Ind. Eng. Chem. Res. 2003, 42, 1761-1772
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Off-Line and On-Line Approach for Optimal Maintenance Management of Continuous Parallel Processes with Decreasing Performance Sebastia´ n E. Sequeira,† Moise` s Graells,‡ and Luis Puigjaner*,† Chemical Engineering Department, Universitat Polite` cnica de Catalunya, ETSEIB, Diagonal 647, Barcelona 08028, Spain, and EUETIB, Comte d’Urgell 187, Barcelona 08036, Spain
The efficiency of equipment units in chemical process plants decreases with time because of fouling, byproduct formation, catalyst deactivation, etc. Therefore, periodic equipment maintenance and/or cleaning are required to restore original operating conditions that allow the desired high productivity rates to be maintained. Maintenance imposes a tradeoff decision between the cost associated with equipment shutdown and the resulting benefit of improvements in productivity. When parallel processing lines are considered in a production facility, the problem becomes more complex because of the presence of material interrelationships. In this work, the solution to this complex maintenance problem is addressed by introducing a simple formulation (NLP) for the planning problem and a procedure for the subsequent implementation of the discrete decisions involved (clean or not, feed assignments) in a real-time environment. This methodology can be considered an extension of the real-time evolution (RTE) approach for continuous processes (Sequeira, S. E.; Graells, M.; Puigjaner, L. Ind. Eng. Chem. Res. 2002, 41 (7), 1815). The main advantage of the proposed methodology is the robustness resulting from the use of on-line information, which reduces the model dependency and, thus, the undesirable effects of variability and plant-model mismatch. The proposed off-line and on-line approaches are tested with benchmark case studies, their performance is evaluated using dynamic simulations, and their long-term behavior is monitored via Monte Carlo simulations. 1. Introduction The problem addressed in this paper is quite common in industrial practice. The efficiency of equipment decreases with running time and/or with the amounts of different materials processed. Therefore, appropriate measures should be taken to reestablish the initial performance level. The resulting decision-making problem consists of determining when such actions should be carried out, thus leading to a tradeoff between the cost of the action itself (resources required, production break, etc.) and the expected benefits it generates (increase in productivity, reduction of operating costs, etc.). The scenario involving a single equipment item has been satisfactorily addressed in different ways.1-4 Moreover, to keep smooth production output, plant equipment might be run in parallel. Typical examples of this situation are catalytic reactors, evaporators, filters, etc. The problem with catalytic reactors is progressive deactivation of the catalyst. In the case of continuous evaporation units, the efficiency decreases because of different factors such as fouling, deposition of insoluble materials, etc. Semibatch filtering efficiency is affected by the continuous increase of the pressure drop through the cake. The existence of parallel lines introduces additional constraints related to the mass balance, as well as possible bounds on the average * Corresponding author: Luis Puigjaner, Universitat Polite`cnica de Catalunya, ETSEIB, Chemical Engineering Department, Av. Diagonal, 647, E-08028 Barcelona, Spain. Telephone: +34-934-016678. Fax: +34-934-010979. E-mail:
[email protected]. † ETSEIB. ‡ EUETIB.
processing rates. Several approaches have been proposed to handle these situations, which usually involve standard planning and scheduling problems formulated as mathematical programs, generally MILP and MINLP.5-8 These approaches allow for the determination of the optimal values for process variables such as the starting time and the duration of cleaning operations and can be coupled to tailored decision-making tools for practical routine work including not only optimization procedures but also “what if” analyses based on simulation.9 Unfortunately, the implementation of the decisions obtained in this way can result in nonoptimal operation because of plant-model mismatch and the inherent variability of plant operating conditions. Otherwise, a particular research effort has been made during the past decade in this field, namely, real-time optimization (RTO) of continuous processes.10,11 The decrease in hardware and software costs has resulted in several implementations of this technology,12-15 showing quite attractive economic results.16,17 Academic research has focused on specific modules of the system, including data acquisition and validation, gross error detection, data reconciliation and model updating, and obviously modeling and optimization. However, little attention has been paid to the on-line management of the discrete decisions involved in processes with decaying performance. Specifically, the fact that the model updating module of an on-line optimization system, or more generally, the distributed control system (DCS), can provide significant information about the process’ degree of performance and its variation with time is usually ignored.
10.1021/ie0202975 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/07/2003
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∫0ts IOFi,j(θ) dθ - Cmi,j i,j
MOFi,j(tsi,j) )
∀i, j
tsi,j + τmi,j
(2)
Let Tfi,j be the time fraction of unit j operation dedicated to the processing of feed i. Then, the average consumption of feed i (Fmi) can be expressed as a function of Tfi,j for each unit j
Fmi ) Figure 1. Scheme for the general case.
In this paper, a step forward toward the integration of the two approaches (conventional mathematical programming and on-line optimization) is presented. A simple formulation (NLP) for the off-line planning problem is provided, together with a procedure for the subsequent implementation of the discrete decisions involved (clean or not, feed assignments) in a real-time environment. The proposed methodology is an extension of the real-time evolution (RTE) approach for continuous processes.18 Some examples are used to illustrate the proposed approach for the planning procedure and the on-line optimization. A dynamic simulator emulates the plant behavior as well as the occurrence of disturbances, while the RTE system is responsible for plant operation performance. This work is organized as follows. The general model considered for off-line calculation is introduced in section 2 and illustrated through examples of increasing complexity. In section 3, an on-line approach is proposed and exemplified through the same case studies. Finally, the results obtained are discussed, and conclusions are summarized. 2. Off-Line Problem Statement
tsi,j i,j
(1)
Then, the corresponding mean objective function (MOF) during the whole subcycle for every feed-unit pair is given by
∀i, j
Tfi,j
+ τmi,j)
(3)
Then, the proposed NLP formulation for this family of problems can be stated as follows: Find
max Z )
∑i ∑j MOFi,j‚Tfi,j
(4)
subject to the following constraints
∫0ts IOFi,j(θ) dθ - Cmi,j i,j
MOF(tsi,j) )
Fmi )
∀i, j (5)
(tsi,j + τmi,j)
∑j
tsi,j Di,j Tfi,j (tsi,j + τmi,j)
∑i Tfi,j ) 1
∀j
Floi e Fmi e Fupi 0 e Tfi,j e 1 0 e tsi,j e tsmax
The general problem to be considered can be stated as follows: There are i ) 1, 2, ..., I feedstocks available to produce the final product P (see Figure 1). The total average consumption of each feed Fmi, should be between a lower and an upper bound, Floi and Fupi, respectively. j parallel units are available to process these feeds, whose performance for processing each feed [instantaneous objective function, IOFi,j(t)] decreases with time. Let the maintenance operations costs and duration be Cmi,j and τmi,j respectively. Additionally, suppose that, whenever there is a feed changeover, maintenance tasks take place, and the operating parameters are set in such a way that the unit returns to operate at the best possible conditions for the next feed. Therefore, the objective of this problem is to find the duration of the cycles (ts) for each feed-unit pair and the corresponding feed flows to be processed. Assume that the unit j operates consuming feed i during time tsi (subcycle time for feed i). During that time, the instantaneous objective function is a function of tsi,j, given by
IOFi,j ) f(tsi,j)
∑j Di,j(ts
∀i
(6)
(7) ∀i
∀i, j ∀i, j
(8) (9) (10)
where MOF is the mean objective function (monetary units/time); IOF is the instantaneous objective function (monetary units/time); Cm is the maintenance cost (monetary units); τm is the time devoted to maintenance (time); Tf is the time fraction of unit j devoted to the processing of feed i (time/time); ts is the operating time for the feed-furnace pair (time); D is the average feed rate of feed i to unit j during the operating time ts (flow basis); and subindexes i and j refer to the feed and the unit, respectively. The problem has 2ij degrees of freedom, corresponding to the 2ij decision variables, namely, the operating times (tsi,j) and the time fractions (Tfi,j), where the latter are related directly to the feed flows. 2.1. One Unit)One Feed: Example I. The simplest case of maintenance problem is illustrated by the following example. There is a single process whose performance decreases with time. At some time ts, the operation must be finished so that some maintenance task that reestablishes the process initial conditions can be performed. The maintenance task consumes the known amount of time τm and its corresponding cost is Cm. It can be seen that the lower the time ts, the higher the performance, but at the expense of the cost and time associated with the maintenance itself. Therefore, there is a tradeoff in the determination of time ts, the question being the time, tsopt, that maximizes the overall performance throughout the operating cycle.
Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1763
U)
kT Xx1 + b‚ts
(14)
where U is the heat exchange coefficient (kcal‚h-1‚ m-2‚°C-1), T is the liquid temperature (°C), X is the product liquid concentration (percentage on a weight basis), and k and b are empirical parameters adjusted experimentally. For this case, the cleaning cost is negligible, and the objective is to find the time between cleaning operations that produces the maximum possible concentration during the cycle. The corresponding data for the specific problem are given in Table 1. The first step is to find IOF(ts), which can be done using the material and energy balances on a pseudostationary basis
Mass Balance Figure 2. IOF and MOF as functions of time during an operation cycle.
FXo ) PX
(15)
F)V+P
(16)
Q ≈ λV
(17)
Energy Balance
Q ) UA‚∆T )
For this particular case, i ) j ) 1, so Tf ) 1. Therefore, the proposed formulation reduces to
∫0tsIOF(θ) dθ - Cm (ts + τm)
IOF(ts) ) X - Xo )
(tsopt + τm)2
∫0ts
kTA‚∆T opt
IOF(θ) dθ
)0 (12)
to give
λFx1 + b‚tsopt
2kTA‚∆T (x1 + b‚tsopt - 1) bλF ) ts + τm
∫0
IOF(θ) dθ - Cm (tsopt + τm)
) MOF(tsopt) (13)
Therefore, using an appropriate model for the IOF, the optimum policy can be found (Figure 2). It should be noted that tsopt corresponds to that time that makes the value of IOF(ts) equal to the value of MOF(ts).19 For illustration purposes, consider a simple evaporator as illustrated in Figure 3. A feed of sugar solution with a concentration Xo arrives at a rate F. The outlet concentration diminishes because of scaling over the outer sides of the evaporator’s tubes according to the following equation20
(21)
or
1 tsopt
(19)
Applying eq 13, one obtains
d[MOF(ts)] |t)topt ) d(ts) IOF(tsopt)‚(tsopt + τm) + Cm -
kTA‚∆T λFx1 + b‚ts
2kTA‚∆T (x1 + b‚tsopt - 1) bλF MOF(ts) ) X - Xo ) ts + τm (20)
(11)
which can be solved analytically, because the optimal maintenance time (tsopt) must satisfy
IOF(tsopt) )
(18)
where P is the liquid product flow rate (kg/h), V is the vapor flow rate (kg/h), λ is the latent heat of the steam (kcal/kg), A is the evaporator heat exchange area (m2), ∆T is the temperature difference between the steam and the liquid (°C), and Q is the heat exchanged (kcal/h). Then
Figure 3. Example I scheme.
maxts Z ) MOF(ts) )
kT A‚∆T x X 1 + b‚ts
x1 + b‚tsopt
2 ( 1 + b‚tsopt - 1) bx ) tsopt + τm
(22)
Note that tsopt depends only on b and τm and can be obtained in a recursive manner, with tsopt equal to 59 h for these conditions. This time corresponds to an average output concentration of 27.39% and an MOF of 12.44%. With the aim of gradually introducing the problem complexity, the following example shows the case of multiple feeds. 2.2. One Unit)Several Feeds: Example II. This example is based on the work reported by Jain and Grossmann.5 The case study considers three different feeds (A, B, and C) that are available for continuous
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Table 1. Data for Example I variable
value
F (kg/h) T (°C) A (m2) ∆T (°C) λ (kcal‚kg-1) k [kcal‚h-1‚m-2‚(°C)-2] b (h-1) τm (h) Xo (wt %)
25 000 103 400 13 520 490 0.0277 14 12.00
Figure 4. Example II scheme. Table 2. Data for Example II
arrival at the corresponding storage tanks at a constant rate. All feedstocks are then processed sequentially in a reactor (the furnace), where, as a consequence of coking, the conversion decreases with time (Figure 4). The rate of arrival of the different feedstocks (i) is a decision variable represented by Fi, which should lie between lower (Floi) and upper (Fupi) bounds. Each feedstock i is processed in the furnace at a rate Di. It is assumed that, after a feed changeover, the furnace is cleaned, and the operating parameters are reset so that the furnace starts operating again at the best possible conversion for the new feed.21-24 The changeover time for feed i is known and given by τmi (sequence-independent). The setup and cleaning costs for each feed i are given by the constants Cmi. Raw material of different grades (A, B, and C) is processed in a furnace to obtain the final product P (Figure 4) at a conversion that depends on the grade i and decreases with time
Xi(t) ) ci + aie
-bit
(23)
where ai, bi, and ci are parameters that are determined experimentally. Additionally, it is assumed that the income is directly proportional to production, with a proportionality constant given by the price parameter Pi. This parameter takes into account the revenues obtained by selling the product and all of the expenses incurred in producing it, except for the cleanup costs. Therefore, the problem consists of determining the operating policy that maximizes the profit. Specific data for the example considered are found in Table 2. Jain and Grossmann5 proposed a problem formulation (F1) using the following variables: ti, the total processing time of feed i in the furnace; ni, the number of subcycles of feed i in the furnace during the total cycle time; Fi, the feeding rate of feed i; ∆ti, the total feeding and cleanup time associated with feed i; Tcycle, the total duration of the cycle; and Z, the average profit during the total cycle, along with the decision variables ti, ni, and Tcycle. Because all of the constraints can be linearized, the resulting MINLP problem can be solved using an NLPbased branch-and-bound approach (taking care to avoid division by zero in the terms with t/n in the objective function). As shown by Jain and Grossmann,5 the global optimum can be found because the constraints are linear and the objective function is pseudoconvex (pseudoconcave). The solution is reported in Table 3. This formulation has some disadvantages. First, the solution is strongly dependent on the arbitrary upper bounds used for Tcycle and ni. The reported solution corresponds to upper bounds of 4 for ni and 140 for Tcycle. For higher values of the Tcycle and ni upper bounds, however, the dependence still continues. For instance, when the bounds for ni and Tcycle are 6 and 200,
feed variable
A
B
C
τmi (day) Di (t/day) ai bi (day-1) ci Pi ($/t) Cmi ($) Floi (t/day) Fupi (t/day)
2 1300 0.20 0.10 0.18 160 100 350 650
3 1000 0.18 0.13 0.10 90 90 300 600
3 1100 0.19 0.09 0.12 120 80 300 600
Table 3. Example II Solution (F1) feed variable
A
B
C
ni ti (day) tsi (day) Tfi Fi (t/day)
4 42.44 10.61 0.3626 396.6
1 41.74 41.74 0.3216 300.0
2 37.94 18.97 0.3159 300.0
Z ($/day) Tcycle (day)
30,430 139.1
respectively, the optimal objective function value increases to 30 507, and so on. Hence, a different formulation (F2) is proposed here instead, in accordance with the general case
max Z )
∑i MOFi‚Tfi
(24)
subject to
[
PiDi ci‚tsi + MOFi )
]
ai (1 - e-bi‚tsi) - Cmi bi
(tsi + τmi)
(25)
tsi Fmi ) Di tsi + τmi
∀i
∑i Tfi ) 1
0 e tsi e tsmax
(26) (27)
∀i
Floi e Fmi e Fupi 0 e Tfi e 1
∀i
∀i ∀i
(28) (29) (30)
In this formulation, the decision variables are the tsi and Tfi (or alternatively Fmi). Now, the optimal solution is summarized in Table 4. From the results given in Tables 3 and 4, the following conclusions can be drawn: (1) The second formulation (F2) provides a better Z value for a given problem. The main reason is that there is no bound constraint related to the finite cyclic operation (the Tcycle variable and
Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1765 Table 4. Example II Solutiona (F2) feed variable
A
B
C
tsi (day) Tfi Fmi (t/day) MOFi ($/day)
9.76 0.374 403.5 53,109
60 0.315 300 10,547
21.34 0.311 300 23,656
Z ($/day)
30,549
(2) The computation time is substantially reduced with the proposed formulation because there are no integer variables to compute. Additionally, it should be mentioned that the lower bounds for feeds C, D, F, and G; the upper bounds for B and E; and tsmax for G-3 are the active constraints. Also note that tsA1 corresponds to the result obtained using eq 13.
a
Starting from random initial points, the same solution is reached.
associated constraints). (2) In both cases, the contribution to profit (MOFi) obtained processing raw materials A and C is substantially higher than that obtained processing B. (3) The lower-bound constraints on feeds B and C are active. The interpretation of these results is quite straightforward. In the absence of lower bounds for FmB and FmC, the obvious solution would be to process only feed A (i.e., the most profitable one) with tsA satisfying eq 13 (the best one) whenever the FmA upper bound allows. As this is not the case, tsB (i.e., the processing time of the least profitable feed) is set to tsmax, thus “forcing” the required feed consumption as soon as possible. In addition, the value of tsC is such that it allows for the minimum required feed to be processed and makes the sum of the time fractions equal to 1. 2.3. Several Units)Several Feeds: Example III. The scenario for this case study corresponds to the general case of the previous example and has been also excerpted from the previously mentioned work.5 Now, four furnaces and seven feeds are considered. The corresponding parameters are given in Table 5. Table 6 shows the solution obtained using F1 (Tcycle of about 35 days and Z of $165,400/day $/d). Alternatively, the proposed formulation (F2) for this case is the general formulation given in eqs 4-10, where MOF becomes
MOFi,j )
[
Pi,jDi,j ci,j‚tsi,j +
]
ai,j (1 - e-bi,j‚tsi,j) - Cmi bi,j
(tsi,j + τmi,j)
∀i,j (31)
Given the nonlinearity of the problem, it was solved from 200 random (uniformly distributed) initial points. It takes about 0.2 s to solve the problem from each initial point on a AMD-K7 processor with 128 MB of RAM at 600 MHz using the CONOPT2 solver in the GAMS environment25 and about 2 s in a spreadsheet environment using an implementation of GRG2.26 The histogram of Figure 5 shows the relative frequencies of the locally optimum objective function values, expressed as percentages of the frequency of the best value. It can be seen that most of the identified optima fall in a favorable region. In fact, using a binomial probability distribution function, one can verify that the probability of reaching the best solution is 99.99% starting only from 10 independent initial points. Table 7 summarizes the results of the best solution (using tsmax ) 20 days). By comparing the solution reached with formulation F2 (Table 7) with that obtained using F1 (Table 6), the following conclusions can be drawn: (1) The objective function value obtained using our formulation (F2) is higher. This result can be explained by the fact that, in the proposed formulation, the cyclic operation corresponds to each furnace rather than to the whole system.
3. On-Line Solution During the process design stage, IOF(t) is usually expressed as a function of the operating conditions, and it is embedded in the remaining equations describing the process. For example, catalyst activity is commonly given by an additional kinetic expression. This expression can be included with other equations describing reactor operation, and therefore, the solution of the whole system of differential equations allows for the computation of the corresponding value of IOF(t). For the sake of simplicity, the previous examples used straightforward empirical relationships that allowed for the direct calculation of IOF(t) (parameters a, b, and c), rather than requiring the solution of a complex set of differential equations. However, for the very complex situations found in practice, the use of such empirical models can fail. Appendix B illustrates the reliability and limitations of the hypothesis of pseudo-steady-state behavior for example I. Furthermore, Schulz et al.15 have shown the effects of using even more refined models (including recycle streams) on the planning decisions made at an existing ethylene plant. Thus, in general, there are two main reasons for disagreement between the modeling results and the actual values obtained in a plant: plant-model mismatch and variability in the model parameters. Plantmodel mismatch can lead to “bad” values of the model parameters (ai, bi, and ci for the considered examples). This behavior might be caused, for instance, by unexpected disturbances or an inadequate model updating procedure and/or frequency. Furthermore, it could arise from some structural error in the model. Also, given that model parameter values are commonly determined by statistical techniques, they are just average values. Therefore, the instantaneous plant behavior will vary from run to run. The magnitude of such variations is approximately given by the degree of deviation observed during the parameter adjustment stage. Hence, the results of applying mathematical programming approaches will vary according to the implementation methodology, because the previous factors will have a greater or lesser influence on the global behavior. Therefore, an alternative approach for finding tsopt and Tfopt on-line is desirable. Fortunately, IOF(t) can be evaluated in another way. The IOF is commonly expressed in terms of profit, measured as the value of the products less the raw material and operating costs. When adequate sensors are available,28 the components of this expression can be computed. For instance, the quantity and quality of production and the consumption of raw materials and utilities are usually available from the distributed control system (DCS) and plant information systems, and therefore, the IOF can be calculated on-line by incorporating the corresponding economic parameters (prices, cost factors, etc.). In this context, an alternative approach for on-line optimal maintenance management is proposed that is based also on plant data rather than
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Table 5. Data for Example III feed i furnace j
A
B
C
D
E
F
G
τmi,j (day)
variable
1 2 3 4
2 3 1 2
3 1 3 1
3 2 1 3
3 2 1 2
1 2 2 2
2 1 1 1
3 1 2 1
Di,j (t/day)
1 2 3 4
1300 1100 900 1200
1200 1050 800 1000
1100 1000 800 800
800 1000 1200 700
1300 1200 1000 1200
300 400 300 400
700 600 850 600
ai,j
1 2 3 4
0.30 0.32 0.31 0.31
0.40 0.38 0.35 0.36
0.35 0.33 0.36 0.35
0.32 0.31 0.36 0.36
0.29 0.28 0.29 0.28
0.35 0.40 0.37 0.39
0.31 0.34 0.31 0.32
bi,j (day-1)
1 2 3 4
0.10 0.20 0.30 0.20
0.20 0.10 0.20 0.20
0.10 0.20 0.30 0.15
0.20 0.25 0.27 0.25
0.23 0.29 0.28 0.29
0.34 0.27 0.29 0.22
0.20 0.30 0.25 0.28
ci,j
1 2 3 4
0.20 0.21 0.19 0.20
0.18 0.19 0.18 0.19
0.21 0.23 0.21 0.21
0.20 0.25 0.23 0.24
0.30 0.31 0.30 0.31
0.26 0.27 0.25 0.26
0.16 0.17 0.18 0.17
Pi,j ($/t)
1 2 3 4
123 114 110 120
105 132 122 125
110 129 120 129
123 114 110 115
105 132 122 115
110 129 120 128
120 113 117 115
Cmi,j ($)
1 2 3 4
100 80 90 80
90 85 90 90
80 75 90 85
75 90 85 80
90 94 93 92
93 78 92 85
78 70 75 72
300 600
400 700
300 600
500 800
500 800
100 400
600 900
Floi (t/day) Fupi (t/day)
Table 6. Example III Solution (F1) F (t/day)
profit (%)a
furnace
ni,j
t (day)
A B C D E
353 700 300 500 800
10.32 26.62 8.69 14.28 24.97
F G
100 600
3.34 11.77
1 2 1 3 1 2 4 4 3 4
1 4 1 3 2 1 1 1 1 2
9.55 23.55 9.57 14.70 9.07 5.65 8.13 8.84 15.62 13.18
feed
Z ($/day) Tcycle (day) a
165,400 35.42
Expressed as a percentage of Z.
only on the decaying performance model. The proposed on-line strategy is introduced next using the previous examples. Basically, the proposed methodology formulates the on-line optimization problem as a set of very simple questions, asked periodically, and is directed toward the identification of eventual incremental improvements. This general approach, termed RTE (real-time evolution), has been successfully applied to the on-line optimization of continuous processes.18 In this work, a variant of this methodology is introduced that takes full advantage of monotonically changing behavior to embed the process model implicitly into the on-line decisionmaking technique. 3.1. Simplest Case: Example I. Let us assume that (1) the information needed to compute the IOF at each time interval k is available on-line and (2) the function IOF(k) is strictly decreasing with time. Under such circumstances, it is known (Figure 2) that there exists a unique optimal time interval, kopt, for performing the corresponding maintenance tasks. Thus, rather than solving the corresponding optimization
Figure 5. Relative frequency of different solutions for example III. Table 7. Example III Solution (F2) feed
Fmi (t/day)
profit (%)
furnace
ts (day)
Tf
A B C
389 700 300
19.3 13.4 8.8
D E
500 800
14.9 28.9
F G
100 600
3.1 11.6
1 2 1 2 3 1 4 4 3 4
8.71 5.25 12.87 8.15 4.48 4.79 7.96 7.52 20.00 6.79
0.368 0.794 0.151 0.206 0.510 0.482 0.294 0.283 0.490 0.423
Z ($/day)
166,500
problem, we can solve a simpler problem by answering the following question at each period k: Should we stop for maintenance at this period or at the next one? The answer can be obtained simply by comparing MOF(k) and MOF(k+1) at each interval k. Naturally, if MOF(k) < MOF(k+1), it is better to wait. Otherwise, if MOF(k) g MOF(k+1), then it is better to stop at period k (see Figure 2).
Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1767
Figure 6. Example I. Ad hoc implementation of off-line results.
Figure 7. Example I. Use of RTE for on-line optimization.
It can be observed that this problem is equivalent to computing
MOF(k) - MOF(k+1) ) 0
(32)
which is the discrete form of eq 12 and, hence, provides the optimal solution. This statement may hold even if Cm and τm are available as functions of the operating time, rather than constant values. An interesting extension of the proposed strategy for continuously obtaining a good estimation of the maintenance time from on-line data is detailed in Appendix B. Regarding the first hypothesis, note that, for this example, the IOF can readily be obtained from on-line concentration measurements (or measurements of any correlated variable). With respect to the second hypothesis, which is intrinsic to the problem (because if the IOF does not decreases with time, the problem does not exist), it is already known that the evaporator performance continuously decreases. To illustrate the application of this technique, consider example I, where the off-line optimal solution for the available data corresponds to ts ) 59 h. Figure 6 shows the performance obtained when implementing this result (ts is fixed in this case, but Xm can be fixed instead) as the strict cleaning policy to be used in a dynamic simulation of the process that includes variability and plant-model mismatch. Aspen Custom Modeler was used29 as simulation tool, mainly because a hybrid discrete-continuous simulation environment was required.30 However, under the same model mismatch and variability, when tsopt is determined on-line according to the proposed RTE strategy, much better performance is achieved, as shown in Figure 7. Note that the on-line determination of the optimality condition (i.e., IOF ) MOF, according to eq 13) might not necessarily lead to the same tsopt value as obtained offline. Next, strictly off-line and on-line approaches were compared using a Monte Carlo simulation, by modeling the empirical parameters b and k in the following way
π/i ) πi[1 + σi‚INVNS(F) + ξ]
(33)
where πi is the nominal parameter value, π/i is the parameter value used in the model (which changes for every subcycle), σ is the standard deviation of the nominal parameter value (to emulate variability), ξ is
Figure 8. Example I. Histogram for tsopt for both ξ and σ equal to 10%.
the relative error in the nominal parameter value (to emulate plant-model mismatch), F is a uniformly distributed random number, and INVNS represents the inverse of the normal standard cumulative distribution. Figure 8 shows the resulting distribution of the online tsopt obtained for ξ ) 10% and σ ) 10%, where the off-line value is included as a reference. A larger set of Monte Carlo simulations for different values of ξ and σ led to the results shown in Figure 9. The difference between the performances (MOFs) of the two approaches is used as a comparative index. It can be seen that the on-line determination of tsopt enhances the process performance. For this particular example, the improvement is about 0.11% in terms of concentration; in absolute terms and according to the given flow rate, the improvement corresponds to more than 200 t of sugar/year. Considering that the evaporation section usually involves several equipment units, the on-line determination of cleaning tasks can be greatly compensated by the resulting savings.31 3.2. One Unit)Several Feeds: Example II. For the second example, the off-line solution offers significant information that also provides an opportunity for on-line optimization. Specifically, the flow assignments resulting from the off-line optimization (model) correspond to a “first-stage” decision. Once such a decision
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Figure 9. Example I. Improvement reached applying RTE for different plant-model mismatch and variability conditions.
is implemented, the optimal operating times (tsi) can be determined as the information from the plant becomes available. Originally, there were six degrees of freedom in this example (three Tf ’s and three ts’s). Assuming that the three Fm’s are identified using the off-line solution, remain only three decision variables. In addition, tsB is the maximum allowed and eq 27 forces the material balance to be satisfied, so that only one degree of freedom remains. This degree of freedom corresponds to the variable tsA, which can be determined on-line using the procedure explained in the previous section. However, it is important to mention that the variability in tsA obtained by its on-line determination will change the corresponding TfA. Therefore, to ensure that the implemented solution satisfies the mass balance, also tsC needs to be recalculated and dampened from cycle to cycle, according to the equations
TfC ) 1 - TfA - TfB and
tsC )
τmC‚FloC [DC‚TfC - FloC]
(34)
where TfA is updated from cycle to cycle via
TfA )
tsA + τmA FmA tsA DA
(see eq 26). To select the next feed to be provided to the furnace after the cleaning is finished, a simple selection rule can be used (just choosing the material having the highest inventory level), and thus, the Tfi decision variables can be implemented automatically. Figure 10 shows the application of the explained strategy to a dynamic model. Additionally, the benefits of the proposed approach when plant-model mismatch and variability are present were tested by means of a long-term Monte Carlo simulation. In this case, the parameters a, b, and c were changed in a manner analogous to that used in the previous example. The results are given in the graphs of Figure 11.
It is noticeable that the influence of the difference between the two approaches (with and without RTE) is more significant for the variability (σ) than for the error (ξ), as should be expected because the variability is the main motivation for on-line optimization systems. Plots of the type shown in Figure 11 are very useful for the economic evaluation of RTE system projects, in that, by properly approximating the variability and the model mismatch. one is able to estimate the benefits of the on-line optimization system. It is also worth mentioning that, often, the lower bounds for the average flows are equal to zero, and that solving both the off-line and online problems of this special case becomes easier, as explained in a previous work.32 3.3. Several Units)Several Feeds: Example III. The application of similar concepts also permits an online approach to this problem, and the larger number of variables and constraints allows for the generalization of the previous concepts. The original formulation has the Tfi,j’s and the tsi,j’s as the decision variables, which means 56 degrees of freedom. However, consideration of the flow assignments and results from the off-line optimization imply 48 additional equations
Fmi )
∑j Di,jts
tsi,j i,j
+ τmi,j
∀i
Tfi,j
Tfi,j ) 0 w tsi,j ) 0 ∀(i, j) ∈ {(A, 2), (A, 3), (A, 4), (B, 1), (B, 3), (C, 3), (C, 4), (D, 1), (D, 2), (D, 4), (E, 2), (E, 3), (F, 1), (F, 2), (F, 3), (G, 1), (G, 2)}
∑i Tfi,j ) 1 tsi,j ) tsmax
∀j
(i, j) ) {(G, 3)}
(35)
which means that the number of degrees of freedom remaining is 8. After some algebraic manipulation,33 all of the dependent variables (Tfi,j’s and tsG4,) can be expressed as functions (mostly mass balances) of an independent set of tsi,j variables: tsA1, tsB2, tsC1, tsC2, tsD3, tsE1, tsE4, and tsF4. In addition, in that reduced space, the optimum point must satisfy
∂Z ) 0 for ∂tsm,n (m, n) ) {(A, 1), (B, 2), (C, 1), (C, 2), (D, 3), (E, 1), (E, 4), (F, 4)} (36) which means
∂[MOFm,n(tsm,n)‚Tf(tsm,n)]
-
{
)
∂tsm,n
∑
i*m j*n
MOFi,j
∂Tfi,j(tsm,n) ∂tsm,n
+
∑
Tfi,j
}
∂MOFi,j(tsm,n)
i*m j*n
∂tsm,n
(37)
where
tsi,j ) tsi,jopt (i, j) * (m, n)
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Figure 10. Example II. Use of RTE for on-line optimization (tsmax ) 30 days).
values can be identified on-line. It should be noted that special care must be taken given that the values of some dependent variables (Tfi,j’s and the remaining tsi,j’s) commonly become negative for lower tsi,j values and, hence, correspond to infeasible solutions. Regarding the on-line implementation of the Tfi,j variables, the reasoning presented in the previous example holds. However, the contribution of each feed to each furnace needs to be considered. One way to easily accomplish this is by splitting the real inventory into “pseudo-inventories” for each furnace. The split fractions (Sfi,j) for this computation are based on the additional terms in eqs 6
Fmi )
Figure 11. Example II. Improvement reached by applying RTE.
This set of equations completes the system for the online solution. Therefore, when feeding furnace n with feed m, the on-line measurement of IOFm,n allows MOFm,n to be obtained continuously. Moreover, Tfm,n can be determined from tsm,n using the mass balances. This allows the LHS of eq 37 to be computed on-line as a function of tsm,n. On the other hand, the RHS of eq 37 is a function of tsm,n that can easily be evaluated numerically. All MOFi,j’s can be obtained from the historical optimal values (using the off-line result as an initial value), and the Tfi,j values (for i, j * m, n) can be computed from the mass balance equations also using the historical tsi,jopt values. Figure 12 shows the evolution of the Zi values with time (different tsi,j) where the plant was not stopped to clearly illustrate the Zi profiles and how the optimum
∑j
tsi,j Di,j Tfi,j ) tsi,j + τmi,j
∑j Sfi,j‚Fmi
tsi,j Tf Di,j tsi,j + τmi,j i,j ∀i, j Sfi,j ) Fmi
∀i (38)
(39)
For its evaluation, the actual values obtained for tsi,jopt are used, and normalization according to
∑j Sfi,j ) 1
∀i
(40)
is performed periodically. The procedure presented is quite general, and its application to the previous example (example II) produces results similar to those obtained in the previous section. However, it is important to note that, in the case of example III, the improvement reached using online optimization is not very significant. This is not surprising, because the solution is reached mainly by the satisfaction of mass balance constraints (lower and
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Figure 12. Example III. On-line determination of tsopt.
upper bounds on Fm), which does not leave enough margin for the on-line manipulation of the ts variables. 4. Conclusions This work represents a step forward toward the integration of classical mathematical formulations for the scheduling of processes with decaying performance and on-line optimization technology. A technique for the off-line calculation of a maintenance schedule and a methodology for the on-line implementation of such scheduling decisions in a coordinated way have been presented. The main advantages of the proposed approach are its simplicity and robustness, which make it attractive for industrial application in process plants provided with information systems. Moreover, the proposed NLP formulation includes general performance relationships and mass balances, which allow its use over a wide range of applications. The variables involved have clear physical meanings, and the computation times required are indeed affordable. More important is the fact that the solutions are easy to implement, and even more, they can be further used for performing on-line optimization. Otherwise, the proposed procedure for on-line optimization is quite simple and robust, and it reduces the effects of plantmodel mismatch and variability. The RTE strategy has been demonstrated to obtain optimal solutions by properly using the information coming from the plant and answering simple questions on-line, rather than performing successive formal optimization, which has been the commonly used strategy. The on-line procedure
seems to be especially worthwhile in cases with highly decreasing performance rates and in cases where the off-line optimal solution is not strongly dictated by the constraints. Acknowledgment Financial support from CICYT (MCYT, Spain) and The European Community is gratefully acknowledged (projects QUI-99-1091 and G1RD-CT-2001-00466). One of the authors (S.E.S.) also acknowledges a grant provided by the Spanish “Ministerio de Ciencia y Tecnologı´a” (FPI). Appendix A: On the Pseudo-Steady-State Behavior Assumption The dynamic behavior of an evaporator such as that in example I can be suitably enough described by the following equations, corresponding to a well-mixed situation
Solid mass balance d(XM) ) F0X0 - FX d(ts) Total mass balance dM ) F0 - F - V d(ts) Assuming that the sensible heat can be neglected and
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Figure 13. About the pseudo-steady-state (PSS) hypothesis. Table 8. Effects of the PSS Assumption on the Optimization Results
Figure 14. On-line forecasting of tsopt value.
model
tsopt (h)
MOFopt (%)
error in ts (%)
PSS D, Tr ) 15 h D, Tr ) 30 h D, Tr ) 60 h
59.1 60.0 61.5 64.5
12.43 12.38 12.31 12.17
0 1.51 3.98 8.74
inventory control is perfect (typically there is a level controller), one can write
V≈
kATp‚∆T
Q C ) ) λ Xx1 + b‚ts Xx1 + b‚ts dM )0 d(ts)
from which the following differential expression of X can be obtained
F0 C dX + (X - X0) ) M d(ts) Mx1 + b‚ts with the initial condition
X(ts)0) ) X0 The graph in Figure 13 shows the profiles obtained by numerically integrating this equation and the corresponding pseudo-steady-state behavior for different values of the evaporator capacity, expressed by its residence time, Tr ) M/F0 (h). After 5 h, in all cases, the pseudo-steady-state assumption acceptably represents the evaporator behavior. However, Table 8 clearly illustrates the error produced in the determination of tsopt using such a representation. Appendix B: A Method for On-Line Forecasting of tsopt From eq 13
[IOF(tsopt)(tsopt + τm)] -
∫0ts
[
IOF(θ) dθ - Cm] ) A1 - A2 ) 0
opt
considering that the IOF is initially greater than the MOF, the RHS of the above expression will be positive
for values of ts lower than tsopt. Using the available information at time k, namely
{i, IOF(i), MOF(i)} ∀i ) 0, ..., k it is possible to obtain the difference between the two terms (DOF)
DOF(i) ) A1(i) - A2(i)
∀i ) 0, ..., k
Then, the DOF can be plotted against i. For most of the cases of decreasing functions analyzed (exponential, quadratic, etc.), the resulting trend plot will be linear. Hence, by using a linear fit it is possible to obtain
DOF(i) ≈ m(k)‚i + b(k) where m and b are functions of k because they can be updated at each k. From this expression, tsopt is the value of ts at which DOF ) 0, and hence
i ) -b(k)/m(k) where i corresponds in this case to the time period of tsopt. The basic strategy is shown in Figure 14. This estimation becomes more accurate with time and is very helpful for the resource planning. Literature Cited (1) O’Donnell, B. R.; Barna, B. A.; Gosling, C. D. Optimize heat exchanger cleaning schedules. Chem. Eng. Prog. 2001, 97 (6), 56. (2) Casado, E. Model optimizes exchanger cleaning. Hydrocarbon Process. 1990, 79 (Aug), 71. (3) Epstein, N. Optimum evaporator cycle with scale formation. Can. J. Chem. Eng. 1979, 57, 659. (4) Zubari, S. M.; Sheikh, A. K.; Budair, M. O.; Badar, M. A. A maintenance strategy for heat transfer equipment subject to fouling: A probabilistic approach. J. Heat Transfer 1997, 119, 575. (5) Jain, V.; Grossmann, I. Cyclic scheduling of continuous parallel process units with decaying performance. AIChE J. 1998, 44, 1623. (6) Georgiadis, M. C.; Papageorgiou, L. G.; Machietto, S. Optimal Cyclic Cleaning Scheduling in Heat Exchanger Networks under Fouling. Comput. Chem. Eng. Suppl. 1999, 23, S203. (7) Tjoa, I. B.; Ota, Y.; Matsuo, H.; Natori, Y. Ethylene Plant Scheduling System Based on an MINLP formulation. Comput. Chem. Eng. Suppl. 1997, 21, S1073. (8) Alle, A.; Pinto, J. M.; Papageorgiou, L. Cyclic Production and Cleaning Scheduling of Multiproduct Continuous Plants. In
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Computer-Aided Chemical Engineering; Grievink J., van Schijndel J., Eds.; Elsevier: Amsterdam, 2002; Vol. 10. (9) Sequeira, S. E.; Graells, M.; Puigjaner, L. Decision-making framework for the scheduling of cleaning/maintenance tasks in continuous parallel lines with time-decreasing performance. In Computer-Aided Chemical Engineering; Gani, R., Jørgensen, S. B., Eds.; Elsevier: Amsterdam, 2001; Vol. 9. (10) Marlin, T. E.; Hrymak, A. N. Real-Time Optimization of Continuous Processes. AIChE Symp. Ser. 1997, 316, 156. (11) Perkins, J. D. Plant-Wide Optimization: Opportunities and Challenges. AIChE Symp. Ser. 1998, 94 (320), 15. (12) Lauks, U. E.; Vasbinder, R. J.; Valkenburg, P. J.; van Leeuween, C. On-line Optimization of an Ethylene Plant. Comput. Chem. Eng. 1992, 16S, S213. (13) Georgiou, A.; Sapre, A. V.; Taylor, P.; Galloway, R. E.; Casey, L. K. Ethylene Optimization System Reaps Operations and Maintenance Benefits. Oil Gas J. 1998, 96 (10), 46. (14) Krist, J. H. A.; Lape`re, M. R.; Wassink, S. G.; Koolen, J. L. A. System for Real Time Optimization. U.S. Patent 5,486,995, 1996. (15) Tvrzska´, M.; Odloak, D. One-Layer Real Time Optimization of LPG Production in the FCC Unit: Procedure, Advantages and Disadvantages. Comput. Chem. Eng. 1998, 22S, 191. (16) White, D. C. On line optimization: What, where and estimating ROI. Hydrocarbon Process. 1997, 43. (17) Yoon, S.; Dasgupta S.; Mijares, G. Real Time Optimization Boosts Capacity of Korean Olefins Plant. Oil Gas J. 1996, 94 (25), 36. (18) Sequeira, S. E.; Graells, M.; Puigjaner, L. Real-Time Evolution for On-line Optimization of Continuous Processes. Ind. Eng. Chem. Res. 2002, 41 (7), 1815. (19) Buzzi, G.; Morbidelli, M.; Forzatti, P.; Carra, S. Deactivation of catalysts III: Mathematical models for the control and optimization of reactors. Int. Chem. Eng. 1984, 24 (3), 441. (20) Honig, P. Principles of Sugar Technology; Elsevier: Amsterdam, 1969; Vol. III. (21) Borio, D. O.; Menendez, M.; Santamaria, J. Simulation and optimization of a fixed bed reactor operating in coking-regeneration cycles. Ind. Eng. Chem. Res. 1992, 31, 2699. (22) Borio, D. O.; Schbib, N. S. Simulation and optimization of a set of catalytic reactors used for dehydrogenation of butene into butadiene. Comput. Chem. Eng. Suppl. 1995, 19S, S345.
(23) Castilla, M.; Gayubo, A. G.; Aguayo, A. T.; Arandes, J. M.; Bilbao, J. Simulation and Optimization of Methanol Transformation into Hydrocarbons in a Isothermal Fixed-Bed Reactor under Reaction-Regeneration Cycles. Ind. Eng. Chem. Res. 1992, 31, 2699. (24) Taskar, M.; Riggs, J. Modeling and Optimization of a Semiregenerative Catalytic Naphtha Reformer. AIChE J. 1997, 43 (3), 740. (25) Brooke, A.; Kendrick, D.; Meeraus, A. GAMS: A Users Guide; Scientific Press: Palo Alto, CA, 1992. (26) Fylstra, D.; Lasdon, L.; Watson, J.; Waren, A. Design and Use of the Microsoft Excel Solver. Interfaces 1998, 28 (5), 29. (27) Schulz, E.; Diaz, S.; Bandoni, A. Interaction between process plant operation and cracking furnaces maintenance policy in a ethylene plant. In Computer-Aided Chemical Engineering; Pierucci, S., Ed.; Elsevier: Amsterdam, 2000; Vol. 8. (28) Sa´nchez, M.; Bagajewicz, M. On the Impact of Corrective Maintenance in the Design of Sensor Networks. Ind. Eng. Chem. Res. 2000, 39 (4), 977. (29) Aspen Custom Modeler: Modeling Language Reference; AspenTech: Cambridge, MA, 1999. (30) Barton, P. I.; Pantelides, C. C. Modeling of combined discrete/continuous processes. AIChE J. 1994, 40, 966. (31) Sequeira, S. E.; Graells, M.; Puigjaner, L. On-line approach for optimal maintenance management of continuous parallel processes. CHISA 2002, Prague, August 2002: paper 49. (32) Sequeira, S. E.; Graells, M.; Puigjaner L. On-line optimisation of maintenance tasks management using RTE approach. In Computer-Aided Chemical Engineering; Grievink, J., van Schijndel, J., Eds.; Elsevier: Amsterdam, 2002; Vol. 10. (33) Lee, W.; Christensen, J. H.; Rudd, D. F. Design Variable Selection to Simplify Process Calculations. AIChE J. 1966, 12 (6), 1104.
Received for review April 19, 2002 Revised manuscript received December 19, 2002 Accepted January 20, 2003 IE0202975
