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Multiperiod synthesis and operational planning of utility systems with environmental concerns
Computers and Chemical Engineering 28 (2004) 745–753
Multiperiod synthesis and operational planning of utility systems with environmental concerns A. P. Oliveira Francisco a , H. A. Matos b,∗ a
Dep. de Engenharia Qu´ımica, Instituto Superior de Engenharia de Lisboa, 1949-014 Lisbon, Portugal b Dep. de Engenharia Qu´ımica, Instituto Superior Técnico, 1049-001 Lisbon, Portugal
Abstract Utility plants supply the required energy demands to industrial processes. Several authors addressed the synthesis and design of those plants. Environmental concerns require to consider minimization of global emissions from utility plants and regional power plant as an additional objective, for which no multiperiod optimization models able to cope with the time varying demands, have been made available yet. This paper presents an extension of the Iyer and Grossmann [Comput. Chem. Eng. 21 (8) (1997) 787; 22 (7–8) (1998) 979] model to synthesis and multiperiod operational planning in order to include the global emissions of atmospheric pollutants issues coming from the fuels burning. A new five-step algorithm is introduced to solve this multi-objective model. One motivation example enables us to compare the different units and fuel selection and also the operation periods of an industrial utility system taking in account the electrical power import/export policy and environmental concerns. © 2004 Elsevier Ltd. All rights reserved. Keywords: Utility systems; Process synthesis; Multiperiod operation; Environmental concerns; Global emissions; Mathematical programming; Multi-objective optimization
1. Introduction Utility plants supply the required energy demands to chemical processes, namely, mechanical, electrical and thermal power (different levels of steam). Changes in specifications, composition of feed and seasonal product demands create several process conditions with the corresponding variation in the utility demands during one annual horizon. Several authors addressed the synthesis and design of utility plants. Among these authors, Papoulias and Grossmann (1983) described a MILP model for the synthesis and design of utility systems, for fixed demands. Iyer and Grossmann (1997, 1998) described models for multiperiod operational planning and synthesis, and operational planning of utility systems. Also, Bruno, Fernandez, Castells, & Grossmann (1998) described a rigorous MINLP model for the optimization of synthesis and operation of utility plants. ∗
Corresponding author. E-mail address: [email protected] (H.A. Matos).
0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2004.02.025
Chang and Wang (1996) described a multi-objective programming approach to waste minimization in the utility systems of chemical process, using the concept of global emissions of gaseous pollutants. This model, merging economic and environmental concerns in the utility system synthesis, was stated for fixed utility demands. Oliveira Francisco (2002) described a methodology for the synthesis and multiperiod operational planning of utility systems in a heat integrated industrial complex. This comprises a multiperiod model for utility systems including environmental concerns. The purpose of the present paper is to show the structure of this modified model and the resolution algorithm, applied to a simple example problem.
2. Problem definition Given a set of time variable (multiperiod) demands of steam at various levels of pressure, electricity and mechanical power, the problem of synthesis and operational
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A.P. Oliveira Francisco, H.A. Matos / Computers and Chemical Engineering 28 (2004) 745–753
Fig. 1. Superstructure for example problem.
planning of the utility system consists in a structural and parametric optimization from a superstructure of alternatives. The superstructure will be decomposed in a set of feasible configurations. For each feasible configuration, the unit sizes are such that allow demands satisfaction in all operation periods. In a single objective problem—only economic optimization—the configuration with the lowest total cost (investment and operational costs) for the horizon of planning should be chosen. If in addition to economic optimization we have to satisfy to other objectives, for example, environmental concerns, the problem arises to a multi-objective problem and a different optimization strategy should be adopted. The superstructure of the utility system adopted in this work is derived from the superstructures described by Iyer and Grossmann (1997, 1998) and Papoulias and Grossmann (1983). Fig. 1 exemplifies the superstructure adopted for the example problem. This superstructure includes several steam headers at various pressure levels (VHP, HP, MP and LP). Steam can be generated in either conventional fired boilers (units 1, 2 and 20) or with waste heat boilers (units 13 and 28) receiving hot gases from gas turbines.
There is a deaerator (unit 11) receiving make-up water and condensates returned from process utilizations. A condenser (unit 8) is provided for condensation of LP steam from unit 4. Power can be generated with several types of steam turbines (units 3, 4 and 23), gas turbines (units 12, 27 and 30). Electrical generators can be driven by steam turbines or gas turbines. Gas turbines can operate in a stand-alone basis (eventually with a regenerator) or associated with waste heat recovery boilers. There are also several types of auxiliary equipment as fans, pressure reducers and pumps.
3. Model formulation Our model for the utility system is an extension of the multiperiod models described by Iyer and Grossmann (1997, 1998) in order to include the concept of global emissions of the gaseous pollutants that came from fuel burning. Following Smith and Delaby (1991), global emissions comprise local emissions derived from the production of utilities in the industrial site and the balance between increasing/lowering of the emissions in the regional power
A.P. Oliveira Francisco, H.A. Matos / Computers and Chemical Engineering 28 (2004) 745–753
station (RPS) due to electricity imports and exports to regional power network by the utility system. The extended mathematical model for the utility system can be formulated as follows: zP =
min
yd ,yt ,d,xt
f0 (yd , d) +
P
ft (xt , yt ) +
t=1
K P
λk EGkt
t=1 k=1
(1) s.t.
ht (d, yd , yt , xt , θt ) ≤ 0,
t = 1, . . . , P
(2)
u
xt − Ω yt ≤ 0,
t = 1, . . . , P
(3)
xt − Ωl yt ≥ 0,
t = 1, . . . , P
(4)
yd ≥ yt , P
t = 1, . . . , P
(5)
yt ≤ αyd
t=1
(6)
EGkt − LEMk
n
d ∈ Rn ,
≤0
Qnrt
(7)
r
xt ∈ Rq × RP ,
yd ∈ {0, 1}n ,
yt ∈ {0, 1}n×P
where yd are integer variables (0–1) defining the selection of units for the design, d the design variables defining the sizes of units, yt integer variables (0–1) that determine the operational status on/off for period t, xt the state and control variables for period t, θ t the parameters (e.g. utility demands) for period t, EGkt the global emissions of pollutant k, in period t, Qnrt the production of level r steam, in the unit n, in period t, as the absorbed heat in the steam generator, LEMk the limit of global emissions of pollutant k, expressed in relation to the absorbed heat in the steam generator, λk the weighted parameter meaning the pollutant k contribution to the objective function, Ωl (·) and Ωu (·) the valid limits, lower and higher, respectively, P the number of periods and α a scalar lower than P. The objective function includes the investment cost for the design (f0 ) and the sum of the operation costs (ft ) for all periods t = 1, . . . , P, as well as the above-referred weighted terms of global emissions of pollutants. Specific integer variables (binary, 0–1) in appropriate restrictions of the mathematical model allow the selection of operation on/off status and operations modes, e.g. extraction or condensing steam turbine. In the example problem given further, we adopted a MILP formulation for the utility system optimization.
4. Decomposition algorithm Iyer and Grossmann (1998) proposed a bilevel decomposition algorithm for solving his model. In the described algorithm, the first step is solving a design problem (DP) obtained from the principal model by making zero all fixed operation
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costs in the objective function, removing constraints of types (4) and (6) and by replacing yt with yd in the constraints of type (3). The solution of model DP gives a lower bound for the objective function of principal model and provides values for design integer variables yd to use in the next step. In a second step, a planning model (OP) is solved by fixing in the main model variables yd obtained in first step. Solution of OP gives a higher bound for the objective function of principal model and provides an operational plan for the next time DP is solved. Iterative procedure with adding of appropriate cuts converges to the solution of main model. Then, a modification of the Iyer and Grossmann algorithm was implemented: (a) Fixing values of weighted parameters λk . These values may be imposed in a basis of legal penalties to pollutants emissions or obtained by an economic optimization procedure referred further. (b) Solving the utility system model with the Iyer and Grossmann (1998) bilevel decomposition algorithm. In this step we obtain a design and an operational plan for the utility system. (c) Fixing the values calculated in step (b) for binary variables yd and yt in constraints and solving the model obtained by replacing the objective function (1) with a second objective function (8): λk EGkt (8) zE = min t
k
This step provides the utilization plan for fuels (binary variables) in the utility system arising to a minimum of global emissions. (d) Fixing binary variables associated with the selected units for design (yd ) and with the utilization plan of fuels obtained in step (c) and solving the model with the objective function (1). Solutions obtained in steps (b)–(d) correspond to diverse importance given to economic and environmental concerns. Less pollutant fuels have higher costs than more pollutant content fuels. Thus, total cost for these three solutions will be (b) < (d) < (c). Since all three solutions will satisfy the global emissions limitations, the selected solution depends from the user criteria. In this step, we can get a value toa parameter POLL defined as the actual value of Pt=1 K t=1 λk EGkt . (e) By applying the Iyer and Grossmann algorithm, finding a Pareto optimal solution (Winston, 1994) for a model comprising the objective function (9): zPA =
min f0 (yd , d) +
yd ,yt ,d,xt
P
ft (xt , yt )
(9)
t=1
subject to constraints (2)–(7) and (10): K P
λk EGkt ≤ POLL
t=1 k=1
where POLL has the value obtained in step (d).
(10)
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(f) (Alternative) Solving only the model described in step (e) for different values of POLL and by representing the total cost—actual value of objective function (9)—as a function of POLL, obtain the Pareto optimal curves. This approach corresponds to the ε-constrained method. Finally chose the solution corresponding to a trade-off value of POLL, for example, POLL = 0. Parameter POLL gives a measure of the environmental impact of global emissions for all pollutants considered. So an acceptable trade-off value for POLL is zero, meaning absence of environmental impact due to the utility system.
This step allows also the optimization of the weighted parameters λk . By varying λk , for a conventional trade-off value of POLL, we obtain different values for total cost. We choose the λk set providing the solution with lower total cost. Fig. 2 represents the modified decomposition algorithm for the utility system. 5. Example problem In order to test the major features of our model, an example problem was “constructed” trying to simulate the
Fig. 2. Modified decomposition algorithm.
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Table 1 Steam/BFW and power demands for the industrial complex for each period Period 1
2
3
4
5
6
7
8
9
10
11
12
55 150 140 350
40 149 145 360
35 143 135 370
60 155 140 350
50 150 120 365
65 150 130 370
35 150 150 370
60 180 170 380
55 180 170 390
55 120 130 300
60 165 150 373
40 120 145 300
BFW for process (t/h)
50
55
55
50
50
50
55
50
50
50
50
55
Power (MW) p=1 p=2
19.7 6
18 4
17 4
17.5 5
18 4.1
19 5
17.2 4.4
19.5 6
17 6
18 4
17 4
16.2 5
Steam (t/h) VHP HP MP LP
p = 1: electrical power; p = 2: mechanical power; VHP: 10 MPa; HP: 5 MPa; MP: 2 MPa; LP: 0.35 MPa.
variable utility demands of a new industrial complex which utility system is to design and plan. The set of multiperiod utility demands of the new industrial complex is shown in Table 1 (1 year with 12 equal periods). The synthesis of the utility system for this site is based in the superstructure represented in Fig. 1. The cost data used in this problem was adapted from Bruno et al. (1998) and Iyer and Grossmann (1997, 1998). This problem was formulated as a MILP and solved with GAMS for two situations: case A—the utility system is not allowed to import electrical power from regional network; case B—electricity imports are allowed. For each case, we solve different models: • Model I.1: Iyer and Grossmann model with local and global emissions calculation. • Model I.2: Iyer and Grossmann objective function with constraints (2)–(7).
• Model II: This paper work model (solution from step (d) in the algorithm). • Model III: Pareto solution from step (e) with POLL obtained in step (d). • Model IV: Pareto solution from step (e) with POLL = 0. 5.1. Choice of weighted coefficients λk In order to solve models II–IV, we need to choice the λk values. In the absence of legal cost penalties for the quantities of the atmospheric pollutants emissions, we adopted a criteria based in the total cost minimization. For this purpose, model IV (case A) was solved for different λk values. By observing the variation of total cost with the λk associated with each pollutant, and fixed λk values for the other(s) pollutant(s) is possible to get the optimal values of the weighted coefficients. Fig. 3 shows the optimization of λk for CO2 and SO2 . The λk values represented are in fact the weighted
Fig. 3. Optimization of the weighted coefficients for CO2 and SO2 .
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A.P. Oliveira Francisco, H.A. Matos / Computers and Chemical Engineering 28 (2004) 745–753 100.00
Case A Case B
Total Cost (M$/Year)
95.00
90.00
85.00
80.00
75.00
70.00 -6000000
-4000000
-2000000
0
2000000
4000000
6000000
8000000
10000000
POLL
Fig. 4. Pareto curves for λCO2 = 0.10 and λSO2 = 451.
parameters multiplied by the operation time in each period (730 h). We obtained a large variation of total cost in the feasible range of λk . The optimal values λSO2 = 451 and λCO2 = 0.10 corresponding to a minimum total cost in case A were also adopted for case B. 5.2. Pareto curves Model III (cases A and B) was solved for diverse values of parameter POLL with the optimal λk values and the Pareto solutions (total cost versus POLL) so obtained were represented in Fig. 4. From Table 2, we can see that solutions of model III have total costs lower than model II for the same values of POLL. So we can take the solutions of model III as our problem solutions. These curves represent the trade-off of total cost and parameter POLL. As we can
see, the curve for case B is much more regular than the corresponding curve for case A. Fig. 5 represents the Pareto curves for non-optimal weighted coefficients: λCO2 = 0.001 and λSO2 = 150. Curves for cases A and B are now much more similar. The optimal Pareto solutions for the utility system model could be obtained without solving the proposed algorithm in this work for the large number of POLL values needed by the Pareto curves, if the criteria used for the choice of parameter POLL is a value in a predefined region. 5.3. Results of model solution Tables 2 and 3 show some results obtained by solving models I–IV. Models II–IV were solved with the optimal λk values. An increase of the total cost is obtained moving
Table 2 Selection and operational plan (# WP) of units for case A/B (λCO2 = 0.10, λSO2 = 451) Unit operation mode
Model I.1
Model I.2
Model II (III)
# WP
# WP
# WP
1 (HP boiler) 2 (MP boiler) 13 (HP boiler) 20 (VHP boiler) 28 (VHP boiler) 3.1 (ST) 3.2 (ST) 4.1 (ST) 12.1(GT) 27.1 (GT) 30.1 (GT) 30.2 (GT)
12/12 8/6 –/1 12/12 1/5 11/– –/– –/– –/1 1/5 –/– –/–
Total cost (million dollars per year) Case A 70.56 – Case B 73.74
12/12 7/5 10/11 12/12 11/9 –/– –/– –/– 10/11 11/9 –/– –/–
Total cost (million dollars per year) Case A 79.70 – Case B 78.26
# WP: number of working periods; ST: steam turbine; GT: gas turbine.
12/12 (12/12) 5/– (8/9) 8/2 (11/11) 12/12 (12/12) 11/11 (11/11) –/– –/– 11/11 (11/11) 8/2 (11/11) 11/11 (11/11) –/– (1/–) –/–
Model IV Total cost (million dollars per year) Case A 93.76 (92.18) – Case B 92.46 (86.84)
# WP 12/12 8/8 11/11 12/12 11/11 –/– –/– 11/11 11/11 11/11 –/– –/–
Total cost (million dollars per year)
Case A 87.04 – Case B 87.13
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751
135.00 130.00 Case A Case B
125.00 120.00
Total Cost (M$/Year)
115.00 110.00 105.00 100.00 95.00 90.00 85.00 80.00 75.00 70.00 -3000000
-2000000
-1000000
0
1000000
2000000
3000000
4000000
POLL
Fig. 5. Pareto curves for λCO2 = 0.001 and λSO2 = 150.
Table 3 Fuel usage and global emissions of CO2 and SO2 for case A/B (λCO2 = 0.10, λSO2 = 451) Fuel
Model I.1
Model I.2
Model II (III)
Model IV
Fuel usage (kt per year) 1 5.9/7.2 2 –/– 3 –/– 4 453.3/416.5 Total 459.2/423.7
79.5/40.4 274.6/351.6 –/– 70.8/27.2 424.9/419.2
397.5/385.8 (267.9/204.8) –/– (165.6/192.5) –/– –/– 397.5/385.8 (433.5/397.3)
210.8/211.0 245.5/251.4 –/– –/– 456.3/462.4
Global emissions (kt per year) CO2 1265/1527 33.7/36.3 SO2
1238/1390 13.4/13.4
642.1/957.5 (671.6/707.3) −1.755/−0.1368 (−1.7173/−0.0814)
705.9/708.0 −0.1565/−0.1570
Fuel: (1) 75.38% C, 0.1% S; (2) 86.47% C, 1.35% S; (3) 87.26% C, 0.84% S; (4) 84.67% C, 3.97% S.
towards the different models I and II, since they take in account more environmental constraints. Moreover, models II–IV show a most intensive exploit of the gas turbines compared with model I.1. The reduction in fuel usage obtained by solving the models is also due to reach optimal power import/export strategy (Oliveira Francisco, 2002). Moving towards the models a decrease in global emissions could be found, but a significant one is achieved in the model III for CO2 and SO2 . The negative values in the SO2 global emissions means that a power is exported from the industrial site causing a necessary lower working level at the regional power station. The model was solved assuming that RPS burns a coal with 74.5% C and 2.0% S. The adding of environmental terms in the objective function and to the set of constraints, show a different choice of fuel with an increase of the utility plant’s annual total cost of about 23–33% for case A and 18–25% for case B. This also corresponds a significant reduction in the global
emissions—about 49% (case A) and 54% (case B)—in CO2 and more than 100% in SO2 . In order to analyse the influence of selection of weighted coefficients in final results, models II–IV were also solved for the non-optimal λCO2 = 0.001 and λSO2 = 150 (Tables 4 and 5). Units selected for the design and the working periods for each unit are very similar for the two sets of weighted coefficients. The main difference resides in global emissions of SO2 , which are much lower for the higher optimal λSO2 = 451.
6. Discussion The formulation and solution of the motivated example presented in this work allows the identification of some weak and strong aspects of the proposed model formulation and algorithm.
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Table 4 Selection and operational plan (# WP) of units for case A/B (λCO2 = 0.001, λSO2 = 150) Unit operation mode
Model II
Model III
# WP
Total cost (million dollars per year)
# WP
Total cost (million dollars per year)
# WP
Total cost (million dollars per year)
1 (HP boiler) 2 (MP boiler) 13 (HP boiler) 20 (VHP boiler) 28 (VHP boiler) 3.1 (ST) 3.2 (ST) 4.1 (ST) 12.1(GT) 27.1 (GT) 30.1 (GT)
12/12 9/9 10/10 12/12 11/9 –/– –/– 11/3 10/10 11/9 2/–
Case A
12/12 9/8 11/11 12/12 11/11 –/– –/– 11/11 11/11 11/11 –/–
Case A
12/12 11/5 11/11 12/12 11/11 –/– –/– 11/11 11/11 11/11 –/–
Case A
92.44
– Case B 92.30
Model IV
87.05
– Case B 88.32
87.40
– Case B 88.81
# WP: number of working periods; ST: steam turbine; GT: gas turbine.
MILP model, last steps of the algorithm implement the ε-constrained method for obtain Pareto optimal solutions without lacking practical interest.
6.1. Strong model features 6.1.1. Model formulation • The inclusion of environmental data (global emissions) in design and operational planning of utility systems allows us the identification of structural and parameter dependence from environmental constraints. • Model formulation as a multiperiod operation accounts for the variability of utility demands in an industrial complex. Uncertainty of demands in each period could be modelled by period splitting considering several scenarios. • Inclusion of environmental costs in the total cost objective function allows us to convert a multi-objective problem into a more treatable single objective one. • The required information in the presented model formulation is almost the same as in more “conventional” methods of design. 6.1.2. Model solution (algorithm) The present work algorithm is an extension of the easy implementation decomposition algorithm described by Iyer and Grossmann. In order to solve the multi-objective Table 5 Fuel usage and global emissions of CO2 and SO2 for case A/B (λCO2 = 0.001, λSO2 = 150) Fuel
Model II
Model III
Model IV
Fuel usage (kt per year) 1 385.2/377.1 2 –/– 3 –/– 4 –/– Total 385.2/377.1
206.7/215.0 264.6/240.0 –/– –/– 471.3/455.0
221.2/237.9 235.3/66.6 –/– –/– 456.5/304.5
Global emissions (kt per year) CO2 975/1020 SO2 0.10/1.03
718/803 0.10/1.04
733.26/744.8 −0.0049/−0.005
Fuel: (1) 75.38% C, 0.1% S; (2) 86.47% C, 1.35% S; (3) 87.26% C, 0.84% S; (4) 84.67% C, 3.97% S.
6.2. Weak model features 6.2.1. Model formulation Parameters λk in this model formulation represent environmental costs for global emissions of each atmospheric pollutant. Values for these parameters should be calculated from environmental legislation for atmospheric emissions or, for example, by the methodologies referred by Allen and Shonnard (2002). In the present work, a more arbitrary way was followed by optimization the λk values compatible with predefined value for the environmental cost function P a K t=1 t=1 λk EGkt . 6.2.2. Model solution (algorithm) The ε-constrained method used in last steps of our algorithm usually requires many sub-problems solution to obtain the Pareto optimal curves. However, if the trade-off value POLL (i.e. ε) was fixed previously a single solution were obtained and the Pareto curve is not necessary anymore.
7. Conclusions The presented formulation is useful for preliminary optimal design of utility plants in an industrial complex working with variable demands. The model could be applied to grass-roots projects or to the revamping of existing ones. The model incorporates the contribution of the environmental parameters concerning the gaseous emissions penalties. With the modification now introduced on the previous other authors models, it is possible to obtain the best utility plant design using the equipment and fuel that minimizes operational and capital costs, but also the pollutant global emissions due to the burning fuels.
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Some weakness in the model derives from the conversion of a multi-objective problem in an easier to solve single objective one and the inclusion of the ε-constrained method in the solution algorithm. An alternative way to exploit in future work is the use of evolutionary algorithms, for example, genetic algorithms.
References Allen, D. T., & Shonnard, D. R. (2002). Green engineering. Environmentally conscious design of chemical process. Upper Saddle River, NJ: Prentice-Hall. Bruno, J. C., Fernandez, F., Castells, F., & Grossmann, I. E. (1998). A rigorous MINLP model for the optimal synthesis and operation of utility plants. Transactions of the IChemE, 76A, 246– 258.
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Chang, C.-T., & Hwang, J.-R. (1996). A multiobjective programming approach to waste minimization in the utility systems of chemical processes. Chemical Engineering Science, 51(16), 3951–3965. Iyer, R., & Grossmann, I. E. (1997). Optimal multiperiod operational planning for utility systems. Computers and Chemical Engineering, 21(8), 787–800. Iyer, R., & Grossmann, I. E. (1998). Synthesis and operational planning of utility systems for multiperiod operation. Computers and Chemical Engineering, 22(7–8), 979–993. Oliveira Francisco, A. P. (2002). Metodologia para a S´ıntese e Planeamento da Operação de Sistemas de Utilidades num Complexo Industrial com Intergração Energética. M.Sc. thesis, I.S.T., U.T.L, Lisbon. Papoulias, S. A., & Grossmann, I. E. (1983). A structural optimization approach in process synthesis. I. Utility systems. Computers and Chemical Engineering, 7(6), 695–706. Smith, R., & Delaby, O. (1991). Targeting flue emissions. Transactions of the IChemE, 69A, 492–505. Winston, W. L. (1994). Operations research—Applications and algorithms. Belmont, CA: Wadsworth.
Multiperiod synthesis and operational planning of utility systems with environmental concerns A. P. Oliveira Francisco a , H. A. Matos b,∗ a
Dep. de Engenharia Qu´ımica, Instituto Superior de Engenharia de Lisboa, 1949-014 Lisbon, Portugal b Dep. de Engenharia Qu´ımica, Instituto Superior Técnico, 1049-001 Lisbon, Portugal
Abstract Utility plants supply the required energy demands to industrial processes. Several authors addressed the synthesis and design of those plants. Environmental concerns require to consider minimization of global emissions from utility plants and regional power plant as an additional objective, for which no multiperiod optimization models able to cope with the time varying demands, have been made available yet. This paper presents an extension of the Iyer and Grossmann [Comput. Chem. Eng. 21 (8) (1997) 787; 22 (7–8) (1998) 979] model to synthesis and multiperiod operational planning in order to include the global emissions of atmospheric pollutants issues coming from the fuels burning. A new five-step algorithm is introduced to solve this multi-objective model. One motivation example enables us to compare the different units and fuel selection and also the operation periods of an industrial utility system taking in account the electrical power import/export policy and environmental concerns. © 2004 Elsevier Ltd. All rights reserved. Keywords: Utility systems; Process synthesis; Multiperiod operation; Environmental concerns; Global emissions; Mathematical programming; Multi-objective optimization
1. Introduction Utility plants supply the required energy demands to chemical processes, namely, mechanical, electrical and thermal power (different levels of steam). Changes in specifications, composition of feed and seasonal product demands create several process conditions with the corresponding variation in the utility demands during one annual horizon. Several authors addressed the synthesis and design of utility plants. Among these authors, Papoulias and Grossmann (1983) described a MILP model for the synthesis and design of utility systems, for fixed demands. Iyer and Grossmann (1997, 1998) described models for multiperiod operational planning and synthesis, and operational planning of utility systems. Also, Bruno, Fernandez, Castells, & Grossmann (1998) described a rigorous MINLP model for the optimization of synthesis and operation of utility plants. ∗
Corresponding author. E-mail address: [email protected] (H.A. Matos).
0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2004.02.025
Chang and Wang (1996) described a multi-objective programming approach to waste minimization in the utility systems of chemical process, using the concept of global emissions of gaseous pollutants. This model, merging economic and environmental concerns in the utility system synthesis, was stated for fixed utility demands. Oliveira Francisco (2002) described a methodology for the synthesis and multiperiod operational planning of utility systems in a heat integrated industrial complex. This comprises a multiperiod model for utility systems including environmental concerns. The purpose of the present paper is to show the structure of this modified model and the resolution algorithm, applied to a simple example problem.
2. Problem definition Given a set of time variable (multiperiod) demands of steam at various levels of pressure, electricity and mechanical power, the problem of synthesis and operational
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A.P. Oliveira Francisco, H.A. Matos / Computers and Chemical Engineering 28 (2004) 745–753
Fig. 1. Superstructure for example problem.
planning of the utility system consists in a structural and parametric optimization from a superstructure of alternatives. The superstructure will be decomposed in a set of feasible configurations. For each feasible configuration, the unit sizes are such that allow demands satisfaction in all operation periods. In a single objective problem—only economic optimization—the configuration with the lowest total cost (investment and operational costs) for the horizon of planning should be chosen. If in addition to economic optimization we have to satisfy to other objectives, for example, environmental concerns, the problem arises to a multi-objective problem and a different optimization strategy should be adopted. The superstructure of the utility system adopted in this work is derived from the superstructures described by Iyer and Grossmann (1997, 1998) and Papoulias and Grossmann (1983). Fig. 1 exemplifies the superstructure adopted for the example problem. This superstructure includes several steam headers at various pressure levels (VHP, HP, MP and LP). Steam can be generated in either conventional fired boilers (units 1, 2 and 20) or with waste heat boilers (units 13 and 28) receiving hot gases from gas turbines.
There is a deaerator (unit 11) receiving make-up water and condensates returned from process utilizations. A condenser (unit 8) is provided for condensation of LP steam from unit 4. Power can be generated with several types of steam turbines (units 3, 4 and 23), gas turbines (units 12, 27 and 30). Electrical generators can be driven by steam turbines or gas turbines. Gas turbines can operate in a stand-alone basis (eventually with a regenerator) or associated with waste heat recovery boilers. There are also several types of auxiliary equipment as fans, pressure reducers and pumps.
3. Model formulation Our model for the utility system is an extension of the multiperiod models described by Iyer and Grossmann (1997, 1998) in order to include the concept of global emissions of the gaseous pollutants that came from fuel burning. Following Smith and Delaby (1991), global emissions comprise local emissions derived from the production of utilities in the industrial site and the balance between increasing/lowering of the emissions in the regional power
A.P. Oliveira Francisco, H.A. Matos / Computers and Chemical Engineering 28 (2004) 745–753
station (RPS) due to electricity imports and exports to regional power network by the utility system. The extended mathematical model for the utility system can be formulated as follows: zP =
min
yd ,yt ,d,xt
f0 (yd , d) +
P
ft (xt , yt ) +
t=1
K P
λk EGkt
t=1 k=1
(1) s.t.
ht (d, yd , yt , xt , θt ) ≤ 0,
t = 1, . . . , P
(2)
u
xt − Ω yt ≤ 0,
t = 1, . . . , P
(3)
xt − Ωl yt ≥ 0,
t = 1, . . . , P
(4)
yd ≥ yt , P
t = 1, . . . , P
(5)
yt ≤ αyd
t=1
(6)
EGkt − LEMk
n
d ∈ Rn ,
≤0
Qnrt
(7)
r
xt ∈ Rq × RP ,
yd ∈ {0, 1}n ,
yt ∈ {0, 1}n×P
where yd are integer variables (0–1) defining the selection of units for the design, d the design variables defining the sizes of units, yt integer variables (0–1) that determine the operational status on/off for period t, xt the state and control variables for period t, θ t the parameters (e.g. utility demands) for period t, EGkt the global emissions of pollutant k, in period t, Qnrt the production of level r steam, in the unit n, in period t, as the absorbed heat in the steam generator, LEMk the limit of global emissions of pollutant k, expressed in relation to the absorbed heat in the steam generator, λk the weighted parameter meaning the pollutant k contribution to the objective function, Ωl (·) and Ωu (·) the valid limits, lower and higher, respectively, P the number of periods and α a scalar lower than P. The objective function includes the investment cost for the design (f0 ) and the sum of the operation costs (ft ) for all periods t = 1, . . . , P, as well as the above-referred weighted terms of global emissions of pollutants. Specific integer variables (binary, 0–1) in appropriate restrictions of the mathematical model allow the selection of operation on/off status and operations modes, e.g. extraction or condensing steam turbine. In the example problem given further, we adopted a MILP formulation for the utility system optimization.
4. Decomposition algorithm Iyer and Grossmann (1998) proposed a bilevel decomposition algorithm for solving his model. In the described algorithm, the first step is solving a design problem (DP) obtained from the principal model by making zero all fixed operation
747
costs in the objective function, removing constraints of types (4) and (6) and by replacing yt with yd in the constraints of type (3). The solution of model DP gives a lower bound for the objective function of principal model and provides values for design integer variables yd to use in the next step. In a second step, a planning model (OP) is solved by fixing in the main model variables yd obtained in first step. Solution of OP gives a higher bound for the objective function of principal model and provides an operational plan for the next time DP is solved. Iterative procedure with adding of appropriate cuts converges to the solution of main model. Then, a modification of the Iyer and Grossmann algorithm was implemented: (a) Fixing values of weighted parameters λk . These values may be imposed in a basis of legal penalties to pollutants emissions or obtained by an economic optimization procedure referred further. (b) Solving the utility system model with the Iyer and Grossmann (1998) bilevel decomposition algorithm. In this step we obtain a design and an operational plan for the utility system. (c) Fixing the values calculated in step (b) for binary variables yd and yt in constraints and solving the model obtained by replacing the objective function (1) with a second objective function (8): λk EGkt (8) zE = min t
k
This step provides the utilization plan for fuels (binary variables) in the utility system arising to a minimum of global emissions. (d) Fixing binary variables associated with the selected units for design (yd ) and with the utilization plan of fuels obtained in step (c) and solving the model with the objective function (1). Solutions obtained in steps (b)–(d) correspond to diverse importance given to economic and environmental concerns. Less pollutant fuels have higher costs than more pollutant content fuels. Thus, total cost for these three solutions will be (b) < (d) < (c). Since all three solutions will satisfy the global emissions limitations, the selected solution depends from the user criteria. In this step, we can get a value toa parameter POLL defined as the actual value of Pt=1 K t=1 λk EGkt . (e) By applying the Iyer and Grossmann algorithm, finding a Pareto optimal solution (Winston, 1994) for a model comprising the objective function (9): zPA =
min f0 (yd , d) +
yd ,yt ,d,xt
P
ft (xt , yt )
(9)
t=1
subject to constraints (2)–(7) and (10): K P
λk EGkt ≤ POLL
t=1 k=1
where POLL has the value obtained in step (d).
(10)
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(f) (Alternative) Solving only the model described in step (e) for different values of POLL and by representing the total cost—actual value of objective function (9)—as a function of POLL, obtain the Pareto optimal curves. This approach corresponds to the ε-constrained method. Finally chose the solution corresponding to a trade-off value of POLL, for example, POLL = 0. Parameter POLL gives a measure of the environmental impact of global emissions for all pollutants considered. So an acceptable trade-off value for POLL is zero, meaning absence of environmental impact due to the utility system.
This step allows also the optimization of the weighted parameters λk . By varying λk , for a conventional trade-off value of POLL, we obtain different values for total cost. We choose the λk set providing the solution with lower total cost. Fig. 2 represents the modified decomposition algorithm for the utility system. 5. Example problem In order to test the major features of our model, an example problem was “constructed” trying to simulate the
Fig. 2. Modified decomposition algorithm.
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Table 1 Steam/BFW and power demands for the industrial complex for each period Period 1
2
3
4
5
6
7
8
9
10
11
12
55 150 140 350
40 149 145 360
35 143 135 370
60 155 140 350
50 150 120 365
65 150 130 370
35 150 150 370
60 180 170 380
55 180 170 390
55 120 130 300
60 165 150 373
40 120 145 300
BFW for process (t/h)
50
55
55
50
50
50
55
50
50
50
50
55
Power (MW) p=1 p=2
19.7 6
18 4
17 4
17.5 5
18 4.1
19 5
17.2 4.4
19.5 6
17 6
18 4
17 4
16.2 5
Steam (t/h) VHP HP MP LP
p = 1: electrical power; p = 2: mechanical power; VHP: 10 MPa; HP: 5 MPa; MP: 2 MPa; LP: 0.35 MPa.
variable utility demands of a new industrial complex which utility system is to design and plan. The set of multiperiod utility demands of the new industrial complex is shown in Table 1 (1 year with 12 equal periods). The synthesis of the utility system for this site is based in the superstructure represented in Fig. 1. The cost data used in this problem was adapted from Bruno et al. (1998) and Iyer and Grossmann (1997, 1998). This problem was formulated as a MILP and solved with GAMS for two situations: case A—the utility system is not allowed to import electrical power from regional network; case B—electricity imports are allowed. For each case, we solve different models: • Model I.1: Iyer and Grossmann model with local and global emissions calculation. • Model I.2: Iyer and Grossmann objective function with constraints (2)–(7).
• Model II: This paper work model (solution from step (d) in the algorithm). • Model III: Pareto solution from step (e) with POLL obtained in step (d). • Model IV: Pareto solution from step (e) with POLL = 0. 5.1. Choice of weighted coefficients λk In order to solve models II–IV, we need to choice the λk values. In the absence of legal cost penalties for the quantities of the atmospheric pollutants emissions, we adopted a criteria based in the total cost minimization. For this purpose, model IV (case A) was solved for different λk values. By observing the variation of total cost with the λk associated with each pollutant, and fixed λk values for the other(s) pollutant(s) is possible to get the optimal values of the weighted coefficients. Fig. 3 shows the optimization of λk for CO2 and SO2 . The λk values represented are in fact the weighted
Fig. 3. Optimization of the weighted coefficients for CO2 and SO2 .
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Case A Case B
Total Cost (M$/Year)
95.00
90.00
85.00
80.00
75.00
70.00 -6000000
-4000000
-2000000
0
2000000
4000000
6000000
8000000
10000000
POLL
Fig. 4. Pareto curves for λCO2 = 0.10 and λSO2 = 451.
parameters multiplied by the operation time in each period (730 h). We obtained a large variation of total cost in the feasible range of λk . The optimal values λSO2 = 451 and λCO2 = 0.10 corresponding to a minimum total cost in case A were also adopted for case B. 5.2. Pareto curves Model III (cases A and B) was solved for diverse values of parameter POLL with the optimal λk values and the Pareto solutions (total cost versus POLL) so obtained were represented in Fig. 4. From Table 2, we can see that solutions of model III have total costs lower than model II for the same values of POLL. So we can take the solutions of model III as our problem solutions. These curves represent the trade-off of total cost and parameter POLL. As we can
see, the curve for case B is much more regular than the corresponding curve for case A. Fig. 5 represents the Pareto curves for non-optimal weighted coefficients: λCO2 = 0.001 and λSO2 = 150. Curves for cases A and B are now much more similar. The optimal Pareto solutions for the utility system model could be obtained without solving the proposed algorithm in this work for the large number of POLL values needed by the Pareto curves, if the criteria used for the choice of parameter POLL is a value in a predefined region. 5.3. Results of model solution Tables 2 and 3 show some results obtained by solving models I–IV. Models II–IV were solved with the optimal λk values. An increase of the total cost is obtained moving
Table 2 Selection and operational plan (# WP) of units for case A/B (λCO2 = 0.10, λSO2 = 451) Unit operation mode
Model I.1
Model I.2
Model II (III)
# WP
# WP
# WP
1 (HP boiler) 2 (MP boiler) 13 (HP boiler) 20 (VHP boiler) 28 (VHP boiler) 3.1 (ST) 3.2 (ST) 4.1 (ST) 12.1(GT) 27.1 (GT) 30.1 (GT) 30.2 (GT)
12/12 8/6 –/1 12/12 1/5 11/– –/– –/– –/1 1/5 –/– –/–
Total cost (million dollars per year) Case A 70.56 – Case B 73.74
12/12 7/5 10/11 12/12 11/9 –/– –/– –/– 10/11 11/9 –/– –/–
Total cost (million dollars per year) Case A 79.70 – Case B 78.26
# WP: number of working periods; ST: steam turbine; GT: gas turbine.
12/12 (12/12) 5/– (8/9) 8/2 (11/11) 12/12 (12/12) 11/11 (11/11) –/– –/– 11/11 (11/11) 8/2 (11/11) 11/11 (11/11) –/– (1/–) –/–
Model IV Total cost (million dollars per year) Case A 93.76 (92.18) – Case B 92.46 (86.84)
# WP 12/12 8/8 11/11 12/12 11/11 –/– –/– 11/11 11/11 11/11 –/– –/–
Total cost (million dollars per year)
Case A 87.04 – Case B 87.13
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751
135.00 130.00 Case A Case B
125.00 120.00
Total Cost (M$/Year)
115.00 110.00 105.00 100.00 95.00 90.00 85.00 80.00 75.00 70.00 -3000000
-2000000
-1000000
0
1000000
2000000
3000000
4000000
POLL
Fig. 5. Pareto curves for λCO2 = 0.001 and λSO2 = 150.
Table 3 Fuel usage and global emissions of CO2 and SO2 for case A/B (λCO2 = 0.10, λSO2 = 451) Fuel
Model I.1
Model I.2
Model II (III)
Model IV
Fuel usage (kt per year) 1 5.9/7.2 2 –/– 3 –/– 4 453.3/416.5 Total 459.2/423.7
79.5/40.4 274.6/351.6 –/– 70.8/27.2 424.9/419.2
397.5/385.8 (267.9/204.8) –/– (165.6/192.5) –/– –/– 397.5/385.8 (433.5/397.3)
210.8/211.0 245.5/251.4 –/– –/– 456.3/462.4
Global emissions (kt per year) CO2 1265/1527 33.7/36.3 SO2
1238/1390 13.4/13.4
642.1/957.5 (671.6/707.3) −1.755/−0.1368 (−1.7173/−0.0814)
705.9/708.0 −0.1565/−0.1570
Fuel: (1) 75.38% C, 0.1% S; (2) 86.47% C, 1.35% S; (3) 87.26% C, 0.84% S; (4) 84.67% C, 3.97% S.
towards the different models I and II, since they take in account more environmental constraints. Moreover, models II–IV show a most intensive exploit of the gas turbines compared with model I.1. The reduction in fuel usage obtained by solving the models is also due to reach optimal power import/export strategy (Oliveira Francisco, 2002). Moving towards the models a decrease in global emissions could be found, but a significant one is achieved in the model III for CO2 and SO2 . The negative values in the SO2 global emissions means that a power is exported from the industrial site causing a necessary lower working level at the regional power station. The model was solved assuming that RPS burns a coal with 74.5% C and 2.0% S. The adding of environmental terms in the objective function and to the set of constraints, show a different choice of fuel with an increase of the utility plant’s annual total cost of about 23–33% for case A and 18–25% for case B. This also corresponds a significant reduction in the global
emissions—about 49% (case A) and 54% (case B)—in CO2 and more than 100% in SO2 . In order to analyse the influence of selection of weighted coefficients in final results, models II–IV were also solved for the non-optimal λCO2 = 0.001 and λSO2 = 150 (Tables 4 and 5). Units selected for the design and the working periods for each unit are very similar for the two sets of weighted coefficients. The main difference resides in global emissions of SO2 , which are much lower for the higher optimal λSO2 = 451.
6. Discussion The formulation and solution of the motivated example presented in this work allows the identification of some weak and strong aspects of the proposed model formulation and algorithm.
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Table 4 Selection and operational plan (# WP) of units for case A/B (λCO2 = 0.001, λSO2 = 150) Unit operation mode
Model II
Model III
# WP
Total cost (million dollars per year)
# WP
Total cost (million dollars per year)
# WP
Total cost (million dollars per year)
1 (HP boiler) 2 (MP boiler) 13 (HP boiler) 20 (VHP boiler) 28 (VHP boiler) 3.1 (ST) 3.2 (ST) 4.1 (ST) 12.1(GT) 27.1 (GT) 30.1 (GT)
12/12 9/9 10/10 12/12 11/9 –/– –/– 11/3 10/10 11/9 2/–
Case A
12/12 9/8 11/11 12/12 11/11 –/– –/– 11/11 11/11 11/11 –/–
Case A
12/12 11/5 11/11 12/12 11/11 –/– –/– 11/11 11/11 11/11 –/–
Case A
92.44
– Case B 92.30
Model IV
87.05
– Case B 88.32
87.40
– Case B 88.81
# WP: number of working periods; ST: steam turbine; GT: gas turbine.
MILP model, last steps of the algorithm implement the ε-constrained method for obtain Pareto optimal solutions without lacking practical interest.
6.1. Strong model features 6.1.1. Model formulation • The inclusion of environmental data (global emissions) in design and operational planning of utility systems allows us the identification of structural and parameter dependence from environmental constraints. • Model formulation as a multiperiod operation accounts for the variability of utility demands in an industrial complex. Uncertainty of demands in each period could be modelled by period splitting considering several scenarios. • Inclusion of environmental costs in the total cost objective function allows us to convert a multi-objective problem into a more treatable single objective one. • The required information in the presented model formulation is almost the same as in more “conventional” methods of design. 6.1.2. Model solution (algorithm) The present work algorithm is an extension of the easy implementation decomposition algorithm described by Iyer and Grossmann. In order to solve the multi-objective Table 5 Fuel usage and global emissions of CO2 and SO2 for case A/B (λCO2 = 0.001, λSO2 = 150) Fuel
Model II
Model III
Model IV
Fuel usage (kt per year) 1 385.2/377.1 2 –/– 3 –/– 4 –/– Total 385.2/377.1
206.7/215.0 264.6/240.0 –/– –/– 471.3/455.0
221.2/237.9 235.3/66.6 –/– –/– 456.5/304.5
Global emissions (kt per year) CO2 975/1020 SO2 0.10/1.03
718/803 0.10/1.04
733.26/744.8 −0.0049/−0.005
Fuel: (1) 75.38% C, 0.1% S; (2) 86.47% C, 1.35% S; (3) 87.26% C, 0.84% S; (4) 84.67% C, 3.97% S.
6.2. Weak model features 6.2.1. Model formulation Parameters λk in this model formulation represent environmental costs for global emissions of each atmospheric pollutant. Values for these parameters should be calculated from environmental legislation for atmospheric emissions or, for example, by the methodologies referred by Allen and Shonnard (2002). In the present work, a more arbitrary way was followed by optimization the λk values compatible with predefined value for the environmental cost function P a K t=1 t=1 λk EGkt . 6.2.2. Model solution (algorithm) The ε-constrained method used in last steps of our algorithm usually requires many sub-problems solution to obtain the Pareto optimal curves. However, if the trade-off value POLL (i.e. ε) was fixed previously a single solution were obtained and the Pareto curve is not necessary anymore.
7. Conclusions The presented formulation is useful for preliminary optimal design of utility plants in an industrial complex working with variable demands. The model could be applied to grass-roots projects or to the revamping of existing ones. The model incorporates the contribution of the environmental parameters concerning the gaseous emissions penalties. With the modification now introduced on the previous other authors models, it is possible to obtain the best utility plant design using the equipment and fuel that minimizes operational and capital costs, but also the pollutant global emissions due to the burning fuels.
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Some weakness in the model derives from the conversion of a multi-objective problem in an easier to solve single objective one and the inclusion of the ε-constrained method in the solution algorithm. An alternative way to exploit in future work is the use of evolutionary algorithms, for example, genetic algorithms.
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