- Email: [email protected]
Optimization of a complex plant by a GRG algorithm
Computers & Chrmral Engmrrring, Printed in Great Elntam
Vol.
3. pp.
597-602.
1979
009&l
354!79/040597-06$02,0/O Pergamon Press Ltd.
Paper 12.4 OPTIMIZATION
OF A COMPLEX GRG ALGORITHM
PLANT
BY A
M. GRAUER,* G. GRUHN and L. POLLMER Technical University Leuna-Merseburg, Department of ChemicalEngineering, DDR-42 Merseburg, Geusaer Strasse, German Democratic Republic
Abstract-In the design and control of chemical systems nonlinear optimization problems often have to be solved. For solving these problems an algorithm based on generalized reduced gradients (GRG) is investigated. The paper illustrates the effectiveness of the algorithm by applying it to the optimal design of a monochlorobenzene plant. The results are compared with the application of a direct search method and a random search method. Scope-In the course of optimizing a chemical engineering system, direct search and random search methods are often used. In this paper, we describe the use of a generalized reduced gradient (GRG) method for solving such problems. GRG has recently been established in a number of investigations [S-9] as one of the most effective optimization algorithms available at the moment. Starting from this fact, a GRG algorithm was developed especially for optimization of complex chemical engineering systems. The algorithm solves nonlinear programs with inequalities, using derivatives which are formed numerically. As a basis for comparison, the GRG algorithm was applied to optimization of a plant for monochlorobenzene production (see Fig. 1). The optimization problem consists in finding a cost optimal design for processing 15 kmol/h benzene to monochlorobenzene. The problem has the dimension eight. The same optimization problem was solved by a direct search method and a random method, in order to compare the effectiveness of these methods with that of the GRG algorithm. From the solution with the GRG algorithm, a sensitivity analysis for the respective design problem becomes possible. This is shown with the monochlorobenzene plant as an example. Conclusions and Significance-From the comparison of the GRG algorithm with a direct search method and a random search method the following conclusions can be drawn : In accordance with the literature [4-lo], the GRG algorithm proves an effective method for the solution of small to medium size problems with nonlinear inequalities. Within the GRG algorithm, the Fletcher-Reeves algorithm proved the most effective for finding the search direction; Golden Search proved best for the one-dimensional search. In comparison with a direct search method and a random search method, the GRG algorithm proves at least twice as effective (see Fig. 3). This statement holds true for the solution as well as during the whole optimization process. Of course, the GRG code has the disadvantage of local convergence. For this type of problem, the GRG method is favorable in comparison with other methods using derivatives. In the first place, it is impossible to give the objective function for this design as an explicit function of the optimization variables, because, after giving a set of the independent variables, the state and output variables are obtained only by solving a complicated non-linear transcendental and/or differential equation system, or after calculating a simulation program system. Moreover, the analytical derivative formation can be successfully bypassed with the GRG algorithm, thus excluding possible sources of error. As the GRG algorithm forms its derivatives numerically, a sensitivity analysis of the problem is possible and instructions for the design of the controllers for the plant may be given. With the design of the monochlorobenzene plant as an example, the GRG algorithm proves to be a very effective method for solving chemical engineering optimization problems. INTRODUCTION The GRG algorithm has been developed the following optimization problem min f(x) XER” I
with g,(x) 5 0 The design problem
i = l(l)m
corresponding
*Author to whom correspondence
.
first presented, and then the GRG algorithm and its developed version are described. The results of the application of the algorithm are compared with the results of the solution of the same problem by a direct search method and a random search method and, finally, sensitivity considerations for the plant are given.
for solving
(1)
I
to the task (1) is
The design problem One of the most
should be addressed. 591
inportant
products
of petro-
598
M. GRAUER et al.
chemical industry is phenol. The chlorobenzene method is one of the three industrial processes for its production. The specific cost of phenol produced in this way is high, so it is of the utmost importance to design an optimal cost plant for producing monochlorobenzene. The design of this plant is based on a general reactor-separator-system (Fig. 1) [l]. The following reaction mechanism may be taken as a basis for monochlorobenzene production
equations are defined in the attached nomenclature.) (1) Structure variables xi and x2 Xl
n3
x2
=T-9
+ Cl, + C,H,Cl,
G(x) = E(x) - K(x) + max
(2) with K(x) costs, as a sum of the costs of the input product benzene, for heating and cooling agents and the costs for the equipment E(x) proceeds from the sale of the output product obtained ; the proceeds from the by-products (dichlorobenzene and hydrogen chloride) are used for the reimbursement of the cost of chlorine c(x) profit X vector of the decision variables (structure variables, apparatus dimensions, process parameters). The vector of the decision variables is made up of the following eight components. (Variables in the defining REACTOR-
SYSTEM
n4
=,.
n6 By the variation of these parameters all variants for the reactors giving sense may be obtained. (2) Structure variable x3 fii,,
x3=,.
+ HCl.
Corresponding to Fig. 1, the reaction is carried out in two reactors Rl and R2, FeCl, acting as a catalyzer. For suppressing the formation of dichlorobenzene, the process is run with a partial conversion of benzene. Supposing that the chlorine concentration can be kept constant and the HCl concentration does not disturb the reaction, the reactions can be treated as a consecutive reaction system, A -+ B + C. This assumption is the basis for the determination of reaction kinetics. The unreacted benzene is separated at the head of column Cl and recycled to the reaction step. In column C2 mono- and dichlorobenzene are separated. The nonlinear, constrained optimization problem consists in designing a plant processing a given input stream of benzene (15 kmol/h) with maximal profit. The corresponding objective function is then
of
n2
C,H, + Cl, + C,H,Cl + HCl C,H,Cl
table
nzo
This variable identifies how much of the recycled raw material stream will be used for regenerative preheating. (3) Mean residence time 71 and 72 for the reactors Rl and R2 x4 =
T1
V
=
s,
xg
=
T2
=
V
2.
v3
V5
(4) Reflux ratio for distillation columns Cl and C2 %8
X6=.r
n14
x,=,.
nzo
(5) Temperature E3
n15
difference in the heat-exchanger
x8 = AT, = T,, - Tz3. For the decision variables, the explicit constraints are : 0. SXiIl. 0. 5 x2 5 1. 0. I x3 5 1. 0. $ x4 5 3.0 h
>
(3)
0. I xg 5 3.0 h 0.8 5 X6 s 1.5 0.8 5 x, s 1.5 0. $x8 s 30. K
The mathematical model of the total plant is described in detail in [lo]. It represents a nonlinear system of equations to be calculated iteratively. DISTILLATION-SYSTEM
16
DlCHLORO BENZENE
Fig. 1. The process scheme to be optimized (Rl, R2-reactors; Cl, C2-distillation E7-heat-exchanger; Ml,. . . , M3-mixer; Sl,. , SS-separator).
columns; El,
.. ,
Optimization
of a complex
along d(k). For increasing the efficiency of the GRG algorithm, different methods of approach in the search of direction and minimization along a line were tested. For the search direction, the Davidon-FletcherPowell-algorithm was not used as in [S] because of the high demand for storage, and numerical instabilities. Test calculations proved the Fletcher-Reevesalgorithm [13] to be more effective than the PolakRibiere-algorithm [12], it was used for the design problem. In the search along a line the algorithm of the golden search and the quadratic-interpolation showed best results. The scaling problem in the numerical formation of gradients was solved by division of the respective independent variable by the corresponding upper explicite bound. For comparison, we used a random search method corresponding to [14], that was also used in[lSJ and [ 161. It is an adaptive random search method with range reduction (RANDOM). From the class of direct search methods, a modification of the COMPLEXmethod [17] by the way of[18] is made. This algorithm uses a so-called weighted COMPLEX (GCOBO). These three optimization procedures, are similar in a program-technical way ; thus the input of objective function, constraints and starting point are identical. The specification of the optimization procedure to be used is given only via a corresponding CALLstatement.
The optimization problem corresponding to task (1) is : Maximize the objective function (2) by variation of the eight decision variables, taking into account the constraints in inequality form (3) and the constraints in equation form given in the mathematical model. This optimization problem is solved by a GRG algorithm and the result obtained is compared with the solutions obtained from other optimization procedures. Optimization procedure The algorithm applied for the solution of the problem (1) was given in detail in [2]. Here it should be mentioned again, that corresponding to [4] and contrary to [12] only inequalities are taken into consideration. The approach is based on the conception of the socalled &-strategy. It uses the projection of the antigradient vectors on the kth iteration step for the directional determination. Starting from the index set of the e-active inequalities L,(k) = {j E L 1-E 5 gj(x(k)) 5 0) the nonsingular projection matrix &c(k)) is formed containing all vectors Vg,(x(k)) with jE L,(k). The projection itself is carried out in the following steps (I) and (II)
o(k) = - CA(x(k))Yj (k) =
‘I’WW 2 - e(k)
0, ifjEL,(k)Auj(k)
uj (k) For obtaining a direction pointed D, another modification is made Sj(k) =
into the interior
6(k), ifjEL,(k)hvj(k)
Results and discussion The results obtained from the optimization of the plant for monochlorobenzene production under application of the three procedures GRG, GCOBO and RANDOM are given in Table 1. The corresponding optimal structure of the plant is shown in Fig. 2. For the solution by the three algorithms, the same starting point was used (see Table 1). It becomes quite obvious from Table 1 and Fig. 3 that, in its effectiveness (number of function evaluations and computing time)
of
2 - e(k)
Yj (k)
with 6(k) < 0. The inverse-transformation vides the direction of search d(k) = - A(k)-‘. The stepsize is obtained
599
plant by a GRG algorithm
then pro-
S(k).
as a result of a minimization
Table 1. Starting point and results of optimization of the monochlorobenzene-plant on a CDC-3300 computer. (G-Objective Function, yearly profit; Nl-Number of iteration; NZF-Number of Function Evaluations; CPU-Standard CPU-time by the Colville-constant [S].) OPTIMAL
SOLUTIOH
STARTING by
POMT
optimizationalgorithm
GRG
RAIYDOM
GCOBO
x1
0.80
1 .oOOoo
1.00000
1.00000
x2
0.80
1.00000
1 .oOOOO
1 00000
=3
0.80
1 .ooooo
1.00000
0.92273
l
1.70
0.08162
0.08163
.0.08192
1.40
0.08161
0.08161
0.08202
=6
1.30
0.80000
0.80000
o.aooo4
x7
1.30
0.80000
0.80000
x4
in
x5
in h
x8
in
G in NI NZP CPU
h
K 10%~
13.00 4.0657
0.0006
20.06003
28.06400
28.30196
7.42884
7.42084
7.42760
31.
1,001.
719.
1,973.
0.221
116. 13,179. 0.474
,
2.550 L
600
M.
GRAUER
REACTOR-SYSTEM
et al.
DISTILLATION-SYSTEM
Fig. 2. The optimal
the GRG algorithm is far superior to the other algorithms. This statement is valid for the solution given in Table 1 as well as for any steps of the solution process which may be taken from the analysis of the history of the optimization process (see Fig. 3). For this example, the conclusion is independent of the stopping criteria used, and the time chosen to stop the solution process. If the CRG algorithm is used, there is consequently at least a doubling of the effectiveness of the search for an optimum compared with the other algorithms. A disadvantage of the GRG is its property of local convergence. The search did not achieve the global solution from all starting points. This disadvantage is specific to the algorithm, but does not seem too serious because, as a rule, there is enough known, as a result of prior investigations on modelling and simulation, to define an admissible starting point in the neighbourhood of the global solution. An advantage of using the GRG algorithm is the existence of numerical derivatives and, thus, the sensitivities of the objective function against changes of the independent variables. In Table 2, some of these sensitivities are given at the optimum of the plant. It can be seen clearly that the optimal design of the plant shows the highest sensitivity to the reflux ratios (x6, x7)
process scheme.
of the columns Cl and C2. Between these control parameters, the reflux ratio (x6) of column Cl has the greatest influence, because it determines mass streams and the energy conditions of the plant in a stronger way than the reflux ratio (x,) of the second column. The second level of sensitivity is found in the residence times (x4, x5) of the two reactors Rl, R2. The objective function is least sensitive to the temperature difference (xs) in the heat-exchanger. These results may be used for model modifications and are a safe basis for designing the controllers for the individual process parameters. They are statements on the accuracy of control of the mass stream, temperature or pressure in question. It should also be mentioned that these sensitivities are not only calculated at the solution point, but are obtained automatically at each point in the optimization if the GRG algorithm is used. The curves of Fig. 4 show when the individual variables reach their optimal value in the course of the solution. This example proves the intuitive comprehension of a hierarchy of the three groups of variables (variables of structure, apparatus dimensions, control parameters). Structure variable x1 is the first to reach its optimal value. The apparatus dimension x5 and the process parameter x8 follow.
0,96
0,56 0,54
0
20
40
Fig. 3. History
60
110 160 120 100 NUMBER OF FUNCTION EVALUATIONS
60
of optimization
process
(for the objective
180
200
function).
220
2LO
Optimization
of a complex
plant by a GRG algorithm
601
Table 2. Sensitivity study in the optimal solution for the monochlorobcnzene-plant. The relative stepsize for the determination of the sensitivities was 10e6. The dimension of the sensitivity coefficients follows from Mark/year per dimension of the independent variable in question t
“4 AG *1 t
X.
-
406.9
x6
x5
-
1,139.3
Fig. 4. History of optimization
-
64,974.O
NOMENCLATURE
projection matrix set of feasible solutions search direction objective function vector of inequality constraints number of iteration step set of inequality constraints set of s-active constraints mass flow rate of the stream I temperature of the stream I vector of the decision variables volume flow rate of the stream I Greek symbols E tolerance residence time REFERENCES
T. Umeda, A. Hirai & A. Ichikawa, Synthesis of optimal processing system by an integrated approach. Chem. Engng Sci. 27,795~804 (1972). H. Sadowski, Studies in application the reduced gradient methods for nonlinear programming. Ph.D. Dissert., Technical University Dresden (1974).
x8
xr
- 25,740.O
process (for selected independent
In addition to the positive conclusions on the GRG algorithm given above, one remark should be added. Combining the GRG algorithm with a random search algorithm or a direct search algorithm makes sense if it becomes necessary to determine admissible starting points in the neighbourhood of the global solution as the first step.
1
I
f
f
I
II
- 2.44
variables).
3. Th. Wolfe, On the convergence of gradient methods under constraints. Res. Paper RC-1752, IBM Watson Research Center, Yorktown Heights, New York (1967). 4. P. Faure & P. Huard, Resolution de programmes mathematiques a fonction nonlineaire par la mithode du gradient reduit. Revue Francaise de Recherche Opcrationcllr 36, 1677206 (1965). 5. A. R. Colville, A comparative study on nonlinear programming codes. IBM New York, Scientific Center, Rep. No. 320-2949 (1968). 6. D. M. Himmelblau, Applied Nonlinear Programming. MacGraw-Hill, New York (1972). 7. L. S. Lasdon, A. D. Waren, A. Jain & M. Ratner, Design and testing of a generalized reduced gradient code for nonlinear programming. ACM Trans. Math. Software 4(l), 3450 (1978). 8. L. J. Lafrance, J. F. Hamilton & K. M. Ragsdell, On the performance of a combined generalized reduced gradient and Davidon-Fletcher-Powell algorithm : GRGDFP. Engng Opt. 2,269-278 (1977). 9. L. A. Krumm, Reduced gradient methods in nonlinear programming. In Optimization Methods (in Russian), pp. 1099157. Irkutsk (1975). 10. G. Gruhn, M. Grauer & L. Pollmer, Eine Verfahrensund Anlagenoptimierung mit einem modifizierten COMPLEX-Verfahren. Chem. Technik (1979) H. 12, pp. 603-607. Generalization of the Wolfe 11. J. Abadie & J. Carpentier, reduced gradient method to the case of nonlinear constraints. In Optimization (Ed. R. Fletcher), Academic Press, London (1969). Computational Methods in Optimization. 12. E. Poljak, Academic Press New York (1971). by 13. R. Fletcher & C. M. Reeves, Function minimization conjugate gradients. Compl. J. 7, 149-154 (1964).
602
M. GRAUER etal.
14. M. W. Heuckroth, J. L. Gaddy & L. D. Gaines, An examination of the adaptive random search technique. AICAE-J. 22(4), 744-750 (1976). 15. J. L. Gaddy, Optimization with flowsheet simulation. Int. Congress Contribution of Computers to the Development of Chemical Engineering and Industrial Chemistry, Paris, paper F 8, p. 27 (1978). 16. B. Ch. Wang & R. Luus, Reliability of optimization
procedures for obtaining global optimum. AIChE-J. 24(4), 619-626 (1978). 17. M. J. Box, A new method of constrained optimization and a comparison with other methods. Camp. J. 8(l), 42 (1965). 18. T. Umeda & A. Ichikawa, A modified complex method for optimization. Ind. Engng Chem. Process Des. Dev. 10(2), 229-236 (1971).
Vol.
3. pp.
597-602.
1979
009&l
354!79/040597-06$02,0/O Pergamon Press Ltd.
Paper 12.4 OPTIMIZATION
OF A COMPLEX GRG ALGORITHM
PLANT
BY A
M. GRAUER,* G. GRUHN and L. POLLMER Technical University Leuna-Merseburg, Department of ChemicalEngineering, DDR-42 Merseburg, Geusaer Strasse, German Democratic Republic
Abstract-In the design and control of chemical systems nonlinear optimization problems often have to be solved. For solving these problems an algorithm based on generalized reduced gradients (GRG) is investigated. The paper illustrates the effectiveness of the algorithm by applying it to the optimal design of a monochlorobenzene plant. The results are compared with the application of a direct search method and a random search method. Scope-In the course of optimizing a chemical engineering system, direct search and random search methods are often used. In this paper, we describe the use of a generalized reduced gradient (GRG) method for solving such problems. GRG has recently been established in a number of investigations [S-9] as one of the most effective optimization algorithms available at the moment. Starting from this fact, a GRG algorithm was developed especially for optimization of complex chemical engineering systems. The algorithm solves nonlinear programs with inequalities, using derivatives which are formed numerically. As a basis for comparison, the GRG algorithm was applied to optimization of a plant for monochlorobenzene production (see Fig. 1). The optimization problem consists in finding a cost optimal design for processing 15 kmol/h benzene to monochlorobenzene. The problem has the dimension eight. The same optimization problem was solved by a direct search method and a random method, in order to compare the effectiveness of these methods with that of the GRG algorithm. From the solution with the GRG algorithm, a sensitivity analysis for the respective design problem becomes possible. This is shown with the monochlorobenzene plant as an example. Conclusions and Significance-From the comparison of the GRG algorithm with a direct search method and a random search method the following conclusions can be drawn : In accordance with the literature [4-lo], the GRG algorithm proves an effective method for the solution of small to medium size problems with nonlinear inequalities. Within the GRG algorithm, the Fletcher-Reeves algorithm proved the most effective for finding the search direction; Golden Search proved best for the one-dimensional search. In comparison with a direct search method and a random search method, the GRG algorithm proves at least twice as effective (see Fig. 3). This statement holds true for the solution as well as during the whole optimization process. Of course, the GRG code has the disadvantage of local convergence. For this type of problem, the GRG method is favorable in comparison with other methods using derivatives. In the first place, it is impossible to give the objective function for this design as an explicit function of the optimization variables, because, after giving a set of the independent variables, the state and output variables are obtained only by solving a complicated non-linear transcendental and/or differential equation system, or after calculating a simulation program system. Moreover, the analytical derivative formation can be successfully bypassed with the GRG algorithm, thus excluding possible sources of error. As the GRG algorithm forms its derivatives numerically, a sensitivity analysis of the problem is possible and instructions for the design of the controllers for the plant may be given. With the design of the monochlorobenzene plant as an example, the GRG algorithm proves to be a very effective method for solving chemical engineering optimization problems. INTRODUCTION The GRG algorithm has been developed the following optimization problem min f(x) XER” I
with g,(x) 5 0 The design problem
i = l(l)m
corresponding
*Author to whom correspondence
.
first presented, and then the GRG algorithm and its developed version are described. The results of the application of the algorithm are compared with the results of the solution of the same problem by a direct search method and a random search method and, finally, sensitivity considerations for the plant are given.
for solving
(1)
I
to the task (1) is
The design problem One of the most
should be addressed. 591
inportant
products
of petro-
598
M. GRAUER et al.
chemical industry is phenol. The chlorobenzene method is one of the three industrial processes for its production. The specific cost of phenol produced in this way is high, so it is of the utmost importance to design an optimal cost plant for producing monochlorobenzene. The design of this plant is based on a general reactor-separator-system (Fig. 1) [l]. The following reaction mechanism may be taken as a basis for monochlorobenzene production
equations are defined in the attached nomenclature.) (1) Structure variables xi and x2 Xl
n3
x2
=T-9
+ Cl, + C,H,Cl,
G(x) = E(x) - K(x) + max
(2) with K(x) costs, as a sum of the costs of the input product benzene, for heating and cooling agents and the costs for the equipment E(x) proceeds from the sale of the output product obtained ; the proceeds from the by-products (dichlorobenzene and hydrogen chloride) are used for the reimbursement of the cost of chlorine c(x) profit X vector of the decision variables (structure variables, apparatus dimensions, process parameters). The vector of the decision variables is made up of the following eight components. (Variables in the defining REACTOR-
SYSTEM
n4
=,.
n6 By the variation of these parameters all variants for the reactors giving sense may be obtained. (2) Structure variable x3 fii,,
x3=,.
+ HCl.
Corresponding to Fig. 1, the reaction is carried out in two reactors Rl and R2, FeCl, acting as a catalyzer. For suppressing the formation of dichlorobenzene, the process is run with a partial conversion of benzene. Supposing that the chlorine concentration can be kept constant and the HCl concentration does not disturb the reaction, the reactions can be treated as a consecutive reaction system, A -+ B + C. This assumption is the basis for the determination of reaction kinetics. The unreacted benzene is separated at the head of column Cl and recycled to the reaction step. In column C2 mono- and dichlorobenzene are separated. The nonlinear, constrained optimization problem consists in designing a plant processing a given input stream of benzene (15 kmol/h) with maximal profit. The corresponding objective function is then
of
n2
C,H, + Cl, + C,H,Cl + HCl C,H,Cl
table
nzo
This variable identifies how much of the recycled raw material stream will be used for regenerative preheating. (3) Mean residence time 71 and 72 for the reactors Rl and R2 x4 =
T1
V
=
s,
xg
=
T2
=
V
2.
v3
V5
(4) Reflux ratio for distillation columns Cl and C2 %8
X6=.r
n14
x,=,.
nzo
(5) Temperature E3
n15
difference in the heat-exchanger
x8 = AT, = T,, - Tz3. For the decision variables, the explicit constraints are : 0. SXiIl. 0. 5 x2 5 1. 0. I x3 5 1. 0. $ x4 5 3.0 h
>
(3)
0. I xg 5 3.0 h 0.8 5 X6 s 1.5 0.8 5 x, s 1.5 0. $x8 s 30. K
The mathematical model of the total plant is described in detail in [lo]. It represents a nonlinear system of equations to be calculated iteratively. DISTILLATION-SYSTEM
16
DlCHLORO BENZENE
Fig. 1. The process scheme to be optimized (Rl, R2-reactors; Cl, C2-distillation E7-heat-exchanger; Ml,. . . , M3-mixer; Sl,. , SS-separator).
columns; El,
.. ,
Optimization
of a complex
along d(k). For increasing the efficiency of the GRG algorithm, different methods of approach in the search of direction and minimization along a line were tested. For the search direction, the Davidon-FletcherPowell-algorithm was not used as in [S] because of the high demand for storage, and numerical instabilities. Test calculations proved the Fletcher-Reevesalgorithm [13] to be more effective than the PolakRibiere-algorithm [12], it was used for the design problem. In the search along a line the algorithm of the golden search and the quadratic-interpolation showed best results. The scaling problem in the numerical formation of gradients was solved by division of the respective independent variable by the corresponding upper explicite bound. For comparison, we used a random search method corresponding to [14], that was also used in[lSJ and [ 161. It is an adaptive random search method with range reduction (RANDOM). From the class of direct search methods, a modification of the COMPLEXmethod [17] by the way of[18] is made. This algorithm uses a so-called weighted COMPLEX (GCOBO). These three optimization procedures, are similar in a program-technical way ; thus the input of objective function, constraints and starting point are identical. The specification of the optimization procedure to be used is given only via a corresponding CALLstatement.
The optimization problem corresponding to task (1) is : Maximize the objective function (2) by variation of the eight decision variables, taking into account the constraints in inequality form (3) and the constraints in equation form given in the mathematical model. This optimization problem is solved by a GRG algorithm and the result obtained is compared with the solutions obtained from other optimization procedures. Optimization procedure The algorithm applied for the solution of the problem (1) was given in detail in [2]. Here it should be mentioned again, that corresponding to [4] and contrary to [12] only inequalities are taken into consideration. The approach is based on the conception of the socalled &-strategy. It uses the projection of the antigradient vectors on the kth iteration step for the directional determination. Starting from the index set of the e-active inequalities L,(k) = {j E L 1-E 5 gj(x(k)) 5 0) the nonsingular projection matrix &c(k)) is formed containing all vectors Vg,(x(k)) with jE L,(k). The projection itself is carried out in the following steps (I) and (II)
o(k) = - CA(x(k))Yj (k) =
‘I’WW 2 - e(k)
0, ifjEL,(k)Auj(k)
uj (k) For obtaining a direction pointed D, another modification is made Sj(k) =
into the interior
6(k), ifjEL,(k)hvj(k)
Results and discussion The results obtained from the optimization of the plant for monochlorobenzene production under application of the three procedures GRG, GCOBO and RANDOM are given in Table 1. The corresponding optimal structure of the plant is shown in Fig. 2. For the solution by the three algorithms, the same starting point was used (see Table 1). It becomes quite obvious from Table 1 and Fig. 3 that, in its effectiveness (number of function evaluations and computing time)
of
2 - e(k)
Yj (k)
with 6(k) < 0. The inverse-transformation vides the direction of search d(k) = - A(k)-‘. The stepsize is obtained
599
plant by a GRG algorithm
then pro-
S(k).
as a result of a minimization
Table 1. Starting point and results of optimization of the monochlorobenzene-plant on a CDC-3300 computer. (G-Objective Function, yearly profit; Nl-Number of iteration; NZF-Number of Function Evaluations; CPU-Standard CPU-time by the Colville-constant [S].) OPTIMAL
SOLUTIOH
STARTING by
POMT
optimizationalgorithm
GRG
RAIYDOM
GCOBO
x1
0.80
1 .oOOoo
1.00000
1.00000
x2
0.80
1.00000
1 .oOOOO
1 00000
=3
0.80
1 .ooooo
1.00000
0.92273
l
1.70
0.08162
0.08163
.0.08192
1.40
0.08161
0.08161
0.08202
=6
1.30
0.80000
0.80000
o.aooo4
x7
1.30
0.80000
0.80000
x4
in
x5
in h
x8
in
G in NI NZP CPU
h
K 10%~
13.00 4.0657
0.0006
20.06003
28.06400
28.30196
7.42884
7.42084
7.42760
31.
1,001.
719.
1,973.
0.221
116. 13,179. 0.474
,
2.550 L
600
M.
GRAUER
REACTOR-SYSTEM
et al.
DISTILLATION-SYSTEM
Fig. 2. The optimal
the GRG algorithm is far superior to the other algorithms. This statement is valid for the solution given in Table 1 as well as for any steps of the solution process which may be taken from the analysis of the history of the optimization process (see Fig. 3). For this example, the conclusion is independent of the stopping criteria used, and the time chosen to stop the solution process. If the CRG algorithm is used, there is consequently at least a doubling of the effectiveness of the search for an optimum compared with the other algorithms. A disadvantage of the GRG is its property of local convergence. The search did not achieve the global solution from all starting points. This disadvantage is specific to the algorithm, but does not seem too serious because, as a rule, there is enough known, as a result of prior investigations on modelling and simulation, to define an admissible starting point in the neighbourhood of the global solution. An advantage of using the GRG algorithm is the existence of numerical derivatives and, thus, the sensitivities of the objective function against changes of the independent variables. In Table 2, some of these sensitivities are given at the optimum of the plant. It can be seen clearly that the optimal design of the plant shows the highest sensitivity to the reflux ratios (x6, x7)
process scheme.
of the columns Cl and C2. Between these control parameters, the reflux ratio (x6) of column Cl has the greatest influence, because it determines mass streams and the energy conditions of the plant in a stronger way than the reflux ratio (x,) of the second column. The second level of sensitivity is found in the residence times (x4, x5) of the two reactors Rl, R2. The objective function is least sensitive to the temperature difference (xs) in the heat-exchanger. These results may be used for model modifications and are a safe basis for designing the controllers for the individual process parameters. They are statements on the accuracy of control of the mass stream, temperature or pressure in question. It should also be mentioned that these sensitivities are not only calculated at the solution point, but are obtained automatically at each point in the optimization if the GRG algorithm is used. The curves of Fig. 4 show when the individual variables reach their optimal value in the course of the solution. This example proves the intuitive comprehension of a hierarchy of the three groups of variables (variables of structure, apparatus dimensions, control parameters). Structure variable x1 is the first to reach its optimal value. The apparatus dimension x5 and the process parameter x8 follow.
0,96
0,56 0,54
0
20
40
Fig. 3. History
60
110 160 120 100 NUMBER OF FUNCTION EVALUATIONS
60
of optimization
process
(for the objective
180
200
function).
220
2LO
Optimization
of a complex
plant by a GRG algorithm
601
Table 2. Sensitivity study in the optimal solution for the monochlorobcnzene-plant. The relative stepsize for the determination of the sensitivities was 10e6. The dimension of the sensitivity coefficients follows from Mark/year per dimension of the independent variable in question t
“4 AG *1 t
X.
-
406.9
x6
x5
-
1,139.3
Fig. 4. History of optimization
-
64,974.O
NOMENCLATURE
projection matrix set of feasible solutions search direction objective function vector of inequality constraints number of iteration step set of inequality constraints set of s-active constraints mass flow rate of the stream I temperature of the stream I vector of the decision variables volume flow rate of the stream I Greek symbols E tolerance residence time REFERENCES
T. Umeda, A. Hirai & A. Ichikawa, Synthesis of optimal processing system by an integrated approach. Chem. Engng Sci. 27,795~804 (1972). H. Sadowski, Studies in application the reduced gradient methods for nonlinear programming. Ph.D. Dissert., Technical University Dresden (1974).
x8
xr
- 25,740.O
process (for selected independent
In addition to the positive conclusions on the GRG algorithm given above, one remark should be added. Combining the GRG algorithm with a random search algorithm or a direct search algorithm makes sense if it becomes necessary to determine admissible starting points in the neighbourhood of the global solution as the first step.
1
I
f
f
I
II
- 2.44
variables).
3. Th. Wolfe, On the convergence of gradient methods under constraints. Res. Paper RC-1752, IBM Watson Research Center, Yorktown Heights, New York (1967). 4. P. Faure & P. Huard, Resolution de programmes mathematiques a fonction nonlineaire par la mithode du gradient reduit. Revue Francaise de Recherche Opcrationcllr 36, 1677206 (1965). 5. A. R. Colville, A comparative study on nonlinear programming codes. IBM New York, Scientific Center, Rep. No. 320-2949 (1968). 6. D. M. Himmelblau, Applied Nonlinear Programming. MacGraw-Hill, New York (1972). 7. L. S. Lasdon, A. D. Waren, A. Jain & M. Ratner, Design and testing of a generalized reduced gradient code for nonlinear programming. ACM Trans. Math. Software 4(l), 3450 (1978). 8. L. J. Lafrance, J. F. Hamilton & K. M. Ragsdell, On the performance of a combined generalized reduced gradient and Davidon-Fletcher-Powell algorithm : GRGDFP. Engng Opt. 2,269-278 (1977). 9. L. A. Krumm, Reduced gradient methods in nonlinear programming. In Optimization Methods (in Russian), pp. 1099157. Irkutsk (1975). 10. G. Gruhn, M. Grauer & L. Pollmer, Eine Verfahrensund Anlagenoptimierung mit einem modifizierten COMPLEX-Verfahren. Chem. Technik (1979) H. 12, pp. 603-607. Generalization of the Wolfe 11. J. Abadie & J. Carpentier, reduced gradient method to the case of nonlinear constraints. In Optimization (Ed. R. Fletcher), Academic Press, London (1969). Computational Methods in Optimization. 12. E. Poljak, Academic Press New York (1971). by 13. R. Fletcher & C. M. Reeves, Function minimization conjugate gradients. Compl. J. 7, 149-154 (1964).
602
M. GRAUER etal.
14. M. W. Heuckroth, J. L. Gaddy & L. D. Gaines, An examination of the adaptive random search technique. AICAE-J. 22(4), 744-750 (1976). 15. J. L. Gaddy, Optimization with flowsheet simulation. Int. Congress Contribution of Computers to the Development of Chemical Engineering and Industrial Chemistry, Paris, paper F 8, p. 27 (1978). 16. B. Ch. Wang & R. Luus, Reliability of optimization
procedures for obtaining global optimum. AIChE-J. 24(4), 619-626 (1978). 17. M. J. Box, A new method of constrained optimization and a comparison with other methods. Camp. J. 8(l), 42 (1965). 18. T. Umeda & A. Ichikawa, A modified complex method for optimization. Ind. Engng Chem. Process Des. Dev. 10(2), 229-236 (1971).