITainan CiM Keelun~ (Keelun~ CilV) Guishuic ITainan County)
trealment level
(CMD)
1320000 S60 000 S4 000 274000 87 SOO 110000 63500 78 000
nrimllrv orimarv
CODsttucUon cost (millsNTS) 7 000 167S
orimsv secon< I3rv secon I3rv
secon I3rv
orimarv
1200 94 2060 38S0"
3500" 440"
year of
bid
1992 1983 1992 1980 1993 1993 1993 1993
IIOI'IIl3hzed
cost
I (millsNTS) 7094 2043 1216 139 2060
CPI index 100.S
86.8 100.S 7l.S 105.9
38S0
IOS.9
3500 440
105.9 IOS.9
" estunated cosls by design consulting firms "" the currency ratio i. 26 NTSII US$ in 1993
To fulfil the pre-optimal analyses, two multiple linear regression analyses need to be conducted separately to identify the construction cost functions, based on a database of six ocean outfall projects and eight large-scale wastewater treatment projects in Taiwan, as listed in Tables 1 and 2. Since the cost data are adapted from different base of time, the consumer price index (CPI) is used to normalize these cost data by the year 1993 in this analysis before the regression analyses are performed. But the normalization for the locational difference is negligible because Taiwan is a spatially small area. SAS software package is used as a computer solver. The regression results appear well suited 10 the proposed models as shown below: Tel
=-1841.75 + 0.16 L + 1113.17 D (-2.082)
TC2
(3.389
(5.379)
=-2641.0338 + 60.4756 S + 0.0058 Q (-10.257)
(4.188)
(11.282)
R 2 =0.9888 R 2 =0.8531
The numbers shown in the above parentheses beneath those model estimators represent t-ratios which are all statistically significant under the 5% level of significance in these two regression practices. In addition, R2 values are also fairly good in both cases. Besides for the identification of the objective functions, other quantitative parameters in the constraint
set should also be assessed before pursuing the optimal solution. According to an investigation conducted by a consulting fum (SINO-Tech Inc., 1989), a dry weather flow between 1.5 and 1.7 CMS of raw sewage is to be collected in the local communities and the estimated BODs cpncentration in the inflow of raw sewage
ranges
from 250 to 350 ppm. The main outfall pipe and its diffuser
have already been designed with the length of 5,014 m and 360 m respectively. A primary treatment process was proposed in an earlier planning report. Traditional engineering criteria suggest that the maximum velocity of discharged sewage in the main outfall pipe should be limited by an upper bound of 1 mls and the velocity of surface current toward the beach is around 0.3 mls. It is assumed that a designated swimming beach is located at several kilometres from the diffuser which is subject to a protection by the stipulated class A standard of sea water quality. The maximum allowable limit
Planning coaslal wastewater lreabnent
27
of BOD, is 2 ppm within the class A sea area, but the background concentration of BOD, is ranged from 0.6 to 1.4 ppm (Yen etnl•• 1983 nod SINO-Tech. 1989). The low water tidal current runs northeast at 0.4-0.8 m/s and the wind rose shows a NE wind to be the most predominant during the winter months (SINO-Tech. 1989). A mild slope of sea bed exists in this coastal area. Investigation shows that the average angle between the sea bed and the surface of ~a is only 0.18 degree. In addition, the upper bound of the number of E. coli after the disinfection process is regulated by the level of 106/100ml, and the mandatory standard for microbial control in the class A sea area is 1,000 E. coli/IOOml in Taiwan. Therefore, a total required dilution, RD, greater than Or equal to 1,000 must be achieved in this case study. For clear coastal waters around Taiwan, a T90 of 2-6 hours is a reasonable value for this parameter (Brooks nod Koh, 1986). A lower bound of 1.8-2 hours is chosen here according to the repons of Yen et aI.• (1983) and SIND-Tech (1989). The solution procedure is handled via an interactive and screening approach to reach the approximately global optimal solution, as shown in Figure 2. GINO software package is chosen as a computer solver for optimization analysis in the case study. The solution procedure proposed by Huang, Baetz, and Patry (1993) for solving the grey or grey fuzzy linear programming model cannot be applied to solve such a nonlinear programming model. This analysis thus initializes a new solution procedure by an interactive fashion associated with the grey arithmetics. The grey optimal solutions, as shown in Table 3, provide a set of appealing and flexible solutions to the problem of planning a large scale wastewater treatment and disposal system. The grey design strategy suggests that an outfall with 2,634 m to 3,183 m in length along with an advanced treatment process is the best combination in this case. TABLE 3 RESULTS OF GREY OPIlMI1AnON ANALYSIS scenario QA QB e!)B e!)A
case 1 case 2
..J.
case 3 case 4
..J
II?L
..J
..J
..J ..J
..J
=rQ L, ~ L] =[2634 m, 3183 ml
..J
IiIL
~
IiITC
2,637
93.7
5,154.66
3,059
97.7
5,224.55
3.183
98.4
5.244.81
2.634
90.0
5,152.02
1I?5 ::: [& 5. e!) 5] ::: [90.0%. 98.4%]
IiITC::: fa TC. e!) Tel::: (5152.02 Mills 1993NT$. 5244.81 Mills 1993NT$] • represents the selection ot such a deslgnated condition
CONCLUDING REMARKS This analysis successfully elaborates an integrated approach between ocean outfall design and the selection of sewage treatment efficiency by a novel class of GNLP model. In a field-proven case study, the total system costs can be minimized and the stipulated environmental quality levels can also be satisfied through a spectrum of environmental and engineering factors. An interval predictions of decision variables may contribute a flexible insight in engineering decision making. It is believed that such a grey analytical framework invariably provides a concise and simple tool for environmental system engineers in planning a large-scale coastal wastewater treatment and disposal system. More sophisticated diffusion models and other environmental quality constraints could be
28
N. -8. CHANG and S. F. WANG
incorporated into the constraint set to improve the soundness of this model Incidentally, if budget is a significant issue in decision making, the cost expression tenns in the objective function can be funher illustrated by fuzzy membership functions easily according to the decision maker's preference. Thus, it allows grey messages related to the input parameter values and fuzzy goals pertaining to the decision maker's aspiration levels to be communicated into the nonlinear optimization processes, and therefore creates a set of more flexible optimal solution.
GNLP Model
Using grey arithmetic to merge grey coeffiCients
Scanning procedure
for solving GNLP
Is the solution global optimal?
no
yes
I GNLP Solullon I Fig. 2 The solution procedure of grey nonlinear programming model.
Planning coastal wastewater treabnent
29
REFERENCES Brooks, N. H. and Koh, R. C. Y. (1986). Outfall Djffuser System for Dischar~e of Effluent from TreaUDent Works on PsytaJja Island. Athens. Hydraulic Desj~n and Water Quality AnaJysis for Near and Intermedjate Fjeld, Final Report submitted by Woodward Clyde Consultants, California, San Francisco, USA. , Bonazountas, M. and Kallidromitou, D. and Dimou, N. (1988). OUTFALL: A Sea Outfall Model, Computer TechnjQues in Enyironmental Studies, Proceedings of the Second International Conference, Porto Carras, Greece, 95-109. Deng, J. (1984). The DeolY and Methods of Socio-economic Grey Systems, Science Press _Beijing, China (in Chinese).
Deng,-!. (1986). Grey Prediction and Decisjon, Huazbong Institute of Technology Wuhan. China (in Chinese).
.
Press
Graves, G. W., Whinston, A. B. and Hatfield, G. B. (1970). Mathematical Pro~rammin~ for Re~ional Water Ouality Mana~ment, Department of Interior, Washington D. C., USA Huang, G.H., Baetz, B. W., and Patry, G. G. (1992). A Grey Linear Programming Approach for Municipal Solid Waste Management Planning Under Uncertainty Civil Engineering Systems& 9.319-335.
Huang, G.H., Baetz, B. W., and Patry, G. G. (1993). A Grey Fuzzy Linear Programming Approach for Municipal Solid Waste Management Planning Under Uncertainty, Civil Engineering Systems& 10. 123-\46. Metcalf & Eddy, Inc. (1979). Wastewater: Engineering: Treatment. Disposal. Reuse. McGraw• Hill, Inc., New York, USA, 2nd edition, 852. Markham, A (1983). Modelling of Sewage Outfalls in the Marine Environment, In: An Introduction to Water OuaUty Model1jn~, 2nd OOn.. ed A. James, John Wiley & Sons, New York, USA, 293-308. Robert, H. V. N. et al. (1975). A Guide to the Selection of Cost-effective Wastewater Treatment Systems, U.S. EPA, July. SINO-.Tech, Inc. (1989). En~jneerin~ Plannin~ of Gujshuic Ocean Outfall System, Taipei, ~8J.wan, R.O.C. (in Chinese). Wallts, L. G. (1979). Ocean Outfall Construction Costs, Journal ofWpcF, May, 951-957, USA. Yen, P. H.! Wen, C. G., et al. (1983). Investi~ations of Gujshujc OutfaJl of Marine Enyjronment. Te~h01<:al Report N~. 61, Institute of Hydraulic Experiment, National Cheng-Kung Umverslty, Taman, T8J.wan, R.O.C. (in Chinese).