Ind. Eng. Chem. Res. 1993,32, 2681-2692
2681
Selection of Process Control Structure Based on Linear Dynamic Economics Lawrence T. Narraway and John D. Perkins’ Centre for Process Systems Engineering, Department of Chemical Engineering and Chemical Technology, Imperial College of Science, Technology and Medicine, London SW7, U.K.
It is recognized widely that the choice of measured and manipulated variables employed in a control system (the control structure) can have a strong effect on the performance of the process control system. This paper outlines a systematic method that can be used to select the economically optimal control structure of a process without designing the process controller, while maintaining good controllability characteristics. This requires examination of the effects of process dynamics on process economics and how changes in the control structure alter these economics. The scope of the problem is limited to selecting economically optimal square regulatory feedback control structures for processes whose operation is dominated by steady-state aspects. As the problem is combinatorial in nature, the method addresses systematic generation and efficient search of the space of potential structures through the use of integer programming techniques. Application of this approach to two case studies demonstrates that the technique leads to an improved understanding of the effects of the control structure on process economics and process controller performance. 1. Introduction
2. Economic Analysis of Dynamics
An important decision in terms of determining the operating performance of a process plant is the choice of a suitable set of measured and manipulated variables for use in the plant control system. In the past, a number of approaches have been proposed to this process synthesis problem (Umeda et al., 1978;Morari and Stephanopoulos, 1980; Nishida et al., 1981; Govind and Powers, 1982). In this paper, we propose a method to make these selections based on economic criteria. Many workers (Maarleveld and Rijnsdorp, 1970; Morari and Stephanopoulos, 1980; Prett and Garcia, 1988; Marlin et al., 1991) have emphasised the role played by constraints limiting the steadystate performance of the plant. The presence of disturbances causes plant personnel to choose operating points removed from the constraints determining optimum steady-state operation in order to accommodate upsets within the feasible operating region. One benefit of control is to mitigate the effect of disturbances so as to maintain feasibility and to allow operation closer to the key operating constraints. Different control structures possess differing abilities to modify the plant dynamics in this way, as well as having different capital and maintenance costs. It is with the trade-off between instrumentation costs and operating benefits that this paper is concerned. In the next section, we propose a modification to the method of Narraway et al. (1991) for the a priori assessment of the effect of disturbances on the economics of a given plant and control structure. Section 4 describes the main contribution of this paper: a branch and bound algorithm for the choice of control structure based on these economic criteria. The algorithm is demonstrated on two case studies (a double-effect evaporator and a flotation circuit) in section 5. The sensitivity of the chosen optimal control structure to assumptions about the costs of instrumentation is investigated in this section. Our conclusions drawn from this work are reported in section 6.
To examine the economic trade-offs involved in the selection of process control structures, it is necessary to have a method for the evaluation of the various components of cost and benefit associated with process dynamics and control. In principle, the quantification of costs associated with any control scheme is straightforward. The capital costs of instrumentation are widely available, and maintenance costs may be accounted for readily. To assess the potential economic benefits of any given control structure, we will make use of a modification of the method proposed by Narraway et al. (1991). The approach consists of identifying the optimum steady-state operating point for the plant and estimating the back off required from constraints active at this optimum to accommodate the effects of disturbances. Applying these back offs gives a modified operating point whose economics may be computed. The back offs are estimated using linearized dynamic models, and to avoid the need to tune controllers the assumption of perfect control is utilized. Thus, whatever measured variables are selected are assumed to be perfectly regulated; the effects of the disturbances on the plant are reflected in the variation of plant variables not chosen as measured variables. In the next subsection, it is shown how the determination of the optimum operating point taking account of the dynamic effects caused by disturbances may be formulated as a linear program. Because the perfect control approximation is used in the analysis, it is important to make an assessment of the controllability characteristics of any candidate structure, as well as the structure’s economics. It is for this reason that we will use our algorithm for control structure selection to find a number of good structures based on the economics at the perfect control asymptote. We will then analyze each of these structures from the controllability point of view to make an assessment of the likelihood of achieving performance close to the limiting case. 2.1. Perfect Control Problem. Consider the following linear system obtained by linearizing a nonlinear process
0888-5885f 93f 2632-268I$O4.0010
0 1993 American Chemical Society
2682 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
model at its steady-state economic optimum: min J = aTxO+ pTuo+ yTz, uod
+ aTd
+ Bu + Cz + D d + Ep 0 = Fx + Gu + Hz + K d + Lp a 1 Mx + N u + Oz + P d + Rp
s.t. k = Ax
d' I d 4 dh u1I u I Uh
where x , u, z , d , and p are the linear deviation vectors of state, manipulated, algebraic, design, and disturbance variables, respectively. Similarly, J i s the linear deviation from the nonlinear steady-state objective function. The vector of constants a is obtained from the Taylor series expansion of the nonlinear inequality constraints, where all active and inactive inequality constraints are included in the linearization. For a given control structure, Narraway et al. (1991) have shown that the maximum variation in the perfectly controlled dynamic variables in response to a vector of disturbances with amplitude p ( w ) at frequency w is given by (the design vector d only makes a steady-state contribution to the problem)
where 1. is the complex magnitude of the elements of the vector or matrix, u is the maximum variation in the slack variables of all of the process constraints, and U is the maximum manipulated variable rangeability required to guarantee perfect disturbance rejection. Similarly, max 1x1 and max IzI are the maximum variations in x and z for the given control structure subject to the process disturbance. The unity elements in the permutation matrix Q correspond to the elements ofu that are not used for control purposes and the elements of z that are perfectly controlled. That is, Q specifies the control structure under examination. The results of the above calculations are the wontcase variations, maximized with respect to the phase shifts, assumed unknown, between different elements of p ( w ) . The optimal operating point (and objective function as a deviation from the nonlinear_steady-stateoptimum) may be obtained by substituting u and U into the following linear program and solving: min Jpc= aTxo+ BTz0 + yTd + STp,
+ Bu, + Cz, + D d 0 = Fx, + Gu, + Hz, + K d a 2 MX,+ NU, + + ~d + ;r 02,
U ~ + U I U I U h - - U
d' Id Idh
3. Control Structure Selection Based on Linear Analysis
This section develops a control structure selection problem incorporating the economic analysis technique developed in the previous section. The perfect control ideas introduced a method for selection of control structures by varying the permutation matrix Q. The permutation matrix can be used to select each of the alternative combinations of measured and manipulated variables. Using Q in this fashion to calculate i,it is possible to assess the dynamic economics for all of the possible control structures and choose the optimum directly. This method is acceptable for problems with a small number of manipulated and measured variables but rapidly becomes unwieldy for problems with a significant number of potential control structures. Therefore, it would be desirable to develop a systematic algorithm for choosing the optimum structure from a large set of manipulated and measured variables. The algorithm should also include screening of the search space to avoid evaluating all of the possible control structures. A hybrid mixed integer linear programming (MILP) method combined with the perfect control dynamic economics from the previous section is presented as a solution to this problem. We start with the linear process model from the previous section and initially consider a nominal disturbance of amplitude p ( w ) at a frequency w. The combined steadystate and dynamic equations can be written as min J = aTxo+ pTu, + yTz, + aTd uo,d
(9) (10) (11)
(14)
+ Bu, + Cz, + D d 0 = Fx, + Gu, + Hz, + K d a 1 Mx,+ Nu, + Oz,+ P d + 1;
(17)
u'du-I2
(18)
d' Id Idh
(20)
0 = Ax,
Wd
0 = Ax,
where subscript 0 is the steady-state deviation of the variable from its nonlinear steady-state value. The solution ?f this problem results in the optimal operating point as u and U correspond to the required steady-state offsets in the process constraints and the manipulated variable bounds to ensure that the process remains in the feasible region under normal operation. Note that the steady-state disturbance term (PO)does not appear in the LP as it is assumed that the expected value of po is zero (i.e., there are no long-term deviations from the nominal disturbance). In the event of infeasible constraint back off, the above LP is infeasible (as it includes linearizations of all of the inactive constraints). This implies that the proposed control structure will not operate in the feasible region and therefore should be rejected and an alternative control structure tested.
(15) (16)
(12) (13)
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2683
SIX = AZ + Ba + CZ + Ep
(21)
0 = Fa + Ga + HZ + Lt,
(22)
o = MZ + NU + oz+ RP + I;
(23)
where i; and ii are approximations to the maximum constraint back off. Settings =j w and considering Z, Z,and a in their complex forms Zr + jz,, etc., decomposing into real and imaginary parts and dropping j , the frequency response equations may be rewritten as
-Epr
+
AZ, Bii, + CZ, + d Z c
+ Gii, + HZ, -Rp, = MZ,+ Nii, + 0 2 , + I;, -Lp, = FZ,
(24)
(28) (29)
Before the problem formulation is complete, the following relationships must be specified (i) the perfect control requirements and (ii) the functional relationships p(ar,;c) and War,ac)3.1. Perfect Control Specifications. There are two possible types of perfect control specifications: (1) a manipulated variable is inactive (unused for control) and (2) a measured variable is controlled perfectly. In both cases, the appropriate element of u or z is constrained to zero. If a perfect control specification is not made on a variable, then that variable is unconstrained (corresponding to uncontrolled measured variables or active manipulatedvariables). We introduce the integer vectors X and X to describe the perfect control specifications, where
X j = 0 if control variable j is inactive
also be used to account for the costs of installing actuators and sensorsfor control purposes by modifyingthe objective function to min J = aTxO+ ,BTzo+ -yTd
uod,X,X
+ eTX + C$TX
(33)
where t and C$ are respectively the actuator and sensor costa. So far, this problem is a mixed integer linear programming (MILP) formulation, as all of the constraints are linear. Problems of this type may be solved using the branch and bound algorithm (Beale, 1377). However, we require the functional relationships for p(w) and n ( w ) before the problem is fully specified. The following discussion will only consider p(w), as the arguments presented also extend to ii(o). 3.2. Linear Lower Bound on i. Ideally, i ( w ) is given by the maximum expected amplitude of oscillation of the slack variables. From the previous section, this is given by I A W lP(0) (34) where A is the appropriate perfect control transfer function between a andp. This calculation requires the evaluation of complex magnitudes, a nonlinear calculation. Use of this constraint will destroy the linear properties of the problem formulation and will result in a complex mixed integer nonlinear program (MINLP) with discontinuous constraints and first derivatives, a particularly difficult problem to solve. Therefore, it is desirable to use some approximation which maintains the linear properties but still allows location of the optimal solution. Two properties may be used to circumvent this problem (Narraway, 1992). 1. Consider the linear programs LP1 and LP2:
min J(LP1) = aTq P
b 2 Aq
+ &q)
(35) (36)
and
X i = 1 if measurement i is perfectly controlled
min J(LPB) = aTq
(37)
bIAq+q
(38)
P
Using these definitions of X and X , and noting that the perfect control requirements may be written as IZil = 0 or Zri = Z , i = 0 for measured variables and lajl = 0 or ar,j = a,: = 0, then the perfect control constraints may be formulated as:
where iis defined as above. If the vector of linear variables q can be constrained such that
-(1 - Xi)$ IZr,i I(1- Xi,$ vi = 1,2, ...,n, (30) -(1- Xi,$IZC1 I(1- Xi)$
where zh is an upper bound on the expected magnitude of 121. a h is a similar bound on la1 and can also be used as an upper bound on the manipulated variable rangeability. To ensure that a square controller is implemented, nu perfect control specifications are required, resulting in the following constraint: (32)
This constraint can be removed if it is desired to consider nonsquare controllers. Finally, the integer vectors can
and it can be shown that the lower bound on q fo_r p y given vector q (4' = rq) is also a lower bound on p ( p I ql), then the following inequality holds for the optimum objective functions of the linear programs: J*(LPl) I J*(LP2) (41) as qi will always be less than p i for an active constraint i at the optimum q*. 2. Whenever an integer solution is generated by the MILP solution procedure, a control structure is fully specified. That is, the permutation matrix Q can be generated from X and X,and therefore the perfectly controlled objective function corresponding to the integer solution (control structure) can be calculated using the methods from the previous section.
2684 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
Using these two properties, an algorithm for the rigorous solution of the control structure selection problem can be proposed. Assuming a linear lower bound on the maximum slack variation can be calculated, set p to this lower bound in the MILP formulation. That is, use p_as an approximation of the maximum slack variation a. This MILP formulation calculates optimistic objective functions (by the first property). When the MILP solution method produces an integer solution, use the perfect control analysis from the previous section to calculate the actual objective function for the given control structure (by the second property). The structure and cost generated represent a candidate solution. Accordingly, we use the actual objective function for comparison with the entry cost for integer solutions in the MILP solution method, where we are only interested in finding structures with economicsbetter than the entry cost (the objective function of the best candidate structure located so far in the MILP solution). This is an incomplete description of the algorithm, as a few more steps are required to ensure the optimal control structure is chosen, but it is sufficient as an initial outline for the justification of the use of the MILP approximation. To maintain the desirable features of an MILP formulation and_still obtain a rigorous solution, a linear lower bound for p must be developed. A possible lower bound is given by the following steps. 1. Consider a single element of the vector obtained by solution of eq 24-29 for an arbitrary disturbance p = pr +jp, (assuming perfect control specifications have been made on elements of x , z, and u to use the remaining nu degrees of freedom). For this arbitrary disturbance, it can be stated that
a,
max Ii(o)l1la(w)l (42) 2. A lower bound on I&)l is obtained from the complex form of a(w) = + j&:
a,
la(w)l 2 lad
(43)
1bJ
(44)
l&)l
3. A linear lower bound on the absolute value of given by
a, is (45)
lad I -a,
(46)
and similar bounds may be written for I&/. For convenience of calculation, the arbitrary disturbance is taken as pr = p(o)and p c = 0. Using the above sequence of lower bounds, we obtain the following linear lower bound on the maximum slack variation p:
- -
P 2
P 1
Q,
- p I -a, - -
p
1 -a,
some screeningtechnique to eliminate infeasible structures without examination. This may be achieved by the introduction of structural connectivity constraints. 3.3. Connectivity Constraints. The current formulation allows selection of any nu perfectly controlled variables as a valid integer solution. This is a combinatorial problem of selecting nu variables out of nu + n,, where
For example, for a problem of selecting 20 degrees of freedom from 40 manipulated and measured variables, the number of integer feasible solutions is 1.378 X loll, which is a large problem. It would be desirable to reduce the feasible integer search space using structural information about the relationships between the inputs and outputs (input/output connectivity). Ideally, we should ensure that the transfer function between the active manipulated variables and the perfectly controlled measurements is structurally nonsingular. However, we have used a set of weaker structural connectivity constraints in this problem representation. In words, these constraints are as follows: For every measured variable selected to be perfectly controlled, at least one manipulated variable which affects the measured variable must be active. Similarly for each active manipulated variable selected, at least one measured variable that it affects must be perfectly controlled. Given the structural transfer function G, generated from the transfer function at frequency o with ijth element gij = 1 if there is a nonzero gain between manipulated variable j and measured variable i and gij = 0 if the gain is zero, it is possible to generate the connectivity constraints. First consider the constraint that for each perfectly controlled measured variable there must be at least one active manipulated variable that affects the measured variable. This constraint may be stated as n"
Cgijxj- x i I o
vi = 1 , 2 , ...,n,
(52)
J=1
The correspondingconstraint that each active manipulated variable must affect at least one of the perfectly controlled measured variables is n,
Cgijxi- xjL o vj = 1 , 2 , ...,nu
(53)
i=l
With these connectivity constraints, the complete MILP formulation of the problem can be written. 3.4. Full MILP Formulation.
(47) (48) (49)
n*
nu
i=l
1=1
c x i- E X j = 0
(50)
where {br,&j are obtained from the solution of eqs 24-29. A similar result holds for ti. Using this, an MILP approximation of the control structure selection problem may be written. However, it would be desirable to include
2: 1 Xi$
- ZCi
(55)
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2685
Vi = 1,2, ...,nz
g . . X . - X i2 0
Et, I
1-1
n.
+ Bu, + Cz, + Dd 0 = Fx, + Gu, + Hz, + Kd a I MX, + NU, + + ~d + I; 0 = Ax,
02,
u1Iu, - ii
uh I u,
+ ii
d‘ Id I dh
- E ~ ( o )= AZ., + Bar+ CZ, + d,Z, - L ~ ( o )= FZ,+ Gar + HZ, -Rp(w) = Ma,
+ Na, + OZ,+ I;,
+ AZ, + BU, + CZ, 0 = FZ, + Gii, + HZ, o = MZ, + ~ i i+, 02, + I;,
0 = -wIZ,
-
-
P Iu.,
-
-
-
-
p I -u,
P I6,
-
-
p 2-0,
B I ii, ii I-fir ii I a,
ii I -a, Breadth first branch and bound (Beale, 1977; Nemhauser and Wolsey, 1988) was found to have adequate performance when applied to the control structure selection MILP, and it has been employed. 3.5. MILP Control Structure Selection Algorithm. Two problems have been posed in this paper: (i) a perfect control problem (section 2.1) whose solution yields an estimate of the attainable dynamic economics for a given control structure and (ii) a MILP problem (section 3.4) that generates optimistic (lower) bounds on the perfect control dynamic economics of partially and fully specified control structures. At this point, it is desirable to
amalgamate these to solve the problem of selecting the optimal perfect control structure. This problem may be solved by using a hybrid MILP technique as follows. Solve the MILP problem (section 3.4) using breadth first branch and bound until an integer solution is found with JMless than the entry cost (including the fully relaxed solution). However, do not continue the branch and bound at this point. Instead, solve the perfect control problem (section 2.1) to generate Jpc as the objective function corresponding to the integer solution and use this to compare with the entry cost. If Jpc is better than the entry cost, then the candidate structure is the best structure located in the search to date, and the entry cost is updated to Jpc.Whether or not the candidate structure replaces the best structure, add an integer cut for the candidate structure to ensure that it is not re-examined, and then repeat the branch which generated the integer solution. If no further integer solutions are generated with objective functions J M less than the entry cost, then proceed with the branch and bound. That is, use J Mto determine if branches should be pruned as normal, and only examine the perfect control problem if there is an integer solution with J M better than the entry cost. The hybrid approach is used as the MILP only generates an optimistic estimate of the perfectly controlled objective function. This leads to a requirement to re-examine branches once an integer solution has been located (true branch and bound never re-examines branches). This is because from some arbitrary branch level the MILP may find an optimal integer solution J L where J L C JO(the current entry cost), with a corresponding perfect control objective function JL. From the same branch, there may also be another feasible integer solution 3 where it also holds that 3M< JOand pM C J L (3M IJM by definition of the optimum), with perfect control objective function In this case, it is possible that ppc IJ L , and either or both may be less than Jo. That is, the optimal linear solution may not correspond to the optimal perfect control solution for any particular branch. This leads to a slight modification of the branch and bound algorithm to ensure that the optimal perfect control solution is found. As a final point, it should be emphasized that the MILP analysis should only be used as a screening tool for prediction of economically sound control structures. The reasons for this are as follows: (i) the analysis does not examine the controllability of the process, (ii) the analysis only calculates an estimate of the dynamic economics, and (iii) the linearization only has a limited accuracy for perturbations from the linearization point. Because of these limitations, it is proposed that the analysis should be used as follows. The design engineer chooses the number of control structures which are to be examined in detail as an input to the analysis program; the program then generates this number of control structures with the best dynamic economics. These structures are then all subjected to controllability analyses or are used as control structures for nonlinear dynamic economic analysis. The results of these analyses should then be used to select the best control structure. 3.6. Implementation. The above algorithm has been implemented using MINOS (Murtaghand Saunders, 1983) as the LP solver and using modified branch and bound routines developed by Shah (1992) for the solution of the MILP problems. All equality relations in the MILP were eliminated to improve speed of solution and have only been included for convenience of representation and completeness. The branch stack was ordered by objective function. A number of rules for the selection of the
r
4,.
2686 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Table I. Steady-State Active Constraint Summary constraint E l liquor vol lower bound E2 liquor vol lower bound product concn VV1 pressure drop VV2 pressure drop VV3 pressure drop
Lamange multiplier 0.0 $/m3 0.0 $/m3 71 430 $/(weight fraction) 5880 $ / P a 8820 $ / P a 12 200 $ / P a
Table 11. Potential Measured Variables on the Double-Effect Evaporator
3.5% A Feed
10% A Product
Figure 1. Double-effect evaporator flow sheet.
branching variable were tested, and a routine based on the solution analysis of Shah (1992) was found to perform adequately. This section has proposed the dynamic economic analysis of Narraway and Perkins (1991) as a basis for a hybrid MILP control structure selection problem, including trade-off between control costs and benefits. The performance of this technique on two case studies is demonstrated in the following section. 4. Case Studies 4.1. Double-Effect Evaporator. This case study considers a concentration operation based on the doubleeffect evaporator problem outlined in the SPEEDUP Casebook (Prosys Technology, 1989). The concentration of a solution of A is to be raised from 3.5% to 10%. The process objective is to optimize the process throughput versus the operating costs. A process flow diagram is given in Figure 1. The liquor is fed from an upstream process, into the first evaporator through the feed pump and feed valve. Utility steam is supplied to the first effect, and steam to the second effect is supplied by the first effect vapor. The liquor product of the first effect is fed to the second effect through the intermediate pump and intermediate valve. The product liquor is then fed through a product pump and valve to a downstream storage facility. LV1, LV2, LV3, VV1, VV2, and VV3 are manual valves, which may be replaced by control valves if required. A dynamic model of the process was developed using SPEEDUP, where the process model, physical characteristics, constraints, and objective function of the plant are given in Narraway (1992). For the steady-state-process optimization, it was assumed that the evaporator plant was available as a scrapped unit. Therefore, it was decided to optimize the annual operating costs, as there was no capital consideration. The process objective function was given by maximize (value of A produced - steam costs pumping costs). A number of constraints were imposed on the operation of the process: (1)The liquor volume in the evaporators must remain within the liquor freeboard (above the steam chest and below the vapor hemisphere of the evaporator: 3.25 Ivolume I 4.7 m3). (2) There must be a 5 O C temperature difference between the steam chest and the process liquor for acceptable heat transfer. (3) The solute A is subject to thermal decay, and as a result the maximum possible temperature in the evaporators is 95 "C. (4) The process must produce at least 3.5 kg/min of A in a solution with a minimum concentration of 10% by weight of A. (5) The pressure drop over all vapor line valves was required
E l steam feed rate E l steam chest temp E l pressure E l liquor vol E2 steam chest pressure E2 temp E2 outlet vapor rate E2 freeboard liquor head LV2 liquor flow rate LV2 solute flow rate L V3 liquor flow rate LV3 solute flow rate
E l steam chest pressure E l temp E l outlet vapor rate E l freeboard liquor head E2 steam chest temp E2 pressure E2 liquor vol LV2 water flow rate LV2 liquor composition LV3 water flow rate L V3 liquor composition
to be at least 5 kPa. (6) The process pumps were required to operate within the design limits of maximum flow, maximum pressure, and minimum pressure across the pump. (7)The squares of the valve coefficientshave simple lower and upper bounds of 5.0 and 1000.0. The optimal profit from the process is $457 000 per annum, with the active constraints and Lagrange multipliers summarized in Table I. 4.1.1. MILP Selection of Optimal Control Structure. Prior to linearization of the nonlinear model, the potential measured and manipulated variables are identified. For this case study, there are 23 potential measured variables, summarized in Table 11. Note that it has been assumed that the feed liquor flow and composition cannot be measured directly and that the liquor composition and/ or component flows can only be measured at the intermediate and product valves. The six valve coefficients were identified as potential manipulated variables. This results in 29C6or approximately 470 000 potential control structures. After identifying the potential measured and manipulated variables, the nonlinear model was linearized at the steady-state optimum using the CDI linearization package provided with SPEEDUP. The six valve coefficients were chosen as inputs of the linear system, and the measured variables, constraint slacks, and objective function were chosen as outputs, yielding a linear system appropriate for MILP analysis of control structures. The next step is costing the manipulated and measured variables for use in the MILP. As the process has been optimized based on annual operating profits, it will be necessary to annualize the installation costs of the control equipment. It will be assumed that maintenance of control equipment will increase its cost by 50% of the annualized installed cost. The plant is expected to run for 15 years at an interest rate of 20% for time value of money calculations. The total annualized cost CAfor equipment with an installed cost of CI is given by (assuming first payment at the end of year 1 and no scrap value)
CA O.32C1 (80) The next step is to assign appropriate costs for installation of sensors and actuators. The cost assignments are summarized in Table 111. The composition measurement cost is based on the assumption that accurate compositions can be readily calculated from the refractive index of the liquor. Component flow measurement cost is assumed to be the sum of composition and flow measurement costs.
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2687 Table 111. Measurement/Value Costs for Double-Effect Evaporator variable installed cost, $ valve 2000 liquor flow 2000 vapor flow 3000 temp 3000 pressure 4000 vol/ head 4000 composition 5000 component flow 7000 Table IV. Base Case Disturbance at disturbance feed pressure feed temp feed composition product liquor pressure steam temp
w
Table V. Optimal Control Structures actuators sensors vv3, LV2 T ,P vv3, LV2 T,W vv3, LV2 P,w vv3, LV2, LV3 T , P,Ft vv3, LV2, LV3 T,Ft, W
= 0.006 cycle/min
magnitude 10 kPa 5O C 0.35 % 5 kPa 0.5 O C
3
2 1.5
le41 7 /
i
I
I
I
Y n
le02
le01
I
I
4
3 2 15
le*M) le 03
cost, $ 8 960 9 280 9 600 10 240 10 560
The last step before solving the MILP is assigning appropriate disturbance magnitudes and frequencies. The disturbance magnitudes are given in Table IV, based on upstream and downstream processing and on utility variation (the variation in steam temperature corresponds to 2% variation in the steam pressure). Given this information, the MILP was solved for the five best control structures, summarized in Table V. T and P are the temperature and pressure of the liquor side in the second effect; W and Ft are the product liquor composition and total flow. The first three control structures consist of the vapor product valve VV3 and the intermediate liquor valve LV2 being used to control either effect 2 temperature and pressure, effect 2 temperature and product composition, or effect 2 pressure and product composition. All three of these control structures are seeking to operate on the product composition constraint and on the constraint on the pressure drop across VV3. The latter fixes the pressure in the second effect and specifying any two of pressure, temperature, and composition automatically fixes the third due to vapor-liquid equilibrium effects. Thus, the three control structures will operate the plant at the same steadystate conditions, with the only differences in economics being the costs of the sensors. By examining the Lagrange multipliers of the constraints (Table I), it can be seen that these control structures correspond to a “good” set of control objectives. The Lagrange multiplier of the constraints is $71 400/weight fraction for composition and $12 2OOlkPa for VV3 pressure drop, with the next largest multiplier $8800/kPa on VV2 pressure drop. Clearly the MILP has selected the measurements which allow control at the constraints with the largest effect on process economics. From this, it can be deduced that the return from controlling the pressure drop VV3 and the product composition justifies the cost of installation of control loops, but controlling the pressure drop over VV2 does not return sufficient benefit to justify installing a control loop. Note that although the liquor holdups have increased to allow for dynamic process variation they have no effect on the process economics, as working capital has been ignored for this problem. While the option of regulating
H
5
IC&
le41
IC&?
RcqucnsY (CPlUlrmn)
Figure 2. Minimum condition number plot for double-effect evaporator.
the levels in the evaporator was open, the optimizer did not exercise it in the best structures. Inherent regulation was found to be sufficient for the disturbances considered. No economic benefit results from direct control of these variables. As discussed previously, the linear dynamic economic analysis requires some controllability analysis to test the validity of the perfect control assumption. Therefore, minimum condition number plots were generated for each control structure (Figure 2). The condition numbers are plotted over a frequency range of 1cycle/s (process noise) to 1 cycle/day (diurnal variation). Additionally, the transmission zeros for all of the control structures were calculated to check for right half-plane zeros (RHPZ). Only the VV3,LV2,LV3,T,P,Ft structure contained a RHPZ at 1.1 X lP, and as this structure was ranked fourth by economics, it was not examined any further. Examining the condition number plots, it can be seen that for the best economic structure, VV3,LV2,TQ, the minimum condition number varies from 2 at low frequency to 100 at high frequency, suggesting that it will be difficult to achieve tight control of the process subject to highfrequency disturbances (process noise). The second-best configuration W3,LV2,T,X would achieve significantly better control at high frequencies, with a maximum condition number of 1.5 over the frequency range. The difference in cost between the two control configurations is $320 per annum-the cost difference between a pressure sensor and a composition sensor. In this case, the design engineer would be able to make a decision that, for the benefits of improved controllability, the second control configuration is significantly better, as it will reject disturbances well across the entire frequency range for a minimal increase in cost. Because it has a minimum condition number close to 1,it should be possible to achieve very near perfect control. 4.1.2. ComputationalResults. It would require about 70 000 Sparc I1cpu seconds for explicit enumeration of all possible solutions (i.e., solution of the perfect control subproblem for each potential control structure). The optimal solution was found in 4600-5000 cpu seconds using the MILP method, and the five best solutions were found within 8500-9900 cpu seconds, resulting in selection of the optimal (first) control structure approximately 15times faster than by explicit enumeration. This is a significant speed up, but further improvements should be possible
2688 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
Collector
Rougher
=
c
Table VI. MIM Steady-State Constraint Summary Lagrange constraint slack multidier, %lo00 0.0 117717% copper product grade 0.0 2.55/ma rougher cell 1vol upper bound 3.646 ms 0.0 rougher cell 2 vol upper bound 0.0 71.58/m9 cleaner cell 1vol upper bound 0.421 m3 0.0 cleaner cell 2 vol upper bound 1.524 ms 0.0 cleaner cell 3 vol upper bound 34.19/ms 0.0 retreater vol upper bound scavenger tailing* min water 0.0 21.3lton water recleaner tailings min water 39.4 ton 0.0 165.9 ton 0.0 retreater tailings min water 0.0 5781air factor rougher aeration upper bound 331lair factor scavenger aeration upper bound 0.0 4242/air factor cleaner aeration upper bound 0.0 0.0 recleaner aeration upper bound 1051air factor 0.0 retreater aeration upper bound 1974/air factor 0.0 collector upper bound 2381% change 0.032 tonWh 0.0 cleaner cell 1tailings valve ~~
e-7
Feed
I
7
R treated
Cleaner
Mill
+ i
I
Concentrate Figure 3. Mount Isa Mines froth flotation circuit.
I
by tightening the problem formulation. Performance issues will be discussed in greater depth in the next case study. 4.2. Mount Isa Mines Froth Flotation Circuit. This case study is based on the Mount Isa Mines froth flotation circuit studied by Fewings, Slaughter, Manlapig, and Lynch (1979), who considered the design of a process control system for the circuit. This case study considers the problem of designing the circuit for optimal steadystate and dynamic performance. The simplifications used in developing the case study as a design problem result in a process which does not reflect all of the properties of the system studied by Fewings et al. but does highlight aspects of the MILP control structure selection algorithm. The mathematical model utilized in this study may be found in Barton and Perkins (1988). The froth flotation circuit is used to concentrate 5 000 000 tons per annum of a 3.2 7% copper chalcopyrite ore. Figure 3 summarizes the process flow sheet. The process is fed from a wet grinding circuit, with collector added to the circuit feed to improve the flotation characteristics of the valuable material. There are five flotation banks: rougher, scavenger, cleaner, recleaner, and retreater. The banks typically consist of 4-12 flotation cells. Each flotation bank has an adjustable aeration rate that sets the rate for all cells in the bank. Makeup water may be added at the entrance to each bank to ensure acceptable slurry flow. All cell tailing flows are set by valves on the tailing lines. The regrind mill serves to reduce the fraction of heavier particulate material and thus improve the mineral flotation characteristics. The full process model is given in Narraway (1992). The process objective is to minimize copper loss while producing a concentrate a t a minimum acceptable grade for downstream processing. The objective function used was of the form minimize (value of copper lost in tailing + production costs + capital costs) The use of the copper loss as a measure of productivity warrants brief discussion. The copper loss in the tailings is not processed in the smelter and thus represents loss of revenue. Therefore, it is appropriate to balance the potential value of copper in the waste stream against the processing costs. Note that it is only a potential value, as no smelter process is able to recover 100% of the metal in an ore, but it still provides a useful measure of process losses for use in the objective function. The optimization problem also requires constraints and the identification of the free optimization variables. The constraints imposed on the process were as follows: (1)
Table VII. Disturbance Magnitudes for MIM Case Study variable magnitude, tonlh water 12 fast floating valuables 2 slow floating valuables 2 fast floating gangue 1 slow floating gangue 31
The circuit concentrate must have a grade (mass percentage of copper in the solids) of a t least 20%. (2) All streams in the process must contain at least 50 % by mass water for acceptable slurry flow. (3)The pulp volumes of all cells must lie within the available range of 20-80 m3. (4) The valve coefficients for tailings flows must lie within the range 0-1500 for the first and second cells in multicell banks and within 0-150 for the last cell in multicell banks and single-cell banks. (5) The aeration rate for each cell must lie in the range 0.5-1.5. The 11tailings valve coefficients, the five aeration rates, the five makeup water flows, and the collector addition rate were identified as free optimization variables. The optimum process design resulted in a minimum processing cost of $19 834 000 discounted to year 0. A constraint summary is given in Table VI. Six potential disturbances to the process were identified: the five component feeds and the feed copper assay. It was decided not to use the copper assay, as the proportion of metal in the valuable component varies slowly and is more appropriately treated as an optimizing control disturbance. As mineral processing plants are subject to frequent upsets as different pockets are mined and due to the variability of upstream processing, a disturbance frequency of 1 cycle/h was used for the MILP analysis (corresponding closely to the dominant time constant of the process). The disturbance magnitudes are summarized in Table VII. An open loop linear disturbance analysis was carried out, and the economic giveaway was estimated at $1 200 000 or roughly 6% of the cost of the plant, indicating a useful potential improvement in economics through addition of a control system. The next step was identification and costing of potential measured and manipulated variables. The measurements and manipulated variables were chosen to resemble those available in the Fewings et al. case study. This resulted in a problem with 11potential manipulated variables and 19 potential measured variables which are summarized in Table VIII. This problem has 30C11= 54 000 000 potential control structures. The problem also initially included linearizations of all 45 constraints used in the steady-state model. However, this slowed the problem solution sig-
Ind. Eng. Chem. Res., Vol. 32,No. 11, 1993 2689 Table VIII. Potential Measured and Manipulated Variables on the MIM Circuit potential manipulated variables rougher exit tailings valve scavenger exit tailings valve cleaner exit tailings valve recleaner exit tailings valve retreater exit tailings valve rougher aeration rate scavenger aeration rate cleaner aeration rate recleaner aeration rate retreater aeration rate collector addition rate
potential measured variables rougher cell 3 vol scavenger cell 3 vol cleaner cell 3 vol recleaner vol retreater vol rougher cell 3 concentrate flow scavenger cell 3 concentrate flow cleaner cell 3 concentrate flow recleaner cell 3 concentrate flow retreater cell 3 concentrate flow retreater concentrate grade retreater tailings grade retreater recovery recleaner tailings grade recleaner recovery process concentrate grade process recovery process tailings grade process loss
Table IX. Optimistic Instrumentation Costs for MIM Circuit variable installed cost, $ slurry valve 2000 10 000 aeration rate controller 1 000 collector addition recovery 15 OOO grade 15 OOO concentrate flow 1000 4000 vol
nificantly, and as a result a reduced constraint set was employed based on those constraints which are either active or near active at the nominal steady-state optimum, which are summarized in Table VI. The final results were then tested to ensure that they satisfied the full constraint set. As a final note, it is assumed that there is sufficient freeboard in the cells to absorb uncontrolled variation in the pulp phase volume, unless the cells volumes are on their upper or lower design limits. That is, the cells are designed to allow for pulp volume variation. This implies that the cell volumes will only be controlled if there is an economic incentive. Note also that as the cells are controlled by square root law valves, they will tend to be self-regulating in practice. For the control equipment, two cases were considered to examine the sensitivity of the solution to instrumentation cost. The costing was carried out on the basis of an installed cost with a factor allowing for maintenance of the form: CT = C I ( ~+ M) where CT is the total instrumentation cost, CIis the installed cost, and Mis the maintenance factor. The two cases and their results are now discussed in detail. 4.2.1. Case l-Optimistic Sensor/Actuator Costs. The first case considered optimistic instrumentation cost estimates, summarized in Table IX. A maintenance factor, M, of 1 was used for all case 1 instrumentation costs. The five best control structures and their economic giveaways (relative to the steady-state optimum) are summarized in Table X, with minimum condition number plots in Figure 4. The condition numbers are plotted over a frequency range corresponding to process noise at 0.5 midcycle to 4 dayslcycle corresponding to variation in mining locations. None of the structures contained right half-plane zeros. The first step in the analysis of results is to note the difference in economics between structures 3 and 5, which differ by a single volume measurement and slurry valve.
Y
: t I 1.80
le-02
le-01
le43
let001
le42
Frcsusncy (cyclUmour)
Figure 4. Minimum condition number plot for MIM case 1. Table X. Optimal Control Structure for Optimistic Costs. I n Each Structure, The Left-Hand Column Shows Actuators and the Right-Hand Column Sensors structure 1 cleaner K, recleaner K, retreater K,
$216 450 cleaner vol product grade retreater vol
rougher K,
rougher vol
structure 2 $225 840 cleaner K, cleaner vol recleaner Ku product grade retreater K, retreater concentrate flow rougher K, rougher vol
,
structure 3 $229 260 structure 4 $238 260 cleaner K, cleaner vol cleaner K, cleaner vol recleaner K, product grade recleaner K, product grade retreater K, retreater vol retreater K, retreater concentrate flow structure 5 cleaner K, recleaner K, retreater K, rougher K,
$244 020 cleaner vol product grade retreater vol scavenger vol
The cost of the additional equipment is (4000 + 2000) X (1+ M) = $12 000. The difference in economics between the structures is 244 020-229 260 = $14 760, which implies that addition of this control loop has actually degraded the control performance by $2800. The reasons for this objective function degradation are explained in the discussion on marginal costs in the followingsection. From this, it can be concluded that control structure 5 is an undesirable structure and will not be examined further. From the remaining four cases, it is clear that there is significant economic benefit to be obtained from controlling the process-potentially, the open-loop economic giveaway can be reduced by 70-80 ?6. The next step is to study the controllability indicators to determine the attainability and ease of achieving this control. Examining the condition number plot (Figure 4),it can be seen that there is practically no difference in condition numbers between the control structures, indicating that the structures have similar controllability properties. Given that there is no significant difference in controllability between the first four structures, it is possible to state for this case that control structure 1 is the best control structure for further development, and as its minimum condition numbers are close to one, very good control should be possible. This is confirmed by considering the
2690 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Table XI. Pessimistic Instrumentation Costs for MIM Circuit variable installed cost, $ maintenance factor slurry valve 10 Ooo 3 aeration rate controller 10 Ooo 1 collector addition 2000 1 30 Ooo 1.5 recovery 30 0o0 1.5 grade concentrate flow 4000 4 8000 4 vol
location of the manipulated and measured variables on the plant. Each valve and measurement lies on a separate bank, where the banks act as buffers for process disturbances and control interaction. This implies that the control should be largely decoupled, and tight control should be possible with a relatively simple control system. However, it must also be remembered that a reduced set of measured and manipulated variables was considered for the MILP, due to constraints on solution time. An extension of this analysis is to use the results of the MILP to determine the desired control objectives and check if any of the unused manipulated and measured variables could be used to improve controller performance. Some of the process control objectives may be determined directly from selected measured variables, in this case controlling the product grade on its constraint and controlling the retreater volume at its maximum, while the remaining objectives must be inferred from the effects of the control structures on the constraint slack variations. Consideringthe first four structures, and control objectives were determined to be: (i) rougher feed cell maximum volume, (ii) cleaner feed cell maximum volume, (iii) retreater maximum volume, and (iv) product minimum grade. From this, it was concluded that it was worth considering the rougher feed cell volume and tailings valve and the cleaner feed cell volume and tailings valve in the economic analysis. Various combinations of these and the existing set of measurements and manipulated variables demonstrated that the feed cell valves could not be used for control, as they would saturate. However, measuring the feed cell volumes could improve process performance marginally. In fact, one further control structure with economics better than structure 1 was located through this method a giveaway of $215 570 using the retreater, cleaner final cell, and recleaner valves to control the product grade, cleaner feed cell volume, and retreater volume. This indicates that careful use of the MILP with a subset of the available control structures may yield enough information to improve the control structure. However, this approach should be used cautiously, and only if solution time is of critical importance, as the optimal control structures may still be missed. 4.2.2. Case 2-Pessimistic SensorIActuator Costs. The second case, summarized in Table XI, considers more realistic plant costs, allowing for the high replacement rates of equipment in contact with slurries and the high maintenance overheads associated with equipment such as X-ray fluorescence composition measurement and magnetic flow meters and densimeters. The slurry valves are given a high maintenance factor as they are manipulating abrasive flows. The low maintenance factor on the grade and recovery measurement does not imply that the maintenance factor is low but that a smaller maintenance factor is needed for an item with a high installed capital cost. The reverse argument holds for the magnetic flow and level meters, which have a low installed cost but require high maintenance for accurate calibration and
Table XII. Optimal Control Structure for Pessimistic Costs structure 1 cleaner K, recleaner K, retreater K,
$441 260 structure 2 $446 260 cleaner vol cleaner K, cleaner vol product grade recleaner K, product grade retreater retreater K, retreater vol concentrate flow
structure 3 cleaner K, recleaner K, retreater K,
$496 840 cleaner vol product grade retreater concentrate flow rougher vol
rougher K,
structure 5 cleaner K, recleaner K, retreater K, retreater air
structure 4 cleaner K, recleaner K, retreater K,
$501 450 cleaner vol product grade retreater vol
rougher K,
rougher vol
$502 510 cleaner vol product grade retreater vol retreater concentrate flow
performance and hence have a high maintenance factor. The aeration control and collector control have low maintenance factors as they are relatively reliable and do not contact the abrasive streams. The aim of this case is to determine if the choice of instrumentation costs can have a significant impact on the choice of the control structure. The five best control structures are given in Table XII. The first four control structures correspond to the first four structures for the optimistic case, but in reverse order, and as a result condition numbers are not plotted for this case. The fifth structure again results in a degradation of control performance over structure 1, with an increase of $1250 in the dynamic giveaway after taking the control costa into account. Therefore, the fifth structure was again determined to be a poor control structure and ignored in further analysis. The result of this case indicates that the cost of the instrumentation is significant in the choice of control structure, as the economic order of the structures has been reversed between cases 1 and 2. This is due to changes in the marginal costs of the instrumentation, where the marginal cost is d e f i e d as the instrumentation cost minus the improvement in economic performance from instrumentation. The MILP minimizes the sum of the marginal costs of the instrumentation. Clearly, the marginal cost is dependent on the instrumentation cost assigned by the engineer, and as the cost of avalve/measurement pair rises for a given control structure, it will reach a point where either its marginal cost is positive or its marginal cost is greater than that of an alternate valve/measurement pair. In the first case, the pair should be removed from the control structure, as it will result in degradation of control performance (aswas the case for structure 5). In the second case, the alternate pair should replace the current pair, corresponding to the order reversal seen for the first four structures between the pessimistic and optimistic cost cases. From this, it can be concluded that if accurate instrumentation costs are not available, then a sensitivity analysis of the results should be carried out to determine if the marginal costa of the controllers will be significant, and if so, a number of potential control structures should continue to be examined until more accurate data are available. In this case, it would be recommended that at least structures 1and4 be examined further, and preferably also structures 2 and 3. As the first four structures are identical to the optimistic cost case, the control objectives will also be the same. Reexaminingthe variable set for alternative control structures
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2691 Table XIII. Major Iterations to Solution and To Prove Optimality iterations for case 2 iterations for case 1 to to prove to to prove structure solution solution solution solution 12 051 295 12 400 1 315 14 756 1103 13 372 2 189 3 448 15 784 1698 28 442 4 229 19 251 3324 30 153 5 605 21 614 1832 30 557
revealed two new structures which outperform structure 1: (i) cleaner final cell valve, recleaner valve, and retreater valve controllingcleaner feed cell volume, retreater volume, and product grade, respectively, yielding a giveaway of $432 170, and (ii) the same valves controlling cleaner feed cell volume, retreater concentrate flow, and product grade, yielding a giveaway of $427 570. The second case corresponds to an improvement of approximately $13 000. This result again emphasizes the possibilities of using the MILP as a method for inferring control objectives. Finally, in both instrumentation cost cases,the economic analysis indicated that control of the process should decrease the open-loop economic giveaway by between 60 ’% and 80% ’ or by approximately 3.5-4.5 ’% of the plant’s steady-state costs. In spite of the fact that this is an optimistic estimate (based on perfect control), this result indicates that there is significant incentive for development of a control system for the plant even if the most pessimistic control costs are used. It is also likely that performance similar to that estimated should be possible due to the low condition numbers of the control structures. A more cautious attitude to control would have been required if the results of the sensitivity analysis had indicated that the pessimistic and optimistic economics were closer to the open-loop economics. 4.2.3. Computational Results. The number of major (MILP) linear programming iterations required to reach the solutions and the number of major iterations to prove optimality are summarized in Table XIII. The MILP solved approximately 2200 master linear programs per day on a dedicated Sparc 11, requiring approximately 5.5 days to prove the optimal solutions for both cases, 10 days to prove the five best structures for case 1, and 14 days to prove the five best structures for case 2. A structure is proved once the MILP predicts a giveaway worse than the perfect control giveaway of the structure. This may seem an extremely long time for solution, but it must be compared with the time required for explicit enumeration of the problem. Using the currently implemented perfect control routine to examine each potential control structure (Le., 54 000 000 perfect control evaluations),it is estimated that it would require 3-6 years to complete, depending on the number of infeasible structures. However, the current perfect control routine is fairly inefficient, and it is estimated that by appropriate variable elimination the time for explicit enumeration can be reduced to between 70 and 100 days of Sparc I1 time. From this, it can be concluded that the current MILP implementation is 1 order of magnitude faster than explicit enumeration at locating the optimum solution and about 5 times faster at locating a useful number of control structures for further design analysis. The current MILP formulation is relatively inefficient, solving only 2200 linear programs per day. By improving the LP description, it should be possible to achieve significant improvements in speed. The second area for improvement is the number of master linear programs
required to prove solutions: in this problem, to prove a solution required from 10-90 times the number of master LP evaluations required to locate the solution. This appears inefficient, suggesting that it should be possible to improve the termination criteria. It should also be possible to improve the MILP formulation to reduce the integrality gap of the problem.
5. Conclusions Considering both case studies, it can be seen that a systematic method for assessing control structures has been developed. It may be summarized as follows: (1) Develop a nonlinear dynamic model of the process. (2) Pose and solve a steady-state optimum design problem. (3) Identify potential manipulated and measured variables and linearize the process at the steady-state optimumusing the manipulated variables as inputs and the measured variables and constraint slacks as outputs. (4) Identify the most significant disturbance frequency and magnitudes and cost the sensors/actuators. (5) Solve the MILP for a number of control structures. (6) Carry out a controllability analysis of these structures and use in conjunction with the economics to choose the best control structure. (7) Carry out a sensitivity analysis as appropriate. (8) If an incomplete measurement and manipulated variable set was employed in the MILP, infer the control objectives and attempt to determine improved control structures. The last point has not been discussed in great depth, but in both case studies it was possible to infer the control objectives from the chosen control structures and examination of the process constraints. This procedure is the reverse of that discussed by Nishida et al. (1981), where control system synthesis is defined as starting with selection of the control objectives and then determining appropriate manipulated and measured variables. This emphasizes the point that it is preferable to determine the control objectives implicitly (as is done in the MILP), rather than explicitly,as an explicit set of control objectives are not guaranteed to be the optimal control objectives. These case studies have demonstrated the MILP formulation to be an effective method for the selection of process control structures. However, while it was concluded that the MILP method is more effective than explicit enumeration, further improvements should be possible from a more efficient implementation and by refining the problem formulation. The need to use controllability indicators to examine the generated control structures was emphasized by the fact that the economically optimal control structure for the evaporator had poor controllability characteristics at high frequency, whereas an alternative control structure had significantly better controllability properties over the entire frequency range for a minimal increase in cost. The MILP analysis demonstrated a sensible selection of economic control objectives, choosing to minimize the variation of the most economically significant constraints using the cheapest sensors available. The case studies also demonstrated that the instrumentation costs for the process are significant in the choice of control structure, as it is the marginal cost (cost of implementing control - improvement in economics from control) that determines the control system economics. In summary, the MILP method has been demonstrated to be an effective method for selecting control structure, although there is still scope for improvement in performance, and it can be used in a systematic manner to gain information on the process dynamics.
2692 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
Nomenclature A-H, K-P,R = state space matrices a, b = vector or scalar constants CA = annualized cost ($1 CI = installed cost ($1
CT = total instrumentation cost ($) d = design parameters gij = structural transfer function element G = structural transfer function I = identity matrix of appropriate dimension J = objective function JM = MILP objective function Jw = perfect control problem objective function j = square root of -1 n, = dimension of vector x P = second-effect pressure p = disturbance variables p ( w ) = disturbance amplitude a t frequency w Q = permutation matrices q , 7 = general LP variables s = Laplace transform parameter T = second-effect temperature U = estimate of the maximum dynamic variation of u u = potential manipulated variables iL = linear approximation to U W = product liquor composition X,X = integer variables x = state variables i = time derivatives of x y = process variables z = potential measured variables a,@, y, 6 = cost coefficients A(s) = transfer function between u and p t, 6 = actuatorlsensor costs r, A = matrices j = linear approximation t o Q = deviation slack variables u = estimate of the maximum constraint back off w = disturbance frequency subscript 0 = steady-state (nominal) value subscript c = complex part subscript r = real part superscript h = simple upper bound superscript 1 = simple lower bound superscript * = optimum value overscore = Laplace transform or frequency response 1. = complex magnitude
a
Literature Cited Barton, G. W.; Perkins, J. D. Experiences with SPEEDUP in the Process Industries. Comput. Chem. Eng. 1988,66 (8),408. Beale, E. M. L. Integer Programming. In The State of the Art in Numerical Analysis; Academic Press: New York, 1977;p 409. Fewings, J. H.; Slaughter,P. J.; Manlapig,E.; Lynch, A. J. Experiences with Process Control of the Chalcopyrite.Flotation Circuit at Mount Isa Mines Limited. In 13th Znt. Min. Proc. Congress, Warsaw; 1979; p 1540. Govind, R.; Powers, G. J. Control System Synthesis Strategies. AIChE J. 1982,28,60. Maarleveld, A.; Rijnsdorp, J. E. Constraint Control on Distillation Columns. Automatica 1970,6, 51. Marlin, T. E.; Perkins, J. D.; Barton, G. W.; Brisk, M. L. Benefits from Process Control: Resulta of a Joint Industry-University Study. J. Process Control 1991,I , 68. Morari, M.; Stephanopoulos, G. Studies in the Synthesis of Control Structures for Chemical Processes. Part I 1 Structural Aepecta and the Synthesis of Alternative Feasible Control Schemes. AZChE J. 1980,26 (2),232. Murtagh, B.A.; Saunders,M. A. MZNOS5.0 Users Guide;Technical Report SOL 83-20;Stanford University, 1983. Narraway, L. T. Selection of Process Control Structure Based on Economics. Ph.D. Thesis, Department of Chemical Engineering, Imperial College, University of London, 1992. Narraway, L. T.; Perkins, J. D.; Barton, G. W. Interaction Between Process Design and Process Control: EconomicAnalysisof Process Dynamics. J. Process Control 1991,I , 243. Nemhauser, G. L.; Wolsey, L. A. Znteger and Combinatorial Optimization; John Wiley and Sons: New York, 1988. Nishida, N.; Stephanopoulos, G.; Westerberg, A. W. A Review of Process Synthesis. AIChE J. 1981,27(4), 423. Prett, D. M.;Garcia, C. E. Fundamental Process Control; Butterworths: Boston, 1988. Prosys Technology. SPEEDUP Case Book; Prosys Technology: Cambridge, 1989. Shah, N. Efficient Scheduling, Planning and Design of Multipurpose Batch Plants. Ph.D. Thesis, Departmentof Chemical Engineering, Imperial College, University of London, 1992. Umeda, T.; Kuriyama, T.; Ichikawa,A. ALogical Structurefor Process Control System Synthesis. In Proceedings, ZFAC Congress; Helsinki; 1978;Pergamon: Oxford, Vol. I.
Received for revieu December 18,1992 Revised manuscript received July 2 , 1993 Accepted July 14, 1993' ~~
e Abstract
1993.
published in Advance ACS Abstracts, October 1,
