Ind. Eng. Chem. Res. 1994,33, 2123-2133
2123
Solution of a Class of Multistage Dynamic Optimization Problems. 2.
Problems with Path Constraints
V. S. Vassiliadis, R. W. H. Sargent, and C. C. Pantelides'
Centre for Process S y s t e m Engineering, Imperial College of Science, Technology and Medicine,
London SW7 2BY, United Kingdom
This paper considers the treatment of general equality and inequality path constraints in the context
of the control vector parametrization approach t o the optimization of dynamic systems described
by mixed sets of differential and algebraic equations (DAEs) of index not exceeding 1. Equality
path constraints are handled by incorporating as many of them as possible within the DAE system
itself without increasing its index. This allows a subset of the control variables to be determined
from the solution of the augmented DAE system. The issues involved in establishing an appropriate
partitioning of the control variable vector are examined. Inequality path constraints are handled
through the combined application of the discretization of these constraints at a finite number of
points, and forcing an integral measure of their violation to zero. Numerical experiments
demonstrating the advantages of this hybrid technique over its individual components are presented.
1. Introduction
The first part of this paper (Vassiliadis et al., 1994)
presented an algorithm for the optimization of systems,
the transient behavior of which is described by multistage
sets of differential and algebraic equations (DAEs) of the
form
f&,
t ) = 0 t E B k - 1 , tk),
k = 1, ..., NS- 1; t E ItNs-1, t J , k NS (1)
where NS is the number of stages and t is time. The sets
f , Y , 4 u,
of differential and algebraic variables are denoted by x ( t )
E X C Rn and y ( t ) E Y C Rm,respectively, with f being
the time derivatives of x ( t ) . u ( t ) E U C R* and u E V
C Rq are sets of time-varying control variables and timeinvariant parameters, respectively, and f k : X X Rn X Y X
U X V X [tk-l, tk) R"+m.
A variety of constraints enforced at a finite set of distinct
time points (corresponding to the stage boundaries) during
the time horizon of interest were considered. These
included stage end-point constraints, as well as general
initial and junction conditions.
In this second part of the paper, we turn our attention
to the handling of equality and inequality constraints that
need to be enforced over finite time intervals. We assume
that the latter coincide entirely with one or more stages,
defining additional artificial stages for this purpose, if
necessary. We are therefore interested in stage path
constraints of the form
-
c F ( x ( t ) , fW, Y W ,
w ,t ) = 0
0,
k = 1,...,NS - 1;
where
cp: X
cF(x(t),
X
Rn
X
Y
X
t
E It,-,,
[tk-l, t k )
We begin this section by considering established approaches to dealing with equality path constraints and
pointing out some numerical problems that may arise from
their application. We then proceed to consider a uniform
manner of treating all path equalities (including the
original DAE system (1))in the optimization problem,
analyze the complications that may arise, and propose a
technique that circumvents them. A simple example
illustrating the application of this technique is presented.
2.1. Established Techniques for Handling Equality
Path Constraints. A widely used method involves the
introduction of integral penalty terms in the objective
function to be minimized (Bryson and Ho, 1975). In the
context of the multistage problem considered here, the
penalty term would be of the form
NP
(4)
-
RXh,
and
w ,Y W , w ,
u, t ) 5 0
k = 1,...,NS - 1;
2. Equality Path Constraints
tk),
t E [tNgl, tr], k = NS (2)
UX VX
Section 2 of this paper considers equality path constraints (2), attempting to establish a unified manner of
treating all path equalities (including both (1)and (2)) in
the problem, while paying special attention to issues
relating to the index of the resulting DAE systems.
Section 3 considers inequality path constraints (3),
proposing a hybrid technique which combines elements
of earlier approaches for handling such constraints.
Numerical results demonstrating the effectiveness of this
technique are presented in section 4.
t E ltk-1, tk),
t E [t,,,, t f ] , k = NS (3)
-
where c$ X X Rn X Y X U X V X [&-I, t k ) Rxk. In
particular, we are concerned with the effective and efficient
handling of such path constraints in the context of control
vector parametrization algorithms.
* Author to whom correspondence should be addressed.
Electronic mail address: [email protected].
where K is a large positive constant.
An alternative approach that seeks to avoid the numerical difficulties which may be caused by the use of the
penalty term involves the conversion of the path constraint
to an equivalent end-point constraint (Sargent and Sullivan, 1977). In the context of the multistage problem,
this would involve the introduction of a new differential
variable Z ( t ) defined through the stage differential equations
3 = Ilcpcxct,, f ( t ) ,Y ( t ) ,u ( 0 ,u, t)l1,2, t E [ t k - , , t k ) ,
k = 1,..., NS - 1; t E [ t N g 1 , tfl, k = NS (5a)
and the initial and junction conditions
1994 American Chemical Society
2124
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
R(tk)
= Z(t,),
k = 1, ...,NS - 1
(5c)
The path constraints (2) are then equivalent to the single
end-point constraint
Z(tf) = 0
(6)
In some cases, if there are significant differences in the
magnitude of the different elements of the path constraint
vector cp(.), and/or of the same element in different
stages, better numerical behavior could be ensured by
disaggregating 2 into measures of the square violation of
individual path constraints in individual stages, and
imposing a separate end-point constraint on each one of
them.
A common characteristic of all the techniques discussed
so far is that the use of square norms implies that the
penalty terms or end-point constraints introduced have
zero gradients with respect to the optimization parameters
at the solution. This, in turn, may result in a reduced
convergence rate near the solution, and as noted by Goh
and Teo (19881, the success of such techniques depends
very strongly on the line-search merit function used by
the optimization algorithm. Interestingly, the approach
described later in this paper for dealing with inequality
path constraints can also be used to alleviate these
problems (see concluding remarks in section 5).
2.2. A Unified Treatment of Path Equalities. An
alternative way of handling equality path constraints (2)
is based on the observation that there is, in principle, no
mathematical difference between these and the system
equations (1) even if, in practice, their origins may be
different. For instance, eqs 1 would normally describe
the underlying physics of the system under consideration,
while constraints 2 could express desired properties to be
attained under the influence of the controls u(t).
We can therefore view the combined set of equations
(1)and (2) as a DAE system to be solved simultaneously.
The addition of hk new equations to the system removes
an equal number of degrees of freedom from the problem,
effectively converting hk elements of the control vector
u ( t ) into algebraic variables, which, like y ( t ) , will be
determined by the solution of the combined DAE system.
The remaining elements of u(t) continue to be treated as
control variables, the time variation of which will be
established by the optimization. We denote the two types
of control variables as &(t) and &(t) respectively. It
should be noted that this partitioning of the u ( t ) may
differ from stage to stage.
However, the above simple approach suffers from a
number of complications:
(a) The combined DAE system may be of index
exceeding unity (see Brenan et al. (1989) for a discussion
of DAE index). In fact, this will always be the case if one
or more of the constraints (2) is a function of the differential
variables x ( t )only. Unfortunately, the numerical solution
of most high index problems is much more difficult than
that of problems of index 0 or 1.
(b) Unless hk = T , there mayhe more than one possible
partitioning of u into and i i k . Different choices may
lead to a different index for the combined DAE system.
(c) General solvability conditions for high index problems have not yet been established. Therefore, it may be
difficult to guarantee that, for a given partitioning of the
u ( t )vector, solutions ( x ( t ) , Y(t)Liik(t))
exist for d l possible
variations of the elements of
(d) In many applications, the space of practically
implementable control functions may be restricted (e.g.,
to be piecewise constant or linear) for certain elements of
u(t). As the solution of DAE systems do not, in general,
belong to such function spaces, the cqrresponding control
variables must be delegated to the i i k partition of u.
In view of the above complications, it may be inevitable
in some cases that some of the constraints (2) must be
handled by the optimization using one of the already
established techniques of section 2.1. Nevertheless, we
seek to establish, for every stage k, a maximal subset of
the constraints that can be treated as part of the DAE
system without giving rise to the above problems. An
appropriate partitioning of u ( t )
(7)
also needs to be established.
Let EL@) be the subset of the control variables u which
can-take any functional form, and denote its cardinality
by T . Clearly, the partition will be a subset of EL, while
i i k will be a superset of its complement. We now form the
matrix
and denote its rank by r k .
Through the application of a set of row and column
pivoting operatings, we effect the transformation
where Lk is a lower triangular matrix with unit diagonal,
k is a rk X rk upper triangular matrix.
and u
Assuming that the DAE system (1)is of index 1, with
the matrix
being nonsingular, we can guarantee that
r,Ln+m
(11)
and it is then possible to select the permutation matrices
Pk and Qk to be of the special structure
where Pk' and Qk' are ( n + m ) X (n m! permutation
matrices, and Pk" is a hk X hk and Qk" a a X T permutation
matrix. This has the effect of ensuring that row pivoting
takes place separately among the rows corresponding to
f k ( ' ) and cp(.),and also that column pivoting takes place
separately among the columns corresponding to {x,y ) on
one hand and to EL on the other. Thus the rows and columns
in the null submatrix 0 in (8)always correspond to subsets
of cp(.) and EL, respectively.
Assuming that Pk and Qk are such that it is everywhere
possible to determine a matrix Lk that will transform Ak
in the manner indicated by (91, then the DAE system
formed from (1) and the first ( r k - n - m) elements of
Pk"cp(') has a global index of 1with respect to a variable
vector comprising x , y , and the first (rk - n - m ) elements
of Qk"G. Therefore, this system can be solved to establish
Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994 2125
a unique time variation of these variables for any given
variation of the rest of the control variables u.
In the context of the control vector_parametrization
algorithm considered here, the last (a - rk + n + n)
elements of &''a together with any elements of u not in
Ei will be manipulated directly by the optimization
procedure. Also the last ( x k - rk + n + m) elements of
cF(.)will have to be treated using one of the techniques
of section 2.1.
2.3. A Simple Example. We consider the following
linear DAE system comprising three equations, and two
differential, one algebraic, and three control variables:
kl
-kl + x1 + x 2 + 2y1 + u1= 0
(13a)
+ 3X1 + 2X2- y1+ 3U1+ 5U2 = 0
(13~)
We now impose the following two equality path constraints on the solution of the above system:
+ x2-
x1
1= 0
(14a)
2x1 + x 2 - 2u1 - u3 = 0
A=
(: 0'
2
1
0
J'
1
!
0
0
0
0
2
0
0
- 2 0
0
")
2
0
-1
0
-1
0
1
3.1. Established Techniques for Handling Inequality Path Constraints. Most of the established approaches for dealing with inequality path constraints are
similar to the techniques for handling equality path
constraints presented in section 2.1. Thus, they rely on
defining a measure of the constraint violation over the
entire horizon, and then penalizing it in the objective
function, or forcing it directly to zero through an endpoint constraint.
One possible measure of the violation of a scalar
constraint (3) is
where the exponent f is typically either 1or 2. Although
f = 1is sufficient, using f = 2 has the advantage that the
function (max(0,w))r has first-order continuity at w = 0.
Vectors of path constraints can be handled by defining
either a separate violation for each element of the vector,
or a combined violation for the entire vector. A possible
measure for the latter case is
(15)
-1
1
0
0
0 0
using the permutation matrices
1
l 70
3. Inequality Path Constraints
0
which can be transformed to
- 1 0
latter will be treated as an algebraic variable to be
determined by the solution of the augmented DAE system.
On the other hand, the optimization procedure will
manipulate directly the remaining control variables, ul(t)
and u3(t),with constraint 14a being handled using one of
the techniques of section 2.1.
(14b)
Furthermore, we assume that no restrictions exist on the
functional form that any of the three control variables can
take. Thus, in this case, Ei 1 u.
To apply the procedure of section 2.2, we form the
Jacobian matrix of the combined system (13) and (14)
with respect to {kl,k2, y1, u1,u2,u3):
- 1 0
P and Q suggests that this augmentation will involve path
constraint 14b and the second control variable u&). The
0
0
where [crlj denotes the j t h element of cp.
It is well recognized that a major common disadvantage
of all the above techniques arises from the use of the max
operator in the violation measures (18) and (19). More
specifically, when the constraint is inactive, both the
measures and their gradients with respect to all optimization parameters are zero. Thus no useful information can
be conveyed to the optimizer regarding the proximity of
the current point to the boundary of the feasible region.
Overall, this may result in inefficient behavior involving
excessive oscillations between feasible and infeasible
choices of the optimization parameters in successive
optimization steps or during the line-search procedure.
An alternative way of handling inequality path constraints (Jacobson and Lele, 1969) involves the introduction of squared slack variables s which convert the
inequalities to equalities. Thus a scalar constraint (3) is
modified to
c F ( x ( t ) , k ( t ) , y(t), u ( t ) , u, t )
0
and the lower triangular matrix
L=
0
)
1
1
(17c)
-4
1
0
0
0 1
Clearly the Rank of A is 4, and hence the original DAE
system (13)can be augmented by one more equation and
one more variable. The form of the permutation matrices
(!
-2
0
0
+ s2 = 0
(20)
In principle, s ( t ) could be treated as a control variable to
be determined by the optimization while (20) could be
appended to the original DAE system (11, effectively
determining the time variation of one of the control
variables u ( t ) .
In practice, this technique suffers from one major
disadvantage. In the very common situation of (3)
involving only differential variables x ( t ) , the augmentation
of (1) by (20) results in a high-index DAE system which
must then be reduced to an index 1problem by a procedure
involving repeated differentiations of (20) (and, possibly,
some of the equations in (l)),
and algebraic manipulations.
2126 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
If the index of the system is K , then ( K - 1)differentiations
are required, introducing time derivatives $, s, ..., s+l).
The latter is then treated as the control variable to be
manipulated by the optimization, while the lower order
derivatives and s ( t ) itself are treated as distinct variables,
related through equations of the form
Zj(0) = 0, j = 1,
e..,
(24b)
Xk
junction conditions
Zj(tk)
= Rj(ti), j = 1,
..e,
X,, k = 1, NS - 1 ( 2 4 ~ )
e..,
and end-point constraints
Zj(tf) Ie, j = 1, ...,i;,
3.2. A Hybrid Approach to Inequality Path Constraints. Integral expressions of the type shown in eq 18
provide a single measure of the violation of an inequality
path constraint over the entire time horizon of interest,
permitting, for instance, the replacement of each such
constraint by a single end-point constraint (cf. eq 6).
The techniques discussed in section 3.1 are particularly
convenient for handling inequality path constraints in the
context of the control parametrization approach to the
solution of dynamic optimization problems. On the other
hand, the complete discretization approach, of the type
advocated, for instance, by Cuthrell and Biegler (1987)
and Logsdon and Biegler (19891, allows a much simpler
treatment of such constraints. Thus, each inequality path
constraint is enforced as a point inequality constraint at
all the points considered by the discretization, in exactly
the same manner as the system equations (1).
It should be noted that, even within control vector
parametrization algorithms, it is in principle possible to
adopt an approach to inequality path constraints similar
to that of complete discretization algorithms. Thus, the
path constraints (3) could be replaced by
i = 1,2, ...,
k = 1, ...,NS (22)
where Tik E [ t k - l , tk] are a given set of points in stage k .
The point constraints (22) and their gradients with respect
to the optimization parameters can readily be evaluated
through the integration of (1)and the sensitivity equations
(see Vassiliadis et al., 1994).
The disadvantage of the approach described above is
that a rather large number of points Tik might be necessary
to ensure that the path constraints (22) are actually not
violated between consecutive Tik. We therefore adopt a
hybrid approach that combines the advantages of the
compactness of representation of (3) afforded by the use
of the integral violation measures together with the
increased information provided to the optimizer by the
point constraints (22).
More specifically, we replace (3) by a set of point
constraints at the stage boundaries only
c F ( x ( T i k ) , *(Ti,),
Y ( T i k ) , u ( T i k ) , u, Ti,)
CF(x(tk), i ( t k ) , Y(tk), U(tk),
50
u, t k ) 5 0
k = 0, NS (23a)
in addition to an end-point constraint forcing the integral
violation of each element of the constraint vector (3) to
zero. This is achieved by introducing a new differential
variable R for each inequality path constraint, defined
through
with initial conditions
(244
The use of a small nonnegative constant e on the righthand side of (24d) to relax the end-point constraint (6)
has been suggested by Walsh (1993), and has been found
to lead to significant performance improvements in some
cases. In any case, it is assumed that the number of
inequality path constraints remains the same from stage
to stage although their functional form may change. The
slight complication arising if this is not the case can readily
be dealt with at the implementation level.
In summary, each inequality path constraint requires
the introduction of an extra simple ordinary differential
equation (ODE) (24a), and is converted to- 2NS point
constraints (23) and one end-point constraint (24b). The
motivation for this is that (23) will provide some guiclance
to the optimization during the search for the solution,
while (24b) will ensure that the final solution satisfies the
constraint everywhere within the DAE integration tolerance and the constant e.
4. Numerical Experiments with Inequality Path
Constraints
The hybrid technique presented in section 3.2 has been
implemented within the DAEOPT dynamic optimization
code (Vassiliadis, 1993a). In this section, we demonstrate
its effectiveness through results obtained by applying
DAEOPT to three dynamic optimization problems involving inequality path constraints.
The first example is derived from a well-known optimal
control test problem, while the other two are chemical
engineering problems concerning the optimal operation
and design of chemical reactors.
4.1. Constrained van der Pol Oscillator. We use
the form of this problem employed by Gritsis (1990).This
can be written as
min x3(5)
(25)
subject to
i1
= (1- x?)xl
2, = x12
- x2 + u
+ x; + u2
@a)
(26~)
with the initial conditions x ( 0 ) = [O,1,OIT,and the control
variable u ( t ) restricted between lower and upper bounds
of -0.3 and 1.0, respectively.
To the above, we add the inequality path constraint
x l ( t ) L - 0.4, t E [O,51
(27)
For the purposes of the solution using DAEOPT, the
time domain [0, 51 was divided into 10 control intervals,
the lengths of which were bounded between 0.05 and 4.5.
A piecewise linear approximation to the control variable
u ( t ) was employed. The optimization was started with a
uniform control profile u ( t ) = 0.7, t E [O, 51.
The above problem was solved both without and with
constraint 27. For the unconstrained problem, an optimal
value of 2.868 75 was obtained for the objective function,
Ind. Eng. Chem. Res., Vol. 33, No. 9,1994 2127
Van der Pol Oscillator
:I
0.90 1.00
0.70
am -
am -
t I
050 0.40 0.60
0.30
-
0.20
-
0.10
-
4.00
-
4.10
-
-
Van der Pol constr.
4.mL I
4.m -
-
1
\
i
4.30
4.30
I
I
I
I
I
0.00
1.M
2.00
3.00
4.00
I 1
0.00
5.00
Figure 1. Control variable profile for unconstrained van der Pol
problem.
I
1.00
I
3.03
I
ZW
I
I
4.00
5.00
t
tirns
Figure 3. Control variable profile for constrained van der Pol
problem.
-
Van der Pol Oscillator
Van der Pol constr.
0.w 1.00
0.80
-
0.70
-
0.60
-
0.M 0.40
-
0.30
-
0.10
0.20
4.00
-
4.10 4.20
-
4.30 4.40-
4.500.00
1.00
2.00
3 .00
4.00
5.00
-.-
1 1
0.00
I
1.00
I
I
zoo
3.00
I
4.00
5.00
Figure 2. Differential variable profiles for unconstrained van der
Pol problem.
Figure 4. Differential variable profiles for constrained van der Pol
problem.
which compares well with the value of 2.868 reported by
Gritsis (1990). The corresponding profiles of the control
and differential variables are shown in Figures 1 and 2,
respectively.
We now consider the solution of the constrained
problem. For comparison purposes, we solve this problem
with three different ways of handling the inequality path
constraint (27). In particular, we employ (a) the hybrid
technique of section 3.2,(b) the integral violation measure
(cf. eqs 24)only, and (c) the discretized constraint at the
11 stage (i.e., control interval) boundaries (cf. eq 23)only.
A value of e = lo-' (cf. eq 24b) was used for (a) and (b).
Run c fails to produce a solution that satisfies the
constraint within the control intervals themselves. On
the other hand, the other two runs produce very similar
results, with the path constraint being satisfied everywhere.
The optimal values of the objective function (2.95474and
2.954 36 for runs a and b, respectively) again compare well
with the value of 2.960 reported by Gritsis (1990). The
corresponding profiles of the control and differential
variables are shown in Figures 3 and 4,respectively.
Detailed computational satistics for the unconstrained
and the three constrained runs are shown in Table 1. By
comparing the results for the two constrained runs a and
b that produce valid solutions to the problem, it can be
seen that the use of the hybrid technique greatly improves
the performance of the algorithm-in fact, the computational cost of (a) is comparable to that of (c) while
ensuring solution feasibility.
A somewhat less expected result is that the cost of solving
the constrained problem using the hybrid technique is
less than that of solving the (theoretically simpler)
2128 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
Table 1. Computational Statistics for van der Pol
Oscillator Problem.
run
QP
LS
20
30
unconstrained
constrained (a)
16
25
constrained (b)
34
51
constrained (c)b
15
20
Table 2. Kinetic and Thermodynamic Data for Batch
Reaction Problem
CPU
105.0
99.8
217.4
94.4
QP, number of quadratic programming subproblems solved by
the optimization; LS, number of function evaluations during line
search; CPU, CPU seconds on SUN SPARCStation 2 workstation.
b Solution fails to satisfy path constraint.
unconstrained problem. This effect has, in fact, been
observed in a large part of the extensive test problem set
examined by Vassiliadis (1993b). One possible explanation
is that the incorporation of the discretized constraints (23)
restricts the feasible region, thus limiting the optimization
search for some problems.
4.2. Optimization of Batch Reactor Operation.
Here we consider the optimal operation of a batch reactor
used to carry out the chemical reaction
A+B-C
in the presence of the side reaction
B+C-D
Both reactions are strongly exothermic, and therefore
direct mixing of the entire necessary amounts of the
reactants A and B must be avoided. Instead, the operation
is performed by initially charging an amount of A in the
reactor vessel, with B being added slowly during the
reaction. The reactor vessel is fitted with a cooling water
jacket which is used to remove the generated heat of
reaction.
Under a suitable set of assumptions, the transient
behavior of the above system is described by the following
mathematical model:
r;lA= -Vr,
(28a)
reactionj
1
2
Reaction Data
Ej (K)
0.008
3000.0
0.002
2400.0
Aj (m3/(mol-s))
A H j (kJ/mol)
-100.0
-75.0
Component Data
(mol/m3) ai (kJ/(moEK)) @i (kJ/(mol.K2))
11250
0.1723
0.474 X 10-9
B
16 OOO
0.2000
0.500X 10-9
C
10 400
0.1600
0.550X 10-9
D
10 OOO
0.1550
0.323 X 10-9
T,r = 298 K
hp = 20 kJ/mol
component i
A
pio
of components i (=A, B, C or D) in the system, while MT
denotes the total molar holdup. The total energy holdup
is denoted by H and the temperature by T. The rates of
the two reactions are denoted by rj, j = 1, 2, and the
corresponding rate constants by kj. The volume of liquid
in the system is denoted by V , and an ideal mixture
assumption is used to calculate the liquid density and
enthalpy (cf. eqs 28i and 281). The values of the necessary
kinetic and thermodynamic constants are given in Table
2.
At the start of processing, the reactor vessel contains
9000 mol of pure A at a temperature of 350 K.
It is desired to operate the above system so as to
maximize the production of component C by manipulating
the rate of addition of B and the external cooling load, i.e.,
FBand Q. Upper bounds of 10 mol/s and 1000 kW are
imposed on these two control variables, respectively. In
order to ensure that the reactions are sufficiently quenched
without excessive subcooling, the final temperature must
lie in the narrow range [295 K, 300 Kl. However, the
overall duration of the reaction is left free to be determined
by the solution of the optimization problem which can be
written as
m u
tf&dt),Q(t),tE[o,
td
MC(tf)
(29a)
subject to the DAE constraints (28) and also
0 IFB(t) I10, t E [o, tf]
(29b)
0 IQ(t)5 1000,t E [O, tfl
(29c)
295 IT(tf) 5 300
(29d)
For safety reasons, we also impose an upper bound of
520 K on the temperature attained during the reaction.
This introduces the path constraint
V =MAa +MBT +MCa +MDT
PA
PB
PC
(28i)
PD
In the above model, Mi, Xi, and Ci denote respectively
the molar holdup, mole fraction, and molar concentration
T(t) 5 520, t E [0, tf]
(29e)
The above problem was solved in DAEOPT without
and with the path constraint (29e). In both cases, five
control intervals were employed, with piecewise constant
controlvariables FB(t) and Q(t). The length of the control
intervals was bounded between 60 and 1800 s. The
optimization was initialized with each of the five control
intervals having a length of 200 8, with &(t) = 5 mol/s and
Q ( t )= 0 for all t in the time horizon.
The solution of the unconstrained problem results in an
optimal processing time of 2690 8, with the final amount
of C being 3804 mol. The control variable profiles are
shown in Figure 5. The optimal operating policy essentially
involves adding a constant flow rate of B for approximately
the first half of the processing time. The supply of B is
then switched off, and a practically constant cooling load
Ind. Eng. Chem. Res., Vol. 33, No. 9,1994 2129
Batch Rcoctor
I10 3
9m
8m
im
6m
sm
4m
3m
100
im
......._......
J. _ ......._...
J.__. A.._.__.
i.....-...-.i _..-... b.10
i
.I
i
om
S
om
1.m
100
~ Q I o l m o o u o z m u o
Batch Reactor
(a) Feed rate of component B
Batch Reactor
6ylm
_..........................
....
:"'
.
aom
~
p-'
nom
p-'
smm
F-.
I
I
a m
!
.-a
!
p-.
-7I
L...
mm
i
!
'-7
j
I-.
..-.
i
-7
C".
...d
som
xom
i
I
j
i
:...A
................i....._...............-..._................. 1.......-......i!L--i
..........!hn 10
am
as
I
.
!
;
IAO
is
am
3
YS. O
(b) Cooling load
Figure 5. Control variable profiles for unconstrained batch reaction
problem.
is applied to bring the temperature of the reactor contents
down to the required level at the end of processing. Figure
6 shows the evolution of the molar holdups and the
temperature. It can be seen clearly from the former that
the reactions effectively terminate as soon as the addition
of B is stopped. Also the reactor temperature satisfies the
end-point constraint (29d), but exceeds the safety limit of
520 K over part of the horizon.
We now re-solve the above problem, this time imposing
the path constraint (29e) handled (a) by the hybrid
technique of section 3.2, and (b) by the simple integral
violation measure on ita own without the use of any
constraint discretization. The objective function values
obtained by the two techniques are identical to four
significant digits, corresponding to the production of 3630
mol of C. Thus the imposition of an upper limit on the
om
1.00
am
Figure 6. Molar holdup and temperature profiles for unconstrained
batch reaction problem.
operating temperature causes a decrease of about 4.6% in
the amount of C produced for a constant initial charge of
A. The optimal processing time is also reduced to
1936 s.
Figure 7 shows the optimal control variable profiles,
and Figure 8 shows the reactor molar holdups and
temperature. It can be seen that, in this case, there is a
period with simultaneous addition of B and a nonzero
cooling load, which practically coincides with the period
during which the inequality path constraint is active.
Some computational statistics for the three runs performed are shown in Table 3. The significant computational advantage of the hybrid technique (a) over its
simpler counterpart (b) is again evident.
4.3. Steady-State Optimization of Tubular Reactor
Design. This problem, which was also considered by
Vasantharajan and Biegler (1990),involves two reactants,
A and B, fed into a tubular reactor of length L in the ratio
1:3 to undergo an exothermic reaction. The reactor is
jacketed, and the heat removed is used to produce steam.
The feed stream is preheated through heat exchange with
the exit stream, as shown in Figure 9.
2130 Ind. Eng. Chem.
Res.,Vol. 33, No.9,1994
Batch R u d o r
BStChRW
.I0
i
I
1'
r
I
s
9
_i
Ec
4-
-.ii
I
!i
r+-
A
i
-4I
I
b.10
om
OJO
im
1
zm
1-
Bakb R u d o r
am
Lo
,
I
I=
u
@) cooling load
Figure 7. Control variable profilea for constrained batch reaction
problem.
Table 3. Compntational Statistics tor Batch m i o n
Problem.
Nn
unconstrained
constrained (a)
constrained (b)
QP
Is
34
35
954.8
42
53
48
60
1266.3
om
OJO
im
1W
Figure 8. Molar holdup and temperature profdea for constrained
bateh reaction problem.
hl
CPU
1664.8
QP. number of quadratic programming subproblems solved by
the optimization; IS,number of function evaluations during line
search;CPU,CPUsefondsonSUN SPARCStation 10141workstation.
a
The variation of the conversion q along the length of
the reactor under steady-state conditions is described by
the equation
q = 0.3(1- q)em1-'lh
(30~)
where q denotes the derivative of q with respect to the
--
10.
..c
Figure 9. Exothermic tubular reactor.
axial position z, and T is the normalized temperature,
defined as the ratio of the temperature to the inlet
temperature TR. T i s determined by the energy balance:
T = 1.5(T- TJT,)
+ (2/3)q
(30b)
Finally, the rate of steam production is proportional to
Ind. Eng. Chem. Res., Vol. 33, No. 9,1994 2131
Table 4. Design Parameters for Reaction Problem
init
est
250.0
462.23
1.0
425.25
oarameter
TP
TR
L
TS
lower
bound
120.0
400.0
0.5
400.0
unconstrained problem
this
Vasantharajan
work
and Biegler
184.0
188.4
500.0
500.0
1.25
1.25
474.0
470.1
-171.4879
-171.2340
upper
bound
1Ooo.o
500.0
1.25
500.0
obj function value
constrained problem
this
Vasantharajan
work
and Biegler
213.6
232.1
500.0
500.0
1.25
1.25
451.5
450.5
-152.8518
-148.0847
the temperature difference between the reactor and the
steam jacket, and the associated profit is assumed to be
given by
P = JL(TRT- T,) dz
which can be written in differential form as
P = TRT- T,
(30~)
The initial conditions associated with equations (30) are
q(0) = 0; T(0)= 1; P(0) = 0
(31)
The operation of the feed preheater is governed by a
simple steady-state energy balance of the form
TR - TF = TRT(L)- Tp
(32a)
where the feed temperature, T F is
, assumed to be 110 O C .
Note that T(L) corresponds to the exit (normalized)
temperature from the reactor. A minimum approach
temperature of 10 "C is also specified through the
constraint
TRT(L)I TR + 10
(32b)
The objective of the optimization is to determine the
values of the four design parameters L, TR,Tp, and Ts
that minimize the cost of the reactor. The latter takes
account of the capital cost, assumed to be proportional to
the length L, and the revenue P, leading to the objective
function:
min L - P(L)
L,TRTP,Ts
(33)
This constitutes a dynamic optimization problem with
four time-invariant parameters and no control variables.
The axial position z and the length of the reactor L
correspond to "time" t and "final time" tf.
The optimization was carried out using the starting point
and bounds on the optimization parameters employed in
the earlier study by Vasantharajan and Biegler (1990).
These and the optimal values obtained at optimization
respectively
and integration tolerances of 10-5 and
are shown in Table 4. The corresponding objective
function is -171.4879. Vasantharajan and Biegler report
an objective function of -171.4600; however, inserting the
optimal values of the parameters that they report into eqs
30and integrating them a t an integration tolerance of
yields a slightly worse objective of -171.2340.
The variation of the conversion and the temperature
along the lengthof reactor for the problem described above
is shown in Figure loa, which clearly demonstrates the
existence of a hot spot. In order to eliminate this hot spot,
we consider the same problem with the additional path
constraint
T(z)I 1.45,
E [O, Ll
(34)
The optimal values of the design parameters for the
constrained case are also listed in Table 4, while Figure
10b shows the axial variation of the conversion and
z
(a) Unconstrained problem
(b) Constrained problem
Figure 10. Axial conversion and temperature variation in tubular
reactor.
temperature. Again the objective function actually corresponding to the optimization parameter values of
Vasantharajan and Biegler is slightly different to that
reported by those authors (-148.4730).
Computational statistics for this problem are shown in
Table 5. In comparison to the simple integral violation
technique (b), the use of the hybrid technique (a) results
2132 Ind. Eng. Chem. Res., Vol. 33, No. 9, 1994
Table 5. Computational Statistics for Tubular Reactor
Problem.
run
unconstrained
constrained (a)
constrained (b)
QP
LS
CPU
7
9
10
6
8
10
29.1
76.8
37.2
QP, number of quadratic programming subproblems solved by
the optimization; LS, number of function evaluations during line
search; CPU, CPU seconds on SUN SPARCStation 2 workstation.
a
in adecrease in both the number of quadratic programming
subproblems solved and the number of function evaluations during the line search. However, in this case, the
decrease is outweighed by the increased cost of solving
the quadratic programs, which now involve more constraints. This results in an increase in the overall CPU
time.
5. Concluding Remarks
This paper has considered general methods for handling
equality and inequality path constraints in the context of
controlvector parametrization algorithms for the dynamic
optimization of multistage DAE systems.
The idea of treating the equality path constraints
together with the underlying system equations is not in
itself new. For instance, Bryson and Ho (1975, Chapter
3) consider means of imposing equality constraints of
various types to systems described by ODES. Although
the concept of DAE index was not known at that time,
they propose a method based on repeated differentiations
and algebraic manipulations that is effectively identical
to reducing the index of the resulting DAE system to unity.
However, the treatment presented in this paper is more
general covering in a unified manner all types of equality
constraints. It also differs in that it incorporates within
the DAE system only those constraints that will not
significantly increase the difficulty of its solution by
increasing its index. The rest of the constraints are left
to be handled by the optimization through conversion to
end-point constraints.
Of course, any end-point constraints derived from
equality path constraints will still suffer from the problem
of their gradients with respect to the optimization
parameters being zero at the solution. However, it is
interesting to note that these problems can be alleviated
by adopting a hybrid approach similar to that for inequality
constraints. Thus, in addition to the end-point constraints,
we enforce each equality constraint as a point constraint
at the stage boundaries.
The proposed approach for handling inequality path
constraints is also a hybrid of techniques proposed earlier
in the literature, namely those based on constraint
discretization at a finite number of points on one hand
and forcing the integral constraint violation to zero on the
other. As already explained, in their combined application,
the former technique plays a key role in guiding the
optimization search, while the latter ensures true feasibility
of the solution obtained.
In addition to the three examples presented here, further
numerical evidence of the relative reliability and efficiency
of the hybrid approach applied to both equality and
inequality path constraints is presented by Vassiliadis
(1993b).
Acknowledgment
This research was supported by a joint SERC/AFRC
grant.
Nomenclature
A, = preexponential Arrhenius constant for reaction j
Ak = matrix defined by eq 8
Bk = matrix defined by eq 9
c p = equality path constraints over stage k
= inequality path constraints over stage k
Ci = concentration of component i
E, = activation energy for reaction j (expressed in K)
f k = equations for stage k
FB = feed flow rate of component B
h~ = specific molar enthalpy of reactor feed stream
hio = specific molar enthalpy of pure component i
H = total energy holdup in reactor
i = standard index for components (=A, B, C, D)
I = unit matrix
j = standard index for inequality path constraints
k = standard stage index
k, = rate constant for reaction j
K = penalty parameter
Lk = lower triangular matrix defined by eq 9
rn = number of algebraic variables y
Mi = molar holdup of component i
n = number of differential variables x
NS = number of stages
Pk = row permutation matrix defined by eq 9
q = number of time-invariant parameters u
Q = reactor cooling load
Q k = column permutation matrix defined by eq 9
rj = rate of reaction j
rk = rank of matrix Ak
s = slack variable
t = time
t o = initial time
tf = final time
t k = end time of stage k
T = temperature
T,,f = reference temperature for enthalpy calculations
u = control variables
a = control variables u,the time variation of which can take
any functional form
i i k = control variables u that will be determined by solution
- of augmented DAE system in stage k
C k = control variables u that will be determined by optimization in stage k
U = domain of u
uk = upper triangular matrix defined by eq 9
u = time-invariant parameters
V = domain of u
V = volume of reactor contents
x = differential variables
f = measure of integral constraint violation
x i = mole fraction of component i
f = time derivatives of differential variables x
X = domain of x
y = algebraic variables
Y = domain of y
cr
Greek Letters
coefficient of linear term in pure component specific
enthalpy expression (28m)
p; = coefficient of quadratic term in pure component specific
enthalpy expression (28m)
AHj = enthalpy of reaction j
e = end-point constraint relaxation parameter (cf. eq 24d)
{ = exponent of constraint violation in integral violation
measure (18)
x k = number of equality path constraints
k k = number of inequality path constraints
y = number of control variables u
?r = number of control variables 1
2
pio = molar density of pure component i
a, =
Ind. Eng. Chem. Res., Vol. 33, No. 9,1994 2133
= points at which inequality path constraints are enforced
in stage k (cf. eq 22)
7ik
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Received for review October 11, 1993
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Accepted June 15,1994.
* Abstract published in Advance ACS Abstracts, August 1,
1994.